A unified pricing of variable annuity guarantees under the optimal stochastic control framework
AA unified pricing of variable annuity guarantees under theoptimal stochastic control framework
Pavel V. Shevchenko and Xiaolin Luo CSIRO Australia, e-mail: [email protected] CSIRO Australia, e-mail: [email protected]
Abstract
In this paper, we review pricing of variable annuity living and death guarantees offered to retailinvestors in many countries. Investors purchase these products to take advantage of market growthand protect savings. We present pricing of these products via an optimal stochastic control frame-work, and review the existing numerical methods. For numerical valuation of these contracts, wedevelop a direct integration method based on Gauss-Hermite quadrature with a one-dimensionalcubic spline for calculation of the expected contract value, and a bi-cubic spline interpolationfor applying the jump conditions across the contract cashflow event times. This method is veryefficient when compared to the partial differential equation methods if the transition density (orits moments) of the risky asset underlying the contract is known in closed form between the eventtimes. We also present accurate numerical results for pricing of a Guaranteed Minimum Accu-mulation Benefit (GMAB) guarantee available on the market that can serve as a benchmark forpractitioners and researchers developing pricing of variable annuity guarantees.
Keywords: variable annuity, guaranteed living and death benefits, guaranteed minimumaccumulation benefit, optimal stochastic control, direct integration method. a r X i v : . [ q -f i n . P R ] M a y Introduction
Many wealth management and insurance companies worldwide are offering investment prod-ucts known as variable annuities (VA) with some guarantees of living and death benefits toassist investors with managing pre-retirement and post-retirement plans. These products takeadvantage of market growth while provide protection of the savings against market downturns.Insurers started to offer these products from the 1990s in United States. Later, these productsbecame popular in Europe, UK and Japan and more recently in Australia. The VA contractcashflows received by the policyholder are linked to the investment portfolio choice and per-formance (e.g. the choice of mutual fund and its strategy) while traditional annuities providea pre-defined income stream in exchange for the lump sum payment. According to LIMRA(Life Insurance and Market Research Association) reports, the VA market is huge: VA sales inUnited States were $158 billion in 2011, $147 billion in 2012 and $145 billion in 2013.The types of VA guarantees (referred in the literature as
VA riders ) offered for investmentportfolios are classified as guaranteed minimum withdrawal benefit (GMWB), guaranteed mini-mum accumulation benefit (GMAB), guaranteed minimum income benefit (GMIB) and guaran-teed minimum death benefit (GMDB). These guarantees, generically denoted as GMxB, providedifferent types of protection against market downturns and policyholder death. GMWB allowswithdrawing funds from the VA account up to some pre-defined limit regardless of investmentperformance during the contract; GMAB and GMIB both provide a guaranteed investmentaccount balance at the contract maturity that can be taken as a lump sum or standard an-nuity respectively.
Guaranteed lifelong withdrawal benefit (GLWB), a specific type of GMWB,allows withdrawing funds at the contractual rate as long as the policyholder is alive. GMDBprovides a specified payment if the policyholder dies. Precise specifications of the productswithin each type can vary across companies and some products may include combinations ofthese guarantees.A good overview of VA products and the development of their market can be found in Baueret al. (2008), Ledlie et al. (2008) and Kalberer and Ravindran (2009). There have been anumber of papers in academic literature considering pricing of these products. Most of theseare focused on pricing VA riders under the pre-determined ( static ) policyholder behaviour inwithdrawal and surrender. Some studies include pricing under the active ( dynamic ) strategywhen the policyholder ‘ optimally ’ decides the amount of withdrawal at each withdrawal datedepending on the information available at that date. Standard Monte Carlo (MC) method caneasily be used to estimate price in the case of pre-defined withdrawal strategy but handlingthe dynamic strategy requires backward in time solution that can be done only via the partialdifferential equation (PDE), direct integration or regression type MC methods.In brief, pricing under the static and dynamic withdrawal strategies via PDE based methodshas been developed in Milevsky and Salisbury (2006), Dai et al. (2008) and Chen and Forsyth(2008). Bauer et al. (2008) develops a unified approach with numerical estimation via MCand direct integration methods. The direct integration method was developed further in Luoand Shevchenko (2015a,b) using Gauss-Hermite quadrature and cubic interpolations. Bacinelloet al. (2011) consider many VA riders under stochastic interest rate and stochastic volatilityif the policyholder withdraws at the pre-defined contractual rate or completely surrenders the2ontract. Their pricing is accomplished either by the ordinary MC or Least-Squares MC toaccount for the optimal surrender. Typically, pricing of VA riders is considered under theassumption of geometric Brownian motion for the risky asset underlying the contract, thougha few papers looked at extensions such as stochastic interest rate and/or stochastic volatility,see e.g. Forsyth and Vetzal (2014), Luo and Shevchenko (2016), Bacinello et al. (2011), Huangand Kwok (2015).Azimzadeh and Forsyth (2014) prove the existence of an optimal bang-bang control for GLWBcontract when the contract holder can maximize contract writer’s losses by only ever performingnon-withdrawal, withdrawal at the contract rate or full surrender. However, they also demon-strate that the related GMWB contract does not satisfy the bang-bang principle other than incertain degenerate cases. Huang and Kwok (2015) developed a regression-based MC methodfor pricing GLWB under the bang-bang strategy in the case of stochastic volatility. GMWBpricing under the bang-bang strategy was studied in Luo and Shevchenko (2015c). The diffi-culty with applying the well known Least-Squares MC introduced in Longstaff and Schwartz(2001) for pricing VA riders under the optimal strategy is due to the fact that the paths of theunderlying VA wealth account are affected by the withdrawals. In principle, one can apply con-trol randomization methods extending Least-Squares MC to handle optimal stochastic controlproblems with controlled Markov processes recently developed in Kharroubi et al. (2014), butthe accuracy and robustness of this method for pricing VA riders have not been studied yet.One common observation in the above mentioned literature is that pricing under the optimalstrategy often leads to prices significantly higher than observed on the market. These studiesrely on the option pricing risk-neutral methodology in quantitative finance to find a fair fee .Here, the fundamental idea is to find the cost of a dynamic self-financing replicating portfoliowhich is designed to provide an amount at least equal to the payoff of the contract. The costof establishing this hedging strategy is the no-arbitrage price of the contract. This is under theassumption that the contract holder adopts an optimal strategy (exercise strategy maximisingthe monetary value of the contract). If the purchaser follows any other exercise strategy, thecontract writer will generate a guaranteed profit if continuous hedging is performed. Of coursethe strategy optimal in this sense is not related to the policyholder circumstances. In pricing VAwith guarantees, it is reasonable to consider alternative assumptions regarding the investor’swithdrawal strategy. This is because an investor may follow what appears to be a sub-optimal strategy that does not maximise the monetary value of the option. This could be due toreasons such as liquidity needs, tax and other personal circumstances. Moreover, mortalityrisk is diversified by the contract issuer through selling many contracts to many people whilethe policyholder cannot do it. Also, there might be no liquid secondary market for VAs onwhich the policy could be sold (or repurchased) at its fair value. The policyholder may actoptimally with respect to his preferences and circumstances but it may be different from theoptimal strategy that maximises the monetary value of the contract. In this case we calculatea fair fee to be deducted in order to finance a dynamic replicating portfolio for the guarantees(options) embedded in the contract under the assumption of a particular exercise strategy. Thereplicating portfolio will provide sufficient funds to meet any future payouts that arise fromwriting the contract.However, the fair fee obtained under the assumption that investors behave optimally to max-3mise the value of the guarantee does offer an important benchmark because it is a worst casescenario for the contract writer. Also, as noted in Hilpert et al. (2014), secondary markets forequity linked insurance products (where the policyholder can sell their contracts) are growing.Thus, third parties can potentially generate guaranteed profit through hedging strategies fromfinancial products such as VA riders which are not priced under the assumption of the opti-mal withdrawal strategy. Knoller et al. (2015) mentions several companies recently sufferinglarge losses related to increased surrender rates, indicating that either charged fees were notsufficiently large or that hedging program did not perform as expected.One way to analyze the withdrawal behavior of VA holder and evaluate the need of theseproducts is to solve the life-cycle utility model accounting for consumption, housing, bequestand other real life circumstances. Developing a full life-cycle model with all preferences andrequired parameters is challenging but there are already several contributions reporting someinteresting findings in this direction: Moenig (2012); Horneff et al. (2015); Gao and Ulm (2012);Steinorth and Mitchell (2015). This topic will not be considered in this paper. It is alsoimportant to note a recent paper by Moenig and Bauer (2015) considering the pricing underthe optimal strategy in the presence of taxes via subjective risk-neutral valuation methodology.They demonstrated that including taxes significantly affects the value of the VA withdrawalguarantees producing results in line with empirical market prices.In this paper we review pricing of living and death benefit guarantees offered with VAs, andpresent a unified optimal stochastic control framework for pricing these contracts. The mainideas have been developed and appeared in some forms in a number of other papers. However,we believe that our presentation is easier to understand and implement. We also present directintegration method based on computing the expected contract values in a backward time-stepping through a high order Gauss-Hermite integration quadrature applied on a cubic splineinterpolation. This method can be applied when transition density of the underlying assetbetween the contract cashflow event dates or its moments are known in closed form. We haveused this for pricing specific financial derivatives and some simple versions of VA guarantees inLuo and Shevchenko (2014, 2015a). Here, we adapt and extend the method to handle pricingVA riders in general. As a numerical example, we calculate accurate prices of GMAB withpossible annual ratchets (reset of the guaranteed capital to the investment portfolio value ifthe latter is larger on anniversary dates) and allowing optimal withdrawals. The contract thatwe consider is very similar in specifications to the real product marketed in Australia, see forexample MLC (2014) and AMP (2014). Numerical difficulties encountered in pricing this VArider are common across other VA guarantees and at the same time comprehensive numericalpricing results for this product are not available in the literature. These results (reported fora range of parameters) can serve as a benchmark for practitioners and researchers developingnumerical pricing of VA riders.In the next section, a general specification of VA riders is given. In Section 3 we discussstochastic models used for pricing these products. Section 4 provides precise specification forsome popular VA riders. In Section 5 we outline the calculation of the fair price and fair fee asa solution of an optimal stochastic control problem. Section 6 presents the numerical methodsand algorithms for pricing VA riders. In Section 7 we present numerical results for the fair feesof GMAB rider. Concluding remarks are given in Section 8.4
