On Mortality and Investment Risk in Life Insurance

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

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

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

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

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

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

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

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

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

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.

Conens Preface Acknowledgemens................................... i i Summary iii Resumé vii 1 Inroducion 1 1.1 Risks in life insurance.............................. 1 1.1.1 Types of risk............................... 2 1.1.2 A qualiaive classificaion of he ypes of risk............. 3 1.1.3 An illusraion of risks in life insurance................ 6 1.2 Tradiional approach o risk in life insurance................. 7 1.3 Financial heory................................. 1 1.3.1 Valuaion and hedging in incomplee markes............. 12 1.4 Applying financial mehods in life insurance.................. 16 1.4.1 Tradiional insurance conracs..................... 16 1.4.2 Uni-linked life insurance........................ 18 1.5 Quanifying he ypes of risk.......................... 2 1.5.1 Equiy risk................................ 21 xi

xii CONTENTS 1.5.2 Ineres rae risk............................. 21 1.5.3 Unsysemaic moraliy risk....................... 22 1.5.4 Sysemaic moraliy risk........................ 23 1.6 Overview and conribuions of he hesis.................... 23 2 Fair Disribuion of Asses in Life Insurance 27 2.1 Inroducion.................................... 27 2.2 The balance shee................................ 3 2.3 The financial model............................... 31 2.4 Capial insurances................................ 32 2.4.1 Disribuion scheme........................... 33 2.4.2 Fair disribuion............................. 35 2.4.3 Buy and hold sraegy.......................... 35 2.4.4 Consan relaive porfolio weighs................... 39 2.4.5 Buy and hold wih sop-loss if solvency is hreaened......... 4 2.4.6 Consan relaive amoun δ in socks unil solvency is hreaened.. 43 2.5 Pure endowmens................................. 44 2.5.1 The model for he insurance porfolio................. 44 2.5.2 The combined model........................... 45 2.5.3 The developmen of he deposi in a 1-period model......... 46 2.5.4 Disribuion scheme........................... 46 2.5.5 Fair disribuion............................. 47 2.5.6 Buy and hold............................... 48 2.5.7 Consan relaive porfolio........................ 49 2.5.8 Buy and hold wih sop-loss if solvency is hreaened......... 5 2.5.9 Consan relaive amoun δ in socks unil solvency is hreaened.. 53

CONTENTS xiii 2.6 Numerical resuls................................. 54 2.6.1 Dependence on invesmen sraegy.................. 54 2.6.2 Dependence on parameers....................... 57 2.6.3 Dependence on iniial disribuion of capial.............. 59 2.6.4 Effec from unsysemaic moraliy risk................ 6 2.7 Impac of alernaive disribuion schemes................... 61 2.8 On he realism and versailiy of he model.................. 62 2.9 Conclusion.................................... 64 2.1 Proofs and echnical calculaions........................ 64 2.1.1 Proof of Proposiion 2.4.4........................ 64 2.1.2 Deermining he limi as U................... 65 2.1.3 Deermining he limi as Y.................... 66 2.1.4 Proof of Proposiion 2.5.8........................ 67 3 Sochasic Moraliy in Life Insurance: Marke Reserves and Moraliy- Linked Insurance Conracs 71 3.1 Inroducion.................................... 71 3.2 Exising lieraure on sochasic moraliy................... 73 3.3 Moraliy inensiy as a sochasic process................... 74 3.3.1 Sochasic versus deerminisic moraliy................ 74 3.3.2 Affine moraliy srucure........................ 76 3.3.3 Model consideraions........................... 78 3.3.4 Forward moraliy inensiies...................... 78 3.4 The model..................................... 79 3.4.1 The financial marke........................... 79 3.4.2 The moraliy inensiy......................... 8 3.4.3 The insurance conrac.......................... 81

xiv CONTENTS 3.4.4 The combined model........................... 81 3.4.5 Change of measure............................ 82 3.4.6 A brief review of financial conceps................... 84 3.4.7 Marke survival probabiliies...................... 85 3.5 Marke Reserves................................. 86 3.6 Moraliy-linked conracs............................ 89 3.6.1 Moivaion................................ 89 3.6.2 Pure endowmen............................. 91 3.7 Securiizaion of sysemaic moraliy risk................... 95 3.7.1 Pricing moraliy derivaives...................... 96 3.7.2 Possible ways of hedging......................... 97 3.7.3 Conracs wih a risk premium..................... 97 3.8 Dynamics of he benefi wih risky invesmens................ 99 4 Valuaion and Hedging of Life Insurance Liabiliies wih Sysemaic Moraliy Risk 11 4.1 Inroducion.................................... 11 4.2 Moivaion and empirical evidence....................... 13 4.3 Modelling he moraliy............................. 15 4.3.1 The general model............................ 15 4.3.2 Deerminisic changes in moraliy inensiies............. 16 4.3.3 Time-inhomogeneous CIR models.................... 16 4.4 The financial marke............................... 17 4.5 The insurance porfolio............................. 11 4.6 The combined model............................... 11 4.6.1 A class of equivalen maringale measures............... 111 4.6.2 The paymen process........................... 113

CONTENTS xv 4.6.3 Marke reserves.............................. 113 4.7 Risk-minimizing sraegies............................ 115 4.7.1 A review of risk-minimizaion...................... 116 4.7.2 Risk-minimizing sraegies for he insurance paymen process.... 117 4.8 Mean-variance indifference pricing....................... 119 4.8.1 A review of mean-variance indifference pricing............ 12 4.8.2 The variance opimal maringale measure............... 121 4.8.3 Mean-variance indifference pricing for pure endowmens....... 122 4.8.4 Mean-variance hedging.......................... 123 4.9 Numerical examples............................... 124 4.1 Proofs and echnical calculaions........................ 128 4.1.1 Proof of Lemma 4.7.1.......................... 128 4.1.2 Calculaion of Var P [N H ]......................... 13 5 A Discree-Time Model for Reinvesmen Risk in Bond Markes 133 5.1 Inroducion.................................... 133 5.2 A bond marke model.............................. 135 5.2.1 A sandard bond marke model..................... 135 5.2.2 A bond marke model wih reinvesmen risk............. 138 5.2.3 Discree-ime rading........................... 142 5.3 Hedging sraegies................................ 144 5.3.1 Super-replicaion............................. 144 5.3.2 Risk-minimizing sraegies........................ 149 5.4 A numerical illusraion............................. 154 6 A Coninuous-Time Model for Reinvesmen Risk in Bond Markes 159 6.1 Inroducion.................................... 159

xvi CONTENTS 6.2 The bond marke model............................. 161 6.2.1 A sandard model............................ 161 6.2.2 Exending he sandard model o include reinvesmen risk..... 162 6.2.3 Model consideraions........................... 167 6.2.4 Trading in he bond marke....................... 169 6.3 Risk-minimizaion................................ 171 6.3.1 A review of risk-minimizaion for paymen processes......... 171 6.3.2 Risk-minimizaion in he presence of reinvesmen risk........ 172 6.3.3 F-risk-minimizing sraegies....................... 176 6.4 A pracical implemenaion of he model.................... 182 7 Valuaion and Hedging of Uni-Linked Life Insurance Conracs Subjec o Reinvesmen and Moraliy Risks 183 7.1 Inroducion.................................... 184 7.2 The sub-models.................................. 185 7.2.1 The financial marke........................... 185 7.2.2 Modelling he moraliy......................... 188 7.2.3 The insurance porfolio......................... 189 7.3 The combined model............................... 19 7.3.1 A class of equivalen maringale measures............... 19 7.3.2 The paymen process........................... 193 7.3.3 Marke reserves.............................. 194 7.3.4 Trading in he financial marke..................... 195 7.4 Risk-minimizaion for uni-linked insurance conracs............. 196 7.4.1 A review of risk-minimizaion...................... 196 7.4.2 Unhedgeable moraliy risk....................... 197 7.4.3 Unhedgeable moraliy and reinvesmen risks............. 199

CONTENTS xvii 7.5 Mean-variance indifference pricing....................... 21 7.5.1 A review of mean-variance indifference pricing............ 21 7.5.2 The variance opimal maringale measure............... 22 7.5.3 Mean-variance indifference pricing for pure endowmens....... 23 7.6 Proofs and echnical calculaions........................ 25 7.6.1 Proof of Lemma 7.4.2.......................... 25 7.6.2 Proof of Proposiion 7.4.5........................ 27 7.6.3 Calculaion of Var P [N H ]......................... 28 8 A Numerical Sudy of Reserves and Risk Measures in Life Insurance 215 8.1 Inroducion.................................... 215 8.2 The Model.................................... 216 8.2.1 The financial marke........................... 217 8.2.2 Modelling he moraliy......................... 219 8.2.3 The insurance porfolio......................... 22 8.2.4 A class of equivalen maringale measures............... 22 8.2.5 The paymen process........................... 221 8.3 Reserving..................................... 222 8.3.1 Marke reserves.............................. 222 8.3.2 Super-replicaion............................. 222 8.3.3 Alernaive approaches o he reinvesmen risk............ 223 8.4 Risk measures................................... 226 8.4.1 Value a Risk............................... 226 8.4.2 Tail condiional expecaion....................... 229 8.5 Numerics..................................... 23 8.5.1 Simulaion of Value a Risk and ail condiional expecaion..... 23

xviii CONTENTS 8.5.2 Parameers................................ 231 8.5.3 Numerical resuls............................. 233 Bibliography 243

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

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. 1.1.1 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

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. 1.1.2 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

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 1.3.1. 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.

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.

6 CHAPTER 1. INTRODUCTION 1.1.3 An illusraion of risks in life insurance In order o illusrae he ideas in Secions 1.1.1 and 1.1.2 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

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

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 1999. 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. 1.2.1 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.

1.2. TRADITIONAL APPROACH TO RISK IN LIFE INSURANCE 9 One such sraegy is o inves in very shor erm bonds. Here, he rae of reurn is r, which by assumpion is larger han or equal o r. Thus, he company is able o generae a sysemaic surplus by obaining an invesmen reurn which always exceeds he echnical ineres rae and by observing a moraliy inensiy in he porfolio larger han he echnical moraliy inensiy. Noe ha he company measures he surplus generaed by he moraliy by comparing he observed moraliy in he porfolio wih he echnical moraliy inensiy, so no disincion is made beween he sysemaic and unsysemaic moraliy risk. In Danish legislaion he so-called conribuion principle saes ha a sysemaic surplus mus be reurned o he group of insured, and he disribuion mechanism should ake he conribuion of each individual o he surplus ino accoun. This reurn of surplus o he insured is usually referred o as bonus, see Norberg 1999. Tradiionally bonus has been used o purchase addiional coverage calculaed using he echnical basis. Thus, he guaraneed benefis are increased during he course of he conrac as bonus is allocaed o he individual insured. However, legislaion does no prescribe when he surplus mus be reurned as bonus, and since he disribuion mechanism is no specified in he conrac eiher, i is lef o he company as a decision parameer wihin some legislaive bounds. Hence, a company may follow a very aggressive conservaive bonus sraegy by reurning bonus immediaely as lae as possible. The surplus no disribued o he individual insured is kep by he company as a porfolio-wide buffer. In he lieraure his buffer is ofen referred o as he bonus reserve and in recen Danish legislaion i is known as he collecive bonus poenial. This buffer serves wo purposes. Firsly, i is used o cover he defici in he case, where he company, in an aemp o maximize he invesmen reurn, invess in socks as well as bonds, and observe an invesmen reurn below he echnical ineres rae. Secondly, he buffer serves o smooh he bonus o he insured over he course of he conrac, such ha he insured observe a seady developmen of he individual accoun. We noe ha he legislaive bounds on he reurn mechanism allows for some redisribuion among he differen generaions. Hence, he acual benefis depend on he invesmen reurn obained by he company, he realized developmen of he insurance porfolio, he compeiion in he marke and he capial of he company a iniiaion of he conrac. Summing up we conclude ha he radiional mehod of risk managemen consiss of collecing premiums, which in all circumsances are sufficien o cover he guaraneed benefis and o redisribue he observed surplus among he insured as bonus. Thus, he radiional risk managemen can be viewed as a saic risk managemen, which works as long as he echnical basis really is o he safe side. However, boh he ineres rae and he number of survivors are sochasic, so a naural quesion is wheher he company really is able o deermine a deerminisic ineres rae and moraliy inensiy, such ha he observed quaniies a all imes are o he safe side. In he case of a pure endowmen his could be obained by using an ineres rae and a moraliy inensiy of zero. However, he guaranee should be of ineres o he insured, since hey oherwise would seek alernaive mehods for saving o reiremen, and his is no he case if he guaranee is calculaed wih ineres rae and moraliy inensiy zero. Thus, compeiion forces he companies o calculae guaranees using sricly posiive echnical elemens, and hese are no o he safe side for all possible fuure scenarios. Moreover, he fac ha he companies ofen allocae some of he surplus as bonus immediaely afer i is observed implies ha even

1 CHAPTER 1. INTRODUCTION hough he premium includes a large safey loading, he company may find iself unable o cover he guaranees laer. This problem is furher enlarged by he use of bonus o calculae addiional guaraneed benefis on he echnical basis. The laer problem seems o be oudaed or a leas reduced, since recen addiional benefis ypically eiher are unguaraneed or calculaed using an ineres rae lower han he echnical ineres rae. The magniude of he risk of a company following a radiional risk managemen approach naurally depends on he echnical basis, he aggressiveness of he bonus sraegy and he invesmen sraegy. 1.3 Financial heory In his secion we describe he approach of modern financial mahemaics o risk. Some recen sandard references are Musiela and Rukowski 1997 and Björk 24. Le T denoe a fixed finie ime horizon and consider a financial marke consising of d+1 raded asses: A savings accoun earning a possibly sochasic rae of ineres and d risky asses socks, bonds, real esae ec.. The price processes, which are given by B = B T and X = X T, respecively, are defined on a probabiliy space Ω, F,P wih filraion F = F T. Here, F can be inerpreed as he informaion available a ime. This covers informaion regarding he price processes, and may in general include oher informaion as well. A rading sraegy is a process ϕ = ϑ,η saisfying cerain inegrabiliy condiions. Here, ϑ is predicable, and η is adaped o he filraion F. The pair ϕ = ϑ,η is inerpreed as he porfolio held a ime. Here, ϑ is a d-dimensional vecor denoing he number of he d risky asses in he porfolio, whereas η is he discouned deposi in he savings accoun. The value process Vϕ associaed wih ϕ is defined by A rading sraegy is called self-financing if V,ϕ = ϑx + ηb, T. V,ϕ = V,ϕ + ϑudxu + ηudbu. 1.3.1 Hence, he value of he porfolio a ime is he iniial value V, ϕ added rading gains, ϑudxu, and ineres on he savings accoun, ηudbu. Thus, no in- or ouflow of capial o/from he porfolio has occurred in, ]. A self-financing sraegy is a socalled arbirage if V,ϕ = and VT,ϕ P-a.s. wih PVT,ϕ > >. Thus, an arbirage is he possibiliy wihou an iniial invesmen o obain a riskfree gain. If he model allows for arbirage possibiliies an invesor has a posiive probabiliy o become infiniely rich, wihou risking any money. Hence, in pracice all invesors would pursue such a sraegy and hereby force prices o correc hemselves, such ha no arbirage possibiliies would exis. Thus, a reasonable model is arbirage free. A coningen claim H wih mauriy T is an FT-measurable random variable, i.e. he value of H is known a ime T. If H only depends on he erminal value of he price

1.3. FINANCIAL THEORY 11 processes i is called a simple coningen claim. The coningen claim H is called aainable if here exiss a self-financing rading sraegy ϕ H such ha VT,ϕ H = H P-a.s. The sraegy ϕ H is called he perfec replicaing hedging sraegy for H. From 1.3.1 we see ha H is aainable if and only if here exiss a self-financing sraegy ϕ H such ha H = V,ϕ H + T ϑ H udxu + T η H udbu. 1.3.2 If on he oher hand no perfec replicaing sraegy exiss, H is called unaainable. If all coningen claims are aainable he model is complee, and oherwise i is called incomplee. Here, we noe ha if he marke is incomplee here are infiniely many unaainable claims. To observe his we assume ha he claim H is unaainable. However, if H is unaainable, hen i follows from 1.3.2 ha ch is unaainable for c R\{}. If his was no he case we could perfecly replicae H by he sraegy ϕ H = ϕ ch /c, where ϕ ch is he perfec replicaing sraegy for ch. The fundamenal pricing principle in financial mahemaics is he no arbirage principle due o Black and Scholes 1973 and Meron 1973. The no arbirage principle saes ha he financial marke sill should be arbirage free afer he inroducion of a new asse. Hence, for an aainable claim H wih replicaing sraegy ϕ H, he unique arbirage free price is given by V,ϕ H, since his is he only price which excludes arbirage possibiliies, see e.g. Møller 22 for a simple argumen. If on he oher hand H is unaainable no perfec replicaing sraegy exiss and hus no unique arbirage free price exiss. In fac is can be shown ha here exiss an inerval of arbirage free prices. We noe ha he no arbirage principle leads o relaive prices only, such ha prices of new asses res heavily on he prices of he original raded asses. Black and Scholes 1973 and Meron 1973 used he no arbirage principle o derive parial differenial equaions for he prices of aainable claims. They observed ha for simple coningen claims he parial differenial equaions differ by heir boundary condiions only. This way Black and Scholes obained he celebraed Black Scholes formula for he price of a so-called European call opion. Addiional insigh in no arbirage pricing was obained by Harrison and Kreps 1979, in discree ime, and Harrison and Pliska 1981, in coninuous ime. They observed a connecion beween on one side he properies of compleeness and absence of arbirage and on he oher side so-called equivalen maringale measures. Recall ha Q is an equivalen maringale measure for he model B,X, F if Q is a probabiliy measure, Q and P are equivalen for all A F: QA = PA = and all discouned price processes associaed wih raded asses are maringales under Q. They observed ha for a complee and arbirage free model here exiss a unique equivalen maringale measure and all prices are given by he expecaion under his unique equivalen maringale measure of he discouned value of claim. Hence, in a complee and arbirage free model he price of H is given by F H = E Q [ BT 1 H ]. 1.3.3 If he model is incomplee here exis infiniely many equivalen maringale measures. In his case he arbirage free prices of an unaainable claim, H, are sill given by 1.3.3. However, now Q is an equivalen maringale measure raher han he unique one. Recall

12 CHAPTER 1. INTRODUCTION ha in a arbirage free and incomplee marke an aainable claim sill has a unique price, since all equivalen maringale measures acually give he same price. The insigh obained by he link o equivalen maringale measures have furher opened for he connecion beween he parial differenial equaions for prices and he pricing formula in 1.3.3 given by he Feynman-Kač sochasic represenaion formula, see e.g. Björk 24. In general a model is arbirage free and complee if i in addiion o a locally riskfree savings accoun includes he same number of risky asses as he number of fundamenal sochasic processes Wiener processes and couning processes accouning for he uncerainy. A simple example of an incomplee marke is if he coningen claims are allowed o depend on a complee financial marke and independen insurance evens. In he following we shall decorae a discouned claim, price process or value process by an aserisk. 1.3.1 Valuaion and hedging in incomplee markes As noed above he principle of no arbirage yields unique prices and perfec replicaing sraegies for aainable claims in boh complee and incomplee markes. However, for unaainable claims he principle gives no unique arbirage free price and replicaing sraegy. Hence, in order o deermine a unique price and a hedging sraegy for an unaainable claim more srucure has o be added. The differen crieria proposed in he lieraure have heir primary focus eiher on he hedging or he pricing aspec and consider he oher quaniy as a secondary informaion. Here, we review several differen principles proposed in he lieraure. The principles considered are naurally inspired by a possible use hroughou he hesis. The review is somewha similar o he one in Møller 22. Super-replicaion The basic idea in super-replicaion super-hedging is o deermine he lowes possible iniial invesmen and he corresponding self-financing sraegy, which eliminaes he shorfall risk of he hedger. Mahemaically his corresponds o min V,ϕ ϕ under he consrain ha PVT,ϕ H = 1. Obviously his crierion is no suiable for deermining prices, since i would inroduce arbirage possibiliies. For insance he super-replicaing price of a capial insurance and a pure endowmen are idenical even hough he capial insurance always pays ou he benefis a ime of mauriy, whereas he pure endowmen only pays ou he benefis in case of survival of he insured. The heory of super-replicaion has been applied in Chaper 5. For more deails on super-replicaion we refer o El Karoui and Quenez 1995.

1.3. FINANCIAL THEORY 13 Quadraic hedging The quadraic hedging approaches focus on hedging in incomplee markes, and as a secondary resul he iniial capial necessary o consruc he opimal hedging sraegy can be inerpreed as a possible price. For a review of quadraic hedging approaches, see Schweizer 21a. Mean-variance hedging The idea of mean-variance hedging was inroduced in Bouleau and Lamberon 1989 and Duffie and Richardson 1991. Wih mean-variance hedging he aim is o deermine he self-financing sraegy, ˆϕ, which minimizes E P [ H VT,ϕ 2]. Since we consider self-financing sraegies only, he sraegy is uniquely deermined by he pair V, ˆϕ, ˆϑ. Here, V, ˆϕ and ˆϑ are known as he approximaion price for H and he mean-variance opimal hedging sraegy, respecively. Risk-minimizaion The crierion of risk-minimizaion was originally proposed by Föllmer and Sondermann 1986 for coningen claims. They considered he special case, where he discouned price processes are P-maringales. The approach was exended o he general semi-maringale case by Schweizer 1991, who inroduced he idea of local risk-minimizaion. Schweizer also observed ha he local risk-minimizing sraegy essenially corresponds o he riskminimizing sraegy under he so-called minimal maringale measure, see also Schweizer 21a. Møller 21c exended he approach in a differen direcion by allowing for paymen processes. Here, we consider a fixed bu arbirary equivalen maringale measure, Q, for he considered model, such ha discouned price processes indeed are Q-maringales. The crierion of risk-minimizaion is closely relaed o he cos process Cϕ defined by C,ϕ = V,ϕ ϑudx u. 1.3.4 From 1.3.4 we observe ha he accumulaed coss C,ϕ a ime are he discouned value V,ϕ of he porfolio reduced by discouned rading gains, ξudx u. A sraegy is called risk-minimizing, if i [ minimizes R,ϕ = E Q CT,ϕ C,ϕ 2 ] F for all wih respec o all no necessarily self-financing sraegies and VT,ϕ = H. The process Rϕ is called he risk process. Föllmer and Sondermann 1986 realized ha he risk-minimizing sraegies are relaed o he so-called Galchouk-Kunia-Waanabe decomposiion of he Q-maringale V,Q = E Q [H F]. The process V,Q is usually called o as he inrinsic value process. Furhermore hey observed ha he cos process is a Q-maringale and ha he discouned value process for he risk-minimizing sraegy coincides wih he inrinsic value process. In his hesis risk-minimizaion is applied in Chapers 4, 5, 6 and 7.

14 CHAPTER 1. INTRODUCTION Uiliy approaches Uiliy funcions have radiionally been applied in micro-economics and non-life insurance o deermine prices, and in recen years hey have be applied o derive prices in incomplee financial markes. Here, we focus on marginal uiliy indifference pricing and a special case of uiliy indifference pricing called mean-variance indifference pricing. Marginal uiliy indifference pricing Davis 1997 proposed o value coningen claims in incomplee markes by a marginal rae of subsiuion argumen. Hence, p is a fair price for he claim H if he maximal achievable expeced erminal uiliy is indifferen o wheher an agen a ime invess a small amoun of capial in he coningen claim. To express he idea mahemaically, Davis 1997 inroduced he funcion T Lδ,c,p = supe [u P c δ + ϑudxu + δ ] ϑ p H, where u is a uiliy funcion, c is he iniial capial of he agen and δ is he capial invesed in H. Now provided ha he parial derivaive of L wih respec o δ exiss a δ = and here exiss a unique soluion, ˆp, o hen ˆp is he price of H. δ Lδ,c,p δ= =, Mean-variance indifference pricing Denoe by A he discouned wealh of he insurer a ime T and consider he meanvariance uiliy-funcions u i A = E P [A ] a i Var P [A ] β i, 1.3.5 i = 1,2, where a i > are so-called risk-loading parameers, and where we ake β 1 = 1 and β 2 = 1/2. I can be shown ha calculaing premiums using he equaions u i A = u i indeed leads o he premiums assigned by he classical acuarial variance i=1 and sandard deviaion premium principle i=2, respecively, see e.g. Møller 21b. These classical acuarial principles have radiionally been applied in non-life insurance. Schweizer 21b proposed o apply he mean-variance uiliy funcions 1.3.5 in an indifference argumen which akes ino consideraion he possibiliy o rade in he financial marke. Denoe by c he insurer s iniial capial a ime. The u i -indifference price v i associaed wih he claim H is defined via supu i c + v i + ϑ T ϑudx u H = supu i c + ϑ T ϑudx u. 1.3.6 The sraegy ϑ which maximizes he lef hand side of 1.3.6 is called he opimal sraegy for H. The opimal sraegies associaed wih he mean-variance indifference prices are derived in Møller 21b. Noe ha he price assigned o a claim using his crierion is no

1.3. FINANCIAL THEORY 15 necessarily arbirage free, see Møller 22 for an example where he price lies ouside he inerval of arbirage free prices. We deermine mean-variance indifference prices in Chapers 4 and 7. The idea of mean-variance indifference pricing is slighly differen from he general se-up in uiliy indifference pricing, since he mean-variance uiliy funcions aken on a random variable reurns deerminisic value. In general one considers sandard uiliy funcions, which sill reurn a random variable. In his case he indifference price for a claim H is he price, which leaves he invesor indifferen beween purchasing he claim or no. Here, he indifference refers o he fac ha he invesor is able o obain he same maximal expeced uiliy of erminal wealh in he wo cases. For an overview of uiliy indifference pricing we refer o Henderson and Hobson 24. We noe ha he indifference price depends on he choice of uiliy funcion. Quanile hedging and minimizaion of expeced shorfall A major disadvanage of he quadraic hedging approaches is ha gains and losses are considered equally unaracive. In an aemp o eliminae his disadvanage, Föllmer and Leuker 1999 proposed he crierion of quanile hedging. Here, wo relaed problems are solved. The firs problem is for a given a fixed iniial capial, say c, o deermine he maximal obainable probabiliy of a successful hedge and he associaed self-financing sraegy. Hence, one has o solve max P [VT,ϕ H] ϕ under he consrain V, ϕ c. The relaed problem is for a given minimal probabiliy of a successful hedge, say 1 ε, o deermine he minimal necessary iniial capial and he associaed self-financing sraegy. Here, he agen is ineresed in deermining min V,ϕ ϕ under he consrain P [VT,ϕ H] 1 ε. Here, one easily observes ha he iniial invesmen converges o he super-replicaing price as ε converges o zero. A naural criicism of quanile hedging is ha he agen in case of a shorfall is indifferen o he size of he shorfall. In order o mee his criicism Föllmer and Leuker 2 and Cvianić 2 inroduced he crierion of minimizing he expeced shorfall. For a given iniial capial hey derived he self-financing sraegy, which minimizes he expeced losses from hedging he considered claim. As an alernaive hey fixed he maximum expeced shorfall and derived he self-financing sraegy, which requires he minimal iniial capial. Föllmer and Leuker 2 inroduced a so-called loss funcion, which is an increasing convex funcion wih l =, such ha hey solved he problem min ϕ E P [ lh VT,ϕ +] under he consrain V, ϕ c. Hence, he mehod of Föllmer and Leuker 2, which hey refer o as efficien hedging, could be referred o as minimizing he expeced

16 CHAPTER 1. INTRODUCTION adjused shorfall. Here, he special case lx = x corresponds o minimizing he expeced shorfall. Even hough he crieria of quanile hedging and especially minimizing he expeced shorfall are advanageous compared o he quadraic approaches hey are no pursued in he hesis, since explici resuls are exremely hard o obain. 1.4 Applying financial mehods in life insurance Even hough he fields of life insurance and finance originally were separae fields he inerplay beween he wo fields have increased over he pas decades, see e.g. Embrechs 2 and Møller 22. New insurance conracs linked direcly o he financial marke have been inroduced and old ones have gained increased populariy. In life insurance, such conracs linked direcly o he financial marke are called uni-linked conracs. Similarly financial conracs linked o insurance evens have been inroduced. We menion caasrophe insurance CAT fuures, caasrophe-linked bonds and moraliy dependen bonds. The increased inerplay has in urn increased he need for comparable mehods for pricing and reserving in he fields of financial and insurance, since prices and solvency requiremens should be independen of wheher he seller is a bank or an insurance company. Since financial mehods are compaible wih observed prices, legislaion has forced life insurance companies o use hese mehods o calculae prices and reserves. Reserves calculaed by he use of financial mahemaics are called marke reserves. 1.4.1 Tradiional insurance conracs As menioned above, legislaion has forced life insurance companies o apply mehods from financial mahemaics o value heir asses and liabiliies. Whereas he value of he asses easily is obained from he prices quoed, he liabiliies represen a greaer problem, since hey involve a mixure of financial and insurance elemens. Persson 1998 inroduced marke reserves in he case of sandard ineres rae risk and unsysemaic moraliy risk. Combining he insurance valuaion principle of diversificaion for he unsysemaic moraliy risk and he financial valuaion principle of no arbirage for he sandard ineres rae risk, he obained unique marke reserves. The valuaion is carried ou in wo seps. Firs, he principle of diversificaion is applied o replace he uncerain course of he insured life by he expeced developmen, and second he resuling purely financial conrac is priced uniquely by he no arbirage principle. For a pure endowmen he approach corresponds o exchanging he claim ITK by exp T µx,dk, where IT is an indicaor funcion denoing wheher he insured is alive a ime T. Now, under he assumpion of he exisence of sufficienly long bonds he only remaining risk is he sandard ineres rae risk. Thus, he marke reserve a ime of he guaraneed

1.4. APPLYING FINANCIAL METHODS IN LIFE INSURANCE 17 benefi, K, is uniquely given by [ ] V Q = E Q e ÊT µx,d BT 1 K = e ÊT µx,d P,TK, 1.4.1 where P,T is he price a ime of a zero coupon bond mauring a ime T. Hence, he marke reserve in 1.4.1 corresponds o he radiional reserve excep now he discoun facor is he price of a zero coupon bond. However, he marke value in 1.4.1 is only one of he infiniely many arbirage free prices for a pure endowmen in he case where we only consider unsysemaic moraliy risk and sandard ineres rae risk. In paricular, if we only apply he no arbirage principle we can define a marke reserve by V Q,g = E Q [ ITBT 1 K ] = e ÊT 1+guµx,udu P,TK 1.4.2 for each choice of sochasic process g > 1 adaped o he filraion generaed by he insurance evens, see Seffensen 2. Here, we essenially calculae he survival probabiliy wih a moraliy inensiy, 1 + guµx,u, which may differ from he real one. Comparing 1.4.1 and 1.4.2 we observe ha 1.4.1 is a special case of 1.4.2 obained by leing g =. This choice of g corresponds o assuming risk-neuraliy wih respec o unsysemaic moraliy risk. In his hesis he principle of no arbirage is applied o deermine marke reserves in he case of a sochasic moraliy inensiy, see Chaper 3. Since we only apply he principle of no arbirage, no unique marke value is obained. In his case appealing o diversificaion argumens is no sufficien o obain a unique marke reserve, since his only eliminaes he pricing uncerainy regarding he unsysemaic moraliy risk, whereas he pricing uncerainy regarding he sysemaic moraliy risk sill remains. Since he insured receive bonus if he invesmen reurn exceeds he echnical ineres rae and he echnical ineres rae serves as a guaranee, he conracs have imbedded an ineres rae opion. Tradiionally hese opions have been ignored, since hey a he ime of issue have been considered highly unlikely o have any effec. For insance a major par of he Danish life insurance conracs wih a echnical rae of 4.5% was issued in he 198 s where he ineres rae was 15-2%. However, he decreasing ineres raes over he pas years have implied ha hese opions have become very valuable. This, in urn, has increased he necessiy for he derivaion of correc prices. However, his is in general no a simple ask, since he benefi of he insured and hus he price of he opions depend on he bonus sraegy of he company. Numerous papers consider he simple case where he invesmen reurn immediaely afer realizaion is disribued beween he individual accouns of he insured and he equiy capial, see e.g. Briys and de Varenne 1997, Aase and Persson 1997, Milersen and Persson 1999 and Bacinello 21. The case including a bonus reserve is sudied in Grosen and Jørgensen 2, Hansen and Milersen 22 and Milersen and Persson 23. These papers differ by considering differen disribuion mechanisms. Anoher feaure encounered in pracice is he possibiliy for he insured o surrender. This has been sudied in Grosen and Jørgensen 2 and Seffensen 22, where he similariies wih American opions from finance are exploied. These similariies also holds for he free-policy opion of he insured, which has been considered in Seffensen 22.

18 CHAPTER 1. INTRODUCTION 1.4.2 Uni-linked life insurance Uni-linked equiy-linked life insurance conracs were inroduced as an alernaive o he radiional life insurance conracs in he Unied Saes, Neherlands and Unied Kingdom in he 195 s, see Turner 1971. In he Unied Saes he uni-linked conracs are known as variable life insurance conracs. In a uni-linked conrac he benefis are linked direcly o he developmen of a specific reference porfolio. The reference porfolio can be chosen by he insured or he company depending on he desire of he insured, and i may change during he insurance period. In recen years life insurance companies a leas in Denmark have experienced an increasing demand for uni-linked conracs. This increase was caalyzed by he explosive developmen of he sock prices in he lae 199 s, since here is a possibiliy o inves more in socks in uni-linked conracs han in radiional life insurance conracs. Furhermore, from he insured s poin of view, uni-linked conracs have he advanages ha he invesmen profile can be adaped o he desire of he individual and ha he insured easily are able o idenify heir individual savings, such ha no disribuion among he differen generaions can ake place. For he company, uni-linked conracs are advanageous, since he financial risk is easily eliminaed using he hedging approach of modern financial mahemaics. However, hey may also prove disadvanageous for he company, since i canno use surplus generaed by he financial risks o cover a possible defici resuling from he moraliy risks and vice versa. Hence, here is an increased need for he insurance company o correcly undersand, model and conrol he individual risks. Assume ha he vecor of risky asses X includes a sock wih price process S = S T and consider a porfolio of uni-linked pure endowmens linked o he developmen of he sock. In his case he oal liabiliy of he company is given by H = n NTfS, where f is a funcion of he enire pah of he sock price, S. Tradiionally he lieraure disinguish beween pure uni-linked life insurance conracs, where fs = ST and unilinked life insurance conracs wih guaranee, where fs = maxst, GT, S. Some possible guaranees include: A mauriy guaranee of a fixed amoun GT,S = K, a periodic guaranee on he reurn of he sock GT,S = K T max 1 + S i S i 1,1 + δ i, S i 1 where δ i is he guaranee in period i and a quanile guaranee i=1 GT,S = α sup S, T where α [,1]. The fixed and periodic guaranees are common in pracice, whereas he quanile guaranee is quie rare. However, we menion ha a produc including a discree

1.4. APPLYING FINANCIAL METHODS IN LIFE INSURANCE 19 version of he quanile guaranee is sold by he Danish life and pension company Danica Pension. In he early years he uni-linked conracs were usually wihou a guaranee, in which case no advanced financial mahemaics is necessary o eliminae he financial risk of he company. However, conracs could include a minimum guaranee, and prior o he inroducion of he no arbirage principle, he early approach in order value he guaranees was o involve saisical mehods for he developmen of he asses and use simulaion sudies o deermine an adequae reserve, see e.g. Turner 1969, Kahn 1971 and Wilkie 1978. Inspired by he inroducion of modern financial mahemaics by Black and Scholes 1973 and Meron 1973, Brennan and Schwarz 1976, 1979a, 1979b considered a large porfolio of uni-linked insurance conracs wih deerminisic moraliy inensiy and a Black Scholes model for he financial marke, such ha hey considered a complee financial marke wih a consan ineres rae. Using a diversificaion argumen hey exchanged he random course of he insured lives by he expeced developmen. Upon his replacemen hey obained a purely financial conrac, which could be priced and hedged using he no arbirage principle. Hence, heir approach essenially corresponds o considering he claim H = ne ÊT µx,udu fs. They considered he case fs = maxst,k and as in Black and Scholes 1973 and Meron 1973, hey obained prices by solving a parial differenial equaion. Delbaen 1986 was he firs o apply he maringale mehods of Harrison and Kreps 1979 and Harrison and Pliska 1981 in order o valuae uni-linked conracs. Since hen numerous papers have used he maringale approach o valuae uni-linked conracs wih guaranees. Bacinello and Oru 1993a consider he case of a consan ineres rae and endogenous guaranees. The case of a sochasic ineres rae has been considered by Bacinello and Oru 1993b, who derive a closed form soluion in he case of a single premium pure endowmen and a so-called Vasiček model for he ineres rae. Nielsen and Sandmann 1995 consider a sochasic ineres rae in associaion wih periodic premiums and periodic guaranees. They observe ha he guaranee inroduces a discreely sampled Asian opion. In order o obain resuls hey use numerical mehods. Aase and Persson 1994 are he firs o consider insananeous deah probabiliies. However, even hough hey consider only one insured, hey also quickly inser he expeced values regarding survival. So far he menioned papers have applied he diversificaion principle a an early poin, such ha he pricing problem reduces o pricing coningen claims in a complee financial marke. Furhermore, he papers considering he hedging aspec all consider he purely financial conrac resuling from he use of he diversificaion principle. Brennan and Schwarz 1976 refer o such a sraegy as riskless, even hough i only eliminaes he financial risk, since he company of course sill is exposed o unsysemaic moraliy risk. Hence, a more appropriae name would be a financially riskless sraegy. In conras o he papers above Møller 1998 does no apply he diversificaion principle a an early poin in he pricing and hedging problem. Hence, he considers an incomplee financial marke consising of a complee financial marke and a couning process coun-

2 CHAPTER 1. INTRODUCTION ing he number of deahs in he porfolio. When assuming risk-neuraliy wih respec o moraliy, he obains prices idenical o hose in he papers above. However, similarly o radiional insurance conracs, here exiss infiniely many arbirage free prices and he one menioned above is jus one paricular arbirage free price. The incomplee marke seing becomes paricularly imporan when discussing hedging sraegies. Here, Møller 1998 considers he crierion of risk-minimizaion and derives risk-minimizing sraegies for uni-linked life insurance conracs payable a a fixed ime. Hence, whereas he previous papers main purpose is o derive prices for uni-linked life insurance conracs he main purpose of Møller 1998 is o deermine a hedging sraegy. The hedging resul obained essenially corresponds o he riskless hedging sraegy of Brennan and Schwarz 1976. However, now he sraegy is adjused coninuously o he expeced number of survivors given he curren developmen of he insurance porfolio. The work is exended o cover paymen processes in Møller 21c such ha a more realisic insurance conracs can be considered. Møller 21a essenially considers a discree ime version of Møller 1998. The use of indifference pricing o value insurance conracs in an incomplee marke have been sudied in Becherer 23, who worked wih exponenial uiliy funcions, and Møller 21b, 23a, 23b, who considered mean-variance indifference uiliy funcions. All of he above papers disregard he sysemaic moraliy risk. In his hesis prices and hedging sraegies are pursued in he presence of sysemaic moraliy risk. We deermine prices using he no arbirage principle only. Similarly o he case of radiional life insurance conracs we refer o hese prices as marke values. Furhermore, we deermine mean-variance indifference prices. For he hedging aspec emphasis is on he crierion risk-minimizaion and he opimal hedging sraegies associaed wih he mean-variance indifference prices. 1.5 Quanifying he ypes of risk The qualiaive descripion of he ypes of risk in Secion 1.1.2 provides valuable knowledge regarding if and how he insurance company is exposed o he differen ypes of risk. However, he main ineres for he company is a quaniaive descripion. In order o obain a quaniaive descripion he company has o specify a model for he sources of risk and he crierion measuring he risk. Hence, he crierion is of grea imporance when defining he sraegy which minimizes he business risk and in order o compare he business risk for differen sraegies. However, he hedging crierion used o deermine he opimal sraegy is no necessarily he crierion used o quanify he business risk. Here, he company ofen use a differen crierion which is more easily inerpreable and perhaps demanded by he regulaors. In order o obain an adequae model for he sources of risk i should involve all he relevan main sources of risk, and he descripion of he ypes of risk should mirror real life as closely as possible. Mehods used o quanify he risk can be found in Secion 1.3.1. In his secion we give an overview of he developmen of he modelling of he differen sources of risk.

1.5. QUANTIFYING THE TYPES OF RISK 21 1.5.1 Equiy risk The modelling of sock prices in coninuous ime was iniiaed by Bachelier 19, who proposed o model he sock price by ds = αd + σdw, where α and σ are consans, and W = W T is a Wiener process. Hence, he change in he sock price in a shor inerval of lengh is independen of earlier changes and follows a normal disribuion wih mean α and variance σ 2. This model however has he undesirable propery ha he sock price may become negaive. This flaw was eliminaed in Samuelson 1965, who proposed o model he sock price by a so-called geomeric Brownian moion, where he dynamics are given by ds = αsd + σsdw. 1.5.1 I was wihin his framework ha Black and Scholes obained heir famous resul. Today his model serves a sandard example and reference in financial mahemaics. A naural exension of 1.5.1 is o allow for ime-dependen funcions α and σ. All resuls obained wih consan parameers are easily exended o his case. Also he exension o a mulidimensional Wiener process is sraighforward. Heson 1993 exended he model o include sochasic volailiy, i.e. he modelled he diffusion parameer σ as a sochasic process. Despie empirical evidence ha an adequae model should allow for jumps mos of he lieraure on sock prices consider diffusion models, i.e. models where he sock prices are driven by Wiener processes. However, a model including jumps was already considered in Meron 1976, who inroduced he jumps in a paricular nice fashion, such ha opion prices are infinie sums of Black Scholes opion prices. Recenly Levy processes have araced aenion, see e.g. Chan 1999, Eberlein 21 and Con and Tankov 24. The advanage of Levy processes is wofold: They consiue a flexible class of models, such ha hey can provide an adequae descripion of he sock prices and a he same ime are hey mahemaically nice. An ineresing simple alernaive can be found in Norberg 23, who considers a marke driven by a finie sae Markov chain. 1.5.2 Ineres rae risk Since i is inconvenien o model bond prices direcly, he sandard approach in he lieraure is o model ineres raes insead. Here, he firs approach was o model he dynamics of he shor rae, r, as a diffusion process, i.e. by dr = α,rd + σ,rdw. Wihin he class of shor rae diffusion models especially hose given by α,r = γ α + δ α r, σ,r = γ σ + δ σ r.

22 CHAPTER 1. INTRODUCTION have received a lo of aenion, since models of his form give an affine erm srucure, which is paricularly nice from a pricing perspecive, see e.g. Björk 24. The class of affine shor rae models includes he famous Vasiček and Cox Ingersoll-Ross CIR models, see Vasiček 1977 and Cox, Ingersoll and Ross 1985. Exensions of boh models o he ime-inhomogeneous case can be found in Hull and Whie 199. A differen approach was proposed in Heah, Jarrow and Moron 1992 who modelled he enire forward rae curve. For a fixed ime of mauriy τ hey modelled forward rae dynamics by df,τ = α,τd + σ,τdw, for some adaped processes α and σ. A relaed work can be found in Musiela 1993, where he forward raes are parameerized wih ime o mauriy insead of ime of mauriy. For an overview of ineres rae models wihou jumps, we refer o Brigo and Mercurio 21. Shirakawa 1991 exended he approach in Heah e al. 1992 o he case, where he forward raes are driven by a Wiener process and Poisson driven jumps of a fixed magniude. A general descripion of a bond marke including jumps can be found in Björk, Di Masi, Kabanov and Runggaldier 1997 and Björk, Kabanov and Runggaldier 1997. All of he above papers ignore he reinvesmen risk by assuming he exisence of sufficienly long bonds. This assumpion is usually jusified, when considering purely financial producs, since hese, as opposed o life insurance conracs, radiionally are shor erm conracs. The firs aemp o include he reinvesmen risk is Sommer 1997. In his hesis models for he reinvesmen risk are proposed in Chapers 5 and 6. 1.5.3 Unsysemaic moraliy risk In general he developmen of a single conrac or a porfolio of similar conracs can be described by a finie sae Markov chain, see Hoem 1969. In he single insurance case he Markov chain describes he differen saes of healh, which are of imporance for he insurance conrac. In he porfolio case he insured are assumed o be a homogeneous group of lives, which are muually independen given he underlying moraliy inensiy. Hence, he Markov chain couns he number of insured in each sae of healh. We noe ha he insured lives no in general are independen, since he underlying moraliy inensiy affecs all he insured. The unsysemaic moraliy risk is now he uncerainy relaed o he developmen of he Markov chain wih known ransiion inensiies. The case of a porfolio of idenical pure endowmens is paricularly simple, since he developmen of he insurance porfolio in his case can be described by a couning process, N = N T, couning he number of deahs in he porfolio. Noe ha while he assumpion of condiional independence of he insured lives in general reasonable when considering he developmen of a porfolio over a long ime horizon, i may no be applicable, when considering he shor erm probabiliy of a large number of deahs, since caasrophes, such as errorism, hurricanes and earhquakes may affec a specific group of persons.

1.6. OVERVIEW AND CONTRIBUTIONS OF THE THESIS 23 1.5.4 Sysemaic moraliy risk In he lieraure he sandard approach has been o consider a deerminisic and imeindependen moraliy inensiy, such ha he model excludes sysemaic moraliy risk. This is in correspondence wih pracice, where he life insurance companies, in a leas Denmark, so far have worked wih a moraliy inensiy, which depends on he age of he insured, only. The companies have been aware ha he moraliy inensiy has decreased over he years and o incorporae his hey have adjused heir moraliy inensiies yearly. However, regardless of he frequency of hese moraliy invesigaions, he esimaion of a ime-homogeneous moraliy inensiy sill only measures he curren level. In order o obain a more accurae predicion of he fuure moraliy inensiy he company has o capure rends in he moraliy inensiy, such ha a ime and age dependen moraliy inensiy is necessary. If company also wans o capure he sochasic naure of he fuure moraliy, he moraliy inensiy should be modelled as a sochasic process. In recen years several papers have considered modelling he fuure moraliy and pricing moraliy derivaives. One of he bes known models for he fuure moraliy is he Lee- Carer model, see Lee and Carer 1992 and Lee 2. Here, he yearly deah raes are modelled by hree facors: Two age-dependen and one ime-dependen. By modelling he ime-dependen facor as a ime-series he model can be used for forecasing. The firs paper o inroduce a sochasic moraliy inensiy is Milevsky and Promislow 21, who consider a so-called mean-revering Gomperz model under he equivalen maringale measure. Milevsky and Promislow 21 are also he firs o consider a model including boh sandard ineres rae and sysemaic moraliy risk. Dahl 24b, see Chaper 3, considers a general diffusion model and discusses he change of measure wih respec o he moraliy. A general affine jump diffusion model for he moraliy inensiy can be found in Biffis and Millossovich 24. These models for he moraliy inensiy are inspired by he shor rae models used o describe sandard ineres rae risk, see Secion 1.5.2. An overview over ineres rae approaches applicable o describe he sysemaic moraliy risk can be found in Cairns, Blake and Dowd 24. An enirely differen approach is aken in Olivieri and Piacco 22, where Baysian mehods are considered. 1.6 Overview and conribuions of he hesis The aim of he hesis is o analyze risks in life insurance. We propose models for he differen ypes of risk and use mehods from financial mahemaics o value and hedge life insurance conracs. The hesis consiss of four main pars. The firs par, Chaper 2, considers he problem of deermining a fair disribuion of asses beween he equiy capial and he porfolio of insured in he case where he insurance conrac includes a periodic ineres rae guaranee. The second par, Chapers 3 and 4, consider he impac of modelling he moraliy inensiy as a sochasic process. Here, Chaper 3 focus on deermining marke reserve and on possible ways o ransfer he sysemaic moraliy risk o he insured or agens in he financial marke. Chaper 4 includes a derivaion of hedging sraegies for life-insurance conracs and a numerical comparison life expecancies using a ime-independen, a ime-dependen and a sochasic moraliy inensiy. In par hree

24 CHAPTER 1. INTRODUCTION we propose models for he reinvesmen risk, and derive opimal hedging sraegies. Here, Chaper 5 conains a discree-ime model, whereas a coninuous-ime model is inroduced in Chaper 6. In he fourh and las par, we essenially combine Chapers 4 and 6, such ha we obain a model including a large number of he risks faced by a life insurance company. Here, Chaper 7 covers he heoreical derivaion of hedging sraegies, whereas Chaper 8 includes a numerical comparison of differen reservaion principles. Now we give a more deailed descripion of he individual chapers. Fair Disribuion of Asses in Life Insurance When issuing life insurance conracs wih a periodic ineres rae guaranee, he equiy capial of he company is exposed o he risk of low or even negaive payoffs a he end of an accumulaion period. In he wors case scenario, where he guaranee can no be covered, all equiy capial is los and he company is declared bankrup. To compensae he owners for he risk of low reurns on equiy capial imposed by he guaranee, he equiy capial should be accumulaed by a rae, which exceeds he riskfree rae in periods, where he invesmen reurn and developmen of he insurance porfolio allows for such a high reurn on equiy capial. In Chaper 2, based on Dahl 24a, we consider an insurance company whose insurance porfolio consiss of eiher capial insurances or pure endowmens wih a periodic ineres rae guaranee. Since he financial marke is given by he complee and arbirage free Black Scholes model, we can for a given invesmen sraegy apply he principle of no arbirage o obain an equaion for he fair addiional payoff o he equiy capial in periods, when such an addiional payoff is possible. 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 order o sudy he magniude of he fair addiional rae of ineres and he dependence on parameer values, iniial disribuion of capial and invesmen sraegy, we supply numerical resuls. Sochasic Moraliy in Life Insurance: Marke Reserves and Moraliy-Linked Insurance Conracs In life insurance, acuaries have radiionally calculaed premiums and reserves using a deerminisic moraliy inensiy, which is a funcion of he age of he insured only. However, he fuure moraliy inensiy is unknown, so i should be modelled as a sochasic process. In Chaper 3, based on Dahl 24b, we model he moraliy inensiy as a diffusion process. This allows us o consider uni-linked conracs in a model including equiy and sandard ineres rae risk, as well as boh ypes of moraliy risks. Wihin his model we derive marke reserves and sudy possible ways of ransferring he sysemaic moraliy risk o oher paries. One possibiliy is o inroduce moraliy-linked insurance conracs. Here, he premiums and/or benefis are linked o he developmen of he moraliy inensiy, hereby ransferring he sysemaic moraliy risk o he insured. Alernaively he insurance company can ransfer some or all of he sysemaic moraliy risk o agens in he financial marke by rading derivaives depending on he moraliy inensiy. We derive a general parial differenial equaion for moraliy derivaives and show an example of how a moraliy derivaive can be used o eliminae or reduce he sysemaic moraliy risk of he company.

1.6. OVERVIEW AND CONTRIBUTIONS OF THE THESIS 25 Valuaion and Hedging of Life Insurance Liabiliies wih Sysemaic Moraliy Risk Chaper 4 considers he problem of valuaing and hedging a porfolio of life insurance conracs ha are subjec o sysemaic moraliy risk as well as he usual sources of risk, namely sandard ineres rae risk and unsysemaic moraliy risk. Since he moraliy risks are unhedgeable hey canno be eliminaed by rading in he financial marke. Furhermore, since he sysemaic moraliy risk is a non-diversifiable risk i canno be reduced by increasing he size of he porfolio and appealing o he law of large numbers. Hence, we propose o apply echniques from incomplee markes in order o hedge and valuae hese conracs. We derive marke reserves and mean-variance indifference prices. The hedging aspec is addressed by deermining risk-minimizing sraegies and opimal hedging sraegies associaed wih he mean-variance indifference prices. The chaper includes empirical evidence supporing he modelling of he moraliy inensiy as a sochasic process, and a numerical example comparing he life expecancies using a ime-independen, a ime-dependen and a sochasic moraliy inensiy. This chaper is based on Dahl and Møller 25. A Discree-Time Model for Reinvesmen Risk in Bond Markes In he lieraure bond markes usually include bonds wih all imes o mauriy. However, in pracice he liquid bonds raded have a fixed maximum ime o mauriy. Hence, a life insurance company selling long erm conracs is exposed o an unhedgeable reinvesmen risk, associaed wih he enry prices of newly issued bonds. In Chaper 5, which is based on Dahl 25b, we propose a discree-ime model for a bond marke, where he reinvesmen risk is presen. The analysis is carried ou in discree ime in order o explain he ideas in a framework, where he echnical deails are kep o a minimum. A each rading ime a bond maures and a new 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. In order o deermine opimal hedging sraegies we consider he crieria of super-replicaion and risk-minimizaion. Furhermore, a link beween super-replicaion and he maximal guaranees for which he shor fall ineres rae risk can be eliminaed is observed. Finally, we consider a numerical example, where we compare our sochasic analysis wih he deerminisic pricing principle of a level long erm yield curve. In his example we also inroduce he alernaive deerminisic pricing principle of a level long erm forward rae curve. A Coninuous-Time Model for Reinvesmen Risk in Bond Markes In Chaper 6, based on Dahl 25a, we propose a coninuous bond marke model including reinvesmen risk. We consider a model, where only bonds wih a limied ime o mauriy are raded in he marke. A fixed imes new bonds wih sochasic iniial prices are inroduced in he marke. Here, he new price is allowed o depend on he exising bond prices and all pas informaion, such ha we obain a flexible model. To quanify and conrol he reinvesmen risk we apply he crierion of risk-minimizaion. Valuaion and Hedging of Uni-Linked Life Insurance Conracs Subjec o Reinvesmen and Moraliy Risks In Chaper 7, based on Dahl 25d, we consider a model covering a large number of he risks faced by a company issuing uni-linked life insurance conracs. Here, he financial

26 CHAPTER 1. INTRODUCTION marke consiss of a bond marke including reinvesmen risk and a sock, whereas he insurance par involves a sochasic moraliy inensiy. Hence, we consider a model, which combine he sysemaic moraliy risk considered in Chapers 3 and 4 wih he reinvesmen risk from Chapers 5 and 6. The valuaion and hedging resuls in Chaper 4 are hen exended o his more refined model. A Numerical Sudy of Reserves and Risk Measures in Life Insurance Reserving and risk managemen are of grea imporance for life insurance companies. In Chaper 8, based ondahl 25c, we provide a numerical invesigaion of differen reservaion principles and risk measures. We consider marke reserves calculaed by he no arbirage principle, only. Furhermore, we consider he following alernaive approaches o pricing he dependence on he reinvesmen risk: Super-replicaion and he principles of a level long erm yield/forward rae curve. Combined wih he no arbirage principle for he remaining risks, hese principles give reserves, which can be compared o he marke reserves. The risk measures considered are Value a Risk and ail condiional expecaion.

