Basic Life Insurance Mathematics. Ragnar Norberg

Transcription

1 Basic Life Insurance Mahemaics Ragnar Norberg Version: Sepember 22

2 Conens 1 Inroducion Banking versus insurance Moraliy Banking Insurance Wih-profi conracs: Surplus and bonus Uni-linked insurance Issues for furher sudy Paymen sreams and ineres Basic definiions and relaionships Applicaion o loans Moraliy Aggregae moraliy Some sandard moraliy laws Acuarial noaion Selec moraliy Insurance of a single life Some sandard forms of insurance The principle of equivalence Prospecive reserves Thiele s differenial equaion Probabiliy disribuions The sochasic process poin of view Expenses A single life insurance policy The general muli-sae policy Muli-life insurances Insurances depending on he number of survivors

3 CONTENTS 2 7 Markov chains in life insurance The insurance policy as a sochasic process The ime-coninuous Markov chain Applicaions Selecion phenomena The sandard muli-sae conrac Selec moraliy revisied Higher order momens of presen values A Markov chain ineres model The Markov model Differenial equaions for momens of presen values Complemen on Markov chains Dependen lives Inroducion Noions of posiive dependence Dependencies beween presen values A Markov chain model for wo lives Condiional Markov chains Rerospecive feriliy analysis Probabiliy disribuions of presen values Inroducion Calculaion of probabiliy disribuions of presen values by elemenary mehods The general Markov mulisae policy Differenial equaions for saewise disribuions Applicaions Reserves Inroducion General definiions of reserves and saemen of some relaionships beween hem Descripion of paymen sreams appearing in life and pension insurance The Markov chain model Reserves in he Markov chain model Some examples Safey loadings and bonus General consideraions Firs and second order bases The echnical surplus and how i emerges Dividends and bonus Bonus prognoses Examples Including expenses

4 CONTENTS Discussions Saisical inference in he Markov chain model Esimaing a moraliy law from fully observed life lenghs Parameric inference in he Markov model Confidence regions More on simulaneous confidence inervals Piecewise consan inensiies Impac of he censoring scheme Heerogeneiy models The noion of heerogeneiy a wo-sage model The proporional hazard model Group life insurance Basic characerisics of group insurance A proporional hazard model for complee individual policy and claim records Experience raed ne premiums The flucuaion reserve Esimaion of parameers Haendorff and Thiele Inroducion The general Haendorff heorem Applicaion o life insurance Excerps from maringale heory Financial mahemaics in insurance Finance in insurance Prerequisies A Markov chain financial marke - Inroducion The Markov chain marke Arbirage-pricing of derivaives in a complee marke Numerical procedures Risk minimizaion in incomplee markes Trading wih bonds: How much can be hedged? The Vandermonde marix in finance Two properies of he Vandermonde marix Applicaions o finance Maringale mehods A Calculus 4 B Indicaor funcions 9 C Disribuion of he number of occurring evens 12

5 CONTENTS 4 D Asympoic resuls from saisics 15 E The G82M moraliy able 17 F Exercises 1 G Soluions o exercises 1

6 Chaper 1 Inroducion 1.1 Banking versus insurance A. The bank savings conrac. Upon celebraing his 55h anniversary Mr. (55) (le us call him so) decides o inves money o secure himself economically in his old age. The firs idea ha occurs o him is o deposi a capial of S = 1 (e.g. one hundred housand pounds) on a savings accoun oday and draw he enire amoun wih earned compound ineres in 15 years, on his 7h birhday. The accoun bears ineres a rae i =.45 (4.5%) per year. In one year he capial will increase o S 1 = S +S i = S (1+i), in wo years i will increase o S = S 1 + S 1 i = S (1 + i) 2, and so on unil in 15 years i will have accumulaed o S 15 = S (1 + i) 15 = = (1.1) This simple calculaion akes no accoun of he fac ha (55) will die sooner or laer, maybe sooner han 15 years. Suppose he has no heirs (or he dislikes he ones he has) so ha in he even of deah before 7 he would consider his savings waised. Checking populaion saisics he learns ha abou 75% of hose who are 55 will survive o 7. Thus, he relevan prospecs of he conrac are: wih probabiliy.75 (55) survives o 7 and will hen possess S 15 ; wih probabiliy.25 (55) dies before 7 and loses he capial. In his perspecive he expeced amoun a (55) s disposal afer 15 years is.75 S 15. (1.2) B. A small scale muual fund. Having hough hings over, (55) seeks o make he following muual arrangemen wih (55) and (55), who are also 55 years old and are in exacly he same siuaion as (55). Each of he hree deposis S = 1 on he savings accoun, and hose who survive o 7, if any, will hen share he oal accumulaed capial 3 S 15 equally. The prospecs of his scheme are given in Table 1.1, where + and signify survival and deah, respecively, L 7 is he number of survivors a age 7, and 5

7 CHAPTER 1. INTRODUCTION 6 Table 1.1: Possible oucomes of a savings scheme wih hree paricipans. (55) (55) (55) L 7 3 S 15 /L 7 Probabiliy S = S = S = S = S = S = S =.47 undefined =.16 3 S 15 /L 7 is he amoun a disposal per survivor (undefined if L 7 = ). There are now he following possibiliies: wih probabiliy.422 (55) survives o 7 ogeher wih (55) and (55) and will hen possess S 15 ; wih probabiliy =.282 (55) survives o 7 ogeher wih one more survivor and will hen possess 1.5 S 15 ; wih probabiliy.47 (55) survives o 7 while boh (55) and (55) die (may hey res in peace) and he will cash he oal savings 3S 15 ; wih probabiliy.25 (55) dies before 7 and will ge nohing. This scheme is superior o he one described in Paragraph A, wih separae individual savings conracs: If (55) survives o 7, which is he only scenario of ineres o him, he will cash no less han he amoun S 15 he would cash under he individual scheme, and i is likely ha he will ge more. As compared wih (1.2), he expeced amoun a (55) s disposal afer 15 years is now.422 S S S 15 =.985 S 15. The poin is ha under he presen scheme he savings of hose who die are bequeahed o he survivors. Thus he oal savings are reained for he group so ha nohing is lef o ohers unless he unlikely hing happens ha he whole group goes exinc wihin he erm of he conrac. This is essenially he kind of solidariy ha unies he members of a pension fund. From he poin of view of he group as a whole, he probabiliy ha all hree paricipans will die before 7 is only.16, which should be compared o he probabiliy.25 ha (55) will die and lose everyhing under he individual savings program. C. A large scale muual scheme. Inspired by he success of he muual fund idea already on he small scale of hree paricipans, (55) sars o play wih he idea of exending i o a large number of paricipans. Le us assume ha a oal number of L 55 persons, who are in exacly he same siuaion as (55), agree o join a scheme similar o he one described for he hree. Then he

8 CHAPTER 1. INTRODUCTION 7 oal savings afer 15 years amoun o L 55 S 15, which yields an individual share equal o L 55 S 15 L 7 (1.3) o each of he L 7 survivors if L 7 >. By he so-called law of large numbers, he proporion of survivors L 7 /L 55 ends o he individual survival probabiliy.75 as he number of paricipans L 55 ends o infiniy. Therefore, as he number of paricipans increases, he individual share per survivor ends o 1.75 S 15, (1.4) and in he limi (55) is faced wih he following siuaion: 1 wih probabiliy.75 he survives o 7 and ges.75 S 15; wih probabiliy.25 he dies before 7 and ges nohing. The expeced amoun a (55) s disposal afer 15 years is S 15 = S 15, (1.5) he same as (1.1). Thus, he beques mechanism of he muual scheme has raised (55) s expecaions of fuure pension o wha hey would be wih he individual savings conrac if he were immoral. This is wha we could expec since, in an infiniely large scheme, some will survive o 7 for sure and share he oal savings. All he money will remain in he scheme and will be redisribued among is members by he loery mechanism of deah and survival. The fac ha L 7 /L 55 ends o.75 as L 55 increases, and ha (1.3) hus sabilizes a (1.4), is precisely wha is mean by saying ha insurance risk is diversifiable. The risk can be eliminaed by increasing he size of he porfolio. 1.2 Moraliy A. Life and deah in he classical acuarial perspecive. Insurance mahemaics is widely held o be boring. Hopefully, he presen ex will no suppor ha prejudice. I mus be admied, however, ha acuaries use o cheer hemselves up wih jokes like: Wha is he difference beween an English and a Sicilian acuary? Well, he English acuary can predic fairly precisely how many English ciizens will die nex year. Likewise, he Sicilian acuary can predic how many Sicilians will die nex year, bu he can ell heir names as well. The English acuary is definiely he more ypical represenaive of he acuarial profession since he akes a purely saisical view of moraliy. Sill he is able o analyze insurance problems adequaely since wha insurance is essenially abou, is o average ou he randomness associaed wih he individual risks. Conemporary life insurance is based on he paradigm of he large scheme (diversificaion) sudied in Paragraph 1.1C. The ypical insurance company

9 CHAPTER 1. INTRODUCTION 8 serves ens and some even hundreds of housands of cusomers, sufficienly many o ensure ha he survival raes are sable as assumed in Paragraph 1.1C. On he basis of saisical invesigaions he acuary consrucs a so-called decremen series, which akes as i saring poin a large number l of new-born and, for each age x = 1, 2,..., specifies he number of survivors, l x. Table 1.2: Excerp from he moraliy able G82M x: l x : d x : q x : p x : Table 1.2 is an excerp of he able used by Danish insurers o describe he moraliy of insured Danish males. The second row in he able liss some enries of he decremen series. I shows e.g. ha abou 65% of all new-born will celebrae heir 7h anniversary. The number of survivors decreases wih age: l x l x+1. The difference d x = l x l x+1 is he number of deahs a age x (more precisely, beween age x and age x + 1). These numbers are shown in he hird row of he able. I is seen ha he number of deahs peaks somewhere around age 8. From his i canno be concluded ha 8 is he mos dangerous age. The acuary measures he moraliy a any age x by he one-year moraliy rae q x = d x l x, which ells how big proporion of hose who survive o age x will die wihin one year. This rae, shown in he fourh row of he able, increases wih he age. For insance, 8.4% of he 8 years old will die wihin a year, whereas 18.9% of he 9 years old will die wihin a year. The boom row shows he one year survival raes p x = l x+1 l x = 1 q x. We shall presen some ypical forms of producs ha an insurance company can offer o (55) and see how hey compare wih he corresponding arrangemens, if any, ha (55) can make wih his bank.

10 CHAPTER 1. INTRODUCTION Banking A. Ineres. Being unable o find his perfec maches (55), (55),..., our hero (55) abandons he idea of creaing a muual fund and resumes discussions wih his bank. The bank operaes wih annual ineres rae i in year = 1, 2,... Thus, a uni S = 1 deposied a ime will accumulae wih compound ineres as follows: In one year he capial increases o S 1 = S + S i 1 = 1 + i 1, in wo years i increases o S 2 = S 1 + S 1 i 2 = (1 + i 1 )(1 + i 2 ), and in years i increases o S = (1 + i 1 ) (1 + i ), (1.6) called he -year accumulaion facor. Accordingly, he presen value a ime of a uni wihdrawn in j years is S j 1 = 1 S j, (1.7) called he j-year discoun facor since i is wha he bank would pay you a ime if you sell o i (discoun) a defaul-free claim of 1 a ime j. Similarly, he value a ime of a uni deposied a ime j < is (1 + i j+1 ) (1 + i ) = S S j, called he accumulaion facor over he ime period from j o, and he value a ime of a uni wihdrawn a ime j > is 1 (1 + i +1 ) (1 + i j ) = S S j, he discoun facor over he ime period from o j. In general, he value a ime of a uni due a ime j is S S j 1, an accumulaion facor if j < and a discoun facor if j > (and of course 1 if j = ). From (1.6) i follows ha S = S 1 (1 + i ), hence i = S S 1 S 1, which expresses he ineres rae in year as he relaive increase of he balance in year. B. Saving in he bank. A general savings conrac over n years specifies ha a each ime =,..., n (55) is o deposi an amoun c (conribuion) and wihdraw an amoun b (benefi). The ne amoun of deposi less wihdrawal a ime is c b. A any ime he cash balance of he accoun, henceforh also

11 CHAPTER 1. INTRODUCTION 1 called he rerospecive reserve, is he oal of pas (including presen) deposis less wihdrawals compounded wih ineres, U = S S 1 j (c j b j ). (1.8) j= I develops in accordance wih he forward recursive scheme = 1, 2,..., n, commencing from U = U 1 (1 + i ) + c b, (1.9) U = c b. Each year (55) will receive from he bank a saemen of accoun wih he calculaion (1.9), showing how he curren balance emerges from he previous balance, he ineres earned meanwhile, and he curren movemen (deposi less wihdrawal). The balance of a savings accoun mus always be non-negaive, U, (1.1) and a ime n, when he conrac erminaes and he accoun is closed, i mus be null, U n =. (1.11) In he course of he conrac he bank mus mainain a so-called prospecive reserve o mee is fuure liabiliies o he cusomer. A any ime he adequae reserve is V = S n j=+1 S j 1 (b j c j ), (1.12) he presen value of fuure wihdrawals less deposis. Similar o (1.9), he prospecive reserve is calculaed by he backward recursive scheme V = (1 + i +1 ) 1 (b +1 c +1 + V +1 ), (1.13) = n 1, n 2,...,, saring from V n =. The consrain (1.11) is equivalen o n S 1 j c j j= n S 1 j b j =, (1.14) j=

12 CHAPTER 1. INTRODUCTION 11 which says ha he discouned value of deposis mus be equal o he discouned value of he wihdrawals. I implies ha, a any ime, he rerospecive reserve equals he prospecive reserve, U = V, as is easily verified. (Inser he defining expression (1.8) wih = n ino (1.11), spli he sum n j= ino j= + n j=+1, and muliply wih S /S n, o arrive a U V =.) C. The endowmen conrac. The bank proposes a savings conrac according o which (55) saves a fixed amoun c annually in 15 years, a ages 55,...,69, and hereafer wihdraws b = 1 (one hundred housand pounds, say) a age 7. Suppose he annual ineres rae is fixed and equal o i =.45, so ha he accumulaion facor in years is S = (1 + i), he discoun facor in j years is S j 1 = (1 + i) j, and he ime value of a uni due a ime j is S S j 1 = (1 + i) j. For he presen conrac he equivalence requiremen (1.14) is 14 j= from which he bank deermines c = (1 + i) j c (1 + i) 15 1 =, (1 + i) j= (1 + i) j =.464, (1.15) Due o ineres, his amoun is considerably smaller han 1/15 =.6667, which is wha (55) would have o save per year if he should choose o uck he money away under his maress. 1.4 Insurance A. The life endowmen. Sill, o (55).464 (four housand six hundred and four pounds) is a considerable expense. He believes in a life before deah, and i should be blessed wih he joys ha money can buy. He alks o an insurance agen, and is delighed o learn ha, under a life annuiy policy designed precisely as he savings scheme, he would have o deposi an annual amoun of only.3743 (hree housand seven hundred and fory hree pounds). The insurance agen explains: The calculaions of he bank depend only on he amouns c b and would apply o any cusomer (x) who would ener ino he same conrac a age x, say. Thus, o he bank he cusomer is really an unknown Mr. X. To he insurance company, however, he is no jus Mr. X, bu he significan Mr. (x) now x years old. Working under he hypohesis ha (x) is one of he l x survivors a age x in he decremen series and ha hey all hold

13 CHAPTER 1. INTRODUCTION 12 idenical conracs, he insurer offers (x) a general life annuiy policy whereby each deposi or wihdrawal is condiional on survival. For he enire porfolio he rerospecive reserve a ime is U p = S S 1 j (c j b j ) l x+j (1.16) j= = U p 1 (1 + i ) + (c b ) l x+. (1.17) The prospecive porfolio reserve a ime is n V p = S S 1 j (b j c j )l x+j (1.18) j=+1 = (1 + i +1 ) 1 ((b +1 c +1 )l x++1 + V p +1 ). (1.19) In paricular, for he life endowmen analogue o (55) s savings conrac, he only paymens are c = c for =,..., 14 and b 15 = 1. The equivalence requiremen (1.14) becomes 14 j= from which he insurer deermines (1 + i) j c l 55+j (1 + i) 15 1 l 7 =, (1.2) c = (1 + i) 15 l 7 14 j= (1 + i) j l 55+j = (1.21) Inspecion of he expressions in (1.15) and (1.21) shows ha he laer is smaller due o he fac ha l x is decreasing. This phenomenon is known as moraliy beques since he savings of he deceased are bequeahed o he survivors. We shall pursue his issue in Paragraph C below. B. A life assurance conrac. Suppose, conrary o he former hypohesis, ha (55) has dependens whom he cares for. Then he migh be concerned ha, if he should die wihin he erm of he conrac, he survivors in he pension scheme will be his heirs, leaving his wife and kids wih nohing. He figures ha, in he even of his unimely deah before he age of 7, he family would need a down paymen of b = 1 (one hundred housand pounds) o compensae he loss of heir bread-winner. The bank can no help in his maer; he benefi of b would have o be raised immediaely since (55) could die omorrow, and i would be meaningless o borrow he money since full repaymen of he loan would be due immediaely upon deah. The insurance company, however, can offer (55) a so-called erm life assurance policy ha provides he waned deah benefi agains an affordable annual premium of c =.171. The equivalence requiremen (1.14) now becomes 14 j= (1 + i) j c l 55+j 15 j=1 (1 + i) j 1 d 55+j 1 =, (1.22)

14 CHAPTER 1. INTRODUCTION 13 from which he insurer deermines c = 15 j=1 (1 + i) j d 55+j 1 14 j= (1 + i) j l 55+j =.171. (1.23) C. Individual reserves and moraliy beques. In he insurance schemes described above he conracs of deceased members are void, and he reserves of he porfolio are herefore o be shared equally beween he survivors a any ime. Thus, we inroduce he individual rerospecive and prospecive reserves a ime, U = U p /l x+, V = V p /l x+. Since we have esablished ha U = V, we shall henceforh be referring o hem as he individual reserve or jus he reserve. For he general pension insurance conrac in Paragraph A we ge from (1.17) ha he individual reserve develops as l x+ 1 U = U 1 (1 + i ) + (c b ) l x+ ( = U d ) x+ 1 (1 + i ) + (c b ). (1.24) l x+ The beques mechanism is clearly seen by comparing (1.24) o (1.9): he addiional erm U 1 (1 + i ) d x+ 1 /l x+ in he laer is precisely he share per survivor of he savings lef over o hem by hose who died during he year. Virually, he moraliy beques acs as an increase of he ineres rae. Table 1.3 shows how he reserve develops for he endowmen conracs offered by he bank and he insurance company, respecively. I is seen ha he insurance scheme requires a smaller reserve han he bank savings scheme. Table 1.3: Reserve U = V for bank savings accoun and for life endowmen insurance : Savings accoun: Life endowmen: For he life assurance described in Paragraph B we obain similarly ha he individual reserve develops as as shown in Table 1.4. D. Insurance risk in a finie porfolio. The perfec balance in (1.2) and (1.22) ress on he hypohesis ha he decremen series l x+ follows he paern of an infiniely large porfolio. In a finie porfolio, however, he facual numbers of survivors, L x+, will be subjec o randomness and will be deermined by

15 CHAPTER 1. INTRODUCTION 14 Table 1.4: Reserve U = V for a erm life assurance of 1 agains level premium in 15 years from age 55 : he survival probabiliies p x+ (some of which are) shown in Table 1.2. The difference beween discouned premiums and discouned benefis, D = 14 j= (1 + i) j.374 L 55+j (1 + i) 15 L 7, will be a random quaniy. I will have expeced value, and is sandard deviaion measures how much insurance risk is lef due o imperfec diversificaion in a finie porfolio. An easy exercise in probabiliy calculus shows ha he 1 sandard deviaion of D/L 55 is L I ends o as L 55 goes o infiniy. For he erm insurance conrac he corresponding quaniy is 1 L , indicaing ha erm insurance is a more risky business han life endowmen. 1.5 Wih-profi conracs: Surplus and bonus A. Wih-profi conracs. Insurance policies are long erm conracs, wih ime horizons wide enough o capure significan variaions in ineres and moraliy. For simpliciy we shall focus on ineres rae uncerainy and assume ha he moraliy law remains unchanged over he erm of he conrac. We will discuss he issue of surplus and bonus in he framework of he life endowmen conrac considered in Paragraph 1.4.A. A ime, when he conrac is wrien wih benefis and premiums binding o boh paries, he fuure developmen of he ineres raes i is uncerain, and i is impossible o foresee wha premium level c will esablish he required equivalence wih 14 j= S j 1 c l 55+j = S l 7, (1.25) S j = (1 + i 1 ) (1 + i j ). If i should urn ou ha, due o adverse developmen of ineres and moraliy, premiums are insufficien o cover he benefi, hen here is no way he insurance company can avoid a loss; i canno reduce he benefi and i canno increase he premiums since hese were irrevocably se ou in he conrac a ime. The only way he insurance company can preven such a loss, is o charge a premium

16 CHAPTER 1. INTRODUCTION 15 on he safe side, high enough o be adequae under all likely scenarios. Then, if everyhing goes well, a surplus will accumulae. This surplus belongs o he insured and is o be repaid as so-called bonus, e.g. as increased benefis or reduced premiums. B. Firs order basis. The usual way of seing premiums o he safe side is o base he calculaion of he premium level and he reserves on a provisional firs order basis, assuming a fixed annual ineres rae i, which represens a wors case scenario and leads o higher premium and reserves han are likely o be needed. The corresponding accumulaion facor is S = (1 + i ). The individual reserve based on he pruden firs order assumpions is called he firs order reserve, and we denoe i by V as before. The premiums are deermined so as o saisfy equivalence under he firs order assumpion. C. Surplus. A any ime we define he echnical surplus Q as he difference beween he rerospecive reserve under he facual ineres developmen and he rerospecive reserve under he firs order assumpion: Q = S S 1 j c l 55+j S j= Sj 1 c l 55+j j= 1 1 = S S 1 j c l 55+j S Sj 1 c l 55+j. j= Seing S = S 1 (1+i ) and S = S 1 (1+i ), wriing 1+i = 1+i (i i ) in he laer, and rearranging a bi, we find ha Q develops as commencing from Q = Q 1 (1 + i ) + V 1 (i i ) l 55+ 1, (1.26) Q =. j= The conribuion o he echnical surplus in year is V 1 (i i ) l 55+ 1, which is easy o inerpre: i is precisely he ineres earned on he reserve in excess of wha has been assumed under he pruden firs order assumpion. The surplus is o be redisribued as bonus. Several bonus schemes are used in pracice. One can repay currenly he conribuion V 1 (i i ) l as so-called cash bonus (a premium deducible), whereby each survivor a ime will receive V 1 (i i ) l /l 55+. Anoher possibiliy is o pospone repaymen unil he erm of he conracs and gran so-called erminal bonus o he survivors (an added benefi), he amoun

17 CHAPTER 1. INTRODUCTION 16 per survivor being b + given by S j=1 S j 1 V j 1 (i j i ) l 55+j = l 7 b +. Beween hese wo soluions here are counless oher possibiliies. In any case, he poin is ha he financial risk can be eliminaed: he insurer observes he developmen of he facual ineres and only in arrears repays he insured so as o resore equivalence on he basis of he facual ineres rae developmen. This works well provided he firs order ineres rae is se on he safe side so ha i i for all. There is a problem, however: Negaive bonus can never be applied. Therefore he insurer will suffer a loss if he facual ineres falls shor of he echnical ineres rae. In his perspecive cash bonus is he mos risky soluion and erminal bonus is he leas risky soluion. If he financial marke is sufficienly rich in asses, hen he ineres rae guaranee ha is hus inheren in he wih-profi policy can be priced, and he insured can be charged an exra premium o cover i. This would ulimaely eliminae he financial risk by diversifying, no only he insurance porfolio, bu also he invesmen porfolio. 1.6 Uni-linked insurance A quie differen way of going abou he financial risk is he so-called uni-linked conrac. As he name indicaes, he idea is o relae paymens direcly o he developmen of he invesmen porfolio, i.e. he ineres rae. Consider he balance equaion for an endowmen of b agains premium c in year = 1,..., 14: 14 S 15 = S 1 c l 55+ b l 7 =. (1.27) A perfec link beween paymens and invesmens performance is obained by leing he premiums and he benefi be inflaed by he index S, and c = S c, b 15 = S Here c is a baseline premium, which is o be deermined. Then (1.27) becomes which reduces o 14 S 15 = S 1 S c l 55+ S 15 l 7 =, 14 = c l 55+ l 7 =,

18 CHAPTER 1. INTRODUCTION 17 and we find c = l 7 14 = l. 55+ Again financial risk has been perfecly eliminaed and diversificaion of he insurance porfolio is sufficien o esablish balance beween premiums and benefis. Perfec linking as defined here is no common in pracice. Presumably, remnans of social welfare hinking have led insurers o modify he uni-linked concep in various ways, ypically by inroducing a guaranee on he sum insured o he effec ha i canno be less han 1 (say). Also he premium is usually no index-linked. Under such modified variaions of he uni-linked policy one canno in general obain balance by he simple device above. However, he problem can in principle be resolved by calculaing he price of he financial claim hus inroduced and o charge he insured wih he needed addiional premium. 1.7 Issues for furher sudy The simple pieces of acuarial reasoning in he previous secions involve wo consiuens, ineres and moraliy, and hese are o be sudied separaely in he wo following chapers. Nex we shall escalae he discussion o more complex siuaions. For insance, suppose (55) wans a life insurance ha is paid ou only if his wife survives him, or wih a sum insured ha depends on he number of children ha are sill alive a he ime of his deah. Or he may demand a pension payable during disabiliy or unemploymen. We need also o sudy he risk associaed wih insurance, which is due o he uncerain developmens of he insurance porfolio and he invesmen porfolio: he deahs in a finie insurance porfolio do no follow he moraliy able (1.2) exacly, and he ineres earned on he invesmens may differ from he assumed 4.5% per year, and neiher can be prediced precisely a he ouse when he policies are issued. In a scheme of he classical muual ype he problem was how o share exising money in a fair manner. A ypical insurance conrac of oday, however, specifies ha cerain benefis will be paid coningen on cerain evens relaed only o he individual insured under he conrac. An insurance company working wih his concep in a finie porfolio, wih imperfec diversificaion of insurance risk, faces a risk of insolvency as indicaed in Paragraph 1.3.D. In addiion comes he financial risk, and ways of geing around ha were indicaed in Secions 1.5 and 1.6. The oal risk has o be conrolled in some way. Wih hese issues in mind, we now commence our sudies of he heory of life insurance. The reader is advised o consul he following auhoriaive exbooks on he subjec: [6] (a good classic sharpen your German!), [4], [29] (lexicographic, reas virually every variaion of sandard insurance producs, and includes a good chaper on populaion heory), [45] (an excellen early ex based on probabilisic models, placing emphasis on risk consideraions), [11], [15] (an

19 CHAPTER 1. INTRODUCTION 18 original approach o he field sharpen your French!), and [23] (he mos recen of he menioned exs, sill classical in is orienaion).

20 Chaper 2 Paymen sreams and ineres 2.1 Basic definiions and relaionships A. Sreams of paymens. Wha is money? In lack of a precise definiion you may add up he face values of he coins and noes you find in your purse and say ha he oal amoun is your money. Now, if you do his each ime you open your purse, you will realize ha he developmen of he amoun over ime is imporan. In he conex of insurance and finance he ime aspec is essenial since paymens are usually regulaed by a conrac valid over some period of ime. We will give precise mahemaical conen o he noion of paymen sreams and, referring o Appendix A, we deal only wih heir properies as funcions of ime and do no venure o discuss heir possible sochasic properies for he ime being. To fix ideas and erminology, consider a financial conrac commencing a ime and erminaing a a laer ime n ( ), say, and denoe by A he oal amoun paid in respec of he conrac during he ime inerval [, ]. The paymen funcion {A } is assumed o be he difference of wo non-decreasing, finie-valued funcions represening incomes and ougoes, respecively, and is hus of finie variaion (FV). Furhermore, he paymen funcion is assumed o be righ-coninuous (RC). From a pracical poin of view his assumpion is jus a maer of convenion, saing ha he balance of he accoun changes a he ime of any deposi or wihdrawal. From a mahemaical poin of view i is convenien, since paymen funcions can hen serve as inegraors. In fac, we shall resric aenion o paymen funcions ha are piece-wise differeniable (PD): where A τ A = A + a τ dτ + <τ A τ, (2.1) = A τ A τ. The inegral adds up paymens ha fall due con- 19

21 CHAPTER 2. PAYMENT STREAMS AND INTEREST 2 inuously, and he sum adds up lump sum paymens. In differenial form (2.1) reads da = a d + A. (2.2) I seems naural o coun incomes as posiive and ougoes as negaive. Someimes, and in paricular in he conex of insurance, i is convenien o work wih ougoes less incomes, and o avoid ugly minus signs we inroduce B = A. Having explained wha paymens are, le us now see how hey accumulae under he force of ineres. There are monographs wrien especially for acuaries on he opic, see [31] and [17], bu we will gaher he basics of he heory in only a few lines. B. Ineres. Suppose money is currenly invesed on (or borrowed from) an accoun ha bears ineres. This means ha a uni deposied on he accoun a ime s gives he accoun holder he righ o cash, a any oher ime, a cerain amoun S(s, ), ypically differen from 1. The funcion S mus be sricly posiive, and we shall argue ha i mus saisfy he funcional relaionship S(s, u) = S(s, ) S(, u), (2.3) implying, of course, ha S(, ) = 1 (pu s = = u and use sric posiiviy): If he accoun holder invess 1 a ime s, he may cash he amoun on he lef of (2.3) a ime u. If he insead wihdraws he value S(s, ) a ime and immediaely reinvess i again, he will obain he amoun on he righ of (2.3) a ime u. To avoid arbirary gains, so-called arbirage, he wo sraegies mus give he same resul. I is easy o verify ha he funcion S(s, ) saisfies (2.3) if and only if i is of he form S(s, ) = S S s (2.4) for some sricly posiive funcion S (allowing an abuse of noaion), which can be aken o saisfy S = 1. Then S mus be he value a ime of a uni deposied a ime, and we call i he accumulaion funcion. Correspondingly, S 1 is he value a ime of a uni wihdrawn a ime, and we call i he discoun funcion. We will henceforh assume ha S is of he form S = e r, S 1 = e r, (2.5) where r is some piece-wise coninuous funcion, usually posiive. (The shorhand exemplified by r = r τ dτ will be in frequen use hroughou.) Accumulaion facors of his form are invariably used in basic banking operaions (loans and savings) and also for bonds issued by governmens and corporaions.

22 CHAPTER 2. PAYMENT STREAMS AND INTEREST 21 by Under he rule (2.5) he dynamics of accumulaion and discouning are given ds = S r d, (2.6) ds 1 = S 1 r d. (2.7) The relaion (2.6) says ha he ineres earned in a small ime inerval is proporional o he lengh of he inerval and o he curren amoun on deposi. The proporionaliy facor r is called he force of ineres or he (insananeous) ineres rae a ime. In inegral form (2.6) reads and (2.7) reads S 1 u S 1 S = S s + = S 1 u = S 1 u + s S τ r τ dτ, s, (2.8) S 1 τ r τ dτ or We will be working wih he expressions s S 1 τ r τ dτ, u. (2.9) S(s, ) = e s r for he general discoun facor when s and S(, u) = e u r for he general accumulaion facor when u. By consan ineres rae r we have S = e r and and S 1 inroducing he he annual ineres rae = e r. Upon i = e r 1, (2.1) whereby he annual accumulaion facor is S 1 = (1+i), and he annual discoun facor we have v = e r = i, (2.11) S = (1 + i), S 1 = v. (2.12) C. Valuaion of paymen sreams. Suppose ha he incomes/ougoes creaed by he paymen sream A are currenly deposied on/drawn from an accoun which bears ineres a rae r a ime. By (2.4) he value a ime of he amoun da τ paid in he small ime inerval around ime τ is e r e τ r da τ.

23 CHAPTER 2. PAYMENT STREAMS AND INTEREST 22 Summing over all ime inervals we ge he value a ime of he enire paymen sream, where e r n U = e r e τ r da τ = U V, e τ r da τ = e τ r da τ (2.13) is he accumulaed value of pas incomes less ougoes, and (recall he convenion B = A) V = e r e τ r db τ = n e τ r db τ (2.14) is he discouned value of fuure ougoes less incomes. This decomposiion is paricularly relevan for paymens governed by some conrac; U is he cash balance, ha is, he amoun held a he ime of consideraion, and V is he fuure liabiliy. The difference beween he wo is he curren value of he conrac. The developmen of he cash balance can be viewed in various ways: Applicaion of (A.8) o (2.13), aking X = e r (coninuous, wih dynamics given by (2.6)) and Y = e τ r da τ, yields By definiion, du = U r d + da, (2.15) U = A. (2.16) Inegraing (2.15) from o, using he iniial condiion (2.16), gives An alernaive expression, U = A + U = A + U τ r τ dτ. (2.17) e τ r A τ r τ dτ, (2.18) is derived from (2.13) upon applying he rule (A.9) of inegraion by pars: e τ r da τ = A + e τ r da τ = A + e r A A A τ e τ r ( r τ ) dτ. The relaionships (2.15) (2.18) show how he cash balance emerges from paymens and earned ineres. They are easy o inerpre and can be read aloud in non-mahemaical erms.

24 CHAPTER 2. PAYMENT STREAMS AND INTEREST 23 I follows from (2.18) ha, if r, hen an increase of A resuls in an increase of U. In paricular, advancing paymens of a given amoun produces a bigger cash balance. Likewise, from (2.14) we derive By definiion, if n <, or, seing B n = B n B n, dv = V r d db. (2.19) V n =, (2.2) V n = B n. (2.21) Inegraing (2.19) from o n, using he ulimo condiion (2.2), gives he following analogue o (2.17): The analogue o (2.18) is V = B n B V = B n B n n V τ r τ dτ. (2.22) e τ r (B n B τ ) r τ dτ. (2.23) The las wo relaionships are valid for n = only if B <. Ye anoher expression is V = e n r (B n B ) + n e τ r (B τ B ) r τ dτ, (2.24) which is obained upon inegraing by pars in (2.23) or, simpler, muliplying B n B wih 1 = e n r n e τ r r τ dτ (a wis on (2.9)) and gahering erms. Again inerpreaions are easy. The relaions (2.22) and (2.23) sae, in differen ways, ha he deb can be seled immediaely a a price which is he oal deb minus he presen value of fuure ineres saved by advancing he repaymen. The relaion (2.24) saes ha repaymen can be posponed unil he erm of he conrac a he expense of paying ineres on he ousanding amouns meanwhile. I follows from (2.24) ha, if r, hen an increase of he ousanding paymens produces an increase in he reserve. In paricular, advancing paymens of a given amoun leads o a bigger reserve. Typically, he financial conrac will lay down ha incomes and ougoes be equivalen in he sense ha U n = or V =. (2.25)

25 CHAPTER 2. PAYMENT STREAMS AND INTEREST 24 These wo relaionships are equivalen and hey imply ha, for any, U = V. (2.26) We anicipae here ha, in he insurance conex, he equivalence requiremen is usually no exercised a he level of he individual policy: he very purpose of insurance is o redisribue money among he insured. Thus he principle mus be applied a he level of he porfolio in some sense, which we shall discuss laer. Moreover, in insurance he paymens, and ypically also he ineres rae, are no known a he ouse, so in order o esablish equivalence one may have o currenly adap he paymens o he developmen in some way or oher. D. Some sandard paymen funcions and heir values. Cerain simple paymen funcions are so frequenly used ha hey have been given names. An endowmen of 1 a ime n is defined by A = ε n (), where ε n () = {, < n, 1, n. (2.27) (The only paymen is A n = 1.) By consan ineres rae r he presen value a ime of he endowmen is e rn or, recalling he noaion in Chaper 1, v n. An n-year immediae annuiy of 1 per year consiss of a sequence of endowmens of 1 a imes = 1,..., n, and is hus given by A = n ε j () = [] n. j=1 By consan ineres rae r is presen value a ime is a n = n j=1 e rj = 1 e rn i, (2.28) see (2.11) (2.1). An n-year annuiy-due of 1 per year consiss of a sequence of endowmens of 1 a imes =,..., n 1, ha is, n 1 A = ε j () = [ + 1] n. j= By consan ineres rae is presen value a ime is n 1 ä n = e rj = (1 + i) a n = (1 + i) 1 e rn i j=. (2.29) An n-year coninuous annuiy payable a level rae 1 per year is given by A = n. (2.3)

26 CHAPTER 2. PAYMENT STREAMS AND INTEREST 25 For he case wih consan ineres rae is presen value a ime is (recall (2.11)) ā n = n e rτ dτ = 1 e rn r. (2.31) An everlasing (perpeual) annuiy is called a perpeuiy. Puing n = in he (2.28), (2.29), and (2.31), we find he following expressions for he presen values of he immediae perpeuiy, he perpeuiy-due, and he coninuous perpeuiy: a = 1 i, ä = 1 + i i, ā = 1 r. (2.32) An m-year deferred n-year emporary life annuiy commences only afer m years and is payable hroughou n years hereafer. Thus i is jus he difference beween an m + n year annuiy and an m year annuiy. For he coninuous version, A = (( m) ) n = ( (m + n)) ( m). (2.33) Is presen value a ime by consan ineres is denoed ā m n and mus be 2.2 Applicaion o loans ā m n = ā m+n ā m = v m ā n. (2.34) A. Basic feaures of a loan conrac. Loans and saving accouns in banks are paricularly simple financial conracs for which ineres is invariably calculaed in accordance wih (2.5). Le us consider a loan conrac sipulaing ha a ime, say, he bank pays o he borrower an amoun H, called he principal ( firs in Lain), and ha he borrower hereafer pays back or amorizes he loan in accordance wih a non-decreasing paymen funcion {A } n called he amorizaion funcion. The erm of he conrac, n, is someimes called he duraion of he loan. Wihou loss of generaliy we assume henceforh ha H = 1 (he principal is proclaimed moneary uni). The amorizaion funcion is o fulfill A = and A n 1. The excess of oal amorizaions over he principal is he oal amoun of ineres. We denoe i by R n and have A n = 1+R n. General principles of book-keeping, needed e.g. for axaion purposes, prescribe ha he decomposiion of he amorizaions ino repaymens and ineres be exended o all [, n]. Thus, A = F + R, (2.35) where F is a non-decreasing repaymen funcion saisfying F =, F n = 1

27 CHAPTER 2. PAYMENT STREAMS AND INTEREST 26 (formally a disribuion funcion due o he convenion H = 1), and R is a non-decreasing ineres paymen funcion. Furhermore, he conrac is required o specify a nominal force of ineres r, n, under which he value of he amorizaions should be equivalen o he value of he principal, ha is, n e τ r da τ = 1. (2.36) There are, of course, infiniely many admissible decomposiions (2.35) saisfying (2.36). A clue o consrains on F and R is offered by he relaionship n e τ r dr τ = n e τ r (1 F τ ) r τ dτ, (2.37) which is obained upon insering (2.35) ino (2.36) and hen using inegraion by pars on he erm n exp ( τ r) df τ = n exp ( τ r) d(1 F τ ). The condiion (2.37) is rivially saisfied if dr = (1 F ) r d, ha is, ineres is paid currenly and insananeously on he ousanding (par of he) principal, 1 F. This will be referred o as naural ineres. Under he scheme of naural ineres he relaion (2.35) becomes da = df + (1 F ) r d, (2.38) which esablishes a one-o-one correspondence beween amorizaions and repaymens. The differenial equaion (2.38) is easily solved: Firs, inegrae (2.38) over (, ] o obain A = F + (1 F τ ) r τ dτ, (2.39) which deermines amorizaions when repaymens ( are given. Second, muliply (2.38) wih exp ) ( r o obain exp ) r da = ( ( d exp ) ) r (1 F ) and hen inegrae over (, n] o arrive a n e τ r da τ = 1 F, (2.4) which deermines (ousanding) repaymens when amorizaions are given. The relaionships (2.39) and (2.4) are easy o inerpre. For insance, since 1 F is he remaining deb a ime, (2.4) is he ime updae of he equivalence requiremen (2.36). When i comes o numerical compuaion, he inegral expressions in (2.39) and (2.4) are no so useful, however. Wheher we wan o compue A for given F or he oher way around, we would use he differenial equaion ( 2.38).

28 CHAPTER 2. PAYMENT STREAMS AND INTEREST 27 B. Sandard forms of loans. We lis some sandard ypes of loans, aking now r consan. I is undersood ha we consider only imes in [, n]. The simples form is he fixed loan, which is repaid in is enirey only a he erm of he conrac, ha is, F = ε n (), he endowmen defined by (2.27). The amorizaion funcion is obained direcly from (2.39): A = ε n () + r. A series loan has repaymens of annuiy form. The coninuous version is given by F = /n, see (2.3). The amorizaion plan is obained from (2.39): A = /n + r(1 /2n). Thus, df /d = 1/n (fixed) and dr /d = r(1 /n) (linearly decreasing). An annuiy loan is called so because he amorizaions, which are he amouns acually paid by he borrower, are of annuiy form. The coninuous version is given by A = /ā n, see (2.36) and (2.31). From (2.4) we easily obain F = 1 ā n /ā n. We find df /d = e r(n ) /ā n (exponenially increasing), and dr /d = (1 e r(n ) )/ā n. Puing n =, he fixed loan and he series loan boh specialize o an infinie loan wihou repaymen. Amorizaions consis only of ineres, which is paid indefiniely a rae r.

29 Chaper 3 Moraliy 3.1 Aggregae moraliy A. The sochasic model. Consider an aggregae of individuals, e.g. he populaion of a naion, he persons covered under an insurance scheme, or a cerain species of animals. The individuals need no be animae beings; for insance, in engineering applicaions one is ofen ineresed in sudying he worklife unil failure of echnical componens or sysems. Having demographic and acuarial problems in mind, we shall, however, be speaking of persons and life lenghs unil deah. Due o differences in inheriance and living condiions and also due o evens of a more or less purely random naure, like accidens, diseases, ec., he life lenghs vary among individuals. Therefore, he life lengh of a randomly seleced new-born can suiably be represened by a non-negaive random variable T wih a cumulaive disribuion funcion F () = P[T ]. (3.1) In survival analysis i is convenien o work wih he survival funcion F () = P[T > ] = 1 F (). (3.2) Fig. 3.1 shows F and F for he moraliy law G82M used by Danish life insurers as a basis for calculaing premiums for insurances on male lives. Find he median life lengh and some oher perceniles of his life disribuion by inspecion of he graphs! We assume ha F is absoluely coninuous and denoe he densiy by f; f() = d d F () = d d F (). (3.3) The densiy of he disribuion in Fig. 3.1 is depiced in Fig Find he mode by inspecion of he graph! Can you already a his sage figure why he median and he mode of F in Fig. 3.1 appear o exceed hose of he moraliy law of he Danish male populaion? 28

30 CHAPTER 3. MORTALITY Figure 3.1: The G82M moraliy law: F broken line, F whole line Figure 3.2: The densiy f for he G82M moraliy law.

31 CHAPTER 3. MORTALITY Figure 3.3: The force of moraliy µ for he G82M moraliy law. B. The force of moraliy. The densiy is he derivaive of F, see (3.3). When dealing wih non-negaive random variables represening life lenghs, i is convenien o work wih he derivaive of ln F, µ() = d d { ln F ()} = f() F (), (3.4) which is well defined for all such ha F () >. For small, posiive d we have µ()d = f()d F () = P[ < T + d] P[T > ] = P[T + d T > ]. (In he second equaliy we have negleced a erm o(d) such ha o(d)/d as d.) Thus, for a person aged, he probabiliy of dying wihin d years is (approximaely) proporional o he lengh of he ime inerval, d. The proporionaliy facor µ() depends on he aained age, and is called he force of moraliy a age. I is also called he moraliy inensiy or hazard rae a age, he laer expression semming from reliabiliy heory, which is concerned wih he durabiliy of echnical devices. Fig 3.3 shows he force of moraliy corresponding o F in Fig Assess roughly he probabiliy ha a years old person will die wihin one year for = 6, 7, 8, 9! Inegraing (3.4) from o and using F () = 1, we obain Relaion (3.4) may be cas as F () = e µ. (3.5) f() = F ()µ() = e µ µ(), (3.6) which says ha he probabiliy f()d of dying in he age inerval (, +d) is he produc of he probabiliy F () of survival o and he condiional probabiliy µ()d of hen dying before age + d. The funcions F, F, f, and µ are equivalen represenaions of he moraliy law; each of hem corresponds one-o-one o any one of he ohers. Since F ( ) =, we mus have µ =. Thus, if here is a finie highes aainable age ω such ha F (ω) = and F () > for < ω, hen µ as ω. If, moreover, µ is non-decreasing, we mus also have lim ω µ() =.

32 CHAPTER 3. MORTALITY 31 C. The disribuion of he remaining life lengh. Le T x denoe he remaining life lengh of an individual chosen a random from he x years old members of he populaion. Then T x is disribued as T x, condiional on T > x, and has cumulaive disribuion funcion F ( x) = P[T x + T > x] = F (x + ) F (x) 1 F (x) and survival funcion F ( x) = P[T > x + T > x] = F (x + ) F (x), (3.7) which are well defined for all x such ha F (x) >. The densiy of his condiional disribuion is f( x) = f(x + ) F (x). (3.8) Denoe by µ( x) he force of moraliy of he disribuion F ( x). I is obained by insering f( x) from (3.8) and F ( x) from (3.7) in he places of f and F in he definiion (3.4). We find µ( x) = f(x + )/ F (x + ) = µ(x + ). (3.9) Alernaively, we could inser (3.5) ino (3.7) o obain F ( x) = e x+ µ(y) x dy = e µ(x+τ)dτ, (3.1) which by he general relaion (3.5) enails (3.9). Relaion (3.9) explains why he force of moraliy is paricularly handy; i depends only on he aained age x +, whereas he condiional densiy in (3.8) depends in general on x and in a more complex manner. Thus, he properies of all he condiional survival disribuions are summarized by one simple funcion of he oal age only. Figs depic he funcions F ( 7), F ( 7), f( 7), and µ( 7) = µ(7 + ) derived from he life ime disribuion in Fig Observe ha he firs hree of hese funcions are obained simply by scaling up he corresponding graphs in Figs by he facor 1/ F (7) over he inerval (7, ). The force of moraliy remains unchanged, however. D. Expeced values in life disribuions. Le T be a non-negaive random variable wih absoluely coninuous disribuion funcion F, and le G : R + R be a PD and RC funcion such ha E[G(T )] exiss and is finie. Inegraing by pars in he defining expression E[G(T )] = G(τ) df (τ),

33 CHAPTER 3. MORTALITY Figure 3.4: Condiional disribuion of remaining life lengh for he G82M moraliy law: F ( 7) broken line, F ( 7) whole line Figure 3.5: Condiional densiy of remaining life lengh f( 7) for he G82M moraliy law Figure 3.6: The force of moraliy µ( 7) = µ(7 + ), >, for he G82M moraliy law.

34 CHAPTER 3. MORTALITY 33 we find Taking G() = k we ge and, in paricular, E[G(T )] = G() + F (τ) dg(τ). (3.11) E[T k ] = k E[T ] = k 1 F () d, (3.12) F ()d. (3.13) The expeced remaining life ime for an x years old person is ē x = F ( x) d. (3.14) From (3.1) i is seen ha F ( x) is a decreasing funcion of x for fixed if µ is an increasing funcion. Then ē x is a decreasing funcion of x. One can easily consruc moraliy laws for which F ( x) and ē x are no decreasing funcions of x. Consider he more general funcion G() = (( b) ( a)) k =, < a, ( a) k, a < b, (b a) k, b, (3.15) ha is, dg() = k( a) k 1 d for a < < b and elsewhere. I is realized ha G(T ) is he kh power of he number of years lived beween age a and age b. From (3.11) we obain E[G(T )] = k b a ( a) k 1 F () d, (3.16) In paricular, he expeced number of years lived beween he ages of a and b is b F a () d, which is he area beween he -axis and he survival funcion in he inerval from a o b. The formula can be moivaed direcly by noing ha F () d is he expeced number of years survived in he small ime inerval (, + d) and using ha he expeced value of he sum is he sum of he expeced values. 3.2 Some sandard moraliy laws A. The exponenial disribuion. Suppose he force of moraliy is µ() = λ, independen of he age. This means here are no wear-ou effecs; each morning when you wake up (if you wake up) life sars anew wih he same prospecs of longeviy as for a new-born. Then he survival funcion (3.5) becomes F () = e λ, (3.17)

35 CHAPTER 3. MORTALITY 34 Figure 3.7: Two exponenial laws wih inensiies λ 1 and λ 2 such ha λ 1 < λ 2 ; F 1 and F 2 whole line, f 1 and f 2 broken line. and he densiy (3.6) becomes f() = λe λ. (3.18) Thus, T is exponenially disribued wih parameer λ. The condiional survival funcion (3.1) becomes F ( x) = e λ, hence F ( x) = F (), (3.19) he same as (3.17). The exponenial disribuion is a suiable model for cerain echnical devices like bulbs and elecronic componens. Unforunaely, i is no so ap for descripion of human lives. One could arrive a he exponenial disribuion by specifying ha (3.19) be valid for all x and, ha is, he probabiliy of surviving anoher years is independen of he age x. Then, from he general relaion (3.7) we ge F (x + ) = F (x) F () (3.2) for all non-negaive x and. I follows by inducion ha for each pair of posiive inegers m and n, F ( m n ) = F ( 1 n )m = F (1) m n, hence F () = F (1) (3.21) for all posiive raional. Since F is righ-coninuous, (3.21) mus hold rue for all >. Puing F (1) = e λ, we arrive a (3.17). Fig. 3.7 shows he survival funcion and he densiy for wo differen values of λ.

36 CHAPTER 3. MORTALITY 35 B. The Weibull disribuion. The inensiy of his disribuion is of he form µ() = βα β β 1, (3.22) α, β >. The corresponding survival funcion is F () = exp ( ( α )β). If β > 1, hen µ() is increasing, and if β < 1, hen µ() is decreasing. If β = 1, he Weibull law reduces o he exponenial law. Draw he graphs of F and f for some differen choices of α and β! We have µ(x + ) = βα β (x + ) β 1, and, by virue of (3.22), F ( x) is no a Weibull law. C. The Gomperz-Makeham disribuion. This disribuion is widely used as a model for survivorship of human lives, especially in he conex of life insurance. Thus, as i will be frequenly referred o, we shall use he acronym G-M for his law. Is moraliy inensiy is of he form µ() = α + βe γ, (3.23) α, β. The corresponding survival funcion is ( ) F () = exp (α + βe γs )ds = exp ( α β(e γ 1)/γ ). (3.24) If β > and γ >, hen µ() is an increasing funcion of. The consan erm α accouns for age-independen causes of deah like cerain accidens and epidemic diseases, and he erm βe γ accouns for all kinds of wear-ou effecs due o aging. We have µ(x + ) = α + βe γx e γ, and so (3.23) shows ha F ( x) is also a G-M survival funcion wih parameers α, βe γx, γ. The special case α = is referred o as he (pure) Gomperz law. The G82M moraliy law depiced in Fig. 3.1 is he G-M law wih α = 5 1 4, β = , γ = ln(1.9144). (3.25) Table E.1 in Appendix E shows µ(), F () and f() for ineger. 3.3 Acuarial noaion A. Acuaries in all counries unie! The Inernaional Associaion of Acuaries (IAA) has laid down a sandard noaion, which is generally acceped among acuaries all over he world. Familiariy wih his noaion is a mus for anyone who wans o communicae in wriing or reading wih acuaries, and we shall henceforh adop i in hose simple siuaions where i is applicable.

37 CHAPTER 3. MORTALITY 36 B. A lis of some sandard symbols. According o he IAA sandard, he quaniies inroduced so far are denoed as follows: q x = F ( x), (3.26) p x = F ( x), (3.27) µ x+ = µ(x + ). (3.28) In paricular, q = F () and p = F (). One-year deah and survival probabiliies are abbreviaed as q x = 1 q x, p x = 1 p x. (3.29) Frequenly used is also he n-year deferred probabiliy of deah wihin m years, n mq x = m+n q x n q x = n p x m+n p x = n p x m q x+n. (3.3) The formulas in Secion 3.1 are easily ranslaed, e.g. p x = exp( µ x+τ dτ), (3.31) f( x) = p x µ x+, (3.32) ē x = p x d. (3.33) Frequenly acuaries work wih expeced numbers of survivors insead of probabiliies. Consider a populaion of l new-born who are subjec o he same law of moraliy given by (3.28). The expeced number of survivors a age x is l x = l x p. (3.34) The funcion {l x ; x > } is called he decremen funcion or, when considered only a ineger values of x, he decremen series. Expressed in erms of he decremen funcion we find e.g. p x = l x+ /l x, (3.35) µ x+ = l x+/l x+, (3.36) f( x) = l x+ /l x, (3.37) ē x = l x+ d/l x. (3.38) The pieces of IAA noaion we have shown here are quie pleasing o he eye and also space-saving; for insance, he symbol on he lef of (3.27) involves hree ypographical eniies, whereas he one on he righ involves six.

38 CHAPTER 3. MORTALITY Selec moraliy A. The insurance porfolio consiss of seleced lives. Consider an individual who purchases a life insurance a age x. In shor, he will be referred o as (x) in wha follows. I is quie common in acuarial pracice o assume ha he force of moraliy of (x) depends on x and in a more complex manner han he simple relaionship (3.9), which resed on he assumpion ha (x) is chosen a random from he x years old individuals in he populaion. The fac ha (x) purchases insurance adds informaion o he mere fac ha he has aained age x; he does no represen a purely random draw from he populaion, bu is raher seleced by some mechanisms. I is easy o hink of examples of such mechanisms. For insance ha poor people can no afford o buy insurance and, o he exen ha longeviy depends on economic siuaion, he moraliy experience for insured people would reflec ha hey are wealhy enough o buy insurance ( survival of he faes ). Judging from exbooks on life insurance, e.g. [4] and [29] and many ohers, i seems ha he underwriing sandards of he insurer are generally held o be he predominan selecive mechanism; before an insurance policy is issued, he insurer mus be saisfied ha he applican mees cerain requiremens wih regard o healh, occupaion, and oher facors ha are assumed o deermine he prospecs of longeviy. Only firs class lives are eligible o insurance a ordinary raes. Thus here is every reason o accoun for selecion effecs by leing he force of moraliy be some more general funcion µ x () or, in oher words, specify ha T x follows a survival funcion F x () ha is no necessarily of he form (3.7). One hen speaks of selec moraliy. B. More of acuarial noaion. The sandard acuarial noaion for selec moraliy is τ q [x]+ = P[T x + τ T x > ], (3.39) τ p [x]+ = P[T x > + τ T x > ], (3.4) hq [x]+ µ [x]+ = lim. (3.41) h h The idea is ha he boh he curren age, x +, and he age a enry, x, are direcly visible in [x] +. From a echnical poin of view selec moraliy is jus as easy as aggregae moraliy; we work wih he disribuion funcion q [x] insead of q x, and are ineresed in i as a funcion of. For insance, τ p [x]+ = +τ p [x] p [x] ( +τ ) = exp µ [x]+s ds. C. Feaures of selec moraliy. There is ample empirical evidence o suppor he following facs abou selec moraliy in life insurance populaions:

39 CHAPTER 3. MORTALITY 38 For insured lives of a given age he rae of moraliy usually increases wih increasing duraion. The effec of selecion ends o decrease wih increasing duraion and becomes negligible for pracical purposes when he duraion exceeds a cerain selec period. The moraliy among insured lives is generally lower han he moraliy in he populaion. There are many possible ways of building such feaures ino he model. For insance, one could modify he aggregae G-M inensiy as µ [x]+ = α() + β() e γ (x+), where α and β are non-negaive and non-decreasing funcions bounded from above. In Secion 7.6 we shall show how he selecion mechanism can be explained in models ha describe more aspecs of he individual life hisories han jus survival and deah.

40 Chaper 4 Insurance of a single life 4.1 Some sandard forms of insurance A. The single-life saus. Consider a person aged x wih remaining life lengh T x as described in he previous secion. In acuarial parlance his life is called he single-life saus (x). Referring o Appendix B, we inroduce he indicaor of he even of survival in years, I = 1[T x > ]. This is a binomial random variable wih success probabiliy p x. The indicaor of he even of deah wihin years is 1 I = 1[T x ], which is a binomial variable wih success probabiliy q x = 1 p x. (We apologize for someimes using echnical erms where hey may sound misplaced.) Noe ha, being or 1, any indicaor 1[A] saisfies 1[A] q = 1[A] for q >. The presen secion liss some sandard forms of insurance ha (x) can purchase, invesigaes some of heir properies, and presens some basic acuarial mehods and formulas. We assume ha he invesmens of he insurance company yield ineres a a fixed rae r so ha accumulaion and discouning ake place in in accordance wih (2.12). B. The pure endowmen insurance. An n-year pure (life) endowmen of 1 is a uni ha is paid o (x) a he end of n years if he is hen sill alive. In oher words, he associaed paymen funcion is an endowmen of I n a ime n. Is presen value a ime is The expeced value of P V e;n, denoed by n E x, is P V e;n = e rn I n. (4.1) ne x = e rn np x. (4.2) For any q > we have (P V e;n ) q = e qrn I n (recall ha I q n = I n), and so he q-h non-cenral momen of P V e;n can be expressed as E[(P V e;n ) q ] = n E (qr) x, (4.3) 39

41 CHAPTER 4. INSURANCE OF A SINGLE LIFE 4 where he op-scrip (qr) signifies ha discouning is made under a force of ineres ha is qr. In paricular, he variance of P V e;n is V[P V e;n ] = n E (2r) x n E 2 x. (4.4) C. The life assurance. A life assurance conrac specifies ha a cerain amoun, called he sum insured, is o be paid upon he deah of he insured, possibly limied o a specified period. We shall here consider only insurances payable immediaely upon deah, and ake he sum o be 1 (jus a maer of noaion). Firs, an n-year erm insurance is payable upon deah wihin n years. The paymen funcion is a lump sum of 1 I n a ime T x. Is presen value a ime is The expeced value of P V i;n is and, similar o (4.3), In paricular, P V i;n = e rtx (1 I n ). (4.5) Ā 1x n = n e rτ τp x µ x+τ dτ, (4.6) E[(P V i;n ) q ] = Ā(qr) 1. (4.7) x n V[P V i;n ] = Ā(2r) 1 x n Ā2 1 x n. (4.8) An n-year endowmen insurance is payable upon deah if i occurs wihin ime n and oherwise a ime n. The paymen funcion is a lump sum of 1 a ime T x n. Is presen value a ime is The expeced value of P V ei;n is P V ei;n = e r(tx n). (4.9) and Ā x n = n e rτ τp x µ x+τ dτ + e rn np x = Ā1 x n + ne x, (4.1) E(P V ei;n ) q = Ā(qr) x n. (4.11) I follows ha V[P V ei;n ] = Ā(2r) x n Ā2 x n. (4.12)

42 CHAPTER 4. INSURANCE OF A SINGLE LIFE 41 D. The life annuiy. An n-year emporary life annuiy of 1 per year is payable as long as (x) survives bu limied o n years. We consider here only he coninuous version. Recalling (2.3), he associaed paymen funcion is an annuiy of 1 in T x n years. Is presen value a ime is The expeced value of P V a;n is P V a;n = ā Tx n = 1 e r(tx n) r ā x n = A more appealing formula is n ā x n = ā τ τ p x µ x+τ dτ + ā n n p x. n. (4.13) e rτ τp x dτ, (4.14) which displays he life annuiy as a coninuum of life endowmens, ā x n = n τ E x dτ. There are several ways of proving (4.14). Using brue force, one can inegrae by pars: ā n n p x = ā p x + = n n e rτ τ p x dτ d dτ āτ τp x dτ + n n ā τ ā τ τ p x µ x+τ dτ. d dτ τ p x dτ Using he brain insead, one realizes ha he expeced presen value a ime of he paymens in any small ime inerval (τ, τ + dτ) is e rτ dτ τ p x, and summing over all ime inervals one arrives a (4.14) ( he expeced value of a sum is he sum of he expeced values ). This kind of reasoning will be omnipresen hroughou he ex, and would also immediaely produce formula (4.6) and (4.1). The recipe is: Find he expeced presen value of he paymens in each small ime inerval and add up. We shall demonsrae below ha E[(P V a;n ) q ] = q r q 1 q ( ) q 1 ( 1) p 1 ā (pr) x n p 1, (4.15) p=1 from which we derive V[P V a;n ] = 2 ( ) ā x n ā (2r) x n ā 2 x n r. (4.16) The endowmen insurance is a combined benefi consising of an n-year erm insurance and an n-year pure endowmen. By (4.9) and (4.13) i is relaed o he life annuiy by P V a;n = 1 P V ei;n r or P V ei;n = 1 rp V a;n, (4.17)

43 CHAPTER 4. INSURANCE OF A SINGLE LIFE 42 which jus reflecs he more general relaionship (2.31). Taking expecaion in (4.17), we ge Ā x n = 1 rā x n. (4.18) Also, since P V i;n = P V ei;n P V e;n = 1 rp V a;n P V e;n, we have Ā 1x n = 1 rā x n n E x. (4.19) The formerly announced resul (4.15) follows by operaing wih he q-h momen on he firs relaionship in (4.17), and hen using (4.12) and (4.18) and rearranging a bi. One needs he binomial formula (x + y) q = q p= ( ) q x q p y p p and he special case q ( q p= p) ( 1) q p = (for x = 1 and y = 1). A whole-life annuiy is obained by puing n =. Is expeced presen value is denoed simply by ā x and is obained by puing n = in (4.14), ha is ā x = e rτ τp x dτ, (4.2) and he same goes for he variance in (4.16) (jusify he limi operaions). E. Deferred benefis. An m-year deferred n-year emporary life annuiy commences only afer m years, provided ha (x) is hen sill alive, and is payable hroughou n years hereafer as long as (x) survives. The presen value of he benefis is P V = P V a;m+n P V a;m = ā Tx (m+n) ā T x m The expeced presen value is = e r(tx m) e r(tx (m+n)) r (4.21) m nā x = ā x m+n ā x m = m+n m e r p x d = m E x ā x+m n. (4.22) The las expression can be obained also by he rule of ieraed expecaion, and we carry hrough his small exercise jus o illusrae he echnique: E [P V ] = E E [P V I m ] = m p x E [P V I m = 1] + m q x E [P V I m = ] = m p x v m ā x+m n. An m-year deferred whole life annuiy is obained by puing n =. The expeced value is denoed by m ā x.

44 CHAPTER 4. INSURANCE OF A SINGLE LIFE 43 Deferred life assurances, alhough less common in pracice, are defined likewise. For insance, an m-year deferred n-year erm assurance of 1 is payable upon deah in he ime inerval (m, m + n]. Is presen value a ime is and is expeced presen value is P V = P V i;m+n P V i;m, (4.23) m+n m nā1 = x Ā1 x m+n Ā1 = me x Ā,. = 1 e rτ τ p x µ x+τ dτ. (4.24) x m x+m n m F. Compuaional aspecs. Disribuion funcions of presen values and many oher funcions of ineres can be calculaed easily; afer all here is only one random variable in play, and finding expeced values amouns jus o forming inegrals in one dimension. We shall, however, no pursue his approach because i will urn ou ha a differen poin of view is needed in more complex siuaions o be sudied in he sequel. Table 4.1: Expeced value (E), coefficien of variaion (CV), and skewness (SK) of he presen value a ime of a pure endowmen (PE) wih sum 1, a erm insurance (TI) wih sum 1, an endowmen insurance (EI) wih sum 1, and a life annuiy (LA) wih level inensiy 1 per year, when x = 3, n = 3, µ is given by (3.25), and r = ln(1.45). PE TI EI LA E CV SK Anyway, by mehods o be developed laer, we easily compue he hree firs momens of he presen values considered above, and find heir expeced values, coefficiens of variaion, and skewnesses shown in Table 4.1. The reader should conemplae he resuls, keeping in mind ha he coefficien of variaion may be aken as a simple measure of riskiness. We inerpose ha numerical echniques will be dominan in our conex. Explici formulas canno be obained even for rivial quaniies like ā x n under he Gomperz-Makeham law (3.23); age dependence and oher forms of inhomogeneiy of basic eniies leave lile room for aesheics in acuarial science. Also relaionships like (4.18) are of limied ineres; hey are cerainly no needed for compuaional purposes, bu may provide some general insigh. 4.2 The principle of equivalence A. A noe on erminology. Like any oher good or service, insurance coverage is bough a some price. And, like any oher business, an insurance company

45 CHAPTER 4. INSURANCE OF A SINGLE LIFE 44 mus fix prices ha are sufficien o defray he coss. In one respec, however, insurance is differen: for obvious reasons he cusomer is o pay in advance. This circumsance is refleced by he insurance erminology, according o which paymens made by he insured are called premiums. This word has he posiive connoaion prize (reward), raher anonymous o price (sacrifice, due), bu he eymological background is, of course, ha premium means firs (French: prime). B. The equivalence principle. The equivalence principle of insurance saes ha he expeced presen values of premiums and benefis should be equal. Then, roughly speaking, premiums and benefis will balance on he average. This idea will be made precise laer. For he ime being all calculaions are made on an individual ne basis, ha is, he equivalence principle is applied o each individual policy, and wihou regard o expenses incurring in addiion o he benefis specified by he insurance reaies. The resuling premiums are called (individual) ne premiums. The premium rae depends on he premium paymen scheme. In he simples case, he full premium is paid as a single amoun immediaely upon he incepion of he policy. The resuling ne single premium is jus he expeced presen value of he benefis, which for basic forms of insurance is given in Secion 4.1. The ne single premium may be a considerable amoun and may easily exceed he liquid asses of he insured. Therefore, premiums are usually paid by a series of insallmens exending over some period of ime. The mos common soluion is o le a fixed level amoun fall due periodically, e.g. annually or monhly, from he incepion of he agreemen unil a specified ime m and coningen on he survival of he insured. Assume for he presen ha he premiums are paid coninuously a a fixed level rae π. (This is admiedly an arificial assumpion, bu i can serve well as an approximae descripion of periodical paymens, which will be reaed laer.) Then he premiums form an m-year emporary life annuiy, payable by he insured o he insurer. Is presen value is πp V a;m, wih expeced value πā x m given by (4.14). We lis formulas for he ne level premium rae for a selecion of basic forms of insurance: For he pure endowmen (Paragraph 4.1.B) agains level premium in he insurance period, π = n E x ā x n. (4.25) For he m-year deferred n-year emporary annuiy (Paragraph 4.1.E) agains level premium in he deferred period, π = m n ā x ā x m = āx m+n ā x m 1. (4.26) For he erm insurance (Paragraph 4.1.C) agains level premium in he insurance period, π = Ā 1x n ā x n. (4.27)

46 CHAPTER 4. INSURANCE OF A SINGLE LIFE 45 For he endowmen insurance (Paragraph 4.1.C) agains level premium in he insurance period, π = Āx n = 1 r, (4.28) ā x m ā x n he las expression following from (4.18). C. The ne economic resul for a policy. The random variables sudied in Secion 4.1 represen he uncerain fuure liabiliies of he insurer. Now, unless single premiums are used, also he premium incomes are dependen on he insured s life lengh and become a par of he insurer s uncerainy. Therefore, he relevan random variable associaed wih an insurance policy is he presen value of benefis less premiums, P V = P V b πp V a;m, (4.29) where P V b is he presen value of he benefis, e.g. P V ei;n in he case of an n-year endowmen insurance. Saed precisely, he equivalence principle lays down ha E[P V ] =. (4.3) For example, wih P V b = P V ei;n (4.3) becomes = Āx n πā xm, which yields (4.28) when m = n. A measure of he uncerainy associaed wih he economic resul of he policy is he variance V[P V ]. For example, wih P V b =P V ei;n and m = n, V[P V ] = V [v Tx n π 1 ] vtx n = (1 + π/r) 2 V[v Tx n ] r ( ) 2 ā x n ā (2r) x n = 1. (4.31) rā 2 x n 4.3 Prospecive reserves A. The case. We shall discuss he noion of reserve in he framework of a combined insurance which comprises all sandard forms of coningen paymens ha have been sudied so far and, herefore, easily specializes o any of hose. The insured is x years old upon issue of he conrac, which is for a erm of n years. The benefis consis of a erm insurance wih sum insured b payable upon deah a ime (, n) and a pure endowmen wih sum b n payable upon survival a ime n. The premiums consis of a lump sum π payable immediaely upon he incepion of he policy a ime, and hereafer an annuiy payable coninuously a rae π per ime uni coningen on survival a ime (, n). As before, assume ha he ineres rae is a deerminisic funcion r.

47 CHAPTER 4. INSURANCE OF A SINGLE LIFE 46 The expeced presen value a ime of oal benefis less premiums under he conrac can be pu up direcly as he sum of he expeced discouned paymens in each small ime inerval: π + n e τ r τ p x {µ x+τ b τ π τ } dτ + b n e n r np x. (4.32) Under he equivalence principle his is se equal o, a consrain on he premium funcion π. B. Definiion of he reserve. The expeced value (4.32) represens, in an average sense, an assessmen of he economic prospecs of he policy a he ouse. A any ime > in he subsequen developmen of he policy he assessmen should be updaed wih regard o he informaion currenly available. If he policy has expired by deah before ime, here is nohing more o be done. If he policy is sill in force, a renewed assessmen mus be based on he condiional disribuion of he remaining life lengh. Insurance legislaion lays down ha a any ime he insurance company mus provide a reserve o mee fuure ne liabiliies on he conrac, and his reserve should be precisely he expeced presen value a ime of oal benefis less premiums in he fuure. Thus, if he policy is sill in force a ime, he reserve is V = n e τ r τ p x+ {µ x+τ b τ π τ } dτ + b n e n r n p x+. (4.33) More precisely, his quaniy is called he prospecive reserve a ime since i looks ahead. Under he principle of equivalence i is usually called he ne premium reserve. We will ake he libery o jus speak of he reserve. Upon insering τ p x+ = e τ µ x+s ds, (4.33) assumes he form V = n e τ (rs+µx+s) ds {µ x+τ b τ π τ } dτ + e τ (rs+µx+s) ds b n. (4.34) Glancing behind a (2.14), we see ha, formally, he expression in (4.34) is he reserve a ime for a deerminisic conrac wih paymens given by B = π (comes from seing (4.32) equal o ), db = (µ x+ b π ) d, < < n and B n = b n, and wih ineres rae r + µ x+. We can, herefore, reuse he relaionships in Chaper 2. For insance, by (2.26) and (2.13), we have he following rerospecive formula for he prospecive premium reserve: V = e (rs+µx+s) ds π + e τ (rs+µx+s) ds (π τ µ x+τ b τ ) dτ. (4.35) This formula expresses V as he surplus of ransacions in he pas, accumulaed a ime wih he benefi of ineres and survivorship.

48 CHAPTER 4. INSURANCE OF A SINGLE LIFE 47 1 ne x n Figure 4.1: The ne reserve for an n-year pure endowmen of 1 agains single ne premium. C. Some special cases. The ne reserve is easily pu up for he various forms of insurance reaed in Secions 4.1 and 4.2. We assume ha he ineres rae is consan and ha premiums are based on he equivalence principle, which can be expressed as V = π. (4.36) Firs, for he pure endowmen agains single ne premium n E x colleced a ime, V = n E x+, < n. (4.37) The graph of V will ypically look as in Fig A poins of disconinuiy a do marks he value of he funcion. If premiums are payable coninuously a level rae π given by (4.25) hroughou he insurance period, hen V = n E x+ πā x+ n = n E x+ n E x ā x n ā x+ n. (4.38) A ypical graph of his funcion is shown in Fig Nex, for an m-year deferred whole life annuiy agains level ne premium π given by (4.26), V = { m ā x+ πā x+ m, < < m, ā x+, m, = ā x+ ā x+ m āx ā x m ā x m ā x+ m = ā x+ āx ā x m ā x+ m (4.39)

49 CHAPTER 4. INSURANCE OF A SINGLE LIFE 48 1 n Figure 4.2: The ne reserve for an n-year pure endowmen of 1 agains level premium in he insurance period. ā x+m m Figure 4.3: The ne reserve for an m-year deferred whole life annuiy agains level premium in he deferred period.

50 CHAPTER 4. INSURANCE OF A SINGLE LIFE 49 n Figure 4.4: The ne reserve for an n-year erm insurance agains level premium in he insurance period 1 n Figure 4.5: The ne reserve for an n-year endowmen insurance wih level premium payable in he insurance period.

51 CHAPTER 4. INSURANCE OF A SINGLE LIFE 5 (wih he undersanding ha ā x m = if > m). A ypical graph of his funcion is shown in Fig For he n-year erm insurance agains level ne premium π given by (4.27), V = Ā 1 x+ n πā x+ n = 1 rā x+ n n E x+ 1 rā x n n E x ā ā x+ n x n = 1 n E x+ (1 n E x )āx+ n ā x n. (4.4) A ypical graph of his funcion is shown in Fig Finally, for he n-year endowmen insurance agains level ne premium π given by (4.28), V = Āx+ n πā x+ n = 1 rā x+ n 1 rā x n ā x n ā x+ n = 1 āx+ n ā x n. (4.41) A ypical graph of his funcion is shown in Fig The reserve in (4.41) is, of course, he sum of he reserves in (4.39) and (4.4). Noe ha he pure erm insurance requires a much smaller reserve han he oher insurance forms, wih elemens of savings in hem. However, a old ages x (where people ypically are no covered agains he risk of deah since deah will incur soon wih cerainy) also he erm insurance may have a V close o 1 in he middle of he insurance period. D. Non-negaiviy of he reserve. In all he examples given here he ne reserve is skeched as a non-negaive funcion. Non-negaiviy of V is no a consequence of he definiion. One may easily consruc premium paymen schemes ha lead o negaive values of V (jus le he premiums fall due afer he paymen of he benefis), bu such paymen schemes are no used in pracice. The reason is ha he holder of a policy wih V < is in expeced deb o he insurer and would hus have an incenive o cancel he policy and hereby ge rid of he deb. (The agreemen obliges he policy-holder only o pay he premiums, and he conrac can be erminaed a any ime he policy-holder wishes.) Therefore, i is in pracice required ha V,. (4.42) E. The reserve considered as a funcion of ime. We will now ake a closer look a he prospecive reserve as a funcion of ime, bearing in mind ha i should be non-negaive. The building blocks are he expeced presen values n E x+, ā x+:n, Ā x+ n, and Ā appearing in he formulas in 1 Secion 4.3. x+ n

52 CHAPTER 4. INSURANCE OF A SINGLE LIFE 51 Firs, n E x+ = e n (r+µx+s) ds is seen o be an increasing funcion of no maer wha are he ineres rae and he moraliy rae. The derivaive is d d n E x+ = n E x+ (r + µ x+ ). We inerpose here ha nohing is changed if r depends on ime. The expressions above show ha, for his pure survival benefi, r and µ play idenical pars in he expeced presen value. Thus, moraliy beques acs as an increase of he ineres rae. Nex consider n ā x+:n = e τ (r+µx+s) ds dτ. The following inequaliies are obvious: ā x+:n 1 r + inf s µ x+s 1 r. The las expression is jus he presen value of a perpeuiy, (2.32). If µ is an increasing funcion, hen We find he derivaive d ā x+:n 1 r + µ x+. dāx+:n = (r + µ x+)ā x+:n 1. I follows ha ā x+:n is a decreasing funcion of if µ is increasing, which is quie naural. You can easily inven an example where ā x+:n is no decreasing. From he ideniy Ā x+ n = 1 rā x+:n we conclude ha Āx+ n is an increasing funcion of if µ is increasing. For Ā = 1 rā 1 x+:n n E x+ x+ n no general saemen can be made as o wheher i is decreasing or increasing. Looking back a he formulas derived in Paragraph C above, we can conclude ha he reserve for he pure life endowmen agains single premium, (4.37), is always increasing. Assume henceforh ha µ is increasing, as is usually he case a ages when people are insured and cerainly holds for he Gomperz- Makeham law. Then also he reserve (4.38) for he pure life endowmen agains level premium during he erm of he conrac is increasing, and he same is he case for he reserve (4.41) of he endowmen insurance. I is lef o he diligen

53 CHAPTER 4. INSURANCE OF A SINGLE LIFE 52 reader o show ha he reserve in (4.39) is increasing hroughou he deferred period and hereafer urns decreasing. This is bes done by examining (4.33) and (4.35) in he cases m and m, respecively. I follows in paricular ha he reserve is non-negaive. The same rick serves also o show ha (4.4) is firs increasing and hereafer decreasing. 4.4 Thiele s differenial equaion A. The differenial equaion. We urn back o he general case wih he reserve given by (4.33) or (4.33), he laer being he more convenien since we can draw on he resuls in Chaper 2. The differenial form (2.19) ranslaes o he celebraed Thiele s differenila equaion, d d V = π b µ x+ + (r + µ x+ ) V, (4.43) valid a each where b, π, and µ are coninuous. The righ hand side expression in (4.43) shows how he fund per surviving policy-holder changes per ime uni a ime. I is increased by he excess of premiums over benefis (which may be negaive, of course), by he ineres earned, rv, and by he fund inheried from hose who die, µ x+ V. When combined wih he boundary condiion V n = b n, (4.44) he differenial equaion (4.43) deermines V for fixed b and π. If he principle of equivalence is exercised, hen we mus add he condiion (4.36). This represens a consrain on he conracual paymens b and π; ypically, one firs specifies he benefi b and hen deermines he premium rae for a given premium plan (shape of π). Thiele s differenial equaion is a so-called backward differenial equaion. This erm indicaes ha we ake our sand a he beginning of he ime inerval we are ineresed in and also ha he differenial equaion is o be solved by a backward scheme saring from he ulimo condiion (2.21). The differenial equaion may be pu up by he direc backward consrucion which goes as follows. Suppose he policy is in force a ime (, n). Use he rule of ieraed expecaion, condiioning on wha happens in he small ime inerval (, + d]: wih probabiliy µ x+ d + o(d) he insured dies, and he condiional expeced value is hen jus b ; wih probabiliy 1 µ x+ d + o(d) he insured survives, and he condiional expeced value is hen π d + e r d V +d. We gaher V = b µ x+ d π d + (1 µ x+ d)e r d V +d + o(d). (4.45) Subrac V +d on boh sides, divide by d and le d end o. Observing ha (e rd 1)/d r as d, one arrives a (4.43)

54 CHAPTER 4. INSURANCE OF A SINGLE LIFE 53 B. Savings premium and risk premium. Suppose he equivalence principle is in use. Rearrange (4.43) as π = d d V rv + (b V )µ x+. (4.46) This form of he differenial equaion shows how he premium a any ime decomposes ino a savings premium, and a risk premium, π s = d d V rv, (4.47) π r = (b V )µ x+. (4.48) The savings premium provides he amoun needed in excess of he earned ineres o mainain he reserve. The risk premium provides he amoun needed in excess of he available reserve o cover an insurance claim. C. Uses of he differenial equaion. In he examples given above, Thiele s differenial equaion was useful primarily as a means of invesigaing he developmen of he reserve. I was no required in he consrucion of he premium and he reserve, which could be pu up by direc prospecive reasoning. In he final example o be given Thiele s differenial equaion is needed as a consrucive ool. Assume ha he pension reay sudied above is modified so ha he reserve is paid back a he momen of deah in case he insured dies during he conrac period, he philosophy being ha he savings belong o he insured. Then he scheme is supplied by an (n + m)-year emporary erm insurance wih sum b = V a any ime (, m + n). The soluion o (4.43) is easily obained as { π s, < < m, V = bā m+n, m < < m + n, where s = (1 + i) τ dτ. The reserve develops jus as for ordinary savings conracs offered by banks. D. Dependence of he reserve on he conrac elemens. A small collecion of resuls due o Lidsone (195) and, in he ime-coninuous se-up, Norberg (1985), deal wih he dependence of he reserve on he conrac elemens, in paricular moraliy and ineres. The saring poin in he ime-coninuous case is Thiele s differenial equaion. For he sake of concreeness, we adop he model assumpions and he conrac described in Secion 4.4 and will refer o his as he sandard conrac. For ease of reference we fech Thiele s differenial from (4.43): d d V = π µ x+ b + (r + µ x+ ) V. (4.49)

55 CHAPTER 4. INSURANCE OF A SINGLE LIFE 54 The boundary condiion following from he very definiion of he reserve is V n = b n. (4.5) Wih premiums deermined by he principle of equivalence, we also have V = π, (4.51) where π is he lump sum premium paymen colleced upon he incepion of he policy (i may be, of course). Now consider a differen model wih ineres r and moraliy µ x+ and a differen conrac wih benefis b and premiums π. This will be referred o as he special conrac. The reserve funcion V under his conrac saisfies Assume ha d d V = π µ x+ b + (r + µ x+ ) V, (4.52) V n = b n, (4.53) V = π. (4.54) π = π, b n = b n. (4.55) We are ineresed in he difference V V, and a few words are in order o moivae his: The reserve is accouned as a liabiliy on he par of he insurance company. To be on he safe side, he company should, a any ime, provide a reserve in excess of wha seems likely o be needed. This is usually obained by using echnical elemens r and µ x+ elemens r and µ x+, and ha produce a reserve V V. Subrac (4.49) from (4.52) o ge ha are differen from he realisic bigger han he realisic where d d (V V ) = η + (r + µ x+) (V V ), (4.56) η = (π π ) + (µ x+ b µ x+ b ) + (r r + µ x+ µ x+) V. (4.57) Inegrae (4.56) from o, using V = V, o obain V V = e s (r +µ ) η s ds. Similarly, inegrae from o n, using V n = Vn, o obain V V = n e s (r +µ ) η s ds.

56 CHAPTER 4. INSURANCE OF A SINGLE LIFE 55 From hese relaions conclude: If here exiss a [, n] such ha η for < >, (4.58) hen V V for all. In paricular, his is he case if η is non-decreasing. The resul remains valid if all inequaliies are made sric. We can now prove he following: (1) For a conrac wih level premium inensiy hroughou he insurance period, and wih non-decreasing reserve, a uniform increase of he ineres rae resuls in a decrease of he reserve. Proof: Now r r = r is a posiive consan, π and π are boh consans, all oher elemens are unchanged, and V increasing. Then η = (π π)+ r V is increasing. (2) Consider an endowmen insurance wih fixed sum insured and level premium rae hroughou he insurance period. Prove ha a change of moraliy from µ o µ such ha µ µ is posiive and non-increasing, leads o a decrease of he reserve. Proof: Now (µ x+ µ x+) is posiive and decreasing (non-increasing), π and π are consans, b = b = b consan, V increasing (his is he case for he endowmen insurance if µ in increasing). Then, since V b, we have η = (π π) (µ x+ µ x+ )(b V ) is increasing. (3) Consider a policy wih no down premium paymen a ime and no life endowmen a ime n. Le he special conrac be he same as he sandard one, excep ha he special conrac charges so-called naural premium, π = b µ x+. Then V = for all, and (4.58) can be used o check wheher he reserve V is non-negaive (as i should be). Proof: Puing π = µ x+b, means premiums covers curren expeced benefis, so here is no accumulaion of reserve; V =. Now η = π + µ x+ b, so if his is increasing, hen = V V. This is he case e.g. if π and b are consans and µ is increasing. The reason why he impac on he reserve of a change in valuaion and/or conrac elemens is a bi involved is ha, under he equivalence principle, he premium is also affeced by he change. However, if we require ha he premium be consan as funcion of, hen (π π) appearing in he expression for η is consan and does no affec he monooniciy properies of η. Noe also ha, since V V sars and ends a, η canno be sricly posiive in some par of (, n) wihou being sricly negaive in some oher par.

