FAST SIMULATION OF EQUITYLINKED LIFE INSURANCE CONTRACTS WITH A SURRENDER OPTION. Carole Bernard Christiane Lemieux


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1 Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. FAST SIMULATION OF EQUITYLINKED LIFE INSURANCE CONTRACTS WITH A SURRENDER OPTION Carole Bernard Chrstane Lemeux Department of Statstcs and Actuaral Scence Unversty of Waterloo Waterloo, ON, N2L 3G1, CANADA ABSTRACT In ths paper, we consder equtylnked lfe nsurance contracts that gve ther holder the possblty to surrender ther polcy before maturty. Such contracts can be valued usng smulaton methods proposed for the prcng of Amercan optons, but the mortalty rsk must also be taken nto account when prcng such contracts. Here, we use the leastsquares Monte Carlo approach of Longstaff and Schwartz coupled wth quasmonte Carlo samplng and a control varate n order to construct effcent estmators for the value of such contracts. We also show how to ncorporate the mortalty rsk nto these prcng algorthms wthout explctly smulatng t. 1 INTRODUCTION Lfe nsurance companes used to sell lfe nsurance contracts that provde a fxed captal at tme of death of the nsured. In the past ten years, ndvduals are more and more wllng to combne nvestment and lfe nsurance. Nowadays popular contracts often nvolve a return lnked to the market. Equty ndex annutes are products that offer nvestors downsde protecton because of the presence of a mnmum maturty beneft and death beneft wth a potental upsde rate n case the ndex performs well (see for nstance Palmer (2006) or Hardy (2003) for more detals on contracts currently offered). In ths paper we nvestgate some of the optons embedded n an equtylnked lfe nsurance contract. We nclude standard optons such as a guaranteed mnmum maturty beneft and a guaranteed mnmum death beneft. But many more complex optons exst (see for nstance Ballotta and Haberman (2003) or Boyle and Hardy (2003) to cte only a few). The possblty of early wthdrawals s nvestgated by Mlevsky and Salsbury (2006). In ths study we focus on the surrender opton, whch gves polcyholders the rght to termnate ther polces before the maturty ndcated n ther contract. It s the possblty of wthdrawng the total value of the contract. Lterature on surrender optons n equtylnked lfe nsurance products s recent snce these optons have been neglected for a long tme. In the 1990s, they were ndeed consdered as very lowrsk optons snce nterest rates n the market were much hgher than the mnmum guaranteed rate and the stock market performed very well. After several bankruptces n the nsurance sector (for nstance around 2001, the Brtsh company, Equtable Lfe went bankrupt due to optons embedded n penson annutes where mortalty rsk and fnancal rsk were underestmated and not hedged), embedded optons are now taken more serously nto account n the valuaton of the contracts. Each opton offers nvestors some addtonal rghts that mght cause some proft reducton for nsurer f optmally exercsed. It s thus of utmost mportance for companes to understand all optons embedded n contracts they sell to better prce and hedge them. Concernng the surrender opton, there are several consequences. Frst, the nsurer mght ncur fnancal losses from early payments caused by surrendered polces due to lqudty of ther assets (they mght have to lqudate deprecated assets to face surrender demand). Indeed companes face a tradeoff between long term nvestment (and hgh returns) but wth few lqudty or nvestng n lqud assets (wth a lower return). Second, n the long run nsurer mght suffer from adverse selecton snce polcyholders that have health problems and thus nsurablty dffcultes wll not surrender. To avod adverse selecton, some contracts (for example contracts that have only a beneft n case of survval) do not allow surrender. To mnmze rsks assocated wth the surrender opton, nsurers can gve hgh penaltes. However nsured are keen on far surrender condtons snce these contracts are often longterm nvestments and ther needs mght change and force them to surrender. Regulators, lawyers and market competton between companes protect nvestors aganst unfar condtons n case of early termnaton. To value ths surrender opton, no unfed framework exsts n prevous lterature. Surrender decson can be of two types: an exogenous surrender (for example, dependng on /08/$ IEEE 444
