On Valuation and Control in Life and Pension Insurance. Mogens Steffensen

On Valuaion and Conrol in Life and Pension Insurance Mogens Seffensen Supervisor: Ragnar Norberg Co-supervisor: Chrisian Hipp Thesis submied for he Ph.D. degree Laboraory of Acuarial Mahemaics Insiue for Mahemaical Sciences Faculy of Science Universiy of Copenhagen May 21

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Preface This hesis has been prepared in parial fulfillmen of he requiremens for he Ph.D. degree a he Laboraory of Acuarial Mahemaics, Insiue for Mahemaical Sciences, Universiy of Copenhagen, Denmark. The work has been carried ou in he period from May 1998 o April 21 under he supervision of Professor Ragnar Norberg, London School of Economics Universiy of Copenhagen unil April 2, and Professor Chrisian Hipp, Universiä Karlsruhe. My ineres in he opics deal wih in his hesis was aroused during my graduae sudies and he preparaion of my maser s hesis. I realized a number of open quesions and waned o search for some of he answers. This search sared wih my maser s hesis and coninues wih he presen hesis. Chaper 2 is closely relaed o pars of my maser s hesis. However, he framework and he resuls are generalized o such an exen ha i can be submied as an inegraed par of his hesis. Each chaper is more or less self-conained and can be read independenly from he res. This prepares a submission for publicaion of pars of he hesis. Some pars have already been published. However, Chapers 3 and 4 build srongly on he framework developed in Chaper 2. For he sake of independence, hey will boh conain a brief inroducion o his framework and a few moivaing examples. Acknowledgmens I wish o hank my supervisors Ragnar Norberg and Chrisian Hipp for heir cheerful supervision during he las hree years. I owe a deb of graiude o Ragnar Norberg for shaping my undersanding of and ineres in various involved problems of insurance and financial mahemaics and for encouraging me o go for he Ph.D. degree. Chrisian Hipp sharpened my undersanding and I hank him for numerous fruiful discussions, in paricular during my six monhs say a Universiy of Karlsruhe. A special hank goes o Professor Michael Taksar, Sae Universiy of New York a Sony Brook, for his hospialiy during my hree monhs say a SUNY a Sony Brook. Despie no supervisory duies, he ook his ime for many valuable discussions on sochasic conrol heory. I also wish o hank my colleagues, fellow sudens, and friends Sebasian Aschenbrenner, Claus Vorm Chrisensen, Mikkel Jarbøl, Svend Haasrup, Bjarne Højgaard, Thomas Møller, Bo Normann Rasmussen, and Bo Søndergaard for ineresing disiii

iv cussions and all heir suppor. Finally, hanks o Jeppe Eksrøm who, under my supervision, prepared a maser s hesis from which he figures in Chaper 3 are aken. Mogens Seffensen Copenhagen, May 21

Summary This hesis deals wih financial valuaion and sochasic conrol mehods and heir applicaion o life and pension insurance. Financial valuaion of paymen sreams flowing from one pary o anoher, possibly conrolled by one of he paries or boh, is imporan in several areas of insurance mahemaics. Insurance companies need heoreically subsaniaed mehods of pricing, accouning, decision making, and opimal design in connecion wih insurance producs. Insurance producs like e.g. endowmen insurances wih guaranees and bonus and surrender opions disinguish hemselves from radiional so-called plain vanilla financial producs like European and American opions by heir complex naure. This calls for a horough descripion of he coningen claims given by an insurance conrac including a saemen of is financial and legislaive condiions. This hesis employs erminology and echniques feched from financial mahemaics and sochasic conrol heory for such a descripion and derives resuls applicable for pricing, accouning, and managemen of life and pension insurance conracs. In he firs par we give a survey of he heoreical framework wihin which his hesis is prepared. We explain how boh radiional insurance producs and exoic linked producs can be viewed as coningen claims paid o and from he insurance company in he form of premiums and benefis. Two main principles for valuaion, diversificaion and absence of arbirage, are briefly described. We give examples of applicaion of sochasic conrol heory o finance and insurance and relae our work o hese applicaions. In he second par we focus on he descripion and he valuaion of paymen sreams generaed by life insurance conracs. We inroduce a general paymen sream wih paymens released by a couning process and linked o a general Markov process called he index. The dynamics of he index is sufficienly general o include boh radiional insurance producs and various exoic uni-linked insurance producs where he paymens depend explicily on he developmen of he financial marke. An implici dependence is presen in a cerain class of insurance producs, pension funding and paricipaing life insurance. However, we describe explici forms which mimic hese producs, and we sudy hem under he name surplus-linked insurance. We also inroduce inervenion opions like e.g. he surrender and free policy opions of a policy holder by allowing him o inervene in he index which deermines he paymens. We develop deerminisic differenial equaions for he marke value of fuure paymens which can be used for consrucion of fair conv

vi racs. In presence of inervenion opions he corresponding consrucive ool akes he form of a variaional inequaliy. In he hird par, we ake a closer look a he opions, in a wide sense, held by he insurance company in he cases of pension funding and paricipaing life insurance. To hese opions belong he invesmen and redisribuion of he surplus of an insurance conrac or of a porfolio of conracs. The dynamics of he surplus is modelled by diffusion processes. I is relevan for he managemen and he opimal design of such insurance conracs o search for opimal sraegies, and sochasic conrol heory applies. Ou saring poin is an opimaliy crierion based on a quadraic cos funcion which is frequenly used in pension funding and which leads o opimal linear conrol here. This classical siuaion is modified in hree respecs: We inroduce a noion of risk-adjused uiliy which remedies a general problem of couner-inuiive invesmen sraegies in connecion wih quadraic objec funcions; we inroduce an absolue cos funcion leading o singular redisribuion of surplus; and we work wih a consrain on he conrol which leads o resuls which are direcly applicable o paricipaing life insurance.

