This page intentionally left blank

Transcription

1

2 This page inenionally lef blank

3 Marke-Valuaion Mehods in Life and Pension Insurance In classical life insurance mahemaics, he obligaions of he insurance company owards he policy holders were calculaed on arificial conservaive assumpions on moraliy and ineres raes. However, he classical approach is being superseded by developmens in inernaional accouning and solvency sandards coupled wih heoreical advances in he undersanding of he principles and mehods for a more marke-based valuaion of risk, i.e. is price if raded in a free marke. The book describes hese new approaches, and is he firs o explain hem in conjuncion wih more radiional mehods. The exposiion inegraes mehods and resuls from financial and insurance mahemaics, and is based on he enries in a life insurance company s marke accouning scheme. Wih-profi insurance conracs are described in a classical acuarial model wih a deerminisic ineres rae and no invesmen alernaives. The classical valuaion based on conservaive valuaion assumpions is explained and an alernaive marke-valuaion approach is inroduced and generalized o sochasic ineres raes and risky invesmen alernaives. The problem of incompleeness in insurance markes is addressed using a variey of mehods, for example risk minimizaion, mean-variance hedging and uiliy opimizaion. The applicaion of mahemaical finance o uni-linked life insurance is unified wih he heory of disribuion of surplus in life and pension insurance. The final chaper provides an inroducion o ineres rae derivaives and heir use in life insurance. The book will be of grea ineres and use o sudens and praciioners who need an inroducion o his area, and who seek a pracical ye sound guide o life insurance accouning and produc developmen. Inernaional Series on Acuarial Science Mark Davis, Imperial College London John Hylands, Sandard Life John McCucheon, Herio-Wa Universiy Ragnar Norberg, London School of Economics H. Panjer, Waerloo Universiy Andrew Wilson, Wason Wya The Inernaional Series on Acuarial Science, published by Cambridge Universiy Press in conjuncion wih he Insiue of Acuaries and he Faculy of Acuaries, will conain exbooks for sudens aking courses in or relaed o acuarial science, as well as more advanced works designed for coninuing professional developmen or for describing and synhesising research. The series will be a vehicle for publishing books ha reflec changes and developmens in he curriculum, ha encourage he inroducion of courses on acuarial science in universiies, and ha show how acuarial science can be used in all areas where here is long-erm financial risk.

4

5 Marke-Valuaion Mehods in Life and Pension Insurance THOMAS MØLLER PFA Pension, Copenhagen MOGENS STEFFENSEN Insiue for Mahemaical Sciences, Universiy of Copenhagen

6 CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge Universiy Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in he Unied Saes of America by Cambridge Universiy Press, New York Informaion on his ile: T. Møller and M. Seffenson 2007 This publicaion is in copyrigh. Subjec o sauory excepion and o he provision of relevan collecive licensing agreemens, no reproducion of any par may ake place wihou he wrien permission of Cambridge Universiy Press. Firs published in prin forma 2007 ISBN ebook (NeLibrary) ISBN ebook (NeLibrary) ISBN hardback ISBN hardback Cambridge Universiy Press has no responsibiliy for he persisence or accuracy of urls for exernal or hird-pary inerne websies referred o in his publicaion, and does no guaranee ha any conen on such websies is, or will remain, accurae or appropriae.

7 Conens Preface page ix 1 Inroducion and life insurance pracice Inroducion The life insurance marke The policy holder s accoun Dividends and bonus Uni-linked insurance and beyond 9 2 Technical reserves and marke values Inroducion The radiional composiion of he liabiliy The marke-based composiion of he liabiliy The liabiliies and principles for valuaion The liabiliy and he paymens The surrender opion The free policy opion 37 3 Ineres rae heory in insurance Inroducion Valuaion by diversificaion revisied Zero coupon bonds and ineres rae heory A numerical example Bonds, ineres and duraion On he esimaion of forward raes Arbirage-free pricing in discree ime Models for he spo rae in coninuous ime Marke values in insurance revisied 98 v

8 vi Conens 4 Bonus, binomial and Black Scholes Inroducion Discree-ime insurance model The binomial model The Black Scholes model Coninuous-ime insurance model Generalizaions of he models Inegraed acuarial and financial valuaion Inroducion Uni-linked insurance The policy holder s accoun Hedging inegraed risks under diversificaion Hedging inegraed risk in a one-period model The muli-period model revisied Hedging inegraed risks Tradiional life insurance Surplus-linked life insurance Inroducion The insurance conrac Surplus Surplus-linked dividends Dividends linear in surplus Bonus Surplus- and bonus-linked dividends Dividends linear in surplus and bonus Ineres rae derivaives in insurance Inroducion Swaps and beyond in coninuous ime Pricing of ineres rae derivaives Swaps and beyond in discree ime A brief inroducion o marke models Ineres rae derivaives in insurance A porfolio of conracs 257

9 Conens vii Appendix 263 A.1 Some resuls from probabiliy heory 263 A.2 The risk-minimizing sraegy 266 A.3 Risk minimizaion for uni-linked conracs 267 A.4 Mean-variance hedging for uni-linked conracs 269 References 272 Index 278

