Life insurance liabilities with policyholder behaviour and stochastic rates

Transcription

1 Life insurance liabiliies wih policyholder behaviour and sochasic raes Krisian Buchard Deparmen of Mahemaical Sciences, Faculy of Science, Universiy of Copenhagen PFA Pension

2 Indusrial PhD Thesis by: Krisian Buchard Øsbanegade 11, 4. v. DK-2100 Copenhagen hp://krisian.buchard.ne Assessmen commiee: Thomas Mikosch (chairman) Professor, Universiy of Copenhagen Ragnar Norberg Professor, Universié Claude Bernard Lyon 1 Mikkel Dahl Parner, Deloie Supervisors: Mogens Seffensen Professor, Universiy of Copenhagen Thomas Møller Adjunc Professor, Universiy of Copenhagen Head of Acuarial Innovaion & Models, PFA Pension Peer Holm Nielsen Appoined Acuary, PFA Pension This hesis has been submied o he PhD School of he Faculy of Science, Universiy of Copenhagen on 31 Ocober The indusrial PhD programme was funded joinly by PFA Pension and he Innovaion Fund Denmark. Chaper 1: c K. Buchard. Chaper 2: c K. Buchard and T. Møller. Chaper 3: c 2014 Taylor & Francis. Chaper 4: c K. Buchard. Chaper 5: c 2014 Elsevier B.V. Chaper 6: c K. Buchard ISBN

3 Conens Preface Lis of papers Summary Resumé iii v vii ix 1 Inroducion Background Overview of he hesis and key ideas Life insurance cash flows wih policyholder behaviour Inroducion Life insurance seup The survival model A general disabiliy Markov model Numerical example A Proofs Cash flows and policyholder behaviour in he semi-markov life insurance seup Inroducion Seup Kolmogorov s forward inegro-differenial equaion Modelling of policyholder behaviour Numerical example A Proof of Kolmogorov s forward inegro-differenial equaion Coninuous affine processes: ransformaions, Markov chains and life insurance 79 i

4 ii CONTENTS 4.1 Inroducion Coninuous affine processes Doubly sochasic decremen Markov chains An applicaion in life insurance Dependen ineres and ransiion raes in life insurance Inroducion Coninuous affine processes The life insurance model Dependen forward raes Modelling ineres and surrender Conclusion A Forward moraliy rae for erm insurances no-so-well defined A sep forward wih Kolmogorov Inroducion The forward parial inegro-differenial equaion The doubly sochasic Markov chain seup Life insurance cash flows Semi-Markov models in life insurance The survival model and he forward moraliy rae Dependen forward raes as expecaions A Compleion of he proof of Theorem Bibliography 159

5 Preface The presen PhD hesis was wrien during he period January 2012 o Ocober 2014, where I underook he ask of sudying for he PhD degree a Deparmen of Mahemaical Sciences, Universiy of Copenhagen. This was done as a par of Innovaion Fund Denmark s indusrial PhD programme, wih PFA Pension as he indusrial parner. The research was supervised by Prof. Mogens Seffensen from he Universiy of Copenhagen, and Adj. Prof. Thomas Møller and Peer Holm Nielsen from PFA Pension. The PhD hesis consiss of an inroducion and five independen manuscrips. They appear as such and small noaional discrepancies exis beween he chapers. Two of he manuscrips are published in inernaional peer reviewed journals a he ime of wriing. I has been a pleasure for me o focus on he mahemaics of reserving in life and pension insurance for he las hree years, and I hope you will enjoy he resul as well. Acknowledgemens I hank PFA Pension for heir ineres and willingness in venuring joinly wih me ino his hree year journey. They have hosed me, and in many ohers ways helped me, for which I m graeful. I hank PFA Pension and he Innovaion Fund Denmark for funding he projec. I hank my supervisors: Mogens Seffensen for educaing me in he differen asks and challenges of becoming a researcher, Thomas Møller for irelessly providing me wih feedback and ideas concerning my research, and Peer Holm Nielsen for coninuously supporing he projec. From February o June 2013 Ann-Sophie and I had grea pleasure in visiing he Universiy of Lausanne, and I hank Hansjörg Albrecher and he Deparmen of Acuarial Science for he kind hospialiy we were shown. Wheher a he H. C. Ørsed Insiue, or a he seminars and conferences I ve aended he las hree years, I ve had numerous fruiful discussions wih many academic colleagues. They helped shape my views on life insurance, probabiliy heory, saisics and financial mahemaics, for which I m graeful. I hank all my colleagues hroughou he period, iii

6 iv PREFACE and i has been a joy coming o work each day, wheher a Dorigny, a he H. C. Ørsed Insiue or a Sundkrogsgade. I hank my family and friends who in one way or anoher suppored me during he las hree years. A las a hearfel hanks o Ann-Sophie for encouraging me o underake he PhD sudies more han hree years ago. Wihou her confidence in me, and her coninued suppor hroughou he las hree years, I would no have been able o complee he projec. Copenhagen, Ocober 2014 Krisian Buchard

7 Lis of papers As Chaper 4: [7] Coninuous affine processes: ransformaions, Markov chains and life insurance. Preprin. Produced beween January and Sepember 2012, wih cerain addiions in As Chaper 5: [8] Dependen ineres and ransiion raes in life insurance. Insurance: Mahemaics & Economics, 55: , Produced beween May 2012 and January As Chaper 3: [11] Cash flows and policyholder behaviour in he semi-markov life insurance seup. Joinly wih T. Møller and K. B. Schmid. To appear in Scandinavian Acuarial Journal. Produced beween December 2012 and July As Chaper 2: [10] Life insurance cash flows wih policyholder behaviour. Joinly wih T. Møller. Preprin. Produced beween December 2012 and Augus As Chaper 6: [9] A sep forward wih Kolmogorov. Preprin. Produced beween May 2014 and Ocober v