Resumé Denne afhandling beskæfiger sig med meoder il finansiel værdiansæelse og sokasisk konrol sam deres anvendelse i livs- og pensionsforsikring. Finansiel værdiansæelse af bealingssrømme mellem o parer, evenuel konrollere af en af parerne eller begge, er vigig i adskillige områder inden for forsikringsmaemaik. Forsikringsselskaber har behov for eoreisk velfunderede meoder il prisfassæelse, regnskabsaflæggelse, besluningsagning og opimal design i forbindelse med forsikringsproduker. Forsikringsproduker som f.eks. oplevelsesforsikringer med garanier og bonus- og genkøbsopioner adskiller sig fra radiionelle såkald plain vanilla finansielle produker som europæiske og amerikanske opioner ved deres komplekse naur. Dee nødvendiggør en grundig beskrivelse af de beingede krav indehold i en forsikringskonrak, herunder en redegørelse for dens finansielle og lovgivningsmæssige beingelser. Denne afhandling anvender erminologi og eknikker hene fra finansmaemaik og sokasisk konroleori il en sådan beskrivelse og udleder resulaer som kan anvendes il prisfassæelse, regnskabsaflæggelse og syring af livs- og pensionsforsikringskonraker. I den førse del gives en oversig over den eoreiske ramme indenfor hvilken denne afhandling er lave. De forklares hvordan både radiionelle forsikringskonraker og eksoiske uni link produker kan opfaes som beingede krav il og fra forsikringsselskabe i form af præmier og ydelser. To hovedprincipper for værdiansæelse, diversifikaion og fravær af arbirage, beskrives kor. Der gives eksempler på anvendelse af sokasisk konroleori i finans og forsikring, og vores arbejde relaeres il disse anvendelser. I den anden del fokuseres på beskrivelsen og værdiansæelsen af bealingssrømme generere af livsforsikringskonraker. Der inroduceres en generel bealingssrøm med bealinger udløs af en ælleproces og knye il en generel Markov proces kalde indekse. Indekses dynamik er ilsrækkelig generel il a inkludere både radiionelle forsikringsproduker og forskellige eksoiske link forsikringsproduker hvor bealingerne afhænger eksplici af udviklingen af de finansielle marked. En implici afhængighed er il sede i en særlig klasse af forsikringsproduker, pension funding og forsikringer med bonus. Eksplicie former som eferligner disse produker beskrives imidlerid, og disse suderes under navne overskudslink forsikring. Der inroduceres også inervenionsopioner som f.eks. forsikringsagerens genkøbsog fripoliceopion ved a illade denne a inervenere i de indeks der besemmer bealingerne. Der udvikles deerminisiske differenialligninger for markedsværdien vii

viii af fremidige bealinger som kan bruges il konsrukion af fair konraker. Ved ilsedeværelse af inervenionsopioner ager de ilsvarende konsrukive redskab form af en variaionsulighed. I den redje del kigges nærmere på opionerne, i bred forsand, eje af forsikringsselskabe i forbindelse med pension funding og livsforsikring med bonus. Til disse opioner hører invesering og ilbageføring af overskud på en forsikringskonrak eller på en porefølje af konraker. Dynamikken af overskudde modelleres ved diffusionsprocesser. De er relevan for syring og opimal design af sådanne forsikringskonraker a søge efer opimale sraegier, og sokasisk konroleori er her e naurlig redskab. Udgangspunke er e opimalieskrierium basere på en kvadraisk absfunkion, som ofe bruges i pension funding og som fører il lineær konrol der. Denne klassiske siuaion er modificere i re henseender: Der inroduceres e begreb kalde risikojusere nye der afhjælper e generel problem med ikke-inuiive inveseringssraegier som ofe opsår i forbindelse med kvadraiske objekfunkioner; der inroduceres en absolu absfunkion som fører il singulær ilbageføring af overskud; og der inroduceres en begrænsning på konrollen som fører il resulaer der er direke anvendelige på livsforsikring med bonus.

Conens Preface Summary Resumé iii v vii I Survey 1 1 A survey of valuaion and conrol 3 1.1 Inroducion................................ 3 1.2 Coninuous-ime life and pension insurance............... 4 1.3 Valuaion................................. 7 1.4 Conrol.................................. 15 1.5 Overview and conribuions of he hesis................ 21 II Valuaion in life and pension insurance 23 2 A no arbirage approach o Thiele s DE 25 2.1 Inroducion................................ 25 2.2 The basic sochasic environmen.................... 27 2.3 The index and he marke........................ 27 2.4 The paymen process and he insurance conrac............ 29 2.5 The derived price process......................... 31 2.6 The se of maringale measures..................... 35 2.7 Examples................................. 39 2.7.1 A classical policy......................... 39 2.7.2 A simple uni-linked policy.................... 4 2.7.3 A pah-dependen uni-linked policy.............. 4 3 Coningen claims analysis 43 3.1 Inroducion................................ 43 3.2 The insurance conrac.......................... 45 3.2.1 The basics............................. 45 ix

x CONTENTS 3.2.2 The main resul.......................... 48 3.3 The general life and pension insurance conrac............ 5 3.3.1 The firs order basis and he echnical basis.......... 5 3.3.2 The real basis and he dividends................. 51 3.3.3 A delicae decision problem................... 52 3.3.4 Main example........................... 53 3.4 The noion of surplus........................... 54 3.4.1 The invesmen sraegy..................... 54 3.4.2 The rerospecive surplus..................... 55 3.4.3 The prospecive surplus..................... 56 3.4.4 Two imporan cases....................... 58 3.4.5 Main example coninued..................... 59 3.5 Dividends................................. 6 3.5.1 The conribuion plan and he second order basis....... 6 3.5.2 Surplus-linked insurance..................... 61 3.5.3 Main example coninued..................... 62 3.6 Bonus................................... 65 3.6.1 Cash bonus versus addiional insurance............. 65 3.6.2 Terminal bonus wihou guaranee............... 66 3.6.3 Addiional firs order paymens................. 66 3.6.4 Main example coninued..................... 68 3.7 A comparison wih relaed lieraure.................. 7 3.7.1 The se-up of paymens and he financial marke....... 7 3.7.2 Prospecive versus rerospecive................. 71 3.7.3 Surplus.............................. 71 3.7.4 Informaion............................ 73 3.7.5 The arbirage condiion..................... 73 3.8 Reserves, surplus, and accouning principles.............. 74 3.9 Numerical illusraions.......................... 75 4 Conrol by inervenion opion 81 4.1 Inroducion................................ 81 4.2 The environmen............................. 83 4.3 The main resuls............................. 88 4.4 The American opion in finance..................... 93 4.5 The surrender opion in life insurance.................. 94 4.6 The free policy opion in life insurance................. 96 III Conrol in life and pension insurance 11 5 Risk-adjused uiliy 13 5.1 Inroducion................................ 13