10

11 Preface Insurance mahemaics and financial mahemaics have converged during he las few decades of he wenieh cenury and his convergence is expeced o coninue in he fuure. New valuaion mehods are added o he radiional valuaion mehods of insurance mahemaics. Valuaion and decision making on he asse side and he liabiliy side of he insurance companies are, o an increasing exen, being considered as wo sides of he same sory. The developmen has wo consequences. Demands are made on pracising acuaries, whose educaion daes back o when financial mahemaics was no considered as an inegraed par of insurance mahemaics. By considering he convergence as i applies o heir daily work, such acuaries should be kep abreas of his convergence. From his saring poin, he ideas, conceps and resuls of finance should be brough ogeher o consruc a pah beween classical acuarial deerminisic paerns of hinking and modern acuarial mahemaics. This is where sochasic processes are brough o he surface in paymen sreams as well as in invesmen possibiliies. A he same ime, presen sudens of acuarial mahemaics need o apply financial mahemaics o classical insurance valuaion problems. These sudens will ypically, and should, mee financial mahemaics in exbooks on pure finance. However, o receive he full benefi of financial mahemaical skills, hese skills need o be inegraed and proven beneficial for classical problems of insurance mahemaics already on a suden level. Inernaional accouning sandards have developed over he years. Denmark has been a he forefron, implemening new accouning mehods o replace (assumed o be) conservaive book values wih real values based on marke informaion. Alhough he inernaional accouning sandards have no ye been seled, he Danish approach o marke valuaion seems o be an imporan sep in he righ direcion. Many aspecs of his approach are underpinned by mehods aken from mahemaical finance. ix

12 x Preface The raionale for his book is ha pracising acuaries need an exposiion of financial mehods and heir applicaions o life insurance from he poin of view of a praciioner. Mehods and applicaions are discussed in erms of he Danish approach o marke valuaion. As a by-produc, he book explains o presen sudens how financial mehods known o hem can be applied o valuaion problems in he life insurance marke. In 1995 and 1996, Tomas Björk and Ragnar Norberg gave courses in financial mahemaics and applicaions o life insurance a he Universiy of Copenhagen. These courses aroused our ineres in he inerplay beween finance and insurance. We sudied he opics in our maser heses, finished in 1996 (T.M.) and 1997 (M.S.), and coninued our sudies in our Ph.D. heses, finished in 2000 and 2001, respecively. Pars of he book (Chaper 2 5) are based on maerial which was developed for a course on marke valuaion in life and pension insurance. This course was organized by he Danish Acuarial Associaion in Each chaper was he maerial for one course module and was wrien more or less independenly of he ohers. The maerial was originally wrien in Danish. In 2002 he maerial was developed furher and ranslaed ino English, and Chaper 1 was added. In 2003, Chapers 6 and 7 were added on he occasion of The Firs Nordic summer school in insurance mahemaics, eniled New Financial Producs in Insurance. In 2004, he maerial was made consisen for noaion and erminology. However, i is sill our inenion ha each chaper should be readable more or less independenly of he ohers. Therefore, some definiions and inroducions of quaniies are repeaed hroughou he book. The book suggess approaches o life insurance marke-valuaion problems. The saring poin is he version of wih-profi insurance provided by Danish life insurance companies. In order o help he reader follow he mahemaical descripion of his ype of produc, Chaper 1, wrien by Mogens Seffensen, provides a non-mahemaical inroducion o life insurance pracise in general. In Chaper 2, also wrien by Mogens Seffensen, he wih-profi insurance conrac is described a firs in a classical acuarial model wih a deerminisic ineres rae and no invesmen alernaives. The classical valuaion based on conservaive valuaion assumpions is explained, and an alernaive markevaluaion approach is inroduced. Here, he pariion of fuure paymens in guaraneed paymens and non-guaraneed paymens is imporan. Paricular aenion is paid o he inervenion opions held by he policy holder, i.e. he surrender and free policy (paid-up policy) opions. Various approaches o hese opions are suggesed. The marke-valuaion mehod inroduced in Chaper 2 is generalized o a sochasic ineres rae in Chaper 3. Boh discree-ime and coninuous-ime

13 Preface xi bond marke heory are inroduced o a level such ha he reader can follow he reasoning behind replacing he discoun facor in he marke-valuaion formulas for guaraneed paymens by zero coupon bond prices. Fundamenal financial conceps, such as arbirage and marke compleeness, are inroduced in a bond marke framework. Difference and differenial equaions for he marke value of guaraneed paymens are derived. Chaper 3 is wrien by Thomas Møller. In Chaper 4 he marke-valuaion mehod inroduced in Chaper 2 is generalized o a siuaion wih one risky invesmen alernaive o he deerminisic ineres rae. Boh discree-ime and coninuous-ime sock marke heory are inroduced o a level such ha he reader can follow calculaions of marke valuaions of non-guaraneed paymens in he case of wo invesmen alernaives. Fundamenal financial conceps, such as arbirage and marke compleeness, are repeaed in a sock marke framework. Difference and differenial equaions for he marke value of he oal paymens, including he non-guaraneed paymens, are derived. Finally, he sock marke is conneced o he sochasic bond marke inroduced in Chaper 3. Chaper 4 is wrien by Mogens Seffensen. The usual ouline of inroducory financial mahemaics is o inroduce he fundamenal financial conceps in a discree and/or coninuous sock marke model. Aferwards hese are repeaed in a discree- and/or coninuous-ime bond marke model. In our exposiion, he cu is differen and is based on enries in a life insurance company s marke accouning scheme. Valuaing he guaraneed paymens, he imporan sochasic generalizaion of he classical acuarial deerminisic financial marke lies a he inroducion of a sochasic bond marke. A sochasic sock marke comes ino play when valuaing he non-guaraneed paymens. An alernaive class of insurance conracs o wih-profi insurance is unilinked insurance conracs, which are sudied in Chaper 5, wrien by Thomas Møller. This class of conracs and heir marke values are analyzed on he basis of he sochasic financial markes inroduced in Chapers 3 and 4. The problem of genuine incompleeness in insurance markes is addressed. The incompleeness in insurance markes semming, for example, from moraliy risk is ofen aken care of in he lieraure by assuming risk neuraliy of he insurance company wih respec o such a risk. Relaxing his assumpion, we sugges various approaches o incomplee marke valuaion. In paricular, valuaion and opimal invesmen mehods based on risk minimizaion, meanvariance hedging and uiliy opimizaion are inroduced and exemplified in he case of uni-linked insurance.