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9 Summary In any life and pension insurance company, i is a cenral ask o calculae he value of he liabiliies oward he policyholders. In he classic model for such valuaions, a coninuous ime Markov chain in a finie sae space describes he sae of he insured, and he ineres rae, moraliy rae, disabiliy rae, and oher ransiion raes are assumed o be deerminisic. Broadly speaking his PhD hesis consiss of various exensions of his model o address he modern needs of life insurance companies. These exensions can be caegorised ino wo ypes: he inclusion of policyholder behaviour in he model, and he modelling of he ineres and ransiion raes as sochasic processes. The forhcoming Solvency II rules from he EU have conribued o an increased focus on he modelling of policyholder behaviour, and in his hesis we focus on he so-called surrender and free policy opions. A surrender is a cancellaion of he policy in reurn for a payou of he echnical value, and a free policy conversion is a cancellaion of all fuure premiums only, whereafer he fuure benefis are reduced o accoun for he missing premiums. Since an exercise of such an opion has an effec on he liabiliies of he life insurance company, i is imporan o ake hem ino accoun when calculaing he liabiliies in he firs place. In his hesis hey are modelled as random ransiions of he Markov chain. Since he modelling of he free policy opion inroduces duraion dependence, his increases he complexiy of he model. The firs main resul is a modificaion of Kolmogorov s forward differenial equaion ha allows us o include modelling of he free policy opion wihou he exra duraion dependence. This resul is presened in boh he Markov and he semi-markov life insurance seup. A cenral par of he forhcoming Solvency II rules is he requiremen o quanify he risk inheren in he life insurance policies. Since a large par of his risk originaes from changes in he underlying ransiion raes, for example he unpredicable decline of he moraliy rae, i is necessary o model he underlying ransiion raes as sochasic processes. This significanly complicaes maers, and one way o deal wih his is o resric oneself o he class of affine processes. The second main resul is a generalisaion of he formulae which make he affine processes mahemaically convenien. This allows us o uilise affine processes for he calculaion of he reserve for more complicaed life vii

10 viii SUMMARY insurance producs. In he general seup wih sochasic ransiion raes, where we are no resriced o affine processes, we presen he hird main resul, namely Kolmogorov s forward parial inegro-differenial equaion. In life insurance companies i is, for hedging purposes, imporan o know he ineres rae sensiiviy of he liabiliies. For his sensiiviy, he expeced cash flows of he policies are ofen calculaed since i is sraighforward o calculae he ineres rae sensiiviy of a cash flow. Considering he general case of sochasic ransiion raes, i is complicaed o calculae he cash flows, since he ransiion probabiliies are required. To find hese, one could, for each fuure ime poin, solve Kolmogorov s backward parial differenial equaion. Here Kolmogorov s forward parial inegro-differenial equaion is advanageous, since i only needs o be solved once in order o obain all he relevan ransiion probabiliies for calculaing he cash flow.

11 Resumé I livsforsikrings- og pensionsselskaber er de en cenral opgave a opgøre værdien af forpligelserne over for forsikringsagerne, også kalde hensæelserne. I den klassiske model angiver en idskoninuer Markovkæde på e endelig ilsandsrum forsikringsagerens ilsand, og renen, dødeligheden, invalidieshyppigheden og andre overgangshyppigheder er anage a være deerminisiske. Grof sag besår denne ph.d.-afhandling af forskellige udvidelser af den klassiske model for a adressere nogle af de moderne krav fra livsforsikringsselskaber. Disse udvidelser kan kaegoriseres i o yper: modellering af policeageradfærd sam modellering af rene og overgangshyppigheder som sokasiske processer. De kommende Solvens II regler fra EU har bidrage il e øge fokus på modellering af policeageradfærd, og i denne afhandling fokuseres speciel på de såkalde genkøbs- og fripoliceopioner. E genkøb er en ophævelse af policen mod a få udbeal den ekniske værdi af denne, og en fripolicekonverering er en annullering af alle fremidige præmier mod a de fremidige ydelser bliver ilsvarende reducere. Da e genkøb eller en fripolicekonverering ændrer værdien af hensæelserne, er de væsenlig a modellere disse opioner i forbindelse med opgørelsen af hensæelserne. I denne afhandling modelleres genkøbs- og fripoliceopionerne som ilfældige overgange i Markovkæden. Da modellering af fripoliceopionen inroducerer varighedsafhængighed kompliceres modellen. De førse hovedresula er en modificere version af Kolmogorov s forlæns differenialligning som illader os a inkludere modellering af fripoliceopionen uden a dee fører il den eksra varighedsafhængighed. Dee resula præseneres både i Markov og semi-markov livsforsikringsseuppe. En cenral del af de kommende Solvens II regler er krave om a kvanificere risikoen i livsforsikringspolicerne. Da en sor del af denne risiko sammer fra ændringer i de underliggende overgangshyppigheder, for eksempel de uforudsigelige fald i dødeligheden, er de nødvendig a modellere overgangshyppighederne som sokasiske processer. Dee komplicerer modellen beydelig, og en måde a håndere dee er a begrænse sig il a modellere overgangshyppigheder med affine processer. De ande ande hovedresula er en generalisering af de formler der gør affine processer speciel fordelagige. Dee illader ix

12 x RESUMÉ os a udnye den affine srukur il a beregne reserver for endnu mere komplicerede livsforsikringsproduker. I den generelle model med sokasiske overgangshyppigheder, hvor vi ikke begrænser os il affine processer, præsenerer vi de redie hovedresula, nemlig Kolmogorov s forlæns parielle inegro-differenialligning. I livsforsikringsselskaber er de, for hedgingformål, væsenlig a kende hensæelsernes renefølsomhed. For a besemme denne følsomhed i praksis udregnes policernes cashflow ofe, da de er simpel a opgøre renefølsomheden af e cashflow. I den generelle model med sokasiske overgangshyppigheder er de komplicere a besemme cashflowe, ide alle overgangssandsynlighederne førs skal findes. For a udregne disse kan man, for hver fremidig idspunk, løse Kolmogorov s baglæns parielle differenialligning. I dee ilfælde er Kolmogorov s forlæns parielle inegro-differenialligning imidlerid fordelagig, da den blo skal løses en enkel gang for a udregne alle de il cashflowe nødvendige overgangssandsynligheder.