VA rider contract specification
Consider a VA contract with some guarantees for living and death benefits purchased by an x -year old individual at time t = 0 with the up-front premium invested in a risky asset (e.g.a mutual fund), denoted as S ( t ) at time t ≥
0. The VA rider specification includes dates whenevents such as withdrawal, ratchet ( step-up ), bonus ( roll-up ), death benefit payment, etc. mayoccur. Precise definitions of these events depend on the contract and corresponding exampleswill be provided in Section 4. We assume that the withdrawal can only take place on the setof the ordered event times T = { t , . . . , t N } , where T = t N is the contract maturity. Also, theset of policy anniversaries when the ratchet may occur is denoted as T r and is assumed to bea subset of T . For simplicity on notation we assume that all other events may only occur onthe withdrawal dates. The value of VA contract with guarantees at time t is determined by thethree main state variables. • Wealth account W ( t ), value of the investment account which is linked to the risky asset S ( t ) and modelled as stochastic process. • Guarantee account A ( t ), also referred in the literature as benefit base . It is not changingbetween event times but can be stochastic via stochasticity in W ( t ) at the event timesdepending on the contract features. • Discrete state variable I n ∈ { , , − } corresponding to the states of policyholder is beingalive at t n , died during ( t n − , t n ], or died before or at t n − correspondingly. Denote thedeath probability during ( t n − , t n ] as q n = Pr[ I n = 0 | I n − = 1], i.e. Pr[ I n = 1 | I n − = 1] =1 − q n . Note that q n depends on the age of the contract holder at t n and thus dependson the age x at t = 0.Other state variables are needed if the interest rate and/or volatility are stochastic but theseare not affected at the contract event times and typically do not enter formulas for the contractcashflows; these will not be considered explicitly. Extra state variable is also required to track atax free base to account for taxes; this will be considered in Section 5.4. In principle, differentguarantees included in VA may have different benefit base state variables. For notationalsimplicity and also from practical perspective, we assume that all guarantees in VA are linkedto the same benefit base account.Initially, W (0) and A (0) are set equal to the upfront premium. The contract holder is allowedto take withdrawal γ n at time t n , n = 1 , . . . , N −
1. Denote the values of the benefit base justbefore and just after t n as A ( t − n ) and A ( t + n ) respectively, and similarly for the wealth account W ( t − n ) and W ( t + n ).The contract product specification determines: • The contractual (guaranteed) withdrawal amount G n for the period ( t n − , t n ] that maydepend on the benefit base A ( t − n ) and/or W ( t − n ). • Jump conditions at the event times relating state variables before and after the event,5ubject to withdrawals γ n belonging to an admissible space A n : W ( t + n ) : = h Wn (cid:0) W ( t − n ) , A ( t − n ) , γ n (cid:1) , (1) A ( t + n ) : = h An (cid:0) W ( t − n ) , A ( t − n ) , γ n (cid:1) , (2) γ n ∈ A n (cid:0) W ( t − n ) , A ( t − n ) (cid:1) , (3)where h Wn ( · ) and h An ( · ) are some functions that may also depend on the fee, penalty andannual step-up parameters. For example, if only a ratchet is possible at t n ∈ T r and noother contract events, then A ( t + n ) = max (cid:0) A ( t − n ) , W ( t − n ) × t n ∈T r (cid:1) . In practice, several events such as withdrawal, ratchet, bonus, etc. may occur at the sametime t n , and the contract specification determines the order of this events. • The payout P T ( W, A ) at the contract maturity if policyholder is alive at t = T . • The payout D n ( W, A ) to the beneficiary at t n in the case of the policyholder death during( t n − , t n ], n = 1 , . . . , N . • The cashflow received by the policyholder (cid:101) f n ( W ( t − n ) , A ( t − n ) , γ n ) at the event times t n , n = 1 , . . . , N −
1, that might be different from γ n due to penalties.The specification details typically vary across different companies and are difficult to extractfrom the very long product specification documents. Moreover, results for specific GMxB riderspresented in academic literature often refer to different specifications.Once the above conditions, i.e. functions h Wn ( · ), h An ( · ), P T ( · ), P D ( · ), (cid:101) f n ( γ n ) and admissiblerange for withdrawal A n are specified by the contract design, and a specific stochastic evolutionof the state variables is assumed within ( t n − , t n ), n = 1 , . . . , N , then pricing of the contractcan be accomplished by numerical methods. In particular, if withdrawals are optimal thenpricing can be accomplished by PDE, direct integration or regression based MC methods. Ifwithdrawals are deterministic, then standard MC along with PDE and direct integration meth-ods can be used. The use of a particular numerical technique is determined by the complexityof the underlying stochastic model. Commonly in the literature, stochastic models for the financial risky asset S ( t ) underlying theVA rider assume that there is no arbitrage in the financial market which means that there isa risk-neutral measure Q under which payment streams can be valued as expected discountedvalues. Moreover, this means that the cost of portfolio replicating the contract is given by itsexpected discounted value under Q . Hence, the fair price of the contract can be expressed as anexpectation of the contract discounted cashflows with respect to Q . Some models considered inthe literature assume that the financial market is complete which means that the risk-neutralmeasure Q is unique. It is also assumed that market has a risk-free asset that accumulates6ontinuously at risk free interest rate. These are typical assumptions in the academic researchliterature on pricing financial derivatives, for a good textbook in this area we refer the readerto e.g. Bj¨ork (2004).Regarding the mortality risk, it is assumed that it is fully diversified via selling the contract tomany policyholders. In the case of systemic (undiversified) mortality risk, the risk-neutral fairvalue can be adjusted using an actuarial premium principle, see e.g. Gaillardetz and Lakhmiri(2011). Another common assumption is that mortality and financial risks are independent.A benchmark model commonly considered in the literature on pricing VA riders is thewell-known Black-Scholes dynamics for the reference portfolio of assets S ( t ) that under therisk-neutral measure Q is known to be dS ( t ) = r ( t ) S ( t ) dt + σ ( t ) S ( t ) dB ( t ) . (4)Here, B ( t ) is the standard Wiener process, r ( t ) is the risk free interest rate and σ ( t ) is thevolatility. Under this model the financial market is complete. Without loss of generality, themodel parameters can be assumed to be piecewise constant functions of time for time discretiza-tion 0 = t < t < · · · < t N = T . Denote corresponding asset values as S ( t ) , . . . , S ( t N ) andrisk free interest rate and volatility as r , . . . , r N and σ , . . . , σ N respectively. That is, σ is thevolatility for ( t , t ]; σ is the volatility for ( t , t ], etc. and similarly for the interest rate.Pricing VA riders in the case of extensions of the above model to the stochastic interest rateand/or stochastic volatility have been developed in e.g. Forsyth and Vetzal (2014), Luo andShevchenko (2016), Bacinello et al. (2011), Huang and Kwok (2015).Regarding mortality modelling, the standard way is to use official Life Tables to estimate thedeath probability q n = Pr[ I n = 0 | I n − = 1] during ( t n − , t n ]. Life Tables provide annual deathprobabilities for each age and gender in a given country; probabilities for time periods withina year can be found by e.g. linear interpolation, see Luo and Shevchenko (2015b). Instead of aLife Table, stochastic mortality models such as the benchmark Lee-Carter model introduced inLee and Carter (1992) can also be used to forecasts the required death probabilities (accountingfor systematic mortality risk).For a given process of risky asset S ( t ), t ≥
0, the value of the wealth account W ( t ) evolvesas W ( t − n ) = W ( t + n − ) S ( t n − ) S ( t n ) e − αdt n ,W ( t + n ) = max( W ( t − n ) − γ n , , n = 1 , , . . . , N, (5)where dt n = t n − t n − and α is the annual fee continuously charged by contract issuer for theprovided guarantee. In the case of S ( t ) following the geometric Brownian motion process (4),we have S ( t n ) = S ( t n − ) e ( r n − σ n ) dt n + σ n √ dt n z n , where z , . . . , z N are independent and identically distributed standard Normal random variables.In practice, the guarantee fee is charged discretely and proportional to the wealth accountthat can easily be incorporated into the wealth process (5). Denoting the discretely charged7ee with the annual basis as (cid:101) α , the wealth process becomes W ( t − n ) = W ( t + n − ) S ( t n − ) S ( t n ) ,W ( t + n ) = max (cid:0) W ( t − n )(1 − (cid:101) αdt n ) − γ n , (cid:1) , n = 1 , , . . . , N. (6)Typically, the difference between continuously and discretely charged fees is not material asobserved in our numerical results given in Section 7.Another popular fee structure corresponds to fees charged as a proportion of the benefitbase, so that W ( t − n ) = W ( t + n − ) S ( t n − ) S ( t n ) ,W ( t + n ) = max (cid:0) W ( t − n ) − A ( t − n ) (cid:101) αdt n − γ n , (cid:1) , n = 1 , , . . . , N. (7)Here, it is assumed that discrete fees are deducted before withdrawal but it can be vice versadepending on the contract specifications.For simplicity, we do not consider management fees α m charged by a mutual fund for man-aging the investment portfolio. If management fees α m is given exogenously, then it will havean impact on the fair fee α that should by charged by the VA guarantee issuer. This can beaccomplished as described in e.g. Forsyth and Vetzal (2014) and can be easily incorporated inthe framework outlined in our paper. Obviously, α will be larger for given α m > α m = 0. The management fees reduce the performance of the investment accountthus increasing the value of the guarantee as reported in e.g. Chen et al. (2008) for GMWBor Forsyth and Vetzal (2014) for GLWB. They commented that insurers wishing to providethe cheapest guarantee could provide the guarantee on the corresponding inexpensive exchangetraded index fund rather than on a managed mutual fund account with extra fees. There are many different specifications for GMWB, GLWB, GMAB, GMIB and GMDB in theindustry and academic literature. In this section we provide a mathematical formulation forsome standard VA rider setups. We assume that the guarantee fee α is charged continuously.If the fee is charged discretely (and before withdrawal and other contract events), then oneshould make the following adjustment to the formulas in this section: W ( t − n ) → W ( t − n )(1 − (cid:101) αdt n ) , if the fee is proportional to the wealth account and W ( t − n ) → max( W ( t − n ) − A ( t − n ) (cid:101) αdt n , , if the fee is proportional to the benefit base. 8 .1 GMWB A VA contract with GMWB promises to return at least the entire initial investment throughcash withdrawals during the policy life plus the remaining account balance at maturity, regard-less of the portfolio performance. Often in academic literature, the studied GMWB type has avery simple structure, where the penalty is applied to the cashflow paid to the contract holder,while the benefit base is reduced by the full withdrawal amount. Specifically, A ( t + n ) := h An ( W ( t − n ) , A ( t − n ) , γ n ) = A ( t − n ) − γ n , (8)with γ n ∈ A n , A n = [0 , A ( t − n )]; and cashflow paid to the contract holder is (cid:101) f n ( W ( t − n ) , A ( t − n ) , γ n ) = (cid:26) γ n , if 0 ≤ γ n ≤ G n ,G n + (1 − β )( γ n − G n ) , if γ n > G n , (9)where β ∈ [0 ,