CONTENTS xi 5.2 The radiional opimizaion problem.................. 15 5.3 Risk-adjused uiliy........................... 16 5.4 Opimal invesmen and consumpion.................. 18 5.4.1 Terminal uiliy.......................... 19 5.4.2 Terminal consrain........................ 11 5.5 Pricing by risk-adjused uiliy...................... 111 5.5.1 Exponenial uiliy........................ 113 5.5.2 Mean-variance uiliy....................... 114 6 Opimal invesmen and consumpion 117 6.1 Inroducion................................ 117 6.2 The general model............................ 119 6.3 A diffusion life and pension insurance conrac............. 121 6.4 Objecives................................. 124 6.4.1 Cos of wealh........................... 125 6.4.2 Cos of consumpion....................... 125 6.5 Consrains................................ 127 6.5.1 A erminal consrain....................... 128 6.6 The dynamic programming equaions.................. 13 6.7 Opimal invesmen............................ 131 6.8 Opimal singular consumpion...................... 133 6.8.1 Finie ime unconsrained consumpion............. 133 6.8.2 Finie ime consrained consumpion.............. 135 6.8.3 Saionary unconsrained consumpion............. 136 6.8.4 Saionary consrained consumpion............... 139 6.9 Opimal classical consumpion...................... 14 6.9.1 Finie ime unconsrained consumpion............. 14 6.9.2 Finie ime consrained consumpion.............. 142 6.9.3 Saionary unconsrained consumpion............. 144 6.9.4 Saionary consrained consumpion............... 146 6.1 Subopimal consumpion......................... 149 6.1.1 Finie ime unconsrained consumpion............. 149 6.1.2 Finie ime consrained consumpion.............. 151 6.1.3 Saionary unconsrained consumpion............. 152 6.1.4 Saionary consrained consumpion............... 153 A The linear regulaor problem 159 B Riccai equaion wih growh condiion 163 C The defecive Ornsein-Uhlenbeck process 165 Bibliography 169

xii CONTENTS

Par I Survey 1

Chaper 1 A survey of valuaion and conrol in life and pension insurance This hesis deals wih valuaion and conrol problems in life and pension insurance. In his inroducory chaper we give a survey of noaion, erminology, and mehodology used hroughou he hesis, and we summarize some of he resuls obained. The chaper conains references o lieraure relaed o he hesis. In some cases noaion and erminology used in he hesis differs from noaion and erminology used in he references. We shall already here use he noaion and erminology of he hesis for he sake of consisence and such ha he chaper can serve o prepare he reader for he remaining chapers. This includes a parial change of noaion when going from valuaion problems o conrol problems. 1.1 Inroducion Life and pension insurance conracs are conracs which sipulae an exchange of paymens beween an insurance company and a policy holder. The paymens are coningen on evens in he life hisory of an insured and possibly oher coningencies. Though i need no be he case, he policy holder and he insured are ofen he same person. By connecing paymens o he life hisory of he insured and possibly oher coningencies, a conrac can be viewed as a be on he life hisory and hese coningencies. Secion 1.2 deals wih he erms of he conrac. Those erms are supposed o be comprehensible wihou any knowledge of probabiliy heory, saisics, or finance. Of course, one canno expec he policy holder o have proficiency in hese areas. Though formulaed in mahemaical erms, Secion 1.2 herefore explains he erms of he insurance conrac wihou use of probabiliy heoreical erminology. Valuaion of he conrac or he be, on he oher hand, builds on assumpions on probabiliy laws governing he life hisory and he coningencies of he insurance conrac. Various principles of valuaion and corresponding probabiliy laws are inroduced in Secion 1.3. Tha secion also inroduces inervenion opions of he 3

4 CHAPTER 1. A SURVEY OF VALUATION AND CONTROL policy holder and discusses briefly heir effec on he valuaion problem. The inervenion opions of he policy holder make up an example of a decision problem imbedded in he insurance conrac. In general, he paymens of an insurance conrac may be raher involved and may conain various imbedded opions held by boh he insurance company and he policy holder. Some of he imbedded decision problems held by he insurance company are brough o he surface in Secion 1.4. Tha secion also relaes hese decision problems o oher decision problems previously reaed in he fields of finance and insurance. 1.2 Coninuous-ime life and pension insurance Classical paymen processes In his secion we specify paymen processes in classical life and pension insurance conracs. References o he mahemaics of classical life and pension insurance conracs are Gerber [25] and Norberg [54]. We le he paymens sipulaed in an insurance conrac be formalized by a paymen process B, where B represens he accumulaed paymens from he policy holder o he insurance company over he ime period [, ]. Thus, paymens ha go from he insurance company o he policy holder appear in B as negaive paymens. We shall specify he paymens in a coninuous-ime framework. In order o formalize he connecion beween paymens and he life hisory of he insured, we inroduce an indicaor process X. The process X indicaes wheher he insured is dead or no in he sense ha X = if he insured is alive a ime and X = 1 if he insured is dead a ime. The process X is illusraed in Figure 1.1. alive 1 dead Figure 1.1: A survival model We also inroduce a couning process N couning he number of deahs of he insured equals or 1 over [, ]. Noe ha N = X in his case. Fixing a ime horizon T for he insurance conrac, mos insurance paymen processes are given by a paymen process B in he form where B = db s, T, 1.1 db = B d1 + b c, X d b d, X dn B T X T d1 T. 1.2 Here B is a lump sum paymen from he policy holder o he insurance company a ime, b c are coninuous paymens from he policy holder o he insurance company,