13 Chaper 1 Inroducion This PhD hesis is broadly abou echniques for valuaion of life insurance liabiliies. In his inroducion we firs ouline some background for he research, and hen we presen he key mahemaical ideas of he hesis and explain he main connecions beween he chapers. Throughou we refer o he differen chapers, and for references o oher lieraure, see he inroducions in he relevan chapers where he research is presened and relaed o exising lieraure. 1.1 Background The classic life insurance seup Given an insurance policy, where cerain fuure premiums and benefis are agreed upon, he insurance company mus calculae he value of his policy in order o know wha is required o honour he policy agreemen. The saring poin for his invesigaion is he classic Markov chain life insurance seup. Here, he sochasic process Z = (Z()) 0 is a coninuous ime Markov chain in a finie sae space J = {0, 1,..., J}, where each sae has an inerpreaion like alive, disabled, dead ec., such ha he random insurance evens can be inerpreed as ransiions beween saes. A classic example is found in Figure 1.1. Paymens are hen associaed wih sojourns in saes, b j () a ime in sae j, and ransiions beween saes, b ij () for a ransiion i j a ime. The paymen a ime can hen be wrien as db() = j 1 {Z()=j} b j () d + i,j b ij () dn ij (), where N ij is a couning process, couning he accumulaed number of ransiions i j. The reserve balance shee. A fundamenal ask of he acuary is o calculae he reserve for he This is given as he expeced presen value of he fuure paymens. 1

14 2 CHAPTER 1. INTRODUCTION 0, acive 1, disabled 2, dead Figure 1.1: The disabiliy Markov model. The arrows represen he possible ransiions. Condiional on being in sae i a ime, his is usually denoed V i ( ), and is given by [ ] V i ( ) = E e r(s) ds db() Z( ) = i, where r() is he ineres rae, which in he classic seup is assumed o be deerminisic. I is well known ha V i () saisfies Thiele s differenial equaion Sochasic ransiion raes Sochasic ineres rae In Denmark and several oher counries, i is mandaory o calculae he reserve in a marke consisen manner. Thus, he ineres rae assumpions used for reserving mus be consisen wih he ineres rae assumpions used for rading ineres rae derivaives on he financial marke. In paricular his means ha he ineres rae r() is sochasic, and he resuls from he classic seup do no seem applicable. Luckily, if he sochasic ineres rae is independen of he Markov chain Z, which is a reasonable assumpion, a simple resul shows ha one can simply replace he ineres rae r() wih he forward ineres rae a ime, f (). This is defined hrough he relaion E [e ] r(s) ds F( ) = e f (s) ds, where (F()) 0 is he filraion generaed by (r()) 0, and he expecaion is aken using he risk neural measure. The reserve V i ( ) is found similar o in he classic seup, bu we mus condiion on F( ) as well. Condiioning on he pah of Z and using he independence, we find ha [ V i ( ) = E [ = E ] ] r(s) ds F( ), Z db() F( ), Z( ) = i ] f (s) ds db() F( ), Z( ) = i. E [e e Thus, using f u () as he ineres rae in he classic seup and considering u as fixed, he usual resuls hold, e.g. Thiele s differenial equaion, and in his way i is simple o exend he classic seup o calculae he reserve wih a marke consisen ineres rae model.

15 1.1. BACKGROUND 3 Wih a sochasic ineres rae assumpion, he reserve is dependen on he forward ineres rae, which changes every day. In he recen years his has been decreasing, which has resuled in an increasing reserve. In order for he life insurance companies o handle his, i is vial o hedge he ineres rae risk, for example by invesing in bonds, which also increase in value when he ineres rae decreases. A simple way o precisely manage his hedge in pracice is o calculae he ineres rae sensiiviy of he liabiliies and inves in asses such ha he asse porfolio has a similar sensiiviy wih respec o he ineres rae. I is hus cenral o calculae he ineres rae sensiiviy of a porfolio of life insurance conracs. The cash flow A compuaionally efficien way of calculaing he ineres rae sensiiviy of a large porfolio of life insurance policies is o calculae he expeced cash flow (which we simply refer o as he cash flow) of each policy. This cash flow can be easily discouned wih differen ineres raes in order o measure he sensiiviy. The cash flow consiss of he expeced paymens a fuure imes >, and condiional on being in sae i a ime, we denoe his by a i (, ), saisfying a i (, ) d = E [ db() Z( ) = i ]. As can be seen from e.g. Chaper 2, a i (, ) can be found by firs calculaing he ransiion probabiliies for Z, by e.g. Kolmogorov s forward differenial equaion. The reserve can hen be calculaed by a simple discouning, V i ( ) = e f (s) ds a i (, ) d. In all of he chapers below, one of he main objecives is o calculae he reserve in he respecive model of he chaper. In Chapers 2, 3 and 6, he primary focus is on efficien calculaion of he cash flow. Sochasic moraliy As he reserves in he balance shee change due o a changing ineres rae, hey also change due o flucuaions in he underlying assumpions of he ransiion raes of Z. In he classic seup he ransiion raes are deerminisic. However, an example is he moraliy, which has been declining he las cenuries a an unpredicable pace. Thus, even hough mos agree ha he moraliy will coninue o decline, i is difficul, if no impossible, o predic how much. Wha is imporan from he insurer s and regulaor s perspecive is o be aware of he risk originaing from he unpredicable moraliy, and a cenral way of modelling his mahemaically is by leing he moraliy ransiion rae be sochasic. In his siuaion, he classic seup again breaks down. In he simple survival model, a forward moraliy rae can be defined, analogously o he forward ineres rae, and he formulae of he classic seup hen hold.

