Calculation of variable annuity market sensitivities using a pathwise methodology

Transcription

1 cuing edge Variable annuiie Calculaion of variable annuiy marke eniiviie uing a pahwie mehodology Under radiional finie difference mehod, he calculaion of variable annuiy eniiviie can involve muliple Mone Carlo imulaion, leading o high compuaional co A pahwie approach reduce hi dramaically, while providing an unbiaed eimae By Carey Hobb, Bala Krihnaraj, Ying Liu and Jay Muelman 1 Inroducion Variable annuiie (VA) wih guaraneed minimum benefi (deah, accumulaion, income, or wihdrawal) repreen produc wih complex embedded derivaive rucure Deermining he marke value and eniiviie ( greek ) of hee produc i imporan in a number of conex However, generaion of hee value can be expenive o compue uing he mo prevalen mehod in he indury In order o evaluae an in-force book in erie, buinee mu employ ignifican compuing reource a coniderable co o generae pricing and eniiviy run For inance, grid enabled compuing farm or high peed compuing faciliie generae ochaic cenario for each conrac Furher, any geography or axonomy employed by he buine o raify he in-force book will need o be modelled, uch a indice or fund group, and hee will ofen be hocked and imulaed o obain componen greek ued for hedging purpoe Addiionally, increaed ale, he growing complexiy of produc, muli-rik facor modelling, and heerogeneiy in rike and ime-o-mauriy pace of aging in-force book can imply addiional imulaed pah o enure convergence and abiliy A uch, any innovaion in he compuaional apec of hee Mone Carlo imulaion could have ignifican impac hi paper will ouline how o develop and apply an alernaive o finie difference baed on he pahwie differeniaion mehod (or infinieimal perurbaion) o calculae marke value eniiviie hi moly em from exenion of Broadie & Glaerman, 1996 for pecific applicaion o VA conrac We begin by reviewing hee reul applied o vanilla equiy opion and exend he mehod o a complex guaraneed minimum wihdrawal benefi (GMWB) conrac Finally, we conclude wih a urvey of applicaion of he mehod in calculaing a wide variey of greek for differen model including hoe wih dynamic behaviour 2 Review of he finie difference mehod A common mehod ued o price VA guaranee i he Mone Carlo imulaion framework, where a number of differen economic cenario, ofen rik neural, are projeced and he value i equal o he dicouned expeced value of he payoff acro all cenario Greek, or marke value derivaive uch a dela, rho, vega, or gamma, are ofen eimaed wih a Mone Carlo imulaion approach uing finie difference approximaion, which are a andard indirec mehod of eimaing a derivaive value uing re-imulaion of pah and perurbing iniial value Addiionally, i i common ha he higher he order of derivaive, he more pah are required o guaranee convergence, creaing coniderable pracical iue involving co and runime he finie difference approximaion for dela i defined a: + h V ( S 0 ) h 2 V ( S 0 ) +K V ( S 0 ) + h V ( S 0 ) 1 2 h 2 V ( S 0 ) K = V S h 2 V ( S 0 ) V S 0 One of he key challenge ariing wih he finie difference cheme, a hown above, i he exience of an error erm over V (S 0 ) ha creae a biaed eimae of he greek hi erm will end o zero a h end o zero, bu here i a rade-off beween elecing a mall enough meh ize o obain a negligible bia and mainaining an accepable level of variance (ee Glaerman, 2003) 2h 40 Life & Penion