14 xii Preface In Chaper 6, he applicaion of mahemaical finance o uni-linked life insurance is unified wih he heory of disribuion of surplus in life and pension insurance. The unificaion is based on a consideraion of disribuion of surplus as an inegraed par of he insurance conrac. The noion of surplus and various dividend and bonus schemes linked o his surplus are sudied. In paricular, explici resuls are obained in he case where dividends and bonus paymens are linear in he surplus. Chaper 6 is wrien by Mogens Seffensen. Typically, insurance companies are faced by insurance liabiliies ha exend up o sixy years ino he fuure, whereas he financial markes ypically do no offer bonds ha exend more han hiry years ino he fuure. Wih marke-based valuaion mehods, he value of boh asses and liabiliies are affeced by changes in he economic environmen. Here, ineres rae derivaives seem o have become an imporan risk-managemen ool for life insurance companies. An inroducion o cerain conceps and insrumens from he area of ineres rae derivaives is herefore given in Chaper 7, wrien by Thomas Møller. Examples are swap raes, swaps, floors, caps, swapions and CMS opions. Various pricing mehods are discussed, and i is demonsraed how he financial impac on a life insurance company of hese insrumens could be assessed. The book sudies approaches for marke valuaion of life insurance liabiliies. The various chapers address specific aspecs of marke-based valuaion and conain inroducions o heoreical resuls from financial mahemaics and sochasic calculus ha are necessary for he applicaions. A brief discussion of he relaion o exising books on financial mahemaics and insurance mahemaics is given in he following lis. Björk, T. (2004). Arbirage Theory in Coninuous Time, 2nd edn (Oxford: Oxford Universiy Press). Mos of he heoreical resuls relaed o financial mahemaics presened here can be found in his book by Tomas Björk. To some exen, he noaion suggesed by Björk is considered as sandard and is herefore used in he presen book. Even he srucural exposiion of cerain opics of financial mahemaics is inspired by Björk s book. Briys, E. and de Varenne, F. (2001). Insurance: From Underwriing o Derivaives: Asse Liabiliy Managemen in Insurance Companies (Chicheser, UK: Wiley). The book discusses he convergence beween he insurance indusry and he capial markes. I is less mahemaical han he curren book and focuses on insiuional aspecs of he inerplay beween he wo fields. In conras, he curren book invesigaes he convergence of he heories of financial mahemaics and insurance mahemaics and heir applicaions o marke-based valuaion.

15 Preface xiii Gerber, H.U. (1997). Life Insurance Mahemaics (Berlin: Springer). This provides an inroducion o classical life insurance mahemaics and can be viewed as a necessary prerequisie for he curren book. The conceps and echniques discussed in Gerber s book (for example, presen values, specific life insurance conracs, decremen series, Thiele s differenial equaion) are also used and explained in he presen book. However, he curren manuscrip has a compleely differen goal and goes considerably beyond he inroducory presenaion in Gerber s book. Koller, M. (2000). Sochasische Modelle in der Lebensversicherung (Berlin: Springer). The book presens a framework where he underlying insurance conracs are modeled by Markov chains and where sochasic ineres raes are allowed. The main difference beween Koller s book and he curren manuscrip is ha we focus more on marke values and he applicaion of heories from financial mahemaics in he area of life insurance; Koller s book deals more wih he underlying Markov chains and on deriving differenial equaions for he corresponding reserves (based on he work by Hoem, Norberg and ohers). We expec our readership o fall ino wo caegories. Firsly, pracising life insurance acuaries who need an updae of he mahemaics of life insurance, an inroducion o financial mahemaics in an insurance conex and an approach o marke valuaion in life insurance. Indeed, he book akes he poin of view of a pracising acuary. Chaper 1 on life insurance pracice will provide he reader wih sufficien insigh ino his pracice. Secondly, i is expeced ha he book will be read by sudens in acuarial science, who have prerequisies in boh life insurance mahemaics and mahemaical finance, bu wan o see how hese disciplines can and will be combined in boh heory and pracice. By aking he viewpoin of a pracising acuary as a saring poin, he suden also sees how aspecs of he classical life insurance mahemaics are implemened in pracice. The level is advanced. Basic knowledge of life insurance mahemaics (such as in Gerber s book) is required. In addiion, basic probabiliy heory is required, such ha he reader can follow he inroducion of filraions, maringales, sochasic differenial equaions, ec. No previous knowledge of financial mahemaics is required. However, he heoreical resuls are a imes developed quickly in order o ge o he applicaions, and a some poins he reader would probably benefi from sudying exbooks on financial mahemaics for more deails and more background informaion; for example, Björk (2004). We would like o hank Tomas Björk and Ragnar Norberg for arousing our curiosiy and for sharpening our undersanding of he mahemaics of life

16 xiv Preface insurance and finance. The combinaion of basic knowledge in boh areas and a provoked curiosiy made our sudies of he inerplay beween hese fields possible, challenging and ineresing. In addiion, we wish o hank Ragnar Norberg, Chrisian Hipp and Marin Schweizer for heir guidance and suppor during our Ph.D. sudies. We would also like o hank Vibeke Thinggard and Mikkel Jarbøl for valuable commens and discussions on earlier versions of his maerial: in 2001 hey were members of he Coninued Professional Developmen Commiee under he Danish Acuarial Associaion and were deeply involved in he organizaion of he course given in 2001.