1.2. CONTINUOUS-TIME LIFE AND PENSION INSURANCE 5 b d is a lump sum paymen a ime of deah from he insurance company o he policy holder, and B X T is a lump sum paymen a ime T from he insurance company o he policy holder. The minus signs in fron of b d and B conform o he ypical siuaion where B and b c are premiums and b d and B are benefis, all posiive. We can now specify he elemens of some sandard forms of benefi paymen processes B =, b c, X b d, X B T X T pure endowmen 1 XT = erm insurance 1 <T,X = endowmen insurance 1 <T,X = 1 XT = emporary life annuiy -1 <T,X= and specify he elemens of some sandard forms of premium paymen processes b d, X = B T X T =, B b c, X single premium 1 level premium 1 <T,X= I is clear ha he even X = in he indicaor funcion of b d, X is redundan since we know ha X = if dn = 1. Neverheless, we choose o expose a dependence on X o prepare for he generalized paymen processes o be inroduced below. Alhough he paymen process in 1.1 formalizes a number of sandard forms of insurances and premiums, here are a number of siuaions which canno be covered by his process. One example is he siuaion where he premium is paid as level premium bu modified such ha no premium is payable during periods of disabiliy. This modificaion is called premium waiver. Premium waiver and differen ypes of disabiliy insurances can be covered by exending X wih a hird sae, disabled. In general, we le X be a process moving around in a finie number of saes J. The case wih a disabiliy sae is illusraed in Figure 1.2. acive 1 disabled ց 2 dead ւ Figure 1.2: A survival model wih disabiliy and possibly recovery Corresponding o he general J sae process X, we inroduce a generalized couning process N, a J-dimensional column vecor where he jh enry, denoed

6 CHAPTER 1. A SURVEY OF VALUATION AND CONTROL by N j, couns he number of jumps ino sae j. Correspondingly, we also generalize b d, X o be a J-dimensional row vecor where he jh enry, denoed by b dj, X, is he paymen due upon a jump from sae X o sae j a ime. Wih he generalized jump process and jump paymens we can specify a number of generalized insurance and premium forms. In he disabiliy model illusraed by Figure 1.2, we can e.g. specify he elemens of some sandard forms of disabiliy benefi paymen processes B = B T X T =, b c, X b d, X disabiliy annuiy -1 <T,X=1,, disabiliy insurance, 1<T,X =, and he elemens of a premium paymen process B = b d, X = B T X T =, level premium wih premium waiver b c, X 1 <T,X= The disabiliy model is a hree sae model, i.e. J = 3. Models wih more saes are relevan for oher ypes of insurances e.g. conracs on wo lives where eiher member of a married pair is covered agains he deah of he oher or muliple cause of deah where paymens depend on he cause of deah. Generalized paymen processes In our consrucion of he paymen process 1.1, we have carefully disinguished beween he process X, deermining a any poin in ime he size of possible paymens, and he process N, releasing hese paymens. So far he purpose of his disincion is no very clear since here is a one-o-one correspondence beween X and N, in he sense ha X deermines N uniquely and vice versa. However, wih he inroducion of e.g. duraion dependen paymens or uni-linked life insurance his simple siuaion changes. Duraion dependen paymens are paymens ha depend, no only on he presen sae of he process X, bu also on he ime elapsed since his sae was enered. Such a consrucion is relevan in e.g. he disabiliy model if he insurance company works wih a so-called qualificaion period. Then he disabiliy annuiy does no sar unil he insured has qualified hrough uninerruped by aciviy disabiliy during a cerain amoun of ime, e.g. hree monhs. Anoher example is a so-called uni-linked insurance conrac which is a ype of conrac where he paymens are linked o some sock index or he value of some more or less specified porfolio. Boh in he case of duraion dependen paymens and in he case of uni-linked insurance, informaion beyond he presen sae of X deermines he possible paymen. We formalize his by allowing of a general index S o deermine he possible paymens. Thus, replacing X by S in he paymen process 1.1, he generalized paymen process becomes db = B d1 + b c, S d b d, S dn B T S T d1 T. 1.3

1.3. VALUATION 7 A specificaion of paymens is obained by a recording of he process S and a specificaion of B and he funcions b c, b d, and B. A special case is, of course, o le S = X, hereby reurning o he classical paymen process given by 1.2. In he case of duraion dependen paymens we pu S = X, Y, where Y equals he ime elapsed since he presen sae X was enered. Considering he disabiliy model illusraed by figure 1.2, we le Y indicae he duraion of disabiliy and see ha he dynamics of Y is given by dy = 1 X=1d Y 1 X =1dN Y 1 X =1dN 2, Y =. An example of elemens of an insurance coverage wih qualificaion period y is given by B = b d, S = B T S T = disabiliy annuiy wih qualificaion period b c, S -1 <T,X=1,Y >y A simple uni-linked insurance conrac can be consruced by puing S = X, Y, where Y is some sock index or he value of some porfolio. Leing G denoe a guaraneed minimum paymen and leing X be he simple wo-sae life deah model illusraed in Figure 1.1, some examples of simple guaraneed unilinked conracs are given by B = b c, S = b d, S B T S T pure endowmen 1 XT = max Y T,G erm insurance 1 <T,X = max Y, G Once he insurance company and he policy holder have agreed on a paymen process, including he recording of he index S, an insurance conrac is specified. Thus, he insurance conrac does no specify any assumpions as o he probabiliy laws for he processes driving he paymens, he ineres rae, and oher feaures of he marke. Such assumpions are invoked by he insurer in he valuaion of he paymens and are needed o answer quesions like: How many unis of level premium wih premium waiver represen a fair price o pay for a simple uni-linked endowmen insurance wih a guaranee? 1.3 Valuaion Valuaion by diversificaion This secion deals wih valuaion of he paymen sreams described in Secion 1.2, and we need for ha purpose he probabilisic apparaus. We assume ha he processes S and N are defined on a probabiliy space Ω, F,F = {F }, P. We assume ha paymens are currenly deposied on or wihdrawn from a bank accoun ha bears ineres. If we denoe by Z he presen value a ime of a uni deposied a ime, we find ha he presen value a ime of a uni

