On Mortality and Investment Risk in Life Insurance

Transcription

1 On Moraliy and Invesmen Risk in Life Insurance Mikkel Dahl Ph.D. Thesis Laboraory of Acuarial Mahemaics Deparmen of Applied Mahemaics and Saisics Insiue for Mahemaical Sciences Faculy of Science Universiy of Copenhagen

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3 On Moraliy and Invesmen Risk in Life Insurance Mikkel Dahl Thesis submied for he Ph.D. degree a he Laboraory of Acuarial Mahemaics Deparmen of Applied Mahemaics and Saisics Insiue for Mahemaical Sciences Faculy of Science Universiy of Copenhagen Ocober 25 Supervisors: Thomas Mikosch Thomas Møller Mogens Seffensen Thesis commiee: Chrisian Hipp, Universiy of Karlsruhe Ragnar Norberg, London School of Economics Rolf Poulsen, Universiy of Copenhagen

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5 Preface This hesis has been prepared in parial fulfillmen of he requiremens for he Ph.D. degree a he Laboraory a Acuarial Mahemaics, Insiue of Mahemaical Sciences, Universiy of Copenhagen, Denmark. The work has been carried ou in he period from November 22 o Ocober 25 under he supervision of Professor Thomas Mikosch, Universiy of Copenhagen, Associae Professor Mogens Seffensen, Universiy of Copenhagen, and Thomas Møller, PFA Pension Assisan Professor a Universiy of Copenhagen unil February 23. In he hesis each chaper is self-conained and can be read independenly of he res of he hesis. This srucure is chosen o ease he submission of pars of he hesis. The independence has resuled in some noaional discrepancies among he differen chapers. The presen version differs from he original version submied for he Ph.D. degree in ha a minor number of misprins have been correced and some saemens, in paricular in Chaper 8, have been clarified. Acknowledgemens Firs I would like o hank Danica Pension, Nordea Pension, Pen-Sam, PFA Pension, PKA and SEB Pension former Codan Pension for financial suppor o wrie his hesis. Also hanks o he Danish Acuarial Associaion for financial help o paricipae in The Nordic Summer School in Insurance Mahemaics 23 and o Knud Højgaards Fond for aid during my say a Universiy of Melbourne. On he personal level I would like o hank Professor Thomas Mikosch for his encouragemen during he las hree years. A remendous hank goes o Thomas Møller and Mogens Seffensen for imporan suggesions and valuable discussions hroughou he erm. Furhermore he inpu from Rolf Poulsen obained hrough numerous discussions is graefully acknowledged. A special hank goes o my friend Carsen Srøh for plowing his way hrough my work in an aemp o improve my English. Furhermore, I am graeful o Professor David Dickson and Professor Ragnar Norberg for heir hospialiy during my visis a Universiy of Melbourne and London School of Economics, respecively. I would also like o hank o my fellow Ph.D. suden Peer Holm Nielsen for many discussions i

6 ii during he pas years and my friend Johannes Müller for answering programming relaed quesions. The willingness of Professor Marin Jacobsen o answer echnical quesions have been remarkable, and I owe him many hanks. Finally, I would also like o ake he opporuniy o personally hank he large number of people from he financially supporing insurance companies, who a one poin or anoher have conribued o discussions in he supervision group: Vivian Weis Byrhol SEB Pension, Torben Dam SEB Pension, Michael Klejs Pen-Sam, Chrisian Kofoed Nordea Pension, Bo Normann Rasmussen PFA Pension, Frank Rasmussen Pen-Sam, Bo Søndergaard Danica Pension, Vibeke Thinggaard PKA. Copenhagen, November 25 Mikkel Dahl

7 Summary This hesis is concerned wih analyzing he risks faced by a life insurance company. In general life insurance companies are exposed o a large number of financial and insurance risks. Usually hese risks are well undersood, and models have been developed and sudied exensively in he lieraure. However, some of he risks have received less aenion boh in he lieraure and in pracice. In his hesis we sudy he modelling of hese risks in deail. An imporan discipline for life insurance companies is o valuae heir liabiliies. We apply mehods from financial mahemaics and in paricular he principle of no arbirage. This principle ress on he reasonable assumpion ha wihou any iniial capial, i is impossible o obain a riskfree gain. In complee financial markes his principle leads o unique prices for all possible conracs. However, since life insurance conracs are no raded in he financial marke, we sudy an incomplee marke, and in his case he no arbirage principle is no sufficien o obain unique arbirage free prices. Hence, in addiion o he no arbirage principle, we consider he mean-variance indifference pricing principles developed in order o obain unique prices in incomplee financial markes. In addiion o valuaing heir liabiliies, life insurance companies are concerned wih possible mehods o decrease heir risk. In his hesis he main emphasis is on he possibiliy of hedging he life insurance conracs in he financial marke. However, oher possibiliies are menioned as well. We apply hedging principles used o deermine opimal hedging sraegies in incomplee financial markes. Here focus is on he crierion of risk-minimizaion and he opimal hedging sraegies associaed wih he mean-variance indifference principles. Risk-minimizing sraegies have he nice propery ha hey decompose he risk associaed wih he conracs ino a hedgeable and an unhedgeable par. In he firs par of he hesis we consider he problem of deermining a fair disribuion of asses beween he equiy capial and he porfolio of insured in he case, where he insurance conracs include a periodic ineres rae guaranee. We sudy a disribuion mechanism, where he equiy capial is accumulaed wih a rae of reurn, which exceeds he riskfree rae, in periods where he combined developmen of he invesmen reurn and he insurance porfolio is favorable. This addiional rae represens he price for he guaranee in he accumulaion period. We consider an insurance company whose insurance porfolio consiss of eiher capial insurances or pure endowmens and a simple financial marke given by he complee and arbirage free Black Scholes model. Given an invesmen sraegy we apply he principle of no arbirage o obain an implici equaion for he fair addiional rae of reurn o he equiy capial in periods, when such an addiional iii