17 1 Inroducion and life insurance pracice 1.1 Inroducion This chaper provides an inroducion o life insurance pracice wih focus on wih-profi life insurance. The purpose is o give he reader sufficien insigh o benefi from he remaining chapers. In life insurance, one pary, he policy holder, exchanges a sream of paymens wih anoher pary, he insurance company. The exchanged sreams of paymens form, in a sense, he basis of he insurance conrac and he corresponding legal obligaions. When speaking of life insurance pracice, we hink of he way his exchange of paymens is handled and seled by he insurer. We ake as our saring poin he idea of he policy holder s accoun. This accoun can be inerpreed as he policy holder s reserve in he insurance company and accumulaes on he basis of he so-called Thiele s differenial equaion. Is formulaion as a forward differenial equaion plays a crucial role, and his chaper explains in words he consrucion and he elemens of his equaion and is role in accouning. Noe, however, ha he policy holder s accoun is no in general a capial righ held by he insured bu a key quaniy in he insurer s handling of his obligaions. 1.2 The life insurance marke In his secion we explain he mos ypical environmens for negoiaion and conracual formulaions for a life insurance policy. We disinguish beween defined benefis and wha we choose o call defined conribuions wih parly defined benefis. Defined conribuions wih parly defined benefis cover he majoriy of life insurance policies. The policy holder agrees wih he insurance company abou a cerain premium o cover a baske of benefis wih, for example, a 1

18 2 Inroducion and life insurance pracice life insurance sum paid ou upon deah before he erminaion of he conrac and/or a pension sum paid ou upon survival unil he erminaion of he conrac. The benefis agreed upon a issuance are se sysemaically low by basing hem on conservaive assumpions on ineres raes, insurance risk and expenses. As he marke ineres raes, he marke insurance risk and he marke coss evolve, a surplus emerges, and his surplus is o be paid back o he policy holder. This ypically happens by increasing one or more benefis. This combinaion of known premiums refleced in guaraneed benefis, which may be increased depending on he developmen in he marke, caegorizes he conracs as defined conribuions wih parly defined benefis. One consrucion is o increase all benefis proporionally such ha he raio beween, for example, he deah sum and he pension sum is mainained. Anoher consrucion is o keep he deah sum consan or regulaed wih some price index while residually increasing he pension sum. Only very rarely is he surplus paid ou in cash or used o decrease he premium insead of increasing he benefis. Such rare consrucions should in principle be caegorized as defined benefis wih parly defined conribuions since hen a baske of known benefis is combined wih guaraneed premiums which may be decreased depending on he developmen in he marke. The policies wih defined conribuions wih parly defined benefis are naurally classified as privae, firm-based, or labor-based. A privae policy is agreed upon by a privae person and he company. The privae person is he policy holder and negoiaes he condiions in he conrac. A firm-based policy is a conrac which is par of an agreemen beween an employer and an insurance company. The employer ypically pays a par of he premium bu receives no benefis. The oal premium paid is ypically a percenage of he salary. Alhough he erms of he conrac are ypically negoiaed beween he employer and he company, he employees are sill he policy holders. The agreemen beween he employer and he company may eiher be compulsory, in which case all employees are forced o paricipae, or opional, in which case i is up o he employees o decide wheher or no o paricipae. Since he employer has no claims and no obligaions besides paying he premium, his premium can in many respecs be inerpreed as salary. A labor-based policy works in many respecs as a firm-based policy excep for he fac ha he employer and he employees are represened a he negoiaion by organizaions raher han he employers and employees hemselves. These organizaions ypically ake care of people employed in he uniformed services or educaion. The resul is an agreemen where he employer is

19 1.2 The life insurance marke 3 obliged o paricipae in he employees policy on erms agreed upon by he organizaions. The policy is hen issued by a company aking care of all he employees in a paricular organizaion. Once he agreemen has been made and he conrac has been issued, i works basically as a firm-based policy wih he employer and he employees as payers of premiums and he employees as receivers of benefis. The oal premium paid is ypically a percenage of he salary. The employees are he policy holders. The labor-based policy is ofen par of a compulsory agreemen, boh he employer and he employed are obliged for which o agree o minimum condiions. The premium par paid by he employer can in many respecs be inerpreed as salary. The defined benefi policies are usually a par of an agreemen beween a firm and an insurance company, are herefore and comparable wih he firm-linked policy described above. The conrac is negoiaed indirecly by selemen of he agreemen. However, insead of sharing he premium defined as a percenage of he salary beween he employer and he employees, only he employees par of he premium is defined as a percenage of he salary. On he oher hand, he benefis are also defined as a percenage of he salary. This leaves a risk on he premium side. This risk is spli beween he employer and he insurance company according o he agreemen. If he risk is lef o he insurance company exclusively, hen neiher he employer nor he policy holder paricipaes in he developmen in he marke bu leaves all risk o he insurance company. In conras, if he risk is lef o he employer exclusively, he insurance company is pure adminisraor and akes no risk. As menioned above, a cerain par of he defined conribuion policies, where he surplus is redisribued as cash or used o decrease he premiums, can acually be considered as defined benefi policies wih parly defined conribuions. The classificaion given above is fairly broad. When discussing deails, here may be a lo of differences in he concree formulaions of he various agreemens and conracs. Policies belonging o differen classes may also be mixed wihin an agreemen and wihin a conrac. In he following, we concenrae on defined conribuions wih parly defined benefis. Alhough mos of he ideas presened in Chapers 2 5 may be applied o defined benefis as well, all examples and inerpreaions ake he defined conribuions wih parly defined benefis policy as a saring poin. as:eksis I should be menioned ha in addiion o he life insurance marke described above, here may exis a se of public insurance schemes. For insance, in Denmark he naional pension scheme is a pay-as-you-go scheme where presen reiremen pensioners are covered by presen ax payers. In addiion, he Danish sae regulaes a couple of paricular funded pension schemes for people who work.