8 CHAPTER 1. A SURVEY OF VALUATION AND CONTROL deposied a ime s equals he amoun Z. We shall assume here exiss a force of Zs ineres or shor rae of ineres r such ha dz = r Z d, Z = 1. 1.4 Conforming o acuarial erminology, a presen value a ime need no be F - measurable. We can now speak of he presen value a ime of a paymen process by adding up he value of all elemens in he paymen process, and we ge he presen value a ime of he paymen process B, T Z db Zs s. The value a ime of a paymen process B is he ne gain a ime which he insurance company faces by issuing he insurance conrac. If he ime of deah and oher coningencies deermining B are known a ime, his gain can be calculaed a ha poin in ime. To avoid gains one should balance he elemens in he paymen process such ha he ne gain equals zero, T 1 Z s db s =. 1.5 However, he ime of deah and oher coningencies deermining B are in general no known a ime. We consider hese coningencies as sochasic variables defined on our probabiliy space such ha he lef hand side of 1.5 becomes a sochasic variable. The quesion is how one should balance he elemens of he insurance conrac in his siuaion. A paricular siuaion arises if he insurance company issues or can issue conracs on a large number n of insured wih idenically disribued paymen processes B i i=1,...,n, B i is independen of B j for i j, he ineres rae and hereby Z is deerminisic. Then he law of large numbers applies and provides ha he gain of he insurance porfolio per insured converges owards he expecaion of he gain of an insured as he number of conracs increases, i.e. 1 n n i=1 T 1 Zs db i s E [ T 1 Zs db s ] as n. To avoid sysemaic gains, one should balance he elemens in he paymen process such ha he expeced ne gain equals zero [ T E ] 1 db s =. 1.6 Z s

1.3. VALUATION 9 This balance equaion formalizes he principle of equivalence which is fundamenal in classical life insurance mahemaics. If one of he hree assumpions above fails, he classical principle of equivalence fails as balancing ool for he paymen process: If he insurance company canno issue a large number of conracs, i makes no sense o draw conclusions from he law of large numbers; if B i and B j are dependen for i j, he law of large numbers does no apply; if he ineres rae is no deerminisic, we canno conclude independence beween T 1 Z s dbs i and T 1 Z s dbs j from he independence beween Bi and B j, i j. I should be menioned, however, ha he firs wo assumpions can be weakened such ha hey are only required o hold in a cerain asympoic sense. So far, we have no said much abou he disribuion of S and N. The principle of equivalence is only based on he assumpion ha paymen processes of differen insured are idenically disribued and independen. We are now going o assume ha here exis deerminisic piecewise coninuous funcions µ j, s such ha N j admis he F S-inensiy process µj, S. This means ha he F S -inensiy of N is a funcion of and S only. In he classical case where he index S is made up by he process X, a consequence of his assumpion is ha X is a Markov process, i.e. Markov wih respec o he filraion generaed by he process iself. In he se-up wih a general index S his need no be he case. However, a consequence is ha X is F S -Markov, i.e. Markov wih respec o he filraion generaed by he index S. Consider he classical siuaion where S = X, assume ha he life hisories of he insured are independen, and assume ha he ineres rae is deerminisic. We can hen use he classical principle of equivalence 1.6 o deermine fair premiums for he sandard forms of insurance inroduced in Secion 1.2. Consider e.g. he calculaion of a fair level premium for an endowmen insurance of 1 in he survival model illusraed by Figure 1.1. Puing µ = µ,, he principle of equivalence saes [ T E 1 Z db ] [ T = E = π T 1 Z π1x=d 1 X =dn 1 XT =d1 T ] e R rs+µ s ds d T = R T π = e rs+µ s ds µ d + e R T rs+µ s ds R T. e rs+µ s ds d e R rs+µ s ds µ d e R T rs+µ s ds Acuaries have developed a special noaion for presen values and expeced presen values of basic paymen sreams. An acuary would wrie he premium formula above on coded form given ha he insured has age x a ime, π = A1 xt + TE x a xt = A xt a xt. The calculaions for disabiliy insurances, premium waiver and deferred benefi policies can be carried ou in he same way, bu hey become, obviously, more involved.

1 CHAPTER 1. A SURVEY OF VALUATION AND CONTROL Valuaion by absence of arbirage A crucial assumpion underlying he principle of equivalence was he independence beween paymen processes. For cerain paymen processes his independence comes from independence beween life hisories and makes sense. We shall now consider a paymen process where his assumpion canno be argued o hold, and we shall reflec on a reasonable valuaion principle in his siuaion. I is clear ha if we canno rely on he law of large of numbers, we have o rely on somehing else. Arbirage pricing heory relies on invesmen possibiliies in a marke and inroduces a principle of absence of arbirage i.e. avoidance of risk-free capial gains. The heory has been one of he mos explosive fields of applied mahemaics over he las decades. The breakhrough of his heory was he opion pricing problem formulaed and solved in Black and Scholes [6] and in Meron [44]. Laer, rigorous mahemaical conen was given o noions like invesmen sraegy, arbirage, and compleeness, and heir connecion o maringale heory was disclosed in Harrison and Kreps [31] and in Harrison and Pliska [32]. We shall only make a few commens on he basic heory and ask he reader o confer he cornucopia of exbooks for furher insigh. A fundamenal heorem in arbirage pricing heory saes ha a sufficien condiion which ensures ha no risk-free capial gains are available is ha he expeced value of gains equals zero, [ T ] E Q 1 db s =, 1.7 Zs where he expecaion is aken wih respec o a so-called maringale measure. A maringale measure is a probabiliy measure Q, equivalen o he measure P, such ha discouned prices of raded asses are maringales under Q. One of he simples illusraions of 1.7 one can hink of, is o find he single premium π of a paymen a ime T, a so-called T-claim, of a sock index Y T where Y is included in S, i.e. b c, S = b d, S = simple claim π Y T B B T S T If he sock index is no available as an invesmen possibiliy, one has no necessarily enough informaion on he probabiliy measure Q o say much abou he price π. If he sock index is available as an invesmen possibiliy, Y is a maringale under Z he valuaion measure Q such ha [ T ] [ ] E Q 1 db Zs s = π E Q YT = ZT [ ] π = E Q YT = Y = Y ZT Z. 1.8 Why is 1.8 a reasonable resul in he case where he sock index is available as an invesmen possibiliy? The issuer can, insead of invesing money in he bank

