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

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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

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.

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 )

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

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) ).

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 ).

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