1] is the penalty parameter for excess withdrawal. The contractual amount isdefined as G n = W (0)( t n − t n − ) /T and the maturity condition is P T ( W ( t − N ) , A ( t − N )) = max( W ( t − N ) , (cid:101) f n ( A ( t − N ))) . Note that the above specification does not allow early surrender which can be included viaextending the withdrawal space A n . Also, there is no death benefit; it is assumed that benefi-ciary will maintain the contract if the case of policyholder death. This contract has only basicfeatures facilitating comparison of results from different academic studies, such as Chen andForsyth (2008), Dai et al. (2008), Luo and Shevchenko (2015c), Luo and Shevchenko (2015a).Specifications common in the industry include cases where the contractual amount G n isspecified to be different from G n = W (0)( t n − t n − ) /T and a penalty is applied to both thewithdrawn amount and the benefit base. For example, specifications used in Moenig and Bauer(2015) to compare with the industry products include: (cid:101) f n ( W ( t − n ) , A ( t − n ) , γ n ) = γ n − δ excess − δ penalty ,δ excess = β e n max ( γ n − min( A ( t − n ) , G n ) , ,δ penalty = β g n ( γ n − δ excess ) × x + t n < . , (10)where x is the age of the policyholder in years at t = 0, β e n and β g n are excess withdrawal andearly withdrawal penalty parameters that can change with time, and γ n ∈ A n , A n = (cid:2) , max (cid:0) W ( t − n ) , min( A ( t − n ) , G n ) (cid:1)(cid:3) . Moenig and Bauer (2015) also considered several specifications for the benefit base jump con-ditions. • Specification 1: A ( t + n ) = (cid:40) max( A ( t − n ) − γ n , , if γ n ≤ G n , max (cid:16) min (cid:16) A ( t − n ) − γ n , A ( t − n ) W ( t + n ) W ( t − n ) (cid:17) , (cid:17) , if γ n > G n . (11)9 Specification 2: A ( t + n ) = (cid:26) max( A ( t − n ) − γ n , , if γ n ≤ G n , max (min ( A ( t − n ) − γ n , W ( t + n )) , , if γ n > G n . (12) • Specification 3: A ( t + n ) = (cid:40) max( A ( t − n ) − γ n , , if γ n ≤ G n , max ( A ( t − n ) − G n , W ( t + n )max( W ( t − n ) − G n , , if γ n > G n . (13)In addition, a ratchet (reset of the benefit base to the wealth account if the latter is higher)can apply at anniversary dates. If it occurs before the withdrawal, then in the above formulasone should make the following adjustment A ( t − n ) → max( A ( t − n ) , W ( t − n )) , if t n ∈ T r . If the reset is taking place after the withdrawal, then one should have A ( t + n ) → max( A ( t + n ) , W ( t + n )) , if t n ∈ T r . GLWB is similar to GMWB but provides guaranteed withdrawal for life; upon death the re-maining wealth account value is paid to the beneficiary. The contractual withdrawal amount G n is typically based on a fixed proportion g of the benefit base A ( t ), i.e. G n = g × A ( t − n )( t n − t n − ).The benefit base can increase via ratchet ( step-up ) or bonus ( roll-up ) features. Bonus featureprovides an increase of the benefit base if no withdrawal is made on a withdrawal date. Com-plete surrender refers to the withdrawal of the whole policy account. The withdrawal can exceedthe contractual amount and in this case the net amount received by the policyholder is subjectto a penalty. Under the typical specification considered e.g. in Huang and Kwok (2015), thecashflow received by the policyholder is (cid:101) f n ( W ( t − n ) , A ( t − n ) , γ n ) = (cid:26) γ n , if 0 ≤ γ n ≤ G n ,G n + (1 − β )( γ n − G n ) , if γ n > G n , (14) γ n ∈ A n , A n = [0 , max( W ( t − n ) , G n )] , where β is the penalty parameter for excess withdrawal. The benefit base jump condition,including ratchets and bonus features, is given by A ( t + n ) = max (cid:0) A ( t − n )(1 + b n ) , W ( t − n )1 t n ∈T r (cid:1) × γ n =0 + max (cid:0) A ( t − n ) , max( W ( t − n ) − γ n , × t n ∈T r (cid:1) × <γ n ≤ G n + max (cid:18) A ( t − n ) W ( t − n ) − γ n W ( t − n ) − G n , ( W ( t − n ) − γ n ) × t n ∈T r (cid:19) × G n <γ n ≤ W ( t − n ) , (15)where b n is the bonus rate parameter that may change in time. Finally, if the policyholder diesduring ( t n − , t n ], the beneficiary receives a death benefit payment D n ( W ( t − n ) , A ( t − n )) = W ( t − n )and t N corresponds to the maximum age beyond which survival is deemed impossible.10 .3 GMAB GMAB rider provides a certainty of capital till some maturity (e.g. 10 or 20 years) and thepotential for a capital growth. Typical GMAB products sold on the market do not imposepenalty on the policyholder withdrawal amount but can penalise the benefit base (protectedcapital balance) under some conditions. It is also common to have a ratchet feature, wherethe protected capital balance increases to the wealth account if the latter is higher on ananniversary date. Withdrawals from the account are allowed subject to a penalty. For example,specifications of the product marketed by MLC (2014) and AMP (2014) in Australia are veryclose to the following formulation: (cid:101) f n ( W ( t − n ) , A ( t − n ) , γ n ) = γ n , (16) A ( t + n ) := h An ( W ( t − n ) , A ( t − n ) , γ n ) = (cid:26) max ( A ( t − n ) , W ( t − n )) − C n ( γ n ) , if t n ∈ T r , max ( A ( t − n ) − C n ( γ n ) , , otherwise , (17)where C n ( γ n ) is a penalty function that can be larger than γ n as defined below, and γ n ∈ A n =[0 , W ( t − n )].The product is offered for the super and pension account types. The super account isdesigned for an investor being in an accumulation phase, while the pension account is fora retired investor in an annuitization phase. The difference between the accounts in termsof technical details is only in the penalty applied to the protected capital after withdrawals;the super account discourages withdrawals more than the pension account. In both casesthe penalty is in the form of a reduction of the protected capital (benefit base) larger thanthe withdrawn amount. The penalty only applies if the wealth account balance is below theprotected capital amount. A super account penalizes any amount of withdrawals, while thepension account only penalize excessive withdrawals.Specifically, for a super account , the function C n ( γ n ) is given by C n ( γ n ) = (cid:26) γ n , if W ( t − n ) ≥ A ( t − n ) ,A ( t − n ) γ n /W ( t − n ) , if W ( t − n ) < A ( t − n ) , (18)and for a pension account , the penalty is C n ( γ n ) = (cid:26) γ n , if W ( t − n ) ≥ A ( t − n ) or γ n ≤ G n ,A ( t − n ) γ n /W ( t − n ) , if W ( t − n ) < A ( t − n ) and γ n > G n . (19)That is, the penalty for the pension account applies only if the wealth account balance isbelow the protected capital amount and the withdrawal is above a pre-determined amount G n .Finally, the terminal condition is given by P T ( W ( t − N ) , A ( t − N )) = max( W ( t − N ) , A ( t − N )) . A total withdrawal of the wealth account balance effectively terminates the contract, as thepenalty mechanism ensures the protected capital is always exhausted to zero by a completewithdrawal. 11 .4 GMIB
At maturity, the holder of GMIB rider can select to take a lump sum of the wealth account W ( T ) or annuitise this amount at annuitization rate ¨ a T current at maturity or annuitize thebenefit base A ( T − ) at pre-specified annuitization rate ¨ a g . Annuitization rate is defined as theprice of an annuity paying one dollar each year. If the account value is below the benefit base,then the customer cannot take A ( T − ) as a lump sum but only as an annuity at pre-specifiedrate. Thus, the payoff of VA with GMIB at time T is P T ( W ( t − N ) , A ( t − N )) = max (cid:18) W ( t − N ) , A ( t − N ) ¨ a T ¨ a g (cid:19) . The benefit base may include roll-ups and ratchets. Again, this rider can be offered jointly withother riders. For example, it can be part of GMWB or GMAB contract maturity conditions.For discussion and pricing of GMIB in academic literature, see Marshall et al. (2010) and Baueret al. (2008).
GMDB rider provides a death benefit if the policy holder death occurs before or at the contractmaturity. Assuming that if the policyholder dies during ( t n − , t n ], then the beneficiary will bepaid an amount D n ( · ) at t n , where some of the common death benefit types are: D n ( W ( t − n ) , A ( t − n )) = max( A ( t − n ) , W ( t − n )) , death benefit type 0 ,W (0) , death benefit type 1 , max( W (0) , W ( t − n )) , death benefit type 2 ,W ( t − n ) , death benefit type 3 . (20)Some providers adjust the initial premium W (0) for inflation in the death benefit. For somepolicies, the death benefit type may change at some age, e.g. death benefit type 1 or type 2may change to type 0, effectively making the death benefit expiring at some age (e.g. at theage of 75 years). The death benefit can be provided on top of some other guarantees and thecontract may provide a spousal continuation option that allows a surviving spouse to continuethe contract. The contract may have accumulation phase where the death benefit may increase,and continuation phase where the death benefit remains constant.Pricing GMDB has been considered in e.g. Milevsky and Posner (2001), B´elanger et al.(2009), Luo and Shevchenko (2015b). Denote the state vector at time t n before the withdrawal as X n = ( W ( t − n ) , A ( t − n ) , I n ) and X = ( X , . . . , X N ). Given the withdrawal strategy γ = ( γ , . . . , γ N − ), the present value of theoverall payoff of the VA contract with a guarantee is a function of the state vector H ( X , γ ) = B ,N H N ( X N ) + N − (cid:88) n =1 B ,n f n ( X n , γ n ) . (21)12ere, H N ( X N ) = P T (cid:0) W ( T − ) , A ( T − ) (cid:1) × I n =1 + D N (cid:0) W ( T − ) , A ( T − ) (cid:1) × I n =0 (22)is the cashflow at the contract maturity and f n ( X n , γ n ) = (cid:101) f n ( W ( t − n ) , A ( t − n ) , γ n ) × I n =1 + D n (cid:0) W ( t − n ) , A ( t − n ) (cid:1) × I n =0 (23)is the cashflow at time t n . Also, B i,j is the discounting factor from t j to t i B i,j = exp (cid:18) − (cid:90) t j t i r ( t ) dt (cid:19) , t j > t i . (24) Let Q t ( W, A ) be the price of the VA contract with a guarantee at time t , when W ( t ) = W , A ( t ) = A and policyholder is alive. For simplicity of notation, if the policyholder is alive, wedrop mortality state variable I n = 1 in the function arguments. Assume that financial risk canbe eliminated via continuous hedging. Also assume that mortality risk is fully diversified viaselling the contract to many people of the same age, i.e. the average of the contract payoffs H ( X , γ ) over L policyholders converges to E I t [ H ( X , γ )] as L → ∞ , where I is the realprobability measure corresponding to the mortality process I , I , . . . . Then the contract priceunder the given withdrawal strategy γ can be calculated as Q ( W (0) , A (0)) = E Q , I t [ H ( X , γ )] . (25)Here, E Q , I t [ · ] denotes an expectation with respect to the state vector X , conditional on in-formation available at time t , i.e. with respect to the financial risky asset process under therisk-neutral probability measure Q and with respect to the mortality process under the realprobability measure I . Then the fair fee value of α to be charged for VA guarantee corre-sponds to Q ( W (0) , A (0)) = W (0). That is, once a pricing of Q ( W (0) , A (0)) for a given α isdeveloped, then a numerical root search algorithm is required to find the fair fee.The withdrawal strategy γ can depend on time and state variables and is assumed to begiven when price of the contract is calculated in (25). The withdrawal strategies are classifiedas static , optimal , and suboptimal . • Static strategy.