20 4 Inroducion and life insurance pracice 1.3 The policy holder s accoun In our seing, he policy holder s accoun or he echnical reserve and is dynamics are he echnical elemens in he handling and adminisraion of life insurance conrac. We classify he changes of he echnical reserve in wo differen ways. The firs way classifies he changes of income and ougo. The oher way classifies he changes in wha was agreed beforehand in a paricular sense and addiional changes made by he insurance company a he discreion of he company. The echnical reserve can in cerain respecs be inerpreed as a bank accoun. The income on a bank accoun consiss of capial injecions and capial reurn provided by he bank from capial gains on invesmen of he accoun. The ougo on a bank accoun consiss of capial wihdrawals. Correspondingly, he income on an insurance accoun consiss of premiums paid by he policy holder and reurn provided by he insurance company from capial gains on invesmen of he accoun. And, correspondingly, he ougo on an insurance accoun consiss of he benefis paid o he policy holder. However, wo addiional erms add o he change of he accoun. One erm is an ougo and covers he expense o he insurance company o adminisrae he policy. Adminisraion expenses on he bank accoun mus also be paid, bu hese are charged indirecly by a reducion of he reurn. The oher addiional erm which adds o he change of he accoun is he so-called risk premium, which can be considered as an income or an ougo depending on is sign. The risk premium is a premium ha he policy holder pays from heir accoun; i may be posiive, in which case i can be considered as an ougo, or negaive, in which case i can be considered as an income. The risk premium is paid o cover he loss o he insurance company in case an insurance even akes place in some small ime inerval. The amoun of he poenial loss is also spoken of as he sum a risk. The premium for his coverage is se o he expeced value of his loss. Considering a so-called erm insurance paying ou a sum upon deah, he loss o he insurance company in case of deah equals he deah sum which has o be paid ou minus he echnical reserve which, on he oher hand, can be cashed. The expeced value of ha loss is he difference beween he deah sum and he echnical reserve imes he probabiliy of dying in some small ime inerval. The deah sum exceeds he echnical reserve such ha he risk premium is posiive and can be considered as an ougo. Considering insead a pure endowmen insurance, he insurance company cashes he echnical reserve upon deah and has no obligaions. The expeced value of his gain is he echnical reserve imes he probabiliy of dying in a small inerval. This resuls in a negaive risk premium, and he

21 1.3 The policy holder s accoun 5 risk premium may be considered as an income. In addiion, he bank accoun can be inerpreed as an insurance conrac where he echnical reserve is simply paid ou upon deah. This gives a poenial loss upon deah of he echnical reserve paid ou minus he echnical reserve cashed, whereby he risk premium equals zero. The erm insurance and he pure endowmen insurance are simple insurance conracs. If we inroduce such hings as disabiliy annuiies, premium waiver and deferred benefis, he picure becomes more blurred, bu he underlying idea is basically he same. Apar from he real incomes and ougoes in form of premiums, reurns and benefis, he policy holder pays or gains, depending on he sign of he risk premium, for he risk imposed on he insurance company. Anoher way of classifying he changes of he echnical reserve is firsly o idenify he echnical change which conforms wih he guaraneed paymens and hen idenify he addiional changes made by he insurance company a he discreion of he company. When an insurance company issues an insurance policy, i guaranees a minimum benefi which is based on a echnical reurn. Furhermore, he minimum benefi is based on a cerain echnical probabiliy of he insurance even, for example he probabiliy of dying in a small ime inerval. Finally, i is based on a echnical amoun for adminisraion expenses. Basically, i simply guaranees o pay ou a benefi which is fair under a cerain se of assumpions on reurn, insurance risk and expenses. However, his se of assumpions is mean o be se so conservaively ha a surplus emerges over ime. This surplus is provided by he policy holder due o conservaism in assumpions and has o be paid back as he real marke condiions evolve. This is achieved by adding dividends o he policy holder s accoun. The law saes ha he surplus mus be paid back o hose who creaed i. The usual way of allocaing he dividends is o change he accoun, no in correspondence wih he echnical assumpions, bu in correspondence wih a se of assumpions ha is more favorable o he policy holder. Then, we can classify, elemen by elemen, he echnical change and he addiional change. Concerning he reurn, he insurance company firsly pays he echnical reurn. Secondly, i pays he difference beween he more favorable reurn and he echnical reurn. Concerning he moraliy, he insurance company firsly collecs a risk premium in correspondence wih he echnical probabiliy of deah. Secondly, i pays back he difference beween he risk premiums in correspondence wih he more favorable probabiliy and he echnical probabiliy. Concerning he expenses, he insurance company firsly wihdraws he echnical amoun for expenses. Secondly, i pays back he difference beween he more favorable amoun and he echnical amoun for expenses. The use of he favorable assumpion is ha, elemen by elemen, he policy holder