2 Bernard and Lemeux personal reasons: famly problem, unemployment, sudden need of lqudty) or an economc or endogenous surrender, lnked to nterest rates fluctuatons, fnancal envronment changes, etc. Actuares estmate exogenous surrender from hstorcal data on the lapse rate. Exogenous surrender rsk can be dversfed. Thus nsurers should be more threatened by endogenous surrender rsk that s dffcult to estmate (n partcular because of lack of data) and leads to systematc rsks. It seems that there were only weak hstorcal evdence of stuatons when surrender was optmal. Any approach based on hstorcal lapses mght underestmate the real cost of surrender optons. Kuo, Tsa, and Chen (2003) study how the lapse rate s nfluenced by unemployment rate and nterest rate fluctuatons. To deal wth the second type of surrender decsons (fnancal, endogenous surrender rsk), the alternatve s to use a pure fnancal approach based on the opton theory. Surrender opton s seen as an Amercan opton. Indeed equtylnked contracts wth a fxed term can be expressed as a portfolo of European optons (Brennan and Schwartz (1976), and Boyle and Schwartz (1977) are the frst ones proposng ths approach). The cost of the surrender opton s then the Amercan premum that s added f these optons can be exercsed before maturty. In the Black and Scholes framework, Grosen and Jørgensen (1997) gve the optmal exercse barrer by usng results from Mynen (1992). Frst studes of the surrender opton market value use numercal schemes to solve partal dfferental equatons (Jensen, Jørgensen, and Grosen 2001)) or bnomal trees (Bacnello 2003, Bacnello 2005). Bacnello takes nto account mortalty rsk, perodcal premums and annual bonus, but both of these approaches are extremely slow. Recently, Shen and Xu (2005) have also nvestgated surrender optons by way of partal dfferental equatons. These works use the noarbtrage prncple and the market value of the surrender opton s the market value of an optmal surrender decson. New accountng rules force nsurers to evaluate ther labltes (consstng manly of the sold contracts) at ther market value even though the contracts are not really traded on a market. There s thus an mportant need for fnancal modelng of nsurance contracts. The no arbtrage approach to value the surrender opton has been crtczed snce ths opton s not traded. Moreover, t mples that f t s optmal for one nsured to surrender, t s thus optmal for all of them to surrender, whch s not realstc even f t s the greatest rsk faced by the ssung company. Albzzat and Geman (1994) propose an nterestng model to address ths ssue. The dea s that the value of the surrender opton at a gven date s known and then t can be splt over dfferent dates usng an exogenous lapse rate. In ths paper, we are also nterested n gvng a new way to tackle ths problem by ntroducng a parameter that wll depends on the ndvdual, and wll nuance hs optmal decson. All ndvduals wll thus not have the same optmal decson. Our approach s a fnancal approach and we want to obtan a market value of the surrender opton, n a smlar way to what has been done n Andreatta and Corradn (2003), Bacnello, Bffs, and Mllossovch (2008a), and Bacnello, Bffs, and Mllossovch (2008b). We use the leastsquares Monte Carlo approach of Longstaff and Schwartz (2001) to perform the valuaton, but we nclude the mortalty rsk usng a dfferent model and a dfferent approach than Bacnello, Bffs, and Mllossovch (2008a), Bacnello, Bffs, and Mllossovch (2008b). In partcular, by makng the usual assumpton that the mortalty rsk s ndependent of the fnancal rsk, we show that survval tmes do not need to be smulated. Ths reduces the varablty of the obtaned estmators for the value of the surrender opton. In addton, we use the value of the European contract as a control varate and perform smulatons usng quasrandom samplng. The rest of ths paper s organzed as follows. In Secton 2, we descrbe the lfe nsurance contract, a pont to pont Equty Indexed Annuty wth a guaranteed mnmum death beneft and a guaranteed mnmum surrender beneft. The surrender decson s frst based exclusvely on fnancal crtera, n other words a polcyholder surrenders