Under this strategy, the policyholder decisions are deterministicallydetermined at the beginning of the contract and do not depend on the evolution of thewealth and benefit base accounts. For example, policyholder withdraws at the contractualrate only. • Optimal strategy.
Under the optimal withdrawal strategy, the decision on the with-drawal amount γ n depends on the information available at time t n , i.e. depends on thestate variable X n . The optimal strategy is calculated as γ ∗ ( X ) = argsup γ ∈A E Q , I t [ H ( X , γ )] , (26)where the supremum is taken over all admissible strategies γ . Any other strategy γ ( X )different from γ ∗ ( X ) is called suboptimal .13iven that the state variable X = ( X , . . . , X N ) is a Markov process and the contractpayoff is represented by the general formula (21), calculation of the contract value (25) underthe optimal withdrawal strategy (26) is a standard optimal stochastic control problem for a controlled Markov process . Note that, the control variable γ n affects the transition law of theunderlying wealth W ( t ) process from t − n to t − n +1 and thus the process is controlled. For a goodtextbook treatment of stochastic control problems in finance, see B¨auerle and Rieder (2011).This type of problems can be solved recursively to find the contract value Q t n ( x ) at t n when X n = x for n = N − , . . . , Bellman equation Q t n ( x ) = sup γ n ∈A n (cid:18) f n ( x, γ n ) + B n,n +1 (cid:90) Q t n +1 ( x (cid:48) ) K t n ( dx (cid:48) | x, γ n ) (cid:19) , (27)starting from the final condition Q T ( x ) = H N ( x ). Here, K t n ( dx (cid:48) | x, γ n ) is the stochastic kernelrepresenting probability to reach state in dx (cid:48) at time t n +1 if the withdrawal ( action ) γ n isapplied in the state x at time t n . Obviously, the above backward induction can also be usedto calculate the fair contract price in the case of static strategy γ ; in this case the space ofadmissible strategies A n consists only one pre-defined value and sup( · ) becomes redundant.For clarity, denote Q t − n ( · ) and Q t + n ( · ) the contract values just before and just after the eventtime t n respectively. Then, after calculating expectation with respect to the mortality statevariable I n +1 in (27), the required backward recursion can be rewritten explicitly as Q t + n ( W, A ) = (1 − q n +1 )E Q t + n (cid:20) B n,n +1 Q t − n +1 (cid:0) W ( t − n +1 ) , A ( t − n +1 ) (cid:1) | W, A (cid:21) + q n +1 E Q t + n (cid:20) B n,n +1 D n +1 (cid:0) W ( t − n +1 ) , A ( t − n +1 ) (cid:1) | W, A (cid:21) (28)with the jump condition Q t − n ( W, A ) = max γ n ∈A n (cid:16) (cid:101) f n ( W, A, γ n ) + Q t + n (cid:0) h Wn ( W, A, γ n ) , h An ( W, A, γ n ) (cid:1)(cid:17) . (29)This recursion is solved for n = N − , N − , . . . ,
0, starting from the maturity condition Q t − N ( W, A ) = P T ( W, A ). Given that the mortality and financial asset processes are assumed independent, and the with-drawal decision does not affect mortality process, one can calculate the expected value of thepayoff (21) with respect to the mortality process, (cid:101) H ( W , A ) = E I t [ H ( X , γ )], and then cal-culate the price under the optimal strategy sup γ E Q t [ (cid:101) H ( W , A )] or under the given strategyE Q t [ (cid:101) H ( W , A )]. It is easy to find that (cid:101) H ( W , A ) = B ,N (cid:18) p N P T (cid:0) W ( T − ) , A ( T − ) (cid:1) + q N p N − D N (cid:0) W ( T − ) , A ( T − ) (cid:1) (cid:19) + N − (cid:88) n =1 B ,n (cid:16) p n (cid:101) f n ( W ( t − n ) , A ( t − n ) , γ n ) + p n − q n D n (cid:0) W ( t − n ) , A ( t − n ) (cid:1)(cid:17) , (30)14here p n = Pr[ τ > t n | τ > t ] and q n p n − = Pr[ t n − < τ ≤ t n | τ > t ] for random death time τ ,i.e. p n = p n − (1 − q n ). Note that, previously we defined q n = Pr[ t n − < τ ≤ t n | τ > t n − ].The payoff (30) has the same general form as the payoff (21). Thus, the optimal stochasticcontrol problem Ψ t ( W (0) , A (0)) = sup γ E Q t [ (cid:101) H ( W , A )] can be solved using Bellman equation(27) leading to the following explicit recursionΨ t + n ( W, A ) = E Q t + n (cid:104) B n,n +1 Ψ t − n +1 (cid:0) W ( t − n +1 ) , A ( t − n +1 ) (cid:1) | W, A (cid:105) , (31)Ψ t − n ( W, A ) = max γ n ∈A n (cid:18) p n (cid:101) f n ( W, A, γ n ) + p n − q n D n ( W, A )+Ψ t + n (cid:0) h Wn ( W, A, γ n ) , h An ( W, A, γ n ) (cid:1) (cid:19) , (32)for n = N − , N − , . . . ,
0, starting from Ψ t − N ( W, A ) = p N P T ( W, A ) + p N − q N D N ( W, A ).It is easy to verify that this recursion leads to the same solution Ψ t ( W, A ) = Q t ( W, A )and the same optimal strategy for γ as obtained from the recursion (28–29), noting thatΨ t − n ( W, A ) = p n Q t − n ( W, A ) + p n − q n D n ( W, A ). The result is somewhat obvious becausesup γ E Q , I t [ H ( X , γ )] = sup γ E Q t (cid:2) E I t [ H ( X , γ )] (cid:3) . (33)Note that, sup γ E Q , I t [ H ( X , γ )] (cid:54) = E I t (cid:2) sup γ E Q t [ H ( X , γ )] (cid:3) . That is, one cannot find the priceunder the optimal strategy conditional on the death time and then average over random deathtimes, that would lead to the result larger than Q t ( W, A ), see Luo and Shevchenko (2015b).
The guarantee fare fee based on the optimal policyholder withdrawal is the worst case scenariofor the issuer, i.e. if the guarantee is hedged then this fee will ensure no losses for the issuer (inother words full protection against policyholder strategy and market uncertainty). Of coursethis is under the given assumptions about stochastic model for the underlying risky asset. Ifthe issuer hedges continuously but investors deviate from the optimal strategy, then the issuerwill receive a guaranteed profit.Any strategy different from the optimal is sup-optimal and will lead to smaller fair fees. Ofcourse the strategy optimal in this sense is not related to the policyholder circumstances. Thepolicyholder may act optimally with respect to his preferences and circumstances but it may bedifferent from the optimal strategy calculated in (29). On the other hand, as noted in Hilpertet al. (2014), secondary markets for equity linked insurance products (where the policyholdercan sell their contracts) are growing. Thus, financial third parties can potentially generateguaranteed profit through hedging strategies from financial products such as VA riders whichare not priced according to the worst case assumption of the optimal withdrawal strategy.Thus the development of secondary markets for VA riders would lead to an increase in thefees charged by the issuing companies. Knoller et al. (2015) undertakes an empirical studyof policyholders behavior in Japanese VA market and they show that the moneyness of theguarantee has the largest explanatory power for the surrender rates.15ne way to introduce a reasonable suboptimal withdrawal model is to assume that thepolicyholder follows a default strategy withdrawing a contractual amount G n at each eventtime t n unless the extra value from optimal withdrawal is greater than θ × G n , θ ≥
0. Setting θ = 0 corresponds to the optimal strategy, while θ (cid:29) A n restricted to the specified strategy. Withdrawals from the VA type contracts may attract country and individual specific govern-ment taxes. Moenig and Bauer (2015) demonstrated that including taxes significantly affectsthe value of VA withdrawal guarantees. They developed a subjective risk-neutral valuationmethodology and produced results in line with empirical market prices. Following closely toMoenig and Bauer (2015), we introduce an extra state variable R ( t ) to present the tax base which is the amount that may still be drawn tax-free, and assume that all event times t n ∈ T are the policy anniversary dates. The initial premium is assumed to be post-tax and taxes areapplied to future investment gains (not the initial investment).Denote a marginal income tax rate as (cid:101) κ and marginal capital gain tax from investmentoutside of VA contract as κ . It is assumed that earnings from VA are treated as ordinaryincome and withdrawals are taxed on a last-in first-out basis. Thus if the wealth account W ( t − n ) exceeds the tax base R ( t − n ), any withdrawal up to W ( t − n ) − R ( t − n ) will be taxed at therate (cid:101) κ and will not affect the tax base; larger withdraws will not be subject to tax but willreduce the tax base. Specifically, the tax base will be changed at withdrawal time t n as R ( t + n ) = R ( t − n ) − max (cid:0) γ t − max( W ( t − n ) − R ( t − n ) , , (cid:1) . The cashflow received by the policyholder will be reduced by taxes tax = (cid:101) κ min (cid:16) (cid:101) f n ( W ( t − n ) , A ( t − n ) , γ n ) , max( W ( t − n ) − R ( t − n ) , (cid:17) , i.e. one has to make the following change in the contract specifications listed in Section 4 (cid:101) f n ( W ( t − n ) , A ( t − n ) , γ n ) → (cid:101) f n ( W ( t − n ) , A ( t − n ) , γ n ) − tax. Using arguments for replicating pre-tax cashlows at t n with post-tax cashflows at t n +1 , itwas shown in Moenig and Bauer (2015) that Q t + n ( W, A, R ) should be found not as the directexpectation (28) but should be found as the solution of the