ON THE RISK-NEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS WITH NUMERICAL METHODS IN VIEW ABSTRACT KEYWORDS 1. INTRODUCTION

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1 ON THE RISK-NEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS WITH NUMERICAL METHODS IN VIEW BY DANIEL BAUER, DANIELA BERGMANN AND RÜDIGER KIESEL ABSTRACT I rece years, marke-cosise valuaio approaches have gaied a icreasig imporace for isurace compaies. This has riggered a icreasig ieres amog praciioers ad academics, ad a umber of specific sudies o such valuaio approaches have bee published. I his paper, we prese a geeric model for he valuaio of life isurace coracs ad embedded opios. Furhermore, we describe various umerical valuaio approaches wihi our geeric seup. We paricularly focus o coracs coaiig early exercise feaures sice hese prese (umerically) challegig valuaio problems. Based o a example of paricipaig life isurace coracs, we illusrae he differe approaches ad compare heir efficiecy i a simple ad a geeralized Black-Scholes seup, respecively. Moreover, we sudy he impac of he cosidered early exercise feaure o our example corac ad aalyze he ifluece of model risk by addiioally iroducig a expoeial Lévy model. KEYWORDS Life isurace; Risk-eural valuaio; Embedded opios; Bermuda opios; Nesed simulaios; PDE mehods; Leas-squares Moe Carlo. 1. INTRODUCTION I rece years, marke-cosise valuaio approaches for life isurace coracs have gaied a icreasig pracical imporace. I 1, he Europea Uio iiiaed he Solvecy II projec o revise ad exed curre solvecy requiremes, he ceral ieio beig he icorporaio of a risk-based framework for adequae risk maageme ad opio pricig echiques for isurace valuaio. Furhermore, i 4 he Ieraioal Accouig Sadards Board issued he ew Ieraioal Fiacial Reporig Sadard (IFRS) 4 (Phase I), which is also cocered Asi Bullei 4(1), doi: 1.143/AST by Asi Bullei. All righs reserved.

2 66 D. BAUER, D. BERGMANN AND R. KIESEL wih he valuaio of life isurace liabiliies. Alhough Phase I jus cosiues a emporary sadard, expers agree ha fair valuaio will play a major role i he fuure permae sadard (Phase II), which is expeced o be i place by 1 (see Ieraioal Accouig Sadards Board (7)). However, so far, mos isurace compaies oly have lile kowledge abou risk-eural valuaio echiques ad, hece, mosly rely o simple models ad brue force Moe Carlo simulaios. This is maily due o he fac ha predomia sofware soluios (e.g. Moses, Prophe, or ALM.IT) were iiially desiged for deermiisic forecass of a isurer s rade accous ad oly subsequely exeded o perform Moe Carlo simulaios. I academic lieraure, o he oher had, here exiss a variey of differe aricles o he valuaio of life isurace coracs. However, here are hardly ay deailed comparisos of differe umerical valuaio approaches i a geeral seup. Moreover, some sudies do o apply mehods from fiacial mahemaics appropriaely o he valuaio of life isurace producs (e.g. quesioable wors-case scearios i Gazer ad Schmeiser (8) ad Klig, Ruß ad Schmeiser (6); see Sec. 3.1 below for deails). The objecive of his aricle is o formalize he valuaio problem for life isurace coracs i a geeral way ad o provide a survey o cocree valuaio mehodologies. We paricularly focus o he valuaio of isurace coracs coaiig early-exercise feaures or ierveio opios (cf. Seffese ()), such as surreder opios, wihdrawal guaraees, or opios o chage he premium payme mehod. While almos all isurace coracs coai such feaures, isurers usually do o iclude hese i heir price ad risk maageme compuaios eve hough hey may add cosiderably o he value of he corac. The remaider of he ex is orgaized as follows: I Secio, we prese our geeric model for life isurace coracs. Subsequely, i Secio 3, we describe differe umerical valuaio approaches. Based o a example of paricipaig life isurace coracs, we carry ou umerical experimes i Secio 4. Similarly o mos prior lieraure o he valuaio of life coigecies from a mahemaical fiace perspecive, we iiially assume a geeral Black-Scholes framework. We compare he obaied resuls as well as he efficiecy of he differe approaches ad aalyze he ifluece of a surreder opio o our example corac. However, as is well-kow from various empirical sudies, several saisical properies of fiacial marke daa are o described adequaely by Browia moio ad, i geeral, guaraees ad opios will icrease i value uder more suiable models. Therefore, we aalyze he model risk for our valuaio problem by iroducig a expoeial Lévy model ad comparig he obaied resuls for our example o hose from he Black-Scholes seup. We fid ha he qualiaive impac of he model choice depeds o he paricular model parameers, i.e. ha here exis (realisic) parameer choices for which eiher model yields higher values. Fially, he las secio summarizes our mai resuls.