57 CHAPTER 4. INSURANCE OF A SINGLE LIFE Probabiliy disribuions A. Moivaion. The basic paradigm being he principle of equivalence, life insurance mahemaics ceners on expeced presen values. The key ool is Thiele s differenial equaion, which describes he developmen of such expeced values and forms a basis for compuing hem by recursive mehods. In Chaper 7 we shall obain analogous differenial equaions for higher order momens, which will enable us o compue he variance, skewness, kurosis, and so on of he presen value of paymens under a fairly general insurance conrac. We shall give an example of how o deermine he probabiliy disribuion of a presen value, which is a he base of he momens and of any oher expeced values of ineres. Knowledge of his disribuion, and in paricular is upper ail, gives insigh ino he riskiness of he conrac beyond wha is provided by he mean and some higher order momens. The ask is easy for an insurance on a single life since hen he model involves only one random variable (he life lengh of he insured). De Pril [14] and Dhaene [16] offer a number of examples. In principle he ask is simple also for insurances involving more han one life or, more generally, a finie number of random variables. In such siuaions he disribuions of presen values (and any oher funcions of he random variables) can be obained by inegraing he finie-dimensional disribuion. B. A simple example. Consider he single life saus (x) wih remaining life ime T x disribued as described in Chaper 3. Suppose (x) buys an n year erm insurance wih fixed sum b and premiums payable coninuously a level rae π per year as long as he conrac is in force (see Paragraphs 4.1.C-D). The presen value of benefis less premiums on he conrac is U(T x ) = be rtx 1[ < T x < n] πā Tx n, where ā = e rτ dτ = (1 e r )/r is he presen value of an annuiy cerain payable coninuously a level rae 1 per year for years. The funcion U is non-increasing in T x, and we easily find he probabiliy disribuion, u < πā n, P[T x > n] ( )], πā n u < b e rn πā n, P[U u] = P [T x > 1r ln br+π ur+π, be rn (4.59) πā n u < b, 1, u b. The jump a πā n is due o he posiive probabiliy of survival o ime n. Similar effecs are o be anicipaed also for oher insurance producs wih a finie insurance period since, in general, here is a posiive probabiliy ha he policy will remain in he curren sae unil he conrac erminaes. The probabiliy disribuion in (4.59) is depiced in Fig. 4.6 for he G82M case wih r = ln(1.45) and µ( x) = (x+) when x = 3, n = 3, b = 1, and π =.4268 (he equivalence premium).

58 CHAPTER 4. INSURANCE OF A SINGLE LIFE 57 1 πā n b Figure 4.6: The probabiliy disribuion of he presen value of a erm insurance agains level premium. 4.6 The sochasic process poin of view A. The processes indicaing survival and deah. In Paragraph A of Secion 4.1 we inroduced he indicaor of he even of survival o ime, I = 1[T x > ], and he indicaor of he complemenary even of deah wihin ime, N = 1 I = 1[T x ]. Viewed as funcions of, hey are sochasic processes. The laer couns he number of deahs of he insured as ime is progresses and is hus a simple example of a couning process as defined in Paragraph D of Appendix A. This moivaes he noaion N. By heir very definiions, I and N are RC. In he presen conex, where everyhing is governed by jus one single random variable, T x, he process poin of view is no imporan for pracical purposes. For didacic purposes, however, i is worhwhile aking i already here as a rehearsal for more complicaed siuaions where sochasic processes canno be dispensed wih. The paymen funcions of he benefis considered in Secion 4.1 can be recas in erms of he processes I and N. In differenial form hey are Their presen values a ime are db e;n = I dε n (), db i;n = 1 (,n] () dn, db a;n = I 1 (,n) () d, db ei;n = db i;n + db e;n. P V e;n = e n r I n, P V i;n = n e τ r dn τ,

59 CHAPTER 4. INSURANCE OF A SINGLE LIFE 58 P V a;n = n e τ r I τ dτ, P V ei;n = V i;n + V e;n. The expressions in (4.14) and (4.1) are obained direcly by aking expecaion under he inegral sign, using he obvious relaions E [I τ ] = τ p x, E [dn τ ] = τ p x µ x+τ dτ. The relaionship (4.19) re-emerges in is more basic form upon inegraing by pars o obain e n r I n = 1 + n e τ r ( r τ )I τ dτ + and seing di = dn in he las inegral. n e τ r di τ,

60 Chaper 5 Expenses 5.1 A single life insurance policy A. Three caegories of expenses. Any firm has o defray expendiures in addiion o he ne producion coss of he commodiies or services i offers, and hese expenses mus be aken accoun of in he prices paid by he cusomers. Thus, he rae of premium charged for a given insurance conrac mus no merely cover he conracual ne benefis, bu also be sufficien o provide for all iems of expendiure conneced wih he operaions of he insurance company. For he sake of concreeness, and also of loyaly o sandard acuarial noaion, we shall inroduce he issue of expenses in he framework of he simple single life policy encounered in Chaper 4. To ge a case ha involves all main ypes of paymens, le us consider a life (x) who purchases an n-year endowmen insurance wih a fixed sum insured, b, and premium payable coninuously a level rae as long as he policy is in force. We recall ha he ne premium rae deermined by he principle of equivalence is π = bāx n ā x n ( 1 = b ā x n ) r, (5.1) and ha he corresponding ne premium reserve o be provided if he insured is alive a ime, is ( ) V = bāx+ n πā x+ n = b 1 āx+ n. (5.2) ā x n The erm ne means ne of adminisraion expenses. When expenses are included in he accouns, one will have o charge he policy wih a gross premium rae π, which obviously mus be greaer han he ne premium rae, and he gross premium reserve V o be provided if he policy is in force a ime will also in general differ from he ne premium reserve. The precise definiions of hese quaniies can only be made afer we 59

61 CHAPTER 5. EXPENSES 6 have made specific assumpions abou he srucure of he expenses, which we now urn o. The expenses are usually divided ino hree caegories. In he firs place here are he so-called α-expenses ha incur in connecion wih he esablishmen of he conrac. They comprise sales coss, including adverising and agen s commission, and coss conneced wih healh examinaion, issue of he policy, enering he deails of he conrac ino he daa files, ec. I is assumed ha hese expenses incur immediaely a ime and ha hey are of he form α + α b. (5.3) In he second place here are he so-called β-expenses ha incur in connecion wih collecion and accouning of premiums. They are assumed o incur coninuously a consan rae β + β π (5.4) hroughou he premium-paying period. Finally, in he hird place here are he so-called γ-expenses ha comprise all expendiures no included in he former wo caegories, such as wages o employees, ren, axes, fees, and mainenance of he business operaions in general. These expenses are assumed o incur coninuously a rae γ + γ b + γ V (5.5) a ime if he policy is hen in force. The consan erms α, β, and γ represen coss ha are he same for all policies. The erms α b, β π, and γ b represen coss ha are proporional o he size of he conrac as measured by he amouns specified in he policy. Typically his is he case for he agen s commission, which may be a considerable porion of he α-expenses on individual insurances sold in an open compeiive marke, and also for he deb collecor s or solicior s commission, which in former days made up he major par of he β-expenses. The erm γ V represens expenses in connecion wih managemen of he invesmen porfolio, which can reasonably be divided beween he policy-holders in proporion o heir curren reserves. B. The gross premium and he gross premium reserve. Upon exercising he equivalence principle in he presence of expenses, one will deermine he gross premium rae π and he corresponding gross premium reserve funcion V. When expenses depend on he reserve, as specified in (5.5), we have o resor o he Thiele echnique o consruc π and V. We can immediaely pu up he following differenial equaion by adding he expenses o he benefis in he se-up of Secion 4.4: d d V = π β β π γ γ b γ V µ x+b + rv + µ x+v. (5.6)

62 CHAPTER 5. EXPENSES 61 The appropriae side condiions are and V n = b, (5.7) V = (α + α b). (5.8) As before, (5.7) is a maer of definiion and relies only on he fac ha he endowmen benefi falls due upon survival a ime n, and (5.8) is he equivalence requiremen, which deermines π for given benefis and expense facors. Gahering erms involving V on he lef of (5.6) and muliplying on boh sides wih e n (r γ +µ) gives d (e n d (r γ +µ) V ) = e n (r γ +µ) {(1 β )π β γ (γ + µ x+ )b}. (5.9) Now inegrae (5.9) beween and n, using (5.7), and rearrange a bi o obain V = n e τ (r γ +µ) {β + γ + (γ + µ x+τ )b (1 β )π } dτ + e n (r γ +µ) b. (5.1) Upon insering = ino (5.1) and using (5.8), we find π = α + α b + n e τ (r γ +µ) {β + γ + (γ + µ x+τ )b}dτ + e n (1 β ) n e τ (r γ +µ) dτ (r γ +µ) b In he special case where γ = we could deermine π and V direcly from he defining relaions wihou using he differenial equaion. Tha goes, in fac, also for he general case wih γ by he following consideraion: By inspecion of he differenial equaion (5.6) and he side condiions, i is realized ha, formally, he problem amouns o deermining he ne premium rae (1 β )π and ne premium reserve V for a policy wih (admiedly unrealisic) benefis consising of a lump sum paymen of α + α b a ime, a coninuous level life annuiy of β + γ + γ b per year, and an endowmen insurance of b, when he ineres rae is r γ. Easy calculaions show ha, when γ =, he gross and ne quaniies are relaed by ( π 1 = 1 β π + α + α ) b + β + γ + γ b, (5.12) ā x n and V = V āx+ n ā x n (α + α b). (5.13). (5.11)

63 CHAPTER 5. EXPENSES 62 I is seen ha π > π, as was anicipaed a he ouse. Furhermore, V < V for < n, which may be less obvious. The relaionship (5.13) can be explained as follows: All expenses ha incur a a consan rae hroughou he erm of he conrac are compensaed by an equal componen in he effecive gross premium rae (1 β )π, see (5.12). Thus, he only expense facor ha appears in he gross reserve is he non-amorized iniial α-cos, which is he las erm on he righ of (5.13). I represens a deb on he par of he insured and is herefore o be subraced from he ne reserve. In Paragraph 4.3.D we have advocaed non-negaiviy of he reserve. Now, already from (5.8) i is clear ha he gross premium ses ou negaive a he ime of issue of he conrac and i will remain negaive for some ime hereafer unil a sufficien amoun of premium has been colleced. The only way o ge around his problem would be o charge an iniial lump sum premium no less han he iniial expense, bu his is usually no done in pracice (presumably) because a subsanial down paymen migh keep cusomers wih liquidiy problems from buying insurance. 5.2 The general muli-sae policy A. General reamen of expenses. Consider now he general muli-sae insurance policy reaed in Chaper 7. Expenses are easily accommodaed in he heory of ha chaper since hey can be reaed as addiional benefis of annuiy and assurance ype. Thus, from a echnical poin of view expenses do no creae any addiional difficulies, and we can herefore suiably end his chaper here. We round off by saying ha expenses are sill of concepual and grea pracical imporance. Assumpions abou he various forms of expenses are par of he echnical basis, which mus be verified by he insurer and is subjec o approval of he supervisory auhoriy. Thus, jus as saisical and economic analysis is required as a basis for assumpions abou moraliy and ineres, horough cos analysis are required as a basis for assumpions abou he expense facors.

64 Chaper 6 Muli-life insurances 6.1 Insurances depending on he number of survivors A. The single-life saus reinerpreed. In he reamen of he single life saus (x) in Chapers 3 4 we were having in mind he remaining life ime T of an x year old person. From a mahemaical poin of view his inerpreaion is no essenial. All ha maers is ha T is a non-negaive random variable wih an absoluely coninuous disribuion funcion, so ha he survival funcion is of he form p x = e µx+τ dτ. (6.1) The fooscrip x serves merely o indicae wha moraliy law is in play. Regardless of he naure of he saus (x) and he noion of lifeime represened by T, he previous resuls remain valid. In paricular, all formulas for expeced presen values of paymens depending on T are preserved, he basic ones being he endowmen, he life annuiy, ā x n = he endowmen insurance, and he erm insurance, n ne x = v n np x, (6.2) v p x d = n E x d, (6.3) Ā x n = 1 rā x n, (6.4) Ā 1 x n = Āx n n E x. (6.5) These formulas demonsrae ha presen values of all main ypes of paymens in life insurance endowmens, life annuiies, and assurances can be raced 63

65 CHAPTER 6. MULTI-LIFE INSURANCES 64 back o he presen value E x of an endowmen and, as far as he moraliy law is concerned, o he survival funcion p x. Once we have deermined p x, all oher funcions of ineres are obained by inegraion, possibly by some numerical mehod, and elemenary algebraic operaions. B. Muli-dimensional survival funcions. Consider a body of r individuals, he j-h of which is called (x j ) and has remaining lifeime T j, j = 1,..., r. For he ime being we shall confine ourselves o he case wih independen lives. Thus, assume ha he T j are sochasically independen, and ha each T j possesses an inensiy denoed by µ xj+ and, hence, has survival funcion p xj = e µx j +τ dτ. (6.6) (The funcion µ need no be he same for all j as he noaion suggess; we have dropped an exra index j jus o save noaion.) The simulaneous disribuion of T 1,..., T r is given by he muli-dimensional survival funcion P [ r j=1{t j > j } ] = or, equivalenly, by he densiy r j=1 r j=1 j p xj = e r j j=1 µx j +τ dτ j p xj µ xj+ j. (6.7) C. The join-life saus. The join life saus (x 1... x r ) is defined by having remaining lifeime T x1...x r = min{t 1,..., T r }. (6.8) Thus, he r lives are looked upon as a single eniy, which coninues o exis as long as all members survive, and erminaes upon he firs deah. The survival funcion of he join-life is denoed by p x1...x r and is p x1...x r = P [ r j=1{t j > } ] = e r j=1 µx j +τ dτ. (6.9) From his survival funcion we form he presen values of an endowmen n E x1...x r, a life annuiy ā x1...x r n, an endowmen insurance Āx 1...x r n, and a erm insurance, Ā 1 x 1...x r n, by jus puing (6.9) in he role of he survival funcion in (6.2) (6.5). By inspecion of (6.9), he moraliy inensiy of he join-life saus is simply he sum of he componen moraliy inensiies, µ x1...x r () = r µ xj+. (6.1) j=1

66 CHAPTER 6. MULTI-LIFE INSURANCES 65 In paricular, if he componen lives are subjec o G-M moraliy laws wih a common value of he parameer c, hen (6.1) becomes µ xj+ = α j + β j e γ (xj+), (6.11) µ x1...x r () = α + β e γ (6.12) wih r r α = α j, β = β j e γ xj, (6.13) j=1 j=1 again a G-M law wih he same γ as in he componen laws. D. The las-survivor saus. The las survivor saus x 1... x r is defined by having remaining lifeime T x1...x r = max{t 1,..., T r }. (6.14) Now he r lives are looked upon as an eniy ha coninues o exis as long as a leas one member survives, and erminaes upon he las deah. The survival funcion of his saus is denoed by p x1...x r. By he general addiion rule for probabiliies (Appendix C), p x1...x r = P [ r j=1{t j > } ] = p xj p xj1 x j ( 1) r 1 p x1...x r. (6.15) j j 1<j 2 This way acuarial compuaions for he las survivor are reduced o compuaions for join lives, which are simple. As explained in Paragraph A, all main ypes of presen values can be buil from (G). Formulas for benefis coningen on survival, obained from (6.2) and (6.3), will reflec he srucure of (G) in an obvious way. Formulas for deah benefis are obained from (6.4) and (6.5). The expressions are displayed in he more general case o be reaed in he nex paragraph. E. The q survivors saus. q The q survivors saus x 1...x r is defined by having as remaining lifeime he (r q + 1)-h order saisic of he sample {T 1,..., T r }. Thus he saus is alive as long as here are a leas q survivors among he original r. The survival funcion can be expressed in erms of join life survival funcions of sub-groups of lives by direc applicaion of he heorem in Appendix C: p q x 1...xr = r ( ) p 1 ( 1) p q p xj1...x q 1 jp. (6.16) j 1<...<j p p=q

67 CHAPTER 6. MULTI-LIFE INSURANCES 66 Presen values of sandard forms of insurances for he q survivors saus are now obained along he lines described in he previous paragraph. Firs, combine (6.16) wih (6.2) o obain ne q x 1...xr = r ( ) p 1 ( 1) p q ne xj1...x q 1 jp, (6.17) j 1<...<j p p=q he noaion being self-explaining. Nex, combine (6.3) and (6.17) o obain ā q x 1...xr n = r ( ) p 1 ( 1) p q ā xj1...x q 1 jp n. (6.18) j 1<...<j p p=q Finally, presen values of endowmen and erm insurances are obained by insering (6.18) and (6.17) in he general relaions (6.4) and (6.5): Ā q x 1...xr n = 1 rā q x 1...xr n, (6.19) Ā 1 q x 1...xr n = Ā q x 1...xr n n E xj1...x jp. (6.2) The following alernaive o he expression in (6.19) has some aesheic appeal as i expresses he insurance by corresponding insurances on join lives: Ā q x 1...xr n = r ( ) p 1 ( 1) p q Ā xj1...x q 1 jp n. (6.21) j 1<...<j p p=q I is obained upon subsiuing (6.18) on he righ of (6.19), hen insering (recall (6.4)) ā xj1...x jp = (1 Āx j1...x jp )/r and using (C.8) in Appendix C. A similar expression for he erm insurance is obained upon subracing (6.17) from (6.21).

68 Chaper 7 Markov chains in life insurance 7.1 The insurance policy as a sochasic process A. The basic eniies. Consider an insurance policy issued a ime for a finie erm of n years. We have in mind life or pension insurance or some oher form of insurance of persons like disabiliy or sickness coverage. In such lines of business benefis and premiums are ypically coningen upon ransiions of he policy beween cerain saes specified in he conrac. Thus, we assume here is a finie se of saes, Z = {, 1,..., r}, such ha he policy a any ime is in one and only one sae, commencing in sae (say) a ime. Denoe he sae of he policy a ime by Z(). Regarded as a funcion from [, n] o Z, Z is assumed o be righ-coninuous, wih a finie number of jumps, and Z() =. To accoun for he random course of he policy, Z is modeled as a sochasic process on some probabiliy space (Ω, H, P). B. Model deliberaions; realism versus simpliciy. On specifying he probabiliy model, wo concerns mus be kep in mind, and hey are ineviably conflicing. On he one hand, he model should reflec he essenial feaures of (a cerain piece of) realiy, and his speaks for a complex model o he exen ha realiy iself is complex. On he oher hand, he model should be mahemaically racable, and his speaks for a simple model allowing of easy compuaion of quaniies of ineres. The ar of modeling is o srike he righ balance beween hese wo concerns. Favouring simpliciy in he firs place, we shall be working under Markov assumpions, which allow for fairly easy compuaion of relevan probabiliies and expeced values. Laer on we shall demonsrae he versailiy of his model framework, showing ha i is capable of represening virually any concepion one migh have of he mechanisms governing he developmen of he policy. We shall ake he Markov chain model presened in [26] as a suiable framework 67

69 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 68 hroughou his ex. A useful basic source is [3]. 7.2 The ime-coninuous Markov chain A. The Markov propery. A sochasic process is essenially deermined by is finie-dimensional disribuions. In he presen case, where Z has only a finie sae space, hese are fully specified by he probabiliies of he elemenary evens p h=1 [Z( h) = j h ], 1 < < p in [, n] and j 1,..., j p Z. Now P [Z( h ) = j h, h = 1,..., p] p = P [Z( h ) = j h Z( g ) = j g, g =,..., h 1], (7.1) h=1 where, for convenience, we have pu = and j = so ha [Z( ) = j ] is he rivial even wih probabiliy 1. Thus, he specificaion of P could suiably sar wih he condiional probabiliies appearing on he righ of (7.1). A paricularly simple srucure is obained by assuming ha, for all 1 < < p in [, n] and j 1,..., j p Z, P [Z( p ) = j p Z( h ) = j h, h = 1,..., p 1] = P[Z( p ) = j p Z( p 1 ) = j p 1 ], (7.2) which means ha process is fully deermined by he (simple) ransiion probabiliies p jk (, u) = P[Z(u) = k Z() = j], (7.3) < u in [, n] and j, k Z. In fac, if (7.2) holds, hen (7.1) reduces o P [Z( h ) = j h, h = 1,..., p] = p p jh 1 j h ( h 1, h ), (7.4) and one easily proves he equivalen ha, for any 1 < < p < < p+1 < < p+q in [, n] and j 1,..., j p, j, j p+1,..., j p+q in Z, P [Z( h ) = j h, h = p + 1,..., p + q Z() = j, Z( h ) = j h, h = 1,..., p] h=1 = P [Z( h ) = j h, h = p + 1,..., p + q Z() = j]. (7.5) Proclaiming he presen ime, (7.5) says ha he fuure of he process is independen of is pas when he presen is known. (Fully known, ha is; if he presen sae is only parly known, i may cerainly help o add informaion abou he pas.) The condiion (7.2) is called he Markov propery. We shall assume ha Z possesses his propery and, accordingly, call i a coninuous ime Markov process on he sae space Z.

70 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 69 From he simple ransiion probabiliies we form he more general ransiion probabiliy from j o some subse K Z, p jk (, u) = P[Z(u) K Z() = j] = k K p jk (, u). (7.6) We have, of course, p jz (, u) = k Z p jk (, u) = 1. (7.7) B. Alernaive definiions of he Markov propery. I is sraighforward o demonsrae ha (7.2), (7.4), and (7.5) are equivalen, so ha any one of he hree could have been aken as definiion of he Markov propery. Then (7.4) should be preceded by: Assume here exis non-negaive funcions p jk (, u), j, k Z, u n, such ha k Z p jk(, u) = 1 and, for any 1 < < p in [, n] and {j 1,..., j p } in Z,... We shall briefly ouline more general definiions of he Markov propery. For T [, n] le H T denoe he class of all evens generaed by {Z()} T. I represens everyhing ha can be observed abou Z in he ime se T. For insance, H {} is he informaion carried by he process a ime and consiss of he elemenary evens, Ω, and [Z() = j], j =,..., r, and all possible unions of hese evens. More generally, H {1,..., p} is he informaion carried by he process a imes 1,..., p. Some ses T of inerval ype are frequenly encounered, and we abbreviae H = H [,] (he enire hisory of he process by ime ), H < = H [,) (he sric pas of he process by ime ), and H > = H (,n] (he fuure of he process by ime ). The process Z is said o be a Markov process if, for any B H >, P[B H ] = P[B H {} ]. (7.8) This is he general form of (7.5). An alernaive definiion says ha, for any A H < and B H >, P[A B H {} ] = P[A H {} ] P[B H {} ], (7.9) ha is, he pas and he fuure of he process are condiionally independen, given is presen sae. In he case wih finie sae space (counabiliy is equally simple) i is easy o prove ha (7.8) and (7.9) are equivalen by working wih he finie-dimensional disribuions, ha is, ake A H {1,..., p} and B H {p+1,..., p+q} wih 1 < < p < < p+1 < < p+q. C. The Chapman-Kolmogorov equaion. For a fixed [, n] he evens {Z() = j}, j Z, are disjoin and heir union is he almos sure even. I follows ha P[Z(u) = k Z(s) = i] = j Z P[Z() = j, Z(u) = k Z(s) = i] = j Z P[Z() = j Z(s) = i] P[Z(u) = k Z(s) = i, Z() = j].

71 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 7 If Z is Markov, and s u, his reduces o p ik (s, u) = j Z p ij (s, )p jk (, u), (7.1) which is known as he Chapman-Kolmogorov equaion. D. Inensiies of ransiion. In principle, specifying he Markov model amouns o specifying he p jk (, u) in such a manner ha he expressions on he righ of (7.4) define probabiliies in a consisen way. This would be easy if Z were a discree ime Markov chain wih ranging in a finie ime se = < 1 < < q = n: hen we could jus ake he p jk ( q 1, q ) as any non-negaive numbers saisfying r k= p jk( p 1, p ) = 1 for each j Z and p = 1,..., q. This simple device does no carry over wihou modificaion o he coninuous ime case since here are no smalles finie ime inervals from which we can build all probabiliies by (7.4). An obvious way of adaping he basic idea o he ime-coninuous case is o add smoohness assumpions ha give meaning o a noion of ransiion probabiliies in infiniesimal ime inervals. More specifically, we shall assume ha he inensiies of ransiion, µ jk () = lim h p jk (, + h) h (7.11) exis for each j, k Z, j k, and [, n) and, moreover, ha hey are piece-wise coninuous. Anoher way of phrasing (7.11) is p jk (, + d) = µ jk ()d + o(d), (7.12) where he erm o(d) is such ha o(d)/d as d. Thus, ransiion probabiliies over a shor ime inerval are assumed o be (approximaely) proporional o he lengh of he inerval, and he proporionaliy facors are jus he inensiies, which may depend on he ime. Wha is shor in his connecion depends on he sizes of he inensiies. For insance, if he µ jk (τ) are approximaely consan and << 1 for all k j and all τ [, + 1], hen µ jk () approximaes he ransiion probabiliy p jk (, + 1). In general, however, he inensiies may aain any posiive values and should no be confused wih probabiliies. For j / K Z, we define he inensiy of ransiion from sae j o he se of saes K a ime as p jk (, u) µ jk () = lim = µ jk (). (7.13) u u k K In paricular, he oal inensiy of ransiion ou of sae j a ime is µ j,z {j} (), which is abbreviaed µ j () = µ jk (). (7.14) k;k j

72 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 71 From (7.7) and (7.12) we ge p jj (, + d) = 1 µ j ()d + o(d). (7.15) E. The Kolmogorov differenial equaions. The ransiion probabiliies are wo-dimensional funcions of ime, and in nonrivial siuaions i is virually impossible o specify hem direcly in a consisen manner or even figure how hey should look on inuiive grounds. The inensiies, however, are one-dimensional funcions of ime and, being easily inerpreable, hey form a naural saring poin for specificaion of he model. Luckily, as we shall now see, hey are also basic eniies in he sysem as hey deermine he ransiion probabiliies uniquely. Suppose he process Z is in sae j a ime. To find he probabiliy ha he process will be in sae k a a given fuure ime u, le us condiion on wha happens in he firs small ime inerval (, + d]. In he firs place Z may remain in sae j wih probabiliy 1 µ j () d and, condiional on his even, he probabiliy of ending up in sae k a ime u is p jk ( + d, u). In he second place, Z may jump o some oher sae g wih probabiliy µ jg () d and, condiional on his even, he probabiliy of ending up in sae k a ime u is p gk ( + d, u). Thus, he oal probabiliy of Z being in sae k a ime u is p jk (, u) = (1 µ j () d) p jk ( + d, u) + µ jg () d p gk ( + d, u) + o(d), (7.16) g;g j Upon puing d p jk (, u) = p jk ( + d, u) p jk (, u) in he infiniesimal sense, we arrive a d p jk (, u) = µ j () d p jk (, u) µ jg () d p gk (, u). (7.17) g;g j For given k and u hese differenial equaions deermine he funcions p jk (, u), j =,..., r, uniquely when combined wih he obvious condiions p jk (u, u) = δ jk. (7.18) Here δ jk is he Kronecker dela defined as 1 if j = k and oherwise. The relaion (7.16) could have been pu up direcly by use of he Chapman- Kolmogorov equaion (7.1), wih s,, i, j replaced by, + d, j, g, bu we have carried hrough he deailed (sill informal hough) argumen above since i will be in use repeaedly hroughou he ex. I is called he backward (differenial) argumen since i focuses on, which in he perspecive of he considered ime period [, u] is he very beginning. Accordingly, (7.17) is referred o as he Kolmogorov backward differenial equaions, being due o A.N. Kolmogorov. A poins of coninuiy of he inensiies we can divide by d in (7.17) and obain a limi on he righ as d ends o. Thus, a such poins we can wrie

73 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 72 (7.17) as p jk(, u) = µ j ()p jk (, u) g;g j µ jg ()p gk (, u). (7.19) Since we have assumed ha he inensiies are piece-wise coninuous, he indicaed derivaives exis piece-wise. We prefer, however, o work wih he differenial form (7.17) since i is generally valid under our assumpions and, moreover, invies algorihmic reasoning; numerical procedures for solving differenial equaions are based on approximaion by difference equaions for some fine discreizaion and, in fac, (7.16) is basically wha one would use wih some small d >. As one may have guessed, here exis also Kolmogorov forward differenial equaions. These are obained by focusing on wha happens a he end of he ime inerval in consideraion. Reasoning along he lines above, we have p ij (s, + d) = p ig (s, ) µ gj () d + p ij (s, )(1 µ j () d) + o(d), hence g;g j d p ij (s, ) = g;g j p ig (s, )µ gj () d p ij (s, )µ j () d. (7.2) For given i and s, he differenial equaions (7.2) deermine he funcions p ij (s, ), j =,..., r, uniquely in conjuncion wih he obvious condiions p ij (s, s) = δ ij. (7.21) In some simple cases he differenial equaions have nice analyical soluions, bu in mos non-rivial cases hey mus be solved numerically, e.g. by he Runge- Kua mehod. Once he simple ransiion probabiliies are deermined, we may calculae he probabiliy of any even in H {1,..., r} from he finie-dimensional disribuion (7.4). In fac, wih finie Z every such probabiliy is jus a finie sum of probabiliies of elemenary evens o which we can apply (7.4). Probabiliies of more complex evens ha involve an infinie number of coordinaes of Z, e.g. evens in H T wih T an inerval, canno in general be calculaed from he simple ransiion probabiliies. Ofen we can, however, pu up differenial equaions for he requesed probabiliies and solve hese by some suiable mehod. Of paricular ineres is he probabiliy of saying uninerrupedly in he curren sae for a cerain period of ime, p jj (, u) = P[Z(τ) = j, τ (, u] Z() = j]. (7.22) Obviously p jj (, u) = p jj (, s) p jj (s, u) for < s < u. consrucion and (7.15) we ge By he backward p jj (, u) = (1 µ j () d) p jj ( + d, u) + o(d). (7.23)

74 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 73 From here proceed as above, using p jj (u, u) = 1, o obain p jj (, u) = e u µj. (7.24) F. Backward and forward inegral equaions. From he backward differenial equaions we obain an equivalen se of inegral equaions as follows. Swich he firs erm on he righ over o he lef and, o obain a complee differenial here, muliply on boh sides by he inegraing facor e u µj : ( d e u µj p ) jk (, u) = e u µj µ jg () d p gk (, u). g;g j Now inegrae over (, u] and use (7.18) o obain δ jk e u µj p jk (, u) = u e u τ µj g; g j µ gj (τ)p jk (τ, u) dτ. Finally, carry he Kronecker dela over o he righ, muliply by e u µj, and use (7.24) o arrive a he backward inegral equaions p jk (, u) = u p jj (, τ) g; g j µ jg (τ)p gk (τ, u) dτ + δ jk p jj (, u). (7.25) In a similar manner we obain he following se of forward inegral equaions from (7.2): p ij (s, ) = δ ij p ii (s, ) + g; g j s p ig (s, τ)µ gj (τ)p jj (τ, )dτ. (7.26) The inegral equaions could be pu up direcly upon summing he probabiliies of disjoin elemenary evens ha consiue he even in quesion. For (7.26) he argumen goes as follows. The firs erm on he righ accouns for he possibiliy of ending up in sae j wihou making any inermediae ransiions, which is relevan only if i = j. The second erm accouns for he possibiliy of ending up in sae j afer having made inermediae ransiions and is he sum, over all saes g j and all small ime inervals (τ, τ + dτ) in (s, ), of he probabiliy of arriving for he las ime in sae j from sae g in he ime inerval (τ, τ + dτ). In a similar manner (7.25) is obained upon spliing up by he direcion and he ime of he firs deparure, if any, from sae j. We now urn o some specializaions of he model peraining o insurance of persons. 7.3 Applicaions A. A single life wih one cause of deah. The life lengh of a person is modeled as a posiive random variable T wih survival funcion F. There are

75 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 74 Alive 1 µ Dead Figure 7.1: Skech of he moraliy model wih one cause of deah. wo saes, alive and dead. Labeling hese by and 1, respecively, he sae process Z is simply Z() = 1[T ], [, n], which couns he number of deahs by ime. The process Z is righconinuous and is obviously Markov since in sae he pas is rivial, and in sae 1 he fuure is rivial. The ransiion probabiliies are p (s, ) = F ()/ F (s). The Chapman-Kolmogorov equaion reduces o he rivial p (s, u) = p (s, )p (, u) or F (u)/ F (s) = { F ()/ F (s)}{ F (u)/ F ()}. The only non-null inensiy is µ 1 () = µ(), and p (, u) = e u µ. (7.27) The Kolmogorov differenial equaions reduce o jus he definiion of he inensiy (wrie ou he deails). The simple wo sae process wih sae 1 absorbing is oulined in Fig. 7.1 B. A single life wih r causes of deah. In he previous paragraph i was, admiedly, he process se-up ha needed he example and no he oher way around. The process formulaion shows i power when we urn o more complex siuaions. Fig. 7.2 oulines a firs exension of he model in he previous paragraph, whereby he single absorbing sae ( dead ) is replaced by r absorbing saes represening differen causes of deah, e.g. dead in acciden, dead from hear disease, ec. The index in he inensiies µ j is superfluous and has been dropped.

76 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 75 Alive µ 1 µ j µ r 1 j r Dead: cause 1 Dead: cause j Dead: cause r Figure 7.2: Skech of he moraliy model wih r causes of deah. Relaion (7.14) implies ha he oal moraliy inensiy is he sum of he inensiies of deah from differen causes, µ() = r µ j (). (7.28) j=1 For a person aged he probabiliy of survival o u is he well-known survival probabiliy p (, u) given by (7.27). The presen enriched model opens possibiliies of expressing ideas abou he relaive imporance of various causes of deah and hus beer moivae a specific moraliy law in he aggregae. For insance, he G-M law in he simple moraliy model may be moivaed as resuling from wo causes of deah, one wih inensiy α independen of age (pure acciden) and he oher wih inensiy βc (wear-ou). The probabiliy ha a years old will die from cause j before age u is p j (, u) = u e τ µ µ j (τ) dτ. (7.29) This follows from e.g. (7.25) upon noing ha p rr (, u) = 1, bu being oally ransparen i can be pu up direcly. Inspecion of (7.28) (7.29) gives rise o a commen. An increase of one moraliy inensiy µ k resuls in a decrease of he survival probabiliy (evidenly) and also of he probabiliies of deah from every oher cause j k, hence (since he probabiliies sum o 1) an increase of he probabiliy of deah from cause k (also eviden). Thus, he increased proporions of deahs from hear diseases and cancer in our imes could be sufficienly explained by he fac ha medical progress has pracically eliminaed moraliy by lunge inflammaion, childbed fever, and a number of oher diseases. The above discussion suppors he asserion ha he inensiies are basic eniies. They are he pure expressions of he forces acing on he policy in

77 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 76 each given sae, and he ransiion probabiliies are resulans of he inerplay beween hese forces. C. A model for disabiliies, recoveries, and deah. Fig. 7.3 oulines a model suiable for analyzing insurances wih paymens depending on he sae of healh of he insured, e.g. sickness insurance providing an annuiy benefi during periods of disabiliy or life insurance wih premium waiver during disabiliy. Many oher problems fi ino he same scheme by mere re-labeling of he saes. For insance, in connecion wih a pension insurance wih addiional benefis o he spouse, saes and 1 would be unmarried and married, and in connecion wih unemploymen insurance hey would be employed and unemployed. For a person who is acive a ime s he Kolmogorov forward differenial (7.2) equaions are p (s, ) = p 1 (s, )ρ() p (s, )(µ() + σ()), (7.3) p 1(s, ) = p (s, )σ() p 1 (s, )(ν() + ρ()). (7.31) (The probabiliy p 2 (s, ) is deermined by he oher wo.) The iniial condiions (7.21) become p (s, s) = 1, (7.32) p 1 (s, s) =. (7.33) (For a person who is disabled a ime s he forward differenial equaions are he same, only wih he firs subscrip replaced by 1 in all he probabiliies, and he side condiions are p (s, s) =, p 11 (s, s) = 1.). a acive σ() ρ() i invalid µ() d dead ν() Figure 7.3: Skech of a Markov chain model for disabiliies, recoveries, and deah.

78 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 77 When he inensiies are sufficienly simple funcions, one may find explici closed expressions for he ransiion probabiliies. Work hrough he case wih consan inensiies. 7.4 Selecion phenomena A. Inroducory remarks. The Markov model (like any oher model) may be accused of being oversimplified. For insance, in he disabiliy model i says ha he prospecs of survival of a disabled person are unaffeced by informaion abou he pas such as he paern of previous disabiliies and recoveries and, in paricular, he duraion since he las onse of disabiliy. One could imagine ha here are several ypes or degrees of disabiliy, some of hem ligh, wih raher sandard moraliy, and some severe wih heavy excess moraliy. In hese circumsances informaion abou he pas may be relevan: if we ge o know ha he onse of disabiliy incurred a long ime ago, hen i is likely ha one of he ligh forms is in play; if i incurred yeserday, i may well be one of he severe forms by which a soon deah is o be expeced. Now, he kind of heerogeneiy effec menioned here can be accommodaed in he Markov framework simply by exending he sae space, replacing he single sae disabled by more saes corresponding o differen degrees of disabiliy. From his Markov model we deduce he model of wha we can observe as skeched in Fig. 7.3 upon leing he sae disabled be he aggregae of he disabiliy saes in he basic model. Wha we end up wih is ypically no longer a Markov model. Generally speaking, by variaion of sae space and inensiies, he Markov se-up is capable of represening exremely complex phenomena. In he following we shall formalize hese ideas wih a paricular view o selecion phenomena ofen encounered in insurance. The ideas are o a grea exen aken from [27]. B. Aggregaing saes of a Markov chain. Le Z be he general coninuous ime Markov chain inroduced in Secion 7.2. Le {Z,..., Z r } be a pariion of Z, ha is, he Z g are disjoin and heir union is Z. By convenion, Z. Define a sochasic process Z on he sae space Z = {,..., r} by Z() = g iff Z() Z g. (7.34) The inerpreaion is ha we can observe he process Z which represens summary informaion abou some no fully observable Markov process Z. Suppose he underlying process Z is observed o be in sae i a ime s. The subsequen developmen of Z can be projeced by condiional probabiliies for he process Z. We have, for s < < u, P[ Z(u) = h Z(s) = i, Z() = g] = 1 p ij (s, )p jzh (, u). p izg (s, ) j Z g

79 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 78 From his expression we obain condiional ransiion inensiies of he aggregae process: P[ lim Z(u) = h Z(s) = i, Z() = g] 1 = p ij (s, )µ jzh (). u u p izg (s, ) j Z g We can no speak of he inensiies since hey would in general be differen if more informaion abou Z were condiioned on. As an example, consider he aggregae of he saes and 1 in he disabiliy model in Paragraph 7.3.C, Z = {, 1}, and pu Z 1 = {2}. Thus we observe only wheher he insured is alive or no. The process Z is Markov, of course (recall he argumen in Paragraph 7.3.A). The survival probabiliy is and he moraliy inensiy a age is p (, ) = p (, ) + p 1 (, ), µ() = p (, )µ() + p 1 (, )ν() p (, ) + p 1 (, ) a weighed average of he moraliy inensiies of acive and disabled, he weighs being he probabiliies of saying in he respecive saes. C. Non-differenial probabiliies. Suppose he ransiion probabiliies p jzh (, u) considered as funcions of j are consan on each Z g, ha is, here exis funcions p gh (, u) such ha, for each < u and g, h Z, p jzh (, u) = p gh (, u), j Z g. (7.35) Then we shall say ha he probabiliies of ransiion beween he subses Z g are non-differenial (wihin he individual subses). The following resul is eviden on inuiive grounds, bu never he less meris emphasis. Theorem 1. If he ransiion probabiliies of he process Z beween he subses Z g are non-differenial, hen he process Z defined by (7.34) is Markov wih ransiion probabiliies p gh (, u) defined by (7.35). If Z possesses inensiies µ jk, hen he process Z has inensiies µ gh given by µ gh = µ jzh, j Z g. (7.36) Proof: By (7.8), we mus show ha, for any even A depending only on { Z(τ)} τ<, P[ Z(u) = h A, Z() = g] = p gh (, u). (7.37) Using firs he fac ha [ Z() = g] = j Zg [Z() = j] is a union of disjoin evens, hen ha A H < and Z is Markov, and finally assumpion (7.35), we,

80 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 79 ge which is equivalen o (7.37). P[A, Z() = g, Z(u) = h] = j Z g P[A, Z() = j, Z(u) Z h ] = j Z g P[A, Z() = j]p jzh (, u). = P[A, Z() = g] p gh (, u), D. Non-differenial moraliy. Le sae r be absorbing, represening deah, and le H = {,..., r 1} be he aggregae of saes where he insured is alive. Assume ha he moraliy is non-differenial, which means ha all µ jr, j H, are idenical and equal o λ, say. Then, by Theorem 1, he survival probabiliy is he same in all saes j H: p jh (, u) = e u λ. (7.38) The condiional probabiliy of saying in sae k H a ime, given survival, is p jk H (, u) = p jk(, u) p jh (, u) = p jk(, u)e u λ. (7.39) Insering p jk (, u) = p jk H (, u)e u λ ino (7.25), we ge for each j, k H ha p jk H (, u)e u λ = u e τ µ j,h {j} τ λ + δ jk e u µ j,h {j} u λ. g H {j} µ jg (τ)p gk H (τ, u)e u τ λ dτ Muliplying wih e s λ, we see ha he condiional probabiliies in (7.39) saisfy he inegral equaions (7.25) for he ransiion probabiliies in he so-called parial model wih sae space H and ransiion inensiies µ j,k, j, k H. Thus, o find he ransiion probabiliies in he full model, work firs in he simple parial model for he saes as alive and muliply he parial probabiliies obained here wih he survival probabiliy. 7.5 The sandard muli-sae conrac A. The conracual paymens. We refer o he insurance policy wih developmen as described in Paragraph 7.1.A. Taking Z o be a sochasic process wih righ-coninuous pahs and a mos a finie number of jumps, he same holds also for he associaed indicaor processes I j and couning processes N jk

81 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 8 defined, respecively, by I j () = 1[Z() = j] (1 or according as he policy is in he sae j or no a ime ) and N jk () = {τ; Z(τ ) = j, Z(τ) = k, τ (, ]} (he number of ransiions from sae j o sae k (k j) during he ime inerval (, ]). The indicaor processes {I j ()} and he couning processes {N jk ()} are relaed by he fac ha I j increases/decreases (by 1) upon a ransiion ino/ou of sae j. Thus di j () = dn j () dn j (), (7.4) where a do in he place of a subscrip signifies summaion over ha subscrip, e.g. N j = k;k j N jk. The policy is assumed o be of sandard ype, which means ha he paymen funcion represening conracual benefis less premiums is of he form (recall he device (A.15)) db() = k I k () db k () + k l b kl () dn kl (), (7.41) where each B k, of form db k () = b k () d + B k () B k ( ), is a deerminisic paymen funcion specifying paymens due during sojourns in sae k (a general life annuiy), and each b kl is a deerminisic funcion specifying paymens due upon ransiions from sae k o sae l (a general life assurance). When differen from, B k () B k ( ) is an endowmen a ime. The funcions b k and b kl are assumed o be finie-valued and piecewise coninuous. The se of disconinuiy poins of any of he annuiy funcions B k is D = {, 1,..., q } (say). Posiive amouns represen benefis and negaive amouns represen premiums. In pracice premiums are only of annuiy ype. A imes / [, n] all paymens are null. B. Ideniies revisied. Here we make an inermission o make a commen ha does no depend on he probabiliy srucure o be specified below. The ideniy (4.18) ress on he corresponding ideniy (4.17) beween he presen values. The laer is, in is urn, a special case of he ideniies pu up in Secion 2.1, from which many ideniies beween presen values in life insurance can be derived. Suppose he invesmen porfolio of he insurance company bears ineres wih inensiy r() a ime. The following ideniy, which expresses life annuiies by endowmens and life assurances, is easily obained upon inegraing by pars, using (7.4): u e τ r I j (τ) db j (τ) = e u r I j (u) B j (u) e r I j ()B j () + + u e τ r B j (τ ) d(n j (τ) N j (τ)). u e τ r I j (τ)b j (τ)r(τ) dτ

82 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 81 C. Expeced presen values and prospecive reserves. A any ime [, n], he presen value of fuure benefis less premiums under he conrac is V () = n e τ r db(τ). (7.42) This is a liabiliy for which he insurer is o provide a reserve, which by saue is he expeced value. Suppose he policy is in sae j a ime. Then he condiional expeced value of V () is V j () = n e τ r k p jk (, τ) db k (τ) + b kl (τ)µ kl (τ) dτ. (7.43) l;l k This follows by aking expecaion under he inegral in (7.42), insering db(τ) from (7.41), and using E[I k (τ) Z() = j] = p jk (, τ), E[dN kl (τ) Z() = j] = p jk (, τ)µ kl (τ) dτ. We expound he resul as follows. Wih probabiliy p jk (, τ) he policy says in sae k a ime τ, and if his happens he life annuiy provides he amoun db k (τ) during a period of lengh dτ around τ. Thus, he expeced presen value r db k (τ). Wih probabiliy p jk (, τ) µ kl (τ) dτ he policy jumps from sae k o sae l during a period of lengh dτ around τ, and if his happens he assurance provides he amoun b kl (τ). Thus, he expeced presen value a ime of his coningen paymen a ime of his coningen paymen is p jk (, τ)e τ r b kl (τ). Summing over all fuure imes and ypes of paymens, we find he oal given by (7.43). Le < u < n. Upon separaing paymens in (, u] and in (u, n] on he righ of (7.43), and using Chapman-Kolmogorov on he laer par, we obain is p jk (, τ) µ kl (τ) dτ e τ V j () = u + e u r k e τ r k p jk (, τ) db k (τ) + b kl (τ)µ kl (τ) dτ l;l k p jk (, u) V k (u). (7.44) This expression is also immediaely obained upon condiioning on he sae of he policy a ime u. Throughou he erm of he policy he insurance company mus currenly mainain a reserve o mee fuure ne liabiliies in respec of he conrac. By saue, if he policy is in sae j a ime, hen he company is o provide a reserve ha is precisely V j (). Accordingly, he funcions V j are called he (sae-wise) prospecive reserves of he policy. One may say ha he principle of equivalence has been carried over o ime, now requiring expeced balance

83 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 82 beween he amoun currenly reserved and he discouned fuure liabiliies, given he informaion currenly available. (Only he presen sae of he policy is relevan due o he Markov propery and he simple memoryless paymens under he sandard conrac). D. The backward (Thiele s) differenial equaions. By leing u approach in (7.44), we obain a differenial form ha displays he dynamics of he reserves. In fac, we are going o derive a se of backward differenial equaions and, herefore, ake he opporuniy o apply he direc backward differenial argumen demonsraed and announced previously in Paragraph 7.2.E. Thus, suppose he policy is in sae j a ime / D. Condiioning on wha happens in a small ime inerval (, + d] (no inersecing D) we wrie V j () = b j () d + µ jk () d b jk () k;k j +(1 µ j () d)e r() d V j ( + d) + k;k j µ jk () d e r() d V k ( + d). Proceeding from here along he lines of he simple case in Secion 4.4, we easily arrive a he backward or Thiele s differenial equaions for he sae-wise prospecive reserves, d d V j() = (r() + µ j ()) V j () b j () k;k j k;k j µ jk () V k () b jk ()µ jk (). (7.45) The differenial equaions are valid in he open inervals ( p 1, p ), p = 1,..., q, and ogeher wih he condiions V j ( p ) = (B j ( p ) B j ( p )) + V j ( p ), p = 1,..., q, j Z, (7.46) hey deermine he funcions V j uniquely. A commen is in order on he differeniabiliy of he V j. A poins of coninuiy of he funcions b j, b jk, µ jk, and r here is no problem since here he inegrand on he righ of (7.43) is coninuous. A possible poins of disconinuiy of he inegrand he derivaive d d V j does no exis. However, since such disconinuiies are finie in number, hey will no affec he inegraions involved in numerical procedures. Thus we shall hroughou allow ourselves o wrie he differenial equaions on he form (7.45) insead of he generally valid differenial form obained upon puing dv j () on he lef and muliplying wih d on he righ. E. Solving he differenial equaions. Only in rare cases of no pracical ineres is i possible o find closed form soluions o he differenial equaions. In pracice one mus resor o numerical mehods o deermine he prospecive

84 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 83 reserves. As a maer of experience a fourh order Runge-Kua procedure works reliably in virually all siuaions encounered in pracice. One solves he differenial equaions from op down. Firs solve (7.45) in he upper inerval ( q 1, n) subjec o (7.46), which specializes o V j (n ) = B j (n) B j (n ) since V j (n) = for all j by definiion. Then go o he inerval below and solve (7.45) subjec o V j ( q 1 ) = (B j ( q 1 ) B j ( q 1 )) + V j ( q 1 ), where V j ( q 1 ) was deermined in he firs sep. Proceed in his manner downwards. I is realized ha he Kolmogorov backward equaions (7.17) are a special case of he Thiele equaions (7.45); he ransiion probabiliy p jk (, u) is jus he prospecive reserve in sae j a ime for he simple conrac wih he only paymen being a lump sum paymen of 1 a ime u if he policy is hen in sae k, and wih no ineres. Thus a numerical procedure for compuaion of prospecive reserves can also be used for compuaion of he ransiion probabiliies. F. The equivalence principle. If he equivalence principle is invoked, one mus require ha V () = B (). (7.47) This condiion imposes a consrain on he conracual funcions b j, B j, and b jk, viz. on he premium level for given benefis and design of he premium plan. I is of a differen naure han he condiions (7.46), which follow by he very definiion of prospecive reserves (for given conracual funcions). G. Savings premium and risk premium. The equaion (7.45) can be recas as b j () d = dv j () r() d V j () + R jk ()µ jk () d. (7.48) where k;k j R jk () = b jk () + V k () V j (). (7.49) The quaniy R jk () is called he sum a risk associaed wih (a possible) ransiion from sae j o sae k a ime since, upon such a ransiion, he insurer mus immediaely pay ou he sum insured and also provide he appropriae reserve in he new sae, bu he can cash he reserve in he old sae. Thus, he las erm in (7.48) is he expeced ne oulay in connecion wih a possible ransiion ou of he curren sae j in (, + d), and i is called he risk premium. The wo firs erms on he righ of (7.48) consiue he savings premium in (, + d), called so because i is he amoun ha has o be provided o mainain he reserve in he curren sae; he incremen of he reserve less he ineres earned on i. On he lef of (7.48) is he premium paid in (, + d), and so he relaion shows how he premium decomposes in a savings par and a risk par. Alhough helpful as an inerpreaion, his consideraion alone canno carry he full undersanding of he differenial equaion since (7.48) is valid also if b j () is posiive (a benefi) or.

85 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 84 H. Inegral equaions. In (7.45) le us swich he erm (r() + µ j ()) V j () appearing on he righ of over o he lef, and muliply he equaion by e (r+µj ) o form a complee differenial on he lef: d (e ) (r+µj ) V j () = d e (r+µj ) µ jk () V k () + b j () + k;k j k;k j b jk ()µ jk (). Now inegrae over an inerval (, u) conaining no jumps B j (τ) B j (τ ) and, recalling ha e τ µj = p jj (, τ), rearrange a bi o obain he inegral equaion V j () = u p jj (, τ) e τ r b j (τ) + k; k j µ jk (τ)(b jk (τ) + V k (τ)) dτ + p jj (, u) e u r V j (u ). (7.5) This resul generalizes he backward inegral equaions for he ransiion probabiliies (7.25) and, jus as in ha special case, also he expression on he righ hand side of (7.5) is easy o inerpre; i decomposes he fuure paymens ino hose ha fall due before and hose ha fall due afer he ime of he firs ransiion ou of he curren sae in he ime inerval (, u) or, if no ransiion akes place, hose ha fall due before and hose ha fall due afer ime u. We shall ake a direc roue o he inegral equaion (7.5) ha acually is he rigorous version of he backward echnique. Suppose ha he policy is in sae j a ime. Le us apply he rule of ieraed expecaions o he expeced value V j (), condiioning on wheher a ransiion ou of sae j akes place wihin ime u or no and, in case i does, also condiion on he ime and he direcion of he firs ransiion. We hen ge V j () = u p jj (, τ) k; k j ( u +p jj (, u) ( τ µ jk (τ) dτ e s e s r b j (s) ds + e u r V j (u ) To see ha his is he same as (7.5), we need only o observe ha u τ p jj (, τ) µ j (τ) = u u s = p jj (, u) e s r b j (s) ds dτ d dτ ( p jj (, τ)) dτe u e s r b j (s) ds + s r b j (s) ds u ) r b j (s) ds + e τ r (b jk (τ) + V k (τ)) ). (7.51) p jj (, s)e s r b j (s) ds.

86 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 85 I. Uses of he differenial equaions. If he conracual funcions do no depend on he reserves, he defining relaion (7.43) give explici expressions for he sae-wise reserves and sricly speaking he differenial equaions (7.45) are no needed for consrucive purposes. They are, however, compuaionally convenien since here are good mehods for numerical soluion of differenial equaions. They also serve o give insigh ino he dynamics of he policy. The siuaion is enirely differen if he conracual funcions are allowed o depend on he reserves in some way or oher. The mos ypical examples are repaymen of a par of he reserve upon wihdrawal (a sae wihdrawn mus hen be included in he sae space Z) and expenses depending parly on he reserve. Also he primary insurance benefis may in some cases be specified as funcions of he reserve. In such siuaions he differenial equaions are an indispensable ool in he consrucion of he reserves and deerminaion of he equivalence premium. We shall provide an example in he nex paragraph. J. An example: Widow s pension. A married couple buys a combined life insurance and widow s pension policy specifying ha premiums are payable a level rae c as long as boh husband and wife are alive, widow s pension is payable a level rae b (as long as he wife survives he husband), and a life assurance wih sum s is due immediaely upon he deah of he husband if he wife is already dead (a benefi o heir dependens). The policy erminaes a ime n. The relevan Markov model is skeched Fig. F.4. We assume ha r is consan. The differenial equaions (7.45) now specialize o he following (we omi he rivial equaion for V 3 () = ): d d V () = (r + µ() + ν()) V () µ()v 1 () ν 2 ()V 2 () + c, (7.52) d d V 1() = (r + ν ()) V 1 () b, (7.53) d d V 2() = (r + µ ()) V 2 () µ () s. (7.54) Consider a modified conrac, by which 5% of he reserve is o be paid back o he husband in case he becomes a widower before ime n, he philosophy being ha couples receiving no pensions should have some of heir savings back. Now he differenial equaions are really needed. Under he modified conrac he equaions above remain unchanged excep ha he erm.5v ()ν() mus be subraced on he righ of (7.52), which hen changes o d d V () = (r + µ() +.5ν()) V () + c µ()v 1 () ν()v 2 (), (7.55) Togeher wih he condiions V j (n) =, j =, 1, 2, hese equaions are easily solved.

87 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 86 1 µ Boh alive Husband dead ν ν 2 µ 3 Wife dead Boh dead Figure 7.4: Skech of a model for wo lives. As a second case he widow s pension shall be analyzed in he presence of adminisraion expenses ha depend parly on he reserve. Consider again he policy erms described in he inroducion of his paragraph, bu assume ha adminisraion expenses incur wih an inensiy ha is a imes he curren reserve hroughou he enire period [, n]. The differenial equaions for he reserves remain as in (7.52) (7.54), excep ha for each j he erm a V j () is o be subraced on he righ of he differenial equaion for V j. Thus, he adminisraion coss relaed o he reserve has he same effec as a decrease of he ineres inensiy r by a. 7.6 Selec moraliy revisied A. A simple Markov chain model. Referring o Secion 3.4 we shall presen a simple Markov model ha offers an explanaion of he selecion phenomenon. The Markov model skeched in Fig. 7.5 is designed for sudies of selecion effecs due o underwriing sandards. The populaion is grouped ino four caegories or saes by he crieria insurable/uninsurable and insured/no insured. In addiion here is a caegory comprising he dead. I is assumed ha each person eners sae as new-born and hereafer changes saes in accordance wih a ime-coninuous Markov chain wih age-dependen forces of ransiion as indicaed in he figure. Non-insurabiliy occurs upon onse of disabiliy or serious illness or oher inervening circumsances ha enail excess moraliy. Hence i is assumed ha λ x > κ x ; x >. (7.56) Le Z(x) be he sae a age x for a randomly chosen new-born, and denoe he ransiion probabiliies of he Markov process {Z(x); x > }. The following

88 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 87 Insurable No Insured ρ x 1 Insurable Insured σ x κ x κ x 4 Dead σ x λ x λ x 3 No insurable No insured 2 No insurable Insured Figure 7.5: A Markov model for occurrences of non-insurabiliy, purchase of insurance, and deah. formulas can be pu up direcly: p 11 (x, x + ) = exp{ p 12 (x, x + ) = x+ p (, x) = exp{ x x+ x exp{ x (σ + κ)}, (7.57) u x (σ + κ)} σ u exp{ x+ u λ} du, (7.58) (σ + κ + ρ)}. (7.59) B. Selec moraliy among insured lives. The insured lives are in eiher sae 1 or sae 2. Those who are in sae 2 reached o buy insurance before hey urned non-insurable. However, he insurance company does no observe ransiions from sae 1 o sae 2; he only available informaion are x and x+. Thus, he relevan survival funcion is p [x] = p 11 (x, x + ) + p 12 (x, x + ), (7.6) he probabiliy ha a person who enered sae 1 a age x, will aain age x +. The symbol on he lef of (7.6) is chosen in accordance wih sandard acuarial noaion, see Secions

89 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 88 The force of moraliy corresponding o (7.6) is µ [x]+ = κ x+ p 11 (x, x + ) + λ x+ p 12 (x, x + ). (7.61) p 11 (x, x + ) + p 12 (x, x + ) (Self-eviden by condiioning on Z(x).) In general, he expression on he righ of (7.61) depends effecively on boh x and, ha is, moraliy is selec. We can now acually esablish ha under he presen model he selec moraliy inensiy behaves as saed in Paragraph 3.4.C. I is suiable in he following o fix x + = y, say, as we are ineresed in how he moraliy a a cerain age depends on he age of enry. C. The selec force of moraliy is an decreasing funcion of he age a enry. Formula (7.61) can be recas as µ [x]+y x = κ y + ζ(x, y)(λ y κ y ), (7.62) where We easily find ha ζ(x, y) = = p 12 (x, y) p 11 (x, y) + p 12 (x, y) (7.63) p 11 (x, y)/p 12 (x, y). (7.64) p 12 (x, y)/p 11 (x, y) = y x σ u exp{ y u (σ + κ λ)} du, which is a decreasing funcion of x. I follows ha µ [x]+y x is a decreasing funcion of x as assered in he heading of his paragraph. The explanaion is simple. Formula (7.61) expresses µ [x]+y x as a weighed average of κ y and λ y, he weighs being (of course) he condiional probabiliies of being insurable and non-insurable, respecively. The weigh aached o λ y, he larger of he wo raes, decreases as x increases. Or, pu in erms of everyday speech: in a body of insured lives of he same age x and duraion = y x, some will have urned non-insurable in he period since enry; he longer he duraion, he larger he proporion of non-insurable lives. In paricular, hose who have jus enered, are known o be insurable, ha is, µ [x] = κ x. D. Comparison wih he moraliy in he populaion. Le µ x denoe he force of moraliy of a randomly chosen life of age x from he populaion. A formula for µ x is easily obained saring from he survival funcion x p = 3 i= p i(, x). I can, however, also be picked direcly from he resuls of he previous paragraph by noing ha he paern of moraliy mus be he same in he populaion as among lives insured as newly-born, i.e. µ y = µ []+y. Then, since µ [x]+y x is a decreasing funcion of x and µ y corresponds o x =, i follows ha µ y > µ [x]+y x for all x < y.

90 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 89 Again he explanaion is rivial; due o he underwriing sandards, he proporion of non-insurable lives will be less among insured people han in he populaion as a whole. 7.7 Higher order momens of presen values A. Differenial equaions for momens of presen values. Our framework is he Markov model and he sandard insurance conrac. The se of ime poins wih possible lump sum annuiy paymens is D = {, 1..., m } (wih = and m = n). Denoe by V (, u) he presen value a ime of he paymens under he conrac during he ime inerval (, u] and abbreviae V () = V (, n) (he presen value a ime of all fuure paymens). We wan o deermine higher order momens of V (). By he Markov propery, we need only he sae-wise condiional momens V (q) j () = E[V () q Z() = j], q = 1, 2,... Theorem 2. The funcions V (q) j are deermined by he differenial equaions d d V (q) j () = (qr() + µ j ())V (q) j () qb j ()V (q 1) j () q ( ) q µ jk () (b jk ()) p V (q p) k (), p k; k j p= valid on (, n)\d and subjec o he condiions D. V (q) j ( ) = q p= Proof: Obviously, for < u < n, ( ) q (B j () B j ( )) p V (q p) j (), (7.65) p V () = V (, u) + e u r V (u), (7.66) For any q = 1, 2,... we have by he binomial formula V q () = q p= ( ) q ( V (, u) p e u q p r V (u)). (7.67) p

91 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 9 Consider firs a small ime inerval (, + d] wihou any lump sum annuiy paymen. Puing u = + d in (7.67) and aking condiional expecaion, given Z() = j, we ge V (q) j () = q p= ( ) [ q ( ] q p E V (, + d) p e r() d V ( + d)) Z() = j. (7.68) p By use of ieraed expecaions, condiioning on wha happens in he small inerval (, + d], he p-h erm on he righ of (7.68) becomes ( ) q (1 µ j () d) (b j () d) p e (q p)r() d V (q p) j ( + d) (7.69) p ( ) q + µ jk () d (b j () d + b jk ()) p e (q p)r() d V (q p) k ( + d). p k; k j (7.7) Le us idenify he significan pars of his expression, disregarding erms of order o(d). Firs look a (7.69); for p = i is for p = 1 i is (1 µ j () d)e qr() d V (q) j ( + d), q b j () d e (q 1)r() d V (q 1) j ( + d), and for p > 1 is o(d). Nex look a (7.7); he facor d (b j () d + b jk ()) p = d p r= ( ) p (b j () d) r (b jk ()) p r r reduces o d (b jk ()) p so ha (7.7) reduces o ( ) q µ jk () d (b jk ()) p e (q p)r() d V (q p) k ( + d). p k; k j Thus, we gaher V (q) j () = (1 µ j () d)e qr() d V (q) j ( + d) + q b j () d e (q 1)r() d V (q 1) j ( + d) q ( ) q + µ jk () d (b jk ()) p e (q p)r() d V (q p) k ( + d). p p= k; k j Now subrac V (q) j ( + d) on boh sides, divide by d, le d end o, and use lim ( e qr() d 1 ) /d = qr() o obain he differenial equaion (7.65).

92 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 91 The condiion (7.65) follows easily by puing d and in he roles of and u in (7.67) and leing d end o. A rigorous proof is given in [38]. Cenral momens are easier o inerpre and herefore more useful han he non-cenral momens. Leing m (q)j denoe he q-h cenral momen corresponding o he non-cenral V (q)j, we have m (1) j () = V (1) j (), (7.71) q ) m (q) j () = p= ( 1) q p ( q p V (p) j () ( V (1) j ()) q p. (7.72) B. Compuaions. The compuaion goes as follows. Firs solve he differenial equaions in he upper inerval ( m 1, n), where he side condiions (7.65) are jus V (q) j (n ) = (B j (n) B j (n )) q (7.73) since V (q) j (n) = δ q (he Kronecker dela). Then, if m > 1, solve he differenial equaions in he inerval ( m 2, m 1 ) subjec o (7.65) wih = m 1, and proceed in his manner downwards. C. Numerical examples. We shall calculae he firs hree momens for some sandard forms of insurance relaed o he disabiliy model in Paragraph 7.3.C. We assume ha he ineres rae is consan and 4.5% per year, r = ln(1.45) =.4417, and ha he inensiies of ransiions beween he saes depend only on he age x of he insured and are µ x = ν x = x, σ x = x, ρ x =.5. The inensiies µ, ν, and σ are hose specified in he G82M echnical basis. (Tha basis does no allow for recoveries and uses ρ = ). Consider a male insured a age 3 for a period of 3 years, hence use µ 2 () = µ 12 () = µ 3+, µ 1 () = σ 3+, µ 1 () = ρ 3+, < < 3 (= n). The cenral momens m (q)j defined in (7.71) (7.72) have been compued for he saes and 1 (sae 2 is unineresing) a imes =, 6, 12, 18, 24, and are shown in Table 7.1 for a erm insurance wih sum 1 (= b 2 = b 12 ); in Table 7.2 for an annuiy payable in acive sae wih level inensiy 1 (= b ); in Table 7.3 for an annuiy payable in disabled sae wih level inensiy 1

93 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 92 (= b 1 ); in Table 7.4 for a combined policy providing a erm insurance wih sum 1 (= b 2 = b 12 ) and a disabiliy annuiy wih level inensiy.5 (= b 1 ) agains level ne premium.1318 (= b ) payable in acive sae. You should ry o inerpre he resuls. D. Solvency margins in life insurance an illusraion. Le Y be he presen value of all fuure ne liabiliies in respec of an insurance porfolio. Denoe he q-h cenral momen of Y by m (q). The so-called normal power approximaion of he upper ε-fracile of he disribuion of Y, which we denoe by y 1 ε, is based on he firs hree momens and is y 1 ε m (1) + c 1 ε m (2) + c2 1 ε 1 m (3) 6 m, (2) where c 1 ε is he upper ε-fracile of he sandard normal disribuion. Adoping he so-called break-up crierion in solvency conrol, y 1 ε can be aken as a

94 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 93 Table 7.1: Momens for a life assurance wih sum 1 Time m (1) = m (1)1 : m (2) = m (2)1 : m (3) = m (3)1 : Table 7.2: Momens for an annuiy of 1 per year while acive: Time m (1) : m (1)1 : m (2) : m (2)1 : m (3) : m (3)1 : Table 7.3: Momens for an annuiy of 1 per year while disabled: Time m (1) : m (1)1 : m (2) : m (2)1 : m (3) : m (3)1 :

95 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 94 Table 7.4: Momens for a life assurance of 1 plus a disabiliy annuiy of.5 per year agains ne premium of.1318 per year while acive: Time m (1) : m (1)1 : m (2) : m (2)1 : m (3) : m (3)1 : minimum requiremen on he echnical reserve a he ime of consideraion. I decomposes ino he premium reserve, m (1), and wha can be ermed he flucuaion reserve, y 1 ε m (1). A possible measure of he riskiness of he porfolio is he raio R = ( y 1 ε m (1)) /P, where P is some suiable measure of he size of he porfolio a he ime of consideraion. By way of illusraion, consider a porfolio of N independen policies, all idenical o he one described in connecion wih Table 7.4 and issued a he same ime. Taking as P he oal premium income per year, he value of R a he ime of issue is for N = 1, 12. for N = 1, 3.46 for N = 1, 1.6 for N = 1, and.332 for N = A Markov chain ineres model The Markov model A. The force of ineres process. The economy (or raher he par of he economy ha governs he ineres) is a homogeneous ime-coninuous Markov chain Y on a finie sae space J Y = {1,..., J Y }, wih inensiies of ransiion λ ef, e, f J Y, e f. The force of ineres is r e when he economy is in sae e, ha is, r() = e I Y e ()r e, (7.74) where Ie Y () = 1[Y () = e] is he indicaor of he even ha Y is in sae e a ime. Figure F.5 shows a flow-char of a simple Markov chain ineres rae model wih hree saes,.2,.5,.8. Direc ransiion can only be made o a neighbouring sae, and he oal inensiy of ransiion ou of any sae is.5, ha is, he ineres rae changes every wo years on he average. By symmery, he long run average ineres rae is.5.

96 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 95 λ 12 =.5 λ 23 =.25 r 1 =.2 r 2 =.5 r 3 =.8 λ21 =.25 λ32 =.5 Figure 7.6: Skech of a simple Markov chain ineres model. B. The paymen process. We adop he sandard Markov chain model of a life insurance policy in Secion 7.2 and equip he associaed indicaor and couning processes wih opscrip Z o disinguish hem from he corresponding eniies for he Markov chain governing he ineres. We assume he paymen sream is of he sandard ype considered in Secion 7.5. C. The full Markov model. We assume ha he processes Y and Z are independen. Markov chain on J Y J Z wih inensiies λ ef (), e f, j = k, κ ej,fk () = µ jk (), e = f, j k,, e f, j k. Then (Y, Z) is a Differenial equaions for momens of presen values A. The main resul. For he purpose of assessing he conracual liabiliy we are ineresed in aspecs of is condiional disribuion, given he available informaion a ime. We focus here on deermining he condiional momens. By he Markov assumpion, he funcions in ques are he sae-wise condiional momens V (q) ej () = E [( 1 v() n q ] v db) Y () = e, Z() = j. Copying he proof in Secion 7.7, we find ha he funcions V (q) ej ( ) are deermined by he differenial equaions d d V (q) ej µ jk () k;k j () = (qr e + µ j () + λ e )V (q) ej q p= ( ) q p (b jk ()) p V (q p) ek () f;f e valid on (, n)\d and subjec o he condiions q ) V (q) ej ( ) = p= ( q p () qb j()v (q 1) () ej λ ef V (q) fj (), (7.75) ( B j ()) p V (q p) ej (), D. (7.76)

97 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 96 For q = 2, 3,..., denoe by m (q) ej () he q-h cenral momen corresponding o V (q) (1) ej (), and define m(1) ej () = V ej (). Having compued he non-cenral momens, we obain he cenral momens of orders q > 1 from m (q) ej () = q p= ( ) q ( ) q p ( 1) q p V (p) ej p () V (1) ej (). B. Numerical resuls for a combined insurance policy. Consider a combined life insurance and disabiliy pension policy issued a ime o a person who is hen aged x, say. The relevan saes of he policy are 1 = acive, 2 = disabled, and 3 = dead. A ime, when he insured is x + years old, ransiions beween hese saes ake place wih inensiies µ 13 () = µ 23 () = (x+), µ 12 () = (x+), µ 21 () =.5. We exend he model by assuming ha he force of ineres may assume hree values, r 1 = ln(1.) = (low in fac no ineres), r 2 = ln(1.45) =.442 (medium), and r 3 = ln(1.9) =.8618 (high), and ha he ransiions beween hese saes are governed by a Markov chain wih infiniesimal marix of he form Λ = λ (7.77) The scalar λ can be inerpreed as he expeced number of ransiions per ime uni and is hus a measure of ineres volailiy. Table 1 displays he firs hree cenral momens of he presen value a ime for he following case, henceforh referred o as he combined policy for shor: he age a enry is x = 3, he erm of he policy is n = 3, he benefis are a life assurance wih sum 1 (= b 13 = b 23 ) and a disabiliy annuiy wih level inensiy.5 (= b 2 ), and premiums are payable in acive sae coninuously a level rae π (= b 1 ), which is aken o be he ne premium rae in sae (2,1) (i.e. he rae ha esablishes expeced balance beween discouned premiums and benefis when he insured is acive and he ineres is a medium level a ime ). The firs hree rows in he body of he able form a benchmark; λ = means no ineres flucuaion, and we herefore obain he resuls for hree cases of fixed ineres. I is seen ha he second and hird order momens of he presen value are srongly dependen on he (fixed) force of ineres and, in fac, heir absolue values decrease when he force of ineres increases (as could be expeced since increasing ineres means decreasing discoun facors and, hence, decreasing presen values of fuure amouns).

98 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 97 Table 7.5: Cenral momens m (q) ej () of orders q = 1, 2, 3 of he presen value of fuure benefis less premiums for he combined policy in ineres sae e and policy sae j a ime, for some differen values of he rae of ineres changes, λ. Second column gives he ne premium π of a policy saring from ineres sae 2 (medium) and policy sae 1 (acive). e, j : 1, 1 1, 2 2, 1 2, 2 3, 1 3, 2 λ π q

99 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 98 I is seen ha, as λ increases, he differences across he hree pairs of columns ge smaller and in he end hey vanish compleely. The obvious inerpreaion is ha he iniial ineres level is of lile imporance if he ineres changes rapidly. The overall impression from he wo cenral columns corresponding o medium ineres is ha, as λ increases from, he variance of he presen value will firs increase o a maximum and hen decrease again and sabilize. This observaion suppors he following piece of inuiion: he inroducion of moderae ineres flucuaion adds uncerainy o he final resul of he conrac, bu if he ineres changes sufficienly rapidly, i will behave like fixed ineres a he mean level. Presumably, he values of he ne premium in he second column reflec he same effec Complemen on Markov chains A. Time-coninuous Markov chains. Le X = {X()} be a ime-coninuous Markov chain on he finie sae space J = {1,..., J}. Denoe by P (, u) he J J marix whose j, k-elemen is he ransiion probabiliy p jk (, u) = P [X(u) = k X() = j]. The Markov propery implies he Chapman-Kolmogorov equaion valid for s u. In paricular P (s, u) = P (s, )P (, u), (7.78) P (, ) = I J J, (7.79) he J J ideniy marix. The inensiy of ransiion from sae j o sae k ( j) a ime is defined as κ jk () = lim d p jk (, + d)/d or, equivalenly, by when he limi exiss. Then, obviously, p jk (, + d) = κ jk () d + o(d), (7.8) p jj (, + d) = 1 κ j ()d + o(d), (7.81) where κ j () = k; k j κ jk() can appropriaely be ermed he oal inensiy of ransiion ou of sae j a ime. The infiniesimal marix M() is he J J marix wih κ jk () in row j and column k, defining κ jj () = κ j (). Wih his noaion (7.8) (7.81) can be assembled in P (, + d) = I + M()d. (7.82) The probabiliies deermine he inensiies. Conversely, he probabiliies are deermined by he inensiies hrough Kolmogorov s differenial equaions, which are readily obained upon combining (7.78) and (7.82). There is a forward equaion, P (s, ) = P (s, )M(), (7.83)

100 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 99 and a backward equaion, P (, u) = M()P (, u), (7.84) each of which deermine P (, u) when combined wih he condiion (7.79). B. Saionary Markov chains. When M() = M, a consan, hen (as is obvious from he Kolmogorov equaions) P (s, ) = P (, s) depends on s and only hrough s. In his case we wrie P () = P (, ), allowing a sligh abuse of noaion. The equaions (7.83) (7.84) now reduce o d P () = P ()M = MP (). (7.85) d The limi Π = lim P () exiss, and he j-h row of Π is he limiing (saionary) disribuion of he sae of he process, given ha i sars from sae j. We shall assume hroughou ha all saes communicae wih each oher. Then he saionary disribuion π = (π 1,..., π J ), say, is independen of he iniial sae, and so Π = 1 J 1 π, (7.86) where 1 J 1 is he J-dimensional column vecor wih all enries equal o 1. Leing in (7.85) and using (7.86), we ge 1 J 1 π M = M1 J 1 π = J J (a marix of he indicaed dimension wih all elemens equal o ), ha is, π M = 1 J, M1 J 1 = J 1. (7.87) Thus, is an eigenvalue of M, and π and 1 J 1 are corresponding lef and righ eigenvecors, respecively. From Paragraph 4.3 of Karlin and Taylor (1975) we gaher he following useful represenaion resul. Le ρ j, j = 1,..., J, be he eigenvalues of M and, for each j, le ψ j and φ j be he corresponding lef and righ eigenvecors, respecively. Le Φ be he J J marix whose j-h column is φ j. Then he j-h row of Φ 1 is jus ψ j, and inroducing R() = diag(eρj ), he ransiion marix P () can be expressed as P () = ΦR()Φ 1 = J e ρj φ j ψ j, (7.88) which is compuaionally convenien. We can ake ρ 1 o be and φ 1 = 1 J 1. Then ψ 1 = π, and we obain P () = 1 J 1 π + j=1 J e ρj φ j ψ j. (7.89) j=2

101 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 1 All he ρ j, j = 2,..., J are sricly negaive, and so he represenaion shows ha he ransiion probabiliies converge exponenially o he saionary disribuion. A his poin we need o make precise ha in (7.85) he d d is o be hough of as an operaor, o be disinguished from he marix P (1) () of derivaives i produces when applied o P (). Now, for λ >, define P λ () = P (λ). (7.9) Upon differeniaing his relaionship and using (7.85), we obain d d P λ() = d d P (λ) = P (1) (λ)λ = P λ ()λm, which shows ha P λ (), which is cerainly a marix of ransiion probabiliies, has infiniesimal marix M λ = λm. (7.91) Thus, doubling (say) he inensiies of ransiion affecs he ransiion probabiliies he same way as a doubling of he ime period. 7.9 Dependen lives Inroducion Acuarial ables for muli-life sauses are invariably based on he assumpion of muual independence beween he remaining lenghs of he individual componen lives. The independence hypohesis is compuaionally convenien or, raher, was so in hose days when ables had o be consruced. In he presen era of scienific compuing such concerns are no so imporan. Le S and T be real-valued random variables defined on some probabiliy space. Being mainly ineresed in survival analysis relaed o life insurance, we shall le S and T represen he remaining life lenghs of wo individuals insured under he same policy, le us say husband and wife, respecively. Thus, we make he convenien (bu no essenial) assumpions ha S and T are sricly posiive wih probabiliy 1 and ha hey possess a join densiy. The variables S and T are sochasically independen if P[S > s, T > ] = P[S > s] P[T > ] for all s and. In paricular, sochasic independence implies ha C(g(S), h(t )) = for all funcions g and h such ha he covariance is well defined. (We le C and V denoe covariance and variance, respecively.) Moraliy saisics sugges ha life lenghs of husband and wife are dependen and, moreover, ha hey are posiively correlaed. I is easy o hink of possible explanaions o his empirical fac. For insance, ha people who marry

102 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 11 do so because hey have somehing in common ( birds of a feaher fly ogeher ), or ha married people share lifesyle and living condiions and herefore also hazards of diseases and accidens, or ha deah of he spouse impairs he living condiions for he survivor ( a grief effec, or maybe he husband jus does no know where he kichen is and sarves o deah shorly afer he loss of he spouse). Correlaion is a raher special measure of dependence essenially i measures linear dependence beween random variables and i is no sufficienly refined for our purposes Noions of posiive dependence There are various noions of posiive dependence beween pairs of random variables, and we will inroduce hree of hem here. A comprehensive reference ex is [5]. Definiion PQD: S and T are posiively quadran dependen, wrien PQD(S, T ), if P[S > s, T > ] P[S > s] P[T > ] for all s and. (7.92) This definiion is symmeric in he wo variables, so PQD(S, T ) is he same as PQD(T, S). The defining inequaliy (7.92) is equivalen o P[S > s T > ] P[S > s], (7.93) which is easy o inerpre: knowing e.g. ha he wife will survive a leas s years improves he survival prospecs of he husband. Definiion AS: S and T are associaed, wrien AS(S, T ), if C(g(S, T ), h(s, T )) (7.94) for all real-valued funcions g and h ha are increasing in boh argumens (and for which he covariance exiss). Also he definiion of AS is symmeric in he wo variables, so AS(S, T ) is he same as AS(T, S). Definiion RTI: S is righ ail increasing in T, wrien RTI(S T ), if P[S > s T > ] is an increasing funcion of for each fixed s. (7.95) The definiion of RTI is no symmeric in he wo variables. To each noion of posiive dependence here is a corresponding noion of negaive dependence. We can reasonably say ha S and T are negaively quadran dependen if he inequaliy (7.92) is reversed. This is he same as PQD( S, T ), see Exercise 21. We can say ha S and T are negaively associaed ( dissociaed

103 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 12 does no have he righ connoaion) if he inequaliy (7.94) is reversed. This is he same as AS( S, T ), see Exercise 21. We say ha S is righ ail decreasing in T, wrien RTD(S T ) if P[S > s T > ] is a decreasing funcion of for each fixed s. This is he same as RTD( S T ), see Exercise 21. Since resuls on posiive dependence hus ranslae ino resuls on negaive dependence, we will henceforh focus on he former. Theorem 1: RTI(S T ) AS(S, T ) PQD(S, T ). Proof (incomplee): The firs implicaion, RTI(S T ) AS(S, T ), is he hard par. The proof is long and echnical and can be found in [2]. The second implicaion AS(S, T ) PQD(S, T ) is easy. For g(s, T ) = 1 (s, ) (S) = 1[S > s] and h(s, T ) = 1 (, ) (T ) = 1[T > ], (7.94) reduces o C(1[S > s], 1[T > ]), (7.96) which is jus a reformulaion of he defining inequaliy (7.92). As a parial compensaion for he absence of proof of he firs implicaion, le us prove he shorcu implicaion RTI(S T ) PQD(S, T ): if RTI(S T ), hen P[S > s T > ] P[S > s T > ] for >, which is he same as (7.93). The following resul is a parial converse o he second implicaion in Theorem 1. I could be formulaed by saying ha posiive quadran dependence is equivalen o marginal associaion. Lemma 1: PQD(S, T ) C(g(S), h(t )) for increasing funcions g and h. Proof : By (7.96) he resul holds for increasing indicaor funcions. Then i holds for increasing simple funcions, g(s) = g + m i=1 g i1[s > s i ] and h(t ) = h + n j=1 h j1[t > j ] (wih consan coefficiens g i and h j and g i >, i = 1,..., m and h j >, j = 1,..., n), as is seen from C(g(S), h(t )) = m i=1 j=1 n g i h j C (1[S > s i ], 1[T > j ]). Then i holds for all increasing funcions g(s) and h(t ) since since any increasing funcion from R o R can be wrien as he limi of a sequence of increasing simple funcions (monoone convergence). In he definiions of PQD, AS, and RTI we could equally reasonably have enered he evens S s and T. Due o he coninuiy propery of probabiliy measures, i does no maer which inequaliies we use, > or. See Exercise 21. for A 1 A 2... lim P[A n ] = P[ n A n ] for A 1 A 2... lim P[A n ] = P[ n A n ]

104 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE Dependencies beween presen values The following able liss formulas for he life lenghs and he survival funcions of he four sauses husband, wife, heir join life, and heir las survivor. Saus (z) Life lengh U Survival funcion P[U > τ] Husband (x) S P[S > τ] Wife (y) T P[T > τ] Join life (x, y) S T P[S > τ, T > τ] Las survivor x, y S T P[S > τ] + P[T > τ] P[S > τ, T > τ] The nex able recapiulae he formulas for presen values and heir expeced values for he mos basic insurance benefis o a saus (z) wih remaining life lengh U. Paymen scheme Presen value Expeced presen value Pure endowmen e rn 1[U > n] ne z = e rn P[U > n] n Life annuiy e rτ 1[U > τ] dτ ā z n = n e rτ P[U > τ] dτ Term insurance e ru 1[U < n] Ā 1x = 1 n E z ā z n n The life lenghs of he sauses lised in he firs able are increasing funcions of boh S and T. From he second able we see ha, for a general saus wih remaining life lengh U, he presen value of a pure survival benefi (life endowmen or life annuiy) is an increasing funcions of U, whereas he presen value of he pure deah benefi is a decreasing funcion of U (we assume he ineres rae r is posiive.) Combining hese observaions and Theorem 1, we can infer he following (and many oher hings): If PQD(S, T ), hen pure survival benefis on any wo sauses are posiively dependen, pure deah benefis on any wo sauses are posiively dependen, and any pure survival benefi and any deah benefi are negaively dependen. We can also draw conclusions abou he bias inroduced in equivalence premiums by erroneously adoping he independence hypohesis. For insance, if PQD(S, T ) and we work under he independence hypohesis, hen he presen value of a survival benefi on he join life will be underesimaed, whereas he presen value of a survival benefi on he las survivor will be overesimaed. For he deah benefi i is he oher way around. Combining hese hings we may conclude e.g. ha, for a deah benefi on he join life agains level premium during join survival, he equivalence premium will be overesimaed A Markov chain model for wo lives I is no easy o creae a given form of dependence beween he life lenghs S and T by direc specificaion of heir join disribuion. However, he process poin

105 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 14 of view, which is a powerful one, quie naurally allows us o express various ideas abou dependencies beween life lenghs of a couple. A suiable framework is he Markov model skeched in Figure F.4. The following formulas are obvious: p (s, ) = e p 1 (s, ) = p 2 (s, ) = s s s µ+ν, The join survival funcion of S and T is e τ s µ+ν µ τ e τ ν dτ, e τ s µ+ν ν τ e τ µ dτ. P[S > s, T > ] = = { p (, ) + p (, s)p 1 (s, ), s, p (, s) + p (, )p 2 (s, ), s >, { e µ+ν + s e τ µ+ν µ τ e τ ν, s, e s µ+ν + s µ+ν ν τ e s τ µ, s >. e τ The marginal survival funcion of T is (pu s = in (7.97)) (7.97) P[T > ] = p (, ) + p 1 (, ) = e µ+ν + e τ µ+ν µ τ e τ ν,. (7.98) I is inuiively obvious ha S and T are independen if µ τ = µ τ and ν τ = ν τ for all τ, and his will follow from Theorem 2 below (see also Exercise 1). I is also inuiively clear ha S and T will become dependen if we le he moraliy raes depend on marial saus. Le us see wha happens if he moraliy rae increases upon he loss of he spouse. Theorem 2: If µ τ µ τ and ν τ ν τ for all τ, hen S and T are posiively dependen in he sense RTI(S T ) (hence AS(S, T ) and PQD(S, T )). If µ τ µ τ and ν τ ν τ for all τ, hen S and T are negaively dependen in he sense RTD(S T ) (hence AS( S, T ) and PQD( S, T )). If µ τ = µ τ and ν τ = ν τ for all τ, hen S and T are independen. Proof : Consider firs he case s. From (7.97) and (7.98) we ge P[S > s T > ] = e µ+ν + s e τ µ+ν µ τ e τ ν dτ e µ+ν + µ+ν µ τ e = 1 e τ s e τ τ ν dτ µ+ν ν µ τ dτ e µ+ν ν + e τ µ+ν ν µ τ dτ. Now we need only o sudy he denominaor in he second erm as a funcion of. Is derivaive is e s µ+ν ν (ν ν ).

