Valuation of Variable Annuities with Guaranteed Minimum Withdrawal and Death Benefits via Stochastic Control Optimization

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1 Valuatio of Variable Auities with Guarateed Miimum Withdrawal ad Death Beefits via Stochastic Cotrol Optimizatio Xiaoli Luo 1, ad Pavel V. Shevcheko 2 arxiv: v2 [q-fi.cp] 7 Apr 2015 Draft, 1st versio 20 November 2014, this versio 10 February The Commowealth Scietific ad Idustrial Research Orgaisatio, Australia; Xiaoli.Luo@csiro.au 2 The Commowealth Scietific ad Idustrial Research Orgaisatio, Australia; Pavel.Shevcheko@csiro.au Correspodig author Abstract I this paper we preset a umerical valuatio of variable auities with combied Guarateed Miimum Withdrawal Beefit (GMWB) ad Guarateed Miimum Death Beefit (GMDB) uder optimal policyholder behavior solved as a optimal stochastic cotrol problem. This product simultaeously deals with fiacial risk, mortality risk ad huma behavior. We assume that market is complete i fiacial risk ad mortality risk is completely diversified by sellig eough policies ad thus the auity price ca be expressed as appropriate expectatio. The computig egie employed to solve the optimal stochastic cotrol problem is based o a robust ad efficiet Gauss-Hermite quadrature method with cubic splie. We preset results for three differet types of death beefit ad show that, uder the optimal policyholder behavior, addig the premium for the death beefit o top of the GMWB ca be problematic for cotracts with log maturities if the cotiuous fee structure is kept, which is ordiarily assumed for a GMWB cotract. I fact for some log maturities it ca be show that the fee caot be charged as ay proportio of the accout value there is o solutio to match the iitial premium with the fair auity price. O the other had, the extra fee due to addig the death beefit ca be charged upfrot or i periodic istalmet of fixed amout, ad it is cheaper tha buyig a separate life isurace. Keywords: Variable Auity, Optimal Stochastic Cotrol, Guarateed Miimum Withdrawal Beefit, Guarateed Miimum Death Beefit, Mortality Risk 1

2 1 Itroductio A variable auity is a fud-liked isurace cotract icludig a variety of fiacial optios o the policy accout value; see e.g. Smith (1982) ad Walde (1985). A recet descriptio of the mai features of variable auity products ad the developmet of their market ca be foud i Ledlie et al. (2008) ad?. The mai features of variable auities are represeted by a variety of possible guaratees, which are briefly referred to as GMxB - Guarateed Miimum x Beefit, where x stads for accumulatio (A), death (D), icome (I) or withdrawal (W). All GMxBs provide a protectio of the policyholder s accout: GMAB durig the accumulatio phase ad GMDB i case of early death, GMIB ad GMWB after retiremet, i particular i the face of high logevity. I this study we focus o Guarateed Miimum Withdrawal Beefit (GMWB) i combiatio with Guarateed Miimum Death Beefit (GMDB), overall referred to as GMWDB. A variable auity cotract with GMWB promises to retur the etire iitial ivestmet through cash withdrawals durig the policy life plus the remaiig accout balace at maturity, regardless of the portfolio performace. Thus eve if the accout of the policyholder falls to zero before maturity, GMWB feature will cotiue to provide the guarateed cashflows. GMWB allows the policyholder to withdraw fuds below or at cotractual rate without pealty ad above the cotractual rate with some pealty. If the policyholder behaves passively ad the withdrawal amout at each withdrawal date is predetermied at the begiig of the cotract, the the behavior of the policyholder is called static. I this case the paths of the accout ca be simulated ad a stadard Mote Carlo simulatio method ca be used to price the GMWB, though i low dimesio problems partial differetial equatio (PDE) or itegratio methods are more efficiet. O the other had if the policyholder optimally decide the amout of withdrawal at each withdrawal date, the the behavior of the policyholder is called dyamic. A ratioal policyholder of GMWB will always choose the optimal withdrawal strategy to maximize the preset value of cashflows geerated from holdig GMWB. Uder the optimal withdrawal strategy of a policyholder, the pricig of variable auities with GMWB becomes a optimal stochastic cotrol problem. There is a rich literature o dyamic programmig to deal with optimal stochastic cotrol problems i geeral, for textbook treatmet see e.g. Powell (2011); Bäuerle & Rieder (2011). This problem caot be solved by a simulatio based method such as the well kow Least-Squares Mote Carlo method itroduced i Logstaff & Schwartz (2001), due to the fact that the paths of the uderlyig variable are altered by the 2

3 optimal withdrawal amouts at all pay dates prior to maturity ad thus they caot be simulated. The variable auities with GMWB feature have bee cosidered i e.g. Milevsky & Salisbury (2006), Bauer et al. (2008), Dai et al. (2008), Che & Forsyth (2008), Baciello et al. (2011) ad Luo & Shevcheko (2014b). I the case of a variable auity cotract with Guarateed Miimum Death Beefit, the beeficiaries obtai a death beefit if the isured dies durig the defermet period. Whe variable auities were itroduced, a very simple form of death beefit was predomiat i the market. Sice the mid 1990s, isurers started to offer a broad variety of death beefit desigs. The basic form of a death beefit is the so-called Retur of Premium Death Beefit. Here, the maximum of the curret accout value at time of death ad the sigle premium is paid. Geerally speakig, give a mortality model or a Life Table, the evaluatio of GMDB is straightforward - a stadard Mote Carlo method will work fie for such a problem ad ofte closed form solutios ca be obtaied. The pricig of GMWB is more ivolved, ad it is eve more challegig uder the dyamic (optimal) policyholder behavior. Milevsky & Salisbury (2006) developed a variety of methods for pricig GMWB products. I their static approach the GMWB product is decomposed ito a Quato Asia put plus a geeric term-certai auity. I their dyamic approach they assume the policyholder ca termiate (surreder) the cotract at the optimal time, which leads to a optimal stoppig problem aki to pricig a America put optio. Bauer et al. (2008) presets valuatio framework of variable auities with multiple guaratees. I their dyamic approach a strategy cosists of a umerous possible withdrawal amouts at each paymet date. They have developed a multidimesioal discretizatio approach i which the Black-Scholes PDE is trasformed to a oe-dimesioal heat equatio ad a quasi-aalytic solutio is obtaied through a simple piecewise summatio with a liear iterpolatio o a mesh. Ufortuately the umerical formulatio cosidered i Bauer et al. (2008) has four dimesios ad the computatio of eve a sigle price of the GMWB cotract uder the optimal policyholder strategy is very costly. It is metioed i their paper that it took betwee 15 ad 40 hours o the stadard desktop PC to obtai a sigle price ad o results forthe dyamic case were show; also it looks like their methodology i the case of death beefit with dyamic GMWB correspods to the upper estimator of the price (i.e. correspods toformula(22)itheext sectio). Dai et al.(2008)developed aefficiet fiite differece algorithm usig the pealty approximatio to solve the sigular stochastic cotrol problem for a cotiuous withdrawal model uder the dyamic withdrawal strategy. They have also developed a fiite differece algorithm for the more 3