following nonlinear equation Q t + n ( W, A, R ) = E Q t + n (cid:2) V ( t − n +1 ) | W, A, R (cid:3) + κ − κ E Q t + n (cid:20) max (cid:2) V ( t − n +1 ) − B n,n +1 Q t + n ( W, A, R ) , (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) W, A, R (cid:21) , (34)16here V ( t − n +1 ) = (1 − q n +1 ) B n,n +1 Q t − n +1 (cid:0) W ( t − n +1 ) , A ( t − n +1 ) , R ( t − n +1 ) (cid:1) + q n +1 B n,n +1 D n +1 (cid:0) W ( t − n +1 ) , A ( t − n +1 ) , R ( t − n +1 ) (cid:1) . (35)This is referred to as subjective valuation from the policyholder perspective and depends on theinvestor current position (including possible offset tax responsibilities) and tax rates. Numericalexamples in Moenig and Bauer (2015) show that the VA guarantee prices accounting for taxesin the above way are lower than ignoring the taxes (not surprisingly, because it lead to thesuboptimal strategy), making the prices overall more aligned with those observed in the market. In the case of realistic VA riders with discrete events such as ratchets and optimal withdrawals,there are no closed form solutions and fair price has to calculated numerically, even in thecase of simple geometric Brownian motion process for the risky asset. In general, one canuse PDE, direct integration or regression type MC methods, where the backward recursion(28–29) is solved numerically. Of course, if the withdrawal strategy is known, then one canalways use standard MC to simulate state variables forward in time till the contract maturity orpolicyholder death and average the payoff cashflows over many independent realizations. Thisstandard procedure is well known and no further discussion is needed.In this section, we give a brief review of different numerical methods that can be used forvaluation of VA riders. Then, we provide detailed description of the direct integration methodthat can be very efficient and simple to implement, when the transition density of the underlyingasset or it’s moments between the event times are known in closed form. Finally, in Section 6.5we present calculation of hedging parameters (referred in the literature as
Greeks ). Simulation based Least-Squares MC method introduced in Longstaff and Schwartz (2001) isdesigned for uncontrolled Markov process problems and can be used to account for the contractearly surrender, as e.g. in Bacinello et al. (2011). However, it cannot be used if an optimalwithdrawal strategy is involved. This is because dynamic withdrawals affect the paths of theunderlying wealth account and one cannot carry out a forward simulation step required forthe subsequent regression in the backward induction. However, it should be possible to applycontrol randomization methods extending Least-Squares MC to handle the optimal stochasticcontrol problems with controlled Markov processes, as was recently developed in Kharroubiet al. (2014). The idea is to first simulate the control (withdrawals) and the state variablesforward in time, where the control is simulated independently from other variables. Then, useregression on the simulated state variables and control to estimate expected value (28) and findthe optimal withdrawal using (29). However, the accuracy and robustness of this method forpricing withdrawal benefit type products have not been studied yet. As usual, it is expected thatthe choice of the basis functions for the required regression step will have significant impact on17he performance. We also note that in some simple cases of the withdrawal strategy admissiblespace such as bang-bang (no withdrawal, withdrawal at the contractual rate, or full surrender),it is possible to develop other modifications of Least-Squares MC such as in Huang and Kwok(2015) for pricing of the GLWB rider.The expected value (28) can also be calculated using PDE or direct integration methods.In both cases, the modeller discretizes the space of the state variables and then calculatesthe contract value for each grid point. The PDE for calculation of expected value (28) underthe assumed risk-neutral process for the risky asset S ( t ) is easily derived using Feynman-Kactheorem; for a good textbook treatment of this topic, see e.g. Bj¨ork (2004). However, theobtained PDE can be difficult or even not practical to solve in the high-dimensional case.In particular, in the case of geometric Brownian motion process for the risky asset (4), thegoverning PDE in the period between the event times is the one-dimensional Black-Scholes PDE,with jump conditions at each event time to link the prices at the adjacent periods. Since thebenefit base state variable A ( t ) remains unchanged within the interval ( t i − , t i ) , i = 1 , , . . . , N ,the contract value Q t ( W, A ) satisfies the following PDE with no explicit dependence on A , ∂Q∂t + σ W ∂ Q∂W + ( r − α ) W ∂Q∂W − rQ = 0 . (36)This PDE can be solved numerically using e.g. Crank-Nicholson finite difference scheme foreach A backward in time with the jump condition (29) applied at the contract event times. Thishas been done e.g. in Dai et al. (2008) and Chen and Forsyth (2008) for pricing GMWB withdiscrete optimal withdrawals. Of course, if the volatility or/and interest rate are stochastic,then extra dimensions will be added to the PDE making it more difficult to solve. Forsythand Vetzal (2014) used PDE approach to calculate VA rider prices in the case of stochasticregime-switching volatility and interest rate.Under the direct integration approach, the expected value (28) is calculated as an integralapproximated by summation over the space grid points, see e.g. Bauer et al. (2008). Moreefficient quadrature methods (requiring less points to approximate the integral) exist. In par-ticular, in the case of a geometric Brownian motion process for the risky asset, it is very efficientto use the Gauss-Hermite quadrature as developed in Luo and Shevchenko (2014) and appliedfor GMWB pricing in Luo and Shevchenko (2015a). Section 6.3 provides detailed descriptionof the method for pricing VA riders in general. This method can be applied when the transitiondensity of the underlying asset between the event times or it’s moments are known in closedform. It is relatively easy to implement and computationally faster than PDE method becausethe latter requires many time steps between the event times. In Luo and Shevchenko (2016),this method was also used to calculate GMWB in the case of stochastic interest rate under theVasicek model.In both PDE and direct integration approaches, one needs some interpolation scheme toimplement the jump condition (29), because state variables located at the grid points of dis-cretized space do not appear on the grid points after the jump event. This will be discussedin detail in Section 6.4. Of course, if the underlying stochastic process is more complicatedthan geometric Brownian motion (4) and does not allow efficient calculation of the transitiondensity or its moments, one can always resort to PDE method.18n our numerical examples of GMAB pricing in Section 7, we adapt a direct intergationmethod based on the Gauss-Hermite integration quadrature applied on a cubic spline inter-polation, hereafter referred to as GHQC . For testing purposes, we also implemented Crank-Nicholson finite difference (FD) scheme solving PDE (36) with the jump condition (29). Both PDE and direct integration numerical schemes start from a final condition for the contractvalue at t = T − . Then, a backward time stepping using (28) or solving corresponding PDEgives solution for the contract value at t = t + N − . Application of the jump condition (29) tothe solution at t = t + N − gives the solution at t = t − N − from which further backward in timerecursion gives solution at t . For simplicity assume that there are only W ( t ) and A ( t ) statevariables. The numerical algorithm then takes the following key steps. Algorithm 6.1 (Direct Integration or PDE method) • Step 1. Generate an auxiliary finite grid A < A < · · · < A J to track the benefitbase balance A . • Step 2. Discretize wealth account balance W space as W < W < · · · < W M to generatethe grid for computing the expectation (28). • Step 3. At t = t N , apply the final condition at each node point ( W m , A j ) , j = 1 , , . . . , J , m = 1 , , . . . , M to get Q t − N ( W, A ) . • Step 4. Evaluate integration (28) for each A j , j = 0 , . . . , J , to obtain Q t + N − ( W, A ) eitherusing direct integration or solving PDE. In the case of direct integration method, thisinvolves one-dimensional interpolation in W space to find values of Q t − N ( W, A ) at theguadrature points different from the grid points. • Step 5. Apply the jump condition (29) to obtain Q t − N − ( W, A ) for all possible values of γ N − and find γ N − that maximizes Q t − N − ( W, A ) . In general, this involves a two-dimensionalinterpolation in ( W, A ) space. • Step 6. Repeat Step 4 and 5 for t = t N − , t N − , . . . , t . • Step 7. Evaluate integration (28) for the backward time step from t to t to obtainsolution Q ( W, A ) at W = W (0) and A (0) , or may be at several points if these areneeded for calculation of some hedging sensitivities such as Delta and
Gamma discussedin Section 6.5.