106 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 15 I follows ha P[S > s T > ] is an increasing funcion funcion of if ν ν and a decreasing funcion funcion of if ν ν. Nex consider he case s >. This is a bi more complicaed. From (7.97) and (7.98) we ge s P[S > s T > ] = e µ+ν + s e τ µ+ν ν τ e s τ µ dτ e µ+ν + e τ µ+ν µ τ e τ ν dτ. By he rule d(u/v) = (v du u dv)/v 2, he sign of P[S > s T > ] is he same as ha of ( e µ+ν + ( e s µ+ν + s ) e τ µ+ν µ τ e ( τ ν dτ e µ+ν ν e s µ ) ) e τ µ+ν ν τ e s τ µ dτ ( e µ+ν ( µ ν ) + e µ+ν µ + ) e τ µ+ν µ τ e τ ν dτ ( ν ). In his expression wo erms cancel in he las parenhesis. Furher, o ge rid of some common facors, le us muliply wih e s µ+ν e µ+ν, which preserves he sign and urns he expression ino ( ) 1 + e τ µ+ν ν µ τ dτ e s µ µ +ν ν ( s ) ( e ) s τ µ µ +ν ν τ dτ ν + e τ µ+ν ν µ τ dτ ν. Subsiuing s e s τ µ µ +ν ν τ dτ = s s + e s τ µ µ +ν (µ τ µ τ + ν τ ) dτ e s τ µ µ +ν (µ τ µ τ) dτ = e s µ µ +ν 1 + and rearranging a bi, we arrive a ( ν + ν e τ µ+ν ν µ τ dτ + ) s e s τ µ+ν ν µ τ dτ (ν ν ). s e s τ µ µ +ν (µ τ µ τ ) dτ e s τ µ µ +ν (µ τ µ τ ) dτ I follows ha P[S > s T > ] is an increasing funcion funcion of if µ µ and ν ν and a decreasing funcion funcion of if µ µ and ν ν.

107 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE Condiional Markov chains Rerospecive feriliy analysis In connecion wih a pension insurance scheme here is an addiional benefi which is a sum insured payable o possible dependen children less han 18 years old a he ime of deah of he insured. In he echnical basis we herefore need o make assumpions abou birhs. We have o disinguish by sex, and in he following we consider female insured only. The Figure below shows a flowchar for possible life hisories wih deah and birhs (a mos J). To keep hings simple, we assume ha he insured eners he scheme in sae a age and ha he process is Markov: for a year old who has given birh o j children, he moraliy rae µ j () and he feriliy rae φ j () are funcions of and j only. Alive birhs φ φ j 1 j φ j φ J 1 J Alive Alive j birhs J birhs µ µ j µ J d. Dead Assume now ha he pas hisory of birhs and deah is observed only upon deah of he insured, when he addiional benefi o he possible dependens is due. Suppose ha he saisical daa comprise only hose who are dead a he ime of consideraion and ha for each of hose here is a complee record of he imes of possible birhs and of deah. In hese daa he observed life hisory of a woman, who enered he scheme u years ago, is governed by a Markov process as described above, bu wih inensiies µ j () = µ 1 j() p jd (, u), (7.99) φ j () = φ j () p j+1,d(, u) p jd (, u). (7.1) We see ha µ j () µ j(), which is easy o explain (we are looking a he moraliy, given deah). We are going o prove a more ineresing resul: If moraliy increases wih he number of birhs, ha is, µ j () µ j+1 (), j =,..., J 1, >, (7.11)

108 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 17 hen φ j () φ j (), j =,..., J 1, >. (7.12) We need o prove ha p j+1,d (, u) p jd (, u), j = 1,..., J 1. I is convenien o work wih p j (, u) = 1 p jd (, u) = J p jk (, u), (7.13) he probabiliy ha a year old wih j birhs will survive o age u, and o prove he hypohesis k=j H j : p k (, u) p j (, u), k = j + 1,..., J, (7.14) for j =,..., J 1. The proof goes by inducion downwards, proving ha H j+1 implies H j. Thus assume H j+1 is rue. By direc reasoning (or an easy calculaion) he moraliy inensiy a age u (> ) associaed wih he survival funcion (F.49) is k j µ j (, u) = p jk(, u)µ k (u) k j p, (7.15) jk(, u) hence p j (, u) = e u µj (,s)ds. (7.16) Two more expressions for p j (, u), boh obvious, are p j (, u) = k j p jk (, τ) p k (τ, u), (7.17) τ u, and p j (, u) = e u (φj+µj ) + By (F.47) and (7.15) we have hence Therefore, from (F.53) we ge u e τ (φj+µj ) φ j (τ) p j+1 (τ, u) dτ. (7.18) µ j (u) µ j+1 (, u), (7.19) e u µj p j+1 (, u). p j (, u) e u φj p j+1 (, u) + u e τ φj φ j (, τ) p j+1 (, τ) p j+1 (τ, u) dτ. (7.11)

109 CHAPTER 7. MARKOV CHAINS IN LIFE INSURANCE 18 Focusing on he las wo facors under he inegral, use in succession (F.49), he inducion hypohesis (7.14), and (F.5), o deduce p j+1 (, τ)p j+1 (τ, u) = Puing his ino (7.11), we obain ( p j (, u) e u u φj + e τ J k=j+1 J k=j+1 = p j+1 (, u). p j+1,k (, τ)p j+1 (τ, u) p j+1,k (, τ)p k (τ, u) ) φj φ j (, τ) dτ p j+1 (, u) = p j+1 (, u). I follows ha H j is rue. Since H J 1 is obviously rue, we are done. Commen: The inequaliy (F.48) means ha he feriliy raes will be overesimaed if one uses he esimaors for he φ j based on diseased paricipans in he scheme. If he inequaliies (F.47) are reversed, hen also he inequaliy (F.48) will be reversed, and he esimaors he φ j will underesimae he feriliy. In paricular i follows ha, under he hypohesis of non-differenial moraliy, he feriliy raes will be unbiasedly esimaed from he seleced maerial of diseased paricipans.

110 Chaper 8 Probabiliy disribuions of presen values Absrac: A sysem of inegral equaions is obained for he saewise probabiliy disribuions of he presen value of fuure paymens on a mulisae life insurance policy under Markov assumpions. They are brough on a differenial form convenien for compuaion and applied o some cases. Key words: Mulisae life insurance, Markov couning process, opional sampling, sochasic ineres. 8.1 Inroducion A. Background and moive of he presen sudy. By radiion, life insurance mahemaics ceners on condiional expeced values of discouned cashflows. A key ool of he heory are Thiele s differenial equaions, which describe he developmen of such expeced values for a mulisae policy driven by a Markov process. In wo recen papers one of he auhors (Norberg, 1994, 1995) obains differenial equaions for higher order momens of presen values and offers examples of heir poenial uses in solvency assessmens and in consrucion of unradiional insurance producs. In coninuance of hose resuls, we underake o deermine he probabiliy disribuions ha are a he base of he momens and of any oher expeced values of ineres. Knowledge of he disribuion of he presen value, and in paricular is upper ail, gives insigh ino he riskiness of he conrac beyond wha is provided by he mean and he higher order momens. B. Conens of he paper. By way of inroducion, Secion 2 deals briefly wih models involving only a finie number of random variables. In such siuaions he disribuions of presen values (and any oher funcions of he random variables) can be obained by 19

111 CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES11 inegraing he finie-dimensional disribuion. This approach comes down overagains more complex siuaions where sochasic processes have o be employed. In Secion 3 we consider a general mulisae insurance policy wih paymens of assurance and annuiy ypes and wih sae-dependen force of ineres. Assuming ha he sae process is Markov, we derive in Secion 4 a se of inegral equaions for he condiional probabiliy disribuion funcion of he presen value of fuure benefis less premiums, given he sae a he ime of consideraion. The equaions are convered o a differenial form ha forms he basis of a compuaional scheme. Examples of applicaions, including numerical illusraions, are given in Secion Calculaion of probabiliy disribuions of presen values by elemenary mehods A. A simple example involving only one life lengh. De Pril (1989) and Dhaene (199) compile liss of disribuions of presen values of sandard single-life insurance benefis ; pure endowmen, erm assurance, endowmen assurance, and life annuiy. To offer an example along his line, consider a life insurance policy specifying ha he sum assured b is o be paid ou immediaely upon he (possible) deah of he insured wihin n years afer he dae of issue of he policy and ha premiums are payable coninuously a level rae c per year as long as he conrac is in force. Suppose ineres accumulaes wih consan inensiy δ so ha he τ years discoun facor is v τ = e δτ. Denoing he remaining life ime of he insured by T, he presen value of benefis less premiums on he conrac is U(T ) = bv T 1 (,n] (T ) cā T n, where ā = vτ dτ = (1 e δ )/δ is he presen value of an annuiy cerain payable coninuously a level rae 1 per year for years. The funcion U is nonincreasing in T and, leing T be a random variable, we easily find he probabiliy disribuion (4.59). The jump a cā n is due o he posiive probabiliy of survival o ime n. Similar effecs are o be anicipaed also for more complex finie erm insurance producs since, in general, here is a posiive probabiliy ha he policy will remain in he curren sae unil he conrac erminaes. Fig. 1 shows he graph of he funcion in (4.59) for he case where δ = ln(1.45) =.4417, P[T > ] = e µ(τ)dτ wih µ() = (3+), n = 3, b = 1, and c =.4268; ineres and moraliy are according o he firs order echnical basis currenly used by mos Danish insurers, supposing he insured is a male who is 3 years old a he ime when he policy is issued, and he given value of c is he ne premium rae. This conrac will henceforh be referred o as he erm insurance policy. Figure 1 abou here. Fig. 1: Probabiliy disribuion of he presen value a ime of he erm in-

112 CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES111 surance policy. B. Models involving a finie number of random variables. The analysis above was sraighforward because he presen value is a funcion of only one random variable. The approach works whenever only a finie number of random variables, T 1,..., T r, are involved, e.g. for a mulilife insurance depending on r lifeimes. The probabiliy disribuion of a presen value U(T 1,..., T r ), say, is obained by inegraing he join probabiliy funcion of he T i -s over he ses {( 1,..., r ); U( 1,..., r ) u} for (in principle) all u. When applicable a all, his procedure would usually be far more complicaed han he one we are going o demonsrae. Thus, wih no furher ado, we now urn o he principal message of he paper. 8.3 The general Markov mulisae policy A. Paymens, ineres, and presen values. Consider a sream of paymens in respec of a conrac issued a ime and erminaing a ime n, say, and denoe by B() he oal amoun paid in he ime inerval [, ]. The paymen funcion {B()} is assumed o be righconinuous and of bounded variaion. Money is currenly invesed in (or borrowed from) a fund ha a any ime yields δ() in reurn per uni of ime and uni amoun on deposi, ha is, δ() is he force of ineres a ime. Then he discouned value a ime of a uni due a ime τ is e τ δ, and so he presen value a ime of he fuure paymens under he conrac is n e τ δ db(τ). (8.1) The shor-hand exemplified by δ = δ(s)ds will be in frequen use hroughou. By convenion, b a means (a,b] if b < and if b =. (a, ) B. The muli-sae insurance policy. We adop he sandard se-up of life insurance mahemaics as presened in Norberg (1994). There is a se of saes, J = {1,..., J}, such ha a any ime [, n] he policy is in one an only one sae. Denoe by X() he sae of he policy a ime. Considered as a funcion of, {X} is aken o be righconinuous, and X() = 1 implying (as a convenion) ha he policy commences in sae 1. Inroduce I j () = 1[X() = j], he indicaor of he even ha he policy is in sae j a ime, and N jk () = {τ; τ (, ], X τ = j, X(τ) = k}, he oal number of ransiions of X from sae j o sae k ( j) by ime. The paymen funcion B is assumed o be of he form (7.41), where each B j is a deerminisic paymen funcion specifying paymens due during sojourns in sae j (a general life annuiy) and each b jk is a deerminisic funcion specifying paymens due upon ransiions from sae j o sae k (a general life assurance). The lef-limi in I j ( ) means ha he sae j annuiy is effecive a ime if

113 CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES112 he policy is in sae j jus prior o (bu no necessarily a) ime. Consisenly we define I j ( ) = 1. We shall allow he force of ineres o depend on he curren sae, ha as in (7.74. C. The ime-coninuous Markov model. I is assumed ha {X()} is a (coninuous-ime) Markov chain. Denoe he ransiion probabiliies by The ransiion inensiies p jk (, u) = P[X(u) = k X() = j]. µ jk () = lim h p jk (, + h) are assumed o exis for all j, k J, j k. The oal inensiy of ransiion ou of sae j is µ j () = k;k j µ jk(). The probabiliy of saying uninerrupedly in sae j during he ime inerval from o u is e u µj. 8.4 Differenial equaions for saewise disribuions A. The saewise probabiliy disribuions. The problem is o deermine he condiional probabiliy disribuion of he liabiliy in (8.1), given he informaion available a ime. Since he Markov assumpion implies condiional independence beween pas and fuure for fixed presen sae of he policy, he relevan funcions are he saewise probabiliy disribuions defined by [, n], u R, j J. [ n P j (, u) = P e τ ] δ db(τ) u I j() = 1, (8.1) B. A sysem of inegral equaions. A simple heurisic argumen will esablish ha he probabiliies in (8.1) saisfy he inegral equaions P j (, u) = k;k j n e s µj µ jk (s) ds s ) P k (s, e δj(s ) u e δj(s τ) db j (τ) b jk (s) + e n µj 1 [ n ] e δj(τ ) db j (τ) u. (8.2)

114 CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES113 In he firs erms on he righ here he facor e s µj µ jk (s)ds is he probabiliy ha he policy says in sae unil ime s (< n) and hen makes a ransfer o sae k ( j) in he small ime inerval [s, s + ds). In his case he annuiy B j is in force during he ime inerval (, s], he lump sum b jk (s) falls due a ime s, and he ineres rae during his ime inerval is δ j, and so he even akes place if n e τ δ db(τ) u (8.3) s or, equivalenly, e δj(τ ) db j (τ) + e δj(s ) b jk (s) + e δj(s ) n n s e τ s δ db(τ) e δj(s ) (u s s e τ s δ db(τ) u ) e δj(τ ) db j (τ) b jk (s). Thus, he corresponding condiional probabiliy is s ) P k (s, e δj(s ) u e δj(s τ) db j (τ) b jk (s). Summing over all imes s and saes k, we obain he firs erms on he righ of (8.2), which hus is he par of he oal probabiliy ha perains o exi from sae j before ime n. Likewise i is realized ha he las erm on he righ of (8.2) is he remaining par of he probabiliy, peraining o he case of no ransiion ou of sae j before ime n. This heurisic argumen is made rigorous by applying Doob s opional sampling heorem o he maringale generaed by he indicaor of he even in (8.3) and he sopping ime defined as he he minimum of n and he ime of he firs ransiion afer from he curren sae j. C. A sysem of differenial equaions. Already (8.2) migh serve as a basis for compuaion of he saewise probabiliy funcions, bu is no convenien since he inegrand on he righ depends on. Inroduce he auxiliary funcions Q j defined by or Q j (, u) = P j (, e δj (u )) e δjτ db j (τ) (8.4)

115 CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES114 P j (, u) = Q j (, e δj u + ) e δjτ db j (τ). (8.5) Muliply by e ( µj in (8.2), inser e δj u ) e δjτ db j (τ) in he place of u, and rearrange a bi o obain e µj Q j (, u) = n e s µj k;k j µ jk (s) ds s Q k (s, e (δj δk)s u + e δkτ db k (τ) s ) e (δj δ k)s e δjτ db j (τ) e δks b jk (s) + e n µj 1 [ n ] e δjτ db j (τ) u. (8.6) Now he inegrand on he righ does no conain, and we are allowed o differeniae along on he righ hand side by simply subsiuing for s in minus he inegrand. Performing his and cancelling he common facor e µj, we arrive a he following main resul, where he side condiion (7.76) comes direcly ou of (8.6) by leing n: Theorem. The funcions Q j in (8.4) are he unique soluions o he differenial equaions d Q j (, u) = µ j ()d Q j (, u) Q k (, e (δj δ k) u + e (δj δ k) n, subjec o he condiions k;k j µ jk () d e δ kτ db k (τ) ) e δjτ db j (τ) e δk b jk (), (8.7) [ n ] Q j (n, u) = 1 e δjτ db j (τ) u. (8.8) Having deermined he auxiliary Q j, we obain he P j from (8.5).

116 CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES115 Remark: The differeniaion is in he Sielje s sense for funcions of bounded variaion and does no require differeniabiliy or any oher smoohness properies of he funcions involved. D. Compuaional scheme. A simple numerical procedure consiss in approximaing he funcions Q j by he funcions Q j obained from he finie difference version of (8.7): Q j ( h, u) = (1 µ j ()h)q j (, u) + h Q k (, e (δj δ k) u + e (δj δ k) k;k j µ jk () e δ kτ db k (τ) ) e δjτ db j (τ) e δk b jk (). (8.9) Saring from (8.8) (wih Q j in he place of Q j), one calculaes firs he funcions Q j (n h, ) by (8.9) and coninues recursively unil he Q j (, ) have been calculaed in he final sep. The Q j (, u) are defined for {, h, 2h,..., n} and u {a, a + h, a + 2h,..., b}, say, where he seplenghs h and h mus be sufficienly small and a and b mus be chosen such ha he suppors of he Q j (, ) are sufficienly well covered by [a, b]. Wha is sufficien mus be decided on in each individual case by judgemen and by rial and error. If wo saes j and k are inercommunicaing, hen Q j (, ) and Q k (, ) have he same suppor. The suppors are finie and usually easy o deermine in siuaions where he number of assurance paymens has a non-random upper bound. The u-argumens in he funcions Q k on he righ of (8.9) mus, of course, be rounded o he neares poin in {a, a + h, a + 2h,..., b}. E. Commens on he mehod. Before urning o applicaions, we pause in his paragraph o offer some moivaion and discussion of our approach. I is no needed in he sequel and may be skipped on he firs reading. The equaion (8.7) is jus he differenial form of he inegral equaion (8.6). I does no require ha he derivaives Q j(, u) exis, which hey do no in general for he obvious reason ha he saewise annuiy funcions on he righ of (8.7) may have jumps. If one should aemp o consruc differenial equaions along he lines of Norberg (1994), he saring poin would be he maringale M defined by [ n ] M() = P e τ δ db(τ) u F, where F = σ{x(τ); τ } is he informaion generaed by he sae process

117 CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES116 up o ime. Using he Markov propery of condiional independence beween pas and fuure, given he presen, we find M() = j I j () P j (, e δ (u )) e τ δ db(τ). Now, he recipe would be o apply he change of variable formula o he expression on he righ and hen o idenify he maringale componen ha is predicable (and of bounded variaion) and hence consan. Accomplishing his wihou caring abou jusificaion, would lead o he firs order parial differenial equaions P j(, u) + u P j(, u)(δ j u b j ()) + µ jk () (P k (, u b jk ()) P j (, u)) =, k;k j valid beween jumps of he conracual annuiy funcions B j, and P j (, u) = P j (, u B j ()), valid a jumps of he B j, and subjec o he condiion ha he P j (n, u) are and 1 according as u < or u. The approach requires ha he funcions P j possess firs order derivaives in boh direcions. As we have seen already in he inroducory example of Secion 2 hey generally do no. This difficuly migh possibly be circumvened by condiioning on he ulimae sae a ime n, bu proving differeniabiliy of he condiional probabiliies would require addiional assumpions and would no be sraighforward. These remarks serve o show ha he mehod developed in Secion 4 is no jus one among several candidae approaches o he problem; i is he only mahemaically sound soluion we are able o offer. One general conclusion we can exrac is ha here is no single general echnique for solving he bulk of problems of he kind considered here; he mehod will have o be designed for each individual problem a hand and will depend on he model assumpions and he funcional of ineres. 8.5 Applicaions A. The Poisson disribuion. In coninuance of he example in Paragraph 6B of Norberg (1994), consider he special case wih wo saes, J = {1, 2}, no ineres, δ 1 = δ 2 =, and he only paymens being an assurance of 1 payable upon each ransiion, b 12 = b 21 = 1.

118 CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES117 Then, aking n = 1, he presen value in (8.1) is jus he number of ransiion in he ime inerval (, 1], N 12 (1) + N 21 (1) N 12 () N 21 (). Furhermore, ake µ 12 = µ 21 = µ, a consan (> ). Then i is seen from he defining relaion (8.4) ha he funcions P j and Q j are all he same. Denoing his funcion by P, we can work wih (8.2), which becomes P (, u) = 1 e µ(s ) µp (s, u 1)ds + e µ(1 ) 1[ u]. (8.1) Using ha P (, u) = for u <, we readily obain from (8.1) ha P (, u) = e µ(1 ) for u < 1. Then, for 1 u < 2, i follows from (8.1) ha P (, u) = µ(1 )e µ(1 ) + e µ(1 ). Proceeding by inducion we obain, for each u, ha P (, u) = [u] i= (µ(1 )) i e µ(1 ), i! which is he Poisson disribuion wih parameer (1 )µ, of course. As a check on he accuracy of he numerical mehod, we lis he compued values of P (, u), u =, 1,..., 8, for = and µ = 1 ogeher wih he exac values of he Poisson probabiliies (in paranheses):.367 (.3679),.7358 (.7358),.922 (.9197),.9813 (.981),.9965 (.9864),.9994 (.9994),.9999 (.9999),.9999 (.9999), 1. (1.). These resuls were obained wih h = 1/2, h = 1/1, a =.5, and (runcaing he infinie suppor) b = 9.5. B. The erm insurance policy. To analyse he erm insurance policy in Paragraph 2A, ake J = {1, 2}, n = 3, µ 12 () = (3+), δ 1 = ln(1.45), b 1 =.4268, b 12 = 1, and all oher inensiies and paymens null. Again, as a check on he accuracy of he numerical mehod, we lis he compued values of P (, u) ogeher wih exac values (in paranheses): P (, u) = for u <.75 ( for u <.79), P (, u) =.8453 for u [.75,.1965) (.8452 for u [.79,.1961)), P (,.2) =.8491 (.849), P (,.4) =.9465 (.9467), P (,.6) =.9774 (.9776), P (,.8) =.9918 (.9919), P (, 1) = 1. (1.). These resuls are based on h = 1/1, h = 1/2, a =.1, and b = 1.1. C. A combined insurance policy. In our final numerical example we consider wha will be referred o as he combined policy, which is he same as he one in Paragraph B, bu wih a disabiliy pension added. More specifically, 1 is payable upon deah, an annuiy wih level inensiy.5 is payable during disabiliy, and premium is payable wih level inensiy in acive sae. The relevan model eniies are J = {1, 2, 3},

119 CHAPTER 8. PROBABILITY DISTRIBUTIONS OF PRESENT VALUES118 n = 3, b 13 = b 23 = 1, b 1 =.1318 (ne premium when he inensiies are as specified below), b 2 =.5, and, adoping he sandard Danish echnical basis excep for he recovery inensiy, δ = ln(1.45) (independen of sae), and µ 13 () = µ 23 () = x, µ 12 () = (3+), µ 21 () =.5, all oher paymens and inensiies being null. Figure 2 abou here Fig. 2: Probabiliy disribuion of he presen value of he combined insurance policy in (a) sae 1 and (b) sae 2. This example is a follow-up of Paragraphs 4B-C in Norberg (1994), where he firs hree momens of he presen value are calculaed for he combined policy. D. Numerical evaluaion of muliple inegrals. Numerical inegraion in higher dimensions is in general complicaed, and here exiss no echnique held o be universally superior. The echnique developed here can be used o evaluae inegrals ha, possibly afer a reinerpreaion, can be recognized as a probabiliy relaed o a presen value for a suiably specified policy. Jus o illusrae he idea, suppose T 1 and T 2 are independen posiive random variables wih cumulaive disribuion funcions F 1 and F 2 wih densiies f 1 and f 2, respecively, and ha we seek P[(T 2 1) (T 1 1) u]. I is realized ha his probabiliy is found as P (, u) for he policy wih J = {1, 2, 3, 4}, µ 12 () = µ 34 () = f 1 ()/(1 F 1 ()), µ 13 () = µ 24 () = f 2 ()/(1 F 2 ()), b 1 () = 1, b 2 () = 1, n = 1, and no ineres. Counless examples of his kind can be consruced.

120 Chaper 9 Reserves Prospecive and rerospecive reserves are defined as condiional expeced values, given some informaion available a he ime of consideraion. Each specificaion of he informaion invoked gives rise o a corresponding pair of reserves. Relaionships beween reserves are esablished in he general se-up. For he prospecive reserve he presen definiion conforms wih, and generalizes, he radiional one. For he rerospecive reserve i appears o be novel. Special aenion is given o he coninuous ime Markov chain model frequenly used in he conex of life and pension insurance. Thiele s differenial equaion for he prospecive reserve is shown o have a rerospecive counerpar. I is poined ou ha he prospecive and rerospecive differenial equaions have, respecively, he Kolmogorov backward and forward differenial equaions as special cases. Pracical uses of he differenial equaions are demonsraed by examples. 9.1 Inroducion A. Skech of he idea. The concep of prospecive reserve is no maer of dispue in life insurance mahemaics. I is defined as he condiional expeced presen value of fuure benefis less premiums on he policy, given is presen sae. A sraighforward generalizaion is obained by condiioning on some oher piece of informaion, e.g. on he policy s saying in some subse of he sae space. I is proposed here o define he rerospecive reserve analogously as he condiional expeced presen value of pas premiums less benefis. B. An example: insurance of a single life. A person aged x buys a life insurance policy specifying ha he sum assured, b, is payable immediaely upon deah before age x + n and ha premiums are o be conribued coninuously wih level inensiy c hroughou he insurance period. Le T x denoe he person s remaining life lengh afer he policy is issued a ime, say. Assume ha he survival funcion p x = P {T x > } is of he form p x = e µx+s ds, wih 119

121 CHAPTER 9. RESERVES 12 coninuous force of moraliy, µ. Finally, assume ha ineres is earned wih a consan, nonrandom inensiy δ so ha v = e δ is he annual discoun facor and i = e δ 1 is he annual rae of ineres (1 + i = v 1 is he annual ineres facor). A any ime [, n] he policy is eiher in sae = alive or in sae 1 = dead. The prospecive reserves in he wo saes, indicaed by subscrips and 1, are V + () = n v τ τ p x+ {bµ x+τ c} dτ (9.1) (by he usual heurisic argumen, he sum of expeced discouned benefis minus premiums in small ime inervals (τ, τ + dτ), < τ <, and, of course, V 1 + () =. (9.2) The saewise rerospecive reserves as defined above are (rivial) and V () = c (1 + i) τ dτ (9.3) V 1 () = 1 1 p x (1 + i) τ {c ( τ p x p x ) b τ p x µ x+τ } dτ (9.4) (use he same kind of argumen as in (9.1) noing ha, condiional on deah wihin ime, he probabiliy of survival o τ is ( τ p x p x )/(1 p x ) and he probabiliy of deah in (τ, τ + dτ) is τ p x µ x+τ dτ/(1 p x ), < τ < ). The sae a ime is X() = 1[T x ], he number of deahs of he person wihin ime. This is he informaion on which he reserves in (9.1) (9.4) are based. Now, suppose he complee prehisory of he policy is currenly recorded, so ha i is known a any ime if he person is alive or dead and, in he laer case, when he died. The informaion available a ime is he pair (X(), min(t x, )). Denoe he reserves correspondingly by a double subscrip. The reserves in sae remain as above, V, ± () = V ± (), and so does he prospecive reserve in sae 1, of course, V + 1,T x () = V 1 + () =. Only he rerospecive reserve in sae 1 is affeced by he addiional informaion on he exac ime of deah. I now becomes simply he value a ime of pas premiums less he benefi paymen, Tx V 1,T x () = c (1 + i) τ dτ b(1 + i) Tx. (9.5)

122 CHAPTER 9. RESERVES 121 The quaniy in (9.1) is wha radiionally is referred o as he prospecive reserve. The noion of rerospecive reserve launched here differs from he radiional one, which in he presen example is V () = 1 p x (1 + i) τ τp x (c bµ x+τ )dτ. (9.6) This quaniy emerges from he principle of equivalence, which requires ha benefis and premiums should balance in he mean a he ouse: n v τ τp x (bµ x+τ c)dτ =. (9.7) Spliing n ino + n in (9.7) and subsiuing from (9.1) and (9.6), yields V () = V + (). (9.8) Thus, he radiional concep of rerospecive reserve is raher a rerospecive formula for he prospecive reserve, valid for b and c saisfying he equivalence principle. For general b and c he quaniy V () is no an expeced value, and i has no probabilisic inerpreaion in he presen model involving one single policy. In an exended (arificial) model, wih m independen replicaes of he policy issued a ime, i may be inerpreed as he almos sure limi of he oal accumulaed surplus per survivor by ime as m ends o infiniy. As compared wih (9.8), he rerospecive reserve inroduced here is relaed o he prospecive reserve under he equivalence principle by he ideniy p x V () + (1 p x )V1 () = p x V + (). (9.9) C. Ouline of he paper. In Secion 2 he presen noions of reserves are defined for quie general sochasic paymen sreams and discouning rules, and cerain relaionships beween hem are esablished. No paricular reference o he insurance conex is made a his sage. In Secions 3 and 4 he framework of he furher discussions is presened: paymens of he life annuiy and life insurance ypes in a coninuous ime Markov chain model. A useful auxiliary resul is ha a Markov process behaves like a composiion of muually independen Markov processes in disjoin inervals when is values a he dividing poins beween he inervals are fixed. This ogeher wih sandard resuls for Markov chains is used in Secion 5 o invesigae he properies of reserves in he Markov chain case. The prospecive reserve, given he sae a he ime of consideraion, is he radiional one, which saisfies he well-known generalized Thiele s differenial equaions (see e.g. Hoem, 1969a), here also referred o as he prospecive differenial equaions. The saewise rerospecive reserves urn ou o saisfy a se of rerospecive differenial equaions, differen from he prospecive ones. I

123 CHAPTER 9. RESERVES 122 is poined ou he differenial equaions for he reserves have he Kolmogorov differenial equaions for he ransiion probabiliies as special cases. Surprisingly, maybe, i is he rerospecive equaions ha generalize he Kolmogorov forward equaions, while he prospecive equaions generalize he Kolmogorov backward equaions. In Secion 6 some more examples are supplied. 9.2 General definiions of reserves and saemen of some relaionships beween hem A. Paymen sreams and heir discouned values. Firs some basic definiions and resuls are quoed from Norberg (199). Consider a sream of paymens commencing a ime. I is defined by a finie-valued paymen funcion A, which for each ime specifies he oal amoun A() paid in [, ]. Negaive paymens are allowed for; i is only required ha A be of bounded variaion in finie inervals and, by convenion, righconinuous. This means ha A = B C, where B and C are non-negaive, nondecreasing, finie-valued, and righ-coninuous funcions represening he ougoes and incomes, respecively, of some business. In he conex of insurance B represens benefis and C represens conribued premiums on an insurance policy (or a porfolio of policies). The paymen funcion exends in a unique way o a paymen measure on he Borel ses, which is also denoed by A. Assuming ha paymens are valuaed by a piecewise monoone and coninuous discoun funcion v, he presen value of A a ime is V (, A) = 1 v(τ) da(τ) (9.1) v() [, ) (he sum of all paymens in small inervals discouned a ime, muliplied by he ineres facor 1/v() for he inerval from o ). Jus o obain ransparen formulas, i will be assumed hroughou ha v is of he form v() = e δ, (9.2) wih piecewise coninuous ineres inensiy δ. (The shorhand exemplified by δ = δ(τ) dτ will be in frequen use hroughou.) B. Definiions of rerospecive and prospecive reserves. The resricion of A o a (measurable) ime se T is he measure A T couning only hose A-paymens ha fall due in T ; A T {S} = A{S T }. A any ime he paymen sream splis ino paymens afer ime and paymens up o and including ime ; A = A (, ) + A [,]. The presen value in (9.1) splis correspondingly ino

124 CHAPTER 9. RESERVES 123 where V (, A) = V + (, A) V (, A), (9.3) V + (, A) = V (, A (, ) ) = 1 v() (, ) v(τ) da(τ) (9.4) is he discouned value of fuure ne ougoes (in he insurance conex benefis less premiums), and V (, A) = V (, A [,] ) = 1 v() [,] v(τ) d( A)(τ) (9.5) is he value, wih accumulaion of ineres, of pas ne incomes (in he insurance conex premiums less benefis). The quaniy V (, A) is observable by ime and can suiable be called he individual rerospecive reserve of he policy a ime. If he fuure developmen of (v, A) were known, hen V + (, A) would be he appropriae amoun o se aside o cover fuure excess of benefis over premiums on he individual policy. However, if he fuure course of (v, A) is uncerain, i is no possible o provide V + (, A) as a prospecive reserve on an individual basis. Assume now ha A and, possibly, also v are sochasic processes on some probabiliy space (Ω, F, P ). An operaional definiion of he prospecive reserve mus depend solely on informaion ha is a hand a he momen when he reserve is o be provided. Le F = {F } be a family of sub-sigmaalgebras of F, F represening some piece of informaion available a ime. The family F may be increasing, ha is, F s F, s <, bu his is no required in general. Reserves are defined as condiional expeced values, given he informaion provided by F. A ime he prospecive F-reserve is and he rerospecive F-reserve is V + F (, A) = E F V + (, A), (9.6) V F (, A) = E F V (, A), (9.7) where he subscrip on he expecaion sign signifies condiioning. The prospecive F-reserve mees he operaionaliy requiremen formulaed above as i is deermined by he curren informaion. Even hough he rerospecive individual reserve in (9.5) is observable by ime, i may be judged relevan o calculae rerospecive reserves wih respec o some more summary informaion F. For a given realizaion of {v(τ)} τ, F may be hough of as a classificaion of

125 CHAPTER 9. RESERVES 124 he policies, whereby all policies wih he same characerisics as specified by F are grouped ogeher. Forming he mean, condiional on F, means averaging over all policies in he same group, roughly speaking. The reserves are condiional means of he presen values V ± (, A). Oher feaures of he condiional disribuions of hese random variables may be of ineres. In paricular, as measures of variabiliy, inroduce he variances V ±(2) F (, A) = V ar F V ± (, A). (9.8) C. Relaionships beween reserves. When only one paymen sream is considered, noaion can be saved by dropping he symbol A from V (, A). Thus, abbreviaions like V () and V ± F () will be frequenly used in he sequel. By (9.1) (9.3), he value of A a ime is relaed o is value a any ime by Taking expecaion gives V () = v()v () = v(){v + () V ()}. E V () = E {v()v ()} (9.9) = E {v()(v + () V ())}. (9.1) The equivalence principle of insurance saes ha premiums and benefis should balance on he average as seen a he ouse, ha is, E V () =. (9.11) I does no imply EV () = for > unless v is a deerminisic funcion, confer (9.9). Taking ieraed expecaions in (9.1), he equivalence requiremen can be cas as if v() is deermined by F, and E {v()v F ()} = E {v()v + F ()} (9.12) E V F () = E V + F () (9.13) if v is deerminisic. Relaion (9.9) is a special case of (9.13). Le F = {F } be some sub-sigmaalgebra represening more summary informaion han F = {F } in he sense ha F F,. The rule of ieraed expexaions yields he following relaionship beween reserves on differen levels of informaion:

126 CHAPTER 9. RESERVES 125 V ± F () = E F V ± F (). (9.14) By he general rule V arx = E V ar Y X + V ar E Y X, variances denoed as in (9.8) are relaed by V ±(2) F () = E ±(2) F {V F () + (V ± F ())2 } (V ± F ())2. (9.15) There exis also useful relaionships beween reserves a differen imes for a fixed source of informaion, F. The discouned values of fuure ne ougoes a wo differen poins of ime, < u, are relaed by Similarly, for s <, v()v + () = (,u] v(τ) da(τ) + v(u)v + (u). v()v () = v(s)v (s) + v(τ) d( A)(τ). (s,] Taking expecaions in hese wo ideniies, yields a pair of basic relaionships under he assumpion ha v is deerminisic. If {A(τ)} τ u depends sochasically only on F u for given F and F u, hen v()v + F () = (,u] v(τ) d E F A(τ) + v(u)e F V + F (u). (9.16) Likewise, if {A(τ)} τ s depends sochasically only on F s for given F s and F, hen v()v F () = v(s)e F V F (s) + (s,] v(τ) d( E F A)(τ). (9.17) D. Righ-coninuiy of he reserve processes. As defined by (9.6) and (9.7) he reserves are righ-coninuous sochasic processes. They could alernaively be made lef-coninuous by leing he inegrals in (9.4) and (9.5) exend over [, ) and [, ), respecively. This would be in keeping wih radiion, bu he righconinuous versions are chosen here since hey fi ino he general apparaus of sochasic inegrals and differenial equaions and hus are he more convenien quaniies o deal wih in anicipaed applicaions of he heory o complex models. Anyway, he righ-coninuous and he lef-coninuous versions differ only a poins of ime where non-null amouns fall due wih posiive probabiliy.

127 CHAPTER 9. RESERVES Descripion of paymen sreams appearing in life and pension insurance A. Specificaion of he insurance reay erms. Having insurances of persons in mind, consider insurance reaies specifying erms of he following general form. There is a se J = {,..., J} of possible saes of he policy. The policy is issued a ime, say. A any ime i is in one and only one of he saes in J, commencing in sae. Paymens are of wo kinds: general life annuiies by which he amoun A g() A g(s) is paid during a sojourn in sae g hroughou he ime inerval (s, ], and general life insurances by which an amoun a gh () is paid immediaely upon a ransiion from sae g o sae h a ime. They comprise benefis, adminisraion expenses of all kinds, and premiums (negaive). The conracual funcions A g and a gh are, respecively, a paymen funcion and a finie-valued, righ-coninuous funcion. B. The form of he paymen funcion. Le X() be he sae of he policy a ime. The developmen of he policy is given by {X()}. This process, regarded as a funcion from [, ) o J, is assumed o be righ-coninuous, wih a finie number of jumps in any finie ime inerval. Le N gh be he process couning he ransiions from sae g o sae h, ha is, N gh () = #{τ [, ]; X(τ ) = g, X(τ) = h}. The sream of ne paymens is of he form A = g {A g + h;h g A gh }, wih da g () = 1[X() = g] da g(), (9.1) da gh () = a gh () dn gh(). (9.2) The behaviour of he rerospecive and prospecive reserves is now o be sudied for some specificaions of F ha are of relevance in insurance. The model framework will be he radiional Markov chain, which yields lucid resuls. 9.4 The Markov chain model A. Model assumpions and basic relaionships. The process {X()} is assumed o be a coninuous ime Markov chain on he sae space J. The ransiion probabiliies are denoed The ransiion inensiies p jk (, u) = P {X(u) = k X() = j}.

128 CHAPTER 9. RESERVES 127 µ jk () = lim u p jk (, u) u are assumed o exis for all and j k. To simplify maers, he funcions µ jk are furhermore assumed o be piecewise coninuous. The oal ransiion inensiy from sae j is µ j = k; k j µ jk. From he Chapman-Kolmogorov equaions, p jk (, u) = g J p jg (, τ) p gk (τ, u), valid for τ u, one obains Kolmogorov s differenial equaions, he forward, p ij(s, ) = g; g j p ig (s, ) µ gj () p ij (s, )µ j (), (9.1) and he backward, p jk(, u) = µ j ()p jk (, u) g; g j µ jg ()p gk (, u). (9.2) Togeher wih he iniial condiions p jk (, ) = δ jk, j, k J, (he Kronecker dela) hey deermine he ransiion probabiliies uniquely. The Kolmogorov equaions are he major ools for consrucing he ransiion probabiliies from he inensiies, which are he basic eniies in he sysem; hey are funcions of one argumen only and, being readily inerpreable, hey form he naural saring poin for specificaion of he model. The condiional probabiliy of saying uninerrupedly in sae j hroughou he ime inerval [, u], given ha X() = j, is (solve a simple differenial equaion) p jj (, u) = e u µj. (9.3) B. Momens of presen values. Throughou he balance of he paper i will be assumed ha he ineres inensiy is nonsochasic. Consider he general annuiies and insurances defined by (9.1) and (9.2). The presen values a ime of heir conribuions in some ime inerval T are

129 CHAPTER 9. RESERVES 128 V (, A gt ) = V (, A ght ) = 1 v() 1 v() T T v(τ)1[x(τ) = g]da g(τ), v(τ)a gh(τ) dn gh (τ). Here follows a lis of formulas for he firs and second order momens of such presen values, condiional on X(s), wih T [s, ). The mean values are E X(s)=i V (, A gt ) = E X(s)=i V (, A ght ) = 1 v() 1 v() T T v(τ)p ig (s, τ) da g(τ), (9.4) v(τ)a gh(τ)p ig (s, τ)µ gh (τ) dτ. (9.5) The noncenral second order momens, for inervals S, T [s, ), are E X(s)=i {V (, A es )V (, A gt )} = 1 v 2 v(ϑ)v(τ){1[ϑ τ]p ie (s, ϑ)p eg (ϑ, τ) () S T +1[ϑ > τ]p ig (s, τ)p ge (τ, ϑ)} da e (ϑ) da g (τ), (9.6)

130 CHAPTER 9. RESERVES 129 E X(s)=i {V (, A efs )V (, A ght )} = 1 v 2 () [ v(ϑ)v(τ){1[ϑ < τ]p ie (s, ϑ)p fg (ϑ, τ) S T +1[ϑ > τ]p ig (s, τ)p he (τ, ϑ)}µ ef (ϑ)µ gh (τ)a ef (ϑ)a gh(τ) dϑ dτ +δ ef,gh v 2 (τ)p ig (s, τ)µ gh (τ)a gh 2 (τ) dτ ], (9.7) S T E X(s)=i {V (, A es )V (, A ght )} = 1 v 2 v(ϑ)v(τ){1[ϑ τ]p ie (s, ϑ)p eg (ϑ, τ) () S T +1[ϑ > τ]p ig (s, τ)p he (τ, ϑ)}da e(ϑ) µ gh (τ)a gh(τ) dτ. (9.8) Variances and covariances are now formed by he rule Cov(X, Y ) = E(XY ) EXEY. The formulas are easily esablished by a heurisic ype of argumen. To moivae e.g. (9.8), pu E X(s)=i { S = v(ϑ)1[x(ϑ) = e] da e (ϑ) S T T v(τ)a gh (τ) dn gh(τ) } v(ϑ)v(τ)e X(s)=i {1[X(ϑ) = e] dn gh (τ)}a gh (τ) da e (ϑ), and calculae he expeced value in he inegrand for ϑ τ and for ϑ > τ. Rigorous proofs will no be given here. Some of he resuls in (9.4) (9.8) are proved for absoluely coninuous A g by Hoem (1969a) and Hoem & Aalen (1978). C. Condiional Markov chains. Viewing reserves as condiional expeced values, one may be concerned wih various sub-sigmaalgebras of he basic sigmaalgebra F. They will ypically be of he form F T = σ{x(τ); τ T }, he sigmaalgebra represening all informaion provided by he process X in he ime se T. Of paricular ineres are F [,] and F {,..., q}. The Markov propery means precisely ha for B F (, ), P {B F [,] } = P {B F {} }, (9.9) ha is, he fuure developmen of he process depends on is pas and presen saes only hrough he presen. An equivalen formulaion is ha, for A F (,) and B F (, ),

131 CHAPTER 9. RESERVES 13 P {A B F {} } = P {A F {} }P {B F {} }, (9.1) which says ha for a fixed presen sae of he process is fuure and pas are condiionally independen. The equivalence of (9.9) and (9.1) is easily esablished in he presen siuaion wih finie sae space. By inducion (9.1) exends o he following: for = < 1 <... < q < q+1 = and A p F (p 1, p), p = 1,..., q + 1, q+1 P { q+1 p=1 A p F {,..., q}} = p=1 P {A p F {p 1, p}}, where F {q, } = F {q}. Thus, condiional on X(), X( 1 ),..., X( q ), he developmens of he process in he inervals ( p 1, p ) are muually independen. So, o sudy he condiional process, given is values a fixed poins, ineres can be focused on he behaviour of he segmen {X(τ)} s<τ< for fixed X(s) and X(), say. For s < τ < ϑ < and A F (s,τ), P {X(ϑ) = h A, X(τ) = g, X(s) = i, X() = j} P {X(s) = i, A, X(τ) = g, X(ϑ) = h, X() = j} = P {X(s) = i, A, X(τ) = g, X() = j} = P {X(s) = i, A, X(τ) = g} p gh(τ, ϑ)p hj (ϑ, ). P {X(s) = i, A, X(τ) = g}p gj (τ, ) Thus, he condiional process is Markov wih ransiion probabiliies p gh ij (τ, ϑ s, ) = p gh(τ, ϑ)p hj (ϑ, ) p gj (τ, ) (9.11) and inensiies µ gh ij (τ s, ) = µ gh (τ) p hj(τ, ) p gj (τ, ), (9.12) boh independen of s and i. As τ, a poin of coninuiy of all he inensiies, he expression in (9.12) ends o if g = j, o µ gh ()µ hj ()/µ gj () if g, h j, and o + if h = j, which reflecs ha he condiional process is forced o end up in sae j a ime. The condiional process {X(τ)} τ, given X() = j, was sudied by Hoem (1969b) in connecion wih saisical analysis of seleced samples in demography.

132 CHAPTER 9. RESERVES Reserves in he Markov chain model A. Individual reserves. There are as many definiions of he reserves in (9.6) and (9.7) as here are possible specificaions of F. Each F represens a way of classifying he policies, and he F-reserves a ime are classwise averages of discouned excess of benefis over premiums in he fuure or accumulaed excess of premiums over benefis in he pas. The simples conceps of reserves are he individual reserves obained by condiioning on he full hisory of he individual policy, ha is, ake F = F [,]. The individual rerospecive reserve is simply he cash value a ime of he ne gains so far, defined by (9.5) and exemplified by (9.5): V F () = V (). (9.1) By he Markov propery, he individual prospecive reserve is V + F () = E X()V + (). Thus, in he Markov case i suffices o sudy he prospecive reserve for given curren sae. B. Reserves in single saes. Now, le F = {F {} } so ha reserves a ime are formed by averaging over policies in each single sae a ha ime. In shor, wrie V j ± () for he reserves V ± F () when X() = j. The prospecive reserves are he same as in Paragraph A. The explici formula is gahered from (9.4) (9.5): V j + () = 1 v(τ) p jg (, τ){da g v() (τ) (, ) g + a gh(τ)µ gh (τ) dτ}, (9.2) h; h g defined on [, ) for j = and on (, ) for j. The rerospecive reserve in sae j is obained by combining he resuls in Paragraphs 4B C. Condiional on X() = j for > (and X() = ), {X(τ)} τ is a Markov chain saring in sae, wih ransiion probabiliies and inensiies given by (9.11) and (9.12). Noing ha p g j (, τ, )µ gh j (τ, ) = p g(, τ)µ gh (τ)p hj (τ, ), p j (, ) he general formulas (9.4) and (9.5) give

133 CHAPTER 9. RESERVES 132 V j () = 1 v(τ) p g (, τ){da v()p j (, ) g(τ)p gj (τ, ) [,] g + a gh (τ)µ gh(τ)p hj (τ, ) dτ}. (9.3) h; h g This expression is valid if p j (, ) >. In paricular, V () = A (), of course, whereas V j () is no defined for j. Formula (9.3) is easy o inerpre by direc heurisic reasoning: condiionally, given X() = and X() = j, he probabiliy of saying in sae g a ime τ is p g (, τ)p gj (τ, )/p j (, ), and he probabiliy of a ransfer from sae g o sae h in he ime inerval (τ, τ + dτ) is p g (, τ)µ gh (τ)dτp hj (τ, )/p j (, ). The condiional variances, V ±(2) j () = V ar X()=j V ± (), are composed from (9.6) (9.8) upon insering he relevan condiional ransiion probabiliies and inensiies. Finally, noe ha he fuure and he pas developmens of he process are independen, hence uncorrelaed, for fixed presen X() = j. C. More general reserves. Referring o Paragraph 2C, le F be as in Paragraph B and le F = {F } be he more summary informaion wih F generaed by he evens X() J l, l = 1,..., m, where {J 1,..., J m } is some pariioning of J. Thus, F -reserves a ime are now condiional on X() J l, l {1,..., m}. For any K J and u pu p jk (, u) = P {X(u) K X() = j} = k K p jk (, u). Applying (9.14) and (9.15), one obains (wih obvious noaion) and V ± J l () = V ±(2) 1 J l () = p Jl (, ) 1 p Jl (, ) p j (, )V ± j () j J l j J l p j (, ){V ±(2) j () + V ± j ()2 } V ± J l () 2, hence F -reserves and F -variances can be composed from he corresponding single-sae quaniies.

134 CHAPTER 9. RESERVES 133 A variey of reserves based on differen choices of F can be analysed by he resuls in Secion 4. For insance, aking F = F {1, 2,...} [,] means ha reserves are formed by averaging over policies ha are known o have visied cerain saes a given epochs in he pas. The reserves and variances are found for each of he inervals and, by condiional independence, simply added o give he required oal reserve and variance. D. Differenial equaions for prospecive reserves in single saes. In he presen Markov chain se-up (9.16) specializes o W + j () = (,u] v(τ) g + g p jg (, τ){da g (τ) + h;h g a gh (τ)µ gh(τ)dτ} p jg (, u)w g + (u), (9.4) where, for each j J, W + j + () = v()v j (). (9.5) Consider he case where all paymens are resriced o a finie inerval [, n]. Assume firs ha he funcions A g are absoluely coninuous beween and n, ha is, da g () = a g () d, (, n). Then all W + g are coninuous in (,n). Assume furhermore ha all he funcions a g, a gh, µ gh, and δ are coninuous in (, n). Then, for u = + d (9.4) becomes W + j () = v(){a j () + g; g j + (1 µ j () d)w + j µ jg ()a jg()} d ( + d) + g; g j µ jg ()dw + g ( + d) + o(d), where o(d)/d as d. Now, subrac W + j ( + d) and divide by d on boh sides, and le d o obain he prospecive differenial equaions, which are he mulisae generalizaion of Thiele s classical equaion: W + j () = v(){a j () + For fixed conracual funcions he V j + (9.6) ogeher wih he condiions g; g j + µ j ()W + j () g; g j µ jg ()a jg ()} µ jg ()W + g (). (9.6) are uniquely deermined by he equaions

135 CHAPTER 9. RESERVES 134 V + j (n ) = A j (n) A j (n ). (9.7) If he principle of equivalence (9.11) is invoked, hen he addiional condiion V + () = A () (9.8) mus be imposed a consrain on he conracual funcions. The procedure is o be modified only slighly if A j has jumps a poins j1,..., j,qj 1 in (, n). Pu j = and jqj = n. The soluion of (9.6) has o be obained in each of he inervals ( j,p 1, jp ), p = 1,..., q j, and he piecewise soluions in adjacen inervals mus be hooked ogeher by he condiions V + j ( jp ) = V + j ( jp) + A j ( jp) A j ( jp ), (9.9) p = 1,..., q j, which comprise (9.7). A commen is needed also on possible disconinuiies of he funcions a jg, µ jg, and δ. A such poins and a he poins gp (if any) where A g jumps and W g + is disconinuous for some g j, he derivaive W + j does no exis. The lef and righ derivaives exis, however, and if he disconinuiies of he kind menioned are finie in number, hey will cause no echnical problem as hey will no affec he inegraions ha have o be performed o obain he soluion of he differenial equaions. Pursuing he previous remark, swich he erm µ j ()W + j () appearing on he righ of (9.6) over o he lef, muliply by he inegraing facor e µj o form he complee differenial {e µj W + j ()} on he lef, and finally inegrae over an inerval (, u) conaining no jumps of A j o obain an inegral equaion. Inser (9.5) and solve (recall (9.2) and (9.3)) V j + () = 1 u v() [ v(τ)p jj (, τ){a j (τ) + g; g j µ jg (τ)(a jg (τ) + V + g (τ))}dτ + v(u)p jj (, u)v + j (u ) ]. (9.1) This expression is easy o inerpre: i decomposes he fuure paymens ino hose ha fall due before and hose ha fall due afer ime u or he ime of he firs ransiion ou of he curren sae, whichever occurs firs. By fixed conracual funcions he inegral equaions in conjuncion wih he condiions (9.9) deermine he V + j. E. Differenial equaions for rerospecive reserves in single saes. The saring poin is (9.17), which now specializes o

136 CHAPTER 9. RESERVES 135 v()v j () = v(s) g v(τ) (s,] g + h; h g Upon muliplying by p j (, ) and inroducing p g (, s)p gj (s, ) Vg (s) p j (, ) p g (, τ) p j (, ) {da g (τ)p gj(τ, ) a gh(τ)µ gh (τ)p hj (τ, ) dτ}. (9.11) j J, (9.11) can be reshaped as W j () = v()p j(, )V j (), (9.12) W j () = Wg (s)p gj (s, ) g v(τ) p g (, τ){da g(τ)p gj (τ, ) (s,] g + a gh (τ)µ gh(τ)p hj (τ, )dτ}. (9.13) h; h g Again, consider he case where all paymens are resriced o a finie inerval [, n], wih da g() = a g() d, (, n), and a g, a gh, µ gh, and δ all coninuous in (, n). Wih and + d in he places of s and, (9.13) becomes W j ( + d) = Wg ()µ gj()d + W j ()(1 µ j()d) g;g j v() p g (, ){a g ()d p gj(, + d) g + h; h g a gh()µ gh ()p hj (, + d)d} + o(d). As p ij (, + d) is µ ij ()d + o(d) for i j and 1 µ j ()d + o(d) for i = j, all erms p ij (, +d) on he righ can be replaced by δ ij (he difference is absorbed in o(d)). Then proceed in he same manner as in he previous paragraph o obain he rerospecive differenial equaions W j () = g;g j W g ()µ gj () W j ()µ j() v(){p j (, )a j () + p g (, )a gj()µ gj ()}. (9.14) g; g j

137 CHAPTER 9. RESERVES 136 For fixed conracual funcions he rerospecive reserves are deermined uniquely by (9.14) ogeher wih he condiions Wj () = δ ja (). (9.15) The equivalence condiion (9.11) can be cas in erms of he rerospecive reserves as j J p j (, n)v j (n) =. (9.16) Possible disconinuiies are accouned for in he same manner as for he prospecive reserves, he rerospecive counerpar of (9.9) being V j ( jp) = V j ( jp ) (A j ( jp) A j ( jp )), (9.17) p = 1,..., q j. Copying essenially he seps leading o (9.1), one obains from (9.14) he inegral equaion V j ( ) = 1 v()p j (, ) [ v(s)p j(, s)p jj (s, )V j (s) v(τ){p j (, τ)p jj (τ, )a j (τ) s + p g (, τ)µ gj (τ)p jj (τ, )(a gj (τ) V g + (τ))}dτ ], g; g j (9.18) valid in inervals (s, ) conaining no jumps of A j. I decomposes he pas paymens ino hose ha fall due before and hose ha fall due afer ime s or he ime of he las ransiion ino he curren sae, whichever is he laer. F. The prospecive and rerospecive differenial equaions are generalizaions of Kolmogorov s backward and forward differenial equaions, respecively. This resul is esablished by considering a simple endowmen in he special case where v() 1 (no ineres). Firs, assume ha he only paymen provided is a uni benefi payable a ime u coningen upon X(u) = k, ha is, A k = ε u, he measure wih a uni mass a u, and all oher A g and all a gh are null. Then, for < u, W + j () in (9.5) reduces o p jk(, u), and he prospecive differenial equaions in (9.6) specialize o he Kolmogorov backward equaions in (9.2). Second, he Kolmogorov forward differenial equaions in (9.1) are obained as a specializaion of he rerospecive differenial equaions in (9.14) by leing he

138 CHAPTER 9. RESERVES 137 only paymen be a uni premium a ime s, coningen upon X(s) = i, whereby W j () in (9.12) reduces o p i(, s)p ij (s, ) for > s. Likewise, inegral equaions for he ransiion probabiliies come ou as special cases. From (9.1) one ges p jk (, u) = u p jj (, τ) g; g j µ jg (τ)p gk (τ, u)dτ + δ jk p jj (, u), (9.19) and from (9.18) p ij (s, ) = δ ij p ii (s, ) + g; g j s p ig (s, τ)µ gj (τ)p jj (τ, )dτ. (9.2) These equaions are easy o inerpre. In a similar manner one may also find differenial and inegral equaions for expeced sojourn imes. For v() 1, da k () = d, and all oher conracual funcions null, V j + () and Vj () are jus he expeced fuure and pas oal sojourn imes in sae k, condiional on X() = j. G. Uses of he differenial equaions. In he case where he conracual funcions do no depend on he reserves, he defining relaions (9.2) and (9.3) are explici expressions for he reserves. The differenial equaions (9.6) and (9.14) are no needed for consrucive purposes hey serve only o give insigh ino he dynamics of he policy. The siuaion is enirely differen if he conracual funcions are allowed o depend on he reserves in some way or oher. The mos ypical examples are repaymen of a par of he reserve upon wihdrawal (a sae wihdrawn mus hen be included in he sae space J ) and expenses depending parly on he reserve. Also he primary insurance benefis may in some cases be specified as funcions of he reserve. In such siuaions he differenial equaions are indispensable ools in he consrucion of he reserves and deerminaon of he equivalence premium. The simples case is when he paymens are linear funcions of he reserve, wih coefficiens possibly depending on ime. Then he operaions leading o (9.1) and (9.18) can basically be reproduced afer collecing all erms involving he reserve in sae j on he lef hand side and muliplying by he appropriae inegraing facor. Such echniques are sandard and frequenly used for he radiional prospecive reserve. They carry over o he rerospecive reserve, as will be illusraed by examples in he final secion. Apar from some very simple siuaions, like he one encounered in Paragraph 1B, he compuaion of he reserves will usually require numerical soluion of he se of differenial equaions (or he equivalen inegral equaions). There is, however, an imporan class of siuaions which allow for more direc compuaion by ieraed numerical inegraions, and a commen shall be rendered on hose. When reurns o formerly visied saes are impossible, µ jg = for

139 CHAPTER 9. RESERVES 138 g < j, say, he equaions (9.1) form he basis of a recursive compuaional procedure. The sum over g on he righ of (9.1) exends only over g > j (void if j = J), and so one can suiably sar by deermining V + J, which is easy since he equaion for V + + J involves no oher V g (ypically he policy is no longer in force in sae J and V + J is idenically ). Then proceed downwards hrough he sae space: having deermined V + j+1,..., V + J, use (9.1) wih u = jp and (9.9) in each inerval ( j,p 1, jp ), saring from ime jqj = n. Similarly, he equaions (9.18) are solved recursively saring wih he equaion for V, which involves no oher Vg + : having deermined V,..., V j 1 +, use (9.18) wih s = j,p 1 and (9.17) in each inerval ( j,p 1, jp ), saring from ime j =. Again, he examples in he following secion are referred o. H. Behaviour of he rerospecive reserves in he viciniy of. In esablishing he rerospecive differenial equaions, he auxiliary funcions W j defined in (9.12) are more convenien o work wih han he reserves hemselves. The V j are defined only for hose where p j (, ) >, whereas he W j are well defined in all of [, ). The W j are righ-coninuous, and heir values in, given by (9.15), are simple iniial condiions for he differenial equaions. The reserves Vj are also righ-coninuous, bu in only V is defined ( A ()). For j he limi V j (+) mus be obained by leing in (9.3). Leaving deails aside, only he inuiively appealing resul is repored. A sae j is said o be immediaely accessible from sae in q seps via he direced pah g = (g 1 g 2... g q 1 ) if µ g () = µ g1 ()µ g1g 2 ()... µ gq 1j() >. Le q j be he minimum of all q for which his propery holds for some pah, and denoe he se of such minimal pahs by P j. Then Vj (+) = A () g P j µ g (){a g 1 () + a g 1g 2 () a g qj 1j ()}. g P j µ g () I. A differen noion of rerospecive reserve. In a recen paper Wolhuis & Hoem (199) have launched a noion of rerospecive reserve quie differen from he one inroduced here. Working in he Markov chain model, hey require ha he saewise rerospecive reserves should saisfy E X()=i V () = v() j J p ij (, ){V + j () V ()} (9.21) for i =, which conforms wih (9.1). Then, imagining ha he policy migh sar from any sae differen from, hey require ha (9.21) be valid for all i, whereby some hypoheical values mus be chosen for he E X()=i V (), i. There exiss no such choice ha can produce he rerospecive reserves (9.3), and so he approach is incompaible wih he one aken here. The same is he case for he approach proposed by Hoem (1988), where (9.21) is required for i = and V + j () is pu equal o V j () for j. j

140 CHAPTER 9. RESERVES Some examples A. Life insurance of a single life (coninued from Paragraph 1B). In his simple siuaion he prospecive differenial equaions (9.6) are W + () = v (c µ x+ b) + µ x+ W + () µ x+w 1 + (), (9.1) W 1 + () =. (9.2) Wih he condiions W + (n ) = W + 1 (n ) = (9.3) hey lead o (9.1) and (9.2), which could be pu up by direc prospecive reasoning. The equivalence premium is c = b n vτ τp x µ x+τ dτ n. vτ τp x dτ Suppose he reay is modified so ha he prospecive reserve is paid ou as an addiional benefi upon deah before ime n; a 1() = b + V + (). Then he reserve and he equivalence premium canno be deermined direcly by prospecive reasoning, and i is necessary o employ he differenial equaions (9.6). Now, insead of (9.1) one ges W + () = v {c µ x+ (b + V + ())} + µ x+w + () µ x+w 1 + () = v {c µ x+ b} µ x+ W 1 + (). (9.4) Equaion (F.38) and he condiions (F.39) remain unchanged. One finds V + 1 () = as before, and n () = v τ (µ x+τ b c) dτ. (9.5) V + The equivalence premium is deermined upon insering = in (9.5) and equaing o : c = b n vτ µ x+τ dτ n. (9.6) vτ dτ These echniques and resuls are classical, and are referred here for he sake of comparison wih wha now follows.

141 CHAPTER 9. RESERVES 14 For he sandard conrac, wih no repaymen of he reserve, he differenial equaions (9.14) become W () = W ()µ x+ + v p x c, (9.7) W 1 () = W ()µ x+ v p x µ x+ b. (9.8) Wih he condiions W () = W 1 () = (9.9) hey lead o (9.3) and (9.4), which could be pu up direcly. Suppose now ha he rerospecive reserve is o be paid ou as an addiional benefi upon deah; a 1 () = b + V (). Then he rerospecive differenial equaions mus be employed. Insead of (9.8) one ges W 1 () = W ()µ x+ v p x µ x+ (b + V = v p x µ x+ b, whereas equaion (9.7) and he condiions (9.9) remain unchanged. One arrives a he same expression for V as before, of course, and V 1 () = b 1 p x The equivalence premium c is deermined by (9.16): n c = b vτ τp x µ x+τ dτ n. np x vτ dτ (1 + i) τ τp x µ x+τ dτ. B. Widow s pension. A married couple buys a widow s pension policy specifying ha premiums are o be paid wih inensiy c as long as boh husband and wife are alive, and pensions are o be paid wih inensiy b as long as he wife is widowed. The policy erminaes a ime n or upon he deah of he wife, whichever occurs firs. The relevan Markov model is skeched in Fig. reffig:wolives. Expressions for he ransiion probabiliies are easily obained by direc reasoning or by use of (9.19). The reserves can be picked direcly from (9.2) and (9.3): n V + () = 1 v(τ){b p 1 (, τ) c p (, τ)}dτ, v() V 1 + () = b n v(τ)p 11 (, τ)dτ, v() V 2 + () = V + () =, 3

142 CHAPTER 9. RESERVES 141 V () = c v() v(τ) dτ, (9.1) V1 () = 1 v(τ) {c p (, τ)p 1 (τ, ) v() p 1 (, ) b p 1 (, τ)p 11 (τ, )}dτ, (9.11) V 2 () = c v() p 2 (, ) V 3 () = 1 v() p 3 (, ) v(τ) p (, τ)p 2 (τ, ) dτ, (9.12) v(τ) {c p (, τ)p 3 (τ, ) b p 1 (, τ)p 13 (τ, )}dτ. (9.13) The differenial equaions are no needed o consruc hese formulas. The rerospecive ones are lised for ease of reference: W () = W ()(µ 1() + µ 2 ()) + v()p (, ) c, (9.14) W 1 () = W ()µ 1() W1 ()µ 13() v()p 1 (, ) b, (9.15) W 2 () = W ()µ 2() W2 ()µ 23(), (9.16) W 3 () = W 1 ()µ 13() + W2 ()µ 23(). (9.17) Consider a modified policy, by which he rerospecive reserve is o be paid back o he husband in case he is widowered before ime n, he philosophy being ha couples receiving no pensions should have heir savings back. Now he rerospecive differenial equaions are needed. The equaions above remain unchanged excep ha he erm v()p (, )V ()µ 2() = W ()µ 2() mus be subraced on he righ of (9.16), which hen changes o W 2 () = W 2 ()µ 23(). (9.18) Togeher wih he condiions Wj easily solved. Obviously, he expressions for V as in (9.1) and (9.11). From (9.18) follows V2 Finally, (9.17) gives () =, j =, 1, 2, 3, hese equaions are () and V1 () remain he same () =, which is also obvious. V3 () = 1 v(τ) {c p (, τ)p 13 (τ,, ) v() p 3 (, ) b p 1 (, τ)p 13 (τ, )} dτ, (9.19)

143 CHAPTER 9. RESERVES 142 where p 13 (τ,, ) = τ p 1 (τ, ϑ)µ 13 (ϑ)dϑ is he probabiliy of passing from sae o sae 3 via sae 1 in he ime inerval [τ, ], given ha X(τ) =. As a final example he widow s pension shall be analysed in he presence of adminisraion expenses ha depend parly on he reserve. Consider again he policy erms described in he inroducion of his paragraph, bu assume ha adminisraion expenses incur as follows. A ime expenses fall due wih inensiy e () + e ()c in sae (he laer erm represens encashmen commision) and wih inensiy e 1 () in sae 1. In addiion, expenses relaed o mainenance of he reserve fall due wih inensiy e() (curren rerospecive reserve) hroughou he enire period [, n]. Insead of (9.14) (9.17) one now ges

144 CHAPTER 9. RESERVES 143 W () = W ()(µ 1() + µ 2 ()) v()p (, ){e () + e ()c + e()v () c}, (9.2) W 1 () = W ()µ 1() W1 ()µ 13() v()p 1 (, ){b + e 1() + e()v1 ()}, (9.21) W 2 () = W ()µ 2() W2 ()µ 23() v()p 2 (, )e()v2 (), (9.22) W 3 () = W 1 ()µ 13() + W2 ()µ 23() v()p 3 (, )e()v3 (), (9.23) and he condiions (9.15) become W j () =, j =, 1, 2, 3. Now a small rick. In each of he equaions (9.2) (9.23), say he one for W j (), here appears a erm v()p j (, )e()v j () = e()w j () on he righ hand side. Swich his over o he lef and muliply on boh sides by e e. Form a complee differenial {e e W j ()} on he lef hand side and absorb everywhere he facor e e ino v() = e δ (remember v() is a facor in W j ()). Wha remains are rerospecive differenial equaions for he he same siuaion as he original one, modified o he effec ha he adminisraion coss relaed o he reserve have vanished and he ineres inensiy δ has been decreased by e. Thus, one can apply he explici expression (9.3) for his modified case, and arrive a he following formulas, where v () = e (δ e) : V () = 1 v () V 1 () = 1 v () p 1 (, ) v (τ){c(1 e (τ)) e (τ)} dτ, p 1 (, τ)p 11 (τ, )(b + e 1 (τ)) ] dτ, V 2 () = 1 v () p 2 (, ) V 3 () = 1 v () p 3 (, ) p 1 (, τ)p 13 (τ, )(b + e 1 (τ)) ] dτ. v (τ) [ p (, τ)p 1 (τ, ){c(1 e (τ)) e (τ)} v (τ)p (, τ)p 2 (τ, ){c(1 e (τ)) e (τ)}dτ, v (τ) [ p (, τ)p 3 (τ, ){c(1 e (τ)) e (τ)} Using he equivalence principle in he form (9.16), one obains he equivalence premium c = n v (τ){p (, τ)e (τ) + p 1(, τ)(b + e 1 (τ))} dτ n v (τ)p (, τ)(1 e (τ)) dτ. (9.24)

145 CHAPTER 9. RESERVES 144 If he cos elemen e()vj + () is replaced by e()v j (), he prospecive differenial equaions mus be used. The procedure above can essenially be repeaed, and jus as for he rerospecive reserves i urns ou ha he prospecive reserves are hose correspending o e() = and discoun funcion v. The equivalence premium remains as in (9.24).

146 Chaper 1 Safey loadings and bonus 1.1 General consideraions A. Bonus wha i is. The word bonus is Lain and means good. In insurance erminology i denoes various forms of repaymens o he policyholders of ha par of he company s surplus ha sems from good performance of he insurance porfolio, a sub-porfolio, or he individual policy. We shall here concenrae on he special form i akes in radiional life insurance. The issue of bonus presens iself in connecion wih every sandard life insurance conrac, characerisic of which is is specificaion of nominal coningen paymens ha are binding o boh paries hroughou he erm of he conrac. All conracs discussed so far are of his ype, and a concree example is he combined policy described in 7.4: upon incepion of he conrac he paries agree on a deah benefi of 1 and a disabiliy benefi of.5 per year agains a level premium of.1318 per year, regardless of fuure developmens of he inensiies of moraliy, disabiliy, and ineres. Now, life insurance policies like his one are ypically long erm conracs, wih ime horizons wide enough o capure significan variaions in inensiies, expenses, and oher relevan economicdemographic condiions. The uncerain developmen of such condiions subjecs every supplier of sandard insurance producs o a risk ha is non-diversifiable, ha is, independen of he size of he porfolio; an adverse developmen can no be counered by raising premiums or reducing benefis, and also no by cancelling conracs (he righ of wihdrawal remains one-sidedly wih he insured). The only way he insurer can safeguard agains his kind of risk is o build ino he conracual premium a safey loading ha makes i cover, on he average in he porfolio, he conracual benefis under any likely economic-demographic developmen. Such a safey loading will ypically creae a sysemaic surplus, which by saue is he propery of he insured and has o be repaid in he form of bonus. 145

147 CHAPTER 1. SAFETY LOADINGS AND BONUS 146 B. Skech of he usual echnique. The approach commonly used in pracice is he following. A he ouse he conracual benefis are valuaed, and he premium is se accordingly, on a firs order (echnical) basis, which is a se of hypoheical assumpions abou ineres, inensiies of ransiion beween policy-saes, coss, and possibly oher relevan echnical elemens. The firs order model is a means of pruden calculaion of premiums and reserves, and is elemens are herefore placed o he safe side in a sense ha will be made precise laer. As ime passes realiy reveals rue elemens ha ulimaely se he realisic scenario for he enire erm of he policy and consiue wha is called he second order (experience) basis. Upon comparing elemens of firs and second order, one can idenify he safey loadings buil ino hose of firs order and design schemes for repaymen of he sysemaic surplus hey have creaed. We will now make hese hings precise. To save noaion, we disregard adminisraion expenses for he ime being and discuss hem separaely in Secion 1.7 below. 1.2 Firs and second order bases A. The second order model. The policy-sae process Z is assumed o be a ime-coninuous Markov chain as described in Secion 7.2. In he presen conex we need o equip he indicaor processes and couning processes relaed o he process Z wih a opscrip, calling hem Ij Z and Njk Z. The probabiliy measure and expecaion operaor induced by he ransiion inensiies are denoed by P and E, respecively. The invesmen porfolio of he insurance company bears ineres wih inensiy r() a ime. The inensiies r and µ jk consiue he experience basis, also called he second order basis, represening he rue mechanisms governing he insurance business. A any ime is pas hisory is known, whereas is fuure is unknown. We exend he se-up by viewing he second order basis as sochasic, whereby he uncerainy associaed wih i becomes quanifiable in probabilisic erms. In paricular, predicion of is fuure developmen becomes a maer of modelbased forecasing. Thus, le us consider he se-up above as he condiional model, given he second order basis, and place a disribuion on he laer, whereby r and he µ jk become sochasic processes. Le G denoe heir complee hisory up o, and including, ime and, accordingly, le E[ G ] denoe condiional expecaion, given his informaion. For he ime being we will work only in he condiional model and need no specify any paricular marginal disribuion of he second order elemens. B. The firs order model. We le he firs order model be of he same ype as he condiional model of second order. Thus, he firs order basis is viewed as deerminisic, and we denoe is elemens by r and µ jk and he corresponding probabiliy measure and expecaion operaor by P and E, respecively. The firs order basis represens a pruden iniial assessmen of he developmen of

148 CHAPTER 1. SAFETY LOADINGS AND BONUS 147 he second order basis, and is elemens are placed on he safe side in a sense ha will be made precise laer. By saue, he insurer mus currenly provide a reserve o mee fuure liabiliies in respec of he conrac, and hese liabiliies are o be valuaed on he firs order basis. The firs order reserve a ime, given ha he policy is hen in sae j, is V j () = E [ n = n e τ r e τ We need Thiele s differenial equaions dvj () = r ()Vj () d db j() g ] r db(τ) Z() = j p jg(, τ) db g (τ) + b gh (τ)µ gh(τ) dτ.(1.1) k; k j h;h g Rjk () µ jk () d, (1.2) where Rjk () = b jk() + Vk () V j () (1.3) is he sum a risk associaed wih a possible ransiion from sae j o sae k a ime. The premiums are based on he principle of equivalence exercised on he firs order valuaion basis, [ n ] E e τ r db(τ) =, (1.4) or, equivalenly, V () = B (). (1.5) 1.3 The echnical surplus and how i emerges A. Definiion of he mean porfolio surplus. Wih premiums deermined by he principle of equivalence (1.4) based on pruden firs order assumpions, he porfolio will creae a sysemaic echnical surplus if everyhing goes well. Quie naurally, he surplus is some average of pas ne incomes valuaed on he facual second order basis less fuure ne ougoes valuaed on he conservaive firs order basis. The porfolio-wide mean surplus hus consrued is [ S() = E e = e r j τ r d( B)(τ) G e τ r j ] j p j (, ) V j () p j (, τ) db j (τ) + b jk (τ)µ jk (τ) dτ k;k j p j (, ) Vj (). (1.6)

149 CHAPTER 1. SAFETY LOADINGS AND BONUS 148 The definiion conforms wih basic principles of insurance accounancy; a any ime he balance is he difference beween, on he debi, he facual income in he pas and, on he credi, he reserve ha by saue is o be provided in respec of fuure liabiliies. In paricular, due o (1.5), and, due o Vj (n) =, [ n S(n) = E S() = (1.7) e n τ ] r d( B)(τ) G n, (1.8) as i ough o be. Noe ha he expression in (1.6) involves only he pas hisory of he second order basis, which is currenly known. B. The conribuions o he surplus. Differeniaing (1.6), applying he Kolmogorov forward equaion (7.2) and he Thiele backward equaion (1.2) o he las erm on he righ, leads o ds() = e r r() d e τ r p j (, τ) db j (τ) + b jk (τ)µ jk (τ) dτ j k;k j p j (, ) db j () + b jk ()µ jk () d j k;k j p g (, ) µ gj () d p j (, ) µ j () d. Vj j g; g j p j (, ) r ()Vj () d db j () j Rjk() µ jk() d. k; k j Reusing he relaion (1.6) in he firs line here and gahering erms, we obain ds() = r() d S() + j p j (, )c j () d, wih c j () = {r() r ()} V j () + k; k j R jk(){µ jk() µ jk ()}. (1.9) Finally, inegraing up and using (1.7), we arrive a S() = e τ r j p j (, τ)c j (τ) dτ, (1.1)

150 CHAPTER 1. SAFETY LOADINGS AND BONUS 149 which expresses he echnical surplus a any ime as he sum of pas conribuions compounded wih second order ineres. One may arrive a he definiion of he conribuions (1.9) by anoher roue, saring from he individual surplus defined, quie naurally, as S ind () = e r e τ r d( B)(τ) j I Z j ()V j (). (1.11) Upon differeniaing his expression, and proceeding along he same lines as above, one finds ha S ind () consiss of a purely erraic erm and a sysemaic erm. The laer is e τ r j IZ j (τ)c j(τ) dτ, which is he individual counerpar of (1.1), showing how he conribuions emerge a he level of he individual policy. They form a random paymen funcion C defined by dc() = j I Z j () c j () d. (1.12) Wih his definiion, we can recas (1.1) as [ ] S() = E e τ r dc(τ) G. (1.13) C. Safey margins. The expression on he righ of (1.9) displays how he conribuions arise from safey margins in he firs order force of ineres (he firs erm) and in he ransiion inensiies (he second erm). The purpose of he firs order basis is o creae a non-negaive echnical surplus. This is cerainly fulfilled if r() r () (1.14) (assuming ha all Vj () are non-negaive as hey should be) and sign {µ jk() µ jk ()} = sign R jk(). (1.15) 1.4 Dividends and bonus A. The dividend process. Legislaion lays down ha he echnical surplus belongs o he insured and has o be repaid in is enirey. Therefore, o he conracual paymens B here mus be added dividends, henceforh denoed by D. The dividends are currenly adaped o he developmen of he second order basis and, as explained in Paragraph 1.1.A, hey can no be negaive. The purpose of he dividends is o esablish, ulimaely, equivalence on he rue second order basis: [ n ] E e τ r d{b + D}(τ) G n =. (1.16)

151 CHAPTER 1. SAFETY LOADINGS AND BONUS 15 We can sae (1.16) equivalenly as [ n ] E e n τ r d{b + D}(τ) G n =. (1.17) The value a ime of pas individual conribuions less dividends, compounded wih ineres, is U d () = e τ r d{c D}(τ). (1.18) This amoun is an ousanding accoun of he insured agains he insurer, and we shall call i he dividend reserve a ime. By virue of (1.8) and (1.13) we can recas he equivalence requiremen (1.17) in he appealing form E[U d (n) G n ] =. (1.19) From a solvency poin of view i would make sense o srenghen (1.19) by requiring ha compounded dividends mus never exceed compounded conribuions: E[U d () G ], (1.2) [, n]. A his poin some explanaion is in order. Alhough he ulimae balance requiremen is enforced by law, he dividends do no represen a conracual obligaion on he par of he insurer; he dividends mus be adaped o he second order developmen up o ime n and can, herefore, no be sipulaed in he erms of he conrac a ime. On he oher hand, a any ime, dividends alloed in he pas have irrevocably been credied o he insured s accoun. These regulaory facs are refleced in (1.2). If we adop he view ha he echnical surplus belongs o hose who creaed i, we should sharpen (1.19) by imposing he sronger requiremen U d (n) =. (1.21) This means ha no ransfer of redisribuions across policies is allowed. The solvency requiremen conforming wih his poin of view, and sharpening (1.2), is U d (), (1.22) [, n]. The consrains imposed on D in his paragraph are of a general naure and leave a cerain laiude for various designs of dividend schemes. We shall lis some possibiliies moivaed by pracice. B. Special dividend schemes. The so-called conribuion scheme is defined by D = C, ha is, all conribuions are currenly and immediaely credied o he accoun of he insured. No dividend reserve will accrue and, consequenly,

152 CHAPTER 1. SAFETY LOADINGS AND BONUS 151 he only insrumen on he par of he insurer in case of adverse second order experience is o cease crediing dividends. In some counries he conribuion principle is enforced by law. This means ha insurers are compelled o operae wih minimal proecion agains adverse second order developmens. By erminal dividend is mean ha all conribuions are currenly invesed and heir compounded oal is credied o he insured as a lump sum dividend paymen only upon he erminaion of he conrac a some ime T afer which no more conribuions are generaed. Typically T would be he ime of ransiion o an absorbing sae (deah or wihdrawal), runcaed a n. If compounding is a second order rae of ineres, hen D() = 1[ T ] T e T τ r dc(τ). Conribuion dividends and erminal dividends represen opposie exremes in he se of conceivable dividend schemes, which are counless. One class of inermediae soluions are hose ha yield dividends only a cerain imes T 1 < < T K n, e.g. annually or a imes of ransiion beween cerain saes. A each ime T i he amoun D(T i ) = T i T i 1 e T i τ r dc(τ) (wih T = ) is enered o he insured s credi. C. Allocaion of dividends; bonus. Once hey have been alloed, dividends belong o he insured. They may, however, be disposed of in various ways and need no be paid ou currenly as hey fall due. The acual payous of dividends are ermed bonus in he sequel, and he corresponding paymen funcion is denoed by B b. The compounded value of credied dividends less paid bonuses a ime is U b () = e τ r d{d B b }(τ). (1.23) This is a deb owed by he insurer o he insured, and we shall call i he bonus reserve a ime. Bonuses may no be advanced, so B b mus saisfy U b () (1.24) for all [, n]. In paricular, since D() =, one has B b () =. Moreover, since all dividends mus evenually be paid ou, we mus have U b (n) =. (1.25) We have inroduced hree noions of reserves ha all appear on he debi side of he insurer s balance shee. Firs, he premium reserve V is provided o mee ne ougoes in respec of fuure evens; second, he dividend reserve U d is provided o sele he excess of pas conribuions over pas dividends; hird, he bonus reserve U b is provided o sele he unpaid par of dividends credied in he pas. The premium reserve is of prospecive ype and is a prediced

153 CHAPTER 1. SAFETY LOADINGS AND BONUS 152 amoun, whereas he dividend and bonus reserves are of rerospecive ype and are indeed known amouns summing up o U d () + U b () = e τ r d{c B b }(τ), (1.26) he compounded oal of pas conribuions no ye paid back o he insured. D. Some commonly used bonus schemes. The erm cash bonus is, quie naurally, used for he scheme B b = D. Under his scheme he bonus reserve is always null, of course. By erminal bonus, also called reversionary bonus, is mean ha all dividends, wih accumulaion of ineres, are paid ou as a lump sum upon he erminaion of he conrac a some ime T, ha is, B b () = 1[ T ] T e T τ r dd(τ). Here we could replace he inegraor D by C since erminal bonus obviously does no depend on he dividend scheme; all conribuions are o be repaid wih accumulaion of ineres. Assume now, wha is common in pracice, ha dividends are currenly used o purchase addiional insurance coverage of he same ype as in he primary policy. I seems naural o le he addiional benefis be proporional o hose sipulaed in he primary policy since hey represen he desired profile of he produc. Thus, he dividends dd(s) in any ime inerval [s, s + ds) are used as a single premium for an insurance wih paymen funcion of he form dq(s){b + (τ) B + (s)}, τ (s, n], where he opscrip + signifies, in an obvious sense, ha only posiive paymens (benefis) are couned. Supposing ha addiional insurances are wrien on firs order basis, he proporionaliy facor dq(s) is deermined by where V + dd(s) = dq(s)v + Z(s) (s), (1.27) Z(s) (s) = E [ n s e τ s ] r db + (τ) Z(s) is he single premium a ime s for he fuure benefis under he policy. Now he bonus paymens B b are of he form db b () = Q()dB + (). (1.28) Being wrien on firs order basis, also he addiional insurances creae echnical surplus. The oal conribuions under his scheme develop as dc() + Q()dC + (), (1.29)

154 CHAPTER 1. SAFETY LOADINGS AND BONUS 153 where he firs erm on he righ sems from he primary policy and he second erm sems from he Q() unis of addiional insurances purchased in he pas, each of which has paymen funcion B + producing conribuions C + of he form dc + () = j IZ j () c+ j () d, wih c + j () = {r() r ()}Vj + () + R + jk (){µ jk () µ jk()}, k; k j R + jk () = b+ + jk () + Vk () V + j (). The presen siuaion is more involved han hose encounered previously since, no only are dividends driven by he conracual paymens, bu i is also he oher way around. To keep hings relaively simple, suppose ha he conribuion principle is adoped so ha he dividends in (1.27) are se equal o he conribuions in (1.29). Then he sysem is governed by he dynamics dc() + Q()dC + () = dq()v + Z() () or, realizing ha V + Z() () is sricly posiive whenever dc() and dc + () are, where G and H are defined by dq() Q() dg() = dh(), (1.3) dg() = dh() = 1 V + Z() () dc+ (), (1.31) 1 V + dc(). Z() () (1.32) Muliplying wih exp( G()) o form a complee differenial on he lef and hen inegraing from o, using Q() =, we obain Q() = 1.5 Bonus prognoses e G() G(τ) dh(τ). (1.33) A. A Markov chain environmen. We shall adop a simple Markov chain descripion of he uncerainy associaed wih he developmen of he second order basis. Le Y (), n, be a ime-coninuous Markov chain wih finie sae space Y = {1,..., q} and consan inensiies of ransiion, λ ef. Denoe he associaed indicaor processes by Ie Y. The process Y represens he economic-demographic environmen, and we le he second order elemens depend on he curren Y -sae: r() = e µ jk () = e I Y e () r e = r Y (), I Y e () µ e;jk() = µ Y ();jk ().