2 cuing edge Variable annuiie 3 Pahwie mehod he pahwie mehod i an aracive alernaive o he finie difference approximaion of deermining greek eniiviie of a financial derivaive; i i a direc mehod ha no only offer reduced compuaional effor, bu i alo unbiaed in i eimae of he derivaive he pahwie mehod leverage he relaionhip beween he ecuriy payoff and he iniial parameer of inere and i obained by differeniaing he expreion for he payoff of he derivaive wih repec o he parameer A uch, higher order eniiviie can be expreed a a funcion of he iniial parameer and eimaed from he iniial pah, perhap generaed during a bae marke valuaion 31 Overview of pahwie mehod o demonrae he pahwie mehod conider he price of a European call opion defined o be: C = E e r max S K where E i he rik-neural expecaion operaor, i he ime o mauriy of he opion, r i he conan inere rae from ime 0 o, K i he rike price, and S i he price of he ock a ime he pahwie dela of he opion i defined o be: C = ( ) = { S S E e r S K E e r I S max S K} 0 0 S 0 where I {S i he indicaor funcion which equal uniy when he K} brackeed condiion i aified and zero oherwie We have inerchanged he expecaion and he derivaive operaor, which i permied for Lipchiz coninuou payoff Auming ha under he rik-neural meaure S evolve wih lognormal dynamic, a in he claic Black-Schole-Meron model, we have So ha S = S 0 e σ 2 ( Z + ( r σ 2 ) ) = S I i clear from he above formula ha in order o obain he dela of he opion uing he pahwie mehod one need o a mo know he value of S on each cenario he value of S i available from he bae marke value Mone Carlo imulaion and, herefore, addiional imulaion i no required o obain he opion dela Broadie & Glaerman how ha hi mehod can be exended o opion which are pah-dependen, ubjec o cerain coninuiy condiion We have uccefully applied hi o a number of more complex rucure However, we alo recommend ha he reader conul he reference cied herein o beer appreciae he limiaion of hi mehod, which involve real analyi and are ouide he cope of hi paper S 0 In he following ecion we demonrae how he mehod can be exended o complex pah dependen financial derivaive uch a a GMWB conrac 4 Pahwie dela for GMWB conrac 41 GMWB conrac pecificaion In wha follow we will chooe a paricularly rich wihdrawal benefi o demonrae he mehod Conider a GMWB conrac where he policy holder can elec o ar aking wihdrawal a any poin unil conrac mauriy Addiionally, he wihdrawal amoun are defined a a percenage of a benefi bae by he iniial wihdrawal he benefi bae will depend on he underlying accoun value he price of hi conrac no only depend on he underlying ae value mechanic, bu i alo dependen on he policy holder behaviour However, a hi poin in he model developmen, we will aume aic behaviour In ecion 44 we how ha hi mehodology work in he preence of dynamic behaviour We define he value of he GMWB conrac a: V = E D p c max( WD AV ) = 0 where E i he rik-neural expecaion of he dicouned payoff of he GMWB policy, D i he dicoun facor from ime 0 o, p c i he probabiliy of urvival o ime given a aic lape aumpion, WD i he wihdrawal a ime, and AV i defined a he accoun value of he conrac a ime he AV will be equal o he accoun value a ime -1 le wihdrawal accumulaed by an equiy reurn r a given by: AV = ( AV 1 WD 1 )e r where WD, i defined a a percenage of benefi bae, BB, a, and AV 0 = P S 0 where P i ued o denoe he iniial premium, and AV 0 can herefore be inerpreed a a noional amoun In many GMWB rucure markeed oday he benefi bae roll up hrough ime and can rache if AV > BB -1 In addiion, many annuiy provider offer feaure where he benefi bae i he high waer mark of he accoun value and he rolled up previou benefi bae We define: BB = max AV, BB 1 WD = w BB where w i he rae of wihdrawal and i aumed o be conan and reaonably bounded We aume wo differen wihdrawal cohor: Cohor 1 i aumed o ar aking wihdrawal a he end of he 1 year and Cohor 2 a he end of he 10h year he marke valuaion model developed exend unil =30 year In general he pecific of he cohor approach or variaion of modelling wihdrawal behaviour are a he dicreion of he modeller wwwlife-penioncom Sepember