In our implementation of the direct integration method based on the Gauss-Hermite quadra-ture for numerical examples in Section 7, we use a one-dimensional cubic spline interpolationrequired to handle integration in Step 4 and bi-cubic spline interpolation to handle jump con-dition Step 5. 19f the model has other stochastic state variables (similar to W ) changing stochasticallybetween the contract event times, such as stochastic volatility and/or stochastic interest rate,then grids for these extra dimensions should be generated and the required integration or PDEto evaluate (28) will have extra dimensions. Also, extra auxiliary state variables (similar to A ) unchanged between the contract event times, such as tax base and/or extra benefit base,will require extra dimensions in the grid and interpolation for the jump condition at the eventtimes.We have to consider the possibility of W ( t ) goes to zero due to withdrawal and marketmovement, thus one has to use the lower bound W = 0. The upper bound W M should be setsufficiently far from the initial wealth at time zero W (0). A good choice of such a boundarycould be based on the high quantiles of distribution of S ( T ). For example, in the case ofgeometric Brownian motion process (5), one can set conservatively W M = W (0) e | mean(ln( S ( T ) /S (0))) | +5 × stdev(ln( S ( T ) /S (0))) . Often, it is more efficient to use equally spaced grid in ln W space. In this case, W cannotbe set to zero and instead should be set to a very small value (e.g. W = 10 − ). Also, for someVA riders, using equally spaced grid in ln A space is also more efficient. To compute Q ( W (0) , A (0)), we have to evaluate the expectations in the recursion (28). As-suming the conditional probability density of W ( t − n ) given W ( t + n − ) is known in closed form (cid:101) p n ( w | W ( t + n − )), the required expectation (28) can be calculated as Q t + n − (cid:0) W ( t + n − ) , A (cid:1) = (cid:90) + ∞ (cid:101) p n ( w | W ( t + n − )) (cid:101) Q t − n ( w, A ) dw, (37)where (cid:101) Q t − n ( w, A ) = B n − ,n (cid:0) (1 − q n ) Q t − n ( w, A ) + q n D n ( w, A ) (cid:1) . The above integral can be estimated using various numerical integration ( quadrature ) meth-ods. Note that, one can always find W ( t − n ) as a transformation of the standard normal randomvariable Z as W ( t − n ) = ψ ( Z ) := F − n (Φ( Z )) , where Φ( · ) is the standard normal distribution, and F n ( · ) and F − n ( · ) are the distribution andits inverse of W ( t − n ). Then, the integral (37) can be rewritten as Q t + n − (cid:0) W ( t + n − ) , A (cid:1) = 1 √ π (cid:90) + ∞−∞ e − z (cid:101) Q t − n ( ψ n ( z ) , A ) dz. (38)This type of integrand is very well suited for the Gauss-Hermite quadrature that for anarbitrary function f ( x ) gives the following approximation (cid:90) + ∞−∞ e − x f ( x ) dx ≈ q (cid:88) i =1 λ ( q ) i f ( ξ ( q ) i ) . (39)20ere, q is the order of the Hermite polynomial, ξ ( q ) i , i = 1 , , . . . , q are the roots of the Hermitepolynomial H q ( x ), and the associated weights λ ( q ) i are given by λ ( q ) i = 2 q − q ! √ πq [ H q − ( ξ ( q ) i )] . This approximate integration works very well if function f ( x ) is without singularities and itcalculates the integral exactly if f ( x ) is represented by a polynomial of degree 2 q − (cid:101) Q t ( w, · ) is known only at the grid points W m , m = 0 , , . . . , M and interpolationis required to estimate (cid:101) Q t ( w, · ) at the quadrature points. From our experience with pricingdifferent VA guarantees, we recommend the use of the natural cubic spline interpolation which issmooth in the first derivative and continuous in the second derivative; and the second derivativeis assumed zero for the extrapolation region above the upper bound.Of course it can be difficult to find the distribution F n ( · ) and its inverse F − n ( · ) in general.In the case of geometric Brownian motion process (5), the transition density (cid:101) p n ( ·|· ) is just alognormal density and W ( t − n ) = ψ n ( Z ) := W ( t + n − ) exp (cid:18) ( r n − α − σ n ) dt n + σ n (cid:112) dt n Z (cid:19) . Then, a straightforward application of the Gauss-Hermite quadrature for the evaluation ofintegral (37) gives Q t + n − (cid:0) W ( t + n − ) , A (cid:1) ≈ √ π q (cid:88) i =1 λ ( q ) i (cid:101) Q t − n (cid:16) ψ n ( √ ξ ( q ) i ) , A (cid:17) , (40)that should be calculated for each grid point W ( t + n − ) = W m , m = 0 , , . . . , M . Often, asmall number of quadrature points is required to achieve very good accuracy; in our numericalexamples in the next section we use q = 9 but very good results are also obtained with q = 5.If the transition density function from W ( t + n − ) to W ( t − n ) is not known in closed form butone can find its moments, then the integration can also be done with similar efficiency andaccuracy by method of matching moments as described in Luo and Shevchenko (2014, 2015a).The method also works very well in the two-dimensional case, see e.g. Luo and Shevchenko(2016) where it was applied for GMWB pricing in the case of stochastic interest rate. Either in PDE or direct integration method, one has to apply the jump condition (29) at theevent times to obtain Q t − n ( W, A ). For the optimal strategy, we chose a value of withdrawal γ n ∈ A n maximizing the value Q t − n ( W, A ).To apply the jump conditions, an auxiliary finite grid 0 = A < A < · · · < A J = W M is usedto track the remaining benefit base state variable A . For each A j , we associate a continuoussolution using (40) and interpolation. In general, as can be seen from (29), the jump conditionmakes it impractical, if not impossible, to ensure the values of W and A after the jump toalways fall on a grid point. Thus a two-dimensional interpolation is required. In this work we21 i W ),,( njiWn AWhW j A k A k A j A W j A A A n tt n tt W ),( AWQ t ),( jit AWQ j A k A k A i W i W m W m W ),,( njiAn AWhA i W i W m W m W i W Figure 1:
Illustration of the application of jump condition. The value Q t ( W i , A j ) at t = t − n and at nodepoint ( W i , A j ) equals to Q t ( W, A ) at t = t + n with W = W i − γ n and A = A j − C ( γ n ). The point ( W, A )is located inside the grid bounded by ( W m , W m +1 ) and ( A k , A k +1 ). adopted the bi-cubic spline interpolation for accuracy and efficiency. Figure 1 illustrates theapplication of jump conditions.It is natural to form a uniform grid in A so that optimal withdrawal strategies can be testedon a constant increment δA = A j +1 − A j , as has been done successfully in Luo and Shevchenko(2015a) for pricing of a basic GMWB specified by (8–9). However, extensive numerical testsshow that if a uniform grid in A is used for pricing GMAB with ratchets and optimal withdrawals(our numerical example in Section 7), then neither linear interpolation nor cubic interpolationin A can achieve an efficient convergence in pricing results. A very fine mesh has to be usedbefore we see a stable solution, which can take up to several hours to obtain a fair fee, in sharpcontrast to basic GMWB where less than one minute computer time is required. On the otherhand, if we make the grid in A uniform in Y = ln A and use linear or cubic interpolation basedon variable Y , then we obtain a very good convergence on a moderately fine grid and the CPUtime for a fair fee is about 30 minutes (a few minutes for a fair price). The CPU used for allthe calculations in this study is Intel(R) Core(TM) i5-2400 @3.1GHz.As we have already mentioned, a two-dimensional interpolation has to be used for applyingthe jump condition. We suggest to use either a bi-linear interpolation or a bi-cubic spline inter-polation, e.g. see (Press et al., 1992, section 3.6), in both cases applied on the log-transformed22tate variables X = ln W and Y = ln A . For numerical examples in this paper, we have adaptedthe more accurate bi-cubic spline interpolation for all the numerical results.For uniform grids, the bi-cubic spline is about five times as expensive in terms of comput-ing time as the one-dimensional cubic spline. Suppose the jump condition requires the value Q ( W, A ) at the point (
W, A ) located inside a grid: W i ≤ W ≤ W i +1 and A j ≤ A ≤ A j +1 . Equiv-alently, the point X = ln W and Y = ln A is inside the grid: ln W i = x i ≤ X ≤ x i +1 = ln W i +1 and ln A j = y j ≤ Y ≤ y j +1 = ln A j +1 . Because the grid is uniform in both X and Y vari-ables, the second derivatives ∂ Q/∂X and ∂ Q/∂Y can be accurately approximated by thethree-point central difference, and consequently the one-dimensional cubic spline on a uniformgrid involves only four neighboring grid points for any single interpolation. For the bi-cubicspline, we can first obtain Q ( · , · ) at four points Q ( W, A j − ), Q ( W, A j ), Q ( W, A j +1 ), Q ( W, A j +2 )by applying the one-dimensional cubic spline on the dimension X = log W for each point andthen we can use these four values to obtain Q ( W, A ) through a one-dimensional cubic splinein Y = log A . Thus five one-dimensional cubic spline interpolations are required for a singlebi-cubic spline interpolation, which involves sixteen grid points neighboring ( W, A ) point.
Calculation of the contract price in (28) under the risk neutral probability measure Q meansthat one can find a portfolio replicating the VA guarantee, i.e. perform hedging eliminatingthe financial risk. Finding correct hedging depends on the underlying stochastic model forthe risky asset. The basic hedging is the so-called delta hedging eliminating randomness dueto stochasticity in the underlying risky asset S ( t ). Here, we use S ( t ) as a tradable assetto hedge the exposure of the guarantee to the wealth account W ( t ). One can construct aportfolio consisting of the money market account and ∆ S ( t ) units of S ( t ), so that ∆ S ( t ) S ( t ) =∆ W ( t ) W ( t ), where ∆ W ( t ) is the number of units of the wealth account referred as Delta . Denotethe value of the VA guarantee as U t ( W, A ) which is just a difference between the contract valuewith the guarantee Q t ( W, A ) and the value of the wealth account W , i.e. Q t ( W, A ) = U t ( W, A ) + W. (41)Then, under the delta hedging strategy, one has to select∆ W ( t ) = ∂U t ( W, A ) ∂W ⇔ ∆ S ( t ) = ∂U t ( W, A ) ∂S WS for time t between the contract event times. Of course if there are extra stochasticities in themodel such as stochastic interest rate and/or stochastic volatility, delta hedging will not elimi-nate risk completely and hedging with extra assets will be required which is model specific. Seee.g. Forsyth and Vetzal (2014), for constructing hedging in the case of regime switching stochas-tic volatility and interest rate. A popular active hedging strategy in the case of extra stochasticfactors is the minimum variance hedging strategy, where ∆ W ( t ) is selected to minimize the vari-ance of portfolio’s instantaneous changes, e.g. applied in Huang and Kwok (2015) for hedgingGLWM in the case of stochastic volatility model. Practitioners also calculate other sensitivities(partial derivatives) of the contract with respect to the interest rate and volatility (referred to23s Rho and
V ega ) and even second partial derivatives such as
Gamma = ∂ U t ( W, A ) /∂W toimprove hedging strategies. Here, we refer the reader to the standard textbooks in the area ofpricing financial derivatives such as Wilmott (2013) or Hull (2006).Numerical estimation of the contract sensitivities (referred to as Greeks ) is more difficultthan estimation of the contract price. A general standard approach to calculate Greeks is toperturb the relevant parameter and re-calculate the price. Then one can use a two-point centraldifference to estimate the first derivatives and a three-point central difference for the secondderivatives. In general, the finite difference PDE (or direct integration) methods generallyproduce superior accuracy in calculating Greeks when compared to Monte Carlo method (atleast for low dimensions when finite difference method is practical or direct integration ispossible). For Delta and Gamma, the finite difference method (or direct integration method)yields second order accurate values without re-calculating price using prices already calculatedat the uniform grid points.More accurate calculation of the main Greeks,
Delta and