155 CHAPTER 1. SAFETY LOADINGS AND BONUS 154 The r e are consans and he µ e;jk () are inensiy funcions, all deerminisic. Wih his specificaion of he full wo-sage model i is realized ha he pair X = (Y, Z) is a Markov chain on he sae space X = Y Z, and is inensiies of ransiion, which we denoe by κ ej,fk () for (e, j), (f, k) X, (e, j) (f, k), are κ ej,fj () = λ ef, e f, (1.34) κ ej,ek () = µ e;jk (), j k, (1.35) and null for all oher ransiions. In his exended se-up he conribuions, whose dependence on he second order elemens was no visualized earlier, can appropriaely be represened as dc() = c() d = e,j I Y e ()I Z j ()c ej () d, where c ej () = {r e r ()} V j () + k; k j R jk(){µ jk() µ e;jk ()}. (1.36) Under he scheme of addiional benefis described in Paragraph 1.4.D a similar convenion goes for C + and c + and, accordingly, (1.31) and (1.32) become dg() = g() d = e,j I Y e ()IZ j ()g ej() d, (1.37) g ej () = c+ ej () V + j (), (1.38) dh() = h() d = e,j I Y e ()IZ j ()h ej() d, (1.39) h ej () = c ej() V + j (). (1.4) B. Preparaory remarks on he issue of bonus prognoses. There is no single funcional of he fuure bonus sream ha presens iself as he relevan quaniy o prognosicae. One could e.g. ake he oal bonuses discouned by some suiable inflaion rae, or he undiscouned oal bonuses, or he rae a which bonus will be paid a cerain imes, and one could apply any of hese possibiliies o he random developmen of he policy or o some represenaive fixed developmen. We shall focus on he expeced value, and in he simples cases also higher order momens, of he fuure bonuses discouned by he sochasic second order ineres. From his we can easily deduce predicors for a number of oher relevan quaniies. We urn now o he analysis of some of he schemes described in Secion 1.4.

156 CHAPTER 1. SAFETY LOADINGS AND BONUS 155 C. Conribuion dividends and cash bonus. This case, where B b = C = D, is paricularly simple since he bonus paymens a any ime depend only on he curren sae of he process. We can hen employ he appropriae version of Thiele s differenial equaion o calculae he sae-wise expeced discouned fuure bonuses (= conribuions), [ n ] W ej () = E e τ r c(τ) dτ X() = (e, j). They are deermined by he appropriae version of Thiele s differenial equaion, d d W ej() = r e W ej () c ej () k;k j f;f e λ ef (W fj () W ej ()) µ e;jk () (W ek () W ej ()), (1.41) subjec o W ej (n ) =, e, j. (1.42) D. Terminal dividend and/or bonus. Under he erminal bonus scheme dividends and bonuses are he same, of course. The problem of predicing he oal bonus paymens discouned wih respec o second order ineres is basically he same as in he previous paragraph since i amouns o adding he oal amoun of compounded pas conribuions, which is known, and he sae-wise predicor of discouned fuure conribuions. Suppose insead ha a ime, he policy sill being in force, i is decided o predic he undiscouned value of he erminal bonus amoun, where W = T e T τ r c(τ) dτ = e τ r c(τ) dτ W () + W (), (1.43) W () = e T W () = r, T We need he sae-wise expeced values e T τ r c(τ) dτ. W e() = E[W () Y () = e], W ej = E[W () X() = (e, j)], o find he sae-wise predicors of W in (1.43), W ej () = e τ r c(τ) dτ W e () + W ej ().

157 CHAPTER 1. SAFETY LOADINGS AND BONUS 156 We shall find hese funcions by he backward consrucion, saring from W () = e r d W ( + d), W () = c() d W () + W ( + d). Condiioning on wha happens in he small ime inerval (, + d], we ge W e() re d = e (1 λ e d) W e( + d) + λ ef () d W f ( + d), and W f; f e ej () = c ej() d W e () + (1 (λ e + µ e;j ()) d) W ej ( + d) + λ ef () d W fj ( + d) f; f e + k; k j µ e;jk () d W ek ( + d). From hese relaionships we easily obain he differenial equaions d d W e () = r e W e () f; f e d d W ej() = c ej () W e() k; k j which are o be solved subjec o λ ef ( W f () W e ()), (1.44) f; f e λ ef ( W fj() W ej() ) µ e;jk () ( W ek() W ej() ), (1.45) W e(n ) = 1, W ej(n ) =, e, j. (1.46) E. Addiional benefis. Suppose we wan o predic he oal fuure bonuses discouned wih respec o second order ineres, W () = n e τ r Q(τ) db + (τ), wih Q defined by (1.33). Recalling (1.37) (1.4), we reshape W () as W () = = = n n e τ τ r e τ r ( e τ r g h(r) dr db + (τ) e r g h(r) dr e τ ) τ g + e τ r g h(r) dr db + (τ) e r g h(r) dr W () + W (), (1.47)

158 CHAPTER 1. SAFETY LOADINGS AND BONUS 157 wih W () = W () = n n Thus, we need he sae-wise expeced values e τ (g r) db + (τ), e τ r W (τ) h(τ) dτ. W ej() = E[W () X() = (e, j)], W ej = E[W () X() = (e, j)], in order o find he sae-wise predicors of W () in (1.47), W ej () = The backward equaions sar from e r g h(r) dr W ej () + W ej (). W () = db + () + e (g() r()) d W ( + d), W () = W () h() d + e r() d W ( + d), from which we proceed in he same way as in he previous paragraph o obain dw ej () = db+ j () + (r e g ej ()) d W ej () λ ef d ( W fj () W ej ()) f; f e k;k j ) µ e;jk () d (b +jk () + W ek () W ej (), (1.48) dw ej() = W ej()h ej () d + r e d W ej() λ ef d ( W fj() W ej() ) f; f e k; k j The appropriae side condiions are W ej(n ) = B + j µ e;jk () d ( W ek() W ej() ). (1.49) (n), W ej(n ) =, e, j. (1.5) F. Predicing undiscouned amouns. If he undiscouned oal conribuions or addiional benefis is wha one wans o predic, one can jus apply he formulas wih all r e replaced by.

159 CHAPTER 1. SAFETY LOADINGS AND BONUS 158 G. Predicing bonuses for a given policy pah. Ye anoher form of prognosis, which may be considered more informaive han he wo menioned above, would be o predic bonus paymens for some possible fixed pursuis of a policy insead of averaging over all possibiliies. Such prognoses are obained from hose described above upon keeping he realized pah Z(τ) for τ [, ], where is he ime of consideraion, and puing Z(τ) = z(τ) for τ (, n], where z( ) is some fixed pah wih z() = Z(). The relevan predicors hen become essenially funcions only of he curren Y -sae and are simple special cases of he resuls above. As an example of an even simpler ype of prognosis for a policy in sae j a ime, he insurer could presen he expeced bonus paymen per ime uni a a fuure ime s, given ha he policy is hen in sae i, and do his for some represenaive selecions of s and i. If Y () = e, hen he relevan predicion is E[c Y (s)i (s) Y () = e] = f p Y ef (, s)c fi(s). 1.6 Examples A. The case. For our purpose, which is o illusrae he role of he sochasic environmen in model-based prognoses, i suffices o consider simple insurance producs for which he relevan policy saes are Z = {a, d} ( alive and dead ). We will consider a single life insured a age 3 for a period of n = 3 years, and le he firs order elemens be hose of he Danish echnical basis G82M for males: r = ln(1.45), µ ad () = µ () = (3+). Three differen forms of insurance benefis will be considered, and in each case we assume ha premiums are payable coninuously a level rae as long as he policy is in force. Firs, a erm insurance (TI) of 1 = b ad () wih firs order premium rae.4268 = b a (). Second, a pure endowmen (PE) of 1 = B a (3) wih firs order premium rae.1469 = b a (). Third, an endowmen insurance (EI), which is jus he combinaion of he former wo; 1 = b ad () = B a (3), = b a (). Jus as an illusraion, le he second order model be he simple one where ineres and moraliy are governed by independen ime-coninuous Markov chains and, more specifically, ha r swiches wih a consan inensiy λ i beween he firs order rae r and a beer rae ɛ i r (ɛ i > 1) and, similarly, µ swiches wih a consan inensiy λ m beween he firs order rae µ and a beer rae ɛ m µ (ɛ m < 1). (We choose o express ourselves his way alhough (1.15) shows ha, for insurance forms wih negaive sum a risk, e.g. pure endowmen insurance, i is acually a higher second order moraliy ha is beer in he sense of creaing posiive conribuions.)

160 CHAPTER 1. SAFETY LOADINGS AND BONUS 159 bb,a r, µ, alive λ i λi gb,a ɛ i r, µ, alive µ () µ (),d λ m λ m dead λ m λ m ɛ m µ () ɛm µ () bg,a r, ɛ m µ, alive λ i λi gg,a ɛ i r, ɛ m µ, alive Figure 1.1: The Markov process X = (Y, Z) for a single life insurance in an environmen wih wo ineres saes and wo moraliy saes. The siuaion fis ino he framework of Paragraph 1.5.A; Y has saes Y = {bb, gb, bg, gg} represening all combinaions of bad (b) and good (g) ineres and moraliy, and he non-null inensiies are λ bb,gb = λ gb,bb = λ bg,gg = λ gg,bg = λ i, λ bb,bg = λ bg,bb = λ gb,gg = λ gg,gb = λ m. The firs order basis is jus he wors-scenario bb. Adoping he device (1.34) (1.35), we consider he Markov chain X = (Y, Z) wih saes (bb, a), (gb, a), ec. I is realized ha all deah saes can be merged ino one, so i suffices o work wih he simple Markov model wih five saes skeched in Figure 1.1. B. Resuls. We shall repor some numerical resuls for he case where ɛ i = 1.25, ɛ m =.75, and λ i = λ m =.1. Prognoses are made a he ime of issue of he policy. Compuaions were performed by he fourh order Runge-Kua mehod, which urns ou o work wih high precision in he presen class of siuaions.

161 CHAPTER 1. SAFETY LOADINGS AND BONUS 16 Table 1.1 displays, for each of he hree policies, he sae-wise expeced values of discouned conribuions obained by solving (1.41) (1.42). We shall be conen here o poin ou wo feaures: Firs, for he erm insurance he moraliy margin is far more imporan han he ineres margin, whereas for he pure endowmen i is he oher way around (he laer has he larger reserve). Noe ha he sum a risk is negaive for he pure endowmen, so ha he firs order assumpion of excess moraliy is really no o he safe side, see (1.15). Second, high ineres produces large conribuions, bu, since high iniial ineres also induces severe discouning, i is no necessarily rue ha good iniial ineres will produce a high value of he expeced discouned conribuions, see he wo las enries in he row TI. The laer remark suggess he use of a discouning funcion differen from he one based on he second order ineres, e.g. some exogenous deflaor reflecing he likely developmen of he price index or he discouning funcion corresponding o firs order ineres. In paricular, one can simply drop discouning and prognosicae he oal amouns paid. We shall do his in he following, noing ha he expeced value of bonuses discouned by second order ineres mus in fac be he same for all bonus schemes, and are already shown in Table 1.1. Table 1.2 shows sae-wise expeced values of undiscouned bonuses for hree differen schemes; conribuion dividends and cash bonus (C, he same as oal undiscouned conribuions), erminal bonus (T B), and addiional benefis (AB). We firs noe ha, now, any improvemen of iniial second order condiions helps o increase prospecive conribuions and bonuses. Furhermore, expeced bonuses are generally smaller for C han for T B and AB since bonuses under C are paid earlier. Differences beween T B and AB mus be due o a similar effec. Thus, we can infer ha AB mus on he average fall due earlier han T B, excep for he pure endowmen policy, of course. One migh expec ha he bonuses for he erm insurance and he pure endowmen policies add up o he bonuses for he combined endowmen insurance policy, as is he case for C and T B. However, for AB i is seen ha he sum of he bonuses for he wo componen policies is generally smaller han he bonuses for he combined policy. The explanaion mus be ha addiional deah benefis and addiional survival benefis are no purchased in he same proporions under he wo policy sraegies. The observed difference indicaes ha, on he average, he addiional benefis fall due laer under he combined policy, which herefore mus have he smaller proporion of addiional deah benefis. C. Assessmen of prognosicaion error. Bonus prognoses based on he presen model may be equipped wih quaniaive measures of he prognosicaion error. By he echnique of proof shown in Secion 1.5 we may derive differenial equaions for higher order momens of any of he predicands considered and calculae e.g. he coefficien of variaion, he skewness, and he kurosis.

162 CHAPTER 1. SAFETY LOADINGS AND BONUS 161 Table 1.1: Condiional expeced presen value a ime of oal conribuions for erm insurance policy (TI), pure endowmen policy (PE), and endowmen insurance policy (EI), given iniial second order saes of ineres and moraliy (b or g). bb gb bg gg TI : PE : EI : Including expenses A. The form of he expenses. Expenses are assumed o incur in accordance wih a non-decreasing paymen funcion A of he same ype as he conracual paymens, ha is, da() = j I Z j () da j () + j k a jk () dn Z jk(). (1.51) I is common in pracice o assume, furhermore, ha expenses of annuiy ype incur wih a lump sum of iniial coss a ime and hereafer coninuously a a rae ha depends on he curren sae, ha is, A () > and da j () = a j () d for >. The ransiion coss a jk () are no explicily aken ino accoun in pracice, bu we include hem here since hey add realism wihou adding mahemaical complexiy. B. Firs order assumpions. The elemens A (), a j (), and a jk () will in general depend on he second order developmen, and he firs order basis mus, herefore, specify pruden esimaes A (), a j (), and a jk (). Denoe he corresponding paymen funcion by A. C. Surplus and conribuions in he presence of expenses. The inroducion of expenses adds a new feaure o he previous se-up in ha also he paymens become dependen on he second order developmen. However, he essenial pars of he analyses in he previous secions carry over wih merely noaional modificaions; all i akes is o replace everywhere he conracual paymen funcion B wih A + B in he pas and A + B in he fuure. One finds, in paricular, ha he firs order equivalence relaion (1.5) now urns ino V () = A () B (), (1.52) he surplus a ime becomes S() = A () A (), (1.53)

163 CHAPTER 1. SAFETY LOADINGS AND BONUS 162 Table 1.2: Condiional expeced value (E) of undiscouned oal conribuions (C), erminal bonus (T B), and oal addiional benefis (AB) for erm insurance policy (TI), pure endowmen policy (PE), and endowmen insurance policy (EI), given iniial second order saes of ineres and moraliy (b or g). bb gb bg gg TI: E C : E T B : E AB : PE: E C : E T B : E AB : EI: E C : E T B : E AB : and he conribuions consis of a jump C() = A () A () (1.54) a ime and hereafer a coninuous par, which is defined upon replacing (1.9) wih where now c j () = {r() r ()} Vj () + {a j () a j()} + {a jk() a jk ()}µ jk () k; k j + k; k j R jk(){µ jk() µ jk ()}, (1.55) R jk() = a jk() + b jk () + V k () V j (). (1.56) Referring o he discussion in Paragraph 1.3.C, we see ha he conribuions emerge from safey margins in all firs order elemens, r, µ jk, and A. D. Predicion in he presence of expenses. The complexiy of he predicion problem depends heavily on he assumpions made abou he second order expenses, and a his poin some new problems may arise. Jus o ge sared, suppose firs ha he expense elemens A (), a j (), and a jk () are deerminisic. Then he mehods in Secion 1.5 carry over wih only rivial modificaions. Presumably, his simplisic model is a he base of he frequenly encounered claim ha adminisraion expenses can be regarded

164 CHAPTER 1. SAFETY LOADINGS AND BONUS 163 as addiional benefis. Unforunaely, real life expenses are of a differen, and ypically less pleasan, naure. An exhausive discussion of his issue could easily exhaus he reader, so we shall be conen wih jus oulining some enaive ideas. The problem is ha expenses are made up of wages, commissions, ren, axes and oher iems ha are governed by he economic developmen. In he framework of he Markov model in Paragraph 1.5.A, one simple way of accouning for such effecs is o make he second order expenses dependen on he curren sae of Y, ha is, A () = e a j () = e a jk () = e I Y e () A e (), I Y e () a ej(), I Y e () a ejk(), wih deerminisic A e, a ej, and a ejk. By enriching sufficienly he sae space of Y, one can in principle creae a fairly realisic model. Perhaps he mos reasonable poin of view is ha expenses are inflaed by some ime-dependen rae γ() so ha we should pu a j () = e γ a j () and a jk () = e γ a jk () wih a j and a jk deerminisic. One possibiliy is o pu he second order force of ineres r in he role of γ. More realisically one should le γ be somehing else, bu sill relaed o r hrough join dependence on a suiably specified Y. We shall no pursue his idea any furher here, bu noe, by way of warning, ha prognosicaion in his kind of inflaion model will presen problems in addiion o hose solved in Secion Discussions A. The principle of equivalence. This principle, as formulaed in (1.4), is basic in life insurance. The expeced value represens averaging over a large (really infinie) porfolio of policies, he philosophy being ha, even if he individual policy creaes a (possibly large) loss or gain, here will be balance on he average beween ougoes and incomes in he porfolio as a whole if he premiums are se by equivalence. The deviaion from perfec balance, which is ineviable in a finie world wih finie porfolios, represens profi or loss on he par of he insurer and has o be seled by an adjusmen of he equiy capial. (The possibiliy of loss, abou as likely and abou as large as he possible profi, migh seem unaccepable o an indusry ha needs o arac invesors, bu i should be kep in mind ha salaries o employees and dividends o owners are accouned as par of he expenses discussed in Secion 1.7.) B. On he noion of second order basis. The definiion of he second order basis as he rue one is slighly a variance wih pracical usage (which is

165 CHAPTER 1. SAFETY LOADINGS AND BONUS 164 no uniform anyway). The various amendmens made o our idealized definiion in pracice are due o adminisraive and procedural bolenecks: The facual developmen of ineres, moraliy, ec. has o be verified by he insurer and hen approved by he supervisory auhoriy. Since his can no be a coninuous operaion, any regulaory definiion of he second order basis mus o some exen involve realisic, sill ypically conservaive, shor erm forecass of he fuure developmen. However, our definiion can cerainly be agreed upon as he inended one. C. Model deliberaions. The Markov chain model is mahemaically racable since sae-wise expeced values are deermined by solving (in mos cases simple) sysems of firs order ordinary differenial equaions. A he same ime, when equipped wih a sufficienly rich sae space and appropriae inensiies of ransiion, i is able o picure virually any conceivable noion of he real objec of he model. The Markov chain model is paricularly ap o describe he developmen of life insurance policy since he pahs of Z are of he same kind as he rue ones. When used o describe he developmen of he second order basis, however, he approximaive naure of he Markov chain is obvious, and i will surface immediaely as e.g. he experienced force of ineres akes values ouside of he finie se allowed by he model. This is no a serious objecion, however, and he nex paragraph explains why. D. The role of he sochasic environmen model. A paramoun concern is ha of esablishing equivalence condiional on he facual second order hisory in he sense of (1.16). Now, in his condiional expecaion he marginal disribuion of he second order elemens does no appear and is, in his respec, irrelevan. Also he conribuions and, hence, he dividends are funcions only of he realized experience basis and do no involve he disribuion of is elemens. Then, wha remains he purpose of placing a disribuion on he second order elemens is o form a basis for prognosicaion of bonus. Subsidiary as i is, his role is sill an imporan par of he play; alhough a prognosis does no commi he insurer o pay he forecased amouns, i should as much as possible be a reliable piece of informaion o he insured. Therefore, he disribuion placed on he second order elemens should se a reasonable scenario for he course of evens, bu i need no be perfecly rue. This is comforing since any view of he mechanisms governing he economic-demographic developmen is o some exen guess-work. When he accouns are evenually made up, every speculaive elemen mus be absen, and ha is precisely wha he principle (1.16) lays down. E. A digression: Which is more imporan, ineres or moraliy? Acuarial wisdom says i is ineres. This is, of course, an empirical saemen based on he fac ha, in he era of conemporary insurance, moraliy raes

166 CHAPTER 1. SAFETY LOADINGS AND BONUS 165 have been smaller and more sable han ineres raes. Our model can add some oher kind of insigh. We shall again be conen wih a simple illusraion relaed o he single life described in Secion 1.6. Table 1.3 displays expeced values and sandard deviaions of he presen values a ime of a erm life insurance and a life annuiy under various scenarios wih fixed ineres and moraliy, ha is, condiional on fixed Y -sae hroughou he erm of he policy. The impac of ineres variaion is seen by reading column-wise, and he impac of moraliy variaion is seen by reading row-wise. The overall impression is ha moraliy is he more imporan elemen by erm insurance, whereas ineres is he (by far) more imporan by life annuiy insurance.

167 CHAPTER 1. SAFETY LOADINGS AND BONUS 166 Table 1.3: Expeced value (E) and sandard deviaion (SD) of presen values of a erm life insurance (TI) wih sum 1 and a life annuiy (LA) wih level inensiy 1 per year, wih ineres r = ɛ i r and moraliy µ = ɛ m µ for various choices of ɛ i and ɛ m. TI LA ɛ m : ɛ i :.5 E : SD : E : SD : E : SD :

168 Chaper 11 Saisical inference in he Markov chain model Think of he insurance company as a car: A he seering wheel sis he managing direcor rying o keep he vehicle seadily on he road. In he fron passenger sea sis he sales manager pushing he speed pedal. A he rear sis he acuary peeping ou he back window and giving he direcions Esimaing a moraliy law from fully observed life lenghs A. Compleely observed life lenghs. Le he life lengh of an individual be represened by a non-negaive random variable T wih cumulaive disribuion funcion of he form F (; θ) = 1 e µ(s;θ) ds. (11.1) The moraliy inensiy µ is assumed o be coninuous (a leas piece-wise) so ha he densiy exiss. f() = µ()(1 F ()) (11.2) A. Righ-censored life imes. Le he oal life lengh of an individual be represened by a non-negaive random variable T wih cumulaive disribuion funcion of he form F () = 1 e µ(s) ds. (11.3) The moraliy inensiy µ is assumed o be coninuous (a leas piece-wise) so ha he densiy f() = µ()(1 F ()), (11.4) exiss. Suppose ha he individual is observed coninually in z years from is birh so ha only he runcaed life lengh W = T z is observed. A echnical erm for his kind of observaional plan is righ-censoring a ime z. The cumulaive disribuion of W is { F (), < < z, P[W ] = 1, z. 167

169 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL168 and he densiy (wih respec o Lebesgue measure on (, z) and he uni mass a z) is (recall (11.4)) { µ()(1 F ()), < < z, g() = 1 F (z), = z. Inroduce o obain he closed expression { 1, < < z, d() = 1 (,z) () =, z, (11.5) g() = µ() d() (1 F ()), < z. (11.6) Denoe he indicaor funcion of survival o age u > by I(u) = 1[U > u]. The indicaor of deah before age z is D = d(t ) = 1 I(z ) = 1 I(z), (11.7) where he las equaliy holds wih probabiliy 1. We will need he following formulas, valid whenever he displayed momens exis: E[D k ] = F (z), k = 1, 2,..., z (11.8) E[T k ] = k k 1 (1 F ())d, k = 1, 2,..., (11.9) E[DT ] = E[T ] z(1 F (z)). (11.1) To verify (11.1), use (11.7) and T = z I()d o wrie DT = z I()d zi(z), and ake expecaion. B. The runcaed exponenial disribuion. We se ou by analyzing he simple case wih consan moraliy inensiy, parly as a moivaing example, bu also because he echniques are a he base of an imporan class of procedures in acuarial life hisory analysis. Thus, assume ha U is exponenially disribued wih cumulaive disribuion funcion F (u; µ) = 1 e µu, u >, (11.11) ha is, µ is consan, independen of age. The expeced life lengh is ν = (1 F (u; µ))du = 1 µ. (11.12) Using (11.8) (11.11) one easily calculaes E[D] = 1 e µz, (11.13) V[D] = e µz (1 e µz ), (11.14) E[T ] = 1 e µz µ, (11.15) V[T ] = 1 2µze µz e 2µz µ 2, (11.16) E[D µt ] =, (11.17) V[D µt ] = 1 e µz. (11.18) C. Maximum likelihood esimaors based on censored exponenial variaes. Le U i, i = 1, 2,..., be independen replicaes of U. Consider he problem of esimaing µ from a sample of n censored life lenghs, T i = U i z i, i = 1,..., n. The inerpreaion is ha a moraliy sudy is carried ou in a populaion during a cerain period of ime erminaing a ime, say, he sample being n individuals born during he period a imes z i, i = 1,..., n.

170 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL169 Referring o (11.7), pu N i = 1[T i < z i ], i = 1,..., n. By (11.6) and (11.11), he likelihood of he observables is where N = W = Λ = Π n i=1 µn i e µt i = µ N e µw = e ln µ N µw, (11.19) n N i, he oal number of deahs occurred, i=1 n T i, he oal ime exposed o risk of deah. i=1 Clearly, (N, W ) is a sufficien saisic. Take he logarihm, and form he derivaives ln Λ = ln µ N µw, µ ln Λ = N µ W, (11.2) 2 µ 2 ln Λ = N µ 2. (11.21) Puing he expression in (11.2) equal o and noing ha he second derivaive is nonposiive, we find ha he maximum likelihood esimaor (MLE) of µ is he so-called occurrenceexposure rae, (OE rae) ˆµ = N W, (11.22) he number of deahs occurred per uni of ime exposed o risk of deah in he sample. I is he empirical counerpar of he moraliy inensiy, which is he expeced number of deahs per ime uni, roughly speaking. The MLE of he expeced life lengh in (11.12) is ˆν = W N, (11.23) defined as + when N =. This esimaor has no mean (and no higher order momens). The expressions for he likelihood in (11.19) and he MLE in (11.22) do no appear o depend on he censoring mechanoism. The censoring is, however, decisive of he probabiliy disribuion of ˆµ and, hence, of is performance as an esimaor of µ. Unforunaely, his probabiliy disribuion is no easy o calculae in general, and we shall herefore have o add assumpions abou he censoring mechanism, ranging from he special case of no censoring, where everyhing is simple and a lo of powerful resuls can be proved, o weak condiions under which only cerain asympoic properies are in reach. D. The special case wih no censoring. Suppose now ha he n lives are compleely observed wihou censoring, ha is, z i = and T i = U i, i = 1,..., n. Then all N i are 1, N = n, W = n i=1 T i, and he likelihood in (11.19) becomes Λ = e ln µ n µw. (11.24) In his simple siuaion i is easy o invesigae he small sample properies of he esimaors. The sum of he life lenghs, W, is now a sufficien saisic. I has a gamma disribuion wih shape parameer n and scale parameer ν = 1/µ, whose densiy is µ n Γ(n) wn 1 e µw, w >. One finds (perform he easy calculaions) for k > n ha E[W k ] = Γ(n + k) Γ(n)µ k,

171 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL17 hence and E[ˆµ] = nµ n 1, n > 1, n 2 µ 2 V[ˆµ] = (n 1) 2 (n 2), n > 2, E[ˆν] = ν, n 1, V[ˆν] = ν2 n, n 1. The esimaor ˆµ is biased and, on he average, overesimaes µ by µ/(n 1), which is negligible for large n. An unbiased esimaor of µ is ˇµ = (n 1)/W. Is variance is µ 2 /(n 2). The esimaor ˆν is now jus he observed average life lengh, he empirical counerpar of ν. I is unbiased, of course. In fac, ˇµ and ˆν are UMVUE (uniformly minimum variance unbiased esimaors) since hey are based on W, which is a sufficien and complee saisic: by (11.24), he disribuion belongs o an exponenial family wih canonical parameer µ varying in he open se (, ). E. Asympoic resuls by uniform censoring. Suppose all z i are equal o z, say. Wriing 1 n n i=1 ˆµ = N i 1 n n i=1 T, i and noing ha E[N i ] = µet i by (11.13) and (11.15), i follows by he srong law of large numbers ha he esimaor is srongly consisen, ˆµ a.s. µ. To invesigae is asympoic disribuion, look a n(ˆµ µ) = 1 n n i=1 (N i µt i ) 1 n n i=1 T i The denominaor of his fracion converges a.s. o E[T i ] given by (11.15). By he cenral limi heorem, he limiing disribuion of he numeraor is normal wih mean (recall (11.17)) and variance given by (11.18). I follows ha ˆµ as N ( µ, µ 2 n(1 e µz ). ). (11.25) Copying he argumens above (or using (D.6) in Appendix D), i can also be concluded ha ˆν defined by (11.23) is srongly consisen and ha ( ) 1 ˆν as N ν, nµ 2 (1 e µz. (11.26) ) No srong conclusions as o opimaliy can be drawn in parallel o hose in he previous paragraph. The reason is seen from (11.19): he disribuion belongs o a general exponenial family wih canonical parameer (ln µ, µ), which does no vary in an open (wo-dimensional) se. Therefore, he sufficien saisic (N, W ) canno be proved o be complee (no he usual way a leas), and sandard heory for inference in regular exponenial families of disribuions canno be employed. F. Asympoic resuls by fairly general censoring. Consider now he general siuaion in Paragraph C wih censoring varying among he individuals. A bi more effor mus now be pu ino he sudy of he asympoic properies of he MLE. I urns ou ha a sufficien condiion for consisency and asympoic normaliy is ha he expeced exposure grows o infiniy in he sense ha n E[T i ], i=1

172 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL171 which by (11.15) is equivalen o n (1 e µz i ), (11.27) i=1 ha is, he expeced number of deahs grows o infiniy. Thus assume ha (11.27) is saisfied. In he following he relaionships (11.13) (11.18) will be used frequenly wihou explici menioning. Firs, o prove consisency, use (11.15) and (11.18) o wrie n i=1 ˆµ µ = (N i µt i ) n i=1 T. i n i=1 = (N ( i µt i ) n i=1 T ) 1 i n i=1 V[N i µt i ] µ n i=1 E[T. (11.28) i] The firs facor in (11.28) has expeced value and variance 1 n i=1 V[N i µt i ] = 1 n i=1 (1 e µz i ), which ends o as n increases. Therefore, his facor ends o in probabiliy. The second facor in (11.28) is he inverse of n i=1 T i/µ n i=1 E[T i], which has expeced value 1/µ and variance equal o 1/µ 2 imes n i=1 V[T n i] ( n i=1 (1 e µz i) ) i=1 = (1 2µz ie µz i e 2µz i) 2 ( n i=1 (1 e µz i) ) 2 = n i=1 a(µz i)(1 e µz i) n i=1 (1 e µz i) 1 n i=1 (1 e µz i), where a is defined as (11.29) a() = 1 2e e 2 1 e,. The funcion a is bounded since i is coninuous and ends o as (use l Hospial s rule) and o 1 as. The firs facor in (11.29) is bounded since i is a weighed average of values of a, and he second facor ends o by assumpion. I follows ha he expression in (11.29) ends o and, consequenly, ha he second facor in (11.28) converges in probabiliy o µ. I can be concluded ha he expression in (11.28) converges in probabiliy o, so ha ˆµ is weakly consisen; ˆµ p µ. Nex, o prove asympoic normaliy, look a n n (1 e µz i=1 i)(ˆµ µ) = (N ( i µt i ) n i=1 T ) 1 i n i=1 i=1 (1 e µz i ) µ n i=1 E[T. i] In he presence of (11.27) he firs facor on he righ converges in disribuion o a sandard normal variae. (Verify ha he Lindeberg condiion is saisfied, see Appendix D.) The second facor has jus been proved o converge in probabiliy o µ. I follows ha ˆµ as N ( µ, µ 2 n i=1 (1 e µz i) ). (11.3) Likewise, i also holds ha ˆν is weakly consisen, and ( ) 1 ˆν as N ν, µ 2 n i=1 (1. (11.31) e µz i)

173 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL172 G. Random censoring. In Paragraph E he censoring ime was assumed o be he same for all individuals. Thereby he pairs (N i, T i ) became sochasic replicaes, and we could invoke simple asympoic heory for i.i.d. variaes o prove srong consisency and asympoic normaliy of MLE-s. In Paragraph F he censoring was allowed o vary among he individuals, bu i urned ou ha he asympoic resuls essenially remained rue, alhough only weak consisency could be achieved. All we required was (11.27), which says ha he censoring mus no urn oo severe so ha informaion deerioraes in he end: here mus be a cerain sabiliy in he censoring paern so ha individuals wih sufficien exposure ime ener he sudy sufficienly frequenly in he long run. One way of securing such sabiliy is o regard he censoring imes as oucomes of i.i.d. random variables. Such an assumpion seems paricularly ap in a non-experimenal conex like insurance. The censoring is no subjec o planning, and he censoring imes are jus as random in heir naure as anyhing else observed abou he individuals. Thus, we henceforh work wih an augmened model where he assumpions in Paragraph F consiue he condiional model for given censoring imes Z i = z i, i = 1, 2,..., and he Z i are independen selecions from some disribuion funcion H wih (generalized) densiy h independen of µ. This way he riples (N i, T i, Z i ), i = 1, 2,..., become sochasic replicaes, and he i.i.d. siuaion is resored wih all is powers. The likelihood of he observaions now becomes Λ = Π n i=1 µn i e µt i h(z i ) = e ln µ N µw Π n i=1 h(z i). (11.32) Maximizing (11.32) wih respec o µ is equivalen o maximizing he likelihood (11.19) in he condiional model for fixed censoring, hence he MLE remains he same as before. Is disribuion is affeced by he srucure now added o he model, however. I is easy o prove ha he resuls in Paragraph E carry over o he presen case, only ha he expression 1 e µz is everywhere o be replaced by 1 E [ e µz], where Z H Parameric inference in he Markov model A. The likelihood of a ime-coninuous Markov process. Consider now he general se-up, whereby he developmen of an insurance policy is represened by a coninuous ime Markov process Z on a finie sae space Z = {, 1,..., J}. As usual, le I g() and N gh () denoe, respecively, he indicaor of he even ha he process is saying in sae g a ime, and he number of ransiions from sae g o sae h in he ime inerval (, ]. The ransiion inensiies µ gh are assumed o exis, and o be piecewise coninuous. Suppose he policy is observed coninuously hroughou he ime period [, ], commencing in sae g a ime. One hen speaks of lef-censoring and righ-censoring a imes and, respecively, and he riple z = (,, g ) will be referred o as observaional design or censoring scheme of he policy. Consider a specific realizaion of he observed par of he process: X(τ) = g, < τ < 1, g 1, 1 + d 1 < τ < 2,... g q 2, q 2 + d q 2 < τ < q 1, g q 1, q 1 + d q 1 < τ <. By he given censoring, he probabiliy of his realizaion is as follows, where =, q =, and µ g = h g µ gh denoes he oal inensiy of ransiion ou of sae g: ( 1 ) ( 2 ) exp µ g µ g g 1 ( 1 )d 1 exp µ g1 µ g1 g 2 ( 2 )d ( ) ( ) exp = q 1 µ gq 2 q 2 q 1 p=1 µ gq 2 g q 1 ( q 1 )d q 1 exp µ gp 1 g p ( p)d p exp ( q 1 p p 1 µ gp 1 q q 1 µ gq 1 )

174 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL173 q 1 = exp ln µ gp 1 g p ( p) p=1 q 1 p p 1 µ gp 1 d 1... d q 1. I follows ha he likelihood of he observables is Λ = exp ln µ gh (τ)dn gh (τ) µ g(τ)i g(τ)dτ g h g = exp {ln µ gh (τ)dn gh (τ) µ gh (τ)i g(τ)dτ}. (11.33) g h B. ML esimaion of parameric inensiies. Now consider a parameric model where he inensiies are of he form µ gh (, θ), wih θ = (θ 1,..., θ s) varying in an open se in he s-dimensional euclidean space, s <. We assume hey are wice coninuously differeniable funcions of θ. Suppose ha inference is o be made abou he inensiies or, equivalenly, he parameer θ on he basis of daa from a sample of n similar policies. Equip all quaniies relaed o he m- h policy by opscrip (m). The processes X (m) are assumed o be sochasically independen replicaes of he process Z described above, bu heir censoring schemes z (m) may be differen. By independence, he likelihood of he whole daa se is he produc of he individual likelihoods: Λ = n m=1 Λ(m). Thus, by (11.33), ln Λ = g h {ln µgh (τ, θ)dn gh (τ) µ gh (τ, θ)i g(τ)dτ }, (11.34) wih n N gh = N (m) gh, Ig = n I g (m). m=1 m=1 The censoring schemes are no visualized in (11.34), and hey need no be if, as a maer of definiion, dn (m) gh () and I(m) g () are aken as for / [ (m), (m) ]. Likewise, inroduce p (m) g () = p (m) g g ((m), )1 [ (m), (m) ] (), he probabiliy ha he censored process Z (m) says in g a ime, by definiion aken as for / [ (m), (m) ]. In he MLE consrucion we need he derivaives of (11.34), of firs order (an s-vecor), θ ln Λ = g h and of second order (an s s marix), θ ln µ gh(τ, θ){dn gh (τ) µ gh (τ, θ)i g(τ)dτ}, (11.35) 2 θ θ ln Λ = { 2 θ θ ln µ gh(τ, θ){dn gh (τ) µ gh (τ, θ)i g(τ)dτ} g h θ ln µ gh(τ, θ) } θ µ gh(τ, θ)i g(τ) dτ (11.36) By (11.35) he MLE ˆθ is he soluion o θ ln µ gh(τ, θ){dn gh (τ) µ gh (τ, θ)i g(τ)dτ} θ=ˆθ= sx1. (11.37) g h A commen on he form of he likelihood (9.31): For each ype of ransiion g h inroduce N gh, he number of ransiions of ha ype, and (if N gh > ) T (1) gh,..., T (N gh) gh, he

175 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL174 imes when such ransiions occurred. In erms of hese quaniies he log likelihood in (9.32) assumes he form N gh g h j=1 and he ML equaions (9.35) become N gh g h j=1 ln µ gh (T (j) gh, ˆθ) g h µ θ gh (T (j) i gh, ˆθ) µ gh (T (j) gh, ˆθ) = g h µ gh (τ, ˆθ) I g(τ)dτ, θ i µ gh (τ, ˆθ) I g(τ)dτ, (11.38) i = 1,..., s. The form (11.38) is explici and is, of course, he one we will work wih when i comes o numerical compuaion of he MLE: The good hing abou he form (9.32) is ha i, by use of he couning processes, wries he sum on he lef as a sum of conribuions from all small ime inervals. This is paricularly useful in he derivaion of he saisical properies of he MLE. (A similar remark could be made abou he benefi of using he couning processes o define he paymen sream for a general insurance policy in Secion 7.5.) by Referring o Appendix D, he large sample disribuion properies of he MLE are given ˆθ as N(θ, Σ(θ)), (11.39) where Σ(θ) is given by is inverse, he so-called informaion marix, [ ] Σ(θ) 1 = E θ θ ln Λ. (11.4) Taking expecaion in (11.36), noing ha he erms have zero means, we obain Σ(θ) 1 = g h 1 µ gh (τ, θ) dn gh (τ) µ gh (τ, θ)i g(τ)dτ ( θ µ gh(τ, θ) ) θ µ n gh(τ, θ) p (m) g (τ, θ)dτ. m=1 (11.41) The expression in parenheses under he inegral sign is an s s marix and all oher quaniies are scalar. I is seen ha he informaion marix ends o infiniy, hence he variance marix of he MLE ends o, if he erms n m=1 p(m) g (τ, θ) grow o infiniy as n increases, roughly speaking, which means ha he expeced number of individuals exposed o risk in differen saes ges unlimied. C. Esimaing he parameers of a Gomperz-Makeham moraliy law. The acuarial office in a life insurance company is o esimae he moraliy law governing he company s porfolio of erm insurance policies. (The moraliy law for he porfolio of life annuiies may be differen since people who (believe hey) are in good healh would probably find a pure survival benefi more useful and profiable han a pure deah benefi. Thus, i seems appropriae o perform a separae moraliy invesigaion for each line of life insurance business. Moreover, since moraliy also depends on sex, he sudy would ypically include only males or only females.) Suppose he saisical daa comprises n individuals who have been insured under he scheme during a cerain period of ime. For each individual No. m (= 1,..., n) here is a policy record wih he following pieces of informaion: x m, he age on enry ino he sudy; y m, he age on exi from he sudy; N m, he number of deahs during he sudy ( or 1).

176 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL175 Here x m would ypically be he age a issue of he policy. If N m = 1, hen y m is he age a deah, and if N m =, hen y m is he age a he ime of censoring, eiher a he erm of he conrac or upon erminaion of he sudy, whichever occurred firs. In any case y m x m is he ime spen under observaion as alive during he sudy. Wih hese definiions x m akes he role of (m) in he general se-up and, for N m =, y m akes he role of (m). The sae space is now jus Z = {, 1} ( alive and dead ). Assume he moraliy law is Gomperz-Makeham so ha he moraliy inensiy a age is wih µ(τ, θ) = α + βe γτ, θ = α β. γ We need he derivaives of he inensiy w.r.. all hree parameers, µ(τ, θ) α = 1, β µ(τ, θ) = eγτ, γ µ(τ, θ) = β eγτ τ, and heir inegrals wih respec o ime, y µ(τ, θ) dτ = y x, x α y x β µ(τ, θ) dτ = eγy e γx, γ y ( e γy y e γx x µ(τ, θ) dτ = β x γ γ The MLE equaions (9.35) specialize o 1 = (y m x m), m; N m=1 ˆα + ˆβeˆγym m eˆγym = eˆγym eˆγxm, m; N m=1 ˆα + ˆβeˆγym ˆγ m eˆγym y m = ( e γy m y m e γxm x m m; N m=1 ˆα + ˆβeˆγym γ m eγy e γx ) γ 2. ) eγym e γxm γ 2. To find he informaion marix (9.38) we need he marix wih he producs of he derivaives, 1 e γτ βe γτ τ µ(τ, θ) µ(τ, θ) = e 2γτ βe 2γτ τ (11.42) θ θ β 2 e 2γτ τ 2 (symmeric) and he probabiliies p (m) (τ, θ), which are ( τ ) p (m) (τ, θ) = exp (α + βe γ s ) ds x m ( ) = exp α(τ x m) β eγτ e γxm, (11.43) γ for τ [x m, y m] (he survival probabiliy) and oherwise. (There is only one kind of ransiion, from o 1, and he summaion over g, h in he informaion marix (9.38) can be dropped.) We see ha all ingrediens in he asympoic variance marix are given by explici formulas, and i remains only o perform a numerical inegraion o find is value for given θ.

177 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL Confidence regions A. An asympoic confidence ellipsoid. From he asympoic normaliy of he MLE i follows ha (ˆθ θ) Σ 1 (θ)(ˆθ θ) as χ 2 s, (11.44) he chi-squared disribuion wih s degrees of freedom. Therefore, denoing he (1 ε)-fracile of his disribuion by χ 2 s, 1 ε, an asympoic 1 ε confidence region is he se of all θ saisfying (θ ˆθ) Σ 1 (θ)(θ ˆθ) χ 2 s, 1 ε. (11.45) The expression on he lef here will ypically be a complicaed funcion of θ, and i is in general no easy o find he values of θ ha saisfy he inequaliy and consiue a confidence region. Now, suppose Σ(θ) can be esimaed by some funcion of he daa, ˆΣ, and ha he esimaor is consisen in he sense ha Then i is easy o show ha also he relaion ˆΣΣ 1 (θ) I. (11.46) (θ ˆθ) ˆΣ 1 (θ ˆθ) χ 2 s, 1 ε (11.47) deermines an asympoic 1 ε confidence region. The relaion (11.47) defines an ellipsoid, which is a fairly simple geomeric figure and, as we shall see in he following paragraph, a convenien basis for deriving oher confidence saemens of ineres. A sraighforward way of consrucing ˆΣ would be o replace θ in Σ(θ) by he consisen esimaor ˆθ, ha is, pu ˆΣ = Σ(ˆθ). This works well if he enries in Σ(θ) are closed expressions in θ. Unforunaely, his is he case only in cerain simple siuaions, ypically when he sae space Z is small and he paern of ransiions is hierarchical. One example is he moraliy sudy wih parameric moraliy law, e.g. of G-M ype. In more complex siuaions we canno in general find closed formulas for he probabiliies p (m) g (τ) involved in Σ, even if he inensiies hemselves are simple parameric funcions. Then a differen consrucion is required. A simple device is o replace he p (m) g (τ) by heir empirical counerpars I g (m) (τ) and pu n m=1 p (m) g (τ) I g(τ). (11.48) B. Simulaneous confidence inervals. The confidence ellipsoid (11.47) can be resolved in simulaneous confidence inervals for all linear funcions of θ in he following way. The Schwarz inequaliy says ha for all vecors a and x in R s, a x a a x x, wih equaliy for a = cx. Thus, noing ha he quadraic form on he lef of (11.47) is (ˆΣ 1/2 (θ ˆθ)) (ˆΣ 1/2 (θ ˆθ)), he confidence saemen can be cas equivalenly as a ˆΣ 1/2 (θ ˆθ) a a χ 2 s, 1 ε, a. (11.49) Since ˆΣ is of full rank, he vecor ˆΣ 1/2 a ranges hrough all of R s as a ranges in R s. Thus, wriing a a = (ˆΣ 1/2 a) ˆΣ(ˆΣ 1/2 a), (11.49) is equivalen o a (θ ˆθ) a ˆΣa χ 2 s, 1 ε, a, ha is, a θ [a ˆθ χ 2 s, 1 ε a ˆΣa, a ˆθ + χ 2 s, 1 ε a ˆΣa], a. (11.5) The inervals in (11.5) are (asympoic) simulaneous confidence inervals for all linear funcions of θ in he sense ha he probabiliy is a leas 1 ε ha hey all hold rue.

178 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL177 C. Confidence band for he G-M moraliy inensiy. Reurning o he moraliy sudy example in Paragraph 11.2.C, le c be aken as known so ha he moraliy inensiy is a linear funcion of he unknown parameer θ = (α, β). The MLE is obained by solving he equaions (11.42) and (11.42), and he appropriae variance marix Σ is obained by invering he upper lef 2 2 block in he informaion marix defined by (11.41), (11.42), and (11.43). From (11.5) we obain simulaneous confidence inervals for all µ(τ) = α + βe γτ, consiuing a confidence band in he space of moraliy inensiy funcions; µ(τ) [ˆα + ˆβe γτ χ 2 2, 1 εˆστ, ˆα + ˆβe γτ χ 2 2, 1 ε ˆστ ], τ >, (11.51) where ( ) ˆσ τ = (1, e γτ 1 )ˆΣ e γτ More on simulaneous confidence inervals A. Simulaneous confidence inervals based on a confidence region. Le X X be some observaion(s) wih disribuion P θ, θ Θ, some s-dimensional se. A 1 ε confidence region for θ is a funcion C from X o he se of subses of Θ such ha he random se C = C(X) saisfies P θ {θ C} = 1 ε, θ Θ. From a confidence region we readily obain confidence inervals for he values g(θ) of all real-valued funcions g : Θ R as follows. For a fixed g, define he random variables g = inf θ C g(θ) and g = sup θ C g(θ). Clearly, θ C implies g g(θ) g for all g, hence Thus, by saing ha P θ {g g(θ) g, g} 1 ε, θ Θ. g(θ) [ g, g ], g, (11.52) all saemens hold rue simulaneously wih a probabiliy no less han 1 ε. In his sense he inervals in (11.52) are simulaneous confidence inervals wih (simulaneous) confidence level 1 ε. A pracical consequence is ha we are allowed o snoop around in he daa se, looking for possible ineresing effecs (wihin he model), and sill keep conrol of he probabiliy of making false saemens. B. Confidence ellipsoid for a normal mean; Scheffé inervals. Le ˆθ is an esimaor of an s-dimensional parameer vecor θ and assume ha ˆθ N(θ, Σ), wih Σ known. A 1 ε confidence region of θ is he ellipsoid defined by (3.2): C = {θ ; (θ ˆθ) Σ 1 (θ ˆθ) χ 2 s, 1 ε }. To consruc he confidence inerval (11.52) for a specific funcion g(θ), we are lef wih he mahemaical problem of finding he exrema of g over he ellipsoid, which may be a difficul ask. For linear funcions i is simple, however, and i goes by he he echnique in Paragraph 3B, which is due o Scheffé. The simulaneous confidence inervals for linear funcions a θ = s p=1 apθp are, wih a bi sloppy noaion, a θ a ˆθ ± σa χ 2 s, 1 ε, a Rs, where σa 2 = a Σ a is he variance of he poin esimaor a ˆθ.

179 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL178 C. Narrowing he confidence inervals. Generally speaking, in he presence of uncerainy, he price we have o pay for making many safe saemens is ha each individual saemen has o be vague. In our siuaion his general ruh akes a very manifes form: for a fixed confidence level 1 ε he lenghs of he confidence inervals increase wih he dimension of θ since χ 2 s, 1 ε is an increasing funcion of s (why?). We can gain precision in erms of lenghs of he inervals by reducing he number of saemens we wan o make. Suppose we are only ineresed in drawing inferences abou linear combinaions of r linearly independen funcions b jθ, j = 1,..., r, r < s. Thus, puing B = (b 1,..., b r), an s r marix, we are only ineresed in linear funcions a θ wih a = Bc for some r-vecor c, ha is, a R(B), he r-dimensional linear space spanned by he b j. Then, sar from B ˆθ N(B θ, B ΣB), and apply he resuls above o obain ha simulaneous confidence inervals for all linear funcions of B θ are given by or, equivalenly, c B θ c B ˆθ ± c B ΣBc χ 2 r, 1 ε, c Rr, (11.53) a θ a ˆθ ± σa χ 2 r, 1 ε, a R(B). I is seen ha, by reducing he dimension of our saemens from s o r, we have gained a reducion of he lenghs of he confidence inervals by a facor χ 2 r, 1 ε /χ2 s, 1 ε. B. Bonferroni inervals for a finie number of parameer funcions. Le g j (θ), j = 1,..., r, be a finie number of real-valued parameer funcions, and assume ha for each g j (θ) we have consruced an individual confidence inerval [ g j, g j ] wih level 1 ε j. Thus, denoing he even g j (θ) [g j, g j ] by A j, we have P θ (A j ) 1 ε j for each j. The probabiliy ha all A j hold rue a he same ime is P θ ( j A j ) = ( 1 P θ j A c ) j r 1 P θ (A c j ) j=1 r 1 ε j. I follows ha he inervals aken ogeher are simulaneous confidence inervals wih simulaneous confidence level no less han 1 ε, where ε = r j=1 ε j. This simple device, due o Bonferroni, represens an aracive alernaive o he approach in Paragraphs A C in siuaions where we ake ineres only in a finie number of parameer funcions. I urns ou ha he Bonferroni inervals ofen are shorer han he Scheffé inervals, which aim a an infinie number of parameer funcions. Bonferroni inervals wih simulaneous confidence level 1 ε for q linear funcions a jθ, j = 1,..., q, are j=1 a j θ a j ˆθ ± σ aj χ 2 1, 1 ε/q, (11.54) (Noe ha χ 2 is jus he (1 ε/2q)-fracile of he sandard normal disribuion, so 1, 1 ε/q we recognize (11.54) as he radiional individual 1 ε/q confidence inerval for a normal mean.) Le r ( s) be he dimension of he space spanned by he coefficien vecors a j. The corresponding Scheffé inervals based on (11.53) are a j θ a j ˆθ ± σ aj χ 2 r, 1 ε. The raio of he lengh of he inervals by he wo consrucions is B/S(q, r, ε) = χ 2 1, 1 ε/q /χ2 r, 1 ε. Clearly, he raio decreases wih r. I increases wih q, and (for r > 1) i sars from q = 1

180 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL179 wih a value smaller han 1 and ends o infiniy as q grows. There will be some value q(r, ε) such ha he raio is 1 for q q(r, ε). Inspecion of a able of he chi-square fraciles shows e.g. ha q(2,.25) = 5, q(4,.1) = 2, q(6,.25) = 5 (approximaely). E. Confidence inervals based on consisen and asympoically normal poin esimaors. The resuls and consideraions in he previous paragraphs carry over o he siuaion in Secion 3, where (3.4) formed he basis for simulaneous inference. Suppose we are ineresed in funcions of a se of parameer funcions f j (θ), j = 1,..., r, r < s. Pu f = (f 1,..., f r). If f is coninuously differeniable, firs order Taylor expansion gives f(ˆθ) as N ( f(θ), Σ f (θ)) ), wih and Σ f (θ) = Df(θ) Σ(θ) Df(θ), Df(θ) = θ f(θ), an s r marix. We obain he asympoic confidence ellipsoid C f = {f ; (f f(ˆθ)) ˆΣ 1 f (f f(ˆθ)) χ 2 r, 1 ε}, where ˆΣ f is some consisen esimaor of Σ f in he sense of (3.3), e.g. ˆΣf = Σ f (ˆθ). Asympoic Scheffé inervals for all funcions g(θ) = h(f(θ)), wih h real-valued and coninuously differeniable, are where g(θ) g(ˆθ) ± ˆσ g χ 2 r, 1 ε, ˆσ g = θ g(θ) Σ(θ) θ g(θ) θ=ˆθ. Asympoic Bonferroni inervals for a finie collecion of funcions g j, j = 1,..., q, are obained upon replacing r and ε wih 1 and ε/q. F. The G-M moraliy inensiy revisied. The confidence inervals (3.8) are infiniely many, so Bonferroni ideas canno help here. If also c is o be esimaed, we obain asympoic confidence inervals by he device above. The same goes for any funcion of acuarial relevance, like he reserve of a life insurance or a porfolio of insurances. Think of examples. Reurning o he moraliy sudy example in Paragraph 2C, le γ be aken as known so ha he moraliy inensiy is a linear funcion of he unknown parameer θ = (α, β). The MLE is obained by solving he eqnarrays (11.42) and (11.42), and he appropriae variance marix Σ is obained by invering he upper lef 2 2 block in he informaion marix defined by (11.41), (11.42), and (11.43). From (11.5) we obain simulaneous confidence inervals for all µ(τ) = α + βe γτ, consiuing a confidence band in he space of moraliy inensiy funcions; µ(τ) [ˆα + ˆβe γτ χ 2 2, 1 εˆστ, ˆα + ˆβe γτ χ 2 2, 1 ε ˆστ ], τ >, where ( ) ˆσ τ = (1, e γτ 1 )ˆΣ e γτ Piecewise consan inensiies A. Piecewise consan inensiies. Le = < 1 < < < r = be some finie pariion of he ime inerval [, ], and assume ha he inensiies are sep funcions of

181 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL18 he form µ gh (τ) = µ gh,q, τ [ q 1, q), q = 1,..., r, r = 1 [q 1, q)(τ)µ gh,q, (11.55) q=1 where he µ gh,q ake values in (, ), wih no relaionships beween hem. The siuaion fis ino he general framework wih θ = (..., µ gh,q,...), a vecor of (ypically high) dimension J J r. B. The MLE esimaors are O-E raes. The log likelihood in (11.34) now becomes ln Λ = r {ln µ gh,q N gh,q µ gh,q W g,q}, (11.56) g h q=1 where N gh,q = W g,q = q q 1 dn gh (τ), (11.57) q q 1 I g(τ)dτ, (11.58) are, respecively, he oal number of ransiions from sae g o sae h and he oal ime spen in sae g during he age inerval [ q 1, q). Since he µ gh,q are funcionally unrelaed, he log likelihood decomposes ino erms ha depend on one and only one of he basic parameers, and finding maximum amouns o maximizing each erm. The derivaives involved in he ML consrucion now become paricularly simple: ln Λ = 1 N gh,q W g,q, (11.59) µ gh,q µ gh,q I follows from (11.59) ha he MLE is 2 1 ln Λ = δ ghq,g µ gh,q µ h q g h,q µ 2 N gh,q. (11.6) gh,q ˆµ gh,q = N gh,q W g,q, (11.61) an O-E rae of he same kind as in he simple model of 11.2.B. Noing ha, by (11.57), q n E[N gh,q ] = µ gh,q p (m) g (τ)dτ, q 1 m=1 we obain from (11.6) ha he asympoic variance marix becomes µ gh,q Σ(θ) = Diag..., q,..., (11.62) g (τ)dτ q 1 n m=1 p(m) a diagonal marix, implying ha he esimaors of he µ gh,q are asympoically independen. An esimaor of Σ is obained upon replacing he parameer funcions appearing on he righ of (11.62) by heir sraighforward esimaors: pu µ gh,q ˆµ gh,q defined by (11.61) and, by he device (11.48), q n q p (m) g (τ)dτ I g(τ)dτ = W g,q, q 1 m=1 q 1 o obain ˆΣ = Diag (..., N gh,q W 2 g,q,... ). (11.63)

182 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL181 C. Smoohing O-E raes. The MLE of he inensiy funcion is obained upon insering he esimaors (11.61) in (11.55). The resuling funcion will ypically have a ragged appearance due o he esimaion error in a finie sample. This is unsaisfacory since he inensiies are expeced o be smooh funcions: for insance, here are a priori reasons o assume ha he moraliy inensiy is a coninuous and non-decreasing funcion of he age. Now, he very assumpion of piecewise consan inensiies is arificial, of course, and he esimaes obained under his assumpion canno serve as an ulimae answer in pracice. In fac, hey represen only he firs sep in a wo-sage procedure, where he second sep is o fi some smooh funcions o he raw esimaes delivered by he O-E raes. The funcions used for fiing consiue he model we have in mind. I may be objeced ha he wo-sage procedure is a deour since, if he inensiies are assumed o be funcions of a smaller se of parameers, one could follow he prescripion in Secion 11.2 and maximize he likelihood direcly. There are wo reasons why he wo-sage procedure never he less meris special reamen: in he firs place, he O-E raes and heir asympoic variance marix are easy o consruc; in he second place, a comparaive plo of he fied funcions and he O-E raes makes i possible o deec sysemaic deviaions beween model assumpions and facs. A commonly used fiing echnique is he so-called generalized leas squares mehod, which amouns o minimizing a posiive definie quadraic form in he deviaions beween he raw esimaes and he fiing funcions. In he following brief ouline of he procedure we focus on one given inensiy and drop he subscrips g, h. For each inerval [ q 1, q) choose a represenaive poin τ q, e.g. he inerval midpoin. Pu ˆµ = (..., ˆµ q,...), he vecor of O-E raes, and (wih a bi sloppy noaion) µ(θ) = (..., µ(τ q, θ),...), he vecor of rue values. Le A = (a pq) be some posiive definie marix of order r r. Esimae θ by θ minimizing (µ(θ) ˆµ) A(µ(θ) ˆµ) = pq a pq(µ(τ p, θ) ˆµ p)(µ(τ q, θ) ˆµ q). (11.64) hen If he inensiy is a linear funcion of θ (like in he G-M sudy wih known γ), µ(θ) = Y (τ)θ, (11.65) ˆθ = (Y AY ) 1 Y Aˆµ. (11.66) The asympoic variance of ˆθ is (Y AY ) 1 Y AΣ(θ)AY (Y AY ) 1. By he Gauss-Markov heorem i is minimized by aking A = Σ(θ) 1, and he minimum is (Y Σ(θ) 1 Y ) 1. Thus, asympoically he bes choice of A is ˆΣ 1, where ˆΣ is some esimae of Σ saisfying (11.46). Wrie Σ = Σ(θ). The symmeric, pd marix Σ has a symmeric pd square roo W such ha Σ = W 2. where = (Y AY ) 1 Y AΣAY (Y AY ) 1 (Y Σ 1 Y ) 1 = (Y AY ) 1 Y AW [ I W Y ((W Y ) (W Y )) 1 (W Y ) ] W AY (Y AY ) 1 = (Y AY ) 1 Y AW H W AY (Y AY ) 1. H = I P (P P ) 1 P. The marix H = I P (P P ) 1 P is symmeric, H = H, and idempoen, H 2 = H, Thus, H = H H and = ( HW AY (Y AY ) 1) ( HW AY (Y AY ) 1) which is indeed pd. More on analyic graduaion - he G-M example: Le us focus on one inensiy ha is o be graduaed and, o fix ideas, assume i is he moraliy inensiy in he simple model wih wo saes alive and dead.. The firs sep is o assume ha he inensiy is piece-wise consan: The log likelihood (9.53) is µ() = µ q, q 1 < q, j = 1, 2,... ln Λ = (ln µ qn q µ qw q), q

183 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL182 where N q and W q are, respecively, he number of deahs and he oal ime spen alive in he age inerval [q 1, q). Each µ q is a parameer which is funcionally unrelaed o all he ohers, so here are many parameers in his model! For insance, if we are ineresed in moraliy up o age 1 and have daa in he age range from o 1, here are 1 parameers, which is quie a lo. Remember, however, ha his model is jus a firs sep in a wo-sage procedure where he second sep is o graduae (smooh) he ML esimaors resuling from he presen naive model wih piece-wise consan inensiy. The ML esimaors are he occurrence-exposure raes, ˆµ q = Nq W q, which are well defined for all q such ha W q > (i.e. in age inervals where here were survivors exposed o risk of deah). The ˆµ q are asympoically (as n increases) normally disribued, muually independen, unbiased, and wih variances given by where he expeced exposure is n q E[W q] = p (m) (τ) dτ, m=1 q 1 σ 2 q = as.v[ˆµq] = µq E[W q], (11.67) p (m) (τ) being he probabiliy ha individual No. m is alive and under observaion a ime τ. The variance σq 2 is inversely proporional o he corresponding expeced exposure. In he presen simple model, wih only one inensiy of ransiion from he sae alive o he absorbing sae dead, we find explici expressions for he expeced exposure. For insance, suppose we have observed each individual life from birh unil deah or unil aained age 1, whichever occurs firs (i.e. censoring a age 1). Then, for τ [q 1, q) wih q = 1,..., 1, we have ( τ ) p (m) (τ) = exp µ(s) ds q 1 = exp µ p (τ (q 1)) µ q, (11.68) hence and p=1 q q 1 E[W q] = n exp µ p (τ (q 1)) µ q dτ q 1 p=1 q 1 = n exp µ p 1 exp ( µq), µ p=1 q σq 2 = 1 µ q ( n exp ). (11.69) q 1 p=1 µp (1 exp ( µ q)) You should look a oher censoring schemes and discuss he impac of censoring on he variance. Take e.g. he case where person No m eners a age z m and is observed unil deah or age 1, whichever occurs firs (all z m less han 1). Esimaors ˆσ 2 q of he variances are obained upon replacing he µ j in (G.55) by he esimaors ˆµ j. Simpler esimaors are obained by jus replacing µ q and E[W q] in (G.53) wih heir sraighforward empirical counerpars: ˆσ 2 q = ˆµ q/w q = N q/w 2 q. Now o graduaion. The occurrence-exposure raes will usually have a ragged appearance. Assuming ha he real underlying moraliy inensiy is a smooh funcion, we will herefore

184 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL183 fi a suiable funcion o he occurrence-exposure raes. Suppose we assume ha he rue moraliy rae is a Gomperz Makeham funcion, µ() = α + βe γ. Then, ake some represenaive age τ q (ypically τ q = q.5) in each age inerval and fi he parameers α, β, γ by minimizing a weighed sum of squared errors Q = q a q (ˆµ q α βe γτq ) 2. This is a maer of non-linear regression. The opimal weighs a q are he inverse of he variances, bu since hese are unknown, we plug in he esimaors and use a q = 1/ˆσ 2 q. The minimizing values α, β, and γ are obained by differeniaing Q wih respec o each of he hree parameers and seing he derivaives equal o. The derivaives are: α Q = q β Q = q γ Q = q a q2 (ˆµ q α βe γτq ) ( 1), a q2 (ˆµ q α βe γτq ) ( e γτq ), a q2 (ˆµ q α βe γτq ) ( βe γτq τ q). Thus α, β, and γ are he soluion o he equaions ( ) a q ˆµ q α β e γ τ q =, q ( ) a q ˆµ q α β e γ τ q e γ τ q =, q ( ) a q ˆµ q α β e γ τ q e γ τ q τ q =. q This is in general a se of non-linear equaions ha does no allow of an explici soluion. Acually hese equaions are jus as involved as he maximum likelihood equaions in Paragraph 9.2C above, which is disappoining since he wo-sage procedure considered here was supposed o be simpler. (Occurrence-exposure raes are easy o find and hey are asympoically independen, which makes i easy o find heir asympoic variances. The graduaion is, however, messy.) Suppose now ha γ is aken o be known. Then only he wo firs equaions above are relevan and hey reduce o a q α + a qe γτq β = a q ˆµ q, q q q a qe γ τ q α a qe 2γ τ q β = a q ˆµ qe γ τ q. q q q This is a linear sysem of equaions wih an explici soluion, which he indusrious reader cerainly will find Impac of he censoring scheme A. The precision of he esimaion. The precision of he MLE depends on he amoun of informaion provided by he censoring scheme of he sudy. Asympoically i is he variance marix Σ(θ) ha deermines everyhing, and in Secion 11.2 i was poined ou ha he size of his marix depends on he censoring scheme only hrough he funcions n m=1 p(m) g (τ), g =,..., J, he expeced numbers of individuals saying in each sae g a ime τ. (I depends also on he parameric srucure of he inensiies, of course.) We shall look a wo censoring schemes frequenly encounered in pracice.

185 CHAPTER 11. STATISTICAL INFERENCE IN THE MARKOV CHAIN MODEL184 B. Longiudinal observaion (cohor sudies). The erm cohor sems from Lain and originally signified a uni division in an ancien Roman legion. In demography i means a class of individuals born in a paricular year or more general period of ime (a generaion ), and a cohor sudy is one where a cohor is observed over a cerain period, possibly unil i is exinc. This was he siuaion in Paragraph 11.2.C. Thus, le he n Markov processes in he general se-up be sochasic replicaes, all commencing in sae a ime and hereafer observed coninuously hroughou he ime inerval [, ]. In his case n p (m) g (τ) = n p (1) g (τ), g =,..., J, and m=1 Σ(θ) = µ(τ, θ) µ(τ, θ)p(1) n g h µ(τ, θ) θ θ g (τ)dτ. This marix ends o as n increases if he inverse marix indicaed exiss. C. Cross-secional observaion. In a cross-secional sudy a populaion is observed over a cerain period of ime. As an example, suppose he G-M moraliy sudy in Paragraph 11.2.C is conduced cross-secionally hroughou a calendar period of duraion, and ha i comprises n individuals a ages (m), m = 1,..., n, a he beginning of he sudy. In his case he facor depending on he design in he informaion marix is n p (m) (τ) = m=1 ( n 1 [ (m), (m) + ] (τ) exp m=1 α(τ (m) ) β eγ(m) (e γτ 1) γ ).

186 Chaper 12 Heerogeneiy models 12.1 The noion of heerogeneiy a wo-sage model The life lengh T of an individual depends on a number of facors like biological inheriance (some are srong and healhy, ohers are weak and sickly), occupaion (mining and foresry have higher acciden raes han office work), leisure aciviies (mounain climbing is more wholesome bu also more dangerous han philaely), nuriion (see he weekly magazines for curren wisdom), smoking and drinking habis. The lis migh be exended endlessly. Le all such individual characerisics be represened by a parameer θ, which may be quie complex, possibly of large dimension comprising numbers and srings of leers. The dependence of T on hese characerisics is accouned for by leing he probabiliy law of T depend on θ. Thus, wrie F θ () for he probabiliy ha a person wih characerisics θ dies wihin age and F θ () for he corresponding survival probabiliy. Assuming ha F θ possesses a densiy f θ, he force of moraliy is µ θ () = f θ ()/ F θ (), and F ( θ () = exp ) µ θ(s) ds, F θ ( x) = ( exp x+ x ) µ θ (s) ds, and so on. Apparenly he aggregae moraliy law reaed in Secion *** conflics wih he myopic viewpoin aken here, bu his is really no so: he aggregae law describes he moraliy paern in he populaion as a whole and represens he prospecs of longeviy for a person when nohing is known as o his/her personal characerisics. To make his precise we le he individual characerisics of a randomly chosen newly born be a random elemen Θ wih a disribuion G, and he condiional disribuion of he life lengh for fixed Θ = θ is F θ. If he value of Θ is observed o be θ, hen F θ is he relevan basis for making predicions abou T. If Θ is no observed, hen predicions abou T can only be based on he uncondiional disribuion funcion F () = E[F Θ ()] = F θ () dg(θ) (12.1) or any of he equivalen funcions in (3.2) (3.4), which now assume he forms F () = F θ () dg(θ), (12.2) f() = f θ () dg(θ), (12.3) µ() = f()/ F () = fθ () dg(θ) Fθ () dg(θ) = µθ () F θ () dg(θ) Fθ () dg(θ) (12.4) = E[µ Θ () T > ]. (12.5) 185

187 CHAPTER 12. HETEROGENEITY MODELS 186 Formula ( 12.5 ) calls for a commen. The join disribuion of T and Θ is given by P [T A, Θ B] = f θ () d dg(θ) = F θ [A] dg(θ). B A B The marginal disribuion of T is given by P[T A] = F [A] = F θ [A] dg(θ), where he inegral sign wihou indicaion of he area signifies inegraion over he enire range of he variable. The condiional disribuion of Θ, given T A, is B G[B T A] = F θ[a] dg(θ) (12.6) Fθ [A] dg(θ) or, in erms of he generalized densiy, dg(θ T A) = F θ[a] dg(θ) Fθ [A] dg(θ ). (12.7) In paricular, he condiional disribuion of Θ, given T >, is given by dg(θ T > ) = F θ () dg(θ) Fθ () dg(θ ), (12.8) and he las expression in (12.4) is recognized as he expeced value of µ Θ () wih respec o his disribuion. Insering A = [, + d] and F θ [A] = f θ d in (12.7) gives (somewha informally) f θ dg(θ) dg(θ T = ) = fθ () dg(θ ). (12.9) The probabiliies in (12.7) have a sraighforward inerpreaion as proporions in a cohor of individuals born a he same ime. They canno in general be inerpreed as proporions in a given populaion, and he reason for his is ha no assumpions have been made as o he birh raes ha would govern he developmen of he age composiion of he populaion over ime. I is cerainly rue ha (12.8) is he disribuion of Θ amongs hose who, a a given momen, are exacly years old in he populaion. I is no he disribuion of Θ amongs hose who, a a given momen, are years or older (as he condiioning on T > migh sugges). Likewise, (12.9) is he disribuion of Θ amongs hose who are known o have died a age in he pas. The condiional probabiliy P[T A T A] may be expressed as F [A A] = F [A A] F [A] Fθ [A S] dg(θ) = F [A] Fθ [A A] F θ [A] dg(θ) = F θ [A] F [A] = F θ [A A] dg(θ T A). (12.1) Formula ( 12.1 ) says ha, when T A is given, he probabiliy of T A is o be formed in he usual way, by aking he average of he probabiliy of T A for fixed Θ over he disribuion of Θ, all disribuions being condiional on T A. In paricular, F ( x) = F θ ( x) dg(θ T > x), (12.11) which resembles (12.2), only ha he disribuions of T and Θ are updaed in regard of he informaion ha he person has survived o age x. Those who adhere o a deerminisic world picure would presumably fancy he idea ha T could be exacly deermined if only he individual and is surroundings could be sufficienly

188 CHAPTER 12. HETEROGENEITY MODELS 187 accuraely described down o he aoms. Then T would be jus a funcion T (Θ) of Θ, and F θ would reduce o a one-poin disribuion in T (θ) which need no longer be explicaed in he model since he moraliy law would simply be GT 1. The model formulaion (12.1) wih F θ a non-degenerae disribuion, expresses he poin of view ha somehing remains unexplained beyond Θ. Two inerpreaions are possible. Eiher ha here exiss such a hing as pure randomness in he world, or ha no all explanaory facors are included in Θ. Now, leaving such speculaions o he philosophers, le us pursue he chosen approach The proporional hazard model The perhaps simples means of describing moraliy variaions is he so-called proporional hazard model, which specifies ha Θ is a posiive random variable, and µ θ () = θµ (), (12.12) where µ () is some baseline force of moraliy. According o his assumpion, he moraliy paern is basically he same for all people, only ha some die faser han ohers a all ages. I does no allow for he possibiliy ha e.g. some people have moraliy above he average in he youh and below he average in he old age. Inroduce he accumulaed baseline inensiy a age, x W (x) = µ (s) ds. (12.13) Under he assumpion (12.11), he condiional survival funcion by fixed Θ = θ is and he condiional densiy is F θ () = e θw (), (12.14) f θ () = θµ () e θw (). (12.15) The funcions in (12.2) (12.5) become F () = e θw () dg(θ), (12.16) f() = µ () θ e θw () dg(θ), (12.17) θ e µ() = µ θw () dg(θ) () e θw () = µ () E(Θ T > ). (12.18) dg(θ) Alhough he uncondiional disribuion F is he only hing ha maers when Θ is unobservable, he presen wo-sage model is of ineres also in such circumsances. In he firs place i is of heoreical ineres due o is explanaory impor, and in he second place i is of pracical value since i produces candidaes for suiable aggregae laws. Assume now ha G is G γ,δ, he gamma disribuion wih shape parameer γ and scale parameer δ 1, whose densiy is g γ,δ (θ) = δγ Γ(γ) θγ 1 e θδ, θ >, (12.19) and γ, δ >. Here Γ is he gamma funcion Γ(γ) = γ 1 e d, which saisfies Γ(γ) = (γ 1) Γ(γ 1), γ > 1, (easy o prove by inegraion by pars) and Γ(1) = 1. I is immediaely seen ha ( E Θ r e Θs) = δ γ Γ(γ) Γ(r + γ), r > γ, s > δ, (s + δ) r+γ = δγ (r + γ 1) (r) (s + δ) r+γ, r =, 1,, s > δ. (12.2)

189 CHAPTER 12. HETEROGENEITY MODELS 188 In paricular, all momens of Θ exis, and E Θ = γ/δ, (12.21) V Θ = γ/δ 2. (12.22) The role of he scale parameer becomes clear by subsiuing θ = θδ in he inegral θ δ γ θδ G γ,δ (θ) = Γ(γ) θγ 1 e θδ 1 dθ = Γ(γ) θ γ 1 e θ dθ = G γ,1 (θδ), (12.23) which is an increasing funcion of δ. By use of ( 12.2 ), i is readily seen ha he presen case (12.16) (12.18) specialize o F () = ( δ ) γ, W () + δ (12.24) f() = µ δ γ γ (), (W () + δ) γ+1 (12.25) µ() = µ γ () W () + δ. (12.26) Observe ha ( ) P [W () > w] = F δ γ (W 1 (w)) =, w + δ ha is, W () + δ is Pareo-disribued wih parameers (δ, γ). Observe also ha µ() is µ () imes a facor ha decreases wih. Noice ha µ is no a force of moraliy in he uncondiional law. The survival funcion for an x year old is obained from ( ) as ( ) F ( x) = F (x + )/ F W (x) + δ γ (x) =, W (x + ) + δ which can be wrien as wih (12.27) ( ) δ(x) γ F ( x) = (12.28) W ( x) + δ(x) W ( x) = W ( + x) W (x) = µ (x + τ) dτ, δ(x) = W (x) + δ. Thus, he survival disribuion of an x-year old is of he same form as ha of a newly born, see ( ), bu wih updaed value of he scale parameer. Despie wha has been said abou Θ and G as elemens of a mere explanaory background of he surface eniies T and F, i may be of ineres o sudy he condiional disribuion ( 12.6 ) for some given aspec T A of he life lengh. As an example, ake he general proporional hazard model and focus aenion a (see ( 12.6 ) θ e θ W () dg(θ ) Ḡ(θ T > ) = P [Θ > θ T > ] = e θ W () dg(θ ). To sudy he dependence of his funcion on, differeniae: Ḡ(θ T > ) = e θ W () dg(θ ) e θ θ W () ( θ µ ()) dg(θ ) e θ W () ( θ µ ()) dg(θ ) e θ θ W () dg(θ ) { e θ W () dg(θ ) } (12.29) 2

190 CHAPTER 12. HETEROGENEITY MODELS 189 By inroducion of he probabiliy measure H given by ( ) can be cas as dh(θ) = e θw (x) dg(θ) e θ W () dg(θ ), µ () { E H (Θ 1[Θ > θ]) + E H Θ E H 1[Θ > θ]} = µ ()Cov H (Θ, 1[Θ > θ]), (12.3) where subscrip H signifies ha he operaors are formed by he disribuion H. Now, Θ and 1[Θ > θ] are boh increasing funcions of Θ, hence associaed and posiively correlaed. I follows ha he expression in ( 12.3 ) is negaive, hence ha he probabiliy G(θ T > ) is a decreasing funcion of and so RT D(Θ T ). Thus, old people are likely o have a small value of θ. Specialize now o he gamma case ( ) again. By ( 12.8), he condiional densiy of Θ, given X > x, is g γ,δ (θ T > ) = cθ γ 1 e θ(w ()+δ), (12.31) where c does no depend on θ. The consan c need no be calculaed since, by inspecion of (12.19) and ( ) i follows ha g γ,δ (θ T > ) = g γ,w ()+δ (θ). (12.32) Formula ( ) is also obained from ( ) upon insering F θ ( x) = exp[ θ{w (x + ) W (x)}] and, from ( ), dg(θ X > x) = g γ,w (x)+δ (θ) dθ and using ( 12.2 ). From a compuaional poin of view his is a deour, bu i uncovers he role of he hidden Θ in he play.