hs polcy f the surrender amount gves hm at least the opportunty to buy the same contract at a lower prce on the market. But ths approach overestmates the surrender opton by lookng at the worst case that practcally never happens: all polcyholders surrender at the same tme (the optmal surrender tme). In Secton 2.3, we address ths ssue and ntroduce a parameter that represents the propensty of the ndvdual to act optmally. In Secton 3, we show how to nclude mortalty rsk n the leastsquares Monte Carlo approach used to prce the contract. Then we dscuss n Secton 4 the two methods we used to mprove the effcency of ths Monte Carlo estmator: a control varate based on the European contract, and quasmonte Carlo samplng to smulate the fnancal paths. Numercal results are gven n Secton 5, and a bref concluson s provded n Secton 6. 2 FRAMEWORK We frst descrbe the contract, a ponttopont Equty Indexed Annuty (EIA) wth a mnmum guarantee at maturty or at tme of death of the nsured. Then, we gve our assumptons for the underlyng ndex dynamcs, and we ntroduce the guaranteed mnmum surrender beneft studed n ths paper. 2.1 The contract We consder a ponttopont EIA wth a Guaranteed Mnmum Maturty Beneft. The polcy has a fxed maturty date of T years. The nsured s ntal nvestment s P. We assume 445
3 Bernard and Lemeux that no addtonal payments are done after the ncepton of the contract. Most EIAs n the U.S. are purchased wth a sngle premum (see Palmer (2006)). The contract s lnked to an ndex, for nstance the S&P 500 ndex. We denote by S t the value of the ndex at tme t. We suppose a percentage α of the ntal premum P s guaranteed at a mnmum annual rate equal to g. These parameters α and g are often constraned by law. For nstance, the Unted States have adopted recently revsed nonforfeture regulatons, statng that the mnmum guarantee under newly ssued contracts must be at least 87.5% of all premums pad, P accumulated at a mnmum nterest rate, here g, between 1% and 3%. About one thrd of EIAs apply ther stated nterest rate guarantee to only 87.5% percent of the premum, one thrd to 90%, and only one ffth to 100%, see Palmer (2006). The fnal payoff of the contract at maturty tme T wrtes as: ( ) ) k V T = αp max (1 + g) T, ( ST S 0. (1) Ths contract s only for the purpose of llustraton and ths study can easly be extended to more complcated contracts. Insurance companes propose untlnked polces or equtyndexed annutes wth addtonal death guarantees. Assumng that we neglect longevty rsk, mortalty rsk can be dversfed. Indeed deaths observed n a portfolo of nsureds mght happen at ndependent dates. The large law of numbers and the central lmt theorem are sutable to estmate rsks. Note that ths s not the case when mortalty s stochastc (as studed, for nstance, by Bacnello, Bffs, and Mllossovch (2008a) and Bacnello, Bffs, and Mllossovch (2008b). Here we assume the equtylnked contract descrbed by (1) s combned wth a Guaranteed Mnmum Death Beneft. We assume the guaranteed mnmum death beneft s pad at the end of the year when the death of the nsured occurs and that t can be wrtten as: ( ( ) ) kd D t = αp max (1 + g d ) t St,, t = 1,...,T, (2) f death occurs between t 1 and t, where we assume there s no penalty but the guaranteed rate g d and the partcpaton k d n the ndex performance mght be dfferent than the ones nvolved n the contract s payoff descrbed by (1). Palmer (2006) note that few EIAs assess a surrender charge f the EIA s cashed n due to the contract owner s death. Unlke the mortalty rsk, the fnancal rsk cannot be hedged by poolng arguments. All polces depend on the same ndex and f t goes up, the nsurer wll have to pay a hgh partcpaton rate for all polces at the same tme. S 0 Ths cannot be hedged by acceptng more polces. Theoretcally, those rsks can be hedged by usng the followng decomposton of the contract s fnal value as ( (ST ) k V T = αp (1 + g) T + max αp (1 + g),0) T, (3) whch s a portfolo made up of a guaranteed amount αp usually yeldng a lower nterest rate g than the rskfree rate r of the market plus a long poston n a call opton wrtten on the underlyng ST k. Usng Black&Scholestype results, companes can adjust ther portfolo contnuously and replcate the above payoff. Ths decomposton and ts prcng n a Black and Scholes framework were frst done n Brennan and Schwartz (1976) and n Boyle and Schwartz (1977). 