A note on the impact of management fees on the pricing of variable annuity guarantees
AA note on the impact of management fees on the pricing ofvariable annuity guarantees
Jin Sun a,b, ∗ , Pavel V. Shevchenko c , Man Chung Fung b a Faculty of Sciences, University of Technology Sydney, Australia b Data61, CSIRO, Australia c Department of Applied Finance and Actuarial Studies, Macquarie University, Australia
Abstract
Variable annuities, as a class of retirement income products, allow equity market expo-sure for a policyholder’s retirement fund with electable additional guarantees to limitthe downside risk of the market. Management fees and guarantee insurance fees arecharged respectively for the market exposure and for the protection from the down-side risk. We investigate the impact of management fees on the pricing of variableannuity guarantees under optimal withdrawal strategies. Two optimal strategies, frompolicyholder’s and from insurer’s perspectives, are respectively formulated and the cor-responding pricing problems are solved using dynamic programming. Our results showthat when management fees are present, the two strategies can deviate significantlyfrom each other, leading to a substantial difference of the guarantee insurance fees.This provides a possible explanation of lower guarantee insurance fees observed in themarket. Numerical experiments are conducted to illustrate our results.
Keywords: variable annuity guarantees, guaranteed minimum withdrawal benefits,management fees, stochastic optimal control, PDE, finite difference
1. Introduction
Variable annuities (VA) with guarantees of living and death benefits are offeredby wealth management and insurance companies worldwide to assist individuals inmanaging their pre-retirement and post-retirement financial plans. These products takeadvantages of market growth while provide a protection of the savings against marketdownturns. Similar guarantees are also available for life insurance policies (Bacinelloand Ortu [1]). The VA contract cash flows received by the policyholder are linked tothe investment portfolio choice and performance (e.g. the choice of mutual fund and itsstrategy) while traditional annuities provide a pre-defined income stream in exchange ∗ Corresponding author
Email addresses: [email protected] (Jin Sun),
[email protected] (Pavel V.Shevchenko),
[email protected] (Man Chung Fung)
Preprint submitted to Elsevier June 25, 2018 a r X i v : . [ q -f i n . P R ] M a y or a lump sum payment. Holders of VA policies are required to pay management feesregularly during the term of the contract for the wealth management services.A variety of VA guarantees, also known as VA riders, can be elected by policyholdersat the cost of additional insurance fees. Common examples of VA guarantees includeguaranteed minimum accumulation benefit (GMAB), guaranteed minimum withdrawalbenefit (GMWB), guaranteed minimum income benefit (GMIB) and guaranteed min-imum death benefit (GMDB), as well as a combination of them. These guarantees,generically denoted as GMxB, provide different types of protection against marketdownturns, shortfall of savings due to longevity risk or assurance of stability of incomestreams. Precise specifications of these products can vary across categories and issuers.See Bauer et al. [2], Ledlie et al. [18], Kalberer and Ravindran [16] for an overview ofthese products.The Global Financial Crisis during 2007-08 led to lasting adverse market conditionssuch as low interest rates and asset returns as well as high volatilities for VA providers.Under these conditions, the VA guarantees become more valuable, and the fulfillment ofthe corresponding required liabilities become more demanding. The post-crisis marketconditions have called for effective hedging of risks associated with the VA guarantees(Sun et al. [26]). As a consequence, the need for accurate estimation of hedging costsof VA guarantees has become increasingly important. Such estimations consist of risk-neutral pricing of future cash flows that must be paid by the insurer to the policyholderin order to fulfill the liabilities of the VA guarantees.There have been a number of contributions in the academic literature consideringthe pricing of VA guarantees. A range of numerical methods are considered, includingstandard and regression-based Monte Carlo (Huang and Kwok [14]), partial differentialequation (PDE) and direct integration methods (Milevsky and Salisbury [21], Dai et al.[8], Chen and Forsyth [4], Bauer et al. [2], Luo and Shevchenko [19, 20], Forsyth andVetzal [10], Shevchenko and Luo [25]). A comprehensive overview of numerical methodsfor the pricing of VA guarantees is provided in Shevchenko and Luo [24].In this article we focus on GMWB which provides a guaranteed withdrawal amountper year until the maturity of the contract regardless of the investment performance.The guaranteed withdrawal amount is determined such that the initial investment isreturned over the life of the contract. When pricing GMWB, one typically assumeeither a pre-determined (static) policyholder behavior in withdrawal and surrender, oran active (dynamic) strategy where the policyholder “optimally” decides the amountof withdrawal at each withdrawal date depending on the information available at thatdate.One of the most debated aspects in the pricing of GMWB with active withdrawalstrategies is the policyholders’ withdrawal