1.3. VALUATION 11 accoun, inves money in he sock index. If he does so, he gain a ime T amouns o Y T Y π Y T, and, obviously, in order o avoid risk-free capial gains, we need o pu π = Y. Indeed, Y T is a paricularly simple T-claim, bu wha abou a European opion Y T K +? Arbirage pricing heory deals wih general claims pricing and invesmen sraegies which in general need o be dynamical as opposed o he saic sraegy above. One of he key resuls is ha 1.7 is sufficien for absence of arbirage. Alhough he pricing formulas 1.6 and 1.7 only differ by a opscrip indicaing he probabiliy measure, one should carefully noe ha hey rely on fundamenally differen properies of he risk in he paymen process. Whereas 1.6 relies on diversificaion, 1.7 relies on absence of arbirage in an underlying marke. The lef hand side of he formulas 1.6 and 1.7 value he fuure paymens of he conrac a ime. For various reasons one may be ineresed in valuing he fuure paymens a any poin of ime before erminaion. Obviously, if one wishes o sell hese fuure paymens one mus se a price. Bu even if one does no wish o sell he fuure paymens, various insiuions may be ineresed in heir value. Owners of he insurance company and oher invesors are ineresed in he value of fuure paymens for he purpose of assessing he value of he company; supervisory auhoriies are ineresed in ensuring ha he paymens are payable by he company and se up solvency requiremens which are o be me; ax auhoriies are ineresed in he curren surplus as a basis for axaion. All hese paries are ineresed in he value of ousanding paymens or liabiliies. In a life insurance company hese liabiliies are called he reserve. Differen insiuions may be ineresed in differen noions of reserve. Whereas he paymen process is more or less uniquely specified by 1.3, he valuaion formulas 1.6 and 1.7 build on a more or less subjecive choice of ineres rae and valuaion probabiliy measure. In paricular, if one does no search for informaion on r and Q on he financial marke, values are cerainly subjecive and possibly no consisen wih absence of arbirage. We call a se of ineres rae and Q-dynamics a valuaion basis because such a se produces one version of he reserve. In Chaper 3, we inroduce various special valuaion bases and sudy he dynamics of he surplus under hese. The acual calculaion of reserves, no giving rise o arbirage possibiliies, relies on he probabiliy law of processes driving he paymen process and on he underlying invesmen possibiliies. So far we have only specified one probabilisic srucure by inroducion of he F S -inensiies for he couning process N. We need some probabilisic srucure on he index S in order o obain applicable pricing formulas. The relaion 1.7 is no worh much if we have no idea of he probabilisic srucure of S. A crucial propery ha one is ap o rely on is he Markov propery. Assuming ha S is a Markov process and requiring ha he reserve is

12 CHAPTER 1. A SURVEY OF VALUATION AND CONTROL F S -Markov leads o appealing compuaional ools in he search for arbirage free reserves and paymen processes. This is due o he close relaion beween expeced values of funcionals of Markov processes and deerminisic differenial equaions. This relaion is ofen used in applied probabiliy, and i is used and parly proved several imes in his hesis. Guaraneed paymens and dividends. In 1.7 he probabiliy measure Q is o some exen deermined by he marke. However, here may be risk presen in S and N which is no priced by he marke and which canno be diversified by independence of paymen processes. The quesion is wha o do wih risk which is neiher diversifiable nor hedgeable. A nice example is he classical case where he only invesmen possibiliy is he bank accoun. We now, realisically, allow he inensiies of N o depend, no only on he life hisory of he individual insured, bu also on demographic, economic, and socio-medical condiions. These condiions are formalized by he index S. Now, he individual paymen processes can no longer be said o be independen. Also he assumpion of deerminisic ineres, which is implici in 1.6, seems unrealisic under ime horizons exending o 5 years. In general, he insurance company may be unwilling o face undiversifiable and unhedgeable risk and needs o do somehing else. One resoluion, developed by life and pension insurance companies, is o add o he firs order paymen process an addiional paymen process of dividends condiioned on a paricular performance of a policy or a porfolio of policies. This dividend can be consrained o be o he benefi of he policy holder or no, depending on he ype of insurance produc. If he dividends are consrained o be o he benefi of he policy holder, he firs order paymens mus represen an overpricing, roughly speaking. In his case he dividends can be seen as a compensaion for his overpricing. One way of producing firs order paymens which represen an overpricing is o use a cerain arificial valuaion basis consising of an arificial rae of ineres rae r driving an arificial risk-free asse Ẑ, and an arificial valuaion measure Q, called a firs order basis, o lay down paymens a he ime of issue. The paymen process produced is called he firs order paymen process B, and i is deermined subjec o he arificial valuaion formula, [ T ] Q E b 1 d B s =. Ẑ s The paymen process of dividends is denoed by B. The firs order paymens and he dividends make up he oal paymens B semming from he conrac, B = B + B. Now he problem of seing fair paymens is ranslaed o he problem of alloing fair dividends. A he end of he day, he insurance company needs o make up