4 realistic discrete withdrawal formulatio. Che & Forsyth (2008) preset a impulse stochastic cotrol formulatio for pricig variable auities with GMWB uder the optimal policyholder behavior, ad develop a sigle umerical scheme for solvig the Hamilto-Jacobi-Bellma variatioal iequality for the cotiuous withdrawal model as well as for pricig the discrete withdrawal cotracts. I Baciello et al. (2011) the static valuatios are performed via ordiary Mote Carlo method, ad the mixed valuatios, where the policyholder is semiactive ad ca decide to surreder the cotract at ay time durig the life of the GMWB cotract, are performed by the Least Squares Mote Carlo method. Recetly we have developed a very efficiet ew algorithm for pricig variable auities with GMWB uder both static ad dyamic policyholder behaviors solvig a equivalet stochastic cotrol problem; see Luo & Shevcheko (2014b). Here the defiitio of dyamic is similar to the oe used by Bauer et al. (2008), Dai et al. (2008) ad Che & Forsyth (2008), i.e. the ratioal policyholder ca decide a optimal amout to withdraw at each paymet date (based o iformatio available at this date) to maximize the expected discouted value of the cashflows geerated from holdig the variable auity with GMWB. The algorithm is either based o solvig PDEs usig fiite differece or o simulatios usig Mote Carlo. It relies o computig the expected optio values i a backward time-steppig betwee withdrawal dates through Gauss-Hermite itegratio quadrature applied o a cubic splie iterpolatio (referred to as GHQC). I Luo & Shevcheko (2014b) it is demostrated that i pricig GMWB uder the optimal policyholder behavior the GHQC algorithm ca achieve similar accuracy as the fiite differece method, but it is sigificatly faster because it requires less umber of steps i time. This method ca be applied whe trasitio desity of the uderlyig asset betwee withdrawal dates or it s momets are kow i closed form ad required expectatios are 1d itegrals. It has also bee successfully used to price exotic optios such as America, Asia, barrier, etc; see Luo & Shevcheko (2014a). So far i the literature GMWB ad GMDB are maily cosidered as two separate cotracts ad it is implicitly assumed the policyholder of a GMWB cotract will always live beyod the maturity date or there is always someoe there to make optimal withdrawal decisios for the etire duratio of the cotract. I reality a elderly policyholder may die before maturity date, especially for a cotract with a log maturity. For example, accordig to the Australia Life Table, Table 6, a male aged 60 will have more tha 57% probability to die before the age of 85. So, for a 60 year old male takig up a GMWB cotract with a maturity of 25 years; the product desig ad pricig 4

5 should certaily cosider the probability of death durig the cotract. Some variable auity products may have expiry for the death beefit guaratee, e.g. at the age 70 or 75. I this paper we formulate pricig GMWDB (GMWB combied with GMDB) as a stochastic cotrol problem, where at each withdrawal date the policyholder optimally decides the withdrawal amout based o iformatio available at this date. It is importat to ote that pricig dyamic GMWDB for a give death time (i.e. coditioal o kowig death time) ad the averagig over possible death times accordig to death probabilities will lead to higher price tha dyamic GMWDB where decisios are based o iformatio available at withdrawal date (this will be discussed i the ext sectio). We first exted the stadard GMWB to allow for termiatio of the cotract due to mortality risk, returig the maximum of the remaiig guaratee withdrawal amout ad the portfolio accout value upo death. I additio, two types of extra death beefits are cosidered: payig the iitial premium or payig the maximum of the iitial premium ad the portfolio accout value upo death. Our recetly developed GHQC algorithm i Luo & Shevcheko (2014b) eables us to perform a comprehesive umerical study o the evaluatio of GMWDB. I the ext sectio we describe the GMWDB product with extra death beefits o top of the usual GMWB features ad outlie the uderlyig stochastic model ad correspodig optimal stochastic cotrol problem. Sectio 3 presets the umerical method ad algorithm for pricig GMWDB uder both static ad dyamic policyholder behaviors. I Sectio 4 we preset umerical results for the fair fees of GMWDB uder a series of cotract coditios. For the case of GMWDB uder the optimal policyholder strategy, the iadequacy of the traditioal fee structure is revealed i the case of additioal death beefit. The fee caot be charged as ay proportio of the accout value there is o solutio to match the iitial premium with the fair auity price. A explaatio of the iadequacy is give through a example which demostrates why there is o solutio for the fair fee for some log maturities if the death beefit payig at least the iitial premium is guarateed to a ratioal policyholder. O the other had, if a fixed upfrot fee or periodic istalmet is charged, the extra fee of addig life isurace to GMWB is cheaper tha holdig a separate life isurace. Cocludig remarks are give i Sectio 5. 5

6 2 Model ad Stochastic Cotrol Formulatio A variable auity cotract with guarateed miimum withdrawal beefit ad death beefit (GMWDB) promises to retur the etire iitial ivestmet through cash withdrawals durig the policy life plus the remaiig accout balace at the cotract maturity T, regardless of the portfolio performace. I additio, if policyholder dies before or at the cotract maturity, the death beefit is paid to the beeficiaries. We assume (commo assumptio i the academic research literature o pricig variables auities) that market is complete i fiacial risk ad mortality risk is completely diversified through sellig eough policies ad thus the auity price ca be expressed as expectatio with respect to appropriate (risk-eutral) probability measure for the risky asset. Below we outlie the cotract setup, model assumptios ad solutio via optimal stochastic cotrol method. 2.1 Assumptios ad GMWDB cotract details Assume that the auity policyholder is allowed to take withdrawals γ at times t 1,...,t N = T. The premium paid upfrot is ivested ito risky asset S(t). Deote the value of correspodig wealth accout at time t as W(t), i.e. upfrot premium is W(0). GMWB is the guaratee of the retur of the premium via withdrawals γ 0 allowed at times t, = 1,2,...,N. Let N w deote the umber of withdrawals i a year (e.g. N w = 12 for a mothly withdrawal), the the total umber of withdrawals N = N w T, where N = deotes the ceilig of a float umber. The withdrawals caot exceed the guaratee balace ad withdrawals ca be differet from cotractual (guarateed) withdrawal amout G = W(0)(t t 1 )/T with pealties imposed if γ > G. Also deote the aual cotractual rate as g = 1/T. Cosider the followig auity cotract details ad model assumptios. Risky asset process. Let S(t) deote the value of the referece portfolio of assets (mutual fud idex, etc.) uderlyig the variable auity policy at time t that follows the risk eutral process ds(t) = r(t)s(t)dt + σ(t)s(t)db(t), (1) whereb(t)isthestadardwieerprocess, r(t)isriskfreeiterestrateadσ(t)is volatility. Hereafter we assume that the model parameters are piecewise costat fuctios of time for time discretizatio 0 = t 0 < t 1 < < t N = T, where t 0 = 0 is today ad T is auity cotract maturity. Deote correspodig asset values 6