191 Chaper 13 Group life insurance 13.1 Basic characerisics of group insurance. A group insurance reay is an arrangemen whereby a group of persons is covered by a single conrac wih an insurer. In is broades conex, group insurance would include a variey of coverages, including life insurance, acciden insurance, healh insurance, annuiies, civil propery insurance, and ohers. The major ypes of groups eligible for group insurance are individual employer groups (he employees of a firm are covered by a conrac beween an insurer and he employer, ypically as a par of a labour managemen negoiaed employee welfare and securiy plan), muliple employer groups (he same as individual employer groups, excep ha he individual firm is exended o wo or more firms/employers, e.g. a rade associaion or an enire indusry), labour union groups (he members of a labour union are covered by a conrac issued direcly o he union), and credior debor groups (life and/or healh insurance is provided for debors hrough a conrac issued o he credior, e.g. a commercial bank; if he borrower dies or is disabled, benefis are paid o he lender o cancel he insured par of he deb). A grea variey of groups beyond he foregoing classificaions are covered by group insurance. Among such miscellaneous groups are associaions of public and privae employees, professional organizaions, fraernal socieies, and many ohers. When group insurance is conrased wih individual insurance, a number of characerisic feaures are eviden, In he firs place, he coverage is offered o all members of he group, usually wihou medical examinaion or oher evidence of individual insurabiliy. Thus, he crieriae by which individuals are recruied o a group are considered o provide a sufficien guaranee agains adverse selecion of high-risk individuals, so-called aniselecion agains he insurer. For insance, i is o be expeced ha he saff of an engineering workshop or publishing house or he membership of an associaion of lawyers or eachers has only a small infusion of impaired lives. To furher preclude he possibiliy of aniselecion, here are usually some requiremens peraining o he minimum number of persons needed o consiue a group and o he minimum proporion covered in he enire group. Anoher feaure characerisic of group insurance is ha he persons insured under a conrac are no paries o he conrac, since legally he conrac is beween he insurer and he policyholder (usually an employer or an organizaion). A hird characerisic of group insurance is ha i is essenially low-cos, mass proecion. Markeing and adminisraion coss are far below he level ypical of individual insurance. As a fourh characerisic, i should be poined ou ha group insurance conracs are of a coninuing naure, in ha he conrac and he plan may las long beyond he lifeime, or membership in he plan, of any one individual. New persons are added o he group from ime o ime, and ohers erminae heir coverage. The conrac is renewed regularly, ypically annually. Therefore, boh conrac erms and premium rae can be currenly adjused in 19

192 15.2. HAZARD MODEL FOR POLICY AND RECORDS 191 accordance wih he observed developmen of coss, risk condiions and oher circumsances influencing he economic resul. This feaure ses a difference of grea principal and pracical imporance beween group conracs and individual conracs in life insurance. An individual life insurance policy usually specifies premiums and benefis ha remain unalerned hroughou he conrac period, which may exend over several decades. Therefore, a subsanial safey loading is usually buil ino he individual premium o mee possible unfavourable fuure developmens of moraliy and expenses. In group life insurance here is no need for his kind of safey loading since he premium rae can be currenly adjused in accordance wih he experiences. The las remark poins direcly o he final feaure of group insurance o be menioned here. To save expenses, usually only very summary characerisics of he groups are observed and used as a basis for he raing of premiums a he ouse. Those risk characerisics ha are no observed may vary considerably beween he groups and give rise o subsanial risk differenials beween hem, despie ha hey appear o be similar. As an example, in group life insurance one may choose o observe only he number of persons insured under he plan of a group and leave oher characerisics such as occupaion and age composiion unobserved. If he groups differ considerably wih respec o hese unobserved risk characerisics, hey will have differen rue underlying risk premiums. These differenials will be refleced by he risk experiences of he individual groups as ime passes and claims saisics accrues. Thus, he individual claims record of a group provides some informaion on is risk profile, which could be aken ino accoun in he curren adjusmen of he premium. When he premium is regulaed his way for each group in regard of is claims experience, one speaks of experience raing. There is ye anoher poin of difference beween individual and group insurance which ough o be menioned because i explains why experience raing is widely used in group life insurance. If you should ask holders of individual life insurance policies if hey find he premiums reasonable, he answers would ypically be I guess so or I don know. They don know and don haggle over he price, simply because hey have no access o saisics from which hey could judge he fairness of he premiums. In group life insurance his is differen. Each maser conrac is managed by a policyholder who can compare premium paymens wih received benefis in he long run. Those policy holders who find ha premiums exceed by far he benefis, will sooner or laer call for a discoun (he ohers will remain silen). Therefore, a compeiive marke will end o enforce experience raing of groups life conracs A proporional hazard model for complee individual policy and claim records Consider a group life porfolio for which saisical records have been mainained during he period (τ, τ ), where τ is he presen momen. The porfolio comprises I maser conracs, labeled by i = 1,..., I. Le (τ i, τ i ) be he period during which conrac i has been in force (τ i < τ if he conrac has been erminaed in is enirey). Le J i be he number of persons currenly or formerly insured under he plan of conrac i. They are labeled by (i, j), j = 1,..., J i. For each individual (i, j) inroduce he following quaniies, which are observable by ime τ : τ ij, he ime of enry ino he group, x ij, he age a eny, T ij, he ime exposed o risk as insured before ime τ i K ij, he number of imes he coverage has been erminaed on an individual basis before M ij, ime τ i, he number of deahs as insured before ime τ i. The pairs (K ij, M ij ) can only assume he values (,), (1,), (,1) (implying ha paricipaion in he group will no be resumed once i has been erminaed). The hidden moraliy

193 15.2. HAZARD MODEL FOR POLICY AND RECORDS 192 characerisics of group i are represened by a laen quaniy Θ i. The T ij, K ij, M ij, and Θ i are viewed as random variables, and he following assumpions are made. (i) Variables belonging o differen groups are sochasically independen and Θ 1,..., Θ I are iid (independen and idenically disribued). (ii) Variables belonging o differen persons wihin one and he same group i are condiionally independen, given Θ i. (iii) All persons in one and he same group follow he same paern of moraliy and erminaion. More specifically, i is assumed ha he Θ i are posiive and ha, condiional on Θ i = θ i, a person who enered group i a age x and is sill a member of he group a age x + (x, > ), hen has a force of erminaion κ i (x, ) (13.1) and a force of moraliy of he form θ i µ(x, ). (13.2) Assumpion (i) corresponds o he idea ha he groups are independen random selecions from a populaion of groups ha are comparable, bu no enirely similar. I is his assumpion, in conjuncion wih (15.2), ha esablishes a relaionship beween he groups and forms he raionale of combining porfoliowide moraliy experience wih he moraliy experience of a given group in an assessmen of he moraliy in ha group. The proporional hazard assumpion (15.2) represens, perhaps, he simples possible way of modelling moraliy variaions beween groups. I saes ha he risk characerisics specific of a group ac on he force of moraliy only hrough a muliplicaive facor, implying ha he moraliy paern is basically he same for all groups. Such an assumpion is no ap for describing more complex moraliy differences, e.g. ha a group may have a moraliy below he average a early ages and above he average owards he end of life. For example, i is hinkable ha such hazardous occupaions as blas furnace operaion and mining arac only physically fi and healhy applicans and ha hose who are employed quickly ge worn ou by he severe working condiions. The saisical daa presenly available from group i are he individual enrance imes, τ ij, enrance ages, x ij, and hisories as insurees, O i = {(K ij, M ij, T ij ); j = 1,..., J i }. The condiional disribuion of (K ij, M ij, T ij ), when Θ i = θ i, is given by P {K ij =, M ij = 1, ij < T ij ij + d ij Θ i = θ i } [ ij ] = θ i µ(x ij, ij ) d ij exp {κ i (x ij, ) + θ i µ(x ij, )} d, < ij < τ i τ ij, P {K ij = 1, M ij =, ij < T ij ij + d ij Θ i = θ i } [ ij ] = κ i (x ij, ij ) d ij exp {κ i (x ij, ) + θ i µ(x ij, )} d, < ij < τ i τ ij,

194 15.2. HAZARD MODEL FOR POLICY AND RECORDS 193 P {K ij = M ij =, T ij = ij Θ i = θ i } [ ij ] = exp {κ i (x ij, ) + θ i µ(x ij, )} d, ij = τ i τ ij. From hese expressions we gaher he following formula for he condiional likelihood of O i : J i (κ i (x ij, T ij ) K ij {θ i µ(x ij, T ij )} M ij j=1 exp[ Tij {κ i (x ij, ) + θ i µ(x ij, )} d]), (13.3) (K ij, M ij ) {(, 1), (1, )} and < T ij < τ i τ ij or (K ij, M ij ) = (, ) and T ij = τ i τ ij, j = 1,..., J i. For each person (i, j) inroduce he cumulaive basic force of moraliy Tij W ij = µ(x ij, ) d. (13.4) I is seen from (15.3) ha a se of sufficien saisics for group i are M i W i = J i j=1 M ij, he oal number of deahs, (13.5) = J i j=1 W ij, he sum of cumulaive basic inensiies, (13.6) and ha he condiional likelihood, considered as a funcion of θ i, is proporional o θ M i i e θ iw i. (13.7) The expression in (15.7) is gamma shaped. Moivaed by he convenience of he analysis in he previous chaper, i is assumed ha he common disribuion of he laen Θ i is he gamma disribuion wih densiy g γ,δ (θ i ) = δγ Γ(γ) θγ 1 i e θ iδ, θ i >. (13.8) The condiional densiy of Θ i, given O, is proporional o he produc of he expressions in (15.7) and (15.8), hence By use of (14.25) (14.27), i follows ha g γ,δ (θ i O i ) = g Mi +γ,w i +δ(θ i ). (13.9) E(Θ i O i ) = M i + γ W i + δ, (13.1) V ar(θ i O i ) = M i + γ (W i + δ) 2, (13.11) ( ) E(e wθ Wi i + δ Mi +γ O i ) =, w > (W i + δ). (13.12) w + W i + δ The condiional mean in (15.1) is he Bayes esimaor Θ i (say) of Θ i wih respec o squared loss. I can be cas as

195 15.3. EXPERIENCE RATED NET PREMIUMS 194 where Θ i = ζ i ˆΘi + (1 ζ i )γ/δ, (13.13) ˆΘ i = M i /W i (13.14) is he maximum likelihood esimaor of θ i in he condiional model, given Θ i = θ i, and ζ i = W i /(W i + δ). (13.15) The expression in (15.13) is a weighed mean of he sample esimaor ˆΘ i and he uncondiional mean, EΘ i = γ/δ. The weigh ζ i aached o he experience of he group, is an increasing funcion of he exposure imes T ij, confer (15.4) Experience raed ne premiums The se of maser conracs in force a he presen momen is I = {i : τ i = τ }, and for each group i I he se of persons presenly covered under he plan of he group is J i = {j : τ ij + T ij = τ }. For each person (i, j) presenly insured le M ij and S ij denoe, respecively, he number of deahs and he sum payable by deah in he nex year, (τ, τ + 1). To preven echnicaliies from obscuring he main poins, disregard ineres and assume ha all he S ij will remain consan hroughou he year. For a group i I he ne annual premium based on he available informaion O i is P A i = S ij j J i The expeced values appearing in (15.16) are E(M ij O i). (13.16) E(M ij O i) = E{E(M ij Θ i, O i ) O i } 1 } = E[1 exp { Θ i µ(x ij, T ij + ) d O i ], which can be calculaed by formula (15.12). Defining and one finds Subsiuing (15.19) ino (15.16) yields 1 w ij = µ(x ij, T ij + ) d (13.17) ( ) Wi + δ Mi +γ Q i,w = 1, (13.18) w + W i + δ E(M ij O i) = Q i,w ij. (13.19)

196 15.4. THE FLUCTUATION RESERVE 195 P A i = S ij Q i,w j J i ij, (13.2) wih Q i,w defined by (15.17) and (15.18). ij As an alernaive o he premium (15.2), which is exac on an annual basis, one could use he insananeous ne premium per ime uni a ime τ, Pi I = lim E ij τ M ij ( τ) O i / τ, (13.21) j J i S where M ij ( τ) is he number of deahs of person (i, j) in he ime inerval (τ, τ + τ). Now, E{M ij i} = E[E{M ij i, O i } O i ] = E{Θ i µ(x ij, T ij ) τ + o( ) M i, W i } = µ(x ij, T ij ) Θ i + o( ), (13.22) he las passage being a consequence of (15.1) and (15.13). Combine (15.21) and (15.22), o obain Pi I = ij µ(x ij, T ij ) Θ i. (13.23) j J i S To see ha Pi I is an approximaion o P i A, apply he firs order Taylor expansion (1+x) α 1 αx o he second erm on he righ of (15.18) and hen approximae w ij in (15.17) by µ(x ij, T ij ), which gives Q i,w ij = 1 {1 + w ij /(W i + δ)} (M i+γ) 1 {1 (M i + γ)w ij /(W i + δ)} = w ij Θ i µ(x ij, T ij ) Θ i. (13.24) Using (15.25) in (15.2), yields Pi A Pi I. The approximaion is good if he w ij are << W i, which is he case for groups wih a reasonably large risk exposure in he pas, and if he µ(x ij, T ij + ) are nearly consan for < < The flucuaion reserve A measure of he uncerainy associaed wih he annual resul for group i is he condiional variance, Vi A = V ar S ij M ij O i) j J i = E S ij S ik M ij M ik O i (P i A )2 j,k J i = S 2 ij E(M ij O i) j J i +2 S ij S ike(m ij M ik O i) (Pi A )2 (13.25) j,k J i ;j<k

197 15.4. THE FLUCTUATION RESERVE 196 (M ij is equal o is square since i is or 1). The expeced values appearing in he second sum in (15.26) are E(M ij M ik O i) = E{E(M ij M ik Θ i, O i ) O i } = E [ {1 exp( Θ i w ij iw ik i = Q i,w ij i,w ik i,w ij ik (13.26) confer (15.12) and (15.18). Now, inser he expressions (15.19) and (15.26) ino (15.25) o obain Vi A = S ij 2 Q i,w + 2 S ij ij S ik (Q i,w + Q ij i,w Q i,w ik ) (P ij +w i A )2 ik j J i j,k J i ; j<k = 2 S ij P i A S ij 2 Q i,w ij j J i j J i 2 S ij S ik Q i,w (P ij +w i A )2, (13.27) ik j,k J i ; j<k ] where Q i,w ij and P A i are given by (15.17), (15.18), and (15.2). By use of he approximaion (15.24), V A i S ij 2 w j J i ij Θ i (P A i )2. (13.28) Charging each group i is ne premium Pi A would only secure expeced equivalence of premium incomes and benefi paymens for he porfolio as a whole. (A any ime groups wih low moraliy will subsidize hose wih high moraliy, bu as ime passes and risk experience accrues, hese ransfers will diminish: evenually each group will be charged is rue risk premium.) To mee unfavourable random flucuaions in he resuls, he company should provide a reserve for he enire porfolio. By approximaion o he normal disribuion, which is reasonable for a porfolio of some size, a flucuaion reserve given by F A = 2.33 V A i i I 1 2, (13.29) wih Vi A defined by (15.27) or (15.28), will be sufficien o cover claim expenses in excess of he oal ne premium, i I P i A, wih 99% probabiliy. To esablish he reserve in (15.29), i may be necessary o charge each group an iniial loading in addiion o he ne premium. Thereafer he reserve can be mainained by ransfer of surplus in years wih favourable resuls for he whole porfolio. (Charging he insurees a oal premium loading equal o F A each year is, of course, no necessary: ha would creae an unlawful profi on he par of he insurer.) The loadings can be deermined in several ways. One reasonable possibiliy is o le each group i conribue o F A by an amoun Fi A proporional o he sandard deviaion (Vi A ) 1 2, ha is, F A i = F A (V A i ) 1 2 / k I(V A k ) 1 2. (13.3) Upon erminaion of a maser conrac, he group should be credied wih he amoun (15.3).

198 15.5. ESTIMATION OF PARAMETERS Esimaion of parameers A ime τ he observaions ha can be uilized in parameer esimaion are O i, i = 1,..., I. I is, of course, only in his connecion ha he daa from erminaed maser conracs come ino play. Assume now ha he basic moraliy law is of Gomperz-Makeham ype and is aggregae, ha is, µ(x, ) = µ(x + ), where µ(y) = α + βc y, y >. (13.31) (In group life insurance here is acually no reason o expec selecional effecs since eligibiliy is no made condiional on he insuree s healh or oher individual risk characerisics.) From (15.3), (15.8) and (15.31) one gahers he following expression for he uncondiional likelihood of he daa: j i J i J i κ j=1 i (x ij, T ij ) K ij θm i i (α + βc x ij +T ij ) M ij j=1 j=1 Ji exp Tij J i Tij κ i (x ij, ) d θ i (α + βc xij+ ) d j=1 j=1 δ γ ] Γ(γ) θγ 1 i e δθ i dθ i. The forces of erminaion do no appear in any of he expressions for premiums and reserves, and so one can concenrae on he esimaion of γ, δ, α, β, c. The essenial par of he likelihood is J I i ( (α + βc x ij+t ij ) M ij δ γ ) I Γ(γ) i=1 j=1 I i=1 Γ(M i + γ) { α J i j=1 T ij + β } J i j=1 cx ij (c T Mi +γ. ij 1)/ ln c + δ The maximum likelihood esimaors γ, δ, α, β, c have o be deermined by numerical mehods, e.g. seepes ascen or Newon-Raphson echniques. Noe ha he number of parameers is essenially only four since a scale parameer in θ i can be absorbed in µ in (15.2). On should, herefore, pu γ = δ or α = 1 or β = 1.

199 Chaper 14 Haendorff and Thiele Haendorff s classical resul on zero means and uncorrelaedness of he losses creaed in disjoin ime inervals by a life insurance policy is an immediae consequence of he very definiion of he concep of loss. Thus, he resul is formulaed and proved here in a seing wih quie general paymens, discoun funcion, and ime inervals, all sochasic. A general represenaion is given for he variances of he losses. They are easy o compue when sufficien srucure is added o he model. The radiional coninuous ime Markov chain model is given special consideraion. A sochasic Thiele s differenial equaion is obained in a fairly general couning process framework Inroducion A. Shor review Haendorff s (1868) heorem saes ha he losses in differen years on a life insurance policy have zero means and are uncorrelaed, hence he variance of he oal loss is he sum of he variances of he per year losses. The loss in a year is defined as he ne ougoes (insurance benefis less premiums) during he year, plus he reserve ha has o be provided a he end of he year, and minus he reserve released a he beginning of he year, all quaniies discouned a ime. This classical resul has had is recen revival wih he adven of modern life insurance mahemaics based on he heory of sochasic processes, noable references being Gerber (1979, 1986), Papariandafylou and Waers (1984), Wolhuis (1987), and Ramlau- Hansen (1988). The firs hree of hese papers deal wih losses in fixed ime inervals bu, apar from ha, he proofs do no really res on any specific model assumpions. The wo las-menioned papers deal wih sochasic periods, namely he oal sojourn imes in differen saes in he framework of he coninuous ime Markov chain model, wih deerminisic conracual benefis and consan ineres rae. The proofs, hence apparenly also he resuls, depend on he paricular srucures of he model. B. Ouline of he presen paper I is shown here ha Haendorff s resuls are valid for virually any paymen sream and rule for accumulaion of ineres, no maer wha sochasic mechanisms govern hem, and for any naural ime periods, random or no. The crux of he maer is he very definiion of he losses: as poined ou by Gerber (1979), hey are he incremens of a maringale. The res follows from he opional sampling heorem. This general resul is esablished in Secion 2. I is only when i comes o formulas for he variances of he losses ha specific model assumpions are crucial. Secion 3 deals wih he siuaion where he evens upon which paymens are coningen are governed by a coninuous ime sochasic process wih finie sae space, essenially a mulivariae couning process. A represenaion heorem for maringales 198

200 CHAPTER 14. HATTENDORFF AND THIELE 199 adaped o a couning process is used o idenify he variance process ha generaes he variances of he losses. As a byproduc, Thiele s differenial equaion, well-known from he case wih Markov couning process and benefis depending only on he presen sae, urns ou o be a special case of a sochasic differenial equaion valid under far more general assumpions. The Markov case wih nonrandom benefis and discoun funcion, is simple enough o allow for easy compuaion of he variances. As an example of wha can be gained by he general formulaion of Haendorff s heorem, i is applied o he periods of ime spen upon he n-h visi o a cerain sae. Benefis depending on he pas developmen of he process can also be deal wih wihou grea difficulies. Relaxing he Markov assumpion is wha makes compuaion of variances cumbersome. For ease of reference, some elemens from he heory of maringales are gahered in he final Secion 4. They are aken as prerequisies hroughou The general Haendorff heorem A. Paymen sreams and discouned values A suiable framework for a general reamen of properies of paymen sreams is se in wo recen papers by he auhor (Norberg, 199, 1991). I is adoped here wih sufficien explanaion o make he presenaion selfconained. Consider a sream of paymens commencing a ime. I is defined by he paymen funcion A, which o each ime specifies he oal amoun A() paid in [, ]. Negaive paymens are allowed for; i is only required ha A be of bounded variaion in finie inervals and, by convenion, righ-coninuous. This essenially means ha A is a difference of wo non-decreasing, finie-valued, and righ-coninuous paymen funcions represening ougoes and incomes, respecively. When he discoun funcion v is used, he presen value a ime of he paymens is V = v da. (14.1) To allow for random developmen of paymens and ineres, he funcions A and v are assumed o be sochasic processes defined on some probabiliy space (Ω, F, P ). They are adaped o a righ-coninuous filraion F = {F }, each F comprising he evens ha govern he developmen of paymens and ineres up o and including ime. B. The noions of reserve and loss A any ime he insurer mus provide a reserve o mee fuure ne liabiliies on he policy. An adequae reserve would be he cash value a ime of fuure ougoes minus incomes, bu his quaniy is in general unobservable by ime. Wha can be enered in he accouns, is is expeced value given he informaion available by ime, he so-called (prospecive) F-reserve, V F () = 1 v() E ( ) v da F. (14.2) (, ) The F-reserve plays a cenral role in insurance pracice since i is aken as a facual liabiliy and, by saue, is o be accouned as a deb in he balance shee of he insurer a ime. In accordance wih his accounancy convenion he insurer s loss in he period (s, ] is defined as L (s,] = v da + v()v F () v(s)v F (s), (14.3) (s,] he expression on he righ being consrued as follows: a he end of he period he ne ougoes hroughou he period have been covered (he firs erm), he new reserve mus be provided (he second erm), and he reserve se aside prior o he period can be cashed (he hird erm). C. Haendorff s heorem generally saed. The basic circumsance underlying Haendorff s heorem and is generalizaions is he fac

201 CHAPTER 14. HATTENDORFF AND THIELE 2 ha he loss as defined by (14.3) is he incremen over (s, ] of he maringale generaed by he value V in (14.1). More precisely, assuming ha E V <, define for each M() = E (V F ) = v da + v()v F (), (14.4) [,] he second equaliy due o F-adapedness of A and v and he definiion (14.2). By inspecion of (14.3) and (14.4), L (s,] = M() M(s). (14.5) Clearly, {M()} is an F-maringale converging o V. I can always be aken o be righconinuous (see e.g Proer, 199, p.8) so ha he opional sampling propery and is consequences apply, confer Secion 4. A general Haendorff heorem is now obained by merely spelling ou some basic properies of maringales and sochasic inegrals quoed in Secion 4, in paricular (F.38), (F.39), and (14.1), wih sopping imes in he roles of he i. Thus, le = T < T 1 < be a nondecreasing sequence of F-sopping imes. This covers he simple case where he T i are fixed (ypically T i = i, he end of year No. i) and all pracically relevan cases where hey are random. Abbreviae L i = L (Ti 1,T i ], i = 1, 2,..., (14.6) and define consisenly he loss a ime as L = A() + V F () E V. (14.7) Theorem 1. Assume ha he value V in (14.1) has finie variance. Then he losses L i defined in (14.6) (14.7) have zero means and are uncorrelaed, and his is also rue condiionally: for i < j < k E (L j F Ti ) =, (14.8) Cov (L j, L k F Ti ) =, (14.9) hence Var L k F Ti = Var (L k F Ti ). (14.1) k=j k=j The variances in (14.1) are of he form Var (L j F Ti ) = E ( M (T j ) M (T j 1 ) F Ti ) (14.11) ( ) = E d M F Ti, (14.12) (T j 1,T j ] where M is he variance process defined by (14.9) in Secion 4. The resuls lised in he heorem are quie independen of he naure of he processes involved. They are rooed in he very definiion of he concep of loss, no in any paricular assumpions as o wha mechanisms hey sem from. There are Haendorff resuls for virually any paymen sream and rule for accumulaion of ineres, in insurance, life as well as non-life, and in finance in general. The periods may be any ime inervals delimied by sopping imes. The qualiaive par of he heorem, (14.8) (14.1), is done wih once and for all. The quaniaive par concerning he form of he variances, (14.11) (14.12) and (14.9), calls for furher examinaion of special models o find compuable expressions. The nex secion is devoed o he case where paymens are generaed by an insurance policy modelled by a mulivariae couning process.

202 CHAPTER 14. HATTENDORFF AND THIELE Applicaion o life insurance A. The general life insurance policy A life or pension insurance reay is ypically of he following general form. There is a se J = {,..., J} of possible saes of he policy. A any ime i is in one and only one of he saes in J, commencing in sae a ime, say. Le X() be he sae of he policy a ime. The developmen of he policy, {X()}, is a sochasic process. Regarded as a funcion from [, ) o J, i is assumed o be righ-coninuous, wih a finie number of jumps in any finie ime inerval, and X() =. For each j J, le I j () = 1[X() = j] be he indicaor of he even ha he process is in sae j a ime, and for each j, k wih j k le N jk () = {τ (, ]; X(τ ) = j, X(τ) = k} be he number of ransiions from sae j o sae k up o and including ime >. Define N jk () = for all j k. The processes X, (I j ) j J, and (N jk ) j k, j,k J correspond muually one-o-one and carry he same informaion, which is he filraion F = {F } generaed by {X()}. In paricular, since he process is in sae j a a given ime if and only if he number of enries ino j exceeds (necessarily by 1) he number of deparures from j by ha ime, I j () = δ j + k; k j (N kj () N jk ()). (14.1) I is assumed ha he process possesses inensiies. The inensiy of ransiion from sae j o sae k, j k, is denoed by λ jk () and is of he form λ jk () = I j ()µ jk (), (14.2) wih µ jk some F-adaped process. Insurance benefis are of wo kinds. In he firs place, he general life annuiy provides he amoun A j () A j (s) during a sojourn in sae j hroughou he ime inerval (s, ]. In he second place, he general assurance provides he lump sum a jk () immediaely upon a ransiion from sae j o sae k a ime. Insurance premiums are couned as negaive benefis. The conracual funcions A j and a jk are, respecively, a paymen funcion as defined in Paragraph 2A and a lef-coninuous, finie-valued funcion, boh predicable wih respec o F, which means ha benefis a any ime may depend on he pas developmen of he policy. The oal sream of paymens A is of he form da() = j I j () da j () + j k a jk () dn jk(). I is assumed hroughou ha he discoun funcion v is coninuous and deerminisic. In general he reserve V F () depends on he enire hisory F, bu i is convenien o suppress his in he noaion and visualize only he dependence on he presen sae of he process. Thus, henceforh wrie V j () for he reserve when X() = j. B. The variance process In he presen model he maringale in (14.4) akes he form M() = A () + (,] v(τ) j I j (τ)da j (τ) + a jk (τ) dn jk(τ) j k + j I j () v()v j () (14.3) Assume ha E( v da) 2 < so ha {M()} is square inegrable, ha is, sup E M 2 () < for all. Then a general represenaion heorem (see Bremaud, 1981, Secion III.3) says ha M is of he form

203 CHAPTER 14. HATTENDORFF AND THIELE 22 M() = M() + (,] j k where he H jk are some predicable processes saisfying H jk (τ) ( dn jk (τ) λ jk (τ)dτ ), (14.4) E Hjk 2 (τ) λ jk(τ)dτ <. (14.5) j k (,] Moreover, he variance process is given by d M () = j k H 2 jk () λ jk()d. (14.6) Thus, o find he variance process, i suffices o idenify hose funcions in (14.3) ha ake he roles of he H jk in (14.4). To simplify noaion, inroduce ã jk () = v() a jk (), (14.7) à j () = v(τ) da j (τ), (14.8) (,] W j () = v()v j (), (14.9) he las wo being righ-coninuous processes wih bounded variaion (define W j () = V j (+) for j ). Noe, in passing, ha he funcion à j + W j is coninuous since à j ( ) + W j( ) = à j ( ) + (à j () à j ( )) + W j() = à j () + W j(). In erms of he funcions in (14.7) (14.9) he relaion (14.3) is M() = A () + I j (τ)dã j (τ) + ã jk (τ) dn jk(τ) (,] j j k + j I j () W j (). (14.1) The las erm on he righ of (14.1) can be reshaped as follows. Inegraion by pars gives I j () W j () = I j () W j () + I j (τ) dw j (τ) + W j (τ )di j (τ). (,] (,] In he las erm here use (14.1) o wrie and obain di j () = k; k j d(n kj () N jk ()) I j () W j () = I j () W j () + I j (τ) dw j (τ) j j j (,] + W j (τ ) d(n kj (τ) N jk (τ)) j (,] k; k j = W () + I j (τ) dw j (τ) (,] j + (W k (τ ) W j (τ ))dn jk (τ). (,] j k

204 CHAPTER 14. HATTENDORFF AND THIELE 23 Upon insering his, (14.1) becomes M() = A () + W () + I j (τ)d(ã j (τ) + W j(τ)) j + (ã jk (τ) + W k(τ ) W j (τ )) dn jk (τ). (14.11) (,] j k Now, idenify he disconinuous pars on he righ of (14.4) and (14.11), o obain he following resul. Theorem 2. For any coninuous discoun funcion and any predicable conracual funcions such ha E( v da) 2 <, he variance process (14.6) is given by H jk () = ã jk () + W k( ) W j ( ), (14.12) wih he elemens on he righ defined by (14.7) and (14.9). The funcions H jk in (14.12) can be expressed as where H jk () = v()r jk (), (14.13) R jk () = a jk () + V k( ) V j ( ) (14.14) is he so-called sum a risk in respec of ransiion from sae j o sae k a ime. C. A sochasic Thiele s differenial equaion. Upon idenifying he coninuous pars of he expressions on he righ of (14.4) and (14.11) and recalling (14.2), he following resul is obained. Theorem 3. For any coninuous discoun funcion and any predicable conracual funcions such ha E( v da) 2 <, he ideniy I j () d(ã j () + W j()) + k; k j H jk () λ jk ()d = (14.15) holds almos surely, he elemens being defined by (14.7) (14.9) and (14.12). This is a generalizaion of he classical Thiele s differenial equaion, well-known from he case wih Markov couning process and deerminisic conracual funcions. The sochasic differenial equaion (14.15) is valid for any couning process possessing inensiies, and for any predicable benefi funcions, including lump sum survival benefis. For echnical purposes he compac form (14.15) is expedien. For ease of inerpreaion and comparison wih he radiional Thiele s equaion, an alernaive form is suiable. Assume ha he discoun funcion is of he form v() = exp( δ(τ)dτ), so ha δ() is he ineres inensiy a ime. Inser (14.8), (14.9), and (14.14) in (14.15), pu dw j () = dv()v j () + v()dv j () = v()δ()dv j () + v()dv j (), divide by v(), and rearrange a bi o obain I j () d( A j )() = I j() (dv j () V j ()δ()d) + k; k j R jk () λ jk ()d. (14.16)

205 CHAPTER 14. HATTENDORFF AND THIELE 24 On he lef of (14.16) is he premium (possibly negaive) paid in sae j during a small ime inerval around. The expression on he righ shows how i decomposes ino a savings premium (he firs erm), which provides he amoun needed for mainenance of he reserve in excess of he ineres i creaes, and a risk premium (he second erm), which covers he ougo relaed o ransiions from he curren sae (sum insured plus new reserve minus old reserve). D. Losses in a given sae In he presen se-up i is meaningful o speak of he oal loss L (j) in a cerain sae j. Le S n (j) and T n (j) denoe he (sopping) imes of arrival and deparure, respecively, by he n-h visi of he policy in sae j (hey are if no such visi akes place). Then L (j) = = = ( M(T (j) n ) ) M(S n (j) ) (14.17) n=1 dm(τ) (14.18) n=1 (S n (j),t n (j) ] I j (τ )dm(τ), he las equaliy due o he righ-coninuiy of X, by which {X( ) = j} = n=1 {S (j) n < T (j) n }. The funcion I j ( ) is lef-coninuous, hence predicable. Then, by a general maringale resul ((14.8) in Secion 4), Var L (j) = E (, ) I j(τ ) d M (τ). Insering (14.6) and recalling (14.2), gives Var L (j) = E I j (τ ) µ jk (τ) Hjk 2 (τ) dτ, (14.19) k; k j E. The Markov chain model revisied To obain feasible expressions for he variance in (14.19), more srucure mus be placed on he model. Thus, assume now ha he process {X()} is a ime-coninuous Markov chain, and denoe he ransiion probabiliies by p jk (, u) = P {X(u) = k X() = j}, u, j, k J. The µ jk appearing in (14.2) are now deerminisic funcions given by p jk (, u) µ jk () = lim. u u For he ime being he conracual funcions A j and a jk are aken o be deerminisic, ha is, he benefis depend only on he curren sae. Then he discouned reserves W j () are deerminisic funcions given by he inegral expressions W j () = v(τ) (, ) g p jg (, τ) da g(τ) + which are easy o compue. I follows ha (14.19) reduces o Var L (j) = p j (, τ) k; k j h; h g a gh (τ) µ gh(τ) dτ, (14.2) µ jk (τ) Hjk 2 (τ) dτ. (14.21)

206 CHAPTER 14. HATTENDORFF AND THIELE 25 The expression (14.21) was obained by Ramlau-Hansen (1988) by use of maringale heorems applied o he compensaed mulivariae Markov couning process in combinaion wih he classical Thiele s differenial equaion, implying ha he life annuiy benefis are absoluely coninuous. As for he qualiaive par of he Haendorff heorem, an examinaion of he proof indicaes ha he special model assumpions are redundan: a he sage where Thiele s differenial equaion is invoked, all erms peraining o inermediae ransiions beween visis o sae j vanish. I follows from Theorem 1 ha no only are he losses creaed in differen saes uncorrelaed, bu so are also he losses creaed in respec of differen visis o one and he same sae. The loss in connecion wih he n-h visi o sae j is L (j) n = M(T n (j) ) M(S n (j) ) = (S n (j),t n (j) ] dm(), he n-h erm on he righ of (14.17) or (14.18). (If here are fewer han n visis o sae j, hen S n (j) = T n (j) = by definiion and L (j) n = since E V < implies lim v da = almos surely.) Repeaing he argumen above, replacing I j ( ) wih 1[S n (j) < T n (j) ] (also a lef-coninuous funcion), one arrives a where Var L (j) n = p (n) j (, τ) µ jk (τ) Hjk 2 (τ) dτ, (14.22) k; k j p (n) j (, ) = P {S(j) n < T n (j) } is he probabiliy of sojourning in sae j for he n-h ime jus before ime. Summing (14.22) over n gives (14.21), a consequence of he general heorem. The resuls above carry over o visis wihin a fixed ime inerval (, u] since for any sopping ime T he runcaed imes (T ) u and (T u) are also sopping imes ( and form maximum and minimum, respecively); jus replace by u. For any sopping ime T he F T -condiional variances of losses in sae j afer ime T are obained upon replacing by T and p(n) j (, τ) by he appropriae condiional probabiliy. F. Compuaional problems in more complex models An immediae exension of he simple se-up in he previous paragraph, which does no desroy he compuabiliy of he variance of he loss, consiss in leing he conracual funcions A j and a jk depend on he pas developmen of he policy. The expressions may become messy, however. An ineresing issue is o sudy compuaional problems under non-markov assumpions, e.g. when he ransiion inensiies are allowed o depend on he duraion of he period ha has elapsed since he policy enered he curren sae. This would complicae maers immensely since inegraions would have o be performed over he imes of ransiions. In principle a numerical procedure can always be arranged. Anoher possibiliy would be o le he couning process be doubly sochasic. Finally, i should be menioned ha he discoun funcion and he conracual funcions could be made sochasic and independen of he developmen of he policy. Then compuaions would sill be feasible by he rule of ieraed expecaions. All he exensions indicaed here presen compuaional problems ha do no perain paricularly o Haendorff s and Thiele s resuls, and his is no he righ place o pursue such echnical issues Excerps from maringale heory A. Definiion of maringale Le (Ω, F, P ) be some probabiliy space. A family F = {F } of sub-sigma-algebras of F is

207 CHAPTER 14. HATTENDORFF AND THIELE 26 called a filraion if i is non-decreasing, ha is, F s F for s. The index represens ime, and F is some collecion of F-evens whose occurrence or non-occurrence is seled by ime. Thus, F represens a descripion of he evoluion of he inernal hisory of he phenomena encounered. I is assumed he F is righ-coninuous, which means ha F s = ;>sf. A sochasic process {M()} is said o be F-adaped if M() is measurable wih respec o F for each. I is an F-maringale if i is F-adaped, E M() exiss for all, and E (M() M(s) F s) = (14.1) for s. (Relaions involving random variables are undersood o hold almos surely.) B. Basic properies of square inegrable maringales In wha follows {M()} is assumed o be a square inegrable maringale, ha is, (14.1) holds and sup E M 2 () <. Le = < 1 < be some pariioning of [, ), and denoe he incremens of M over he inervals by j M = M( j ) M( j 1 ). (14.2) As sraighforward consequences of he maringale propery, he incremens have condiional zero means and covariances: for i < j < k, E ( j M F i ) = E ( E ( j M F j 1 ) F i ) =, (14.3) Cov ( j M, k M F i ) = E ( j M k M F i ) = E ( j M E( k M F k 1 ) F i ) =. (14.4) I follows ha, for i < j < l and for H any square inegrable F-adaped process, and l E H( k 1 ) k M F i k=j l = E H( k 1 )E( k M F k 1 ) F i k=j =, (14.5) l Var H( k 1 ) k M F i k=j l = E l H( k 1 )H( k 1)E ( k M k M F k 1 k 1 ) F i k=j k =j l = E H 2 ( k 1 )E ( ( k M) 2 ) F k 1 Fi. (14.6) k=j Inegral analogues of (14.5) and (14.6), obained by passing o he limi, are valid wih cerain qualificaions. Denoe by F he hisory as specified by F up o, bu no including, ime. I is a sub-sigma-algebra of F. Assume ha he process H is predicable in he sense ha H() is F -measurable for each. Lef-coninuiy cerainly implies predicabiliy. Now, le s = i < i+1 < < l = be a pariioning of [s, ]. Refine i indefiniely o obain from (14.5) ha

208 CHAPTER 14. HATTENDORFF AND THIELE 27 and from (14.6) ha E ( ) H dm F s =, (14.7) (s,] Var ( ) H dm F s = E (s,] ( ) H 2 d M F s, (14.8) (s,] where M is he so-called variance process of M defined (informally) by d M () = E ( (dm()) 2 F ). (14.9) As a special case, ake H() = 1 (j 1, j ](), a lef-coninuous funcion of. Then (14.8) reduces o Var ( j M F i ) = E ( M ( j ) M ( j 1 ) F i ) ( ) = E d M F i. (14.1) ( j 1, j ] C. Sopping imes and opional sampling So far he ime poins i have been aken as fixed. I urns ou ha all resuls saed above carry over o cerain random imes T i, now o be defined. A nonnegaive random variable T is an F-sopping ime if {T } F,, which means ha he informaion provided by he descripion of he hisory by F suffices o ascerain a any ime wheher or no T has occurred. The collecion of all evens ha are observable by ime T is denoed by F T. Formally, hese are he evens F F such ha F {T } F,. Obviously hey form a sigma-algebra. Doob s fundamenal opional sampling heorem saes ha (14.1) remains valid if s and are replaced by F-sopping imes S and T. More precisely, if {M()} is a righ-coninuous F-maringale, which converges o a random variable M( ) wih finie mean, hen E (M(T ) M(S) F S ) = (14.11) for S and T F-sopping imes such ha S T. Now, since all he resuls above res on he maringale propery (14.1), hey remain valid if he i are replaced by F-sopping imes T i in he presence of (14.11). By necessiy he presen brief survey is imprecise a some poins. Recommendable rigorous exposiions of maringale heory are e.g. he readable ex by Gihman and Skorohod (1979) or he recen one by Proer (199). Acknowledgemen My hanks are due o Chrisian Max Mller for many useful discussions during he progress of his work.

209 CHAPTER 14. HATTENDORFF AND THIELE 28 References Bremaud, P. (1981). Poin processes and queues. Springer-Verlag, New York, Heidelberg, Berlin. Gerber, H.U. (1979). An Inroducion o Mahemaical Risk Theory. Huebner Foundaion, Univ. of Pennsylvania, Philadelphia. Gerber, H.U. (1986). Lebensversicherungsmahemaik. Springer-Verlag, Berlin, Heidelberg, New York. Gihman, I.L. and Skorohod, A.V. (1979). The Theory of Sochasic Processes Vol. 3. Springer-Verlag, Heidelberg, New York. Haendorff, K. (1868). Das Risiko bei der Lebensversicherung. Masius Rundschau der Versicherungen 18, Norberg, R. (199). Paymen measures, ineres, and discouning an axiomaic approach wih applicaions o insurance. Scand. Acuarial J. 199, Norberg, R. (1991). Reserves in life and pension insurance. Scand. Acuarial J. 1991, Papariandafylou, A. and Waers, H.R. (1984). Maringales in life insurance. Scand. Acuarial J. 1984, Proer, P. (199). Sochasic Inegraion and Differenial Equaions. Springer- Verlag, Berlin, New York. Ramlau-Hansen, H. (1988). Haendorff s heorem: a Markov chain and couning process approach. Scand. Acuarial J. 1988, Wolhuis, H. (1987). Haendorff s heorem for a coninuous-ime Markov model. Scand. Acuarial J. 1987, Addendum o Haendorff s Theorem and Thiele s Differenial Equaion generalized Ragnar Norberg A. Purpose of his noe Any specific applicaion of he heory in Secion 3 of he paper would demand ha he saewise reserves V j be precisely defined. There is some laiude a his poin, however, and i urns ou ha Theorem 3 as saed may require ha an appropriae definiion be used. Paragraph B of he presen noe adds rigour on his issue. Paragraph C offers some guidance as o how o consruc and compue he reserves in nonrivial cases. Some echnical lemmas are placed in he final Paragraph D. Noaion and resuls from he background paper are used wihou furher explanaion (formula numbers in ha paper indicae secion and formula wihin he secion separaed by decimal poin, in conras o he ineger formula numbers inernal o he presen noe). We shall feel free also o employ cerain basic conceps and resuls in he heory of poin processes and heir maringales as presened in Paragraphs II.1-4 of Andersen & al. A number of saemens abou pahs of sochasic processes are acily undersood o be wih he qualificaion almos surely. We shall also frequenly use he fac ha we are dealing only wih processes of bounded variaion wihou explicily saying so. B. Supplemen o Secion 3

210 CHAPTER 14. HATTENDORFF AND THIELE 29 The hisory of he sae process hroughou he ime inerval (, ] is compleely described by he presen sae, X(), he ime of las enry ino he presen sae, S(), and he sric pas made up by he imes of (possible) previous ransiions and he sae arrived a upon each of hose ransiions (define he firs ransiion o ake place a ime, aking he process ino sae ). Therefore, fairly generally, he reserve (2.2) is of he form V F() = f(s(),, U(), X()), (14.12) where U() is some (finie-dimensional) predicable semimaringale. Now, (14.12) may be cas as V F() = j I j()f(s(),, U(), j) = j I j()f(s j(),, U(), j), (14.13) where S j() = I j()s() + (1 I j()) (14.14) is S() during sojourns in sae j and oherwise. The saewise reserves inroduced a he end of Paragraph 3A may convenienly be defined as V j() = f(s j(),, U(), j), (14.15) and, accordingly, W j() = v()v j(). Obviously, he definiion of S j() a imes when I j() = is immaerial. The choice (14.14) reflecs he fac ha, even if we canno predic imes of ransiion of he policy, we know jus prior o any ransiion wha S() will be and hence wha new value he reserve will assume upon he ransiion. The semimaringale S j() is coninuous a (he random) imes of ransiion ino sae j and also everywhere else excep a imes of ransiion ou of sae j, where i jumps from S() (< ) o. We are now in a posiion o explicae he argumens following (3.11) and leading o Theorems 2 and 3. Firs, noing ha k I k 1, pu I j()dw j() = I k ( )I j()(w j() W j( )) + I j( )I j()dw j() k;k j = dn kj ()v()(f(,, U(), j) f(,, U( ), j)) (14.16) k;k j +I j( )I j()dw j(). (14.17) where we have used ha, as a maer of definiion, S j(τ) = τ for all τ in he nondegenerae inerval [S( ), ] if he policy eners sae j a ime >. By Lemma 1 below (in Paragraph D), he predicable process f(,, U(), j) has no jumps coinciding wih jumps of he couning processes, and so he sum in (14.16) is. By Lemma 3 below, he erm in (14.17) is predicable (or, more presicely, he incremen of a predicable process). I follows ha I j()dw j() is predicable. Finally, by Lemma 2 below, I j()dã j () is predicable since dã j () is. Comparison of he predicable (bounded variaion) pars of (3.4) and (3.11) leads o Theorems 2 and 3.

211 CHAPTER 14. HATTENDORFF AND THIELE 21 Wih he presen definiion of he saewise reserves i makes no difference if we replace W k ( ) wih W k () in (3.12) (and V k () wih V k ( ) in (3.14)). This observaion seles he apparen disagreemen wih he resul (3.5) in Møller (1993). He considers he case where he conracual paymen funcions, he ransiion inensiies and, hence, he reserve depend only on and S() or, equivalenly, on and S() (he uses U() o denoe he laer). C. Consrucion of he reserve Usually Thiele s differenial equaions are mobilized when he reserve canno be pu up by a direc prospecive argumen, ypically when he paymens depend on he curren reserve. However, as long as he paymens and he ransiion inensiies do no depend on he pas, he saewise reserves remain funcions of only and can, herefore, be deermined by a se of simple differenial equaions. Real difficulies arise only when paymens and ransiion inensiies are allowed o depend on he pas in a more or less complex manner; hen he reserves will be funcions of several variables and i is raher obvious ha hey canno be deermined by jus a se of firs order ordinary differenial equaions. Le us look briefly a he case menioned a he balance of he previous paragraph, where paymens, inensiies, and hence he reserve depend on S() (and ). We seek he wo-dimensional saewise discouned reserve-funcions W j(s, ) (say), s n, j J, where n is he ime of expiry of he conrac. Condiioning on wheher or no here is a ransiion ou of sae j wihin ime n and, in case here is, he ime and he direcion of he firs such ransiion, we obain he inegral equaion W j(s, ) = n e τ µ j (s,u)du k;k j µ jk (s, τ)dτ (à j (s, τ) à j (s, ) + ã jk(s, τ) + W k (τ, τ)) (14.18) + e n µ j (s,u)du (à j (s, n) à j (s, )). Again, we see ha W k (τ, τ) can be replaced wih W k (τ, τ ) since he inegraion wih respec o dτ annihilaes he a mos counable number of differences beween hem. To deermine he reserve funcions, solve firs he W j(, ) from he inegral equaions wih s =, and hen solve he W j(s, ) from he general equaions. To obain a differenial form of he inegral equaion (14.18), add à j (s, ) and muliply by e µ j (s,u)du on boh sides, hen differeniae wih respec o, and finally divide by he common facor e µ j (s,u)du, which gives he following equivalen of (3.15): d (W j(s, ) + à j (s, )) = k;k j µ jk (s, )d(ã jk(s, ) + W k (, ) W j(s, )). (14.19) D. Some auxiliary resuls Lemma 1. If H is a predicable process, hen H(τ )dn jk (τ) = H(τ)dN jk (τ). (14.2)

212 CHAPTER 14. HATTENDORFF AND THIELE 211 Consequenly, H has no jumps in common wih he couning processes. Proof: Le M jk be he maringale defined by dm jk () = dn jk () λ jk ()d. The difference beween he wo inegrals in (14.2) is (H(τ) H(τ ))dn jk (τ) = (H(τ) H(τ ))λ jk (τ)dτ + (H(τ) H(τ ))dm jk (τ). The firs erm on he righ is zero because he inegrand differs from a mos a a counable se of poins. Likewise, he second erm is also because is variance is E (H(τ) H(τ )) 2 λ jk (τ)dτ. The las asserion is an obvious implicaion of he argumen above. Lemma 2. If A is a predicable semimaringale, hen so is also he process H defined by dh() = I j()da(). Proof: By he rule of inegraion by pars, we have he ideniies dh() = di j()a( ) + I j()da() = di j()a() + I j( )da(). Forming he difference beween he las wo expressions and using (3.1), gives I j()da() = I j( )da() + (A() A( ))(dn kj () dn jk ()). k;k j The firs erm on he righ is predicable and he second erm is by Lemma 1. Lemma 3. If G is a semimaringale, hen he process H defined by dh() = I j( )I j()dg() is predicable. Proof: Insering he Doob-Meyer decomposiion dg() = dg () + i k G ik ()(dn ik () λ ik ()d), wih G () and he G ik () predicable, and noing ha I j( )I j()dn ik () =, we ge dh() = I j( )I j()dg () i k G ik ()λ ik ()d. Here he firs erm is predicable by Lemma 2, and he second erm is cerainly predicable as i is coninuous.

213 Chaper 15 Financial mahemaics in insurance 15.1 Finance in insurance Finance was always an essenial par of insurance. Trivially, one migh say, because any business has o aend o is money affairs. However, for a leas wo reasons, insurance is no jus any business. In he firs place insurance producs are no physical goods or services, bu financial conracs wih obligaions relaed o uncerain fuure evens. Therefore, pricing is no jus a piece of accounancy involving he four basic arihmeical operaions, bu requires assessmen and managemen of risk by sophisicaed mahemaical models and mehods. In he second place, insurance conracs are more or less long erm (in life insurance for up o several decades), and hey are ypically paid in advance (hence he erm premium for price derived from French prime for firs ). Therefore, he insurance indusry is a major accumulaor of capial, and insurance companies especially pension funds are major insiuional invesors in oday s sociey. I follows ha he financial operaions (invesmen sraegy) of an insurance company may be as decisive of is revenues as is insurance operaions (design of producs, risk managemen, premium raing, procedures of claims assessmen, and he pure randomness in he claims process). Accordingly one speaks of asses risk or financial risk and liabiliy risk or insurance risk. We anicipae here ha financial risk may well be he more severe: Insurance risk creaed by random deviaions of individual claims from heir expeced is diversifiable in he sense ha, by he law of large numbers, i can be eliminaed in a sufficienly large insurance porfolio. This noion of diversificaion does no apply o caasrophe coverage and i does no accoun for he risk associaed wih long erm conracs insurance ic insurance risk Financial risk creaed random economic evens are by booms, recessions and, rare bu disasrous, crashes in he marke as a whole is no diversifiable; i is he uncerain evens are par of our he world hisory nonreplicable averaged ou in any meaningful sense. Financial risk creaed by he day-oday rises and falls of individual socks, is no diversifiable 212

214 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 213 on large random ups and downs is held o be indiversifiable since he enire porfolio is affeced by he developmen of he economy. On his background one may ask why insurance mahemaics radiionally ceners on measuremen and conrol of he insurance risk. The answer may parly be found in insiuional circumsances: The insurance indusry used o be heavily regulaed, solvency being he primary concern of he regulaory auhoriy. Possible adverse developmens of economic facors (e.g. inflaion, weak reurns on invesmen, low ineres raes, ec.) would be safeguarded agains by placing premiums on he safe side. The comforable surpluses, which would ypically accumulae under his regime, were redisribued as bonuses (dividends) o he policyholders only in arrears, afer ineres and oher financial parameers had been observed. Furhermore, he insurance indusry used o be separaed from oher forms of business and proeced from compeiion wihin iself, and severe resricions were placed on is invesmen operaions. In hese circumsances financial maers appeared o be somehing he radiional acuary did no need o worry abou. Anoher reason why insurance mahemaics used o be void of financial consideraions was, of course, he absence of a well developed heory for descripion and conrol of financial risk. All his has changed. Naional and insiuional borders have been downsized or eliminaed and regulaions have been liberalized: Mergers beween insurance companies and banks are now commonplace, new insurance producs are being creaed and pu on he marke virually every day, by insurance companies and oher financial insiuions as well, and wihou prior licencing by he supervisory auhoriy. The insurance companies of oday find hemselves placed on a fiercely compeiive marke. Many new producs are direcly linked o economic indices, like uni-linked life insurance and caasrophe derivaives. By so-called securiizaion also insurance risk can be pu on he marke and hus open new possibiliies of inviing invesors from ouside o paricipae in risk ha previously had o be shared solely beween he paricipans in he insurance insurance schemes. These developmens in pracical insurance coincide wih he adven of modern financial mahemaics, which has equipped he acuaries wih a well developed heory wihin which financial risk and insurance risk can be analyzed, quanified and conrolled. A new order of he day is hus se for he acuarial profession. The purpose of his chaper is o give a glimpse ino some basic ideas and resuls in modern financial mahemaics and o indicae by examples how hey may be applied o acuarial problems involving managemen of financial risk Prerequisies A. Probabiliy and expecaion. Taking basic measure heoreic probabiliy as a prerequisie, we represen he relevan par of he world and is uncerainies by a probabiliy space (Ω, F, P). Here Ω is he se of possible oucomes ω, F is a sigmaalgebra of subses of Ω represening he evens o which we wan o assign probabiliies, and P : F [, 1] is a probabiliy measure. A se A F such ha P[A] = is called a nullse, and a propery ha akes place in all of Ω, excep possibly on a nullse, is said o hold almos surely (a.s.). If more han one probabiliy measure are in play, we wrie nullse (P) and a.s. (P) whenever emphasis is needed. Two probabiliy measures P and P are said o be equivalen, wrien P P, if hey are defined on he same F and have he same nullses.

215 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 214 Le G be some sub-sigmaalgebra of F. We denoe he resricion of P o G by P G; P G[A] = P[A], A G. Noe ha also (Ω, G, P G) is a probabiliy space. A G-measurable random variable (r.v.) is a funcion X : Ω R such ha X 1 (B) G for all B R, he Borel ses in R. We wrie X G in shor. The expeced value of a r.v. X is he probabiliy-weighed average E[X] = X dp = X(ω) dp(ω), provided his inegral is well defined. Ω The condiional expeced value of X, given G, is he r.v. E[X G] G saisfying E{E[X G] Y } = E[XY ] (15.1) for each Y G such ha he expeced value on he righ exiss. I is unique up o nullses (P). To moivae (15.1), consider he special case when G = σ{b 1, B 2,...}, he sigma-algebra generaed by he F-measurable ses B 1, B 2,..., which form a pariion of Ω. Being G-measurable, E[X G] mus be of he form k b k1[b k ]. Puing his ogeher wih Y = 1[B j] ino he relaionship (15.1) we arrive a E[X G] = j 1[B j] B j X dp P[B j] as i ough o be. In paricular, aking X = 1[A], we find he condiional probabiliy P[A B] = P[A B]/P[B]. One easily verifies he rule of ieraed expecaions, which saes ha, for H G F, E { E[X G] H} = E[X H]. (15.2) F. Change of measure. If L is a r.v. such ha L a.s. (P) and E[L] = 1, we can define a probabiliy measure P on F by P[A] = L dp = E[1[A]L]. (15.3) If L > a.s. (P), hen P P. The expeced value of X w.r.. P is A, Ẽ[X] = E[XL] (15.4) if his inegral exiss; by he definiion (15.3), he relaion (15.4) is rue for indicaors, hence for simple funcions and, by passing o limis, i holds for measurable funcions. Spelling ou (15.4) as X d P = XL dp suggess he noaion d P = L dp or d P dp = L. (15.5) The funcion L is called he Radon-Nikodym derivaive of P w.r.. P. Condiional expecaion under P is formed by he rule To see his, observe ha, by definiion, Ẽ[X G] = E[XL G] E[L G]. (15.6) Ẽ{Ẽ[X G] Y } = Ẽ[XY ] (15.7)

216 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 215 for all Y G. The expression on he lef of (15.7) can be reshaped as The expression on he righ of (15.7) is E{Ẽ[X G] Y L} = E{Ẽ[X G] E[L G] Y }. E[XY L] = E{E[XL G] Y }. I follows ha (15.7) is rue for all Y G if and only if which is he same as (15.6). For X G we have showing ha Ẽ[X G] E[L G] = E[XL G], Ẽ G[X] = Ẽ[X] = E[XL] = E {X E[L G]} = EG {X E[L G]}, (15.8) d P G dp G = E[L G]. (15.9) B. Sochasic processes. To describe he evoluion of random phenomena over some ime inerval [, T ], we inroduce a family F = {F } T of sub-sigmaalgebras of F, where F represens he informaion available a ime. More precisely, F is he se of evens whose occurrence or non-occurrence can be ascerained by ime. If no informaion is ever sacrificed, we have F s F for s <. We hen say ha F is a filraion, and (Ω, F, F, P) is called a filered probabiliy space. A sochasic process is a family of r.v.-s, {X } T. I is said o be adaped o he filraion F if X F for each [, T ], ha is, a any ime he curren sae (and also he pas hisory) of he process is fully known if we are currenly provided wih he informaion F. An adaped process is said o be predicable if is value a any ime is enirely deermined by is hisory in he sric pas, loosely speaking. For our purposes i is sufficien o hink of predicable processes as being eiher lef-coninuous or deerminisic. C. Maringales. An adaped process X wih finie expecaion is a maringale if E[X F s] = X s for s <. The maringale propery depends boh on he filraion and on he probabiliy measure, and when hese need emphasis, we shall say ha X is maringale (F, P). The definiion says ha, on he average, a maringale is always expeced o remain on is curren level. One easily verifies ha, condiional on he presen informaion, a maringale has uncorrelaed fuure incremens. Any inegrable r.v. Y induces a maringale {X } defined by X = E[Y F ], a consequence of (15.2). Abbreviae P = P F, inroduce L = d P dp,

217 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 216 and pu L = L T. By (15.9) we have which is a maringale (F, P). L = E[L F ], (15.1) D. Couning processes. As he name suggess, a couning process is a sochasic process N = {N } T ha commences from zero (N = ) and hereafer increases by isolaed jumps of size 1 only. The naural filraion of N is F N = {F N } T, where F N = σ{n s; s } is he hisory of N by ime. This is he smalles filraion o which N is adaped. The sric pas hisory of N a ime is denoed by F. N An F N -predicable process {Λ } T is called a compensaor of N if he process M defined by M = N Λ (15.11) is a zero mean F N -maringale. If Λ is absoluely coninuous, ha is, of he form Λ = λ s ds, hen he process λ is called he inensiy of N. We may also define he inensiy informally by λ d = P [dn = 1 F ] = E [dn F ], and we someimes wrie he associaed maringale (15.11) in differenial form, dm = dn λ d. (15.12) A sochasic inegral w.r.. he maringale M is an F N -adaped process H of he form H = H + h s dm s, (15.13) where H is F N -measurable and h is an F N -predicable process such ha H is inegrable. The sochasic inegral is also a maringale. A fundamenal represenaion resul saes ha every F N maringale is a sochasic inegral w.r.. M. I follows ha every inegrable F N measurable r.v. is of he form (15.13). If H (1) = H (1) + h(1) s dm s and H (2) = H (2) + h(2) s dm s are sochasic inegrals wih finie variance, hen an easy heurisic calculaion shows ha and, in paricular, Cov[H (1) T, H(2) T F] = E [ T [ T ] Var[H T F ] = E h 2 sλ s ds F. h (1) s h (2) s λ s ds F ], (15.14) H (1) and H (2) are said o be orhogonal if hey have condiionally uncorrelaed incremens, ha is, he covariance in (15.14) is null. This is equivalen o saying ha H (1) H (2) is a maringale.

218 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 217 The inensiy is also called he infiniesimal characerisic if he couning process since i enirely deermines i probabilisic properies. If λ is deerminisic, hen N is a Poisson process. If λ depends only on N, hen N is a Markov process. A comprehensive exbook on couning processes in life hisory analysis is [3]. E. The Girsanov ransform. Girsanov s heorem is a celebraed one in sochasics, and i is basic in mahemaical finance. We formulae and prove he couning process variaion: Theorem (Girsanov). Le N be a couning process wih (F, P)-inensiy λ, and le λ be a given non-negaive F-adaped process such ha λ = if and only if λ =. Then here exiss a probabiliy measure P such ha P P and N has (F, P)-inensiy λ. The likelihood process (15.1) is ( L = exp (ln λ ) s ln λ s) dn s + (λ s λ s) ds. Proof: We shall give a consrucive proof, saring from a guessed L in (15.5). Since L mus be sricly posiive a.e. (P), a candidae would be L = L T, where ( ) L = exp φ s dn s + ψ s ds wih φ predicable and ψ adaped. In he firs place, L should be a maringale (F, P). By Iô s formula, dl = L ψ d + L (e φ 1) dn ( ) ) ) = L (ψ + e φ 1 λ d + L (e φ 1 dm. The represenaion resul (15.13) ells us ha o make L a maringale, we mus make he drif erm vanish, ha is, ) ψ = (1 e φ λ, (15.15) whereby ) dl = L (e φ 1 dm, In he second place, we wan o deermine φ such ha he process M given by d M = dn λ d (15.16) is a maringale (F, P). Thus, we should have Ẽ[ M F s] = M s or, by (15.6), [ ] E ML F s = E [L F M s. s] Using he maringale propery (15.1) of L, his is he same as [ ] E ML F s = M sl s

219 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 218 i.e. ML should be a maringale (F, P). Since d( M L ) = ( λ d)l + M (e φ 1)L ( λ d) ( + ( M + 1)L e φ M ) L ) dn ) ( = L d ( λ + e φ λ + ( M + 1)L e φ M ) L ) dm. we conclude ha he maringale propery is obained by choosing φ = ln λ ln λ. The mulivariae case goes in he same way; jus replace by vecor-valued processes A Markov chain financial marke - Inroducion A. Moivaion. The heory of diffusion processes, wih is wealh of powerful heorems and model variaions, is an indispensable oolki in modern financial mahemaics. The seminal papers of Black and Scholes [1] and Meron [32] were crafed wih Brownian moion, and so were mos of he almos counless papers on arbirage pricing heory and is bifurcaions ha followed over he pas quarer of a cenury. A main course of curren research, iniiaed by he maringale approach o arbirage pricing ([24] and [25]), aims a generalizaion and unificaion. Today he core of he maer is well undersood in a general semimaringale seing, see e.g. [13]. Anoher course of research invesigaes special models, in paricular various Levy moion alernaives o he Brownian driving process, see e.g. [18] and [4]. Pure jump processes have been widely used in finance, ranging from plain Poisson processes inroduced in [33] o quie general marked poin processes, see e.g. [8]. And, as a pedagogical exercise, he marke driven by a binomial process has been inensively sudied since i was launched in [12]. The presen paper underakes o sudy a financial marke driven by a coninuous ime homogeneous Markov chain. The idea was launched in [39] and reappeared in [19], he conex being limied o modelling of he spo rae of ineres. The purpose of he presen sudy is wo-fold: In he firs place, i is insrucive o see how well esablished heory urns ou in he framework of a general Markov chain marke. In he second place, i is worhwhile invesigaing he feasibiliy of he model from a heoreical as well as from a pracical poin of view. Poisson driven markes are accommodaed as special cases. B. Preliminaries: Noaion and some useful resuls. Vecors and marices are denoed by in bold leers, lower and upper case, respecively. They may be equipped wih opscrips indicaing dimensions, e.g. A n m has n rows and m columns. We may wrie A = (a jk ) k K j J o emphasize he ranges of he row index j and he column index k. The ranspose of A is denoed by A. Vecors are invariably aken o be of column ype, hence row vecors appear as ransposed. The ideniy marix is denoed by I, he vecor wih all enries equal o 1 is denoed by 1, and he vecor wih all enries equal o is denoed by. By D j=1,...,n(a j ), or jus D(a), is mean he diagonal marix wih he enries of a = (a 1,..., a n ) down he principal diagonal. The n-dimensional Euclidean space is denoed by R n, and he linear subspace spanned by he columns of A n m is denoed by R(A).

220 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 219 A diagonalizable square marix A n n can be represened as A = Φ D j=1,...,n(ρ j) Φ 1 = n ρ jφ j ψ j, (15.1) where he φ j are he columns of Φ n n and he ψ j are he rows of Φ 1. The ρ j are he eigenvalues of A, and φ j and ψ j are he corresponding righ and lef eigenvecors, respecively. Eigenvecors (righ or lef) corresponding o eigenvalues ha are disinguishable and non-null are muually orhogonal. These resuls can be looked up in e.g. [3]. The exponenial funcion of A n n is he n n marix defined by exp(a) = p= j=1 1 p! Ap = Φ D j=1,...,n(e ρ j ) Φ 1 = n e ρ j φ j ψ j, (15.2) where he las wo expressions follow from (15.1). The marix exp(a) has full rank. If Λ n n is posiive definie symmeric, hen ζ 1, ζ 2 Λ = ζ 1 Λζ 2 defines an inner produc on R n. The corresponding norm is given by ζ Λ = ζ, ζ 1/2. If Λ Fn m has full rank m ( n), hen he, Λ -projecion of ζ ono R(F) is where he projecion marix (or projecor) P F is j=1 ζ F = P Fζ, (15.3) P F = F(F ΛF) 1 F Λ. (15.4) The projecion of ζ ono he orhogonal complemen R(F) is ζ F = ζ ζ F = (I P F)ζ. Is squared lengh, which is he squared, Λ -disance from ζ o R(F), is ζ F 2 Λ = ζ 2 Λ ζ F 2 Λ = ζ Λ(I P F)ζ. (15.5) The cardinaliy of a se Y is denoed by Y. For a finie se i is jus is number of elemens The Markov chain marke A. The coninuous ime Markov chain. A he base of everyhing (alhough slumbering in he background) is some probabiliy space (Ω, F, P). Le {Y } be a coninuous ime Markov chain wih finie sae space Y = {1,..., n}. We assume ha i is ime homogeneous so ha he ransiion probabiliies p jk = P[Y s+ = k Y s = j] depend only on he lengh of he ransiion period. This implies ha he ransiion inensiies λ jk = lim p jk, (15.1)

221 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 22 j k, exis and are consan. To avoid repeiious reminders of he ype j, k Y, we reserve he indices j and k for saes in Y hroughou. We will frequenly refer o Y j = {k; λ jk > }, he se of saes ha are direcly accessible from sae j, and denoe he number of such saes by n j = Y j. Pu λ jj = λ j = k;k Y j λ jk (minus he oal inensiy of ransiion ou of sae j). We assume ha all saes inercommunicae so ha p jk > for all j, k (and > ). This implies ha n j > for all j (no absorbing saes). The marix of ransiion probabiliies, and he infiniesimal marix, P = (p jk ), Λ = (λ jk ), 1 are relaed by (F.41), which in marix form reads Λ = lim (P I), and by he backward and forward Kolmogorov differenial equaions, Under he side condiion P = I, (15.2) inegraes o In he represenaion (15.2), d P = PΛ = ΛP. (15.2) d P = Φ D j=1,...,n(e ρ j ) Φ 1 = P = exp(λ). (15.3) n e ρj φ j ψ j, (15.4) he firs (say) eigenvalue is ρ 1 =, and corresponding eigenvecors are φ 1 = 1 and ψ 1 = (p1,..., p n ) = lim (p j1,..., p jn ), he saionary disribuion of Y. The remaining eigenvalues, ρ 2,..., ρ n, are all sricly negaive so ha, by (15.4), he ransiion probabiliies converge exponenially o he saionary disribuion as increases. Inroduce j=1 I j = 1[Y = j], (15.5) he indicaor of he even ha Y is in sae j a ime, and N jk = {s; < s, Y s = j, Y s = k}, (15.6) he number of direc ransiions of Y from sae j o sae k Y j in he ime inerval (, ]. For k / Y j we define N jk. Taking Y o be righ-coninuous, he same goes for he indicaor processes I j and he couning processes N jk. As is seen from (15.5), (15.6), and he obvious relaionships Y = ji j, I j = I j + (N kj N jk ), j k;k j

222 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 221 he sae process, he indicaor processes, and he couning processes carry he same informaion, which a any ime is represened by he sigma-algebra F Y = σ{y s; s }. The corresponding filraion, denoed by F Y = {F Y }, is aken o saisfy he usual condiions of righ-coninuiy (F = u>f u) and compleeness (F conains all subses of P-nullses), and F is assumed o be he rivial (, Ω). This means, essenially, ha Y is righ-coninuous (hence he same goes for he I j and he N jk ) and ha Y deerminisic. The compensaed couning processes M jk, j k, defined by dm jk = dn jk I j λ jk d (15.7) and M jk =, are zero mean, square inegrable, muually orhogonal maringales w.r.. (F Y, P). We now urn o he subjec maer of our sudy and, referring o inroducory exs like [9] and [43], ake basic noions and resuls from arbirage pricing heory as prerequisies. B. The coninuous ime Markov chain marke. We consider a financial marke driven by he Markov chain described above. Thus, Y represens he sae of he economy a ime, F Y represens he informaion available abou he economic hisory by ime, and F Y represens he flow of such informaion over ime. In he marke here are m + 1 basic asses, which can be raded freely and fricionlessly (shor sales are allowed, and here are no ransacion coss). A special role is played by asse No., which is a locally risk-free bank accoun wih sae-dependen ineres rae r = r Y = I j r j, j where he sae-wise ineres raes r j, j = 1,..., n, are consans. process is ( ) ( ) B = exp r s ds = exp r j Is j ds, j wih dynamics db = B r d = B r j I j d. j Thus, is price (Seing B = 1 a jus a maer of convenion.) The remaining m asses, henceforh referred o as socks, are risky, wih price processes of he form S i = exp j α ij Is j ds + j β ijk N jk k Y j, (15.8) i = 1,..., m, where he α ij and β ijk are consans and, for each i, a leas one of he β ijk is non-null. Thus, in addiion o yielding sae-dependen reurns of he same form as he bank accoun, sock No. i makes a price jump of relaive size ( γ ijk = exp β ijk) 1

223 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 222 upon any ransiion of he economy from sae j o sae k. By he general Iô s formula, is dynamics is given by ds i = S i α ij I j d + γ ijk dn jk. (15.9) j j k Y j Taking he bank accoun as numeraire, we inroduce he discouned sock prices S i = S/B i, i =,..., m. (The discouned price of he bank accoun is B 1, which is cerainly a maringale under any measure). The discouned sock prices are S i = exp (α ij r j ) j I j s ds + j β ijk N jk k Y j, (15.1) wih dynamics d S i = S i j (α ij r j )I j d + j γ ijk dn jk, (15.11) k Y j i = 1,..., m. We sress ha he heory we are going o develop does no aim a explaining how he prices of he basic asses emerge from supply and demand, business cycles, invesmen climae, or whaever; hey are exogenously given basic eniies. (And God said le here be ligh, and here was ligh, and he said le here also be hese prices.) The purpose of he heory is o derive principles for consisen pricing of financial conracs, derivaives, or claims in a given marke. C. Porfolios. A dynamic porfolio or invesmen sraegy is an m + 1-dimensional sochasic process θ = (η, ξ ), where η represens he number of unis of he bank accoun held a ime, and he i-h enry in ξ = (ξ 1,..., ξ m ) represens he number of unis of sock No. i held a ime. As i will urn ou, he bank accoun and he socks will appear o play differen pars in he show, he laer being he more visible. I is, herefore, convenien o cosume he wo ypes of asses and heir corresponding porfolio enries accordingly. To save noaion, however, i is useful also o work wih double noaion θ = (θ,..., θ m ), wih θ = η, θ i = ξ i, i = 1,..., m, and work wih S = (S,..., S m ), S = B. The porfolio θ is adaped o F Y (he invesor canno see ino he fuure), and he shares of socks, ξ, mus also be F Y -predicable (he invesor canno, e.g. upon a sudden crash of he sock marke, escape losses by selling socks a prices quoed jus before and hurry he money over o he locally risk-free bank accoun.)

224 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 223 The value of he porfolio a ime is V θ = η B + m ξs i i = η B + ξ S = θ S i= Henceforh we will mainly work wih discouned prices and values and, in accordance wih (15.1), equip heir symbols wih a ilde. The discouned value of he porfolio a ime is Ṽ θ = η + ξ S = θ S. (15.12) The sraegy θ is self-financing (SF) if dv θ = θ ds or, equivalenly, dṽ θ = θ d S = m ξ i d S i. (15.13) We explain he las sep: Pu Y = B 1, a coninuous process. The dynamics of he discouned prices S = Y S is hen d S = dy S + Y ds. Thus, for Ṽ θ = Y V θ, we have dṽ θ = dy V θ i=1 + Y dv θ = dy θ S + Y θ ds = θ (dy S + Y ds ) = θ d S, hence he propery of being self-financing is preserved under discouning. The SF propery says ha, afer he iniial invesmen of V θ, no furher invesmen inflow or dividend ouflow is allowed. In inegral form: Ṽ θ = Ṽ θ + θ s d S s = Ṽ θ + ξ s d S s. (15.14) Obviously, a consan porfolio θ is SF; is discouned value process is Ṽ θ = θ S, hence (15.13) is saisfied. More generally, for a coninuous porfolio θ we would have dṽ (θ) = dθ S + θ d S, and he self-financing condiion would be equivalen o he a budge consrain dθ S =, which says ha any purchase of asses mus be financed by a sale of some oher asses. We urge o say ha we shall ypically be dealing wih porfolios ha are no coninuous and, in fac, no even righ-coninuous so ha dθ is meaningless (inegrals wih respec o he process θ are no well defined). D. Absence of arbirage. An SF porfolio θ is called an arbirage if, for some >, or, equivalenly, V θ < and V θ a.s. P, Ṽ θ < and Ṽ θ a.s. P. A basic requiremen on a well-funcioning marke is he absence of arbirage. The assumpion of no arbirage, which appears very modes, has surprisingly far-reaching consequences as we shall see. A maringale measure is any probabiliy measure P ha is equivalen o P and such ha he discouned asse prices S are maringales (F, P). The fundamenal heorem of arbirage pricing says: If here exiss a maringale measure, hen here is

225 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 224 no arbirage. This resul follows from easy calculaions saring from (15.14): Forming expecaion Ẽ under P and using he maringale propery of S under P, we find E[Ṽ θ ] = Ṽ θ + E[ ξ s d S s] = Ṽ θ (he sochasic inegral is a maringale). I is seen ha arbirage is impossible. We reurn now o our special Markov chain driven marke. Le Λ = ( λ jk ) be an infiniesimal marix ha is equivalen o Λ in he sense ha λ jk = if and only if λ jk =. By Girsanov s heorem, here exiss a measure P, equivalen o P, under which Y is a Markov chain wih infiniesimal marix Λ. Consequenly, he processes M jk, j = 1,..., n, k Y j, defined by d M jk = dn jk I j λ jk d, (15.15) jk and M =, are zero mean, muually orhogonal maringales w.r.. (F Y, P). Rewrie (15.11) as d S i = S i α ij r j + I j d + γ ijk jk d M, (15.16) j j k Y j k Y j γ ijk λjk i = 1,..., m. The discouned sock prices are maringales w.r.. (F Y, P) if and only if he drif erms on he righ vanish, ha is, α ij r j + k Y j γ ijk λjk =, (15.17) j = 1,..., n, i = 1,..., m. From general heory i is known ha he exisence of such an equivalen maringale measure P implies absence of arbirage. The relaion (15.17) can be cas in marix form as j = 1,..., n, where 1 is m 1 and α j = r j 1 α j = Γ j λj, (15.18) ( α ij), Γ j = (γ ijk) k Y j ( λjk ), λj =. i=1,...,m i=1,...,m k Y j The exisence of an equivalen maringale measure is equivalen o he exisence of a soluion λ j o (15.18) wih all enries sricly posiive. Thus, he marke is arbiragefree if (and we can show only if) for each j, r j 1 α j is in he inerior of he convex cone of he columns of Γ j. Assume henceforh ha he marke is arbirage-free so ha (15.16) reduces o d S i = S i j k Y j γ ijk d M jk, (15.19) where he M jk defined by (15.15) are maringales w.r.. (F Y, P) for some measure P ha is equivalen o P.

226 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 225 Insering (15.19) ino (15.13), we find ha θ is SF if and only if dṽ θ = j k Y j i=1 m ξ i S γ i ijk d implying ha Ṽ θ is a maringale w.r.. (F Y, P) and, in paricular, M jk, (15.2) Ṽ θ = Ẽ[Ṽ θ T F ]. (15.21) Here Ẽ denoes expecaion under P. (Noe ha he ilde, which in he firs place was inroduced o disinguish discouned values from he nominal ones, is also aached o he equivalen maringale measure and cerain relaed eniies. This usage is moivaed by he fac ha he maringale measure arises from he discouned basic price processes, roughly speaking.) E. Aainabiliy. A T -claim is a conracual paymen due a ime T. Formally, i is an FT Y -measurable random variable H wih finie expeced value. The claim is aainable if i can be perfecly duplicaed by some SF porfolio θ, ha is, Ṽ θ T = H. (15.22) If an aainable claim should be raded in he marke, hen is price mus a any ime be equal o he value of he duplicaing porfolio in order o avoid arbirage. Thus, denoing he price process by π and, recalling (15.21) and (15.22), we have π = Ṽ θ = Ẽ[ H F ], (15.23) or [ π = Ẽ e T r H ] F. (15.24) By (15.23) and (15.2), he dynamics of he discouned price process of an aainable claim is d π = m ξ i S γ i ijk d M jk. (15.25) j i=1 k Y j F. Compleeness. Any T -claim H as defined above can be represened as H = Ẽ[ H] + T j ζ jk k Y j d M jk, (15.26) where he ζ jk are F Y -predicable and inegrable processes. Conversely, any random variable of he form (15.26) is, of course, a T -claim. By virue of (15.22), and (15.2), aainabiliy of H means ha H = Ṽ θ + = Ṽ θ + T T dṽ θ ξ i S γ i ijk d j k Y j i M jk. (15.27)

227 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 226 Comparing (15.26) and (15.27), we see ha H is aainable iff here exis predicable processes ξ 1,..., ξ m such ha m i=1 ξ i S i γ ijk = ζ jk, for all j and k Y j. This means ha he n j -vecor ζ j = (ζjk ) k Y j is in R(Γ j ). The marke is complee if every T -claim is aainable, ha is, if every n j -vecor is in R(Γ j ). This is he case if and only if rank(γ j ) = n j, which can be fulfilled for each j only if m max j n j Arbirage-pricing of derivaives in a complee marke A. Differenial equaions for he arbirage-free price. Assume ha he marke is arbirage-free and complee so ha prices of T -claims are uniquely given by (15.23) or (15.24). Le us for he ime being consider a T -claim of he form H = h(y T, S l T ). (15.1) Examples are a European call opion on sock No. l defined by H = (S l T K) +, a caple defined by H = (r T g) + = (r Y T g) +, and a zero coupon T -bond defined by H = 1. For any claim of he form (15.1) he relevan sae variables involved in he condiional expecaion (15.24) are, Y, S l, hence π is of he form π = n I j f j (, S l ), (15.2) j=1 where he [ f j (, s) = Ẽ e T r H ] Y = j, S l = s (15.3) are he sae-wise price funcions. The discouned price (15.23) is a maringale w.r.. (F Y, P). Assume ha he funcions f j (, s) are coninuously difereniable. Using Iô on we find d π = e r j +e r j I j π = e r n I j f j (, S l ), (15.4) j=1 ( r j f j (, S l ) + f j (, S l ) + s ) f j (, S l )S l α lj d k Y j ( f k (, S l (1 + γ ljk )) f j (, S l ) ) dn jk

228 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 227 = e r j + I j ( r j f j (, S l ) + f j (, S l ) + s f j (, S l )S l α lj ) {f k (, S (1 l + γ ljk )) f j (, S )} λ l jk d k Y j +e ( ) r f k (, S (1 l + γ ljk )) f j (, S ) l d j k Y j M jk. (15.5) By he maringale propery, he drif erm mus vanish, and we arrive a he nonsochasic parial differenial equaions r j f j (, s) + f j (, s) + s f j (, s)sα lj + ( f k (, s(1 + γ ljk )) f j (, s)) λjk =, (15.6) k Y j j = 1,..., n, which are o be solved subjec o he side condiions j = 1,..., n. In marix form, wih f j (T, s) = h(j, s), (15.7) R = D j=1,...,n(r j ), A l = D j=1,...,n(α lj ), and oher symbols (hopefully) self-explaining, he differenial equaions and he side condiions are Rf(, s) + f(, s) + sal s f(, s) + Λf(, s(1 + γ)) =, (15.8) f(t, s) = h(s). (15.9) B. Idenifying he sraegy. Once we have deermined he soluion f j (, s), j = 1,..., n, he price process is known and given by (15.2). The duplicaing SF sraegy can be obained as follows. Seing he drif erm o in (15.5), we find he dynamics of he discouned price; d π = e r j ( ) f k (, S (1 l + γ ljk )) f j (, S ) l d k Y j M jk. (15.1) Idenifying he coefficiens in (15.1) wih hose in (15.25), we obain, for each sae j, he equaions m ξs i γ i ijk = f k (, S (1 l + γ ljk )) f j (, S ) l, (15.11) i=1 k Y j. The soluion ξ j = (ξ i,j ) i=1,...,m (say) cerainly exiss since rank(γ j ) m, and i is unique iff rank(γ j ) = m. Furhermore, i is a funcion of and S and is hus predicable. This simplisic argumen works on he open inervals beween he jumps of he process Y, where d M jk = I j λ jk d. For he dynamics (15.1) and (15.25) o

229 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 228 be he same also a jump imes, he coefficiens mus clearly be lef-coninuous. We conclude ha n ξ = I ξ j, which is predicable. Finally, η is deermined upon combining (15.12), (15.23), and (15.4): ( ) η = e n m r I j f j (, S l ) I j ξ i,j S i. j=1 j=1 i=1 C. The Asian opion. As an example of a pah-dependen claim, le us consider an Asian opion, which ( T +, essenially is a T -claim of he form H = Sl τ dτ K) where K. The price process is [ ( π = Ẽ e T + ] T r Sτ l dτ K) F Y = n I j f (, j S l, j=1 ) Sτ l dτ, where f j (, s, u) = Ẽ [ e T ( T r S l τ + u K) + Y = j, Sl = s ]. The discouned price process is π = e r n I j f (, j S l, j=1 Ss l ). We obain parial differenial equaions in hree variables. The special case K = is simpler, wih only wo sae variables. D. Ineres rae derivaives. A paricularly simple, bu sill imporan, class of claims are hose of he form H = h(y T ). Ineres rae derivaives of he form H = h(r T ) are included since r T = r Y T. For such claims he only relevan sae variables are and Y, so ha he funcion in (15.3) depends only on and j. The equaion (15.6) reduces o d d f j = r j f j and he side condiion is (pu h(j) = h j ) k Y j (f k f j ) λ jk, (15.12) f j T = hj. (15.13) In marix form, d d f = ( R Λ)f,

230 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 229 subjec o The soluion is f T = h. f = exp{( Λ R)(T )}h. (15.14) I depends on and T only hrough T. In paricular, he zero coupon bond wih mauriy T corresponds o h = 1. We will henceforh refer o i as he T -bond in shor and denoe is price process by p(, T ) and is sae-wise price funcions by p(, T ) = (p j (, T )) j=1,...,n; p(, T ) = exp{( Λ R)(T )}1. (15.15) For a call opion on a U-bond, exercised a ime T (< U) wih price K, h has enries h j = (p j (T, U) K) +. In (15.14) (15.15) i may be useful o employ he represenaion shown in (15.2), say. exp{( Λ R)(T )} = Φ D j=1,...,n(e ρ j(t ) ) Φ 1, (15.16) 15.6 Numerical procedures A. Simulaion. The homogeneous Markov process {Y } [,T ] is simulaed as follows: Le K be he number of ransiions beween saes in [, T ], and le T 1,..., T K be he successive imes of ransiion. The sequence {(T n, Y Tn )} n=,...,k is generaed recursively, saring from he iniial sae Y a ime T =, as follows. Having arrived a T n and Y Tn, generae he nex waiing ime T n+1 T n as an exponenial variae wih parameer λ Yn (e.g. ln(u n)/λ Yn, where U n has a uniform disribuion over [, 1]), and le he new sae Y Tn+1 be k wih probabiliy λ Ynk/λ Yn. Coninue in his manner K + 1 imes unil T K < T T K+1. B. Numerical soluion of differenial equaions. Alernaively, he differenial equaions mus be solved numerically. For ineres rae derivaives, which involve only ordinary firs order differenial equaions, a Runge Kua will do. For sock derivaives, which involve parial firs order differenial equaions, one mus employ a suiable finie difference mehod, see e.g. [46] Risk minimizaion in incomplee markes A. Incompleeness. The noion of incompleeness perains o siuaions where a coningen claim canno be duplicaed by an SF porfolio and, consequenly, does no receive a unique price from he no arbirage posulae alone. In Paragraph 15.4F we were dealing implicily wih incompleeness arising from a scarciy of raded asses, ha is, he discouned basic price processes are incapable of spanning he space of all maringales w.r.. (F Y, P) and, in paricular, reproducing he value (15.26) of every financial derivaive (funcion of he basic asse prices).

231 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 23 Incompleeness also arises when he coningen claim is no a purely financial derivaive, ha is, is value depends also on circumsances exernal o he financial marke. We have in mind insurance claims ha are caused by evens like deah or fire and whose claim amouns are e.g. inflaion adjused or linked o he value of some invesmen porfolio. In he laer case we need o work in an exended model specifying a basic probabiliy space wih a filraion F = {F } conaining F Y and saisfying he usual condiions. Typically i will be he naural filraion of Y and some oher process ha generaes he insurance evens. The definiions and condiions laid down in Paragraphs 15.4C-E are modified accordingly, so ha adapedness of η and predicabiliy of ξ are aken o be w.r.. (F, P) (keeping he symbol P for he basic probabiliy measure), a T -claim H is F T measurable, ec. B. Risk minimizaion. Throughou he remainder of he paper we will mainly be working wih discouned prices and values wihou any oher menion han he noaional ilde. The reason is ha he heory of risk minimizaion ress on cerain maringale represenaion resuls ha apply o discouned prices under a maringale measure. We will be conen o give jus a skechy review of some main conceps and resuls from he seminal paper of Föllmer and Sondermann [21]. Le H be a T -claim ha is no aainable. This means ha an admissible porfolio θ saisfying = H Ṽ θ T canno be SF. The cos, Cθ, of he porfolio by ime is defined as ha par of he value ha has no been gained from rading: C θ = Ṽ θ ξ τ d S τ. The risk a ime is defined as he mean squared ousanding cos, [ ] R = Ẽ ( C T θ C θ ) 2 F. (15.1) By definiion, he risk of an admissible porfolio θ is [ R θ = Ẽ ( H T ] Ṽ θ ξ τ d S τ ) 2 F, which is a measure of how well he curren value of he porfolio plus fuure rading gains approximaes he claim. The heory of risk minimizaion akes his eniy as is objec funcion and proves he exisence of an opimal admissible porfolio ha minimizes he risk (15.1) for all [, T ]. The proof is consrucive and provides a recipe for how o acually deermine he opimal porfolio. One ses ou by defining he inrinsic value of H a ime as Ṽ H [ ] = Ẽ H F. Thus, he inrinsic value process is he maringale ha represens he naural curren forecas of he claim under he chosen maringale measure. By he Galchouk-Kunia- Waanabe represenaion, i decomposes uniquely as Ṽ H = Ẽ[ H] + ξ H d S + L H,

232 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 231 where L H is a maringale w.r.. (F, P) which is orhogonal o S. The porfolio θ H defined by his decomposiion minimizes he risk process among all admissible sraegies. The minimum risk is [ T ] R H = Ẽ d L H τ F. C. Uni-linked insurance. As he name suggess, a life insurance produc is said o be uni-linked if he benefi is a cerain predeermined number of unis of an asse (or porfolio) ino which he premiums are currenly invesed. If he conrac sipulaes a minimum value of he benefi, disconneced from he asse price, hen one speaks of uni-linked insurance wih guaranee. A risk minimizaion approach o pricing and hedging of uni-linked insurance claims was firs aken by Møller [34], who worked wih he Black-Scholes- Meron financial marke. We will here skech how he analysis goes in our Markov chain marke, which conforms well wih he life hisory process in ha hey boh are inensiy-driven. Le T x be he remaining life ime of an x years old who purchases an insurance a ime, say. The condiional probabiliy of survival o age x + u, given survival o age x + ( < u), is u p x+ = P[T x > u T x > ] = e u µ x+s ds, (15.2) where µ y is he moraliy inensiy a age y. We have Inroduce he indicaor of survival o age x +, and he indicaor of deah before ime, d u p x+ = u p x+ µ x+ d. (15.3) I = 1[T x > ], N = 1[T x ] = 1 I. The process N is a (very simple) couning process wih inensiy I µ x+, ha is, M given by dm = dn I µ x+ d (15.4) is a maringale w.r.. (F, P). Assume ha he life ime T x is independen of he economy Y. We will work wih he maringale measure P obained by replacing he inensiy marix Λ of Y wih he maringalizing Λ and leaving he res of he model unalered. Consider a uni-linked pure endowmen benefi payable a a fixed ime T, coningen on survival of he insured, wih sum insured equal o one uni of sock No. l, bu guaraneed no less han a fixed amoun g. This benefi is a coningen T -claim, H = (S l T g) I T. The single premium payable as a lump sum a ime is o be deermined. Le us assume ha he financial marke is complee so ha every purely financial derivaive has a unique price process. Then he inrinsic value of H a ime is Ṽ H = π I T p x+,

233 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 232 where π is he discouned price process of he derivaive S l T g. Using Iô and insering (15.4), we find dṽ H = d π I T p x+ + π I T p x+ µ x+ d + ( π T p x+) dn = d π I T p x+ π T p x+ dm. I is seen ha he opimal rading sraegy is ha of he price process of he sum insured muliplied wih he condiional probabiliy ha he sum will be paid ou, and ha dl H = T p x+ π dm. Consequenly, R H = T = T p x+ T T sp 2 x+s Ẽ [ π s 2 ] F s p x+ µ x+s ds Ẽ [ π s 2 ] F T sp x+s µ x+s ds. (15.5) 15.8 Trading wih bonds: How much can be hedged? A. A finie zero coupon bond marke. Suppose an agen faces a coningen T -claim and is allowed o inves only in he bank accoun and a finie number m of zero coupon bonds wih mauriies T i, i = 1,..., m, all pos ime T. For insance, regulaory consrains may be imposed on he invesmen sraegies of an insurance company. The quesion is, o wha exen can he claim be hedged by self-financed rading in hese available asses? An allowed SF porfolio has discouned value process Ṽ θ of he form dṽ θ = m i=1 ξ i j k Y j ( p k (, T i) p j (, T i))d M jk where ξ is predicable, Mj = ( he non-null enries in he j-h row of M = ( where M jk = j d( M j ) F j ξ, ) k Yj is he n j -dimensional row vecor comprising ), and F j = Y j F M jk F = ( p j (, T i)) i=1,...,m j=1,...,n = ( p(, T1),, p(, Tm)), (15.1) and Y j is he n j n marix which maps F o ( p k (, T i) p j (, T i)) i=1,...,m. If e.g. k Y j Y n = {1,..., p}, hen Y n = (I p p, p (n p 1), 1 p 1 ). The sub-marke consising of he bank accoun and he m zero coupon bonds is complee in respec of T -claims iff he discouned bond prices span he space of all maringales w.r.. (F Y, P) over he ime inerval [, T ]. This is he case iff, for each j, rank(f j ) = n j. Now, since Y j obviously has full rank n j, he rank of F j is deermined by he rank of F in (15.1). We will argue ha, ypically, F has full rank. Thus, suppose c = (c 1,..., c m) is such ha F c = n 1.