Chaper 2 Fair Disribuion of Asses in Life Insurance This chaper is an adaped version of Dahl 24a When a life insurance company disribues asses beween he equiy capial and he porfolio of insured, possible periodic guaranees o he insured mus be covered whenever possible. Hence, depending on he developmen of he financial marke and he porfolio of insured, he equiy capial may experience periods wih low or even negaive payoffs. In he wors case scenario, where he guaranee can no be covered, he company is declared bankrup, and he enire equiy capial is los. To compensae he owners for he risk of low reurns on equiy capial, he equiy capial should be accumulaed by a rae, which exceeds he riskfree rae in periods, where he invesmen reurn and developmen of he insurance porfolio allows for such a high reurn on equiy capial. We consider an insurance company wih a very simple insurance porfolio: I consiss of eiher capial insurances or pure endowmens. The financial marke is described by a Black Scholes model. Given an invesmen sraegy for he company, he principle of no arbirage gives an equaion for he fair addiional payoff o he equiy capial in periods, when such an addiional payoff is possible. 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. To invesigae he magniude of he fair addiional rae of ineres and he dependence on parameer values, iniial disribuion of capial and invesmen sraegy, we supply numerical resuls. 2.1 Inroducion When issuing life insurance conracs wih a guaranee, he insurance companies are exposed o a risk, since he guaranee mus be covered whenever possible. The wo mos common ypes of guaranees are: A mauriy guaranee, where he company guaranees a 27

28 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE minimal oal accumulaion for he enire duraion of he conrac, and guaraneed periodic accumulaion facors guaraneed periodic ineres raes, where he company guaranees a minimal accumulaion facor for each period. Even hough he mos common ype of guaranee in Denmark is a mauriy guaranee, we consider he case of guaraneed periodic accumulaion facors, since i allows us o consider each accumulaion period independenly. When guaraneeing periodic accumulaion facors, he equiy capial of he company migh experience low or even negaive payoffs in periods wih low reurns on invesmens and/or an adverse developmen of he insurance porfolio. In he exreme case, where he guaranee can no be covered, he ineres of he insured ake precedence over he ineress of he company, and all asses are paid o he insured. Guaraneed periodic accumulaion facors implicily inroduce a sring of European call opions on he invesmen gain in he insurance conrac. Hisorically, he guaranees have in pracice been chosen far ou of he money, and herefore hey have been ignored when pricing he insurance conracs. However, he decreasing ineres raes in recen years has caused he guaranees o become an imporan elemen of some old conracs. This, in urn, has increased he imporance for correc pricing of he opions imbedded in he insurance conracs, see e.g. Briys and de Varenne 1997, Aase and Persson 1997, Milersen and Persson 1999 and Bacinello 21. In pracice, insurance companies use a bonus accoun for undisribued surplus in order o smooh he accumulaion facors over ime. When including a bonus accoun, he price of an insurance conrac depends on he bonus mechanism. For some differen possible bonus mechanisms and heir impac on prices, see Grosen and Jørgensen 2, Hansen and Milersen 22 and Milersen and Persson 23. Anoher feaure encounered in pracice is he possibiliy for he insured o surrender, which is included e.g. in Grosen and Jørgensen 2. The bankrupcy of major life insurance companies in England and Japan have also underlined he imporance of including he risk of he company defauling. This is done inbriys and de Varenne 1997. The main purpose of he above menioned papers is essenially o obain he arbirage free price of an insurance conrac by considering he developmen of he insurance conrac unil erminaion. The aim of he presen chaper is slighly differen from ha of pricing individual conracs. Here, he goal is o deermine a fair disribuion of asses beween he owners of he insurance company and he porfolio of insured a he end of each accumulaion period. Thus, he model considered is essenially a 1-period model wih one accumulaion period. In he model he accumulaion facor, announced by he company prior o he accumulaion period, is viewed as an exogenous parameer. Hence, we avoid he modelling of he announced accumulaion facor, which is quie difficul since compeiion seems o play a major role in he decision process. In conras o many companies, we do no view he announced accumulaion facor as binding. Thus, he acual and announced accumulaion facors may differ when experiencing poor invesmen reurns and/or an adverse developmen of he insurance porfolio. To deermine he disribuion of he asses beween he deposi, he bonus reserve and he equiy capial a he end of he accumulaion period, we define a disribuion scheme. Wihin his scheme, he only unknown parameer is he ineres rae used, in addiion o he riskfree ineres rae, o accumulae he equiy capial in periods when possible. We assume ha he company is allowed o inves in a financial marke described by a Black-Scholes model. This marke is

2.1. INTRODUCTION 29 known o be complee and arbirage free. A disribuion scheme is considered as fair, if i does no inroduce arbirage possibiliies for he owners or he insurance porfolio. When considering a porfolio of capial insurances, he disribuion scheme depends enirely on he developmen of he financial marke, and since he financial marke is complee and arbirage free, we can derive a simple equaion, which has o be fulfilled by a disribuion scheme in order no o inroduce arbirage possibiliies. Thus, we are able o find an equaion for he unique fair addiional ineres rae. For a porfolio of pure endowmens he disribuion scheme depends on boh he financial marke and he developmen of he insurance porfolio. Hence, we are in an incomplee marke. Thus, infiniely many equivalen maringale measures exis, such ha he principle of no arbirage yields infiniely many possible equaions from which o derive a fair disribuion. However, for a fixed equivalen maringale measure, we again have a unique equaion for he fair addiional ineres rae. Since he equaions derived for he fair addiional ineres rae are implici equaions, we have o use numerical echniques o derive he resul. Hence, in conras o oher papers including bonus accouns, no simulaion is necessary. We poin ou ha he resuls in his chaper for he fair addiional ineres rae are based on a simple financial model wih consan ineres rae and a deerminisic moraliy inensiy. Hence, we only ake he financial risk associaed wih invesmens in socks and he unsysemaic moraliy risk ino accoun. The fair addiional ineres rae would be larger if we were o add ineres rae risk and/or sysemaic moraliy risk o he model. Noe ha we disinguish beween sysemaic moraliy risk, referring o he fuure developmen of he underlying moraliy inensiy, and unsysemaic moraliy risk, referring o a possible adverse developmen of he insured porfolio wih known moraliy inensiy, see Chaper 3. Furhermore expenses and he associaed risk have been disregarded in he sudy. In addiion o he measurable risks menioned above one could consider operaional risk as well. Thus, he fair addiional ineres rae deermined in his chaper serves as a lower bound for he fair addiional ineres rae in pracice. The chaper is organized as follows: In Secion 2.2, a simplified balance shee and a shor descripion of he accouns are given. The financial model and he relevan financial erminology is inroduced in Secion 2.3. In Secion 2.4, a company wih an insurance porfolio of capial insurances is considered. Given differen invesmen sraegies, we decompose he erminal equiy capial ino payoffs from sandard opions, such ha each invesmen sraegy leads o an equaion for he fair addiional ineres rae. Secion 2.5 sudies he case of a porfolio of pure endowmens. In his case, he value for he fair addiional ineres rae depends on he chosen equivalen maringale measure. Since he equaions obained in Secions 2.4 and 2.5 for he he fair addiional ineres rae are implici equaions only, we supply numerical resuls in Secion 2.6. In Secion 2.7 we discuss some possible changes o he disribuion mechanism and heir impac on he resuls. A discussion on he realism and versailiy of he model is given in Secion 2.8, whereas Secion 2.9 conains a conclusion. Proofs and calculaions of some echnical resuls can be found in Secion 2.1

3 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE 2.2 The balance shee To describe he asses and liabiliies of he insurance company we use he following simplified balance shee. Asses A A Liabiliies V U E A The asse side consiss of he accoun A only, while he liabiliy side is comprised of hree accouns: V, U and E. The boom line of he balance shee jus saes ha he asses and liabiliies mus balance, i.e. V + U + E = A. We now give a deailed descripion of he individual accouns. Accoun V he deposi is he oal deposi of he insurance porfolio. The deposi is allocaed o he insured on an individual basis. In case of a capial insurance or a pure endowmen, he individual deposi a ime of erminaion is he sum paid o he insured. Whenever an insurance conrac saes a guaraneed periodic accumulaion facor, he guaranee applies o he deposi. Capial allocaed o he deposi belongs o he individual owning he acual accoun, and canno be ransferred o he deposi of anoher insured or oher accouns on he liabiliy side. Accoun U he bonus reserve is he undisribued surplus allocaed o he insurance porfolio as a whole. I is used by he company o smooh deposi accumulaion facors over ime. Capial allocaed o he bonus reserve canno freely be ransferred o he equiy capial. Such a ransfer may only ake place as a paymen o he equiy capial for he risk associaed wih he insurance conracs. Accoun E he equiy capial is he capial belonging o he owners of he company. Accoun A he asses describes he value of he asses of he insurance company. We assume ha he insurance company invess in he financial marke described in Secion 2.3. In order o consider he risk associaed wih he insurance conracs only, we assume ha he company invess he amoun E in he savings accoun, and he amoun V + U in an admissible sraegy ϕ = ϑ,η wih value process Vϕ. Thus, a ime, [,T], we have A = e r E + V ϕ V ϕ V + U. I now follows from he following argumen ha we wihou loss of generaliy may assume ha V + U = V ϕ 2.2.1

2.3. THE FINANCIAL MODEL 31 such ha A = e r E + V ϕ. Assume ha 2.2.1 does no hold. Then he self-financing sraegy given by ϕ = V + U V ϕ ϕ = V + U V ϕ ϑ, V + U V ϕ η fulfills e r E + V ϕ = e r E + V + U V ϕ V ϕ = A, [,T]. A similar simplified balance shee is used in Grosen and Jørgensen 2, Hansen and Milersen 22 and Milersen and Persson 23. However, he number of accouns on he liabiliy side of he balance shee, and heir inerpreaion varies. 2.3 The financial model We consider a financial marke described by he sandard Black Scholes model. Here, he marke consiss of wo raded asses: A risky asse wih price process S and a riskfree asse wih price process B. The risky asse is usually referred o as a sock and he riskfree asse as a savings accoun. The price processes are defined on a probabiliy space Ω, F,P, and he P-dynamics of he price processes are given by ds = αs d + σs d W, S >, 2.3.1 db = rb d, B = 1, where W T is a Wiener process on he inerval [,T] under P, wih T being a fixed finie ime horizon. The coefficien σ is a sricly posiive consan, while α and r are nonnegaive consans. The filraion G = G T is he P-augmenaion of he naural filraion generaed by B,S, i.e. G = G + N, where N is he σ-algebra generaed by all P-null ses and G + = σ{b u,s u,u } = σ{s u,u } = σ{ W u,u }. Here, we have used he sric posiiviy of σ in he las equaliy. We inerpre α as he mean rae of reurn of he sock, σ as he sandard deviaion of he rae of reurn and r as he shor rae of ineres. The consan ν defined by ν = α r σ is known as he marke price of risk associaed wih S. I is well-known, see e.g. Musiela and Rukowski 1997, ha in he Black Scholes model, he probabiliy measure Q defined by dq dp O T = e ν W T 1 2 ν2 T is he unique equivalen maringale measure. Hence, Q is a probabiliy measure equivalen o P under which all discouned price processes on he financial marke are local maringales. The Q -dynamics of he price processes are ds = rs d + σs dw, S >, 2.3.2 db = rb d, B = 1,

32 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE where W T is a Wiener process on he inerval [,T] under Q. A rading sraegy is an adaped process ϕ = ϑ,η saisfying cerain inegrabiliy condiions. The pair ϕ = ϑ,η is inerpreed as he porfolio held a ime. Here, ϑ and η, respecively, denoe he number of socks and he discouned deposi in he savings accoun in he porfolio a ime. The value process Vϕ associaed wih ϕ is given by A sraegy ϕ is called self-financing if V ϕ = V ϕ + V ϕ = ϑ S + η B. ϑ u ds u + η u db u. Thus, he value a any ime of a self-financing sraegy is he iniial value added rading gains from invesing in socks and ineres earned on he deposi in he savings accoun; wihdrawals and addiional deposis are no allowed during, T. A self-financing sraegy ϕ = ϑ,η is called admissible if ϑ,η, which guaranees ha V ϕ P-a.s. for all [,T]. We resric he invesmen sraegies of he insurance company o admissible sraegies. A self-financing sraegy is a so-called arbirage if V ϕ = and V T ϕ P-a.s. wih PV T ϕ > >. A coningen claim or a derivaive in he model B,S, G wih mauriy T is a G T -measurable, Q -square inegrable random variable X. A coningen claim is called aainable if here exiss a self-financing sraegy such ha V T ϕ = X, P-a.s. An aainable claim can hus be replicaed perfecly by invesing V ϕ a ime and invesing during he inerval [,T] according o he self-financing sraegy ϕ. Hence, a any ime, here is no difference beween holding he claim X and he porfolio ϕ. In his sense, he claim X is redundan in he marke, and from he assumpion of no arbirage i follows ha he price of X a ime mus be V ϕ. Thus, he iniial invesmen V ϕ is he unique arbirage free price of X. If all coningen claims are aainable, he model is called complee and oherwise i is called incomplee. I is wellknown from he financial lieraure, see e.g. Björk 24, ha he Black Scholes model is complee and arbirage free, and ha he discouned value process associaed wih any self-financing rading sraegy is a Q -maringale. Throughou he chaper, we denoe by S he discouned sock price and by V ϕ he discouned value process. Furhermore we use he aserisk o denoe ha a consan or funcion has been muliplied by e rt, i.e. discouned from ime T o ime. 2.4 Capial insurances Consider a life insurance company whose insurance porfolio consiues capial insurances exclusively. A capial insurance pays ou a sum insured a a specified ime, wheher he insured is alive or no. For simpliciy we assume ha no paymens beween he company and he insured ake place during,t. In his case, we can disregard he individual conracs and focus on he oal insurance porfolio. The aim of his secion is, while respecing he general erms of he conracs, o deermine an arbirage free disribuion of he asses a ime T among he accouns on he liabiliy

2.4. CAPITAL INSURANCES 33 side. We shall refer o such a disribuion as fair, see Secion 2.4.2 for more deails. We assume ha all insured are promised he same accumulaion facor G T on he deposi in he period,t. In pracice, we ofen have G T 1. The consequence of he guaranee is ha he oal deposi should be a leas G T V a ime T whenever possible. If he company is unable o cover he guaranee, all asses are allocaed o he deposi and paid o he insured in cash, while he company is declared bankrup. Remark 2.4.1 Two possible choices for he guaraneed accumulaion facor are 1 + gt and e gt, depending on wheher g is expressed in erms of a periodical or a coninuously compounding rae. If T = 1 and ime is measured in years, hen G 1 = 1 + g corresponds o a guaraneed annual ineres rae of g. Remark 2.4.2 For he company o survive in he long run, we should have G T e rt. However, since we are ineresed in shor erm condiions only, also he reverse siuaion is relevan. To be consisen wih common pracice, he company a ime announces a deposi accumulaion facor K T, K T G T, by which hey inend o accumulae he deposi a ime T. In conras o G T, we do no consider K T as legally binding. Hence, a ime T he company is allowed o use an accumulaion facor differen from K T for he acual accumulaion. However, using an accumulaion facor differen from K T affecs he credibiliy of he company, and hus, i is no done frequenly in pracice. In order o model his relucancy in a simple way wihou removing he possibiliy of using an accumulaion facor differen from K T, we assume ha he company uses K T unless he value of he risky invesmens a ime T, V T ϕ, is less han K T V 1+γ. Here, he facor γ, γ, is he proporion of he deposi which is he arge for he minimal bonus reserve, as decided by he managemen of he insurance company. To compensae he equiy capial for he exposure o he financial risk inheren in capial insurances, we inroduce he parameer ρ, which represens he ineres rae credied o he equiy capial in addiion o he riskfree ineres rae, whenever such an addiional reurn is possible. 2.4.1 Disribuion scheme The disribuion scheme used by he company o disribue asses a ime T beween he hree accouns on he liabiliy side depends on he developmen of he asses of he company and hence on he financial marke. We disinguish beween he following hree siuaions for he developmen of he asses: 1. A T < G T V : In his case, he company does no have sufficien asses o cover he guaranee. Since he ineres of he insured akes prioriy over he ineres of he

34 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE owners of he company, all capial is allocaed o he deposi, and he equiy is se o, ha is V T = A T, E T =, U T =. 2. G T V A T < K T V 1 + γ + e rt E : Here, he asses are sufficien o mee he guaraneed accumulaion of he deposi. However, using he announced deposi accumulaion facor would leave he company wih a bonus reserve less han he arge for he minimal bonus reserve, γv T. Hence, he company chooses o accumulae he deposi by he guaraneed accumulaion facor G T. This way, he company obains he maximal possible bonus reserve, which in some cases exceeds γv T. The equiy capial a ime T is given by he equiy capial a ime accumulaed wih he ineres rae r + ρ or he oal asses deduced he deposi a ime T, whichever is smalles. The bonus accoun is calculaed residually as he asses subraced he deposi and he equiy capial. This leads o he following disribuion: V T = G T V, E T = min e r+ρt E,A T V T, U T = A T V T E T. 3. e rt E + K T V 1 + γ A T : This oucome leaves he company wih a bonus reserve larger han γv T afer accumulaing he deposi wih K T. The disribuion is given by an expression similar o he one in case 2 wih G T subsiued by K T. Thus V T = K T V, E T = min e r+ρt E,A T V T, U T = A T V T E T. Noe ha in he disribuion scheme we firs use he bonus reserve o cover he accumulaion of he deposi, and if his is insufficien, we hen use he equiy capial. In he disribuion scheme, he only unknown parameer is ρ. Hence, deermining he fair disribuion scheme reduces o deermining he fair value of ρ. Since E T e r+ρt E, a necessary requiremen for a disribuion scheme o be arbirage free is ρ. Hence, he referral o ρ as he addiional rae of reurn. Furhermore, we immediaely observe from he disribuion scheme ha E T is non-decreasing in ρ for all ω. If furher A T is sochasic, i.e. if he company invess some capial in he risky asse, hen PA T V T e r+ρt E > for all finie ρ. Hence he se of ω s for which E T is sricly increasing in ρ has a posiive probabiliy. We hus have ha he fair value of ρ, if i exiss, is unique.

2.4. CAPITAL INSURANCES 35 2.4.2 Fair disribuion A disribuion scheme is said o be fair if i does no inroduce arbirage opporuniies for he insurance company or he insurance porfolio. Since he size of he accouns on he liabiliy side of he balance shee depends on he developmen of he financial marke only, we can view E T and V T + U T as coningen claims in he complee and arbirage free marke B,S, G. Hence, he claims E T and V T + U T have unique prices. Thus, he disribuion scheme is fair, if and E = e rt E Q [E T ], 2.4.1 V + U = e rt E Q [V T + U T ]. 2.4.2 Noe ha since we are ineresed in he disribuion of asses beween he insurance porfolio as a whole and he company, and no beween he insured individuals, we do no disinguish beween he deposi and he bonus reserve in 2.4.2. Depending on he bonus sraegy of he company, he individual conracs may or may no be fair, bu for he insured porfolio as a whole he conracs are fair if 2.4.2 is fulfilled. Since he asses are invesed in a self-financing porfolio we have E Q [e rt A T ] = A, such ha 2.4.1 is saisfied if and only if 2.4.2 is saisfied. Hence, deermining he fair value of ρ, if i exiss, amouns o solve 2.4.1 wih respec o ρ. 2.4.3 Buy and hold sraegy Consider a buy and hold sraegy, which is he simples example of a self-financing sraegy. In he buy and hold sraegy he company invess ϑs and η, respecively, in he risky and he riskfree asse a ime and no rading akes place unil ime T. Hence, he value a ime T of he risky porfolio is V T ϕ = ϑs T + ηe rt. Assume he company follows a buy and hold sraegy wih ϑ >, i.e. wih some invesmens in he risky asse. We now derive an implici expression for he fair value of ρ by decomposing he value of he equiy capial a ime T ino payoffs from sandard European opions on he sock. Define he quaniies s 1 and s 2 as he values of S T which solve he wo equaions and G T V = e rt E + ϑs T + ηe rt, 2.4.3 K T V 1 + γ = ϑs T + ηe rt, 2.4.4

36 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE respecively. Hence, s 1 is he lowes sock price a ime T, which does no lead o bankrupcy of he insurance company, while s 2 is he lowes sock price for which, he company uses K T as accumulaion facor. Solving 2.4.3 and 2.4.4 for S T we ge and s 1 = G TV ηe rt e rt E, 2.4.5 ϑ s 2 = K TV 1 + γ ηe rt ϑ. 2.4.6 Noe ha even hough he sock price is posiive, s 1 and s 2 migh be negaive. If s 1 is negaive, he capial invesed in he savings accoun is sufficien o ensure ha he company is no bankruped, whereas a negaive value for s 2 corresponds o he case, where he capial invesed in he savings accoun is sufficien o ensure ha he company always uses K T o accumulae he deposi. Using s 1 and s 2, we can rewrie he value of he equiy capial a ime T as E B T = 1 S T <s 1 E B T + 1 s 1 S T <s 2 E B T + 1 s 2 S T E B T EB1 T + EB2 T + EB3 T. Here, he superscrip B indicaes ha we are working wih a buy and hold sraegy. The expressions for he equiy capial in he differen siuaions can be found in Secion 2.4.1. Since E B1 T is he equiy capial in case of bankrupcy, i is equal o. In order o decompose E B2 T E B2 T, we firs recall ha = 1 s1 S T <s 2 min e r+ρt E, V T ϕ + e rt E G T V. 2.4.7 In order o calculae 2.4.7, we deermine s 3 which is he maximum value of S T for which V T ϕ + e rt E G T V e r+ρt E. 2.4.8 Hence s 3 is he larges value for he sock price a ime T for which he asses are insufficien o accumulae he equiy capial wih ineres rae r + ρ, if we accumulae he deposi wih G T. Solving 2.4.8 we ge Rewriing s 3 as s 3 = eρt 1e rt E + G T V ηe rt ϑ s 3 = s 1 + er+ρt E, ϑ. 2.4.9 and using ha minr,ρ > and ϑ > we observe ha s 3 > s 1, such ha insering in 2.4.7 gives E B2 T = 1 s1 S T <mins 2,s 3 e rt E + V T ϕ G T V + 1mins2,s 3 S T <s 2 e r+ρt E = 1 s1 S T <mins 2,s 3 ϑ S T s 1 + 1 mins2,s 3 S T <s 2 e r+ρt E = ϑ S T s 1 + S T mins 2,s 3 + mins 2,s 3 s 1 1 mins2,s 3 <S T + e r+ρt E 1mins2,s 3 S T 1 s2 S T. 2.4.1

2.4. CAPITAL INSURANCES 37 Thus, ET B2 can be decomposed ino wo erms. The firs erm is he number of socks muliplied by he difference beween he payoff from wo European call opions wih srikes s 1 and mins 2,s 3 subraced he payoff from a so-called binary cash call opion wih srike mins 2,s 3 and cash mins 2,s 3 s 1. The second erm is he equiy capial accumulaed wih ineres rae r + ρ muliplied by he difference beween he payoff from wo binary cash call opions wih srikes mins 2,s 3 and s 2. For a descripion of he hese and oher opions menioned in his chaper see Musiela and Rukowski 1997. In order o decompose ET B3 we firs deermine s 4, which is he larges value of S T for which Solving for S T we ge e rt E + V T ϕ K T V e r+ρt E. s 4 = eρt 1e rt E + K T V ηe rt. 2.4.11 ϑ The inerpreaion of s 4 is similar o ha of s 3, however here he deposi is accumulaed wih K T. Calculaions similar o hose for E B2 give E B3 T = 1 s2 S T min e r+ρt E,e rt E + V T ϕ K T V = 1 s2 S T <maxs 2,s 4 ϑ S T s 5 + 1 maxs2,s 4 S T e r+ρt E = ϑ S T s 2 + S T maxs 2,s 4 + + 1 s2 S T s 2 s 5 1 maxs2,s 4 S T maxs 2,s 4 s 5 where we have used he noaion T + 1 maxs2,s 4 S T e r+ρt E, 2.4.12 s 5 = K TV ηe rt e rt E. 2.4.13 ϑ Hence, ET B3 can be decomposed ino wo erms as well. The firs erm is he number of socks muliplied by he payoff from known European opions, and he second erm is he equiy capial accumulaed wih ineres rae r + ρ muliplied by he payoff from a binary cash call opion. Denoe by BCC and C, respecively, he price of a binary cash call and a call opion. I is well-known ha BCC and C are given by and [ BCCx,S,σ = E Q e rt ] e 1 rt Φ x ST = log S x +r 1 2 σ2 T σ T, x >, e rt, x, [ Cx,S,σ = E Q e rt S T x +] log S x +r+ 1 2 S = Φ σ2 T log S x +r σ e rt 1 2 xφ σ2 T T σ, x >, T S e rt x, x,

38 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE where Φ denoes he disribuion funcion for he sandard normal disribuion. To simplify noaion, we use he shor hand noaion BCCx and Cx in expressions involving many opion prices wih he same iniial value and volailiy. Applying crierion 2.4.1 we obain he following proposiion Proposiion 2.4.3 If a company invess according o a buy and hold sraegy he fair value of ρ saisfies E = e r+ρt E BCCmins 2,s 3 BCCs 2 + BCCmaxs 2,s 4 + ϑ Cs 1 Cmins 2,s 3 + Cs 2 Cmaxs 2,s 4 mins 2,s 3 s 1 BCCmins 2,s 3 + s 2 s 5 BCCs 2 maxs 2,s 4 s 5 BCCmaxs 2,s 4, where s 1 s 5 are given by 2.4.5, 2.4.6, 2.4.9, 2.4.11 and 2.4.13 and all opion prices are calculaed using iniial value S and volailiy σ. If ϑ =, all asses are invesed in he savings accoun. Hence, he value a ime T of he asses is deerminisic and equal o A T = e rt A. In his case we obain he following resul for he fair value of ρ. Proposiion 2.4.4 If a company invess in he savings accoun only, a fair value of ρ mus saisfy 1. If e rt A < G T V hen no values of ρ exis for which he disribuion scheme fair. 2. If G T V e rt A < K T V 1 + γ + e rt E, hen he disribuion scheme is fair, if eiher of he following apply a e rt E < e rt A G T V and ρ =. b G T = e rt V +U V and ρ. 3. If K T V 1+γ+e rt E < e rt A, hen he disribuion scheme is fair, if eiher of he following apply a e rt E < e rt A K T V and ρ =. b K T = e rt V +U V and ρ. Proof of Proposiion 2.4.4: See Secion 2.1.1.

2.4. CAPITAL INSURANCES 39 Proposiion 2.4.4 has he following inerpreaion: If he asses and hence he accouns on he liabiliy side are deerminisic a ime T he disribuion scheme is fair if only if E T = e rt E. Since his is inuiively clear, he proposiion is no paricularly ineresing and saed for compleeness, only. We end his secion wih a resul for he probabiliy of ruin of he company a ime T. Proposiion 2.4.5 The probabiliy, p ruin ϕ, ha a company, using he buy and hold sraegy ϕ, is ruined a ime T is given by p ruin ϕ = Φ log s1 S α 1 2 σ2 T σ T. Proof of Proposiion 2.4.5: The company is ruined a ime T if A T < G T V T. Hence, [ p ruin ϕ = P [A T < G T V ] = P S T < G TV T ηe rt e rt ] E = P [S T < s 1 ]. ϑ Here, we have used he definiion of s 1 from 2.4.5. The resul now follows by insering he soluion, S T = S e α σ2 /2T+σ W T, o he sochasic differenial equaion for he dynamics of S under P given in 2.3.1. 2.4.4 Consan relaive porfolio weighs Now consider he case where he company coninuously adjuss he invesmen porfolio, such ha a all imes, [, T], he proporion δ [, 1] of he porfolio value is invesed in socks. Hence, ϑ S = δv ϕ and η B = 1 δv ϕ. In his case he dynamics under Q of he value process of he self-financing sraegy are dv ϕ = ϑ ds + η db = ϑ rs d + σs dw + η rb d = rv ϕd + δσv ϕdw. We noe ha he dynamics of he value process are of he same form as he dynamics of he sock price. For δ > calculaions similar o hose for a buy and hold sraegy give Proposiion 2.4.6 When invesing in a porfolio wih consan relaive porfolio weighs he fair value of ρ solves he following equaion E = e r+ρt E BCCminv 2,v 3 BCCv 2 + BCCmaxv 2,v 4 + Cv 1 Cminv 2,v 3 + Cv 2 Cmaxv 2,v 4 minv 2,v 3 v 1 BCCminv 2,v 3 + v 2 v 5 BCCv 2 maxv 2,v 4 v 5 BCCmaxv 2,v 4,

4 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE where v 1 = G T V e rt E, v 2 = K T V 1 + γ, v 3 = e ρt 1e rt E + G T V, v 4 = e ρt 1e rt E + K T V, v 5 = K T V e rt E, and all opion prices are calculaed wih iniial value V + U and volailiy δσ. If δ = we are in exacly he same siuaion as in he buy and hold sraegy wih ϑ =, so Proposiion 2.4.4 applies. Noe ha under P he dynamics of he value process for a self-financing sraegy wih consan relaive porfolio weighs are dv ϕ = ϑ ds + η db = ϑ αs d + σs d W + η rb d = r + δα rv ϕd + δσv ϕd W. This leads o he following proposiion for he probabiliy of ruin a ime T. Proposiion 2.4.7 The probabiliy of ruin, p ruin ϕ, is given by p ruin ϕ = Φ log v1 V +U r + δα r 1 2 δσ2 T δσ T. 2.4.5 Buy and hold wih sop-loss if solvency is hreaened Consider he case where he regulaory insiuions se a solvency requiremen for he insurance company. As in pracice, he requiremen considered here is a requiremen on he equiy capial. Afer accumulaing he deposi a ime T, he equiy capial should be a leas a proporion β of he deposi, i.e. E T βv T. Since he solvency requiremen mus be saisfied a he end of each accumulaion period we know ha E βv. Oherwise he company would have been declared insolven already. If furher e rt E βk T V and A e rt G T V 1 + β he company may avoid insolvency by rebalancing he risky porfolio o include invesmens in he savings accoun only, if he value of he asses reaches he lower boundary A = e rt G T V 1 + β. 2.4.14 Now assume he company invess in a buy and hold sraegy as inroduced in Secion 2.4.3, unil a possible inervenion. In his case we can wrie 2.4.14 in erms of he discouned

2.4. CAPITAL INSURANCES 41 sock price S = e rt G T V 1 + β η E ϑ H. Remark 2.4.8 The sop-loss crierion in 2.4.14 is jus one of many possible crierions. If V + U e rt K T V 1 + γ he alernaive crierion V ϕ = e rt K T V 1 + γ in addiion o solvency also guaranees ha K T is used as accumulaion facor. Decomposing he equiy capial we firs disinguish beween wheher he company has inervened or no E BS T = 1 inf T S H E BS T + 1 inf T S >H E BS T E BS1 T + E BS2 T. Here, he superscrip BS indicaes ha he company uses a buy and hold sraegy wih sop-loss. When inf T S H he asse value is deerminisic and equal o G T V 1+β, such ha E BS1 T = 1 inf T S H min e r+ρt E,βG T V = 1 inf T S H βg T V. Here, we have used ha e rt E βk T V βg T V in boh equaliies and ρ in he las equaliy. We recognize his as he payoff from a down-and-in barrier opion on he discouned sock price wih he deerminisic payoff βg T V when knocked in. When inf T S > H i holds in paricular ha S T > G TV 1 + β ηe rt e rt E ϑ s β 1. The assumpions on he equiy capial and he fac ha ρ gives ha s 3 s β 1. Hence, calculaions similar o hose leading o 2.4.1 and 2.4.12 gives E BS2 T = 1 inf T S >H e r+ρt E 1 mins 2,s 3 S T 1 s 2 ST + 1 maxs 2,s 4 S T + ϑ e rt + ST sβ, 1 S T mins 2,s 3 + + ST s 2 + ST maxs 2,s 4 + + s β 1 s 1 1 s β, 1 <ST mins 2,s 3 s 1 1 mins 2,s 3 <S T + s 2 s 5 1 s 2 S T 1 maxs 2,s 4 S T maxs 2,s 4 s 5. Thus, he equiy capial can be wrien in erms of payoffs from barrier opions on he discouned sock price. To indicae ha an opion is wrien on he discouned sock price, we equip he opion price by an aserisk. When working wih barrier opions we equip he noaion for he corresponding European opion, or 1 in case of a deerminisic value, wih wo leers as a sub- or superscrip depending on wheher we are dealing wih a down

42 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE or an up barrier opion. The firs leer is he barrier and he second describe wheher we are dealing wih an ou, denoed O, or an in, denoed I, opion. Using Björk 24, Theorem 18.8 we are able o wrie prices of he relevan barrier opions on he discouned sock price in erms of prices of European opions. For S > H we obain he following opion prices: A down-and-ou opion wih payoff 1 [ ] 1 HOS,σ = E Q e rt 1 inf T S >H log S e = rt Φ H 1 2 σ2 T σ S T H e rt log H S Φ 1 2 σ2 T σ, H >, T e rt, H, a down-and-ou binary cash call opion [ BCCHO x,s,σ = E Q e rt 1 inf T S >H 1 x S T BCC x,s,σ S H BCC x, H2 S,σ, < H x, = 1 H S,σ, max,x H, BCC x,s,σ, H < x, e rt, maxx,h, and a down-and-ou call opion ] [ CHO x,s,σ = E Q e rt 1 inf T S >H ST x+] C x,s,σ S H C x, H2 S,σ, < H x, C = x,s,σ, H < x, e rt S x, maxh,x, C H,S,σ S H C H, H2 S,σ + H x1 HO S,σ, max,x H. Here he prices BCC and C can be calculaed from he formulas for BBC and C using BCC x,s,σ = BCCe rt x,s,σ, C x,s,σ = e rt Ce rt x,s,σ. For S H all down-and-ou opions have a price equal o. For a down-and-in opion wih payoff 1 we have for all S 1 HIS,σ = e rt 1 HOS,σ. The following proposiion now follows from applying crierion 2.4.1. Proposiion 2.4.9 If a company follows a buy and hold sraegy wih sop-loss in case solvency is hreaened

2.4. CAPITAL INSURANCES 43 he fair value of ρ mus saisfy E = 1 HI βg TV + e r+ρt E BCC HO min s 2,s 3 BCC HO s 2 + BCC HO maxs 2,s 4 + ϑ e rt C HO s β, 1 + s β 1 s 1 BCCHO CHO min s 2,s 3 + CHO s 2 CHO maxs 2,s 4 s β, 1 mins 2,s 3 s 1 BCCHO min s 2,s 3 + s 2 s 5 BCC HO s 2 maxs 2,s 4 s 5 BCC HO maxs 2,s 4, where all opion prices are calculaed wih iniial value S and volailiy σ. 2.4.6 Consan relaive amoun δ in socks unil solvency is hreaened Now assume ha a company, whose asses a ime fulfill A e rt G T V 1+β, iniially invess in a porfolio wih consan relaive porfolio weighs as described in Secion 2.4.4. As in Secion 2.4.5 he company rebalances he invesmen porfolio o include he riskfree asse only, he firs ime 2.4.14 holds. Wrien in erms of he discouned value process of he invesmen porfolio he company rebalances he porfolio if V ϕ = e rt G T V 1 + β E H. As in Secion 2.4.5 we know ha E βv and furher assume ha e rt E βk T V. The following proposiion now follows from Proposiion 2.4.9 in he same way as Proposiion 2.4.6 followed from Proposiion 2.4.3 Proposiion 2.4.1 For a company invesing in a porfolio wih consan relaive porfolio weighs unil solvency is hreaened he fair value of ρ mus saisfy E = 1 HI βg TV + e r+ρt E BCC HO minv 2,v3 BCC HO v 2 + BCC HO maxv 2,v4 + e rt C HO v β, 1 C HO min v 2,v3 + C HO v 2 C HO maxv 2,v4 + v β 1 v 1 min v 2,v 3 v 1 BCC HO min v 2,v 3 BCC HO v β, 1 + v 2 v 5 BCC HO v 2 maxv 2,v 4 v 5 BCC HO maxv 2,v 4, where all opion prices are calculaed wih iniial value V + U and volailiy δσ and v β 1 = G TV 1 + β e rt E.

44 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE 2.5 Pure endowmens We now consider a company whose insurance porfolio consiss of pure endowmens. To carry ou he sudy we firs exend he probabilisic model o include he developmen of a porfolio of insured lives. This is done following he approach in Møller 1998. 2.5.1 The model for he insurance porfolio Consider an insurance porfolio consising a ime of Y insured lives of he same age, say x. We assume ha he individual remaining lifeimes a ime of he insured are described by a sequence T 1,...,T Y of i.i.d. non-negaive random variables defined on Ω, F,P. We furher make he naural assumpion ha he disribuion of T i is absoluely coninuous and PT i > T >, such ha he moraliy inensiy µ x+ is well-defined on [,T]. The survival probabiliy from ime o, [,T] for one individual in he insurance porfolio is given by p x PT i > = e Ê µ x+udu. Denoe by q x he probabiliy of deah from ime o, i.e. q x = 1 p x. Now define he processes Y = Y T and N = N T by Y Y Y = 1 Ti > and N = i=1 i=1 1 Ti. Then Y and N, respecively, denoe he number of survivors and he number of deahs in he insurance porfolio a ime. The filraion H = H T is he P-augmenaion of he naural filraion generaed by N, i.e. H = H + N, where H + = σ{n u,u }. Since he probabiliy of wo individuals dying a he same ime is, hen N is a 1- dimensional couning process. The i.i.d. assumpion on he remaining lifeimes furher gives ha N is an H-Markov process. The sochasic inensiy process λ = λ T of N under P can now be informally defined by λ d E P [dn H ] = Y N µ x+ d. Thus, he probabiliy of experiencing a deah in he porfolio in he nex shor inerval is he number of survivors muliplied by he probabiliy of one person dying. I is well-known ha he process M defined by M = N is an H-maringale under P. λ u du = N Y N u µ x+u du, T,

2.5. PURE ENDOWMENTS 45 2.5.2 The combined model Now inroduce he filraion F = F T for he combined model of he economy and he insurance porfolio by F = G H. Assume ha he economy is sochasically independen of he developmen of he insurance porfolio, i.e. G and H are independen. This ensures ha he properies of M and W are inheried in he larger filraion F. We now address he choice of equivalen maringale measure in he combined model. For any F-predicable funcion h, h > 1, we can define a likelihood process L = L T by dl = L h dm, L = 1, and consruc a new measure equivalen o P by dq h dp = O TL T. 2.5.1 We noe ha h = corresponds o Q defined in Secion 2.3. The measure Q h defined by 2.5.1 is a probabiliy measure if E Q [L T ] = 1, or equivalenly, if E P [O T L T ] = 1. To preserve he independence beween G and H under Q h we resric ourselves o funcions h which are H-predicable. Under his addiional assumpion, all measures Q h defined by 2.5.1 are equivalen maringale measures if E P [L T ] = 1, see Møller 1998 for he necessary calculaions. Girsanov s heorem for poin processes, see e.g. Andersen, Borgan, Gill and Keiding 1993, gives ha he sochasic inensiy process λ h = λ h T for N under Q h is given by λ h = 1 + h λ = Y N 1 + h µ x+. Hence, changing measure from Q o Q h can be inerpreed as changing he moraliy inensiy from µ x+ o 1 + h µ x+. Wih his inerpreaion he survival probabiliy under Q h is given by p h x = Qh T i > = e Ê 1+huµ x+udu. The probabiliy of deah under Q h is given by q h x = 1 p h x. We noe ha if h is on he form h,n hen N is a Markov process under Q h as well as under P. Since no unique equivalen maringale measure exiss for he combined model, no all coningen claims in B,S, F have unique prices. However, since B,S, G is complee, all coningen claims depending only on he financial marke sill have unique prices. To find unique prices for coningen claims depending on he developmen of he insurance porfolio, we henceforh consider a fixed, bu arbirary, equivalen maringale measure Q h. Moivaed by he srong law of large numbers, he measure Q, corresponding o risk neuraliy wih respec o unsysemaic moraliy risk, is frequenly used in he lieraure o price insurance conracs wih financial risk, see e.g. Aase and Persson 1994 and Møller 1998. Møller 1998 also recognizes Q as he minimal maringale measure for he considered model.

46 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE 2.5.3 The developmen of he deposi in a 1-period model Now assume all insured in he porfolio inroduced in Secion 2.5.1 have purchased idenical pure endowmens wih erminaion a ime T or laer. If premiums are paid before or a ime and he porfolio of insured lives develop exacly as expeced, he porfolio-wide deposi a ime T is given by V de T = E T V de. Here, E T {G T,K T } is he deposi accumulaion facor for he inerval,t], and he superscrip de refers o a deerminisic developmen of he insured porfolio. Dividing by he number of expeced survivors we obain an expression for he developmen of he deposi of one insured surviving o ime T V ind T = E T V ind 1. Tp x Thus, he porfolio-wide deposi a ime T is given by V T = Y T V ind T = Y T E T V ind 1. 2.5.2 Tp x 2.5.4 Disribuion scheme Using 2.5.2 we define a disribuion scheme, similar o he disribuion scheme from Secion 2.4.1, which is used by he company in case of a porfolio of pure endowmens: 1 1. A T < Y T G T V ind T p x : Here, he asses are insufficien o mee he guaraneed deposi a ime T for all he survivors in he insured porfolio. Hence, he company is declared bankrup and all capial is allocaed o he deposi. 2. Y T G T V ind 1 T p x A T < Y T K T V ind 1 V T = A T, E T =, U T =. T p x 1 + γ + e rt E : In his case he asses are sufficien o mee he guaranee. However, accumulaing wih he announced accumulaion facor leaves he company wih a bonus reserve less han he minimal arge, γv T. Thus, as in he case of capial insurances he company uses G T o accumulae. Similarly o Secion 2.4.1 he capial is disribued as follows V T = Y T G T V ind 1, Tp x E T = min e r+ρt E,A T V T, U T = A T V T E T.

2.5. PURE ENDOWMENTS 47 1 3. e rt E + Y T K T V ind T p x 1 + γ A T : Here, he invesmens and he developmen of he insurance porfolio allow he company o accumulae using he announced deposi rae and sill have a bonus reserve above he minimal arge. The disribuion is similar o he one above wih G T replaced by K T V T = Y T K T V ind 1, Tp x E T = min e r+ρt E,A T V T, U T = A T V T E T. Noe ha we by he above disribuion scheme implicily consider he moraliy inensiy as guaraneed, since i is used even if he porfolio of insured behaves worse han anicipaed. Thus, in he presen siuaion he addiional ineres rae ρ is a compensaion for boh financial and unsysemaic moraliy risk. As in he case of capial insurances, he equiy capial is only used o cover he accumulaion of he deposi if he payoff generaed by he deposi and bonus reserve is insufficien. 2.5.5 Fair disribuion From Secion 2.5.4 we noe ha E T and V T +U T can be viewed as coningen claims in he combined model B,S, F. As in he case of capial insurances, we define he disribuion scheme as fair if i does no include an arbirage possibiliy for eiher he company or he porfolio of insured, i.e. if and The relaion E = e rt E Qh [E T ], 2.5.3 V + U = e rt E Qh [V T + U T ]. 2.5.4 E Qh [e rt A T ] = A, now ensures ha 2.5.3 holds if and only if 2.5.4 holds, such ha we may consider 2.5.3 only. Using he law of ieraed expecaions we can wrie 2.5.3 as E Qh [E T ] = E Qh E Qh [E T H T ] Y [ Y = T p hx n T q hx Y n E Qh n n= + 1 ngt V ind 1 A T <ng T V ind 1 T px A T <nk T V ind 1 T px 1+γ+erT E min + 1 nkt V 1 T px 1+γ V T ϕ min px 1 T e r+ρt E,A T nk T V ind e r+ρt E,A T ng T V ind 1 Tp x 1 Tp x ]. 2.5.5

48 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE Recall ha wih respec o he financial marke all measures Q h are idenical. Thus, he expecaion can be viewed as a weighed average of Y + 1 porfolios of capial insurances wih iniial deposi nv ind 1 T p x, n =,1,...,Y, respecively. Hence, mos calculaions necessary o derive an implici equaion for ρ are idenical o hose already carried ou in Secion 2.4. Remark 2.5.1 Noe ha since all insured have idenical conracs, he individual conracs are fair if he bonus reserve a he ime of purchase was and a possible bonus reserve a ime of erminaion is disribued among he survivors in he insurance porfolio. 2.5.6 Buy and hold When he company follows a buy and hold sraegy he fair value of ρ is given by he following proposiion Proposiion 2.5.2 If an insurance company, whose porfolio consiss of Y pure endowmens, follows a buy and hold sraegy, hen he fair value of ρ saisfies Y E = n= Y T p h n x n T qx h Y n + BCCmins n 2,s n 3 BCC s n 2 + ϑ e r+ρt E BCCmaxs n 2,s n 4 Cs n 1 Cmins n 2,s n 3 + Cs n 2 Cmaxs n 2,s n 4 mins n 2,s n 3 s n 1BCCmins n 2,s n 3 + s n 2 s n 5BCCs n 2 maxs n 2,s n 4 s n 5BCC maxs n 2,s n 4, where s n 1 = ng TV ind s n 2 = nk TV ind s n 3 = 1 T p x ηe rt e rt E, ϑ 1 T p x 1 + γ ηe rt, ϑ e ρt 1 e rt E + ng T V ind ϑ e ρt s n 4 = 1 e rt E + nk T V ind ϑ s n 5 = nk TV ind 1 T p x ηe rt e rt E ϑ 1 T p x ηe rt 1, T p x ηe rt, Here, all opion prices are calculaed using iniial value S and volailiy σ..

2.5. PURE ENDOWMENTS 49 Again we are ineresed in he probabiliy ha he company is ruined a ime T. Proposiion 2.5.3 The probabiliy of ruin, p ruin ϕ, a ime T for a company following a buy and hold sraegy is Y Y p ruin ϕ = T p x n T q x Y n Φ log sn 1 S α 1 2 σ2 T n σ. T n= Proof of Proposiion 2.5.3: Using ieraed expecaions we ge [ ] p ruin ϕ = P A T < Y T G T V ind 1 Tp x [ ]] = E [P P A T < Y T G T V ind 1 Tp x H T Y [ ] Y = T p x n T q x Y n P A T < ng T V ind 1. n Tp x n= The resul now follows immediaely from Proposiion 2.4.5 and he definiion of s n 1. 2.5.7 Consan relaive porfolio In he case of invesmens in a porfolio wih consan relaive porfolio weighs we obain he following proposiion from 2.5.5. Proposiion 2.5.4 For a company invesing in a porfolio wih consan relaive porfolio weighs he fair value of ρ is he soluion o he following equaion Y Y E = T p h x n n T qx h Y n e r+ρt E BCCmaxv 2 n,vn 4 n= + BCCminv2 n,vn 3 BCCvn 2 + Cv n 1 Cminv n 2,v n 3 + Cv n 2 Cmaxv n 2,v n 4 minv n 2,vn 3 vn 1 BCCminvn 2,vn 3 + v n 2 v n 5BCCv n 2 maxv n 2,v n 4 v n 5 BCCmaxv n 2,v n 4,

5 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE where v n 1 = ng TV ind v n 2 = nk T V ind 1 e rt E, 2.5.6 Tp x 1 1 + γ, 2.5.7 Tp x 1, Tp x 1, Tp x 1 e rt E. 2.5.8 Tp x v n 3 = e ρt 1 e rt E + ng T V ind v n 4 = e ρt 1 e rt E + nk T V ind v n 5 = nk T V ind All opion prices above are calculaed using iniial value V + U and volailiy δσ. Calculaions similar o he case of invesmens in a buy and hold sraegy gives he following resul for he ruin probabiliy. Proposiion 2.5.5 If a company, whose insurance porfolio consiss of pure endowmens, invess in a porfolio wih consan relaive porfolio weighs, hen he probabiliy of ruin a ime T is given by Y p ruin ϕ = n= Y n T p x n T q x Y n Φ log vn 1 V +U r + δα r 1 2 δσ2 T δσ. T 2.5.8 Buy and hold wih sop-loss if solvency is hreaened Assume he solvency requiremen deermined by he regulaory insiuions is given by E T βy T VT ind. Hence, E βy V ind, since he company oherwise would be insolven already a ime. Here, we furher assume ha he iniial asses of he company fulfills A e rt Y G T V ind Tp h x 1 + β. Tp x To avoid accumulaing wih K T in siuaions where his leads o insolvency, we require ha e rt E βk T Y V ind 1 1 T p x. Thus, he facor T p x makes he assumpion on he iniial equiy capial sronger han in he case of capial insurances. A ime he company invess in a buy and hold sraegy. However, o decrease he probabiliy of insolvency he company rebalances he invesmen porfolio o include invesmens in he savings accoun only, if he asses hi he lower boundary [ ] A = E Qh e rt Y T G T V ind 1 1 + β = e rt Y G T V ind Tp h x 1 + β. 2.5.9 Tp x Tp x Thus, disregarding he informaion a ime abou he developmen of he insurance porfolio he company rebalances he porfolio if he value of he solvency requiremen is

2.5. PURE ENDOWMENTS 51 equal o he asses. The advanage of 2.5.9 is ha i can be wrien as S = e rt Y G T V ind T p h x T p x 1 + β η E ϑ Z. Hence, as in Secion 2.4 he requiremen on he asses can be ransformed ino a barrier problem for he discouned sock price wih a consan barrier. Remark 2.5.6 A naural exension of 2.5.9 is o ake he developmen of he insurance porfolio ino accoun. This gives he crierion A = E Qh [ e rt Y T G T V ind ] 1 1 + β Tp x F = e rt Y G T V ind T p h x+ 1 + β. Tp x 2.5.1 However, his crierion does no allow us o wrie he problem as a consan barrier problem. Boh crierion 2.5.9 and 2.5.1 leave he company wih a posiive probabiliy of insolvency. To avoid insolvency almos surely, we could assume ha A e rt Y G T V ind 1 1 + β, Tp x and use he inervenion crierion A = e rt Y G T V ind 1 1 + β, Tp x which corresponds o assuming ha all insured persons, which are alive a ime survive o ime T. In order o use 2.5.5 we consider a fixed number of survivors, say n. Given he number of survivors he equiy capial can be decomposed ino a erm E n,bs1 T, which is differen from if he company has inervened and a erm E n,bs2 T, which is non-zero if he company

52 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE has no inervened. For E n,bs1 T we obain E T = 1 inf T S Z + 1 ngt V ind min + 1 nkt V ind 1 Y px G T V ind T ph x T px 1+β<nG T V ind 1 T 1 T px Y G T V ind T ph x T px 1+β<nK T V ind 1 T px 1+γ+erT E e r+ρt E,Y G T V ind Tp h x 1 + β ng T V ind 1 Tp x Tp x 1 T px 1+γ Y G T V ind T ph x T px 1+β ert E min e r+ρt E,Y G T V ind Tp h x 1 + β nk T V ind 1 Tp x Tp x = 1 inf T S 1 Z v6 <v1 n + 1 v n 1 v 6 <v2 n min e r+ρt E,v 6 v1 n + 1 v n 2 v 6 min e r+ρt E,v 6 v5 n, where v n 1, vn 2 and vn 5 are given by 2.5.6, 2.5.7 and 2.5.8, respecively, and For E n,bs2 T v 6 = Y G T V ind Tp h x 1 + β e rt E. Tp x he calculaions in Secion 2.4.5 applies. Thus, we ge Proposiion 2.5.7 In he siuaion wih sop-loss he fair value of ρ mus saisfy Y Y E = T p h n x n T qx h Y n 1 ZI 1 v n 1 v 6 <v2 n min e r+ρt E,v 6 v1 n where n= + 1 v n 2 v 6 min e r+ρt E,v 6 v5 n + e r+ρt E + BCCZO min s n, 2,sn, 3 BCC ZO s n, 2 + ϑ e rt C ZOs β,n, 1 C ZO BCC ZO maxs n, 2,sn, 4 min s n, 2,sn, 3 + C ZO s n, 2 C ZOmaxs n, 2,sn, 4 + s β,n 1 s n 1BCCZOs β,n, 1 min s n 2,s n 3 s n 1BCCZO min s n, 2,sn, 3 + s n 2 sn 5 BCC ZO sn, 2 maxsn 2,sn 4 sn 5 BCC ZO maxs n, 2,sn, 4, 1 s β,n 1 = ng TV ind T p x 1 + β ηe rt e rt E. ϑ Here, all opion prices are calculaed wih iniial value S and volailiy σ.

2.5. PURE ENDOWMENTS 53 The probabiliy of insolvency for a company following he invesmen sraegy described above is given in he following proposiion Proposiion 2.5.8 For a company following a buy and hold sraegy wih sop-loss he probabiliy of insolvency is given by Y Y p ins ϕ = T p x n T q x Y n 1 n YT p h x <n n= Z + S 2α r σ 2 1 1 Φ log Φ log s β,n, 1 S Z 2 α r 1 2 σ2 T σ T s β,n, 1 S Z 2 α r 1 2 σ2 T σ T. Proof of Proposiion 2.5.8: See Secion 2.1.4. 2.5.9 Consan relaive amoun δ in socks unil solvency is hreaened Consider he same se-up as in Secion 2.5.8. The only difference is ha he company invess in a sraegy wih consan relaive porfolio weighs unil a possible inervenion. Wrien in erms of he discouned value of he porfolio including risky invesmens he rebalancing akes place he firs ime V ϕ = e rt Y G T V ind Tp h x 1 + β E Tp Z. x The resul now follows from calculaions similar o hose in Secion 2.5.8. Proposiion 2.5.9 In he siuaion wih sop-loss he fair value of ρ mus saisfy Y E = n= Y T p h x n n T qx h Y n 1 ZI 1 v n 1 v 6 <v2 n min e r+ρt E,v 6 v1 n + e r+ρt E + 1 v n 2 v 6 min + BCC ZO e r+ρt E,v 6 v5 n BCC ZO min v n, 2,vn, 3 + e rt C vβ,n, ZO 1 C ZO min v n, 2,vn, 3 v n, 2 BCC ZO + v β,n 1 v1 n vβ,n, BCC ZO 1 min v2 n,vn 3 vn 1 BCC ZO maxv n, 2,v n, 4 + C ZO vn, 2 maxvn, C ZO 2,vn, 4 min v n, 2,vn, 3 + v2 n v5bcc n vn, ZO 2 maxv2 n,v4 n v5bcc n ZO maxv n, 2,vn, 4,

54 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE where 1 = ng T V ind 1 1 + β e rt E. Tp x v β,n Here, all opion prices are calculaed wih iniial value V + U and volailiy δσ. Now calculaions similar o hose leading o Proposiion 2.5.8 give Proposiion 2.5.1 For a company following a sraegy wih consan relaive porfolio weighs wih sop-loss he probabiliy of insolvency is given by p ins ϕ Y Y = T p x n T q x Y n 1 n YT p h x <n Φ n= + Z V + U 2δα r δσ 2 1 1 Φ log log v β,n, 1 V +U Z 2 δα r 1 2 δσ2 T δσ T v β,n, 1 V +U Z 2 δα r 1 2 δσ2 T δσ T. 2.6 Numerical resuls Since we obain implici equaions for ρ only, we now resor o numerical echniques o obain fair values of ρ. We rewrie he expressions for he fair value of ρ on he form ρ = fρ for some funcion f and use ieraions o find fix poins for f. For all numerical calculaions we assume ha ime is measured in years and le T = 1. For an overview of he noaion used in his chaper we refer o Table 2.6.1. 2.6.1 Dependence on invesmen sraegy In his secion we fix he parameers r =.6, σ =.2, G T = 1.45, K T = 1.6 and γ =.1 and consider he dependence of ρ on he invesmen sraegy. For now we assume he iniial capial is disribued as follows: V = 1, U = 1 and E = 1. Figure 2.6.1 hen shows he dependence of ρ on he relaive iniial invesmen in socks for a buy and hold sraegy and a consan relaive porfolio. The relaive iniial invesmen in socks is given by κ = ϑs /V ϕ for he buy and hold sraegy and by δ for he consan relaive porfolio. The observaions o be made from Figure 2.6.1 are wofold. Firsly, ρ is an increasing funcion of he relaive iniial invesmen in socks for boh invesmen sraegies. This is no surprising, since ρ is a measure for he risk of he insurance company and invesing in socks increases he risk. Secondly, comparing

2.6. NUMERICAL RESULTS 55 Symbol V U E S Vϕ ρ r σ G T K T γ T β ϑ δ Y h Inerpreaion Porfolio-wide deposi Bonus reserve Equiy capial Sock price Value of invesmen porfolio ϕ Fair addiional rae of reurn o equiy capial Riskfree ineres rae Volailiy of sock Guaraneed accumulaion facor Announced accumulaion facor Targe for minimal bonus reserve per deposi Lengh of accumulaion period Solvency requiremen on equiy capial per deposi Number of socks held in a buy and hold sraegy Consan proporion invesed in socks Number of survivors in insurance porfolio Marke aiude owards unsysemaic moraliy risk Table 2.6.1: Overview of noaion ρ..1.2.3.4 Consan relaive porfolio Buy and hold..2.4.6.8 1. κ or δ Figure 2.6.1: ρ as a funcion of he relaive iniial invesmen in socks.

56 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE he wo invesmen sraegies, we observe ha for a relaive iniial invesmen in socks beween.2 and.7 he fair value of ρ is slighly higher when invesing in a consan relaive porfolio raher han following a buy and hold sraegy. This may be explained by he fac ha when invesing in a porfolio wih consan relaive porfolio weighs a decrease in he sock price leads o addiional invesmens in socks and hence an increase in he capial a risk. Comparing he sraegies we also noe ha he values of ρ coincide in he exremes where none or all capial is invesed in socks. This relies on he fac ha he sraegies coincide in hese wo siuaions. ρ.5.1.15.2 Buy and hold wih sop loss Buy and hold wihou sop loss..2.4.6.8.1 Figure 2.6.2: ρ as a funcion of β for κ =.5. β In order o invesigae he dependence of β we consider a buy and hold sraegy wih soploss. The iniial disribuion of capial is changed such ha U = 5, since he dependence is more obvious in his case. The dependence of ρ on he required solvency margin β is now shown in Figure 2.6.2 for κ =.5. We observe ha ρ is a decreasing funcion of β. This is also inuiively clear since increasing β, wihin he resricions given in Secion 2.4.5, increases he minimum payoff o he equiy capial and hence decreases he risk of he company. For comparison Figure 2.6.2 also includes a horizonal line showing he fair value for an ordinary buy and hold sraegy. Comparing he wo sraegies we observe ha for low values of β he sop-loss sraegy leads o higher values of ρ han he sraegy wihou sop-loss. The reason for his is, ha for low values of β he equiy capial receives a payoff in case of inervenion which is so low ha a he ime of a possible inervenion he expeced increase in he payoff from coninuing he buy and hold sraegy ouweighes he risk of an even smaller payoff.

2.6. NUMERICAL RESULTS 57 2.6.2 Dependence on parameers For a company following a buy and hold sraegy we now consider he dependence of ρ on he parameers r, σ, G T, K T and γ for a fixed iniial disribuion of capial. To sudy ρ..5.1.15 κ =.1 κ =.25 κ =.5.4.6.8.1.12.14 Figure 2.6.3: ρ as a funcion of he shor rae of ineres. r he dependence on r we le σ =.2, G T = 1.45, K T = 1.6, γ =.1, V = 1, U = 1 and E = 1. Figure 2.6.3 hen shows he dependence on r for κ {.1,.25,.5}. The values of κ are chosen o illusrae a company wih a conservaive, a moderae and an aggressive invesmen sraegy, respecively. We observe ha ρ is a decreasing funcion of r for all values of κ. This is also expeced since increasing he riskfree ineres rae lowers he probabiliy of invesmen reurns below he guaraneed/announced accumulaion facor, hence decreasing he risk of he insurance company. Fixing r =.6 and leing U = 5 and E = 5, we now urn o he dependence on he guaraneed accumulaion facor, G T. The low values of E and U are chosen in order o observe a dependence on G T for low values of κ. Figure 2.6.4 now shows he dependence on G T for he same values of κ as above, i.e. κ {.1,.25,.5}. We observe ha ρ is an increasing funcion of G T for all hree values of κ. The posiive dependence of ρ on G T is inuiively clear, since he larger he guaranee o he insured, he more risky he conrac is for he company. For a company invesing in a consan relaive porfolio he consans δ and σ only ener he implici equaions for ρ as δσ, hence varying σ is idenical o varying δ. Thus, we observe from Figure 2.6.1 ha ρ is an increasing funcion of σ. This seems inuiively clear since

58 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE ρ..5.1.15.2.25.3 κ =.1 κ =.25 κ =.5 1. 1.1 1.2 1.3 1.4 Figure 2.6.4: ρ as a funcion of he guaraneed accumulaion facor. G T increasing he volailiy of he socks increases he risk of he company. Invesigaing he dependence of ρ on γ, we find ha ρ essenially is independen of γ. However, a sligh negaive dependence has been observed for high levels of volailiy, low values of γ and an equiy capial which is large compared o he bonus reserve. Tha ρ is a decreasing funcion of γ may be explained by he fac ha increasing γ increases he probabiliy of accumulaing using G T. Hence for some oucomes of he sock price here is a small increase in he payoff o he equiy capial, whereas all oher oucomes give he same payoff. Since he dependence is very small and in mos cases non-exisen, we have lef ou a figure illusraing his. Regarding he relaionship beween ρ and K T we find ha ρ only depends on K T if V and E are large compared o U and he invesmen sraegy is quie risky. In his case ploing ρ as a funcion K T shows a shape similar o a 2. order polynomial wih branches poining downwards. The dependence may be explained by he fac ha, when increasing K T he payoff o he insurance porfolio increases if K T is used as accumulaion facor, bu a he same ime he probabiliy of accumulaion wih K T decreases. Thus, he risk of he company is a radeoff beween wo facors working in opposie direcions, such ha he value of K T for which he maximum value of ρ is obained depends on V and U. Since he equiy capial in pracice is much smaller han he deposi, we conclude ha ρ for pracical purposes is independen of K T, and doing so, we leave ou a graph showing he unineresing case where a dependence is found.

soc 2.6. NUMERICAL RESULTS 59 2.6.3 Dependence on iniial disribuion of capial To sudy he dependence of ρ on he iniial disribuion of capial we fix he parameers r =.6, σ =.2, G T = 1.45, K T = 1.6 and γ =.1 and consider an insurance company invesing according o a buy and hold sraegy wih κ =.25. Since he value of ρ is indifferen o scaling of he iniial disribuion of capial, we furher fix V = 1 and allow E and U o vary. Figure 2.6.5 now shows he dependence of ρ on U for differen ρ..1.2.3.4 E = 2 E = 5 E = 1 E = 2 E = 1 5 1 15 2 U Figure 2.6.5: ρ as a funcion of he iniial bonus reserve for differen values of he iniial equiy capial. values of E. Comparing he graphs for he differen values of E, we observe ha ρ is a decreasing funcion of he equiy capial. However, since ρ is an addiional ineres rae o he enire equiy capial, we sill observe an increase in he nominal paymen for he increased risk even hough ρ is decreasing. A decrease in ρ should hus be inerpreed as a decrease in he average risk of one uni of equiy capial in he company. Furhermore, we observe ha ρ is a decreasing funcion of U for all values of E. In Secion 2.1.2 i is shown ha ρ as U. Since he resuls are indifferen o scaling of he iniial capial, hen increasing V is similar o decreasing E and U. Hence, since ρ is a decreasing funcion of E and U we have ha i, as expeced, is an increasing funcion of V.

6 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE 2.6.4 Effec from unsysemaic moraliy risk Now consider an insurance company whose insurance porfolio consiss of idenical pure endowmens for a group of persons of age 5. To model he possible deahs of he insured individuals we use a so-called Gomperz Makeham form for he moraliy inensiy. Here, he moraliy inensiy can be wrien as µ x+ = a + bc x+. Here, he parameers, as in he Danish G82 moraliy able for males, are given by a =.5, b =.75858 and c = 1.9144. To invesigae he dependence on he number of insured and he choice of equivalen maringale measure we assume he company follows a buy and hold sraegy wih κ =.25 and keep he parameers and iniial capial fixed as r =.6, G T = 1.45, K T = 1.6, σ =.2, γ =.1, V = 1, U = 5 and E = 5. Recall ha V = Y V ind, so he oal deposi is held consan while he number of insured individuals increases by decreasing he individual deposis accordingly. From ρ.3.35.4.45.5 h =.5 h =.2 h =.1 h = h =.1 h =.2 5 1 15 2 Y Figure 2.6.6: ρ as a funcion of he number of insured for differen values of h. Figure 2.6.6 we see ha ρ is a decreasing funcion of he number of insured. This is in correspondence wih our inuiion, since increasing he size of he insurance porfolio decreases he unsysemaic moraliy risk. Furhermore, we observe ha ρ is a decreasing funcion of h, and ha he dependence on h is an increasing funcion of he number of insured. Tha ρ is a decreasing funcion of h is inuiively clear since decreasing h corresponds o decreasing he marke moraliy inensiy, and hence increase he survival probabiliy in he derivaion of ρ. The increasing dependence on h can be explained by he

2.7. IMPACT OF ALTERNATIVE DISTRIBUTION SCHEMES 61 srong law of large numbers, which says ha as he number of insured increases he number of survivors concenrae increasingly around he moraliy inensiy. Hence he moraliy inensiy used o deermine ρ becomes increasingly imporan as he size of he insured porfolio is increased. I can be shown, see Secion 2.1.3, ha if he number of insured ends o infiniy hen ρ converges downwards o he soluion in case of capial insurances T p wih G T and K T replaced by G h x T p T T p x and K h x T T p x, respecively. Hence considering he case h =, we see ha adding unsysemaic moraliy risk o a finie insurance porfolio leads o a fair value of ρ, which is higher han he fair value of ρ,.322, obained for capial insurances. 2.7 Impac of alernaive disribuion schemes In his secion we discuss how possible changes in he disribuion scheme impac he resuls for he fair value of ρ. A major possible change in he disribuion scheme would be no o allow any ransfer of capial from he bonus reserve o he equiy capial. In he case where G T V A T < K T V 1 + γ + e rt E his would lead o he following expression for he equiy capial E T = max,min e r+ρt E,A T V T,A T V + U, A similar change of course applies o he siuaion where e rt E + K T V 1 + γ A T. Here he las erm, which ensures ha capial is no ransferred from he bonus reserve o he equiy capial, migh be negaive and hence he maximum operaor is necessary o ensure ha he equiy capial is non-negaive. Using his model increases he fair values of ρ, since he exposure of he equiy capial o risk is larger. The increase is easily seen from he fac, ha for a fixed ρ he new model would give an equiy capial a ime T which always is less or equal o he equiy capial in he original model. Hence, a fair value of ρ mus be higher. This model has been invesigaed in deail in he case where he solvency requiremen applies o he sum of he equiy capial and bonus reserve. Two imporan differences beween he model above and he model considered in his chaper are: Firsly, as U ends o infiniy ρ converges o a sricly posiive number, and secondly he dependence on he solvency parameer is more complex as he equiy capial migh receive he same negaive payoff in case of inervenion for differen values of β. Anoher possibiliy is o change he disribuion scheme, such ha he company use K T o accumulae he deposi if A T K T V 1 + γ. Thus, he company uses he accumulaion facor K T, providing ha his leaves i wih a minimum of γk T V in he sum of he bonus reserve and equiy capial. This crierion is closely relaed o a solvency requiremen of βv T on he sum of he equiy capial and bonus reserve. Here, however he requiremen on he sum of he bonus reserve and equiy capial is se by he board of he company and no by legislaion. Using his crierion in associaion wih he model above we obain a srange hump around E = 2 for low values of U, when invesigaing he dependence on he size of he equiy capial. This may be explained by he fac ha wih he proposed

62 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE crierion he accumulaion facor for he deposi depends on he iniial equiy capial. Hence, for some oucomes of he invesmens differen values for he iniial equiy capial leads o differen accumulaion facors. Since applying a higher accumulaion facor for he same invesmen reurn obviously increases he risk of he company, his leads o a posiive dependence on he equiy capial. The hump around E = 2 for low values of U shows ha here his effec is more dominan han he oherwise predominan effec ha increasing he equiy capial decreases ρ. For γ = he crierion corresponds o he case where he company views he announced accumulaion facor as binding unless using K T insead of G T would bankrup he company. In his case we would obviously expec an increase in he fair value of ρ. If he company views he announced accumulaion facor as legally binding he company is bankrup if A T < K T V, and for A T K T V he deposi is accumulaed using K T. Applying he proper changes o he disribuion scheme all necessary calculaions are similar o hose already presened. Since viewing K T as binding obviously increases he risk for he equiy capial, his change should lead o higher fair values of ρ. 2.8 On he realism and versailiy of he model In his secion we commen on he chosen model. Firs we commen on he chosen probabilisic model and he requiremen on he invesmen sraegy. Then we discuss he advanages and versailiy of he 1-period model. To end he secion we discuss possible exensions. The assumpion ha he financial marke can be described by a Black Scholes model is no very realisic, since boh he ineres rae and he volailiy changes sochasically over ime. However, if he accumulaion period is relaively small he model is likely o be an accepable approximaion o realiy. Hence, working wih a more advanced financial model would make he resuls unnecessarily complicaed. In he model we assume ha he moraliy inensiy is deerminisic, such ha only he unsysemaic moraliy risk is considered. By unsysemaic moraliy risk we refer o he risk associaed wih he random developmen of an insured porfolio wih known moraliy inensiy. Thus, he unsysemaic moraliy risk is he diversifiable par of he moraliy risk. For a more realisic model we could inroduce a sochasic moraliy inensiy as in Chaper 3. This would allow us o consider he sysemaic moraliy risk, referring o he risk associaed wih changes in he underlying moraliy inensiy, as well. Since changes in he underlying moraliy inensiy affec all insured, he sysemaic moraliy risk is non-diversifiable. On he conrary i increases as he number of similar conracs in he porfolio of insured increases. Hence, if we were o add sysemaic moraliy risk o he model he impac on he fair value of ρ would increase as a funcion of he lengh of he accumulaion period, T, and he number of insured, Y. Since we consider one accumulaion period only, he assumpion of deerminisic moraliy inensiy is very close o realiy and sufficien for our purpose.