3 ON THE RISK-NEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS 67. GENERIC CONTRACTS We assume ha fiacial ages ca rade coiuously i a fricioless ad arbirage-free fiacial marke wih fiie ime horizo T. 1 Le ( W F, F F, Q F, F = ( F F )! [,T ] ) be a complee, filered probabiliy space, where Q F is a pricig measure ad F is assumed o saisfy he usual codiios. I his probabiliy space, we iroduce he q 1 -dimesioal, locally bouded, adaped Markov F F, process (Y )! [,T ] = (Y (1) F, (q,, Y 1 ))! [,T ], ad call i he sae process of he fiacial marke. Wihi his marke, we assume he exisece of a locally risk-free asse (B )! [,T ] wih B = exp {# F r u du}, where r = r(,y ) is he shor rae. Moreover, we allow for! oher risky asses ( A ) )! [,T ], 1 i, raded i he (i marke wih ( i ) ( i) = F ), A A (, Y 1 # i #. I order o iclude he moraliy compoe, we fix aoher probabiliy space ( W M, G M, P M ) ad a homogeous populaio of x-year old idividuals a icepio. Similar o Biffis (5) ad Dahl (4), we assume ha a q -dimesioal, locally bouded Markov process ( Y M )! [,T ] = ( Y M, (q 1 +1 ),, Y M, ( q ) )! [,T ], q = q 1 + q, o ( W M, G M, P M ) is give. Now le m(, ) : + q " + be a posiive coiuous fucio ad defie he ime of deah T x of a idividual as he firs jump ime of a Cox process wih iesiy ( m ( x +, Y M ) )! [,T ], i.e. T = if' m( x + s, Y M ) ds $ E1, x # s where E is a ui-expoeially disribued radom variable idepede of ( Y M )! [,T ] ad muually idepede for differe idividuals. Also, defie subfilraios M = ( F M )! [,T ] ad = ( H )! [,T ] as he augmeed subfilraios geeraed by ( Y M )! [,T ] ad ( 1 {Tx })! [,T ], respecively. We se G M = F M H ad M = ( G M )! [,T ]. Isurace coracs ca ow be cosidered o he combied filered probabiliy space _ W, GQ,, = ( G )! [, T ] i, where W = W M W F, G = F F G M, G = F F G M, ad Q = Q F 7 P M is he produc measure of idepede fiacial ad biomeric eves. We furher le 1 I acuarial modelig, i is commo o assume a so-called limiig age meaig ha a fiie ime horizo aurally suffices i view of our objecive. We deoe by A (i) oly asses which are o solely subjec o ieres rae risk, e.g. socks or immovable propery. The price processes of o-defaulable bods raded i he marke are implicily give by he shor rae process.

4 68 D. BAUER, D. BERGMANN AND R. KIESEL = ( F )! [, T ], where F = F F F M. A sligh exesio of he resuls by Lado (1998) ( Prop. 3.1 ) ow yields ha for a F -measurable payme C, we have for u 3 Q B 9B C u -1 1{ T x > } Gu Q -1 { T Y x > u} Bu < B C exp& m( x s, s) ds F u u = C # F, which ca be readily applied o he valuaio of isurace coracs. For oaioal coveiece, we iroduce he realized survival probabiliies p : = 8 B = &- m( x+ s, Y ) ds, () Q x 1{T x > } F H exp () () px -u px+ u = = exp& ( u) -# m + Ys u p u x : ( x s, ) ds, # u #, # s as well as he correspodig oe-year realized deah probabiliy q () () x 1 = 1- p () + - x+ - 1 = 1-1 px+ -1 :. While Q F is specified as some give equivale marigale measure, here is some flexibiliy i he choice of P M. I a complee fiacial marke, i.e. if Q F is uique, wih a deermiisic evoluio of moraliy ad uder he assumpio of risk-euraliy of a isurer wih respec o moraliy risk ( cf. Aase ad Persso (1994) ), Møller (1) pois ou ha if P M deoes he physical measure, Q as defied above is he so-called Miimal Marigale Measure ( see Schweizer (1995) ). This resul ca be exeded o icomplee fiacial marke seigs whe choosig Q F o be he Miimal Marigale Measure for he fiacial marke ( see e.g. Rieser (6) ). However, Delbae ad Schachermayer (1994) quoe he use of moraliy ables i isurace as a example ha his echique [chage of measure] i fac has a log hisory i acuarial scieces, idicaig ha he assumpio of risk-euraliy wih respec o moraliy risk may o be adequae. The, he measure choice depeds o he availabiliy of suiable moraliy-liked securiies raded i he marke ( see Dahl, Melchior ad Møller (8) for a paricular example ad Blake, Cairs ad Dowd (6) for a survey o moraliy-liked securiies ) ad/or he isurer s prefereces ( see Bayrakar ad Ludkovski (9), Becherer (3) or Møller (3) ). I wha follows, we assume ha he isurer has chose a measure P M for valuaio purposes, so ha a paricular choice for he valuaio measure Q is give. 3 I wha follows, we wrie m( x +, Y ) := m( x +, Y M ) ad r(, Y ) := r(, Y F ), where ( Y )! [, T ] := ( Y F, Y M )! [, T ] is he sae process.

5 ON THE RISK-NEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS 69 To obai a model for our geeric life isurace corac, we aalyze he way such coracs are admiisraed i a isurace compay. A impora observaio is ha cash flows, such as premium paymes, beefi paymes, or wihdrawals, are usually o geeraed coiuously bu oly a discree pois i ime. For he sake of simpliciy, we assume ha hese discree pois i ime are he aiversaries! {,,T } of he corac. Therefore, he value V of some life isurace corac a ime uder he assumpio ha he isured i view is alive by he risk-eural valuaio formula is: T Q -1 = B / Bm Cm m = V 9 F C, where C m is he cash flow a ime m, m T. Sice he value a ime oly depeds o he evoluio of moraliy ad he fiacial marke, ad as hese agai oly deped o he evoluio of ( Y s ) s! [, ], we ca wrie: V = VY u(,, s [, ]). s! Bu savig he eire hisory of he sae process is cumbersome ad, foruaely, uecessary: Wihi he bookkeepig sysem of a isurace compay, a life isurace corac is usually maaged ( or represeed ) by several accous savig releva iformaio abou he hisory of he corac, such as he accou value, he cash-surreder value, he curre deah beefi, ec. Therefore, we iroduce m! virual accous ( D )! [,T ] = ( D ( 1 ),, D ( m ) )! [,T ], he so-called sae variables, o sore he releva hisory. I his way, we obai a Markovia srucure sice he releva iformaio abou he pas a ime is coaied i ( Y, D ). Furhermore, we observe ha hese virual accous are usually o updaed coiuously, bu adjusmes, such as crediig ieres or guaraee updaes, are ofe oly made a cerai key daes. Also, policyholders decisios, such as wihdrawals, surreders, or chages o he isured amou, ofe oly become effecive a predeermied daes. To simplify oaio, we agai assume ha hese daes are he aiversaries of he corac. Therefore, o deermie he corac value a ime if he isured i view is alive, i is sufficie o kow he curre sae of he sochasic drivers ad he values of he sae variables a SV = max{! }, i.e. he value of he geeric life isurace corac ca be described as follows: V = VY (,, D ) = VY (,, D ), [, T]. We deoe he se of all possible values of ( Y, D ) by Q. This framework is geeric i he sese ha we do o regard a paricular corac specificaio, bu we model a geeric life isurace corac allowig