16 4 CHAPTER 1. INTRODUCTION Bu in more advanced models, e.g. in he disabiliy model from Figure 1.1, he forward moraliy rae approach does no direcly work if he moraliy as acive and as disabled differs. I is worh noing ha he rules on marke consisen reserving in principle also apply for he moraliy elemen. However, as here is ye o be a liquid marke for rading moraliy risk, here does no exis a unique risk neural measure for he moraliy elemen. In ha case, expecaion is aken using he physical measure, which is a bes esimae, and some sor of safey margin may be added insead. Sysemaic risk According o he forhcoming Solvency II rules from he EU, he insurance company mus measure 1-year risk and have enough capial o wihsand a loss corresponding o he 99.5% quanile. For large insurance companies, he main par of his loss originaes from he so-called sysemaic risk, which can be hough of as he risk affecing all policies in he same way. As he ransiion raes of Z in pracice change during he year, his becomes he main conribuion o he sysemaic risk. The reserve in one year, V ( + 1), is dependen on his developmen. Thus i is sochasic, and i is he 99.5% quanile in V ( + 1) ha consiues he quanificaion of he sysemaic risk in a Solvency II conex. For handling his problem mahemaically, i is essenial o model he ransiion raes as sochasic processes, and be able o calculae he reserves in such a model. In Chaper 4 we see how he class of affine processes can be uilised for modelling dependen sochasic ineres and ransiion raes. This sudy is coninued in Chaper 5, where we also consider he Solvency II capial requiremen Policyholder behaviour In he recen years, in par because i is required by he forhcoming Solvency II rules, here has been a focus on correcly modelling some of he policyholders opions inheren in life insurance policies. Two of hese opions, he surrender and free policy opions, have aained a paricular focus in Denmark. The surrender opion is a righ of he policyholder o cancel all fuure premiums, and in reurn have he (echnical) value of he policy paid ou immediaely. This occurs ofen in Danish labour marke pensions when a policyholder ransfers his policy from one pension fund o anoher, ypically in connecion wih a job change. The free policy opion is a righ of he policyholder o cancel all fuure premiums, where in reurn all fuure benefis are reduced o accoun for he missing premiums. By considering he surrender opion as a random ransiion of he Markov chain Z, i is readily handled in he classic seup, since he paymen upon surrender ypically is a deerminisic funcion of ime. The free policy opion can also be modelled as a random ransiion of Z if he classic seup is exended o include dependence on he duraion

17 1.1. BACKGROUND 5 3, surrender 0, acive 1, disabled 2, dead J 7, surrender free policy 4, acive free policy 5, disabled free policy 6, dead free policy J f Figure 1.2: The 8-sae Markov model, wih disabiliy, surrender and free policy. Arrows illusrae he possible ransiions, and each arrow can be associaed wih erms in Kolmogorov s forward differenial equaion. The free policy opion can be handled by modifying a erm of he ransiion 0 4. since he ime of he free policy conversion, which brings us o he so-called semi-markov seup. Modelling of he policyholder s opions can hen be included in he disabiliy model from Figure 1.1 by an exension o he sae space from Figure 1.2. The main challenge of his seup is ha he paymens are reduced upon he free policy conversion, i.e. he ransiion 0 4, and his reducion is dependen on he ime of conversion. Therefore he paymens in he free policy saes are duraion dependen, which is no par of he classic seup, bu can be handled by he semi-markov seup. The semi-markov life insurance seup is an exension of he classic life insurance seup o include he duraion in cerain saes, such ha he ransiion raes and he paymens are allowed o depend on his duraion. However, he semi-markov seup is more complicaed han he classic Markov life insurance seup, and in paricular he compuaional ime required is higher. The main resul of Chapers 2 and 3 is a modificaion of Kolmogorov s forward differenial equaion in order o accoun for his duraion dependence and avoid dealing wih he complicaions of he semi-markov formulae. In Chaper 5 we also consider surrender modelling, bu here we le he surrender rae be sochasic and dependen on he ineres rae. In his model, we firs see ha he usual forward ineres rae is no applicable due o he dependence. We inroduce so-called dependen forward raes. Furher, we examine he Solvency II capial requiremen and presen a numerical example.

18 6 CHAPTER 1. INTRODUCTION 1.2 Overview of he hesis and key ideas The classic seup is well sudied in he lieraure. In broad erms, his PhD hesis deals wih differen ways of generalising he classic seup, such ha some of he challenges discussed above can be handled. The focus is primarily on calculaion of he reserve and he cash flow in pracice. Each following chaper of he hesis consiss of a sand-alone paper. The papers are presened here no in chronological order of producion 1, bu in he order which he auhor hinks is bes suied for presening a combined sory. Chapers 2 and 3 mainly deal wih he quesion of including he modelling of policyholder behaviour, which is done in a seup wih deerminisic ransiion raes, and he semi-markov seup is also included. Chapers 4 and 5 mainly deal wih modelling he ineres and ransiion raes as sochasic and dependen processes, in paricular as affine processes, wih he aim of finding efficien formulae for calculaing he life insurance reserve. Chaper 6 somewha unifies hese wo opics in he sense ha Kolmogorov s forward differenial equaion is generalised, such ha general (i.e. boh affine and non-affine) diffusion processes can be used o model he ransiion raes. Boh he semi-markov seup and he seup wih sochasic ransiion raes can be considered as special cases of his seup. The main conribuions of his hesis can a a srech be formulaed as hree mahemaical ideas. In he res of he inroducion, we ry o explain hese hree ideas in an inuiive way. In addiion, we highligh oher relevan pars and connecions in he hesis. Some mahemaical precision is sacrificed, and for a precise reamen he reader is referred o he respecive chapers. The firs mahemaical idea is he duraion eliminaion allowing us o efficienly model he free policy opion. This is inroduced in Secion below. In Chaper 2 his idea is made precise in he classic Markov seup, and in Chaper 3 he resul is presened in he semi-markov seup. The second mahemaical idea is he generalisaion of cerain expecaions of ransformaions of affine processes ha allows for a more general applicaion of affine processes in life insurance. This is inroduced in Secion below. In Chaper 4 we presen he heory of affine processes and he resuls, and in Chaper 5 he resuls are applied more horoughly in life insurance, including consideraions on he Solvency II risk quanificaion and a numerical example. The hird mahemaical idea is he generalisaion of Kolmogorov s forward differenial equaion o handle e.g. he doubly sochasic Markov chain seup. This is inroduced in Secion below and is made precise in Chaper 6. 1 The chronological order of he chapers is: 4, 5, 3, 2 and 6.