22 6 Inroducion and life insurance pracice should no be pu ino a worse posiion han if he echnical assumpion had been used. The favorable developmen of he accoun including dividends may be used o reach a higher value o be paid ou as a pension sum a he erminaion of he conrac. However, he policy holder may also wish ha his favorable developmen provides capial for an increase of benefis and/or a decrease of premiums hroughou he erm of he policy. This has a feedback effec on he dynamics of he accoun, since premiums, benefis and risk premiums need o be adjused in he ligh of such a change of paymens. One consrucion is o le he deah sum and he pension sum increase proporionally, such ha he raio beween he wo benefis is consan. However, he mos ypical consrucions are o le he deah sum be consan or regulaed by some price index and hen use he residual dividends o increase he pension sum. We should menion an alernaive applicaion of dividends which has gained ground in recen years. Insead of increasing benefis and/or decreasing premiums, one may keep he guaraneed paymens and insead change he underlying echnical assumpions. In his way, dividends are added o he policy holder s accoun wihou changing he guaraneed paymens. One may hen ask: where did he money go and does allocaion of dividends really pu he policy holder in a beer posiion? The poin is ha paying ou dividends leads o wha seems o be less favorable echnical condiions. However, he guaraneed paymens are no changed and can herefore no be less favorable. Furhermore, he consequence of less favorable echnical condiions is higher surplus conribuions in he fuure. And since hese surplus conribuions evenually have o be redisribued and refleced in paymens, he posiion is indeed favorable. By his consrucion, allocaion of dividends in a way pospone he incremen of guaraneed benefis wihou posponing he incremen of he accoun. 1.4 Dividends and bonus The premiums agreed upon a he ime of issuance of an insurance policy are oo high compared wih he benefis ha are guaraneed a he ime of issuance. This disproporion is he source of surplus, and i is saed by law ha his surplus should be paid back o hose policy holders who creaed i. In pracice, his happens in wo seps. Firsly, he surplus is disribued among he owners of he insurance company and he group of policy holders, and, secondly, he surplus disribued o he policy holders is disribued among he policy holders.

23 1.4 Dividends and bonus 7 So, why should he owners of he company ake par in he surplus ha was creaed by policy holders? The problem is ha oo high may no be high enough. The insurance company is allowed o inves no only in fixed income asses, bu also in socks. Invesmen in socks is, however, a risky business, and he insurance company may end up in a siuaion where i is no possible o increase he policy holder s accoun by he echnical ineres rae by means of capial gains. In ha siuaion he owners of he insurance company mus sill provide capial for he echnical incremens of he echnical reserve. Also, concerning moraliy and expenses he insurance company may experience a siuaion worse han ha considered as he wors possible case a he ime of issuance. Concerning moraliy and oher kinds of insurance risk, medical, sociological and demographic uncerainies play a differen role. The insurance company may need o help by injecing capial in he echnical reserve in order o live up o he echnical condiions. The owners of he insurance company mus evenually cover he loss on he insurance porfolio. The risk ha hings may go wrong, leading o he owners having o pay, is he reason why hey, when hings go righ, deserve a share in he surplus creaed by he insurance porfolio. However, he disribuion of surplus beween owners and policy holders has o be fair in some sense. One of he purposes of his book is o provide he insurance companies wih ools and ideas o make disribuions ha, o an increasing exen, are fair. The par of he surplus disribued o he policy holders is deposied in he so-called undisribued reserve. Tha is, his reserve is disribued o he policy holders as a group bu is no ye disribued among he policy holders. The disribuion among policy holders akes place by deciding on he favorable se of assumpions inroduced above. This mechanism ransfers money from he undisribued reserve o he individual policy holder s accoun. As was required from he disribuion beween owners and policy holders, he muual disribuion beween policy holders is also required o be fair. Fairness is here given by he saemen ha he surplus should be redisribued o hose who earned i. A redisribuion of he surplus o hose who earned i has wo consequences. The firs consequence is ha he insurance company is no allowed o grow large undisribued reserves. This would sysemaically redisribue surplus from he pas and presen policy holders o he fuure policy holders. Thus, he insurance company needs o assign he undisribued reserve o he individual echnical reserves in due ime. Here, due ime is, of course, closely conneced o he risk of he insurance company owners o evenually suffer a loss on he porfolio, which again connecs o he owner s share in

24 8 Inroducion and life insurance pracice he surplus. If he undisribued reserve is high, hen his reserve can ake a big loss before he owners have o ake over. Then he insurance porfolio pays a small, possibly zero, premium o he owners for aking risk. If he undisribued reserve is low, hen his reserve may easily run ou, and he insurance porfolio mus pay a larger premium o he owners. This shows ha he soluion o a fair disribuion of he surplus beween he owners and he policy holders ineracs subsanially wih he soluion o a fair muual redisribuion amongs policy holders over ime. The second consequence is ha, given a redisribuion o he presen policy holders, his mus happen in a way ha reflecs which presen policy holders have conribued a lo o he surplus and which have conribued less. Such a mechanism can be imposed by favorable assumpions on ineres raes, moraliy and expenses. The reurn is proporional o he echnical reserve, he risk premium is proporional o he sum a risk, and he expense is ypically formalized as a par of he premium. Therefore he individual echnical reserve, he sum a risk and he premium deermine he individual share in he oal disribuion. Once a redisribuion from he undisribued reserve among he policy holders is eleced o happen now insead of laer, he se of favorable assumpions mus o some exen reflec he presen policy holder s conribuions o he undisribued reserves. Depending on he bonus scheme, he policy holder may experience he redisribuion in differen ways. The ypical consrucion is o increase he benefis proporionally or o increase he pension sum residually, for example, afer a price index regulaion of he deah sum. The redisribuion may also be paid ou as cash. The redisribuion of he surplus beween he owners of he company and he policy holders and he muual redisribuion beween policy holders are regulaed by law and overseen by he supervisory auhoriies. Thus, hey are no direcly specified in he conrac. However, hey make up a par of he legislaive environmen in which he conrac has been agreed upon, and herefore hey can be considered, in many respecs, as par of he conrac iself. On he oher hand, he conversion of dividends ino paymens on he individual policy is a par of he individual policy condiions. Therefore, his conversion is direcly negoiable beween he insurance company and he policy holder or, in he case of a firm-linked or labor-linked conrac, beween he company and he firm or labor organizaion, respecively. I is imporan o realize how he legislaive environmen and he conrac, in combinaion, make up he condiions for all changes ha are made over ime by he insurance company o he premiums and benefis agreed upon a issuance. Firsly, he disribuion beween owners and policy holders (regulaed by law);