6 7 D. BAUER, D. BERGMANN AND R. KIESEL for paymes ha deped o he isured s survival. While more geeral coracs depedig o he survival of a secod life ( muliple life fucios ) or paymes depedig e.g. o he healh sae of he isured ( muliple decremes ) are o explicily cosidered i our seup, heir iclusio would be sraighforward aki o he classical case. Similar frameworks i coiuous ime have bee e.g. proposed by Aase ad Persso (1994) ad Seffese (), where he value or, more precisely, he marke reserve of a geeric corac is described by a geeralized versio of Thiele s Differeial Equaio. I coras, we limi our cosideraios o discree paymes sice ( a ) his is cohere wih acuarial pracice as poied ou above ad ( b ) he case of coiuous paymes may be approximaed by choosig he ime iervals sufficiely small. Hece, we do o believe ha hese limiaios resric he applicabiliy of our seup. I paricular, may models for he marke-cosise valuaio of life isurace coracs preseed i lieraure fi io our framework. For example, Brea ad Schwarz (1976) price equiy-liked life isurace policies wih a asse value guaraee. Here, he value of he corac a ime oly depeds o he value of he uderlyig asse which is modeled by a geomeric Browia moio, i.e. we have a isurace corac which ca be described by a oe-dimesioal sae processes ad o sae variables. Paricipaig life isurace coracs are characerized by a ieres rae guaraee ad some bous disribuio rules, which provide he possibiliy for he policyholder o paricipae i he earigs of he isurace compay. Furhermore, hese coracs usually coai a surreder opio, i.e. he policyholder is allowed o lapse he corac a ime! {1,, T }. Such coracs are, for isace, cosidered i Briys ad de Varee (1997), Grose ad Jørgese () ad Milerse ad Persso (3). All hese models ca be represeed wihi our framework. Moreover, he seup is o resriced o he valuaio of eire isurace coracs, bu i ca also be used o deermie he value of pars of isurace coracs, such as embedded opios. Clearly, we ca deermie he value of a arbirary opio by compuig he value of he same corac i- ad excludig ha opio, ceeris paribus. The differece i value of he wo coracs is he margial value of he opio. For example, he geeric model ca be used i his way o aalyze paid-up ad resumpio opios wihi paricipaig life isurace coracs such as i Gazer ad Schmeiser (8) or exchage opios such as i Nordahl (8). Aleraively, he value of a cerai embedded opio may be deermied by isolaig he cash-flows correspodig o he cosidered guaraee ( see Bauer, Kiesel, Klig ad Ruß (6) ). Bauer, Klig ad Ruß (8) cosider Variable Auiies icludig socalled Guaraeed Miimum Deah Beefis ( GMDBs ) ad/or Guaraeed Miimum Livig Beefis ( GMLBs ). Agai, heir model srucure fis io our framework; hey use oe sochasic driver o model he asse process ad eigh sae variables o specify he corac.

7 ON THE RISK-NEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS A SURVEY OF NUMERICAL METHODS The coracs uder cosideraio are ofe relaively complex, pah-depede derivaives, ad i mos cases, aalyical soluios o he valuaio problems cao be foud. Hece, oe has o resor o umerical mehods. I his secio, we prese differe possibiliies o umerically ackle hese valuaio problems Moe Carlo simulaios Moe Carlo simulaios are a simple ad ye useful approach o he valuaio of isurace coracs provided ha he cosidered corac does o coai ay early exercise feaures, i.e. policyholders cao chage or ( parially ) surreder he corac durig is erm. We call such coracs Europea. I his case, we ca simulae K pahs of he sae process ( Y )! [,T ], say ( Y ( k ) )! [,T ], k = 1,, K, ad compue he uméraire process, he realized survival probabiliies as well as he sae variables a each aiversary of he corac. The, he value of he corac for pah k, V ( k ), 1 k K, is give as he sum of discoued cash flows i pah k, ad, by he Law of Large Numbers ( LLN ), he risk-eural value of he corac a icepio V may be esimaed by he sample mea for K sufficiely large. However, if he corac icludes early exercise feaures, he problem is more delicae sice he value of he opio or guaraee i view depeds o he policyholder s acios. The quesio of how o icorporae policyholder behavior does o have a sraigh-forward aswer. From a ecoomic perspecive, oe could assume ha policyholders will maximize heir persoal uiliy, which would lead o a o-rivial corol problem similar as for he valuaio of employee sock opios ( see Carpeer (1998), Igersoll (6), or refereces herei ). However, he assumpio of homogeous policyholders does o seem proximae. I paricular, he implied asserio ha opios wihi coracs wih he same characerisics are exercised a he same ime does o hold i pracice, ad i is o clear how o iclude heerogeeiy amog policyholders. Aleraively, i is possible o assess he exercise behavior empirically. For such a approach, our framework provides a coveie seup: A regressio of hisorical exercise probabiliies o he sae variables could yield cohere esimaes for fuure exercise behavior. However, aside from problems wih rerievig suiable daa, whe adopig his mehodology isurers will face he risk of sysemaically chagig policyholder behavior, which has had severe cosequeces i he pas. For example, he UK-based muual life isurer Equiable Life, he world s oldes life isurace compay, was closed o ew busiess due o solvecy problems arisig from a misjudgme of policyholders exercise behavior of guaraeed auiy opios wihi idividual pesio policies.