234 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 233 Recalling (15.15), his is he same as m c i exp{( Λ R)T i}1 =, i=1 or, by (15.16) and since Φ has full rank, D j=1,...,n( m c ie ρj T i ) Φ 1 1 =. (15.2) i=1 Since Φ 1 has full rank, he enries of he vecor Φ 1 1 canno be all null. Typically all enries are non-null, and we assume his is he case. Then (15.2) is equivalen o m c ie ρj T i =, j = 1,..., n. (15.3) i=1 Using he fac ha he generalized Vandermonde marix has full rank, we know ha (15.3) has a non-null soluion c if and only if he number of disinc eigenvalues ρ j is less han m, see Secion 15.9 below. In he case where rank(f j ) < n j for some j we would like o deermine he Galchouk-Kunia-Waanabe decomposiion for a given FT Y -claim. The inrinsic value process has dynamics d H = j ζ jk k Y j d M jk = j d( M j ) ζ j. (15.4) We seek a decomposiion of he form dṽ = ξ i d p(, T i) + ψ jk jk d M i j k Y j = ξ i ( p k (, T i) p j (, T i)) d j i = j j Y j d( M j ) F j ξ j + j d( M j ) ψ j, M jk + j ψ jk k Y j d M jk such ha he wo maringales on he righ hand side are orhogonal, ha is, I j (F j ξ j ) Λj ψ j =, j k Y j where Λ j = D( λ j ). This means ha, for each j, he vecor ζ j in (15.4) is o be decomposed ino is, Λj projecions ono R(F j ) and is orhocomplemen. From (15.3) and (15.4) we obain F j ξ j = Pj ζ j, where P j = F j (F j Λj F j ) 1 F j Λj, hence ξ j = (Fj Λj F j ) 1 F j Λj ζ j. (15.5)

235 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 234 Furhermore, ψ j = (I Pj )ζ j, (15.6) and he risk is T j p Yj s λ jk (ψ jk s ) 2 ds. (15.7) k Y j The compuaion goes as follows: The coefficiens ζ jk involved in he inrinsic value process (15.4) and he sae-wise prices p j (, T i) of he T i-bonds are obained by simulaneously solving (15.6) and (15.12), saring from (15.9) and (15.12), respecively, and a each sep compuing he opimal rading sraegy ξ by (15.5) and he ψ from (15.6), and adding he sep-wise conribuion o he variance (15.7) (he sep-lengh imes he curren value of he inegrand). B. Firs example: The floorle. For a simple example, consider a floorle H = (r r T ) +, where T < min i T i. The moivaion could be ha a ime T he insurance company will ascribe ineres o he insured s accoun a curren ineres rae, bu no less han a prefixed guaraneed rae r. Then H is he amoun ha mus be provided per uni on deposi and per ime uni a ime T. Compuaion goes by he scheme described above, wih he ζ jk = f k f j obained from (15.12) subjec o (15.13) wih h j = (r r j ) +. C. Second example: The ineres guaranee in insurance. A more pracically relevan example is an ineres rae guaranee on a life insurance policy. Premiums and reserves are calculaed on he basis of a pruden so-called firs order assumpion, saing ha he ineres rae will be a some fixed (low) level r hroughou he erm of he insurance conrac. Denoe he corresponding firs order reserve a ime by V. The (porfolio-wide) mean surplus creaed by he firs order assumpion in he ime inerval [, +d) is (r r ) + p xv d. This surplus is currenly credied o he accoun of he insured as dividend, and he oal amoun of dividends is paid ou o he insured a he erm of he conracs a ime T. Negaive dividends are no permied, however, so a ime T he insurer mus cover H = T Ts e r (r r s) + sp xv s ds. The inrinsic value of his claim is [ T ] H = Ẽ e s r (r r s) + sp xvs ds F = e s r (r r s) + sp xv s ds + e r j I j f j, where he f j ime, are he sae-wise expeced values of fuure guaranees, discouned a f j [ T ] = Ẽ e s r (r r s) + sp xvs ds Y = j.

236 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 235 Working along he lines of Secion 15.5, we deermine he f j subjec o d d f j = (r r j ) + p xv + r j f j The inrinsic value has dynamics (15.4) wih ζ jk = f k f j. From here we proceed as described in Paragraph A. by solving k Y j (f k f j ) λ jk, f j T =. (15.8) D. Compuing he risk. Consrucive differenial equaions may be pu up for he risk. As a simple example, for an ineres rae derivaive he sae-wise risk is R j = T Differeniaing his equaion, we find d d R j = k;k j g p jg τ and, using he backward version of (15.2), k;k g ( ) 2 λ gk ψτ gk dτ. ( ) 2 T λ jk ψ jk d + d pjg τ g k;k g d d pjg s = λ jh p hg jg s + λ j ps, h;h j ( ψ gk τ ) 2 dτ, we arrive a d d R j = k;k j ( ) 2 λ jk ψ jk λ jk R k + λ j R j. k;k j 15.9 The Vandermonde marix in finance A. The Vandermonde marix. Le A n denoe he generic n n marix of he form ) j=1,...,n A n = (e α iβ j i=1,...,n, (15.1) where α 1,..., α n and β 1,..., β n are reals. This is a classic in marix heory, known as he generalized Vandermonde marix (usually is elemens are wrien in he form x β j i wih x i > ). I is well known ha i is non-singular iff all α i are differen and all β j are differen, see Ganmacher [22] p. 87. B. Purpose of he sudy. The marix A n in (15.1) and is close relaive A n 1 n1 n = ( ) j=1,...,n e α iβ j 1 i=1,...,n, (15.2)

237 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 236 arise naurally in zero coupon bond prices based on spo ineres raes driven by cerain homogeneous Markov processes. I urns ou ha, in such bond markes, he issue of compleeness is closely relaed o he rank of he wo archeype marices. Roughly speaking, non-singulariy of marices of ypes (15.1) or (15.2) ensures ha any simple T -claim can be duplicaed by a porfolio consising of he risk-free bank accoun and a sufficienly large number of zero coupon bonds. The non-singulariy resuls are proved in Secion 15.1, and applicaions o bond markes are presened in Secion Two properies of he Vandermonde marix A. The main resul. We ake he opporuniy here o provide a shor proof of he quoed resul on nonsingulariy of he Vandermonde marix in (15.1), and will supply a similar resul abou is relaive defined in (15.2). Theorem (i) If he α i are all differen and he β j are all differen, hen A n is non-singular. (ii) If, furhermore, he α i and he β j are all differen from, hen A n 1 n1 n is non-singular. Proof: The proof goes by inducion. Le H n be he hypohesis saed in he wo iems of he lemma. Trivially, H 1 is rue. Assuming ha H n 1 is rue, we need o prove H n. Addressing firs iem (i) of he he hypohesis, i suffices o prove ha de(a n). Recas his deerminan as where de(a n) = = ( n j=1 ( n j=1 e αnβ j e αnβ j ) de n 1 i=1 e (α 1 α n)β 1 e (α 1 α n)β n e (α n 1 α n)β 1 e (α n 1 α n)β n 1 1 e (α i α n)β n ) de ( An 1 1 n 1 1 n 1 1 ) (15.1) A n 1 = (e ) (α j=1,...,n 1 i α n)(β j β n). (15.2) i=1,...,n 1 The deerminan appearing in (15.1) remains unchanged upon subracing he n-h row of he marix from all oher rows, which gives ( ) ( ) An 1 1 n 1 An 1 1 n 11 de 1 n 1 n 1 = de n n 1 1 = de ( A n 1 1 n 11 n 1). (15.3)

238 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 237 Now, since he α i are all differen and also he β j are all differen, he marix A n 1 in (15.2) is of he form required in iem (ii) of he lemma and so, by he assumed hypohesis H n 1, de(a n 1 1 n 11 n 1). I follows from (15.1) and (15.3) ha de(a n), hence iem (i) of H n holds rue. Nex, we urn o iem (ii) of H n. Preparing for an ad absurdum argumen, assume ha A n is as specified in iem (ii) of he lemma and ha A n 1 n1 n is singular. Then here exiss a vecor c = (c 1,..., c n) n such ha Inroducing he funcion A nc = 1 n1 nc. (15.4) f(α) = n c je αβ j, j=1 and puing α =, we can spell ou (15.4) as f(α ) = f(α 1) = = f(α n), (15.5) ha is, f assumes he same value a n + 1 disinc values of α. Since f is coninuously differeniable, Rolle s heorem implies ha he derivaive f of f is a n disinc values α 1,..., α n (say) of α. Now, f (α) = n c jβ je αβ j, j=1 and since some c j are differen from and all β j are differen from, i follows ha ( ) j=1,...,n he marix A n = e α i β j should be singular. This conradics he previously i=1,...,n esablished iem (i) under H n, showing ha he assumed singulariy of A n 1 n1 n is absurd. We conclude ha also iem (ii) of H n holds rue. B. Remarks. In fac, if α 1 < < α n and β 1 < < β n, hen de(a n) > (see [22]). If we ake his fac for graned, (15.1) and (15.3) show ha also de(a n 1 n1 n) >, implying ha he laer is non-singular under he hypohesis of iem (ii) in he heorem. The sign of a general Vandermonde deerminan is, of course, he produc of he signs of he row and column permuaions needed o order he α i and he β j by heir size Applicaions o finance A. Zero coupon bond prices. A zero coupon bond wih mauriy T, or jus T -bond in shor, is he simple coningen claim of 1 a ime T. Taking an arbirage-free financial marke for graned, he price process {p(, T )} [,T ] of he T -bond is [ p(, T ) = Ẽ e ] T r u du F, (15.1) where Ẽ denoes expecaion under some maringale measure, and F is he informaion available a ime.

239 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 238 We will provide some examples where he resuls in Secion 15.1 are insrumenal for esablishing linear independence of price processes of bonds wih differen mauriies. The issue is non-rivial only in cases where he bond prices are governed by more han one source of randomness, of course, so we have o look ino cases where he spo rae of ineres is driven by more han one maringale. B. Markov chain ineres rae. Referring o Chaper 7, le us model he spo rae of ineres {r } as a coninuous ime, homogeneous, recurren Markov chain wih finie sae space {r 1,..., r n }. We are working under some maringale measure given by an infiniesimal marix Λ = ( λ jk ) of he Markov chain, ha is, he ransiion inensiies are λ jk, j k, and λ jj = k;k j λ jk. The price a ime T of a zero coupon bond wih mauriy T is where I j = 1[r = r j ] and The vecor of sae-wise prices, is given by (15.15), p(, T ) = p j (, T ) = Ẽ n I j p j (, T ), j=1 [ e T r u du r = r j]. p(, T ) = (p j (, T )) j=1,...,n, p(, T ) = exp{( Λ R)(T )}1 = Φ Diag(e ρ j(t ) ) Ψ1, where R = Diag(r j ) is he n n diagonal marix wih he enries r j down he principal diagonal, 1 is he n-vecor wih all enries equal o 1, ρ j, j = 1,..., n, are he eigenvalues of Λ R, and Φ and Ψ are he n n marices formed by he righ and lef eigenvecors, respecively. The price processes of m zero coupon bonds wih mauriies T 1 < < T m are linearly independen only if he marix (e ρj T i ) j=1,...,n i=1,...,m has rank m. From iem (i) in he heorem in Paragraph 15.1A we conclude ha his is he case if here are a leas m disinc eigenvalues ρ j. I( also follows ha he ) marke consising of he bank accoun wih price process exp rs ds and he m zero coupon bonds is complee for he class of all FT r 1 -claims only if boh he number of disinc eigenvalues and he number of bonds are no less han he maximum number of saes ha can be direcly accessed from any single sae of he Markov chain. C. Mixed Vaciček ineres rae. The Vasiček model akes he spo rae of ineres o be an Ornsein-Uhlenbeck process given by dr = α (ρ r ) d + σ d W. (15.2)

240 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 239 Here ρ is he saionary mean of he process, α is a posiive mean reversion parameer, σ is a posiive volailiy parameer, and W is a sandard Brownian moion under a maringale measure. The dynamics of he discouned T -bond price, p(, T ) = e r u du p(, T ), (15.3) is d p(, T ) = p(, T ) σ α ( ) e α (T ) 1 d W, (15.4) see e.g. [9]. Obviously, any F W T claim can be duplicaed by a self-financing porfolio in he T -bond and he bank accoun, and so he compleeness issue is rivial in his model. To creae an example where one bond is no sufficien o complee he marke, le us concoc a mixed Vasiček model by puing r = n r j, j=1 where he r j are independen Ornsein-Uhlenbeck processes, dr j = α j (ρ j r j ) d + σ j d W j, j = 1,..., n, and he W j are independen sandard Brownian moions. We assume ha he α j are all disinc (oherwise we could gaher all processes r j wih coinciding mean reversion parameer ino one Ornsein-Uhlenbeck process). The mixed Vasiček process is no mean-revering in he same simple sense as he radiional Vasiček process. I is saionary, however, and is ap o describe ineres ha is subjec o several random phenomena, each of mean-revering ype. By he assumed independence, he price of he T -bond is jus [ where p j (, T ) = Ẽ e T r u du p(, T ) = n p j (, T ), j=1 ] r j, and he discouned price is p(, T ) = n p j (, T ), j=1 where p j (, T ) is he j-analogue o (15.3). By virue of (15.4), we conclude ha he discouned T -bond price has dynamics d p(, T ) = p(, T ) n j=1 σ j ( ) e αj (T ) 1 d α W j j. (15.5) Now, consider he marke consising of he bank accoun and m zero coupon bonds wih mauriies T 1 < < T m. From (15.5) i is seen ha his marke is complee for he class of F W 1,..., W n T 1 -claims if and only if he marix ( ) i=1,...,m e αj (T i ) 1 j=1,...,n (15.6)

241 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 24 has rank n. By virue of iem (ii) in he heorem in Paragraph 15.1A, we conclude ha his is he case if m n. D. Mixed Poisson-driven Ornsein-Uhlenbeck ineres rae. Referring o [4], le us replace he Brownian moions in Paragraph C above wih independen compensaed Poisson processes, ha is, d W j = dn j λ j d, where each N j is a Poisson process wih inensiy λ j. Insead of (15.5) we obain d p(, T ) = p(, T ) n j=1 ( { σ j ( exp e αj (T ) 1) } ) 1 d α W j j. (15.7) I is seen from (15.7) ha he marke consising of he bank accoun and m zero coupon bonds wih mauriies T 1 < < T m is complee for he class of F Ñ1,...,Ñn T 1 - claims if and only if he marix ( { σ j ( exp e αj (T ) 1) } ) i=1,...,m 1 α j j=1,...,n has rank n. By iem (ii) in he heorem in Paragraph 15.1A, we know ha he marix (15.6) has full rank. Thus, compleeness of a marke consising of he bank accoun and a leas n bonds would be esablished and we would be done if we could prove ha he n m marix (e γ ji 1) has full rank whenever (γ ji) has full rank. Wih his conjecure our sudy of hese problems will have o hal for he ime being Maringale mehods A. Preliminaries. Referring o Secions 5.4 and 7.3 we shall see examples of how maringale mehods can be used o prove resuls ha hroughou he ex have been obained by he direc backward argumen. The policy was described as a ime-coninuous Markov chain Z wih finie sae space and ransiion inensiies µ jk, j k. We inroduced I j() = 1[Z() = j], he indicaor of he even ha Z is in sae j a ime, and N jk () = {s; < s, Z(s ) = j, Z(s) = k}, he number of direc ransiions of Z from sae j o sae k ( j) in he ime inerval (, ]. Taking Z o be righ-coninuous, he same goes for he indicaor processes I j and he couning processes N jk. Le F Z = σ{z s; s },, be he filraion generaed by Z. The compensaed couning processes M jk, j k, defined by dm jk () = dn jk () I j() µ jk d (15.8) and M jk () =, are zero mean, square inegrable, muually orhogonal maringales w.r.. he filraion.

242 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 241 We need a couple of general resuls: 1. Le X be a real-valued random variable such ha E X <. Then he process M defined by M() = E[X F ] is a maringale. This follows by he rule of ieraed expecaion and he filraion propery, F s F for s < : E[M() F s] = E{E[X F ] F s} = E[X F s] = M(s). 2. A maringale M wih pahs ha are (almos surely) coninuous and of finie variaion in every finie inerval is consan as a funcion of ime; M() = M() for all. This is seen as follows. Since M has finie variaion, i obeys he rules of ordinary calculus and, in paricular, M 2 () = M 2 () + 2 M(s) dm(s). Since M is coninuous, i is also predicable so ha he inegral 2M(s) dm(s) is a maringale. I follows ha E [ M 2 () ] = M 2 (). Since E[M()] = M(), we conclude ha hence M is consan. Now o he maringale echnique: Var[M()] =, B. Firs example: Thiele s differenial equaion. Consider he sandard muli-sae insurance policy defined in Paragraph 5.4.A and recall he form of he paymen funcion, db() = j I j() db j() + j k b jk () dn jk (), (15.9) where he B j and he b jk are deerminisic funcions. Moivaed by Secion 5.6, we allow he ineres rae o depend on he curren sae of he Markov process: r() = r Z() = j I j() r j. (15.1) (This does no complicae maers.) Define he maringale [ n ] M() = E e τ r db(τ) F [ = e τ r db(τ) + e n ] r E e τ r db(τ) F = e τ r db(τ) + e r I j()v j(). (15.11) j

243 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 242 (Recall he shor-hand τ r = τ r(s) ds.) The las sep follows from he Markov propery of Z and he fac ha he paymens a any ime are (funcionally) independen of he pas: [ n ] [ E e τ n ] r db(τ) F = E e τ r db(τ) Z = V Z() () = j I j()v j(). Now apply Iǒ s formula o (15.11): dm() = e r db() + e r ( r() d) j I j() V j() + e r j I j() dv j() + e r j k dn jk () (V k () V j( )). The las erm on he righ akes care of he jumps of he Markov process: upon a jump from sae j o sae k he las erm in (15.11) changes immediaely from he discouned value of he reserve in sae j jus before he jump o he value of he reserve in sae k a he ime of he jump. Since he sae-wise reserves are deerminisic funcions wih finie variaion, hey have a mos a counable number of disconinuiies a fixed imes. The probabiliy ha he Markov process jumps a any such ime is. Therefore, we need no worry abou possible common poins of disconinuiy of he V j() and he I j(). For he same reason we can also disregard he lef limi in V j( ) in he las erm. We proceed by insering he expressions (15.9) for db(), (15.1) for r(), and he expression dn jk () = dm jk () I j() µ jk () d obained from (15.8), and gaher dm() = e r I j() db j() r j V j() d + dv j() + µ jk d R jk () j k; k j + e r j k R jk () dm jk (), (15.12) where R jk () = b jk () + V k () V j() is recognized as he sum a risk in respec of ransiion from j o k a ime. Since he las erm on he righ of (15.12) is he incremen of a maringale, he firs erm of he righ is he difference beween he incremens of wo maringales and is hus iself he incremen of a maringale. This maringale has finie variaion and, as will be explained below, is also coninuous, and mus herefore be consan. For his o be rue for all realizaions of he indicaor funcions I j, we mus have db j() r j V j() d + dv j() + µ jk d R jk () =. (15.13) k; k j This is nohing bu Thiele s differenial equaion. We also obain ha dm() = e r j k R jk () dm jk (), which displays he dynamics of he maringale M.

244 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 243 Finally, we explain why he (15.13) is he incremen a of a coninuous funcion. The d erms are coninuous incremens, of course. Ouside jump imes of he B j boh he B j hemselves and he V j are coninuous. A any ime where here is a jump in some B j he reserve V j jumps by he same amoun in he opposie direcion since V j( ) = B j() + V j(). Thus, B j + V j is indeed coninuous. C. Second example: Revisiing Exercise 56, he geomeric Poisson price process. Maringale mehods are cerainly no needed in his case since he Poisson process is sufficienly well srucured o allow of direc compuaion of all funcions asked for in he exercise. Bu jus for he sake of he example: Inroduce V () = E[S()]. Le F,, be he filraion generaed by he Poisson process N. Fix a ime T > and inroduce he maringale M() = E [ S(T ) F ] [ ] = e α+βn() E e α(t )+β(n(t ) N() F = S() V (T ). Here we have made use of he fac ha he Poisson process has saionary and independen incremens. Wrie V = d V and apply Iǒ: d dm() = S() α d V (T ) + S() V (T )( d) + (S() S( )) V (T ) (15.14), he las erm accouning of jumps. Rewrie S() S( ) = e α+βn() e α+βn( ) = e α+β(n( )+ N()) e α+βn( ) ( ) = S( ) e β N() 1 ( ) = S( ) e β 1 dn(), where he las wo sep is due o he zero-or-one naure of he incremens of couning process N. Upon insering his ino (15.14) and inroducing he maringale dm N() = dn() λ d, we ge [ dm() = S() α V (T ) V (T ) + ( ) + S( ) e β 1 dm N(), ( ) ] e β 1 λ V (T ) d Here we have used he fac ha S( ) d = S() d which is o be undersood in inegral form: f( ) d = f() d since he inegral is no affeced by a change of he inegrand a an a mos counable se of poins. Arguing as in he previous example, he drif erm in he dynamics of M mus vanish, and we arrive a he differenial equaion [ ( ) ] V () = α + e β 1 λ V (),

245 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 244 The soluion, subjec o he obvious condiion V () = 1, is [( ( ) ) ] V () = exp α + e β 1 λ. To rehearse he echnique, you should use i o solve Iem (c) in exercise 56. Sar from he maringale [ T ] M() = E S 1 (τ) dτ F = S 1 (τ) dτ + e α βn() V (T ), where [ ] V () = E S 1 (τ) dτ. D. Third example: Revisiing pricing of he uni-linked erm insurance wih guaranee, pages We need o deermine [ n ( π = E 1 e ) ] τ r g f(τ) dτ. Le F = σ{y s; s },, be he filraion generaed by he economy process Y. Sar from he maringale [ ( M() = E 1 e ) ] τ r g f(τ) dτ F ( = 1 e ) [ τ r g f(τ) dτ + e n ( r E e r e ) ] τ r g f(τ) dτ F where = ( 1 e ) τ r g f(τ) dτ + e r e [ n W e(, u) = E ( u e ) τ r g I e() W e(, e τ r ), ] f(τ) dτ Y () = e. Assuming ha he parial derivaives We(, u) and We(, u) exis, apply Iǒ: u dm() = ( 1 e ) r g f() d + e r ( r() d) e + e r e I e() ( We(, e τ r ) d + + e r e f dn ef () (W f (, e τ r ) W e(, e τ r )). I e() W e(, e τ r ) ) u We(, e τ r τ ) e r r() d Subsiue dn ef () = dm ef () + λ ef d,

246 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE 245 where he M ef are maringales, and pu U() = e τ r. Arguing along he lines of Paragraph B, we conclude ha ( 1 U() 1 g ) f() U() 1 r e W e(, U()) ( ) + U() 1 We(, U()) + We(, U()) U() re() u + U() 1 λ ef (W f (, U()) W e(, U())) =. f; f e for all realizaions of Y. We end up wih he parial differenial equaion ( (u g) f() r e W e(, u) + We(, u) + ) We(, u) u re() u + λ ef (W f (, u) W e(, u)) =, f; f e which are o be solved subjec o he obvious condiions W e(n, u) =. E. Remark on he echnique. In all hree examples we needed o deermine he expeced value of some random variable W ha depends on he developmen of a sochasic process. Here is an ouline of he mehod: Sar from he maringale M() = E [ W F ]. Inspec W and ry o wrie i in he form W = W (T (), U()), where T () depends only on he fuure developmen of he process and U() depends only on he pas hisory F. The random variable U() is he called he sae variable(s) (i may be muli-dimensional). How o proceed from here depends on he properies of he driving sochasic process. In our siuaions (le us keep he example in Paragraph D in mind) we use he Markov propery of Y o conclude ha he condiional expeced value M() mus be a funcion only of he curren ime, sae, and value of he sae variable: M() = F Y () (, U()) = e I e() F e(, U()). Assuming coninuous differenibiliy of he funcions F e wih respec o and u, use Iǒ o form he dynamics of M: dm() = ( I e() Fe(, U()) d + ) u Fe(, U()) du c () e + e [I e() F e(, U()) I e( ) F e(, U( ))]. Here U c denoes he coninuous par of U. The jump par may have conribuions from possible jumps of U ouside jump imes for Y, bu hese will always cancel ou and vanish in he end, however, see he example in Paragraph B above. In any case he jump par will consis of he following erms due o jumps of Y : dn ef () [F f (, U()) F e(, U( ))]. f e

247 CHAPTER 15. FINANCIAL MATHEMATICS IN INSURANCE Now, inser dn ef () = dm ef ()+λ ef d and idenify he maringale par and he drif par wih he facor d in he expression for dm(). The drif par mus vanish, and we arrive a a se of consrucive non-sochasic differenial equaions from which he funcions F e(, u) can be solved (a leas numerically).

248 Bibliography [1] Aase, K.K. and Persson, S.-A. (1994). Pricing of uni-linked life insurance policies. Scand. Acuarial J., 1994, [2] Aczel J. Lecures on Funcional Equaions and heir Applicaions. Academic Press, [3] Andersen, P.K., Borgan, Ø., Gill, R.D., Keiding, N. (1993). Saisical Models Based on Couning Processes. Springer-Verlag, New York, Berlin, Heidelberg. [4] Anderson, J.L. and Dow, J.B. (1948). Acuarial Saisics, Vol. II: Consrucions of Moraliy and oher Tables. Cambridge Universiy Press. [5] Barlow, R.E. and Proschan, F (1981): Saisical Theory of Reliabiliy and Life Tesing, Hol, Reinhar and Winson Inc. [6] Berger, A. (1939): Mahemaik der Lebensversicherung. Verlag von Julius Springer, Vienna. [7] Bibby J.M., Mardia, K.V., and Ken J.T. Mulivariae Analysis. Academic Press, [8] Björk, T., Kabanov, Y., Runggaldier, W. (1997): Bond marke srucures in he presence of marked poin processes. Mahemaical Finance, 7, [9] Björk, T. (1998): Arbirage Theory in Coninuous Time, Oxford Universiy Press. [1] Black, F., Scholes, M. (1973): The pricing of opions and corporae liabiliies. J. Poli. Economy, 81, [11] Bowers, N.L. Jr., Gerber, H.U., Hickman, J.C., and Nesbi, C.J. (1986). Acuarial Mahemaics. The Sociey of Acuaries. Iasca, Illinois. [12] Cox, J., Ross, S., Rubinsein, M. (1979): Opion pricing: A simplified approach. J. of Financial Economics, 7, [13] Delbaen, F., Schachermayer, W. (1994): A general version of he fundamenal heorem on asse pricing. Mahemaische Annalen, 3, [14] De Pril, N. (1989). The disribuions of acuarial funcions. Mieil. Ver. Schweiz. Vers.mah., 89, [15] De Vylder, F. and Jaumain, C. (1976). Exposé moderne de la héorie mahémaique des opéraions viagère. Office des Assureurs de Belgique, Bruxelles. [16] Dhaene, J. (199). Disribuions in life insurance. ASTIN Bull. 2, [17] Donald D.W.A. Compound Ineres and Annuiies Cerain. Heinemann, London,

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251 Appendix A Calculus A. Piecewise differeniable funcions. Being concerned wih operaions in ime, commencing a some iniial dae, we will consider funcions defined on he posiive real line [, ). Thus, le us consider a generic funcion X = {X } and hink of X as he sae or value of some process a ime. For he ime being we ake X o be real-valued. In he presen ex we will work exclusively in he space of so-called piecewise differeniable funcions. From a mahemaical poin of view his space is iny since only elemenary calculus is needed o move abou in i. From a pracical poin of view i is huge since i comforably accommodaes any idea, however sophisicaed, ha an acuary may wish o express and analyse. I is convenien o ener his space from he ouside, saring from a wider class of funcions. We firs ake X o be of finie variaion (FV), which means ha i is he difference beween wo non-decreasing, finie-valued funcions. Then he lef-limi X = lim s X s and he righ-limi X + = lim s X s exis for all, and hey differ on a mos a counable se D(X) of disconinuiy poins of X. We are paricularly ineresed in FV funcions X ha are righ-coninuous (RC), ha is, X = lim s X s for all. Any probabiliy disribuion funcion is of his ype, and any sream of paymens accouned as incomes or ougoes, can reasonably be aken o be FV and, as a convenion, RC. If X is RC, hen X = X X, when differen from, is he jump made by X a ime. For our purposes i suffices o le X be of he form X = X + x τ dτ + (X τ X τ ). (A.1) <τ The inegral, which may be aken o be of Riemann ype, adds up he coninuous incremens/decremens, and he sum, which is undersood o range over disconinuiy imes, adds up incremens/decremens by jumps. We assume, furhermore, ha X is piecewise differeniable (PD); A propery holds piecewise if i akes place everywhere excep, possibly, a a finie number of poins in every finie inerval. In oher words, he se of excepional poins, if no empy, mus be of he form {, 1,...}, wih < 1 <, and, in case i is infinie, lim j j =. Obviously, X is PD if boh X and x are piecewise coninuous. A any poin 4

252 APPENDIX A. CALCULUS 5 / D = D(X) D(x) we have d X = x, ha is, he funcion X grows (or decays) d coninuously a rae x. As a convenien noaional device we shall frequenly wrie (A.1) in differenial form as dx = x d + X X. (A.2) A lef-coninuous PD funcion may be defined by leing he sum in (A.1) range only over he half-open inerval [, ). Of course, a PD funcion may be neiher righconinuous nor lef-coninuous, bu such cases are of no ineres o us. B. The inegral wih respec o a funcion. Le X and Y boh be PD and, moreover, le X be RC and given by (A.2). The inegral over (s, ] of Y wih respec o X is defined as s Y τ dx τ = s Y τ x τ dτ + s<τ Y τ (X τ X τ ), (A.3) provided ha he individual erms on he righ and also heir sum are well defined. Considered as a funcion of he inegral is iself PD and RC wih coninuous incremens Y x d and jumps Y (X X ). One may hink of he inegral as he weighed sum of he Y -values, wih he incremens of X as weighs, or vice versa. In paricular, (A.1) can be wrien simply as X = X s + s dx τ, (A.4) saying ha he value of X a ime is is value a ime s plus all is incremens in (s, ]. By definiion, Y τ dx τ = lim Y τ dx τ = Y τ dx τ Y (X X ) = Y τ dx τ, s s r s a lef-coninuous funcion of. Likewise, Y τ dx τ = lim Y τ dx τ = Y τ dx τ + Y s(x s X s ) = r s a lef-coninuous funcion of s. r s s (s,) [s,] Y τ dx τ, C. The chain rule (Iô s formula). Le X = (X 1,..., X m ) be an m-variae funcion wih PD and RC componens given by dx i = x i d + (X i X ). i Le f : R m R have coninuous parial derivaives, and form he composed funcion f(x ). On he open inervals where here are neiher disconinuiies in he x i nor jumps of he X i, he funcion f(x ) develops in accordance wih he well-known chain rule for scalar fields along recifiable curves. A he excepional poins f(x ) may change (only) due o jumps of he X i, and a any such poin i jumps by f(x ) f(x ). Thus, we gaher he so-called change of variable rule or Iô s formula, which in our simple funcion space reads df(x ) = m i=1 f x i (X) xi d + f(x ) f(x ), (A.5)

253 APPENDIX A. CALCULUS 6 or, in inegral form, f(x ) = f(x s) + s m i=1 x i f(xτ ) xi τ dτ + s<τ {f(x τ ) f(x τ )}. (A.6) Obviously, f(x ) is PD and RC. A frequenly used special case is (check he formulas!) d(x Y ) = X y d + Y x d + X Y X Y = X dy + Y dx + (X X )(Y Y ) = X dy + Y dx. (A.7) If X and Y have no common jumps, as is cerainly he case if one of hem is coninuous, hen (A.7) reduces o he familiar d(x Y ) = X dy + Y dx. (A.8) The inegral form of (A.7) is he so-called rule of inegraion by pars: s Y τ dx τ = Y X Y sx s s X τ dy τ. (A.9) Le us consider hree special cases for which (A.9) can be obained by direc calculaion and specialises o well-known formulas. Seing s = (jus a maer of noaion), (A.9) can be cas as Y X = Y X + Y τ dx τ + X τ dy τ, (A.1) which shows how he produc of X and Y a ime emerges from is iniial value a ime plus all is incremens in he inerval (, ]. Assume firs ha X and Y are boh discree. To keep noaion simple assume X = [] j= xj and Y = [] j= yj. Then X Y = [] x i i= [] y j j= [] = x y + i=1 j= [] j 1 i y j x i + x i y j j=1 i= [] [] = X Y + Y i x i + X j 1 y j = X Y + i=1 Y τ dx τ + j=1 X τ dy τ, which is (A.1). We see here ha he lef limi on he righ of (A.1) is essenial. This case is basically nohing bu he rule of changing he order of summaion in a double sum, he only new hing being ha we formally consider he sums X and Y as funcions of a coninuous ime index; only he values a ineger imes maer, however.

254 APPENDIX A. CALCULUS 7 Assume nex ha X and Y are boh coninuous, ha is, X = X + Take X = Y = for he ime being. Then X Y = = = = = x σ dσ <τ σ σ x τ dτ, Y = Y + y τ dτ y τ dτ x σ dσ + y τ dτ x σ dσ + Y σx σ dσ + Y τ dx τ + y τ dτ. <σ<τ τ X τ y τ dτ X σ dy σ, x σ dσ y τ dτ x σ dσ y τ dτ which also conforms wih (A.1): he lef limi in he nex o las line disappeared since an inegral wih respec o dτ remains unchanged if we change he inegrand a a counable se of poins. The resul for general X and Y is obained by applying he formula above o X X and Y Y. Finally, le one funcion be discree and he oher coninuous, e.g. X = [] j= xj and Y = Y + yτ dτ. Inroduce We have X = X [] and ŷ = Y, ŷ j = j j 1 y τ dτ, j = 1,..., []. X Y = X [] Y [] + X [] (Y Y [] ) = [] x i i= [] j= ŷ j + X [] y τ dτ. [] (A.11) Upon applying our firs resul for wo discree funcions, he firs erm in (A.11) becomes [] X Y + i=1 j= [] j 1 i ŷ j x i + x i ŷ j j=1 i= [] [] j = X Y + Y j x i + X j 1 = X Y + i=1 Y τ dx τ + j=1 [] j 1 X τ dy τ y τ dτ The second erm in (A.11) is [] Xτ dyτ. Thus, also in his case we arrive a (A.1). Again he lef-limi is irrelevan since dy τ = y τ dτ. The general formula now follows from hese hree special cases by he fac ha he he inegral is a linear operaor wih respec o he inegrand and he inegraor.

255 APPENDIX A. CALCULUS 8 D. Couning processes. Le 1 < 2 < be a sequence in (, ), eiher finie or, if infinie, such ha lim j j =. Think of j as he j-h ime of occurrence of a cerain even. The number of evens occurring wihin a given ime is N = {j ; j } = j 1 [ j, )() or, puing =, N = j for j < j+1. The funcion N = {N } hus defined is called a couning funcion since i currenly couns he number of occurred evens. I is a paricularly simple PD and RC funcion commencing from N = and hereafer increasing only by jumps of size 1 a he epochs j, j = 1, 2,... The change of variable rule (A.6) becomes paricularly simple when X is a couning funcion. In fac, for f : R R and for N defined above, f(n ) = f(n s) + {f(n τ ) f(n τ )} (A.12) s<τ = f(n s) + {f(n τ + 1) f(n τ )}(N τ N τ ) = f(n s) + s<τ Basically, wha hese expressions sae, is jus s f(j) = f() + {f(n τ + 1) f(n τ )} dn τ. j {f(i) f(i 1)}. i=1 (A.13) (A.14) Sill hey will prove o be useful represenaions when we come o sochasic couning processes. Going back o he general PD and RC funcion X in (A.1), we can associae wih i a couning funcion N defined by N = {τ (, ]; X τ X τ }, he number of disconinuiies of X wihin ime. Equipped wih our noion of inegral, we can now express X as dx = x c d + x d dn, (A.15) where x c = x is he insananeous rae of coninuous change and x d = X X is he size of he jump, if any, a. Generalizing (A.14), we have d f(x ) = f(x s) + s dx f(xτ ) xc τ dτ + s {f(x τ + x d τ) f(x τ )} dn τ. (A.16)

256 Appendix B Indicaor funcions A. Indicaor funcions in general spaces. Le Ω be some space wih generic poin ω, and le A be some subse of Ω. The funcion I A : Ω {, 1} defined by { 1 if ω A, I A(ω) = if ω A c, is called he indicaor funcion or jus he indicaor of A since i indicaes by he value 1 precisely hose poins ω ha belong o A. Since I A assumes only he values and 1, (I A) p = I A for any p >. Clearly, I =, I Ω = 1, and I A c = 1 I A, (B.1) where A c = Ω\A is he complemen of A. For any wo ses A and B (subses of Ω), and I A B = I AI B I A B = I A + I B I AI B. (B.2) (B.3) The las wo saemens are displayed here only for ease of reference. They are special cases of he following resuls, valid for any finie collecion of ses {A 1,..., A r}: r I r j=1 A j = I Aj, j=1 (B.4) I r j=1 A j = j I Aj j 1 <j 2 I Aj1 I Aj ( 1) r 1 I A1 I Ar. (B.5) The relaion (B.4) is obvious. To demonsrae (B.5), we need he ideniy r (a j + b j) = j=1 r a j1 a jp b jp+1 b jr, p= r\p (B.6) 9

257 APPENDIX B. INDICATOR FUNCTIONS 1 where r\p signifies ha he sum ranges over all ( r p) differen ways of dividing {1,..., r} ino wo disjoin subses {j 1,..., j p} ( when p = ) and {j p+1,..., j r} ( when p = r). Combining he general relaion wih (B.1) and (B.4), we find and arrive a (B.5) by use of (B.6). { αa α} c = αa c α, (B.7) r I r j=1 A j = 1 I r j=1 A c = 1 (1 I Aj ), j B. Furher aspecs of indicaors. The algebraic expressions in (B.4) and (B.5) apply only o he finie case. For any collecion {A α} of ses indexed by α ranging in an arbirary space, possibly uncounable, j=1 I αa α = inf α IAα (B.8) and I αaα = sup I Aα. (B.9) α In fac, inf and sup are aained here, so we can wrie min and max. In accordance wih he laer wo resuls one may define sup α A α = αa α and inf α A α = αa α. The relaion (B.7) ress on elemenary logical operaions, bu also follows from 1 sup α I Aα = inf α(1 I Aα ). The represenaion of ses by indicaors suppors he undersanding of some convenions and definiions in se heory. For insance, if {A j} j=1,2,... is a disjoin sequence of ses, some auhors wrie j Aj insead of jaj. This is moivaed by I j A j = j I Aj, valid for disjoin ses. For any sequence {A j} of ses one wries lim sup A j for he se of poins ω ha belong o infiniely many of he A j, ha is, lim sup A j = j k j A k (for all j here exiss some k j such ha ω belongs o A k ). By lim inf A j is mean he se of poins ω ha belong o all bu possibly finiely many of he A j, ha is, lim inf A j = j k j A k (here exiss a j such ha for all k j he poin ω belongs o A k ). This usage is in accordance wih I j k j A k = inf sup I Ak j k j and I j k j A k = sup j obained upon combining (B.8) and (B.9). inf k j IA, k

258 APPENDIX B. INDICATOR FUNCTIONS 11 C. Indicaors of evens. Le (Ω, F, P) be some probabiliy space. The indicaor I A of an even A F is a simple binomial random variable; I follows ha I A Bin(1, P[A]). E [I A] = P[A], V [I A] = P[A](1 P[A]). (B.1) Since we ofen will need o equip indicaor funcions wih subscrips, we will use he noaion 1[A] and I A inerchangeably.

259 Appendix C Disribuion of he number of occurring evens A. The main resul. Le {A 1,..., A r} be a finie assembly of evens, no necessarily disjoin. Inroduce he shor-hand I j = I Aj. We seek he probabiliy disribuion of he number of evens ha occur ou of he oal of r evens, Q = r I j. j=1 I urns ou ha his disribuion can be expressed in erms of he probabiliies of inersecions of selecions from he assembly of ses. Inroduce Z p = P [ ] A j1 A jp, p = 1,..., r, (C.1) j 1 <...<j p and define in paricular Z = 1. Theorem The probabiliy disribuion of Q can be expressed by he Z p in (C.1) as ( ) r P[Q = q] = ( 1) p q p Z p, q =,..., r, (C.2) p q p=q ( ) r P[Q q] = ( 1) p q p 1 Z p, q = 1,..., r. (C.3) p q Proof: Obviously, p=q {Q = q} = r\q A j1 A jq A c j q+1 A c j r. The elemens in he union are muually disjoin, and so I {Q=q} = r\q I j1 I jq (1 I jq+1 ) (1 I jr ). 12

260 APPENDIX C. DISTRIBUTION OF THE NUMBER OF OCCURRING EVENTS13 Saring from his expression, he generaing funcion of he sequence {I {Q=p} } p=,...,r can be shaped as follows by repeaed use (B.6): r s p I {Q=p} = p= = = = = r I j1 I jp (1 I jp+1 ) (1 I jr ) s p p= r\p r si j1 si jp (1 I jp+1 ) (1 I jr ). p= r\p r (si j + 1 I j) j=1 r ((s 1)I j + 1) j=1 r (s 1)I j1 (s 1)I jp 1 r p, p= r\p where he firs erm corresponding o p = is o be inerpreed as 1. Thus, r r s p I {Q=p} = (s 1) p Y p, (C.4) where Y p = p= p= j 1 <...<j p I j1 I jp, p = 1,..., r, (C.5) and Y = 1. Upon differeniaing (C.4) q imes wih respec o s and puing s =, we ge r q!i {Q=q} = p (q) ( 1) p q Y p, p=q hence, noing ha p (q) /q! = ( ) ( p q = p p q), ( ) r I {Q=q} = ( 1) p q p Y p. p q p=q (C.6) Taking expecaion, we arrive a (C.2). To prove (C.3), inser I {Q=p} = I {Q p} I {Q p+1} on he lef of (C.4) and rearrange as follows: r s p ( ) r r I {Q p} I {Q p+1} = s p I {Q p} s p 1 I {Q p} p= p= = 1 + Thus, recalling ha Y = 1, (C.4) is equivalen o r s p 1 I {Q p} = p=1 p=1 r (s 1)s p 1 I {Q p}. p=1 r (s 1) p 1 Y p. p=1

261 APPENDIX C. DISTRIBUTION OF THE NUMBER OF OCCURRING EVENTS14 Differeniaing here q 1 imes wih respec o s and puing s =, gives r (q 1)!I {Q q} = (p 1) (q 1) ( 1) p q Y p, p=q hence which implies (C.3). I {Q q} = ( ) r ( 1) p q p 1 Y p, p q p=q (C.7) B. Commens and examples. Seing all A j equal o he sure even Ω, all he indicaors I j become idenically 1. Thus Y p defined by (C.5) becomes ( r p), and (C.7) specializes o ( )( ) r 1 = ( 1) p q p 1 r. (C.8) p q p p=q For q = r, he heorem reduces o he rivial resul P [Q = r] = P[Q r] = P [A 1 A r]. Puing q = 1 in (C.3) and noing ha [Q 1] = ja j, yields P [ ja j] = P[A j] P[A j1 A j2 ] j j 1 <j ( 1) r 1 P[A 1 A r], which also resuls upon aking expecaion in (B.5). This is he well-known general addiion rule for probabiliies, called so because i generalizes he elemenary rule P[A B] = P[A] + P[B] P[A B] (see (B.3)). The heorem saes he mos general resuls of his ype. As a non-sandard example, le us find he probabiliy of exacly wo occurrences among hree evens, A 1, A 2, A 3. Puing r = 3 and q = 2 in (C.2), gives P[Q = 2] = P[A 1 A 2] + P[A 1 A 3] + P[A 2 A 3] 3P[A 1 A 2 A 3]. From (C.3) we obain he probabiliy of a leas wo occurrences, P[Q 2] = P[A 1 A 2] + P[A 1 A 3] + P[A 1 A 3] 2P[A 1 A 2 A 3]. The usefulness of he heorem is due o he decomposiion of complex evens ino more elemenary ones. The observan reader may have asked why inersecions raher han unions are aken as he elemenary evens. The reason for doing so is apparen in he case of independen evens, since hen P [ p i=1 Aj ] = p i i=1 P[Aj i ], and he expressions in (C.1) (C.3) can be compued from he P[A j] by elemenary algebraic operaions. Noe ha he resuls in he heorem are independen of he probabiliy measure involved; hey res enirely on he se-relaions (C.6) and (C.7).

262 Appendix D Asympoic resuls from saisics A. The cenral limi heorem Le X 1, X 2,... be a sequence of random variables wih zero means, E[X i] =, and finie variances, σ 2 i = V[X i], i = 1, 2,... (D.1) Define b 2 n = n σi 2, n = 1, 2,... (D.2) The celebraed Lindeberg/Feller cenral limi heorem says ha if n i=1 E [ X 2 i 1[X 2 i > εb 2 n] ] i=1 b 2 n, ε >, (D.3) hen n i=1 Xi b n d N(, 1). (D.4) B. Asympoic properies of MLE esimaors The asympoic disribuions derived in Chaper 11 could be obained direcly from he following sandard resul, which is cied here wihou proof. Le X 1, X 2,... be a sequence of random elemens wih join disribuion depending on a parameer θ ha varies in an open se in he s-dimensional euclidean space. Assume ha he likelihood funcion of X 1, X 2,..., X n, denoed by Λ n(x 1, X 2,..., X n, θ), is wice coninuously differeniable wih respec o θ and ha he equaion ln Λn(X1, X2,..., Xn, θ) = s 1 θ has a unique soluion ˆθ n(x 1, X 2,..., X n), called he MLE (maximum likelihood esimaor). Then, if he marix Σ(θ) = ( [ ]) 2 1 E ln Λn (D.5) θ θ 15

263 APPENDIX D. ASYMPTOTIC RESULTS FROM STATISTICS 16 ends o s s as n, he MLE is asympoically normally disribued, ˆθ as N(θ, Σ(θ)). C. The dela mehod Assume ha ˆθ is a consisen esimaor of θ Θ, an open se in R s, and ha ˆθ as N(θ, Σ). If g : R s R r is a wice coninuously differeniable funcion of θ, hen ( g(ˆθ) as N g(θ), θ g(θ)σ ) θ g(θ). (D.6) The resul follows easily by inspecion of he firs order Taylor expansion of g(ˆθ) around θ.

264 Appendix E The G82M moraliy able Table E.1: The moraliy able G82M x µ x f(x) l x = 1 5 F (x) dx q x

265 APPENDIX E. THE G82M MORTALITY TABLE 18 x µ x f(x) l x = 1 8 F (x) dx q x

266 APPENDIX E. THE G82M MORTALITY TABLE 19 x µ x f(x) l x = 1 8 F (x) dx q x

267 APPENDIX E. THE G82M MORTALITY TABLE 2 x µ x f(x) l x = 1 8 F (x) dx q x

268 Appendix F Exercises Exercise 1 Acquain yourself wih he sofware Turbo and he program ode-1.pas. Use he program o solve he following simple problems: (a) Compue he inegrals 1 2 d and 1 1/2 d and check he answers. (b) Produce a able of values of he sandard normal disribuion funcion Φ(x) = 1 x exp( 2 2π ) d. (Noe ha Φ( x) = 1 Φ(x) so you only need o compue he 2 inegral x.) (c) Consider he paymen funcion A =, [, 1], and le he rae of ineres be consan r =.5. Compue he cash balance U by a forward scheme and he reserve V (which is here negaive of course) by a backward scheme for =, 1,..., 1. Exercise 2 Draw a skech of he paymen funcion A and he cash balance U for he following deerminisic paymen sreams, assuming ha he ineres rae is 4.5% p.a. (r = ln(1.45)): (a) A single paymen (endowmen) of 1 a ime 5. (b) An annuiy-due wih paymens of 1 a imes, 1,..., 9. (c) An immediae annuiy wih paymens of 1 a imes 1,..., 1. (d) An annuiy paid coninuously a rae 1 per ime uni in he ime inerval [, 1]. Compare he graphs in (b) (d). (e) Now consider he random paymen funcion generaed by premiums less benefis on a emporary erm assurance wih sum 1, payable immediaely upon deah wihin 1 years, agains premium payable coninuously a level rae.212 during he insurance period. (This is he equivalence premium if he insured is 3 years old upon issue of 1

269 APPENDIX F. EXERCISES 2 he conrac, and we use he Danish moraliy law.) Draw a skech of he paymen funcion and he cash balance in he case where he insured dies a ime = 5 and in he case where he insured survives hroughou he erm of 1 years. Exercise 3 Draw he graphs in Figures in he lecure noes (Basic Life Insurance Mahemaics). Use he program ode-1.pas o do he compuaions. Exercise 4 Find expressions for f(), F (), µ(), f( x), and F ( x) and skech graphs of hese funcions for he following moraliy laws, all of which have finie ω: (a) De Moivre s uniform moraliy law, (b) The more general law given by F () = 1 F () = ω ω. α >. Look in paricular a he case α = 1/2. ( 1 ω ) α, ω (c) The law given by F () = ln(1 + ( ω)) ln(1 + ω). Exercise 5 Populaion saisics shows ha more han half of all new-born are boys (e.g. in Norway some 51.2% a he presen). I also shows ha he moraliy law depends on he sex. Denoe by s m he probabiliy ha a new-born is male and by s f = 1 sm he probabiliy ha a new-born is female. Le p m and µ m be he survival funcion and moraliy inensiy, respecively, for males and le p f and µf be he corresponding funcions for females. (a) Find he probabiliy s m ha a randomly chosen years old person is a male. (b) Find he survival funcion p and he moraliy inensiy µ in he populaion (of males and females). If µ m and µ f are boh of Gomperz-Makeham form, does i hen follow ha also µ is Gomperz-Makeham? (c) Assume ha female moraliy is lower han male moraliy a all ages. (This is ypically he case, and a common way of modeling his difference is by age reducion, e.g. ake µ f = µ m 5.) Show ha under his assumpion s m is a decreasing funcion. Give a sufficien condiion for s m o end o as ends o.

270 APPENDIX F. EXERCISES 3 (d) Express he following quaniies peraining o he populaion in erms of s m x, s f x, and he corresponding quaniies specific for males and females, respecively: p x, m nq x, ne x, and, in general he ne premium reserve a any ime during he erm of an insurance conrac. Exercise 6 A survival funcion of polynomial form can be fied arbirarily well o a given (finie) se of (non-conradicory) consrains if we choose he degree of he polynomial high enough. Obviously, he maximum aainable age ω mus be finie for a polynomial survival funcion. Consider henceforh a rinomial survival funcion, [, ω]. F () = a + a 1 + a a 3 3, (a) Find he general expressions for he densiy f() and he moraliy inensiy µ(). Verify direcly, by inspecion of he expression, ha lim ω µ() =. Observe ha F ( x), considered as funcion of, is also rinomial. (b) Having four parameers, he coefficiens (a,..., a 3), a rinomial survival funcions can be fied o four independen consrains. I should cerainly be required ha F () = 1 and i is quie naural o fix an ω, hence require ha F (ω) =. Two more consrains should be added, e.g. f(ω) = and ha he expeced life lengh be equal o a given age e (, ω). Spell ou he equaions ha he coefficiens mus saisfy. One mus, of course, check ha he soluion produces a F ha is decreasing and hus is a genuine survival funcion. Exercise 7 We adop he Danish echnical basis in (3.25) and consider a policy issued o a life a age x = 3 and wih erm n = 3. Find and draw a skech of he probabiliy disribuion of he presen value a ime of he following benefis: (a) A pure endowmen benefi of 1; (b) A erm insurance wih sum 1; (c) An endowmen insurance wih sum 1; (d) A life annuiy of 1 per year. Exercise 8 (a) Prove and explain he following relaionships: m+ne x = me x ne x+m, m nā x = me x ā x+m n, m nāx = mex Āx+m n. (b) A more general rule for expeced presen values of deferred benefis can be formulaed in erms of he prospecive reserve: Verify ha V = V (,u] + u E x+ V u,

271 APPENDIX F. EXERCISES 4 where V (,u] is he expeced presen value a ime of benefis less premiums payable in he ime inerval (, u]. (c) Find formulas (analogous o hose derived in Secion 4.1) for he variances of he presen values wih expeced values lised in iem (a) above. Exercise 9 Le T > be he ime of occurrence of some even and le N = 1 [T, ) () be he (very simple) couning funcion ha, for each ime, couns how many imes he even has occurred by ime (none before ime T and once a and afer ime T ). Applicaion of he rule of inegraion by pars o he produc N 2 = N N, gives N 2 = N 2 + N τ dn τ + N τ dn τ. (F.1) Check ha his relaion is rue for all by finding he value of each individual erm for < T (rivial) and for T (almos rivial). The lef-limi in he las inegral on he righ hand side of (F.1) is now essenial since he wo facors in he produc cerainly have a common jump. Wha happens if you ignore he lef-limi? Explain ha he firs inegral on he righ hand side of (F.1) is he same as (Nτ + 1) dnτ and ha rivial fac ha N = dnτ. Exercise 1 Nτ dnτ = for all. Thus (F.1) boils down o he Using he program ode-1.pas, do Iems (a), (b), and (c) below; for each iem run he program and prin he oupu file. (a) Compue he survival funcion p for he moraliy law in (3.25) in BL and oupu values for =, 1,..., 1. (b) Modify he program so ha i compues p 7 and oupus is values for =,1,...,3. (Inser he following saemens in he appropriae places: x := 7; erm := 3; oup := 3;) (c) Now consider he expeced life lengh in years for a new-born, ē : = τ p dτ. Make he program compue and oupu values for =, 1,..., 1. Exercise 11 Sudy he program hiele1.pas, which solves Thiele s differenial equaion numerically for a fairly general single-life policy. (a) Check he accuracy of he program by compuing some funcions ha possess closed formulas (e.g. ā 3 = (1 e 3r )/r by seing n = 3, µ =, ba = 1, and all

272 APPENDIX F. EXERCISES 5 oher paymens null). In he following use he Danish echnical basis given in he program. (b) Compue 3 E 3+, ā 3+ 3, Ā 1, Ā 3+ 3, (15 )+ 15 ( 15) + ā for =, 5,..., 3. (c) Compue he quaniies lised in (b), now wih r =, o find 3 p 3+, ē 3+ 3, 3 q 3+, 1 (!), and (15 )+ 15 ( 15) + ē 3+. Compare wih he resuls in (b) o ge an impression of he impac of ineres. (d) Seing insead he moraliy rae µ equal o, compue he quaniies in (b) again (some of hem are now rivial), idenify heir proper names and symbols in he world of deerminisic paymens, and compare wih he quaniies obained in (b) o ge an impression of he impac of moraliy. (e) Do he same exercise again, now assuming ha boh r and µ are, which means ha you jus have o wrie down he oal (no discouned)paymens under some simple deerminisic conracs. Cerainly you do no need he program o find he answers. Compare he resuls wih hose obained in he previous iems. (f) Going back o he full model wih moraliy and ineres, compue he ne premium level π and ne premium reserve V a imes =, 5, 1, 15, 2, 25, 3 for he sandard forms of insurance reaed in Chaper 3, firs for he (rare) case wih a single premium a ime (has already been done in Iem (b) above), and hen for he case wih coninuous premium a level rae. For he deferred annuiy, ake m = n = 15. Draw he missing graphs in Figures (g) Compue he variances of he random variables whose expeced values are lised in (b). Exercise 12 Revisi Exercise 11 Iem (b), which liss he reserves for four conracs wih sandard benefis agains single premium a ime. Compue he corresponding reserves under he modified scheme where k V is paid back o he insured (i.e. his or her dependens) upon deah during he erm of he conrac. Use k =.5. (You have already he numbers for he cases k = and k = 1, and you should be able o explain why ha is so.) Exercise 13 Prove ha, if he moraliy inensiy is an increasing funcion of age, hen he ne premium reserve for a deferred level benefi life annuiy agains level equivalence premium in he deferred period (boh coninuously paid) is an increasing funcion of ime hroughou he deferred period, and hereafer is a decreasing funcion. Exercise 14

273 APPENDIX F. EXERCISES 6 Consider he endowmen insurance agains coninuously payable premium, which comprises he hree main ypes of coningen paymens: a deah benefi of b a ime (, n), a life endowmen of b n a ime n, and a life annuiy wih inensiy π a ime (, n). Thiele s differenial equaion for he reserve V is (, ), wih side condiion d V = π b µx+ + (r + µx+) V, d (F.2) V n = b n. We remind of he direc consrucion of he differenial equaion a he beginning of Secion 4.4 and add here some furher deails: Denoe by P V (,u] he presen value a ime of benefis less premiums in he ime inerval (, u]. By definiion, V = E[P V (,n] T x > ]. We have P V (,n] = P V (,+d] + e r d P V (+d,n]. (F.3) Take expecaion in (F.3), given ha he policy is in force a ime. On he lef we ge V. On he righ we use he rule of ieraed expecaion, condiioning on wha happens in he small ime inerval (, + d]: wih probabiliy µ x+ d he insured dies, and he condiional expeced value is hen jus b ; wih probabiliy 1 µ x+ d he insured survives, and he condiional expeced value is hen π d + e r d V +d. We gaher V = µ x+ d b + (1 µ x+ d)( π d + e r d V +d ). Subracing V +d on boh sides, dividing by d, and leing d end o, we arrive a (F.2). (a) Use he same echnique o obain a differenial equaion for he condiional second order momen, V (2) = E[(P V (,n] ) 2 T x > ]. Sar by squaring (F.3), (P V (,n] ) 2 = (P V (,+d] ) P V (,+d] e r d P V (+d,n] + e 2r d (P V (+d,n] ) 2, and work along he lines above. Having obained a differenial equaion for V (2), you find also one for he condiional variance M (2) = V (2) V 2 upon differeniaing: d d M (2) = d d V (2) 2 V d d V. (b) Explain how you can compue he variance funcion M (2) hiele1.pas. by exending he program

274 APPENDIX F. EXERCISES 7 Exercise 15 Poisson processes, which are oally memoryless, can of course be generaed from coninuous ime Markov chains, which are more general. For insance, le {Z()} be a Markov chain on he sae space {1, 2} wih inensiies of ransiion µ 12() = µ 21() = µ. Then {N()} defined by N() = N 12() + N 21() (he oal number of ransiions in (, ]) is a Poisson process wih inensiy µ; ransiions couned by N occur wih inensiy µ a any ime regardless of he pas hisory of he process. Two independen Poisson processes, {N 1()} wih inensiy µ 1 and {N 2()} wih inensiy µ 2, can be generaed by leing {Z()} be a Markov chain on he sae space {1, 2, 3, 4} wih inensiies µ 12() = µ 21() = µ 34() = µ 43() = µ 1, µ 13() = µ 31() = µ 24() = µ 42() = µ 2, µ 14() = µ 41() = µ 23() = µ 32() =, and defining N 1() = N 12()+N 21()+N 34()+N 43() and N 2() = N 13()+N 31()+N 24()+N 42(). Three independen Poisson processes can be generaed from a Markov chain wih 8 saes (work ou he deails), and, in general, k independen Poisson processes can be generaed from a Markov chain wih 2 k saes. In he following le {Z()} be a Markov chain on he sae space {1, 2} and ake µ 12() = µ 21() = 1. The oal ime spen by Z in sae 1 during he ime inerval (, n] is T 1(, n] = n I 1(τ) dτ, and he oal number of ransiions made from sae 1 o sae 2 in ha inerval is N 12(, n] = N 12(n) N 12(). (These quaniies can be viewed as presen values of benefis of annuiy ype and assurance ype, respecively, for a wo-sae policy wih no ineres.) (a) Assume Z() = 1. Wha are he inerpreaions of he random variables T 1(, u] and N 12(, u] in erms of he Poisson process N() = N 12() + N 21()? (b) Find, by solving he relevan differenial equaions analyically, explici expressions for he firs wo sae-wise condiional momens V (q) j () = E[T q 1 (, n] Z() = j], W (q) j () = E[N q 12(, n] Z() = j], j = 1, 2, q = 1, 2, and find also he corresponding variances. (You should obain e.g. E[T 1(, n] Z() = 1] = 1 ( ) 1 + 2µ e 2µ, 4µ Var[T 1(, n] Z() = 1] = 1 16µ 2 ( 1 + 4µ (2 e 2µ ) 2). (c) Using he program prores1.pas, solve he differenial equaions also numerically and compare he resuls wih he exac soluions obained in (b). Exercise 16 Consider he Markov chain model for deah, sickness, and recovery skeched in Figure 7.3, which is ap o describe insurances wih paymens ha depend on he sae of

275 APPENDIX F. EXERCISES 8 healh of he insured (e.g. disabiliy pension or waiver of premium during disabiliy). Apar from he names of he saes and he symbols for he inensiies, we adop sandard noaion for he ransiion probabiliies p jk (, u) and he occupancy probabiliies p jj (, u). (a) Prove, boh by a forward and a backward argumen, ha ( u ) p aa(, u) = exp (σ(s) + µ(s)) ds. (b) Given sar in sae a a ime, wrie up he probabiliy ha he process remains in a during he ime inerval [, 1), hen jumps o sae i in [ 1, 1 + d 1), hen remains in i during he ime inerval [ 1 + d 1, 2), hen jumps o sae a in [ 2, 2 + d 2), hen remains in a during he ime inerval [ 2 + d 2, 3), and finally jumps o sae d in he ime inerval [ 3, 3 + d 3). This is he probabiliy of one paricular full specificaion of he hisory of he process. (c) Wrie down he forward Kolmogorov differenial equaions for p aa(s, ) and p ai(s, ) and heir appropriae iniial condiions a ime = s. (Since p ad (s, ) = 1 p aa(s, ) p ai(s, ), here is no need o work wih he differenial equaion for p ad (s, ). This is a useful propery of he forward equaions.) (d) Find an explici soluion o he differenial equaions in (c) in he case where he inensiies are consan. Since he ransiion probabiliies in his case depend only on s, we can se s = and simplify noaion o p jk (, ) = p aa(). (Hin: Differeniae he differenial equaion for e.g. p aa(), and subsiue expressions for p ai() and is firs order derivaive o obain a second order differenial equaion involving only p aa(). This equaion can be solved by a sandard echniques.) Exercise 17 Coninuing Exercise 16: We consider an insurance policy purchased by an x years old person in acive sae. We are now ineresed in he life hisory afer age x, condiional on he sae hen being a. Therefore, o visualize he insured s age a enry and he ime elapsed since issue of he policy, we pu σ() = σ x+, ρ() = ρ x+, µ() = µ x+, ν() = ν x+ and p aa x = p aa(x, x + ) ec. From Exercise 16 (c) we fech he (forward) differenial equaions for p aa x = p aa(, ) and p ai x = p ai(, ) and heir side condiions. (a) Find closed form expressions for p aa x, p ai x, and p ii x in he case wihou recovery (ρ x+ ). You can eiher solve he differenial equaions or pu up he expressions by direc reasoning, see Chaper 7. (b) The remaining lifeime T x of (x) has survival funcion p [x] = p aa x + p ai x. Wha are he condiional probabiliies, p aa x and p ai x, of being acive and invalid, respecively, a ime, given survival o ime (and sar as acive a ime )? Find he moraliy inensiy µ [x]+ of he survival funcion p [x]. Discuss he expression for µ [x]+, observe ha i is selec in general, and give a verbal explanaion of he selecive mechanism which is a work here.

276 APPENDIX F. EXERCISES 9 (c) Assume now ha µ x+ = ν x+, usually referred o as he case of non-differenial moraliy. In his case µ [x]+ = µ x+, of course. Verify ha he condiional probabiliies p aa x and p ai x are he corresponding ransiion probabiliies in he parial model skeched in Figure F.1. (Check ha hey are soluions o he forward differenial equaions for he ransiion probabiliies in he parial model.) a acive σ x+ i invalid ρ x+ Figure F.1: A parial Markov chain model for sickness and recovery, no deah. (d) Consider he following case: Age a enry is x = 3, erm of conrac is n = 3, he benefi is a life insurance of 1, premium is payable a level rae while acive (waiver of premium during disabiliy), ineres rae and ransiion inensiies are as given in he sandard version of he program prores1. Fill in appropriae saemens in he program prores1 o make i compue he sae-wise reserves a imes =, 1,..., 3. Exercise 18 Le Z(), be a ime-coninuous Markov chain on a finie sae space J = 1,..., J, saring in sae 1 a ime ; Z() = 1. Assume ha i possesses ransiion inensiies, and adop sandard noaion for basic eniies: inensiies of ransiion µ jk (), ransiion probabiliies p jk (, u), and probabiliies of uninerruped sojourns p jj (, u). (a) Le 1 r r+1 be imes in [, ) and le j, j 1,..., j r, j r+1 be saes in J. Express he following probabiliies in erms of he basic eniies: [ r+1 ] P Z( i) = j i ; i= [ r ] P Z( i) = j i Z( ) = j, Z( r+1) = j r+1. (F.4) i=1 (b) Le s and be fixed imes such ha s <, and le i and j be fixed saes. Use he resul in (F.4) (if you go i righ) o show ha, condiional on Z(s) = i and Z() = j, he process Z(τ), τ [s, ], is a Markov chain. (c) Suppressing he dependence on s, and i, j from he noaion, denoe he condiional Markov chain by Z(τ), τ [s, ], and denoe is ransiion probabiliies and inensiies by p gh (τ, ϑ) and µ gh (τ), respecively. Deermine hese probabiliies and inensiies.

277 APPENDIX F. EXERCISES 1 (d) Wha is he limi of µ gh (τ) as τ ends o? Disinguish beween he cases where (i) g and h are boh differen from he desinaion sae j, (ii) g j and h = j, (iii) g = j and h j. Give a verbal explanaion of he resuls. (e) Now specialize o he disabiliy model in Figure 7.3, and condiion on Z() = a and Z() = i. Wrie ou he expression for σ(τ) and he value of µ(τ) (rivial). Find an explici expression for σ(τ) in he case wih consan inensiies and no recovery, and discuss i wih paricular aenion o he limiing value as τ ends o. Exercise 19 (a) Describe in words he single life insurance policy for which Thiele s differenial equaion is d V d = r V + c µx+(1 V), < < n, subjec o he condiion V n = 1. Inegrae he equaion, using he echnique wih inegraing facor, o obain he direc prospecive expression for he reserve V. Express he reserve in sandard acuarial noaion. Explain how o deermine c so as saisfy he equivalence requiremen V =. (b) Prove ha if µ x+ is an increasing funcion of [, n], hen V is an increasing funcion of [, n]. Consruc an example where V is no an increasing funcion of for all (, n). Exercise 2 (a) Le T be a random lifeime wih coninuous survival funcion p, and le G be a non-decreasing real-valued funcion such ha E[G(T )] is finie. Prove he formula E[G(T )] = G() + p dg(). (F.5) Le w(), defined on [, ), be a coninuous and sricly posiive funcion such ha w(s) ds =. Define and W () = w(s) ds ( ) γ δ p = = (1 + W ()/δ) γ. (F.6) W () + δ (b) Show ha p is a survival funcion, and find he corresponding moraliy inensiy µ. Find he condiional survival funcion p x and observe ha i is of he same form as p, only wih differen W and δ. Prove ha W (T ) + δ is Pareo-disribued wih parameers (γ, δ).

278 APPENDIX F. EXERCISES 11 (c) Consider he special case where w() = 2, and γ > 1. Use (F.5) o find an explici expression for E[T 2 ]. (d) The parameers γ and δ are o be esimaed from daa on n independen lives observed from birh unil aained age z or deah, whichever occurs firs. Thus, if T m is he life lengh of person No. m, he observaions are T m z, m = 1,..., n. Wrie up he log likelihood funcion, derive he likelihood equaions ha deermine he ML esimaors ˆγ and ˆδ, and explain wihou doing calculaions how o find heir asympoic covariance marix. (e) Assume ha δ is known. Then here is an explici expression for ˆγ. Find his expression and use (F.5) o show ha ˆγ converges o γ wih probabiliy 1. (f) Assume now ha he daa available are he occurrence-exposure raes in years j = 1,..., z. Explain how o esimae γ and δ by he echnique of analyic graduaion by weighed leas squares. Exercise 21 An x years old person buys a permanen disabiliy pension policy specifying ha premium is paid coninuously a fixed rae c per ime uni in acive sae and pension is payable coninuously a fixed rae b per ime uni in disabled sae. The policy erminaes z years afer issue. The relevan Markov model is skeched in Fig. F.2. Assume ha he ineres rae r is consan. a acive σ x+ i disabled µ x+ ν x+ d dead Figure F.2: A Markov chain model for permanen disabiliy insurance. (a) Derive he differenial equaions and heir side condiions for he reserves in he saes a and i. (b) Suppose he reserve is paid back upon he insured s deah as acive before ime z. How does his change he differenial equaions in Iem (a)?

279 APPENDIX F. EXERCISES 12 (c) Describe in words he produc for which he reserves saisfy he differenial equaions d Va() d = rva() + c σx+(vi() Va()) µx+(1 Va()), d Vi() d = r Vi() b νx+(1 Vi()), wih side condiions V a(z ) = 1 and V i(z ) = 1. (d) Wrie down, wihou proofs, explici expressions for he probabiliies p aa(, ), p ii(, ), and p ai(, ). (e) Assume ha n independen policies, idenical o he one described above, are observed in z years, and ha he inensiies are of he form: µ x+ = α + β e γ (x+), ν x+ = α + β e γ (x+), σ y+ = σ (consan). Explain how o esimae he parameers α, β, γ, γ, and σ from he daa. Wrie up he log likelihood funcion and derive he likelihood equaions ha deermine he ML esimaor. Exercise 22 Use he program acuary.pas o sudy how he (fixed) premium rae and he reserve per survivor depend on he age of he insured, he lengh of he premium paymen period, he erm of he conrac, and he ineres rae for various forms of insurance. In he case of life annuiies, compare wih he corresponding bank savings soluion. Exercise 23 Prove ha an FV funcion has a mos a counable number of disconinuiies. (Hin: Consider a non-decreasing and finie-valued funcion X. Is se of disconinuiies is D(X) = m=1 n=1 D m,n, where D m,n = { ; n 1 < n, X + X 1 }. Show ha Dm,n is finie, possibly m empy. Thus, D(X) is a counable union of finie ses, and is hus a mos counable.) Exercise 24 Define ε u() = 1 [u, ) (), which for fixed u is a very simple RC and PD funcion of. Apply (A.7) o he very ransparen case where X = ε 1 (),and Y = ε 2 (), < 1 2, and compare he resuls of he formal calculaions o wha you can pu up by direc reasoning. Exercise 25

280 APPENDIX F. EXERCISES 13 Le X be an RC and PD funcion and le N coun is disconinuiies. Prove ha ( ) q 1 dx q = qx q 1 x c q d + X p p (x d ) q p dn, (F.7) p= which generalizes he well-known rule dx q = qx q 1 dx for coninuous X. Exercise 26 Le N be a couning process and consider a process S defined by S = exp(a + bn ), where a and b are consans. Use Iô s formula o show ha he dynamics of S is given by ( ) ds = S a d + (e b 1)dN. In inegral form S = 1 + S τ ( a dτ + (e b 1)dN τ ). Assume now ha N is a Poisson process wih inensiy λ. By he independen incremens propery of N, we can heurisically conclude ha X τ and dn τ are independen, so E[S τ dn τ ] = E[S τ ]E[dN τ ]. Use his ogeher wih E[S τ ] = E[S τ ] o obain an inegral equaion for E[S ]. Solve i o find ( ( ) ) E[S ] = exp a + e b 1 λ. Verify he resul by direc calculaion using ha N has a Poisson disribuion wih parameer λ. Exercise 27 Verify (2.4). (A recommended ex on funcional equaions is [2].) Exercise 28 ( ) (a) Apply (A.8) o (2.14), aking X = exp rs ds and Y = n exp ( τ rs ds) db τ, o obain (2.19) and inegrae over (, n] o obain (2.22). (b) Prove he relaionship (2.24). by applying inegraion by pars (A.9) o n (i is convenien o pu db τ = d(b τ B )). Give a verbal inerpreaion of (2.24). Then prove (2.23) by using (2.9). e τ r db τ (c) Observe ha if B consiss only of a uni payable a ime n, hen, essenially, all he relaionships (2.22), (2.23), and (2.24) reduce o (2.9).

281 APPENDIX F. EXERCISES 14 Exercise 29 (a) Le A = B and A = B be wo paymen funcions wih rerospecive and prospecive reserves denoed by U and V and U and V, respecively. Assuming ha he ineres rae is always posiive, verify he following raher obvious asserions: If A A for all, hen U U for all. If, moreover, A < A for some, hen U τ U τ for all τ. Similarly, if B n is finie and B n B B n B for all, hen V V for all. (b) In paricular, any advancemen of deposis will produce an increase of he rerospecive reserve if he ineres rae is posiive. This is he general circumsance underlying resuls like he following abou ordering of he presen values in (2.28), (2.29), and (2.31): a n < ā n < ä n. Exercise 3 Le he paymen sream A represen deposis less wihdrawals on an n years savings accoun ha bears ineres wih sricly posiive ineres rae. I is required ha U for all, wih sric inequaliy for some, and ha U n =, see (1.1) and (1.11). Prove ha A n <, and explain his resul. Exercise 31 We refer o he heory of loans in Subsecion 2.2. In pracice amorizaions are ypically due a whole years afer he loan is paid ou and, accordingly, ineres is calculaed on an annual basis. Thus, he paymen funcions A, F, and R are pure jump funcions wih jumps a, f, and ρ, respecively, a imes = 1,..., n. (The symbol ρ is chosen since r is reserved for he insananeous ineres rae, which is somehing differen.) Denoe he annual ineres rae in year by i and he corresponding discoun facor by ( v = (1 + i ) 1. They are relaed o he insananeous ineres rae r by 1 + i = exp ). 1 rτ dτ (a) Rewrie he relaionships appearing in Subsecion 2.2 in erms of he A, a, F, f, R, ρ, i, and v, resricing o discree imes =,..., n. In paricular, dynamical relaions like (2.38) are o be wrien ou as recursive equaions in he syle of Chaper 1. (b) Use he program acuary.pas on L:\KUFML (se moraliy equal o ) o sudy numerically he properies of various forms of loans; fixed loan (f = for = 1,..., n 1 and f n = 1), series loan (f = f for = 1,..., n), and annuiy loan (a = a for = 1,..., n). Exercise 32 By inspecion of Fig. 3.1, esimae roughly he expeced number of years spen in he age inervals (, 1], (1, 2],..., (9, 1] for a new-born. Compue he exac figures.

282 APPENDIX F. EXERCISES 15 Exercise 33 Saring from a given aggregae moraliy inensiy µ x, here are uncounably many ways of invening a selec moraliy law µ [x], wih selec period of a given lengh s and ulimae moraliy law µ x+, s. We shall propose four possibiliies: (a) A duraion dependen weighed moraliy inensiy, µ [x], = w() ( w() νx+ + 1 w() ) µ x+, w() where ν is some moraliy inensiy such ha ν x < µ x for all x >, and w is some non-increasing funcion such ha w() > and w() = for s. For insance, one could ake ν x = µ x a for some a > (assuming µ is increasing) and w() = (1 /s) q wih q >. If q > 1, hen he derivaive of w a = s is, making µ smooh here. Sugges anoher choice of w. (b) A moraliy inensiy obained by reducing age in he selec period, wih w as in iem (a) above. (c) A simple Cox regression model, wih w as in iem (a) above. (d) A piecewise consan inensiy, j = 1, 2,..., k = 1, 2,..., s. µ [x], = µ x+ w() µ [x], = exp(β 1w() + β 2w 2 ())µ x+ µ [x], = λ j + κ k, j 1 x + < j, k 1 < k, (e) Discuss in general erms how he selec moraliy laws in iems (a) - (d) can be esimaed from daa. Maers may simplify in (a) - (c) if you assume ha he ulimae moraliy law is known. Carry ou a more formal analysis for he case (d). Exercise 34 The bank proposes a savings conrac according o which (55) saves a fixed amoun c annually in 15 years, a ages 55,...,69, and hereafer wihdraws a fixed amoun of b = 1 anually in 1 years, a ages 7,...,79. The annual ineres rae is i =.45. The balance equaion is 14 j= (1 + i) 24 j c 24 j=15 (1 + i) 24 j b =,

283 APPENDIX F. EXERCISES 16 which gives 24 j=15 c = vj 14 =.381 b. j= vj Find he annual premium for he corrresponding life annuiy wih benefis payable coningen on survival. Exercise 5 So-called Loss of profis insurance covers losses suffered by firms during cessaion of producion caused by fire or oher accidenal evens. Suppose ha a firm a any ime is eiher in sae a = acive (normal producion) or i = inacive (no producion). The insurance can be purchased when he firm is in acive sae. The indemnificaion (benefi) is paid coninuously a consan rae 1 per ime uni in inacive sae, and premium is paid as a single amoun upon issue of he policy a ime (say). Denoe he erm of he conrac by n. Le Z() denoe he sae of he firm a ime. Under hese convenions he oal indemnificaion in respec of he policy is he ime spen in inacive sae during he erm of he conrac, T n = n I {Z(τ)=i} dτ. Assume ha Z is a ime-homogeneous Markov chain as skeched in Figure F.3; he inensiies σ and ρ are consan. Pu u n and j, k {a, i}. p jk (, u) = P[Z(u) = k Z() = j], a Acive σ ρ i Inacive (a) Show ha Figure F.3: A Markov chain model for loss of profi insurance. p ai(, u) = σ ( 1 e (σ+ρ)(u )), σ + ρ and p aa(, u) = 1 p ai(, u). The probabiliies p ia(, u) and p ii(, u) are given by he same expressions when σ and ρ change roles. Explain why he probabiliies p jk (, u) depend only on he lengh of he ime inerval, u. Noice ha and explain why he resul is reasonable. lim pai(, u) = σ u σ + ρ,

284 APPENDIX F. EXERCISES 17 (b) Find he single premium π n = E[T n] for he loss of profi insurance described above; Employ he principle of equivalence assuming no ineres. Show ha he premium per year, π n/n, is an increasing funcion of n, wih upper limi σ/(σ + ρ) (compare wih he las formula in (a)). (c) The inensiies σ and ρ are o be esimaed from saisical daa from m independen, idenical firms, which have been insured and observed over he same period of ime, [, n]. Assume firs ha for each firm here is a complee record of ransiions beween saes in he observaion period. Wrie up he likelihood funcion, explain how he maximum likelihood (ML) esimaors of he parameers are derived, and find heir asympoic variances. Examine hese variances as funcions of n for fixed mn = w (oal ime under observaion), and commen. (d) iem For s 1 s r u n and j 1,..., j r {a, i}, find he condiional probabiliy p jk (, u) = P[Z(u) = k Z(s 1) = j 1,..., Z(s r) = j r, Z() = j, Z(n) = i], a funcion of j, k,, and u. Conclude ha, condiional on Z(n) = i, he process Z behaves like a Markov chain Z wih ransiion probabiliies p jk (, u). Find he ransiion inensiies σ() and ρ() for Z. (e) Assume now ha ρ = and ha he daa available are occurrence-exposure raes in years j = 1,..., n for hose firms ha are inacive a ime n and no informaion abou oher firms in he porfolio. Using he resuls in (a) and (d), explain how o esimae σ by he echnique of analyic graduaion. (Do no ry o find an explici expression for he esimae of σ.) Exercise 51 (a) Consider he Markov model for deah, sickness, and recovery skeched in Figure 7.3. Apar from he special symbols for he inensiies, we adop sandard noaion and assumpions: Z() is he sae a ime, he process sars from acive sae, Z() = a, p jk (, u) = P [Z(u) = k Z() = j], (F.8) and p jj (, u) = P [Z(τ) = j, τ [, u] Z() = j]. (F.9) (a) iem Derive forward differenial equaions for he probabiliy p (1) ai (, ) of being disabled for he firs ime a ime and for he probabiliy p (1) aa (, ) of being acive a ime afer having been disabled once. Proceed in he same manner o derive forward differenial equaions for he probabiliies p (k) ai (, ) and p(k) aa (, ) of being disabled and acive, respecively, a ime afer exacly k onses of disabiliy, for k = 2, 3,... Wha is (, )? k=1 p(k) ai (b) Express he probabiliy P [Z(τ) = i; τ [ q, ] Z() = a] in erms of he basic eniies defined in (F.4) and (F.9).

285 APPENDIX F. EXERCISES 18 (c) A ime an acive person buys an insurance policy which specifies ha a pension benefi is o be paid coninuously wih fixed inensiy 1 as long as he insured is disabled and has been so uninerrupedly for a leas q years. (The erm q is called he qualifying period.) The policy expires upon deah or, a he laes, a ime n (> q). A single premium π is paid a ime. Deermine he premium π by he principle of equivalence, assuming ha he ineres rae r is consan. Find he reserve a ime < n q for an insured who is disabled and is currenly receiving he disabiliy benefi. (d) Consider he Markov chain model in Figure 7.3. Fill in appropriae saemens in he enclosed program prores1 o make i compue he ransiion probabiliies p ai(, 1) and p ii(, 1) for =, 1,..., 1 in he case where all he inensiies are consan: µ = ν = σ =.1 and ρ =. Exercise 52 (a) We adop he usual noaion and assumpions of he heory of muli-life insurance policies and consider wo independen lives (x) and (y) wih remaining life lenghs T x and T y, respecively. Assume ha he benefi is an assurance of 1 payable a ime T y if T x + n/2 < T y < n and ha premium is payable a consan rae π unil ime min(t x, T y, n/2), where n is he erm of he conrac (fixed). Deermine he equivalence premium π. Exercise 53 The employees of a firm are auomaically members of a pension scheme wih salary dependen premiums and benefis defined as follows. Consider an employee (x), who eners he scheme x years old a ime (say), reires a pensionable age 65 a ime m = 65 x, and earns salary a rae S() per ime uni a ime [, m]. Coningen on survival, premium is payable coninuously a rae πs() a ime [, m], and pension is received as an endowmen of 5S(m) upon reiremen a ime m. Assume ha he economy is governed by a coninuous ime Markov chain Y (),, wih sae space J = {1,..., J}, consan inensiies of ransiion λ jk, j k, and iniial sae Y () = i, say. A any ime he accumulaion facor U() of he invesmen porfolio is given by ( ) U() = exp r(s) ds, r(s) = I j(s)r j, j and he salary rae is given by ( ) S() = exp a(s) ds, a(s) = j I j(s)a j. Here, I j() = 1[Y () = j], and he r j and he a j are known, fixed numbers; r j is he ineres rae and a j is he rae a which salary increases per ime uni and per uni of salary when he economy is in sae j.

286 APPENDIX F. EXERCISES 19 (a) Demonsrae how o deermine he premium rae π ha makes expeced discouned benefis less premiums equal o a ime ; Consruc differenial equaions for benefis and for premiums and specify appropriae side condiions. (b) Explain ha, if a j = r j for all saes j so ha salary is perfecly linked o invesmens, hen equivalence can be aained in a large (in principle infiniely large) porfolio of idenical policies. Exercise 54 (a) Describe briefly he main characerisics of wih-profi insurance and uni-linked insurance. (b) Consider a pure endowmen benefi a ime n (he erm of he conrac) agains a single premium a ime. Assume ha he invesmen porfolio of he insurance company bears ineres a rae r() a ime, where r is driven by a Markov chain as described in Quesion 6. Show how o deermine he single premium for a uni-linked conrac wih sum insured max(u(n), g), where U(n) is he index of he invesmen porfolio given by U() as defined in Quesion 6, and g is a guaraneed minimum sum insured. Exercise 55 We adop he usual noaion and assumpions of he heory of muli-life insurance policies and consider wo independen lives (x) and (y) wih remaining life lenghs T x and T y, respecively. (a) Assume ha he benefi is an assurance of 1 payable a ime T y if 2T x < T y < n and ha premium is payable a consan rae π unil ime min(t x, T y, n/2), where n is he erm of he conrac (fixed). Deermine he equivalence premium π. (b) Propose a mehod for compuing he premium numerically. (Hin: One possibiliy is o rea /2 p x as a survival funcion p x wih inensiy µ x+, which you would need o express in erms of µ, and hen solve a Thiele differenial equaion numerically.) (c) Deermine he reserve a any ime, assuming ha he insurer currenly knows he complee pas hisory of he wo lives. You need o disinguish beween various cases, wheher (y) is alive or dead, wheher is before or afer ime n/2, and wheher x is alive or dead and, if dead, when. Is he reserve always non-negaive? (d) Wha is he variance of he presen value of he benefi? SURPLUS, BONUS, WITH PROFIT, GUARANTEES, UNIT-LINKED A. Preliminaries. We are going o resae he heory of surplus and bonus and relaed problems in he framework of he simple, sill fairly general, single life conrac reaed in Secion 4.4 and add maerial on ineres guaranees and uni-linked insurance.