19 1.2. OVERVIEW OF THE THESIS AND KEY IDEAS Modelling of he free policy opion When he free policy opion is modelled as a random ransiion of he Markov chain Z, as in he sae space shown in Figure 1.2, he paymens in he free policy saes (4, 5 and 6) become dependen on he ime of he free policy conversion. This can be handled by he more complex semi-markov seup. The firs resul is ha we don have o adop he semi-markov seup for modelling he free policy opion, bu can do somehing which in comparison is simple. The idea behind he resul is o consider and modify Kolmogorov s forward differenial equaion for he ransiion probabiliies of Z. Upon a free policy conversion a ime, all fuure benefis are reduced by a deerminisic facor ρ(). Considering paymens in he free policy saes only, he cash flow can be wrien as a i (, ) = [ 1{Z()=j} ρ( W ()) Z( ) = i ] b j () + µ jl ()b jl (). j J f E Here W () is he ime since he free policy conversion, hus W () is he ime of he ransiion 0 4. Also, µ jl () are he ransiion raes of Z. If ρ() = 1 for all, he expecaion is simply he ransiion probabiliy for sae i j from ime o, and his can be calculaed wih Kolmogorov s forward differenial equaion, d d p ij(, ) = p ij (, ) l;l j l;l j µ jl () + p il (, )µ lj (). The rick is o see ha he expecaion is essenially he ransiion probabiliy, bu muliplied wih ρ() a he ime of he ransiion 0 4. Inerpreing Kolmogorov s differenial equaion, we idenify he firs of he wo erms, which is negaive, as probabiliy mass leaving sae j a ime o any sae l. The second erm is idenified as probabiliy mass enering sae j a ime from any sae l. Thus if we replace he erm wih he erm p i0 (, )µ 04 (), p i0 (, )µ 04 ()ρ(), we have effecively muliplied all he probabiliy mass enering sae 4 by he value of ρ a he ime of ransiion. The resuling modified ransiion probabiliies are hen in all free policy saes reduced by ρ a he ime of he ransiion 0 4, and his is essenially he resul. Chaper 2 consiss of wo pars and a numerical example. In he second par, we show he main resul, which is he effecive eliminaion of he duraion. This is done in he l;l j

20 8 CHAPTER 1. INTRODUCTION classic 3-sae disabiliy model. In he firs par of he chaper, we show a simple way of manipulaing exising cash flows wihou inheren policyholder behaviour in order o include policyholder behaviour. This is done in a survival model, however applying he mehod for cash flows from a disabiliy model can be hough of as an approximaion which is easy o handle. In he end of he paper a numerical example is sudied, where he approximaion is compared wih he correc formulae from he second par of he paper. In Chaper 3 he ouse is he semi-markov seup. The purpose of his chaper is wo-fold. Firs, we presen Kolmogorov s forward inegro-differenial equaion, which is he generalisaion of Kolmogorov s forward differenial equaion o he semi-markov seup, and we discuss how o solve i in pracice. Second, we apply he same duraion eliminaion rick as above o he semi-markov model; modelling free policy behaviour in he semi-markov seup sill inroduces an exra duraion dependence, in pracice giving us a double duraion seup, and applying he resul from above we can include he modelling of free policy behaviour wihou he exra duraion dependence. Some pars of Chaper 2 can be hough of as a pedagogical version of Chaper 3, where we can discuss he resuls wihou having o deal wih he complicaions of he semi-markov seup. In paricular, he conen of Secion 2.4 consiues a special case of Chaper Affine processes in life insurance One of he disadvanages of he policyholder behaviour model from Chapers 2 and 3 is ha he surrender and free policy opions occur randomly and independen of he ineres rae. A simple, reasonable conjecure is ha if he marke ineres rae is high, hen a low ineres rae guaranee inheren in a life insurance policy is of lile value, and surrender raes will be high. On he oher hand, if he marke ineres rae becomes lower han he guaraneed ineres rae, hen he guaranee will be of high value, and he likelihood of surrender may be low. This effec can be modelled wih sochasic and posiively correlaed ineres and surrender raes, and his is a moivaion for modelling he ineres and ransiion raes as sochasic processes. Oher reasons for modelling he ineres and ransiion raes as sochasic processes include risk measuremen of he sysemaic risk, e.g. in a Solvency II conex, and also he sudy of hedging possibiliies. In Chapers 4 and 5 we sudy he applicaion of affine processes for valuaion of life insurance liabiliies. Assume X = (X()) 0 is an affine sochasic process, possibly mulidimensional, and ha he ineres and ransiion raes are funcions of X(): r(, X()) and µ ij (, X()). Condiional on he whole pah of X, we assume ha Z is a Markov chain. Uncondiionally Z is no longer a Markov chain, and he resuls from he classic life insurance seup do no hold. In paricular, we can use neiher he classic

21 1.2. OVERVIEW OF THE THESIS AND KEY IDEAS 9 version of Thiele s differenial equaion nor he classic versions of Kolmogorov s forward and backward differenial equaions. The idea is o consider hierarchical models (or decremen models) where you canno reurn o a sae afer you have lef i. In ha case, boh he reserve and he ransiion probabiliies can be wrien as expecaions of inegral expressions. For example, for a erm insurance wih payou 1 and level premium π, one can show ha he reserve, condiional on being alive, can be wrien as [ T V () = E e ] s (r(u,x(u))+µ 01(u,X(u))) du (µ 01 (s, X(s)) π) ds X(), using an appropriae marke-consisen measure. Inerchanging expecaion and inegraion we see ha we mus calculae he following expecaions, E [e ] s (r(u,x(u))+µ 01(u,X(u))) du X(), (1.2.1) E [e ] s (r(u,x(u))+µ 01(u,X(u))) du µ 01 (s, X(s)) X(). (1.2.2) For more complicaed models, e.g. a erm insurance in a disabiliy model, we also need o calculae expressions similar o E [e s (r(u,x(u))+µ 01(u,X(u))+µ 02 (u,x(u))) du µ 01 (s, X(s)) e ] v s (r(u,x(u))+µ 12(u,X(u))) du µ 12 (v, X(v)) X(), (1.2.3) for < s < v. I is well known ha if r(, x) and µ ij (, x) are affine funcions of x, hen (1.2.1) can be wrien as an exponenial affine funcion in X(), E [e s (r(u,x(u))+µ 01(u,X(u))) du ] X() = e φ(,s)+ψ(,s) X(), where φ and ψ are soluions o a se of Riccai differenial equaions. This relaion is closely coupled wih he definiion of affine processes. Furhermore, a similar relaionship for (1.2.2) exiss, E [e ] s (r(u,x(u))+µ 01(u,X(u))) du µ 01 (s, X(s)) X() ( ) = e φ(,s)+ψ(,s) X() A 01 (, s, s) + B 01 (, s, s) X(), (1.2.4) where A and B are soluions o a se of linear differenial equaions. The main resul of Chaper 4 is wofold. Firs, we presen a new proof of (1.2.4). Second, his proof is consrucive, and he idea can be applied o obain a similar relaion for