8 7 D. BAUER, D. BERGMANN AND R. KIESEL Hece, i compliace wih ideas from he ew solvecy ad fiacial reporig regulaios, we ake a differe approach ad cosider a valuaio of embedded opios as if hey were raded i he fiacial marke. While from he isurer s perspecive he resulig value may exceed he acual or realized value, i is a uique supervaluaio i he sese ha policyholders have he possibiliy ( or he opio ) o exercise opimally wih respec o he fiacial value of heir corac. Moreover, he resulig superhedgig sraegy for aaiable embedded opios is uique i he same sese. This is i lie wih Seffese (), where a quasi-variaioal iequaliy for he value of life isurace coracs coaiig coiuously exercisable opios is derived uder he same paradigm. I order o deermie his value, we eed o solve a opimal corol problem. To illusrae i, le us cosider a life isurace corac wih surreder opio. The opio is mos valuable if he policyholder behaves fiacially raioal, i.e. - 1 Q -1 () -1 () ( + 1) ( ) x,y / ( + 1 x x +! Y = V = sup > B p C(, D ) + B ) p q f ( + 1, Y, D ) F H + 1 where C(, y, d ) is he surreder value a ime ad f (, y, d 1 ) is he deah beefi upo deah i [ 1, ) if he sae process ad he sae variables ake values y ad d ( d 1 a = 1 ), respecively, ad Y deoes he se of all soppig imes i {1,, T }. Clearly, maximizig he exercise value over each sigle sample pah ad compuig he sample mea, as e.g. pursued i Gazer ad Schmeiser (8) ad Klig e al. (6) for differe ypes of coracs, overesimaes his value. To deermie a Moe Carlo approximaio of his value, which we refer o as he corac value i wha follows, we eed o rely o so-called esed simulaios. We do o allow for surreders a icepio of he corac, so we defie C(, y, d ) :=. By he Bellma equaio ( see e.g. Bersekas (1995) for a iroducio o dyamic programmig ad opimal corol ) he corac value a ime,! {,, T 1}, is he maximum of he exercise value ad he coiuaio value. The laer is he weighed sum of he discoued expecaio of he corac value give he iformaio ( y, d )! Q, i.e. V(, y, d ) Q = max% C(, y, d ), B 9B p V( + 1, Y, D ) ( Y D ) = ( y, d ) C -1 ( + 1) + 1 x ( + 1) + 1 x , Q + B 9B q f( + 1, Y, D ) ( Y, D ) = ( y, d ) C/. + 1 We ow geerae a ree wih T ime seps ad b! braches ou of each ode. We sar wih iiial value Y ad he geerae b idepede successors Y 1 1,, Y 1 b. From each ode we geerae agai b successors ad so o. To

9 ON THE RISK-NEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS 73 simplify oaio, le X l 1 l l = ( Y l 1 l l, D l 1 l l ). Wih his oaio, a esimaor for V,! {,, T }, a ode Y l 1 l is Z l1... l b l1f l B l1f ll -1 l1f ll l ll ] 1f max) C(, X ), b / ( B + 1 ) px + V + 1 ] l = 1 ] ( (, {,..., 1} ] ] l1f l C(, X ), = T, ] \ l1... l b l1fl B l1f ll -1 l1fll l1f ll l1fl V : = [ + b / B + 1 ) qx + f + 1, Y + 1, D ) 3! T- l = 1 l... l l... l l where B 1 l 1 1 ad p x + -1( q x + -1) deoe he values of he bak accou ad l he oe-year survival ( deah ) probabiliy a = i sample pah ( Y, Y 1 1,, l Y 1 l ), respecively. Usig K replicaios of he ree, we deermie he sample mea V ( K, b ), ad by he LLN we ge V ( K, b ) " Q [V ] as K " 3 almos surely. Hece, fixig b, we ca cosruc a asympoically valid ( 1 d ) cofidece ierval for Q [V ]. Bu his esimaor for he risk-eural value V = V(, X ) is biased high ( see Glasserma (3), p. 433 ), i.e. Q 7 A $ (, ), V V X where, i geeral, we have a sharp iequaliy. However, uder some iegrabiliy codiios, he esimaor is asympoically ubiased ad hece, we ca reduce he bias by icreasig he umber of braches b i each ode. I order o cosruc a cofidece ierval for he corac value V(, X ), followig Glasserma (3), we iroduce a secod esimaor. I differs from he esimaor iroduced above i ha all bu oe replicaio are used o decide wheher o exercise he opio or o, whereas i case exercisig is o decided o be opimal, he las replicaio is employed. More precisely, we defie for! {,, T 1} l1f l u Z l1f l B b ] l1f l l1f ll -1 l1f ll l1f ll C(, X ), if / ( B + 1 ) p b - 1 x + ] + 1 l= 1, l! u ] ] B l1f l b l f ll -1 fll ( B q 1, + 1 ) x + 1, b Y l f ] + / + f (+ D l= 1, l! u ] l1f l : = [ # C(, X ) ] ] l1f l l1f lu -1 l1f lu l1f lu ] B ( B + 1 ) px + + 1,oherwise. ] ] l1f l l1f lu -1 l1f lu l1f lu l1f l + B ( B + 1 ) qx + f (+ 1, Y + 1, D ) ] \ l l ll l1f )

10 74 D. BAUER, D. BERGMANN AND R. KIESEL The, averagig over all b possibiliies of leavig ou oe replicaio, we obai Z 1 b l1f l l l ] /., {,..., T 1} 1f : b u 1 - = u! = [ ] l1 l C(, X f ), = T. \ Agai usig K replicaios of he ree, we obai a secod esimaor for he corac value by he sample mea ( K, b ), which is ow biased low, ad we ca cosruc a secod asympoically valid ( 1 d ) cofidece ierval, his ime for Q [ v ]. Takig he upper boud from he firs cofidece ierval ad he lower boud from he secod oe, we obai a asympoically valid ( 1 d )-cofidece ierval for V : sv ( Kb, ) e ( Kb, )- z d 1, V ( Kb, ) + z K d - 1- s V ( Kb, ) o, K where z 1 - d is he ( 1 d )-quaile of he sadard ormal disribuio. s V ( K, b ) ad s v ( K, b ) deoe he sample sadard deviaios of he K replicaios for he wo esimaors. The drawback for o-europea isurace coracs is ha he umber of ecessary simulaio seps icreases expoeially i ime. Sice isurace coracs are usually log-erm ivesmes, he compuaio of he value usig esed simulaios is herefore raher exesive ad ime-cosumig. Moreover, for differe opios wih several ( or eve ifiiely may ) admissible acios, such as wihdrawals wihi variable auiies, he complexiy will icrease dramaically. 3.. A PDE approach P( I )DE mehods bear cerai advaages i compariso o he Moe Carlo approach. O oe had, hey iclude he calculaio of cerai sesiiviies ( he so-called Greeks ; see e.g. Hull (), Chaper 13 ), which are useful for hedgig purposes. O he oher had, hey ofe prese a more efficie mehod for he valuaio of o-europea isurace coracs. The idea for his algorihm is based o solvig he correspodig corol problem o a discreized sae space ad, for special isurace coracs, was origially preseed i Grose ad Jørgese () ad Taskae ad Lukkarie (3). I order o apply heir ideas i he curre seup, for he remaider of his subsecio, we work uder he addiioal assumpio ha he sae process ( Y )! [, T ] is a Lévy process. The value V of our geeric isurace corac depeds o, Y, ad he sae variables D. However, bewee wo policy aiversaries 1 ad,