8 iv rae of reurn is possible. In he case of a porfolio of pure endowmens he equaion depends on he marke s aiude owards unsysemaic moraliy risk. The invesmen sraegies considered are: A buy and hold sraegy and a sraegy wih consan relaive porfolio weighs, boh wih and wihou sop-loss in case solvency is hreaened. In he second par we focus on he so-called sysemaic moraliy risk, which is he uncerainy associaed wih he fuure moraliy inensiy. In order o describe his uncerainy we model he moraliy inensiy as a sochasic process. We noe ha he relaive impac of sysemaic moraliy risk canno be reduced by increasing he size of he porfolio. Hence, we canno use he well-esablished acuarial pricing principle of diversificaion o price life insurance conracs in he presence of sysemaic moraliy risk. Insead we apply he no arbirage principle o derive marke reserves. Since he life insurance conracs are no raded in he financial marke, we do no obain a unique marke reserve. In paricular we have ha he marke reserves depend on he marke s aiude owards he sysemaic moraliy risk. In order o obain a unique reserve we apply he mean-variance indifference pricing principles. We sudy differen mehods for he company o lower he exposure o he sysemaic moraliy risk. One possibiliy is o rade in he financial marke. Here, we consider he crieria of risk-minimizaion and he opimal sraegies associaed wih he mean-variance indifference prices. Alernaively, he company can rade so-called moraliy derivaives, i.e. conracs which depend on he developmen of he moraliy inensiy. As a las opion we discuss he possibiliy of ransferring he sysemaic moraliy risk o he insured by issuing conracs, where he premiums and/or benefis are linked o he developmen of he moraliy inensiy. In pracice only bonds wih a limied ime o mauriy are raded in he marke. Hence, companies issuing long erm conracs are exposed o an uncerainy associaed wih he iniial price of a new bond issued in he marke. In he lieraure his risk is usually ignored, since he bond marke is assumed o include bonds wih all imes o mauriy. The hird par of hesis is devoed o he modelling of his so-called reinvesmen risk. For financial conracs he reinvesmen risk is usually non-exising due o he shor erm of he conracs. However, for life insurance companies his risk is of imporance, since life insurance conracs usually are very long erm conracs. We propose a discree-ime model for he reinvesmen risk. A each rading ime a bond maures and a new long erm bond is inroduced in he marke. The enry price of he new bond depends on he prices of exising bonds and a sochasic erm independen of he exising bond prices. Wihin his purely financial model we deermine risk-minimizing sraegies. Danish legislaion force he life insurance companies o value heir long erm liabiliies using a level long erm yield curve. In a numerical example we compare his principle o he relaed principle of a level long erm forward rae curve and he financial principle of super-replicaion. In addiion o he discree-ime model, we also propose a coninuous-ime model wih fixed imes of issue. Here, he uncerainy of he iniial prices of bonds issued in he marke is modelled by leing he exension of he forward rae curve be sochasic. In his case we also derive risk-minimizing sraegies. In he fourh and las par, we consider a model including a large number of he risks faced by a life insurance company. In paricular, his model includes he sysemaic moraliy

9 risk and he reinvesmen risk. Wihin his refined model we deermine marke reserves and mean-variance indifference prices for life insurance conracs. Furhermore he hedging aspec is addressed by he derivaion of risk-minimizing sraegies and he opimal hedging sraegies associaed wih he mean-variance indifference principles. A numerical sudy of marke reserves and he alernaive principles of a level long erm yield curve, a level long erm forward rae curve and super-replicaion of reinvesmen risk is carried ou. This numerical sudy also includes he risk measures of Value a Risk and ail condiional expecaion. v