287 APPENDIX F. EXERCISES 2 The erms of he conrac are se ou in he expression for he prospecive reserve, V = n and he equivalence relaion e τ (r u+µ x+u ) du (µ x+τ b τ π τ ) dτ + e n (r u+µ x+u ) du b n, V = π, (F.1) where π is he lump sum premium paymen colleced upon he incepion of he policy (i may be, of course). The equivalence relaion (F.1) can be cas as π + e τ (r u+µ x+u ) du (π τ µ x+τ b τ ) dτ e (r u+µ x+u ) du V =. (F.11) The firs wo erms in (F.11) are he expeced presen value a ime of premiums less benefis up o and including ime. The second erm is minus he expeced presen value a ime of benefis less premiums afer ime. Thus, he equivalence principle ensures ha, a any ime, ne incomes in he pas provide precisely he amoun needed o mee ne liabiliies in he fuure. Thiele s differenial equaion is wih he boundary condiion d V = π µx+ b + (r + µx+) V, d (F.12) V = b n. (F.13) B. Wih profi conracs (paricipaing policies); Surplus and Bonus. Insurance policies are long erm conracs, wih ime horizons wide enough o capure significan variaions in ineres and moraliy. Therefore, a ime when he conrac is wrien wih benefis and premiums binding o boh paries, he fuure developmen of (r, µ x+), >, is uncerain, and i is impossible o foresee which premium level will saisfy (F.11) and esablish equivalence in he end. If i should urn ou ha, due o adverse developmen of ineres and moraliy, premiums are insufficien o cover benefis, hen here is no way he insurance company can avoid a loss; i canno reduce he benefis and i canno increase he premiums since hese were irrevocably se ou in he conrac a ime. The only way he insurance company can preven such a loss, is o charge a premium on he safe side, high enough o be adequae under all likely scenarios. Then, if everyhing goes well, a surplus will accumulae. This surplus belongs o he insured and is o be repaid as so-called bonus, e.g. as increased benefis or reduced premiums. The usual way of seing premiums o he safe side is o base he calculaion of he premium level and he reserves on a provisional firs order basis, (r, µ x+), >, which represens a wors case scenario and leads o higher premium and reserves han are likely o be needed. We follow common pracice and ake he firs order ineres rae o be consan, r. (From a mahemaical poin of view his is jus a maer of noaion.) The reserve based on he pruden firs order assumpions is called he firs order reserve, and we denoe i by V. I saisfies Thiele s differenial equaion d d V = π + r V µ x+(b V ), (F.14) subjec o he naural side condiion V n = b n. The premiums are deermined so as o saisfy he firs order equivalence relaion V = π.

288 APPENDIX F. EXERCISES 21 Taking our sand a a given ime afer he incepion of he policy, he developmen of ineres and moraliy in he pas, (r τ, µ x+τ ), τ, is now known and can be invoked in an updaed calculaion of he ne incomes up o ime. The fuure developmen of ineres and moraliy, (r τ, µ x+τ ), τ >, remains uncerain, however, so for he assessmen of fuure liabiliies one mus sick o he conservaive firs order basis, (r, µ x+τ ), τ >. Thus, insead of he balance equaion (F.11), which canno be se up since i involves unknown fuure raes of ineres and moraliy, we have he following expression for he mean surplus per policy a ime : S = π + e τ (r u+µ x+u ) du (π τ µ x+τ b τ ) dτ e (r u+µ x+u ) du V. (F.15) If he facual ineres and moraliy in he pas were more favourable han he pessimisic firs order basis, hen his surplus is posiive. C. Emergence of surplus. To see how he surplus is emerges, we need o sudy he dynamics of S. Differeniaing (F.15), using (F.14), and rearranging erms, we obain where c = (r r ) V d (r S = u+µ x+u ) e du c, d + (µ x+ µ x+)(b V ). (F.16) Obviously, c is he rae a which surplus emerges per survivor and per ime uni a ime. Inerpre he wo erms on he righ hand side of (F.16) as surplus emerging from safey margins in he ineres rae and in he moraliy rae, respecively. We can reasonably say ha a firs order elemen is se on he safe side if he corresponding conribuion o he surplus is posiive. By inspecion of he firs erm on he righ of (F.16), we see ha r is on he safe side as long as i is less han he rue r (provided ha he firs order reserve is posiive, as i should be for any meaningful conrac). By inspecion of he second erm on he righ of (F.16), we see ha he sign of he sum a risk by deah, b V, deermines how o se firs order moraliy o he safe side: If he sum a risk is posiive (e.g. erm assurance or endowmen assurance), hen µ x+ is on he safe side if i is bigger han µ x+. If he sum a risk is negaive (as is he case for e.g. a pure endowmen, a deferred annuiy, or some oher savings insurance wih b = ), hen µ x+ is on he safe side if i is less han µ x+. D. Redisribuion of surplus as bonus. The word bonus is Lain and means good. In insurance erminology i denoes various forms of repaymens o he policyholders of ha par of he company s surplus ha sems from good performance of he insurance porfolio, a sub-porfolio, or he individual policy. In he presen conex of life insurance i denoes he repaymens of surplus semming from favourable developmen of ineres and moraliy. Le us denoe such repaymens by b in general. For he sake of concreeness, suppose bonuses are paid back coninuously a rae b per survivor for < < n and possibly wih a lump sum b n per survivor a ime = n. By saue, surplus is o be repaid in is enirey, which means ha equivalence is o be re-esablished on basis of he rue ineres and moraliy condiions when hese are ulimaely known a he erm of he conrac: n e τ (r u+µ x+u ) du c τ dτ = n e τ (r u+µ x+u ) du bτ dτ + e n (r u+µ x+u ) du bn. (F.17)

289 APPENDIX F. EXERCISES 22 In he following Paragraphs E - G we will sudy some commonly used bonus schemes. E. Cash Bonus. This means ha surplus is being repaid coninually as i emerges, i.e. b = c, < < n, and b n =. I may for insance ake he form of a premium deducible payable a rae c as long as he insured is alive during he conrac period. In he case of a erm assurance conrac i could reasonably ake he form of an addiional paymen ˆb upon deah a ime (, n), and a naural choice is ˆb = c /µ x+. Exercise 1-12 Verify ha he wo cash bonus schemes described above comply wih he ulimae equivalence requiremen (G.36). Consruc a scheme ha is a combinaion of he wo proposed here. F. Terminal Bonus. This means ha surplus is repaid as a lump sum b n o survivors a he end of he erm, and b =, < < n. Exercise 1-13 Deermine b n by (G.36). G. Purchase of Addiional Insurance. Under his bonus scheme he surplus is spen on purchase of addiional insurance. Addiional insurance is wrien on he firs order basis and will herefore also generae surplus, which in is urn will be used for furher purchase of addiional insurance, and so on. The scheme is non-rivial and requires a bi of heoreical reasoning: For a policy in force a ime le V + denoe he expeced presen value, on he firs order basis, of fuure benefis only; V + = n I saisfies he Thiele s differenial equaion e τ (r +µ x+u ) du µ x+τ b τ dτ + e n (r +µ x+u ) du b n. d d V + = r V + µ x+(b V + ), (F.18) wih naural side condiion V + n = b n. The quaniy V + is he single premium payable a ime if he insured hen were o purchase an addiional insurance for he balance of he erm, wih he same benefis as in he original conrac. Spending he surplus c d generaed in [, +d) as a single premium for addiional benefis of he form specified in he original conrac, will buy he insured a fracion q d of fuure benefis given by c = q V +. (F.19) A any ime (, n) he deah benefis from he original conrac and from he addiional benefis purchased during (, ] oal (1 + Q ) b, (F.2) where Q = q τ dτ. (F.21)

290 APPENDIX F. EXERCISES 23 Likewise, he oal endowmen benefi a he erm of he conrac is (1 + Q n) b n. (F.22) A ime he oal surpluses from he original conrac and he addiional benefis purchased during (, ] emerge a rae c = (r r ) (V + Q V + ) + (µ x+ µ ( x+) (1 + Q )b V Q V + ). (F.23) Now all elemens needed are in place, and a dynamic compuaion will deliver he soluion. Firs, a he ime of he incepion of he conrac, he funcions V and V + and he equivalence premium π are deermined by use of he program prores1.pas (or prores2.pas ). The compuaion goes backwards saring from he side condiions V = b n and V + = bn. Then, as ime goes by and surpluses are being observed and disposed of, one compues simulaneously he funcions V and V + (again) and he random funcion Q as soluions o he differenial equaions (F.14), (F.18), and (rewrie (F.19)) d d Q = 1 V + c, (F.24) wih c given by (F.23). The compuaion goes forwards, saring from ime = wih he iniial condiions V =, (picked from he firs compuaion), and V + = V + Q =. Having deermined Q, he benefis under his bonus scheme are now given by (F.2) and (F.22). H. Prognosicaion of bonus. A regular imes (ypically annually) he cusomer receives a saemen of his policy accoun, informing abou bonus earned from surplus in he pas and also predicing fuure bonuses based on a qualified guess as o he fuure developmen of ineres and moraliy. Exercise 1-14 Ouline such a saemen wih hese pieces of informaion for he sandard conrac considered so far, including he relevan formulas, and basing he prognosicaion of fuure surplus on he assumpion ha r τ = r + r and µ x+τ = µ x+τ µ for some given posiive r and µ. Exercise 1-15 Apply he presen heory o a pension insurance policy for which benefis are an m year deferred life annuiy payable a level rae 1 per year in n years, and premiums are payable a level rae during he deferred period. Wrie ou all relaions and formulas ha differ from he corresponding ones above. Will surplus emerge also in he benefi period [m, m + n)?

291 APPENDIX F. EXERCISES 24 Exercise 1-16 Exend he heory so as o include expenses. Consider an endowmen insurance wih consan sum insured b and consan gross premium rae π (no down paymen π a ime ). Assume ha rue expenses incur wih a lump sum α + α b a ime and hereafer coninuously a rae β + β π + γ + γ b + γ V a ime (, n) as long as he policy is in force. Here V denoes he gross premium reserve on firs order basis. Firs order assumpions specify ha expenses incur wih a lump sum α b a ime and hereafer coninuously a consan rae β π + γ b. Discuss how he firs order elemens can be se on he safe side. Observe ha here is a lump sum conribuion o surplus a ime. I. Sochasic ineres. The uncerain developmen of he second order elemens can be buil ino he model by describing he ineres rae and he (parameers of he) moraliy rae as sochasic processes. To keep hings simple, we will focus on ineres, which is he more imporan of he wo, and assume ha he moraliy is perfecly prediced by he firs order basis: µ x+ = µ x+. This means ha conribuions o surplus sem only from ineres gains, so ha c = (r r )V. (F.25) As a simple, bu flexible, model for sochasic ineres, we will assume ha {r } is generaed by he Markov chain model in Secion 7.8 of BL, see also Exercise 2. To save space, we will wrie Y and r insead of Y () and r() and, since subscrips are now used for he ime variable, denoe he sae-wise ineres rae by r e. The saemen of accoun, which is regularly sen o he insured, usually comes wih a prognosis of fuure bonuses on he insurance. Such a prognosis mus be based on a qualified guess abou he fuure developmen of he facual valuaion basis in our simplified siuaion abou r. This guess may be exogenous o he model, e.g. based on combined opinions of expers in he finance deparmen of he company. Having adoped a sochasic model for r, he insurer can make an endogenous, model-based forecas of fuure bonus paymens. Thus, consider a policy which is sill in force a ime, and suppose he insurer wans o inform he insured abou he condiional expeced value of fuure bonuses, given ha he curren ineres rae is r = r e (which means Y = e, assuming ha all r e are differen). We will consider a few examples. J. Cash bonus: The rae a which bonus will be paid a some fixed fuure ime u, provided he insured is hen alive, is W = (r u r )V u. A ime < u, given r = r e, W is prediced by is condiional expeced value W e() = E[W r = r e ]. Exercise 1-17 Show ha he funcions W e() are he soluion o he differenial equaions d d We() = l;f e λ ef (W e() W f ()),

292 APPENDIX F. EXERCISES 25 subjec o he condiions e = 1,..., J Y. W e(u) = (r e r )V u, K. Terminal bonus: Bonus payable as a lump sum a he erm of he conrac n, provided he insured is hen alive, is where W = n = W nτ r e s ds (r τ r )Vτ dτ τ r e s ds (r τ r )Vτ dτ + W, W = e n r s ds, W = n e nτ r s ds (r τ r )V τ dτ. The random variables W and W, which are unknown a ime, are prediced by W e() = E[W r = r e ], W e () = E[W r = r e ]. Exercise 1-18 Wriing W = e r d W +d, W = W (r r ) V d + W +d, show ha he funcions W e() and W e () are he soluion o he differenial equaions d d W e() = r e W e() + f;f e λ ef (W e() W f()), d d W e () = W e()(r e r )V + λ ef (W e () W f ()), f;f e subjec o he condiions e = 1,..., J Y. W e(n ) = 1, W e (n ) =, L. Addiional benefis: A a fixed fuure ime u bonus is paid as a muliple Q u of he conracual benefis provided he insured is hen alive. A ime we decompose Q u ino Q, which is known, and Q u Q, which is unknown, and we need o predic he laer. Recalling (F.19) and (F.25), sar from he differenial equaion d d Q = (r r ) ( V + Q ) V +

293 APPENDIX F. EXERCISES 26 and use he echnique wih inegraing facor o obain where Q u = W Q + W, W = e u (r s r ) ds, W = u uτ (r e s r ) ds (r τ r ) V τ Vτ + dτ. Exercise 1-19 Derive differenial equaions for he sae-wise predicions W e() and W e () of W and W. Exercise 1-2 (a) Predic discouned fuure cash bonuses given survival o n, n e τ (b) Predic discouned fuure cash bonuses, n r s ds (r τ r )V τ dτ. e τ (r s+µ x+s ) ds (r τ r )V τ dτ. (c) Find differenial equaions for he condiional variance, given r = r e, of he fuure cash bonuses. You may ry your hand also on he condiional variance of fuure erminal bonus. As we have said before, he Markov model proposed here can hardly be 1 per cen realisic. Now, he usefulness of a model depends on is purpose. The sole purpose of he ineres rae model is o provide he insured wih a reasonable guess as o his fuure prospecs of bonus, and for ha purpose a rough model can cerainly be adequae. Anyway, a he end of he day he bonus paymens will be deermined enirely by he facual ineres rae and will no depend on he assumpions in our model. M. Guaraneed ineres. Recall he basic rules of he wih profi insurance conrac: On he one hand, any surplus is o be redisribued o he insured. On he oher hand, benefis and premiums se ou in he conrac canno be alered o he insured s disadvanage. This means ha negaive surplus, should i occur, canno resul in negaive bonus. Thus, he wih profi policy comes wih an ineres rae guaranee o he effec ha bonus is o be paid as if facual ineres were no less han firs order ineres, roughly speaking. For insance, cash bonus is o be paid a rae (r r ) +V per survivor a ime, hence he insurer has o cover (r r ) +V. (F.26)

294 APPENDIX F. EXERCISES 27 Similarly, erminal bonus (ypical for e.g. a pure endowmen benefi) is o be paid as a lump sum ( n ) nτ r e s ds (r τ r )Vτ dτ + per survivor a ime n, hence he insurer has o cover ( n ) nτ r e s ds (r r τ )Vτ dτ. (F.27) (We wrie a + = max(a, ) = a.) An ineres guaranee of his kind represens a liabiliy on he par of insurer. I canno be offered for free, of course, bu has o be compensaed by a premium. This can cerainly be done wihou violaing he rules of game for he paricipaing policy, which lay down ha premiums and benefis be se ou in he conrac a ime. Thus, for simpliciy, suppose a single premium is o be colleced a ime for he guaranee. The quesion is, how much should i be? Being brough up wih he principle of equivalence, we migh hink ha he expeced discouned value of he liabiliy is an agreeable candidae for he premium. However, he raionale of he principle of equivalence, which was o make premiums and benefis balance on he average in an infiniely large porfolio, does no apply o financial risk. Ineres rae variaions canno be eliminaed by increasing he size of he porfolio; all policy-holders are faring ogeher in one and he same boa on heir once-in-a-lifeime voyage hrough he roubled waers of heir chaper of economic hisory. This risk canno be averaged ou in he same way as he risk associaed wih he lenghs of he individual lives. None he less, in lack of anyhing beer, le us find he expeced discouned value of he ineres guaranee, and jus anicipae here ha his acually would be he correc premium in an exended model specifying a so-called complee financial marke. Those who are familiar wih basic arbirage heory know wha his means. Those who are no should jus imagine ha, in addiion o he bank accoun wih he ineres rae r, here are some oher invesmen opporuniies, and ha any fuure financial claim can be duplicaed perfecly by invesing a cerain amoun a ime and hereafer jus selling and buying available asses wihou any furher infusion of capial. The iniial amoun required o perform his duplicaing invesmen sraegy is, quie naurally, he price of he claim. I urns ou ha his price is precisely he expeced discouned value of he claim, only under a differen probabiliy measure han he one we have specified in our physical model. Wih hese reassuring phrases, le us proceed o find he expeced discouned value of he ineres guaranee. (a) Cash bonus wih guranee given by (F.26): Given ha r = r e (say), he price of he oal claims under he guaranee, averaged over an infiniely large porfolio, is [ n ] E e τ r (r r τ ) +Vτ τ p x dτ r = re. (F.28) A naural saring poin for creaing some useful differenial equaions by he backward consrucion is he price of fuure claims under he guaranee in sae e a ime, [ n ] W e() = E e τ r (r r τ) +Vτ τ p x dτ r = re, (F.29) e = 1,..., J Y, n. The price in (F.28) is precisely W e(). +

295 APPENDIX F. EXERCISES 28 Condiioning on wha happens in he ime inerval (, + d] and neglecing erms of order o(d) ha will disappear in he end anyway, we find ( ) W e() = (1 λ e d) (r r e ) +V p x d + e red W e( + d) + λ ef d W f (). From here we easily arrive a he differenial equaions f; f e f; f e d d We() = (r r e ) +V p x + r e W e() λ ef d (W f () W e()), which are o be solved subjec o he condiions (F.3) W e(n ) =. (F.31) (b) Terminal bonus a ime n given by (F.27): Given r = r e, he price of he claim under he guaranee, averaged over an infiniely large porfolio, is [ ( E e n ) ] n r s ds nτ r e s ds (r r τ )Vτ dτ np x r = r e = E [ ( n ) e τ r (r r τ )Vτ dτ + + r = re ] np x. (F.32) Le us ry and copy he mehod of Iem (a) and look a he price of he claim a ime, which should be he condiional expeced discouned value of he claim, given wha we know a he ime: [ ( E e n ) ] n r s ds nτ r e s ds (r r τ)vτ dτ np x r τ ; τ where = E [ ( U + n e τ U = ) r (r r τ )Vτ dτ + + τ e r (r r τ )Vτ dτ. ] rτ ; τ np x. (F.33) The quaniy in (F.33) is more involved han he one in (F.29) since i depends effecively on he pas hisory of ineres rae hrough U. We can, herefore, no hope o end up wih he same simple ype of problem as in Iem (a) above and in all oher siuaions encounered so far, where we essenially had o deermine he condiional expeced value of some funcion depending only on he fuure course of he ineres rae. Which was easy since, by he Markov propery, we could look a sae-wise condiional expeced values W e(), e = 1,..., J Y, say. These are deerminisic funcions of he ime only and can be deermined by solving ordinary differenial equaions. Le us proceed and see wha happens. Due o he Markov propery (condiional independence beween pas and fuure, given he presen) he expression in (F.33) is a funcion of, r and U. Dropping he unineresing facor np x, consider is value for given U = u and r = r e, [ ( n ) ] W e(, u) = E u + e τ r (r r τ )Vτ dτ r = re. +

296 APPENDIX F. EXERCISES 29 Use he backward consrucion: W e(, u) = (1 λ e d)e + f; f e [ ( n ) u + (r r e )V d + e re d e τ r +d (r r τ )Vτ dτ +d λ ef d W f (, u) = (1 λ e d)e re d W e( + d, e re d u + (r r e )V d) + Inser here e ±re d = 1 ± r e d + o(d), W e( + d, e re d u + (r r e )V d) = f; f e + λ ef d W f (, u). W e(, u) + We(, u) d + u We(, u)(u re + (r r e )V ) d + o(d), and proceed in he usual manner o arrive a he parial differenial equaions We(, u)+(ure +(r r e )V ) u We(, u) re W e(, u)+ These are o be solved subjec o he condiions W e(n, u) = u +, f; f j ] r +d = r e λ ef (W f (, u) W e(, u)) =. e = 1,..., J Y. Since he funcions we are ineresed in involved boh and U, we are lead o sae-wise funcions in wo argumens and, herefore, quie naurally end up wih parial differenial equaions for hose. N. Uni linked insurance. We have been discussing he paricipaing (or wih profi) policy, characerisic of which is ha benefis and premiums are se ou in nominal amouns in he conrac a ime. Thus, For he fairly general conrac described in he inroducion o his noe, he funcions b and π would be deerminisic, no dependen on he developmen of he ineres rae over he erm of he conrac. Inroduce U = e r u du, which is he value a ime of a uni deposied in he invesmen porfolio a ime. We may call i he price index of he invesmen porfolio. Recas he equivalence relaion (F.11) as π + n Uτ 1 τ p x(µ x+τ b τ π τ ) dτ + Un 1 np x b n =. (F.34) Wih b and π fixed a ime here is no way one can make hem fulfill (F.34) for all possible fuure courses of he ineres rae process. Depending on he economic developmen here will be inequaliy in he one or he oher direcion. The financial risk hus inroduced is hedged (hopefully perfecly) by seing premiums on a pruden firs order basis, i.e. replacing he unknown r in (F.34) by some r se o he safe side.

297 APPENDIX F. EXERCISES 3 An alernaive scheme for managemen of financial risk in life insurance is known as uni linked insurance (also called variable life insurance). The idea of his concep is o link benefis and premiums o he performance of he invesmen porfolio, ha is, le conracual paymens be inflaed by he index U insead of being fixed nominal amouns. Under a perfec uni linked conrac we would have b = U b and π = U π for some baseline benefis b and premiums π, [, n], deermined a ime. Insering his ino (F.34) gives π + n τ p x(µ x+τ b τ π τ ) dτ + np x b n =. (F.35) We see ha, for a given baseline benefi funcion b, he equivalence relaion can be fulfilled by a suiable choice of baseline premium rae π. The fuure course of he ineres rae process has disappeared from he relaion upon discouning he indexed paymens and, hus, he problem wih financial risk has been resolved by he perfec uni linked device. However, in pracice uni-linked conracs are usually no perfec in he sense described above. Typically, only he benefis are linked o he invesmen index, whereas premiums are no. Furhermore, he uni linked conrac is ypically equipped wih a guaranee specifying ha he benefi canno fall below a cerain pre-specified nominal minimum. Such modificaions o he perfec linking re-inroduce financial risk, of course. Before we reurn o mahemaics, we dare o sugges ha guaranees, wheher hey apply o benefis under uni linked conracs or o ineres under wih profi conracs, are remains of he social securiy concern ha radiionally was paramoun in life insurance. They inroduce a discriminaion beween various forms of saving; unlike hose who inves in socks, bonds, or real esae, hose who inves in life or pension insurance are graned he privilege of gaining from booms wihou loosing from recessions. However, pariy can be resored by leing he insured pay for he guaranee. Thus we proceed o deermine is righ price. For an example, le us ry and deermine he single premium payable a ime for a erm insurance wih sum b = (U g) a ime (, n), where g is he guaraneed minimum sum insured specified a ime. The premium is [ n π = E e ( τ r τ ) e r g τ p xµ x+τ dτ where we have abbreviaed ] = E f = p xµ x+. [ n ( 1 e τ r g ) ] f τ dτ, Following he recipe in Iem (b) of Paragraph M, consider he price of fuure claims a ime, [ n E e ( τ r τ ) ] e r g f τ dτ rτ; τ [ n ( = E U e ) ] τ r g f τ dτ rτ ; τ. (F.36) Arguing as before, he expression in (F.36) is a funcion of, r and U. Consider is value a ime for given U = u, and r = r e, [ n ( W e(, u) = E u e ) ] τ r g f τ dτ r = re.

298 APPENDIX F. EXERCISES 31 When r = r e, he premium we seek is W e(, 1) Now use he backward consrucion, his ime leaving deails aside: W e(, u) = (1 λ e d)e + f; f e = (1 λ e d) [ n (u g) f d + e re d λ ef d W f (, u) Inser e ±re d = 1 ± r e d + o(d) and +d ( e red u e ) ] τ r +d g f τ dτ r +d = r e ( ) (u g) f d + e red W e( + d, e red u) + W e( + d, e re d u) = W e( + d, u + ur e d) + o(d) f; f e λ ef d W f (, u). = W e(, u) + We(, u) d + u We(, u) u re d + o(d), and fill in some deails o arrive a he parial differenial equaions (u g)f r e W e(, u)+ We(, u)+ We(, u) u re+ λ ef (W f (, u) W e(, u)) =. u f; f e These are o be solved subjec o he condiions e = 1,..., J Y. W e(n, u) =, O. Salary dependen premiums and benefis. The employees of a firm are enrolled in a pension scheme wih salary dependen premiums and benefis. Consider an employee (x), who eners he scheme x years old a ime (say), reires a pensionable age 65 a ime m = 65 x, earns salary a rae S() per ime uni a any ime < m, and will receive pension coninuously a level rae Q (ye o be deermined) for n years afer reiremen. Le us firs work under he assumpion ha he ineres rae r is consan and known for he enire erm of he conrac up o ime m + n. (a) We will firs consider a defined conribuions scheme under which a fixed proporion of he salary is used as premium for addiional pension benefis. I will urn ou ha equivalence is auomaically aained regardless of he developmen of he salary. In any small ime inerval [, + d), < m, he insured earns S() d. A fixed proporion πs() d, < π < 1, of his salary is used as a single premium for a pension of q() d per ime uni in he ime inerval [m, m + n]. By he principle of equivalence, q is given by π S() d = q() d m n ā x+, ha is, S() S() q() = π = π. m nā x+ m E x+ ā x+m n The oal rae of pension per ime uni purchased by a survivor a ime m is Q = m q(τ) dτ = π m S(τ) m τ E x+τ ā x+m n dτ.

299 APPENDIX F. EXERCISES 32 I should be fairly obvious ha he equivalence requiremen is fulfilled by his scheme since, no maer how much or lile salary he insured will earn and no maer if i can be prediced or no a he ouse, he benefis are enirely deermined by he salary-dependen conribuions. Le us, however, jus check: For any given salary funcion, S, he expeced discouned premiums are π m e τ (r+µ x+s ) ds S(τ) dτ = π and he expeced discouned benefis are Q m n ā x = π m S(τ) m τ E x+τ ā x+m n m τ E x S(τ) dτ, m dτ me x ā x+mn = π τ E x S(τ) dτ, where we have used he well-known ideniy me x = τ E x m τ E x+τ. (b) Suppose now ha, insead of leing he conribuions deermine he benefis, he benefis are linked o he salary whereas he premiums are no. More specifically, suppose pension is payable coninuously a rae.75s(m) (i.e. 75% of he salary rae a he ime of reiremen) for n years afer reiremen, and ha premium is payable a a prefixed level rae π while acive (boh coningen on survival, of course). To deermine he premium level π a ime, we now need o make assumpions abou he fuure developmen of he salary. Le us also abandon he unrealisic assumpion ha he fuure developmen of he ineres rae is known: Assume ha he economy is governed by a coninuous ime Markov chain Y (),, wih sae space J = {1,..., J}, consan inensiies of ransiion λ jk, j k, and iniial sae Y () = i, say. A any ime he accumulaion facor U() of he invesmen porfolio is given by ( ) U() = exp r(s) ds, r(s) = I j(s)r j, (F.37) j and he salary rae is given by ( ) S() = exp a(s) ds, a(s) = j I j(s)a j. Here, I j() = 1[Y () = j], and he r j and he a j are known, fixed numbers; r j is he ineres rae and a j is he rae a which salary increases per ime uni and per uni of salary when he economy is in sae j. Exercise 1-21 Deermine he premium rae π ha makes expeced discouned benefis less premiums equal o a ime ; Consruc differenial equaions for benefis and for premiums and specify appropriae side condiions. Exercise 1-22

300 APPENDIX F. EXERCISES 33 Consider he Markov chain ineres model oulined in Figure F.5. Fill in appropriae saemens in he [ program ( prores2 o make i compue he sae-wise expeced discoun facors E exp ) 5 r(s) ds Y () = j ], j = 1, 2, 3. Exercise 1-22 Le he benefi be an n years uni-linked pure life endowmen wih guaraneed sum insured max(u(n), g), and le premium be payable coninuously a level rae π during he insurance period. Show how o deermine π. Exercise 56 Le {N()} be Poisson process wih inensiy λ. This is a couning process of even simpler ype han he couning processes associaed wih a Markov chain; N is no only Markov, bu also has independen incremens. Thus, in any small ime inerval [, + d) he process N makes a jump of 1 wih probabiliy λ d regardless of he pas hisory of he process in [, ). Le he price S() of a share of sock a ime be modelled as a so-called geomeric Poisson process wih drif, S() = exp (α + βn()),. If β =, hen S() is jus he accumulaion facor for a bank accoun wih fixed ineres rae. The Poisson erm in he exponen adds jumps a random imes, and a jump a ime makes he sock price jump from S( ) o S() = S( ) e β. Thus, γ = e β 1 is he relaive change (S() S( ))/S( ) in he sock price a he jump ime. Beween he jumps he sock price increases a fixed rae of ineres α. (a) Find he expeced value E[S()] a ime of he sock price a ime, and do his in wo ways: Firs, work direcly wih he Poisson disribuion of N() and, second, solve a differenial equaion obained by he direc backward consrucion (condiion on wha happens in he small ime inerval [, d) ). Explain ha, having deermined he expeced value, higher order momens are easily obained. (b) Find he dynamics ds() of he sock price by applying he change of variable rule, see Appendix A. (c) Using he direc backward consrucion, show ha he expeced presen value of a perpeuiy (an everlasing annuiy), is [ ] E S 1 (τ) dτ = (α + λ(1 e β )) 1. (d) Le {N 1()} and {N 2()} be independen Poisson processes wih inensiies λ 1 and λ 2, respecively. Le he price S() of a share of sock a ime be S() = exp (α + β 1N 1() + β 2N 2()),. Leing β 1 and β 2 have opposie signs, we have creaed a sock price which may increase or decrease wih insananeous jumps. Wrie ou he sraighforward analogs

301 APPENDIX F. EXERCISES 34 of he formulas in Iems (a) - (c) for his more general model. Exercise 57 (a) Le A and A be wo paymen funcions wih rerospecive reserves denoed by U and U, respecively. Assuming ha he ineres rae is always posiive, verify he following raher obvious asserion: If A A for all, hen U U for all. (In paricular, any advancemen of deposis will produce an increase of he rerospecive reserve if he ineres rae is posiive. This is he general circumsance underlying resuls like he following abou ordering of expeced presen values: a n ā n ä n.) (b) Le he paymen sream A represen deposis less wihdrawals on an n years savings accoun ha bears ineres wih sricly posiive ineres rae. I is required ha U for all, wih sric inequaliy for some, and ha U n =. Prove ha A n <, and explain his resul. Exercise 58 (a) Show ha, for s u, p jj (, u) = p jj (, s) p jj (s, u), which is obvious. (b) Given sar in sae a a ime, wrie up he probabiliy ha he process remains in a during he ime inerval [, 1), hen jumps o sae i in [ 1, 1 + d 1), hen remains in i during he ime inerval [ 1 + d 1, 2), hen jumps o sae a in [ 2, 2 + d 2), hen remains in a during he ime inerval [ 2 + d 2, 3), and finally jumps o sae d in he ime inerval [ 3, 3 + d 3). This is he probabiliy of one paricular full specificaion of he hisory of he process. Exercise 59 Read Secions 9.1 and 9.2 in Basic Life Insurance Mahemaics. Suppose n independen lives, which follow he same Gomperz-Makeham moraliy law wih inensiy µ() = α + β exp(γ), are observed from birh unil deah. Find he equaions for he Maximum likelihood esimaors for he parameers α, β, and γ, and find he asympoic properies of he esimaors. Exercise 6 In he siuaion of Paragraphs 9.1.B-E, consider he problem of esimaing µ from he D i alone, he inerpreaion being ha i is only observed wheher survival o z akes place or no. Show ha he likelihood based on D i, i = 1,..., n, is q N (1 q) n N, wih q = 1 e µz, he probabiliy of deah before z. (Trivial: i is a binomial siuaion.) Noe ha N is now sufficien, and ha he class of disribuions is a regular exponenial class. The MLE of q is q = N n

302 APPENDIX F. EXERCISES 35 wih he firs wo momens Eq = q, Varq = q(1 q) n I is UMVUE in he class of esimaors based on he D i. The MLE of µ = ln(1 q)/z is µ = ln(1 q )/z. Apply sandard asympoic resuls abou MLE o show ha µ as N ( µ, ) q. nz 2 (1 q) The asympoic efficiency of ˆµ relaive o µ is ( ) asvarµ asvarˆµ = e µz 2 e µz 2 ( 2 sinh(µz/2) = µz µz/2 (sinh is he hyperbolic sine funcion defined by sinh(x) = (e x e x )/2). This funcion measures he loss of informaion suffered by observing only deah/survival by age z as compared o inference based on complee observaion hroughou he ime inerval (, z). I is 1 and increases from 1 o as µz increases from o. Thus, for small µz, he number of deahs is all ha maers, whereas for large µz, he life lenghs are all ha maers. Reflec over hese findings. Exercise 61 We refer o he disabiliy model. (a) Consider an x years old insured who eners an insurance scheme a ime. The probabiliy p aa(, ) = exp( (µx+s + σx+s) ds) can be viewed as he probabiliy p () aa (, ) of being acive a ime afer having been disabled imes. Derive forward differenial equaions for he probabiliy p (1) ai (, ) of being disabled for he firs ime a ime and for he probabiliy p (1) aa (, ) of being acive a ime afer having been disabled once. (b) Find he probabiliy of being disabled for he firs ime a ime and ha he disabiliy has lased for a leas q years. (c) A ime an acive person aged x buys a disabiliy pension insurance wih he following erms: The benefi is a pension payable a level rae 1 during he firs disabiliy, bu only afer i has lased for a leas q years (he qualifying period). Premium is payable a level rae π as long as he insured is acive and has no ye been disabled, bu no afer ime n q, where n is he conrac period (n > q). Deermine he premium π by he principle of equivalence, assuming ha he ineres rae r is consan. Find he reserve a ime < n q for an insured who is disabled for he firs ime and is currenly receiving he disabiliy benefi.. ) 2 Exercise 62 The wo-sae Markov chain Z skeched in Fig. F.1 can be given many inerpreaions; i could describe ransiions of a person ino and ou of he work-force ( acive means

303 APPENDIX F. EXERCISES 36 employed, invalid means unemployed, and moraliy is disregarded), or he ransiions of a person beween marial saes ( acive means single, invalid means married), or he ransiions of a machine or mechanical device beween saes of funcioning ( acive means inac, invalid means ou of order), and so on. Le us ake he model as a descripion of a lamp, which is always swiched on (i is insalled in a submarine), and which is acive when he bulb is inac and invalid or inacive when he bulb is burn-ou. One can assume ha he life-ime of a bulb is exponenially disribued, and i also seems reasonable o assume ha he lapse of ime from a bulb burns ou unil he failure is discovered and he bulb is replaced, is exponenially disribued. Then he inensiies σ and ρ are consan (σ is he moraliy inensiy of he life lengh of a bulb or he expeced number of deahs per ime uni for an burning bulb, and ρ is he expeced number of mainenance inspecions per ime uni.). We follow he lamp from ime when i is acive. Le N ai() and N ia() denoe he number of failures and replacemens of bulb, respecively, in he ime inerval [, ], and le I a() and I i() be he indicaors of he evens acive a ime and inacive a ime, respecively. Of course, I i()+i a() = 1, and N ia() N ai() is eiher or 1. (a) Find E[I i()] = p ai(, ) by solving Kolmogorov s forward equaions, using p aa(, ) = 1 p ai(, ). (b) Find he expeced ime as inacive in [, ], E[ Ii(τ) dτ]. You may jus inegrae he funcion p ai(, τ) in (a). Explain why. (c) Find E[N ai()]. You may also find he answer by jus inegraing he funcion p aa(, τ) in (a) and muliplying wih σ. Explain why. (d) Divide he expeced values in (b) and (c) by and find he limi as. Discuss he expressions as funcions of σ and ρ. (e) Find he failure inensiy σ(), < < n, for he Markov chain Z, condiional on Z(n) = i. (f) If σ = ρ, hen N ai() + N ia() is a Poisson process wih inensiy σ. Explain why. Wha is hen N ai() in erms of he Poisson process? Exercise 63 The siuaion is as described in Exercise 62. Suppose daa are available for m independen lamps observed over he same ime inerval [, n], all acive a ime. (a) Assume complee hisories have been recorded for each lamp. Find he MLE for σ and find is asympoic disribuion. Exercise 64 (b) The parameer ρ is subjec o conrol since i is he frequency wih which he lamps are being checked by mainenance personnel. Show ha he asympoic variance of he MLE in (a) is a decreasing funcion of ρ.

304 APPENDIX F. EXERCISES 37 (c) Suppose ha, a ime n, daa are available only for lamps ha are inacive a he ime. Then i is he condiional process in 7.8e which is he relevan sochasic model for he individual hisories. Find he MLE in his siuaion and find also is asympoic variance. Exercise 65 Go hrough Paragraphs 9.1 A-E and G in BL and add all deails in proofs. Exercise 66 A your disposal are daa from m independen lives insured under one and he same scheme. You know for each individual he ime of enry ino he scheme, he age upon enry, he lengh of he observaion period, wheher he/she died during he observaion period, and - if died - he age a deah. Find he ML esimaors under he assumpion of piece-wise consan inensiies and explain how hey can be fied by analyic graduaion by a Gomperz Makeham funcion. Exercise 67 Consider he coninuous ime version of an n-year endowmen insurance of 1 (say) agains level premium π during he insurance period. Assume ha he ineres rae r is consan. Prove ha he premium rae π is a decreasing funcion of n and ha he premium reserve V for fixed is a decreasing funcion of n for n >. Noe ha he resuls are valid for all specificaions of ineres rae and moraliy rae. Exercise 68 We refer o Basic Life Insurance Mahemaics (BL), Secion 7.5, Paragraph J, Widow s pension. The siuaion is depiced in Figure F.4. A ime = (say) he husband (x) is x years old and he wife (y) is y years old. Their remaining life lenghs afer ime are denoed by S and T, respecively. The join disribuion of S and T is deermined by he moraliy raes named in he figure: for insance, a any ime, (x) has moraliy rae µ() if (y) is sill alive and µ () if (y) is dead. (a) Find inegral expressions for he marginal survival probabiliies P[S > s], P[T > ], and for he join survival probabiliy P[S > s, T > ]. (b) Prove ha, if he moraliy raes are independen of marial saus, µ = µ and ν = ν, hen S and T are sochasically independen: P[S > s, T > ] = P[S > s] P[T > ]. Adop he independence assumpion in (b) and assume ha x = y = 3 and ha (x) and (y) are subjec o he same law of moraliy, G82M, irrespecive of marial saus. Thus, a any ime, µ() = ν() = µ () = ν () = (F.38)

305 APPENDIX F. EXERCISES 38 The couple buys a combined life insurance and widow s pension policy specifying ha a pension is o be paid wih inensiy b =.5 as long as he wife is alive and husband is dead, a life assurance wih sum s = 1 is due immediaely upon he deah of he husband if he wife is already dead (a benefi o heir dependens), and he equivalence premium is o be paid wih level inensiy c as long as boh husband and wife are alive. The policy erminaes a ime n = 3. Ineres is earned on he reserve a consan rae r = ln(1.45). (c) Using he direc backward argumen, derive he differenial equaion for he noncenral momens of firs and second order of he discouned fuure benefis in sae. (The firs order momen is jus he reserve.) (d) Using he program prores1.pas, compue he hree firs momens of discouned benefis less premiums a imes,5,...,3. Do he same for a modified conrac, by which 5% of he curren reserve (in sae ) is o be paid back immediaely o he husband in case he wife dies before him during conrac period. (e) Le us now drop he independence assumpion in (b) and insead assume ha, for each pary, he moraliy rae increases upon he deah of he spouse (a grief effec ): µ > µ, ν > ν. (F.39) In a forhcoming noe Dependen lives i will be proved ha, under he assumpion (F.39), S and T are posiively dependen in he sense ha P[S > s, T > ] > P[S > s] P[T > ]. Give some numerical evidence in suppor of his resul by e.g. compuing P[S > 3, T > 3], P[S > 3], and P[T > 3] in he case where µ and ν are as in (F.38) and µ () = µ() +.5 and ν () = ν() +.5. (f) Under he assumpions of Iem (e) find he covariance beween he presen values a ime of a erm insurance of 1 in 3 years on (x) and a similar insurance on (y). (The relaionship Cov(X + Y ) = 1 [Var(X + Y ) Var(X) Var(Y )] may be useful.) 2 Exercise 69 The uncerain developmen of ineres can be accouned for by leing he ineres rae r() a any ime depend on he sae of he economy Y () modeled as a sochasic process. We will assume ha {Y ()} is a coninuous ime Markov chain wih finie sae space J Y = {1,..., J Y }. The probabiliy of ransiion from sae e o sae f in he ime inerval from o u is denoed by p ef (, u) = P[Y (u) = f Y () = e]. (F.4) We assume ha he process is homogeneous, which means ha he ransiion probabiliies p ef (, u) depend on and u only hrough u, he lengh of he ime inerval. This implies ha he process has consan inensiies of ransiion, p ef (, + d) λ ef = lim. (F.41) d d Jus like he moraliy inensiy, λ ef is a probabiliy of ransiion per ime uni. The oal inensiy of ransiion ou of sae e a ime is denoed by λ e = λ ef. f; f e

306 APPENDIX F. EXERCISES 39 1 µ Boh alive Husband dead ν ν 2 µ 3 Wife dead Boh dead Figure F.4: Skech of a Markov model for wo lives. λ 12 =.5 λ 23 =.25 r 1 =.2 r 2 =.5 r 3 =.8 λ21 =.25 λ32 =.5 Figure F.5: Skech of a simple Markov chain ineres model. (a) Use he direc backward argumen o derive he so-called backward Kolmogorov differenial equaions for he ransiion probabiliies: The side condiions are p ef (, u) = g;g e λ eg(p ef (, u) p gf (, u)). p ef (u, u) = δ ef, i.e. 1 if e = f and oherwise (he Kroenecker dela). These differenial equaions can easily be solved numerically wih a suiable variaion of he program prores1.pas or prores2.pas. Sochasic ineres is now modeled by leing he ineres rae assume J Y possible values r 1,..., r J Y and, a any ime, he ineres rae is r() = r Y () (dependen on he sae of he economy). Figure F.5 shows a flow-char of a simple Markov chain ineres rae model wih hree saes,.2,.5,.8. Direc ransiion can only be made o a neighbouring sae, and he oal inensiy of ransiion ou of any sae is.5, ha is, he ineres rae changes once in wo years on he average. By symmery, he long run average ineres rae is.5.

307 APPENDIX F. EXERCISES 4 (b) Use he direc backward argumen o derive differenial equaions for he sae-wise expeced discoun facors [, n), Wha are he side condiions? W e() = E[e n r(s) ds Y () = e], Remark: I has been said ha all models are wrong, bu some are useful. The Markov model proposed here is, of course, unable o mimic perfecly he developmen of he rue ineres rae. We can, however, by judicious choice of he sae space (sufficienly big) and he ransiion inensiies, make i cach he basic feaures of he real world ineres fairly well. (c) Consider he model in Fig. F.5. Fill in appropriae saemens in he [ program exp prores2.pas o make i compue he sae-wise expeced discoun facors E e = 1, 2, 3. ( 5 r(s) ds ) Y () = e ], Exercise 7 Poisson processes, which are oally memoryless, can of course be generaed from coninuous ime Markov chains, which are more general. For insance, le {Y ()} be a Markov chain on he sae space {1, 2} wih inensiies of ransiion µ 12() = µ 21() = λ. Then {N()} defined by N() = N 12() + N 21() (he oal number of ransiions in (, ]) is a Poisson process wih inensiy λ; ransiions couned by N occur wih inensiy λ a any ime regardless of he pas hisory of he process. Two independen Poisson processes, {N 1()} wih inensiy λ 1 and {N 2()} wih inensiy λ 2, can be generaed by leing {Y ()} be a Markov chain on he sae space {1, 2, 3, 4} wih inensiies µ 12() = µ 21() = µ 34() = µ 43() = λ 1, µ 13() = µ 31() = µ 24() = µ 42() = λ 2, µ 14() = µ 41() = µ 23() = µ 32() =, and defining N 1() = N 12()+N 21()+N 34()+N 43() and N 2() = N 13()+N 31()+N 24()+N 42(). Three independen Poisson processes can be generaed from a Markov chain wih 8 saes (work ou he deails), and, in general, k independen Poisson processes can be generaed from a Markov chain wih 2 k saes. (a) Le {Y ()} be a Markov chain on he sae space {1, 2}, and ake µ 12() = µ 21() = 1. Denoe he corresponding indicaor processes and couning processes be I e() and N ef (). The oal ime spen by Y in sae 1 during he ime inerval (, n] is T 1(, n] = n I 1(τ) dτ, and he oal number of ransiions made from sae 1 o sae 2 in ha inerval is N 12(, n] = N 12(n) N 12(). (These quaniies can be viewed as presen values of benefis of annuiy ype and assurance ype, respecively, for a wo-sae policy wih no ineres.) (a) Assume Y () = 1. Wha are he inerpreaions of he random variables T 1(, u] and N 12(, u] in erms of he Poisson process N() = N 12() + N 21()?

308 APPENDIX F. EXERCISES 41 (b) Find, by solving he relevan differenial equaions analyically, explici expressions for he firs wo sae-wise condiional momens V e (q) () = E[T q 1 (, n] Y () = e], W e (q) () = E[N q 12 (, n] Y () = e], e = 1, 2, q = 1, 2, and find also he corresponding variances. (You should obain e.g. E[T 1(, 1] Y () = 1] = 1 ( 1 + 2µ e 2µ ), 4µ Var[T 1(, 1] Y () = 1] = 1 16µ 2 ( 1 + 4µ (2 e 2µ ) 2). (c) Using he program prores1.pas (or prores2.pas), solve he differenial equaions also numerically and compare he resuls wih he exac soluions obained in (b). Exercise 71 Le {N()} be Poisson process wih inensiy λ. This is a couning process of even simpler ype han he couning processes associaed wih a Markov chain; N is no only Markov, bu also has independen incremens. Thus, in any small ime inerval [, + d) he process N makes a jump of 1 wih probabiliy λ d regardless of he pas hisory of he process in [, ). Le he price S() of a share of sock a ime be modelled as a so-called geomeric Poisson process wih drif, S() = exp (α + βn()),. If β =, hen S() is jus he accumulaion facor for a bank accoun wih fixed ineres rae. The Poisson erm in he exponen adds jumps a random imes, and a jump a ime makes he sock price jump from S( ) o S() = S( ) e β. Thus, γ = e β 1 is he relaive change (S() S( ))/S( ) in he sock price a he jump ime. Beween he jumps he sock price increases a fixed rae of ineres α. (a) Find he expeced value E[S()] a ime of he sock price a ime, and do his in wo ways: Firs, work direcly wih he Poisson disribuion of N() and, second, solve a differenial equaion obained by he direc backward consrucion (condiion on wha happens in he small ime inerval [, d) ). Explain ha, having deermined he expeced value, higher order momens are easily obained. (b) Find he dynamics ds() of he sock price by applying he change of variable rule, see Appendix A. (c) Using he direc backward consrucion, show ha he expeced presen value of a perpeuiy (an everlasing annuiy), is [ ] E S 1 (τ) dτ = (a + (1 e β )) 1. (d) Le {N 1()} and {N 2()} be independen Poisson processes wih inensiies λ 1 and λ 2, respecively. Le he price S() of a share of sock a ime be S() = exp (α + β 1N 1() + β 2N 2()),

309 APPENDIX F. EXERCISES 42. Leing β 1 and β 2 have opposie signs, we have creaed a sock price which may increase or decrease wih insananeous jumps. Wrie ou he sraighforward analogs of he formulas in Iems (a) - (c) for his more general model. Exercise 72 (a) Using your Turbo Pascal program wih he Danish basis, compue he ne premium rae π and he ne premium reserve V (a seleced imes ) for an endowmen insurance wih age a enry x = 3, erm n = 3, sum insured b = b n = b = 1, and premium payable coninuously a consan rae hroughou he conrac period. (b) Compue he gross premium rae π and he gross premium reserve V assuming ha adminisraion expenses consis of a lump sum cos of b a ime, coss incurring coninuously a rae π +.5 V a any ime in he insurance period, a cos of.2 due immediaely upon possible paymen of he deah benefi, and a cos of.1 due a ime n upon possible paymen of he endowmen benefi. Compare wih he quaniies ne of expenses found in Iem (a). Exercise 73 In connecion wih a pension insurance here is an addiional benefi which is a sum insured o possible dependen children less han 18 years old a he ime of deah of he insured. I he echnical basis we herefore need o make assumpions abou birhs. We have o disinguish by sex, and in he following we are going o consider insured women only. The Figure below shows a flowchar describing a life hisory wih birhs (a mos J). To keep hings simple, we will assume ha he process is Markov, ha all paricipans ener he insurance scheme in sae a age, and ha he individual life hisories are independen replicaes of he process. Assume furhermore ha he moraliy inensiy µ j() and he birh inensiy φ j() for a year old woman, who has given birh o j children, are given by µ j() = α j + βc, j =,..., J, (F.42) φ j() = η jf(), j =,..., J 1, (F.43) where c and he funcion f are known and he parameers α j, β, and η j are unknown. (a) Assume for he ime being ha, a he ime of consideraion, we have complee informaion abou he pas hisory for all individuals ha have previously been or currenly are insured under he scheme (i.e. we know he exac imes of possible birhs and deah). Pu up he equaions for deermining he ML (maximum likelihood) esimaors for he unknown parameers and, o he exen possible, find explici expressions for he esimaors. Explain also how o deermine he asympoic covariance marix. (b) In our model here is non-differenial moraliy if α = = α J. (F.44) Derive he likelihood raio es for he null hypohesis (F.44) and deermine he rejecion limi such ha he asympoic level is 1%.

310 APPENDIX F. EXERCISES 43 Assume now ha he pas hisory of birhs and deah is being observed only upon deah of he insured, when he addiional benefi o he possible dependens is due. Suppose ha he saisical daa comprise only hose who are dead a he ime of consideraion and ha for each of hose here is a complee record of he imes of possible birhs and of deah. In hese daa he observed life hisory of a woman, who enered he scheme u years ago, is governed by a Markov process as described above, bu wih inensiies µ 1 j () = µ j() p jd (, u), φ j () = φ j() p j+1,d(, u) p jd (, u) (F.45). (F.46) I is o be proved ha, if he moraliy increases wih he number of birhs, ha is, µ j() µ j+1(), >, j =,..., J 1, (F.47) hen Inroduce φ j () φ j(), >, j =,..., J 1. p j(, u) = 1 p jd (, u), (F.48) (F.49) he probabiliy ha a year old wih j birhs will survive o age u. Build he proof as follows: (c) Prove ha, for τ u, p j(, u) = k j p jk (, u) = k j p jk (, τ)p k (τ, u). (F.5) (d) Prove ha p j(, u) = e u µ j (,ϑ)dϑ, (F.51) where µ j(, u) = k j p jk(, u)µ k (u) k j p, (F.52) jk(, u) he moraliy inensiy a age u for a woman who is in sae j a ime < u). (e) Show (preferably by direc reasoning), ha p j(, u) = e u (φ j +µ j ) + u e τ (φ j +µ j ) φ j(τ)p j+1(τ, u) dτ. (d) Use he resuls (F.5) (F.53) o show ha (F.47) implies p j+1(, u) p j(, u), j =,..., J 1, (F.53) (F.54) which by (F.46) and (F.49) implies (F.48). The relaionship (F.54) is easy o prove for j = J 1, and he resul hen follows by inducion.

311 APPENDIX F. EXERCISES 44 Alive birhs φ φ j 1 j φ j J φ J 1 Alive Alive j birhs J birhs µ µ j µ J d. Dead Commmen: The inequaliy (F.48) means ha he feriliy raes will be overesimaed if one uses he esimaors for he φ j based on diseased paricipans in he scheme. If he inequaliies (F.47) are reversed, hen also he inequaliy (F.48) will be reversed, and he esimaors he φ j will underesimae he feriliy. In paricular i follows ha, under he hypohesis of non-differenial moraliy, he feriliy raes will be unbiasedly esimaed from he seleced maerial of diseased paricipans. Exercise 74 We adop he usual noaion and assumpions of he heory of muli-life insurance policies and consider wo independen lives (x) and (y) wih remaining life lenghs T x and T y, respecively. (a) Assume ha he benefi is an assurance of 1 payable a ime T y if 2T x < T y < n and ha premium is payable a consan rae π unil ime min(t x, T y, n/2), where n is he erm of he conrac (fixed). Deermine he equivalence premium π. (b) Propose a mehod for compuing he premium numerically. (Hin: One possibiliy is o rea /2 p x as a survival funcion p x wih inensiy µ x+, which you would need o express in erms of µ, and hen solve a Thiele differenial equaion numerically.) (c) Deermine he reserve a any ime, assuming ha he insurer currenly knows he complee pas hisory of he wo lives. You need o disinguish beween various cases, wheher (y) is alive or dead, wheher is before or afer ime n/2, and wheher x is alive or dead and, if dead, when. Is he reserve always non-negaive? (d) Wha is he variance of he presen value of he benefi? Exercise 75 Find an expression for he moraliy inensiy of he life lengh of he las-survivor saus x 1... x r. Do his by direc reasoning and also he hard way by calculaing µ x1...x r () = d d ln px 1...x r.

312 APPENDIX F. EXERCISES 45 Exercise 76 An example of an insurance policy wih benefis ha are pah-dependen, ha is, dependen on he pas hisory of he policy: (a) Consider wo independen lives (x) and (y). Find he expeced presen value a ime of a life annuiy payable coninuously a rae 1 from ime T xy unil ime max(t xy + 2, T xy + 1). In words, paymens sar a he ime of he firs deah and coninues hereafer for a erm of 2 years or unil 1 years afer he deah of he survivor, whichever is he longer period. (b) Suppose premium is payable coninuously a level rae π from ime unil ime T xy. Wha is he reserve for his policy? This quesion is difficul and will be addressed laer, bu you can sar hinking abou i. Exercise 77 Anoher example of pah-dependen paymens: Consider hree independen lives (x), (y), and (z). Find he expeced presen value of a sum insured of 1 payable a ime T x if T x < min(t z, T y + 2). Sae in words wha his conracual benefi is. Exercise 78 Use he program prores1.pas o compue π = Ā 2,2,n ā 2,2,n, assuming ha boh lives follow he moraliy law G82M and ha he ineres rae is r = ln(1.5). Wha are he erms of he policy for which π is he level equivalence premium rae? Exercise 8 We refer here o Secion 7.9 Dependen lives in BL. (a) Prove he raher obvious saemens PQD(T, T ), AS(T, T ), and RTI(T T ). (b) Prove ha he Definiions PQD RTI are equivalen o he modified definiions obained upon replacing he sric inequaliies S > s and T > wih he non-sric inequaliies S s and T. For insance, for PQD prove ha (7.92) is equivalen o P[S s, T ] P[S s] P[T ] for all s and. (c) Negaive dependence in he PQD sense: Prove ha PQD( S, T ) is equivalen o P[S > s, T > ] P[S > s] P[T > ] for all s and. (d) Negaive dependence in he AS sense: Prove ha AS( S, T ) is equivalen o Cov(g(S, T ), h(s, T )) for all real-valued funcions g and h ha are decreasing in S and increasing in T (and for which he covariance exiss). (e) Negaive dependence in he RTI sense: Prove ha RTI( S T ) is equivalen o RTD(S T ).

313 APPENDIX F. EXERCISES 46 (f) Consider he model in Figure 7.6, and le µ = µ and ν = ν. We have shown in Exercise 1 ha S and T are hen independen. Now, add a cause of simulaneous deah (due o caasrophe ) wih inensiy µ 3 = κ. Does i follow ha RTI(S T )? Assume insead, maybe more reasonably, ha caasrophe risk is presen independenly of he sae of he marriage: µ 1() = µ(), µ 2() = ν(), µ 3() = κ(), µ 13() = ν() + κ(), µ 23() = µ() + κ(). Prove ha RTI(S T ), hence PQD(S, T ). Exercise 81 We refer o Chaper 8 of BL and Exercise paper No. 6. Noe he following correcion o Exercise paper No. 6, page 8, line 2 from op: Recalling (1) and (14),.... A 3 years old buys a pure life endowmen of 1 in n = 3 years agains premium payable coninuously a level rae as long as he policy is in force. The firs order basis specifies G82M moraliy and ineres a insananeous rae.2. The second order basis specifies he same moraliy G82M and sochasic ineres as given in Fig. 1 of Exercise 2. (a) See Exercise 2a. Suppose surpluses are o be repaid immediaely as cash bonus. Compue he sae-wise expeced presen values (under second order ineres) a ime of fuure bonuses, given ha he insured survives 3 years. Use he program prores2.pas. I may be a good idea o define wo alive saes which do no communicae, one for compuaion of he firs order reserve, he oher for compuaion of he expeced presen values of bonuses. (b) See Exercise 19. Suppose insead ha ha surpluses are currenly spen on purchase of addiional benefis. Compue he sae-wise expeced values a ime of he addiional benefi Q 3. OK TO HERE (b) Give an alernaive proof of (3.16) along he following lines: Wrie where I = I {T >}. By Iô s formula, (T b) (T a) = ( b k I d) = a b a b a I d, ( k 1 I s ds) I d. a Use I si = I for s < and I k = I, and ake expecaion. Exercise 2 Find he q-h non-cenral momen of P V a;m n in (4.21). Sar from ( ) (V a;m n ) q = 1 q q v p(tx m) v (q p)(t (m+n)). r q p p= Exercise 4 Dependence of expeced presen values on age, duraion, and echnical basis.

314 APPENDIX F. EXERCISES 47 Commence wih he pure endowmen reaed in Paragraph A. The expeced presen value in (4.2) can be wrien as E x = e r µ x+s ds. (F.55) The dependence of E x on he deferred period, he age x, and he echnical basis r and µ is o be discussed. Some qualiaive aspecs are obvious by inspecion of (F.55) and also on inuiive grounds: E x decreases wih increasing ; E x decreases wih increasing x if µ is an increasing funcion; E x decreases wih increasing r (or ineres rae i); E x decreases by a general increase of µ. A closer sudy, also of he quaniaive aspecs, can be based on he derivaives (he derivaive of µx+s ds = x+ µ y dy x Ex = Ex (r + µx+), (F.56) Ex x = Ex (µx+ µx) (F.57) Ex = Ex, r (F.58) µy dy wih respec o x is µx+ µx ), and, if µ x = µ(x, α) is a differeniable funcion of some finie-dimensional parameer α = (α 1,..., α r), E x = E x µ(x + s, α) ds. (F.59) α j α j In paricular, if he force of moraliy is of G-M ype, hen One easily finds (check he deails) c µ x = α + βc x, µ x+s ds = α + βc x (c 1)/ ln c α β = α + (µ x+ µ x)/ ln c. µ x+s ds =, µ x+s ds = cx (c 1) ln c { µ x+s ds = βc x 1 (x + )c x ln c {( 1 = x 1 c ln c ln c = µx+ µx, β ln c } c 1 ln 2 c ) (µ x+ µ x) + (µ x+ α) }.

315 APPENDIX F. EXERCISES 48 Thus, in he G-M case (F.59) specializes o Ex = Ex, (F.6) α β Ex = 1 Ex (µx+ µx), β ln c (F.61) c Ex = 1 Ex c ln c {( x 1 ) } (µ x+ µ x) + (µ x+ α). (F.62) ln c The expressions in (F.59) and (F.58) are he same, of course, since E x depends on r and α only hrough heir sum r + α. In general, a consan change in he force of moraliy is equivalen o a change in he force of ineres. A comparison of (F.57) and (F.61) reveals ha a change of β is essenially he same as a change of x. In fac, replacing β by β = βc h (say) is equivalen o a shif from x o x + h. The expression in (F.62) is no so ransparen, bu one may say ha a change of c o c = c h (say) amouns o an expansion or conracion in age by a facor h. All righ hand side expressions in (F.56) (F.62) include E x as a facor. Division by E x gives he derivaive of ln E x, which is he relaive decrease of E x (in unis of E x) by a uni increase in he argumen. The facors muliplying E x form a basis for comparing he argumens wih respec o he impor of changes. Roughly speaking, for common values of he argumens, hey can be ordered as follows according o heir impacs on he value of E x, saring wih he less imporan: x, and α, β, and c. In fac, a change in is generally of greaer imporance han a similar change in x. This resul is immediaely meaningful since and x are measured in he same unis. Oher comparisons ha can be made are no so lucid since he argumens are no commensurable. For insance, even hough changes in β are more imporan han changes in x in he sense ha β Ex > x Ex (for common values of he argumens, ha is), i mus be kep in mind ha he relevan changes in hese wo argumens are of quie differen order of magniude. Usually β is confined o values below 1 3, whereas x ypically ranges beween 2 and 7. To accoun for his aspec, one should raher compare he wo differenials β Ex db and x Ex dx for represenaive changes db and dx, and (presumably) conclude ha x produces he greaer changes in E x. Here is anoher commen along he same line: Despie he fac ha r and α play equivalen roles in E x from a purely mahemaical poin of view, he ineres rae is widely held o he mos imporan elemen in he echnical basis. The reason is, of course, ha he ineres rae nowadays varies by percenages whereas he moraliy is fairly sable and varies by less han per milles. Now, consider an n-year emporary level life annuiy wih expeced presen value given by (4.14). Taking derivaives under he inegral sign, one obains resuls abou ā x n by jus insering he expressions from (F.56) (F.62). The dependence on n is obained immediaely from (4.14). The resuls are (F.65) āx n n = nex, (F.63) x āx n = Ā1 + x n µxāx n = {1 (r + µ x) ā x n ne x}, (F.64) r āx n = āx n = (Ī ā)x n, α

316 APPENDIX F. EXERCISES 49 i āx n = (Ī ā) x n v, (F.66) β āx n = 1 c āx n = 1 c ln c {1 (r + µx) āx n nex}, β ln c (F.67) {( x 1 ) (Ā1 ln c ( x 1 ln c = 1 c ln c [ x n µx āx n ) + ( Ī Ā) x x n α ( Ī ā ) x n ) {1 (r + µ x)ā x n ne x} + ā x n (r + α)(īā)x n n nex]. (F.68) Finally, he dependence of Ā1 (and Āx n ) on he argumens is obained from he x n resuls above upon puing Ā1 = 1 rāx n nex. Work ou he deails. x n Clearly, since he expressions in (F.56) (F.62) are all negaive when µ is increasing, so are also he expressions in (F.63) (F.68). As a by-produc one obains ha 1 > (r + µ x)ā x n + ne x, which can easily be esablished by direc calculaion. Exercise 2 In he siuaion of he presen paragraph consider he problem of esimaing µ from he D i alone, he inerpreaion being ha i is only observed wheher survival o z akes place or no. Show ha he likelihood based on D i, i = 1,..., n, is q N (1 q) n N, wih q = 1 e µz, he probabiliy of deah before z. (Trivial: i is a binomial siuaion.) Noe ha N is now sufficien, and ha he class of disribuions is a regular exponenial class. The MLE of q is } wih he firs wo momens q = N n Eq = q, Varq = q(1 q) n I is UMVUE in he class of esimaors based on he D i. The MLE of µ = ln(1 q)/z is µ = ln(1 q )/z. Apply (D.6) in Appendix D o show ha ( µ as N µ, ) q. nz 2 (1 q) The asympoic efficiency of ˆµ relaive o µ is ( ) asvarµ asvarˆµ = e µz 2 e µz 2 ( 2 sinh(µz/2) = µz µz/2 (sinh is he hyperbolic sine funcion defined by sinh(x) = (e x e x )/2). This funcion measures he loss of informaion suffered by observing only deah/survival by age z as compared o inference based on complee observaion hroughou he ime inerval. ) 2

317 APPENDIX F. EXERCISES 5 (, z). I is 1 and increases from 1 o as µz increases from o. Thus, for small µz, he number of deahs is all ha maers, whereas for large µz, he life lenghs are all ha maers. Reflec over hese findings. Exercise 3 Use he general heory of he presen secion o prove he special resuls in Secion Exercise 4 Work ou he deails leading o (11.42) (11.42). Exercise 5 An insurance company is o carry ou a moraliy sudy based on complee records for n life insurance policies wih unlimied erm period. Policy No. i was issued z i years ago o a person who was hen aged x i. The acuary ses ou o maximize he likelihood n ( xi µ(x i + T i, θ) D +T i ) i exp µ(s, θ)ds, x i i=1 where he noaion is selfexplaining. One employee in he deparmen objecs ha he mehod represens a neglec of informaion; i is known ha he insured have survived, no only he period hey were insured, bu also he period from birh unil enry ino he scheme. Thus, he claims, he appropriae likelihood is raher n ( xi µ(x i + T i, θ) D +T i ) i exp µ(s, θ)ds. i=1 Sele his apparen paradox. (Hin: A suiable framework for discussing he problem is an enriched model wih hree saes, uninsured, insured, and dead.) Exercise 6 (a) Modify he formulas o he siuaion where person No. i enered he sudy z i years ago a age x i. (b) Find explici expressions for he enries of he asympoic covariance marix of he MLE. Exercise 1 Prove (C.6) in he heorem by inducion: Verify ha i is rue for r = 1 and, assuming i is rue for a given r, prove ha i is rue also for r + 1. Exercise 2 Derive he binomial disribuion by applying he heorem o he siuaion where A 1,..., A r are independen and equally probable, P[A j] = p, j = 1,..., r.

318 APPENDIX F. EXERCISES 51 Exercise 3 Use he heorem o find E[Q] and V[Q] expressed in erms of he Z p. Exercise 4 Find he probabiliy ha a leas 3 ou of 4 evens occur. Le he price a ime of a sock be S() = e α+βn(), where N() is a Poisson process wih inensiy λ. The money marke accoun bears ineres a spo rae r. A ime our hero (x) purchases an n-year uni linked pure life endowmen wih sum insured S(n) g agains a single premium π. Here g is he guaraneed minimum sum insured inroduced o proec he insured agains poor performance of he sock; if β is negaive (in which case α should cerainly be posiive), hen a Poisson even a ime represens a sudden drop in he sock price from S( ) o S() = S( )e β (a crash in he sock marke if he absolue value of β is big). Combining basic principles in finance (no arbirage) and insurance (equivalence), π should be he expeced discouned value of he claim under a suiable probabiliy measure (equivalen maringale measure for he marke and physical measure for he life lengh): π = E [ e rn (S(n) g) 1[T x > n] ] = E [S(n) g] e rn np x. (F.69) We need o find he expeced value appearing in he las expression. Sar as usual from he condiional expeced value of he random variable S(n) g, given everyhing ha is known by ime : E[S(n) g N(τ); τ ] [ ] = E e αn+βn(n) g N(τ); τ [ ] = E e α+βn() e α(n )+β(n(n) N()) g N(τ); τ. Here we have separaed ou wha perains o he pas (known under he condiioning) and wha perains o he fuure (remains random under he condiioning), and i is seen ha we can work wih he funcion [ ] W (, u) = E u e α(n )+β(n(n) N()) g. Preparing for a backward consrucion, wrie [ ] W (, u) = = E u e αd+β(n(+d) N()) e α(n d)+β(n(n) N(+d)) g, and proceed as usual, condiioning on wha happens in (, + d): ( W (, u) = (1 λ d) W + d, ue αd) ( + λ d W + d, ue αd+β) + o(d) ( = W + d, ue αd) ( λ d W (, u) + λ d W, ue β) + o(d) ( = W ( + d, u + u α d) λ d W (, u) + λ d W, ue β) + o(d) = W (, u) + W (, u) d + W (, u) u α d ( u λ d W (, u) + λ d W, ue β) + o(d).

319 APPENDIX F. EXERCISES We arrive a W (, u) + which is o be solved subjec o he condiion ( u W (, u) u α λ W (, u) + λ W, ue β) =, W (n, u) = u g. Remark: We could have wrien (F.69) as [ π = E e βn(n) ge α n] e (α r) n np x, and, redefining W (, u) accordingly, essenially ge rid of e α. We have chosen he presen approach since i gives us an opporuniy o see he differen roles of he (non-sochasic) smooh funcion e α and he (sochasic) jump process N().

320 Appendix G Soluions o exercises Exercise 1 (a) Exac value 1 2 d = 1/3 = Can be compued by solving numerically he ordinary differenial equaion v () = a() + b() v() (G.1) by Ode-1.pas. Take v() = s 2 ds, v () = 2, hence a() = 2 and b() =. Compue forwards ( F ) saring from v() =. Inser he following saemens in he program: (*SPECIFY DIMENSION OF v!*) dim := 1; (*SPECIFY DIRECTION OF COMPUTATION - FORWARD ( F ) OR BACKWARD ( B ):!*) BF := F ; (*SPECIFY THE TIME INTERVAL [,T] BY INSERTING T IN erm AND - IF NEEDED - THE AGE OF THE LIFE AT TIME! *) erm := 1; (*SPECIFY BOUNDARY CONDITION(S) AT TIME = IF FORWARD (BF = F ) AND AT = T if BACKWARD (BF = B ):!*) v[1] := ; (*SPECIFY THOSE COEFFICIENTS a AND B AND DERIVATIVES a and B THAT ARE NOT!*) a[1] := ; a1[1] := 2 ; 1

321 APPENDIX G. SOLUTIONS TO EXERCISES 2 Similar for 1 1/2 d =.5. Take v() = s 1/2 ds, v () = 1/2, hence a() = 1/ and b() =. Saemens as above, excep (*SPECIFY THOSE COEFFICIENTS a AND B AND DERIVATIVES a and B THAT ARE NOT!*) a[1] := 1/sqr{}; a1[1] := -.5 a[1]/; (b) Since Φ( x) = 1 Φ(x), we need only compue Φ(x) for posiive x. Moreover, Φ() =.5, hence Φ(x) = x ( ) exp 2 d. 2π 2 Pu v() = Φ(), v () = 1 ( ) exp 2, 2π 2 and use Ode-1.pas for (G.1) wih a() = 1 2π exp( 2 ) and b() =, working forwards ( F ) saring from v() =.5. Saemens as above, 2 excep (*SPECIFY THOSE COEFFICIENTS a AND B AND DERIVATIVES a and B THAT ARE NOT!*) a[1] := (1/sqr{2 pi}) exp(- /2); a1[1] := a[1] (-); (c) U : Use Ode-1.pas forwards for he differenial equaion U = ru + 1. V : Use Ode-1.pas backwards for he differenial equaion V = rv 1. Exercise 3 Fig 3.1: The funcions we are ineresed in are ( v 1() = F ) () = exp µ(s) ds v 2() = f() = F () µ() = v 1() v 3(), v 3() = µ() = α + β e γ. ( ) = exp α β eγ 1, γ They can be compued direcly since hey are given by explici formulas. Thus, he program Ode-1.pas is no really needed, bu i is sill useful since i can produce a nicely arranged oupu. One could drop everyhing ha has o do wih he difference scheme and jus pu in he following saemens beginning from line 27: dim := 3; (*auxiliary quaniies from Danish life able :*) alpha :=.5; (*Gomperz-Makeham parameers*) bea := ;

322 APPENDIX G. SOLUTIONS TO EXERCISES 3 gamma :=.38*ln(1); for l := o 1 do (*Do no confuse he leer l wih he number 1*) begin wrieln; wrieln(odeou); (*Line shif*) := l; v[3] := alpha + bea exp(gamma ); v[1] := exp ( - alpha - bea ( exp(gamma ) - 1)/gamma ); v[2] := v[1] v[3]; wrie(:4, ); wrie(odeou,:4, ); for j := 1 o dim do begin wrie(,v[j]); wrie(odeou,,v[j]); end; end; close(odeou); end. Alernaively, one could use he program as i is o compue F and add saemens o compue f and µ. We have where v 1() = b 11() v 1(), b 11() = α β exp(γ )). Having compued b 11() and v 1(), we compue Saemens: (*SPECIFY DIMENSION OF v!*) dim := 1; v 3() = b 11(), v 2() = v 1() v 3(). (* SPECIFY DIRECTION OF COMPUTATION - FORWARD ( F ) OR BACK- WARD ( B ):!*) BF := F ; (*auxiliary quaniies from Danish life able :*) alpha :=.5; (*Gomperz-Makeham parameers*) bea := ; gamma :=.38*ln(1); (*SPECIFY THE TIME INTERVAL [,T] BY INSERTING T IN erm AND - IF NEEDED - THE AGE OF THE LIFE AT TIME! *) x:= ; erm:= 1; (*SPECIFY STEPS IN DIFFERENCE SCHEME AND OUTPUT!: *) seps := 1; (*number of seps in difference mehod*)

323 APPENDIX G. SOLUTIONS TO EXERCISES 4 oup := 1; (*number of values in oupu*) h := erm/seps; (*seplengh*) coun := seps/oup -.5; (* SPECIFY BOUNDARY CONDITION(S) AT TIME = IF FORWARD (BF = F ) AND AT = erm if BACKWARD (BF = B ):!*) v[1] := 1; (*SPECIFY THOSE COEFFICIENTS a AND B AND DERIVATIVES a and B THAT ARE NOT!*) B[1,1] := - alpha - bea exp(gamma (x+)); B1[1,1] := - bea exp(gamma*(x+)) gamma; (Then comes he difference scheme, and we do no need o do anyhing unil we come o he oupu a imes =,1,... :) if counou > coun hen begin v[3] := - B[1,1]; v[2] := v[1] v[3]; (*his comes exra*) counou := ; wrieln; wrieln(odeou); wrie(:4, ); wrie(odeou,:4, ); for j := 1 o 3 do (*NB! 3 insead of dim*) begin wrie(,v[j]); wrie(odeou,,v[j]); end; end; end; (*difference scheme*) close(odeou); end. Exercise 4 (a) A special case of (b) - jus pu α = 1. (b) Wrie ou he defining expression in each inerval where here is one analyic expression:,, F () = 1 ( ) 1 α, < < ω, ω 1, ω. 1 (,, F () = ω ) α, < < ω, ω, ω. { ( α ω ) α 1 ( ) ( f() = ω 1 ω = α ω ) α 1, < < ω, ω ω else. The moraliy inensiy is µ() = f()/(1 F ()), defined for all such ha he denominaor is posiive: { α ( ω ) 1 = µ() = α, < < ω, ω ω ω undefined, ω.

324 APPENDIX G. SOLUTIONS TO EXERCISES 5 Noe ha µ() + as ω (as is always he case if ω < and µ is nondecreasing). For x and x + boh in (, ω), F ( x) = F (x + ) F (x) = ( ) α ω x, ω x he same ype of disribuion as F, only wih ω replaced by ω x. Thus we find f( x) and µ( x) by jus replacing ω wih ω x in he formulas above. (c) Do as above. Exercise 5 (a) p = P[T > ] = P[M] P[T > M] + P[F ] P[T > F ] = s m p m + s f p f. s m = P[M T > ] = P[M (T > )] P[T > ] = P[M] P[T > M] P[T > ] s m p m = s m p m +. sf p f (b) µ = d d p p = s m µ m + s f µ f, = sm d d pm + s f d d pf s m p m + sf p f = sm p m µ m + s f p f µf s m p m + sf p f a weighed average of he moraliies a age for males and females, he weighs being he condiional probabiliies of being male and female, respecively, given survival o age. (c) s m = s m s m + sf p f /pm = s m s m [ ], + sf exp (µm s µ f s ) ds a decreasing funcion of if µ m µ f > for all >. If (µm s µ f s ) ds =, hen s m as. (d) p x = x+p = sm x+p m + s f x+p f xp s m xp m + = s m sf xp f x p m x + s f x p f x. (From his expression we could have derived he moraliy inensiy µ x upon forming lim (1 p x)/, which would give he resul in (b).) Similarly m nq x = s m x m nqx m + s f x m nqx f, ne x = s m x ne m x + s f x ne f x.

325 APPENDIX G. SOLUTIONS TO EXERCISES 6 Exercise 6 F () = a + a 1 + a a 3 3, [, ω], a 3,. (a) Proof of f() = F () = a 1 2a 2 3a 3 2, [, ω]. µ() = f() F () = a1 + 2a2 + 3a3 2, [, ω]. a + a 1 + a a 3 3 f() lim µ() = lim ω ω F () = : (G.2) We know ha lim ω F () =. If lim ω f(), hen (G.2) holds. If lim ω f() =, hen we have a / expression in he limi and use l Hospial s rule: f () lim µ() = lim ω ω f(). If lim ω f (), hen (G.2) holds. If lim ω f () =, use l Hospial again: f () lim µ() = lim ω ω f (). Bu f () = 6a 3, and so (G.2) holds. (b) Observe ha F ( x), considered as funcion of, is also rinomial. ω F () = 1 : a = 1, F (ω) = : a + a 1 ω + a 2 ω 2 + a 3 ω 3 =, f(ω) = : a 1 + 2a 2ω + 3a 3 ω 2 =, F () d = e : a ω + a1 2 ω2 + a2 3 ω3 + a3 4 ω4 = e. Exercise 7 The probabiliy disribuion of he random variable T x is given by P[T x ] = p x,. We are ineresed in he probabiliy disribuion of a presen value P V (T x) which is jus a real-valued funcion of he random variable T x: P[P V (T x) u], u (, ). Thus for each value of u we need o deermine he se of values of T x ha make P V (T x) u and deermine is probabiliy. I may be helpful o draw a graph of he funcion, wih T x on he horizonal axis and P V (T x) on he verical axis, and for each given u on he verical axis deermine he se on he horizonal axis where he graph is under u. We summarize he resuls: (a) Pure endowmen benefi of 1: P V (T x) = e rn 1[T x > n] = {, Tx n, e rn, T x > n.

326 APPENDIX G. SOLUTIONS TO EXERCISES 7 A very simple non-decreasing funcion wih jus wo values., u <, P[P V (T x) u] = 1 np x, u < e rn, 1, e rn u. (b) Term insurance wih sum 1: P V (T x) = e r Tx 1[T x n] = { e r T x, T x n,, T x > n. A simple non-increasing funcion saring from is maximum value 1 a T x =, decreasing exponenially for T x (, n], dropping o a T x = n and remaining hereafer., u <, np x, u < e rn, P[P V (T x) u] = ln u / rp x, e rn u < 1, 1, 1 u. The hird line on he righ is he only one ha akes a small bi of calculaion: For e rn u < 1 solve e r = u o find = ln u / r, and conclude ha P V (T x) u is equivalen o T x. (c) Endowmen insurance wih sum 1: Same problem as (b), only simpler:, u < e rn, P[P V (T x) u] = ln u / rp x, e rn u < 1, 1, 1 u. (d) Life annuiy of 1 per year:, u <, P[P V (T x) u] = 1 ln(1 ru) / r p x, u < ā n, 1, ā n u. Exercise 8 (a) There are several ways of proving hese hings. We will skech wo: Firs mehod works direcly on he expressions for he expeced values: Life endowmen: or jus m+ne x = e m+n (r+µ x+u ) du = e m (r+µ x+u ) du e m+n m Deferred life annuiy: m+n m nā x = m m+ne x = v m+n m+np x = v m mp x v n np x+m. (r+µ x+u) du m+n v p x d = v m mp x v m mp x+m d = me x ā x+m n, he las sep by subsiuing τ = m in he inegral. m

327 APPENDIX G. SOLUTIONS TO EXERCISES 8 Second mehod works wih he indicaor funcions I = 1[T x > ] and he rule of ieraed expecaion E[X] = EE[X Y ]. (Seems like shooing sparrows wih cannons in hese simple examples, bu serves o illusrae a echnique ha may be useful): Life endowmen: and (jus o illusrae) m+ne x = E [ v m+n I m+n ] = v m v n E [I m+n], E [I m+n] = E [E[I m+n I m]] = mp x E[I m+n I m = 1] + mq x E[I m+n I m = ] Deferred annuiy: = mp x np x+m. m nā x = E = v m E [ m+n m [ E = v m mp xe ] v I d m+n m [ E [ +v m mq xe E ] v ( m) I d Im ] v ( m) I d Im = 1 ] v ( m) I d Im = m+n = v m mp x ā x+m n. Similar for deferred endowmen insurance. m m+n (b) Here we use he second mehod, which is general. Le P V (,u] denoe he random presen value a ime of benefis less premiums in (, u], and abbreviae P V = P V (, ]. We have m P V = P V (,u] + v u P V u. (G.3) Then V = E[P V I = 1] = E[P V (,u] I = 1] + v u E[P V u I = 1]. Here E[P V (,u] I = 1] = V (,u], and E[P V u I = 1] = E [ E[P V u I = 1, I u] I = 1] = u p x+ V u + u q x+. From his you gaher he saed resul. (c) Second mehod is he superior one. We sar from (G.3) for paymens deferred in m years: P V = v m P V m. Denoe variance by V. We have and V[P V ] = v 2m V[P V m] V[P V m] = V [ E[P V m I m]] + E [ V[P V m I m]] = V [ I m E[P V m I m = 1]] + E [ I m V[P V m I m = 1]] = V[I m] (E [P V m I m = 1]) 2 + E[I m] V [P V m I m = 1],

328 APPENDIX G. SOLUTIONS TO EXERCISES 9 hence V[P V ] = v { 2m mp x mq x (E [P V m I m = 1]) 2 + mp x V [P V m I m = 1] } ( ) = me x (2r) mex 2 (E [P V m I m = 1]) 2 + me x (2r) V [P V m I m = 1]. (G.4) For insance, for an m year deferred n year annuiy (G.4) gives he variance (see (4.16) in BL) me x (2r) 2 ( ) ā x+m n ā (2r) x+m n mex 2 ā 2 x+m n. r Apply he resul o he pure life endowmen (a deferred benefi by is very definiion) and o a deferred life insurance. (You could pu up he expressions immediaely using he rick shown in Chaper 4.) Exercise 9 N 2 = N 2 + N τ dn τ + N τ dn τ. For < T : All erms on boh sides are, so he equaion holds. For T : On he lef N 2 = 1. On he righ N 2 =, Nτ dnτ = 1, and Nτ dnτ =, so he equaion holds. Ignoring he lef limi gives 2 on he righ when > T (he jump of N a T ) has been couned wice ). The relaionship N τ dn τ = (N τ + 1) dn τ is rue for any couning process, no only he simple one considered here. By definiion N τ dn τ = τ N(τ)(N(τ) N(τ )) The sum on he righ ranges effecively over he (finie number of) ime poins τ where N jumps, and a such a ime τ i jumps (by 1) from he value i had jus before he jump, N(τ ), o he value a he jump ime, N(τ) = N(τ ) + 1. In paricular, if N has only one jump, hen N(τ ) = a he ime jump ime τ, and he inegral is herefore. Exercise 1 (c) Se v 1() = p, v 2() = ē :. They saisfy he differenial equaions wih side condiions v 1() = 1, v 2() =. Saemens: (*SPECIFY DIMENSION OF v!*) dim := 2; v 1() = v 1() µ, v 2() = v 2() v 1(), (* SPECIFY DIRECTION OF COMPUTATION - FORWARD ( F ) OR BACK- WARD ( B ):!*)

329 APPENDIX G. SOLUTIONS TO EXERCISES 1 BF := F ; (*auxiliary quaniies from Danish life able :*) alpha :=.5; (*Gomperz-Makeham parameers*) bea := ; gamma :=.38*ln(1); (*SPECIFY THE TIME INTERVAL [,T] BY INSERTING T IN erm AND - IF NEEDED - THE AGE OF THE LIFE AT TIME! *) x:= ; erm:= 1; (*SPECIFY STEPS IN DIFFERENCE SCHEME AND OUTPUT!: *) seps := 1; (*number of seps in difference mehod*) oup := 1; (*number of values in oupu*) h := erm/seps; (*seplengh*) coun := seps/oup -.5; (* SPECIFY BOUNDARY CONDITION(S) AT TIME = IF FORWARD (BF = F ) AND AT = erm if BACKWARD (BF = B ):!*) v[1] := 1; v[2] := ; (*SPECIFY THOSE COEFFICIENTS a AND B AND DERIVATIVES a and B THAT ARE NOT!*) B[1,1] := - alpha - bea exp(gamma (x+)); B1[1,1] := - bea exp(gamma*(x+)) gamma; B[2,1] := 1; Exercise 13 Premium payable a rae π in he deferred period < m and pension payable a rae 1 in he pension period m < < m+n, say. Use Thiele s differenial equaion: Inspec i direcly, or inegrae i from o and from o n o obain he rerospecive formula for in he deferred period and he prospecive formula in he benefi paymen period: { π ( ) V = exp (r + µx+u) du dτ, < m, τ m+n exp ( τ (r + µx+u) du) dτ, m < m + n. I is seen ha V is an increasing funcion for < m, no maer if µ is increasing or no; increasing gives a bigger inegrand inegraed over a longer inerval. We know from he heory in Chaper 4 of BL ha, for m, V = ā x+ m+n is decreasing if µ is increasing. Exercise 14 (a) V = E [ P V (,n] T x > ] [ ] = E P V (,+d] + e r d P V (+d,n] Tx > [ ] = (1 µ x+ d) E P V (,+d] + e r d P V (+d,n] Tx > + d

330 APPENDIX G. SOLUTIONS TO EXERCISES 11 [ + µ x+ d E = (1 µ x+ d) ] P V (,+d] + e r d P V (+d,n] < Tx < + d [ π d + e r d V +d ] + µ x+ d [ b + e r d ] + o(d) = π d + (1 µ x+ d)(1 r d)(v + d V d) + µx+ d b + o(d) d = π d + V (µ x+ + r) d V + d V d + µx+ d b + o(d) d Cancel V, divide by d, and le d go o, o arrive a Thiele s diff. eq. (b) = E [ P V 2 (,n] T x > ] [ ] = E (P V (,+d] ) P V (,+d] e r d P V (+d,n] + e 2r d (P V (+d,n] ) 2 Tx > [ ] = (1 µ x+ d) E (P V (,+d] ) P V (,+d] e r d P V (+d,n] + e 2r d (P V (+d,n] ) 2 Tx > + d [ ] + µ x+ d E (P V (,+d] ) P V (,+d] e r d P V (+d,n] + e 2r d (P V (+d,n] ) 2 < Tx < + d [ ] = (1 µ x+ d) ( π d) 2 + 2( π d)e r d V +d + e 2r d V (2) +d [ +µ x+ d b 2 + 2b e r d + e 2r d 2] + o(d) = (1 µ x+ d) 2 ( π d) (1 r d) (V + dd ) ( V d + (1 µ x+ d)(1 2r d) V (2) + d ) d V (2) d V (2) +µ x+ d b 2 + o(d) = 2π d V + (1 (µ x+ + 2r) d)v (2) + d d V (2) d + µ x+ d b 2 + o(d) Cancel V (2), divide by d, le d go o, and rearrange a bi o arrive a d d V (2) = 2π V + (µ x+ + 2r)V (2) µ x+ b 2.