3 cuing edge Variable annuiie 42 Pahwie dela for he GMWB conrac he pahwie dela for he GMWB conrac pecified i defined a: PW = V = E D p c max WD AV S 0 = 0 Auming aic behaviour he derivaive will ac excluively on he hird erm reuling in he following expreion max WD AV = I WD > AV { } WD AV Secion 44 deal wih derivaive on he behaviour funcion when i i dynamic he reulan PW of he GMWB conrac i he dicouned (ime 0) rik neural expecaion acro he variou cenario wih he appropriae derivaive for WD and AV numerically calculaed in he imulaion I i worh noing ha derivaive involving he guaranee ake on recurive form ha lend hemelve o imple coding: AV = AV 1 S 0 which erminae wih he condiion WD WD 1 er = P Furher, = BB w = AV max( AV, BB 1 ) w = I BB S R + I 1 0 S NR 0 w he indicaor funcion in hi la expreion exi o rack wheher or no he accoun value racheed a ime, R, or no, NR 43 Pahwie key rae duraion for he GMWB conrac o exend he mehod o a more complex problem we will apply he pahwie mehod o compuing key rae duraion (rho), or imply KRD Rho eniiviie are of paricular relevance o hee produc deign ince i become imporan o recognie ae liabiliy managemen need and poibly hedge he long daed inere rae rik of he produc KRD are preferable in hee cae ince hey permi effecive hedging a numerou enor of, for inance, he wap curve, and ake ino accoun poible non-parallel yield curve dynamic (ee, for inance, uckman, 2002) We will begin wih a review of key rae eniiviie he KRD i defined o be he vecor of n fir order inere rae eniiviie a each enor uch ha he reulan change in liabiliy ariing from he inere rae impac i: KRD 1 δr 1 δ [ Liabiliy] = KRD δr = KRD 2 M KRD n δr 2 M δr n where δ[] indicae a change over ome period, and r i he po rae a ime he liabiliy arie due o he embedded opion from he guaranee in he GMWB conrac I i change in hee liabiliie ha he praciioner would eek o hedge, perhap wih wap radiionally, he vecor KRD involve performing finie differencing a each Clearly finie differencing can conume coniderable runime when numerou hocked key rae are required Wih he pahwie mehod one only require he bae valuaion run o ge inere rae eniiviie a every enor Le he pahwie KRD be defined a: KRD PW = V r = V E D p c max( WD AV ) r = 0 We are now concerned wih wo derivaive wihin he expecaion, pecifically, he dicoun facor and he guaranee I i clear ha any change in he po rae r will affec he adjacen forward a ime and +1 Hence, D r = D r + D r where we have now inroduced he forward rae, f I i worh noing ha derivaive involving po and forward rae ake he following form: D = D d δ where d = 025 year in our model and δ i he Kronecker dela equal o one if = Furher, = f r r = r r d d r ( d ) + 1 Derivaive of he guaranee ake he form: r max WD AV = I WD > AV { } WD r d AV r I i imporan o noe ha for < he above expreion i zero Similar o he cae of dela, recurive relaionhip arie when derivaive on he guaranee are compued Specifically, AV = + AV f AV + r f r r + 1 where for k=, +1 he accoun value varie in forward rae by AV = AV k k WD 1 1 and wihdrawal vary in forward rae by WD k = BB k k 1 rk e + AV rk ( 1 WD 1) e d MAW = AV max( AV, BB 1 ) w = I BB R + I k NR k k w Where again he indicaor funcion rack wheher or no he accoun value racheed a ime, R, or no, NR 42 Life & Penion