7 as S(t 0 ),...,S(t N ) ad risk free iterest rate ad volatility as r 1,...,r N ad σ 1,...,σ N respectively. That is, σ 1 is the volatility for (t 0,t 1 ]; σ 2 is the volatility for (t 1,t 2 ], etc. ad similarly for iterest rate. Wealth accout. For clarity, deote the value of the wealth accout at time t before withdrawal as W(t ) ad after withdrawal as W(t + ). The, for the risky asset process (1), the value of wealth accout W(t) evolves as W(t ) = W(t+ 1 ) S(t 1 ) S(t )e αdt = W(t + 1)e (rα 1 2 σ2 )dt+σ dtz, (2) W(t + ) = max ( W(t )γ,0 ), = 1,2,...,N, (3) where dt = t t 1, z are idepedet ad idetically distributed stadard Normal radom variables, ad α is aual fee charged by isurace compay. If the wealth accout balace becomes zero or egative, the it will stay zero till maturity. For the process (2), the trasitio desity from W(t + 1 ) to W(t ) is logormal desity, i.e. lw(t + 1) is from Normal distributio with the mea lw(t )+(r α 1 2 σ2 )dt ad variace σ 2 dt. Guaratee accout. Deote the value of guaratee accout at time t as A(t) with A(0) = W(0), ad the value of the accout at t before withdrawal as A(t ) ad after withdrawal as A(t + ). The guaratee accout evolves as A(t + ) = A(t )γ = A(t + 1)γ, = 1,2,...,N (4) with A(T + ) = 0 ad W(0) = A(0) γ 1 + +γ N ad A(t + 1) N k= γ k. Also ote that guaratee accout is uchaged withi (t 1,t ), i.e. A(t + 1) = A(t ). Some real products iclude step-up arragemet that will icrease guaratee accout balace A(t ) to max(a(t ),W(t )) o aiversary dates, i.e. i the case of good ivestmet performace. For simplicity we do ot cosider this feature explicitly but it is ot difficult to icorporate this ito the algorithm preseted i this paper. Pealty. The cashflow received by the policyholder at t, {1,...,N} if alive is give by C (γ ) = { γ, if 0 γ G, G +(1β)(γ G ), if γ > G, (5) 7

8 where G is cotractual withdrawal amout. That is, pealty is applied if withdrawal γ exceeds cotractual amout G, i.e. β [0,1] is the pealty applied to the portio of withdrawal above G. Todiscourageexcessive withdrawalsbeyodg, somecotractsmayicludereset provisio o the guaratee level A(t + ) = mi(a(t ),W(t ))γ if γ > G. We do ot cosider this explicitly but it is ot a problem to icorporate this feature i the pricig algorithm preseted i this paper. Death process. Deote the time of policyholder death, a radom variable, as τ with coditioal death probabilities q = Pr[t 1 < τ t τ > t 1 ]. It is assumed that death time τ ad asset process S(t) are idepedet. The policyholder age at t 0 is eeded to fid these probabilities from Life Tables. We assume that these probabilities are kow ad to simplify otatio we do ot use policyholder age variable explicitly i the formulas. I our umerical examples we assume age 60 for male ad female policyholders ad use the curret Australia Life Table, see Table 6. Cosider the correspodig Markov process defied by discrete radom variables at t 1,t 2,... 1, if policyholder is alive at t, I = 0, if policyholder died durig (t 1,t ], 1, if policyholder died before, i.e. τ t 1, with trasitio desity from I 1 at t 1 to I at t specified by probabilities Pr[I = 1 I 1 = 1] = 1q ; Pr[I = 1 I 1 = 0] = 0; Pr[I = 1 I 1 = 1] = 0; Pr[I = 0 I 1 = 1] = q ; Pr[I = 0 I 1 = 0] = 0; Pr[I = 0 I 1 = 1] = 0; Pr[I = 1 I 1 = 1] = 0; Pr[I = 1 I 1 = 0] = 1; Pr[I = 1 I 1 = 1] = 1. For example, if death time is betwee t 3 ad t 4, the realizatio of the process for = 0,1,2,... is I = {1,1,1,1,0,1,1,...}. Note that this variable I is ot affected by withdrawal at t. Death beefit. Ifdeathtimeτ isaftercotractmaturityt,theatthematurity the policyholder takes the maximum betwee the remaiig guaratee withdrawal et of pealty charge ad the remaiig balace of the persoal accout, i.e. the fial payoff is P T (W(T ),A(T )) = max ( C N ( A(T ) ),W(T ) ). (7) (6) 8

9 If death time τ occurs before or at cotract maturity T the the payoff take by the beeficiary at death time slice t d (the first time slice larger or equal τ) is the death beefit P D (W(t d ),A(t d )). We cosider three types of death beefit deoted as DB0, DB1 ad DB2 as follows: max ( A(t d ),W(t d )), death beefit DB0, P D (W(t d ),A(t d )) = W(0), death beefit DB1, max ( (8) W(0),W(t d )), death beefit DB2. I the above, iitial premium W(0) is sometimes adjusted for iflatio which is a trivial extesio. I some policies the death beefit may chage at some age, for example DB2 or DB1 may chage to DB0 at the age of 75 years (effectively makig the death beefit guaratee expirig at the specified age). If death beefit expirig correspods to switchig to the stadard GMWB, the it ca be hadled by settig death probabilities q to zero after death beefit expiry till the cotract maturity. Payoff. Give withdrawal strategy γ = (γ 1,...,γ N ), the preset value of the overall payoff of the auity cotract is a fuctio of state variables correspodig to wealth accout W = (W(t 0 ),...,W(t N )), guaratee accout A = (A(t 0 ),...,A(t N )) ad death state I = (I 0,...,I N ). Deote the state vector at timet beforethewithdrawalasx = (W(t ),A(t ),I )adx = (X 1,...,X N ). The the auity payoff is where H 0 (X,γ) = B 0,N h N (X N )+ N1 =1 B 0, f (X,γ ), (9) h N (X N ) = P T (W(T ),A(T ))1 {IN =1} +P D (W(T ),A(T ))1 {IN =0} (10) is the cashflow at the cotract maturity ad f (X,γ ) = C (γ )1 {I=1} +P D (W(t ),A(t ))1 {I=0} (11) is the cashflow at time t. Here, 1 { } is idicator fuctio equals 1 if coditio i { } is true ad zero otherwise, ad B i,j is discoutig factor from t j to t i ( tj ) ( j ) B i,j = exp r(t)dt = exp r dt, t j > t i. (12) t i 9 =i+1

