# Basic Life Insurance Mathematics. Ragnar Norberg

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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 =,

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

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