1.3. VALUATION 13 is mind abou he assessmen of he value of non-diversifiable and non-hedgeable risk and balance dividends by he corresponding equivalence relaion E Q [ T 1 Zs db s ] =. 1.9 However, by he inroducion of dividend paymens, i is o some exen possible for he insurance company o ransfer a par of he risk from he insurance company o he policy holders. Hereby he insurance company is less exposed o risk han in a siuaion wihou dividends, of course depending on how hese are deermined. We shall no go deeper ino he inerpreaion of dividend disribuion as a risk managemen insrumen now bu conen ourselves wih a simple illusraive example. Assume e.g. ha dividends are only paid ou a ime T and ha his dividend paymen is a funcion of he performance of he firs order paymens. Then, by inroducing he process Z Z s d B s in he index S, we can define for a some funcion f, B = b c, S = b d, S =, erminal dividends B T S T T ZT f d B Zs s This dividend plan leads o a oal gain of T 1 Zs db s = T 1 Zs d B s 1 f ZT T ZT d Z B s s. If e.g. he insurance company is allowed o choose as funcion f he ideniy funcion he gain is zero and all risk is ransferred o he policy holder. This is, of course, an exreme and exremely unineresing case, bu i illusraes wha is mean by ransferring risk o he policy holder. Anoher funcion f, which moreover ensures ha dividends are o he benefi of he policy holder, is T f ZT d Z B s s = q T ZT + d Z B s s, where q is a consan. In he case of no consrains on he dividends, we shall speak of pension funding, and in he case where dividends are consrained o be o he policy holder s benefi, we shall speak of paricipaing life insurance. Chaper 3 deals wih valuaion bases, surplus, and dividends. The relaion beween expeced values and deerminisic differenial equaions gives a consrucive ool for calculaion of fair sraegies for invesmen and repaymen of surplus hrough dividends. Numerical resuls shall illusrae his ool. Valuaion under inervenion opions I is implicily assumed in all valuaion formulas above ha he insurance company and he policy holder have no influence on he performance of he insurance conrac,

14 CHAPTER 1. A SURVEY OF VALUATION AND CONTROL hereunder he dynamics of he index. In pracice, here are a number of inervenion opions ha may or may no affec he valuaion of paymens. One example of an inervenion opion is he exercise opion of an American opion. The exercise opion allows he owner of an American opion o exercise he conrac a any poin in ime up o he expiraion dae T. For a life insurance conrac he mos imporan inervenion opion is probably he surrender opion of he policy holder. Holding his opion, he can a any poin in ime up o T close he conrac and conver all fuure paymens ino an immediae paymen of he surrender value. Also he issuer of an insurance conrac may hold inervenion opions. E.g. he bankrupcy opion of he owners of he insurance company can be considered as an inervenion opion held by he insurance company. I urns ou ha a very convenien way of modelling hese inervenion opions is o allow he policy holder and/or he insurance company o inervene in he index S in some specified way. This enables us o capure exacly he ypes of inervenion opions in which we are ineresed. Disregarding all inervenion opions held by he insurance company bu aking ino consideraion inervenion opions of he policy holder, arbirage argumens lead o a valuaion formula on he form [ T sup E QI I I 1 Z I s db I s ], 1.1 where opscrip I indicaes ha he quaniy is dependen on a cerain admissible inervenion sraegy aken by he policy holder and he supremum is aken over all admissible inervenion sraegies. The resuls building on opimal inervenion represen one approach o he problem of valuaion, aking ino accoun inervenion opions. This approach expecs he policy holder o behave financially opimal. Whereas his assumpion may be reasonable for shor-erm pure financial conracs, a simple example demonsraes ha one should follow his approach wih care in connecion wih long-erm insurance conracs. Consider an insured holding a erm insurance and assume he possibiliy of saring o smoke wih an increasing effec on moraliy. We model his siuaion by inroducing an index indicaing wheher he insured is a smoker or no. Before saring o smoke, he moraliy is µ, and afer saring o smoke i increases o µ, 1. We disregard he possibiliy of sopping smoking. The quesion is now on which moraliy rae should he insurance company base he premium calculaion and he reservaion if he new cusomer ells ha he is a non-smoker. The insured can advance his deah occurrence and hereby maximize his expeced benefi paymens by saring smoking, and, in fac, he valuaion formula 1.1 ells he insurance company o use he high moraliy rae assuming ha he does so immediaely. However, he insured may ake oher hings ino consideraion han he benefis from he insurance conrac and conclude ha, afer all, i is opimal no o advance deah whereafer he chooses no o sar smoking. This is a oy example which, neverheless, shows ha he insurance company

1.4. CONTROL 15 should use a valuaion formula based on opimizaion wih care. The policy holder may have oher objecives han increasing he value of his insurance conrac, and when i comes o paymens linked o his life hisory, he probably will. Neverheless, in Chaper 4 we give mahemaical conen o inervenion opions and work wih he valuaion formula 1.1. In he case of no inervenion opions, he relaion beween expeced values and deerminisic differenial equaions is demonsraed in Chaper 2. In presence of inervenion opion, 1.1 relaes o a so-called quasivariaional inequaliy. This is a consrucive ool for deermining fair conracs under inervenion opions and is derived in Chaper 4. 1.4 Conrol In Secion 1.3 we discussed valuaion of paymen sreams. A he end of ha secion we unveiled one conrol problem imbedded in he paymen process, namely he conrol by inervenion of he policy holder. Furhermore we consruced guaraneed paymens and dividends, and we argued ha his consrucion of paymens allows he insurance company o ransfer risk o he policy holder. In fac, he design of he dividend paymen process B can be considered as a genuine conrol problem on he par of he insurance company. E.g. one could simply formulae an objecive of risk reducion in some sense and hen look for an opimal dividend process. We shall now consider a framework frequenly used in finance and insurance decision problems. Wihin his framework we recall some classical decision problems in finance and non-life insurance, and we consider how he decision problem of he life insurance company also has been approached wihin his framework in he lieraure on life and pension insurance. The approach o he life insurance decision problem sudied in Chapers 5 and 6 is a modificaion of his classical framework. A he end of his secion we explicae his. We will now parially change noaion in order o conform o Chapers 5 and 6. Consider he wealh reserve, value, or surplus of an agen consumer or insurance company wih he following dynamics dx = α θ, X d + σ θ, X dw du, 1.11 X = x, where x is he iniial wealh, and W is a sandard Brownian moion defined on a probabiliy space Ω, F,F = {F }, P. The parameer θ is chosen by he agen o balance drif and diffusion in he wealh process hrough he funcions α and σ. In his secion, θ will represen he proporion of wealh invesed in a risky asse and/or a parameer indicaing he exen of cover by some ype of reinsurance. U is he amoun wihdrawn from he wealh eiher for consumpion or as dividend disribuion unil ime. The agen needs an objecive for his decisions, and he wishes o choose θ, U so