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11 Resumé I denne afhandling analyseres de forskellige risici som e livsforsikringsselskab er eksponere for. Generel er livsforsikringsselskaber eksponere for e sor anal finansielle og forsikringsmæssige risici. Som regel er der e udbred kendskab il og en indgående forsåelse af disse risici, og der er udvikle modeller, som er sudere dealjere i lierauren. Enkele risici har dog ikke fåe samme opmærksomhed, hverken i lierauren eller i praksis. I denne afhandling foreages e dealjere sudie af modelleringen af disse risici. En vigig opgave for livsforsikringsselskaber er a værdiansæe deres forpligigelser. Vi anvender meoder fra finansmaemaikken og speciel princippe om fravære af arbiragemuligheder. Dee princip bygger på den rimelige anagelse om, a man uden sarkapial ikke kan opnå en risikofri gevins. I fuldsændige finansielle markeder fører dee princip il enydige arbiragefri priser for alle konaker. Da livsforsikringskonraker ikke handles på de finansielle marked, berager vi e ufuldsændig marked, og i dee ilfælde er princippe om fravær af arbirage ikke ilsrækkelig il a sikre enydige arbiragefri priser. Vi berager derfor også mean-variance indifferens prisfassæelses principper udvikle med henblik på a opnå enydige priser i ufuldsændige finansielle markeder. Udover a værdiansæe deres forpligigelser er livsforsikringsselskaber opage af mulige meoder il a mindske deres risiko. I denne afhandling er hovedfokus på muligheden for a hedge afdække livsforsikringskonraker i de finansielle marked, men andre muligheder vil også blive nævn. Vi anvender afdækningsprincipper, som normal anvendes il a besemme opimale handelssraegier i ufuldsændige finansielle markeder. Her er fokus på krierie risiko-minimering og på de opimale handelssraegier forbunde med mean-variance indifferens principperne. Risiko-minimerende sraegier har den pæne egenskab, a de dekomponerer risikoen forbunde med konrakerne i en del som kan elimineres ved a handle på de finansielle marked, og en del som ikke kan elimineres. I den førse del af afhandlingen berager vi probleme med a besemme en fair fordeling af akiverne mellem egenkapialen og forsikringsporeføljen i de ilfælde, hvor forsikringskonakerne indeholder en renegarani. Vi berager en fordelingsmekanisme, hvor egenkapialen forrenes med en rene, der er højere end den risikofri rene i perioder, hvor den samlede udvikling af inveseringerne og forsikringsporeføljen er favorabel. Denne eksra forrenning af egenkapialen repræsenerer prisen for garanien i perioden. Vi berager e forsikringsselskab, hvis forsikringsporefølje udelukkende besår af enen kapialforsikringer eller rene overlevelsesforsikringer og e simpel finansiel marked beskreve ved den fuldsændige og arbiragefri Black Scholes model. For en given inveseringssraegi vii

12 viii anvender vi princippe om fravær af arbirage il a besemme en implici ligning for den eksra forrenning af egenkapialen i perioder, hvor en sådan eksra rene er mulig. I ilfælde hvor forsikringsporeføljen besår af rene oplevelsesforsikringer, afhænger ligningen af markedes aiude il usysemaisk dødsrisiko. Vi berager følgende inveseringssraegier: En buy and hold sraegi og en sraegi med konsane relaive poreføljevæge. I begge ilfælde berages både ilfælde med og uden sop-loss, hvis selskabes solvens er rue. I den anden del fokuserer vi på den såkalde sysemaiske dødsrisiko, som er usikkerheden forbunde med den fremidige dødelighed. For a kunne beskrive denne usikkerhed modellerer vi dødeligheden som en sokasisk proces. Vi bemærker a den relaive effek af den sysemaiske dødsrisiko ikke kan reduceres ved a øge sørrelsen af forsikringsporeføljen. Vi kan derfor ikke benye de veleablerede akuar prisfassæelsesprincip, diversifikaion, il a prisfassæe livsforsikringskonraker i forbindelse med sysemaisk dødsrisiko. I sede anvender vi princippe om fravær af arbirage il a udlede markedsreserver. Da livsforsikringskonraker ikke handles på de finansielle marked, giver dee ikke en enydig markedsreserve. Speciel gælder de, a markedsreserverne afhænger af markedes aiude il sysemaisk dødsrisiko. For a opnå en enydig reserve anvender vi mean-variance indifferens prisfassæelses principperne. Vi berager forskellige meoder for selskabe il a mindske eksponeringen il den sysemaiske dødsrisiko. En mulighed er a handle i de finansielle marked. Her berager vi krierie risiko-minimering og de opimale handelssraegier forbunde med mean-variance indifferens priserne. Alernaiv kan selskabe handle med såkalde dødelighedsderivaer, som er konraker, der afhænger af udviklingen i dødeligheden. Som en sidse mulighed diskuerer vi muligheden for a overføre den sysemaiske dødsrisiko il de forsikrede ved a udsede konraker, hvor præmierne og/eller ydelserne er afhængige af udviklingen i dødeligheden. I praksis handles kun obligaioner med en begrænse løbeid på de finansielle marked. Selskaber, der udseder konraker med lang løbeid, er derfor eksponere il en usikkerhed forbunde med inrodukionsprisen, når nye obligaioner udsedes på de finansielle marked. I lierauren ignoreres denne risiko ypisk, da man anager a obligaioner med alle løbeider handles i markede. Den redje del af afhandlingen behandler modelleringen af denne såkalde geninveseringsrisiko. For ren finansielle konraker er geninveseringsrisikoen normal ikke eksiserende, da de som regel har en kor idshorison. For livsforsikringskonraker er denne risiko imidlerid af sor vigighed, da livsforsikringskonraker generel har en mege lang idshorison. Vi opsiller en diskre-ids model for geninveseringsrisikoen. På ehver handelsidspunk udløber en obligaion og en ny obligaion med lang løbeid udsedes i markede. Udsedelsesprisen afhænger af prisen på de eksiserende obligaioner og e sokasisk led uafhængig heraf. I denne ren finansielle model besemmer vi risiko-minimerende sraegier. Ifølge dansk lovgivning skal livsforsikringsselskaber anvende en renekurve, som er flad ved lange løbeider, il a værdiansæe deres forpligigelser med lang idshorison. I e numerisk eksempel sammenligner vi dee princip med de relaerede princip om a anvende en forwardrenekurve, som er flad for lange løbeider, og de finansielle princip super-replikering. Udover diskreids modellen opsilles en model i koninuer id med fase udsedelsesidspunker. Her modellerer vi usikkerheden forbunde med udsedelsesprisen på nye obligaioner, ved a