10 GMWDB static case. Give the above assumptios ad defiitios the auity price uder the give static strategy γ 1,γ 2,...,γ N ca be calculated as Q 0 (W(0),A(0),I 0 = 1) = E X t 0 [H 0 (X,γ)], (13) where E X t 0 [ ] deotes expectatio with respect to process X coditioal o iformatio available at time t 0. I the case of static strategy, A is determiistic ad thus expectatio is with respect to W ad I processes. GMWDB dyamic case. Uder the optimal dyamic strategy the auity price is Q 0 (W(0),A(0),I 0 = 1) = supe X t 0 [H 0 (X,γ)], (14) γ where γ = γ (X ) is a fuctio of state variable X at time t, i.e. ca be differet for differet realizatios of X. Note that i this case, A process is stochastic via stochasticity i γ. GMWB case. The stadard GMWB cotract correspods to the above payoff forgmwdbifdeathprobabilitiesaresettozero, q 1 = = q N = 0, i.e. h N (X N ) ad f (X,γ ) i GMWDB payoff H 0 (X,γ) give i (9) simplify to h N (X N ) = P T (W(T ),A(T )) ad f (X,γ ) = C (γ ). The static ad dyamic GMWB prices are give by (13) ad (14) respectively where expectatios are calculated with respect to W process. 2.2 Solvig Optimal Stochastic Cotrol Problem Give that state variable X = (X 1,...,X N ) is Markov process, it is easy to recogize that the auity valuatio uder the optimal withdrawal strategy (14) is optimal stochastic cotrol problem for Markov process that ca be solved recursively to fid auity value Q t (x) at t, = N 1,...,0 via backward iductio ) Q t (x) = sup (f (X,γ (X ))+B,+1 Q t+1 (x )K t (dx x,γ ) 0 γ A(t ) startig from fial coditio Q T (x) = h N (x). Here K t (dx x,γ ) is the stochastic kerel represetig probability to reach state i dx at time t +1 if the withdrawal (actio) γ is applied i the state x at time t. For a good textbook treatmet of stochastic cotrol problem i fiace, see Bäuerle & Rieder (2011). (15) 10

11 Explicitly, this backward recursio ca be rewritte as Q t + (W,A,I ) = B,+1 E X ( ] +1 t [Q t +1 W(t +1 ),A(t +1 ),I +1) W,A,I, ( Q t (W,A;I ) = max C (γ )1 {I=1} +P D (W,A)1 {I=0} +Q t + ((W γ 0 γ A,0),Aγ,I ) ) for = N 1,N 2,...,0 startig from maturity coditio at t N = T ( Q t W(t N N ),A(t N ),I N) = PT (W(T ),A(T ))1 {IN =1} +P D (W(T ),A(T ))1 {IN =0}, (16) where for clarity we deote Q t ( ) ad Q t + ( ) the auity values at time t before ad after withdrawal respectively. Takig expectatio with respect to death variable I +1 ad usig Q t + (W,A,I = 0) = Q t + (W,A,I = 1) = Q t (W,A;I = 1) = 0 ad Q t (W(t ),A(t ),I = 0) = P D (W(t ),A(t )), itsimplifiestotherecursioequatios Q t + (W,A,I = 1) = (1q )B,+1 E X +1 t [Q t +1 ( W(t +1 ),A(t +1),I +1 = 1 ) W,A,I = 1] +q B,+1 E X +1 t [ PD (W(t +1 ),A(t +1 )) W,A,I = 1 ] (17) with jump coditio ( Q t (W,A;I = 1) = max C (γ )+Q t + ((W γ 0 γ A,0),Aγ,I = 1) ), (18) ad maturity coditio ( Q t W(t N N ),A(t N ),I N = 1 ) = P T (W(T ),A(T )). (19) Note that expectatios i (17) are with respect to W(t +1 ) oly because A(t+ ) = A(t +1). 2.3 Dyamic GMWDB Upper ad Lower Estimators There are may lower estimators that ca be costructed for dyamic GMWDB price Q t0 ( ) give by (14). I particular, ay static (determiistic) strategy will produce lower estimator give by equatio (13). I Sectio 4 we will utilize oe of such lower estimators. Oe of the possible upper estimators of the dyamic GMWDB (14) ca be foud via calculatios of GMWDB coditioal o death time (i.e. perfect forecast of the 11

12 death time). For easier uderstadig ad otatioal coveiece, istead of dealig with the death process I i the followig we cosider the correspodig death time radom variable τ ad let t d be the first withdrawal time equal or exceedig τ ad Ñ = mi(d,n) (if τ > T the without loss of geerality set d = N + 1). To avoid cofusio, we deote this GMWDB coditioal o τ by V 0 ( ;t d ). Coditioal o τ (i.e. coditioal o t d ), it is give by V 0 (W(0),A(0);t d ) = max γ 1,...,γÑ1 E W t 0 B 0, Ñ h(tñ)+ Ñ1 =1 B 0, C (γ ), (20) h(tñ) = P T (W(t N ),A(t N ))1 {τ>t} +P D (W(t d ),A(t d ))1 {τ T}, where E W t 0 [ ] is expectatio with respect to wealth process W(t ), = 0,1,...,N. It ca be solved usig backward recursio V t + (W,A;t d ) = e r(t +1t ) E W(t +1 ) ( ] t [V t +1 W(t +1 ),A(t +1 );t d) W,A;td, ( V t (W,A;t d ) = max C (γ )+V t + ((W γ 0 γ A,0),Aγ ;t d ) ) for = Ñ 1,Ñ 2,...,0 startig from maturity coditio at Ñ = mi(d,n) ( ) W(t Ñ ),A(t);t Ñ d = h(tñ). (21) V t Ñ The the expectatio with respect to death time gives Q (u) 0 (W(0),A(0),I 0 = 1) = E τ t 0 [V 0 (W(0),A(0);t d )] = Pr[τ > T]V 0 (W(0),A(0);t N+1 ) N + Pr[t 1 < τ t ]V 0 (W(0),A(0);t ). (22) =1 Note that V 0 (W(0),A(0);t N+1 ) correspods to the stadard GMWB (i.e. GMWDB coditioal o death after cotract maturity). Also ote that the death probabilities p = Pr[t 1 < τ t ]aredifferetfromcoditioaldeathprobabilitiesq = Pr[t 1 < τ t τ > t 1 ] i the death process (6); both p ad q are easily foud from Life Tables or mortality process models. Q (u) 0 ( ) is the upper estimator for dyamic GMWDB Q t0 ( ), give by (14), because it is based o optimal strategy coditioal o perfect forecast for time of death for each death process realizatio. Formally, Q (u) 0 (W(0),A(0),I 0 = 1) = E τ t 0 [sup γ ] E W t 0 [H 0 (X,γ(W,A)) τ] supe τ,w t 0 [H 0 (X,γ(X))] = Q 0 (W(0),A(0),I 0 = 1). (23) γ 12