Gamma , can be achieved usingthe so-called likelihood method as follows. The contract price at t is calculated in the lasttime step ( t , t ) in backward induction as an integral (37). Differentiating (37) with respectto W (0) = w , Delta can be found as ∂Q t +0 ( w , A ) ∂w = (cid:90) (cid:101) Q t − ( w, A ) ∂ (cid:101) p ( w | w ) ∂w dw = (cid:90) (cid:101) Q t − ( s, A ) ∂ ln (cid:101) p ( w | w ) ∂s (cid:101) p ( w | w ) dw = E Q t +0 (cid:20) (cid:101) Q t − ( W, A ) ∂ ln (cid:101) p ( W | w ) ∂w (cid:12)(cid:12)(cid:12)(cid:12) w , A (cid:21) . (42)Thus it can be calculated using the same direct integration method as used for Q + t ( s , A ) withthe factor ∂ ln (cid:101) p ( W | w ) /∂w added to the integrand. Similarly, the required derivative tocalculate Gamma can be found as ∂ Q t +0 ( w , A ) ∂w = (cid:90) (cid:101) Q t − ( w, A ) ∂∂w (cid:20) ∂ ln (cid:101) p ( w | w ) ∂w (cid:101) p ( w | w ) (cid:21) dw = (cid:90) (cid:101) Q t − ( w, A ) (cid:34)(cid:18) ∂ ln (cid:101) p ( w | w ) ∂w (cid:19) + ∂ ln (cid:101) p ( w | w ) ∂w (cid:35) (cid:101) p ( w | w ) dw = E Q t +0 (cid:34) (cid:101) Q t − ( W, A ) (cid:34)(cid:18) ∂ ln (cid:101) p ( w | w ) ∂w (cid:19) + ∂ ln (cid:101) p ( w | w ) ∂w (cid:35) (cid:12)(cid:12)(cid:12)(cid:12) w , A (cid:35) . (43)Note, the above integrations for Delta and Gamma are only required for the ( t , t ) time stepand for a single grid point W (0) = w . Here, t should be understood as the current contractvaluation time rather than time when the contract was initiated. Equivalently, for PDE ap-proach using finite difference method, one can sometimes derive the corresponding PDEs for theGreeks and solve these PDEs for the last time step, see e.g. Tavella and Randall (2000). Simi-larly for Monte Carlo method, simulations used to calculate the price can be used to calculate Delta and
Gamma weighted with extra factors under the expectations in (42) and (43).24t is illustrative to show how to derive hedging portfolio under the basic Black-Scholes model.Here, we assume that the underlying risky asset S ( t ) follows dS ( t ) = µ ( t ) S ( t ) + σ ( t ) S ( t ) dB P ( t ) , (44)where B P ( t ) is the standard Brownian motion under the physical ( real ) probability measure,and µ ( t ) is the real drift. Then the wealth account evolution is dW ( t ) = ( µ ( t ) − α ) W ( t ) + σ ( t ) W ( t ) dB P ( t ) . Here, we assume a continuously charged fee proportional to the wealth account but it is notdifficult to deal with the case of discretely charged fees.To hedge, the guarantee writer takes a long position in ∆ S units of S ( t ), i.e. forms a portfolioΠ t = − U t ( W, A ) + ∆ S × S By Ito’s lemma, the changes of portfolio within ( t n − , t n ), n = 1 , . . . , N are d Π t = − (cid:18) ∂U t ∂t + ∂U t ∂W dW + 12 σ S ∂ U t ∂W (cid:19) dt − ∆ S dS + αW dt, (45)where the last term αW dt is the fee amount collected over dt . Setting∆ S = ∂U t ( W, A ) ∂W WS = (cid:18) ∂Q t ( W, A ) ∂W − (cid:19) WS (46)eliminates all the random terms in (45), making the portfolio locally riskless. This means thatthe portfolio earns at risk free interest rate r ( t ), i.e. d Π t = r Π t dt , leading to the PDE ∂U t ∂t + ( r − α ) W ∂U t ∂W + 12 σ S ∂ U t ∂W − rU t − αW = 0 . (47)Substituting U t ( W, A ) = Q t ( W, A ) − W in the above gives the PDE for Q t ( W, A ), the total valuefor the contract with the guarantee, i.e. the same as (36). Recalling Feynman-Kac theorem, itis easy to see that the stochastic process for W corresponding to this PDE is the risk-neutralprocess (4). Numerical solutions for pricing VA riders involve many complicated numerical procedures andfeatures. These are more involved when compared to pricing of most exotic derivatives infinancial markets. It is important that these solutions are properly tested and validated. Asa numerical example for illustration, using direct integration method (GHQC), we calculateaccurate prices of GMAB with possible annual ratchets and allowing optimal withdrawals asspecified in Section 4.3. With these features, the GMAB is very close to the real productmarketed in Australia by e.g. MLC (2014) and AMP (2014). We assume geometric Brownianmotion model for the risky asset (4). When applicable, we compare results with MC and finite25ifference PDE methods. Numerical difficulties encountered in pricing this GMAB rider arecommon across other VA guarantees. Also, comprehensive numerical pricing results for thisparticular product are not available in the literature. These validated results (reported for arange of parameters) can serve as a benchmark for practitioners and researchers developingnumerical pricing of VAs with guarantees. We consider four GMAB types:1. GMAB with the annual ratchets but no withdrawals. In this case, a standard MC methodcan be used to compare with GHQC results – this is a good validation of the implementednumerical functions related to the ratchet feature, in addition to validating the numericalintegration by Gauss-Hermite quadrature.2. GMAB with the annual ratchets and a regular withdrawals of a fixed percentage of thewealth account. In this case a standard MC method can also be used to compare withGHQC results – this is a good validation of implemented numerical functions related tojump conditions due to both ratchet and withdrawal features. In addition, in order totest the numerical functions related to the application of penalties, we assume a pensionaccount where the static withdrawal rate is set above the penalty threshold.3. GMAB with the optimal quarterly withdrawals and the annual ratchets for a super ac-count , where the penalty (18) is applied on any withdrawal γ n when W ( t − n ) < A ( t − n ).4. GMAB with the optimal quarterly withdrawals and the annual ratchets for a pensionaccount , where the penalty (19) is applied if the withdrawal γ n is above the penaltythreshold G n and if W ( t − n ) < A ( t − n ).As a comparison, results from our PDE finite difference implementation will also be shownfor Case 4, the most complicated example among the four listed above. In addition, we willalso calculate results for Case 4 in the case of the guarantee fee charged discretely (quarterly).All reported GHQC results are based on q = 9 quadrature points. We did not observe anymaterial difference in results if q is increased further. Results based on q = 5 are also veryaccurate. Consider a GMAB rider with the annual ratchet and no withdrawals. In this case a standard MCmethod can be used to compare with GHQC results which is a good validation of implementednumerical functions related to the ratchet feature. Table 1 compares GHQC and MC resultsfor the fare fee α of GMAB with the annual ratchet for the interest rate r ranging from 1% to7% and the volatility σ = 10% and 20%. The maximum relative difference between the twomethods is 0 .
76% at interest rate r = 5% and σ = 10%. The maximum absolute differencebetween the two methods is one basis point at the lowest interest rate r = 1% where the feeis the highest. On average, the relative difference is 0 .
52% and the absolute difference is 0.5basis point, which is 5 cents per year on a one thousand dollar account. The GHQC resultsare obtained with the mesh size M = 400 and J = 200, and on the average it takes 22 seconds26 nterest rate, r σ = 10% σ = 20%GHQC, bp MC, bp (cid:101) δ GHQC, bp MC, bp (cid:101) δ
1% 337.2 338.2 0 .
30% 998.7 999.8 0 . .
43% 637.1 637.7 0 . .
43% 458.0 458.5 0 . .
47% 346.9 347.5 0 . .
76% 271.1 271.6 0 . .
60% 216.3 216.7 0 . .
68% 175.1 175.3 0 . Table 1:
Fair fee α in bp (1 bp=0.01%) as a function of the interest rate r for the GMAB riderwith the annual ratchet and no withdrawal. The contract maturity is T = 10 years. (cid:101) δ is the relativedifference between Monte Carlo (MC) and GHQC method results. per price (calculation of the fare fee requires iterations over several prices). The MC results areobtained using 20 million simulations and it takes about 62 seconds per price.The agreement between the two methods at σ = 20% is also very good. In absolute terms,the maximum difference between the two methods is 1.1 basis point at the lowest interest rate r = 1%. In relative terms, the maximum difference between the two methods is 0 .
34% at thehighest interest rate r = 7%. On average, the relative difference is 0 .
17% which is significantlysmaller than the corresponding case at σ = 10%. Consider a GMAB rider with the annual ratchet and a regular quarterly withdrawals of afixed percentage of the wealth account. In this case a standard MC method can also be used tocompare with the GHQC results which is a good validation of implemented numerical functionsrelated to jump conditions due to both ratchet and withdrawal features. Here, we consider apension type account with the penalty given by (19). In this case, regular withdrawals at apre-determined percentage level are allowed. In order to test the numerical functions relatedto the application of penalty, we also consider the static withdrawal above a pre-determinedthreshold level that will attract a penalty. We set the withdrawal threshold at 15% of the wealthaccount per annum, and the withdrawal frequency is quarterly, i.e. the quarterly withdrawalthreshold is G n = 3 .
75% of the wealth account.In the first test we allow a regular quarterly withdrawal of 3 .
75% of the wealth accountbalance, i.e. γ n = G n and there is no penalty on the withdrawals. Table 2 compares GHQCand MC results for the fare fee α . In relative terms, the maximum difference between the twomethods is 0 .
08% at interest rate r = 5%. On average, the relative difference is 0 .
06% andthe absolute difference is 0.3 basis point. The GHQC results are obtained with the mesh size M = 400 and J = 400, and the MC results are obtained with 20 million simulations per price.Comparing with Table 1, the fair fee for the static withdrawal is about 8% higher thanthe corresponding no-withdrawal case at the lowest interest rate r = 1%, but it is about 13%lower than the corresponding no-withdrawal case at r = 7%. We have also tested static 2 . nterest rate, r
15% annual withdrawal 16% annual withdrawalGHQC, bp MC, bp (cid:101) δ GHQC, bp MC, bp (cid:101) δ
1% 1084 1085 0 .
09% 185.3 185.3 < . .
06% 152.9 152.9 < . .
06% 126.6 126.6 < . .
06% 105.1 105.1 < . .
08% 87.54 87.51 0 . < .
01% 73.21 73.14 0 . .
07% 61.40 61.36 0 . Table 2:
Fair fee α in bp as a function of interest rate r for the GMAB with the annual ratchet and staticquarterly withdrawal of 3 .
75% and 4% of the wealth account (15% and 16% annually respectively). Thepenalty threshold (pension type account) is set at 15% annually. The contract maturity is T = 10 yearsand volatility is σ = 20%. (cid:101) δ is the relative difference between Monte Carlo (MC) and GHQC results. quarterly withdrawals and obtained the same pattern: at lower interest rate the fair fee ofstatic withdrawal (which is also below the penalty threshold) is higher than the correspondingno-withdrawal case, and at higher interest it is the opposite. These differences in the fair feesat relatively low and high interest rates can be broadly interpreted as follows. At lower interestrates, where the expected capital growth is relatively slow, it is better to perform a regularwithdrawal at or below the penalty threshold and take the protected capital at the maturity.However, at higher interest rates, where the expected capital growth rate is also high, it isbeneficial not to carry out a regular withdrawal and keep the capital to grow.The above test also demonstrates that the MC and GHQC methods agree very well forpricing GMAB with a static withdrawal not exceeding the penalty threshold. This confirmsthe accuracy and efficiency of our numerical implementation of the jump condition using abi-cubic interpolation in GHQC method.In the second test of static withdrawal, we allow a regular quarterly withdrawal of 4% (16%per annum), i.e. the annual withdrawal rate is slightly higher than the penalty threshold of15% per annum and there is a penalty applied for each withdrawal. The GHQC and MC resultsfor this test are also presented in Table 2. In absolute terms, the maximum difference betweenthe two methods is only 0.07 basis point at interest rate r = 6%, which is less than 1 cent peryear for a one thousand dollar account. In relative terms, the maximum difference between thetwo methods is 0 .