22 10 CHAPTER 1. INTRODUCTION (1.2.3), E [e s (r(u,x(u))+µ 01(u,X(u))+µ 02 (u,x(u))) du µ 01 (s, X(s)) e ] v s (r(u,x(u))+µ 12(u,X(u))) du µ 12 (v, X(v)) X() ( ) = e X()( φ(,v)+ψ(,v) A 01 (, v, s) + B 01 (, v, s) X() ( ) ) A 12 (, v, v) + B 12 (, v, v) X() + C(, v, s, v) + D(, v, s, v) X(), where he funcions C and D are soluions o a se of differenial equaions. The idea can be reapplied o obain relaions for similar expecaions, where r(, X()) and µ ij (, X()) appear inside an exponenial funcion as erms and ouside an exponenial funcion as facors. Using hese hree relaions, we calculae ransiion probabiliies and life insurance reserves using affine ineres and ransiion raes, and hese mehods are explored in Chapers 4 and 5. In Chaper 4, he heory is presened wih resuls and proofs. Furher generalisaions are discussed, and a numerical example of a doubly sochasic Markov chain is carried ou o illusrae he numerical advanage compared wih solving a parial differenial equaion or applying Mone Carlo mehods. In Chaper 5, we apply he relaions in life insurance and discuss several aspecs of he heory. Firs, from relaion (1.2.4) i is naural o define so-called dependen forward raes, which generalise he concep of he forward ineres rae and he forward moraliy rae o he case of dependen raes. This is discussed and relaed o oher proposals of forward raes in a seup wih dependen ineres and ransiion raes. Second, we sudy an example wih surrender modelling. Leing he ineres and surrender rae be dependen, he effec of he dependence on he reserve is calculaed numerically. In he las par of Chaper 5, we sudy he sysemaic risk in a Solvency II conex, and show how he affine processes can be exploied in measuring his risk. We also sudy numerically how he dependence of he ineres and surrender rae affecs he sysemaic risk, defined as he 99.5% quanile in he loss over one year Generalisaion of Kolmogorov s forward differenial equaion In he life insurance seup wih affine sochasic ransiion raes in Chapers 4 and 5, here are wo main resricions. Firs, he raes mus be an affine ransformaion of an affine process X. However, we would like o consider he case where he ransiion raes are no necessarily affine. Second, we need a hierarchical model in order o apply he mehods, so for example in he classic 3-sae disabiliy model as shown in Figure 1.1, we canno include recovery o he original acive sae. In Chaper 6 we generalise

23 1.2. OVERVIEW OF THE THESIS AND KEY IDEAS 11 Kolmogorov s forward differenial equaion, such ha i handles he general case where he ransiion raes are any diffusion process. Also, here is no resricion o hierarchical models. This resul is of paricular use for calculaing cash flows, and i generalises Kolmogorov s forward differenial equaions from he classic Markov and he semi-markov life insurance seup, wherein he ransiion raes are deerminisic. The key idea is o examine Kolmogorov s forward differenial equaions. To obain a consisen noaion in his inroducion, he noaion differs slighly from he noaion used in some of he following chapers. In he classic seup where Z is a Markov chain wih ransiion raes µ ij (), i reads d d p ij(, ) = p ij (, ) µ jl () + p il (, )µ lj (), (1.2.5) l J l J l j l j where p ij (, ) = P(Z() = j Z( ) = i). We inerpre he righ hand side as consising of essenially wo pars: wha he locaion of he process is, and where i is headed. The probabiliy densiy p ij (, ) conains informaion abou where Z() is, and he ransiion raes µ Z( )j () deermine where Z() is headed. The firs erm, which is negaive, is ransiions ou of sae j o any sae l, and he second erm is ransiions from any sae l o sae j. The ransiion raes µ Z( )j () depend on he locaion of Z( ), hus an idea is ha his consrucion is possible because Z is a Markov chain: knowledge of he curren locaion of Z( ) yields he ransiion raes ha deermine where Z() is headed. In he more complex semi-markov case, he ransiion probabiliies are soluions o Kolmogorov s forward inegro-differenial equaion, d d p ij (, u,, D() ) D() ( = p ij, u,, dz ) µ j (, z) 0 + l J l j u+ 0 p il (, u,, dz ) µ lj (, z), (1.2.6) where p ij (, u,, D() ) = P(Z() = j, U() D() Z( ) = i, U( ) = u), and D() = d +, see Theorem This differenial equaion yields he ransiion probabiliies for he process (Z, U). We can again inerpre he righ hand side as consising of he wo pars: he locaion of he process, and where he process is headed. Again, he ransiion densiy p ij (, u,, dz) deermines he locaion in he sae space. The main difference from above is ha he ransiion raes µ Z( )j (, U( )) deermining where he process is headed are dependen on he curren duraion U( ), hus making i a requiremen o know he locaion of U( ) in he sae space. In paricular, i does no