13 3 Numerical algorithm I the literature, the optimal stochastic cotrol problem for pricig GMWB with discrete optimal withdrawals has oly bee successfully dealt with by solvig the oedimesioal PDE equatio usig a fiite differece method preseted i Dai et al. (2008) ad Che & Forsyth(2008). Simulatio based method such as the Least Squares Mote Carlo method caot be applied for such problems due to the dyamic behavior of the policyholder affectig the paths of the uderlyig wealth accout. Recetly, Luo & Shevcheko (2014b) have cosidered a alterative method without solvig PDEs or simulatig paths. The ew approach relies o computig the expected optio values i a backward time-steppig betwee withdrawal dates through a high order Gauss-Hermite itegratio quadrature applied o a cubic splie iterpolatio. I this paper we adopt this method to calculate GMWDB. 3.1 Numerical quadrature to evaluate the expectatio TopricevariableauitycotractwithGMWDB,i.e. tocomputeq 0 (W(0),A(0),I 0 = 1), we have to evaluate expectatios i (17). Assumig the coditioal probability desity of W(t ) give W(t+ 1) is kow as p (w W(t + 1)), i the case of process (2) it is just logormal desity, (17) ca be evaluated as where ( Q t + 1 W(t + 1 ),A,I = 1 ) + = B 1, p (w W(t + 1)) Q (w)dw, (24) 0 Q (w) = B 1, ( (1q )Q t (w,a,i = 1)+q P D (w,a) ). We use Gauss-Hermite quadrature for the evaluatio of the above itegratio over a ifiite domai. The required cotiuous fuctio Q t (w, ) will be approximated by a cubic splie iterpolatio o a discretized grid i the W space. The wealth accout domai is discretized as W mi = W 0 < W 1,...,W M = W max, where W mi ad W max are the lower ad upper boudary, respectively. For pricig GMWDB, because of the fiite reductio of W at each withdrawal date, we have to cosider the possibility of W goes to zero, thus the lower boud W mi = 0. The upper boud is set sufficietly far from the spot asset value at time zero W(0). A good choice of such a boudary could be W max = W(0)exp(5σT). The idea is to fid auity values at all these grid poits at each time step t through itegratio (24), startig at maturity t N = T. At each time step we evaluate the itegral (24) for every grid poit by a high accuracy umerical quadrature. 13

14 The auity value at t = t is kow oly at grid poits W m, m = 0,1,...,M. I order to approximate the cotiuous fuctio Q t (w, ) from the values at the discrete grid poits, we use the atual cubic splie iterpolatio which is smooth i the first derivative ad cotiuous i the secod derivative; ad secod derivative is zero for extrapolatio regio. The process for W(t) betwee withdrawal dates is a simple process give i (2), the coditioal desity of W(t ) give W(t + 1) is from a logormal distributio. To apply Gauss-Hermite umerical quadrature for itegratio over a ifiite domai, we itroduce a ew variable Y(t ) = l( W(t )/W(t+ 1) ) (r α 1 2 σ2 )dt σ dt, (25) ad deote the fuctio Q (w) after this trasformatio as Q (y) (y). The the itegratio becomes ( Q t + 1 W(t + 1 ),A,I = 1 ) = 1 + 2π e 1 2 For a arbitrary fuctio f(x), the Gauss-Hermite quadrature is + e x2 f(x)dx q j=1 y2 Q(y) (y)dy. (26) λ (q) j f(ξ (q) j ), (27) where q is the order of the Hermite polyomial, ξ (q) i,i = 1,2,...,q are the roots of the Hermite polyomial H q (x), ad the associated weights λ () j are give by λ (q) i = 2q1 q! π q 2 [H q1 (ξ (q) i )] 2. Applyig a chage of variable x = y/ 2 ad usig the Gauss-Hermite quadrature to (26), we obtai ( Q t + 1 W(t + 1 ),A,I = 1 ) 1 π q i=1 λ (q) i Q (y) ( 2σ dt ξ () j ). (28) If we apply the chage of variable (25) ad the Gauss-Hermite quadrature (26) to every grid poit W m, m = 0,1,...,M, i.e. let W(t + 1 ) = W m, the the optio values at time t = t + 1 for all the grid poits ca be evaluated through (28). If the trasitio desity fuctio from W(t + 1 ) to W(t ) is ot kow i closed form but oe ca fid its momets, the itegratio ca also be doe by matchig momets as described i Luo & Shevcheko (2014a,b). 14

15 3.2 Jump coditio applicatio Ay chage of A(t) oly occurs at withdrawal dates. After the amout γ is draw at t, the auity accout reduces from W(t ) to W(t+ ) = max(w(t )γ,0), ad the guaratee balace drops from A(t ) to A(t + ) = A(t ) γ. The jump coditio of Q t (W,A,I = 1) across t is give by Q t (W,A,I = 1) = max 0 γ A [Q t + (max(w γ,0),aγ,i = 1)+C(γ )]. (29) For the optimal strategy, we chose a value for γ uder the restrictio 0 γ A to maximize the fuctio value Q t (W,A,I = 1) i (29). To apply the jump coditios, a auxiliary fiite grid 0 = A 1 < A 2 < < A J = W(0) is itroduced to track the remaiig guaratee balace A, where J is the total umber of odes i the guaratee balace amout coordiate. For each A j, we associate a cotiuous solutio from (28) ad the cubic splie iterpolatio. We ca limit the umber of possible discrete withdrawal amouts to be fiite by oly allowig the guaratee balace to be equal to oe of the grid poits 0 = A 1 < A 2 < < A J = W(0). This implies that, for a give balace A j at time t, the withdraw amout γ takes j possible values: γ = A j A k, k = 1,2,...,j ad the jump coditio (29) takes the followig form Q t (W m,a j,i = 1) = max 1 k j [Q t + (max(w m A j +A k,0),a k,i = 1)+C(A j A k )]. Fortheoptimalstrategy,wechoseavaluefor1 k j tomaximizeq t (W m,a j,i = 1)i(30). Notethatalthoughthejumpamoutγ = A j A k, k = 1,2,...,j isidepedet of time t ad accout value W m, the value Q t + (max(w m A j +A k,0),a k,i = 1) depeds o all variables (W m,a j,t ) ad the jump amout. It is worth poitig out that part of the good efficiecy of the GHQC algorithm for pricig GMWB or GMWDB uder ratioal policyholder behavior is due to that fact that the same cubic splie iterpolatio is used for both umerical itegratio (24) ad the applicatio of jump coditio (30). A clear advatage of this umerical algorithm over PDE based fiite differece approach is that sigificatly smaller umber of time steps are required because the trasitio desity over the fiite time period i (24) is kow. The fiite differece method requires dividig the period betwee two cosecutive withdrawal dates ito fier time steps for a good accuracy due to the fiite differece approximatio to the partial derivatives. 15 (30)