11 ON THE RISK-NEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS 75! {1,, T }, he evoluio of V depeds o ad Y oly sice he sae variables remai cosa. Cosequely, give he sae variables D 1 = d 1 ad he value fucio a some ime! [ 1, ) provided ha he isured i view is alive, V, he value fucio o he ierval [ 1, ] is VY (,, d ) Q - 1 {T exp x > } ' s = = 1 - # r ds1v G, T x > G Q Q { Tx # } # s # s + > 1 exp' - r ds1 = exp' - r ds1 f (, Y, d ) F G G, T > H - 1 x Q = = exp' - rs + m( x+ s, Ys ) ds1v F G Q # = : F(, Y ) # s # s + > d1 -exp'- m ( x+ s, Y ) ds1exp& - r ds f (, Y, d ) F H. - 1 (1) Here, F(,Y ) ca be ierpreed as he par of he value V ha is aribuable o paymes i case of survival uil ime whereas he secod par correspods o beefis i case of deah i [, ]. I paricular, F(, y ) = V (, y, d 1 ). Applyig Iô s formula for Lévy processes ( see e.g. Prop i Co ad Takov (3) ), we obai df(, Y ) k(, Y, F(, Y )) d dm = - - +, wih drif erm k(,y, F(,Y ) ) ad local marigale par M. Boh erms srogly deped o he paricular model choice ad, herefore, cao be specified i more deail. Sice, by cosrucio, # cexp' - r + m( x + s, Y ) ds1f(, Y ) m s s! [ -1, ] is a ( closed ) Q-marigale, he drif eeds o be zero Q-almos surely. This is a sadard echique aki o he well-kow Feyma-Kac formula. We hus obai a P( I )DE for he fucio F : (, y ) 7 F (, y ): -ry (, ) Fy (, )- m( y, ) Fy (, ) + k (, y, Fy (, )) = () wih ermial codiio F (, y) V (, yd, ). = - 1 A he policy aiversary, o he oher had, he value fucio is lef-coiuous for " by o-arbirage argumes ( see Taskae ad Lukkarie

12 76 D. BAUER, D. BERGMANN AND R. KIESEL (3)) ad sice dyig a he isa is a zero-probabiliy eve. Moreover, V = sup f! F V (, h f ( Y, d 1 ) ) by he priciples of dyamic programmig ( Bellma equaio ) ad o-arbirage, where F is he se of all opios ha may be exercised a = ad h f : Q " Q deoes he rasiio fucio which describes how he sae variables chage a = if opio f is exercised. Hece, all i all, V (, yd, 1) " sup V(, h ( yd, )) as ". (3) - f -1 f! F Sice he value fucio a mauriy T is kow for all ( y, d )! Q T, we ca use Equaios ( 1 ), ( ), ad ( 3 ) o cosruc a backwards algorihm o obai he value fucio o he whole ierval [, T ]: For = T u, u! {1,, T }, evaluae he P( I )DE ( ) for all possible d T u wih ermial codiio ( cf. ( 3 ) ) FT ( - u+ 1, y T- u+ 1) = sup VT ( - u+ 1, h ( yt u 1, T u)). (4) The, se ( cf. ( 1 ) ) V(T - u, y, ) T-u d T -u = F(T -u, y ) Q T-u # ft-u+ 1! FT-u+ 1 ft u d T- u+ 1 ( T- u+ 1) r s ds qx+ T-u f T u 1 yt- u+ 1 dt-u) FT- u T-u + < exp& - ( - +,, F. I he special case of a life isurace corac wih surreder opio aki o he previous subsecio, F cosiss of oly wo elemes, say {SUR, NO-SUR}, correspodig o surrederig ad o surrederig he corac, respecively. I he case of a surreder, he rasiio resuls i he value fucio coicidig wih he surreder value C( + 1, h SUR ( y + 1, d ) ), whereas o surrederig will resul i he value fucio V( + 1, h NO-SUR ( y + 1, d ) ). Therefore, ( 4) simplifies o F^T- u+ 1, y h T- u+ 1 = max$ V _ T- u+ 1,, ii C_ T- u+ 1, y, d ii.. h NO - SUR _ yt- u+ 1 dt-u, hsur_ T- u+ 1 T-u I order o apply he algorihm, he sae spaces Q, =,, T, are discreized ad ierpolaio mehods are employed o deermie he righ-had sides of ( 4 ) if he argumes are off he grid. I paricular, i is ecessary o solve he P( I )DE for all sae variables o he grid, so ha he efficiecy of he algorihm highly depeds o he evaluaio of he P( I )DEs. I Taskae ad Lukkarie (3), he classical Black-Scholes model ad a deermiisic evoluio of moraliy are assumed. I his case, he resulig