24 12 CHAPTER 1. INTRODUCTION seem possible o obain a differenial equaion for Z alone. By iself, Z is no a Markov process, and we migh again infer ha he Markov assumpion of he process is essenial. Consider now a diffusion process X, saisfying he sochasic differenial equaion dx() = β(, X()) d + σ(, X()) dw (), where β(, x) and W () are d-dimensional and σ(, x) is (d d)-dimensional. This is also a Markov process, and we know ha given cerain regulariy condiions, he probabiliy densiy saisfies he Fokker-Planck parial differenial equaion, p(, x,, x) = i ( βi (, x)p(, x,, x) ) x i i,j 2 x i x j ( ρij (, x)p(, x,, x) ), (1.2.7) where p(, x,, x) dx = P (X() = dx X( ) = x ) and ρ(, x) = σ(, x)σ(, x). Again, he righ hand side can be inerpreed as consising of he wo pars, locaion and direcion. The ransiion densiy appears on he righ hand side, ogeher wih he drif β(, x) and he diffusion erm ρ(, x), which, hrough he sochasic differenial equaion, deermine where X() is headed. We can repea, ha o deermine he value of β(, x) and ρ(, x), we need o know where in he sae space X() is. Common for he hree examples above is ha he Markov assumpion seems necessary if we wan o deermine where he process is headed solely based on informaion abou where he process is now. In he general case of a doubly sochasic Markov chain, he ransiion raes of Z are dependen on X, and we wrie µ Z( )j (, X()). In his seup, (Z, X) is a Markov process, and he idea is ha we should be able o deermine where (Z(), X()) is headed since, if we know he locaion of (Z( ), X()), we know he ransiion raes µ Z( )j (, X()) of Z, as well as he drif and diffusion erm of X. I urns ou ha his is indeed doable, and we refer o he resul as Kolmogorov s forward parial differenial equaion, p(, k, x,, k, x) = d i= l J l k l J l k ( βk (, x)p(, k, x,, k, x) ) x i d i,j=1 2 ( ρij (, x)p(, k, x,, k, x) ) x i x j µ lk (, x)p(, k, x,, l, x) µ kl (, x)p(, k, x,, k, x), (1.2.8)

25 1.2. OVERVIEW OF THE THESIS AND KEY IDEAS 13 where p(, k, x,, k, x) dx = P (X() = dx, Z() = k X( ) = x, Z( ) = k ). This is a key version of he main resul of Chaper 6. We see ha he righ hand side is essenially (1.2.5) and (1.2.7) combined. In Chaper 6 we firs consider a general jump-diffusion, and assuming cerain regulariy condiions, we obain a forward parial inegro-differenial equaion for he ransiion probabiliy. Similar resuls seem o exis in he lieraure, however he auhor has no been able o find such in he form presened and proven in Chaper 6. We hen consider he doubly sochasic Markov chain wih diffusion driven ransiion raes, and in paricular obain (1.2.8) as a special case of he general forward parial inegro-differenial equaion. We relae he resuls o he life insurance seup and see ha we can easily calculae he cash flow of a life insurance policy in he general doubly sochasic seup. This is he naural generalisaion of he seup excluding policyholder behaviour from Chapers 2 and 3. However, he auhor believes ha he duraion eliminaion resul from Chapers 2 and 3 can be generalised o he doubly sochasic seup; his is posponed for furher research. We remark ha (1.2.8) does no seem direcly conneced o (1.2.6), which is because he semi-markov seup is no a special case of he doubly sochasic Markov chain seup. However, he semi-markov process is a special case of a general jump-diffusion, and (1.2.6) is indeed a special case of he general forward parial inegro-differenial equaion ha is presened in Secion 6.2; his connecion is sudied in Secion 6.5. Forward raes as expecaions In he las par of Chaper 6 we show ha he forward moraliy rae f (), defined hrough he relaion E [e ] µ(s,x(s)) ds X() = e f (s) ds, (1.2.9) can be represened as he expecaion of he moraliy rae µ(, X()), condiional on survival, f () = E [ µ(, X()) Z() = 0, X( ) ]. (1.2.10) This represenaion follows by comparing (1.2.8) wih he derivaive of (1.2.9). Furher, his is generalised o he dependen forward raes from Chaper 5, and we find ha when Z is a doubly sochasic Markov chain in a survival model wih muliple causes of deah, he expecaion in (1.2.10) equals he dependen forward rae.

26

27 Chaper 2 Life insurance cash flows wih policyholder behaviour This chaper is based on he paper [10], wrien joinly wih T. Møller. Absrac The problem of valuaion of life insurance paymens wih policyholder behaviour is sudied. Firs a simple survival model is considered, and i is shown how cash flows wihou policyholder behaviour can be modified o include surrender and free policy behaviour by calculaion of simple inegrals. In he second par, a more general disabiliy model wih recovery is sudied. Here, cash flows are deermined by solving a modified Kolmogorov forward differenial equaion. This mehod has recenly been suggesed in a more general semi-markov seup in Buchard e al. [11]. We conclude he paper wih numerical examples illusraing he mehods proposed and he impac of policyholder behaviour. 2.1 Inroducion In a classic Markov chain muli-sae life insurance seup, we show how policyholder behaviour can be included in cash flow projecions. We sudy wo approaches. Firs, we consider he survival model and show how simple inegral expressions can solve he problem. Second, we consider he disabiliy model, and presen cerain ordinary differenial equaions ha solve he problem. We discuss how he inegral expressions originaing from he survival model can be used o approximae he more correc modelling in he disabiliy model, for a very simple ye effecive ype of policyholder behaviour modelling. 15