behaviors (Cramer et al. [6], Chen andForsyth [4], Moenig and Bauer [22], Forsyth and Vetzal [10]). It is often customaryto refer to the withdrawal strategy that maximizes the expected liability, or the hedg-ing cost, of the VA guarantee as the “optimal” strategy. Even though such a strategyunderlies the worst case scenario for the VA provider with the highest hedging cost, itmay not coincide with the real-world behavior of the policyholder. Nevertheless, the2rice of the guarantee under this strategy provides an upper bound of hedging cost fromthe insurer’s perspective, which is often referred to as the “value” of the guarantee. Thereal-world behaviors of policyholders often deviate from this “optimal” strategy, as isnoted in Moenig and Bauer [22]. Different models have been proposed to account forthe real-world behaviors of policyholders, including the reduced-form exercise rules ofHo et al. [13], and the subjective risk neutral valuation approach taken by Moenig andBauer [22]. In particular, it is concluded by Moenig and Bauer [22] that a subjectiverisk-neutral valuation methodology that takes different tax structures into considerationis in line with the corresponding findings from empirical observations.Similar to the tax consideration in Moenig and Bauer [22], the management fee isa form of market friction that would affect policyholders’ rational behaviors. However,management fees are rarely considered in the VA pricing literature. When the man-agement fee is zero and deterministic withdrawal behavior is assumed, Hyndman andWenger [15] and Fung et al. [11] show that risk-neutral pricing of guaranteed withdrawalbenefits in both a policyholder’s and an insurer’s perspectives will result in the samefair insurance fee. Few studies that take management fees into account in the pricingof VA guarantees include B´elanger et al. [3], Chen et al. [5] and Kling et al. [17]. Inthese studies, fair insurance fees are considered from the insurer’s perspective with thegiven management fees. The important question of how the management fees as a formof market friction will impact withdrawal behaviors of the policyholder, and hence thehedging cost for the insurer, is yet to be examined in a dynamic withdrawal setting.The main goal of the paper is to address this question.The paper contributes to the literature in three aspects. First, we consider twopricing approaches based on the policyholder’s and the insurer’s perspective. In theliterature it is most often the case that only an insurer’s perspective is considered,which might result in mis-characterisation of the policyholder’s withdrawal strategies.Second, we characterize the impact of management fees on the pricing of GWMB,and demonstrate that the two afore-mentioned pricing perspectives lead to differentfair insurance fees due to the presence of management fees. In particular, the fairinsurance fees from the policyholder’s perspective is lower than those from the insurer’sperspective. This provides a possible justification of lower insurance fees observed inthe market. Third, the sensitivity of the fair insurance fees to management fees underdifferent market conditions and contract parameters are investigated and quantifiedthrough numerical examples.The paper is organized as follows. In Section 2 we present the contract details ofthe GMWB guarantee together with its pricing formulation under a stochastic optimalcontrol framework. Section 3 derives the policyholder’s value function under the risk-neutral pricing approach, followed by the insurer’s net liability function in Section 4. InSection 5 we compare the two withdrawal strategies that maximize the policyholder’svalue and the insurer’s liability, respectively, and discuss the role of the managementfees in their relations. Section 6 demonstrates our approaches via numerical examples.Section 7 concludes with remarks and discussion.3 . Formulation of the GMWB pricing problem We begin with the setup of the framework for the pricing of GMWB and describethe features of this type of guarantees. The problem is formulated under a generalsetting so that the resulting pricing formulation can be applied to different GMWBcontract specifications. Besides the general setting, we also consider a very specificsimple GMWB contract, which will be subsequently used for illustration purposes innumerical experiments presented in Section 6.The VA policyholder’s retirement fund is usually invested in a managed wealthaccount that is exposed to financial market risks. A management fee is usually chargedfor this investment service. In addition, if GMWB is elected, extra insurance feeswill be charged for the protection offered by the guarantee provider (insurer). Weassume the wealth account guaranteed by the GMWB is subject to continuously chargedproportional management fees, paid to an independent wealth manager other than theinsurer. This assumption implies that the management fees cannot be used to fundthe hedging portfolio for the GMWB guarantee. The sole purpose of managementfees is to compensate for the fund management services provided, and should not beconfused with the hedging cost of the guarantee. The cost of hedging, on the the hand,is paid by proportional insurance fees continuously charged to the wealth account. Thefair insurance fee rate, or the fair fees in short, refers to the minimal insurance fee raterequired to fund the hedging