2.8. ON THE REALISM AND VERSATILITY OF THE MODEL 63 Throughou chaper we assume ha he company disinguishes beween he invesmens belonging o he equiy capial and he invesmens belonging o he insurance porfolio. Furhermore we assume ha he asses belonging o he equiy capial are invesed in he savings accoun o keep possible risky invesmens on behalf of he owners aside from he risk associaed wih he insurance conracs. If he company does no make his disincion when invesing, we may obain he desired disincion by assuming ha he equiy capial is invesed in he savings accoun and define he value of he risky porfolio residually as V ϕ = A e r E. Now he resuls in he chaper apply immediaely for buy and hold sraegies for A, whereas an invesmen sraegy for A wih consan relaive porfolio weighs would lead o minor modificaions of he resuls. Using a model wih only one accumulaion period has several advanages. Firsly, we, as seen above, can jusify working wih a relaively simple probabilisic model. Secondly, we are able o define a disribuion scheme wih only one endogenously given parameer, since we do no have o specify a formula used o anicipae how he company chooses K T. This is of imporance, since in pracice he choice of K T is widely influenced by he compeiion, and hus, i is difficul o model. As for he versailiy of he model we are paricularly ineresed in answers o he following wo quesions: Does repeaed use of he 1-period model yield fairness in a muli-period seing? And if so, wha insigh does he company gain by repeaed use of he model? To answer he firs quesion we consider an arbirary sequence of accumulaion imes = T < T 1 <... < T n. For he disribuion of he asses o be fair in he muli period model i mus hold ha E Qh [ e rt n E Tn ] = E, for an arbirary, bu fixed, equivalen maringale measure, Q h. If we a each accumulaion ime, T i, condiion on he informaion F Ti we obain a sring of 1-period models. Thus, if we deermine he fair value of ρ in each 1-period model we obain: E Q [ e rtn E Tn ] = E Q [ e rtn T n 1 E Q [ e rt n 1 E Tn FTn 1 ] ] = E Q [e rt n 1 E Tn 1 ] =... = E. Here, he only resricion is, ha he iniial disribuion of capial in one period is he erminal disribuion in he preceding period. Hence, i even holds if he model parameers r, σ, γ and β and he invesmen sraegy varies for differen accumulaion periods. Thus, repeaed use of he 1-period crierion for fairness yields fairness in a muli-period seing. Using he model in a muli-period seing he company can obain confidence bands for he developmen of he balance shee and long erm ruin probabiliies by simulaing he developmen of he financial marke and he insurance porfolio. However, using he model for simulaion purposes we need o specify a formula, from which he company deermines he announced accumulaion facor in each period. Furhermore he assumpions abou consan parameers in he financial marke and a deerminisic moraliy inensiy are less realisic on a long erm basis. This however, could be remedied by applying sochasic models o deermine he consan ineres rae and volailiy and deerminisic moraliy inensiy for he nex accumulaion period.

64 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE Some possible exensions of he model are o include differen ypes of insurance conracs, insured of differen ages and paymens during he accumulaion period. However, exending he model o include differen ypes of conracs and differen age groups increases he possibiliy of a sysemaic redisribuion of capial from one group of insured o anoher. 2.9 Conclusion For a company issuing insurance conracs wih guaraneed periodic accumulaion facors we consider he problem of disribuing he asses fairly beween he accouns of he insured and he equiy capial. To derive a fair disribuion we consider a 1-period model represening one accumulaion period. In he model he only free parameer in he disribuion scheme is he ineres rae ρ, paid o he equiy capial in addiion o he riskfree ineres rae, when such an addiional rae is possible. Using he principle of no arbirage, we are able o derive an implici equaion for he fair value of ρ given one of four differen invesmen sraegies. Invesigaing he dependence of ρ on he invesmen sraegy, we observe ha a consan relaive porfolio is slighly more risky han a buy and hold sraegy, and ha ρ is an increasing funcion of he relaive iniial invesmen in socks. In he case of a solvency requiremen and sop-loss sraegies we find ha ρ is a decreasing funcion of β. Considering he dependence of ρ on he parameers, we observe a posiive dependence on he volailiy and he guaraneed accumulaion facor and a negaive dependence on he riskfree ineres rae. As for he announced deposi rae and he parameer γ we found ha he dependence for pracical purposes is non-exisen. When considering he iniial disribuion of capial we find ha ρ is an increasing funcion of he iniial deposi and a decreasing funcion of he equiy capial and he bonus reserve. Exending he model o include moraliy obviously increases he fair value of ρ, since i adds more uncerainy o he model. As expeced we observe ha in he case of risk neuraliy wih respec o unsysemaic moraliy risk he fair value of ρ is a decreasing funcion of he number of insured ending o he fair value in he case wihou moraliy. Furhermore we observe ha he influence of he marke aiude owards moraliy risk on he fair value of ρ increases as he number of insured increases. 2.1 Proofs and echnical calculaions 2.1.1 Proof of Proposiion 2.4.4 If he company invess in he savings accoun only, he value of he asses a ime T is A T = e rt A. Since he value of he asses is deerminisic, he disribuion scheme is fair if and only if E T = e rt E. Considering he differen inervals in he disribuion scheme for he possible oucomes of A T, we ge 1. If e rt A < G T V hen E T =, so we canno have E = e rt E T if E > since

2.1. PROOFS AND TECHNICAL CALCULATIONS 65 r <. Thus, no value of ρ gives a fair disribuion scheme. 2. If G T V e rt A < K T V 1+γ+e rt E hen each of he wo erms in he minimum operaor may be he smalles, and we have o consider each of he possibiliies. a If e r+ρt E e rt A G T V hen a fair value of ρ saisfies E = e rt e r+ρt E, i.e. ρ =. b If e rt A G T V e r+ρt E hen we mus have E = e rt e rt A G T V, i.e. G T = e rt V +U V. Thus, fair values of ρ mus saisfy e rt E e r+ρt E, i.e. ρ. 3. If K T V 1+γ+e rt E e rt A hen each of he wo erms in he minimum operaor may be he smalles, and we have o consider each of he possibiliies. a If e r+ρt E e rt A K T V hen a fair value of ρ saisfies E = e rt e r+ρt E, i.e. ρ =. b If e rt A K T V e r+ρt E hen we mus have E = e rt e rt A G T V, i.e. K T = e rt V +U V. Thus, fair values of ρ mus saisfy e rt E e r+ρt E, i.e. ρ. 2.1.2 Deermining he limi as U In his secion we derive he fair value of ρ as he bonus reserve ends o. For simpliciy we consider he case of capial insurances. Taking he limi as U in crierion 2.4.1 gives [ E = e rt lim U EQ 1 GT V A T <K T V 1+γ+e rt E min e r+ρt E,A T G T V ] + 1 KT V 1+γ V T ϕ min e r+ρt E,A T K T V 2.1.1 Assuming ρ < dominaed convergence allows us o inerchange limi and expecaion. Since we consider admissible invesmen sraegies only, we have ha lim U V T =. Hence i holds ha and lim 1 G U T V A T <K T V 1+γ+e rt E = lim 1 K U T V 1+γ<V T ϕ = 1. Furhermore we have for E T {G T,K T } ha lim min e r+ρt E,A T E T V = e r+ρt E <. U

66 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE Hence, in he limi we obain he following equaion E = e rt e r+ρt E, such ha in he limi ρ =. This is also inuiively clear, since increasing he bonus reserve decreases he probabiliy of he equiy capial suffering a loss, and in he limi where he bonus reserve is infiniely large he equiy capial bears no risk and obviously i should no receive an addiional paymen compared o he riskfree ineres rae. We end his secion by noing ha he assumpion ρ < does no impose a resricion, since ρ = canno be a soluion o 2.1.1. In order o do so we assume ρ = solves 2.1.1. This in urn would lead o E = e rt lim U EQ [ 1 GT V A T <K T V 1+γ+e rt E A T G T V + 1 KT V 1+γ V T ϕ A T K T V e rt =, lim U EQ [ 1 KT V 1+γ V T ϕ A T K T V where we have used monoone convergence o inerchange limi and inegraion in he las equaliy and consideraions similar o hose above o deermine he limi. However, since E < we have a conradicion, such ha ρ = can no be he soluion. ] ] 2.1.3 Deermining he limi as Y We now deermine he convergence of ρ as Y ends o, while keeping V = Y V ind fixed. Taking he limi in 2.5.3 we ge E = e rt min lim Y EQh + 1 YT K T V ind [ 1 YT G T V ind e r+ρt E,A T Y T G T V ind 1 T px 1+γ V T ϕ min 1 T px A T <Y T K T V ind 1 T px 1+γ+erT E 1 Tp x e r+ρt E,A T Y T K T V ind Assuming ha ρ < we can use dominaed convergence o inerchange limi and inegral. Using he srong law of large numbers we have for an arbirary accumulaion facor E T : lim Y T E T V ind 1 Y Tp x YT 1 Tp h x = lim E T V = E T V, Y Y Tp x Tp x 1 Tp x ] Q h a.s.

2.1. PROOFS AND TECHNICAL CALCULATIONS 67 Since Q h is idenical o Q wih respec o he financial marke for all h, we obain he following equaion in he limi E = e rt E [1 Q min GT V T ph x T px A T <K T V T ph x T px 1+γ+erT E + 1 KT V T ph x T px 1+γ V T ϕ min e r+ρt E,A T K T V Tp h x Tp x e r+ρt Tp h x E,A T G T V Tp x ]. T p This is exacly he equaion in he case of capial insurances wih G T replaced by G h x T T p x T p and K T replaced by K h x T T p x. Noe in paricular, ha assuming risk neuraliy wih respec o unsysemaic moraliy risk, i.e. h = gives he same resuls in he limi as in he case wihou moraliy. The calculaions above are carried ou for an arbirary h. However in he limi he measures Q h and P are singular raher han equivalen if h. Thus, using a Q h wih h in an aemp o derive a fair value of ρ for an infiniely large insurance porfolio would hus resul in inroducing an arbirage possibiliy in he model. However, even hough he limi resul for h has no economic inerpreaion, i sill provides useful insigh for he dependence of ρ on h for a large porfolio. Furhermore solving he limi equaion gives an approximaion o he fair value in he case of a large porfolio of pure endowmens. 2.1.4 Proof of Proposiion 2.5.8 In order o prove Proposiion 2.5.8, we firs noe ha he probabiliy of insolvency can be wrien as: [ ] p ins ϕ = P E T < Y T βvt ind Y Y [ ] = T p x n T q x Y n P E T < nβvt ind n n= Y Y = n n= [ + P [ T p x n T q x Y n P E T < nβvt ind, inf T S Z Y Y = n n= [ + P T p x n T q x Y n A T < n1 + βg T V ind [ P E T < nβvt ind, inf ] ] T S > Z A T < n1 + βg T V ind 1, inf Tp x ] T S Z. 1, inf Tp x ] T S > Z Here, we have use ieraed expecaions in he second equaliy, and in he hird we spli he probabiliy according o wheher he company inervenes or no. The fourh equaliy

68 CHAPTER 2. FAIR DISTRIBUTION OF ASSETS IN LIFE INSURANCE follows from he relaionship e rt E βk T Y V ind T p x, since his ensures ha he company never is insolven if he deposi is accumulaed wih he facor K T or if he equiy capial a ime T is given by E T = e r+ρt E. Now we inser s β,n, 1 and he deerminisic value of A T in case of inervenion o obain Y Y p ins ϕ = n n= [ + P Y = n= [ ] T p x n T q x Y n P ST < s β,n, 1, inf T S > Z Y G T V ind Y T p x n T q x Y n n [ + 1 YT p h x<np 1 Tp h x 1 + β < ng T V ind 1 1 + β, inf Tp x Tp x [ P ST < sβ,n, 1, inf inf T S Z ] T S > Z ] T S Z ]. 2.1.2 From 2.1.2 we observe ha if he number of survivors if greaer ha he Q h expecaion hen he company is insolven in case of inervenion, whereas his is no necessarily he case in he siuaion wihou inervenion. Calculaions similar o hose in he proof of Björk 24, Theorem 18.8 give P [ ] [ ] ST < s β,n, 1, inf T S > Z = E P 1 S T <s β,n, 1 1 inf T S >Z = E P [ 1 Z<S T <s β,n, 1 ] 2α r Z σ 2 1 [ ] E P 1 S Z< S, T <sβ,n, 1 where S is a process wih he same dynamics as S, bu wih iniial value S = Z2 S. Invesigaing each erm separaely we ge [ ] E P 1 Z<S T <s β,n, 1 [ ] = 1 Z<s β,n, P S 1 T < sβ,n, 1 P[ST Z] s = 1 YT p h x<n Φ log β,n, 1 S α r 1 2 σ2 T σ T Φ log Z S α r 1 2 σ2 T σ T, and ] [ E P 1 Z< S T <sβ,n, 1 = 1 YT p h x<n Φ log s β,n, 1 S Z 2 α r 1 2 σ2 T σ T Φ log S Z α r 1 2 σ2 T σ T.

2.1. PROOFS AND TECHNICAL CALCULATIONS 69 Similarly [ ] [ ] P inf T S Z = 1 E P 1 inf T S >Z Combining he resuls we ge n= [ ] 2α r 1 2 σ2 Z [ = 1 E P σ 1 Z<S T + 2 E P 1 S Z<S = Φ log Z S α r 1 2 σ2 T σ T 2α r Z + S Y Y p ins ϕ = T p x n T q x Y n 1 n YT p h x <n Z + S 2α r σ 2 1 1 Φ σ 2 1 1 Φ log S Z log Φ log T ] α r 1 2 σ2 T σ. T s β,n, 1 S Z 2 α r 1 2 σ2 T σ T s β,n, 1 S Z 2 α r 1 2 σ2 T σ T.

Chaper 3 Sochasic Moraliy in Life Insurance: Marke Reserves and Moraliy-Linked Insurance Conracs This chaper is an adaped version of Dahl 24b In life insurance, acuaries have radiionally calculaed premiums and reserves using a deerminisic moraliy inensiy, which is a funcion of he age of he insured only. Here, we model he moraliy inensiy as a sochasic process. This allows us o capure wo imporan feaures of he moraliy inensiy: Time dependency and uncerainy of he fuure developmen. The advanage of inroducing a sochasic moraliy inensiy is wofold. Firsly i gives more realisic premiums and reserves, and secondly i quanifies he risk of he insurance companies associaed wih he underlying moraliy inensiy. Having inroduced a sochasic moraliy inensiy, we sudy possible ways of ransferring he sysemaic moraliy risk o oher paries. One possibiliy is o inroduce moraliy-linked insurance conracs. Here he premiums and/or benefis are linked o he developmen of he moraliy inensiy, hereby ransferring he sysemaic moraliy risk o he insured. Alernaively he insurance company can ransfer some or all of he sysemaic moraliy risk o agens in he financial marke by rading derivaives depending on he moraliy inensiy. 3.1 Inroducion Tradiionally, acuaries have been calculaing premiums and reserves using a deerminisic moraliy inensiy, which is a funcion of he age only, and a consan ineres rae rep- 71

72 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS resening he payoff of he invesmens made by he companies. However, since neiher he ineres rae nor he moraliy inensiy is deerminisic, life insurance companies are essenially exposed o hree ypes of risk when issuing conracs: Financial risk, sysemaic moraliy risk and unsysemaic moraliy risk. Here, we disinguish beween sysemaic moraliy risk, referring o he fuure developmen of he underlying moraliy inensiy, and unsysemaic moraliy risk, referring o a possible adverse developmen of he insured porfolio. So far he life insurance companies have deal wih he financial and sysemaic moraliy risks by choosing boh he ineres rae and he moraliy inensiy o he safe side, as seen from he insurers poin of view. When he real moraliy inensiy and invesmen payoff are experienced over ime, his usually leads o a surplus, which, by he so-called conribuion principle, mus be redisribued among he insured as bonus, see Norberg 1999. Since insurance conracs ofen run for 3 years or more, a moraliy inensiy or ineres rae, which seems o be o he safe side a he beginning of he conrac, migh urn ou no o be so. This phenomenon has in paricular been observed for he ineres rae during recen years, where we have experienced large drops in sock prices and low reurns on bonds. However, he sysemaic moraliy risk is of a differen characer han he financial risk. While he asses on he financial marke are very volaile, changes in he moraliy inensiy seem o occur more slowly. Thus, he financial marke poses an immediae problem, whereas he level of he moraliy inensiy poses a more long erm, bu also more permanen, problem. This difference could be he reason why emphasis so far has been on he financial markes. We hope o urn some of his aenion owards he uncerainy associaed wih he moraliy inensiy by modelling i as a sochasic process. In order o obain a more accurae descripion of he liabiliies of life insurance companies, marke reserves have been inroduced, see Seffensen 2 and references herein. Here, he financial uncerainy as well as he uncerainy semming from he developmen of an insurance porfolio wih known moraliy inensiy is considered. By modelling he moraliy inensiy as a sochasic process, marke reserves can be furher exended o include he uncerainy associaed wih he fuure developmen of he moraliy inensiy. This should allow for an even more accurae assessmen of fuure liabiliies, since possible rends in he moraliy inensiy and he marke aiude owards sysemaic moraliy risk can be aken ino accoun. In addiion, a sochasic moraliy inensiy allows for a quanificaion of he sysemaic moraliy risk of he insurance companies. Having quanified he sysemaic moraliy risk, we invesigae how he insurance companies could manage he risk. As a firs possibiliy, we inroduce a new ype of conracs called moraliy-linked conracs. The basic idea is o link and currenly adap benefis and/or premiums o he developmen of he moraliy inensiy in general, and hereby ransfer he sysemaic moraliy risk from he insurance company o he group of insured. A second possibiliy is o ransfer he sysemaic moraliy risk o oher paries in he financial marke. Here, he idea is o inroduce cerain raded asses, which depend on he developmen of he moraliy inensiy. This chaper is organized as follows: Secion 3.2 conains a review of exising lieraure on sochasic moraliy. Secion 3.3 deals wih he modelling of he moraliy inensiy as a sochasic process, and Secion 3.4 inroduces he model considered in he res of

3.2. EXISTING LITERATURE ON STOCHASTIC MORTALITY 73 his chaper. An expression for he marke reserve for a general paymen sream is given in Secion 3.5. In Secion 3.6, we inroduce he concep of a moraliy-linked insurance conracs, whereas Secion 3.7 includes a discussion of how he sysemaic moraliy risk could be ransferred o oher agens in he financial marke. Finally, he derivaion of he dynamics of he benefi for a moraliy-linked pure endowmen in he case of risky invesmens is given in 3.8. 3.2 Exising lieraure on sochasic moraliy In his secion we give a brief review on exising lieraure concerning he uncerainy associaed wih he fuure developmen of he moraliy inensiy. For furher references see he referred papers. Olivieri 21 assumes ha he insurance companies ake possible rends in he moraliy inensiy ino accoun by esimaing a moraliy inensiy, which is a funcion of boh ime and age. Hence he companies obain more realisic premiums and reserves han by using a funcion of age only. However, he esimaed survival funcion, no maer how good i is, is only one possible fuure developmen. Thus, Olivieri uses he observed moraliy inensiies o generae wo addiional survival funcions, which represen very high and very low fuure survival probabiliies, respecively. Using hese hree possible scenarios for he fuure survival funcion, Olivieri illusraes he impac of sysemaic moraliy risk by calculaing variances of presen values. Marocco and Piacco 1998 model he yearly moraliy raes via a bea disribuion wih age and ime-dependen parameers. Hence, hey are able o quanify he moraliy risk inheren in an insurance porfolio, since he number of survivors follows a binomial-bea disribuion. The approach in Olivieri and Piacco 22 is somewha differen. They describe he fuure survival funcion by a parameerized family of possible fuure survival funcions. However, since he fuure is unknown, he parameer is a random variable. In order o obain prices and assess he risk hey apply Baysian mehods o describe he disribuion funcion for he parameer. Wihin his model hey are able o disinguish beween he unsysemaic moraliy risk semming from he randomness for a given parameer survival funcion, and he sysemaic moraliy risk semming from he uncerainy associaed wih he parameer survival funcion. In he above papers no explici financial marke has been inroduced and all calculaions are carried ou using a consan ineres rae. Models involving boh ineres rae risk and sysemaic moraliy risk are proposed in Milevsky and Promislow 21. For a fixed equivalen maringale measure hey propose boh a discree and coninuous ime model for he moraliy and ineres rae. Wihin he proposed models hey are able o obain prices and deermine hedging sraegies for claims ha are coningen on he moraliy and ineres rae. The conribuion of he presen chaper is as follows: Inspired by ineres rae modelling we model he moraliy inensiy by a fairly general diffusion model, which include he

74 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS Mean revering Brownian Gomperz model proposed by Milevsky and Promislow 21 as a special case. Taking he incomplee model comprised by he financial marke, moraliy inensiy and insurance conrac as a saring poin, we hen noe ha here exis infiniely many equivalen maringale measures corresponding o differen marke aiudes owards sysemaic and unsysemaic moraliy risk. Hence, conracs involving an insurance elemen canno be priced uniquely using a no arbirage approach. For a fixed bu arbirary equivalen maringale measure we derive inegral expressions and parial differenial equaions for marke reserves in he presence of sochasic moraliy. These resuls show how marke reserves depend on he expecaion o he fuure moraliy inensiy and he marke aiude owards sysemaic moraliy risk. The laer seems o be a new resul. Furhermore we inroduce a new ype of conracs called moraliy-linked insurance conracs as a way o ransfer he sysemaic moraliy risk o he insured. Finally a general parial differenial equaion for moraliy derivaives is derived, and i is shown how he company may use such derivaives o ransfer he sysemaic moraliy risk o he financial marke. 3.3 Moraliy inensiy as a sochasic process 3.3.1 Sochasic versus deerminisic moraliy In acuarial pracice, saisical mehods are usually used o esimae a moraliy inensiy, which is a funcion of he age, x +, only. In Denmark, life insurance companies use a socalled Gomperz Makeham model for he moraliy inensiy. Here, he moraliy inensiy can be wrien as µ x+ = a + bc x+. In his chaper more realism is added by viewing he moraliy inensiy as a sochasic process, which is adaped o some filraion F. We model he moraliy inensiies as diffusion processes such ha for every fixed x he moraliy inensiy has dynamics of he form dµ [x]+ = α µ,x,µ [x]+ d + σ µ,x,µ [x]+ d W, 3.3.1 where W is a Wiener process sandard Brownian moion wih respec o he filraion F. Here and hroughou we have borrowed he selec moraliy noaion of Norberg 1988. Since he dynamics of µ depend on he presen sae of he process only, hen µ is a Markov process. In 3.3.1, we have assumed ha he person is of age x a ime which is an arbirary calendar ime. The parameer hen describes he ime ha has passed since ime. Resuls similar o hose presened in his chaper can be obained in he case, where he moraliy inensiy is driven by a finie sae Markov process, see Dahl 22. Remark 3.3.1 By modelling moraliy inensiies by 3.3.1, we have made he following wo raher unrealisic assumpions: Firsly, all sudden changes in he moraliy inensiies are of he same ype and affec all ages/cohors and secondly, he moraliy inensiy for

3.3. MORTALITY INTENSITY AS A STOCHASTIC PROCESS 75 each age is a Markov process. A more realisic model would recognize ha he moraliy inensiies are affeced by many differen so-called moraliy facors and ha hese facors affec he moraliy inensiies differenly. Some moraliy facor affec all ages/cohors while ohers affec only some ages/cohors. Furhermore even moraliy facors affecing he same ages/cohors may have a differen impac on he moraliy inensiies. An appropriae model for he dependence on he differen moraliy facors could be a hierarchical model. Assume for example ha he dynamics of he moraliy inensiies are given by he following exension of 3.3.1: dµ [x]+ = α µ,x,µ d + σ µ,x,µ r d W, where µ denoes he infinie dimensional vecor conaining he moraliy inensiies a ime for all x. Moreover, σ µ,x,µ and W are d-dimensional column vecors, and a r denoes vecor a ransposed. Now a hierarchical srucures is obained, if we inerpre each Wiener process as he impac of a specific moraliy facor and define σ such ha he Wiener process only affecs he appropriae ages/cohors. However, since we are working wih one value of x only and because no furher insigh is gained from working wih a muli-dimensional Wiener process, we resric ourselves o he simple 1-dimensional case given by 3.3.1. Remark 3.3.2 Insead of modelling he moraliy inensiy as a diffusion process of he form in 3.3.1, we could assume a Gomperz Makeham srucure and model he parameers a, b and c as sochasic processes. Dahl 22 includes some examples, where a and b are modelled by sochasic processes. In he case wih known moraliy inensiy he survival probabiliy from ime o T for a person of age x a ime is given by e ÊT µ [x]+u du. However, since we do no know he fuure developmen of he moraliy inensiy, his should be replaced by an expeced value, condiioning on he known developmen up o ime, represened by F. Here, F = F T is he filraion for he model describing all he randomness observed, which in paricular conains informaion abou he developmen of he sochasic process µ. Informally, F is he informaion available o he insurer a ime. Noe ha we hereby assume ha µ is observable, which corresponds o assuming ha he porfolio of observed lives is sufficienly large, such ha he moraliy inensiy can be esimaed correcly. Using ha µ is a Markov process and ieraed expecaions, we see ha S,x,µ [x]+,t, defined by [ ] S,x,µ [x]+,t := E P e ÊT µ [x]+τ dτ µ [x]+, is he survival probabiliy from ime o T for a person of age x + given he informaion unil ime. Remark 3.3.3 Noe ha he moraliy inensiy, in conras o he e.g. he ineres rae, is modelled under he objecive measure P. For he ineres rae modelling usually akes

76 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS place direcly under some equivalen maringale measure Q. For fixed x we now define a P-maringale M by [ ] M,x := E P e ÊT µ [x]+τdτ F = e Ê µ [x]+τdτ S,x,µ [x]+,t, where µ is defined by 3.3.1. The quaniy M,x can be inerpreed as he probabiliy of survival from ime o T for a person of age x a ime given he developmen of he moraliy inensiy unil ime. Provided ha S is coninuously differeniable in and wice coninuously differeniable in µ, we can use Iô s formula on he maringale M, such ha we for fixed x obain he following parial differenial equaion PDE for S,x,µ,T on [,T] R + : = S,x,µ,T + α µ,x,µ µ S,x,µ,T + 1 2 σµ,x,µ 2 µµ S,x,µ,T µs,x,µ,t, 3.3.2 which should be solved wih he boundary condiion ST,x,µ,T = 1. Here, we have used he noaion S = S, µs = µ S and µµs = 2 S, which will µ 2 be used hroughou he chaper, whenever he derivaives exis. The differenial equaion 3.3.2 is analogous o he differenial equaion for zero coupon bonds obained when working wih a sochasic ineres rae, see e.g. Björk 1997, Proposiion 3.4. 3.3.2 Affine moraliy srucure We now concenrae on a special moraliy srucure, which will be referred o as an affine moraliy srucure. The following definiion of an affine moraliy srucure is almos analogous o he definiion of an affine erm srucure, see e.g. Björk 1997, Definiion 3.1: Definiion 3.3.4 Affine moraliy srucure If, for fixed x, he survival probabiliies are given by S,x,µ [x]+,t, where S has he form S,x,µ [x]+,t = e A,x,T B,x,Tµ [x]+, 3.3.3 for deerminisic funcions A,x,T and B,x,T, hen he model for he moraliy inensiy is said o possess an affine moraliy srucure for cohor x. If 3.3.3 holds for all x, hen he model is simply said o possess an affine moraliy srucure. Affine moraliy srucures are of ineres, since hey allow survival probabiliies o be expressed by he relaively simple expression in 3.3.3. However, explici expressions for A and B may be quie complicaed or even impossible o find.

3.3. MORTALITY INTENSITY AS A STOCHASTIC PROCESS 77 Example 3.3.5 A naural quesion is, wheher Definiion 3.3.4 includes models wih deerminisic moraliy inensiy. This is indeed he case, as can be seen for example if we choose A and B by T A,x,T = µ [x]+τ dτ, B,x,T =, respecively, such ha he survival probabiliy is given by S,x,µ [x]+,t = e ÊT µ [x]+τ dτ. If he deerminisic moraliy inensiy only depends on x and hrough x +, his is recognized as he radiional survival probabiliy, T p x+. The definiion of an affine moraliy srucure does no give a way o deermine wheher a given model for he moraliy inensiy possesses an affine srucure. One has o find he expression for he survival probabiliies and deermine wheher hey can be wrien on he desired form in 3.3.3. This is no of much help, since he reason for checking wheher we have an affine moraliy srucure or a leas an affine moraliy srucure for some cohors x exacly is, ha i yields expression 3.3.3 for he probabiliies. The following heorem, which also appears in Björk 1997 for zero coupon bond prices, gives sufficien condiions for a moraliy srucure o be affine for cohor x. In addiion, i yields a se of differenial equaions for he funcions A and B for fixed x. Theorem 3.3.6 Sufficien condiions for an affine moraliy srucure Assume ha α µ and σ µ are of he form: α µ,x,µ [x]+ = δ α,xµ [x]+ + ζ α,x, σ µ,x,µ [x]+ = δ σ,xµ [x]+ + ζ σ,x, for some deerminisic funcions δ α, ζ α, δ σ and ζ σ. Then he model has an affine moraliy srucure for cohor x, where A and B for fixed x saisfy he sysem and B,x,T = δ α,xb,x,t + 1 2 δσ,xb,x,t 2 1, 3.3.4 BT,x,T =, A,x,T = ζ α,xb,x,t 1 2 ζσ,xb,x,t 2, 3.3.5 AT,x,T =. Proof of Theorem 3.3.6. The proof is analogous o he one given in Björk 1997, Proposiion 3.5 for an affine erm srucure.

78 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS For fixed x an affine srucure for α µ and σ µ 2 in is hus sufficien for an affine moraliy srucure for cohor x. If in addiion α µ and σ µ are ime independen, he condiion is necessary as well, see Duffie 1992. Thus, provided we can solve 3.3.4 and 3.3.5, an affine moraliy srucure for cohor x gives a closed form expression for he survival probabiliies for cohor x. 3.3.3 Model consideraions In his secion, properies for he moraliy inensiy are discussed, and a specific model for he moraliy inensiy is considered. In ineres rae modelling posiiviy of he ineres rae is a desirable propery. For he moraliy inensiy his is no only a desirable, bu mandaory, propery. While one could imagine having ineres rae, he moraliy inensiy should be sricly posiive, since a moraliy inensiy equal o corresponds o a survival probabiliy of 1, and his is no realisic for any ime inerval. One model which fulfills he requiremen of a sricly posiive moraliy inensiy, is he following analogue o he so-called exended Cox Ingersoll Ross model, which was firs considered by Hull and Whie 199 as a model for he ineres rae. Applying he exended Cox Ingersoll Ross model for he modelling of moraliy inensiies leads o he following dynamics for fixed x dµ [x]+ = β µ,x γ µ,xµ [x]+ d + ρ µ,x µ [x]+ d W, 3.3.6 where β µ,x, γ µ,x and ρ µ,x are posiive bounded funcions. I can be shown ha he exended Cox Ingersoll Ross model ensures sric posiiviy of he moraliy inensiy for cohor x provided ha for fixed x we have 2β µ,x ρ µ,x 2, for all [, T], see Maghsoodi 1996. Furhermore, he model is mean revering around β µ,x γ µ,x he ime and cohor dependen level. Theorem 3.3.6 shows ha he moraliy inensiy given by 3.3.6 admis an affine moraliy srucure. Provided ha we are able o solve he PDEs for A and B, we are hus able o find closed form expressions for he survival probabiliies. I would hus be ineresing o see wheher saisical daa suppors modelling he dynamics of he moraliy inensiy by an exended Cox Ingersoll Ross model, such ha he desirable properies for he moraliy inensiy are obained. 3.3.4 Forward moraliy inensiies When modelling ineres raes, imporan quaniies are forward raes defined by or equivalenly f,t := T log p,t, T, p,t = e ÊT f,udu. Here p,t is he price a ime of a zero coupon bond mauring a ime T. The forward rae f,u can hus be inerpreed as he riskfree rae of ineres, conraced a ime,

3.4. THE MODEL 79 over he infiniesimal inerval [u, u + du. Analogously o he concep of forward raes, we define he forward moraliy inensiy for cohor x a ime under he rue measure P for he ime T, by f µ,x,t := T log S,x,µ [x]+,t. Equivalenly, we can express he relaion beween he forward moraliy inensiies and survival probabiliies by S,x,µ [x]+,t = e ÊT f µ,x,udu. Thus, he forward moraliy inensiy funcion for cohor x, f µ,x,u u T is he adaped moraliy inensiy funcion, which makes he survival probabiliy a ime for a x + year old equal o e Êτ fµ,x,udu for all < τ T. Insead of modelling he moraliy inensiy direcly, one could imagine ha he life insurance companies would model he forward moraliy inensiy. This could be done by replacing he ime homogeneous deerminisic funcion, which hey are using oday, wih a funcion of x,, T and he observed moraliy inensiy µ [x]+. Noe ha he forward moraliy inensiies are sochasic processes, since he forward moraliy inensiy for cohor x, f µ τ,x,t, a τ is no known in general a, if < τ. 3.4 The model Le Ω, F,P be a probabiliy space wih a filraion F = F T saisfying he usual condiions of righ-coninuiy, i.e. F = F u, and compleeness, i.e. F conains all P- u> null ses. Here, T is a fixed ime horizon. Throughou, F describes he oal informaion available a ime. Below we inroduce he hree componens which consiue he model: The financial marke, he moraliy inensiy and he insurance conrac. 3.4.1 The financial marke We consider a financial marke consising of wo raded asses only: A risky asse wih price process S and a locally riskfree asse wih price process B. The risky asse is usually referred o as a sock and he locally riskfree asse as a savings accoun. The price processes are defined on he above inroduced probabiliy space Ω, F,P, and he P-dynamics of he price processes are given by ds = α s,s S d + σ s,s S dw, S >, 3.4.1 db = r,s B d, B = 1, 3.4.2 where r is non-negaive, σ s is uniformly bounded away from and W T is a Wiener process on he inerval [,T] under P. Throughou he chaper we also use he shorhand noaion exemplified by r = r,s for coefficien funcions from sochasic differenial

8 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS equaions ha have already been inroduced. The filraion G = G T is he P- augmenaion of he naural filraion generaed by B,S, i.e. G = G + N, where N is he σ-algebra generaed by all P-null ses and G + = σ{b u,s u,u } = σ{s u,u } = σ{w u,u }, 3.4.3 since W accouns for all he randomness in he model defined by 3.4.1 3.4.2. We noe ha he las equaliy in 3.4.3 only holds if σ s does no ake he value, which is he case since we have assumed ha σ s is uniformly bounded away from. Assuming ha α s and σ s fulfill cerain regulariy condiions, see Kloeden and Plaen 1992, Theorem 4.5.3, he sochasic differenial equaion 3.4.1 has a unique soluion. Henceforh i is assumed ha hese condiions are fulfilled and ha T r τdτ exiss and is finie almos surely, such ha he funcion B is defined for all [,T]. In he model given by 3.4.1 3.4.2, he process α s is inerpreed as he mean rae of reurn of he sock and σ s as he sandard deviaion of he rae of reurn. The process r is known as he shor rae of ineres. Le furher he process ν be defined by ν,s = α,s r,s σ,s. Hence, ν measures he excess reurn of he sock over he riskfree ineres rae divided by he risk associaed wih he sock as measured by σ s. In he lieraure, ν is known as he marke price of risk associaed wih S. In he following we assume ha ν saisfies he so-called Novikov condiion see Duffie 1992, Appendix D. E P [ e 1 2ÊT ν2,s d ] <, 3.4.2 The moraliy inensiy Le he moraliy inensiy process be defined on he above inroduced probabiliy space Ω, F,P. From here on all dependence on x is lef ou of he noaion excep in µ [x]+, since we only consider one fixed, bu arbirary, value of x. As in Secion 3.3, we le he P-dynamics of he moraliy inensiy be given by dµ [x]+ = α µ,µ [x]+ d + σ µ,µ [x]+ d W, 3.4.4 where α µ and σ µ are non-negaive and W T is a Wiener process on he inerval [,T] under P. Noe ha he coefficiens are funcions of he curren value of he moraliy inensiy only, such ha he moraliy inensiy is a Markov process. The filraion I = I T is he P-augmenaion of he naural filraion generaed by he moraliy inensiy. Thus, we have I = I + N, where I + = σ{µ [x]+u,u }. In order o ensure he exisence of a soluion o 3.4.4, we assume ha he coefficiens fulfill he regulariy condiions in Yamada and Waanabe 1971, see also Karazas and Shreve 1991, Chaper 5, Proposiion 2.13. This proposiion is more general han Kloeden and Plaen 1992, Theorem 4.5.3 since i allows for square roo diffusions.

3.4. THE MODEL 81 3.4.3 The insurance conrac Le he developmen of he life insurance conrac be described by an F-adaped righconinuous Markov process Z = Z T on a finie sae space J = {,1,...,J}. We assume ha Z has a mos a finie number of jumps, and le be he iniial sae of he process, i.e. Z = a.s. For example, J could consis of wo saes describing wheher he insured is alive or dead. The associaed indicaor funcions I j are defined by I j = 1 {Z =j}. In addiion, we inroduce he mulivariae couning process N = N jk j k defined by N jk = #{u u,],z u = j,z u = k}. The process N jk couns he number of ransiions direcly from sae j o k. Moreover, assume ha he Markov process admis ransiion raes λ jk given by λ jk = I j µjk [x]+, where µ jk [x]+ are sochasic processes. In his chaper we resric ourselves o models where he ransiion inensiies depend on he moraliy inensiy only, i.e. we resric ourselves o he siuaion µ jk [x]+ = Rjk,µ [x]+, where R jk is a deerminisic funcion. However, we could equally well have worked wih a muli-dimensional process µ = µ jk j k. We obain he following maringales wih respec o P M jk = N jk λ jk u du = Njk I j u µjk [x]+udu, T. By consrucion, he processes N jk do no have simulaneous jumps, hence he maringales M jk are orhogonal. The filraion H = H T is defined as he P-augmenaion of he naural filraion generaed by he insurance conrac, i.e. H = H + N, where H + = σ{z u,u } = σ{n u,u }. Noe ha he above model can be used o describe boh he developmen of he insurance conrac for one insured individual and for a whole porfolio of insured individuals of he same age x a ime. 3.4.4 The combined model We assume ha he filraion F = F T inroduced earlier is given by F = G H I. Thus, F is he filraion for he combined model of he economy, he moraliy inensiy and he insurance conrac. Moreover, we assume ha he economy is sochasically

82 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS independen of he developmen of he insurance conrac and he moraliy inensiy, i.e. G and H, I are independen. We noe ha he combined model is on he general index-form sudied in Seffensen 2. However, Seffensen 2 conains no explici remarks or calculaions regarding a sochasic moraliy inensiy. 3.4.5 Change of measure In his secion, we discuss he choice of equivalen maringale measure in he combined model. An equivalen maringale measure fulfills hree requiremens. Firsly, i is equivalen o P. Secondly, all discouned price processes on he financial marke are maringales under he new measure and lasly i is a probabiliy measure. To consruc a new measure Q we define a likelihood process by dλ = Λ h s dw + h µ d W + g jk Λ = 1. j,k:j k dm jk Here h s and h µ are adaped processes, and g = g jk j k is a predicable process. We assume ha h s, h µ and g are chosen such ha E P [Λ T ] = 1 and such ha g jk > 1 for all j k. Here, h s changes he drif erm of S, h µ changes he drif erm of µ, and g jk changes he inensiy for a ransiion from j o k for Z. We can now define a measure Q by, dq dp = Λ T. 3.4.5 Remark 3.4.1 We emphasize ha Q defined above only changes measure for one value of x. If we were o consider a porfolio including differen ages, we would model he moraliy inensiy by a d-dimensional Wiener process as proposed in Remark 3.3.1. Hence changing measure for he moraliy inensiy requires h µ o be a d-dimensional Girsanov kernel. In addiion we noe ha he maringales M jk implicily depends on x. Thus, we would need a differen maringale M jk and hence a new g jk for each value of x in he porfolio. However, since we only consider one value of x, his is no necessary here. Girsanov s heorem shows ha under he measure Q defined by 3.4.5, W Q and W Q = W hs u du = W hµ udu are independen Q-Wiener processes. If we consider he financial model only, i is well-known ha he discouned price process of he sock is a Q-maringale if and only if h s = r,s α s,s σ s,s = ν,s, 3.4.6

3.4. THE MODEL 83 see e.g. Duffie 1992, Chaper 7. In our model, he value of h s in 3.4.6 sill allows us o express he dynamics of he discouned price process of he sock under Q in erms of he Q-maringale W Q, such ha Q indeed is a maringale measure for he combined model. We see ha h s is a funcion of ime and he presen value of he sock only, such ha he price process of he sock is a Markov process under Q as well. Noe ha all discouned price processes of asses radeable on he marke have o be maringales under he equivalen maringale measure. However, since conracs coningen on he moraliy inensiy or ransiions of Z are no raded on he financial marke, his requiremen does no give furher condiions on h µ and g jk han he ones already given by E P [Λ T ] = 1 and g > 1. We do, however, impose he furher condiion, ha h µ and g mus be of he form h µ,µ [x]+ and g jk,µ [x]+. This preserves he independence beween G and H, I under Q, and ensures ha µ and Z are Markov processes under Q. The dynamics of µ under Q are given by dµ [x]+ = α µ,µ [x]+ + σ µ,µ [x]+ h µ,µ [x]+ d + σ µ,µ [x]+ d W Q = α µ,q,µ [x]+ d + σ µ,µ [x]+ d W Q, 3.4.7 where we have defined α µ,q,µ [x]+ := α µ,µ [x]+ + σ µ,µ [x]+ h µ,µ [x]+. Using Girsanov s heorem for poin processes, see e.g. Andersen e al. 1993, we find ha jk he ransiion inensiy of Z from j o k under Q is given by λ = 1 + g jk λ jk. Hence, he above assumpion g jk > 1 is needed in order o ensure ha λ jk >. Changing he measure from P o Q yields some new naural Q-maringales: M jk,q = N jk λ jk u du = Njk 1 + g jk u I j u µjk [x]+u du. This shows ha he P-maringales M jk coincide wih he corresponding Q-maringales M jk,q if and only if g jk =. Remark 3.4.2 The sign of g jk does no have o be he same for all. In he model, where we only observe wheher one insured individual is alive or dead, represened by saes and 1, respecively, we could for example expec g 1 > for low ages low values of x + and g 1 < a large ages large values of x +. This leads o a moraliy inensiy which is oo high a low ages and oo low a high ages, such ha he moraliy inensiy a all imes is chosen o he safe side as seen from he insurance companies poin of view, if he insurance companies sell erm insurance coverage a low ages and life annuiies saring a large ages. Remark 3.4.3 In he res of he chaper we will be working under some arbirary, bu fixed maringale measure Q, and herefore i is of imporance o be able o find expressions for he marke survival probabiliies, see Secion 3.4.7 for a definiion. Modelling he moraliy inensiy by an exended Cox Ingersoll Ross model under P, we are ineresed

84 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS in choices of h µ ha lead o an exended Cox Ingersoll Ross model under Q as well. We hus need he dynamics under Q o be of he form dµ [x]+ = β µ,q γ µ,q µ [x]+ d + ρ µ,q µ [x]+ d W Q, where β µ,q, γ µ,q and ρ µ,q are funcions of, which saisfy he condiions given in Secion 3.3.3 in order o ensure sric posiiviy of he moraliy inensiy. A comparison of he dynamics under P and Q shows ha h µ mus be of he form h µ,µ [x]+ = δ µ [x]+ + δ µ[x]+ 3.4.8 for some deerminisic funcions δ and δ. Since µ [x]+ >, his leads o he following equaions ρ µ,q = ρ µ, 3.4.9 β µ,q = β µ + ρ µ δ, 3.4.1 γ µ,q = γ µ ρ µ δ. 3.4.11 This shows ha given an exended Cox Ingersoll Ross model under P and a Girsanov kernel h µ of he form 3.4.8, hen he Q-dynamics are in accordance wih an exended Cox Ingersoll Ross model, wih coefficiens given by 3.4.9, 3.4.1 and 3.4.11. Moreover, if we have sric posiiviy of he moraliy inensiy under P, hen he condiion δ ensures sric posiiviy under Q as well. 3.4.6 A brief review of financial conceps In his secion some conceps from he financial lieraure are inroduced wihin he presen framework. Under some equivalen maringale measure Q inroduced in Secion 3.4.5, he dynamics of he price processes under Q are given by ds = r,s S d + σ s,s S dw Q, S >, db = r,s B d, B = 1, where W Q T is a Wiener process on he inerval [,T] under Q. A rading sraegy is an adaped process ϕ = ϑ,η saisfying cerain inegrabiliy condiions. The pair ϕ = ϑ,η is inerpreed as he porfolio held a ime. Here, ϑ and η, respecively, denoe he number of socks and he discouned deposi on he savings accoun in he porfolio a ime. The value process Vϕ associaed wih ϕ is given by A sraegy ϕ is called self-financing if V ϕ = V ϕ + V ϕ = ϑ S + η B. ϑ u ds u + η u db u.

3.4. THE MODEL 85 Thus, he value a any ime of a self-financing sraegy is he iniial value added ineres on he savings accoun and rading gains; wihdrawals and deposis are no allowed during,t. A coningen claim or a derivaive wih mauriy T is an F T -measurable, Q- square inegrable random variable H. Hence, he class of coningen claims depends on he equivalen maringale measure Q. If H can be wrien as ΦS T,µ [x]+t,z T for some funcion Φ : R 2 + J R, i is called a simple coningen claim. A coningen claim is called aainable if here exiss a self-financing sraegy such ha V T ϕ = H, P-a.s. An aainable claim can hus be replicaed perfecly by invesing V ϕ a ime and invesing during he inerval [,T] according o he self-financing sraegy ϕ. Hence, a any ime, here is no difference beween holding he claim H and he porfolio ϕ. In his sense, he claim H is redundan in he marke and from he assumpion of no arbirage i follows ha he price of H mus be V ϕ a any ime. Thus, he iniial invesmen V ϕ is he unique arbirage free price of H. Noe ha if ϕ = ϑ,η is a self-financing porfolio replicaing he coningen claim H, hen H has he following represenaion T T H = V T ϕ = V ϕ + ϑ ds + η db. If all coningen claims are aainable, he model is called complee and oherwise i is called incomplee. A self-financing sraegy is a so-called arbirage if V ϕ = and V T ϕ P-a.s. wih PV T ϕ > >. I is well known from he financial lieraure, see e.g. Björk 24, ha he model B,S, G is complee and arbirage free under he assumpions on he coefficiens given in Secion 3.4.1. Thus, a coningen claim specifying he amoun ΦS T o be paid ou a ime T has a unique arbirage free price process π,s T, which can be characerized by he following PDE on [,T] R + : π,s + r,ss s π,s + 1 2 σs,s 2 s 2 ss π,s r,sπ,s =, 3.4.12 wih boundary condiion πt,s = Φs. When we inroduce oher sources of randomness in he model, which are no radeable on he marke, we ge an incomplee marke. This will be he case for B,S, F. Here, we are sill able o replicae claims which only depend on he randomness from B, S, whereas claims conaining an elemen of insurance are no replicable. Thus, insurance conracs canno be priced uniquely by a no arbirage argumen. However, for each admissible choice of h µ and g, we ge an equivalen maringale measure Q, which can be used o derive possible prices for coningen claims, which are consisen wih absence of arbirage. One possible choice of Q is obained by leing h µ = and g jk =. Here, he marke is said o be risk neural wih respec o sysemaic and unsysemaic moraliy risk. 3.4.7 Marke survival probabiliies Here, we derive a PDE for he marke survival probabiliies. Le he dynamics of µ be given by 3.4.7. Now consider he case where J = {,1}, wih corresponding o he policyholder being alive and 1 o he policyholder being dead. Using he noaion µ [x]+

86 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS and g insead of µ 1 [x]+ o T for an x + year old by and g1 we can define he marke survival probabiliy from ime S Q,µ [x]+,t := E Q [ IT ] Z [ =,µ [x]+ = E Q e ÊT Using Iô s formula and he fac ha M given by M = E Q [ e ÊT 1+gτµ [x]+τdτ ] F, 1+gτµ [x]+τdτ is a Q-maringale, we can obain he following PDE on [,T] R + : = S Q,µ,T + α µ,q,µ µ S Q,µ,T wih boundary condiion ] µ [x]+. + 1 2 σµ,µ 2 µµ S Q,µ,T 1 + g,µµs Q,µ,T, 3.4.13 S Q T,µ,T = 1. This PDE differs from he one given in 3.3.2 for he survival probabiliies by he coefficien α µ,q and he loading facor g appearing in he las erm only. Remark 3.4.4 Analogously o he forward moraliy inensiy we can now define he marke forward moraliy inensiy by f µ,q,t := T log S Q,µ [x]+,t. 3.5 Marke Reserves In radiional lieraure on life insurance he prospecive reserve is deermined as he expeced value of fuure discouned benefis less premiums under a echnical probabiliy measure, which is subjecively chosen and in general differen from P and Q. In he presen conex, we are working wih a marke reserve, which is he price a which he insurance conrac could be sold on he financial marke. In order o exclude arbirage possibiliies, he marke reserve is he expeced value of discouned fuure benefis less premiums under some arbirary, bu fixed, marke measure Q. Consider a general paymen sream A, where paymens are allowed o depend on he developmen of he financial marke. More precisely, A is assumed o be of he form da = I k dak + a kl dnkl, 3.5.1 k l:l k where da k = ak d + Ak Ak = ak d + Ak. 3.5.2

3.5. MARKET RESERVES 87 We hus consider a paymen sream, where paymens are coningen on he developmen of he underlying insurance conrac as described by he Markov process Z, see Secion 3.4.3. According o 3.5.1 3.5.2, we allow for 3 differen ypes of paymens, all of which may be linked o he sock S. Firsly, here are amouns a jk = a jk,s payable immediaely upon ransiion from sae j o sae k. These are called general life insurances. Secondly, here are general sae-wise annuiies payable coninuously a rae a j = aj,s a ime, coningen on he policy sojourning in sae j. Lasly, we allow for lump sum paymens A j,s. However, for noaional convenience, we resric lump sum paymens o he iniial ime and he erminal ime T only, i.e. A k = if / {,T }. We noe ha since all paymens are assumed o be funcions of he curren value of he sock only, we exclude pah dependen paymen funcions, such as so-called Asian and Russian opions; for a reamen of hese and oher exoic opions see Musiela and Rukowski 1997. Assume ha,s a j,s,,s a jk,s and s A j T,s are measurable funcions, and ha E Q [ T B 1 u ajk u λ jk u du ] <, j k. Then he processes B 1 a jk dm jk,q are Q-maringales, see Brémaud 1981, Lemma L3 p. 24. We use he convenion, which is sandard in acuarial lieraure, ha he reserve a ime is he value of fuure paymens afer paymens due a ime. Le posiive amouns represen benefis and negaive amouns represen premiums. The marke reserve for a conrac wih paymen sream A described above and erminaion a ime T can hen be wrien as V,S,µ [x]+,z = E Q [ T e Êτ rudu da τ F ], < T, and V T,S T,µ [x]+t,z T =. Noe ha V is a funcion of he sae of he insurance porfolio and he curren value of he sock and moraliy inensiy only. This is due o he resricions on h µ and g, which ensure ha he processes are Markov under Q, and he fac ha he paymen funcions are resriced o depend on he presen value of he sock only. Since he presen sae of he policy is known, he relevan quaniies are he sae-wise marke reserves. Using ha S and µ are Markov processes under Q and insering he definiion of A from 3.5.1 3.5.2,

88 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS we ge he following expression for he sae-wise reserves for < T: V j,s,µ [x]+ = E Q [ T = E Q T + E Q [ = k J + k J + k J T e Êτ rudu da τ Z = j,s,µ [x]+ ] e Êτ rudu k J e ÊT r udu k J E Q [ e Êτ l:l k T [ E Q e ÊT rudu a k τ E Q [ e Êτ I τ k a k τdτ + a kl τ dnτ kl l:l k Z = j,s,µ [x]+ ] IT A k k T Z = j,s,µ [x]+ r udu A k T S ]E [ Q Iτ k Z = j,µ [x]+ ]dτ rudu a kl τ ] [ S E Q Iτ k 1 + g kl τ S ]E [ ] Q IT k Z = j,µ [x]+. µ kl [x]+τ ] Z = j,µ [x]+ dτ Here, we have used he Q-compensaors for N jk and he fac ha B 1 a jk dm jk,q are Q- maringales in he second equaliy. Moreover, we have used he Q-independence beween S and Z,µ. Disregarding he random course of he policy, he quaniies a j u,s u, a jk u,s u and A j T,S T are simple coningen claims in he financial marke given by B, S, G. Since his marke is complee, see Secion 3.4.6, he claims can be uniquely priced. Using ha S is a Markov process he corresponding unique arbirage free price processes are for u T given by [ ] F j,s,u = E Q e Êu rτdτ a j u,s u S, [ ] F jk,s,u = E Q e Êu rτdτ a jk u,s u S, [ ] F j,s,t = E Q e ÊT r τdτ A j T,S T S. Defining he funcions H k,j,µ [x]+,u and H kl,j,µ [x]+,u by [ ] H k,j,µ [x]+,u := E Q Iu k Z = j,µ [x]+, u T, and H kl,j,µ [x]+,u := E Q [ I k u 1 + g kl u µ kl [x]+u ] Z = j,µ [x]+, u T, k l,

3.6. MORTALITY-LINKED CONTRACTS 89 he sae-wise marke reserves for < T can be wrien as V j,s,µ [x]+ = k J + k J T F k,s,τh k,j,µ [x]+,τdτ l:l k T F kl,s,τh kl,j,µ [x]+,τdτ + k J F k,s,th k,j,µ [x]+,t. For similar calculaions under deerminisic ransiion inensiies, see Møller 21c. Using eiher maringale mehods as in Møller 21c or he generalized Thiele differenial equaion in Seffensen 2, we obain he following sysem of PDEs for he marke reserves on [,T R 2 + for all j J : a j,s r,sv j,s,µ + V j,s,µ + α µ,q,µ µ V j,s,µ + 1 2 σµ,µ 2 µµ V j,s,µ + r,ss s V j,s,µ + 1 2 σs,s 2 s 2 ss V j,s,µ + a jk,s + V k,s,µ V j,s,µ 1 + g jk,µ µ jk =, 3.5.3 k:k j wih boundary condiions V j T,s,µ = A j T,s. Since he sysem of PDEs in general does no have an analyic soluion, we have o resor o numerical echniques in order o solve for V j,s,µ in 3.5.3. Example 3.5.1 We address he special case where he sae space for Z is J = {,1}, wih corresponding o he policyholder being alive and 1 corresponding o he policyholder being dead. In his case we have ha S Q,µ [x]+,u = H,,µ [x]+,u for u T. Consider a uni-linked endowmen insurance paid by a lump sum premium a ime. An endowmen insurance pays ou a specified amoun in case he policyholder dies or survives o ime T whichever happens firs. Here he sae-wise marke reserves for < T are given by V,S,µ [x]+ = T V 1,S,µ [x]+ =. F 1,S,τH 1,,µ [x]+,τdτ + F,S,TS Q,µ [x]+,t, 3.6 Moraliy-linked conracs 3.6.1 Moivaion In Danish life insurance pracice, premiums or benefis, depending on whichever is chosen in he conrac, are deermined applying he principle of equivalence wih a deerminisic

9 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS moraliy inensiy, known as he firs order moraliy inensiy. The firs order moraliy inensiy is chosen o be on he safe side as seen from he company s poin of view. This is usually obained by working wih a moraliy inensiy which is believed o be oo high a low ages and oo low a large ages, see for example he Danish moraliy able G82. Of course, he fuure moraliy inensiy is unknown, so his has o be based on he informaion available a he ime of signing of he conrac. If he moraliy inensiy behaves as expeced, and he insured porfolio behaves according o he general moraliy inensiy, his approach leads o a sysemaic surplus for he enire porfolio. If he porfolio is large enough, he srong law of large numbers applies, which implies ha he porfolio behaves according o he general moraliy inensiy, provided ha no selecion mechanism has been applied. By he so-called conribuion principle, he sysemaic surplus mus be redisribued among he insured as bonus by aking ino consideraion o which exen he insured has aken par in generaing he surplus. The companies ypically use he bonus o buy addiional insurance cover, similar o he ones already sipulaed in he conrac. This procedure is unproblemaic as long as he real moraliy inensiy does no behave worse, as seen from he insurer s poin of view, han he chosen deerminisic moraliy inensiy. However, no maer how safe he deerminisic moraliy inensiy is chosen, here is always a risk ha he moraliy inensiy behaves worse, even hough his risk may be very small. According o he insurance conracs, he companies canno allocae negaive bonus o he insured, i.e. hey canno reduce benefis or, equivalenly, increase premiums. Thus, he companies are subjec o a sysemaic moraliy risk relaed o he fuure developmen of he moraliy inensiy. One possible way for he insurance companies o reduce his risk is o ransfer some or all of i o he insured or oher agens or companies. For example, one could currenly adap premiums or benefis o he developmen of he moraliy inensiy. We shall refer o such conracs as moraliy-linked conracs. More precisely, one could link he premiums or benefis o he developmen of some large group of reference individuals. This group migh consis of he enire Danish populaion, he enire porfolio of he insurance company or a mixed porfolio from all Danish insurance companies. One could hen agree on a specific esimaion procedure from which he rue moraliy inensiy is deermined in order o avoid misuse from he companies and possible misrus from he insured. In his sense, one can view he rue moraliy inensiy as an observable quaniy. The main idea wih moraliy-linked insurance conracs is ha equivalence beween premiums and benefis is esablished by using he informaion available a ime. A ime, he sae-wise rerospecive and prospecive reserves are calculaed using he informaion currenly available. In order for he expeced sae-wise rerospecive and prospecive reserves o be equal here are wo adjusmen possibiliies: The premiums and he benefis. Adjusmen of he premium is only a possibiliy if he conrac is no enirely paid by a lump sum premium. This approach would reduce he companies moraliy risk o he risk associaed wih changes in he moraliy inensiy ha have occurred afer he las adjusmen and o unsysemaic risk. If he adjusmen is done sufficienly ofen, he sysemaic moraliy risk can be considered negligible.