28 16 CHAPTER 2. CASH FLOWS WITH POLICYHOLDER BEHAVIOUR In his paper, he policyholder behaviour consiss of wo policyholder opions. Firs, he surrender opion, where he policyholder may surrender he conrac cancelling all fuure paymens and insead receiving a single paymen corresponding o he value of he conrac on a echnical basis. Second, he free policy opion 1, where he policyholder may cancel he fuure premiums, and have he benefis reduced according o he echnical basis. Policyholder modelling has a significan influence on fuure cash flows. If he echnical basis differs considerably from he marke basis, policyholder behaviour can also have a subsanial impac on he marke value of he conrac. The policyholder behaviour is modelled as random ransiions in a Markov model as in [35] and [26], and raionaliy behind surrender and free policy modelling is hus disregarded. An empirical analysis of policyholder behaviour in he German marke and furher references on policyholder modelling can be found in [18]. In conras, one can consider surrender and free policy exercises as raional, where hey purely occur if i is beneficial for he policyholder wih some objecive measure, see e.g. [45]. For an inroducion o policyholder modelling and a comparison of various approaches, see [38] and references herein. Aemps o couple he wo approaches have been made for surrender behaviour, where surrender occurs randomly, bu where he probabiliy is somewha conrolled by raional facors, e.g. [23] and [8]. From a Solvency II poin of view, he modelling of policyholder behaviour is required, see Secion 3.5 in [12]. In he firs par of he paper a simple survival model is considered. We calculae cash flows wihou policyholder behaviour as inegral expressions. Then we exend he model by including firs surrender behaviour and hen boh surrender and free policy behaviour. We see ha hese exensions can be obained via simple modificaions of he cash flows wihou policyholder behaviour. This can be viewed as a formula for exending curren cash flows wihou policyholder behaviour. However, his modificaion of he cash flows is only correc for he survival model, and no for e.g. a disabiliy model. If his mehod is applied o cash flows from a disabiliy model, i could be viewed as an approximaion o a more correc way of modelling policyholder behaviour. Also, we show ha he cash flows wih policyholder behaviour can be derived from cash flows wih surrender behaviour. This mehod can be used in he case where one has access o cash flows wih surrender behaviour bu no free policy behaviour. In pracice, many life insurance companies work wih cash flows wihou policyholder behaviour, hence, he proposed mehod may be viewed as a simple alernaive o full, combined modelling of policyholder behaviour and insurance risk. The qualiy of hese formulae as an approximaion is assessed numerically in he las par of he paper. This issue is also sudied numerically in [26], where hey examine ways o simplify he calculaions when modelling policyholder behaviour. In he second par, we consider he more correc way of modelling policyholder behaviour 1 he free policy opion is someimes referred o as a paid-up policy in he lieraure.

29 2.1. INTRODUCTION 17 in a muli-sae life insurance seup. This model is presened in [11] for he general semi-markov seup, and here we presen he special case of a Markov process for he disabiliy model wih recovery. Wihin his seup, he ransiion probabiliies are firs calculaed using Kolmogorov s forward differenial equaion, and hen he cash flow can be deermined. When including policyholder behaviour, duraion dependence is inroduced since he fuure paymens are affeced by he ime of he free policy conversion. This complicaes calculaions significanly. We presen he main resul from [11] ha allows us o effecively dismiss he duraion dependence and calculae cash flows wih policyholder behaviour by simply calculaing a slighly modified version of Kolmogorov s forward differenial equaion. The complexiy of he calculaions is herefore no increased significanly by inclusion of policyholder behaviour. In he hird par of he paper a numerical example is sudied, which illusraes, in par he imporance of including policyholder modelling when valuaing cash flows, and in par he qualiy of he approximaing cash flows obained by applying he inegral expressions from he firs par o cash flows wihou policyholder behaviour from a disabiliy model. We see ha he srucure of he cash flows changes significanly in our example, and he dollar duraion measuring ineres rae risk is reduced by abou 66%, when including policyholder behaviour. For hedging of ineres rae risk, i is hus essenial o consider policyholder behaviour. We compare he approximae mehod wih he correc approach of solving he modified Kolmogorov differenial equaions, and find cash flows wih policyholder behaviour in a disabiliy model. We find ha in our example he approximaion is very precise. Since he resuls obained in he second par of he paper can be viewed as a special case of he ones presened in he more general semi-markov framework in [11], we briefly describe he main differences beween he wo presenaions. As menioned above, in [11] he Kolmogorov forward inegro-differenial equaion in he semi-markov framework is sudied and a modified version is presened ha allows for he inclusion of policyholder behaviour in an efficien manner. The presen paper conains hree pars. In he firs par, we discuss a simple approach o modelling policyholder behaviour, which is based on a modificaion of he underlying cash flows wihou policyholder behaviour. This consrucion provides simple pedagogical inerpreaions for he various new erms ha arise in he cash flows when we inroduce policyholder behaviour. Similar resuls are presened in [26], who compare wih alernaive modificaions of he cash flows in more absrac models, such as he disabiliy model. The second par presens he modified Kolmogorov equaion in he classic Markov model. We believe ha he presenaion in his par could be accessible o a wider audience han [11], since we can avoid he more echnical issues relaed o he semi-markov framework wih duraion dependence. This leads o simpler resuls ha are more easy o inerpre, implemen and more direcly applicable han he semi-markov framework. Moreover, he proofs are more direc and

30 18 CHAPTER 2. CASH FLOWS WITH POLICYHOLDER BEHAVIOUR should be easy o follow for readers familiar wih he classic Markov models as presened in e.g. [31]. 2.2 Life insurance seup The general seup is he classic muli-sae seup in life insurance, consising of a Markov process, Z, in a finie sae space J = {0, 1,..., J} indicaing he sae of he insured, see [27]. We associae paymens wih sojourns in saes and ransiions beween saes, and his specifies he life insurance conrac. We go hrough he seup and basic resuls; for more deails, see e.g. [39], [31] or [38]. Assume ha Z is a Markov process in J, and ha Z(0) = 0. The ransiion probabiliies are defined by p ij (s, ) = P (Z() = j Z(s) = i), for i, j J and s. Define he ransiion raes, for i j, 1 µ ij () = lim h 0 h p ij(, + h), µ i. () = j J j i µ ij (). We assume ha hese quaniies exis. Define also he couning processes N ij (), for i, j J, i j, couning he ransiions beween sae i and j. They are defined by N ij () = # {s (0, ] Z(s) = j, Z(s ) = i}, where we have used he noaion f( ) = lim h 0 f( h). The paymens consis of coninuous paymen raes during sojourns in saes, and single paymens upon ransiions beween saes. Denoe by b i () he paymen rae a ime if Z() = i, and le b ij () be he paymen upon ransiion from sae i o j a ime. Then, he accumulaed paymens a ime are denoed B(), and are given by db() = i J 1 {Z()=i} b i ()d + i,j J i j b ij ()dn ij (). (2.2.1) Posiive values of he paymen funcions b i () and b ij () correspond o benefis, while negaive values corresponds o premiums. I is also possible o include single paymens during sojourns in saes, bu ha is for noaional simpliciy omied here.