13 ix lade forsæelsen af forwardrenekurven være sokasisk. I dee ilfælde udledes også risiko-minimerende sraegier. I den fjerde og sidse del berager vi en model, der inkluderer e sor anal af de risici, som e livsforsikringsselskab er eksponere for. Speciel indeholder modellen sysemaisk dødsrisiko og geninveseringsrisko. I denne forfinede model besemmes markedsreserver og mean-variance indifferens priser for livsforsikringskonraker. Yderligere er afdækningsproblemaikken belys ved udledningen af risiko-minimerende sraegier og de opimale handelssraegier forbunde med mean-variance indifferens principperne. Der foreages e numerisk sudie af markedsreserver og de alernaive principper om en flad renekurve for lange idshorisoner, en flad forwardrenekurve for lange idshorisoner og super-replikering af geninveseringsrisiko. Dee numeriske sudie inkluderer også risikomålene Value a Risk and ail condiional expecaion.

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15 Conens Preface Acknowledgemens i i Summary iii Resumé vii 1 Inroducion Risks in life insurance Types of risk A qualiaive classificaion of he ypes of risk An illusraion of risks in life insurance Tradiional approach o risk in life insurance Financial heory Valuaion and hedging in incomplee markes Applying financial mehods in life insurance Tradiional insurance conracs Uni-linked life insurance Quanifying he ypes of risk Equiy risk xi

16 xii CONTENTS Ineres rae risk Unsysemaic moraliy risk Sysemaic moraliy risk Overview and conribuions of he hesis Fair Disribuion of Asses in Life Insurance Inroducion The balance shee The financial model Capial insurances Disribuion scheme Fair disribuion Buy and hold sraegy Consan relaive porfolio weighs Buy and hold wih sop-loss if solvency is hreaened Consan relaive amoun δ in socks unil solvency is hreaened Pure endowmens The model for he insurance porfolio The combined model The developmen of he deposi in a 1-period model Disribuion scheme Fair disribuion Buy and hold Consan relaive porfolio Buy and hold wih sop-loss if solvency is hreaened Consan relaive amoun δ in socks unil solvency is hreaened.. 53

17 CONTENTS xiii 2.6 Numerical resuls Dependence on invesmen sraegy Dependence on parameers Dependence on iniial disribuion of capial Effec from unsysemaic moraliy risk Impac of alernaive disribuion schemes On he realism and versailiy of he model Conclusion Proofs and echnical calculaions Proof of Proposiion Deermining he limi as U Deermining he limi as Y Proof of Proposiion Sochasic Moraliy in Life Insurance: Marke Reserves and Moraliy- Linked Insurance Conracs Inroducion Exising lieraure on sochasic moraliy Moraliy inensiy as a sochasic process Sochasic versus deerminisic moraliy Affine moraliy srucure Model consideraions Forward moraliy inensiies The model The financial marke The moraliy inensiy The insurance conrac

18 xiv CONTENTS The combined model Change of measure A brief review of financial conceps Marke survival probabiliies Marke Reserves Moraliy-linked conracs Moivaion Pure endowmen Securiizaion of sysemaic moraliy risk Pricing moraliy derivaives Possible ways of hedging Conracs wih a risk premium Dynamics of he benefi wih risky invesmens Valuaion and Hedging of Life Insurance Liabiliies wih Sysemaic Moraliy Risk Inroducion Moivaion and empirical evidence Modelling he moraliy The general model Deerminisic changes in moraliy inensiies Time-inhomogeneous CIR models The financial marke The insurance porfolio The combined model A class of equivalen maringale measures The paymen process

19 CONTENTS xv Marke reserves Risk-minimizing sraegies A review of risk-minimizaion Risk-minimizing sraegies for he insurance paymen process Mean-variance indifference pricing A review of mean-variance indifference pricing The variance opimal maringale measure Mean-variance indifference pricing for pure endowmens Mean-variance hedging Numerical examples Proofs and echnical calculaions Proof of Lemma Calculaion of Var P [N H ] A Discree-Time Model for Reinvesmen Risk in Bond Markes Inroducion A bond marke model A sandard bond marke model A bond marke model wih reinvesmen risk Discree-ime rading Hedging sraegies Super-replicaion Risk-minimizing sraegies A numerical illusraion A Coninuous-Time Model for Reinvesmen Risk in Bond Markes Inroducion

20 xvi CONTENTS 6.2 The bond marke model A sandard model Exending he sandard model o include reinvesmen risk Model consideraions Trading in he bond marke Risk-minimizaion A review of risk-minimizaion for paymen processes Risk-minimizaion in he presence of reinvesmen risk F-risk-minimizing sraegies A pracical implemenaion of he model Valuaion and Hedging of Uni-Linked Life Insurance Conracs Subjec o Reinvesmen and Moraliy Risks Inroducion The sub-models The financial marke Modelling he moraliy The insurance porfolio The combined model A class of equivalen maringale measures The paymen process Marke reserves Trading in he financial marke Risk-minimizaion for uni-linked insurance conracs A review of risk-minimizaion Unhedgeable moraliy risk Unhedgeable moraliy and reinvesmen risks