25 1.5 Uni-linked insurance and beyond 9 secondly, he disribuion muually among policy holders (regulaed by law); and hirdly, by changing he erms in he individual policy (regulaed by he conrac). 1.5 Uni-linked insurance and beyond Several feaures characerize he paricipaing policy as explained above. The policy holder paricipaes in a muual fund, so o speak, ogeher wih he oher policy holders and ogeher wih he owners of he insurance company. The legislaive environmen ses he condiions for his cooperaion. However, i may be difficul for he individual policy holder o undersand wheher he condiions are followed, in paricular concerning he several layers of fair disribuions. Even by represenaion of heir ambassadors in he cooperaion in he form of he supervisory auhoriies, his may be a difficul ask. One way of avoiding he problems wih fair disribuion is ha each and every single policy holder forms heir own individual fund. This is wha happens in uni-linked insurance. In a uni-linked insurance conrac, he policy holder does no paricipae in a muual fund bu decides on heir own invesmens o some exen. The paricipaing policies hold a very srong posiion in many counries and he uni-linked marke has been long in coming, bu since he beginning of he weny-firs cenury life insurance companies in hese counries have sared o offer uni-linked insurance conracs. The uni-linked insurance conrac can be decoraed wih many differen kinds of guaranees, and insurance companies have shown some creaiviy on ha poin. However, he marke is sill young, and here is sill a lo of space for new developmens and improvemens. When giving up he invesmen cooperaion and enering ino uni-linked conracs, policy holders ypically also give up cerain feaures of he paricipaing policy. By working wih an undisribued reserve, one achieves a smoohing effec of he marke condiions. The undisribued reserve proecs he underlying echnical reserves, and hereby he guaraneed paymens, from shocks in he marke condiions. The echnical reserves hen only experience a smoohed effec from such shocks. However, i is imporan o realize ha hese smoohing effecs do no rely paricularly on he policy holders paricipaion in an invesmen cooperaion. There is, in principle, no problem in mainaining he smoohing effec in a uni-linked insurance policy. This is only a maer of a proper definiion of he uni o which he paymens of he conrac are linked. Some insurance companies have inroduced advanced

26 10 Inroducion and life insurance pracice uni-linked producs which mainain he smoohing effec. When speaking of such producs as uni-linked conracs, our characerizaion of uni-linked producs is ha he invesmen game is an individual maer. If he invesmen game is individualized, hen he uni-linked conrac says uni-linked, no maer he complexiy of he uni, even when including any kind of smoohing effec. One may argue ha a uni-linked insurance conrac endowed appropriaely wih smoohing effecs and guaranees is close, boh in spiri and in paymens, o a paricipaing policy. On he oher hand, one may also argue ha i makes a huge difference wheher he condiions for smoohing effecs and he guaranees are saed in he conrac and individualized or are given in he legislaive environmen by somewha more vague saemens on fairness. One challenge is o incorporae he paricipaing policies in an environmen of finance heory, as has successfully been achieved for uni-linked policies. However, a proper descripion of uni-linked producs in erms of finance heory requires an enlargemen of his environmen. Furhermore, an appropriae enlargemen of his environmen is definiely needed o deal wih he complex naure of paricipaing conracs and he special condiions of he life and pension insurance marke in general. One of he aims of he remaining chapers of his book is o provide he reader wih a box of ools ha can be applied for working wih his challenge wih heoreical subsaniaion.

27 2 Technical reserves and marke values 2.1 Inroducion This chaper deals wih some aspecs of valuaion in life and pension insurance ha are relevan for accouning a marke values. The purpose of he chaper is o demonsrae he rerospecive accumulaion of he echnical reserve and o formalize an approach o prospecive marke valuaion. We explain and discuss he principles underpinning his approach. The exposiion of he maerial disinguishes iself from scienific exposiions of he same subjec, see, for example, Norberg (2000) or Seffensen (2001). By considering firsly he rerospecive accumulaion of echnical reserves, secondly he prospecive approach o marke valuaion and hirdly he underpinning principles, hings are urned somewha upside down here. The aim is o mee he pracical reader a a saring poin wih which he is familiar. The erms prospecive and rerospecive play an imporan role. The idea of a liabiliy as a rerospecively calculaed quaniy needs revision when going from he radiional composiion of he liabiliy o a marke-based composiion of he liabiliy. This is an imporan sep owards comprehending boh he marke-valuaion approach presened here and he generalizaion and improvemen hereof, aking ino consideraion more realisic acuarial and financial modeling. Throughou he chaper, we consider all calculaions peraining o he primary example of an insurance conrac. This primary example is an endowmen insurance wih premium inensiy, pension sum guaraneed a ime 0, b a 0, and guaraneed deah sum, b ad. Bonus is paid ou by increasing he pension sum. The insurance conrac is issued a ime 0 when he insured is x years old and wih a erm of n years. Generalizaions o oher insurance conracs are lef o he reader. Throughou he chaper expenses are disregarded. 11