13 ON THE RISK-NEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS 77 PDE is he well-kow Black-Scholes PDE, which ca be rasformed io a oe-dimesioal hea equaio, from which a iegral represeaio ca be derived whe he ermial codiio is give. If a modified Black-Scholes model wih sochasic ieres raes is assumed as i Zaglauer ad Bauer (8), he siuaio ges more complex: The PDE is o loger aalyically solvable ad oe has o resor o umerical mehods. For a geeral expoeial Lévy process drivig he fiacial marke, PIDEs wih o-local iegral erms mus be solved. Several umerical mehods have bee proposed for he soluio, e.g. based o fiie differece schemes ( see e.g. Aderse ad Adrease () ad Co ad Volchkova (5) ), based o wavele mehods (Maache, vo Peersdorff ad Schwab (4) ), or Fourier rasform based mehods (Jackso, Jaimugal ad Surkov (8), Lord, Fag, Bervoes ad Ooserlee (8)). While i compariso o Moe Carlo simulaios he complexiy does o icrease expoeially i ime, he high umber of P( I )DEs eedig o be solved may slow dow he algorihm cosiderably A leas-squares Moe Carlo approach The leas-squares Moe Carlo ( LSM ) approach by Logsaff ad Schwarz (1) was origially preseed for pricig America opios bu has recely also bee applied o he valuaio of isurace coracs ( see e.g. Adreaa ad Corradi (3) ad Nordahl (8) ). We prese he algorihm for life isurace coracs wih a simple surreder opio. Subsequely, problems for he applicaio of his mehod o more geeral embedded opios as well as poeial soluios are ideified. As poied ou by Cléme, Lambero ad Proer (), he algorihm cosiss of wo differe ypes of approximaios. Wihi he firs approximaio sep, he coiuaio value fucio is replaced by a fiie liear combiaio of cerai basis fucios. As he secod approximaio, Moe Carlo simulaios ad leas-squares regressio are employed o approximae he liear combiaio give i sep oe. Agai, le C(, y, d ) be he payoff a ime! {1,, T } if he sochasic drivers ad he sae variables ake values y ad d, respecively, ad he opio is exercised a his ime. Furhermore, le C( s,, y s, d s ), < s T, describe he cash flow a ime s give he sae process y s ad he sae variables d s, codiioal o he opio o beig exercised prior or a ime, ad he policyholder followig he opimal sraegy accordig o he algorihm a all possible exercise daes s! { + 1,, T } assumig ha he policyholder is alive a ime. The coiuaio value g(,y, D ) a ime is he sum of all expeced fuure cash flows discoued back o ime uder he iformaio give a ime, i.e. Q T / # g(, Y, D) = > exp& - ru du C_ s,, Ys, Ds i F H. s = + 1 s

14 78 D. BAUER, D. BERGMANN AND R. KIESEL To deermie he opimal sraegy a ime =, i.e. o solve he opimal soppig problem, i is ow sufficie o compare he surreder value o he coiuaio value ad choose he greaer oe. Hece, we obai he followig discree valued soppig ime := 1 : T = T * (5) = 1{ C(, Y, D ) $ g(, Y, D )} { C(, Y, D ) < g(, Y, D )} 1 # # T -1, ad he corac value ca be described as V_, Y, D i = < exp& - r du p C(, Y, D ) Q # () u F x F - 1 Q + 1 () ( + 1) / # u x x v + 1, + = + > exp& - r du p q f ( + 1, Y D ) F H. (6) Followig Cléme e al. (), we assume ha he sequece ( L j ( Y, D ) ) j is oal i he space L ( s( ( Y, D ) ) ), = 1,, T 1, ad saisfies a liear idepedece codiio ( cf. codiios A 1 ad A i Cléme e al. () ) such ha g(,y, D ) ca be expressed as g(, Y, D ) = / 3 a ( ) L ( Y, D ), (7) j = for some a j ( )!, j! {}. For he firs approximaio, we replace he ifiie sum i ( 7) by a fiie sum of he firs J basis fucio. We call his approximaio g ( J ). Similarly o ( 5 ) ad ( 6 ), we ca ow defie a ew soppig ime ( J ) ad a firs approximaio V ( J ) for he corac value by replacig g by g ( J ). However, i geeral he coefficies ( a j ( ) ) j J = 1 are o kow ad eed o be esimaed. We use K! replicaios of he pah ( Y, D ), T, ad deoe hem by ( Y ( k ), D ( k ) ), 1 k K. The coefficies are he deermied by a leas-squares regressio. We assume ha he opimal sraegy for s + 1 is already kow ad hece, for each replicaio he cash flows C( s,, Y s ( k ), D s ( k ) ), s! { + 1,, T }, are kow. Uder hese assumpios, he leas-squares esimaor for he coefficies is K T (K ) s ( = arg mi exp - ru du C s, k) * / / & # Ys D J s a( )! k = 1 s = + 1 J / aj ( ) L j ( Y, D ) H 4. j = a ( ) = (,, Replacig ( a j ( ) ) j J = 1 by ( a j ( K ) ( ) ) j J = 1, we obai he secod approximaio g ( J, K ) ad agai, we defie he soppig ime ( J, K ) ad aoher approximaio V ( J, K ) of he value fucio by replacig g by g ( J, K ). j j )

15 ON THE RISK-NEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS 79 Wih he help of hese approximaios, we ca ow cosruc a valuaio algorihm for our isurace corac: Firs, simulae K pahs of he sae process up o ime T ad compue he sae variables uder he assumpio ha he surreder opio is o exercised a ay ime. Sice he corac value, ad hece, he cash flow a mauriy T is kow for all possible saes, defie he followig cash flows: (, - 1, T, T ) ( T) ( T ) x+ T- T T ) x+ T- T T- 1 CTT Y D = p 1C( T, Y, D + q 1 f( T, Y, D ), 1 # k # K. For = T u, u! {1,, T 1}, compue g ( J, K ) as described above ad deermie he opimal sraegy i each pah by comparig he surreder value o he coiuaio value. The, deermie he ew cash flows. 4 For s! {T u + 1,, T }, we have C(, s T -u-1, Ys, Ds ), if he opio is exercised a T- u = * ( T- u) p C(, s T - u, Y, D ),oherwise, x+ T-u-1 ad for s = T u we se ( T-, T- -1, T- u, T- u) ( T- u) x+ T-u-1 - T-u T-u ( T- u) x+ T-u-1 T- u T-u-1 ( T- u) x+ T-u-1 + T- u T-u-1 s s C u u Y D Z p C( T u, Y, D ), if he opio is exercised ] = [ + q f( T+ u, Y, D ) a T -u ] q f( T u, Y, D ), oherwise. \ A ime =, discou he cash flows i each pah ad average over all K pahs, i.e. wih V K ( JK, ) (, 1 J K, k) (, Y, D) : = K / V (, Y, D) k = 1 T ( J, K, k ) s (,, D) : = / exp& -# u (,, s, Ds s = 1 V Y r du C s Y ). 4 Noe ha we do o use he esimaed coiuaio value bu he acual cash flows for he ex regressio. Oherwise he esimaor will be biased ( cf. Logsaff ad Schwarz (1) ).