2.2 Model For the ease of exposton, here we neglect all types of costs, and assume the market s complete and perfectly lqud. We set ourselves n Black and Scholes framework, although our methodology could easly be appled to more complex models. The rskfree nterest rate s constant and denoted by r. Snce the market s complete, there s a unque rskneutral measure, Q. Dynamcs of the ndex S under the Qmeasure wrte as: ds t S t S 0 = rdt + σdw t where σ s the volatlty of the ndex and W t s a Qstandard Brownan moton. The far value of the contract s obtaned by takng the expectaton under the rskneutral probablty Q of the dscounted cashflows: ξ (S 0,g,k,T ) = e rt E [V T ] (4) wherev T s gven by (1). After straghtforward computatons n the Black and Scholes framework, we obtan a closedform formula for the European contract: ( ) ξ (S 0,g,k,T ) = e (g r)t S 0 Φ g k r σ 2 2 T kσ ( + S 0 e (k 1)rT +k(k 1) σ2 T 2 Φ α + kσ ) T, where Φ( ) s the CDF of a standard normal varable. Ths s the rskneutral prce the ntal amount of money needed to perfectly hedge the fnal cashflows and provde the payoff (1) to the nsured for the European verson of the contract where surrender s not allowed and there s 446
4 Bernard and Lemeux no death beneft. Ths formula does not nclude the cost of hedgng, transacton costs and the dscrete hedgng error that wll obvously occurred n practce. We now nclude the mnmum guaranteed death beneft defned earler n (2). Let us assume the probablty that a polcyholder of age x+t wll de before the end of the year s q x+t, for t = 0,...,T 1, and we let p x+t = 1 q x+t. More precsely, f we nclude the mortalty rsk n the formula (4), then we have that the European contract s value at tme 0, denoted V 0,e, s gven by T 1 V 0,e = T p x ξ (S 0,g,k,T )+ t p x q x+t ξ (S 0,g d,k d,t +1), t=0 (5) where ξ s defned by the formula (4) and t p x = t 1 j=0 p x+ j s the probablty that an ndvdual of age x survves at least t years. 2.3 The surrender opton We are nterested n a contract where there s a Guaranteed Mnmum Surrender Beneft. That s, we assume that the above polcy also offers a mnmum surrender guarantee, and that the rght to surrender can only be exercsed at the end of each year untl the maturty of the contract. Hence ths s a Bermudantype opton rather than a truly Amercan one. In practce, ths mnmum guarantee s often ndependent of the return of the ndex, and thus we make the assumpton that the polcyholder receves the followng amount L t f he surrenders the polcy at tme t: L t = (1 β t ) αp (1 + h) t, (6) where 0 h < r s the guaranteed rate, and β t s the penalty charged for surrenderng at tme t, for t = 1,...,T 1. A standard penalty would be for nstance a decreasng rate over years. Examples of penalty functons are gven n Palmer (2006). Sometmes a market value adjustment s appled when market performs poorly. By law, the polcy s cash surrender value cannot fall below the guaranteed mnmum value (h g). The ndvdual wll surrender because of fnancal reasons f L t > C t, where C t s the market value at tme t of the contract. In other words, f the nsured has an opportunty to make a proft f he surrenders and that at the same tme he can buy exactly the same polcy. However, and as mentoned n the ntroducton, t s plausble to assume that a polcyholder may not make a decson that s optmal from the purely fnancal pont of vew when choosng to surrender or not. For ths reason, we assume there s an extra decson parameter λ 1 such that the contract s surrendered only f L t > λc t. In some sense, t means that some agents wll react faster to the surrender crtera than others. Informed agents wll have λ = 1. For unnformed agents that do not worry about ther nvestments, they wll surrender f the surrender condton s really nterestng and thus f L t s sgnfcantly hgher than the market value of the contract C t. It should be clear that as λ ncreases, the value of the (fnancal) surrender opton goes to 0, a fact that we verfy n our experments of Secton 5. 