3.6. MORTALITY-LINKED CONTRACTS 91 In he following secion we presen he idea of moraliy-linked conracs by means of a simple example. More general resuls concerning moraliy-linked conracs will be presened elsewhere. 3.6.2 Pure endowmen The simples non-rivial conrac is a pure endowmen paid by a single premium π a ime for some policy-holder aged x. Benefis are described by an adaped sochasic process K T, which deermines he sum o be paid ou a ime T in case of survival unil age x + T. In paricular K is he sum insured calculaed a ime. A ime he sum insured is given by K, which may be smaller or bigger han K. The exac size of K will depend on he developmen of he underlying moraliy inensiy and he financial marke in a specified way described below. In he following, we use he noaion µ [x]+ and g insead of µ 1 [x]+ model for he insurance conrac and one policyholder. The principle of equivalence under Q gives he premium π for he pure endowmen wih sum insured K : and g1, indicaing ha we are working wih a wo sae Markov π = K p,t S Q,µ [x],t, 3.6.1 where he processes S Q,µ [x]+,t and p,t are defined by E Q [e ÊT 1+guµ[x]+udu µ [x]+ ] and E Q [e ÊT r udu G ] respecively. Thus, he premium is he marke price of benefis. Remark 3.6.1 Using he marke forward moraliy inensiy, he premium can be wrien as π = K p,te ÊT fµ,q,udu. The advanage of rewriing he premium his way is wofold. Firsly, we observe ha calculaing premiums using he equivalence principle under he marke measure is done using a known moraliy inensiy. Secondly, working wih f µ,q insead of f µ shows ha he fair premium is deermined by using a measure, which reflecs he marke s aiude owards boh he sysemaic and he unsysemaic moraliy risk. Inspired by Norberg 1991 we work wih sae-wise rerospecive reserves a ime, T, defined by V i,rero = π U U K T 1 {i=} 1 {=T }, 3.6.2 where U T is a sochasic process wih Q-dynamics of he form du = α U d + σu dw Q. Here α U and σ U are G-adaped and hus F-adaped processes. In order for he conrac o be fair he process U mus fulfill a condiion, which will be given laer. The sae-wise

92 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS rerospecive reserves are hus he accumulaed value of premiums less benefis in [, ], given he presen sae of he policy. Noe ha he accumulaion is done wih an arbirary accumulaion facor U U,. For differen choices of α U and σ U we hus have differen sae-wise rerospecive reserves. One possibiliy is o choose α U = r U and σ U =, where r is some rae of reurn. Possible choices of r are he acual ineres rae or some rae averaging ou he rue ineres rae or he invesmen reurn of he company over ime. If all individual conracs are fair, as measured by he no arbirage principle, his can be hough of as he deposi rae used in pracice. We recall ha he prospecive reserves are equal o he marke reserves defined in Secion 3.5. Le V i,pro denoe he prospecive reserve a ime [ given he insured is in sae i and he sum insured is K ] = E Q e ÊT r udu K IT Z = i, I G, < T. 3.6.3 V i,pro As a crierion in order o [ calculae he adaped benefis, K, we use ] E Q V Z,pro [ ] G I = E Q V Z,rero G I. 3.6.4 We noe ha he expecaion operaor only refers o he possible saes of he insurance conrac, i.e. wheher Z is or 1. For he conrac o be fair he expeced discouned value under Q of he acual paymens should be, i.e. E Q [ π I T e ÊT rudu K T ] =, 3.6.5 which means ha he principle of equivalence under Q should apply. As we shall see below, his leads o a condiion on he accumulaion process U. Firs, we express K T in erms of U: A ime T we have by definiion ha whereas 3.6.2 gives and V,pro T V,rero T V 1,rero T Crierion 3.6.4 applied a ime T hus gives = V 1,pro T =, = π U T U K T = π U T U. U T 1 K T = π U. e ÊT 1+guµ [x]+udu Insering his ino 3.6.5 and using ieraed expecaions we find ha he process U mus fulfill: [ ] = E Q π ITe ÊT rudu K T = E Q [ π I T e ÊT rudu π U T U 1 = π 1 E Q [ e ÊT ruduu T U e ÊT 1+guµ [x]+udu ], ]

3.6. MORTALITY-LINKED CONTRACTS 93 which is equivalen o [ ] E Q e ÊT ruduu T = 1. 3.6.6 U Hence he expeced discouned value of accumulaion facor from o T should be 1. As expeced, we noe ha if U = eê rudu, i.e. if we accumulae wih he real ineres rae, hen U fulfills 3.6.6. Having resolved he problem of defining a fair conrac, we now urn our aenion owards he developmen of benefis. Applying crierion 3.6.4 a ime < T gives e Ê 1+guµ [x]+udu K p,ts Q,µ [x]+,t = π U U. Insering he expression for he premium from 3.6.1 we find he following relaionship beween he benefis decided a ime and ime : K K = e Ê p,ts Q,µ [x],t U U 1+guµ[x]+udu p,ts Q,µ [x]+,t. 3.6.7 We see ha he raio beween he new sum insured and he old sum insured is he raio beween he marke value a ime of a pure endowmen conrac wih expiraion T accumulaed o ime using he accumulaion facor U U, and he marke value a ime of a pure endowmen wih expiraion T muliplied by he a ime known marke survival probabiliy from ime o. For simpliciy we resric ourselves o he siuaion where r is deerminisic and U U is equal o eê rudu. We can hus consider he impac of he moraliy inensiy only. This implies ha 3.6.7 reduces o K K = S Q,µ [x],t e Ê 1+guµ [x]+udu S Q,µ [x]+,t To see how he benefi evolves in connecion wih changes in he moraliy inensiy, we derive he dynamics for K. Firs noe ha K can be wrien as K = K S Q,µ [x],teê 1+guµ [x]+udu 1 S Q,µ [x]+,t. In he following we use he simplified noaion S Q = S Q,µ [x]+,t. Using he parial differenial equaion 3.4.13 for S Q 1 we find he dynamics of : S Q 1 d S Q = 1 S Q 1 + g µ [x]+ + σ µ µ S Q 2 d S Q σ µ µ S Q S Q d W Q. 3.6.8.

94 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS Using 3.6.8 we arrive a he following dynamics of K under P: 2 dk = σ µ µ S Q S Q h µ + σ µ µ S Q S Q K d σ µ µ S Q S Q K d W. 3.6.9 The benefi hus increases or decreases by a fracion which is proporional o he curren benefi. This proporion facor consiss of wo erms: The firs erm comes from changing measure wih respec o he moraliy inensiy. This erm is he risk associaed wih he relaive change in marke survival probabiliy muliplied by h µ, which is minus 1 imes he marke price of sysemaic moraliy risk. The second drif erm is he squared relaive change in marke survival probabiliy associaed wih a change in moraliy inensiy. The las erm in he expression is he produc of he relaive change in marke survival probabiliy, he presen benefi and he change in he Wiener process driving he moraliy inensiy. Since σ µ µs Q d W is a local P-maringale, he sochasic exponenial formula S Q gives ha he differenial equaion 3.6.9 has he soluion K = K exp σ µ u µ S Q u S Q u h µ u + 1 2 σ µ u µ S Q u S Q u 2 du σ u µ µ Su Q Su Q d W u Noe ha he dynamics are expressed under P insead of Q, since hese are he dynamics o be observed by he insured and he insurer. Developmen of he benefi when allowing for risky invesmens Seffensen 21 works wih he value of pas conracual paymens accumulaed by he developmen of he invesmen porfolio of he insurance company. Using his idea, we assume ha he insurance company invess in a self-financing porfolio ϕ = ϑ,η, which leads o he sricly posiive value process Vϕ. Choosing U = V ϕ he rerospecive reserves are calculaed using he accumulaion facor obained by he risky invesmens of he company, i.e. we have V i,rero = π V ϕ V ϕ K T1 {i=} 1 {=T }. 3.6.1 Using he rerospecive reserve in 3.6.1 ogeher wih crierion 3.6.4 leaves he insured wih boh he risk associaed wih he developmen of he financial marke and he sysemaic moraliy risk, hence leaving he insurance company wih he unsysemaic moraliy risk only. In Secion 3.8 we show ha in his case he P-dynamics of K are

3.7. SECURITIZATION OF SYSTEMATIC MORTALITY RISK 95 given by dk = σ s S + h s + σ s s p,t 2 + σ µ µ S Q p,t σ s S s p,t σ s p,t ϑ S V ϕ σs S S Q ϑ S V ϕ s p,t p,t 2 σ s + h µ σ µ µ S Q K dw ϑ S V ϕ σs S σ µ S Q µ S Q S Q s p,t p,t K d K d W. 3.6.11 The drif hus consiss of five erms. The second and las drif erm relae o he moraliy inensiy and are recognized as he drif in 3.6.9. The firs erm is he square of he relaive change in zero coupon prices, and he hird erm comes from he correlaion beween he invesmen porfolio and he zero coupon prices. This erm is posiive negaive if he ineres rae has a posiive negaive dependence of he sock price. The fourh erm is minus he marke price of financial risk, h s, muliplied by he sum of he relaive change in zero coupon prices and minus he relaive change in he value of he invesmen porfolio. The las wo erms in he dynamics are relaed o he Wiener processes driving he sock and moraliy inensiy, respecively. 3.7 Securiizaion of sysemaic moraliy risk As a way o conrol he moraliy risk inheren in an insurance porfolio he company may purchase reinsurance cover. Reinsurance conracs usually consider he specific insurance porfolio of he company, and hence provide coverage for boh sysemaic and unsysemaic moraliy risk. An example of a moraliy dependen reinsurance conrac sold in pracice is a so-called moraliy swap. Prices of reinsurance conracs concerning boh sysemaic and unsysemaic moraliy risk can be found using he mehods already esablished in Secion 3.5. However i seems ha many life insurance companies are hesian o buy long erm reinsurance coverage. One reason could be ha he riskiness of he reinsurance business would leave he insurance companies wih a subsanial credi risk. As an alernaive o reinsurance we consider securiizaion. Here, he company rades conracs on he financial marke, which depend on he developmen of he moraliy inensiy. An imporan difference beween reinsurance and securiizaion is, ha moraliy conracs sold on he financial marke depend on he general developmen of he moraliy inensiy, and hence only offer proecion for he sysemaic moraliy risk. Inroducing producs coningen on he moraliy inensiy naurally raises quesions regarding he esimaion of he moraliy inensiy. Since, hese quesions are similar o hose in he case of moraliy-linked conracs, we refer o he discussion in Secion 3.6. The advanages of securiizaion over radiional reinsurance is he possible lower cos when sandardizing producs and he larger capaciy of he financial marke. More deails on securiizaion of moraliy risk can be found in Lin and Cox 25. For reamens of securiizaion of

96 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS caasrophe losses, which seems o be he mos developed area of securiizaion, see Chrisensen 2, Cox, Fairchild and Pedersen 2 and references herein. In his secion we firs derive a PDE for he price process of a wide class of derivaives on he moraliy inensiy. Then we examine differen possibiliies for an insurance company, which is ineresed in hedging a pure endowmen, and finally we invesigae conracs wih a risk premium. 3.7.1 Pricing moraliy derivaives Inspired by Björk 24, Chaper 8 we consider derivaives of he moraliy inensiy wih a payoff of he form ΦT,µ [x]+t,ψ 1 T,Ψ 2 T, where he processes Ψ i, i = 1,2, are given by Ψ i = q i τ,µ [x]+τ dτ, for posiive funcions q i. The noaion above indicaes ha he derivaive is payable a ime T, and ha i may depend on he moraliy inensiy a expiraion ime T and on he inegral over,t] of wo differen funcions of he moraliy inensiy. This ype of conrac covers sandard European and Asian opions, and hus includes mos conracs. Using he independence beween he financial marke and he moraliy inensiy, he price process can be wrien as π,s,µ [x]+,ψ 1,Ψ2 = p,teq [ ΦT,µ [x]+t,ψ 1 T,Ψ2 T I ]. Given an expression for p, T i is hus sufficien o derive a PDE for he Q-maringale Υ defined by Υ,µ [x]+,ψ 1,Ψ2 = EQ [ ΦT,µ [x]+t,ψ 1 T,Ψ2 T I ]. Using Iô s formula and he produc rule, we can now find he dynamics of Υ. Since Υ is a Q-maringale, he drif erm mus be, such ha we ge he following PDE on [,T] R 3 + : = Υ,µ,ψ 1,ψ 2 + α µ,q,µ µ Υ,µ,ψ 1,ψ 2 + q 1,µ ψ 1Υ,µ,ψ 1,ψ 2 + q 2,µ ψ 2Υ,µ,ψ 1,ψ 2 + 1 2 σµ,µ 2 µµ Υ,µ,ψ 1,ψ 2, 3.7.1 wih boundary condiion ΥT,µ,ψ 1,ψ 2 = ΦT,µ,ψ 1,ψ 2.

3.7. SECURITIZATION OF SYSTEMATIC MORTALITY RISK 97 3.7.2 Possible ways of hedging The fair premium for a pure endowmen conrac wih sum insured K can be wrien as π = KE Q [ e ÊT 1+guµ [x]+udu ] p,t. In he following we examine some possibiliies for hedging/conrolling he sysemaic moraliy risk associaed wih a pure endowmen on he financial marke. One possibiliy is o buy a derivaive wih payou Ke ÊT µ [x]+udu a ime T. The price for such a derivaive a ime is π,µ [x],ψ = KE Q [ e ÊT µ [x]+udu ] p,t, 3.7.2 where he process Ψ = Ψ T is given by Ψ = µ [x]+u, i.e. q,µ [x]+ = µ [x]+. This derivaive hedges he financial risk and sysemaic moraliy risk and leaves he company wih he unsysemaic moraliy risk only. From 3.7.2, we see ha he price of he derivaive is larger han he premium obained from he insured if and only if E Q [e ÊT µ [x]+udu ] > E Q [e ÊT 1+guµ[x]+udu ]. Since he companies wan a premium in order o carry a risk, he above hedging possibiliy only becomes ineresing if he price of he derivaive is less han he premium paid by he insured. Ofen he companies are ineresed in carrying pars of he sysemaic moraliy risk hemselves. In his case he companies can buy a call opion on he survival probabiliy wih srike C. The payoff from he call opion is given by ΦT,µ [x]+t,ψ T = e Ψ T C +. 3.7.3 Here, as in 3.7.2, he process Ψ = Ψ T is defined by Ψ = µ [x]+udu. The derivaive wih payoff 3.7.3 leads o a paymen if he real survival probabiliy is above some predefined level C. This leaves he insurance company wih he sysemaic moraliy risk up o a cerain level. Here, he srike C could be he survival probabiliy calculaed by using some known moraliy inensiy, for example he marke forward moraliy inensiy. The price process π,s,µ [x]+,ψ for he call opion can be found by solving 3.7.1 wih boundary condiion ΥT, µ, ψ = ΦT, µ, ψ and muliplying by p, T. 3.7.3 Conracs wih a risk premium Assume ha he company calculaes he premium of a pure endowmen wih sum insured K using some specified moraliy inensiy µ [x]+u u T, which saisfies e ÊT µ [x]+u du > e ÊT fµ,q,udu. The moraliy inensiy µ [x]+u u T can be inerpreed as he firs order moraliy inensiy used in pracice. Using µ [x]+u u T he company charges a premium π, which is larger han he fair premium π, given by he marke price under Q. This is similar o

98 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS charging a risk premium. The sysemaic surplus generaed by pricing wih µ [x]+u u T insead of f µ,q,u u T mus be reurned o he policyholders, and his could be obained by increasing benefis if he moraliy inensiy behaves as expeced. For example, he company could pay + K T = K 1 + a e ÊT µ [x]+u du e ÊT µ [x]+udu, 3.7.4 if he person survives. Here, a,1 is he proporion of he surplus which is paid o he policyholder. A naural resricion for conracs of he form 3.7.4 is ha hey are fair as measured by he marke measure. This gives he following equaion [ π = EQ IT K Te ÊT rudu] [ [ = E Q ITKe ÊT rudu] ] + + E Q ITaKe ÊT rudu e ÊT µ [x]+u du e ÊT µ [x]+udu ] + = π + akp,te Q E [I Q T e ÊT µ [x]+u du e ÊT [x]+udu µ I T [ ] + = π + akp,te Q e ÊT 1+guµ [x]+udu e ÊT µ [x]+u du e ÊT µ [x]+udu. Here, [ ] + p,te Q e ÊT 1+guµ [x]+udu e ÊT µ [x]+u du e ÊT µ [x]+udu is he price a ime, henceforh denoed π,s,µ [x],,, for a derivaive wih he following payoff a ime T where ΦT,µ [x]+t,ψ 1 T,Ψ 2 T = e Ψ1 T Ψ 1 = 1 + g u µ [x]+u du and Ψ 2 = e ÊT µ [x]+u du e Ψ2 T +, µ [x]+u du. Hence, he price a ime can be found by solving 3.7.1 wih he boundary condiion ΥT,µ,ψ 1,ψ 2 = ΦT,µ,ψ 1,ψ 2 and muliply by p,t. We obain he following expression for he fair value of a a = π π Kπ,S,µ [x],,. This formula can be inerpreed in he following way: The benefi is increased wih a number of pu opions on he survival probabiliy, which corresponds o he excess premium over he fair premium divided by he price of he pu opion.

3.8. DYNAMICS OF THE BENEFIT WITH RISKY INVESTMENTS 99 3.8 Dynamics of he benefi wih risky invesmens In his secion we derive 3.6.11. We assume ha he insurance company invess in a self-financing porfolio ϕ = ϑ,η, which leads o he value process Vϕ given by V ϕ = ϑ S + η B. We require ha V ϕ > for all. For u T, he raio Vϕ V uϕ describes he value a ime of one uni deposied a ime u. In he presen case, he sae-wise rerospecive reserves are given by V i,rero = π V ϕ V ϕ K T1 {i=} 1 {=T }. The sae-wise prospecive reserve given he insured is alive is sill given by 3.6.3. A ime < T he benefi mus saisfy he following equaion K = π eê 1+guµ V [x]+udu ϕ V ϕp,ts Q,µ [x]+,t = π eê 1+guµ[x]+udu 1 1 V ϕ V ϕ p,t S Q,µ [x]+,t. 3.8.1 Noe ha π and V ϕ are deermined a ime, and hus hey are independen of. In order o find he dynamics for K, we need o find he dynamics of each of he las four facors and possible quadraic covariaions. Since ϕ is self-financing, he dynamics of he value process are For 1 p,t we obain Firs we find dp, T dp,t = dv ϕ = ϑ ds + η db = ϑ r S d + σ s S dw Q + η r B d 1 d p,t = r V ϕd + σ s ϑ S dw Q. 1 = p,t 2dp,T + 1 d p,t. 3.8.2 p,t 3 p,t + r S s p,t + 1 2 σs 2 S 2 ssp,t d + σ s S s p,tdw Q = r p,td + σ s S s p,tdw Q, 3.8.3 where we have used 3.4.12. The predicable quadraic variaion is given by d p,t = σ s S s p,t 2 d. 3.8.4

1 CHAPTER 3. MARKET RESERVES AND MORTALITY-LINKED CONTRACTS Insering 3.8.3 and 3.8.4 ino 3.8.2, we ge 1 1 d = p,t p,t 2 r p,td + σs s s p,tdw Q 1 + p,t 3 σs S s p,t 2 d = 1 r d + σ s s p,t 2 p,t S d p,t 1 σ s s p,t p,t S dw Q p,t. 1 The dynamics of S Q,µ [x]+,t are given in 3.6.8. Since W Q and W Q are independen, we only need o find [ ] 1 d V ϕ, = ϑ σ s 1 p,t S σ s s p,t p,t S d. p,t Iô s formula gives he following dynamics of K : dk = π V ϕ V 1 1 ϕ p,t S Q,µ [x]+,t d eê 1+guµ [x]+udu + π eê 1+guµ 1 1 [x]+udu V ϕ p,t S Q,µ [x]+,t dv ϕ + π eê 1+guµ[x]+udu 1 1 V ϕ V ϕ S Q,µ [x]+,t d p,t + π eê 1+guµ[x]+udu 1 V ϕ V ϕ p,t d 1 S Q,µ [x]+,t [ ] V ϕ,. + π V ϕ eê 1+guµ [x]+udu 1 S Q,µ [x]+,t d Insering he above expressions and using 3.8.1, we ge 1 p,t dk = K 1 + g µ [x]+ d + K r d + K σ s ϑ S V ϕ dw Q + K r d + σ s S s p,t 2 d K σ s p,t S s p,t dw Q p,t + K 1 + g µ [x]+ + σ µ µ S Q 2,µ [x]+,t d S Q,µ [x]+,t K σ µ µ S Q,µ [x]+,t S Q,µ [x]+,t d W Q K σ s ϑ S s p,t V ϕ σs S d. p,t Simplifying, rearranging erms and changing o P-maringales now gives 3.6.11.

Chaper 4 Valuaion and Hedging of Life Insurance Liabiliies wih Sysemaic Moraliy Risk This chaper considers he problem of valuaing and hedging life insurance conracs ha are subjec o sysemaic moraliy risk in he sense ha he moraliy inensiy of all policy-holders is affeced by some underlying sochasic processes. In paricular, his implies ha he insurance risk canno be eliminaed by increasing he size of he porfolio and appealing o he law of large numbers. We propose o apply echniques from incomplee markes in order o hedge and valuae hese conracs. We consider a special case of he affine moraliy srucures considered in Chaper 3, where he underlying moraliy process is driven by a ime-inhomogeneous Cox-Ingersoll-Ross CIR model. Wihin his model, we sudy a general se of equivalen maringale measures, and deermine marke reserves by applying hese measures. In addiion, we derive risk-minimizing sraegies and mean-variance indifference prices and hedging sraegies for he life insurance liabiliies considered. Numerical examples are included, and he use of he sochasic moraliy model is compared wih deerminisic models. 4.1 Inroducion During he pas years, expeced lifeimes have increased considerably in many counries. This has forced life insurers o adjus expecaions owards he underlying moraliy laws used o deermine reserves. Since he fuure moraliy is unknown, a correc descripion requires a sochasic model, as i has already been proposed by several auhors, see e.g. Marocco and Piacco 1998, Milevsky and Promislow 21, Dahl 24b see Chaper 3, Cairns e al. 24, Biffis and Millossovich 24 and references herein. For a survey on curren developmens in he lieraure and heir relaion o our resuls, we refer he reader o Secion 3.2. The main conribuion of he presen chaper is no he inroducion 11

12 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING of a specific model for he moraliy inensiy, bu raher he sudy of he problem of valuaing and hedging life insurance liabiliies ha are subjec o sysemaic changes in he underlying moraliy inensiy. In Chaper 3, a general class of Markov diffusion models are considered for he moraliy inensiy, and he affine moraliy srucures are recognized as a class wih paricular nice properies. Here, we sudy a special case of he general affine moraliy srucures and demonsrae how such models could be applied in pracice. As saring poin we ake some smooh iniial moraliy inensiy curve, which is esimaed by sandard mehods. We hen assume ha he moraliy inensiy a a given fuure poin in ime a a given age is obained by correcing he iniial moraliy inensiy by he oucome of some underlying moraliy improvemen process, which is modelled via a ime-inhomogeneous Cox-Ingersoll-Ross CIR model. Our model implies ha he moraliy inensiy iself is described by a imeinhomogeneous CIR model as well. As noed in Chaper 3, he survival probabiliy can now be deermined by using sandard resuls for affine erm srucures. Wihin his seing, we consider an insurance porfolio and assume ha he individual lifeimes are affeced by he same sochasic moraliy inensiy. In paricular, his implies ha he lifeimes are no sochasically independen. Hence, he insurance company is exposed o sysemaic as well as unsysemaic moraliy risk. Here, as in Chaper 3, sysemaic moraliy risk refers o he risk associaed wih changes in he underlying moraliy inensiy, whereas unsysemaic moraliy risk refers o he risk associaed wih he randomness of deahs in a porfolio wih known moraliy inensiy. The sysemaic moraliy risk is a non-diversifiable risk, which does no disappear when he size of he porfolio is increased, whereas he unsysemaic moraliy risk is diversifiable. Since he sysemaic moraliy risk ypically canno be raded efficienly in he financial markes or in he reinsurance markes, his leaves open he problem of pricing insurance conracs. Here, we follow Chaper 3 and apply financial heories for pricing he conracs, and sudy a fairly general se of maringale measures for he model. We work wih a simple financial marke, consising of a savings accoun and a zero coupon bond and derive marke reserves for general life insurance liabiliies. These marke reserves depend on he marke s aiude owards sysemaic and unsysemaic moraliy risk. Based on an invesigaion of some Danish moraliy daa, we propose some pragmaic parameer values and calculae marke reserves by solving appropriae versions of Thiele s differenial equaion. Furhermore, we invesigae mehods for hedging and valuaing general insurance liabiliies in incomplee financial markes. One possibiliy is o apply risk-minimizaion, which has been suggesed by Föllmer and Sondermann 1986 and applied for he handling of insurance risks by Møller 1998, 21a, 21c. We demonsrae how risk-minimizing hedging sraegies may be deermined in he presence of sysemaic moraliy risk. These resuls generalize he resuls in Møller 1998, 21c, where risk-minimizing sraegies were obained wihou allowing for sysemaic moraliy risk. In addiion, his can be viewed as an exension of he work in Secion 3, where marke reserves were derived in he presence of sysemaic moraliy risk, bu wihou considering he hedging aspec. Uiliy indifference valuaion and hedging has gained considerable ineres over he las

4.2. MOTIVATION AND EMPIRICAL EVIDENCE 13 years as a mehod for valuaion and hedging in incomplee markes, see e.g. Schweizer 21b and Becherer 23 and references herein. These mehods have been applied for he handling of insurance conracs by e.g. Becherer 23, who worked wih exponenial uiliy funcions, and by Møller 21b, 23a, 23b, who worked wih mean-variance indifference principles. We derive mean-variance indifference prices wihin our model and compare he resuls wih he ones obained in Møller 21b. The presen chaper is organized as follows. Secion 4.2 conains a brief analysis of some Danish moraliy daa. In Secion 4.3, we inroduce he model for he underlying moraliy inensiy and derive he corresponding survival probabiliies and forward moraliy inensiies. The financial marke used for he calculaion of marke reserves, hedging sraegies and indifference prices is inroduced in Secion 4.4, and he insurance porfolio is described in Secion 4.5. Secion 4.6 presens he combined model, he insurance paymen process and he associaed marke reserves. Risk-minimizing hedging sraegies are deermined in Secion 4.7, and mean-variance indifference prices and hedging sraegies are obained in Secion 4.8. Numerical examples are provided in Secion 4.9, and Secion 4.1 conains proofs and calculaions of some echnical resuls. 4.2 Moivaion and empirical evidence We briefly describe ypical empirical findings relaed o he developmen in he moraliy during he las couple of decades. The resuls in his secion are based on Danish moraliy daa, which have been compiled and analyzed by Andreev 22. A more deailed saisical sudy is carried ou in Fledelius and Nielsen 22, who applied kernel hazard esimaion. From he daa maerial, we have deermined he exposure imes 7 74 78 76 8 84 196 197 198 199 2 196 197 198 199 2 Figure 4.2.1: To he lef: Developmen of he expeced lifeime of 3 year old females doed line a he op and 3 year old males solid line a he boom from year 196 o 23. To he righ: Expeced lifeimes for age 65. Esimaes are based on he las 5 years of daa available a calendar ime. Wy,x and number of deahs N y,x for each calendar year y, and age x, and calculaed he occurrence-exposure raes µ y,x = N y,x /W y,x. For each fixed y, we have deermined a smooh Gomperz-Makeham curve µ y,x = α y + β y c y x based on he las 5 years of daa available a calendar ime by using sandard mehods as described in Norberg 2. We have visualized in Figure 4.2.1 he developmen in he oal expeced lifeime of 3-

Reserv 14 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING..1.2 3 4 5 6 7 8..1 3 4 5 6 7 8 Figure 4.2.2: Esimaed moraliy inensiies for males o he lef and females o he righ. Solid lines are 197-esimaes, dashed lines correspond o 198, doed lines 199 and do-dashed lines 23. and 65-year old males and females based on he hisorical observaions from he year 196 o 23. These numbers are based on raw occurrence-exposure raes. The figure shows ha his mehod leads o an increase in he remaining lifeime from 198 o 23 of approximaely 2.5 years for males and 1.5 years for females aged 3. Using his mehod, he expeced lifeime in 23 is abou 75.3 years for 3 year old males and 79.5 for 3 year old females. If we alernaively use only one year of daa we see an increase from 72.5 o 75.5 for males and from 77.9 o 79.9 for females. Figure 4.2.2 conains he esimaed Gomperz-Makeham moraliy inensiies µ y,x for males and females, respecively, for 197, 198, 199 and 23. These figures show how he moraliy inensiies have decreased during his period. A closed sudy of he parameers α y,β y,c y indicae ha α y has decreased. The esimaes for β y increase from 196 o 199, where he esimaes for c y decrease. In conras, β y decreases and c y increases from 199 o 23. This approach.6 1..8 1. 198 1985 199 1995 2 age 3 198 1985 199 1995 2 age 65 Figure 4.2.3: Changes in he moraliy inensiy from 198 o 23 for males solid lines and females doed lines a fixed ages. The numbers have been normalized wih he 198 moraliy inensiies and are based direcly on he occurrence-exposure raes. does no involve a model ha akes changes in he underlying moraliy paerns ino consideraion. Anoher way o look a he moraliy inensiies is o consider changes in he moraliy inensiies a fixed ages, for example age 3 and 65, see Figure 4.2.3. For boh ages, we see periods where he moraliy increases and periods where i decreases. However, he general rend seems o be ha moraliy decreases. Moreover, we see ha he moraliy behaves differenly for differen ages and for males and females. Finally, we consider he siuaion, where we fix he iniial age and compare he fied Gomperz- Makeham curve for 198 wih he subsequen ones as he age increases wih calendar

4.3. MODELLING THE MORTALITY 15.75.9.88.94 1. 198 1985 199 1995 2 age 3 198 1985 199 1995 2 age 65 Figure 4.2.4: Changes in he moraliy inensiy from 198 o 23 for males solid lines and females doed lines as age increases. The numbers have been normalized wih he 198 moraliy inensiies and are based on he esimaed Gomperz-Makeham curves. ime. This is presened in Figure 4.2.4. Again, we see periods where he raio beween he curren moraliy and he 198-esimae increases and periods where i decreases. 4.3 Modelling he moraliy 4.3.1 The general model We ake as saring poin an iniial curve for he moraliy inensiy a all ages µ,g x for age x and gender g =male, female. I is assumed ha µ,g x is coninuously differeniable as a funcion of x. We neglec he gender aspec in he following, and simply wrie µ x. For an individual aged x a ime, he fuure moraliy inensiy is viewed as a sochasic process µx = µx, [,T] wih he propery ha µx, = µ x. Here, T is a fixed, finie ime horizon. In principle, one can view µ = µx x as an infiniely dimensional process. We model changes in he moraliy inensiy via a sricly posiive infinie dimensional process ζ = ζx, x, [,T] wih he propery ha ζx, = 1 for all x. Here and in he following, we ake all processes and random variables o be defined on some probabiliy space Ω, F,P equipped wih a filraion F = F [,T], which conains all available informaion. In addiion, we work wih several sub-filraions. In paricular, he filraion I = I [,T] is he naural filraion of he underlying process ζ. The moraliy inensiy process is hen modelled via: µx, = µ x + ζx,. 4.3.1 Thus, ζx, describes he change in he moraliy from ime o for a person of age x +. The rue survival [ probabiliy is defined by ] Sx,,T = E P e ÊT µx,τdτ [ ] I = E P e ÊT µ x+τ ζx,τdτ I, 4.3.2 and i is relaed o he maringale S M x,,t = E P [ e ÊT µx,τdτ I ] = e Ê µx,τdτ Sx,,T. 4.3.3

16 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING In general, we can consider survival probabiliies under various equivalen probabiliy measures. This is discussed in more deail in secion 4.6.1. 4.3.2 Deerminisic changes in moraliy inensiies As a special case, assume ha ζx, = e γx, where γx is fixed and consan. Thus, he moraliy inensiy a ime of an x + year old is defined by changing he known moraliy inensiy a ime of an x + -year old by he facor exp γx. If γx >, his model implies ha he moraliy improves by he facor exp γx each year. In paricular, aking all γx equal o one fixed γ means ha all inensiies improve/increase by he same facor. If µ corresponds o a Gomperz-Makeham moraliy law, i.e. hen he moraliy inensiy µ is given by µ x + = α + β c x+, 4.3.4 µx, = αe γx + β c x ce γx, 4.3.5 which no longer is a Gomperz-Makeham moraliy law. 4.3.3 Time-inhomogeneous CIR models The empirical findings in Secion 4.3.1 indicae ha he deerminisic ype of model considered above is oo simple o capure he rue naure of he moraliy. We propose insead o model he underlying moraliy improvemen process via dζx, = γx, δx,ζx,d + σx, ζx,dw µ, 4.3.6 where W µ is a sandard Brownian moion under P. This is similar o a so-called imeinhomogeneous CIR model, originally proposed by Hull and Whie 199 as an exension of he shor rae model in Cox e al. 1985, see also Rogers 1995. We assume ha 2γx, σx, 2 such ha ζ is sricly posiive, see Maghsoodi 1996. Here, γ, δ and σ are assumed o be known, coninuous funcions. I now follows via Iô s formula ha where dµx, = γ µ x, δ µ x,µx,d + σ µ x, µx,dw µ, 4.3.7 γ µ x, = γx,µ x +, 4.3.8 δ µ x, = δx, d d µ x + µ x +, 4.3.9 σ µ x, = σx, µ x +. 4.3.1

4.4. THE FINANCIAL MARKET 17 This shows ha µ also follows an ime-inhomogeneous CIR model, a propery which was also noed by Rogers 1995. In paricular, we noe ha γ µ x,/σ µ x, 2 = γx,/σx, 2, such ha µ is sricly posiive as well. If γx,/σx, 2, and hus γ µ x,/σ µ x, 2, is independen of, hen numerical calculaions can be simplified considerably, see Jamshidian 1995. The following proposiion regarding he survival probabiliy follows e.g. from Björk 24, Proposiion 22.2; see also Chaper 3. Proposiion 4.3.1 Affine moraliy srucure The survival probabiliy Sx,,T is given by where Sx,,T = e Aµ x,,t B µ x,,tµx,, Bµ x,,t = δ µ x,b µ x,,t + 1 2 σµ x, 2 B µ x,,t 2 1, 4.3.11 Aµ x,,t = γ µ x,b µ x,,t, 4.3.12 wih B µ x,t,t = and A µ x,t,t =. The dynamics of he survival probabiliy are given by dsx,,t = Sx,,T µx,d σ µ x, µx,b µ x,,tdw µ. Forward moraliy inensiies Inspired by ineres rae heory we inroduced he concep of forward moraliy inensiies in Chaper 3. In an affine seing, he forward moraliy inensiies are given by f µ x,,t = T log Sx,,T = µx, T Bµ x,,t T Aµ x,,t. 4.3.13 The imporance of forward moraliy inensiies is underlined by wriing he survival probabiliy on he form Sx,,T = e ÊT f µ x,,udu. 4.4 The financial marke In his secion, we inroduce he financial marke used for he calculaions in he following secions. The financial marke is essenially assumed o exis of wo raded asses: A savings accoun and a zero coupon bond wih mauriy T. The price processes are given by B and P,T, respecively. The uncerainy in he financial marke is described via a ime-homogeneous affine model for he shor rae. Hence, he shor rae dynamics under P are dr = α r rd + σ r rdw r, 4.4.1

18 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING where α r r = γ r,α δ r,α r, σ r r = γ r,σ + δ r,σ r. Here, W r is a sandard Brownian moion under P and γ r,α, δ r,α, γ r,σ and δ r,σ are consans. Denoe by G = G [,T] he naural filraion generaed by W r. The dynamics under P of he price processes are given by db = rbd, 4.4.2 dp,t = r + ρ,rp,td + σ p,rp,tdw r, 4.4.3 where ρ,r = σ p c,r σ r r + cσr r. 4.4.4 Here, c and c are consans saisfying cerain condiions given in Remark 4.4.1. Wih his choice of ρ, σ p is uniquely deermined from sandard heory for affine shor rae models, see 4.4.11. If we resric he model o he filraion G, he unique equivalen maringale measure for he financial marke is dq dp = ΛT, 4.4.5 where d Λ = Λh r dw r, Λ = 1, and where h r = ρ,r σ p,r = c σ r r + cσr r. 4.4.6 Under Q given by 4.4.5 he dynamics of he shor rae are given by dr = γ r,α,q δ r,α,q r d + γ r,σ + δ r,σ rdw r,q, 4.4.7 where W r,q is a sandard Brownian moion under Q and γ r,α,q = γ r,α cγ r,σ c, δ r,α,q = δ r,α + cδ r,σ. Since he drif and squared diffusion erms in 4.4.7 are affine in r, we have an affine erm srucure, see Björk 24, Proposiion 22.2. Thus, he bond price is given by where A r,t and B r,t solves P,T = e Ar,T B r,tr, Br,T = δ r,α,q B r,t + 1 2 δr,σ B r,t 2 1, 4.4.8 Ar,T = γ r,α,q B r,t 1 2 γr,σ B r,t 2, 4.4.9

4.4. THE FINANCIAL MARKET 19 wih B r T,T = and A r T,T =. The bond price dynamics under Q can be deermined by applying Iô s formula: dp,t = rp,td σ r rb r,tp,tdw r,q, 4.4.1 which in urn gives ha σ p,r = σ r rb r,t. 4.4.11 Remark 4.4.1 Recall ha if δ r,σ and γ r,α δ r,σ + δr,α γ r,σ δ r,σ 2 < 1 2, 4.4.12 hen Pr = >. Hence, we immediaely ge from 4.4.4 ha c = in his case. If δ r,σ and 4.4.12 does no hold, hen, exploiing he resuls of Cheridio, Filipović and Kimmel 23, gives ha 4.4.5 defines an equivalen maringale measure if c γ r,α + δr,α γ r,σ δ r,σ δr,σ 2. 4.4.13 No resricions apply o c in any case or o c if δ r,σ =. Remark 4.4.2 If δ r,σ = he shor rae is described by a Vasiček model, see Vasiček 1977. In his case he funcions A r and B r are given by B r,t = 1 δ r,α,q 1 e δr,α,q T, A r,t = Br,T T + γ r,α,q δ r,α,q 1 2 γr,σ δ r,α,q 2 γr,σ B r,t 2 4δ r,α,q. Leing γ r,σ =, we ge a ime-homogeneous CIR model for he shor rae, see Cox e al. 1985, which gives he following expressions for A r and B r 2 e ξr,q T 1 B r,t = ξ r,q + δ r,α,q e ξr,q T 1 + 2ξ r,q, A r,t = 2γr,α,Q 2ξ r,q e ξr,q +δ r,α,q T 2 δ r,σ log ξ r,q + δ r,α,q e ξr,q T 1, + 2ξ r,q where ξ r,q = δ r,α,q 2 + 2δ r,σ. For boh models, he funcions A r and B r depend on and T via he difference T, only.

11 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING 4.5 The insurance porfolio Consider an insurance porfolio consising of n insured lives of he same age x. We assume ha he individual remaining lifeimes a ime of he insured are described by a sequence T 1,...,T n of idenically disribued non-negaive random variables. Moreover, we assume ha PT 1 > IT = e Ê µx,sds, T, and ha he censored lifeimes T i = T i 1 {Ti T } + T1 {Ti >T }, i = 1,...,n, are i.i.d. given IT. Thus, given he developmen of he underlying process ζ, he moraliy inensiy a ime s is simply µx,s. Now define a couning process Nx = Nx, T by Nx, = n i=1 1 Ti, which keeps rack of he number of deahs in he porfolio of insured lives. We denoe by H = H T he naural filraion generaed by Nx. I follows ha Nx is an H I-Markov process, and he sochasic inensiy process λx = λx, T of Nx under P can be informally defined by λx,d E P [dnx, H I] = n Nx, µx,d, 4.5.1 which is proporional o he produc of he number of survivors and he moraliy inensiy. I is well-known, ha he process Mx = Mx, T defined by is an H I, P-maringale. dmx, = dnx, λx,d, T, 4.5.2 4.6 The combined model The filraion F = F T inroduced earlier is given by F = G H I. Thus, F is he filraion for he combined model of he financial marke, he moraliy inensiy and he insurance porfolio. Moreover, we assume ha he financial marke is sochasically independen of he insurance porfolio and he moraliy inensiy, i.e. GT and HT, IT are independen. In paricular, his implies ha he properies of he underlying processes are preserved. For example, Mx is also an F, P-maringale, and he F,P-inensiy process is idenical o he H I,P-inensiy process λx. We noe ha he combined model is on he general index-form sudied in Seffensen 2. However, Seffensen 2 conains no explici remarks or calculaions regarding a sochasic moraliy inensiy.

4.6. THE COMBINED MODEL 111 4.6.1 A class of equivalen maringale measures If we consider he financial marke only, i.e. if we resric ourselves o he filraion G, we found in Secion 4.4 ha given some regulariy condiions here exiss a unique equivalen maringale measure. This is no he case when analyzing he combined model of he financial marke and he insurance porfolio, see e.g. Møller 1998, 21c for a discussion of his problem. In he presen model, we can also perform a change of measure for he couning process Nx and for he underlying moraliy inensiy; we refer o Chaper 3 for a more deailed reamen of hese aspecs. Consider a likelihood process on he form dλ = Λ h r dw r + h µ dw µ + gdmx,, 4.6.1 wih Λ = 1. We assume ha E P [ΛT] = 1 and define an equivalen maringale measure Q via dq dp = ΛT. In he following, we describe he erms in 4.6.1 in more deail. The process h r, which is defined in 4.4.6, is relaed o he change of measure for he underlying bond marke. I is uniquely deermined by requiring ha he discouned bond price process is a Q-maringale. The erm involving h µ leads o a change of measure for he Brownian moion which drives he moraliy inensiy process µ. Hence, dw µ,q = dw µ h µ d defines a sandard Brownian moion under Q. Here, we resric ourselves o h µ s of he form h µ,ζx, = βx, ζx, σx, β x, + σx, ζx, 4.6.2 for some coninuous funcions β and β. In his case, he Q-dynamics of ζx, are given by where dζx, = γ Q x, δ Q x,ζx, d + σx, ζx,dw µ,q, γ Q x, = γx, + β x,, 4.6.3 δ Q x, = δx, + βx,. 4.6.4 Hence, ζ also follows a ime-inhomogeneous CIR model under Q. A necessary condiion for he equivalence beween P and Q is ha ζ is sricly posiive under Q. Thus, we observe from 4.6.3 ha we mus require ha β x, σx, 2 /2 γx,. The Q-dynamics of µx are now given by dµx, = γ µ,q x, δ µ,q x,µx,d + σ µ x, µx,dw µ,q, 4.6.5 where γ µ,q x, and δ µ,q x, are given by 4.3.8 and 4.3.9 wih γx, and δx, replaced by γ Q x, and δ Q x,, respecively. If h µ =, i.e. if he dynamics of ζ and hus µ are idenical under P and Q, we say he marke is risk-neural wih respec o sysemaic moraliy risk.

112 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING The las erm in 4.6.1 involves a predicable process g > 1. This erm affecs he inensiy for he couning process. More precisely, i can be shown, see e.g. Andersen e al. 1993, ha he inensiy process under Q is given by λ Q x, = 1 + gλx,. Using 4.5.1, we see ha λ Q x, = n Nx, 1 + gµx,, such ha µ Q x, = 1 + gµx, can be viewed as he moraliy inensiy under Q. Hence he process M Q x = M Q x, T defined by dm Q x, = dnx, λ Q x,d, T, 4.6.6 is an F,Q-maringale. If g =, he marke is said o be risk-neural wih respec o unsysemaic moraliy risk. This choice of g can be moivaed by he law of large numbers. In his chaper, we resric he analysis o he case, where g is a deerminisic, coninuously differeniable funcion. Combined wih he definiion of h r in 4.4.6 and he resriced form of h µ in 4.6.2, his implies ha he independence beween GT and HT, IT is preserved under Q. Now define he Q-survival probabiliy and he associaed Q-maringale by [ ] S Q x,,t = E Q e ÊT µ Q x,τdτ ζx, and S Q,M x,,t = E Q [ e ÊT µq x,τdτ ] ζx, = e Ê µq x,τdτ S Q x,,t. Calculaions similar o hose in Secion 4.3.3 give he following Q-dynamics of µ Q x dµ Q x, = γ µ,q,g x, δ µ,q,g x,µ Q x,d + σ µ,q,g x, µ Q x,dw µ,q, where γ µ,q,g x, = 1 + gγ µ,q x,, δ µ,q,g x, = δ µ,q x, d d g 1 + g, σ µ,q,g x, = 1 + gσ µ x,. Since he drif and squared diffusion erms for µ Q x, are affine in µ Q x,, we have he following proposiion Proposiion 4.6.1 Affine moraliy srucure under Q The Q-survival probabiliy S Q x,,t is given by S Q x,,t = e Aµ,Q x,,t B µ,q x,,t1+gµx,, where A µ,q and B µ,q are deermined from 4.3.11 and 4.3.12 wih γ µ x,, δ µ x, and σ µ x, replaced by γ µ,q,g x,, δ µ,q,g x, and σ µ,q,g x,, respecively. The dynamics of he Q-maringale associaed wih he Q-survival probabiliy are given by ds Q,M x,,t = 1 + gσ µ x, µx,b µ,q x,,ts Q,M x,,tdw µ,q. 4.6.7

4.6. THE COMBINED MODEL 113 Similarly o he forward moraliy inensiies, he Q-forward moraliy inensiies are given by f µ,q x,,t = T log SQ x,,t = µ Q x, T Bµ,Q x,,t T Aµ,Q x,,t. 4.6.8 4.6.2 The paymen process The oal benefis less premiums on he insurance porfolio is described by a paymen process A. Thus, da are he ne paymens o he policy-holders during an infiniesimal inerval [, + d. We ake A of he form da = nπdi { } + n Nx,T A TdI { T } + a n Nx,d + a 1 dnx,, 4.6.9 for T. The firs erm, nπ is he single premium paid a ime by all policyholders. The second erm involves a fixed ime T T, which represens he reiremen ime of he insured lives. This erm saes ha each of he surviving policy-holders receive he fixed amoun A T upon reiremen. The hird erm involves a piecewise coninuous funcion a = π c 1 { <T } + a p 1 {T T }, where π c are coninuous premiums paid by he policy-holders as long as hey are alive and a p corresponds o a life annuiy benefi received by he policy-holders. Finally, he las erm in 4.6.9 represens paymens immediaely upon a deah, and we assume ha a 1 is some piecewise coninuous funcion. 4.6.3 Marke reserves In he following we consider an arbirary, bu fixed, equivalen maringale measure Q from he class of measures inroduced in Secion 4.6.1 and define he process [ ] V,Q = E Q e Êτ rudu daτ [,T] F, 4.6.1 which is he condiional expeced value, calculaed a ime, of discouned benefis less premiums, where all paymens are discouned o ime. Using ha he processes A and r are adaped, and inroducing he discouned paymen process A defined by we see ha V,Q = [,] da = e Ê rudu da, [ e Êτ rudu daτ + e Ê rudu E Q e Êτ,T] rudu daτ ] F = A + e Ê rudu Ṽ Q. 4.6.11

114 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING In he lieraure, he process V,Q is called he inrinsic value process, see Föllmer and Sondermann 1986 and Møller 21c. The process Ṽ Q inroduced in 4.6.11 represens he condiional expeced value a ime, of fuure paymens. We shall refer o his quaniy as he marke reserve. We have he following resul: Proposiion 4.6.2 The marke reserve Ṽ Q is given by Ṽ Q = n Nx,V Q,r,µx,, 4.6.12 where V Q,r,µx, = T P,τS Q x,,τ a τ + a 1 τf µ,q x,,τ dτ + P,TS Q x,,t A T. 4.6.13 This can be verified by using mehods similar o he ones used in Møller 21c and Chaper 3. A skech of proof is given below. Some commens on his resuls: The quaniy V Q,r,µx, is he marke reserve a ime for one policy-holder who is alive, given he curren level for he shor rae and he moraliy inensiy. The marke reserve has he same srucure as sandard reserves. However, he usual discoun facor has been replaced by a zero coupon bond price P,T and he usual deerminisic survival probabiliy of he form exp τ µ x,udu has been replaced by he erm S Q x,,τ. In addiion, he Q-forward moraliy inensiy, f µ,q x,,τ, now appears insead of he deerminisic moraliy inensiy µ x,τ in connecion wih he sum a 1 τ payable upon a deah. Skech of proof of Proposiion 4.6.2: The proposiion follows by exploiing he independence beween he financial marke and he insured lives. In addiion, we use ha for any predicable, sufficienly inegrable process g, is an F,Q-maringale. For example, his implies ha gsdnx,s λ Q x,sds 4.6.14 [ T ] E Q e Êτ rudu a 1 τdnx,τ F [ T ] = E Q e Êτ rudu a 1 τλ Q x,τdτ F = T P,τa 1 τe Q [ n Nx,τµ Q x,τ F ] dτ. 4.6.15

4.7. RISK-MINIMIZING STRATEGIES 115 Here, he second equaliy follows by changing he order of inegraion and by using he independence beween r and N,µ. By ieraed expecaions, we ge ha E Q [n Nx,τ F] = E Q [ E Q [n Nx,τ F IT] ] F [ ] = E Q n Nx,e Êτ µq x,udu F = n Nx,S Q x,,τ, 4.6.16 where he second equaliy follows by using ha, given IT, he lifeimes are i.i.d. under Q wih moraliy inensiy µ Q x, and he hird equaliy is he definiion of he Q-survival probabiliy. Similarly, we have ha E Q [ n Nx,τµ Q x,τ F ] = E Q [ E Q [ n Nx,τµ Q x,τ ] ] F IT F [ ] = E Q n Nx,µ Q x,τe Êτ µq x,udu F = n Nx, τ SQ x,,τ = n Nx,S Q x,,τf µ,q x,,τ. 4.6.17 Here, he hird equaliy follows by differeniaing S Q x,,τ under he inegral. The resul now follows by using 4.6.15 4.6.17. We emphasize ha he marke reserve depends on he choice of equivalen maringale measure Q. In he remaining of he paper we work under he following assumpion Assumpion 4.6.3 V Q,r,µ C 1,2,2, i.e. V Q,r,µ is coninuously differeniable wih respec o and wice differeniable wih respec o r and µ. 4.7 Risk-minimizing sraegies The discouned insurance paymen process A is subjec o boh financial and moraliy risk. This implies ha he insurance liabiliies ypically canno be hedged and priced uniquely by rading on he financial marke. Møller 1998 applied he crierion of riskminimizaion suggesed by Föllmer and Sondermann 1986 for he handling of his combined risk for uni-linked life insurance conracs. This analysis led o so-called riskminimizing hedging sraegies, ha essenially minimized he variance of he insurance liabiliies calculaed wih respec o some equivalen maringale measure. Here, we follow Møller 21c, who exended he approach of Föllmer and Sondermann 1986 o he case of a paymen process. Furher applicaions of he crierion of risk-minimizaion o insurance conracs can be found in Møller 21a, 22.

116 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING 4.7.1 A review of risk-minimizaion Consider he financial marke inroduced in Secion 4.4 consising of a zero coupon bond expiring a T and a savings accoun. We denoe by X = P,T he discouned price process of he zero coupon bond. A sraegy is a process ϕ = ξ,η, where ξ is he number of zero coupon bonds held and η is he discouned deposi on he savings accoun. The discouned value process V ϕ associaed wih ϕ is defined by V,ϕ = ξx + η, and he cos process Cϕ is defined by C,ϕ = V,ϕ ξudxu + A. 4.7.1 The accumulaed coss C, ϕ a ime are he discouned value V, ϕ of he porfolio reduced by discouned rading gains he inegral and added discouned ne paymens o he policy-holders. A sraegy is called risk-minimizing, if i minimizes [ R,ϕ = E Q CT,ϕ C,ϕ 2 ] F 4.7.2 for all wih respec o all sraegies, and a sraegy ϕ wih V T,ϕ = is called - admissible. The process Rϕ is called he risk process. Föllmer and Sondermann 1986 realized ha he risk-minimizing sraegies are relaed o he so-called Galchouk-Kunia- Waanabe decomposiion, V,Q = E Q [A T F] = V,Q + ξ Q udxu + L Q, 4.7.3 where ξ Q is a predicable process and where L Q is a zero-mean Q-maringale orhogonal o X. I now follows by Møller 21c, Theorem 2.1 ha here exiss a unique -admissible risk-minimizing sraegy ϕ = ξ,η given by ϕ = ξ,η = ξ Q, V,Q ξ Q X A. 4.7.4 In paricular, i follows ha he cos process associaed wih he risk-minimizing sraegy is given by C,ϕ = V,Q + L Q. 4.7.5 The risk process associaed wih he risk-minimizing sraegy, he so-called inrinsic risk process, is given by R,ϕ = E Q [ L Q T L Q 2 F ]. 4.7.6 I follows from 4.7.4 ha V,ϕ = V,Q A, i.e. he discouned value process associaed wih he risk-minimizing sraegy coincides wih he inrinsic value process reduced by he discouned paymens. Noe ha he risk-minimizing sraegy depends on he choice of maringale measure Q. In he lieraure, he minimal maringale measure has been applied for deermining risk-minimizing sraegies, since his essenially corresponds o he crierion of local risk-minimizaion, which is a crierion in erms of P, see Schweizer 21a.

4.7. RISK-MINIMIZING STRATEGIES 117 4.7.2 Risk-minimizing sraegies for he insurance paymen process As noed in Secion 4.6.3, he inrinsic value process V,Q associaed wih he paymen process A is given by V,Q = A + n Nx,B 1 V Q,r,µx,, 4.7.7 where V Q,r,µx, is defined by 4.6.13. The Galchouk-Kunia-Waanabe decomposiion of V,Q is deermined by he following lemma: Lemma 4.7.1 The Galchouk-Kunia-Waanabe decomposiion of V,Q is given by where and V,Q = V,Q + ξ Q τdp τ,t + L Q, 4.7.8 V,Q = nπ + nv Q,r,µx,, 4.7.9 L Q = ν Q τdm Q x,τ + κ Q τds Q,M x,τ,t, 4.7.1 T ξ Q B r,τp,τ = n Nx, B r,tp,t SQ x,,τ a τ + a 1 τf µ,q x,,τ dτ + Br,TP,T B r,tp,t SQ x,,t A T, 4.7.11 ν Q = B 1 a 1 V Q,r,µx,, 4.7.12 T κ Q = n Nx, P B µ,q x,,τs Q x,,τ,τ B µ,q x,,ts Q,M x,,t a τ + a 1 τ f µ,q τ x,,τ Bµ,Q x,,τ B µ,q dτ x,,τ + P,T Bµ,Q x,,ts Q x,,t B µ,q x,,ts Q,M x,,t A T. 4.7.13 Proof of Lemma 4.7.1: See Secion 4.1.1. In he decomposiion obained in Lemma 4.7.1, he inegrals wih respec o he compensaed couning process M Q x and he Q-maringale S Q,M x,,t associaed wih he Q-survival probabiliy comprise he non-hedgeable par of he paymen process. The facor ν Q appearing in he inegral wih respec o M Q x in 4.7.1 represens he discouned

118 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING exra cos for he insurer associaed wih a deah wihin he porfolio of insured lives. I consiss of he discouned value of he amoun a 1 o be paid ou immediaely upon deah, reduced by he discouned marke reserve of one policy-holder B 1 V Q,r,µx,. In radiional life insurance, ν Q is known as he discouned sum a risk associaed wih a deah in he insured porfolio a ime, see e.g. Norberg 21; in Møller 1998, a similar resul is obained wih deerminisic moraliy inensiies. Changes in he moraliy inensiy lead o new Q-survival probabiliies, and his affecs he expeced presen value under Q of fuure paymens. This sensiiviy is described by he process κ Q appearing in 4.7.1, which can be inerpreed as he change in he discouned value of expeced fuure paymens associaed wih a change in he Q-maringale associaed wih he Q-survival probabiliy. I follows from 4.7.6 ha he inrinsic risk process is given by [ T 2 ] R,ϕ = E Q ν Q udm Q x,u + κ Q uds x,u,t Q,M F [ T = E Q ν Q u 2 d M Q x,u + κ Q u ] 2 d S Q,M x,u,t F = E Q[ T ν Q u 2 n Nx,u 1 + guµx,udu + κ Q u1 + guσ µ x,u 2 ] µx,ub µ,q x,u,ts Q,M F x,u,t du. Here, we have used he square bracke processes, ha M Q x is an adaped process wih finie variaion, and ha S Q,M x,,t is coninuous hence predicable, such ha he maringales are orhogonal. Using he general resuls on risk-minimizaion and Lemma 4.7.1, we ge he following resul. Theorem 4.7.2 The unique -admissible risk-minimizing sraegy for he paymen process 4.6.9 is ξ,η = ξ Q,n Nx,B 1 V Q,r,µx, ξ Q P,T, where ξ Q is given by 4.7.11. This resul is similar o he risk-minimizing hedging sraegy obained in Møller 21c, Theorem 3.4. However, our resuls differ from he ones obained here in ha he marke reserves depend on he curren value of he moraliy inensiy. The fac ha he sraegies are similar is reasonable, since we are essenially adding a sochasic moraliy o he model of Møller, and his does no change he marke in which he hedger is allowed o rade. As in Møller 21c, he discouned value process associaed wih he risk-minimizing sraegy ϕ is V,ϕ = n Nx,B 1 V Q,r,µx,, 4.7.14

4.8. MEAN-VARIANCE INDIFFERENCE PRICING 119 where V Q,r,µx, is given by 4.6.13. This shows ha he porfolio is currenly adjused, such ha he value a any ime is exacly he marke reserve. Insering 4.7.9 and 4.7.1 in 4.7.5 gives C,ϕ = nv Q,r,µx, nπ + ν Q τdm Q x,τ + κ Q τds Q,M x,τ,t. Hence he hedger s loss is driven by M Q x and S Q,M x,,t. The firs hree erms are similar o he ones obained by Møller 21c. The las erm, which accouns for coss associaed wih changes in he moraliy inensiy, did no appear in his model, since he worked wih deerminisic moraliy inensiies. Example 4.7.3 Consider he case where T = T, and where all n insured purchase a pure endowmen of A T paid by a single premium a ime. In his case, he Galchouk- Kunia-Waanabe decomposiion 4.7.8 of V,Q is deermined via ξ Q = n Nx, S Q x,,t A T, ν Q = P,TS Q x,,t A T, κ Q = n Nx, P,TeÊ µq x,udu A T, since V Q,r,µx, = P,TS Q x,,t A T, and S Q x,,t S Q,M x,,t = eê µq x,udu. This gives he -admissible risk-minimizing sraegy ξ = n Nx, S Q x,,t A T, η = Nx, Nx, P,TS Q x,,t A T. The risk-minimizing sraegy has he following inerpreaion: The number of bonds held a ime is equal o he Q-expeced number of bonds needed in order o cover he benefis a ime T, condiional on he informaion available a ime. The invesmens in he savings accoun only differ from if a deah occurs a ime, and in his case i consiss of a wihdrawal loan equal o he marke reserve for one insured individual who is alive. 4.8 Mean-variance indifference pricing Mehods developed for incomplee markes have been applied for he handling of he combined risk inheren in a life insurance conrac in Møller 21b, 22, 23a, 23b wih focus on he mean-variance indifference pricing principles of Schweizer 21b. In his secion, hese resuls are reviewed and indifference prices and opimal hedging sraegies are derived.

12 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING 4.8.1 A review of mean-variance indifference pricing Denoe by K he discouned wealh of he insurer a ime T and consider he meanvariance uiliy-funcions u i K = E P [K ] a i Var P [K ] β i, 4.8.1 i = 1,2, where a i > are so-called risk-loading parameers and where we ake β 1 = 1 and β 2 = 1/2. I can be shown ha he equaions u i K = u i indeed lead o he classical acuarial variance i=1 and sandard deviaion principle i=2, respecively, see e.g. Møller 21b. Schweizer 21b proposes o apply he mean-variance uiliy funcions 4.8.1 in an indifference argumen which akes ino consideraion he possibiliies for rading in he financial markes. Denoe by Θ he space of admissible sraegies and le G T Θ be he space of discouned rading gains, i.e. random variables of he form T ξudxu, where X is he price process associaed wih he discouned raded asse. Denoe by c he insurer s iniial capial a ime. The u i -indifference price v i associaed wih he liabiliy H is defined via sup u i c + v i + ϑ Θ T T ϑudxu H = sup u i c + ϑ Θ ϑudxu, 4.8.2 where H is he discouned liabiliy. The sraegy ϑ which maximizes he lef side of 4.8.2 will be called he opimal sraegy for H. In order o formulae he main resul, some more noaion is needed. We denoe by P he variance opimal maringale measure and le ΛT = d P dp. In addiion, we le π be he projecion in L2 P on he space G T Θ and wrie 1 π1 = T βudxu. I follows via he projecion heorem ha any discouned liabiliy H allows for a unique decomposiion on he form H = c H + T ϑ H udxu + N H, 4.8.3 where T ϑh dx is an elemen of G T Θ and where N H is in he space IR + G T Θ. From Schweizer 21b and Møller 21b we have ha he indifference prices for H are: v 1 H = E P [H ] + a 1 Var P [N H ], 4.8.4 v 2 H = E P [H ] + a 2 1 Var P [ ΛT]/a 2 2 Var P [N H ], 4.8.5 where 4.8.5 is only defined if a 2 2 VarP [ ΛT]. The opimal sraegies associaed wih hese wo principles are: ϑ 1 = ϑ H + 1 + VarP [ ΛT] β, 4.8.6 2a 1 ϑ 2 = ϑ H 1 + Var P [ ΛT] + Var a 2 1 P [N H ] β, 4.8.7 Var P [ ΛT]/a 2 2

4.8. MEAN-VARIANCE INDIFFERENCE PRICING 121 where 4.8.7 is only well-defined if a 2 2 > VarP [ ΛT]. For more deails, see Møller 21b, 23a, 23b. 4.8.2 The variance opimal maringale measure In order o deermine he variance opimal maringale measure P we firs urn our aenion o he minimal maringale measure, which loosely speaking is he equivalen maringale measure which disurbs he srucure of he model as lile as possible, see Schweizer 1995. The minimal maringale measure is obained by leing h µ = and g =. Hence, we have from Secion 4.4 ha he Radon-Nikodym derivaive ΛT for he minimal maringale measure is given by where h r is defined by 4.4.6. T ΛT = exp h r udw r u 1 2 T h r u 2 du, In general, he variance opimal maringale measure P and he minimal maringale measure ˆP differ. However, we find below ha hey coincide in our model. Since h r is G-measurable, he densiy ΛT is GT-measurable, and herefore i can be represened by a consan D and a sochasic inegral wih respec o P,T, see e.g. Pham, Rheinländer and Schweizer 1998, Secion 4.3. Thus, we have he following represenaion of ΛT T ΛT = D + ζudp u,t. 4.8.8 Schweizer 1996, Lemma 1 gives ha ΛT is he densiy for he variance opimal maringale measure as well, i.e. d P dp = ΛT, such ha ˆP = P. Hence, under he equivalen maringale measure P, he dynamics of he moraliy inensiy and he inensiy of he couning process Nx are unalered. For laer use, we inroduce he P-maringale Λ := E P [ ΛT F] = E P [ ΛT G]. Noe ha ΛT = ΛT. If h r is consan, calculaions similar o hose in Møller 23b for he Black-Scholes case give ha Λ Λ = e hr 2 T.