331 APPENDIX G. SOLUTIONS TO EXERCISES 12 Exercise 15 (a) Le τ 1, τ 2,... denoe he imes of ransiion of Z lised in chronological order. The process N = i 1[τi ],, which couns he oal number of ransiions, is a homogeneous Poisson process wih inensiy µ. (Wrie µ insead of λ o make he formulas given in iem (b) meaningful.) The imes τ 1, τ 3,... are he imes where N akes odd values, and τ 2, τ 4,... are he imes where N akes even values. Thus N 12(, u] is he number odd numbered occurrences ] of Poisson evens beween ] ime and ime u. In paricular, N 12() =. Likewise, N 21() =. T 1(, u] is he oal ime [ N()+1 2 in (, u] wih odd value of N. [ N() 2 (b) We could derive he backward differenial for he funcions V (q) j () and W (q) j (), bu hey are really special cases of Thiele s differenial equaion for he reserve and is generalizaion o higher order momens: T 1(, n] is he presen value a ime of an n-year annuiy of 1 per ime uni running in sae 1 in he presen simple Markov model, and wih no ineres. Similarly, N 12(, n] is he presen value a ime of an n-year insurance 1 payable upon every ransiion from sae 1 o sae 2. Thiele for he annuiy: subjec o d d V (1) 1 () = 1 µ (V (1) 2 () V (1) 1 ()), (G.5) d d V (1) 2 () = µ (V (1) 1 () V (1) 2 ()), (G.6) V (1) 1 (n) = V (1) 2 (n) =. (G.7) There are many ways of solving hese equaions and we menion a few: (1) Differeniae (G.5) and subsiue expressions for he firs order derivaives from (G.5) and (G.6) o obain a second order differenial equaion in V (1) 1 () subjec o condiions V (1) 1 (n) = and d d V (1) 1 (n) = 1, he laer obained from (G.5). There are sandard mehods for his problem, see soluion o Exercise No 15 in Exercise paper No 5. (2) Add (G.5) and (G.6) o obain d (1) (V 1 + V (1) 2 )() = 1, d subjec o (V (1) 1 + V (1) 2 )(n) =. Soluion: Subrac (G.6) from (G.5) o obain V (1) 1 + V (1) 2 () = n. (G.8) d (1) (V 1 V (1) 2 )() = 1 2 µ (V (1) 1 V (1) 2 )(), d subjec o (V (1) 1 V (1) 2 )(n) =. Soluion (we recognize he differenial equaion and side condiion for he reserve on a deerminisic annuiy of 1 a ineres rae 2 µ): V (1) 1 () V (1) 2 () = n e 2 µ(τ ) dτ µ (n ) 1 e 2 dτ =. 2 µ

332 APPENDIX G. SOLUTIONS TO EXERCISES 13 We find V (1) 1 () = n 2 + µ (n ) 1 e 2, (G.9) 4 µ V (1) 2 () = n 2 µ (n ) 1 e 2. 4 µ (3) Already before we arrived a relaion (G.8) we ough o have realized he following: Given Z() = 1, T 1(, n] is he ime ha remains o spend in he curren sae. Given Z() = 2, T 1(, n] is he ime ha remains o spend in he oher sae. Given Z() = 1, T 2(, n] = (n ) T 1(, n] is he ime ha remains o spend in he oher sae. Due o symmery, he wo las menioned cases are probabilisically idenical, so we can conclude ha V (1) 2 () = (n ) V (1) 1 (), which is (G.8). Subsiuing his ino (G.5), we ge d d V (1) 1 () = 1 + µ ((n ) 2 V (1) 1 ()), which is o be solved subjec o V (1) 1 (n) =. We easily arrive a he soluion above. Now o non-cenral second order momens, and we proceed wih mehod (3) using he general differenial equaion, which specializes o d d V (2) 1 () = µ V (2) 1 () 2 V (1) 1 () µ V (2) 2 (), subjec o V (2) 1 (n) =. Using he symmery again, we realize ha V (2) 2 () = (n ) 2 2 (n ) V (1) 1 () + V (2) 1 (). Subsiuing his and reorganizing a bi, he differenial equaion above becomes d d V (2) 1 () = 2 (µ (n ) 1)V (1) 1 () µ (n ) 2. Insering he expression (G.9), we ge n ( ( ) ) V (2) n τ µ (n τ) 1 e 2 1 () = 2 (µ (n τ) 1) + µ (n τ) 2 dτ. 2 4 µ Subsiuing n τ, we are lef wih he simple ask of inegraing some sandard funcions. One should finally arrive a ) 2 Var[T 1(, n] Z() = 1] = V (2) 1 () (V (1) 1 ()) 2 = 1 1 (2 e 2µ (n ) n +. 4µ 4µ Exercise 16 (a) Obviously, for < s < u, p aa (, u) = p aa (, s)p aa (s, u).

333 APPENDIX G. SOLUTIONS TO EXERCISES 14 Deails: P[Z(τ) = a, τ (, u] Z() = a] = P[Z(τ) = a, τ (, s] Z() = a]p[z(τ) = a, τ (s, u] Z() = a, Z(τ) = a, τ (, s]] ; The firs facor here is p aa(, s), and he second facor is (due o he Markov propery) p aa (s, u). For insance forward argumen (pu u and u + du in he roles of s and u): p aa(, u + du) = p aa(, u)p aa(u, u + du) = p aa(, u)(1 (µ(u) + σ(u))du). From his we ge paa(, u) = paa(, u)(µ(u) + σ(u)). u Inegraing, using he side condiion p aa(, ) = 1, we arrive a he answer. (b) Using he Markov propery: p aa(, 1) σ( 1) d 1 p ii( 1 + d 1, 2) ρ( 2) d 2 p aa( 2 + d 2, 3) µ( 3) d 3 (plus o(d 1) + o(d 2) + o(d 3), sricly speaking). Due o he facors d i we can replace he argumens i + d i appearing here by i. (d) The Kolmogorov forward equaions wih consan inensiies, leing differeniaion w.r.. be denoed by primes: p aa() = p aa()(µ + σ) + p ai()ρ, p ai() = p aa()σ p ai()(ν + ρ). (G.1) (G.11) Side condiions: p aa() = 1, p ai() =. (G.12) Differeniae (G.1): p aa() = p aa()(µ + σ) + p ai()ρ. (G.13) Subsiue here p ai() from (G.11): p aa() = p aa()(µ + σ) + (p aa()σ p ai()(ν + ρ))ρ. (G.14) Now solve p ai() from (G.1) and subsiue ino (G.14) and rearrange a bi o arrive a p aa() + p aa()((µ + σ) + (ν + ρ)) + p aa()((µ + σ)(ν + ρ) σρ) =. (G.15) This is a simple homogeneous second order ordinary differenial equaion, which is o be solved subjec o he condiions p aa() = 1, p aa() = (µ + σ), (G.16) he laer obained by seing = s in (G.1). The general soluion o (G.15) - (G.16) is p aa() = c 1e r 1 + c 2e r 2,

334 APPENDIX G. SOLUTIONS TO EXERCISES 15 when r 1 and r 2 are disinc soluions o r 2 + r((µ + σ) + (ν + ρ)) + ((µ + σ)(ν + ρ) σρ) =. Here c 1 and c 2 are consans ha are o be deermined so as o mach he side condiions (G.16): c 1 + c 2 = 1, c 1r 1 + c 2r 2 = (µ + σ). If he roos r 1 and r 2 coincide, r 1 = r 2 = r (say), hen he general soluion is of he form (c 1 + c 2)e r. You find r 1 r 2 } Fill in he deails yourself. = (µ + σ) + (ν + ρ) ± ((µ + σ) (ν + ρ)) 2 + 4σρ 2 Exercise 17 Recall he forward equaions: Side condiions paa(, ) = paa(, )(µx+ + σx+) + pai(, )ρx+, (G.17) pai(, ) = paa(, )σx+ pai(, )(νx+ + ρx+). (G.18) p aa(, ) = 1, p ai(, ) =. (G.19) Noe he following: The forward differenial equaions for he ransiion probabiliies are usually easier o work wih han he backward differenial equaions. Under he forward consrucion we work wih he funcions p jk (, ), k = 1, 2,..., J, for fixed j and. These funcions sum o one and herefore we need only solve J 1 equaions. Under he backward consrucion we work wih he funcions p jk (, u), j = 1, 2,..., J, for fixed k and u, which do no sum o 1 or anyhing else ha could be helpful. If one should never he less wan o use he backward equaions in he presen siuaion, one would face a quasi-difficuly which is due o he noaion: e.g. in p aa x, which means p aa(x, x + ), x appears in boh ime variables. To apply he backward consrucion one mus paramerize ime as in he general heory, leing saring ime and ending ime be funcionally unrelaed. (a) When ρ x+ =, (G.17) reduces o paa(, ) = paa(, )(µx+ + σx+), which subjec o he firs condiion in (G.19) inegraes o ( ) p aa(, ) = exp (µ x+s + σ x+s) ds,

335 APPENDIX G. SOLUTIONS TO EXERCISES 16 he same as he occupancy probabiliy p aa (, ), of course. The equaion (G.18) reduces o pai(, ) = paa(, )σx+ pai(, ) νx+, or pai(, ) + pai(, ) νx+ = paa(, )σx+, where he funcion on he righ is now known. Muliply by inegraing facor exp inegrae from o, use he second condiion in (G.19), and arrive a ( τ ) ( ) p ai(, ) = exp (µ x+s + σ x+s) ds σ x+τ exp ν x+s ds dτ, or p ai(, ) = τ p aa x σ x+τ τ p ii x+τ dτ. τ ( νx+s ds ), This expression can be read aloud; Under he inegral sign is he probabiliy ha he policy says in a unil an inermediae ime τ, hen makes he ransfer o sae i in he ime inerval (τ, τ + dτ], and hereafer says in sae i unil ime, and he inegral sums up he probabiliies of hese muually exclusive evens. See Secion 7.F of BL. The reason why we ge explici expressions here is ha here is no reurn o a sae once he policy has lef i. I should be made clear, however, ha compuaion of he probabiliies goes by numerical soluion o he differenial equaions, which is jus as easy wih recovery as wihou. The explici expressions are useful mainly because hey can be direcly undersood and also because hey give us a possibiliy of discussing how he ransiion probabiliies depend on he inensiies. In he same manner as for p aa(, ) we find in he case wihou recovery ha ( ) p ii(, ) = exp ν x+s ds. (b) The condiional probabiliy of being acive a ime, given alive a ime (and sar as acive a ime ), is p aa P[Z() = a Z() {a, i}] x = P[Z() = a Z() {a, i}] = P[Z() {a, i}] p aa(, ) = p aa(, ) + p ai(, ). Likewise, p ai p ai(, ) x = p aa(, ) + p ai(, ). Differeniaing p [x] = p aa(, ) + p ai(, ) and using (G.17) and (G.18), one finds µ [x]+ = p [x] p [x] = paa(, ) + pai(, ) p aa(, ) + p ai(, ) paa(, )(µx+ + σx+) + pai(, )ρx+ + paa(, )σx+ pai(, )(νx+ + ρx+) = p aa(, ) + p ai(, ) paa(, )µx+ + pai(, )νx+ = p aa(, ) + p ai(, ) = p aa x µ x+ + p ai x ν x+,

336 APPENDIX G. SOLUTIONS TO EXERCISES 17 a weighed average of moraliy for acive and moraliy for invalid, he weighs being he condiional probabiliies of being acive and invalid, respecively, given survival. The selec mechanism is in his case due o he fac ha he insured is known o be, no jus alive and x years old a ime, bu also in acive sae. (c) If µ x+ = ν x+ for all, hen µ [x]+ = µ x+, of course, and We have p aj x ( p [x] = exp ) µ x+s ds. ( ) = p aj x / p [x] = p aj x exp µ x+s ds, j = a, i. Differeniaing w.r.., and rearranging a bi, you will see ha he p aj x, j = a, i, saisfy he differenial equaions (G.17) - (G.18) wih ρ =, which are he Kolmogorov forward equaions for in he parial model. They also saisfy he side condiions (G.19). Thus, in he case of non-differenial moraliy, he ransiion probabiliies can be obained by firs deermining he ransiion probabiliies in he simpler parial model wih only wo saes, and hen muliplying hem wih he survival probabiliy. (d) Saemens: (* SPECIFY NON-NULL PAYMENTS AT TIME! *) bi[1,3] := 1; bi[2,3] := 1; ca[1] := 1; (* SPECIFY MAXIMUM ORDER OF MOMENTS AND NUMBER OF STATES! *) q := 1; (*momens*) JZ := 3; (*number of saes of he policy*) (* SPECIFY TRANSITION INTENSITIES FOR POLICY Z! Here Danish basis exended wih recovery; Saes 1 = acive, 2 = disabled, 3 = dead:*) alpha[1,3] :=.5; bea[1,3] := ; gamma[1,3] :=.38*ln(1); alpha[2,3] :=.5; bea[2,3] := ; gamma[2,3] :=.38*ln(1); alpha[1,2] :=.4; bea[1,2] := ; gamma[1,2] :=.6*ln(1); alpha[2,1] :=.5; (*SPECIFY AGE x, TERM, INTEREST RATES AND NON-NULL LIFE ENDOW- MENTS! *) x:= 3; (*age*) := 3; (*erm*) r := ln(1+.45); (*ineres rae*)

337 APPENDIX G. SOLUTIONS TO EXERCISES 18 be[1] := ; be[2] := ; (*endowmens a erm of conrac*) (*SPECIFY LUMP SUM PREMIUM AT TIME : PUT c := 1 IF ALL OTHER PREMIUMS ARE AND ONLY MOMENTS OF BENEFITS ARE WANTED! *) c := ; b := ; (*SPECIFY NUMBER OF STEPS IN RUNGE-KUTTA (OPTIONAL)! *) seps := 3; (*number of seps*) h := /seps; (*seplengh*) coun := 3; (*number of oupu imes*) coun := seps/coun -.5; Resul: sae: 1 2 ime: 3. NC 1:.. ime: 29. NC 1: ime: 28. NC 1: ime: 27. NC 1: ime: 15. NC 1: ime: 1. NC 1: ime:. NC 1: pi =.4336 Exercise 18 The process mus sar somewhere, so le us say Z() = 1. (a) P [ r+1 i= Z( i) = j i ] [ r ] [ r ] = P Z( i) = j i P Z( r+1) = j r+1 Z( i) = j i i= i=

338 APPENDIX G. SOLUTIONS TO EXERCISES 19 [ r ] = P Z( i) = j i p jr,jr+1 ( r, r+1). i= We have here used he Markov propery of he process. Repeaing his, we obain [ r+1 ] r+1 P Z( i) = j i = p 1j (, ) p ji 1,j i ( i 1, i). (G.2) i= i=1 Nex, using (G.2), [ r ] P Z( i) = j i Z( ) = j, Z( r+1) = j r+1 i=1 = P [ r+1 i= Z(i) = ji ] P [Z( ) = j, Z( r+1) = j r+1] = p1j (, r+1 ) i=1 pj i 1,j i ( i 1, i) p 1j (, ) p j,j r+1 (, r+1) r+1 i=1 = pj i 1,j i ( i 1, i). (G.21) p j,j r+1 (, r+1) (b) For s = < 1 < < r < r+1 = : [ ] r 1 P Z( r) = j r Z( i) = j i, Z(s) = i, Z() = j i=1 [ ] r 1 = P Z( r) = j r Z( i) = j i, Z( ) = j, Z( r+1) = j r+1 = i=1 P [ r+1 i= Z(i) = ] [ ji ] P i=,...,r 1,r+1 Z(i) = ji = p 1j (, ) r+1 i=1 pj i 1,j i ( i 1, i) p 1j (, ) i=1,...,r 1 pj i 1,j i ( i 1, i) p jr 1,j r+1 ( r 1, r+1) = pj r 1,j r ( r 1, r) p jr,j( r, ) p jr 1,j( r 1, ). (G.22) For given r =, j r+1 = j (and = s, j = i) his is jus a funcion p jr 1,j r ( r 1, r) (say) of j r 1, j r, r 1, and r, showing ha he condiional Markov chain is iself Markov. The inensiies of he condiional Markov chain are p gh (τ, τ + dτ) µ gh (τ) = lim dτ dτ lim µ gh(τ) = τ = lim dτ p gh (τ, τ + dτ) p hj (τ + dτ, ) dτ p gj(τ + dτ, ) = µ gh (τ) p hj(τ, ) p gj(τ, ). µ gh () µ hj () µ gj () g j, h j,, g j, h = j,, g = j, h j. (G.23)

339 APPENDIX G. SOLUTIONS TO EXERCISES 2 The expression in he case g j, h j is obained upon wriing p hj (τ, ) p gj(τ, ) = p hj(τ, )/( τ) p gj(τ, )/( τ) before aking he limi. Think abou hese resuls - hey are quie naural. (e) Consan inensiies and no recovery: pii(τ, ) σ(τ) = σ(τ) p ai(τ, ), µ(τ) = µ(τ) p di(τ, ) p ai(τ, ) =. ν ( τ) e σ(τ) = σ τ e (µ+σ)(s τ) σ e ν ( s) ds = 1 { ν µ σ =, ν µ σ, e (ν µ σ)( τ) 1 1, ν µ σ =. τ τ e(ν µ σ)(s τ) ds Exercise 19 (a) An n-year endowmen insurance wih sum insured 1 agains level premium c per ime uni, coninuous ime, age of insured upon issue of conrac is x. (b) Inegraion is book-work, see lecure noes. One obains V = n e τ (r+µ x+s ) ds (µ x+τ c) dτ + e n (r+µ x+s ) ds = Āx+ n c ā x+ n. Deermine c by equivalence requiremen V = (no down paymen a ime): (b) One mus firs prove c = Āx n ā x n. Ā x+ n = 1 r ā x+ n, which is book-work (here are several ways). See lecure noes. Then wrie V = 1 r ā x+ n c ā x+ n = 1 (r + c) ā x+ n. Thus, he problem reduces o proving ha if µ x+ is an increasing funcion of, hen ā x+ n is a decreasing funcion of. This is book-work, see lecure noes.

340 APPENDIX G. SOLUTIONS TO EXERCISES 21 To consruc an example where V is no an increasing funcion, i.e ā x+ n is no a decreasing funcion, we mus obviously ake a µ x+ ha is no increasing. Fix < n and wrie ā x n = = n e τ (r+µ x+s ) ds dτ e τ (r+µ x+s ) ds dτ + e (r+µ x+s ) ds = ā x + E xā x+ n. n e τ (r+µ x+s ) ds dτ Keep µ x+s fixed for s n. By increasing µ x+s for s, we can obviously make ā x and E x arbirarily small. In paricular we can arrange ha ā x <.5 ā x+ n and E x <.5, hence ā x n < ā x+ n. Exercise 2 (a) Book-work. See lecure noes. (b) p is a survival funcion since i is decreasing and p = 1, p =. The moraliy inensiy is µ = d d p ( δ + W (x) δ + W (x + ) = = γw() p δ + W (). ) γ ( = ) γ. (G.24) p x = x+p δ + W (x) = xp (δ + W (x)) + (W (x + ) W (x) This is of he same form as p, only wih W () and δ replaced by W (x + ) W (x) and δ + W (x), respecively. ( ) γ P[W (T ) + δ > x] = P[T > W 1 δ (x δ)] = = W (W 1 (x δ)) + δ ( ) γ δ, x x > δ, a Pareo disribuion wih shape parameer γ and runcaion parameer δ. (c) Use (F.5) wih G() = 2, hence dg() = 2 d E[T 2 ] = 2 1 (1 + 2 /δ) γ d. Subsiue u = /δ. We have du = 2 /δ d and u is an increasing funcion of, u = 1 for =, and u = for =. Thus E[T 2 ] = δ One could also observe direcly ha 1 u γ du = δ γ 1. 2 ( /δ ) γ δ d = d ( /δ ) γ+1 γ + 1 δ [ (1 = + 2 /δ ) γ+1 ( /δ ) ] γ+1 γ + 1 δ = 1 γ.

341 APPENDIX G. SOLUTIONS TO EXERCISES 22 (d) Recall (G.24). We need ln µ = ln γ + ln w() ln(δ + W ()) and T z µ d = Log likelihood is ln Λ = T z γ w() d = γ (ln(δ + W (T z)) ln δ). δ + W () m; T m< z m [ln γ + ln w(t m) ln(δ + W (T m))] γ m [ln(δ + W (T m z)) ln δ]. Firs order derivaives: γ ln Λ = m; T m< z m γ 1 δ ln Λ = 1 δ + W (T γ m; T m< z m m) m [ln(δ + W (T m z)) ln δ], m ( 1 δ + W (T m z) 1 ) δ Pu N(z) = {m; T m < z}, he number of deahs wihin age z. The ML equaions are N(z) ˆγ =, (G.25) m ln(1 + W (Tm z)/ˆδ) 1 m; T m< z m ˆδ + W (T + ˆγ ( ) 1 m) m ˆδ + W (T m z) 1ˆδ =. (G.26) These equaions have o be solved numerically. Inser he expression (G.25) for ˆγ ino (G.26) and solve he laer w.r.. ˆδ. Subsiue he soluion ino (G.25) o find ˆγ. To find he asympoic variance marix, form he second derivaives 2 ln Λ, γ 2 2 ln Λ, 2 ln Λ, find heir expeced values, change sign, and inver he informaion δ 2 γ δ marix. (e) If δ is known, hen (G.25) wih ˆδ replaced by δ is an explici expression for ˆγ. Divide by n in denominaor and numeraor, le n go o, and use he law of large numbers o conclude ha From (F.6) we have ˆγ P[T 1 z] E[ln(1 + W (T 1 z)/δ)]. P[T 1 z] = 1 p ( ) γ δ = 1. W (z) + δ. (G.27) Use (F.5) wih p given by (F.6) and G() = ln(1 + W ( z)/δ), for which G() = and { w() dg() = d, < < z, δ+w () > z.

342 APPENDIX G. SOLUTIONS TO EXERCISES 23 We find z ( ) γ δ w() E[ln(1 + W (T 1 z)/δ)] = W () + δ δ + W () d z = δ γ w() d (W () + δ) γ+1 ( ) = δ γ 1 γ (W () + δ) 1 γ (W (z) + δ) γ ( ( ) γ ) δ = γ 1 W (z) + δ Insering hese expressions in (G.27), we find ha he limi is γ. (f) OE rae in year j is ˆµ = Nj W j, where N j = {m; j 1 < T m j} and W j = m ((Tm j) (j 1)). Choose represenaive age τ j (j 1, j] and weighs w j and minimize Q = n w j (µ(τ j; γ, δ) ˆµ j) 2 = j=1 n j=1 ( ) 2 γw(τj) δ + W (τ j) ˆµj. Minimize by forming he parial derivaives, γ Q = δ Q = n j=1 n j=1 w j 2 w j 2 ( ) γw(τj) w(τj) δ + W (τ j) ˆµj δ + W (τ, j) ( ) γw(τj) γ w(τj) δ + W (τ j) ˆµj (δ + W (τ, j)) 2 seing hem equal o zero and solving for γ and δ Exercise 21 (a) Direc backward argumen, condiioning on wha happens in he small ime inerval (, + d) and spliing paymens in (, z] ino paymens in (, + d] and paymens in ( + d, z]: V a() = (1 µ x+ d σ x+ d)( c d + e r d V a( + d)) + σ x+ d (O(d) + e r d V i( + d)) + µ x+ d O(d) + o(d), where O(d) is of order d and o(d) is of order less han d (i.e o(d)/d as d ). Inser (Taylor expansion o firs order) e r d = 1 r d + o(d), V j( + d) = V j() + d Vj() d + o(d), j = a, i, d

343 APPENDIX G. SOLUTIONS TO EXERCISES 24 and collec all erms of order d 2 or less in o(d): V a() = c d + (1 µ x+ d σ x+ d r d) V a() + d Va() d + σx+ d Vi() + o(d). d Subrac V a() on boh sides, divide by d and le d go o zero, o arrive a d Va() = rva() + c σx+(vi() Va()) + µx+va(). d Similarly d Vi() = r Vi() b + νx+vi(). d Side condiions V a(z) = V i(z) =. (b) The differenial equaions now become d Va() = rva() + c σx+(vi() Va()). d d Vi() = r Vi() b + νx+ Vi()). d Side condiions remain he same. (c) Same produc as in (a) plus an endowmen insurance wih sum 1. (d) p aa(s, ) = e s (µ x+u +σ x+u ) du, p ii(s, ) = e s ν x+u du, p ai(s, ) = s e τ s (µ x+u +σ x+u ) du σ x+τ e τ ν x+u du dτ, p ad (s, ) = 1 p aa(s, ) p ai(s, ). (e) As usual, le I j() and N jk (d) denoe, respecively, he number of policies in sae j a ime and he oal number of ransiions j k up o and including ime. The log likelihood funcion is ln Λ = z z ln σ dn ai() + 1s derivaives: α ln Λ = β ln Λ = z (α + βe γ(x+) + σ) I a() d z z z z ln(α + βe γ(x+) ) dn ad () + z 1 α + βe dn ad() + γ(x+) (I a() + I i()) d, e γ(x+) α + βe γ(x+) dn ad() + z (α + βe γ (x+) ) I i() d. z z (e γ(x+) I a() + e γ (x+) I i()) d, ln(α + βe γ (x+) ) dn id () 1 α + βe γ (x+) dn id() e γ (x+) α + βe γ (x+) dn id()

344 APPENDIX G. SOLUTIONS TO EXERCISES 25 γ ln Λ = γ ln Λ = σ ln Λ = z z z βe γ(x+) (x + ) α + βe γ(x+) dn ad () βe γ (x+) (x + ) dn α + βe γ (x+) id () 1 σ dnai() z I a() d. z z βe γ(x+) (x + )I a() d, βe γ (x+) (x + )I i() d, Se equal o and solve for he ML esimaors. Exercise 23 Assume ha X is non-decreasing. Noe ha D(X) = m=1 n=1 D m,n, where D m,n = { ; n, X + X 1 }. Since m > X n X + D m,n (X + X ) > X + 1 m Dm,n, we conclude ha D m,n is finie. Thus, being a counable union of finie ses, D(X) is (a mos) counable. Exercise 25 dx q ( ) = qx q 1 x c d + (X + x d ) q X q dn. Exercise 26 (See also Exercise 56. Here noaion is changed: a and b are called α and β) ds() = e α+βn() αd + dn() (e α+β(n( )+1) e α+βn( )) ( ) = e α+βn( ) αd + dn()e α+βn( ) e β 1 ( ( ) ) = S( ) α d + e β 1 dn() ( ( )) ( ) = S( ) α + λ e β 1 d + S( ) e β 1 dm(), where M() = N() λ, a so-called maringale (a process wih condiionally zero mean and uncorrelaed incremens, here acually independen incremens). The res is sraighforward. Exercise 34 Balance equaion 14 (1 + i) 24 j c l 55+j 24 j= j=15 (1 + i) 24 j b l 55+j =

345 APPENDIX G. SOLUTIONS TO EXERCISES 26 gives c = 24 j=15 (1 + i) j l 55+j 14 j= (1 + i) j l 55+j =.381. Exercise 5 (a) Kolmogorov forward, using he obvious p aa(, u) = 1 p ai(, u): p ai(, u + du) = p ai(, u)(1 ρ du) + (1 p ai(, u))σ du, pai(, u) + pai(, u)(ρ + σ) = σ u ( p ai(, u)e (ρ+σ)u) = σe (ρ+σ)u u Inegrae from o u using p ai(u, u) = o find p ai(, u)e (ρ+σ)u = σ ρ + σ ( e (ρ+σ)u e (ρ+σ)), hence he claimed formula. I depends only on u due o homogeneiy. (b) Eiher direc inegraion π n = n p ai(, τ) dτ = σ ρ + σ or use Thiele s differenial equaions. Single premium per uni of ime insured is The funcion π n n = σ ρ + σ (n 1 ) e (ρ+σ)n, ρ + σ (1 1 ) e (ρ+σ)n. (ρ + σ)n has derivaive g(x) = 1 e x x g (x) = e x x 1 e x x 2 which is <. I follows ha, as n +, π n n σ ρ + σ. = e x 1 + x e x x 2, Reasonable: Increasing funcion of σ, decreasing funcion of ρ. (G.28) (c) Likelihood Λ = σ N ai ρ N ia e σwa ρw i, where W a = m n, he oal ime spen in acive sae (and W i = nm W a he oal ime spen in inacive sae). l=1 T (m) Nai ln Λ = σ σ Wa,

346 APPENDIX G. SOLUTIONS TO EXERCISES 27 Nai ln Λ = σ2 σ, 2 plus similar expressions for derivaives w.r.. ρ, and 2 2 σ ρ ln Λ =. Thus, he ML esimaors are he occurrence-exposure raes ˆσ = Nai W a, ˆρ = Nia W i, which are asympoically independen, normally disribued and unbiased, and where we have used EN ai = m n as.varˆσ = p aa(, τ)σ dτ = mσ see iem (b). For fixed w = mn (i.e. m = w/n) σ2 σ = EN ai mn(1 π, n/n) as.varˆσ = n (1 p ai(, τ)) dτ = mnσ(1 πn n ), σ w(1 πn n ), which is an increasing funcion of he ime span n by he resuls in iem (b). We also find as.varˆρ = ρ2 = ρ, EN ia where we have used EN ia = m n w πn n p ai(, τ)ρ dτ = mnρ πn n. We see ha as.varˆρ is a decreasing funcion of he ime span n for fixed w. Commen: The asympoic variance of an inensiy esimaor is beer he longer he oal expeced ime spen in he sae from which he he relevan ransiion is made. All policies sar from sae a a ime. For fixed oal exposure he esimaion of σ will be good for many policies observed over a shor ime (when hey are likely o remain acive), and esimaion of ρ will be good for few policies observed over a long ime when hey can make i o inacive sae. (d) We find p jk (, u) = P[Z(s1) = j1,..., Z(sr) = jr, Z() = j, Z(u) = k, Z(n) = i] P[Z(s 1) = j 1,..., Z(s r) = j r, Z() = j, Z(n) = i] = paj 1(, s 1) p jr 1 j r (s r 1, s r)p jrj(s r, )p jk (, u)p ki (u, n) p aj1 (, s 1) p jr 1 j r (s r 1, s r)p jrj(s r, )p ji(, n) = p jk(, u)p ki (u, n) p ji(, n).

347 APPENDIX G. SOLUTIONS TO EXERCISES 28 Corresponding inensiies: and, similarly, p ai(, u) p ii(u, n) pii(, n) σ() = lim = σ u u p ai(, n) p ai(, n) pai(, n) ρ() = ρ p ii(, n), wih p ai(, n) and p ii(, n) given by expressions in iem (a). Full expression for σ() is useful laer: ) 1 (1 ρ e (σ+ρ)(n ) σ+ρ σ() = σ σ (1 = σ + ρe (σ+ρ)(n ). σ+ρ e (σ+ρ)(n ) ) 1 e (σ+ρ)(n ) (e) When ρ =, σ σ() = 1 e. σ(n ) Le he occurrence-exposure rae in year j = 1,..., n be ˆσ j = Nai;j W a;j, where N ai;j and W a;j are he number of claims and he oal ime a risk in year j (ime inerval [j 1, j)). Graduaion means esimaing σ by minimizing he weighed squares differences beween he observed raes ˆσ j and he heoreical inensiies σ(j.5), wih weighs equal o he inverse esimaed variances Varˆσ ˆ j = N ai;j/wa;j: 2 n j=1 Wa;j 2 ( ) 2 σ ˆσ j. N ai;j 1 e σ(n j+.5) Exercise 51 (a) leads o p (1) ai (, + d) = p(1) (, )(1 (ν() + ρ()) d) + paa(, ) σ() d d wih side condiion d p(1) ai ai (, ) = p(1) (, )(ν() + ρ()) + paa(, ) σ(), ai p (1) ai (, ) =. (Inegraing gives he following inegral expression, which could be pu up by direc reasoning: Nex, leads o p (1) ai (, ) = exp( s (µ + σ))σ(s) ds exp( s ai (ν + ρ)).) p (1) aa (, + d) = p (1) aa (, )(1 (µ() + σ()) d) + p (1) (, ) ρ() d d d p(1) aa (, ) = p (1) aa (, )(µ() + σ()) + p (1) ai (, ) ρ(),

348 APPENDIX G. SOLUTIONS TO EXERCISES 29 wih side condiion p (1) aa (, ) =,. (Inegral expression, which could be pu up by direc reasoning: and p (1) aa (, ) = p (1) Repeaing he argumen for k = 2, 3,...: d d p(k) ai ai (, s) ρ(s) ds exp( (µ + σ)).) (, ) = p(k) (, )(ν() + ρ()) + p(k 1) aa (, ) σ(), ai p (k) ai (, ) =, d d p(k) aa (, ) = p (k) aa (, )(µ() + σ()) + p (k) ai (, ) ρ(), p (k) aa (, ) =. Inroducing p () aa (, ) = p aa(, ) would save work. (b) (c) Obviously, k=1 p (k) ai (, ) = pai(, ). P [Z(τ) = i; τ [ q, ] Z() = a] = p ai(, q)p ii ( q, ). π = n q e rτ p ai(, τ q)p ii (τ q, τ) dτ. Reserve: +q e r(τ ) p ii (, τ) dτ + n +q e r(τ ) p ii(, τ q)p ii (τ q, τ) dτ. s Exercise 52 Expeced PV a ime of benefis is n n/2 e rτ (1 τ n/2 p x) τp yµ y+τ dτ. Expeced PV a ime of premiums is π imes n/2 e rτ τ p xτ p y dτ. Equivalence premium π is he raio beween hese expressions. Exercise 53 (a) Expeced presen value a ime of benefis is 5 imes

349 APPENDIX G. SOLUTIONS TO EXERCISES 3 Consider he funcions [ W b = E e W b j () = E m ( r(s)+a(s) µ(x+s)) ds [ e m ( r(s)+a(s) µ(x+s)) ds ] Y () = i. (G.29) ] Y () = j, [, m], j = 1,..., J. We need o deermine W b = Wi b (). Backward consrucion: Wj b () = (1 λ j d)e ( r j+a j µ(x+)) d Wj b ( + d) + λ jk dwk() b + o(d) k;k j leads o d d W b j () = (λ j + r j a j + µ(x + ))W b j () k; k j λ jk W b k(). Solve backwards subjec o condiions j = 1,..., J. W b j (m) = 1, Expeced presen value a ime of premiums is π imes [ m ] W c τ ( r(s)+a(s) µ(x+s)) = E e ds dτ Y () = i. (G.3) Consider he funcions [ m Wj c () = E e τ ( r(s)+a(s) µ(x+s)) ds dτ ] Y () = j. [, m], j = 1,..., J. We need o deermine W c = Wi c (). Backward consrucion: ( ) Wj c () = (1 λ j d) d + e ( r j+a j µ(x+)) d Wj c ( + d) + λ jk dwk() c + o(d) leads o k;k j d d W j c () = 1 + (λ j + r j a j + µ(x + ))Wj c () Solve backwards subjec o condiions W c j (m) =, k; k j λ jk W c k (). j = 1,..., J. Solve boh sysems numerically by e.g. prores2 and deermine π = W b /W c. (b) If a j = r j for all j, hen a() = r() for all and hey cancel ou of he expressions for he presen values when we drop he expecaion. Thus, equivalence in he sense defined can be aained.

350 APPENDIX G. SOLUTIONS TO EXERCISES 31 Exercise 54 (a) Wih profi conrac: Sipulaing benefis and premiums in nominal values, binding o boh paries. Charge a premium on he safe side, ypically by using conservaive echnical (firs order) valuaion basis. If everyhing goes well, surplus will accumulae. This surplus belongs o he insured and is o be repaid as so-called bonus, e.g. as increased benefis or reduced premiums. Uni linked: Uni linked conrac: Sipulaing benefis and (possibly) premiums in unis of a share of he invesmen porfolio, ha is, le conracual paymens be inflaed by he price index of he invesmen porfolio raher han being fixed nominal amouns. Wih all paymens perfecly linked, he financial risk will be eliminaed in a large porfolio. (b) Single premium, given Y () = i, is [ π = E e ] n r (U(n) g) Y () = i np x. Disregarding he unineresing erm np x henceforh, we need o deermine [ W = E (1 e ] n r g) Y () = i. The saring poin is he price of he claim a ime, given he curren informaion abou he pas, [ E e ] [ ( n r (U(n) g) Y (τ); τ = E U() e ) ] n r g Y (τ); τ. Due o he Markov propery his expression is a funcion of, Y () and U(). Consider is value a ime for given U() = u, and Y () = j, [ ( W j(, u) = E u e ) ] n r g Y () = j. The premium we seek is W i(, 1) Now use he backward consrucion, disregarding erms of order o(d): [ ( W j(, u) = (1 λ j d)e u e r j d e ) ] τ r +d g Y ( + d) = j + λ jk d W k (, u) k; k j [ ( = (1 λ j d) e rjd E u e r j d e ) ] τ r +d g Y ( + d) = j + = (1 λ j d) e r j d W j( + d, ue r j d ) + k; k j λ jk d W k (, u) k; k j λ jk d W k (, u) Inser e ±r j d = 1 ± r jd + o(d) and W j( + d, e r j d u) = W j( + d, u + ur jd) + o(d) = W j(, u) + Wj(, u) d + Wj(, u) u rj d + o(d), u and fill in some deails o arrive a he parial differenial equaions r jw j(, u) + Wj(, u) + Wj(, u) u rj + λ jk (W k (, u) W j(, u)) =. u k; k j

351 APPENDIX G. SOLUTIONS TO EXERCISES 32 These are o be solved subjec o he condiions j = 1,..., J. W j(n, u) = (u g), Exercise 55 (a) Expeced PV a ime of benefis: n e rτ (1 τ/2 p x) τ p yµ y+τ dτ. Expeced PV a ime of premiums is π imes n/2 e rτ τ p xτ p y dτ. Equivalence premium π is he raio beween hese expressions. (b) (Do no spend oo much ime on his.) A sraighforward mehod for compuaion is o define v 1() = 1 /2 p x, v 2() = p y, v 3() = e rτ (1 τ/2 p x) τ p yµ y+τ dτ. and solve numerically he sysem of differenial equaions v 1() = µ x+/2 (1/2) v 1(), v 2() = µ y+ v 2(), v 3() = e r v 1() v 2() µ y+, by a forward difference scheme saring from he condiions v 1() =, v 2() = 1, v 3() =. A more sophisicaed mehod hined a in he problem: Observe ha /2p x = exp ( /2 µ(x + s) ds ) ( 1 = exp 2 µ(x + 1 ) s) ds, 2 formally a survival funcion wih inensiy µ() = µ(x + /2)/2. Then he single premium is he difference beween he single premiums of wo well-known simple producs, which may be compued by solving heir Thiele differenial equaions numerically. Or compue by e.g. he program prores1 he expeced discouned value of an assurance of 1 payable upon ransiion from sae 1 o sae 3 in a four saes Markov model on {, 1, 2, 3}, saring from sae, wih ransiion inensiies µ 1() = µ 23() = µ(x + /2)/2, µ 2() = µ 13() = µ(y + ), and all oher inensiies. (c) Reserve V a ime [, n] depends on wha is currenly known abou (x) and (y): (y) dead: V =. (y) alive, n/2, T x > n/2: V =.

352 APPENDIX G. SOLUTIONS TO EXERCISES 33 (y) alive, n/2, T x n/2: V = n 2T e r(τ ) x τ p y+µ y+τ dτ. (y) alive, < n/2, T x : V = n 2T x e r(τ ) τ p y+ µ y+τ dτ. (y) alive, < n/2, T x > : V = n 2 e r(τ ) (1 τ/2 p x+) τ p y+µ y+τ dτ π n/2 (d) In general, for a uni due a some random ime, he non-cenral 2nd momen of presen value is he same as he expeced value, only wih 2r insead of r. Exercise 21 (b) The expeced presen value a ime of he benefis is.75 imes [ W b = E S(m) [ m = E e m+n m m+n a(s) ds ] U(τ) 1 τ p x dτ Y () = i ] e τ (r(s)+µ(x+s)) ds dτ Y () = i. m Proceed along he lines of Problems The condiional expeced value of he discouned benefis, given wha is known a ime [, m + n], is.75 imes [ W b m m+n ] a(s) ds () = E e e τ (r(s)+µ(x+s)) ds dτ Y (τ); τ. m The gis of he maer in he backward consrucion is o separae ou wha relaes only o he pas and wha relaes o he fuure here, and o work wih he condiional expeced values of he laer, given he relevan pieces of curren informaion. Firsly, for m m + n, W b m () = e a(s) ds e τ (r(s)+µ(x+s)) ds dτ m m a(s) ds + e [ ] (r(s)+µ(x+s)) ds E W b () Y (τ); τ, e r(τ ) τ p x+ τ p y+ dτ. where W b () = m+n e τ (r(s)+µ(x+s)) ds dτ. In he expression for W b () we need o deermine he condiional expeced value of W b () in he las erm. Due o he Markov propery (condiional independence beween pas and fuure, given he presen), his condiional expeced value depends only on he ime and he curren sae of he economy, Y (). Therefore, we can resric aenion o he sae-wise condiional expeced values, [ ] W b j () = E W b () Y () = j, m m + n, j = 1,..., J. These are he soluion o he differenial equaions d d W j b () = (r j + µ(x + )) W j b () 1 subjec o he condiions k; k j λ jk ( W b k() W b j ()) =, (G.31) W b j (m + n) =, (G.32)

353 APPENDIX G. SOLUTIONS TO EXERCISES 34 j = 1,..., J. Ouline of deails (which you shouldn spell ou unless you are explicily asked o do so): W b () = +d e τ (r(s)+µ(x+s)) ds dτ + e +d (r(s)+µ(x+s)) ds m+n +d e τ +d (r(s)+µ(x+s)) ds dτ = e O(d) d + e (r()+µ(x+)) d W b ( + d) + o(d), where O(d) is of order d and o(d) is of order (d) 2, hence W b () = d + (1 (r() + µ(x + )) d) W b ( + d) + o(d). (We have lumped negligible erms ino o(d): For insance, by Taylor expansion, d exp(o(d)) = d (1 + O(d)) = d + o(d) and d W k( b + d) = d ( W k() b + O(d)) = d W k()+o(d).) b Now, condiion on wha happens in he small ime inerval (, +d): [ W j b () = (1 λ j d) E d + (1 (r() + µ(x + )) d) W ] b ( + d) + o(d) Y (τ) = j, τ [, + d] + [ λ jk d E d + (1 (r() + µ(x + )) d) W ] b ( + d) + o(d) Y () = j, Y ( + d) = k k; k j = (1 λ j d) [ d + (1 (r j + µ(x + )) d) W b j ( + d)] + = d + W b j ( + d) (r j + µ(x + ) + λ j ) d W b j () + k; k j k; k j λ jk d W b k() + o(d) λ jk d W b k() + o(d). (The firs sep in he above derivaion is made here jus o make sure you undersand he argumen - i need no be given in an answer o he Problem.) Subracing W j b ( + d) on boh sides, dividing by d, and leing d go o, we arrive a (G.31). Anoher way o do i is o se W b j ( + d) = W b j () + d d W b j () d + o(d), b cancel W j () on boh sides, and finally divide by d. Secondly, for m, W b [ ] (a(s) r(s) µ(x+s)) () = e ds E W b () Y (τ); τ, where now W b () = e m (a(s) r(s) µ(x+s)) ds m+n m e τ m (r(s)+µ(x+s)) ds dτ. The sae-wise condiional expeced values of his funcion are [ ] W j b () = E W b () Y () = j, m, j = 1,..., J. These are he soluion o he differenial equaions d d W j b () = (r j + µ(x + ) a j) W j b () k; k j λ jk ( W b k() W b j ()) =. (G.33)

354 APPENDIX G. SOLUTIONS TO EXERCISES 35 A ime = m he defining expressions for W b () in [, m] and [m, m + n] coincide, so we jus proceed backwards in [, m] from he values W b j (m) obained afer having compleed he backward scheme in [m, m + n]. Brief ouline of deails of he derivaion of he differenial equaions: W b +d () = e (a(s) r(s) µ(x+s)) ds W b ( + d) = (1 + (a() r() µ(x + )) d) W b ( + d) ; W j b () = (1 λ j d) (1 + (a j r j µ(x + )) d) W j b ( + d) + λ jk d W k( b + d) + o(d), k; k j and he res is rivial. The expeced presen value of fuure benefis a ime, when he economy sars in sae i, is W b = W i b (). The expeced presen value a ime of he premiums is π imes [ m ] W c = E U(τ) 1 τp x dτ Y () = i [ m ] = E e τ (r(s)+µ(x+s)) ds dτ Y () = i. The siuaion is now simpler han in he case of he benefis. For [, m] i suffices o look a he sae-wise condiional expeced values [ m ] Wj c () = E e τ (r(s)+µ(x+s)) ds dτ Y () = j, m, j = 1,..., J. These we deermine as he soluion o he differenial equaions (copy argumens for he case [m, m + n] above) d d W j c () = (r j + µ(x + )) Wj c () 1 subjec o he condiions k; k j λ jk (W c k() W c j ()) =, (G.34) W c j (m) =, (G.35) j = 1,..., J. The expeced presen value a ime, when he economy sars in sae i, is W c = W c i (). Finally, deermine π from he equivalence relaion π W c i () =.75 W b i (): π =.75 W b i ()/W c i (). This soluion was filled wih deails ha would no make any good in an exam; you should jus wrie he essenial seps. To check your undersanding of hese hings,

355 APPENDIX G. SOLUTIONS TO EXERCISES 36 ry your hand on he modified conrac where he benefi is a lump sum of 4 S(m) payable upon reiremen a ime m. (A simpler problem, of course.) Problem 4.15 This is a small perurbaion of Problem 4.11, and acually i is simpler. We rea only he benefis (for premiums we do as in Problem 4.13). Suppose Y () = i. The expeced discouned value of he benefis is np x imes [ W = E e ] [ n r (U(n) g) Y () = i = E (1 e ] n r g) Y () = i. A suiable saring poin is he price of he claim a ime, given he curren informaion abou he pas, [ E e ] [ ( n r (U(n) g) Y (τ); τ = E U() e ) ] n r g Y (τ); τ. Arguing as in Problem 4.1 of he Exercises, we realize ha his expression is a funcion of, Y () and U(). Consider is value a ime for given U() = u, and Y () = j, [ ( W j(, u) = E u e ) ] n r g Y () = j. The value W we seek is W i(, 1). Now use he backward consrucion, disregarding erms of order o(d): [ ( W j(, u) = (1 λ j d)e u e r j d e ) ] τ r +d g Y ( + d) = j + λ jk d W k (, u) k; k j [ ( = (1 λ j d) e rjd E u e r j d e ) ] τ r +d g Y ( + d) = j + = (1 λ j d) e r j d W j( + d, ue r j d ) + k; k j λ jk d W k (, u). k; k j λ jk d W k (, u) Now do as in 4.11 and fill in some deails o arrive a he parial differenial equaions r jw j(, u) + Wj(, u) + Wj(, u) u rj + λ jk (W k (, u) W j(, u)) =. u k; k j These are o be solved subjec o he condiions W j(n, u) = (u g), j = 1,..., J. The expeced discouned premiums are found as in Problem Exercise 57 (a) Direc from he Poisson disribuion: E[S()] = e α E = e α [e βn()] n= e βn (λ) n e λ n!

356 APPENDIX G. SOLUTIONS TO EXERCISES 37 = e α e λ (e β λ) n n! n= ( ) = e α e λ exp e β λ ( ) = exp α + (e β 1)λ. Backward consrucion, paricularly simple here since α + bn() has saionary and independen incremens: V () = E [e α+βn()] = (1 λd)e αd V ( d) + λ d e αd+β V ( d) + o(d) = e αd V ( d) λ d e αd V ( d) + λ d e αd+β V ( d) + o(d) = (1 + α d)(v () d V () d) d λ d e αd V ( d) + λ d e α d+β V ( d) + o(d). Cancel V (), divide by d and le d go o o obain a simple differenial equaion, o be solved subjec o V () = 1. For non-cenral q-h momens, jus replace α and β wih qα and qβ. (b) ds() = e α+βn() αd + dn() (e α+β(n( )+1) e α+βn( )) ) ( = e α+βn( ) αd + dn()e α+βn( ) e β 1 ( ( )) = S( ) α + λ e β 1 d + S( ) ( e β 1 ) dm(), where M() = N() λ, a so-called maringale (a process wih condiionally zero mean and uncorrelaed incremens, here acually independen incremens). (c) Replace α and β in (a) wih α and β, and inegrae he expression for he expeced value, o obain he claimed answer (misprin: a λ is missing). (d) This is now rivial. BEGIN From 35exsol Exercise 55 (a) Expeced PV a ime of benefis: n e rτ (1 τ/2 p x) τ p yµ y+τ dτ. Expeced PV a ime of premiums is π imes n/2 e rτ τ p xτ p y dτ.

357 APPENDIX G. SOLUTIONS TO EXERCISES 38 Equivalence premium π is he raio beween hese expressions. (b) (Do no spend oo much ime on his.) A sraighforward mehod for compuaion is o define v 1() = 1 /2 p x, v 2() = p y, v 3() = e rτ (1 τ/2 p x) τ p yµ y+τ dτ. and solve numerically he sysem of differenial equaions v 1() = µ x+/2 (1/2) v 1(), v 2() = µ y+ v 2(), v 3() = e r v 1() v 2() µ y+, by a forward difference scheme saring from he condiions v 1() =, v 2() = 1, v 3() =. A more sophisicaed mehod hined a in he problem: Observe ha /2p x = exp ( /2 µ(x + s) ds ) ( 1 = exp 2 µ(x + 1 ) s) ds, 2 formally a survival funcion wih inensiy µ() = µ(x + /2)/2. Then he single premium is he difference beween he single premiums of wo well-known simple producs, which may be compued by solving heir Thiele differenial equaions numerically. Or compue by e.g. he program prores1 he expeced discouned value of an assurance of 1 payable upon ransiion from sae 1 o sae 3 in a four saes Markov model on {, 1, 2, 3}, saring from sae, wih ransiion inensiies µ 1() = µ 23() = µ(x + /2)/2, µ 2() = µ 13() = µ(y + ), and all oher inensiies. (c) Reserve V a ime [, n] depends on wha is currenly known abou (x) and (y): (y) dead: V =. (y) alive, n/2, T x > n/2: V =. (y) alive, n/2, T x n/2: V = n 2T e r(τ ) x τ p y+µ y+τ dτ. (y) alive, < n/2, T x : V = n 2T x e r(τ ) τ p y+ µ y+τ dτ. (y) alive, < n/2, T x > : V = n 2 e r(τ ) (1 τ/2 p x+) τ p y+µ y+τ dτ π n/2 e r(τ ) τ p x+ τ p y+ dτ. (d) In general, for a uni due a some random ime, he non-cenral 2nd momen of presen value is he same as he expeced value, only wih 2r insead of r. Exercise 75 We work under he independence hypohesis (should have been saed in he exercise ex.) Firs he brue force mehod: p x1...x r = 1 j (1 p xj ).

358 APPENDIX G. SOLUTIONS TO EXERCISES 39 Using he rule for differeniaing a produc (special case of Iǒ), d d px 1...x r I follows ha = d d px k (1 p xj ) = p xk µ xk + k j; j k k µ x1...x r () = k px k µx k+ j; j k (1 px ) j 1 j (1 px ). j j; j k (1 p xj ). Second, direc reasoning: The condiional probabiliy ha he las survivor dies in (, + d), given ha here are survivors a ime, is d imes he expression above. Exercise 76 (a) Apology: Problems of his kind are one of he favorie spors of classical acuaries, no because hey are so common in pracice (marke share in he per mille range), bu raher because hey can enerain and simulae he brains of acuaries. The proposed produc is no oally inconceivable, however: i migh be useful for a couple ha needs o secure economically he las survivor and also heir children afer he possible early deah of he las survivor. I is also of some heoreical ineres beyond ha of mere parlor games as i is an example of a produc where paymens are dependen on he pas hisory of he driving process. This is seen clearly if he problem is formulaed in he se-up of he Markov chain models for wo lives. Now o work: As is almos always he case, he bes mehod is o find he expeced value of he discouned paymen in each small ime inerval (τ, τ +dτ) and hen sum over all imes. For τ 2 he benefi is running if (and only if) T xy < τ. For τ > 2 he benefi is running if τ 2 < T xy < τ or if T xy < τ 2 and τ 1 < T xy < τ. We gaher he following expeced value of fuure discouned paymens a ime : e rτ (1 τ p x τ p y) dτ 2 2 e rτ [ τ 2p x τ 2p y τ p x τ p y] dτ e rτ [(1 τ 2p x) ( τ 1p y τ p y) + (1 τ 2p y) ( τ 1p x τ p x)] dτ. (b) The premium rae π is he raio beween he expeced presen value in iem (a) and he expeced presen value e rτ τp x τ p y dτ. The reserve is a long and edious sory. One mus, a each ime of consideraion, disinguish beween all possible pas hisories of he wo lives along. For insance, if T xy >, hen he reserve is simply he firs expression above minus π imes he second expression above, wih x and y replaced by x + and y +. Exercise 77 2 e rτ τ p x µ x+τ τ p z dτ + 2 e rτ τ p x µ x+τ τ 2p y τ p z dτ.

359 APPENDIX G. SOLUTIONS TO EXERCISES 4 A benefi of 1 payable immediaely upon he deah of (x) if (z) is hen sill alive and (y) was alive 2 years ago. Exercise 78 Le us say n = 3. Use he se-up of he four saes Markov chain for wo lives. Se age x = (we could have aken x = 2 since, accidenally, he wo sar a same age, bu he x in he program, which refers o a single life, doesn really apply o he general case). Accoun of saring age of he wo by wriing α + βe γ (2+) = α + β e γ wih β = β e γ 2 : (* SPECIFY NON-NULL PAYMENTS AT TIME! *) bi[2,4] := 1; bi[3,4] := 1; ca[1] := 1; (* SPECIFY MAXIMUM ORDER OF MOMENTS AND NUMBER OF STATES! *) q := 1; (*momens*) JZ := 4; (*number of saes of he policy*) (* SPECIFY TRANSITION INTENSITIES FOR POLICY Z! *) alpha[1,2] :=.5; gamma[1,2] :=.38*ln(1); bea[1,2] := *exp(gamma[1,2]*2); alpha[1,3] :=.5; gamma[1,3] :=.38*ln(1); bea[1,3] := *exp(gamma[1,3]*2); alpha[2,4] :=.5; gamma[2,4] :=.38*ln(1); bea[2,4] := *exp(gamma[2,4]*2); alpha[3,4] :=.5; gamma[3,4] :=.38*ln(1); bea[3,4] := *exp(gamma[3,4]*2); (*SPECIFY AGE x, TERM, INTEREST RATES AND NON-NULL LIFE ENDOW- MENTS! *) x:= ; (*age*) := 3; (*erm*) r := ln(1+.45); (*ineres rae*) be[1] := ; be[2] := ; (*endowmens a erm of conrac*) (*SPECIFY LUMP SUM PREMIUM AT TIME : PUT c := 1 IF ALL OTHER PREMIUMS ARE AND ONLY MOMENTS OF BENEFITS ARE WANTED! *) c := ; b := ; Exercise 1-12 Pure verificaion - jus inser he appropriae expressions on he righ hand side. Any combinaion of cash bonus a rae b = α c and addiional deah benefi of ˆb = (1 α ) c /µ x+, α 1, produces a righ hand side equal o he lef hand

360 APPENDIX G. SOLUTIONS TO EXERCISES 41 side. Exercise 1-13 bn mus solve hence n e τ (r u+µ x+u ) du c τ dτ = e n (r u+µ x+u ) du bn, bn = n (G.36) e nτ (r u+µ x+u ) du c τ dτ. (G.37) Exercise 1-14 Relaion (7) becomes c = r V + µ(b V ). Here are some examples of ime prognosis of fuure bonuses, assuming ha he insured will survive he erm of he conrac: 1. Rae of cash bonus paymens a ime u (, n) is jus c u defined above. 2. Presen value of fuure cash bonuses: n 3. Value of erminal bonus (no discouned): n e (r + r)(τ ) c τ dτ, e nτ (µ x+s + µ + r + r) ds c τ dτ. Exercise 1-15 All ha is needed is o pu m + n in he role of n and work wih he general formulas. Here V is given by d d V = (µ x+ + r ) V + π, < < m, d d V = (µ x+ + r ) V 1, m < < m + n. This is he only place where he pariculars of he conrac maer: Thiele s differenial equaion is needed for he compuaion of V alongside ha of c. Since V > for all (m, m + n), also c > hroughou his ime inerval. Exercise 1-16 Expenses can be reaed as benefis in addiion o hose specified in he conrac (see Chaper 5). We need he differenial equaion for he firs order gross reserve, (wih side condiion V n d d V = V r + π β π γ b µ x+(b V ) (G.38) = b), and he equivalence relaionship, V = α b,

361 APPENDIX G. SOLUTIONS TO EXERCISES 42 which deermines π. The discouned mean surplus per policy a ime is now S = (α + α b) + I is seen ha e τ (r s+µ x+s ) ds (π µ x+τ b β τ β τ π γ τ γ τ b γ e (r s+µ x+s ) ds V. S = (α + α b) V = α b (α + α b), τ Vτ which is he surplus arising immediaely upon issue of he conrac due o pruden firs order assumpions abou he iniial cos. I is posiive (and indeed pruden) if α b > (α + α b), which means ha he firs order iniial cos is se on he safe side. (This canno be achieved for all b > if α > ; one hen has o assume ha b is greaer han a cerain minimum, which is cerainly he case in pracice.) The dynamics of he surplus is ds = e (r s+µ x+s ) ds (π µ x+ b β β π γ γ b γ V + e (r u+µ x+u ) du (r + µ x+) V e (r u+µ x+u ) du dv. Insering dv = d V d d from (G.38), we gaher where ds = e (r s+µ x+s ) ds c d, c = (r r )V + (β π β β π ) + (γ V γ γ b γ V ) + (µ x+ µ x+)(b V ) is he mean conribuion o surplus per survivor a ime. This conribuion decomposes ino gains semming from safey loadings in he various firs order elemens ineres, expenses of β ype, expenses of γ ype, and moraliy and how hese elemens can be se on he safe side. Jus as for he iniial cos, here is a problem wih he safey loading on expenses of β and γ ype: if e.g. γ >, hen here will ineviably be a loss on he γ expenses for small since he gross reserve sars from a negaive value. This loss has o be compensaed by seing oher firs order elemens sufficienly o he safe side o make c (or a leas S ) non-negaive for all (, n). ) d ) dτ Exercise 1-17 This is a rivial one, and he same goes for he condiional expeced value of any random variable ha depends only on he sae of Y a some fixed fuure ime. Saring from W e() = (1 λ e d)w e( + d) + λ ef d W f ( + d) + o(d), we ge l;f e W e() W e( + d) = λ e W e( + d) + λ ef d W f ( + d) + o(d), l;f e

362 APPENDIX G. SOLUTIONS TO EXERCISES 43 and, dividing by d and leing d, we arrive a he answer. The side condiions are obvious (as always). Exercise 1-18 We sar wih W e and supply deails (o be precise, add a erm o(d) on he righ of he wo expressions given for W and W in he exercise): [ ] W e() = (1 λ e d)e e r d W +d Y τ = e, τ τ + dτ + [ ] λ ef d E e r d W +d Y = e, Y +d = f + o(d) f;f e = (1 λ e d)e re d W e( + d) + f;f e λ ef d e O(d) W f( + d) + o(d), where O(d) signifies a erm of order d (i.e. such ha O(d)/d is bounded as d ). Insering he Taylor expansions e re d = 1 + r e d + o(d), W e( + d) = W e() + d d W e() d + o(d), e O(d) = 1 + O(d), W f( + d) = W f () + O(d), muliplying ou, gahering all o(d) erms and rearranging a bi, one obains he differenial equaions for he funcions W e. Nex, he W e, a bi more skechy and gahering o(d) erms currenly as hey arise wihou furher menioning: W e () = (1 λ ( e d) W e() (r e r ) V d + W e ( + d) ) + λ ef d W e () + o(d). f;f e Proceeding as above, we obain he differenial equaions for he funcions W e. The side condiions are obvious. Exercise 1-19 Goes along he lines of Exercise 18. Exercise 1-2 (a) Same hing again. Inroduce and wrie W = n e τ r s ds (r τ r )V τ dτ, W = (r r )V d + e r d W +d + o(d).

363 APPENDIX G. SOLUTIONS TO EXERCISES 44 Apply he direc backward consrucion o W e() = E[W Y = e]. Sar from ( ) W e() = (1 λ e d) (r e r )V d + e re d W e( + d) + λ ef d W f () + o(d), f;f e and do a small piece of paper-work o arrive a d d We() = We() re (r e r ) V Side condiions are: W e(n ) =, e = 1,..., J Y. (b) Basically he same exercise as (a). f;f e λ ef (W f () W e()). (c) For discouned cash bonuses use he general formula for higher order momens of presen values of paymen sreams wih sae-dependen paymen inensiy and ineres inensiy. Forge abou he variance of erminal bonus (oo cumbersome). Exercise 8 Iems (a) - (e) are raher heoreical and no ypical exam quesions in ST35. Anyway, we offer somehing for sudens who accep only saemens ha have been firmly proved. (a) RTI(T T ) is easy o prove: P[T > s T > ] = P[T > max(s, )] P[T > ] = { P[T >s] P[T >], < s, 1, s. This is obviously an increasing (non-decreasing) funcion of for fixed s. PQD(T, T ) and AS(T, T ) hen follow. We could also prove PQD(T, T ) direcly: P[T > s, T > ] = P[T > max(s, )] P[T > s] P[T > ]. A direc proof of AS(T, T ) goes as follows: Le g(s, ) and h(s, ) be increasing funcions in boh argumens. Then g() = g(, ) and h() = h(, ) are increasing funcions in. Thus, marginal associaion is enough, see noes depend-l.pdf. (b) Proof of he coninuiy propery of probabiliies: Le A n, n = 1, 2,... be an increasing sequence of evens, ha is, A n A n+1 for all n. Wrie j=1a j = A 1 (A j A c j 1), which is a union of muually exclusive ses. (Here A c denoes he complemen of he even A.) Then P [ j=1a j ] = P[A 1] + j=2 (P[A j] P[A j 1]) j=2

364 APPENDIX G. SOLUTIONS TO EXERCISES 45 = lim n ( P[A 1] + ) n (P(A j) P[A j 1]), hence P [ ] j=1a j = lim P[An]. n Le A n, n = 1, 2,..., be a decreasing sequence of evens, ha is, A n A n+1 for all n. Then P [ ] j=1a j = lim P[An]. n The wo saemens are equivalen. For insance, he laer follows by applying he former o he increasing sequence A c n: P [ ] j=1a j = 1 P [( ) c ] [ j=1a j = 1 P j=1 Aj] c = 1 lim P n [Ac n] Now, P[S > s, T > ] = P hence and P[S s, T ] = P hence = 1 lim (1 P[An]) = lim n [ n P[S > s]p[t > ] = [ n P[S s]p[t ] = j=2 P[An]. n ( S s + 1 n, T + 1 ) ] [ = lim n P S s + 1 n n, T + 1 ], n [ lim P S s + 1 ] [ P T + 1 ], n n n ( S s 1 n, T 1 ) ] [ = lim n P S s 1 n n, T 1 ], n [ lim P S > s 1 ] [ P T > 1 ]. n n n I follows ha he sric inequaliies > in he definiion of PQD and RTI can be replaced by. (The definiion of AS is no issue here.) (c) PQD( S T ) means P[ S > s, T > ] P[ S > s] P[T > ] for all s (or all s, which is he same, of course) and all. This is he same as which is he same as which is he same as P[S < s, T > ] P[S < s] P[T > ], P[T > ] P[S s, T > ] (1 P[S s]) P[T > ], P[S s, T > ] P[S s] P[T > ]. Due o he resul in (b), his is he same as he assered resul.

365 APPENDIX G. SOLUTIONS TO EXERCISES 46 (d) Misprin in he exercise ex: should be. Now, AS( S, T ) means ha C(g( S, T ), h( S, T )) for all g and h ha are increasing in boh argumens. Bu his is equivalen o he assered resul. (e) RTI( S T ) means ha P[ S > s T > ] is increasing in for fixed s. Rewriing P[ S > s T > ] = 1 P[S s T > ] and recalling he resul in (b), we arrive a he assered resul. (f) The Markov model is skeched in he figure below. Only small amendmens are needed in he calculaions made in he heory ( depend-l.pdf ), bu he resuls are a bi surprising. We will discuss he maer under he more general assumpion ha µ µ and ν ν. Firs he case s : P[S > s T > ] = e µ+ν+κ + τ s e µ+ν+κ µ τ e τ ν dτ e µ+ν+κ + e τ µ+ν+κ µ τ e τ ν dτ s τ = 1 e µ+ν+κ ν µ τ dτ e µ+ν+κ ν + e τ µ+ν+κ ν µ τ dτ. µ 1 Boh alive Husband dead κ ν ν 2 µ 3 Wife dead Boh dead We need o discuss his expression as a funcion of, which appears only in he denominaor. The derivaive of he denominaor is e s µ+ν+κ ν (ν ν κ ). I follows ha, in he presence of a posiive κ, P[S > s T > ] is no in general an increasing funcion funcion of if ν ν. For ν = ν i is acually decreasing. We have hus already answered he quesion and need no look ino he case s <. The second par of he quesion is now easily sored ou in he case s < by seing µ = µ + κ and ν = ν + κ in he resul above. We find ha P[S > s T > ] is

366 APPENDIX G. SOLUTIONS TO EXERCISES 47 independen of for s <. Therefore, he RTI issue is so far unseled and we need o invesigae he case s > : s P[S > s T > ] = e µ+ν+κ + s τ e µ+ν+κ ν τ e s µ+κ τ dτ e µ+ν+κ + e τ µ+ν+κ µ τ e ν+κ τ dτ = e s κ P [S > s T > ], where P denoes probabiliy under he independence hypohesis µ = µ and ν = ν (see expression in depend-l.pdf ). This is an increasing funcion of, and we have proved RTI(S T ). Exercise 81 See EXERC22A.pas on public folder. END From 35exsol Exercise 57 (a) Inspec (2.16) in BL. I should be quie obvious ha he cash balance a any ime will ge bigger if a any ime a bigger amoun has been deposied on he accoun (when ineres is posiive). In paricular his is rue if a given amoun of deposis is being advanced, i.e. payed earlier. (b) If U > for some (, n), hen, by righ-coninuiy of U, U τ > for τ [, +ɛ), some non-degenerae inerval, hence +d U τ r τ dτ > if r is sricly posiive. If, moreover, U for all (, n), i follows ha n Uτ rτ dτ >. Then, if Un =, i follows from (2.15) ha = A n + n U τ r τ dτ, and so A n <. Think of a savings accoun: Deposis are made firs, ineres is earned on hese, and a he end one can wihdraw more han he oal deposied. For insance, a uni deposied a ime grows wih ineres o exp( n r) in n years, which can hen be wihdrawn o make he balance nil a ime n. In his case A = 1 for < n and A n = 1 exp( n r), which is negaive if ineres is posiive. Exercise 58 (b) p aa(, 1) σ( 1) d 1 p ii ( 1, 2) ρ( 2) d 2p aa( 2, 3) µ( 3) d 3 + o(d), where d = max(d 1, d 2, d 3). We have used he differeniabiliy of he ransiion probabiliies o wrie e.g. p ii ( 1 + d 1, 2) = p ii ( 1, 2) + o(d 1). Exercise 61 (a) leads o p (1) ai (, + d) = p(1) (, )(1 (ν() + ρ()) d) + paa(, ) σ() d d d p(1) ai ai (, ) = p(1) ai (, )(ν() + ρ()) + p aa(, ) σ(),

367 APPENDIX G. SOLUTIONS TO EXERCISES 48 wih side condiion p (1) ai (, ) =. Inegraing gives he following inegral expression, which could be pu up by direc reasoning: Nex, leads o p (1) ai (, ) = exp( s (µ + σ))σ(s) ds exp( s ai (ν + ρ)). p (1) aa (, + d) = p (1) aa (, )(1 (µ() + σ()) d) + p (1) (, ) ρ() d d d p(1) wih side condiion aa (, ) = p (1) aa (, )(µ() + σ()) + p (1) (, ) ρ(), p (1) aa (, ) =,. Inegral expression, which could be pu up by direc reasoning: p (1) aa (, ) = p (1) ai (, s) ρ(s) ds exp( (µ + σ)). As an exercise, repea he argumen for k = 2, 3,... o find differenial equaions for he probabiliy p (k) ai (, ) of being disabled for he k-h ime and he probabiliy p (k) aa (, ) of being acive afer having been disabled k imes. One could aack his problem by redefining he sae-space of he process as indicaed in Figure G.1, where he noaion speaks for iself: s ai σ x+ ρ x+ σ x+ a() i(1) a(1) i(> 1) µ x+ ν x+ µ x+ ν x+ ρ x+ σx+ d µ x+ a(> 1) (b) p (1) Figure G.1: Problem 7.7 q ai (, q)p ii ( q, ) = p aa (, τ) σ x+τ p ii (τ, ) dτ. You should be able o inerpre he inegral expression as a sum of probabiliies of muually exclusive favourable evens.

368 APPENDIX G. SOLUTIONS TO EXERCISES 49 (c) n q e rτ p (1) n ai π = (, τ q)p ii (τ q, τ) dτ. e rτ p aa(, τ) dτ Commen: The premium plan is unaccepable in pracice since i will produce a negaive reserve if he insured is in premium paying sae afer ime n q (hen, for sure, no benefis will be received, bu premium will sill be paid). Therefore, he premiums should be paid only over a shorer period and cerainly no afer ime n q. Reserve: +q e r(τ ) p ii (, τ) dτ + n +q e r(τ ) p ii(, τ q)p ii (τ q, τ) dτ. Exercise 62 (a) Use Kolmogorov forward, aking advanage of he obvious relaionship p aa(, u) = 1 p ai(, u): p ai(, u + du) = p ai(, u)(1 ρ du) + (1 p ai(, u))σ du. You obain a (very simple) differenial equaion for p ai(, ), which you inegrae from o u using p ai(u, u) =, and find p ai(, u) = Then calculae p aa(, u) = 1 p ai(, u): p aa(, u) = σ ρ + σ σ ρ + σ e (ρ+σ)(u ). ρ ρ + σ + σ ρ + σ e (ρ+σ)(u ). (G.39) (G.4) The ransiion probabiliies depend only on u due o homogeneiy. Discuss he probabiliies as funcions of u and look a he limis as u ends o +. (b) Thiele is acually no a good idea (i is doable, of course, bu being a backward equaion i does no make use of he fac ha p aa(, u) + p ai(, u) = 1). Since we have an explici expression for p ai(, u), he easies way is o calculae [ ] E I i(τ) dτ = E[I i(τ)] dτ = (c) [ ] E [N ai()] = E dn ai() = p ai(, τ) dτ = p aa(, τ) σ dτ = (d) From (G.41) he expeced proporion of inacive ime in years is The funcion σ ρ + σ σ ( 1 e (ρ+σ)) (G.41). (ρ + σ) 2 ρ σ ρ + σ + σ 2 ( 1 e (ρ+σ)) (G.42). (ρ + σ) 2 σ ρ + σ σ 1 e (ρ+σ). (G.43) (ρ + σ) 2 g(x) = 1 e x x (G.44)

369 APPENDIX G. SOLUTIONS TO EXERCISES 5 is 1 for x = (l Hospial). I has derivaive g (x) = e x x 1 e x x 2 = e x e x 1 x x 2, which is < for x > (Taylor expansion), hence g is decreasing. Thus, as +, he proporion in (G.43) is increasing from o σ ρ + σ. Reasonable: Increasing funcion of σ, decreasing funcion of ρ. From (G.42) he expeced number of onses of invalidiy per ime uni in years is ρ σ ρ + σ + σ 2 1 e (ρ+σ). (ρ + σ) 2 As increases from o + he value of his expression decreases from σ (of course!) o (ρ σ)/(ρ+σ). I is ineresing o noe ha he limiing expression here is symmeric in ρ and σ. You should ry and figure why. (e) If ρ were, we could pick he resul from Iem 7.2 (e) above, seing µ = ν = : In general we have σ() = σ, < < n. (G.45) 1 e σ (n ) pii(, n) σ() = σ p ai(, n) = σ + ρe (ρ+σ)(n ), < < n. (G.46) 1 e (ρ+σ)(n ) Here we have used (wih a view o (G.4), jus swich roles of σ and ρ) p ii(, u) = σ ρ + σ + ρ ρ + σ e (ρ+σ)(u ). (f) Consan inensiy of ransiion (wheher a i or i a) means ha he ransiions are generaed by a Poisson process. N ai and N ia coun every second ransiion: N ai couns ransiion No. 1,3,... and so on, i.e. N ai() is disribued as [N() + 1]/2, where N() is a Poisson variae wih parameer σ. N ia couns ransiion No. 2,4,... and so on, i.e. N ia() is disribued as [N()]/2, where N() is a Poisson variae wih parameer σ. Exercise 64 Problem 9.3 (a) To simplify noaion, pu N ai = N ai(n) and N ia = N ia(n). The likelihood is Λ = σ N ai ρ N ia e σwa ρw i, where W a = m n, he oal ime spen in acive sae (and W i = nm W a he oal ime spen in inacive sae). l=1 T (m) Nai ln Λ = σ σ Wa,

370 APPENDIX G. SOLUTIONS TO EXERCISES 51 Nai ln Λ = σ2 σ, 2 plus similar expressions for derivaives w.r.. ρ, and 2 2 σ ρ ln Λ =. Thus, he ML esimaors are he occurrence-exposure raes ˆσ = Nai W a, ˆρ = Nia W i, which are asympoically independen, unbiased, and normally disribued wih asympoic variances By (G.4), as.varˆσ = EN ai = m = m σ = m σ n σ2, as.varˆρ = ρ2. EN ai EN ia p aa(, τ) σ dτ ( ρ ρ + σ + ( ρ ρ + σ n + n σ ) ρ + σ e (ρ+σ)τ dτ σ ( 1 e (ρ+σ)n)), (ρ + σ) 2 (G.47) (G.48) (G.49) hence as.varˆσ = m ( ρ n + σ (ρ + σ) σ ρ+σ (1 e (ρ+σ)n ) ). (G.5) Similarly as.varˆρ = m σ ρ (ρ + σ) ( n 1 (1 ρ+σ e (ρ+σ)n ) ). (G.51) (b) One can discuss he explici expression (G.51), bu here is an easier way: By (G.47) and (G.48) as.varˆσ = m n σ, (G.52) paa(, τ) dτ so i suffices o show ha p aa(, τ) is an increasing funcion of ρ or, wha is he same, an increasing funcion of ρ + σ. By (G.4) p aa(, τ) = ρ ρ + σ + σ ρ + σ e (ρ+σ) τ = 1 σ τ 1 e (ρ+σ) τ, (ρ + σ) τ so you need only recall ha he funcion g in (G.44) is decreasing. (c) General commen: The heory of condiional Markov chains worked ou in Problem 7.2 is ofen needed in saisical analysis of insurance daa where one does no have

371 APPENDIX G. SOLUTIONS TO EXERCISES 52 access o he complee life hisories of he policies. I is ofen he case ha one mus work wih daa ha are seleced somehow. For insance, suppose daa on disabiliies can be seen only from he claims records of hose ha are currenly disabled. Then he relevan probabiliies and inensiies are he condiional one, given ha he process is now in disabled sae. In he presen siuaion you mus herefore work wih he inensiies in 7.8 (e). Exercise 66 To conform wih he noaion in Basic Life Insurance Mahemaics, le us call he number of lives n insead of m. We will hroughou refer o formulas in he general heory in Basic Life Insurance Mahemaics. Assume piece-wise consan moraliy inensiy, see (9.58): The log likelihood (9.53) is µ() = µ q, q 1 < q, j = 1, 2,... ln Λ = q (ln µ qn q µ qw q), where N q and W q are, respecively, he number of deahs and he oal ime spen alive in he age inerval [q 1, q). Each µ q is a parameer which is funcionally unrelaed o all he ohers, so here are many parameers in his model! For insance, if we are ineresed in moraliy up o age 1 and have daa in he age range from o 1, here are 1 parameers, which is quie a lo. Remember, however, ha his model is jus a firs sep in a wosage procedure where he second sep is o graduae (smoohen) he ML esimaors resuling from he presen naive model wih piece-wise consan inensiy. The ML esimaors are he occurrence-exposure raes ˆµ q = Nq W q, which are well defined for all q such ha W q > (i.e. in age inervals where here were survivors exposed o risk of deah). The ˆµ q are asympoically (as n increases) normally disribued, muually independen, unbiased, and wih variances given by σ 2 q = as.v[ˆµ q] = µq E[W q], (G.53) where he expeced exposure is E[W q] = n m=1 q q 1 p (m) (τ) dτ, p (m) (τ) being he probabiliy ha individual No. m is alive and under observaion a ime τ. The variance σ 2 q is inversely proporional o he corresponding expeced exposure. In he presen simple model, wih only one inensiy of ransiion from he sae alive o he absorbing sae dead, we find explici expressions for he expeced exposure.

372 APPENDIX G. SOLUTIONS TO EXERCISES 53 For insance, suppose we have observed each individual life from birh unil deah or unil aained age 1, whichever occurs firs (i.e. censoring a age 1). Then, for τ [q 1, q) wih q = 1,..., 1, we have p (m) (τ) = exp ( τ ) µ(s) ds ( ) q 1 = exp µ p (τ (q 1)) µ q, (G.54) p=1 hence E[W q] = n q q 1 ( ) q 1 exp µ p (τ (q 1)) µ q dτ p=1 ( ) q 1 = n exp µ p p=1 1 exp ( µ q) µ q, and σ 2 q = 1 n µ 2 q ( exp ) q 1 p=1 µp (1 exp ( µ q)), (G.55) You should look a oher censoring schemes and discuss he impac of censoring on he variance. Take e.g. he case where person No m eners a age z (m) and is observed unil deah or age 1, whichever occurs firs (all z (m) less han 1). Esimaors ˆσ 2 q of he variances are obained upon replacing he µ j in (G.55) by he esimaors ˆµ j. Simpler esimaors are obained by jus replacing µ q and E[W q] in (G.53) wih heir sraighforward empirical counerpars: ˆσ 2 q = ˆµ q/w q = N q/w 2 q. Now o graduaion. The occurrence-exposure raes will usually have a ragged appearance. Assuming ha he real underlying moraliy inensiy is a smooh funcion, we herefore will fi a suiable funcion o he occurrence-exposure raes. Suppose we assume ha he rue moraliy rae is a Gomperz Makeham funcion, µ() = α+βe γ. Then, ake some represenaive age ξ q (ypically ξ q = q.5) in each age inerval and fi he parameers α, β, γ by minimizing a weighed sum of squared errors Q = q a q (ˆµq α βe γ) 2. This is a maer of non-linear regression. Opimal weighs a q are he inverse of he variances, bu since hese are unknown, we plug in he esimaors and use a q = 1/ˆσ 2 q. Belongs o wha?: For fixed w = mn we have n = w/m and σ as.varˆσ = w(1 π w/m ), w/m which is a decreasing funcion of m by he resuls in iem (b). We also find as.varˆρ = ρ2 ρ =, EN ia w π w/m w/m

373 APPENDIX G. SOLUTIONS TO EXERCISES 54 where we have used EN ia = m n p ai(, τ)ρ dτ = mnρ πn n. We see ha as.varˆρ is an increasing funcion of m for fixed w. Commen: The asympoic variance of an inensiy esimaor is beer he longer he oal expeced ime spen in he sae from which he he relevan ransiion is made. All policies sar from sae a a ime. For fixed oal exposure he esimaion of σ will be good for many policies observed in a shor ime (when hey are likely o remain acive), and esimaion of ρ will be good for few policies observed over a long ime when hey can make i o inacive sae. Exercise 67 If r is consan, hen π = 1 r, ā x n which is a decreasing funcion of n. For < n he premium reserve is V = 1 āx+ n = 1 1 x+ n ā x. ā x n E x ā x + x+ n ā x For fixed his is a decreasing funcion of n. In paricular, a whole life insurance has smaller premium han an n-year emporary endowmen insurance, and i has also smaller reserve for < n. Exercise 3 Follows from () and he fac ha U is righ-coninuous, hence sricly posiive in some inerval. The resul says ha oal wihdrawals exceed oal deposis, which is due o earned ineres. Exercise 3 (a) Prove (3.11) along he following lines: By definiion, Observe ha E[G(T )] = G()F () + Inegrae by pars o obain n G() df () = G() d F () = G(n) F (n) G() F () G() df (). G()d F (τ). n F ( ) dg(). Assuming firs ha G is non-negaive and non-decreasing, we have (G.56) (G.57) (G.58) G(n) F (n) = G(n) df () n n G() df (),

374 APPENDIX G. SOLUTIONS TO EXERCISES 55 hence G(n) F (n), as n. The same holds rue for an inegrable G of finie variaion, which is he difference beween wo inegrable non-decreasing funcions. Thus, leing n in (G.58), we obain G() d F () = G() F () Now combine (G.56) (G.59) o arrive a (3.11). F ( ) dg(). (G.59) Exercise 1 Assume ha X is non-decreasing. Noe ha D(X) = m=1 n=1 D m,n, where D m,n = { ; n, X + X 1 }. Since m > X n X + D m,n (X + X ) > X + 1 m Dm,n, we conclude ha D m,n is finie. Thus, being a counable union of finie ses, D(X) is (a mos) counable. Exercise 2 dx q ( ) = qx q 1 x c d + (X + x d ) q X q dn.