21 CONTENTS xvii 7.5 Mean-variance indifference pricing A review of mean-variance indifference pricing The variance opimal maringale measure Mean-variance indifference pricing for pure endowmens Proofs and echnical calculaions Proof of Lemma Proof of Proposiion Calculaion of Var P [N H ] A Numerical Sudy of Reserves and Risk Measures in Life Insurance Inroducion The Model The financial marke Modelling he moraliy The insurance porfolio A class of equivalen maringale measures The paymen process Reserving Marke reserves Super-replicaion Alernaive approaches o he reinvesmen risk Risk measures Value a Risk Tail condiional expecaion Numerics Simulaion of Value a Risk and ail condiional expecaion

22 xviii CONTENTS Parameers Numerical resuls Bibliography 243

23 Chaper 1 Inroducion In his hesis we focus on he risks o which an insurance company is exposed when selling life insurance conracs. Here, we use he erm risk o describe a source of uncerainy, even hough i may lead o a surplus as well as a loss. We are ineresed in idenifying and modelling he sources of risk in order o measure and conrol he risk of he insurance company, and o value life insurance conracs. The exposiion relies heavily on mehods from financial mahemaics. In paricular we apply he no arbirage principle and mehods used for valuaion and hedging in incomplee markes. 1.1 Risks in life insurance A life insurance conrac specifies a sream of paymens beween he insured and he insurance company coningen on some predeermined insurance evens. Paymens from he insured are called premiums, and paymens o he insured are referred o as benefis. The premiums usually consis of a lump sum premium a iniiaion of he conrac and coninuous premiums paid unil reiremen as long as he insured is alive and acive. Sandard exbook examples of benefis are: Pure endowmen, erm insurance and emporary or whole life annuiy. For an explanaion of hese insurance conracs and an inroducion o life insurance in general, we refer o Gerber 1997, in discree ime, and Norberg 2, in coninuous ime. When enering he conrac he qualiaive naure of he premiums and benefis is agreed upon by he insured and he insurance company. Furhermore, he insured specifies eiher he premiums or benefis quaniaively, and i is lef o he insurance company o calculae he remaining quaniy. Hence, boh he quaniaive and he qualiaive naure of benefis and premiums are saed in he insurance conrac. Typically he quaniaive specificaions serve as a guaranee o he insured leaving he company unable o lower benefis, or equivalenly increase premiums, if i observes an adverse developmen of he financial marke and/or he insurance porfolio. Thus, since he company is unable o 1

24 2 CHAPTER 1. INTRODUCTION aler he specificaions in he conrac in order o ake an unfavorable developmen of he financial marke and/or he insurance porfolio ino accoun, i is of imporance for he company o undersand he risks associaed wih enering he insurance conrac. Hence, he company should be able o idenify and adequaely model he major sources of risk, such ha i is able o price he conrac correcly. However, an adequae descripion of he risks is no only of imporance when pricing he conrac. I is imporan hroughou he course of he conrac, boh for inernal conrol purposes and for measuring he impac of differen scenarios as described in he so-called raffic ligh sysem inroduced by he regulaory auhoriies in Denmark. Furhermore i is believed ha fuure solvency rules will require he company o consanly monior and measure he risks of he company. Having measured he risks i is naural for he company o consider mehods o reduce he risk, and hereby lower he effec of he differen scenarios in he raffic ligh sysem and in he fuure he solvency requiremens. Here, some possibiliies are rading in he financial marke and purchasing reinsurance. In his chaper we consider he case where he insured specify he benefis quaniaively, and he company has o calculae he premiums. This is no loss of generaliy, since he alernaive case can be handled similarly. Throughou he chaper we resric calculaions o he case of a porfolio of pure endowmens paid by single premiums, since hese are he simples life insurance conrac involving a dependence on he deah or survival of he insured. In paricular his allows us o consider benefis a a fixed ime only, such ha we avoid considering paymen processes. However, all qualiaive saemens in his chaper hold for paymen processes as well Types of risk We focus on wo main ypes of risk for he insurance company: Financial risk and moraliy risk. In he lieraure moraliy risk is someimes referred o as insurance risk. The company is naurally exposed o oher ypes of risk as well. We menion operaional risk and risk associaed wih fuure adminisraion coss, such as wages, purchase of compuer sysems, ren and general mainenance of business operaions. The Basel Commiee s definiion of operaional risk is he risk of losses resuling from inadequae or failed inernal processes, people and sysems or from exernal evens. Hence, he operaional risk covers all losses resuling from errors conneced o running he business. This includes boh human and sysem errors. For a deailed descripion of, and an approach o modelling, operaional risk we refer o King 21 and Cruz 22. Here, we furher spli he financial risk ino equiy risk and ineres rae risk. Hence, in his exposiion we disregard oher ypes of risk, such as credi risk, which is he risk associaed wih he defaul of he counerpary in a financial ransacion. For a deailed descripion of credi risk see e.g. Lando 24. The equiy risk covers he uncerainy associaed wih risky invesmens excep bonds, and ineres rae risk covers uncerainy associaed wih fuure ineres raes and hence bond prices. Here, we furher divide he ineres rae risk ino sandard ineres rae risk, which is uncerainy associaed wih he developmen of he currenly raded bonds he currenly observable yield curve and reinvesmen risk, which