122 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING 4.8.3 Mean-variance indifference pricing for pure endowmens Le T = T and consider a porfolio of n individuals of he same age x each purchasing a pure endowmen of A T paid by a single premium a ime. Thus, he discouned liabiliy is given by H = n Nx,TBT 1 A T. Explici expressions for he mean-variance indifference prices can be obained under addiional inegrabiliy condiions. More precisely, we need ha cerain local P-maringales considered in he calculaion of Var P [N H ] are rue P-maringales. In his case we have he following proposiion. Proposiion 4.8.1 The indifference prices are given by insering he following expressions for E P [H ] and Var P N H in 4.8.4 and 4.8.5: and E P [H ] = np,tsx,,t A T, 4.8.9 T T Var P [N H ] = n Υ 1 Υ 2 d + n 2 Υ 1 Υ 3 d, 4.8.1 where [ ] Λ Υ 1 = E P Λ P,T A T 2, Υ 2 = E P[ Sx,,T 2 e Ê µx,udu µx, Υ 3 = E P [ σ µ x, µx,b µ x,,tsx,,te Ê µx,udu 2 ]. ] 1 + σ µ x,b µ x,,t 2 1 e Ê µx,udu, Idea of proof of Proposiion 4.8.1: The independence beween r and N,µ under P immediaely gives 4.8.9. The expression for he variance of N H in 4.8.1 follows from calculaions similar o hose in Møller 21b. For compleeness he calculaions are carried ou in Secion 4.1.2 under cerain addiional inegrabiliy condiions. We see from 4.8.1 ha he variance of N H can be spli ino wo erms. The firs erm, which is proporional o he number of insured, sems from boh he sysemaic and unsysemaic moraliy risk. Møller 21b also obained a erm proporional o he number of insured in he case wih deerminisic moraliy inensiy and hence only unsysemaic moraliy risk. The second erm which is proporional o he squared number of survivors sems solely from he sysemaic moraliy risk. Hence, he uncerainy associaed wih he fuure moraliy inensiy becomes increasingly imporan, when deermining indifference prices for a porfolio of pure endowmens, as he size of he porfolio increases.

4.8. MEAN-VARIANCE INDIFFERENCE PRICING 123 There are wo reasons for his. Firsly, changes in he moraliy inensiy, as opposed o he randomness associaed wih he deahs wihin he porfolio, are non-diversifiable; in paricular hey affec all insured individuals in he same way. Secondly, his risk is no hedgeable in he marke. Proposiion 4.8.2 The opimal sraegies are given by insering 4.8.1 and he following expression for ϑ H in 4.8.6 and 4.8.7: where ϑ H = ξ P ζ 1 ν P udmx,u + κ P uds M x,u,t, 4.8.11 Λu ξ P = n Nx, Sx,,T A T, ν P = P,TSx,,T A T, 4.8.12 κ P = n Nx, P,TeÊ µx,udu A T. 4.8.13 Proof of Proposiion 4.8.2: Expression 4.8.11 follows from Schweizer 21a, Theorem 4.6 Theorem 4.1.1, which relaes he decomposiion in 4.8.3 o he Galchouk- Kunia-Waanabe decomposiion of he P-maringale V, P = E P [H F] given in Example 4.7.3. 4.8.4 Mean-variance hedging We now briefly menion he principle of mean-variance hedging used for hedging and pricing in incomplee financial markes. This shor review follows a similar review in Møller 21b. Wih mean-variance hedging, he aim is o deermine he self-financing sraegy ˆϕ = ˆϑ, ˆη which minimizes E P [ H V T,ϕ 2]. The main idea is hus o approximae he discouned claim H as closely as possible in L 2 P by he discouned erminal value of a self-financing porfolio ϕ. Since we consider self-financing porfolios only, he opimal porfolio is uniquely deermined by he pair V, ˆϕ, ˆϑ, where V, ˆϕ is known as he approximaion price for H and ˆϑ is he meanvariance opimal hedging sraegy. Schweizer 21a, Theorem 4.6 gives ha V, ˆϕ = E P [H ] and ˆϑ = ϑ H. Thus, we recognize he approximaion price and he mean-variance hedging sraegy as he firs par of he mean-variance indifference prices and opimal hedging sraegies, respecively. Noe ha even hough he minimizaion crierion is in erms of P, he soluion is given parly in erms of P.

124 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING 4.9 Numerical examples In his secion, we presen some numerical examples wih calculaions of he marke reserves of Proposiion 4.6.2. Furhermore, we invesigae wo differen parameerizaions wihin he class of ime-inhomogeneous CIR models and compare hese o he 23 moraliy inensiies and a deerminisic projecion for he moraliy inensiies. Calculaion mehod A useful way of evaluaing he expression 4.6.13 is o define auxiliary funcions V Q, = T e Êτ fr,u+f µ,q x,,udu a τ + a 1 τf µ,q x,,τ dτ + e ÊT fr,u+f µ,q x,,udu A T, 4.9.1 where he zero coupon bond price and he Q-survival probabiliies are expressed in erms of he relevan forward raes and Q-forward moraliy inensiies. Noe moreover, ha we have inroduced he addiional parameer. We noe ha V Q,r,µx, = V Q,, whereas hese wo quaniies differ if. I follows immediaely, ha on he se,t T,T, V Q, saisfies for fixed he differenial equaion V Q, = f r, + f µ,q x,, V Q, a a 1 f µ,q x,,, 4.9.2 subjec o he erminal condiion V Q T, = and wih V Q T, = A T + V Q T,. Alernaively, he expression 4.6.13 can be deermined by solving he following parial differenial equaion on,t T,T IR IR + = V Q,r,µ + γ µ,q x, δ µ,q x,µ µ V Q,r,µ + 1 2 σµ x, 2 µ 2 µ 2V Q,r,µ + γ r,α,q δ r,α,q r r V Q,r,µ + 1 2 γr,σ + δ r,σ r 2 r 2V Q,r,µ rv Q,r,µ + a + 1 + gµa 1 V Q,r,µ, wih erminal condiion V Q T,r,µ = and wih V Q T,r,µ = A T + V Q T,r,µ. The parial differenial equaion follows eiher as a byproduc from he proof of Lemma 4.7.1 in Secion 4.1.1 or as a special case of he generalized Thiele s differenial equaion in Seffensen 2. A similar parial differenial equaion can be found in Chaper 3. Parameers for financial marke We now presen he parameers which will be used in he numerical examples. The financial marke will be described via a sandard Vasiček model wih parameers γ r,α =.8, δ r,α =.2, γ r,σ =.1, δ r,σ =, c =.3 and r =.25. Given hese

4.9. NUMERICAL EXAMPLES 125.25.35.45 1 2 3 4 5 Figure 4.9.1: Forward rae curve for he Vasiček model. parameers, he mean reversion level for he shor rae is γ r,α /δ r,α =.4 under P and γ r,α c/δ r,α =.55 under Q. The shor rae volailiy is given by γ r,σ =.1 and he speed of mean reversion is δ r,α =.2. The parameer c/δ r,α =.15 can be inerpreed as he ypical difference beween he long and shor erm zero coupon yield, see Poulsen 23 for more deails. The iniial shor rae is given by r =.25, which corresponds o he presen shor rae level. The forward rae curve f r,τ can be found in Figure 4.9.1. Parameers for insurance porfolio We have fied he parameers for he underlying Gomperz-Makeham disribuions a various ime. In Table 4.9.1 below, we presen he numbers for 198 and 23 which have been obained by sandard mehods. We now lis some parameers for he under- Males Females Year α β c α β c 198.233.658 1.959.22.197 1.163 23.134.353 1.12.8.163 1.174 Table 4.9.1: Esimaed Gomperz-Makeham parameers for 198 and 23. lying moraliy improvemen process ζ defined by 4.3.6, which is supposed o capure he variaion presen in Figure 4.2.4 from Secion 4.2. We consider wo differen parameerizaions, Case I and Case II, see Table 4.9.2. Case I: We ake δx, = δ consan δx, γx, σx, Case I δ δe γ σ 1 Case II γ 2 σ2 σ Table 4.9.2: Paramerizaion for he underlying process ζ. and assume ha γx, = δe γ, where log1 + γ represens he expeced yearly relaive decline in he moraliy inensiy. Thus, e γ is he ime-dependen level o which he

126 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING process ζ adaps and δ conrols how fas i adaps o his level. Finally, we propose o le σx, = σ, which describes he noise. Case II: Here, we le δx, = γ, γx, = 1 2 σ2 and σx, = σ. This means ha we expec a relaive yearly decline in ζ of approximaely γ. Noe ha Case II has one parameer less han case I. The choice γx, = 1 2 σ2 in Case II ensures ha ζ remains sricly posiive. Quaniles for he moraliy improvemen δ γ σ 5% 25% 5% 75% 95% Case I.2.8.2.838.867.887.97.937 1.8.2.837.85.859.868.881.2.8.3.814.856.886.917.962 1.8.3.827.846.859.872.892 Case II.8.2.726.81.854.99.99 Table 4.9.3: Quaniles for he moraliy improvemen process ζ for ime horizon 2 years. Numbers are based on 1 simulaions wih 1 seps per year in an Euler scheme. The resuls are indisinguishable if we alenaively use a Milsein scheme. process ζ for he wo parameerizaions can be found in Table 4.9.3. In Case II, he mean reversion level is γ,x/δ,x = 1 2 σ2 / γ. Wih γ =.8 and σ =.2, his leads o he mean reversion level of.25, which is negligible. A comparison of he moraliy for 23 1 2 3 4..4.8 4 6 8 1 12 4 6 8 1 12 1 2 3 4..4.8 4 6 8 1 12 4 6 8 1 12 Figure 4.9.2: Top picures are Case I and boom picures are Case II. To he lef: Moraliy inensiy curve for 3 year old males for 23 solid line, exponenially correced wih facor exp γ dashed line and forward moraliy inensiies doed line. To he righ: The corresponding survival probabiliies. for males and he corresponding forward moraliy inensiies in Case I wih parameers δ, γ, σ =.2,.8,.2 can be found a he op of Figure 4.9.2. The figure shows a raher limied difference beween he forward moraliy inensiies and he exponenially

or 4.9. NUMERICAL EXAMPLES 127 correced inensiies hey essenially coincide, whereas here is a big difference beween hese wo curves and he 23 esimae for he moraliy inensiies. For Case II, here is a more subsanial difference beween he forward moraliy inensiies and he exponenially correced moraliy inensiies a very high ages. Expeced lifeimes Figure 4.9.3a shows he hisogram for he expeced lifeime of a policyholder aged 3 for case I wih parameers.2,.8,.3. As a comparison, he expeced lifeime for a male policyholder aged 3 is 75.8, 79. and 78.6 if we use he 23 esimae, he exponenially correced moraliy and he forward moraliy inensiies, respecively. The variabiliy in he figure reflecs he uncerainy relaed o changes in he fuure moraliy inensiies. The hisogram shows ha here is a relaively small uncerainy associaed wih he expeced lifeime in Case I. This is explained by he fac ha he model for he moraliy improvemen process is mean-revering wih a relaively small volailiy. If we insead. 1. 77. 77.5 78. 78.5 79. 79.5 8. a..15.3 76 78 8 82 84 b Figure 4.9.3: Hisograms for he expeced lifeime for a policy-holder aged 3 for Case I wih parameers.2,.8,.3 figure a and Case II figure b. Hisograms are based on 1 simulaions wih 1 seps per year in an Euler scheme. consider case II, he expeced lifeime changes o 79.2, and we now ge subsanially bigger variaion ino he expeced lifeimes, see he hisogram for he expeced lifeime in Figure 4.9.3b. Marke reserves In Figure 4.9.4, we have ploed he funcions V Q, x for fixed = = as a funcion of age x in he case where Q is he minimal maringale measure. Here, we have added an x o he funcion V Q in order o underline is dependence on he iniial age x. We have considered Case I wih parameers.2,.8,.3 and sudied a life annuiy saring a age 65. Moreover, we have compared his wih he reserves obained by using he 23 esimae wihou any correcion for fuure moraliy improvemens, and he moraliy inensiies obained by reducing he moraliy inensiies exponenially. For each iniial age x, we have calculaed he relevan forward moraliies and solved he differenial equaion for V Q,. We see only very lile difference beween he reserves obained by using he forward moraliy inensiies and he exponenially correced moraliy inensiies. Risk-minimizing sraegies and mean-variance indifference pricing The risk-minimizing sraegies and mean-variance indifference hedging sraegies obained

128 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING 2 4 6 8 1 4 6 8 1 12 Figure 4.9.4: Reserves for a life annuiy saring a age 65. Reserve based on 23- esimae for males solid line, exponenially correced moraliy inensiies dashed line and forward moraliies doed line. in Secion 4.7 and 4.8 can also be deermined numerically. Møller 21b conains a secion wih numerical examples for a similar conrac wihou sysemaic moraliy risk, where he sraegies have been deermined for a couple of simulaions. In addiion, he mehods lised here may be used for deermining he mean-variance indifference prices of Proposiion 4.8.1. 4.1 Proofs and echnical calculaions 4.1.1 Proof of Lemma 4.7.1 Recall from 4.7.7 ha he Q-maringale V,Q can be wrien as V,Q = A + n Nx,B 1 V Q,r,µx,. Differeniaing under he inegral gives and r V Q,r,µ = T B r,τp,τs Q x,,τ a τ + a 1 τf µ,q x,,τ dτ B r,tp,ts Q x,,t A T, 4.1.1 T µ V Q,r,µ = 1 + g P,τB µ,q x,,τs Q x,,τ a τ + a 1 τ f µ,q τ x,,τ Bµ,Q x,,τ B µ,q dτ x,,τ + P,TB µ,q x,,ts Q x,,t A T, 4.1.2

4.1. PROOFS AND TECHNICAL CALCULATIONS 129 where we have used µ fµ,q x,,τ = 1 + g τ Bµ,Q x,,τ. Inegraion by pars used on n Nx,B 1 V Q,r,µx, yields V,Q = A + nv Q,r,µx, + + n Nx,uV Q u,ru,µx,udbu 1 Bu 1 n Nx,u dv Q u,ru,µx,u Bu 1 V Q u,ru,µx,udnx,u. 4.1.3 In order o calculae he fourh erm in 4.1.3, we need o find dv Q u,ru,µx,u. Recall from 4.4.7 and 4.6.5 ha he dynamics of r and µx under Q are given by where dr = α r,q rd + σ r,rdw r,q, dµx, = α µ,q,µx,d + σ µ,µx, µx,dw µ,q, α r,q r = γ r,α,q δ r,α,q r, α µ,q,µx, = γ µ,q x, δ µ,q x,µx,. In he res of he proof we use he shorhand noaion V Q u = V Q u,ru,µx,u. Furhermore we only include explicily he ime argumen in he coefficien funcions. The assumpion V Q C 1,2,2 allows us o apply Iô s formula. We obain dv Q u = u V Q u + α µ,q u µ V Q u + 1 2 σµ u 2 µx,u 2 µ 2V Q u +α r,q u r V Q u + 1 2 σr u 2 2 r 2V Q u du + σ r u r V Q udw r,q u + σ µ u µx,u µ V Q udw µ,q u = u V Q u + α µ,q u µ V Q u + 1 2 σµ u 2 µx,u 2 µ 2V Q u +α r,q u r V Q u + 1 2 σr u 2 2 r 2V Q r u du V Q u B r u,tp u,t dp u,t µ V Q u 1 + gub µ,q x,u,ts Q,M x,u,t dsq,m x,u,t. In he firs equaliy we have used he dynamics of r and µx and ha he Brownian moions W r,q and W µ,q are independen, such ha we do no ge any mixed second order erms. In he second equaliy we use 4.1.1 and 4.1.2 ogeher wih he dynamics of

13 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING S Q,M x,,t and P,T given in 4.6.7 and 4.4.1, respecively. Rewriing A in erms of he Q-maringale M Q x we ge A = nπ + + Bτ 1 a τn Nx,τ + a 1 τn Nx,τ µ Q x,τ dτ BT 1 n Nx,T A TdI {τ T } + Bτ 1 a 1 τdm Q x,τ. Collecing he erms from 4.1.3 involving inegrals wih respec o P,T, S Q,M x,,t and M Q x, respecively, we ge he las hree erms in 4.7.8. Since hese hree erms and V,Q are Q-maringales, he remaining erms consiue a Q-maringale as well. Since his process is coninuous hence predicable and of finie variaion, i is consan. Insering = we immediaely ge ha V,Q = nπ + nv Q,r,µx,. Thus, we have proved he decomposiion in 4.7.8. 4.1.2 Calculaion of Var P [N H ] The following heorem due o Schweizer 21a, Theorem 4.6 relaes he decomposiion in 4.8.3 o he Galchouk-Kunia-Waanabe decomposiion of he P-maringale V, P = E P [H F]; see also Møller 2. Theorem 4.1.1 Assume ha H L 2 FT,P and consider he Galchouk-Kunia-Waanabe decomposiion of V, P given by V, P = E P [H ] + ξ P udp u,t + L P, T. 4.1.4 We can now express c H, ϑ H and N H from 4.8.3 in erms of decomposiion 4.1.4 by c H = E P [H ], ϑ H = ξ P ζ T N H = ΛT 1 Λu dl P u. 1 Λu dl P u, Since L P = ν P udmx,u + κ P udsx,u,t, where ν P and κ P are given by 4.8.12 and 4.8.13, respecively, Theorem 4.1.1 gives he following expression for N H : N H = ΛT T 1 Λ dl P = ΛT T 1 ν P dmx, + κ P ds M x,,t. Λ

4.1. PROOFS AND TECHNICAL CALCULATIONS 131 Since E P [N H ] =, we firs noe ha Var P [N H ] = E P [N H 2 ] = E P [ ] 2 ΛT LT + RT = E P [ ΛT LT 2 + 2 ΛT LT RT + ΛT RT 2], 4.1.5 where we have defined L = ν P u Λu dmx,u and R = κ P u Λu dsm x,u,t. The hree erms appearing in 4.1.5 can be rewrien using Iô s formula. For he firs erm we ge ΛT LT 2 = T and for he las erm we find ha ΛT RT 2 = T T L 2 d Λ + 2 Λ L d L + Λ = T T + T The mixed erm becomes ΛT RT LT = T + R 2 d Λ + 2 R 2 d Λ + 2 T Λ T T T Λ Rd R + Λd R Λ Rd R 2 ν P dnx,, Λ κ P Λ σµ x, 2 µx,b µ x,,ts M x,,t d. T Λ Rd L + Ld[ R, Λ]. T L Rd Λ + Λ Ld R Assuming all he local maringales are maringales, and using ha he Brownian moions driving r and µ are independen, we ge Var P [ [ ] N H] T = E P ν P 2 dnx, Λ T κ P σ µ x, 2 µx,b µ x,,ts M x,,t + E P d. 4.1.6 Λ We now invesigae he wo erms in 4.1.6 separaely. The firs erm can be rewrien as [ ] T E P ν P 2 dnx, Λ [ ] P,T A T 2 [ ] = E P Sx,,T 2 n Nx, µx, d Λ = T E P T E P [ Λ Λ P,T A T 2 ] [ ] E P Sx,,T 2 n Nx, µx, d,

132 CHAPTER 4. SYSTEMATIC MORTALITY RISK: VALUATION AND HEDGING where we have used he expression for ν P from 4.8.12 and he independence beween r and N,µ. The second erm is given by T κ P σ µ x, 2 µx,b µ x,,ts M x,,t E P d Λ T n Nx, P,Tσ µ x, 2 µx,b µ x,,tsx,,t A T = E P d Λ [ ] T = E P Λ Λ P,T A T 2 [ E P n Nx, σ µ x, ] 2 µx,b µ x,,tsx,,t d, where we have used he expression for κ P from 4.8.13 and once again he independence beween r and N,µ. Using ha condiioned on I he number of survivors a ime is binomially disribued wih parameers n,e Ê µx,udu under P, we ge [ E P n Nx, σ µ x, ] 2 µx,b µ x,,tsx,,t [ = E P E P n Nx, σ µ x, ] 2 µx,b x,,tsx,,t µ I [ = E P σ µ x, 2 µx,b µ x,,tsx,,t E P [ n Nx, 2 ] ] I [ = E P σ µ x, 2 µx,b µ x,,tsx,,t ne Ê µx,udu 1 e Ê µx,udu ] + n 2 e Ê µ 2 x,udu, and [ ] [ ] E P Sx,,T 2 n Nx, µx, = E P E P Sx,,T 2 n Nx, µx, I [ ] = E P Sx,,T 2 µx,e P [n Nx, I] = ne P [ Sx,,T 2 µx,e Ê µx,udu]. Collecing he erms proporional o n and n 2, respecively, we arrive a 4.8.1.

Chaper 5 A Discree-Time Model for Reinvesmen Risk in Bond Markes This chaper is an adaped version of Dahl 25b In his chaper we propose a discree-ime model wih fixed maximum ime o mauriy of raded bonds. A each rading ime, a bond maures and a new bond is inroduced in he marke, such ha he number of raded bonds is consan. The enry price of he newly issued bond depends on he prices of he bonds already raded and a sochasic erm independen of he exising bond prices. Hence, we obain a bond marke model for he reinvesmen risk, which is presen in pracice, when hedging long erm conracs. In order o deermine opimal hedging sraegies we consider he crieria of super-replicaion and risk-minimizaion. 5.1 Inroducion In he lieraure, bond markes are usually assumed o include all bonds wih ime of mauriy less han or equal o ime of mauriy of he considered claim. However, his is in conras o pracice, where only bonds wih a limied sufficienly shor ime o mauriy are raded. Hence, a sandard model is he correc framework for pricing and hedging so-called shor erm conracs, where he payoff depends on bonds wih ime o mauriy less han or equal o he longes raded bond. However, when considering long erm conracs, i.e. conracs, whose payoffs depend on bonds wih longer ime o mauriy han he longes raded bond, he bond marke does no in general include bonds, which a all imes allow for a perfec hedge of he conrac. Thus, in pracice, an agen ineresed in pricing and hedging a long erm conrac, such as a life insurance conrac, where he paymens may be due 5 years or more ino he fuure, is in general no able o eliminae 133

134 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK he reinvesmen risk associaed wih he conrac. Since he reinvesmen risk is ignored in sandard bond marke models, hey do no seem o be he righ framework for pricing and hedging long erm conracs. Here, we propose a model, which behaves similarly o a sandard model when hedging and pricing shor erm conracs, and a he same ime i includes reinvesmen risk, when hedging long erm conracs. A firs idea in order o inroduce reinvesmen risk would be o consider a sandard binomial model for he bond prices and resric he invesmen sraegies o bonds wih a limied ime o mauriy only. However, his simple approach does no inroduce reinvesmen risk, since a long erm bond can be perfecly replicaed by a dynamic rading sraegy, where we a all imes inves in wo shor erm bonds. Hence, we have o exend he sandard model o include an addiional unhedgeable sochasic erm, whose uncerainy deermines he reinvesmen risk. To describe he reinvesmen risk, we propose a discree-ime bond marke model, where he raded bonds have a fixed maximum ime o mauriy, T. Hence, a ime all bonds wih ime of mauriy v, v {1,..., T } are raded. A any ime, he bond wih mauriy maures and a new bond wih ime o mauriy T is inroduced in he marke. Thus, afer he issue of he new bond, he model is similar o he one a ime. A any ime, he enry price of he new bond depends on all pas informaion, curren prices of bonds already raded and an addiional sochasic erm. In his model he class of aainable claims depends on ime. Hence, a claim which is unaainable a ime may be aainable a ime +1. Consider for example a claim of 1 a ime + T +1, which is unaainable a ime, whereas i is clearly aainable a ime + 1, where a bond wih ime of mauriy + T + 1 is issued. A ime + 1 he unique arbirage free price is equal o he price of he bond wih mauriy + T + 1, and he replicaing sraegy consiss of purchasing exacly one such bond. The idea of fixing he maximum ime o mauriy of he raded asses and inroducing new asses as ime passes can also be found in Neuberger 1999, who considers a marke for fuures on oil prices. To model he iniial price of he new fuure, Neuberger assumes ha i is a linear funcion of he prices of raded fuures and a normally disribued error erm. To he auhor s knowledge, he only oher papers o consider he problem of modelling he prices of newly issued bonds are Sommer 1997 and Dahl 25a see Chaper 6, who boh consider models in coninuous ime. In Sommer 1997, new bonds are issued coninuously, whereas in Dahl 25a new bonds are issued a fixed imes only, since his is he case in pracice. In order o conrol and quanify he reinvesmen risk, boh auhors consider he crierion of risk-minimizaion. The chaper is organized as follows: In Secion 5.2, a bond marke model including reinvesmen risk is inroduced. This is done in wo seps: Firs we describe a complee and arbirage free sandard bond marke model. Then we exend he model o include reinvesmen risk. Since he exended model is incomplee, here exis infiniely many equivalen maringale measures. We idenify he equivalen maringale measures for he exended model and define he considered price processes, which for noaional convenience are differen from he bond prices. Given he considered price processes we review

5.2. A BOND MARKET MODEL 135 he relevan financial erminology. Opimal hedging sraegies wih respec o he crieria of super-replicaion and risk-minimizaion are deermined in Secion 5.3. Here, we also remark on he relaionship beween he crierion of super-replicaion and he maximal guaranees for which he shorfall risk can be eliminaed. The chaper is concluded by a numerical illusraion in Secion 5.4. The numerical illusraion includes a comparison wih pracice in Danish life insurance, where long erm conracs are common. 5.2 A bond marke model Le T N be a fixed ime horizon and Ω, F,P a probabiliy space wih a filraion F = F {,1,..., T } saisfying he usual condiion of compleeness, i.e. F conains all P-null ses. 5.2.1 A sandard bond marke model Prior o he inroducion of he bond marke model wih reinvesmen risk, we now describe a sandard discree-ime bond marke model. For a horough descripion of discree-ime bond marke models we refer o Jarrow 1996. Consider a bond marke where rading akes place a imes =,1,..., T, for a fixed ime horizon T N, T < T. A ime we assume ha all zero coupon bonds wih mauriy v =,..., T are raded in he bond marke. For {,..., T } and v {,..., T } we denoe by P,v he price a ime of a zero coupon bond mauring a ime v. To avoid arbirage we assume P, v is sricly posiive and P, = 1 for all. For non-negaive ineres raes P,v is a decreasing funcion of v for fixed. An imporan quaniy when modelling bond prices is he forward rae, f,v, conraced a ime for he period [v,v+1] defined by f,v = or, saed differenly, P,v P,v + 1 1, {,..., T 1} and v {,..., T 1}, 5.2.1 1 P,v = v 1 i= 1 + f,i, {,..., T 1} and v { + 1,..., T }. 5.2.2 The forward rae f,v can be inerpreed as he riskfree ineres rae conraced a ime for he inerval [v,v + 1]. Now inroduce he shor rae process r = r {,1,..., T 1} given by r = f,. Since 5.2.1 and 5.2.2 esablish a one-o-one correspondence beween forward raes and bond prices, modelling he developmen of he bond prices and he forward raes is equivalen. As i is sandard in he lieraure, we model he forward raes. Le f = r,f,+1,...,f, T 1 denoe he T -dimensional forward rae vecor a ime. To model he developmen of he forward rae vecor we assume f = g f,...,f 1,ρ, {1,..., T 1}, 5.2.3

136 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK for some funcion g : R T R T 1 R T 1 {u,d} R T and an i.i.d. sequence ρ 1,...,ρ T 1 of random variables wih disribuion Pρ 1 = u = 1 Pρ 1 = d = p, p,1. A naural resricion would be o consider sricly posiive forward raes, only. 1 In his case we would have g : R T + R T + R T 1 + {u,d} R T +. We observe from 5.2.3 ha coningen on he developmen of he forward raes unil ime 1, he forward rae vecor a ime akes one of wo possible values: g f,...,f 1,u or g f,...,f 1,d. If f = g f,...,f 1,u we say ha he forward raes have moved up, and likewise, if f = g f,...,f 1,d we say hey have moved down. We noe from 5.2.2 ha he bond prices move in he opposie direcion of he forward raes. The developmen of he forward raes and bond prices can be represened by non-recombining binomial ree, see Figure 5.2.1 for a visualizaion of he firs hree possible values of he forward raes. r f,1. f, T 1 r u 1 f u 1,2. f u 1, T 1 r d 1 f d 1,2. f d 1, T 1 r u,u 2 f u,u 2,3. f u,u 2, T 1 r u,d 2 f u,d 2,3. f u,d 2, T 1 r d,u 2 f d,u 2,3. f d,u 2, T 1 r d,d 2 f d,d 2,3. f d,d 2, T 1 Figure 5.2.1: Developmen of he forward rae vecor. Remark 5.2.1 If g only depends on f,...,f 1 hrough f 1, hen he forward rae vecor is a discree ime-inhomogeneous Markov chain.

5.2. A BOND MARKET MODEL 137 The naural filraion G = G {,..., T } generaed by he forward raes is given by G = {,Ω} and G = σ{ρ 1,...,ρ T 1 }, {1,..., T }. Inroduce he noaion ξ, ξ Ξ = {all possible sequences of u s and d s of lengh }. This allows us o denoe he generic value of for insance he forward rae vecor a ime by f ξ and he forward rae vecor a ime + 1 given ρ +1 = d by f ξ,d +1. Risk-neural probabiliies I is well known ha he bond marke model described above is arbirage free if here exiss a so-called equivalen maringale measure Q. Recall ha an equivalen maringale measure is a probabiliy measure equivalen o P, such ha all discouned bond prices are maringales. The discouned bond prices are Q-maringales if for {,..., T 1} and v { + 1,..., T } i holds ha P,v = 1 1 + r E Q [P + 1,v G ]. 5.2.4 If furher he equivalen maringale measure Q is unique, he model is complee; see also Secion 5.2.3 for he definiion of arbirage and compleeness. Denoe by q ξ +1 he Q- probabiliy of he even ρ +1 = u given he presen informaion ξ. Since 5.2.4 is rivially fulfilled for v = + 1, we have T + 1 equaions for q ξ +1, {,..., T 2}. Thus, if a soluion exiss, i is unique, provided here exiss a v { + 2,..., T }, such ha P ξ,d + 1,v P ξ,u + 1,v. For {,..., T 2}, solving 5.2.4 gives he following expressions for q ξ +1 : +1 = P ξ,d + 1,v 1 + r ξ P ξ,v P ξ,d + 1,v P ξ,u + 1,v, v { + 2,..., T }. 5.2.5 q ξ Here, we have used he noaion r ξ and P ξ,v o denoe explicily he dependence on he pas. From 5.2.5 we observe ha he Q-probabiliy of ρ +1 = u depends on ξ and hence in general differs for differen oucomes of ρ 1,...,ρ. Furhermore, 5.2.5 gives ha he Q-probabiliy of an upward movemen is small large if he difference P ξ,d + 1,v 1+r ξ P ξ,v is small large compared o he difference P ξ,d +1,v P ξ,u +1,v. The measure Q given by 5.2.5 for all ensures ha all discouned bond price processes are Q-maringales. If furher q ξ +1,1 for all and ξ, hen P and Q are equivalen measures, such ha Q indeed is an equivalen maringale measure. From 5.2.5 we ge ha Q and P are equivalen if for all {,..., T 2} and ξ i holds ha P ξ,u + 1,v < 1 + r ξ P ξ,u,v < P ξ,d + 1,v, v { + 2,..., T }. 5.2.6 Here, we have used ha P ξ,u + 1,v < P ξ,d + 1,v, since an upward movemen of he forward raes corresponds o a downward move of he bond prices. Using 5.2.2 one can alernaively express 5.2.5 and 5.2.6 in erms of he forward raes. Condiion 5.2.6 can be inerpreed as follows: No bond wih ime o mauriy larger han one mus dominae or be dominaed by he 1-period bond. If his was he case we could make arbirage by rading in he paricular bond and he 1-period bond.

138 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK 5.2.2 A bond marke model wih reinvesmen risk Now, we exend he sandard model in Secion 5.2.1 o include reinvesmen risk. The idea is as follows: Assume ha a any ime only bonds wih ime o mauriy less han or equal o T are raded, and he developmen of he bond prices from ime o + 1 can be described by a binomial model. Hence, a ime he one period developmen of he raded bonds is he same as in he sandard model inroduced in Secion 5.2.1. A ime + 1 he bond wih mauriy + 1 maures and a new bond wih ime o mauriy T is issued, such ha afer he inroducion of he new bond he model considered is similar o he one a ime. To model he reinvesmen risk we assume ha condiional on he pas and he prices a ime + 1 of he bonds raded a ime, he enry price a ime + 1 of he new bond wih ime o mauriy T can ake wo differen values. Consider a bond marke where rading akes place a imes =,1,..., T. In his bond marke no all zero coupon bonds wih mauriy less han or equal o T are raded a all imes =,1,..., T. Insead we fix he maximum ime o mauriy, T, for bonds raded in he marke. Hence, he zero coupon bond prices P,v are defined for {,..., T } and v {,...,+ T T}. In addiion o T and T we inroduce he fixed ime horizon T N, which is he ime of mauriy of he considered conrac. Figure 5.2.2 shows he possible orderings of T, T and T. Wihou loss of generaliy we assume ha T = T + T, such ha Today Mauriy of claim Longes ime o mauriy of bonds Las rading ime Today Longes ime o mauriy of bonds Mauriy of claim Las rading ime T a T T T b T T Figure 5.2.2: Possible orderings of T, T and T. In case a all fixed claims wih mauriy T are aainable. In b hey are unaainable. he bond marke a all imes, {,...,T }, includes he T bonds wih ime of mauriy v, v { + 1,..., + T }. From 5.2.1 we observe ha he forward raes are defined for {,..., T 1} and v {,...,+ T 1 T 1}, so he forward rae vecor a ime which we sill denoe by f is given by f = r,f,+1,...,f,+ T 1 T 1. Define he u + 1 T-dimensional vecor f,u of forward raes defined a ime wih ime of mauriy less han or equal o u. Now assume ha f,+ T 2 T 1 = g f,...,f 1,ρ, 5.2.7 where g : R T R T T 1... R T T 1 {u,d} R, and ρ 1,...,ρ T 1, similarly o Secion 5.2.1, is an i.i.d. sequence of random variables wih disribuion Pρ 1 = u = 1 Pρ 1 = d = p, p,1. The filraion G = G {,..., T } is now given by G = {,Ω} and G = σ{ρ 1,...,ρ T 1 }, {1,..., T }.

5.2. A BOND MARKET MODEL 139 A ime he mauriies of he forward raes given by 5.2.7 are hose where a forward rae wih he same mauriy is defined a ime 1. Hence, he forward raes a ime given by 5.2.7 deermine he bond prices a ime for bonds wih ime of mauriy v, v {+1,...,+ T 1 T }, which are he bonds raded a ime 1 when disregarding he bond mauring a ime. Thus, he uncerainy associaed wih he developmen of he forward raes bond prices from ime 1 o is described by ρ. However, he uncerainy associaed wih he price of he new bond wih ime o mauriy T inroduced in he marke a ime, {1,...,T }, canno be described enirely by ρ ; i depends on an addiional source of risk. In order o model his addiional uncerainy we assume ha a ime, {1,...,T }, he T 1-period forward rae, f,+ T 1, is given by f,+ T 1 = c f,...,f 1, f,+ T 2,ε 5.2.8 for some funcion c : R T R T 1 {h,l} R and an i.d.d. sequence ε 1,...,ε T of random variables independen of ρ {1,..., T 1}. The disribuion of ε 1 is given by Pε 1 = h = 1 Pε 1 = l = p, p,1. Hence, for {1,...,T } i holds ha given he pas forward raes and he T 1-dimensional forward rae vecor f,+ T 2 = r,f,+1,...,f,+ T 2 a ime, he T 1-period forward rae, f,+ T 1, can aain wo differen values: c f,...,f 1, f,+ T 2,h and c f,...,f 1, f,+ T 2,l. We refer o hese values as he high and low value, respecively. Analogously o ξ we now inroduce λ, λ Λ = {all possible sequences of h s and l s of lengh } for all =,...,T. Thus, λ keeps rack of wheher he pas values of he T 1-period forward rae has aained he high or he low value. Hence, ξ and λ T deermine he developmen of he enire forward rae vecor unil ime, {1,..., T 1}, such ha we can denoe he generic value of he forward rae vecor a ime by f ξ,λ T. Now inroduce he filraion H = H {,..., T } by H = {,Ω} and H = σ{ε 1,...,ε T }, {1,..., T }. We now assume he filraion F = F {,1,..., T } inroduced earlier is given by F = G H. Hence, F is he filraion for he exended bond marke. We noe ha i is sufficien o consider he sae space for ω given by Ω = {u,d} T 1 {h,l} T and he σ-algebra F = F T = F T 1. A ime, {T + 1,..., T } he developmen of he bond marke is essenially idenical o he binomial model in Secion 5.2.1, whereas he model is non-sandard a ime, {1,...,T }. Here, we have ha coningen on ξ 1 and λ 1 here are four possible forward rae vecors a ime and hence 4 possible saes a ime. Thus, for T he developmen of he forward rae vecor can be represened using a non-recombining quadrinomial ree, see Figure 5.2.3 for a visualizaion of he forward raes wih T = 2. From 5.2.7 we observe ha he forward rae a ime wih mauriy τ, τ {,..., + T 2 T 1}, is allowed o depend on all pas forward raes, such ha he T 1-period

14 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK forward rae a ime 1 may influence he enire forward rae vecor a ime. Hence, he forward rae f,τ, τ {,...,+ T 2 T 1}, is F 1 G -measurable, which in urn gives ha P,τ also is F 1 G -measurable for τ { + 1,..., + T 1 T }. For an illusraion of he dependence of he forward raes on he ρ s and ε s we again refer o Figure 5.2.3, where he dependence is shown explicily. Noe ha if we coningen on he oucome of he vecor ε 1,...,ε T, he developmen can be described by a binomial model, and hence he condiional model is complee, such ha in he condiional model zero coupon bonds wih all mauriies have unique prices even before hey are raded. Hence, in he condiional model we, a all imes, have a forward rae vecor for all mauriies. However, in he uncondiional model he fuure values of ε +1,...,ε T are unknown a ime, such ha i is uncerain which of he condiional forward rae vecors in rerospec will urn ou o have been he correc one, when ε +1,...,ε T have been observed a ime T. Thus, he reinvesmen risk can be inerpreed as he uncerainy associaed wih which of he condiional forward rae vecors in rerospec has urned ou o have been he correc one. This in urn gives ha he magniude of he reinvesmen risk is relaed o how much he condiional forward rae vecors differ. Example 5.2.2 Consider he case where T = 2 and T = 3. Hence, he ime o mauriy of he longes bond in he marke is 2 and he ime of mauriy of he considered claim is 3. The developmen of he forward rae vecor can be visualized by Figure 5.2.3. Here, he superscrips denoe he dependence of he forward raes on he oucome of he variables ρ and ε. As an example he noaion r uu,l 2 denoes he shor rae in period 2 if ρ 1 = u, ρ 2 = u and ε 1 = l. We end he example by noing ha all examples in his chaper are one coninuing example. Risk-neural probabiliies We now aim a deermining he equivalen maringale measures in he exended model. Here, he uncerainy is generaed by ρ {1,..., T 1} and ε {1,...,T }, such ha he measure Q is uniquely deermined by q ξ 1,λ 1 T q ξ 1,λ 1 T {1,..., T 1} and qξ,λ 1 {1,...,T }, where denoes he probabiliy of ρ = u given ξ 1 and λ 1 T, and q ξ,λ 1 denoes he probabiliy of ε = h given ξ and λ 1. Recall ha for {,..., T 1} and v { + 1,..., + T T }, a necessary condiion for Q o be an equivalen maringale measure is P,v = 1 1 + r E Q [P + 1,v F ], 5.2.9 such ha he discouned bond prices are maringales. Now noe ha 5.2.9 is rivially fulfilled if v = + 1. Since P + 1,v in 5.2.9 is F G +1 -measurable, i.e. independen of ε +1, hen 5.2.9 yields T 2 T 1 equaions for q ξ,λ T +1. Hence, if here exiss an equivalen maringale measure, hen q ξ,λ T +1 is unique for all {,..., T 2},

5.2. A BOND MARKET MODEL 141 r f,1 r u 1 f u,h 1,2 r u 1 f u,l 1,2 r d 1 f d,h 1,2 r d 1 f d,l 1,2 r uu,l 2 f uu,lh 2,3 r uu,l 2 f uu,ll 2,3 r ud,l 2 f ud,lh 2,3 r ud,l 2 f ud,ll 2,3 r udu,lh 3 f udu,lhh 3,4 r udu,lh 3 f udu,lhl 3,4 r udd,lh 3 f udd,lhh 3,4 r udd,lh 3 f udd,lhl 3,4 Figure 5.2.3: Developmen of he forward raes in he exended model wih T = 2. provided here for each exiss a v { +2,..., + T T }, such ha P ξ,λ,d + 1,v P ξ,λ,u + 1,v. For {,...,T 1} no informaion regarding q ξ +1,λ +1 can be derived from 5.2.9, so any Q for which q ξ,λ T +1 fulfills 5.2.9 for all {,..., T 2} ensures ha he discouned bond prices are maringales. If furher boh q ξ,λ T +1 and q ξ +1,λ +1 lie in he inerval,1, hen Q is an equivalen maringale measure. If q ξ +1,λ +1 = p, we say he marke is risk-neural wih respec o reinvesmen risk. This measure is known as he minimal maringale measure for he exended model, i.e. he equivalen maringale measure which disurbs he srucure of he model as lile as possible, see Schweizer 1995. Here, we resric ourselves o Q s given by q ξ +1,λ +1 = q +1 λ, such ha under Q he disribuion of he ε s is independen of he realizaion of he ρ s. Henceforh we consider a fixed, bu arbirary, equivalen maringale measure Q. Remark 5.2.3 Noe ha for {,..., T 1} and v { + 1,..., + T T } repeaed use of 5.2.9 gives he equaion P,v = 1 [ ] E Q 1 E Q [P + 2,v F +1 ] 1 + r 1 + r +1 F. 5.2.1 Hence, since P+2,v is F G +2 -measurable i seems as if 5.2.1 gives an equaion from which o deermine q ξ +1,λ +1 for {,...,T 1}. However, his is no he case, since he F G +1 -measurabiliy of P + 1,v ensures ha E Q [P + 2,v F +1 ] is independen of ε +1.

142 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK Model consideraions Consider he case where we for fixed model he forward rae f,u, u {,..., + T 2 T 1} by f,u = g,u f,u,..., f 1,u,ρ, {1,..., T 1}, 5.2.11 where g,u : R u+1 T R u +2 T {u,d} R. Hence, he developmen of he forward raes given by 5.2.11 is a special case of 5.2.7, where we have resriced he possible dependence on he pas forward raes. Here, he forward rae a ime wih mauriy τ, τ {,..., + T 2 T 1}, is allowed o depend on he pas forward raes wih mauriy less han or equal o τ only. This in urn gives ha f,τ is independen of ε v for v > τ T + 1. Wih his resricion we have ha he price a ime of a bond wih ime of mauriy τ {,...,+ T 1 T } is F τ T G -measurable. Here, and hroughou he chaper, we adop he convenion ha F τ = F for τ N. Wihin his model we have ha once a bond is inroduced in he marke, he developmen of he price process is enirely described by he oucome of he ρ s. Hence, a ime we essenially are in he complee and arbirage free model from Secion 5.2.1 when considering he filraion G and he ime horizon + T 1 T. 5.2.3 Discree-ime rading Since he bonds raded in he bond marke depend on he ime considered, i is inconvenien o define rading sraegies in erms of he bonds. Hence, we define T new price processes S k k=1,..., T, which are defined for all =,1,...,T, by S k = 1 and S k = 1 P, 1 + k P 1, 1 + k Sk 1 = i= Pi + 1,i + k, {1,...,T }. Pi,i + k 5.2.12 We noe ha unil ime T hese price processes include exacly he same informaion as he original bond prices. The price process S k is generaed by invesing 1 uni a ime in bonds wih ime o mauriy k and a imes = 1,...,T selling he bonds wih ime o mauriy k 1 purchased a ime 1 and reinvesing he money in bonds wih ime o mauriy k. Hence, for k {1,..., T } he price process S k is he value process generaed by a roll-over sraegy in bonds wih ime o mauriy k. Such a value process is usually referred o as a rolling-horizon bond, see Rukowski 1999. Recall ha in a discree-ime model, he 1-period bond is equal o a savings accoun, so he price process S 1 corresponds o invesing in a savings accoun wih a locally riskfree ineres rae. Noe ha given F 1, he fuure value of he price process vecor a ime, S k k {1,..., T }, depends on ρ only, such ha i is sufficien for hedging purposes o consider any wo of he rolling-horizon bonds defined by 5.2.12. Here, we consider he savings accoun, henceforh denoed B, and S T, henceforh simply denoed S. We shall refer o he asse wih price process S as he risky asse. Noe ha he measurabiliy condiions on he bond prices give ha B and S, respecively, are F 2 G 1 - and F 1 G -measurable.

5.2. A BOND MARKET MODEL 143 A rading sraegy wih respec o B,S is an adaped wo-dimensional process ϕ = ϑ,η. Hence, ϑ and η are F -measurable for all. The pair ϕ = ϑ,η is inerpreed as he porfolio esablished a ime and held unil ime + 1. Here, ϑ denoes he number of risky asses held in he porfolio, and η is he discouned deposi in he savings accoun. The value process associaed wih he rading sraegy ϕ is denoed Vϕ. Here, V ϕ, which is he value a ime of holding he porfolio ϑ,η, is given by V ϕ = ϑ S + η B. 5.2.13 Wih he definiion of ϑ and η above he value process is seen o be he value afer any in- or ouflow of capial a ime. A rading sraegy is called self-financing if for all 1 1 V ϕ = V ϕ + ϑ u S u+1 + η u B u+1, 5.2.14 u= where we have inroduced he noaion exemplified by S u = S u S u 1. Thus, he value a ime of a self-financing sraegy is he iniial value added ineres and invesmen gains from rading in he bond marke. Hence, wihdrawals or deposis are no allowed a inermediae imes = 1,...,T 1. A self-financing sraegy is a so-called arbirage if V ϕ = and V T ϕ P-a.s. wih PV T ϕ > >. A coningen claim or a derivaive wih mauriy T is an F T -measurable random variable H. A coningen claim is called aainable if here exiss a self-financing sraegy ϕ such ha V T ϕ = H P-a.s. An aainable claim can hus be replicaed perfecly by invesing V ϕ a ime and adjusing he porfolio a imes = 1,...,T 1, according o he self-financing sraegy ϕ. Hence, a any ime, here is no difference beween holding he claim H and he porfolio ϕ. In his sense, he claim H is redundan in he marke and from he assumpion of no arbirage i follows ha he price of H a ime mus be V ϕ. Thus, he iniial invesmen V ϕ is he unique arbirage free price of H. Noe ha if ϕ is a self-financing porfolio replicaing he coningen claim H, hen 5.2.14 gives he following represenaion for H: T 1 T 1 H = V T ϕ = V ϕ + ϑ u S u+1 + η u B u+1. 5.2.15 u= If all coningen claims are aainable, he model is called complee and oherwise i is called incomplee. Throughou he chaper, we denoe by S, V ϕ and H he discouned price process of he risky asse, he discouned value process and he discouned claim, respecively. Remark 5.2.4 The definiion of rading sraegies in discree ime is no uniform in he lieraure. Harrison and Kreps 1979, Jarrow 1996 and Musiela and Rukowski 1997 define rading sraegies as adaped processes, whereas Harrison and Pliska 1981 and Björk 24 consider predicable processes. The differen measurabiliy condiions lead o one significan difference, namely, wheher he value process defined by 5.2.13 denoes he value before or afer a possible wihdrawal or deposi. A hird possibiliy is he definiion in Föllmer and Schweizer 1988. They consider a predicable process ϑ and an adaped process η. Hence, he porfolio a ime is given by he number of risky asses held in he porfolio from ime 1 o and he discouned deposi in he savings u= u=

144 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK accoun afer a possible wihdrawal or deposi. Since hey define he value process afer a possible wihdrawal or deposi heir value process coincides wih he value process in he presen chaper. Hence, we have he following connecion beween our definiion of rading sraegies and he Föllmer Schweizer definiion: Here, ϑ FS ϑ = ϑ FS +1 5.2.16 η = η FS + ϑ FS ϑ FS +1 S. 5.2.17,η FS denoes he porfolio a ime using he Föllmer Schweizer definiion. Example 5.2.5 If T = 2 and T = 3 hen he price processes for he savings accoun and he risky asse are given by 1 B = 1, B = 1 + r i, {1,2,3} and S = 1, S = i= 1 i= Pi + 1,i + 2, {1,2,3}, Pi,i + 2 respecively. Here, one easily observes ha, as noed above, B is F 2 G 1 -measurable and S is F 1 G -measurable. 5.3 Hedging sraegies Consider a company ineresed in hedging he claim H wih mauriy T. If H only depends on bonds wih ime of mauriy a ime T or earlier, i has a unique arbirage free price and can be replicaed perfecly leaving he company wihou any risk. However, if H depends on bonds mauring afer ime T, hen H does in general no have a perfec replicaing sraegy, and hence in general i does no have a unique arbirage free price. For unaainable claims we deermine he opimal hedging sraegies for he crieria of super-replicaion and risk-minimizaion. 5.3.1 Super-replicaion A sraegy ϕ is called super-replicaing for he claim H wih mauriy T if he value process is of he form 1 1 V ϕ = V ϕ + ϑ u S u+1 + η u B u+1 U, 5.3.1 u= where U is a non-decreasing process, and he erminal value of he value process saisfies V T ϕ H P-a.s. Here, he process U is he accumulaed ouflow of capial when using he sraegy ϕ. Thus, when following a super-replicaing sraegy no inflow of capial is needed in addiion o he iniial invesmen in order o guaranee ha a ime of mauriy, u=

5.3. HEDGING STRATEGIES 145 he value of he porfolio is a leas as large as he considered claim. Hence, following a super-replicaing sraegy allows he hedger o eliminae he risk of falling shor of he claim. The smalles iniial value needed a ime o consruc a super-replicaing sraegy is referred o as he super-replicaing price a ime, henceforh denoed ˆπ H. Hence, he super-replicaing price a ime is he smalles iniial invesmen a ime allowing he company o hedge he considered claim wihou any risk of falling shor. For more deails on super-replicaion see El Karoui and Quenez 1995 and Föllmer and Schied 22. Now define he super-replicaing price process as he process of he super-replicaing prices, i.e. he value of he super-replicaing price process a ime is exacly he super-replicaing price a ime. A any ime he opimal super-replicaing sraegy is defined as he super-replicaing sraegy corresponding o he super-replicaing price process. Prior o he general resul for he super-replicaing price process and opimal super-replicaing sraegy for a claim H wih mauriy T, we firs consider super-replicaion in a 1-period model. Lemma 5.3.1 A ime, {,...,T 1}, he opimal super-replicaing sraegy, ˆϕ = ˆϑ, ˆη, for a claim H wih ime of mauriy + 1 is given by ˆϑ = Ĥd Ĥu S d +1 Su +1 and ˆη = ĤuSd +1 ĤdSu +1 B +1 S d +1 Su +1, where The super-replicaing price is Ĥρ +1 = maxhρ +1,h,Hρ +1,l, ρ +1 {u,d}. ˆπ H = 1 1 + r q +1 Ĥu + 1 q +1 Ĥd. Proof of Lemma 5.3.1: Consider an agen holding he porfolio ϑ,η a ime. Before any adjusmens a ime + 1 he value of he porfolio can ake one of wo values: ϑ S u +1 + η B +1 or ϑ S d +1 + η B +1. Hence, he value of he porfolio is he same in he saes u,h and u,l, as well as in d,h and d,l. Thus, for ϑ,η o be superreplicaing i mus hold ha ϑ S u +1 + η B +1 maxhu,h,hu,l, ϑ S d +1 + η B +1 maxhd,h,hd,l, where he Hi,j denoes he payoff from H if ρ +1 = i and ε +1 = j, where i {u,d} and j {h,l}. Define he coningen claim Ĥ wih payoff and noe ha he sraegy Ĥρ +1 = maxhρ +1,h,Hρ +1,l, ρ +1 {u,d}, ϑ = Ĥd Ĥu S d +1 Su +1 and η = ĤuSd +1 ĤdSu +1 B +1 S d +1 Su +1

146 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK replicaes Ĥ. A no arbirage argumen now gives ha he replicaing sraegy and he unique arbirage free price for Ĥ is he opimal super-replicaing sraegy and superreplicaing price, respecively. Lemma 5.3.1 has he following inerpreaion: The dependence of H on ε +1 is unhedgeable. Hence, for each oucome of ρ +1 we assume he oucome of ε +1 which leads o he highes value of H and replicae his claim. The replicaing sraegy and he unique arbirage free price of his wors scenario claim are hen equal o he opimal superreplicaing sraegy and super-replicaing price, respecively. Remark 5.3.2 The main resul in Alipranis, Polyrakis and Tourky 22 saes ha in a 1-period model he opimal super-replicaing sraegy shall be found among he replicaing sraegies in he complee sub-models arising from eliminaing saes of he world. Hence, Lemma 5.3.1 can be seen as a special case, where he opimal super-replicaing sraegy is easily idenifiable. Theorem 5.3.3 Consider a claim H wih ime of mauriy T. For {,...,T 1} he porfolio, ˆϕ = ˆϑ, ˆη held in he opimal super-replicaing sraegy is given by ˆϑ ξ,λ = ˆπξ,d,λ +1 H ˆπ ξ,u,λ +1 H S ξ,d,λ +1 S ξ,u,λ +1 and ˆη ξ,λ = ˆπξ,u,λ +1 S ξ,d,λ B ξ,λ 1 +1 +1 ˆπ ξ,d,λ S ξ,d,λ +1 S ξ,u,λ +1 +1 S ξ,u,λ +1, where ˆπ ξ,ρ +1,λ +1 H = max ˆπ ξ,ρ +1,λ,h +1 H, ˆπ ξ,ρ +1,λ,l +1 H, ρ +1 {u,d}. Saring wih he erminal value ˆπ ξ T,λ T T H = H, he super-replicaing price process a ime, {,..., T 1}, is given by he following recursive formula 1 ˆπ ξ,λ H = 1 + r ξ,λ q ξ,λ 1 +1 ˆπξ,u,λ +1 H + 1 q ξ,λ +1 ˆπξ,d,λ +1 H. Proof of Theorem 5.3.3: Firs noe ha a ime T he super-replicaing price is rivial and equal o H. A ime, {,...,T 1} we may consider he super-replicaing price a ime + 1, ˆπ +1 H, as he payoff from a coningen claim wih mauriy + 1. Thus, Lemma 5.3.1 gives he super-replicaing price and opimal super-replicaing sraegy a ime in erms of he super-replicaing price a ime + 1.

5.3. HEDGING STRATEGIES 147 Noe ha we in Theorem 5.3.3 explicily denoe he dependence on he pas hrough ξ and λ in order o emphasize he dependence of he opimal super-replicaing sraegy and super-replicaing price process on he pas. For sufficienly nice claims, such as fixed claims, he following corollary allows for an easy calculaion of he super-replicaing price process and he opimal super-replicaing sraegy. Corollary 5.3.4 If for each, {,...,T 1}, i holds, for fixed k +1 {h,l} ha ˆπ ξ,ρ +1,λ +1 H = ˆπ ξ,ρ +1,λ,k +1 +1 H for all ξ, ρ +1 and λ, hen he super-replicaing price and opimal super-replicaing sraegy a ime τ are, respecively, he unique arbirage free price and he replicaing sraegy in he condiional model given ε τ+1,...,ε T = k τ+1,...,k T. Denoe by Û he process U from 5.3.1 associaed wih he opimal super-replicaing sraegy. Hence, Û denoes he accumulaed ouflow of capial, when using he opimal super-replicaing sraegy. Combining 5.3.1 and Theorem 5.3.3 gives he following explici expression for he change in Û a ime Ûξ 1,ρ,λ 1,ε = ˆπ ξ 1,ρ,λ 1 H ˆπ ξ 1,ρ,λ 1,ε H. 5.3.2 Invesigaing 5.3.2 we observe ha he wihdrawal is he difference beween he value a ime of he opimal super-replicaing porfolio purchased a ime 1 and he superreplicaing price a ime. Hence, when using he opimal super-replicaing sraegy he wihdrawal from he porfolio a ime depends on he oucome of he wo random variables observed a ime, ρ and ε. Given 5.3.2, one easily derives he condiional expecaion under P of Ûξ 1,ρ,λ 1,ε given F 1, namely, ] F 1 [ˆπ ] = E P ξ 1,ρ,λ 1 H ˆπ ξ 1,ρ,λ 1,ε H F 1 = p ˆπ ξ 1,u,λ 1 H pˆπ ξ 1,u,λ 1,h H + 1 pˆπ ξ 1,u,λ 1,l H + 1 p ˆπ ξ 1,d,λ 1 H pˆπ ξ 1,d,λ 1,h H + 1 pˆπ ξ 1,d,λ 1,l H. E P [ Ûξ 1,ρ,λ 1,ε Thus, he expeced wihdrawal from he opimal super-replicaing porfolio is he probabiliy of an upward jump muliplied by he expeced wihdrawal coningen on an upward jump added he probabiliy of a downward jump muliplied by he expeced wihdrawal in his case. Example 5.3.5 Le T = 2, T = 3 and H = 1. Since H is aainable a ime 1 and ˆπ ρ 1 1 = ˆπ ρ 1,l 1, we have from Corollary 5.3.4 ha he super-replicaing price and he superreplicaing sraegy a ime corresponds o using he condiional forward rae vecor

148 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK given ε 1 = l. Hence, we obain he following super-replicaing price process, expressed in erms of bond prices: ˆπ ξ 3,λ 3 3 1 = 1, ˆπ ξ 2,λ 2 2 1 = ˆπ ξ 1,λ 1 1 1 = 1 1 + r ξ 2,λ 1 2 1 1 + r ξ 1 1 q ξ 2,λ 2 3 + 1 q ξ 2,λ 2 3 q ξ 1,λ 1 2 P ξ 1,u,λ 1 2,3 + ˆπ 1 = 1 1 + r q 1 P u,l 1,3 + 1 q 1 P d,l 1,3 1 = 1 + r ξ 2,λ 1 = P ξ 2,λ 1 2,3, 2 1 q ξ 1,λ 1 2 P ξ 1,d,λ 1 2,3 = P ξ 1,λ 1 1,3, The opimal super-replicaing sraegy is given by P,2P ˆϑ d,l 1,3 P u,l 1,3, ˆη = P d 1,2 P u, P d 1,2P u,l 1,3 P u 1,2P d,l 1,3 1,2 1 + r P d 1,2 P u, 1,2 1,λ 1 ˆϑξ 1, ˆη ξ 1,λ 1 P,2P ξ 1,λ 1 1,3 1 = P ξ, and 2,λ 2 ˆϑξ 1 1,2 2, ˆη ξ 2,λ 2 1 2 =,.. B ξ 2,λ 1 3 Relaion o guaranees Apar from he nice propery of allowing he hedger o eliminae he shorfall risk he super-replicaing price process relaes o he maximal possible guaranees for which he risk of falling shor can be eliminaed. Here, we consider wo ypes of guaranees: Mauriy guaranees and periodic ineres rae guaranees. Given a deposi a ime he mauriy guaranee is he minimal possible payoff a ime T, whereas he periodic ineres guaranee is he minimum ineres earned on he deposi in each period unil ime T. We shall refer o he maximal guaranees for which he shor fall risk can be eliminaed as he maximal riskfree mauriy guaranee and maximal riskfree periodic ineres rae guaranee. Proposiion 5.3.6 Given an iniial deposi of 1 a ime, he maximal riskfree mauriy guaranee, G T, a ime T is given by The maximal riskfree periodic ineres rae guaranee is g T = G T = 1 ˆπ 1. 5.3.3 1 1 T 1. 5.3.4 ˆπ 1 Proof of Proposiion 5.3.6: A ime he super-replicaing price of 1 uni a ime T is given by ˆπ 1. Hence, by invesing 1 a ime we may purchase 1/ˆπ 1 unis of he super-replicaing sraegy. This guaranees a payoff a ime T of a leas 1/ˆπ 1. Hence,

5.3. HEDGING STRATEGIES 149 since he super-replicaing price per definiion is he lowes iniial deposi for which a cerain payoff is guaraneed, he maximal riskfree mauriy guaranee a ime T is given by 5.3.3. Now, he maximal riskfree periodic ineres rae guaranee is he consan shor rae which gives a payoff of G T a ime T, when deposiing 1 uni a ime. Hence, g T is he unique soluion greaer han 1 o 1 + g T T = G T. Insering 5.3.3 and isolaing g T now gives 5.3.4. Proposiion 5.3.6 is of imporance o for insance life insurance companies, since i gives he maximal guaranees, which he companies should promise he insured a iniiaion of he conrac. 5.3.2 Risk-minimizing sraegies As an alernaive o he hedging crierion of super-replicaion we now consider riskminimizaion. Here, we give a brief review of risk-minimizaion and deermine riskminimizing sraegies in he presence of reinvesmen risk. We noe ha since we define rading sraegies differenly han Föllmer and Schweizer 1988 and Møller 21a our resuls canno be compared direcly o heir resuls. A brief review of risk-minimizaion In his secion we review he crierion of risk-minimizaion inroduced in discree ime by Föllmer and Schweizer 1988. The presenaion is based on Møller 21a. The idea of risk-minimizaion is closely relaed o he inroducion of he cos process defined by 1 C ϕ = V ϕ ϑ u Su+1. 5.3.5 Thus, he cos process is he discouned value of he porfolio reduced by discouned rading gains. The cos process measures he accumulaed discouned cos of an agen following he sraegy ϕ. Comparing 5.2.15 and 5.3.5 we noe ha he cos process is consan P-a.s. if and only if he sraegy ϕ is self-financing. To measure he risk associaed wih he sraegy ϕ we inroduce he risk process defined by u= R ϕ = E Q [ C T ϕ C ϕ 2 F ]. 5.3.6 Hence, he risk process is he condiional expeced value of he fuure coss associaed wih he sraegy ϕ. A rading sraegy ϕ is called risk-minimizing for he coningen claim H if for all {,...,T } i minimizes R ϕ over all rading sraegies wih V T ϕ = H.