16 8 D. BAUER, D. BERGMANN AND R. KIESEL The wo covergece resuls i Secio 3 of Cléme e al. () esure ha, uder weak codiios, he algorihm gives a good approximaio of he acual corac value whe choosig J ad K sufficiely large. The LSM algorihm ca be coveiely implemeed for isurace coracs coaiig a simple surreder opio sice he ew fuure cash flows ca be easily deermied: If he surreder opio is exercised a! {1,, T 1}, he cash flow C(, 1, Y, D ) equals he surreder value ad all fuure cash flows are zero. If we have more complex early exercise feaures, he derivaio of he fuure cash flows could be more ivolved sice he corac may o be ermiaed. For example, if a wihdrawal opio i a corac icludig a Guaraeed Miimum Wihdrawal Beefi is exercised, his will chage he saes variables a ha ime. However, he fuure cash flows for he ew sae variables will o be kow from he origial sample pahs, i.e. i is ecessary o deermie he ew fuure cash flows up o mauriy T. This may be very edious if i is a log-erm isurace corac ad he opio is exercised relaively early. I paricular, if he opio ca be exercised a every aiversary ad if he wihdrawal is o fixed bu arbirary wih cerai limis, his may icrease he complexiy of he algorihm cosiderably. A poeial soluio o his problem could be employig he discoued esimaed codiioal expecaio for he regressio isead of he discoued fuure cash flows. However, his will lead o a biased esimaor ( see Logsaff ad Schwarz (1), Sec. 1 ). Bu eve if his bias is acceped, aoher problem regardig he qualiy of he regressio fucio may occur. I he LSM algorihm, we deermie he coefficies of he regressio fucio wih he help of sample pahs ha are geeraed uder he assumpio ha o opio is exercised a ay ime, i.e. he approximaio of he coiuaio value will be good for values which are close o he used regressors. Bu g ( J, K ) may o be a good esimae for coracs wih, e.g., high wihdrawals because wihdrawals reduce he accou balace, ad hece, he ew sae variables will o be close o he regressors. A idea of how o resolve his problem migh be he applicaio of differe samplig echiques: For each period, we could deermie a cerai umber of differe iiial values, simulae he developme for oe period, compue he corac value a he ed of his period ad use he discoued corac value as he regressad. However, deermiig hese iiial values, agai, is o sraigh-forward. We leave he furher exploraio of his issue o fuure research. Aside from hese problems, he LSM approach bears profoud advaages i compariso o he oher approaches: O oe had, he umber of simulaio seps icreases liearly i ime ad, o he oher had, i avoids solvig a large umber of P( I )DEs. Also, i coras o he P( I )DE approach, he LSM approach is idepede of he uderlyig asse model: The oly par ha eeds o be chaged i order o icorporae a ew asse model is he Moe Carlo simulaio.

17 ON THE RISK-NEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS EXAMPLE: A PARTICIPATING LIFE INSURANCE CONTRACT I his secio, we compare he resuls obaied wih he hree differe umerical approaches for a Germa paricipaig life isurace corac icludig a surreder opio The corac model We cosider he paricipaig erm-fix corac from Bauer e al. (6) ad Zaglauer ad Bauer (8). While his corac is raher simple ad, i paricular, does o deped o biomeric eves, i preses a coveie example o illusrae advaages ad disadvaages of he preseed approaches ad o compare hem based o umerical experimes. We use a simplified balace shee o model he isurace compay s fiacial siuaio ( see Table 1 ). TABLE 1 SIMPLIFIED BALANCE SHEET Asses Liabiliies A L R A A Here, A deoes he marke value of he isurer s asse porfolio, L is he policy holder s accou balace, ad R = A L is he bous reserve a ime. Disregardig ay charges, he policyholder s accou balace a ime zero equals he sigle up-fro premium P, ha is L = P. Durig is erm, he policyholder may surreder her corac: If he corac is lapsed a ime! {1,, T }, he policyholder receives he curre accou balace L. Furhermore, we assume ha divideds are paid o shareholders a he aiversaries i order o compesae hem for adoped risk. As i Bauer e al. (6) ad Zaglauer ad Bauer (8), we use wo differe bous disribuio schemes, which describe he evoluio of he liabiliies: The MUST-case describes wha isurers are obligaed o pass o o policyholders accordig o Germa regulaory ad legal requiremes, whereas he IS-case models he ypical behavior of Germa isurace compaies i he pas; his disribuio rule was firs iroduced by Klig, Richer ad Ruß (7) The MUST-case I Germay, isurace compaies are obligaed o guaraee a miimum rae of ieres g o he policyholder s accou, which is currely fixed a.5%.