25 1.1. RISKS IN LIFE INSURANCE 3 measures he addiional uncerainy associaed wih he enry prices, when new bonds are issued in he marke. The reinvesmen risk is naurally only of ineres if bonds wih sufficienly long ime o mauriy are no raded a he ime of consideraion. This use of he erm reinvesmen risk differs from he one of e.g. Luenberger 1998, who uses i o describe he risk associaed wih he unknown rae of reurn, when currenly owned bonds maure in he fuure, and he capial is reinvesed in he bond marke. Hence, Luenberger 1998 does no disinguish beween wheher or no he bonds in which he capial is reinvesed were raded a he ime of purchase of he firs bonds. In our erminology, he reinvesmen risk only covers he case, where no bonds wih sufficienly long ime horizon are raded iniially, whereas he risk associaed wih he fuure rae of reurn of bonds presenly raded is covered by he sandard ineres rae risk. The moraliy risk consiss of wo fundamenally differen sources of risk: Sysemaic and unsysemaic moraliy risk. Here, he unsysemaic moraliy risk refers o he risk associaed wih he random developmen of an insurance porfolio wih known moraliy inensiy. From he srong law of large numbers we know ha he relaive impac of he unsysemaic moraliy risk is a decreasing funcion of he number of insured, and if he insurance porfolio is infiniely large, he unsysemaic moraliy risk is eliminaed. Thus, he unsysemaic moraliy risk is diversifiable. The sysemaic moraliy risk refers o he uncerainy associaed wih changes in he underlying moraliy inensiy. Since changes in he underlying moraliy inensiy affec all insured, he sysemaic moraliy risk is an increasing funcion of he number of insured wih similar conracs. Hence, in conras o he unsysemaic moraliy risk he sysemaic moraliy risk is non-diversifiable. However, a reducion eliminaion of he sysemaic moraliy risk is possible, if he company boh sells conracs, where he payoff is coningen on survival, and conracs, where he payoff is coningen on deah. Noe ha similar consideraions can be made for oher ransiion inensiies, e.g. disabiliy, recovery ec. Hence, we can inerpre he moraliy risk as covering all biomeric risks A qualiaive classificaion of he ypes of risk In order o obain a qualiaive descripion of he ypes of risk we classify hem according o he exposure of he company. Firs we concenrae on he conrac and classify he ypes of risk according o wheher he company is exposed o he risk as a consequence of enering he conrac. Hence, he differen ypes of risk are divided ino he following wo classes: Conracual risks: The ypes of risk o which he insurance company is exposed as a consequence of enering he conrac. Non-conracual risks: The ypes of risk, which are no conracual risks. Noe ha if enering he conrac does no expose he company o a ype of risk, i is a non-conracual risk. Wihin he class of conracual risks we furher disinguish beween

26 4 CHAPTER 1. INTRODUCTION wheher he company is able o eliminae he ype of risk by rading in he financial marke. We say ha a ype of risk can be eliminaed if all uncerainy associaed wih he ype of risk can be eliminaed by rading in he financial marke. To deermine wheher his is he case, we consider he coningen model where he paricular ype of risk accouns for all uncerainy. Now all uncerainy can be eliminaed if he company can inves a fixed iniial amoun and rade in he financial marke, such ha i always has exacly he desired amoun. Hence, he class of conracual risks consiss of he following sub-classes: Hedgeable conracual risks: The ypes of conracual risk for which he company, given a cerain fixed iniial invesmen, can eliminae all uncerainy by rading in he financial marke. Unhedgeable conracual risks: The ypes of conracual risk for which he company, given a cerain fixed iniial invesmen, canno eliminae all uncerainy by rading in he financial marke. The definiions above are closely relaed o he definiion of hedging in financial heory, see Secion 1.3 for more deails. Here, i is imporan o noe ha in he coningen model i may be possible o eliminae he so-called shor-fall risk, which is he risk of holding insufficien funds, relaed o an unhedgeable conracual risk by invesing a sufficienly large amoun a iniiaion of he conrac. However, since he company in his case has a large posiive probabiliy of holding more han required o cover he benefis, he risk is no eliminaed. Hence, one canno urn an unhedgeable conracual risk ino a hedgeable conracual risk by invesing a sufficienly large amoun. The idea of eliminaing he shorfall risk is closely relaed o so-called super-replicaing super-hedging sraegies, see Secion The classes and sub-classes above are conneced o he conrac only, so i can be inerpreed as a classificaion of he risks on he liabiliy side. However, i is imporan o noe ha he effec of he differen ypes of risk on he balance shee depends on boh he considered insurance conrac and he invesmen sraegy. Hence, in order o correcly describe he exposure of he company o he differen ypes of risk, one should involve he asse side as well. Here, he asses only refer o he asses associaed wih he liabiliies, whereas he asses corresponding o he equiy capial is disregarded. The imporance of including he asses has also been observed by he life insurance companies, which in general devoe a large amoun of effor o ALM asse liabiliy modelling/managemen. The necessiy o involve he asse allocaion arises since he value of he asses and liabiliies may increase or decrease a he same ime. Hence, in some cases he company may be able o reduce a ype of unhedgeable conracual risk by raded wisely in he financial marke. On he oher hand he company may decide no o eliminae he uncerainy associaed wih a hedgeable conracual risk. I may even expose he balance shee o non-conracual risks. In order o describe he ypes of risk o which he insurance company is exposed, when aking he asse allocaion ino accoun, we inroduce: Business risks: The ypes of risk o which he combined balance shee of he company is exposed.