15 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK The consrucion of risk-minimizing sraegies is based on he Q-maringale V = E Q [H F ], known as he inrinsic value process. Using he so-called Kunia Waanabe decomposiion for maringales, V can be uniquely decomposed as V = V + ϑ H u S u + LH, 5.3.7 u=1 where ϑ H is predicable, and L H is a zero-mean Q-maringale orhogonal o S, i.e. S L H is a Q-maringale as well. For more deails on he Kunia Waanabe decomposiion we refer o Föllmer and Schied 22. Shifing he index in 5.3.7 and defining he adaped process ϑ H by ϑ H u = ϑh u+1 we have he following decomposiion V = V 1 + u= ϑ H u Su+1 + L H. 5.3.8 Comparing 5.2.15 and 5.3.8 we observe ha H is aainable if and only if L H T = Q-a.s. Using 5.2.16 and 5.2.17 we obain he following heorem, due o Föllmer and Schweizer 1988, which relaes he Kunia Waanabe decomposiion o he risk-minimizing sraegy. Theorem 5.3.7 There exiss a unique risk-minimizing sraegy, ϕ, wih V T ϕ = H given by ϑ,η = ϑ H +1,V ϑ H +1S. 5.3.9 Insering 5.3.9 in 5.3.5 and using he Kunia-Waanabe decomposiion from 5.3.7 we obain he following expression for he cos process associaed wih he risk-minimizing sraegy: C ϕ = V 1 u= ϑ H u+1 S u+1 = V + LH. 5.3.1 Combining 5.3.1 and 5.3.6 now gives he following expression for he so-called inrinsic risk process, which is he risk process associaed wih he risk-minimizing sraegy: [ R ϕ = E Q L H T L H 2 ] F. 5.3.11 Noe ha when deermining he risk-minimizing sraegy we consider all admissible sraegies. This is in conras o many oher quadraic hedging crieria, such as mean-variance hedging, where only self-financing sraegies are allowed. From 5.3.11 we observe ha risk-minimizing sraegies are no self-financing for non-aainable claims. However, hey are mean-self-financing, i.e. he corresponding cos processes are Q-maringales. Since 5.3.6 involves an expecaion wih respec o Q, he risk-minimizing sraegy depends on he chosen equivalen maringale measure. Furhermore we observe from 5.3.6

5.3. HEDGING STRATEGIES 151 ha he crierion of risk-minimizaion, like oher quadraic hedging crieria, penalizes gains and losses equally. This is of course disadvanageous, however, when using a crierion penalizing only losses, explici resuls are hard o obain; see he discussion in Møller 21a and references herein. In general, he risk-minimizing sraegy is given by he predicable Q-expecaion of he replicaing sraegy given he unhedgeable uncerainy, see Schweizer 1994 for a proof in a coninuous-ime seup. A paricular simple risk-minimizing sraegy is obained in Møller 21a, since he considers an unhedgeable risk, which is sochasically independen of he financial marke. As we shall see below in Theorem 5.3.9, he expression for he risk-minimizing sraegy is slighly more complicaed in he presen model han in Møller 21a, since he unhedgeable risk is in he financial marke. Risk-minimizing sraegies in he presence of reinvesmen risk We now urn o he derivaion of risk-minimizing sraegies in he presen model including reinvesmen risk. In order o deermine he Kunia Waanabe decomposiion of V we inroduce he Q-maringales = E Q [ ] 1 ε1,...,ε T =λ T F = E Q [ ] 1 ε1,...,ε T =λ T H M λ T 5.3.12 for all λ T Λ T. Here, we have used ha under Q he disribuion of he ε s is independen of he filraion G. Using he quaniies defined in 5.3.12, we ge he following expression for V : V = E Q [H F ] = E Q [ E Q [H F H T ] F ] = λ T Λ T M λt π λ T, H, 5.3.13 where π λ T, H is he unique discouned arbirage free price for H given ε 1,...,ε T = λ T. Using 5.3.13 we obain he following expression for he developmen of V from ime 1 o, V = V V 1 = M λt π λ T, λ T Λ T = λ T Λ T = λ T Λ T = λ T Λ T = λ T Λ T M λ T H π λ T, M λt 1 πλ T, 1 H λ T Λ T H M λ T 1 πλ T, 1 H M λ T M λ T 1 π λ T, H + M λ T 1 π λ T, H π λ T, 1 H π λ T, π λ T, H M λ T H M λ T + M λ T 1 πλ T, H + M λ T 1 ϑλ T 1 S,

152 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK where ϑ λ T is he number of risky asses in he replicaing sraegy in he complee model given ε 1,...,ε T = λ T. Hence, we have he following decomposiion of V : V = V + Su + H M λ T u. 5.3.14 u=1 M λt u 1 ϑλ T u 1 λ T Λ T u=1 π λt, u λ T Λ T In order o show ha 5.3.14 acually is he Kunia Waanabe decomposiion of V, we firs noe ha λ T Λ T M λ T 1 ϑλ T 1 is F 1-measurable, such ha he process ϑ H defined by ϑ H = is predicable. Now define he process L H by L H = u=1 M λt 1 ϑλ T 1 λ T Λ T π λt, u λ T Λ T Using he law of ieraed expecaions we see ha E Q [ L H ] F 1 = E Q π λt, H M λ T λ T Λ T F 1 = [ ] H M λ T F 1 λ T Λ T E Q = =, λ T Λ T E Q [ π λ T, π λ T, H M λ T u. 5.3.15 [ ] ] HE Q M λ T G H 1 F 1 since M λ T is a maringale sochasically independen of he filraion G. Hence, L H is a Q-maringale. To show ha ha L H S is a Q-maringale we firs observe ha L H S = LH S LH 1 S 1 = LH 1 S + S 1 LH + L H S. Thus, since L H and S are Q-maringales, i is sufficien o show ha E Q [ L S F 1 ] = E Q π λt, H M λ T S λ T Λ T F 1 = [ ] H M λ T F 1 λ T Λ T E Q = =. λ T Λ T E Q Hence, we have proved he following. [ π λ T, π λ T, S [ ] ] H S E Q M λ T G H 1 F 1

5.3. HEDGING STRATEGIES 153 Lemma 5.3.8 For a claim H wih ime of mauriy T he Kunia Waanabe decomposiion is given by V = V + u=1 M λt u 1 ϑλ T u 1 λ T Λ T S u + u=1 π λt, u λ T Λ T H M λ T u. Combining Lemma 5.3.8, Theorem 5.3.7 and he expression for he inrinsic risk process in 5.3.11 we obain he following heorem regarding he risk-minimizing sraegy and he inrinsic risk process. Theorem 5.3.9 The risk-minimizing sraegy, ϕ, for H is given by ϑ,η = M λt λ T Λ T ϑ λ T, The inrinsic risk process is given by R ϕ = E Q M λt π λ T, λ T Λ T T u=+1 π λt, u λ T Λ T H M λt λ T Λ T H M λ T 2 u F. ϑ λ T S. Thus, he number of risky asses held in he risk-minimizing sraegy a ime is he average under Q of he replicaing sraegies for H in he condiional models given he oucome of ε 1,...,ε T. The deposi in he savings accoun is adjused each period according o he realizaion of he unhedgeable variables, such ha he discouned value process is equal o he inrinsic value process. Insering 5.3.15 in 5.3.1 gives he following expression for he cos process associaed wih ϕ : C ϕ = V + u=1 π λt, u λ T Λ T H M λ T u. 5.3.16 From 5.3.16 we see ha he change in he cos process a ime for an agen following he risk-minimizing sraegy depends on he change in he Q-maringales M λ T associaed wih he oucome of ε. If he claim is aainable a some ime prior o T, hen he cos process is consan P-a.s. from ime, and hence, he inrinsic risk process is zero from ime. Example 5.3.1 Le T = 2, T = 3 and H = 1. For he fixed Q-measure given by q λ 1 = q,1 we now obain he risk-minimizing sraegy from Theorem 5.3.9. A

154 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK ime he risk-minimizing sraegy is given by ϑ = P,2 q P d,h 1,3 P u,h 1,3 P d 1,2 P u + 1 q P d,l 1,3 P u,l 1,3 1,2 P d 1,2 P u, 1,2 η = p P d 1,2P u,h 1,3 P u 1,2P d,h 1,3 1 + r P d 1,2 P u 1,2 + 1 p P d 1,2P u,l 1,3 P u 1,2P d,l 1,3 1 + r P d 1,2 P u, 1,2 whereas i a ime 1 and 2 is given by ϑ ξ 1,λ 1, 1,η ξ 1,λ 1, P,2P ξ 1,λ 1 1,3 1 = P ξ, 1 1,2 and ϑ ξ 2,λ 2, 2,η ξ 2,λ 2, 2 =, 1 B ξ 2,λ 1 3 We noe ha since he claim is aainable from ime 1, he risk-minimizing sraegy and super-replicaing sraegies coincide a imes 1 and 2. Moreover, since he sraegies coincide so do he super-replicaing price and he value of he porfolio held in he riskminimizing sraegy.. 5.4 A numerical illusraion Here, he purpose is o provide some numbers in he coninuing example considered in Secions 5.2 and 5.3. Hence, T = 2, T = 3 and H = 1. Now assume ha given he iniial forward rae vecor r,f,1 he forward raes a ime, {1,2,3}, are given by r ξ,λ 1 = r a1 1 ρi =u + a 2 1 ρi =d i=1 f ξ,λ,+1 = rξ,λ 1 1 i=1 b1 1 ε=h + b 2 1 ε=l, a3 1 εi =h + a 4 1 εi =l, where a 1,...,a 4,b 1,b 2 are posiive consans, and u i=1 is inerpreed as 1 if u =. The consans a 1 and a 2 describe he movemen of he forward rae vecor due o he oucome of he ρ s, whereas a 3 and a 4 describe he dependence of he forward rae vecor on pas values of he ε s, and finally b 1 and b 2 describe he unhedgeable uncerainy associaed wih he newly issued bonds. In his simple model he dependence on ξ is given by he number of u s and no by he ordering of he u s. Hence, he number of saes a ime 2 is reduced from 16 o 12. However, his is sill a large number of saes compared o he 4 in a binomial model 3 if he binomial model is recombining. In conras o an addiive srucure, he muliplicaive srucure above ensures ha he forward raes are sricly posiive. Now le he iniial forward rae curve and he consans be given by r =.3, f,1 =.31, a 1 = 1.25, a 2 =.8, a 3 = 1.1, a 4 =.99, b 1 = 1.325 and b 2 = 1.15. Recall from Examples 5.3.5 and 5.3.1 ha he opimal super replicaing and risk-minimizing sraegies for H = 1 depend on ξ 2 and λ 1, only. Thus, Figure 5.4.1 shows he forward

5.4. A NUMERICAL ILLUSTRATION 155 raes relevan for deermining he hedging sraegies. Furhermore, Example 5.3.5 gives ha he super-replicaing price a ime corresponds o he zero coupon bond price in he condiional model given ε 1 = l, which in urn corresponds o a 2-period forward rae of.3142. Here, and in he remaining of he secion, all numbers are given wih 4 significan digis..3.31.375.3872.375.386.24.2478.4734.33.4641.297.33.1939.24.2436.297.191 Figure 5.4.1: Relevan forward raes a ime, 1 and 2. A ime and 1 he vecor shows he shor rae and 1-period forward rae, whereas a ime 2 only he shor rae is relevan. From Example 5.3.1 we furhermore noe ha he risk-minimizing sraegy depends on q. Thus, o obain some numbers we have o specify Q. Henceforh, we le p =.5 and consider risk-minimizaion under he minimal maringale measure, i.e. q = p. The opimal super-replicaing and risk-minimizing sraegies and he corresponding prices are illusraed in Figure 5.4.2. Here, he firs column gives he super-replicaing price, ˆϑ and ˆη, and he second column shows he risk-minimizing price, ϑ and η. Here, and henceforh we refer o he value of he risk-minimizing sraegy as a price, since i is he arbirage free price under he chosen equivalen maringale measure. Since he super-replicaing price is an upper bound for he inerval of arbirage free prices, he price using he crierion of risk-minimizaion is obviously lower han or equal o he super-replicaing price. In paricular i is sricly lower if here is a reinvesmen risk, i.e. if b 1 b 2. In addiion o he hedging sraegies we may apply Proposiion 5.3.6 o obain he maximal riskfree

156 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK mauriy guaranee G 3 = 1.95 and he maximal riskfree periodic ineres rae guaranee g 3 =.381..913 1.8397.9267.9128 1.847.9342.9279.968.9285.9353.9529.9474.9533.9478.9279.968.9285.9353.9529.9474.9533.9478.9548.8935.976.983.9557.8943.9712.988.976.922.981.931.9712.928.9813.934.9548.8935.976.983.9557.8943.9712.988.976.922.981.931.9712.928.9813.934 Figure 5.4.2: Hedging sraegies and associaed prices. Firs column: Super-replicaing price, ˆϑ and ˆη. Second column: Price using risk-minimizaion, ϑ and η. Now we are ineresed in how he prices using he crieria of super-replicaion and riskminimizaion are affeced by changing b 1 and b 2, which deermine he shape of he forward rae curve a ime, {1,2,3}. Invesigaing Table 5.4.1 we observe ha he price a ime using risk-minimizaion is decreasing in boh b 1 and b 2. This is inuiively clear since a seeper posiive slope leads o lower bond prices and hence a smaller iniial invesmen. The super-replicaing price is also decreasing in b 2, however, in conras o he risk-minimizing price, i is independen of b 1. The independence can be explained by he fac ha he crierion of super-replicaion considers he wors scenario only. Furhermore, we observe ha, as anicipaed above, he risk-minimizing and super-replicaing prices coincide when b 1 = b 2, i.e. when here is no reinvesmen risk. A comparison wih pracice in Danish life insurance The Danish life insurance companies are forced by legislaion o disregard he reinvesmen risk and value heir long erm liabiliies using a yield curve, which is level beyond 3 years. Here, we consider he similar principle of a level yield curve beyond he ime of mauriy of he longes raded bond. We shall refer o his approach as he principle of a level long

5.4. A NUMERICAL ILLUSTRATION 157 b 1 b 2 Risk-minimizaion Super-replicaion 1.5 1.15.9125.913 1.325 1.15.9128.913 1.15 1.15.913.913 1.325 1.15.9128.913 1.325 1.913.9134 1.325.99.9131.9137 1.325.98.9132.914 Table 5.4.1: A comparison of prices a ime using risk-minimizaion and superreplicaion. Top: Dependence on b 1. Boom: Dependence on b 2. erm yield curve even hough we in discree ime have a yield vecor raher han a yield curve. In his seing wih discree compounding he yield a ime of a zero coupon bond wih mauriy is defined by 1 1 y, = 1. P, Here, he yield vecor a ime is given by y,1,y,2 =.3,.35. Thus, he principle of a level long erm yield curve corresponds o assuming y,3 = y,2 =.35, which leads o a price of.9138. In addiion o he level long erm yield curve principle we inroduce he analogous principle of a level long erm forward rae curve, where we price using a forward rae curve, which is level beyond he ime of mauriy of he longes raded bond. Here, his leads o he price.9134. We noe ha boh principles only depend on he presen forward rae curve, and hus hey are independen of he possible fuure developmens. Furhermore none of he principles are based on he no arbirage principle. We now urn o he relaionship beween he yield vecor and he forward rae vecor. When he yield vecor is increasing decreasing he forward rae vecor lies above below he yield vecor. Thus, if we have a level long erm yield vecor, he long erm forward rae vecor is level and equal o he yield vecor. On he oher hand an increasing decreasing forward rae vecor which is level for long imes o mauriy corresponds o a yield vecor which increases decreases and ends owards he forward rae vecor as he ime o mauriy increases. The increase decrease in he yield vecor on he inerval, where he forward rae vecor is level, is given by 1 1 + f, 1 y, y, 1 = 1 + y, 1 1. 1 + y, 1 In his example he forward rae vecor a ime is increasing, such ha he principle of a level long erm forward rae curve leads o a lower price han he level long erm yield curve principle. Now we are ineresed in wheher he principles lead o prices in he inerval of arbirage free prices. In his simple example, where we consider a fixed claim and he ime horizons

158 CHAPTER 5. A DISCRETE-TIME MODEL FOR REINVESTMENT RISK T = 2 and T = 3 a principle leads o a price in he inerval of arbirage free prices if and only if he value of f,2 implied by he principle lies above he 2-period forward rae implied by he super-replicaing price and below he 2-period forward rae implied by he bes scenario price which is a lower bound for he inerval of arbirage free prices. From Table 5.4.2 we observe ha if we allow for increasing forward rae vecors only, boh b 1 b 2 Bes scenario Super-replicaion Level forward Level yield 1.5 1.15.325.3142.31.35 1.325 1.15.3196.3142.31.35 1.15 1.15.3142.3142.31.35 1.325 1.15.3196.3142.31.35 1.325 1.3196.396.31.35 1.325.99.3196.365.31.35 1.325.98.3196.334.31.35 Table 5.4.2: Values of f,2 implied by, respecively, he bes scenario price, he superreplicaing price and he principles of a level long erm forward rae/yield curve. Top: Dependence on b 1. Boom: Dependence on b 2. principles lead o a 2-period forward rae below he one implied by he super-replicaing price, and hence hey lead o a price higher han he super-replicaing price. Thus, in his case boh principles clearly overesimae he price. If we allow for a level or decreasing forward rae vecor, he 2-period forward rae implied by he super-replicaing price is lower han he one implied by a he principle of a level long erm forward rae curve, and if he possible decrease is sufficienly large also lower han he one implied by using a level long erm yield curve, such ha he principles lead o prices, which lie in he inerval of arbirage free prices. However, especially he price obained using a level long erm yield curve is in he high end of he inerval of arbirage free prices. Noe ha he same informaion also could have been observed from Table 5.4.1. Based on he discussion above we conclude ha he principles should no be used in siuaions where a decreasing forward rae curve is very unlikely. If one uses one of he principles anyhow, we recommend using he level long erm forward rae principle and a he same ime o keep in mind ha he price mos likely is overesimaed. In siuaions where a decreasing forward rae vecor is more likely, he principles are more likely o be accurae. The accuracy depends heavily on he siuaion and in paricular on he correspondence beween he presen forward rae vecor and he condiional forward rae vecors. The conclusion regarding he principles is ha even hough hey are easy o use, heir resuls should be used as guidelines only.

Chaper 6 A Coninuous-Time Model for Reinvesmen Risk in Bond Markes This chaper is an adaped version of Dahl 25a We propose a bond marke model, where, as in pracice, only bonds wih a limied ime o mauriy are raded in he marke. As ime passes, new bonds wih sochasic iniial prices are inroduced in he marke. Hence, we are able o model he reinvesmen risk presen in pracice, when considering long erm conracs. To quanify and conrol he reinvesmen risk we apply he crierion of risk-minimizaion. 6.1 Inroducion In he lieraure, bond markes are usually assumed o include all bonds wih ime of mauriy less han or equal o he ime of mauriy of he considered claim. However, in pracice only bonds wih a limied sufficienly shor ime o mauriy are raded. Hence, sandard models are only adequae o describe pricing and hedging of so-called shor erm conracs, where he payoff depends on bonds wih ime o mauriy less han or equal o he longes raded bond. When considering long erm conracs, where he payoff depends on bonds wih longer ime o mauriy han he longes raded bond, he bond marke does no in general include bonds which a all imes allow for a perfec hedge of he conrac. Thus, in pracice, an agen ineresed in pricing and hedging long erm conracs is exposed o a reinvesmen risk, which is ignored in sandard bond marke models. Here, he reinvesmen risk refers o he uncerainy associaed wih he obainable rae of reurn, when reinvesing in bonds no ye raded in he marke. An example of long erm conracs sold in pracice are life insurance conracs, where he liabiliies of he insurance companies ofen exend 5 years, or more, ino he fuure. In 159

16 CHAPTER 6. A CONTINUOUS-TIME MODEL FOR REINVESTMENT RISK his chaper, we propose a model, where pricing and hedging of shor erm conracs is similar o a sandard bond marke model, whereas he model includes reinvesmen risk, when considering long erm conracs. In order o describe he reinvesmen risk, we iniially consider a sandard coninuous-ime bond marke model wih some fixed finie ime horizon, which is less han or equal o he ime horizon of he considered paymen process. A fixed imes new bonds are issued in he marke, such ha we immediaely afer he issue of new bonds consider a sandard model idenical o he iniial one. The enry prices of he new bonds depend on he prices of he bonds already raded and a sochasic erm. As is sandard in bond marke lieraure, we model he forward raes raher han he bond prices hemselves. Beween he imes of issue, he forward raes follow a sandard Heah Jarrow Moron model, see Heah e al. 1992. When new bonds are issued, he forward rae curve is exended. We assume ha a each ime of issue he exension is coninuous and depends on a single random variable. The idea of fixing he maximum ime o mauriy of he raded asses and inroducing new asses as imes passes can also be found in Neuberger 1999, who considers a marke for fuures on oil prices. Neuberger 1999 models he iniial price of he new fuure as a linear funcion of prices on raded fuures and a normally disribued error erm. To he auhor s knowledge he only oher papers considering he problem of modelling he prices of newly issued bonds are Sommer 1997 and Dahl 25b see Chaper 5. Dahl 25b considers a discree-ime model for he reinvesmen risk, whereas Sommer 1997 considers a coninuous-ime bond marke. A major difference beween Sommer 1997 and his chaper is he way new bonds are issued and priced in he marke. While Sommer 1997 considers he case where new bonds are issued coninuously, his chaper, as is he case in pracice, considers a se of fixed imes, where new bonds are issued. Hence, he presen model should be more ap o describe pracice. Wihin his seup Sommer derives condiions on he forward rae dynamics in order o have sufficienly smooh forward rae curves and risk-minimizing sraegies. To quanify and conrol he reinvesmen risk associaed wih long erm conracs, we apply he crierion of risk-minimizaion inroduced by Föllmer and Sondermann 1986 for coningen claims and exended in Møller 21c o he case of paymen processes. The derivaion of he risk-minimizing sraegies are based on he ideas of Schweizer 1994 regarding risk-minimizaion under resriced informaion. Hence, he risk-minimizing sraegies are given in erms of he replicaing sraegies in he case wihou reinvesmen risk. The chaper is organized as follows: In Secion 6.2 a bond marke model including reinvesmen risk is inroduced. This is done in wo seps: Firs we describe a sandard bond marke model, and hen he model is exended o include reinvesmen risk. In his secion we also inroduce he considered class of equivalen maringale measures and he relevan financial erminology. Risk-minimizing sraegies are derived in Secion 6.3, and we conclude he chaper by describing a possible implemenaion of he model in Secion 6.4.

6.2. THE BOND MARKET MODEL 161 6.2 The bond marke model Le T be a fixed finie ime horizon and Ω, F,P a probabiliy space wih a filraion F = F saisfying he usual condiions of righ-coninuiy, i.e. F T = F u, and compleeness, i.e. F conains all P-null ses. u> 6.2.1 A sandard model Consider anoher fixed ime horizon T, T T, and a bond marke, where a ime, T all zero coupon bonds wih mauriy τ, τ T are raded. Le P,τ denoe he price a ime of a zero coupon bond mauring a ime τ. To avoid arbirage we assume ha P,τ is sricly posiive and P, = 1 for all. For non-negaive ineres raes he price P,τ is a decreasing funcion of τ for fixed. An imporan quaniy when modelling bond prices is he insananeous forward rae wih mauriy τ conraced a ime defined by or, saed differenly, f,τ = log P,τ, 6.2.1 τ P,τ = e Êτ f,udu. 6.2.2 The forward rae f,τ can be inerpreed as he riskfree ineres rae, conraced a ime over he infiniesimal inerval [τ,τ + dτ. The shor rae process r T is defined as r = f,. Since i is inconvenien o model he dynamics of bond prices direcly, he common approach in he lieraure is o model ineres raes. Here, we ake he approach of Heah e al. 1992 where he dynamics of no only he shor rae bu he enire forward rae curve are modelled. The connecion beween he forward raes and bond prices esablished in 6.2.1 and 6.2.2 hen gives he dynamics of he bond prices. For fixed τ, τ T, he P-dynamics of he forward raes are given by df,τ = α P,τd + σ,τdw P, 6.2.3 where W P is a Wiener process under P. For simpliciy W P is assumed o be 1-dimensional. The processes α P and σ are adaped o he filraion G = G T, which is he P- augmenaion of he naural filraion generaed by he Wiener process, i.e. G = G + N, where N is he σ-algebra generaed by all P-null ses and G + = σ { W P u,u }. Using Björk 24, Proposiion 2.5 we obain he following P-dynamics for he price process of a bond wih mauriy τ: τ dp,τ = r α P,udu + 1 τ 2 σ, udu P,τd 2 τ σ,udup,τdw P, τ T. 6.2.4

162 CHAPTER 6. A CONTINUOUS-TIME MODEL FOR REINVESTMENT RISK In addiion o he bonds, we assume ha he financial marke includes a savings accoun earning he shor rae r. The dynamics of he savings accoun are db = r B d, B = 1. 6.2.5 Remark 6.2.1 The exisence of a savings accoun wih drif r can be proven if we allow for invesmens in infiniely many differen bonds. In his case, invesing in a roll-over sraegy in jus-mauring bonds produces a value process, whose dynamics are given by 6.2.5, see Björk, Kabanov and Runggaldier 1997. For any G-adaped process h we may define a likelihood process Λ T by Λ = e 1 2Ê h2 u du+ê hudw P u. I is well known, see e.g. Musiela and Rukowski 1997 and Björk 24, ha if here exiss an h such ha E P [Λ T ] = 1 and, for all τ T, he Heah Jarrow Moron HJM drif condiion τ α P,τ = σ,τ σ,udu h 6.2.6 holds, hen here exiss a unique equivalen maringale measure Q given by dq dp = Λ T. 6.2.7 Here, i is imporan ha for fixed, 6.2.6 holds simulaneously for all τ, τ T. Recall ha an equivalen maringale measure fulfills hree requiremens: Firsly, i is equivalen o P. Secondly, all discouned price processes are maringales under he new measure and lasly, i is a probabiliy measure. If here exiss a unique equivalen maringale measure he model is arbirage free and complee, see e.g. Björk 24, Chaper 1. 6.2.2 Exending he sandard model o include reinvesmen risk We now exend he sandard model in Secion 6.2.1 o include reinvesmen risk. The idea is as follows: Assume ha a ime he bond marke can be described by he sandard model inroduced in Secion 6.2.1. A some predeermined imes new bonds are issued in he marke, such ha immediaely afer he issue he bond marke is given by a sandard model idenical o he one a ime. To inroduce reinvesmen risk in he model he iniial prices of he new bonds issued depend on a random variable independen of he observable bond prices. In order o exend he bond marke we inroduce he fixed ime horizon T, T T T. The inerpreaion of he ime horizons is as follows: T is he las ime where rading is

6.2. THE BOND MARKET MODEL 163 possible in he bond marke, i.e. T may be hough of as he end of he world, T is he las ime a which we allow paymens and T is he upper limi for he ime o mauriy of a bond raded in he marke. Hence, a any ime he ime o mauriy of he longes raded bond is less han or equal o T. Now define he sequence = T < T 1 <... < T n T of imes, where new bonds are issued in he marke. A ime T i new bonds are issued such ha all bonds wih ime o mauriy less han or equal o T are raded. To ensure ha a any ime, bonds are raded in he marke, we assume ha T max i=1,...,n T i T i 1 and T = T n + T. The illusraion in Figure 6.2.1 shows one possible ordering of T 1,...,T n, T and T in he case n = 3. Issue of new bonds wih mauriy τ T, T 1 + T] Issue of new bonds wih mauriy τ T 1 + T, T 2 + T] Terminal ime of paymen process End of he world Today Mauriy of longes bond raded oday Mauriy of longes bond issued a ime T 1 Mauriy of longes bond issued a ime T 2 T = T 1 T T 2 T 1 + T T = T 3 T 2 + T T = T + T Figure 6.2.1: Illusraion of T 1, T 2, T 3, T, T and T. For fixed we define i = sup { i n T i }, such ha T i is he las ime new bonds are issued prior o ime ime included. Thus, a ime he ime of mauriy, τ, of he bonds raded in he bond marke saisfies τ T i + T. For an illusraion of T i see Figure 6.2.2. Issue of new bonds wih mauriy τ T, T 1 + T] Fixed ime considered Issue of new bonds wih mauriy τ T 1 + T, T 2 + T] T 1 = T i T 2 Figure 6.2.2: Illusraion of T i. When he forward raes are defined, i.e. for τ T i + T, heir dynamics are given

164 CHAPTER 6. A CONTINUOUS-TIME MODEL FOR REINVESTMENT RISK by df,τ = α P,τd + σ,τdw P, where he processes α P and σ are F-adaped, and W P is a 1-dimensional Wiener process under P. As in Secion 6.2.1 he filraion G = G T is he P-augmenaion of he naural filraion generaed by he Wiener process. Noe ha G and he shor rae process r are defined unil ime T. T As noed above, forward raes for all mauriies are no defined a ime. They are inroduced a he imes of issue of new bonds. To model he iniial value of he new forward raes a ime T i, i {1,...,n}, we inroduce a sequence Y = Y i i=1,...,n of muually independen random variables wih disribuion funcions F P i i=1,...,n. Assume ha Y and W P are independen as discussed below his is no resricion. Here, Y i, which is revealed a ime T i, describes he uncerainy independen of he observed bond prices associaed wih he iniial prices of bonds issued a ime T i. The filraion H = H T is defined as he P-augmenaion of he naural filraion generaed by he random variables Y i i=1,...,n, i.e. H = H + N, where H + = σ{y i i=1,...,i }. We now assume ha F is he oal filraion generaed by he bond marke, such ha F = G H. For T i 1 + T < τ T i + T we model he forward raes by ft i,τ = ft i,t i 1 + T + τ T i 1 + T γ i udu, 6.2.8 where γ i is an F Ti -measurable funcion, i.e. each γu i is F T i -measurable for u T i 1 + T,T i + T]. The inerpreaion is ha he new forward raes inroduced a ime T i depend on he pas forward raes and some noise represened by he random variable Y i. The assumed independence beween W P and Y is no resricion, since we oherwise could define a vecor of random variables Ỹ = Ỹi i=1,...,n independen of W P and funcions γ 1,..., γ n, such ha ft i,τ given by 6.2.8 wih γ and Y replaced by γ and Ỹ, respecively, has he same disribuion as ft i,τ for T i 1 + T < τ T i + T. We noe from 6.2.8 ha he forward rae curve is coninuous a all imes. In addiion o F we consider he filraions F T i, i {,1,...,n}, given by F T i = F T i T = H T i F T. We immediaely noe ha F = F T = F. For i 1 he inerpreaion of he filraion F T i is ha he sequence Y j j=1,...,i is known a ime. When considering F Tn he enire vecor Y is known a ime, so he model is complee. Furhermore, we noe ha if we for i {,...,n 1} consider he filraion F T i and he ime inerval [,T i+1, hen he model is complee. In he exended bond marke we have ha for any i he oucome of he random variable Y i affecs he iniial prices of bonds issued a ime T i, and once i is realized i may affec he

6.2. THE BOND MARKET MODEL 165 drif and he volailiy of he forward raes. Thus, prior o ime T i, where Y i is realized, we are unable o rade in asses depending on he oucome of Y i. Hence, he vecor Y is unhedgeable. Once he forward raes are inroduced, he dynamics of he bonds are driven solely by W P. Thus, he model can be viewed as a series of complee models on [T i,t i+1 and a vecor of independen random variables realized a imes T i, i {1,...,n}. Remark 6.2.2 As noed above he model is complee when considering F Tn. Hence, coningen on he oucome of Y, all zero coupon bonds have unique prices a all imes even before hey are raded. Thus, a ime, < T n, where he uncondiional model is incomplee, we have a forward rae curve for all mauriies in he condiional model. Here, we noe ha all condiional forward rae curves, of which here may be infiniely many, are idenical unil ime T i + T. However, in he uncondiional model, he fuure values of Y i+1,...,y n are unknown. Hence, i is uncerain which of he condiional forward rae curves will urn ou in rerospec o have been he correc one when Y i+1,...,y n have been observed a ime T. Thus, we can inerpre he reinvesmen risk as he uncerainy associaed wih which of he condiional forward rae curves in rerospec has urned ou o have been he correc one. This in urn gives ha he magniude of he reinvesmen risk is relaed o how much he condiional forward rae curves differ. We now derive an expression for fuure forward raes, and in paricular fuure shor raes, in erms of he presen forward raes and he fuure uncerainy. For fixed τ we define { ī τ = inf i n T i + T τ such ha T īτ is he firs ime a bond wih mauriy τ is raded. Hence, he iniial ime a forward rae wih mauriy τ is defined. For u τ T i + T we have he well-known relaion fu,τ = f,τ + u }, dfs,τ. 6.2.9 However, as can be seen from he following proposiion, he relaionship beween he forward raes is in general more involved, since he fuure forward raes depend on he enry prices of bonds ye o be issued. In he proposiion, and hroughou he chaper, we inerpre l k=j as if l < j. Proposiion 6.2.3 For u τ we have he following relaion beween he forward raes: fu,τ = f,t i + T τ + ī τ 1 Tk+1 τ Tk+1 + T dfs,t k + T + γs k+1 ds k=i T k T k + T u + dfs,τ. 6.2.1 T īτ

166 CHAPTER 6. A CONTINUOUS-TIME MODEL FOR REINVESTMENT RISK In paricular i holds for he shor rae ha ī r u = f,t i + T u 1 Tk+1 u + dfs,t k + T u Tk+1 + T + γs k+1 ds k=i T k T k + T u + dfs,u. 6.2.11 T īu Proof of Proposiion 6.2.3: Formula 6.2.1 follows by repeaed use of relaions 6.2.8 and 6.2.9, whereas he expression for he shor rae in 6.2.11 is obained by seing τ = u in 6.2.1. A class of equivalen maringale measures In his secion we inroduce he considered class of equivalen maringale measures. Firs we deermine he unique Girsanov kernel wih respec o he Wiener process and define he equivalen maringale measure corresponding o a change of measure wih respec o he Wiener process only. We hen consider a change of measure wih respec o Y as well. Since Y is unhedgeable here exis infiniely many equivalen maringale measures. Here, we consider a class of measures wih paricular nice properies. Similarly o he sandard model we observe ha he exisence of an equivalen maringale measure depends on he exisence of an F-adaped process h, such ha for all τ T i + T he HJM drif condiion τ α P,τ = σ,τ σ,udu h is saisfied, and he likelihood process Λ = Λ T Λ = e 1 2Ê h2 u du+ê hudw P u defined by fulfills E P [Λ T ] = 1. Hence, if such an h exiss, we may define an equivalen maringale measure Q by dq dp = Λ T. 6.2.12 However, he equivalen maringale measure Q is no unique. In paricular we can define anoher likelihood process U = U T by U = i j=1 1 + u j Y j, for some funcions u j, j {1,...,n}, saisfying u j y > 1 for all y in he suppor of Y j and E P [u j Y j ] =. Here, and henceforh, l j=k is inerpreed as 1 if k > l. If

6.2. THE BOND MARKET MODEL 167 E Q [U T ] = 1 or equivalenly EP [Λ T U T ] = 1, we can define an equivalen maringale measure Q by dq = dq U T. 6.2.13 Girsanov s heorem gives ha for any Q of he form 6.2.13, he process W Q = W P h u du 6.2.14 is a Wiener process. Moreover, he disribuion funcion of Y i, i {1,...,n}, under Q, F Q i, is given by F Q i y = y 1 + u i zdfi P z. Here, we resric ourselves o he case, where h is G-adaped, such ha he measures considered are paricularly simple, since Y and W Q are independen under Q and he muual independence of he Y i s is preserved under Q. Using 6.2.14 we find ha he dynamics of he forward raes under Q are given by where we have defined df,τ = α Q,τd + σ,τdw Q, 6.2.15 τ α Q,τ = σ,τ σ, udu. 6.2.16 Now, Björk 24, Proposiion 2.5 gives he following bond price dynamics for τ T i + T under Q: dp,τ = r P,τd τ σ,udup,τdw Q. 6.2.17 Remark 6.2.4 I can be shown ha Q defined by 6.2.12 is he so-called minimal maringale measure for he exended model, i.e. he equivalen maringale measure which disurbs he srucure of he model as lile as possible, see Schweizer 1995. 6.2.3 Model consideraions In his secion we commen on he model specificaion in Secion 6.2.2. A any ime he prices of bonds wih mauriy τ, < τ T i + T, mus saisfy boh P,τ = e Êτ f,udu and P,τ = E Q [ e Êτ rudu F ]. 6.2.18

168 CHAPTER 6. A CONTINUOUS-TIME MODEL FOR REINVESTMENT RISK Furhermore insering 6.2.15 and 6.2.16 in Proposiion 6.2.3 gives he following expression for he shor rae a a ime u, u T i + T: r u = f,u + = f,u + u u dfs,u σs, u u s σs,vdv ds + u σs,udw Q s. Hence, since σ is F-adaped, we have ha in addiion o he presen informaion F and he fuure developmen of W Q he fuure shor rae a ime u, r u, may depend on he fuure oucome of Y j, j {i + 1,...,i u }. Thus, a a firs glance i seems as if he expecaion in 6.2.18 depends on he disribuion of Y j, j {i + 1,...,i τ } under Q, such ha he disribuion of Y j, j {i + 1,...,i Ti } under Q may be parly given a ime by + T 6.2.18. However, as we shall see below, his is no he case. Firs observe ha and τ u σs, u u s τ u Now, use ha provided τ s σs,vdv ds du = = 1 2 = 1 2 σs,udw Q s du = τ u τ τ τ τ τ 1 u 2 σs,vdv ds du 2 u s u 2 σs,vdv duds s u s τ 2 σs,vdv ds s s σs,ududw Q s. σs,udu is sufficienly inegrable i holds for fixed τ ha E Q [ e 1 2Êτ Êτ s σs,udu2 ds Êτ Êτ s σs,udu dw Q s F ] = 1, such ha E Q [ e Êτ rudu F ] = e Êτ f,udu. Hence, no undesirable resricions on he class of equivalen maringale measures occur when σ is F-adaped. Since σ is F-adaped he volailiy and hence he drif under Q of all forward raes a ime may depend on Y i if T i <. Hence, he iniial bond prices of he newly issued bonds a ime T i may influence he fuure prices of no only he newly issued bonds, bu also bonds wih shorer ime o mauriy. As an alernaive model consider he case where he dependence on Y i is resriced o he volailiy of he forward raes bond prices inroduced a ime T i and laer. In his case he fuure developmen of he bond prices depend on he informaion a he ime of issue and he fuure developmen of W Q only. As a las example we menion he quie resricive case where σ is G-adaped, such ha he informaion gahered from he issue of new bonds does no influence he volailiy of he forward raes bond prices.

6.2. THE BOND MARKET MODEL 169 6.2.4 Trading in he bond marke When rading in he exended bond marke inroduced in Secion 6.2.2 wo problems arise: Firsly, a any ime infiniely many bonds are raded in he bond marke and secondly, he bonds raded a ime depend on he ime considered. Regarding he firs problem we noe ha since he forward raes are driven by a 1-dimensional Wiener process only, i is sufficien if we a all imes are allowed o inves in wo asses, which are no linearly dependen. Furhermore, we noe ha he second problem may be overcome by considering a new se of price processes defined for all including he same informaion as he original price processes. Thus, boh problems are solved by considering he following wo asses: A savings accoun wih dynamics given by 6.2.5 and an asse, wih price process X, generaed by invesing 1 uni a ime and a ime, T, invesing in he longes bond raded in he marke. The dynamics of X are given by dx = X dp,t i + T P,T i + T, X = 1. Insering he bond price dynamics from 6.2.17 we ge he following Q-dynamics of X dx = r X d Ti + T σ,udux dw Q. 6.2.19 The price process X can be seen as he bes available approximaion o he value process generaed by a roll-over sraegy in bonds wih ime o mauriy T. Such a value process is usually referred o as a rolling-horizon bond, see Rukowski 1999. The idea of roll-over sraegies is closely relaed o he Musiela paramerizaion of forward raes, see Musiela 1993, where he forward raes are parameerized by ime o mauriy insead of ime of mauriy. In a coninuous-ime seing where bonds wih all mauriies are raded, a rolling-horizon bond requires invesmens in infiniely many differen bonds. However, here we only adjus he porfolio, when new bonds are issued, such ha X requires a finie number of bonds, n + 1, only. Following he ideas of Møller 21c we now define rading in he presence of paymen processes. Henceforh fix an arbirary equivalen maringale measure Q for he model B,X, F, ha is, we are working wih he probabiliy space Ω, F,Q and he filraions F T i i {,...,n}. We noe ha for all i, i {,...,n}, he discouned price process X is a Q, F T i -maringale. Here, and hroughou he chaper, we use an aserisk o denoe discouned price processes. Le X denoe he predicable quadraic variaion process for X associaed wih Q and F Tn, i.e. he unique predicable process such ha X 2 X is a Q, F Tn -maringale. For any i we now inroduce he space L 2 Q X, F T i of F T i -predicable processes ϑ saisfying [ ] T E Q ϑ 2 ud X u <. An F T i -rading sraegy is any process ϕ = ϑ,η, where ϑ L 2 Q X, F T i and η is F T i -adaped such ha he value process Vϕ defined by V ϕ = ϑ X + η B, T,

17 CHAPTER 6. A CONTINUOUS-TIME MODEL FOR REINVESTMENT RISK is RCLL Righ Coninuous wih Lef Limis and V ϕ L 2 Q for all [, T]. The pair ϕ = ϑ,η is inerpreed as he porfolio held a ime. Here, ϑ denoes he number of asses wih price process X, and η denoes he discouned deposi in he savings accoun. A paymen process is an F-adaped process A = A T describing he liabiliies of he seller of a conrac owards he buyer. Noe ha A is defined on [,T] only, such ha no paymens ake place afer ime T. Moreover, we noe ha since A is F-adaped, i is F T i -adaped for all i {1,...,n}. We assume ha A is square inegrable, i.e. ha E Q [A 2 ] < for all, and RCLL. For s T, we le A A s be he oal ougoes less incomes in he inerval s, ]. In he following we shall consider he discouned paymen process A defined by da = e Ê rudu da. The cos process associaed wih he pair ϕ,a is given by C ϕ = V ϕ ϑ u dx u + A. 6.2.2 Thus, he cos process is he discouned value of he porfolio reduced by discouned rading gains and added he oal discouned ougoes less incomes of he paymen process. The cos process is inerpreed as he seller s accumulaed discouned coss during [, ]. The cos process is square inegrable due o he square inegrabiliy of he paymen process A and he assumpions on he sraegy ϕ and X. Furhermore he cos process is adaped o he same filraion as he rading sraegy. We say ha a sraegy ϕ is F T i -self-financing for he paymen process A, if he cos process is consan Q-a.s. wih respec o F T i. In conras o he classical definiion of self-financing sraegies, we hus allow for exogenous deposis and wihdrawals as represened by A. The wo definiions of self-financing sraegies are equivalen if and only if he paymen process is consan Q-a.s. wih respec o he considered filraion. The inerpreaion of a selffinancing sraegy in he presence of paymen processes is ha all flucuaions of he value process are eiher rading gains/losses or due o he paymen process. A paymen process is called F T i -aainable, if here exiss an F T i -self-financing sraegy ϕ for A such ha V T ϕ = Q-a.s. wih respec o FT i. A paymen process is hus F T i -aainable, if invesing he iniial amoun C ϕ according o he rading sraegy ϕ leaves us wih a porfolio value of afer he selemen of all liabiliies. Hence, he unique arbirage free price in B,X, F T i of an F T i -aainable paymen process is C ϕ. A any ime, here is no difference beween receiving he fuure paymens of he F T i -aainable paymen process A and holding he porfolio ϕ and invesing according o he F T i -replicaing sraegy ϕ. Thus, a no arbirage argumen gives ha a any ime he price of fuure paymens from A in B,X, F T i mus be V ϕ. I can be shown ha he paymen process A is aainable if and only if he coningen claim H = A T wih mauriy T is classically aainable. If all coningen claims, and hence all paymen processes, are aainable, he model is called complee and oherwise i is called incomplee.

6.3. RISK-MINIMIZATION 171 6.3 Risk-minimizaion As noed above, an F T i -aainable paymen process has a unique arbirage free price C ϕ in B,X, F T i. However, for a non-aainable paymen process, we do no have a unique arbirage free price. Thus, for non-aainable processes, quanifying and conrolling he risk becomes imporan. Here, we apply he crierion of risk-minimizaion. We give a review of risk-minimizaion and deermine risk-minimizing sraegies in he presence of reinvesmen risk. 6.3.1 A review of risk-minimizaion for paymen processes In his secion we review he concep of risk-minimizaion inroduced by Föllmer and Sondermann 1986 for coningen claims, and furher developed in Møller 21c o cover paymen processes. For more deails we refer o Møller 21c. Throughou his secion, we consider a fixed bu arbirary filraion F T i, such ha we are working wih he filered probabiliy space Ω, F,Q, F T i. For a given paymen process A we define he F T i -risk process associaed wih ϕ by [ R T i ϕ = EQ C T ϕ C ϕ 2 F T i ], 6.3.1 where he cos process is defined in 6.2.2. Thus, he risk process is he condiional expecaion of he discouned squared fuure coss given he curren available informaion. We will use his quaniy o measure he risk associaed wih ϕ,a. An F T i -rading sraegy ϕ = ϑ,η is called F T i -risk-minimizing if for any [,T] i minimizes R T i ϕ over all F T i -rading sraegies wih he same value a ime T. Wih he inerpreaion of he cos process in mind, we noe ha V ϕ is he discouned value of he porfolio ϕ afer possible paymens a ime. In paricular, VT ϕ is he discouned value of he porfolio ϕ T upon selemen of all liabiliies. Thus, a naural resricion is o consider so-called -admissible sraegies which saisfy V Tϕ =, Q-a.s. The consrucion of risk-minimizing sraegies is based on he so-called Galchouk Kunia Waanabe decomposiion for maringales. Define he Q, F T i -maringale V Ti, by [ ] = E Q A T F T i, T. 6.3.2 V T i, The process V Ti,, which is known as he inrinsic value process wih respec o F T i, can now be uniquely decomposed using he Galchouk Kunia Waanabe decomposiion V T i, = V T i, + ϑ T i,a u dxu + L T i,a. 6.3.3 Here, L T i,a is a zero-mean square inegrable Q, F T i -maringale which is orhogonal o X, i.e. he process X L T i,a is a Q, F T i -maringale, and ϑ T i,a is an F T i -predicable process

172 CHAPTER 6. A CONTINUOUS-TIME MODEL FOR REINVESTMENT RISK in L 2 Q X, F T i. We noe ha if A is F T i -aainable, hen V T i, is he discouned unique arbirage free price in B,X, F T i a ime of he fuure paymens specified by he paymen process A and L Ti,A = Q-a.s. wih respec o F T i. The following heorem relaes he riskminimizing sraegy and he associaed risk process o he Galchouk Kunia Waanabe decomposiion. Theorem 6.3.1 Møller 21c There exiss a unique -admissible F T i -risk-minimizing sraegy ϕ T i = ϑ T i,η T i for A given by ϑ T i,ηt i = ϑ T i,a,v T i, A ϑt i,a X, T. The associaed F T i -risk process is given by R T i ϕt i = E Q [ L T i,a T L T i,a 2 F T i ]. 6.3.4 When deermining he risk-minimizing sraegy, we minimize over all admissible sraegies. This is in conras o many oher quadraic hedging crieria such as mean-variance indifference principles and mean-variance hedging, where only self-financing sraegies are allowed. For more deails on and a comparison of hese crieria see Møller 21b. As noed earlier, risk-minimizing sraegies are no self-financing for non-aainable paymen processes. However, hey can be shown o be mean-self-financing, i.e. he corresponding cos processes are Q-maringales wih respec o he considered filraion, see Møller 21c, Lemma A.4. Noe ha he risk-minimizing sraegy depends on he choice of equivalen maringale measure Q. In he lieraure, he minimal maringale measure has been applied for deermining risk-minimizing sraegies, since his, in he case where X is coninuous, essenially corresponds o he crierion of local risk-minimizaion, which is a crierion in erms of P, see Schweizer 21a. Remark 6.3.2 We noe ha since F T i and F T j coincide for maxt j,t i so do he inrinsic value processes. This is also inuiively clear, since for maxt j,t i he addiional informaion a ime in he larger of he filraions has been revealed, and hus is included in he σ-algebra in he smaller filraion as well. 6.3.2 Risk-minimizaion in he presence of reinvesmen risk From Secion 6.3.1 i follows ha if we deermine he Galchouk Kunia Waanabe decomposiion of V T i,, hen he unique -admissible F T i -risk-minimizing sraegy, i {,...,n}, is given by Theorem 6.3.1. However, since i is ofen difficul o deermine he Galchouk Kunia Waanabe decomposiion we ake a differen approach. We apply he main resul in Schweizer 1994 regarding risk-minimizaion under resriced informaion in order o obain he following heorem, allowing us o deermine he F T i -risk-minimizing sraegy in erms of he F Tn -risk-minimizing sraegy.

6.3. RISK-MINIMIZATION 173 Theorem 6.3.3 The unique -admissible F T i -risk-minimizing sraegy ϕ T i = ϑ T i,η T i for A given by [ ] ϑ T i,ηt i = E Q ϑ Tn F T i,v T i, A ϑ T i X, T, and he process L T i,a is given by L T i,a = V T i T i, V T i, + E Q [ ] ϑ Tn u dxu F T i ϑ T i u dx u. 6.3.5 Proof of Theorem 6.3.3: Since F T i F Tn for all and X is F-adaped wih XT being F T i T -measurable, Schweizer 1994, Theorem 3.1 gives he FT i -risk-minimizing sraegy for any F T i T -measurable coningen claim in erms of he FTn -risk-minimizing sraegy. Since boh X and A are F T i -adaped for all i, he resul for coningen claims carries over o he presen framework wih paymen processes. Using he fac ha X furhermore is F T i - predicable for all i, we have from Schweizer 1994, Secion 4, ha he F T i -risk-minimizing sraegy is given by [ ] ϑ T i = E Q ϑ Tn F T i and η T i = E Q [ V Tn, = E Q [ V Tn, = V T i, A ϑ T i F T i A ϑ T i X. X F T i ] ] A ϑ T i X Here, we have used ha ϑ T i is F T i -predicable, and ha A and X are F T i -adaped in he second equaliy, and ieraed expecaions in he hird. To derive an expression for L Ti,A we firs noe ha V Tn, T = V T i, T, see Remark 6.3.2. Insering he expressions from 6.3.3 and isolaing L T i,a T we obain T L T i,a T = V Tn, V T i, + ϑ T n u ϑ T i u dx u. Using ha L Ti,A is a Q, F T i -maringale we ge [ ] L T i,a = E Q L T i,a T F T i = E Q [ V Tn, V T i, + T ϑ T n u [ = V T i T i, V T i, + E Q ϑ Tn ϑt i u dx u F T i ] u dx u FT i ] ϑ T i u dx u. Here, we have used ha X is a maringale and ϑ T j lies in L 2 Q X, F T j such ha he inegral E Q [ T ϑ T j u dxu FT i ] = for all j. Furhermore, we have used ha F T i = F T i T i

174 CHAPTER 6. A CONTINUOUS-TIME MODEL FOR REINVESTMENT RISK o obain [ ] E Q V Tn, [ ] F T i = E Q V Tn, [ ] F T i T i = E Q V Tn, F T i T i = V T i T i,. From Theorem 6.3.3 we ge ha he F T i -risk-minimizing sraegy is he predicable condiional expecaion of he risk-minimizing sraegy in he complee model B,X, F Tn given he presen informaion, F T i. Thus, Theorem 6.3.3 provides an alernaive o Theorem 6.3.1 when deermining he F T i -risk-minimizing sraegy. The advanage of Theorem 6.3.3 is ha, since B,X, F Tn is a complee model, he F Tn -risk-minimizing sraegy for any paymen process coincides wih he F Tn -replicaing sraegy. Hence, using Theorem 6.3.3 o deermine he F T i -risk-minimizing sraegy requires he derivaion of a replicaing sraegy in a complee model and a condiional expecaion insead of he derivaion of a Galchouk Kunia Waanabe decomposiion for a non-aainable paymen process. Remark 6.3.4 Invesigaing he risk minimizing sraegies in Theorem 6.3.3 we observe ha since F T i and F T j coincide for maxt i,t j, hen he F T i - and F T j -risk-minimizing sraegies coincide for > maxt i,t j. The inuiive inerpreaion is ha he sraegies are based on he same informaion, and hence hey are idenical. Corollary 6.3.5 If we resric ourselves o paymen processes for which ϑ Tn is uniformly bounded hen L T i,a is given by L T i,a = V T i T i, V T i, + ϑ T i T i u ϑ T i u dxu. Proof of Corollary 6.3.5: Since ϑ Tn is uniformly bounded we may use sochasic Fubini, see Proer 24, Chaper IV, Theorem 64, o inerchange he order of inegraion in 6.3.5. The expression for L Ti,A in Corollary 6.3.5 has he following nice inerpreaion: A any ime he unhedgeable par of V Ti, consiss of wo erms. The firs erm is he difference beween he iniial deposi given he informaion a ime and he curren informaion, respecively, whereas he second erm is he difference beween he rading gains generaed by he risk-minimizing sraegy given he presen informaion regarding Y and he F T i - risk-minimizing sraegy. In paricular we noe from Corollary 6.3.5 ha for < T i+1, we have ha L T i,a =. This is also inuiively clear, since no addiional informaion concerning he Y j s has been revealed. Corollary 6.3.6 In he case where ϑ Tn is uniformly bounded we have he following alernaive expression for he process L T i,a : L T i,a = i j=i+1 V T j, T j E Q [ V Tn, T j ] T F j 1 T j.

6.3. RISK-MINIMIZATION 175 This leads o he following expression for he F T i -risk process associaed wih ϕ T i T R T i ϕt i = E Q [ ] 2 V T j, T j E Q V Tn, T T j F j 1 T j. 6.3.6 j=i +1 FT i Proof of Corollary 6.3.6: From Corollary 6.3.5 we have he following expression for L T i,a L T i,a = V T i T i, V T i, + ϑ T i T i u ϑ T i u dxu. Now, wrie V T i T i, V T i, and ϑ T i T i u ϑ T i u as elescoping sums and use ha, as noed above, he F T i - and F T j -risk-minimizing sraegies coincide for > maxt i,t j o obain L T i,a = = i j=i+1 i j=i+1 V T j, V T j 1, + ϑ T j u ϑ T j 1 u dxu V T j, V T Tj j 1, + ϑ T j u ϑ T j 1 u dxu. Using ieraed expecaions in order o express all quaniies as expecaions of he respecive F Tn -quaniies, we ge L T i,a = i j=i+1 Tj + The resul now follows from L T i,a = = = = i j=i+1 Tj + i j=i+1 i j=i+1 i j=i+1 [ E Q V Tn, F T j E Q [ ϑ Tn u E Q [ V Tn, E Q [ ϑ Tn u T j ] ] F T j E Q [ V Tn, ] [ F T j u E Q ϑ Tn u E Q [ V Tn, ] [ F T j T j E Q ϑ Tn u [ Tj E Q V Tn, + ϑ Tn [ E Q V Tn, T T j F j V T j, T j T j ] E Q [ V Tn, T j ] F T j 1 T j F T j 1 T j ] u dxu FT j T j E Q [ V Tn, T j F T j 1 T j ]. ] dxu ] F T j 1 F T j 1 u ] dxu. [ Tj E Q V Tn, + ϑ Tn ] T F i 1 T j u dxu FT j 1 T j Here, we have used E Q [V Tn, F T j ] = EQ [V Tn, F T j ] for < T j+1 and E Q [ϑ Tn u FT j u ] = E Q [ϑ Tn u FT j ] for u < T j+1 in he firs equaliy. The uniform boundedness of ϑ Tn : ]

176 CHAPTER 6. A CONTINUOUS-TIME MODEL FOR REINVESTMENT RISK allows us o use sochasic Fubini, see Proer 24, Chaper IV, Theorem 64, o inerchange he order of inegraion in he second equaliy. Furhermore we have used ha ϕ Tn is self-financing in he hird equaliy and he definiion of V T j, in he las equaliy. The expression for he F T i -risk process associaed wih ϕ T i in 6.3.6 is now obained by insering he expression for L T i,a in 6.3.4. From Corollary 6.3.6 we observe ha L T i,a, which measures he deposis or wihdrawals o/from he risk-minimizing porfolio in addiion o hose generaed by he paymen process, only changes value a imes T j, j > i. Hence, he risk-minimizing sraegy is self-financing beween he imes of issue. A ime T j he informaion revealed by he issued bonds, i.e. he observed value of Y j, affecs he weighs given o he differen oucomes of Y, and hence i leads o a change in L T i,a. 6.3.3 F-risk-minimizing sraegies We now derive he F-risk-minimizing sraegy for a general paymen process of he form da = A d1 + a d + A T d1 T. Here, A is a consan, whereas a is F -measurable for all, and A T is F T -measurable. We noe ha he paymen process is F- and F Tn -adaped. In order o derive he riskminimizing sraegy we consider he discouned paymen process A = A + e Ês rudu da s = A + e Ês rudu a s ds + e ÊT rsds A T 1 =T. Since he model B,X, F Tn is complee, we have he following expression for A : s T A = A + F Tn,s + ϑ Tn,s u dxu ds + F Tn, T + ϑ Tn, T u dxu 1 =T, where ϑ Tn,s s T and ϑ Tn, T are he replicaing sraegies in B,X, F Tn for a s s T and A T, respecively, and we for s T have defined [ ] [ ] F Tn,s = E Q e Ês rudu a s F T n and F Tn, T = E Q e ÊT r udu A T F T n. Hence, F Tn,s and F Tn, T are he unique arbirage free prices a ime in he model B,X, F Tn for he claims a s and A T, respecively. Now use ha B,X, F Tn is complee o obain V Tn, = E Q [ A T = E Q [A + ] F Tn T F Tn,s + s T [ T = A + F Tn,s ds + F Tn, T + E Q ϑ Tn,s u dxu ds + F Tn, T s T + T ϑ Tn,s u dx uds + ϑ Tn, T u dxu ϑ Tn, T u F Tn dxu FTn ] ].