4 cuing edge Variable annuiie A hi poin we have idenified all of he perinen expreion required o generae pahwie dela and KRD eniiviie for he pecified GMWB conrac wih aic behaviour 44 Dynamic behaviour Dynamic behaviour i a complex funcion of moneyne above aic bae lape aumpion (imilar o a morgage prepaymen funcion) Ofen, buinee and academic reearcher employ complicaed proprieary funcion derived for exiing produc or from experience udie he augmened GMWB conrac dela in he preence of dynamic behaviour i: V = V + p E D p S = S S V he augmened GMWB conrac KRD in he preence of behaviour i: V = D E + + V p p V p V r r r r D = 0 V where we have inroduced he (now dynamic) p Le he probabiliy of urvival be defined a p = p p 1 uch ha p depend on ome (o be defined) funcion φ(s 0 ) p = 1 φ( S 0 ) Derivaive in he behaviour funcion will involve differeniable approximaion of he indicaor funcion, ince i conain expreion for AV and lape will occur when he benefi i ou-of-he-money For he implemenaion of hi cheme we employed he following approximaion: anh x ε H x able 1: Comparion of ime 0 dela a a percenage of iniial premium for a GMWB conrac wih no policyholder behaviour Cohor 1 Cohor 2 Finie difference dela -7589% -2526% Pahwie dela -7592% -2550% able 2: Comparion of ime 0 dela a a percenage of iniial premium for a GMWB conrac wih policyholder behaviour Cohor 1 Cohor 2 Finie difference dela -6302% -3390% Pahwie dela -6333% -3399% 45 Implemenaion Implemenaion wa performed wih excel/vba calling malab funcion and run were performed on everal ordinary PC One of he main challenge in hi implemenaion i o keep rack of he variou indicaor hrough ime in he imulaion hee indicaor would evenually be ued o calculae greek a he end of he imulaion Addiionally, higher order or more complex greek, uch a vanna or key rae vega, rely on careful coding and derivaion of he perinen derivaive he reducion in runime due o no re-imulaion valy ouweighed he increaed complexiy of racking hee indicaor While exac number are highly dependen on coding language, produc pecific, yem, and any opimizaion, we oberved run-ime reducion of 44% for he inrumen modelled for hi paper In all imulaion 5000 rik-neural pah were ued for each cohor 46 Dicuion of reul able 1 and 2 how comparion of dela uing he pahwie mehod o ha of he radiional finie difference mehod We find ha he pahwie mehod of obaining dela give u aifacory value wih and wihou dynamic behaviour In addiion, Figure 1 how ha he dela uing boh mehod are roughly idenical a he GMWB conrac age In addiion, he pahwie mehod i een o be robu when re eed In Figure 2, we how he impac of changing moneyne on he dela uing boh he mehod We find ha he dela calculaed uing pahwie coincide wih ha of he finie difference mehod he andard error i impaced wih hi approach a hown in able 3 In general he pahwie mehod will have le andard error relaive o finie difference for differen h and N hi relaionhip beween h and N can be beneficial wih higher order greek where convergence may be dependen on hock ize able 4 how a imilar comparion Figure 1 Dynamic behaviour dela Dela 5% 0% -5% -10% -15% -20% -25% -30% Valuaion year Finie difference dela Pahwie dela Comparion of dela for a GMWB conrac ample pah during he fir 10 year wih dynamic behaviour Figure 2 Variable moneyne dela Dela 5% 0% -5% -10% -15% -20% Moneyne Finie difference dela Pahwie dela Comparion of dela for a GMWB conrac (Cohor 1) when moneyne (AV 0 /BB 0 ) i varied 00% 080% -20% 070% 060% -40% 050% -60% 040% -80% 030% -100% 020% -120% 010% -140% 000% Valuaion year Finie difference rho error Pahwie rho error Finie difference rho Pahwie rho Rho Figure 3 KRD projecion Comparion of KRD projecion for a GMWB conrac (Cohor 1) Sandard error wwwlife-penioncom Sepember