1% at interest rate r = 6%. On average the relative difference is only 0 . r = 1%, and it is reduced by about 65%at r = 7%. Thus, a regular withdrawal above the penalty threshold is a very bad strategyregardless the interest rate level. In this instance, the penalty takes away the protected capitalon a regular basis. Note that, the penalty is applied in terms of the whole withdrawal amount,not just on the exceeded part, see penalty function (19). Thus a slight excess over the penaltythreshold can cause a large change in the price or in the fair fee, as observed in the second test.The above two tests show that a regular static withdrawal is only slightly beneficial at verylow interest rate and only when the withdrawal rate does not exceed the penalty threshold. Inthe next section, it will be demonstrated by numerical results that an optimal withdrawal isalways beneficial regardless the interest rate level and penalties. Consider a GMAB for a super account with the annual ratchet and assume that the policyholdercan exercise an optimal withdrawal strategy quarterly. For super account, any withdrawal willpenalize the protected capital amount (benefit base) if the wealth account is below the benefitbase according to (17) and (18).
Interest rate, r σ = 10% σ = 20%Fee (b.p) (cid:101) ε Fee (b.p) (cid:101) ε
1% 370.7 10% 1235 23 . .
8% 700.1 9 . .
2% 478.8 4 . .
9% 355.5 2 . .
1% 275.2 1 . .
48% 218.8 1 . .
36% 176.9 1 . Table 3:
Fair fee α in bp (1 bp=0.01%) as a function of interest rate r for the GMAB on asuper account when withdrawals are optimal. The contract maturity is T = 10 years. (cid:101) ε is thepercentage difference between the fair fee in the case optimal withdrawal and the fair fee in thestatic case (no withdrawal) from Table 1. Table 3 shows the fair fee for a super account as a function of the interest rate at σ = 10%and σ = 20%. The columns under (cid:101) ε show an extra percentage value in the fee due to optimalwithdrawal when compared to the static case of no withdrawal in Table 1. The results show,the extra fee is only about one or two percent for most cases, except at the low interest rate andhigh volatility. This extra fee due to optimal withdrawal is insignificant for the super accountin most cases, mainly due to the heavy penalties applied. As will be shown in the next section,if the penalty is less severe as in the case of pension account, the extra fee becomes much moresignificant. If the penalty is completely removed, then numerical experiments show that the29xtra fee will be several times, e.g. 300%, larger than the static case, demonstrating the fullvalue of the optimal strategy. Consider a GMAB for a pension account with the annual ratchet and assume that the poli-cyholder can exercise an optimal withdrawal strategy quarterly. For a pension account, anywithdrawal above a pre-defined withdrawal level G n will penalize the protected capital amount A ( t ) if the wealth account W ( t ) is below A ( t ) according to (17) and (19). Here, we set the an-nual withdrawal limit at 15% of the wealth account, i.e. G n = 0 . ×
15% = 3 .
75% is quarterlywithdrawal threshold.
Interest rate, r σ = 10% σ = 20%Fee (b.p) (cid:101) ε Fee (b.p) (cid:101) ε Fee-FD (b.p)1% 472.6 39% 1474 (1479) 46% 14662% 227.7 21% 836.1 (836.3) 30% 833.73% 135.4 15% 552.8 (553.6) 20% 551.74% 88.15 13% 399.1 (399.7) 14% 398.65% 60.24 11% 304.3 (304.7) 12% 304.06% 42.58 10% 239.6 (239.9) 10% 239.47% 30.63 9% 192.5 (192.8) 9% 192.4
Table 4:
Fair fee α in bp (1 bp=0.01%) as a function of interest rate r for the GMAB for a pensionaccount when withdrawals are optimal. The contract maturity is T = 10 years and quarterlywithdrawal limit is G n = 0 . × (cid:101) ε is the percentage difference between the fair fee in the caseoptimal withdrawal and the fair fee in the static case (no withdrawal) from Table 1. Table 4 shows the fair fee of GMAB for a pension account as a function of the interest rate r when σ = 10% and 20%.The columns under (cid:101) ε show an extra percentage value in the fee due to optimal withdrawalwhen compared to the static case of no withdrawal in Table 1. The results show, the extra feeranges from about 9% at the highest interest rate r = 7% to about 39% at the lowest interestrate r = 1%. This extra fee is much more significant than in the case of super account, seeTable 3, apparently due to reduced penalties. At σ = 20%, the extra fee ranges from about9% at the highest interest rate r = 7% to about 46% at the lowest interest rate r = 1%. Thisextra fee is higher than in the case of lower volatility σ = 10%, in both percentage and absoluteterms.Also, in Table 4, the numbers in the parentheses next to the continuous fair fee values α are the GHQC results for the discretely charged fair fee α d = − log(1 − (cid:101) αdt n ) /dt n , where atthe end of each quarter t n , the policyholder wealth account is charged a fee proportional tothe account value (cid:101) αdt n W ( t − n ), see the wealth process (6). Results show only little differencebetween the continuous fee α and the discrete fee α d . On average, the relative difference is0 . . r using results from Table 4, in comparison with the static case (no withdrawal)from Table 1.For some comparison, the market fees offered by MLC (2014) are 1.75% for a 10 year capitalprotection of a “balanced portfolio” and 0.95% for a “conservative growth portfolio”; and feesoffered by AMP (2014) are 1.3% for a 10 year capital protection of a “balanced strategy”portfolio and 0.95% for a “moderately defensive strategy” portfolio. Though the values ofvolatility are not known for these market portfolios, it seems that market prices are significantlylower than the fair fee , which is also observed in the literature before; e.g. see Milevsky andSalisbury (2006), Bauer et al. (2008) and Chen et al. (2008). F a i r f ee α ( b p ) Interest rate r
Pension account, σ =0.1 Optimal withdraw (GHQC) No wirhdraw (GHQC) No withdraw (MC)
100 200 300 400 500 600 700 800 900 0.02 0.03 0.04 0.05 0.06 0.07 F a i r f ee α ( b p ) Interest rate r
Pension account, σ =0.2 Optimal withdraw (GHQC) No withdraw (GHQC) No withdraw (MC)
Figure 2:
Fair fee α in bp as a function of the interest rate r at σ = 10% and σ = 20% for theGMAB on a pension account when withdrawals are optimal, in comparison with the static case whereno withdrawal is allowed. Values are taken from Table 4 and Table 1, respectively. In this paper we have reviewed pricing VA riders and presented a unified pricing approach via anoptimal stochastic control framework. We discussed different models and numerical proceduresapplicable in general to most of the VA riders with various contractual specifications. To pricethese VA riders under the geometric Brownian motion model for the risky asset, often assumedin practice, we have extended and generalized the direct integration method based on theGauss-Hermite quadrature, introduced earlier in Luo and Shevchenko (2015a) for some specificand simpler product specifications.As an example, we presented a numerical valuation of capital protection guarantees (GMABriders), with specifications matching closely the real market products offered in Australia bye.g. MLC (2014) and AMP (2014). Numerical valuation of this guarantee involves all the main31umerical difficulties encountered in pricing other VA riders, such as ratchets and optimalwithdrawals. Numerical results have been validated by MC and finite difference PDE methodsand can serve as a benchmark for practitioners and researchers developing numerical pricingof VA riders. As expected, we observed that the extra fee that has to be charged to counterthe optimal policyholder behavior is most significant at lower interest rate and higher volatilitylevels, and it is very sensitive to the penalty withdrawal threshold.As we have already discussed in Section 5.3, the fee based on the optimal policyholderwithdrawal is the worst case scenario for the issuer, i.e. if the guarantee is hedged then this feewill ensure no losses for the issuer (in other words full protection against policyholder strategyand market uncertainty). If the issuer hedges continuously but investors deviate from theoptimal strategy, then the issuer will receive a guaranteed profit. Any strategy different from theoptimal is sup-optimal and will lead to smaller fair fees. Of course the strategy optimal in thissense is not related to the policyholder circumstances. The policyholder may act optimally withrespect to his preferences and circumstances which may be different from the optimal strategymaximizing losses for the policy issuer. Life-cycle modelling can be undertaken to analyze andestimate sub-optimality of policyholder behavior. However, development of secondary marketsfor insurance products may expose the policy issuers to some significant risk if a fee for theguarantees is not charged to cover the worst case scenario.It is important to note that the guarantee could be written on more that one asset (severalmutual funds). In this case it is still common for practitioners to use a single-asset proximodel to calculate the price and hedging parameters. Obviously such approach has significantdrawbacks (e.g. the sum of geometric Brownian motions is not a geometric Brownian motion).PDE and direct integration methods are not practical in high-dimensions and thus one has torely on the MC methods to treat multi-asset case accurately. In the case of static withdrawal, itis not difficult to consider full multi-asset model and calculate the price using standard MC as inNg and Li (2013). However, in the case of optimal withdrawal strategies, numerical valuation inthe multi-asset case will require development of regression type MC solving backward recursionfor processes affected by the withdrawals. One could apply control randomization methodsextending Least-Squares MC developed in Kharroubi et al. (2014), but the accuracy robustnessof this method for pricing VA riders have not been studied yet.The specification details of VA riders typically vary across different companies and are dif-ficult to extract and compare from the very long product specification documents. Moreover,results for specific GMxB riders presented in academic literature often refer to different spec-ifications. As a result, cross-validation and benchmark research studies are rare. Given thatnumerical solutions used for pricing of VA riders are complex, it is important that these solu-tions are properly tested and validated. Moreover, new products are appearing in VA marketregularly with increasing complexity that raises an important question, as discussed in Carlinand Manso (2011) and mentioned in Moenig and Bauer (2015), whether new complex productsare designed to suite the policyholder needs better or introduced for the purpose of obfuscation .32
Acknowledgement
This research was supported by the CSIRO-Monash Superannuation Research Cluster, a col-laboration among CSIRO, Monash University, Griffith University, the University of WesternAustralia, the University of Warwick, and stakeholders of the retirement system in the interestof better outcomes for all. This research was also partially supported under the AustralianResearch Council’s Discovery Projects funding scheme (project number: DP160103489). Wewould like to thank Man Chung Fung for many constructive comments.
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