6.3. RISK-MINIMIZATION 177 Here, we resric ourselves o paymen processes for which ϑ Tn,s s T are uniformly bounded, such ha we may use sochasic Fubini, see Proer 24, Chaper IV, Theorem 64, o inerchange he order of inegraion above. Hence, we obain he following Galchouk Kunia Waanabe decomposiion of V Tn, : V Tn, = V Tn, + E Q [ T T + u = V Tn, = V Tn, + T ϑ Tn,s u u ds dx u + ϑ Tn,s u ds dx u + ϑ Tn,A u dx u, T ϑ Tn, T u ϑ Tn, T u dxu dxu FTn ] where T ϑu Tn,A = u ϑ Tn,s u ds + ϑ Tn, T u. 6.3.7 Recall ha ϑ Tn = ϑ Tn,A. The F-risk-minimizing sraegy ϕ = ϑ,η and he associaed risk process are now given by insering 6.3.7 in Theorem 6.3.3. F-risk-minimizing sraegies when Y has finie suppor Consider he case where Y i, i {1,...,n}, has finie suppor, hence Y i {y1 i,...,yi m i }. Le K = n i=1 m i denoe he possible number of oucomes of he vecor Y. To simplify he expression for he risk-minimizing sraegies we inroduce he noaion = E Q [ ] 1 Y1,...,Y n=δ k F = E Q [ ] 1 Y1,...,Y n=δ k H, M δ k where δ 1,...,δ K are he possible oucomes of he vecor Y 1,...,Y n. Here, we have used he Q-independence beween Y and W Q in he second equaliy. If we furher inroduce he noaion ϑ δ k and V δ k, o denoe, respecively, he replicaing sraegy and he inrinsic value process given Y = δ k, hen he F-risk-minimizing sraegy is given by ϑ,η = K k=1 M δ k ϑδ k, K k=1 M δ k V δ k, A ϑ X, T. 6.3.8

178 CHAPTER 6. A CONTINUOUS-TIME MODEL FOR REINVESTMENT RISK In his case he expression for he process L,A in Corollary 6.3.6 simplifies o L,A = = = = = i j=1 i j=1 i j=1 i V T j, T j E Q [ V Tn, T j [ E Q V Tn, T T j F j K k=1 K j=1 k=1 K k=1 T j ] M δ k T j V δ k, T j ] T F j 1 T j E Q [ V Tn, T j K k=1 V δ k, T j M δ k T j M δ k T j 1 V δ k, u dm δ k u. M δ k T j 1 V δ k, T j ] T F j 1 Here, we have used ha he probabiliies change a imes T i, i = 1,...,n, only, in he las equaion, and ha we are allowed o inerchange summaion and inegraion. Hence, in he case where Y has finie suppor we have he following simple Galchouk Kunia Waanabe decomposiion: V, = K k=1 M δ k V δ k, + K k=1 M δ k u ϑδ k u dx u + K k=1 T j V δ k, u dm δ k u. 6.3.9 Example 6.3.7 Consider he case where Y = Y 1 follows a binomial disribuion, i.e. Y {,1} wih 1 PY = = PY = 1 = p, p,1. Now he goal is o deermine he risk-minimizing sraegy under he minimal maringale measure, Q. In his simple example wih jus wo possible oucomes of Y we have δ i = i 1, i {1,2}. We do no specify he paymen process, he forward rae dynamics and γ. Here, he quaniies M δ k simplify o M δ [ ] 1 = E Q 1Y =δ1 F = 1 p1 <T1 + 1 Y 1 T1 T, 6.3.1 M δ [ ] 2 = E Q 1Y =δ2 F = p1 <T1 + Y 1 T1 T, 6.3.11 where we have used ha he disribuion of Y is unaffeced by he change o he minimal maringale measure. Furhermore, he inrinsic value process V, given by V, = 1 <T1 1 pv δ 1, + pv δ 2, + 1 T1 T 1 Y V δ 1, + Y V δ 2,. 6.3.12 Insering 6.3.1 and 6.3.11 ino 6.3.8 gives he following risk-minimizing sraegy ϑ = 1 T1 1 pϑ δ 1 + pϑ δ 2 + 1 T1 < T 1 Y ϑ δ 1 + Y ϑ δ 2, 6.3.13 and η = V, A 1 T1 1 pϑ δ 1 + pϑ δ 2 + 1 T1 < T 1 Y ϑ δ 1 + Y ϑ δ 2 X,

6.3. RISK-MINIMIZATION 179 where V, is given by 6.3.12. Now, insering 6.3.1 6.3.13 in 6.3.9 gives he following Galchouk Kunia Waanabe decomposiion V, = 1 pv δ 1, + pv δ 2, +1 T1 <u T + 1 Y ϑ δ 1 u + Y ϑδ 2 u 1 u T1 1 pϑ δ 1 u + pϑ δ 2 u dx u + 1 T 1 TY p V δ 2, T 1 V δ 1, T 1. Example 6.3.8 We now exend Example 6.3.7 by specifying he paymen process, he forward rae dynamics and he funcion γ. Hence, we sill assume ha Y = Y 1 {,1} wih 1 PY = = PY = 1 = p, p,1. Consider a company, which a ime wans o hedge a claim of 1 a ime T, i.e. A T = 1. Wihou loss of generaliy we assume T = T + T 1. To model he dynamics of he forward raes, we le σ be given by σ,τ = ce aτ, for some posiive consans c and a. Here, as in pracice, flucuaions of he forward raes dampen exponenially as a funcion of ime o mauriy. Using ha τ τ σ,udu = ce au du = c 1 e aτ, a we obain he following forward rae dynamics under Q for τ T i + T: df,τ = c2 a e aτ 1 e aτ d + ce aτ dw Q. 6.3.14 To model he exension of he forward rae curve a ime T 1 we assume γ is given by γ s = 1 T T k 1Y + k 2 1 Y, 6.3.15 for some consans k 1 and k 2. Thus, he forward rae curve is coninued by a sraigh line wih slope k 1 /T T or k 2 /T T. In order o obain he F-risk-minimizing sraegy under Q we now consider he complee model B,S, F T 1, where Y is known. Proposiion 6.2.3 gives he following expression for he shor rae r = { f, + dfs,, T, f, T + T 1 dfs, T + T γ sds + T 1 dfs,, > T. 6.3.16 We noe from 6.3.16 ha r depends on Y for > T. Insering 6.3.14 and 6.3.15 in 6.3.16 gives { m + r = ce a s dws Q, T, m,y + T 1 ce a s dws Q, > T, where we have defined m = f, + c2 2a 2 1 e a 2, T,

18 CHAPTER 6. A CONTINUOUS-TIME MODEL FOR REINVESTMENT RISK and m,y = f, T + c2 2a 2 1 e a T 2 1 e a T 2 T 1 + 1 e a T 2 1 + T T T k 1Y + k 2 1 Y + T1 ce a T s dw Q s, > T. Using Iô s formula we now obain he shor rae dynamics { φ ar dr = d + cdw Q, T, φ,y ar d + cdw Q, > T, where φ = am + m and φ,y = am,y + m,y. 6.3.17 Hence, given Y he shor rae follows an exended Vasiček model under Q. The resul is well-known for T, where r is independen of Y, see e.g. Musiela and Rukowski 1997. From 6.3.17 we observe ha he drif and he squared diffusion boh are affine in r, such ha an exended Vasiček model for he shor rae leads o an affine erm srucure, see e.g. Björk 24, Proposiion 22.2. Thus, in he condiional model we have he following expression for he unique arbirage free price a ime for 1 uni a ime T: wih A,T,Y and B,T given by B,T = 1 a A,T,Y = T P δ Y +1,T = expa,t,y B,Tr, 6.3.18 1 e at, 6.3.19 T 1 T 2 c2 B 2 s,tds φsbs,tds φs,y Bs,Tds. T Even hough B in general is allowed o depend on Y, i is no he case here, so we have omied Y in he noaion for B. Noe ha we have used he noaion P δ Y +1,T even hough he bond is no raded. Applying Iô s formula o 6.3.18 and using he differenial equaions for A and B from Björk 24, Proposiion 22.2, we obain he following Q-dynamics for he price process P δ Y +1,T: dp δ Y +1,T = r P δ Y +1,Td cb,tp δ Y +1,TdW Q. 6.3.2 Combining 6.2.19, 6.3.19 and 6.3.2 gives dx = r X d cb,t i + TX dw Q. 6.3.21 Noe ha a ime, > T, we have ha X depends on Y hrough he shor rae process. However, if we consider he discouned price process, X, he dynamics are given by dx = cb,t i + TX dw Q,

6.3. RISK-MINIMIZATION 181 such ha X is independen of Y. Comparing 6.3.2 and 6.3.21 we find ha given Y he replicaing sraegy is given by ϑ δ Y +1 = B,TP δ Y +1,T B,T i + TX = B,TP δy +1,,T B,T i + TX. 6.3.22 Since V δ Y +1, = e Ê rudu P δ Y +1,T = P δ Y +1,,T, we have V, = 1 <T1 1 pp δ1,,t + pp δ2,,t + 1 T1 T 1 Y P δ1,,t + Y P δ2,,t. 6.3.23 Insering 6.3.22 and 6.3.23 in he resuls from Example 6.3.7 gives he following riskminimizing sraegy: ϑ = 1 X 1 T1 + 1 T1 < T B,T B,T i + T 1 pp δ1,,t + pp δ2,,t 1 Y P δ1,,t + Y P,T δ2,, 6.3.24 and 1 pp δ1,,t + pp δ2,,t 1 pϑ δ 1 + pϑ δ 2 X, < < T 1, η = 1 Y P δ1,,t + Y P δ2,,t 1 pϑ δ 1 + pϑ δ 2 X, = T 1,, T 1 < T. 6.3.25 Invesigaing 6.3.24 and 6.3.25 we noe ha for > T 1 he risk-minimizing sraegy consiss of P δ Y +1,,T/X unis of he risky asse, which a his ime corresponds o invesing in bonds wih mauriy T. Hence, holding P δ Y +1,,T/X unis of he risky asse is equivalen o holding one bond wih mauriy T, which in urn is he replicaing sraegy. The Galchouk Kunia Waanabe decomposiion is given by V, = 1 pp δ1,,t + pp δ2,,t 1 Bu,T + Xu 1 u T1 Bu,T iu + T 1 pp δ1, u,t + pp δ2, u,t +1 T1 <u T 1 Y P δ1, u,t + Y P δ2, u,t dxu + 1 T1 TY p P δ2, T 1,T P δ1, T 1,T.

182 CHAPTER 6. A CONTINUOUS-TIME MODEL FOR REINVESTMENT RISK 6.4 A pracical implemenaion of he model In his secion we discuss a possible implemenaion of he model. Wihou loss of generaliy we assume ha new bonds are issued a ime, such ha a ime he ime o mauriy of he longes raded bond is T. A ime we observe he bond prices in he marke. Assuming he forward raes are given by a parameric model, wih parameer θ Θ, we esimae he value of θ, say θ, which gives he bes correspondence wih he observed bond prices. For a possible paramerizaion we refer o Svensson 1995, who considers an exension of he so-called Nelson Siegel paramerizaion; see Nelson and Siegel 1987 for he original Nelson Siegel paramerizaion. Now le he iniial forward rae curve a ime, f,τ τ T, be given by he esimaed forward rae curve f θ,τ τ T. In addiion o he iniial forward rae curve we, for laer purpose, use θ o esimae f θ,t 1 + T. Given a model for he forward rae dynamics we simulae he forward rae vecor ft 1,τ T1 τ T and he poin f θ T 1,T 1 + T. The forward rae ft 1,T 1 + T is now drawn from a disribuion esimaed from hisorical daa wih mean f θ T 1,T 1 + T. To obain he forward rae curve a ime T 1 afer he issue of new bonds we combine ft 1, T and ft 1,T 1 + T by a mehod giving a smooh exension of he forward rae curve. One possibiliy is he mehod of cubic splines, see e.g. Press, Flannery, Teukolsky and Veerling 1986. Using he parameric forward rae model, we now esimae he parameer θ 1, which gives he bes correspondence wih he forward rae curve a ime T 1. The esimaed parameer is only used o esimae f θ 1 T 1,T 2 + T. Saring from he forward rae curve a ime T 1, ft 1,T 1 + τ, and τ T fθ 1 T 1,T 2 + T he procedure above is repeaed o deermine he forward rae curve a ime T 2, and in urn he forward rae curve a any fuure ime. When implemening he model as described above we have he sandard problems of esimaing he iniial forward rae curve from he observed bond prices and modelling he forward raes. In addiion we have he model relaed problem of deermining he disribuion of ft i,t i + T, i {1,...,n}. We noe, however, ha we avoid a direc modelling of γ i. Insead γ i is given indirecly by f θ i 1 T i,t i + T, he esimaed disribuion wih mean f θ i 1 T i,t i + T and he chosen smoohing mehod.

Chaper 7 Valuaion and Hedging of Uni-Linked Life Insurance Conracs Subjec o Reinvesmen and Moraliy Risks This chaper is an adaped version of Dahl 25d This paper considers he problem of valuaing and hedging a porfolio of uni-linked life insurance conracs, which are subjec o several hedgeable and unhedgeable sources of risk. In Chaper 4 we consider a porfolio of life insurance conracs wih deerminisic payoffs which are subjec o hedgeable ineres rae risk as well as unhedgeable sysemaic and unsysemaic moraliy risk. Here, we exend his seup by considering a porfolio of uni-linked life insurance conracs, which are subjec o boh hedgeable and unhedgeable financial risk, as well as unhedgeable sysemaic and unsysemaic moraliy risk. The unhedgeable financial risk is he reinvesmen risk, described in Chaper 6, which is presen in bond markes, where only bonds wih a limied ime o mauriy are raded. In addiion o he bond marke, he financial marke consiss of a sock, whose price process is correlaed o he bond prices. To model he underlying moraliy inensiy we apply a ime-inhomogeneous Cox Ingersoll Ross model, as proposed in Chaper 4. Wihin he combined model, we sudy a general se of equivalen maringale measures and deermine marke reserves by applying hese measures. As an alernaive o he marke reserves we derive mean-variance indifference prices. To quanify and conrol he risk of he insurance company, we derive risk-minimizing sraegies and he opimal sraegies associaed wih he mean-variance indifference prices. 183

184 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS 7.1 Inroducion In Chaper 4 we derive opimal hedging sraegies and marke values for sandard life insurance conracs wih fixed paymens, in he presence of sysemaic and unsysemaic moraliy risk. Here, we exend his work by considering uni-linked life insurance conracs, which in addiion o hedgeable financial risk and unhedgeable sysemaic and unsysemaic moraliy risk, are subjec o an unhedgeable financial risk. This unhedgeable financial risk is he so-called reinvesmen risk presen in bond markes, where only bonds wih a limied ime o mauriy are raded. The exension o uni-linked life-insurance conracs wihou reinvesmen risk is rivial, since he financial marke remains complee afer he addiion of a sock. Hence, he main conribuion of his paper is he inclusion of he unhedgeable reinvesmen risk, such ha we obain a more refined model for he uncerainy associaed wih uni-linked life insurance conracs. In order o model he reinvesmen risk we apply he model proposed in Chaper 6. Hence, we iniially consider a sandard coninuous-ime bond marke model wih some fixed finie ime horizon, which is smaller han he ime horizon of he considered paymen process. A fixed imes new bonds are issued in he marke, such ha we immediaely afer he issue of new bonds consider a sandard model similar o he iniial one. The enry prices of he new bonds depend on he prices of exising bonds and some independen random variable, whose oucome deermines he exension of he forward rae curve. In addiion o he bonds, he financial marke consiss of a sock, which is correlaed o he bonds. As in Chaper 4 we model he moraliy inensiy by a ime-inhomogeneous Cox Ingersoll Ross CIR model, such ha we obain an affine moraliy srucure, see Chaper 3. Wihin his seing, we apply financial heory for pricing and hedging he paymen process generaed by a porfolio of uni-linked life insurance conracs. We sudy a fairly general se of equivalen maringale measures for he model and derive marke reserves, which depend on he marke s aiude owards sysemaic and unsysemaic moraliy risk as well as reinvesmen risk. Similarly o Chaper 4 we derive risk-minimizing sraegies and mean-variance indifference prices and opimal hedging sraegies. The derivaion of risk-minimizing sraegies consiss of a wo-sep procedure. Firs we disregard he reinvesmen risk and derive he risk-minimizing sraegies in he case of a complee financial marke. The sraegies obained here are essenially idenical o he ones in Chaper 4. The second sep is o apply he resul of Schweizer 1994 for risk-minimizaion under resriced informaion o derive he risk-minimizing sraegies in he case, where we also consider reinvesmen risk. This wo-sep procedure has also been applied in Chaper 6. The paper is organized as follows: In Secion 7.2 we inroduce he various sub-models. These include he financial marke and he moraliy and insurance porfolio. Secion 7.3 inroduces he combined model, he paymen process, marke reserves and he financial erminology necessary o define rading sraegies in he presen model. Risk-minimizing sraegies are obained in Secion 7.4, and in Secion 7.5 we derive mean-variance indifference prices and opimal hedging sraegies for a porfolio of uni-linked pure endowmens. Proofs and calculaions of some echnical resuls can be found in Secion 7.6.

7.2. THE SUB-MODELS 185 7.2 The sub-models Le T be a fixed ime horizon and Ω, F,P a probabiliy space wih a filraion F = F T saisfying he usual condiions of righ-coninuiy and compleeness. In addiion o he filraion F, which conains all available informaion, we shall consider several sub-filraions. 7.2.1 The financial marke The model for he financial marke consiss of he bond marke model in Chaper 6 wih he inclusion of a sock. Here, we firs give a brief inroducion o he model. A deailed review of he bond marke model is hen given in Secion 7.2.1. Consider a financial marke consising of hree raded asses: A savings accoun wih price process B and wo risky asses wih price processes Z and S. Here, Z is a price process generaed by invesing in bonds see Secion 7.2.1 for more deails, and S is he price process for a sock. The P-dynamics of he raded asses are db = rbd, B = 1, 7.2.1 dz = r + h f σ z Zd σ z ZdW f, Z = 1, 7.2.2 ds = r + ρ s Sd σ s SdW f + β s SdW s, S >, 7.2.3 where ρ s = σ s h f β s h s. 7.2.4 Here, W f and W s T T are independen Wiener processes under P on he inerval [, T]. In 7.2.2 7.2.4 he process r is he sochasic rae of ineres. The dynamics of r are assumed o be driven by W f only. Furhermore we inroduce he noaion X = Z,S r, where a r denoes he vecor a ransposed, and le G x = G x T be he filraion generaed by X, i.e. by W f and W s. In addiion o he uncerainy generaed by he raded asses, we observe a sequence Y = Y i i=1,...,m of muually independen random variables independen of W f and W s. The observaion imes of Y are given by he sequence = T < T 1 <... < T m T, where T i is he observaion ime for Y i and T, T T, is he erminal ime of he considered paymen process, see Secion 7.3.2. Le G y be he naural filraion generaed by Y, i.e. G y = σ{y i i=1,...,m,t i }. The oucome of he Y i s influences he fuure values of he sochasic shor rae r hrough he coefficien funcions in he dynamics of r. We are now in a posiion o define G, which is he oal filraion generaed by he financial marke, i.e. G = G y G x. In 7.2.2 7.2.4 he processes σ z, σ s and β s are G-adaped, whereas h f and h s are G x -adaped. In addiion o he filraions above we shall consider he enlarged filraions G T i = G T i = G T Gy T i T, i {,...,m}.

186 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS We immediaely noe ha G = G T = G. For i {1,...,m} he inerpreaion of he filraion G T i is ha Y j j=1,...,i are known a ime. Furhermore, we noe ha if we consider he filraion G T i and he ime inerval [,T i+1 hen he financial marke is complee. Hence in paricular, when considering G Tm he financial marke is complee. Remark 7.2.1 The fac ha h f and h s are G x -adaped is an assumpion, which we impose in order o simplify he calculaions in Secion 7.5. The bond marke Here, we review he bond marke model including reinvesmen risk proposed in Chaper 6. Le P,τ denoe he price a ime of a zero coupon bond mauring a ime τ. To avoid arbirage we assume P,τ is sricly posiive and P, = 1 for all. An imporan quaniy when modelling bond prices is he insananeous forward rae wih mauriy dae τ conraced a ime defined by or, saed differenly, f,τ = log P,τ, 7.2.5 τ P,τ = e Êτ f,udu. 7.2.6 The forward rae f,τ can be inerpreed as he riskfree ineres rae, conraced a ime over he infiniesimal inerval [τ,τ + dτ. The shor rae process r T is defined as r = f,. Now inroduce wo addiional fixed ime horizons T and T, where T T T. Here, T, T and T, respecively, describe he upper limi for he ime o mauriy of a bond raded in he marke, he erminal ime of he considered paymen process and he las ime where rading is possible in he bond marke, i.e. he end of he world. Thus, a any ime he ime o mauriy of he longes raded bond is less han or equal o T. The sequence = T < T 1 <... < T m T describes he imes, where new bonds are issued in he marke. A ime T i new bonds are issued such ha all bonds wih ime o mauriy less han or equal o T are raded. To ensure ha a all imes, bonds are raded in he marke, we assume ha T max i=1,...,m T i T i 1 and T = T m + T. The illusraion in Figure 7.2.1 shows one possible ordering of T 1,...,T m, T and T in he case m = 3. For fixed we define i = sup { i m T i }, 7.2.7 such ha T i is he las ime new bonds are issued prior o ime ime included. Thus, he ime of mauriy, τ, of he bonds raded in he bond marke a ime saisfy τ T i + T.

7.2. THE SUB-MODELS 187 Issue of new bonds wih mauriy τ T, T 1 + T] Issue of new bonds wih mauriy τ T 1 + T, T 2 + T] Terminal ime of paymen process End of he world Today Mauriy of longes bond raded oday Mauriy of longes bond issued a ime T 1 Mauriy of longes bond issued a ime T 2 T = T 1 T T 2 T 1 + T T = T 3 T 2 + T T = T + T Figure 7.2.1: Illusraion of T 1, T 2, T 3, T, T and T. Since i is inconvenien o model he bond prices direcly, we model he forward rae dynamics, as proposed in Heah e al. 1992. The connecion beween forward raes and bond prices esablished in 7.2.5 and 7.2.6 hen gives he dynamics of he bond prices. For τ T i + T we assume ha he forward rae dynamics under P are given by τ df,τ = σ f,τ σ f,udu h f d + σ f,τdw f, 7.2.8 where σ f is G-adaped. As noed above forward raes for all mauriies are no defined a ime. They are inroduced a he imes of issue of new bonds. To model he iniial value of he forward raes inroduced a ime T i, i {1,...,m}, we assume ha for T i 1 + T < τ T i + T i holds ha ft i,τ = ft i,t i 1 + T + τ T i 1 + T γ i udu. 7.2.9 Here, γ i is an GT i -measurable funcion, i.e each γ i u is GT i -measurable for u T i 1 + T,T i + T], and Y = Y i i=1,...,m is sequence of muually independen random variables independen of W f wih disribuion funcions Fi P i=1,...,m. Hence, Y i describes he unhedgeable uncerainy associaed wih he iniial prices of bonds issued a ime T i. Using Björk 24, Proposiion 2.5 we obain he following P-dynamics for he price process of a bond wih mauriy τ, τ T i + T: dp,τ = r + h f σ p,τ P,τd σ p,τp,τdw f, 7.2.1 where we have defined σ p,τ = τ σ f,udu. When rading in he bond marke i is sufficien o consider invesmens in a savings accoun wih dynamics 7.2.1, and an asse wih price process Z generaed by invesing 1 uni a ime and a imes T invesing in he longes bond raded in he marke. The dynamics of Z are given by dz = Z dp,t i + T, Z = 1. P,T i + T

188 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS Insering he bond price dynamics from 7.2.1 and defining we ge he dynamics in 7.2.2. σ z = σ p,t i + T, 7.2.2 Modelling he moraliy In order o model he uncerainy associaed wih he fuure moraliy inensiies we use he model proposed in Chaper 4. Le µ = µ x x be a given iniial curve for he moraliy inensiy a all ages. I is assumed ha µ x is coninuously differeniable as a funcion of x. Here, and in he following we neglec he gender aspec. For an individual aged x a ime, he fuure moraliy inensiy is viewed as a sochasic process µx = µx, T wih he propery ha µx, = µ x. In principle, one can view µ = µx x as an infiniely dimensional process. We model changes in he moraliy inensiy via a sricly posiive infinie dimensional process ζ = ζx, x, [, T] wih he propery ha ζx, = 1 for all x. The filraion I = I [, T] is he naural filraion of he underlying process ζ. The moraliy inensiy process is hen modelled via µx, = µ x + ζx,. 7.2.11 Thus, ζx, describes he relaive change in he moraliy inensiy from ime o for a person of age x +. The rue survival probabiliy is defined by [ Sx,,T = E P e ÊT and he relaed maringale is given by µx,τdτ ] I, 7.2.12 S M x,,t = E P [ e ÊT µx,τdτ I ] = e Ê µx,τdτ Sx,,T. 7.2.13 In general, we can consider survival probabiliies under various equivalen probabiliy measures. This is discussed in more deail in Secion 7.3.1. The process ζx is modelled via a so-called ime-inhomogeneous CIR model dζx, = γ ζ x, δ ζ x,ζx, d + σ ζ x, ζx,dw µ, 7.2.14 where γ ζ, δ ζ and σ ζ are known funcions and W µ is a Wiener process under P on he inerval [, T]. Here, and in he following, we assume ha 2γ ζ x, σ ζ x, 2, such ha ζ is sricly posiive, see Maghsoodi 1996. I now follows via Iô s formula ha dµx, = γ µ x, δ µ x,µx, d + σ µ x, µx,dw µ, 7.2.15

7.2. THE SUB-MODELS 189 where γ µ x, = γ ζ x,µ x +, 7.2.16 δ µ x, = δ ζ x, d d µ x + µ x +, 7.2.17 σ µ x, = σ ζ x, µ x +. 7.2.18 This shows ha µ also follows an ime-inhomogeneous CIR model. Furhermore µ is sricly posiive as well. Since we have an affine moraliy srucure, see Theorem 3.3.6, he survival probabiliy is given by where Sx,,T = e Aµ x,,t B µ x,,tµx,, Bµ x,,t = δ µ x,b µ x,,t + 1 2 σµ x, 2 B µ x,,t 2 1, 7.2.19 Aµ x,,t = γ µ x,b µ x,,t, 7.2.2 wih B µ x,t,t = and A µ x,t,t =. In his case he forward moraliy inensiies are given by f µ x,,t = T log Sx,,T = µx, T Bµ x,,t T Aµ x,,t. 7.2.21 7.2.3 The insurance porfolio Consider an insurance porfolio consising of n insured lives of he same age x. We assume ha he individual remaining lifeimes a ime of he insured are described by a sequence D 1,...,D n of idenically disribued non-negaive random variables. Moreover, we assume ha PD 1 > I T = e Ê µx,sds, T, and ha he censored lifeimes Di c = D i1 Di T + T1 Di > T, i = 1,...,n, are i.i.d. given I T. Thus, given he developmen of he underlying process ζx, he moraliy inensiy a ime s is µx,s. Now define a couning process Nx = Nx, T by Nx, = n i=1 1 Di. Hence, Nx keeps rack of he number of deahs in he porfolio of insured lives. We denoe by H = H T he naural filraion generaed by Nx. I follows ha Nx

19 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS is an H I-Markov process. The sochasic inensiy process λx = λx, T of Nx under P can now be informally defined by λx,d E P [dnx, H I] = n Nx, µx,d, 7.2.1 which is given by he produc of he number of survivors and he moraliy inensiy. I is well-known ha he compensaed couning process Mx = Mx, T defined by is an H I, P-maringale. dmx, = dnx, λx,d, T, 7.2.2 7.3 The combined model Assume ha he filraion F = F T inroduced earlier is given by F = G H I. Thus, F is he filraion for he combined model of he financial marke, he moraliy inensiy and he insurance porfolio. Moreover, we assume ha he financial marke is sochasically independen of he developmen of he insurance porfolio and he moraliy inensiy, i.e. G T and H T, I T are independen. In paricular, his implies ha he properies of he underlying processes are preserved. For example, Mx is also an F,P- maringale, and he F, P-inensiy process is idenical o he H I, P-inensiy process λx. 7.3.1 A class of equivalen maringale measures The combined model allows for infiniely many equivalen maringale measures, such ha he model is arbirage free, bu no complee, see e.g. Björk 24, Chaper 1. In order o perform a simulaneous change of measure for he Wiener processes W f, W s and W µ, and he couning process Nx, we consider he likelihood process dλ = Λ h f dw f + h s dw s + h µ dw µ + gx,dmx,, 7.3.1 wih Λ = 1. In addiion o Λ, we define he likelihood process O, which leads o a change of measure for Y, by O = i j=1 1 + o j Y j, 7.3.2 for some funcions o j, j {1,...,m}, saisfying o j y > 1 for all y in he suppor of Y j and E P [o j Y j ] =. Here, i is defined in 7.2.7 and i j=1 is inerpreed as 1 if < T 1

7.3. THE COMBINED MODEL 191 and hus i =. We assume ha E P [Λ T O T ] = 1 and define an equivalen maringale measure Q via dq dp = Λ TO T. 7.3.3 In he following, we describe he erms in 7.3.1 and 7.3.2 in more deail. The processes h f and h s are relaed o he change of measure for he financial marke. Girsanov s heorem gives ha under Q defined by 7.3.1 7.3.3, W f,q = W f hf udu and W s,q = W s hs udu are independen Wiener processes, such ha he Q- dynamics of Z and S are given by dz = rzd σ z ZdW f,q, 7.3.4 ds = rsd σ s SdW f,q + β s SdW s,q. 7.3.5 Hence, he specificaion of he financial marke ensures ha under any Q given by 7.3.3 he discouned price processes are Q-maringales. The erm involving h µ leads o a change of measure for he Wiener process which drives he moraliy inensiy process µ. Hence, W µ,q = W µ hµ udu defines a Wiener process under Q. Here, as in Chaper 4, we resric ourselves o h µ s of he form h µ,ζx, = βx, ζx, σ ζ x, + β x, σ ζ x, ζx, 7.3.6 for some coninuous funcions β and β. In his case he Q-dynamics of ζx are given by dζx, = γ ζ,q x, δ ζ,q x,ζx, d + σ ζ x, ζx,dw µ,q, where γ ζ,q x, = γ ζ x, + β x,, 7.3.7 δ ζ,q x, = δ ζ x, + βx,. 7.3.8 Hence, ζ follows a ime-inhomogeneous CIR under Q as well. For 7.3.1 and 7.3.2 o define an equivalen maringale measure i mus hold ha ζ is sricly posiive under Q. Thus, we observe from 7.3.7 ha a necessary condiion is β x, σ ζ x, 2 /2 γ ζ x,. The Q-dynamics of µx are given by dµx, = γ µ,q x, δ µ,q x,µx, d + σ µ x, µx,dw µ,q, 7.3.9 where γ µ,q x, and δ µ,q x, are given by 7.2.16 and 7.2.17 wih γ ζ x, and δ ζ x, replaced by γ ζ,q x, and δ ζ,q x,, respecively. If h µ =, i.e. if he dynamics of ζ and µ are idenical under P and Q, we say he marke is risk-neural wih respec o sysemaic moraliy risk. The las erm in 7.3.1 involves a predicable process gx > 1. This erm affecs he inensiy for he couning process. More precisely, i can be shown, see e.g. Andersen

192 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS e al. 1993, ha he inensiy process under Q is given by λ Q x, = 1 + gx,λx,. Using 7.2.1, we see ha λ Q x, = n Nx, 1 + gx,µx,, such ha µ Q x, = 1+gx,µx, can be inerpreed as he moraliy inensiy under Q. Hence, he process M Q x = M Q x, T defined by dm Q x, = dnx, λ Q x,d, T, 7.3.1 is an F,Q-maringale. If gx =, he marke is said o be risk-neural wih respec o unsysemaic moraliy risk. This choice of g can be moivaed by he law of large numbers. In his paper, we resric he analysis o he case, where gx is a deerminisic, coninuously differeniable funcion. Wih O given by 7.3.2 he disribuion funcion of Y i, i {1,...,m}, under Q is given by F Q i y = y 1 + o i zdfi P z. If o i = for all i he marke is called risk-neural wih respec o reinvesmen risk. The measures considered are paricularly simple, since he independence beween G x T, G y T and H T, I T as well as he muual independence of he Y i s are preserved under Q. Now define he Q-survival probabiliy and he associaed Q-maringale by [ ] S Q x,,t = E Q e ÊT µ Q x,τdτ ζx, and S Q,M x,,t = E Q [ e ÊT µq x,τdτ ] ζx, = e Ê µq x,τdτ S Q x,,t. Calculaions similar o hose in Secion 7.2.2 give he following Q-dynamics of µ Q x: dµ Q x, = γ µ,q,g x, δ µ,q,g x,µ Q x, d + σ µ,g x, µ Q x,dw µ,q, 7.3.11 where γ µ,q,g x, = 1 + gx,γ µ,q x,, δ µ,q,g x, = δ µ,q x, gx, 1 + gx,, σ µ,g x, = 1 + gx,σ µ x,. Since he drif and squared diffusion erms in 7.3.11 are affine in µ Q x, we have an affine moraliy srucure under Q. Hence, we have he following expression for he Q- survival probabiliy: S Q x,,t = e Aµ,Q x,,t B µ,q x,,t1+gx,µx,,

7.3. THE COMBINED MODEL 193 where A µ,q and B µ,q are deermined from 7.2.19 and 7.2.2 wih γ µ x,, δ µ x, and σ µ x, replaced by γ µ,q,g x,, δ µ,q,g x, and σ µ,g x,, respecively. Furhermore, he dynamics of S Q,M x,,t are given by ds Q,M x,,t = 1 + gx,σ µ x, µx,b µ,q x,,ts Q,M x,,tdw µ,q, 7.3.12 and he Q-forward moraliy inensiies by f µ,q x,,t = T log SQ x,,t = µ Q x, T Bµ,Q x,,t T Aµ,Q x,,t. 7.3.13 7.3.2 The paymen process The oal benefis less premiums on he insurance porfolio is described by a paymen process A, where da are he ne paymens o he policy-holders during an infiniesimal inerval [, + d. For T we le A be of he form da = nπd1 + n Nx,T A Td1 T + a n Nx,d + a 1 dnx,, 7.3.14 where π is a consan, a and a 1 are G-adaped processes and A T is GT- measurable. The firs erm, nπ is he single premium paid a ime by all policyholders. The second erm involves a fixed ime T T, which represens he reiremen ime of he insured. This erm saes ha each of he surviving policy-holders receive he amoun A T upon reiremen. The hird erm involves he process a given by a = π c 1 <T + a p 1 T T, where π c are coninuous premiums paid by he policy-holders as long as hey are alive and a p corresponds o a life annuiy benefi received by he policy-holders. Finally, he las erm saes ha a 1 is paid immediaely upon a deah. Henceforh we consider an arbirary bu fixed equivalen maringale measure Q from he class of measures inroduced in Secion 7.3.1. Since he paymens a u and a 1 u are Gu- measurable for all u [,T] and A T is GT-measurable we can define he arbirage free prices wih respec o he filraion G T i under he fixed equivalen maringale measure Q by [ ] F Ti,,u = E Q e Êu rτdτ a u G T i, [ ] F Ti,1,u = E Q e Êu rτdτ a 1 u G T i, [ ] F Ti,,T = E Q e ÊT rτdτ A T GT i. We noe ha since he model B,X, G Tm is complee he funcions F Tm,,u u T, F Tm,1,u u T and F Tm,,T are unique for all ; in paricular hey are independen

194 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS of he fixed equivalen maringale measure Q. Here, we resric ourselves o paymen processes, where F Tm,,u u T, F Tm,1,u u T and F Tm,,T are funcions of and X, only. Henceforh, we shall apply he noaion ϑ Ti,c = ϑ Ti,c,z,ϑ T i,c,s = z F Ti,c, s F T i,c = x F T i,c for c {,1, }. 7.3.3 Marke reserves For each i, i {,...,m}, we define he process V Ti, by [ ] V Ti, = E Q e Êτ rudu daτ [,T] FT i. 7.3.15 Hence, V T i, is he condiional expeced value wih respec o he filraion F T i, calculaed a ime, of discouned benefis less premiums, where all paymens are discouned o ime. Using ha he processes A and r are F T i -adaped for all i, and inroducing he discouned paymen process A defined by we see ha V Ti, = [,] da = e Ê rudu da, [ e Êτ rudu daτ + e Ê rudu E Q e Êτ,T] rudu daτ ] FT i = A + e Ê rudu Ṽ T i. 7.3.16 In he lieraure, he process V T i, has been called he inrinsic value process wih respec o F T i, see Föllmer and Sondermann 1986 and Møller 21c. The process Ṽ T i represens he condiional expeced value wih respec o F T i, of fuure discouned paymens. We shall refer o his quaniy as he F T i -marke reserve. Using mehods similar o hose in Møller 21c and Chapers 3 and 4, we obain he following resul. Proposiion 7.3.1 The F T i -marke reserve, Ṽ T i, is given by where V T i = T Ṽ T i = n Nx, V T i, 7.3.17 S Q x,,τ F T i,,τ + f µ,q x,,τf T i,1,τ dτ + S Q x,,tf T i,,t. 7.3.18

7.3. THE COMBINED MODEL 195 Here, he process V T i is inerpreed as he individual F T i -marke reserve given he insured is alive. In he remaining of he paper we work under he following assumpion Assumpion 7.3.2 V Tm,x,µ C 1,2,2, i.e. V Tm,x,µ is coninuously differeniable wih respec o and wice differeniable wih respec o x and µ. 7.3.4 Trading in he financial marke As in Chaper 6 we follow he ideas of Møller 21c and define rading in he presence of paymen processes. Since we consider a fixed arbirary equivalen maringale measure Q for he model B,X, F, we are working wih he probabiliy space Ω, F,Q and he filraions F T i i {,...,m}. An F T i -rading sraegy is a process ϕ = ϑ,η saisfying cerain inegrabiliy condiions, where ϑ is F T i -predicable and η is F T i -adaped. The value process Vϕ associaed wih ϕ is defined by V,ϕ = ϑx + ηb, T. The pair ϕ = ϑ,η is inerpreed as he porfolio held a ime. Here, ϑ = ϑ z,ϑ s is a vecor denoing, respecively, he number of asses wih price process Z and S, and η denoes he discouned deposi in he savings accoun. The cos process associaed wih he pair ϕ,a is given by C,ϕ = V,ϕ ϑudx u + A. 7.3.19 Here, and hroughou, we denoe by V and X, respecively, he discouned value process and he discouned price process of he risky asses. Thus, he cos process is he discouned value of he invesmen porfolio reduced by discouned rading gains and added he oal discouned ne paymens o he policy-holders. The cos process is inerpreed as he company s accumulaed discouned coss during [,]. We say ha a sraegy ϕ is F T i -self-financing for he paymen process A, if he cos process is consan Q-a.s. wih respec o F T i. In conras o he classical definiion of self-financing sraegies, we hus allow for exogenous deposis and wihdrawals as represened by A. The wo definiions of self-financing sraegies are equivalen if and only if he paymen process is consan Q-a.s. wih respec o F T i. The inerpreaion of a self-financing sraegy in he presence of paymen processes is ha all flucuaions of he value process are eiher generaed by he rading sraegy or due o he paymen process. The paymen process A is called F T i -aainable, if here exis an F T i -self-financing sraegy ϕ for A such ha V T,ϕ = Q-a.s. wih respec o F T i. Thus, A is F T i -aainable, if invesing he iniial amoun C,ϕ according o he F T i -rading sraegy ϕ leaves us wih a porfolio value of

196 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS afer he selemen of all liabiliies. Hence, if A is F T i -aainable he unique arbirage free price in B,X, F T i is C,ϕ. A any ime, here is no difference beween receiving he fuure paymens of he F T i -aainable paymen process A and holding he porfolio ϕ and invesing according o he F T i -replicaing sraegy ϕ. Thus, a no arbirage argumen gives ha a any ime he price of fuure paymens from A in B,X, F T i mus be V,ϕ. I can be shown ha he paymen process A is aainable if and only if he coningen claim H = AT wih mauriy T is classically aainable. If all coningen claims, and hence all paymen processes, are aainable, he model is called complee and oherwise i is called incomplee. 7.4 Risk-minimizaion for uni-linked insurance conracs The insurance paymen process A may be subjec o boh unhedgeable reinvesmen and moraliy risks. This implies ha A ypically canno be replicaed perfecly and priced uniquely by rading in he financial marke. In order o quanify and conrol he risk associaed wih A we apply he crierion of risk-minimizaion proposed by Föllmer and Sondermann 1986 for coningen claims and exended in Møller 21c o paymen processes. Here, we give a review of risk-minimizaion and derive risk-minimizing sraegies in he presen se-up. The derivaion consiss wo seps. Firs, we derive risk-minimizing sraegies in he case of a complee financial marke and unhedgeable sysemaic and unsysemaic moraliy risk; a sudy which also was carried ou in Chaper 4 in a slighly differen financial marke. Second, we exend o he case where A also is subjec o unhedgeable reinvesmen risk. 7.4.1 A review of risk-minimizaion Throughou his secion, we consider a fixed bu arbirary filraion F T i, such ha we are working wih he filered probabiliy space Ω, F,Q, F T i. The F T i -risk process associaed wih ϕ is defined by R T i,ϕ = E Q [ CT,ϕ C,ϕ 2 F T i ], 7.4.1 where he cos process is defined in 7.3.19. Thus, he F T i -risk process is he condiional expecaion of he discouned squared fuure coss given he curren available informaion given by F T i. We shall use his quaniy o measure he risk associaed wih ϕ,a. An F T i -rading sraegy ϕ = ϑ,η is called F T i -risk-minimizing if for any [,T] i minimizes R T i,ϕ over all F T i -rading sraegies wih he same value a ime T. Since V T,ϕ is he discouned value of he porfolio ϕt upon selemen of all liabiliies a naural resricion is o consider so-called -admissible sraegies which saisfy V T,ϕ =, Q-a.s.

7.4. RISK-MINIMIZATION FOR UNIT-LINKED INSURANCE CONTRACTS 197 The consrucion of risk-minimizing sraegies is based on he Galchouk Kunia Waanabe decomposiion of he inrinsic value process V T i, = V T i, + ϑ T i,a udx u + L T i. 7.4.2 Here, L T i is a zero-mean square inegrable Q, F T i -maringale which is orhogonal o X, i.e. he process X L T i is a Q, F T i -maringale, and ϑ T i,a is an F T i -predicable process. We noe ha if A is F T i -aainable, hen V T i, is he discouned unique arbirage free price a ime in B,X, F T i of fuure paymens, and L T i = Q-a.s. wih respec o F T i. The following heorem relaes he risk-minimizing sraegy and he associaed risk process o he Galchouk Kunia Waanabe decomposiion. Theorem 7.4.1 Møller 21c There exiss a unique -admissible F T i -risk-minimizing sraegy ϕ T i given by = ϑ T i,η T i for A ϑ T i,η T i = ϑ T i,a,v T i, A ϑ T i,a X. From Theorem 7.4.1 we immediaely ge ha V,ϕ T i = V T i, A 7.4.3 such ha he discouned value process associaed wih he F T i -risk-minimizing sraegy coincides wih he F T i -inrinsic value process reduced by he discouned paymens. Insering 7.4.1 and 7.4.3 in 7.3.19 i follows ha he cos process associaed wih he F T i -risk-minimizing sraegy is given by C,ϕ T i = V T i, + L T i. 7.4.4 Hence, he cos process associaed wih he F T i -risk-minimizing sraegy is an F T i,q- maringale. Insering 7.4.4 in 7.4.1 we ge ha he so-called F T i -inrinsic risk process, which is he risk process associaed wih he F T i -risk-minimizing sraegy, is given by [ R T i,ϕ T i = E Q L T i T L T i ] 2 F T i. 7.4.5 Noe ha he risk-minimizing sraegy depends on he equivalen maringale measure Q. In he lieraure, he so-called minimal maringale measure, see Secion 7.5.2, has been applied in order o deermine risk-minimizing sraegies, since his essenially corresponds o he crierion of local risk-minimizaion, which is a crierion in erms of P, see Schweizer 21a. 7.4.2 Unhedgeable moraliy risk In his secion we consider risk-minimizaion wih respec o F Tm. Hence, for now we disregard he reinvesmen risk, such ha he only unhedgeable risks are he sysemaic and unsysemaic moraliy risk. In his case we have he following resul wih respec o he Galchouk Kunia Waanabe decomposiion of V Tm,.

198 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS Lemma 7.4.2 The Galchouk Kunia Waanabe decomposiion of V Tm, is given by where V Tm, = V Tm, + ϑ Tm,A τdx τ + L Tm, 7.4.6 V Tm, = nπ + n V Tm, 7.4.7 L Tm = ν Tm τdm Q x,τ + κ Tm τds Q,M x,τ,t, 7.4.8 and T ϑ Tm,A = n Nx, S Q x,,τ ϑ Tm,,τ + f µ,q x,,τϑ Tm,1,τ dτ + S Q x,,tϑ Tm,,T, 7.4.9 ν Tm = B 1 a 1 V Tm, 7.4.1 κ Tm = n Nx, B 1 T F Tm,,τ + F Tm,1,τ B µ,q x,,τs Q x,,τ B µ,q x,,ts Q,M x,,t f µ,q x,,τ τ Bµ,Q x,,τ B µ,q x,,τ + Bµ,Q x,,ts Q x,,t B µ,q x,,ts Q,M x,,t F Tm,,T. 7.4.11 dτ Proof of Lemma 7.4.2: The proof, which is similar o he proof of Lemma 4.7.1, is carried ou in Secion 7.6.1. The Galchouk Kunia Waanabe decomposiion of V Tm, in Lemma 7.4.2 is essenially he same as he one obained in Lemma 4.7.1. The process L Tm describes he unhedgeable risk associaed wih he paymen process. The inegral wih respec o he Q-compensaed couning process, M Q x, is relaed o he unsysemaic moraliy risk, whereas he inegral wih respec o he maringale associaed wih he Q-survival probabiliy, S Q,M x,,t, is relaed o he sysemaic moraliy risk. Combining Theorem 7.4.1 and Lemma 7.4.2 we ge he following resul regarding he F Tm -risk-minimizing sraegy for he paymen process A in 7.3.14. Theorem 7.4.3 For he paymen process given by 7.3.14, he unique -admissible F Tm -risk-minimizing sraegy ϕ Tm is ϑ T m,η Tm = ϑ Tm,A,n Nx,B 1 T V m ϑ Tm X, where ϑ Tm,A is given by 7.4.9.

7.4. RISK-MINIMIZATION FOR UNIT-LINKED INSURANCE CONTRACTS 199 The imporance of Lemma 7.4.2 and Theorem 7.4.3 is wofold. Firsly, hey give he risk-minimizing sraegy and he process L Tm for he paymen process A in he case of a complee financial marke and unhedgeable sysemaic and unsysemaic moraliy risk. Secondly, hey are of imporance, since he F-risk-minimizing sraegies deermined in Secion 7.4.3 are given in erms of he F Tm -risk-minimizing sraegies. 7.4.3 Unhedgeable moraliy and reinvesmen risks Now consider he case where he company, in addiion o he unhedgeable moraliy risks, is exposed o reinvesmen risk. We can now apply Schweizer 1994, Theorem 3.1 o he case of paymen processes o obain he F-risk-minimizing sraegies in erms of he F Tm -risk-minimizing sraegies. Hence, calculaions similar o hose leading o Theorem 6.3.3 give he following heorem, which we sae wihou given a proof. Theorem 7.4.4 The unique -admissible F-risk-minimizing sraegy ϕ = ϑ,η for A is given by ϑ,η = E Q [ ϑ Tm F ],n Nx,B 1 V ϑ X. In he following we shall use he quaniies given by ϑ T i = E Q [ ϑ Tm F T i ], 7.4.12 ν T i = E Q [ ν Tm τ F T i ], 7.4.13 κ T i = E Q [ κ Tm τ F T i ], 7.4.14 and he noaion exemplified by i ϑ = ϑ T i ϑ T i 1. The Galchouk Kunia Waanabe decomposiion of V, is given by he following proposiion. Proposiion 7.4.5 If ϑ T i, ν T i and κ T i are sufficienly inegrable for all i {,...,m} hen he Galchouk Kunia Waanabe decomposiion of V, is given by where and V, = V, + V, = nπ + n V, L = M y,q + M y,q = i i=1 Ti + i V + ν τdm Q x,τ + Ti i κτds Q,M x,τ,t ϑ τdx τ + L, 7.4.15 Ti i ϑτdx τ +. κ τds Q,M x,τ,t, 7.4.16 i ντdm Q x,τ

2 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS Proof of Proposiion 7.4.5: See Secion 7.6.2 Invesigaing he expression for L in 7.4.16 we observe ha he wo inegrals, which are associaed wih he moraliy risks are similar o ones from Lemma 7.4.2. However, he sum, M y,q, which is relaed o he reinvesmen risk, is new. I describes he addiional in- or ouflow o/from he invesmen sraegy upon he realizaion of he prices of newly issued bonds bonds. Example 7.4.6 Consider he case where T = T and all n insured have purchased a pure endowmen of A T paid by a single premium a ime. In his case, he Galchouk Kunia Waanabe decomposiion of V, is deermined via he processes ϑ T i τ = n Nx,τ S Q x,τ,tϑ T i, τ, 7.4.17 ν T i τ = Bτ 1 S Q x,τ,tf T i, τ,t, 7.4.18 κ T i τ = n Nx,τeÊτ µq x,udu Bτ 1 F T i, τ,t. 7.4.19 Using Theorem 7.4.4 we obain he -admissible F-risk-minimizing sraegy ϑ = n Nx, S Q x,,tϑ,, η = n Nx,S Q x,,tb 1 F,,T ϑ X. Insering 7.4.17 7.4.19 in Proposiion 7.4.5 gives he following expressions for he erms in he Galchouk Kunia Waanabe decomposiion of V, : and where M y,q = L = M y,q i + i=1 Ti + Ti V, = nπ + ns Q x,,tf,,t Bτ 1 S Q x,τ,tf, τ,tdm Q x,τ n Nx,τeÊτ µq x,udu Bτ 1 F, τ,tds Q,M x,τ,t, ns Q x,,t i F,T + Ti Bτ 1 S Q x,τ,t i F τ,tdm Q x,τ n Nx,τ S Q x,τ,t i ϑ τdx τ n Nx,τeÊτ µq x,udu Bτ 1 i F τ,tds Q,M x,τ,t. 7.4.2

7.5. MEAN-VARIANCE INDIFFERENCE PRICING 21 Example 7.4.7 Assume each of he enries in he vecor Y has finie suppor, i.e. Y i akes values in y1 i,...,yi c i, i = 1,...,m, for some c i N. Now inroduce he Q-maringales M δk,q = E Q [ ] 1 Y1,...,Y m=δ k F = E Q [ 1 Y1,...,Y m=δ k G y ], where δ 1,...,δ K are he possible oucomes of he vecor Y 1,...,Y m, such ha K = m i=1 c i. Here, we have used ha Y is Q-independen of all oher sources of randomness in he second equaliy. Throughou his example we use he noaion exemplified by ϑ δ k for ϑ Tm in he case where Y = δ k. This allows us o wrie he F-risk-minimizing sraegy as Since dm δ k,q o L = K k=1 ϑ = K M δk,q ϑ δ k, 7.4.21 k=1 K η = n Nx,B 1 M δk,q V δ k ϑ X. k=1 = for / {T 1,...,T m } he expression for L in Proposiion 7.4.5 simplifies V δ k, τdm δ k,q τ + ν τdm Q x,τ + κ τds Q,M x,τ,t. 7.5 Mean-variance indifference pricing The mean-variance indifference pricing principles of Schweizer 21b have been applied for he handling of he combined risk inheren in life insurance conracs in Møller 21b, 22, 23a, 23b and Dahl and Møller 25 see Chaper 4. In his secion, we presen a review of mean-variance indifference pricing almos idenical o he one in Chaper 4 and derive indifference prices and opimal hedging sraegies for a porfolio of uni-linked pure endowmens. 7.5.1 A review of mean-variance indifference pricing Denoe by K he discouned wealh of he insurer a ime T and consider he meanvariance uiliy funcions u i K = E P [K ] a i Var P [K ] β i, 7.5.1 i = 1,2, where a i > are so-called risk-loading parameers and where we ake β 1 = 1 and β 2 = 1/2. I can be shown ha he equaions u i K = u i indeed lead o he classical acuarial variance i=1 and sandard deviaion principle i=2, respecively, see e.g. Møller 21b.

22 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS Schweizer 21b proposes o apply he mean-variance uiliy funcions in 7.5.1 in an indifference argumen, which akes he possibiliy for rading in he financial marke ino consideraion. Denoe by Θ he space of admissible sraegies and le GT,Θ be he space of discouned rading gains, i.e. random variables of he form T ϑudx u, where X is he discouned price process associaed wih he risky asses. Denoe by c he insurer s iniial capial a ime. The u i -indifference price v i associaed wih he liabiliy H is defined via sup u i c + v i + ϑ Θ T T ϑudx u H = sup u i c + ϑ Θ ϑudx u, 7.5.2 where H is he discouned liabiliy. The sraegy ϑ which maximizes he lef side of 7.5.2 will be called he opimal sraegy for H. In order o formulae he main resul, some more noaion is needed. We denoe by P he variance opimal maringale measure and le ΛT = d P dp F T. In addiion, we le π be he projecion in L 2 P on he space GT,Θ and wrie 1 π1 = T βudx u. I follows via he projecion heorem ha any discouned liabiliy H allows for a unique decomposiion on he form T H = c H + ϑ H udx u + N H, 7.5.3 where T ϑh dx is an elemen of GT,Θ, and N H lies in he space R + GT,Θ. From Schweizer 21b and Møller 21b we have ha he indifference prices for H are: v 1 H = E P [H ] + a 1 Var P [N H ], 7.5.4 v 2 H = E P [H ] + a 2 1 Var P [ ΛT]/a 2 2 Var P [N H ], 7.5.5 where 7.5.5 only is defined if a 2 2 VarP [ ΛT]. The opimal sraegies associaed wih hese wo principles are: ϑ 1 = ϑ H + 1 + VarP [ ΛT] β, 7.5.6 2a 1 ϑ 2 = 1 + Var P [ ΛT] ϑh + Var a 2 1 P [N H ] β, 7.5.7 Var P [ ΛT]/a 2 2 where 7.5.7 only is well-defined if a 2 2 > VarP [ ΛT]. For more deails, see Møller 21b, 23a, 23b. 7.5.2 The variance opimal maringale measure In order o deermine he variance opimal maringale measure P we firs urn our aenion o he minimal maringale measure, P, which loosely speaking is he equivalen maringale measure which disurbs he srucure of he model as lile as possible,

7.5. MEAN-VARIANCE INDIFFERENCE PRICING 23 see Schweizer 1995. I is easily seen ha he minimal maringale measure is obained by leing h µ =, g = and o i = for all i. Hence, he likelihood process for he change of measure o he minimal maringale measure is given by Λ = exp h f udw f u + h s udw s u 1 h f u 2 + h s u 2 du. 2 Since h f u and h s u are assumed o be G x u-measurable, he densiy Λ is G x - measurable, and herefore i can be represened by a consan C and a sochasic inegral wih respec o X, see e.g. Pham e al. 1998, Secion 4.3. Thus, we have he following represenaion of Λ Λ = C + ζudx u. 7.5.8 Schweizer 1996, Lemma 1 now gives ha Λ T is he densiy for he variance opimal maringale measure as well, such ha P = P and ΛT = ΛT. For laer use, we inroduce he P-maringale Λ by Λ [ ] = E P Λ T [ ] F = E P Λ T G x. Hence, even hough he variance opimal maringale measure P and he minimal maringale measure P in general differ, hey coincide in our model. 7.5.3 Mean-variance indifference pricing for pure endowmens Le T = T and consider a porfolio of n individuals of he same age x each purchasing a uni-linked pure endowmen of A T paid by a single premium a ime. Thus, he discouned liabiliy of he company is given by H = n Nx,TBT 1 A T. Explici expressions for he mean-variance indifference prices can be obained under addiional inegrabiliy condiions. More precisely, we need ha cerain local P-maringales considered in he calculaion of Var P [N H ] are rue P-maringales. In his case we have he following proposiion. Proposiion 7.5.1 The indifference prices are given by insering he following expressions for E P [H ] and Var P N H in 7.5.4 and 7.5.5: E P [H ] = nsx,,tf,,t, 7.5.9 and T Var P [N H ] = n + n T Υ 1 Υ 2 d + n 2 Υ 1 Υ 3 d m m Υ 4 T i + n 2 Υ 5 T i, 7.5.1 i=1 i=1

24 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS where [ ] Λ Υ 1 = E P Λ B 1 F,,T 2, [ Υ 2 = E P Sx,,T 2 e Ê µx,udu µx, 1 + σ µ x,b µ x,,t 2 1 e Ê µx,udu], Υ 3 = E P [ µx, σ µ x,b µ x,,ts M x,,t 2 ], [ ΛTi Υ 4 T i = E P ΛT i µx,τ Ti 1 + e Êτ µx,udu Sx,τ,T 2 Bτ 1 i F τ,t 2 1 e Êτ µx,τ σ µ x,τb µ x,τ,t 2 + 1 e Êτ µx,τ 2 i ϑ,z τ i ϑ,s τσ z τσ s τz τs τ ] + σ s τ 2 + β s τ 2 i ϑ,s τs τ 2 + i ϑ,z τσ z τz τ 2 dτ, [ ΛTi Υ 5 T i = E P Sx,,T i F ΛT,T Ti 2 + S M x,τ,t 2 i i ϑ,z τσ z τz τ 2 + σ s τ 2 + β s τ 2 i ϑ,s τs τ 2 + 2 i ϑ,z τ i ϑ,s τσ z τσ s τz τs τ ] + µx,τ Bτ 1 i F τ,tσ µ x,τb µ x,τ,t 2 dτ. Idea of proof of Proposiion 7.5.1: The P-independence beween he financial marke and he insurance elemens immediaely gives 7.5.9. The expression for he variance of N H in 7.5.1 follows from calculaions similar o hose in Møller 21b and Chaper 4. For compleeness he calculaions are carried ou in Secion 7.6.3 under cerain addiional inegrabiliy condiions. The firs wo erms in 7.5.1 are essenially he same as hose obained in Proposiion 4.8.1. The firs erm, which is proporional o he number of insured, sems from boh he sysemaic and unsysemaic moraliy risk, whereas he second erm, which is proporional o he squared number of survivors, sems solely from he sysemaic moraliy risk. The las wo erms are relaed o he reinvesmen risk. Each of hese erms involve a sum measuring adjusmens of he inrinsic value process upon he realizaion of he Y i s. The reinvesmen risk a ime T i is he difference beween he inrinsic value process before and afer he observaion of Y i, as measured by M P,y T i. Invesigaing he erms

7.6. PROOFS AND TECHNICAL CALCULATIONS 25 in M P,y we find ha he difference in he iniial invesmens, he difference in rading gains/losses due o differen rading sraegies and he difference in gains/losses generaed by changes in he survival probabiliy all conribue o he erm proporional o n 2. The las wo sources along wih he difference in gains/losses generaed by he developmen of he compensaed couning process Mx give rise o he erms proporional o n. The reason for collecing he erms wih respec o n and n 2, respecively, is ha i enables us o disinguish he imporance of he erms as he size of he porfolio of insured increases. Concerning he opimal sraegies we have he following proposiion. Proposiion 7.5.2 The opimal sraegies are given by insering 7.5.1 and he following expression for ϑ H in 7.5.6 and 7.5.7: ϑ H = ϑ ζ 1 Λu dm P,y u + ν udmx,u + κ uds M x,u,t, 7.5.11 where and M P,y is given by M P,y = i i=1 Ti + Ti ν τ = Bτ 1 Sx,τ,TF, τ,t, 7.5.12 κ τ = n Nx,τeÊτ µx,udu Bτ 1 F, τ,t, 7.5.13 nsx,,t i F,T + Ti Bτ 1 Sx,τ,T i F τ,tdmx,τ n Nx,τ Sx,τ,T i ϑ τdx τ n Nx,τeÊτ µx,udu Bτ 1 i F τ,tds M x,τ,t. 7.5.14 Proof of Proposiion 7.5.2: Expression 7.5.11 follows from Schweizer 21a, Theorem 4.6 Theorem 7.6.1, which relaes he decomposiion in 7.5.3 o he Galchouk Kunia Waanabe decomposiion of he P-maringale V, = E P [H F] given in Example 7.4.6. 7.6 Proofs and echnical calculaions 7.6.1 Proof of Lemma 7.4.2 Recall from 7.3.16 and 7.3.17 ha he Q-maringale V Tm, can be wrien as V Tm, = A + n Nx,B 1 V T m,x,µx,. 7.6.1

26 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS Here, we explicily denoe he dependence of V Tm on X and µ. Differeniaing under he inegral gives T x V Tm,x,µ = S Q x,,τ x F Tm,,τ + f µ,q x,,τ x F Tm,1,τ dτ + S Q x,,t x F Tm,,T, 7.6.2 and T µ V Tm,x,µ = 1 + gx, B µ,q x,,τs Q x,,τ F Tm,,τ + F Tm,1,τ + B µ,q x,,ts Q x,,tf Tm,,T f µ,q x,,τ τ Bµ,Q x,,τ B µ,q x,,τ dτ, 7.6.3 where we have used µ fµ,q x,,τ = 1 + gx, τ Bµ,Q x,,τ. Using inegraion by pars on n Nx,B 1 V T m,x,µx, allows us o wrie 7.6.1 as V Tm, = A + n V Tm,X,µx, + + n Nx,u V Tm u,xu,µx,udbu 1 Bu 1 n Nx,u d V Tm u,xu,µx,u Bu 1 V T m u,xu,µx,udnx,u. 7.6.4 In order o calculae he fourh erm in 7.6.4, we need o find d V Tm u,xu,µx,u. Recall from 7.3.9 ha he dynamics of µx, under Q are given by where dµx, = α µ,q,µx,d + σ µ,µx, µx,dw µ,q, α µ,q,µx, = γ µ,q x, δ µ,q x,µx,. In he res of he proof we reurn o he shorhand noaion V Tm u. Furhermore in he coefficien funcions we explicily include he ime argumen only. The assumpion

7.6. PROOFS AND TECHNICAL CALCULATIONS 27 V Tm C 1,2,2 allows us o apply Iô s formula o obain d V Tm u = u V Tm u + α µ,q u µ V Tm u + rusu s V Tm u + ruzu z V Tm u + 1 2 σµ u 2 µx,u 2 µ 2 V Tm u + 1 2 σ s u 2 + β s u 2 Su 2 2 s V Tm 2 u du + 1 2 σz uzu 2 2 z 2 V Tm u + σ s uσ z usuzu 2 z s V Tm u + σ µ u µx,u µ V Tm udw µ,q u σ z uzu z V Tm udw f,q u σ s usu s V Tm udw f,q u + β s usu s V Tm udw s,q u = u V Tm u + α µ,q u µ V Tm u + rusu s V Tm u + ruzu z V Tm u + 1 2 σµ u 2 µx,u 2 µ 2 V Tm u + 1 2 σ s u 2 + β s u 2 Su 2 2 s V Tm 2 u du + 1 2 σz uzu 2 2 z 2 V Tm u + σ s uσ z usuzu 2 z s V Tm u + Bu z V Tm udz u + Bu s V Tm uds u µ V Tm u 1 + gx,ub µ,q x,u,ts Q,M x,u,t dsq,m x,u,t. In he firs equaliy we have used he dynamics of Z, S and µ, whereas we in he second use 7.6.2 and 7.6.3 ogeher wih he dynamics of Z, S and S Q,M x,,t. Rewriing A in erms of he Q-maringale M Q x we ge A = nπ + + Bτ 1 a τn Nx,τ + a 1 τn Nx,τ µ Q x,τ dτ BT 1 n Nx,T A Td1 τ T + Bτ 1 a 1 τdm Q x,τ. Now collec he erms from 7.6.4 involving inegrals wih respec o X, S Q,M x,,t and M Q x, respecively. Since hese hree erms and V Tm, are Q-maringales, he remaining erms consiue a Q-maringale as well. Since his process is coninuous hence predicable and of finie variaion, i is consan. Insering = we immediaely ge ha V Tm, = nπ + n V Tm,X,µx,. Thus, we have proved he decomposiion in 7.4.6. 7.6.2 Proof of Proposiion 7.4.5 The expression for V, follows immediaely from 7.3.16 and 7.3.17. Now, use ha V Tm, T = V, T and he Galchouk Kunia Waanabe decomposiion of V Tm, in

28 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS Lemma 7.4.2 o ge T L T = V Tm, + T ϑ Tm τdx τ + T + κ Tm τds Q,M x,τ,t Since L is an F,Q-maringale we ge L = E Q [V Tm, V, + = = T + κ Tm τds Q,M x,τ,t T m E [V Q Ti, V Ti 1, + i=1 T + + m T i=1 + + + V, + ν Tm τdm Q x,τ T ϑ τdx τ. T ϑ Tm τ ϑ τdx τ + ν Tm τdm Q x,τ ] F T ϑ T i τ ϑ T i 1 τdx τ ν T i τ ν T i 1 τ T dm Q x,τ + κ T i τ κ T i 1 τ ds Q,M x,τ,t T ν τdm Q x,τ + V T i T i, V T i 1 T i, + ] κ τds Q,M x,τ,t F ν T i T iτ ν T i 1 T iτ dm Q x,τ κ T i T iτ κ T i 1 T iτ ds Q,M x,τ,t ν τdm Q x,τ + κ τds Q,M x,τ,t. ϑ T i T iτ ϑ T i 1 T iτdx τ Here, he second equaliy follows by wriing he differences as elescoping sums using he quaniies defined in 7.4.12 7.4.14. In he hird equaliy we use he assumpion ha ϑ T i, ν T i and κ T i are sufficienly inegrable for all i {,...,m} o ensure ha all he considered local Q-maringales are Q-maringales. Furhermore, we use ieraed expecaions ogeher wih he srucure of he filraions F T i. The resul now follows by observing ha for i > i all erms in he sum are zero and ha F T i τ = F T i 1 τ for τ T i. 7.6.3 Calculaion of Var P [N H ] The following heorem due o Schweizer 21a, Theorem 4.6 relaes he decomposiion in 7.5.3 o he Galchouk-Kunia-Waanabe decomposiion of he P-maringale V, = E P [H F]; see also Møller 2.