5 cuing edge Variable annuiie for KRD for = 1, 3, 5, 7, 10 uing he pahwie mehod and he radiional finie difference mehod he projecion in Figure 3 of one KRD10 pah alo how conien agreemen wih finie difference along a 10 year projecion I i worh noing ha he periodic piking in he error, while mall in abolue magniude, i he reul of he annual rache feaure rendering he opion AM hi being only one conrac, he porfolio effec will end o mooh hi feaure, bu imilar o dela of a book of pah dependen opion, here may be non-rivial convexiy co o hedging a porfolio of hee conrac if here i any ype of heerogeneiy in conrac anniverarie due o pike in ale or eaonaliy hi undercore he value of he pahwie mehod in ha he abiliy o perform projecion and need hedge imulaion are very imporan for proper rik managemen and produc developmen of long daed produc wih hee complex feaure Rache and wihdrawal end o diplay complex behaviour and impac greek ued in hedging hi lend ielf o he noion ha i i imporan and good rik managemen o be able o calculae a va number of complex greek, and projec hem for behaviour and imulaion purpoe o beer underand hedge impac and raegy deign We have ofen found ha i i compuing ime and reource ha make hi challenging o manage 5 Exenion of mehodology he power of he pahwie mehodology increae a more greek are calculaed uing he bae marke valuaion imulaion We have exended hi framework o calculae oher greek, uch a key rae vega, gamma, and componen dela and gamma By exenion he mahemaic become more cumberome emming from manipulaing he variable in he fir order expreion, uch a he indicaor funcion Performing ucceful differeniaion may involve numerical or funcional approximaion ha have cerain limi of appropriae ue or involve more complex mahemaic, like generalied funcion, which are beyond he cope of hi paper We have uccefully calculaed complex GMIB and GMWB greek wih hi mehod, however, i i imporan o noe ha many payoff funcion generally encounered may no be C 2 and oher mehod, like imporance ampling, may need o be employed Laly, hi mehod can admi more ochaic rik facor, hould hey be warraned, uch a ochaic rae or volailiy in he bae pricing run 6 Concluion In hi paper we have derived and implemened a pahwie greek mehodology for a complex GMWB conrac he moivaion emmed mainly from he deire o reduce he expenive compuing ime for compuing marke value eniiviie and he reulan addiional reource ha may be employed for hedging, raegy developmen, ae-liabiliy managemen and reearch hi i of pracical ue for many marke paricipan, epecially hedging or raegy R&D eam he mehod produce an unbiaed eimae of greek We have uccefully exended hi mehod o oher produc and oher greek However, we would like o end on a cauionary noe o fully underand he limiaion of hi mehod, when i i appropriae o ue, and o alway have anoher, more brue force mehod of validaing and checking reul readily available, uch a finie difference L&P he auhor are member of Derivaive Sraegie eam wihin ING USFS Marke Rik Managemen Addiionally, we wih o hank he reviewer for heir helpful commen able 3: Comparion of dela andard error for a GMWB conrac (Cohor 1) for variou h and N Sandard error (h=0001) Sandard error (h=00001) Finie difference dela (N = 500) Finie difference dela (N = 5000) Pahwie dela (N = 500) Pahwie dela (N = 5000) able 4: Comparion of KRD value a a percenage of iniial premium for a GMWB conrac wihou policyholder behaviour Key rae Finie difference KRD Pahwie KRD Difference 1 805% 804% 001% % 1035% 003% 5 879% 878% 001% 7 739% 742% -003% % -400% 000% he view expreed herein are hoe of he auhor and no of ING Group or any of i global buine uni Reference 1 Broadie, Mark & Glaerman, Paul Eimaing Securiy Price Derivaive Uing Simulaion Managemen Science, V42, No 2, , February Glaerman, Paul Mone Carlo Mehod in Financial Engineering Springer, uckman, Bruce Fixed Income Securiie Wiley Finance, 2nd Ed 2002, Page Life & Penion welcome ubmiion o i peer-reviewed Cuing Edge ecion Aricle hould be en o lauriecarver@inciivemediacom Submiion guideline are available a hp://wwwlife-penioncom/ public/howpagehml?page= Life & Penion