3 LEASTSQUARES MONTE CARLO WITH MORTALITY Let us frst revew the leastsquares Monte Carlo approach as proposed n Longstaff and Schwartz (2001) appled to the problem of prcng an equtylnked contract when there s no mortalty rsk. The method uses n realzaton paths {St,t = 0,1,...,T ; = 1,...,n} of the ndex, and then estmates for each path when s the optmal exercse tme t. Ths s done by proceedng backward from T as follows: set t = T, and the contract s value on path to VT, whch s the value of V T as n (1), but wth S T replaced by the fnal value ST of the ndex on path. Then at tme t = T 1,T 2,...,1, set t = t and Vt = Lt f Lt > Ĉt, where Ĉt s an estmate of the contnuaton value of the contract at tme t gven St, gven by e r E(V t+1 St), and Lt s the surrender value at tme t on path. Ths estmate Ĉt s obtaned by regresson of the dscounted contract s value at the next tme step aganst the current value of the ndex. More precsely, a fnte set of multvarate bass functons {ψ l ( ),l = 0,1,...,M} s chosen, and the regresson coeffcents are estmated as ( ˆβ 0,..., ˆβ M ) T = (Ψ T Ψ) 1 Ψ T (y 1,...,y n ) T, where y = e r Vt+1, and Ψ,l = ψ l (St) for = 1,...,n,l = 0,...,M. Then Ĉt = M ˆβ l=0 l ψ l (St). When Lt Ĉ(t,St), then we smply update the contract s value on path by dscountng t for one more step,.e., we set Vt = e r Vt+1. Once the optmal exercse tmes t are estmated for each path, the contract s value s approxmated by the lowbased estmator ˆV 0, f n = 1 n n =1 where we used the conventon that L T = V T. e rt L t, (7) 447
5 Bernard and Lemeux We now want to nclude the mortalty rsk n the above prcng methodology, as well as the surrender parameter λ descrbed n the prevous secton. When decdng whether to surrender or not, the polcyholder must take nto account the fact that he/she mght de n the comng year, thus recevng D t+1 at tme t + 1 rather than holdng on to the contract. That s, the contnuaton value can be wrtten as E(V t+1 S t) = e r (q x+t E(V t+1 S t,death) +p x+t E(V t+1 S t,survval)) = q x+t e r E(D t+1 S t) + p x+t C t = q x+t ξ (S t,g d,k d,1) + p x+t C t. where ξ s defned by (4). Hence at tme t, the contract s surrendered f L t > C t := q x+t e r D t+1 + p x+t Ĉ t, where Ĉ t s the contnuaton value condtoned on survval untl t + 1, and s calculated as before. In ths case, we set V t = L t. However the contract s value V t must be updated dfferently when the surrender value L t does not exceed the weghted contnuaton value C t. More precsely, we have LsMCEqLnkMort() for 1 to n t () T generate S1,...,S T V [] VT for j T 1 downto 1 q q x+t p = 1 q compute ˆβ 0,..., ˆβ M for 1 to n Ct M 1 l=0 ˆβ l, j ψ l (St) C t e r qξ (St,g d,k d,1) + pct f Lt > λ Ct then t [] t V [] Lt else V [] e r (qξ (St,g d,k d,1) + pv []) for 1 to n A[] t []p x e rt [] Lt [] for t = 0 to t [] A[] A[] + e rt t p x q x+t Dt+1 return n =1 A[]/n V t = e r (q x+t ξ (S t,g d,k d,1) + p x+t V t+1 ). Once we have an estmate of the optmal exercse tme t on each path, then we must take nto account the mortalty rsk by replacng the estmator (7) by where A = ( t p x e rt L t ˆV 0 = 1 n t 1 + n =1 A, (8) t p x q x+t e r(t+1) Dt+1 t=0 and t p x = t 1 j=0 p x+ j s the probablty that an ndvdual of age x survves at least t years. The dea s that f the polcyholder des before the estmated optmal surrender tme, then a death beneft wll nstead be pad at the end of the year of death. Summng up, we proceed as n the pseudocode gven n Fgure 1. 4 EFFICIENCY IMPROVEMENT A frst way to mprove the qualty of the estmator (8) s to use the value of the correspondng European contract as a control varate. We know the European contract value when the mortalty s ncluded. It s gven by V 0,e defned earler by (5). Hence the control varate estmator s gven ), Fgure 1: Pseudocode descrbng regressonbased Monte Carlo for an equtylnked contract wth mortalty by where ˆV 0,e s gven by where ˆV 0,cv = ˆV 0 + ˆγ(V 0,e ˆV 0,e ), (9) ˆV 0,e = 1 n E = e rt T p x VT T 1 + t=0 n =1 E, e r(t+1) t p x q x+t D t+1. The coeffcent ˆγ n (9) s gven by the estmated value of Cov(E,A )/Var(E ). The surrender opton can then be estmated as ˆX 0,cv = ˆV 0,cv V 0,e. (10) Secondly, we can use randomzed quasmonte Carlo methods to smulate the n paths of the ndex. Ths was done n the context of the leastsquares Monte Carlo algorthm n Lemeux (2004), among others. Here, we do ths usng 448