27 1.1. RISKS IN LIFE INSURANCE 5 As noed above i holds ha even for a company, which is aware of he conracual risks o which i is exposed, he mere possibiliy o eliminae or reduce a ype of risk by rading in he financial marke is no equivalen o he fac ha he company acually decides o do so. Hence, in some cases he company exposes he balance shee o risks ha could have been avoided. This behavior can be explained by he fac ha he company follows an invesmen sraegy which also focuses on he expeced rae of reurn. In paricular, he belief ha he long erm reurn is higher on socks han on bonds encourages many insurance companies o inves in socks even when he financial risk associaed wih he conrac only consiss of ineres rae risk. For a specific conrac or porfolio of conracs and a given invesmen sraegy he business risks consis of he following hree sub-classes: Non-hedged hedgeable conracual risks: The ypes of hedgeable conracual risk, which he company has no eliminaed. Unhedgeable conracual risks: The ypes of conracual risk for which he company, given a cerain fixed iniial invesmen, canno eliminae all uncerainy by rading in he financial marke. Gambling risks: Non-conracual risks o which he company is exposed as a consequence of he invesmen sraegy. The mehod available o he company in order o eliminae/conver a cerain ype of business risk depends on he sub-class o which he risk belongs. A non-hedged hedgeable conracual risk can by definiion be eliminaed by rading in he financial marke, whereas i is impossible o eliminae an unhedgeable conracual risk once he conrac is signed. However, risks, which oherwise would be unhedgeable conracual risks may be ransferred o he insured and hus hey may be convered ino non-conracual risks by designing he conrac cleverly. The so-called moraliy-linked conracs inroduced in Chaper 3 is an example of a ype of conracs designed o conver an unhedgeable conracual risk ino a non-conracual risk. Here, he sysemaic moraliy risk is ransferred from he insurance company o he insured. The gambling risks can naurally be eliminaed simply by alering he invesmen sraegy, such ha is does no include invesmens in he asses which expose he company o he non-conracual risk. The classificaion of he risks is of imporance when pricing and reserving for he conrac, as well as for risk managemen. The conracual risks influence prices and reserves, whereas he business risks influence he sensiiviy o he differen scenarios in e.g. he raffic ligh sysem and in he fuure possibly he solvency requiremens. I could be argued ha he invesmen sraegy, and hence he business risks, also should be of imporance when pricing he conrac, since a company which follows a risky invesmen sraegy has a larger risk of defaul and hence exposes he policy-holder o a larger credi risk. However, we ignore his aspec, since he raffic ligh sysem and he solvency rules essenially should eliminae he credi risk of he policy-holders.

28 6 CHAPTER 1. INTRODUCTION An illusraion of risks in life insurance In order o illusrae he ideas in Secions and we now idenify and classify qualiaively he differen ypes of risk in a simple example. Consider a porfolio of n insured of age x all purchasing a pure endowmen of K paid by a single premium π a ime. Hence, a ime he company receives he premium π from each of he insured, such ha he oal premiums received are nπ, and a he ime of mauriy, T, he surviving policy-holders receive K. Le NT denoe he number of deahs in he porfolio unil ime T. Hence, he number of survivors is given by n NT, such ha he oal benefis o he policy-holders are H = n NTK. Here, we firs idenify and classify he ypes of risk associaed wih he conracual paymens described above. In order o idenify possible financial risks, we assume he random course of he insured lives are known, such ha he number of survivors a ime T, and hence he benefis, are known a ime. Since K is a fixed benefi, no specific dependence on socks is saed in he conrac, so he equiy risk is a non-conracual risk. Hence, among he financial risks only he ineres rae risks may be conracual risks. To classify he ineres rae risks we disinguish beween wheher he ime of mauriy of he insurance conrac lies before or afer he ime of mauriy of he longes bond raded a ime. In he firs case he sandard ineres rae risk is a conracual risk and he reinvesmen risk is a non-conracual risk, whereas boh ineres rae risks are conracual risks in he second case. The company is able o eliminae he conracual ineres rae risks if i can inves a fixed amoun a ime and rade in he financial marke, such ha i is cerain o hold n NTK a ime T. This is he case if here exiss a so-called zero coupon bond a bond, which always pays one a ime of mauriy wih he same ime of mauriy as he insurance conrac, since purchasing n NTK zero coupon bonds a ime leaves he company wih exacly n NTK a ime T. This siuaion corresponds o he firs case wih sufficienly long bonds. Hence, in his case he sandard ineres rae risk is a hedgeable conracual risk and he reinvesmen risk is a non-conracual risk. If, on he oher hand, boh he sandard ineres rae risk and he reinvesmen risk are conracual risks, hen he company is unable o pursue an invesmen sraegy which guaranees exacly n NTK a ime T. Hence, in his case a leas one of he ineres rae risks is an unhedgeable conracual risk. Considering he wo ypes of ineres rae risk separaely, we find ha he sandard ineres rae risk sill is hedgeable, since he risk associaed wih he movemen of he bond prices beween he imes of issue of new bonds can be eliminaed by rading in he bonds. On he conrary he reinvesmen risk canno be eliminaed by rading in bonds already in he marke, such ha i is an unhedgeable conracual risk. In order o idenify and classify he moraliy risks we consider he coningen model where he fuure sock and bond prices are known. As menioned in Secion 1.1.1, he uncerainy regarding he number of survivors a ime T can be sli ino unsysemaic and sysemaic moraliy risk. Here, we firs urn our aenion o he unsysemaic moraliy risk. Hence, we assume ha he fuure moraliy inensiy is known and consider he