28 12 Technical reserves and marke values Assume ha deah occurs wih a deerminisic moraliy inensiy s a age x + s, keeping in mind ha his s of course depends on x hen, and assume ha discouning is based on a deerminisic ineres rae r. An imporan quaniy is he discoun facor from u o, exp ( u r s ds ), which we abbreviae by exp ( u r ). If he ineres rae is consan, his discoun facor equals exp r u = v u, where v = exp r. Anoher imporan quaniy is he survival probabiliy from ime o u corresponding o age x+ o age x + u, exp ( u s ds ), which we abbreviae by exp ( u ). The acuarial noaion for such a survival probabiliy is u p x+. We remind he reader abou he following noaion for he capial value a ime of one uni of a pure endowmen insurance, a emporary erm insurance and a emporary life annuiy also used as a premium paymen annuiy: n E x+ = v n n p x+ = e n r+ n A 1 x+n = v s s p x+ s ds = a x+n = n v s s p x+ ds = n n e s r+ s ds e s r+ ds In principle, he quaniies a x+ n and A 1 x+ n should be decoraed wih a bar, bu since only coninuous-ime versions appear in his chaper, his noaion is omied. All quaniies appear, on he oher hand, wih he decoraions and, for example n Ex+ and a x+ n. These refer o he corresponding fundamenal quaniies r and and he conens are obvious from he conex. 2.2 The radiional composiion of he liabiliy The echnical reserve and he second order basis In his secion we sudy he echnical reserve and how i arises from he second order basis. The second order basis is a pair ( r ) conaining he second order ineres rae and he second order moraliy rae by which he echnical reserve accumulaes. We consider an accumulaion of he echnical reserve by he second order basis. This corresponds o a difference equaion wih iniial condiion as follows: V = r V + R (2.1) V 0 = 0

29 2.2 Tradiional composiion of he liabiliy 13 where R is he sum a risk given by R = b ad V and where is he ime uni chosen for he accumulaion. Acually, Equaion (2.1) is a discree-ime version of a corresponding differenial equaion which can be obained by dividing he difference equaion by and leing go o zero. The corresponding differenial equaion wih iniial condiion reads as follows: d d V = r V + R (2.2) V 0 = 0 This differenial equaion can be solved over he ime inerval 0 and he iniial condiion leads o he rerospecive form: V = 0 e s r + ( b ad s ) ds (2.3) The differenial equaion, Equaion (2.2), solved over he ime inerval n leads o he prospecive form, V = b ad A 1 x+ n + V n n E x+ a x+n (2.4) where V n is he erminal value of he echnical reserve. The second order basis is a decision variable held by he insurer ha is o be chosen wihin cerain legislaive consrains and marke condiions. One noes ha in he prospecive form, Equaion (2.4), he second order basis over n appears ogeher wih he erminal value of he echnical reserve V n. Since hese may be unknown a ime, Equaion (2.4) is a represenaion of he soluion o Equaion (2.2), bu no a consrucive ool for calculaion of he echnical reserve. Neverheless, if he erminal value V n is inerpreed as a erminal benefi, he prospecive form expresses he echnical reserve as a prospecive value of all paymens valuaed under he fuure second order basis The echnical reserve and he firs order basis In his secion we sudy he echnical reserve and how i arises from he firs order basis. The firs order basis is a pair r conaining he firs order ineres rae and he firs order moraliy rae under which he guaraneed benefis are se according o he equivalence principle.

30 14 Technical reserves and marke values We consider an accumulaion of he echnical reserve by he firs order basis. This corresponds o a differenial equaion wih iniial condiion as follows: d d V = r V + R + (2.5) V 0 = 0 where he dividend rae is given by = ( r r ) V + ( ) R (2.6) The rae of dividends is deermined such ha he echnical reserve in Secion coincides wih he echnical reserve in his secion. Given a second order basis he rae of dividends is deermined by Equaion (2.6). On he oher hand, given a rae of dividends, any pair ( r ) conforming wih Equaion (2.6) is a candidae for he second order basis. The differenial equaion, Equaion (2.5), solved over 0 leads o he rerospecive form, V = 0 e s r + ( b ad s + s ) ds (2.7) On he basis of he echnical reserve, he pension sum guaraneed a ime is calculaed in accordance wih he equivalence principle, b a = V + a x+ n bad A 1 n E x+ x+ n Hereafer, we can wrie he prospecive form as follows: (2.8) V = b ad A 1 x+ n + ba n E x+ a x+ n (2.9) Noe ha from Equaion (2.8) wegeb a n = V n such ha he echnical reserve a he erminal ime n coincides wih he pension sum a ha ime. This moivaes he inerpreaion of he echnical reserve V n as he erminal benefi a he end of he Secion In he following, we wrie b a n insead of V n when i is appropriae o hink of V n as he pension sum. Noe ha Equaion (2.8) also ses he guaraneed pension sum a ime 0: b a 0 = a x n bad A 1 x n Since b a is calculaed on he basis of he rerospecively derived echnical reserve, we see ha Equaion (2.9) is a represenaion of V bu no a consrucive ool for is derivaion. Neverheless, he prospecive ne x