18 8 D. BAUER, D. BERGMANN AND R. KIESEL Furhermore, accordig o he regulaio abou miimum premium refuds i Germa life isurace, a miimum paricipaio rae d of he earigs o book values has o be passed o o he policyholders. Sice earigs o book values usually do o coicide wih earigs o marke values due o accouig rules, we assume ha earigs o book values amou o a porio y of earigs o marke values. The earigs o marke values equal A A + 1, where A ad A + = max{a d, L } describe he marke value of he asse porfolio shorly before ad afer he divided paymes d a ime, respecively. The laer equaio reflecs he assumpio ha e.g. uder Solvecy II, he marke-cosise embedded value should be calculaed eglecig he isurer s defaul pu opio, i.e. ha shareholders cover ay defici. Therefore, we have - + L = ( 1+ g) L- 1+ [ dy( A -A- 1) -gl- 1 ], 1 # # T. (8) Assumig ha he remaiig par of earigs o book values is paid ou as divideds, we have { dy( A - A-1) > gl-1} { y( A A-1) gl-1 y( A A-1)} d = ( -d) y( A -A ) 1 + [ y( A -A )- gl ] d - # # -. (9) The IS-case R L A + - L L I he pas, Germa isurace compaies have ried o gra heir policyholders sable bu ye compeiive reurs. I years wih high earigs, reserves are accumulaed ad passed o o policyholders i years wih lower earigs. Oly if he reserves dropped beeah or rose above cerai limis would he isurace compaies decrease or icrease he bous paymes, respecively. I he followig, we give a brief summary of he bous disribuio iroduced i Klig e al. (7), which models his behavior. The reserve quoa x is defied as he raio of he reserve ad he policyholder s accou, i.e. x = =. Le z! [, 1] be he arge ieres rae of he isurace compay ad a! [, 1] be he proporio of he remaiig surplus afer he guaraeed ieres rae is credied o he policyholder s accou ha is disribued o he shareholders. Wheever he arge ieres rae z leads o a reserve quoa bewee specified limis a ad b wih L = ( + z) L d = a( z-g) L + - A = A -d, + R = A -L, 1-1,, - 1 he exacly he arge ieres rae z is credied o he policyholder s accou.

19 ON THE RISK-NEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS 83 If he reserve quoa drops below a or exceeds b whe crediig z o he policyholder s accou, he he rae is chose such ha i exacly resuls i a reserve quoa of a or b, respecively. However, ( 8) eeds o be fulfilled i ay case. Hece, by combiig all cases ad codiios, we obai ( see Zaglauer ad Bauer (8) ): = max& 8d B L ( g) L y( A A ) gl, ( z- g) L {((1 + a)(1 + z) + a( z- g) L-1# A #((1 + b)(1 + z) + a( z-g)) L-1} A ( 1 g)( 1 a) L 1 + a + a A -( 1+ g)( 1+ b) L 1 + b + a 9-1 C - {(1 + a)(1 + g) L-1< A < ((1 + a)(1 + z) + a( z-g)) L-1} C {((1 + b)(1 + z) + a( z-g)) L-1 < A } 1, ad = max& a 8d B d y( A A ) gl, a( z- g) L {(( 1+ a)( 1+ z) + a( z- g) L-1# A #(( 1+ b)( 1+ z) + a( z-g)) L-1} a - + A -( 1+ g)( 1+ a) L 1 + a + a {( 1+ a)( 1+ g) L-1< A < (( 1+ a)( 1+ z) + a( z-g)) L-1} a - + A ( 1 g)( 1 b) L b + a C - {(( 1+ b)( 1+ z) + a( z-g)) L-1 < A } C 4.. Asse models ( I ) We cosider wo differe asse models, amely a geomeric Browia moio wih deermiisic ieres rae ( cosa shor rae r ), ad a geomeric Browia moio wih sochasic ieres raes give by a Vasicek model ( see Vasicek (1977) ). I he firs case, we have he classical Black-Scholes ( BS ) seup, so he asse process uder he risk-eural measure Q evolves accordig o he SDE: da = ra d + saa dw, A = P( 1 + x), where r is he cosa shor rae, s A > deoes he volailiy of he asse process A, ad W is a sadard Browia moio uder Q. Sice we allow for divided paymes a each aiversary of he corac, we obai - + sa = - 1 expd - + sa( W A A r W ).

20 84 D. BAUER, D. BERGMANN AND R. KIESEL I he secod case, we have a geeralized Black-Scholes model wih da = ra d + rs AAdW+ 1- r saad Z, A = P( 1+ x), dr = k( z- r ) d + s dw, r >, r where r! [ 1, 1] describes he correlaio bewee he asse process A ad he shor rae r, s r is he volailiy of he shor rae process, ad W ad Z are wo idepede Browia moios. z ad k are cosas. Hece, - + sa r s rs 1 1 A dw - - s -1 # # # A = A- 1 expe ds r s A dz o. s We refer o his model as he exeded Black-Scholes ( EBS ) model. Accordig o he risk-eural valuaio formula, he value for our paricipaig life isurace corac icludig a surreder opio is give by: 5 V ; & # dul F E. (1) NON-EUR Q = sup exp - r u! Y For a discussio of he problems occurrig whe implemeig a suiable hedgig sraegy as well as poeial soluios, we refer o Bauer e al. (6) ad Zaglauer ad Bauer (8) Choice of parameers ad regressio fucio To compare resuls, we use he same parameers as i Bauer e al. (6) ad Zaglauer ad Bauer (8). We le he guaraeed miimum ieres rae g = 3.5%, 6 he miimum paricipaio rae d = 9%, ad he miimal proporio of marke value earigs ha has o be ideified as book value earigs i he balace shee y = 5%. Moreover, he reserve corridor is defied o be [a, b] = [ 5%, 3% ], he proporio of earigs ha is passed o o he shareholders is fixed a a = 5%, ad he volailiy of he asse porfolio is assumed o be s A = 7.5%. The correlaio bewee asse reurs ad moey marke reurs is se o r =.5. We cosider a corac wih mauriy T = 1 years. The iiial ivesme is P = 1,, he isurer s iiial reserve quoa is x = 1%, ad he iiial ( or cosa ) ieres rae r = r is se o 4%. I he Vasicek model, he volailiy of he shor rae process s r is chose o be 1%, he mea reversio rae is k =.14, ad he mea reversio level z = 4%. A crucial poi i he LSM approach for o-europea coracs is he choice of he regressio fucio as a fucio of he sae process ad he 5 Y is he se of all soppig imes i {1,, T }. 6 The larges Germa isurer Alliaz Lebesversicherugs-AG repors a average guaraeed ieres rae of approximaely 3.5% i 6 ( see Alliaz Lebesversicherugs-AG (6), p. 19 ).

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