29 1.2. TRADITIONAL APPROACH TO RISK IN LIFE INSURANCE 7 uncerainy associaed wih he number of survivors. In his case, he survival probabiliy of each individual is known, and we know from a diversificaion argumen ha in a large porfolio he number of survivors is approximaely equal o he expeced number of survivors given by he produc of he survival probabiliy and he number of insured. However, since he size of he porfolio is finie he number of survivors is no exacly equal o he expeced number of survivors. Hence, in his case he unsysemaic moraliy risk accouns for he uncerainy associaed wih he number of survivors a ime T given he underlying moraliy inensiy. In addiion o he unsysemaic moraliy risk he company is exposed o a risk associaed wih he acual developmen of he moraliy inensiy, he so-called sysemaic moraliy risk. Here, he company will experience a surplus loss if he moraliy inensiy increases decreases more han expeced, such ha he realized expeced number of survivors is lower higher han he expeced number of survivors calculaed a ime. Since we assume ha he financial marke only consiss of bonds and socks, he company is unable o eliminae he uncerainy associaed wih he number of survivors by rading in he financial marke. Hence, he moraliy risks are unhedgeable conracual risks. Thus, when considering a porfolio of pure endowmens wih fixed benefis, he conracual risks include boh ypes of moraliy risk and sandard ineres rae risk, whereas he equiy risk in a non-conracual risk. Wheher he reinvesmen risk is a conracual or non-conracual risk depends on he ime o mauriy of he bonds compared o he ime o mauriy of he conracs. The sandard ineres rae risk is a hedgeable conracual risk and he moraliy risks are unhedgeable conracual risks. If he reinvesmen risk is a conracual risk i is an unhedgeable conracual risk. As noed in Secion 1.1.2, he class of business risks depends on he invesmen sraegy. In order o illusrae his dependence we consider wo differen invesmen sraegies. Firs consider he case where he company eliminaes he sandard ineres rae risk by invesing in bonds. In his case he business risks consiss of he unhedgeable conracual risks: The moraliy risks and possibly he reinvesmen risk. The sandard ineres rae risk, which is he only hedgeable conracual risk has been eliminaed so here are no non-hedged conracual risks, and since he company does no inves in socks, here are no gambling risks. As a second example we consider he case where he company invess in a mixure of bonds and socks, such ha is does no enirely eliminae he sandard ineres rae risk, and hus, since he classificaion is qualiaive, he sandard ineres rae risk is a non-hedged conracual risk. The unhedgeable conracual risks are independen of he invesmen sraegy, so he sub-class is unalered. The invesmen in socks inroduces equiy risk as a gambling risk. Hence, in his second example all ypes of risk are business risks. 1.2 Tradiional approach o risk in life insurance Tradiional life insurance conracs include some guaraneed benefis. The wo mos common ypes of guaranees are mauriy guaranees and periodic ineres rae guaranees. A

30 8 CHAPTER 1. INTRODUCTION mauriy guaranee saes a minimal benefi, whereas a periodic ineres rae guaranee saes a minimum reurn in each accumulaion period. In order o calculae he premiums, he insurance company ypically applies he principle of equivalence using a consan ineres rae, r, and a deerminisic moraliy inensiy, µ, which is independen of calender ime henceforh referred o as ime-independen. The pair r, µ is usually referred o as he echnical basis or he firs order basis, see e.g. Norberg The principle of equivalence saes ha he expeced value of he discouned guaraneed benefis and premiums mus be equal. Hence, in he case of deerminisic guaraneed benefis he calculaions depend on he specificaion of he fuure ineres rae and moraliy inensiy. Since he echnical basis is deerminisic, he derivaion of he premiums is paricularly simple. For a conrac wih deerminisic benefis he calculaions necessary o deermine a lump sum premium simply requires he company firs o replace he uncerain course of he random life by he expeced developmen using he echnical moraliy inensiy and second o deermine he presen value of he resuling deerminisic benefis using he echnical ineres rae. Consider a porfolio consising of n pure endowmens wih guaraneed benefis K paid by a single premium. In his case, he individual premium calculaed by he principle of equivalence using he echnical basis is given by π = e ÊT µx+udu e rt K Here, µx + is he echnical moraliy inensiy a ime for a person aged x a ime, where he conrac was issued. Hence, exp T µx + udu is he survival probabiliy for a person of age x from ime o T using he echnical basis. The basic idea in radiional risk managemen in life insurance is o choose he echnical basis o he safe side, as seen from he company s poin of view, such ha he fuure ineres rae and porfolio-wide moraliy inensiy never behaves worse again seen from he company s poin of view han he echnical basis. In he case of a pure endowmen his corresponds o applying a echnical ineres rae and moraliy inensiy, which are oo low. Thus, a any ime he reserve V = e ÊT µx+udu e rt K, is on average more han sufficien o cover he guaraneed benefi K a ime T given survival unil ime. However, he company receives he porfolio-wide premium nπ a ime, which i invess in he financial marke. If we le r denoe he rae of reurn obained by he company, he porfolio-wide asses a ime are given by nπ exp rudu. If we furher denoe by µx,u he observed moraliy inensiy in he porfolio a ime u for an insured of age x a ime, hen he observed number of survivors a ime is given by n exp µx,udu. So he asses per survivor are π exp ru + µx,udu. Now he choice of echnical basis ensures ha he company is able o choose an invesmen sraegy, such ha he individual asses are sufficien o cover he reserve calculaed wih he echnical basis, i.e. πeê ru+ µx,udu V.