7.6. PROOFS AND TECHNICAL CALCULATIONS 29 Theorem 7.6.1 Assume ha H L 2 FT,P and consider he Galchouk Kunia Waanabe decomposiion of V, given by V, = E P [H ] + ϑ udx u + L, T. 7.6.5 Then c H, ϑ H and N H from 7.5.3 are given in erms of decomposiion 7.6.5 by Here, we have L = c H = E P [H ], ϑ H = ϑ ζ T N H = ΛT dm P,y u + 1 Λu dl u. ν udmx,u + 1 Λu dl u, κ uds M x,u,t, where ν, κ and M P,y are given by 7.5.12 7.5.14. Thus, Theorem 7.6.1 gives he following expression for N H N H = ΛT T 1 Λ dm P,y + ν dmx, + κ ds M x,,t. Since E P [N H ] =, we firs noe ha Var P [N H ] = E P [ [ N H 2] = E P ΛT My T + LT + RT ] 2 = E P [ ΛT My T 2 + ΛT LT 2 + ΛT RT 2 + 2 ΛT M y T RT +2 ΛT LT RT + 2 ΛT M y T RT ], 7.6.6 where we have defined M y = 1 Λu dm P,y u, L = ν u Λu dmx,u and R = κ u Λu dsm x,u,t. The six erms appearing in 7.6.6 can be rewrien using Iô s formula, see Jacod and Shiryaev 23 for a version ha applies in his seing. For he firs erm we ge ΛT M y T 2 = = T T + M y 2 d Λ + M y 2 d Λ + 2 i=1 m ΛT i i=1 2 M P,y T i ΛT i m ΛT i M y T i 1 M y T i i=1 2 m ΛT M P,y T i i, ΛT i 2 M P,y T i 1 ΛT i

21 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS where he second equaliy follows from rearranging he erms. For he second erm similar rearrangemens gives ΛT LT 2 = T T L 2 d Λ + 2 whereas we for he las erm find ha ΛT RT 2 = = T T + T i=1 R 2 d Λ + 2 R 2 d Λ + 2 T T T Λ L d L + T Λ Rd R + Λ Rd R Λ ν 2 dnx,, Λ Λd R κ Λ Λ σµ x, 2 µx,b µ x,,ts M x,,t d. Assuming all he local maringales are maringales, and using ha he Wiener processes are independen, we ge Var P [ [ m ] N H] = E P M P,y T i 2 [ T ν + E P 2 ] dnx, ΛT i Λ + E P T κ σ µ x, µx,b µ x,,ts M x,,t Λ 2 d. 7.6.7 Noe ha given IT he number of survivors a ime, n Nx,, follows a binomial disribuion under P and P wih parameers n,e Ê µx,udu. Hence, calculaions similar o hose in Chaper 4 give E P and E P [ T = n T T ν 2 ] dnx, Λ E P [ Λ Λ B 1 F,,T 2 κ σ µ x, 2 µx,b µ x,,ts M x,,t d Λ [ ] T Λ = n E P Λ B 1 F,,T 2 ] [ ] E P Sx,,T 2 e Ê µx,udu µx, d [ E P Sx,,T 2 e Ê µx,udu µx,σ µ x,b µ x,,t 2 1 e Ê µx,udu] d [ ] T Λ [ + n 2 E P Λ B 1 F,,T 2 E P µx, σ µ x,b µ x,,ts M x,,t ] 2 d.

7.6. PROOFS AND TECHNICAL CALCULATIONS 211 The firs erm, which relaes o he reinvesmen risk is new, so we invesigae his erm in more deail. Firs we inroduce he noaion M y 1 T i = nsx,,t i F,T, M y 2 T i = M y 3 T i = M y 4 T i = Ti Ti Ti n Nx,τ Sx,τ,T i ϑ τdx τ, Bτ 1 Sx,τ,T i F τ,tdmx,τ, n Nx,τeÊτ µx,udu Bτ 1 i F τ,tds M x,τ,t. Thus, he firs erm in 7.6.7 can be wrien as E P = = = [ m i=1 m i=1 m i=1 m i=1 ] M P,y T i 2 ΛT i E P [ ΛTi ΛT i M P,y T i 2 ] E P [ ΛTi ΛT i M P,y 1 T i + M P,y 2 T i + M P,y 3 T i + M P,y 4 T i 2 [ ΛTi E P M P,y 1 T i ΛT 2 + M P,y 2 T i 2 + M P,y 3 T i 2 + M P,y 4 T i 2]. i ] Here, he hird equaliy follows by assuming ha all local maringales are maringales. Hence, we can invesigae he four erms separaely. For he firs erm we immediaely obain [ ΛTi E P nsx,,t i F ΛT,T ] [ 2 ΛTi = n 2 Sx,,T 2 E P i F i ΛT,T ] 2. i For he second erm we firs use he process X, which makes he process X 2 X a Q-maringale, o obain [ ΛTi E P Ti 2 ] n Nx,τ Sx,τ,T i ϑ ΛT τdx τ i [ ΛTi = E P Ti σ n Nx,τ Sx,τ,T ΛT 2 s τ 2 + β s τ 2 i ϑ,s τs τ 2 i ] + i ϑ,z τσ z τz τ 2 + 2 i ϑ,z τ i ϑ,s τσ z τσ s τz τs τ dτ.

212 CHAPTER 7. A MODEL WITH REINVESTMENT AND MORTALITY RISKS Now he following resul follows from he use of ieraed expecaions [ ΛTi E P Ti 2 ] n Nx,τ Sx,τ,T i ϑ ΛT τdx τ i [ ΛTi = ne P ΛT i Ti e Êτ µx,udu 1 e Êτ µx,udu Sx,τ,T 2 σ s τ 2 + β s τ 2 i ϑ T i,,s τs τ 2 + i ϑ,z τσ z τz τ 2 +2 i ϑ,z τ i ϑ,s τσ z τσ s τz τs τ [ ΛTi + n 2 E P Ti S ΛT M x,τ,t 2 σ s τ 2 + β s τ 2 i ϑ,s τs τ 2 i + i ϑ,z τσ z τz τ 2 + 2 i ϑ,z τ i ϑ,s τσ z τσ s τz τs τ Similar calculaions give E P [ ΛTi ΛT i Ti [ ΛTi = E P Ti ΛT i 2 ] n Nx,τeÊτ µx,udu Bτ 1 i F τ,tds M x,τ,t dτ n Nx,τeÊτ µx,udu Bτ 1 i F τ,t σ µ x,τ 2 µx,τb µ x,τ,ts x,τ,t M dτ ] dτ ]. [ ΛTi = ne P Ti µx,τe Êτ ΛT µx,udu 1 e Êτ µx,udu i 2 Bτ 1 i F τ,tσ µ x,τb x,τ,tsx,τ,t µ dτ + n 2 E P ΛT i ΛT i Ti µx,τ 2 Bτ 1 i F τ,tσ µ x,τb µ x,τ,ts x,τ,t M dτ, and [ ΛTi E P Ti 2 ] Bτ ΛT 1 Sx,τ,T i F τ,tdmx,τ i [ ΛTi = E P Ti Bτ ΛT 1 Sx,τ,T i F τ,t ] 2 n Nx,τ µx,τdτ i ] [ ΛTi = ne P Ti Bτ ΛT 1 Sx,τ,T i F τ,t 2 e Êτ µx,udu µx,τdτ i.

7.6. PROOFS AND TECHNICAL CALCULATIONS 213 Collecing he erms in he sum wih respec o n and n 2, respecively, complees he proof.

Chaper 8 A Numerical Sudy of Reserves and Risk Measures in Life Insurance This chaper is an adaped version of Dahl 25c In his chaper we sudy differen mehods for calculaing reserves for life insurance conracs wih deerminisic benefis in a sligh simplificaion of he model in Secion 7. Hence, he model considered includes he equiy, sandard ineres rae and reinvesmen risks on he financial side and he sysemaic and unsysemaic moraliy risks on he insurance side. We consider marke reserves calculaed by he no arbirage principle, only. Furhermore, we consider he following alernaive approaches o pricing he dependence on he reinvesmen risk: Super-replicaion and he principles of a level long erm yield/forward rae curve. Combined wih he no arbirage principle for he remaining risks, hese principles give reserves, which can be compared o he marke reserves. Moreover, he risk measures of Value a Risk and ail condiional expecaion are considered. These differen reservaion principles and he relaionship o he risk measures are compared numerically. 8.1 Inroducion In recen years legislaion has forced life insurance companies o value boh asses and liabiliies a marke value. Here, he value of he asses is easily obained from he financial marke. Life insurance conracs, on he oher hand, are no raded in he financial marke, so deermining marke values for he liabiliies represens a greaer problem. We consider a model including a large number of risks faced by a life insurance company. The model, which is a simplificaion of he model in Chaper 7 o he case of deerminisic coefficien funcions, consiss of wo independen pars: A financial marke and an 215

216 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES insurance porfolio. In he financial marke he company is allowed o inves in a savings accoun, bonds wih a limied ime o mauriy and a sock. Hence, he following hree financial risks are included in he model: Equiy risk, sandard ineres rae risk and reinvesmen risk. In he insurance porfolio he moraliy inensiy is modelled as a sochasic process, so we consider boh sysemaic and unsysemaic moraliy risk, see Chaper 3 for an explanaion of he differen ypes of moraliy risk. Wihin his model we sudy differen reservaion principles. As a firs approach we apply he no arbirage principle from financial mahemaics in order o obain marke reserves. Since he insurance conracs are no raded in he financial marke he marke reserve depends on he marke s aiude owards reinvesmen risk as well as sysemaic and unsysemaic moraliy risk. Danish legislaion force he life insurance companies o disregard he reinvesmen risk and value heir long erm liabiliies using a yield curve, which is level beyond 3 years. Here, we consider he similar principle of a level yield curve beyond he ime of mauriy of he longes raded bond. Combined wih he no arbirage principle for he remaining risks his principle yields a semi marke reserve. Moreover, we consider wo alernaives o he principle of a level long erm yield curve. The firs alernaive is he principle of a level long erm forward rae curve, which was firs considered in Dahl 25b see Chaper 5 in discree ime. The second alernaive is o super-replicae he reinvesmen risk. In addiion o deermining reserves he life insurance companies are concerned wih measuring he riskiness of heir business. Here, we consider wo measures for he riskiness of he insurance porfolio. Firsly, we consider he Value a Risk, which for a given invesmen sraegy describes he iniial capial necessary o mee he liabiliy wih a cerain probabiliy, and secondly, we consider he ail condiional expecaion which measures he average necessary iniial capial given i exceeds a cerain hreshold. We emphasize ha he main focus of his chaper is no of heoreical naure. On he conrary we keep all echnicaliies o an absolue minimum o improve he readabiliy. Hence, he aim is o provide an easily readable overview of differen reservaion principles and risk measures in he presence of a large number of risks. A main par of he insigh is gained hrough a numerical secion, where we compare he differen principles and illusrae he impac of he marke s aiude oward he differen unhedgeable risks. The chaper is organized as follows: Secion 8.2 conains an inroducion of he model. In Secion 8.3 he differen reservaion principles are considered. The risk measures Value a Risk and ail condiional expecaion are considered in Secion 8.4. Secion 8.5 conains he numerical resuls. Furhermore, his secion conains an explanaion of he simulaion procedure used o calculae he risk measures, and an overview of and moivaion for he parameers used in he numerical calculaions. 8.2 The Model Since he model considered in his chaper is a sligh simplificaion of he one in Chaper 7, we refer he reader o ha chaper for deails. In general he simplificaion consiss

8.2. THE MODEL 217 of resricing all parameer processes o deerminisic funcions and o consider a specific model for he developmen of he moraliy inensiy. Furhermore, he presen exposiion deviaes from Chaper 7 by a paricularly simple approach o he modelling of bond prices a he ime of issue. An approach, which is suiable for numerical calculaions. 8.2.1 The financial marke Le P,u denoe he price a ime of a zero coupon bond mauring a ime u, f,u he forward rae wih mauriy dae u conraced a ime and r = f, he shor rae of ineres. Now consider wo fixed ime horizons T and T, where T T. Here, T and T, respecively, describe he upper limi for he ime o mauriy of raded bonds and he erminal ime of he considered paymen process. Thus, a any ime he ime o mauriy of he longes raded bond is less han or equal o T. In addiion o T and T, we consider he sequence = T < T 1 <... < T m = T T, which describes he imes, where new bonds are issued in he marke. A ime T i new bonds are issued, such ha all bonds wih ime o mauriy less han or equal o T are raded. To ensure ha a all imes, bonds are raded in he marke, we assume ha T max i=1,...,m T i T i 1. Now inroduce a sequence Y = Y i i=1,...,m of muually independen and idenically disribued random variables wih finie suppor. Wihou loss of generaliy we assume he suppor is given by {1,...,b} and le p j = PY 1 = j,1. To model he iniial price of new bonds issued a ime T i, we assume he forward rae curve and hence he zero coupon bond price curve generaed by exising bonds is coninued in a nice coninuous fashion. Here, he oucome of Y i deermines he coninuaion a ime T i. Insead of modelling he coninuaion direcly as in Chapers 6 and 7, we presen an indirec approach, which is paricularly suiable for obaining numerical resuls. Le δ 1,...,δ b m denoe he b m differen possible oucomes of he vecor Y. Given Y = δ k he bond marke is complee since he developmen of he bond prices is assumed o be driven by one Wiener process, see 8.2.1. Hence, we are able o obain zero coupon bond prices P δ k, and forward raes f δ k, for all mauriies T. Since all bonds wih ime of mauriy less han or equal o T are raded a ime, all condiional forward rae curves are idenical for T. Furhermore, all condiional forward rae curves condiioned on he same values of Y 1,...,Y i are idenical for T i + T. Now assume ha he dynamics under P of he condiional forward rae curves are given by τ df δ k,τ = σ f,τ σ f,udu h f d + σ f,τdw f, 8.2.1 where σ f,τ = ce aτ 8.2.2 for some consans a and c. Here, W f is a Wiener process under P independen of Y. Hence, he dynamics of he condiional forward rae curves are idenical, such ha he

218 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES only difference beween he condiional forward rae curves is he iniial long erm forward raes. Wih σ f given by 8.2.2 he flucuaions of he forward raes, as in pracice, dampen exponenially as a funcion of ime o mauriy. Furhermore, i is well known ha he condiional shor raes follow an exended Vasiček model, see Musiela and Rukowski 1997. A ime T i he value of Y i is observed and he exension of he forward rae curve is given by f δ kt i,u Ti 1 + T u T i + T, where he observed oucome of Y 1,...,Y i are he firs i values in he vecor δ k. We noe ha all b m i values of k for which he observed oucome of Y 1,...,Y i are he firs i values give he same exension of he forward rae curve a ime T i. One way o inerpre his model is ha an invesor knows ha he iniial price of long erm bonds should be calculaed wih one of b m differen forward rae vecors. A imes T i more informaion is revealed regarding which curve iniially was he correc one. This addiional informaion uniquely deermines par of he iniial forward rae curve and narrows he possibiliies for he remaining par of he iniial forward rae curve. For fixed we define i = sup { i m T i }, such ha T i is he las ime new bonds are issued prior o ime ime included. Thus, he ime of mauriy, τ, of he bonds raded in he bond marke a ime saisfy τ T i + T. When rading in he bond marke i is sufficien o consider invesmens in a savings accoun wih price process B earning he sochasic rae of ineres r, and an asse wih price process Z generaed by invesing 1 uni a ime and a imes T invesing in he longes bond raded in he marke. Now assume ha he financial marke includes a sock wih price process S, whose developmen is correlaed wih he developmen of he bond marke. The P-dynamics of he raded asses are where db = rbd, B = 1, dz = r + h f σ z Zd σ z ZdW f, Z = 1, ds = r + ρ s Sd σ s SdW f + β s SdW s, S = 1, ρ s = σ s h f β s h s, Ti σ z + T = σ f,udu = c 1 e at i + T, a and h f, h s, σ s and β s are known funcions. Here, W s is a Wiener process under P independen of W f and Y. A rading sraegy is a hree-dimensional vecor process ϕ = ϑ s,ϑ z,η. The riple ϕ = ϑ s,ϑ z,η is inerpreed as he porfolio held a ime. Here, ϑ s and ϑ z

8.2. THE MODEL 219 denoe, respecively, he number of asses wih price process S and Z, whereas η is he discouned deposi in he savings accoun. The value process Vϕ associaed wih he sraegy ϕ is defined by V,ϕ = ϑ s S + ϑ z Z + ηb, T. In his chaper, we resric ourselves o so-called self-financing sraegies, where no in- or ouflow of capial o/from he porfolio is allowed. 8.2.2 Modelling he moraliy Le µx, denoe he moraliy inensiy a ime of an insured of age x a ime. If we furher le µ describe he iniial moraliy curve for all ages, hen i holds ha µx, = µ x. For a fixed iniial age x we assume he moraliy inensiy follows he following ime-inhomogeneous Cox Ingersoll Ross model where dµx, = γ µ x, δ µ x,µx, d + σ µ x, µx,dw µ, 8.2.3 γ µ x, = 1 2 σ2 µ x +, δ µ x, = δ d d µ x + µ x +, σ µ x, = σ µ x +. Here, σ and δ are non-negaive consans, W µ is a Wiener process under P independen of he financial marke, δ µ x, is he ime-dependen speed of mean-reversion and γ µ x,/δ µ x, is he ime-dependen level of mean-reversion. Noe ha 2γ µ x, σ µ x, 2, such ha he moraliy inensiy is sricly posiive, see Maghsoodi 1996. Now define he survival probabiliy by [ Sx,,T = E P e ÊT µx,udu ] µx,. From Proposiion 4.3.1 we have he following expression for he survival probabiliy where Sx,,T = e Aµ x,,t B µ x,,tµx,, Bµ x,,t = δ µ x,b µ x,,t + 1 2 σµ x, 2 B µ x,,t 2 1, 8.2.4 Aµ x,,t = γ µ x,b µ x,,t, 8.2.5 wih B µ x,t,t = and A µ x,t,t =. The forward moraliy inensiies are given by f µ x,,t = µx, T Bµ x,,t T Aµ x,,t.

22 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES 8.2.3 The insurance porfolio Consider an insurance porfolio consising of n insured lives of he same age x, and le he moraliy inensiy in Secion 8.2.2 describe he probabiliy of he deah of an insured in a small ime inerval. The insured lives are obviously no independen, since he survival/deah of all insured depend on he developmen of he moraliy inensiy. However, condiioned on he developmen of he moraliy inensiy, we assume he insured lives are independen. To keep rack of he number of deahs in he insurance porfolio we inroduce he couning process N. The sochasic inensiy process λ of N under P, which describes he probabiliy of experiencing a deah in he porfolio wihin he nex small ime inerval, is given by he number of survivors muliplied by he moraliy inensiy. The independence beween he moraliy inensiy and he financial marke hen ensures ha he insurance porfolio is independen of he financial marke as well. 8.2.4 A class of equivalen maringale measures In he model described in Secions 8.2.1 8.2.3 here exiss infiniely many equivalen maringale measures, such ha he model is arbirage free, bu no complee, see e.g. Björk 24, Chaper 1. Here, we only consider a specific class of equivalen maringale measures. The class is paricularly nice, since any independence under P is preserved under Q and he Q-properies of N, µ and Y are closely relaed o he P-properies. For all equivalen maringale measures i holds ha he discouned price processes of raded asses are Q-maringales. To accoun for he unsysemaic moraliy risk we le he inensiy process for N under Q be given by λ Q x, = 1 + gλx,, for some consan g > 1. This essenially corresponds o changing he moraliy inensiy o µ Q x, = 1 + gµx,. If g =, he marke is called risk-neural wih respec o unsysemaic moraliy risk. This choice of g can be moivaed by he law of large numbers. Now inroduce he consans β and β, which affec he marke price of sysemaic moraliy risk, and le he Q-dynamics of µ Q x be given by dµ Q x, = γ µ,q,g x, δ µ,q,g x,µ Q x, d + σ µ,g x, µ Q x,dw µ,q, 8.2.6 where 1 γ µ,q,g x, = 1 + g 2 σ2 + β µ x +, 8.2.7 δ µ,q,g = δ + β d d µ x + µ x +, σ µ,g x, = 1 + g σ µ x +. 8.2.8 Hence, β affecs he level of mean-reversion, whereas β affecs boh he level and speed of mean-reversion. If β and β are equal o zero, we say he marke is risk-neural wih

8.2. THE MODEL 221 respec o sysemaic moraliy risk. From 8.2.7 and 8.2.8 we ge ha µ Q is sricly posiive under Q if and only if β. Now define he Q-survival probabiliy by [ S Q x,,t = E Q e ÊT µ Q x,udu ] µ Q x,. Since 8.2.6 has he same form as 8.2.3 we have an affine moraliy srucure under Q as well. Hence, we have he following expression for he Q-survival probabiliy: S Q x,,t = e Aµ,Q x,,t B µ,q x,,tµ Q x,, where A µ,q and B µ,q are deermined from 8.2.4 and 8.2.5 wih γ µ x,, δ µ x, and σ µ x, replaced by γ µ,q,g x,, δ µ,q,g x, and σ µ,g x,, respecively. Furhermore, he Q-forward moraliy inensiies are given by f µ,q x,,t = µ Q x, T Bµ,Q x,,t T Aµ,Q x,,t. Under all considered equivalen maringale measures i sill holds ha Y 1,...,Y m are idenically disribued. For any j {1,...,b} i holds ha under he equivalen maringale measure he probabiliy of Y 1 = j is changed o from p j o q j, where q j,1. If q j = p j for all j he marke is called risk-neural wih respec o reinvesmen risk. 8.2.5 The paymen process The oal benefis less premiums on he insurance porfolio is described by a paymen process A. Thus, da are he ne paymens o he policy-holders during an infiniesimal inerval [, + d. We ake A of he form da = nπd1 + n NT A Td1 T + a n Nd + a 1 dn, 8.2.9 for T. The firs erm nπ is he single premium paid a ime by all policyholders. The second erm involves a fixed ime T T, which represens he reiremen ime of he insured. This erm saes ha each of he surviving policy-holders receive he fixed amoun A T upon reiremen. Hence, A T corresponds o a pure endowmen. The hird erm involves a piecewise coninuous funcion a = π c 1 <T + a p 1 T T, where π c are coninuous premiums paid by he policy-holders as long as hey are alive, and a p corresponds o a life annuiy benefi received by he policy-holders. Finally, he las erm in 8.2.9 represens paymens immediaely upon a deah, and we assume ha a 1 is some piecewise coninuous funcion.

222 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES 8.3 Reserving In his secion we consider differen reservaion principles applicable in life insurance. We noe ha all reserves calculaed a ime are calculaed afer a possible iniial premium, so an obained reserve a ime can be inerpreed as he iniial premium implied by he considered crierion. 8.3.1 Marke reserves For any equivalen maringale measure Q from he class of measures considered in Secion 8.2.4 we can define a marke reserve by [ T ] V Q = E Q e Êu rsds dau F, where F represens all informaion available a ime. Calculaions similar o hose in Chaper 4 give he following simplificaion of he marke reserve in Chaper 7: where V Q,i = b m k=1 V Q = n NV Q,i, T QY = δ k P δ k,us Q x,,u a u + a 1 uf µ,q x,,u du + P δ k,ts Q x,,t A T. Here, he quaniy V Q,i is he individual marke reserve a ime for a policy-holder who is alive. 8.3.2 Super-replicaion The super-replicaing super-hedging price for a liabiliy is he minimal iniial capial necessary o guaranee he exisence of a rading sraegy, which always leaves he company wih sufficien capial o cover he liabiliy. Hence, for a coningen claim H he superreplicaing price, F sr,h, is given by F sr,h = inf PVT,ϕ H = 1. V,ϕ In he case of a paymen process he super-replicaing price is given by T F sr,a = inf P VT,ϕ V,ϕ VT, ϕ Vu,ϕ dau = 1. 8.3.1

8.3. RESERVING 223 Here, i is sufficien o consider he erminal ime only, since an iniial capial and a rading sraegy, which ensures sufficien capial o cover he accumulaed paymens a erminal ime T also is sufficien a ime o cover he accumulaed benefis and he ime super-replicaing price. In 8.3.1 he paymens are accumulaed wih he rae of reurn obained by he super-replicaing sraegy, since hey can be inerpreed as in- or ouflow o/from he porfolio. Noe ha he super-replicaing price is upper boundary for he open inerval of arbirage free prices. Hence, he super-replicaing price can be inerpreed as he lowes price which allows he seller of he conrac o make arbirage. In he considered model wih deerminisic benefis and a finie number of condiional forward rae curves he super-replicaing prices are paricularly simple o calculae, if he benefis only are coningen on eiher survival or deah. Proposiion 8.3.1 The super-replicaing price for a pure endowmen of A T a ime T and a emporary life annuiy wih a coninuous ime-dependen rae a p from ime T o T is given by F sr,a pe,a = max k {1,...,b m } P δ k,t A T + T T P δ k,ua p udu. For a erm insurance of a consan amoun a 1 payable upon deah prior o ime T he super-replicaing price is F sr,a i = a 1. The super-replicaing prices above can be inerpreed as follows: If he benefis are coningen on survival we assume he insured never dies, and if he benefis are coningen on deah, we assume he insured dies immediaely. This corresponds o using a survival probabiliy over any ime inerval of 1 or, respecively. The resuling purely financial claim is now priced in he condiional models wihou reinvesmen risk. The super-replicaing price is hen he maximum of hese condiional prices. In general he crierion of super-replicaion is no suiable o deermine reserves for life insurance conracs, and hence i shall no be pursued furher in his chaper. However, for a financial risk, such as he reinvesmen risk, he crierion of super-replicaion may provide valuable informaion. Hence, his idea is pursued in Secion 8.3.3. 8.3.3 Alernaive approaches o he reinvesmen risk Insead of calculaing reserves by he no arbirage principle only, we now consider hree alernaive approaches o handling he reinvesmen risk. Combined wih he no arbirage principle for he remaining risks hese principles yield reserves, which serve as alernaives o he marke reserves deermined in Secion 8.3.1. We noe ha he reserves in his secion also depend on he marke s aiude owards sysemaic and unsysemaic moraliy risk, and hence he considered equivalen maringale measure Q.

224 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES Super-replicaion of reinvesmen risk Consider he case where he company deermines reserves using he crierion of superreplicaing for he reinvesmen risk. In his case, he company for a fixed Q deermines he marke reserves in he condiional models wihou reinvesmen risk. The reserve based on he crierion super-replicaion of he reinvesmen risk is hen he maximum of he condiional marke reserves. Mahemaically his corresponds o T F srr,a = max n P δ k,us Q x,,u a u + a 1 uf µ,q x,,u du k {1,...,b m } + P δ k,ts Q x,,t A T. Level long erm yield and forward rae curves In order o handle he reinvesmen risk we, inspired by Danish legislaion, consider he principle of a level long erm yield curve. Here, he companies value heir liabiliies using a yield curve, which is level afer ime of mauriy of he longes bond currenly raded in he marke. In addiion we also consider he relaed principle of a level long erm forward rae curve, where reserves are obained using a forward rae curve which is level beyond he ime of mauriy of he longes raded bond. This principle was inroduced in discree ime in Dahl 25b see Chaper 5. Denoe he bond prices using a level long erm yield and forward rae curve by P y, and P f,, respecively. For a fixed Q he reserves using hese principles are given by T F c,a = n P c,us Q x,,u a u + a 1 uf µ,q x,,u du + P c,ts Q x,,t A T, where c {y, f}. Noe ha he reserves calculaed by hese principles no necessarily lie in he inerval of arbirage free prices. Connecion beween yield and forward rae curves In order o compare he principles of a level long erm yield curve and a level long erm forward rae curve, we sudy he shape of he forward rae yield curve implied by a level long erm yield forward rae curve. The yield from ime u o is defined as he consan rae of ineres, yu,, implied by he price a ime u of a zero coupon bond mauring a ime. Hence, yu, is given by yu, = log Pu,, u

8.3. RESERVING 225 or saed differenly Pu, = e yu, u. Björk 24 refers o he yield for he period [u,] as he coninuously compounded spo rae for he period [u,]. For u we have he following connecion beween yields and forward raes e yu, u = e Ê u fu,sds. 8.3.2 Since we are ineresed in applying he principles of a level long erm yield or forward rae curve o obain reserves a ime, we henceforh resric ourselves o he case u =. Firs consider he yield curve corresponding o a level long erm forward rae curve. To be more specific we assume he forward rae curve is level from ime T, such ha we for any T have e y, = e Ê T f,sds f, T T. Using 8.3.2 wih = T and isolaing y, gives y, = f, T + y, T f, T T. 8.3.3 From 8.3.3 we observe ha if y, T is smaller larger han f, T hen y, converges upwards downwards o f, T as. Now consider he implicaions on he forward rae curve of a level long erm yield curve. Assuming he yield curve is level from ime T gives he following equaion for any T: Again we apply 8.3.2 wih = T o obain e y, T = e Ê f,sds. e y, T T = e Ê T f,sds. 8.3.4 Since 8.3.4 holds for all T we ge f, = y, T. Hence, he long erm forward rae curve is level as well and equal o he yield curve. We noe ha if y, T f, T hen he forward rae curve is disconinuous a T, which is couner inuiive and in conras o sandard financial lieraure. Figure 8.3.1 conains an illusraion of he principles of a level long erm forward rae curve and a level long erm yield curve. In his example he principle of a level long erm yield curve would lead o a disconinuiy in he forward rae curve. Regarding he general relaionship beween he yield curve and he forward rae curve i holds ha when he yield curve is increasing decreasing he forward rae curve lies above below he yield curve. Furhermore i holds ha if he forward rae curve is increasing decreasing for all mauriies hen i lies above below he yield curve.

226 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES..1.2.3.4.5.6.7 Level long erm forward rae curve Level long erm yield curve 5 1 15 2 25 3 Mauriy Figure 8.3.1: An illusraion of he principles of a level long erm forward rae curve and a level long erm yield curve. 8.4 Risk measures In order o measure he risk of he company associaed wih he insurance porfolio we consider he risk measures of Value a Risk and ail condiional expecaion. We noe ha boh risk measures are calculaed afer a possible iniial premium. In his exposiion we follow Arzner, Delbaen, Eber and Heah 1999 and define he risk measures in erms of he iniial capial insead of he erminal capial. The mean excess funcion from acuarial lieraure he expeced shor fall in he financial lieraure measures he overshoo for a given level. However, since we consider he necessary iniial capial and no jus he addiional necessary iniial capial exceeding a specific level we prefer he erm ail condiional expecaion, which sems from Arzner e al. 1999. For an overview of Value a Risk wrien especially for praciioners we refer o Duffie and Pan 1997. 8.4.1 Value a Risk Given a rading sraegy he erminal Value a Risk a confidence level κ, V ar κ, is he iniial capial necessary o mee a given liabiliy wih probabiliy κ. Hence, for a coningen claim H wih mauriy T we have V arκ ϕ,h = inf P VT,ϕ H κ. 8.4.1 V,ϕ

8.4. RISK MEASURES 227 However, i is no enough for a company o hold sufficien funds a he ime of mauriy of he claim, i should hold sufficien funds hroughou he course of he conrac. Hence, in addiion o he erminal Value a Risk given by 8.4.1 we consider he barrier Value a Risk, V arκ b, given by V arκ b ϕ,h = sup inf P V,ϕ V,H κ. 8.4.2 T V,ϕ Here, V,H describes a capial requiremen a ime for he claim H. Hence, V,H could be a solvency requiremen or a marke reserve. From 8.4.1 and 8.4.2 we observe ha V arκ bϕ,h V ar κ ϕ,h, which also is inuiively clear since he barrier Value a Risk should be sufficien o mee requiremens a any ime prior o and including mauriy, while he erminal Value a Risk is sufficien o fulfill a requiremen a mauriy only. Since we consider a paymen process we need slighly differen formulas han hose in 8.4.1 and 8.4.2 o calculae he Value a Risk. For a paymen process he erminal and barrier Value a Risk are given by and T V arκϕ,a = inf P VT,ϕ V,ϕ V arκ b ϕ,a = sup T inf P V,ϕ V,ϕ respecively. Here, a naural idea is o le V,A = E Q [ T VT, ϕ V,ϕ da κ, 8.4.3 V, ϕ dau + V,A κ, Vu,ϕ 8.4.4 ] e Êu rsds dau for some equivalen maringale measure Q. Noe ha, as in 8.3.1, he paymens in 8.4.3 and 8.4.4 are accumulaed wih he relaive reurn on he invesmens. In pracice companies calculae he shor erm Value a Risk for heir liabiliies. Here, he fixed shor erm ime-horizon usually lies beween one day and one year. Hence, in addiion o 8.4.3 and 8.4.4 we define he ime s erminal and barrier Value a Risk for s < T, by and V ar,s κ ϕ,a = inf V,ϕ V arκ b,s ϕ,a = sup inf s V,ϕ P Vs,ϕ s P V,ϕ Vs, ϕ da + V s,a V,ϕ κ, V, ϕ dau + V,A κ. Vu,ϕ Example 8.4.1 Consider he case, where a company follows eiher a buy and hold sraegy or a sraegy wih consan relaive porfolio weighs. In a buy and hold sraegy no adjusmens are made o he iniial porfolio during he considered ime-period, whereas a

228 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES company following a sraegy wih consan relaive porfolio weighs coninuously adjuss he invesmen porfolio, such ha a all imes, [,T] he same proporion of he value process is invesed in he differen asses. We observe ha in boh cases he dynamics under P of he value process are of he form dv,ϕ = r + ρ V V,ϕd + σ V V,ϕdW f + β V V,ϕdW s, for some funcions ρ V, σ V and β V. Now we are ineresed in calculaing V ar κ for a capial insurance of K a ime T. Hence, we have o deermine inf b m V,ϕ k=1 inf PVT,ϕ K κ V,ϕ PY = δ k PV δ k T,ϕ K κ, 8.4.5 where V δ kϕ is he value process in he condiional model given Y = δ k. In order o deermine PV δ kt,ϕ K we noe ha he condiional model given Y = δ k he shor rae is given by Hence, we have r δ k u = f δ k,u + c2 2a 21 e au 2 + log V δ k,ϕ = log V,ϕ + + u = log V,ϕ + u f δ k,u + α V u du ce au s dw f sdu + f δ k,u + α V u du c + 1 e a u dw f u + a ce au s dw f s. σ V udw f u + σ V udw f u + β V udw s u β V udw s u, where we have defined α V u = c2 2a 2 1 e au 2 + ρ V u 1 2 σv u 2 1 2 βv u 2 and used Fubini s heorem for sochasic processes o inerchange inegrals in he double inegral. This gives log V δ k,ϕ Nlog V,ϕ + α δ k,σ 2, where α δ k = f δ k,u + c2 2a 21 e au 2 + ρ V u 1 2 σv u 2 1 2 βv u 2 du, c σ = 2 1 e a u + σ a V u + β V u 2 du.

8.4. RISK MEASURES 229 Thus, i holds ha PV δ k T,ϕ K = P log V δ k T,ϕ logk log V,ϕ + α δ k T logk = Φ, 8.4.6 σt where Φ is he sandard normal disribuion funcion. Insering 8.4.6 in 8.4.5 we obain he following implici expression for V arκ for a capial insurance of K a ime T b m log V,ϕ + α δ k T logk inf PY = δ k Φ κ. 8.4.7 V,ϕ σt k=1 Similar argumens give ha in he case wihou sysemaic moraliy risk and reinvesmen risk he V arκ for a porfolio of n idenical pure endowmens wih sum insured A T and mauriy T is given by inf V,ϕ n n Tpx n l 1 T p x l log V,ϕ + αt log n l A T Φ κ. l σt l= where T p x = exp T µx,udu. 8.4.8 We noe ha in hese simple cases no simulaion is necessary, since we a leas numerically are able o calculae he value a risk explicily from 8.4.7 and 8.4.8. 8.4.2 Tail condiional expecaion The V ar κ s measure he iniial capial necessary o cover a fuure liabiliy and possible inermediae requiremens wih probabiliy κ. However, he crierion provides no informaion regarding he magniude of he necessary capial in he cases, which occur wih probabiliy 1 κ, where his iniial capial is insufficien. Thus, in addiion o he Value a Risk we now consider he ail condiional expecaion, which for a fixed rading sraegy measures he average iniial invesmen necessary o cover he liabiliies provided ha a given iniial invesmen is insufficien. Le ω denoe a possible sae of he world. For each ω we now define he iniial capial necessary o cover he ime s value of he benefis by s V,s Vs, ϕ min,ϕ,ω = inf Vs,ϕ da + V s,a. V,ϕ V,ϕ For a given iniial invesmen, u, we are now able o define he ime s erminal ail condiional expecaion by [ ] Vu,s,ϕ = EP V,s min,ϕ,ω V,s min,ϕ,ω > u.

23 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES If s = T we simply refer o he value as he erminal ail condiional expecaion. Similarly we can define V b,s V, ϕ min,ϕ,ω = sup inf V,ϕ dau + V,A s V,ϕ Vu,ϕ and he ime s barrier ail condiional expecaion a level u by V b,s u,ϕ = EP [ V b,s min,ϕ,ω V b,s min,ϕ,ω > u ]. If we le u be equal o V ar κ for some κ hen we obain he so-called ail Value a Risk. Remark 8.4.2 Noe ha since all calculaions for he Value a Risk and ail condiional expecaion are carried ou under P he only possible dependence on Q is hrough he capial requiremen V,A. 8.5 Numerics 8.5.1 Simulaion of Value a Risk and ail condiional expecaion Here, we explain he simulaion procedure used o calculae he Value a Risk and ail condiional expecaion for a fixed rading sraegy and paymen process. A ime-sep j he necessary capial o cover he accumulaed benefis and he reserve for he fuure liabiliies is given by Vj,ϕ = j i=1 Ai Vj,ϕ Vi,ϕ + V j,a 8.5.1 where Ai = Ai Ai 1 and V j,a is he capial requiremen. We assume he capial requiremen is given by he marke reserve calculaed wih risk-neuraliy wih respec o all unhedgeable sources of risk reinvesmen, sysemaic and unsysemaic moraliy risks, i.e. under he so-called minimal maringale measure, see Schweizer 1995. Now le V 1,ϕ be he value of 1 invesed a ime. The necessary iniial invesmen o cover he requiremen a ime-sep j is hen given by V j,ϕ = j i=1 Ai V1 j,ϕ V 1 i,ϕ + V j,a V 1. 8.5.2 j,ϕ Hence, in order o calculae 8.5.2 we keep rack of he value process generaed by invesing 1 a ime, and of he pas benefis accumulaed by he rae of reurn obained by he invesmen sraegy. In each simulaion we can now for a fixed ime-horizon s calculae V b,s min,ϕ = max V j,ϕ. j {,...,s/ }

8.5. NUMERICS 231 and V,s min,ϕ = V s/,ϕ. For any ime horizon s he barrier and erminal Value a Risk and ail condiional expecaion for differen κ s and u s can now be calculaed from he vecors conaining V b,s min,ϕ and V,s min,ϕ from each simulaion. Noe ha in he simple case of a pure endowmen he firs erm on he righ hand side in 8.5.1 disappears, such ha he calculaions simplify considerably. 8.5.2 Parameers In his secion we moivae he choice of parameers in he numerical calculaions. I is imporan o noe ha we have no empirical ambiions, bu he parameers should be reasonable. All condiional shor raes follow an exended Vasiček model wih he same speed of meanreversion, a, and volailiy, c. Hence, we may use he values used in Poulsen 23 for a sandard Vasiček model. Furhermore, he marke price of sandard ineres rae risk, h f, is deermined such ha i is idenical o he one in Poulsen 23, namely h f =.3125. Consider a so-called Nelson Siegel paramerizaion, see Nelson and Siegel 1987, for he iniial forward rae curve f, = α + α 1 exp + α 2 τ τ exp. τ Here, α, α 1, α 2 and τ are some consans of which α and τ sricly posiive. As a basis for he iniial condiional forward rae curves we esimae he parameers in he Nelson Siegel paramerizaion from he prices of Danish governmen bonds early 25. Le Y = δ 1 correspond o he condiional iniial curve, where Y i = 1 for all i. This curve is obained by muliplying he esimaed forward raes f, by ψ d i for T i 1 + T < T i + T. We now resric ourselves o he case b = 2 and obain he remaining condiional iniial forward rae curves by muliplying he forward raes f δ 1, Ti 1 + T< T i + T by ψu j if j of he i h firs values in δ k is equal o 2. To describe he probabilisic naure of he Y i s, we le p = PY 1 = 2 =.5 and q = QY 1 = 2. In general, empirical evidence shows ha here is a posiive correlaion beween bonds and socks, such ha increasing ineres raes lead o a decrease in boh bond and sock prices. We calibrae he parameers σ s and β s, such ha he correlaion, σ s / σ s 2 + β s 2, is equal o.5 and he volailiy of he sock, σ s 2 + β s 2, is.2. This gives σ s =.2 and β s.19. Furhermore, we le h s =.2, such ha he addiional expeced rae of reurn on he sock compared o he long erm bonds is approximaely.3, which seems reasonable, see e.g. Graham and Harvey 25. The iniial moraliy curve is described by a so-called Gomperz Makeham curve, where he moraliy inensiy is of he form µ x + = a + b c x+, for some consans

232 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES a, b and c. We use he parameers esimaed in Chaper 4 for males in year 23. The sochasic model including parameers for he fuure developmen of he moraliy inensiy is idenical o so-called case II -model in Chaper 4. Table 8.5.1 provides an overview of he inerpreaion of he differen parameers and he values used in he numerical calculaions. Parameer Inerpreaion Value T Terminal ime of paymens 3 varies T Time of reiremen 3 α Parameer Nelson Siegel paramerizaion of forward raes.44556 α 1 Parameer Nelson Siegel paramerizaion of forward raes -.224 α 2 Parameer Nelson Siegel paramerizaion of forward raes -.231 τ Time-parameer Nelson Siegel 1.97184 a Speed of mean-reversion of r.25 c Volailiy of r.12 h f Marke price of ineres rae risk.3125 T Maximum ime o mauriy of raded bonds 2 T i Time beween issue of new bonds 2 varies m Number of issues of new bonds 2 p Probabiliy under P of Y i = 2.5 q Probabiliy under Q of Y i = 2.5 varies σ s Volailiy parameer sock.2 β s Volailiy parameer sock.19 h s Relaed o marke price of risk associaed o he sock -.2 a Gomperz Makeham parameer.134 b Gomperz Makeham parameer.353 c Gomperz Makeham parameer 1.12 x Iniial age 3 δ Speed of mean-reversion for µ under P.8 σ Volailiy parameer for µ.2 β Affecs level and speed of mean-reversion for µ under Q varies β Affecs level of mean-reversion for µ under Q g Affecs marke price of unsysemaic moraliy risk ψ u Muliplicaion facor up 1.5 varies ψ d Muliplicaion facor down.99 varies Table 8.5.1: Parameers used in he numerical calculaions. A number followed by varies means ha where nohing else is saed he parameer is equal o he value, and in a leas one case his is no he case.

8.5. NUMERICS 233 8.5.3 Numerical resuls In his secion we, unless saed oherwise, consider a porfolio of pure endowmens paid by a single iniial premium. Furhermore, in order o ease comparison all numbers in his secion, unless explicily explained, are scaled by he marke reserve under he minimal maringale measure, i.e. where q = p and β = β = g =. Dependence on marke s aiude owards reinvesmen risk From Figure 8.5.1 we observe ha he marke reserve is a decreasing funcion of q. This relies on he fac ha we consider a pure endowmen paid by a single iniial premium. In his case, he condiional reserve is an increasing funcion of he condiional price of a zero coupon bond mauring a ime T, and since increasing q increases he weigh assigned o he low condiional bond prices he marke reserve is decreasing in q. The alernaive reservaion principles are independen of he choice of maringale measure wih respec o reinvesmen risk, so hey are independen of q. Noe ha as q he marke reserve converges o he reserve calculaed by he principle of reinvesmen risk super-replicaion. This is also inuiively obvious since he weigh assigned o he larges condiional zero coupon bond price, which is exacly he price used o calculae he reinvesmen risk super-replicaing reserve, approaches 1. The iniial forward rae curve is increasing for all mauriies, so he principle of a level long erm yield curve gives a larger reserve han he reserve calculaed wih a level long erm forward rae curve. Furhermore, we observe ha in his case he principle of reinvesmen risk super-replicaing gives a larger reserve han he principle of a level long erm forward rae curve. A necessary requiremen for his is ha a leas one condiional forward rae curve is decreasing for some mauriies. Invesigaing Figure 8.5.2 we observe ha he relaive magniude of he dependence on q depends on he number of issues of new bonds. This is also inuiively clear since increasing he number of issues for fixed ψ u and ψ d increases he diversiy beween he condiional bond prices. Hence, he weighs assigned o he differen condiional bond prices become increasingly imporan. We noe ha since he reserves depend on he number of issues he scaling facors differ for he hree lines i Figure 8.5.2, so he figure can only be used o observe he impac of he number of issues on he relaive dependence on q. Dependence on marke s aiude owards moraliy risk For he considered porfolio of pure endowmens he reserves are given by he produc of he number of insured, he Q-survival probabiliy, he fixed benefi per insured and he price of a zero coupon bond mauring a ime T under he assumpions imposed by he reservaion principle. Hence, he relaive impac of changing β is he same for all four reservaion principles. This fac is easily observed form Figure 8.5.3. Furhermore Figure 8.5.3 shows ha he reserves have a posiive dependence on β. This is also inuiively

Measure of 234 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES Reserve.9.95 1. 1.5 1.1 1.15 Marke reserve Level long erm forward rae curve Level long erm yield curve Super replicaion reinvesmen risk..2.4.6.8 1. Figure 8.5.1: Iniial marke reserve as a funcion of marke s aiude o reinvesmen risk. For comparison he reserves calculaed by he principles of a level long erm yield/forward rae curve and super-replicaion of reinvesmen risk are ploed as well. q

8.5. NUMERICS 235 Reserve.9.95 1. 1.5 1.1 T i = 1 T i = 2 T i = 5..2.4.6.8 1. Figure 8.5.2: Iniial marke reserves as a funcion of marke s aiude o reinvesmen risk for hree differen ime inervals beween he ime of issue of new bonds. q clear, since increasing β increases he speed of mean-reversion and decreases he level of mean-reversion for µ under Q, such ha he Q-survival probabiliy increases. Similarly we have ha increasing β increases he level of mean reversion of µ under Q, such ha he reserves decrease. Since β = is he lowes possible value ensuring a sricly posiive moraliy inensiy under Q, oher values of β would lead o lower reserves. The parameer g, which is relaed o he unsysemaic moraliy risk changes he level of meanreversion and he volailiy of µ under Q. However, since he level of mean-reversion is very small he effec of g for reasonable values is negligible. Dependence on he muliplicaion facors From Table 8.5.2 we observe ha for a fixed Q he marke reserve is a decreasing funcion of boh ψ u and ψ d. This corresponds o our inuiion, since increasing ψ u and/or ψ d decreases he condiional bond prices and hence he marke reserve. Since he principle of super-replicaion of reinvesmen risk only considers he larges condiional bond price, his reserve is independen of ψ u and a decreasing funcion of ψ d. The principles of a level long erm yield/forward rae curve are independen of he long erm condiional bond prices and hus of ψ u and ψ d. We observe ha if ψ d is small enough.96 in his case hen he principle of reinvesmen risk super-replicaion gives a reserve larger han he principle of a level long erm yield curve. Likewise we have ha if ψ d is large enough 1 in

236 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES Reserve.95 1. 1.5 1.1 1.15 Marke reserve q=.5 Level long erm yield curve Level long erm forward rae curve Super replicaion reinvesmen risk.5..5.1 Figure 8.5.3: Iniial reserves as a funcion of β, which is associaed wih marke s aiude o sysemaic moraliy risk. β ψ u ψ d Marke reserve Super-replicaion Level forward Level yield 1.5.99 254.63 263.16 259.75 271.59 1.4.99 256.35 263.16 259.75 271.59 1.3.99 258.7 263.16 259.75 271.59 1.2.99 259.77 263.16 259.75 271.59 1.5 1 251.1 259.73 259.75 271.59 1.5.99 254.63 263.16 259.75 271.59 1.5.98 258.21 266.54 259.75 271.59 1.5.97 261.74 269.87 259.75 271.59 1.5.96 265.22 273.15 259.75 271.59 Table 8.5.2: Dependence of iniial reserves on he muliplicaion facors ψ u and ψ d. Top: Dependence on ψ u. Boom: Dependence on ψ d. All values wih wo decimals. No scaling is applied in his able.

8.5. NUMERICS 237 his case he principle of reinvesmen risk super-replicaing gives a smaller reserve han he principle of a level long erm forward rae curve. Value a Risk and ail condiional expecaion We now urn o he risk measures of Value a Risk and ail condiional expecaion. Since we only consider he ail condiional expecaion a levels given by V ar κ s, we, as explained in Secion 8.4.2, refer o he considered ail condiional expecaions as ail Value a Risk. All resuls in his secion are based on invesmen sraegies wih consan relaive porfolio weighs and 1 simulaions using he Euler mehod. Measure of risk.5 1. 1.5 2. 2.5 3. Barrier Value a Risk Barrier ail Value a Risk Terminal Value a Risk Terminal ail Value a Risk..2.4.6.8 1. Figure 8.5.4: Terminal/barrier ail Value a Risk as a funcion of κ. κ Figure 8.5.4 shows he dependence of he erminal and barrier Value a Risk and ail Value a Risk on κ for a fixed invesmen sraegy wih 4% invesed in socks and bonds, respecively, and 2% in he savings accoun. From Figure 8.5.4 we immediaely noe ha for fixed κ he barrier ail Value a Risk is larger han he erminal ail Value a Risk and ha he ail Value a Risk is larger han he corresponding Value a Risk. Furhermore, we observe ha in his case he barrier Value a Risk is larger han he erminal ail Value a Risk. For any κ i holds ha he barrier Value a Risk is greaer han or equal o he marke reserve under he minimal maringale measure, which corresponds o a horizon line a 1. This is due o he fac ha he marke reserve under he minimal maringale measure is equal o he barrier resricion. For all four risk measures we observe a seep slope for very large values of κ.

238 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES Measure of risk.9 1. 1.1 1.2 1.3 1.4 1.5 1 year Value a Risk 1 year ail Value a Risk 1 year barrier Value a Risk 1 year barrier ail Value a Risk..2.4.6.8 1. Figure 8.5.5: 1-year erminal/barrier ail Value a Risk as a funcion of κ. κ Figure 8.5.5 is essenially idenical o Figure 8.5.4, excep here he risk measures are considered on a one year ime horizon. We observe ha on a one year ime-horizon he barrier Value a Risk lies below he erminal ail Value a Risk. Apar from he erminal Value a Risk for small κ s we observe a considerably smaller magniude of he risk measures on a one year ime horizon han when considering he ime horizon of he conrac. This is also inuiively clear since he amoun of uncerainy on a one year scale is considerably smaller han on a long erm scale. As an example we menion ha on a one year ime-horizon, boh he barrier and erminal Value a Risk a level.99 are approximaely 1.35 imes he marke reserve under he minimal maringale measure, whereas on a 3 year ime-horizon his only corresponds o a barrier Value a Risk a level.35 and a erminal Value a Risk a level.85. To consider he dependence of he ail Value a Risk on he invesmen sraegy we invesigae Tables 8.5.3 and 8.5.4. Here, we have resriced ourselves o sraegies wihou shor selling and borrowing. We observe ha he proporion in he savings accoun is unchanged by moving one cell up and o he righ. Invesigaing Table 8.5.3 we find ha for a fixed proporion invesed in socks, all risk measures decrease as he proporion in bonds increases. This is no surprising, since he bonds mos closely resembles he financial naure of he benefis. Furhermore we observe ha for fixed proporion in bonds, all risk measures, excep he barrier ail Value a Risk, decrease, when he proporion in socks increases. The barrier ail Value a Risk also indicae ha i is beer o hold some socks han none. For a fixed proporion in he

8.5. NUMERICS 239 Socks Bonds.2.4.6.8 1 4.84 5.84 3.1 3.68 2.64 3.42 2.44 3.46 2.52 3.96 2.81 5.11 4.54 5.59 2.66 3.26 1.97 2.71 1.58 2.47 1.44 2.6 1.44 3.23 3.1 3.43 2.12 2.32 1.69 1.8 1.42 1.54 1.24 1.37 1.14 1.29 3.1 3.43 2.15 2.35 1.73 1.84 1.48 1.59 1.32 1.43 1.22 1.35.2 3.16 3.64 2.3 2.69 2.15 2.32 2.44 3.46 2.19 3.33 2.95 3.48 1.95 2.35 1.97 2.71 1.58 2.47 1.15 2.2 2.15 2.32 1.69 1.78 1.69 1.8 1.42 1.54 1.1 1.22 2.15 2.32 1.71 1.8 1.73 1.84 1.48 1.59 1.16 1.27.4 2.27 2.49 1.77 2. 1.72 2.8 1.76 2.38 2.1 2.35 1.45 1.72 1.18 1.57 1.8 1.67 1.67 1.74 1.37 1.43 1.19 1.25 1.7 1.15 1.67 1.75 1.4 1.45 1.23 1.27 1.12 1.19.6 1.75 1.84 1.48 1.63 1.44 1.7 1.61 1.72 1.19 1.36.97 1.25 1.37 1.41 1.17 1.21 1.5 1.11 1.38 1.42 1.19 1.23 1.8 1.13.8 1.42 1.46 1.26 1.38 1.25 1.3.95 1.9 1.17 1.2 1.3 1.7 1.18 1.21 1.5 1.8 1 1.23 1.27 1.1 1.3 1.3 1.6 1.4 1.7 Table 8.5.3: The risk measures as a funcion of he proporion invesed in he differen asses for κ =.75. Firs line for a fixed invesmen sraegy: Barrier Value a Risk barrier ail Value a Risk. Second line: Terminal Value a Risk erminal ail Value a Risk. Third line: 1-year erminal Value a Risk 1-year erminal ail Value a Risk and fourh line: 1-year barrier Value a Risk 1-year barrier ail Value a Risk. All values wih wo decimals.

24 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES Socks Bonds.2.4.6.8 1 7.98 8.9 4.89 5.15 5.61 6.33 5.53 7.27 7.45 12.6 11.49 18.5 7.9 8.72 4.39 4.82 4.71 5.56 4.38 5.67 4.97 9.75 7.53 12.95 4.27 4.44 2.67 2.76 2. 2.8 1.74 1.8 1.59 1.69 1.53 1.65 4.27 4.44 2.7 2.77 2.4 2.12 1.78 1.83 1.62 1.72 1.56 1.71.2 4.5 5. 3.38 3.78 3.66 4.39 4.99 6.62 6.6 9.9 4.38 4.82 3.15 3.55 3.1 3.62 3.83 5.51 4.84 8.5 2.61 2.71 1.91 1.99 1.62 1.66 1.43 1.49 1.4 1.52 2.62 2.71 1.92 2.1 1.64 1.67 1.47 1.52 1.47 1.55.4 2.98 3.1 2.47 2.81 2.83 3.7 3.76 4.54 2.86 3.1 2.25 2.58 2.3 2.51 3.3 3.52 1.88 1.92 1.53 1.55 1.35 1.38 1.28 1.33 1.89 1.92 1.54 1.56 1.37 1.39 1.3 1.34.6 2. 2.5 1.89 1.97 2.21 2.36 1.94 1.99 1.64 1.73 1.77 2.2 1.48 1.51 1.28 1.3 1.21 1.23 1.48 1.52 1.28 1.3 1.22 1.24.8 1.52 1.53 1.62 1.72 1.38 1.41 1.36 1.45 1.25 1.27 1.12 1.15 1.15 1.28 1.13 1.15 1 1.35 1.38 1.7 1.8 1.11 1.12 1.12 1.13 Table 8.5.4: The risk measures as a funcion of he proporion invesed in he differen asses for κ =.99. Firs line for a fixed invesmen sraegy: Barrier Value a Risk barrier ail Value a Risk. Second line: Terminal Value a Risk erminal ail Value a Risk. Third line: 1-year erminal Value a Risk 1-year erminal ail Value a Risk and fourh line: 1-year barrier Value a Risk 1-year barrier ail Value a Risk. All values wih wo decimals.

8.5. NUMERICS 241 savings accoun i is hard o say anyhing in general. From Table 8.5.4 we observe ha for a fixed proporion in bonds, he erminal and barrier ail Value a Risk indicae ha holding some socks usually lead o a lower risk measure han holding no socks. However, a large invesmen in socks is clearly more dangerous han none or few socks. This is especially obvious from he ail Value a Risks. The 1-year risk measures are almos all decreasing as a funcion of he proporion in socks. For a fixed proporion in socks all risk measures are decreasing as a funcion of he proporion in bonds and for a fixed proporion in he savings accoun, hey are decreasing as a funcion of he proporion in bonds. Life annuiies Now consider a porfolio of life annuiies, where he insured coningen on survival receives a coninuous benefi from age 6 o 9. In order o illusrae he dependence of he risk measures on κ, we, as in he case of a pure endowmen, consider an invesmen sraegy wih 4% invesed in socks and bonds, respecively, and 2% in he savings accoun. Invesigaing Figures 8.5.6 and 8.5.7 we observe he same behavior for he risk measures as in Figures 8.5.4 and 8.5.5 for he pure endowmen. Comparing he relaive magniude of he risk measures for he wo differen conracs, we observe ha for all risk measures he relaive size is larger for he annuiy han for he pure endowmen. This indicaes ha he annuiy is a more risky conrac han he pure endowmen, which corresponds o our inuiion, since he longer ime horizon of he annuiy exposes he company o more risk. Measure of risk 1 2 3 4 Barrier Value a Risk Barrier ail Value a Risk Terminal Value a Risk Terminal ail Value a Risk..2.4.6.8 1. Figure 8.5.6: Terminal/barrier ail Value a Risk for a life annuiy as a funcion of κ. κ

242 CHAPTER 8. A NUMERICAL STUDY OF RESERVES AND RISK MEASURES Measure of risk.9 1. 1.1 1.2 1.3 1.4 1.5 1.6 1 year Value a Risk 1 year ail Value a Risk 1 year barrier Value a Risk 1 year barrier ail Value a Risk..2.4.6.8 1. Figure 8.5.7: 1-year erminal/barrier ail Value a Risk for a life annuiy as a funcion of κ. κ

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