6 Bernard and Lemeux a randomly dgtally shfted Sobol pont set, and apply the Brownan brdge technque (Caflsch and Moskowtz 1995) n order to reduce the effectve dmenson of the problem. More precsely, what we need here s a T dmensonal pont set P n n [0,1) T, obtaned usng the frst n ponts of the Sobol sequence (Sobol 1967) wth the drecton numbers gven n Bratley and Fox (1988). Then we randomze ths pont set usng a random dgtal shft (L Ecuyer and Lemeux 2002). By repeatng the randomzaton process m tmes ndependently, we thus obtan m estmators ˆV 0,l of the form (8) for the contract (wth the surrender opton), but where the path S1,...,S T s based on the randomzed pont ũ,l, obtaned by addng the lth dgtal shft to the th pont of P n, for =,1...,n and l = 1,...,m. The applcaton of the Brownan brdge technque n ths case amounts to use the frst coordnate of ũ,l to generate ST, when constructng the lth estmator ˆV 0,l, then the second coordnate s used to generate S T /2, the thrd one for S T /4, and so on. Once we have these m estmators ˆV 0,l, we can construct a 95% confdence nterval for V 0 wth halfwdth m(m 1) m l=1 ( ˆV 0,l V 0 ) 2, where V 0 = m l=1 ˆV 0,l /m. A smlar approach can be used for the control varate estmator. 5 RESULTS In what follows, we gve results for dfferent equtylnked contracts. In each case, we report the value of the contract, ˆV 0, the smulated value of the contract wthout a surrender opton, gven by ˆV 0,e, and the value ˆX 0,cv of the surrender opton. Note that we also report the exact value of the contract n the lne startng wth exact. For each estmate, we report results usng ether Monte Carlo (MC) or randomzed quasmonte Carlo (RQMC) samplng, and gve below each estmate the halfwdth of a 95% confdence nterval for the value that we are tryng to estmate. To prce the contract, we run smulatons ether wth a control varate as gven n lnes MCcv and RQMCcv or wthout as gven n lnes MCnocv and RQMCnocv. Note that snce the estmate ˆX 0,cv for the surrender opton dffers only by a constant from the one V 0,cv for the contract tself, the halfwdth of the confdence nterval s the same for both estmators. The default (benchmark) values for the parameters are: T = 10 years, σ = 20%, P = 100, α = 0.85, r = 4%, the mnmum guaranteed rates g, h and g d are all set to 2%, and the partcpatng coeffcent k and k d to 90%. The penaltes are as follows: β 1 = 0.05, β 2 = 0.04, β , β 4 = 0.01, β t = 0 for t 5. We also assume that the polcyholder s ratonal and perfectly nformed: λ = 1. For mortalty, we use a smple parametrc model: the Makeham s model. The survval functon s(x), whch gves the probablty that an ndvdual of age 0 wll survve at least x years, s gven by ( x ) s(x) = exp µ(s)ds = exp( Ax B(c x 1)) 0 where µ(x) = A + Bc x wth B > 0,A B,c > 1,x 0. We can then calculate the probablty to lve and to de as follows: t p x = s(x +t), tq x = 1 t p x. s(x) Melnkov and Romanuk (2006) calbrated Makeham s model for the mortalty n the USA and gve the followng estmates of the parameters A, B and c: A = , B = , c = All experments have been performed by generatng 25..d. groups of 8192 smulatons, for a total of 204,800 smulaton runs. Ths s about 10 tmes less than the number of smulatons 19,000 smulatons wth 140 seeds, for a total of 2, runs used n Bacnello, Bffs, and Mllossovch (2008a), Bacnello, Bffs, and Mllossovch (2008b), who get results wth about the same precson as our RQMC estmators wth the control varate (but use stochastc mortalty, as explaned before). For the bass functons of the regresson, we use the frst four Legendre polynomals. Table 1: Result wth the default parameters ˆV 0 ˆV 0,e ˆX 0 MCnocv MCcv RQMCnocv RQMCcv exact In Table 1, we report our results wth the benchmark parameters. The European contract has the value Note that the ntal premum of the contract s P = 100 and thus the market value of the contract s lower than the premum charged at tme 0. Ths s consstent wth real contracts that often nclude commssons and fees for dfferent servces and optons. Includng the surrender opton ncreases the value of the contract by approxmately