16 CHAPTER 1. A SURVEY OF VALUATION AND CONTROL as o maximize [ τ E ] υ, X, du, d + Υ τ, X τ, 1.12 for some uiliy funcions υ and Υ and some sopping ime τ. Noe ha he compac noaion for curren uiliy only makes sense for paricular funcions υ. Examples will be given below. We use he erms uiliy and disuiliy funcions for general no necessarily increasing, concave/concave and coninuously differeniable reward and cos funcions, respecively. Conrol in finance Suppose here is a marke in which wo asses are raded coninuously. One asse is he bank accoun inroduced in 1.4, Z. The oher asse is a risky asse denoed by Z 1 wih a price process modelled as a geomeric Brownian moion wih drif, i.e. dz = rz d, Z = 1, dz 1 = µz 1 d + σz 1 dw, 1.13 Z 1 = z 1. Now he wealh process of an agen following he proporional invesmen sraegy θ and he consumpion sraegy U can be shown o have he dynamics given by 1.11, wih α θ, X = r + θ µ r X, σ θ, X = θ σx. One class of problems are so-called invesmen-consumpion problems where one requires consumpion o be posiive and absoluely coninuous wih respec o he Lebesgue measure such ha u = du exiss and where uiliies are ypically d given by υ, X, du, d = e γ υ u d, Υ τ, X τ = e γτ Υ X τ. Meron iniiaed he sudy of his problem in Meron [42] and [43] and found explici soluions for some paricular uiliy funcions. Primary examples of uiliy funcions in his case are he logarihmic and he power funcions. Anoher class of problems are hedging problems conneced wih a coningen claim Y τ. One approach is o le he consumpion be fixed a zero and le he uiliy of X τ depend on Y τ such ha for e.g. a quadraic loss funcion, υ, X, du, d =, Υ τ, X τ = X τ Y τ 2.

1.4. CONTROL 17 In his way deviaions from he τ-claim are punished and an opimal hedging sraegy can be searched for. In his hedging problem, one may also consider he saring poin of our wealh, x, as a decision variable and speak of he opimal x as some kind of price of Y. This approach o opimal invesmen sraegies and prices is called mean-variance hedging. Of course, he idea of mean-variance hedging is no resriced o he simple marke given by 1.13 bu can sudied for general non- Markovian markes as well, see e.g. Schweizer [61]. Conrol in non-life insurance Consider a non-life insurance company receiving premiums coninuously and paying ou claims. The company balances is gains by an exen θ of cover of some ype of reinsurance where he company pays premiums coninuously and receives some compensaion for claims. Furhermore, he company decides o pay ou dividends o share-holders. We emphasize ha dividend here is he share of profis paid o share-holders as opposed o he dividend inroduced in Secion 1.3 which goes o he policy holders. If we denoe he rae of premiums ne of he reinsurance premium by π θ, he number of claims received up o ime by N, he ime for occurrence of claim number i by τ i, and he size of he ih claim ne of he reinsurance compensaion by Y i θ, hen X, he company s reserve ne of reinsurance paymens a ime is given by X = x + N π θ sds Y i θ τ i U. We remark ha he noion of reserve in non-life insurance has a differen meaning han in life insurance. If N is a Poisson process wih inensiy λ and he claims are i.i.d., hen X can be approximaed by he process given by 1.11 wih i=1 α θ,x = π θ λe [Y i θ ], σ θ,x = λe [Yi 2 θ ]. Usually one les τ in 1.12 be he ime of ruin, i.e. he firs ime he reserve his zero, and wishes o maximize discouned dividends, υ, X, du, d = e γ du. One argumen for using he ideniy funcion as uiliy funcion is ha he value of he firm may be represened by expeced discouned dividends. This argumen, however, seems criicizable since i is by no means clear which discoun facor and which measure o use. For opimal proporional reinsurance see Højgaard and Taksar [34]. In he non-life insurance model above, we have disregarded capial gains, bu opimal conrol of reinsurance can, of course, be combined wih opimal invesmen.

18 CHAPTER 1. A SURVEY OF VALUATION AND CONTROL Conrol in life and pension insurance The lieraure on conrol in life and pension insurance has unil now concenraed primarily on conrol of pension funds. For references o lieraure on conrol of pension funds, see Cairns [12], which is parly a survey aricle gahering resuls of several auhors. The conrol parameers are usually a proporion in risky asses and/or he level of premiums/benefis. The insiuional condiions for pension funds may be raher involved, and i is by no means clear how he objecives of he fund manager, he employer pays he premium, and he employed receives he benefis should be refleced in he objecive funcion of he conrol problem. Here, we shall briefly demonsrae how pension funds are modelled and conrolled in coninuous-ime as exposed in Cairns [12]. Assume ha he pension fund receives premiums and pays benefis. The infiniesimal ne ougo of he fund is normally disribued wih expeced value u d and variance β 2 d, independenly of he financial marke. The mean rae of he ne ougo is conrollable by he fund manager who can adjus his according o he performance of he fund. The employer and he employed, respecively, experience his conrol by changes in he premium level in he case of defined benefis and in he benefi level in he case of defined conribuions, respecively. We assume ha he money in he fund is invesed in he marke described by 1.13. Then he dynamics of he fund is given by 1.11 wih α θ, X = r + θ α r X, σ θ, X = θ 2 σ 2 X 2 + β 2, du = u d. So far he problem only differs from he invesmen-consumpion problem of finance by he erm β. This indicaes a connecion beween invesmen-consumpion problems and he conrol problems of he life insurance company. This connecion is briefly menioned in Cairns [12] and will be clarified in Chapers 5 and 6. Now we inroduce an objecive which rewards a cerain kind of sabiliy of he pension fund and in he paymens. This is done by working wih a quadraic disuiliy he expeced oal of which is now o be minimized, υ, X, du, d = e γ a X x 2 + b u û 2 + c X x u û d. 1.14 This disuiliy funcion punishes disance beween X and x and disance beween u and û. The punishmens are weighed by a, b, c. This disuiliy funcion is clearly somewha conneced o he quadraic approaches o hedging of coningen claims in finance like mean-variance hedging, bu he one canno be considered as a special case of he oher. However, he quadraic approaches share a couner-inuiive conclusion on invesmen which is easily explained: If one wishes o have X close o x or evenually X τ close o Y τ,