16 3.3 Death probabilities Give a Life Table, such as Table 6, estimatig the death probabilities q ad p required i (6) ad (22) is straightforward. The Life Table tabulates the umber of people survivig to the exact age startig with 100,000 at the age zero for each sex ad goes beyod 100 years. Deote the umber of people still alive at age k years as a list L(k), k = 0,1,...,K. Deote the age of policyholder at the start of the cotract (i.e. at time t 0 = 0) as k 0. To estimate the coditioal death probabilities q = Pr[t 1 < τ t τ > t 1 ] ad p = Pr[t 1 < τ t τ > t 0 ], we oly eed to kow the umber of people alive at time t = t 1 ad t = t. Because i the Life Table the umber of people L(k) is oly give at iteger umber k, we estimate the umber of people alive L(k 0 +t) for a arbitrary time t usig liear iterpolatio, i.e. assumig a uiform distributio for the death time withi a year. Of course a more elaborate iterpolatio is also possible. The, for k t+k 0 k +1, L(k 0 +t) is calculated as L(k 0 +t) = (k +1tk 0 ) L(k)+(t+k 0 k) L(k +1). Havig obtaied L(k 0 +t 1 ) ad L(k 0 +t ) usig the above formula (ote t 1 ad t may ot ecessarily fall withi the same year), the coditioal death probabilities are estimated as q = Pr[t 1 < τ t τ > t 1 ] L(k 0 +t 1 )L(k 0 +t ), L(k 0 +t 1 ) p = Pr[t 1 < τ t τ > t 0 ] L(k 0 +t 1 )L(k 0 +t ). L(k 0 +t 0 ) Istead of a Life Table, stochastic mortality models such as frequetly used bechmark Lee-Carter model (?) forecastig mortality rate usig stochastic process (ad typically accoutig for systematic mortality risk) ca also be used for estimatig death probabilities q ad p. (31) 3.4 Overall algorithm descriptio Startigfromafialcoditioatt = T (justimmediatelybeforethefialwithdrawal), abackward time steppig usig (24) gives solutioup to timet = t + N1. Applyig jump coditio(29) tothesolutioatt = t + N1 we obtaithesolutioatt = t N1 fromwhich further backward time steppig gives us solutio at t = t + N2, etc till t 0. The umerical algorithm takes the followig key steps 16

17 Algorithm 3.1 (GMWDB pricig) Step 1. Geerate a auxiliary fiite grid 0 = A 1 < A 2 < < A J = W(0) to track the guaratee accout A. Step 2. Discretize wealth accout W space as W 0,W 1,...,W M which is a grid for computig (24). Step 3. At t = t N = T apply the fial coditio at each ode poit (W m,a j ), j = 1,2,...,J, m = 1,2,...,M to get payoff Q T (W,A,I = 1). Step 4. Evaluate itegratio (24) (for each of the A j ) to obtai Q t + (W,A,I = N1 1). Step 5. Apply the jump coditio (29) to obtai Q t (W,A,I = 1) for all N1 possible jumps γ ad fid the jump that maximizes Q t (W,A,I = 1). N1 Step 6. Repeat Step 4 ad 5 for t = t N2,t N3,...,t 1. Step 7. Evaluate itegratio (24) for the backward time step from t 1 to t 0 for the sigle ode value A = A J = W(0) to obtai solutio Q 0 (W,A J,I 0 = 1) ad take the value Q t0 (A J,A J,I 0 = 1) as the auity price at t = t 0. Of course if the cotract is re-evaluated at some time after it started, oe will eed to take solutio at the ode correspodig to A ad W at that time. We use Gauss-Hermite quadrature itegratio (28) with cubic splie i Step 4. Oe ca also perform itegratio usig momet matchig if the desity is ot kow i closed form but its momets are available. For static case, Step 1 is ot eeded because oly a sigle solutio is required ad the jump coditio applies to the sigle solutio itself. 4 Numerical Results The accuracy ad efficiecy of GHQC method i pricig GMWB uder optimal withdraw is well demostrated i Luo & Shevcheko (2014a). From umerical poit of view, oce the problem is correctly formulated, pricig GMWDB (with combied GMWB ad GMDB features) uses the same key algorithm compoets as those for pricig GMWB, such as umerical quadrature itegratio for the expectatios ad cubic splie iterpolatio for applyig the jump coditios. 17

18 To the best of our kowledge, there is o umerical results available i the literature for variable auity cotracts with combied GMWB ad GMDB features uder the optimal withdrawal strategy, ad oly some very limited results are available i the literature for GMWB uder the optimal withdrawal strategy, amely from Dai et al. (2008) ad Che & Forsyth (2008). To validate our implemetatio for GMWDB, we have made the followig efforts: Implemeted a efficiet fiite differece (FD) algorithm with the same mathematical formulatio ad fuctioality, so GHQC results for GMWDB ca always be compared with FD. For the limitig case where the death probability is zero, i.e. q = 0, = 1,...,N, the GMQDB price should reduce to the stadard GMWB price exactly (still uder optimal withdrawal), ad we ca compare our GMWDB results with GMWB results foud i the literature. For the static withdraw case, GMWDB cotract ca be evaluated by Mote Carlo (MC) method, so we ca compare our GHQC results for GMWDB with those of MC usig a very large umber of simulatios. The close agreemet of results betwee the two algorithms offers a reassurig validatio for both methods. Below we preset ad discuss umerical results for GMWDB. I our umerical examples we assume policyholder age 60 for male ad female ad use the curret Australia Life Table, see Table 6, to fid correspodig death probabilities. 4.1 Results for dyamic GMWB Before showig umerical results for GMWDB, we first validate our umerical algorithms by testig the limitig case of zero death probability. I such a case we ca compare our GMWDB results with some fiite differece results foud i the literature for GMWB. I Che & Forsyth (2008), the fair fees for the discrete withdrawal model with g = 10% for the yearly (N w = 1) ad half-yearly (N w = 2) withdrawal frequecy at σ = 0.2 ad σ = 0.3 were preseted i a carefully performed covergece study, with the same values for other iput parameters (r = 5%, β = 10%). Table 1 compares GHQC (Algorithm 3.1) results with those of Che & Forsyth (2008). The values of Che & Forsyth (2008) quoted i Table 1 correspod to their fiest mesh grids ad time steps at M = 2049 grid poits for W coordiate, J = 1601 grid poits for A coordiate ad 1920 steps for time, while our GHQC values were obtaied usig M = 400, 18