7 Bernard and Lemeux Thus, the surrender opton represents about 1.9% of the total value of the (European) contract. We now experment wth dfferent values of λ, partcpatng parameters, and model parameters. Table 2: Varyng λ ˆV 0 ˆV 0,e ˆX 0 λ = 1.05 MCnocv MCcv RQMCnocv RQMCcv exact λ = 1.15 MCnocv MCcv RQMCnocv RQMCcv exact We frst ncrease the value of λ to 1.05 or 1.15 and we report the results n Table 2. We observe that when λ s ncreasng, the polcyholder becomes less and less nterested n the surrender opton. Its value becomes almost zero. In fact, nsurers rely a lot on ths fact. Polcyholders wll not have a fnancally optmal behavour and wll be reluctant to surrender ther polces. A stuaton where λ > 1 as t s the case n Table 2 s thus more realstc. Note that t would be nterestng to look at the rsk of a portfolo of polces wth a pool of polcyholders wth dfferent parameter λ. We can also look at the mpact of changes n the partcpatng coeffcent k and k d, or n the mnmum guaranteed rates g, g d, and h. Our results suggest that the surrender opton s less valuable when the mnmum guarantees at maturty, for death, and for surrender are hgher (that s, g = h = g d s hgher) or f the partcpatng coeffcent k at maturty or k d at tme of death ncreases. Indeed, the surrender opton represents 0.82/96.73 = 0.85% of the value of the European contract when the guaranteed rates ncrease to g = h = g d = 3% nstead of the benchmark case of g = h = g d = 2%, and 1.68/ % when the partcpatng coeffcents k and k d are 95% nstead of 90%, nstead of a rato of 1.95% wth the default parameters. In the second case, we expect that an ncrease n the partcpaton at maturty and n case of death are ncentve Table 3: Varyng partcpaton parameters ˆV 0 ˆV 0,e ˆX 0 g = h = 0.03 MCnocv MCcv RQMCnocv RQMCcv exact k = k d = 0.95 MCnocv MCcv RQMCnocv RQMCcv exact h = 0.03 MCnocv MCcv RQMCnocv RQMCcv exact to not surrender. But n the frst case, there s a tradeoff between the ncreased guarantee at maturty and death whch are ncentve for not exercsng and the ncrease n the guarantee n case of surrender. Our results suggest that the ncrease at maturty and death offset the ncrease n the surrender payoff. However, f only the guaranteed rate h for the surrender opton ncreases to 3%, then, as expected, the opton s value ncreases from about 1.8 to about Fnally, we can analyze the mpact of the fnancal assumptons and n partcular the volatlty of the underlyng ndex. We can compare the value of the surrender opton equal to 1.8 when volatlty s 20% (benchmark parameter n Table 1), wth the values n Table 4, gven by 1.47 f σ = 10% or 1.91 f σ = 30%. The value of the surrender opton thus appears to be ncreasng wth the volatlty σ. Ths s consstent wth the wellknown fact that the value of an opton ncreases wth the volatlty of the underlyng ndex. 450
8 Bernard and Lemeux Table 4: Varyng fnancal model ˆV 0 ˆV 0,e ˆX 0 σ = 0.1 MCnocv MCcv RQMCnocv RQMCcv exact exact σ = 0.3 MCnocv MCcv RQMCnocv RQMCcv exact CONCLUSIONS Ths paper extends the Least Squares Monte Carlo method proposed by Longstaff and Schwartz (2001) to the valuaton of surrender benefts embedded n lfe nsurance ndex lnked contracts. Wth a relatvely small number of smulatons, we get a qute precse approxmaton of the surrender beneft usng quasmonte Carlo smulatons and ncorporatng a powerful control varate. We show how to nclude mortalty rsk when ths rsk s modeled usng fxed probabltes that are the same for everybody. In practce, polcyholders who decde to surrender a lfe nsurance contract are healther than the average. The surrender opton wll ntroduce adverse selecton (worse rsks stay wth the nsurer and healthy people leave the portfolo) and thus wll ncrease a lot the potental value of the surrender benefts. We beleve that the proposed method could easly be extended, for nstance, to the case where the optmal decson of the polcyholder s taken condtonally to hs health state. We plan to study ths model n the near future. ACKNOWLEDGMENTS Both authors are grateful to the Natonal Scence and Engneerng Research Councl (NSERC) of Canada for ts fnancal support through ndvdual Dscovery Grants. We would lke to thank Phelm Boyle, Mary Hardy and Harry Panjer for useful comments and suggestons on ths work. REFERENCES Albzzat, M.O., and H. Geman Interest rate rsk management and valuaton of the surrender opton n lfe nsurance polces. The Journal of Rsk and Insurance 61: Andreatta, G., and S. Corradn Far value of lfe labltes wth embedded optons: an applcaton to a portfolo of Italan nsurance polces. Workng Paper, RAS Spa, Panfcazone Reddtvtá d Gruppo. Bacnello, A. R Prcng guaranteed lfe nsurance partcpatng polces wth annual premums and surrender opton. North Amercan Actuaral Journal 7 (3): Bacnello, A. R Endogenous model of surrender condtons n equtylnked lfe nsurance. Insurance: Mathematcs and Economcs 37 (2): Bacnello, A. R., E. Bffs, and P. Mllossovch. 2008a. Prcng lfe nsurance contracts wth early exercses features. Journal of Computatonal and Appled Mathematcs. In press. Bacnello, A. R., E. Bffs, and P. Mllossovch. 2008b. Regressonbased algorthms for lfe nsurance contracts wth surrender guarantees. Workng Paper. Ballotta, L., and S. Haberman Valuaton of guaranteed annuty converson optons. Insurance: Mathematcs and Economcs 33 (1): Boyle, P. P., and M. R. Hardy Guaranteed annuty optons. Astn Bulletn 33 (2): Boyle, P. P., and E. S. Schwartz Equlbrum prces of guarantees under equtylnked contracts. Journal of Rsk and Insurance 44: Bratley, P., and B. L. Fox Algorthm 659: Implementng Sobol s quasrandom sequence generator. ACM Transactons on Mathematcal Software 14 (1): Brennan, M. J., and E. S. Schwartz The prcng of equtylnked lfe nsurance polces wth an asset value guarantee. Journal of Fnancal Economcs 3: Caflsch, R. E., and B. Moskowtz Modfed Monte Carlo methods usng quasrandom sequences. In Monte Carlo and QuasMonte Carlo Methods n Scentfc Computng, ed. H. Nederreter and P. J.S. Shue, Number 106 n Lecture Notes n Statstcs, New York: SprngerVerlag. Grosen, A., and P. L. Jørgensen Valuaton of early exercsable nterest rate guarantees. Journal of Rsk and Insurance 64 (3): Hardy, M. R Investment Guarantees: Modellng and Rsk Management for EqutyLnked Lfe Insurance. Wley. 451
9 Bernard and Lemeux Jensen, B., P. L. Jørgensen, and A. Grosen A fnte dfference approach to the valuaton of pathdependent lfe nsurance labltes. The Geneva Papers on Rsk and Insurance Theory 26 (1): Kolkewcz, A., and K. Tan Untlnked lfe nsurance contracts wth lapse rates dependent on economc factors. Annals of Actuaral Scence 1 (1): Kuo, W., C. Tsa, and W.K. Chen An emprcal study on the lapse rate: the contegraton approach. Journal of Rsk and Insurance 70 (3): L Ecuyer, P., and C. Lemeux Recent advances n randomzed quasmonte Carlo methods. In Modelng Uncertanty: An Examnaton of Stochastc Theory, Methods, and Applcatons, ed. M. Dror, P. L Ecuyer, and F. Szdarovszk, : Kluwer Academc Publshers, Boston. Lemeux, C Randomzed quasmonte Carlo methods: a tool for mprovng the effcency of smulatons n fnance. In Proceedngs of the 2004 Wnter Smulaton Conference, : IEEE Press. Longstaff, F., and E. Schwartz Valung Amercan optons by smulaton: a smple leastsquares approach. The Revew of Fnancal Studes 14 (1): Melnkov, A., and Y. Romanuk Evaluatng the performance of Gompertz, Makeham and LeeCarter mortalty models for rsk management wth untlnked contracts. Insurance: Mathematcs and Economcs 39: Mlevsky, M., and T. S. Salsbury Fnancal valuaton of guaranteed mnmum wthdrawal benefts. Insurance: Mathematcs and Economcs 38 (1): Mynen, R The prcng of the Amercan opton. Annals of Appled Probablty 2:1 23. Palmer, B. A Equtyndexed annutes: Fundamental concepts and ssues. Workng Paper. Shen, W., and H. Xu The valuaton of untlnked polces wth or wthout surrender optons. Insurance: Mathematcs and Economcs 36 (1): Sobol, I. M On the dstrbuton of ponts n a cube and the approxmate evaluaton of ntegrals. U.S.S.R. Comput. Math. and Math. Phys. 7: from the Unversté de Montréal. Her research nterests are quasmonte Carlo methods, and how they can replace random samplng n a varety of problems. Her emal address s AUTHOR BIOGRAPHIES CAROLE BERNARD s an Assstant Professor n the Department of Statstcs and Actuaral Scence at the Unversty of Waterloo. She obtaned her Ph.D. n 2006 from Unversté de Lyon I, for whch she was rewarded wth the Best Fnance Thess Award from France the same year. Her research nterests are fnancal and nsurance modelng. Her emal address s CHRISTIANE LEMIEUX s an Assocate Professor n the Department of Statstcs and Actuaral Scence at the Unversty of Waterloo. She obtaned her Ph.D. n
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