19 J = 100 ad with q = 9 for the umber of quadrature poits (ad the umber of time steps is the same as the umber of withdrawal dates N). As show i Table 1, the maximum absolute differece i the fair fee rate betwee the two umerical studies is oly 0.3 basis poit, ad the average absolute differece of the four cases i the table is less tha 0.2 basis poit. As oe basis poit is 0.01%, the average differece is i the order of 20 cets for a oe thousad dollar accout. I relative terms, the maximum differece is less tha 0.15%, ad the average magitude of the relative differeces betwee the two studies is less tha 0.08%. Che & Forsyth (2008) did ot provide CPU umbers for their calculatios of fair fees. I our case each calculatio of the fair fee i Table 1, which ivolves a Newto iterative method of root fidig, took about 5 secods. I Luo & Shevcheko (2014b), a detailed compariso shows the GHQC algorithm is sigificatly faster tha the FD i pricig dyamic GMWB cotract, especially for lower withdrawal frequecies. All our calculatios i this study were performed usig stadard desktop PC (Itel(R) Core(TM) withdrawal frequecy volatility, σ Fair fee, α Fair fee, α Che & Forsyth (2008) GHQC yearly half-yearly yearly half-yearly Table 1: Fair fee α i bp (1bp = 0.01%) for GMWB auity with optimal policyholder withdrawals (i.e. equivalet to GMWDB with death probability set to zero). Compariso of results obtaied by GHQC method ad those from fiite differece by Che & Forsyth (2008). The iput parameters are g = 10%, β = 10% ad r = 5%. 4.2 Results for GMWDB The stadard GMWB auity without cosideratio of death evet assumes that the policyholder will be alive durig the cotract or there is beeficiary to cotiue withdrawals util maturity. We refer to this case as GMWDB without death beefit; it is equivalet to GMWDB with death probability set to zero. There are several more or less atural cosideratios of GMWDB cotact payoff i the case of death before or at cotract maturity. We cosider death beefit types summarised i (8), though may other modificatios are possible. A atural cotract coditio i the case of death evet is to pay the maximum of the remaiig guarateed accout ad the wealth 19

20 accout values (deoted as DB0). A alterative is to payback the iitial premium at death(deoted as DB1), which is similar to a term lifeisurace payig a fixed amout equal to the premium upo death (the amout ca be adjust for iflatio, etc). O the other had, the Guarateed Miimum Death Beefit (GMDB), i its basic form, offers to pay the maximum of the wealth accout value ad the premium (deoted as DB2). GMWDB cotract with DB2 has both features of GMWB ad GMDB but requirig oly a sigle premium. All three types of death beefits ca be added to either static GMWB or dyamic GMWB. All the followig results of GMWDB are based o death probabilities calculated from the Life Table for the male populatio i Australia, except those i Table 5 which have used the Life Table for the female populatio i Australia; also see Table 6. We assume that policyholder is 60 years old Static GMWDB Table 2 shows static case results of GMWDB with DB0, DB1 ad DB2 death beefits, ad stadard GMWB(i.e. GMWDB with zero death probability). Here we also preset MC ad FD results for DB0 case. The umber of simulatios for MC is 20 millio ad correspodig stadard error i calculatio of fair fees is withi 0.2 basis poit. As show i Table 2, the maximum differece of fair fee betwee GHQC ad MC methods is 0.2 basis poit, ad the maximum differece betwee GHQC ad FD methods is 0.1 basis poit i the case of DB0. We have also calculated DB1 ad DB2 cases usig FD ad MC methods ad the differece betwee the methods is virtually the same as i the case of DB0 so we do ot preset these results. The close agreemet betwee MC ad GHQC methods is a reassurig validatio for both algorithms. Comparig with results for static GMWB (i.e. GMWDB with death probability set tozero), addigdb0deathbeefitrequireslesstha10extrabasispoitsithefairfee. The fair fee icreases sigificatly i the case of DB2 death beefit. As expected, the fair fee for DB2 at all maturities is higher tha for DB1 ad the differece decreases as maturity icreases. At g = 4% (T = 25 years), the fair fee for DB1 is actually egative, meaig the cotract is ot appropriate. This seems to be odd at first, but it actually makes sese: the g = 4% withdrawal rate is lower tha the expected growth rate of r = 5%, ad returig oly the premium at ay time of death is a loss to the policyholder, ad a possible gai ca oly be obtaied if the policyholder survives beyod the maturity which is ot eough to offset the loss due to the probability of death. At g = r = 5%, the fair fee for DB1 becomes positive but it is still lower tha for DB0, ad it is eve lower tha a static GMWB. 20

21 Real product desig may impose expiry date for the death beefit guaratee (e.g. at the age 70 or 75) that will reduce the cost of the death beefit guaratee itself ad will help avoid the case of egative fee such as i Table 2 for log cotract maturity T = 25 years. If the cotract will switch from DB2 or DB1 to DB0 (effectively makig the death beefit expirig at the specified age), the it ca be hadled by the same algorithm described i this paper with adjustmet to the death beefit fuctio (8). If the death beefit expirig correspods to switchig to the stadard GMWB, the it ca be hadled by settig death probabilities q to zero after death beefit expiry till the cotract maturity. cotractual rate maturity GHQC GHQC FD MC GHQC GHQC g T = 1/g o death DB0 DB0 DB0 DB1 DB2 4% % % % % % % % Table 2: Fair fee α i bp (1bp = 0.01%) of GMWDB with DB0, DB1, ad DB2 death beefits for the static case with a quarterly withdrawal frequecy (N w = 4) as a fuctio of aual cotractual rate g. o death correspods to GMWB (i.e. GMWDB with death probability set to zero). Other parameters are r = 5% ad σ = 20% Dyamic GMWDB Table 3 shows results for dyamic GMWDB with DB0, DB1 ad DB2 death beefits ad dyamic GMWB calculated usig GHQC method. Here we also preset FD method results for DB0 case. The maximum differece of fees betwee GHQC ad FD methods is 0.4 basis poit occurrig at the shortest maturity, which has the maximum magitude i the fee. I relative terms this is less tha 0.2%. For other cases the differece betwee methods is similar, i.e. very small ad we show results calculated usig GHQC method oly. Comparig with results for GMWB, addig DB0 death beefit to a dyamic GMWB oly requires a maximum of 10 extra basis poits i the fair fee, similar to the static case. As easily see from the table, addig DB1 or DB2 death beefit (returig 21

22 at least the iitial premium) chages the situatio dramatically. At g = 15%, the shortest maturity, the fair fee of GMWDB with DB1 or DB2 death beefit more tha doubled from the value of GMWB. Moreover, the fair fee icreases rapidly as the maturity icreases, while i all the other cases so far show i this paper the fair fee is a decreasig fuctio of maturity. This icrease i the fair fee is so rapid that o solutio for the fair fee exists for g 7% (i.e. for T = 1/g 14.29) for either DB1 or DB2 meaig that eve chargig fee at 100% is still ot eough to cover the risks. The last colum i the table is the upper boud of the fair fee correspodig to the upper estimator Q (u) 0 (W(0),A(0),I 0 = 1) calculated usig equatio (22), i.e. estimator calculatig GMWDB for give death time (coditioal o kowig death time) ad the averagig over possible death times with correspodig death probabilities. Figure 1 plots the fair fee of GMWDB with DB2 as a fuctio of the cotractual withdrawal rate g, showig the rapid icrease of fair fee as the cotractual rate decreases or the maturity icreases. Betwee DB1 ad DB2, the differece i the fair fee is very small, ulike the static case where the differece i the fair fee betwee DB1 ad DB2 is sigificat. Also show i Figure 1 (dashed lie) is the the upper boud of the fair fee from Table 3. The upper boud of the fee is much higher tha the fee correspodig to the optimal withdraw strategy based o iformatio up to the withdrawal date, maifestig the value of kowig the future. cotractual rate maturity GHQC GHQC FD GHQC GHQC GHQC g T = 1/g o death DB0 DB0 DB1 DB2 DB2 4% N/A N/A N/A 5% N/A N/A N/A 6% N/A N/A N/A 7% N/A N/A N/A 8% % % % Table 3: Fair fee α i bp (1bp = 0.01%) of GMWDB with DB0, DB1 ad DB2 death beefits for the dyamic case with a quarterly withdrawal frequecy (N w = 4). The other parameters are r = 5%, σ = 20% ad β = 10%. o death correspods to GMWB (GMWDB with death probability set to zero). The last colum, DB2, results are fair fees correspodig to the auity upper estimator based o the optimal strategy i the case of perfect death time forecast give by equatio (22). 22

23 Now let us look at the reaso for the o-existece of solutio at loger maturity by cosiderig the followig simple pre-defied strategy (sufficietly good strategy but ot ecessarily optimal thus correspodig to the lower price estimator). Assume the policyholder of a dyamic GMWDB with DB2 withdraws all guaratee amout A(0) at the first withdrawal date (if survivig the first time period) ad waitig for the possible collectio of the death beefit. The wealth accout W(t ) evolves accordig to (2) with γ 1 = A(0), γ = 1, = 2,...,N. The expected preset value of payoff (expectatio with respect to death time τ ad wealth process W(t )) received by the policyholder usig the above strategy is E X t 0 [H 0 (X,γ)] = E W t 0 [B 0,N max(w(t ),0)Pr[τ > T] +(1p 1 )C 1 (A(0))B 0,1 + ] N i=1 p max(w(0),w(t ))B 0, (1p 1 )C 1 (A(0))B 0,1 +W(0) N i=1 p ib 0,i Q (L) 0, where p = Pr[t 1 < τ < t ] is the probability of death occurrig durig the -th paymet period coditioal o the policyholder is alive at the begiig of the cotract, C 1 (A(0)) = G 1 +(W(0)G 1 )(1β) is the full withdraw amout mius pealty if the policyholder survives the first paymet periodtoexecute thestrategy, adq (L) 0 deotes a lower boud of the expected payoff for this simple strategy. The above estimate of the payoff for the strategy is idepedet of the fee charged by the isurace compay - because the fee oly affects the accout value, ot the guaratee amout, or the miimum death beefit. Usig the same strategy, the above lower boud also applies to DB1. Figure 2 shows the lower boud Q (L) 0 as a fuctio of the cotractual rate g. Clearly, at g 7 ad β = 10% the strategy always yields a cashflow greater tha the iitial premium, irrespective of the fee charged, thus explaiig why o solutio of fair fee exists for g 7. Qualitatively, GMWDB with the miimum death beefit of returig the premium allows a ratioal policyholder of a log maturity cotract to get almost all the iitial premium back while havig a high probability of collectig the death beefit as well. Oe remedy for the above problem of o-existece of solutio is simply icreasig the pealty rate to discourage excessive withdrawals above the cotractual rate. Also show i Figure 2 is aother payoff curve i the case of pealty rate icreased to β = 30% i this istace the estimated payoff is well below the iitial premium at all withdrawal rates g 4% ad solutio for the fair fee exists. Also, as already metioed i previous sectios, real product desig may impose expiry date for the death beefit guaratee (e.g. at the age 70 or 75) that will reduce 23

24 GHQC FD Upper boud g Figure 1: Fair fee α i bp (1bp = 0.01%) of GMWDB with DB2 death beefit for the dyamic case with a quarterly withdrawal frequecy (N w = 4). The other parameters are r = 5%, σ = 20%, β = 10%. The Upper boud is the fair fee associated with the auity price upper estimator calculated usig equatio (22) based o optimal strategy if death time is kow. the cost of the death beefit guaratee itself ad will help to avoid the case of oexistece of solutio Dyamic GMWDB with fixed istallmet fees for death beefit The above problem of o-existece of fair fee solutio for dyamic GMWDB i some cases, ca also be dealt with by a more reasoable fee structure. So far the extra cost of addig DB1 or DB2 death beefit is absorbed i the cotiuous fee α which is liked to the wealth accout value W(t), ot to the guarateed premium i the death beefit. As the fud value is reduced by each withdrawal, the charge base is also reduced. A better fee structure is to keep the cotiuous fee the same as the dyamic GMWB but charge the extra fee due to addig DB1 or DB2 death beefit upfrot or i fixed istalmets. I practice the later is usually preferred. Deote the fair fee of a dyamic GMWB (i.e. GMWDB with death probabilities set to zero) as α (e.g. colum 3 i Table 3), ad the fair price of a GMWDB (with either DB1 or DB2) uder the same fee as Q(α ) Q 0 (A(0),W(0),I 0 = 1), the obviously Q(α ) > W(0) ad the fair upfrot fee is the differece = Q(α )W(0). To work 24