portfolio, or the replicating portfolio, so that the guaranteeprovider can eliminate the market risk associated with the selling of the guarantees.We consider the situation where a policyholder purchases the GMWB rider in orderto protect his wealth account that tracks an equity index S ( t ) at time t ∈ [0 , T ], where 0and T correspond to the inception and expiry dates. The equity index account is mod-elled under the risk-neutral probability measure Q following the stochastic differentialequation (SDE) dS ( t ) = S ( t ) ( r ( t ) dt + σ ( t ) dB ( t )) , t ∈ [0 , T ] , (1)where r ( t ) is the risk-free short interest rate, σ ( t ) is the volatility of the index, whichare made time-dependent and can be stochastic, and B ( t ) is a standard Q -Brownianmotion modelling the uncertainty of the index. Here, we follow standard practices inthe literature of VA guarantee pricing by modelling under the risk-neutral probabilitymeasure Q , which allows the pricing of stochastic cash flows as taking the risk-neutralexpectation of the discounted cash flows. The risk-neutral probability measure Q existsif the underlying financial market satisfies certain “no-arbitrage” conditions. For detailson risk-neutral pricing, see, e.g., Delbaen and Schachermayer [9] for an account undervery general settings.The wealth account W ( t ) , t ∈ [0 , T ] over the lifetime of the GMWB contract isinvested into the index S , subject to management fees charged by a wealth manager atthe rate α m ( t ). An additional charge of insurance fees at rate α ins ( t ) for the GMWBrider is collected by the insurer to pay for the hedging cost of the guarantee. Bothfees are deterministic, time-dependent and continuously charged. Discrete fees may be4odelled similarly without any difficulty. The wealth account in turn evolves as dW ( t ) = W ( t ) (( r ( t ) − α tot ( t )) dt + σ ( t ) dB ( t )) , (2)for any t ∈ [0 , T ] at which no withdrawal of wealth is made. Here, α tot ( t ) = α ins ( t ) + α m ( t ) is the total fee rate. The GMWB contract allows the policyholder to withdrawfrom a guarantee account A ( t ) , t ∈ [0 , T ] on a sequence of pre-determined contract eventdates, 0 = t < t < · · · < t N = T . The initial guarantee A (0) usually matches theinitial wealth W (0). The guarantee account stays constant unless a withdrawal is madeon one of the event dates, which changes the guarantee account balance. We assumethat the GMWB contract will be taken over by the beneficiary if the policyholderdies before the maturity T , so that no early termination of the contract is possible,nor is there any death benefit included in the contract. Additional features such asearly surrender and death benefits can be included straightforwardly but will not beconsidered in this article, in order to better illustrate the impact of management feeswithout unnecessary complexities.To simplify notations, we denote by X ( t ) the vector of state variables at t , given by X ( t ) = ( r ( t ) , σ ( t ) , S ( t ) , W ( t ) , A ( t )) , t ∈ [0 , T ] . (3)We denote by E Q t [ · ] the risk-neutral expectation conditional on the state variables at t , i.e., E Q t [ · ] := E Q [ ·| X ( t )]. Here, we assume that all state variables follow Markovprocesses under the risk-neutral probability measure Q , so that X ( t ) contains all theinformation available at t . For completeness, we include the index value S ( t ) in thevector of state variables which under the current model may seem redundant, due tothe scale-invariance of the geometric Brownian motion type model (1). In general,however, S ( t ) may determine the future dynamics of S in a nonlinear fashion, as is thecase under, e.g., the minimal market model described in Platen and Heath [23].On event dates t n , n = 1 , . . . , N −
1, the policyholder may choose to withdraw anominal amount γ n ≤ A ( t n ). The real cash flow received by the policyholder is denotedby C n ( γ n , X ( t − n )), where t − refers to the time “just before” t . As a specific example, C n ( γ n , X ( t − n )) may be given by C n ( γ n , X ( t − n )) = γ n − β max( γ n − G n , , (4)where the contractual withdrawal G n is a pre-determined withdrawal amount specifiedin the GMWB contract, and β is the penalty rate applied to the part of the withdrawalexceeding the contractual withdrawal G n . The policyholder may decide the withdrawalamount γ n based on all current state variables, i.e., γ n = Γ( t n , X ( t − n )) , (5)where the mapping Γ( · , · ) is defined as a withdrawal strategy . Given the assumed Marko-vian structure of the state variables X , the withdrawal strategy (5) uses all currentinformation. 5pon withdrawal, the guarantee account is changed by the amount D n ( γ n , X ( t − n )),that is, A ( t n ) = A ( t − n ) − D n ( γ n , X ( t − n )) . (6)For example, if D n ( γ n , X ( t − n )) = γ n , then A ( t n ) = A ( t − n ) − γ n . (7)The guarantee account stays nonnegative, that is, γ n can only be taken such that D n ( γ n , X ( t − n )) ≤ A ( t − n ). The wealth account is reduced by the amount γ n upon with-drawal and remains nonnegative. That is, W ( t n ) = max( W ( t − n ) − γ n , , (8)where W ( t − n ) is the wealth account balance just before the withdrawal. It is assumedthat γ = 0, i.e., no withdrawals at the start of the contract.The function of policyholder’s remaining value at time t is denoted by V ( t, X ( t )) , t ∈ [0 , T ], which corresponds to the risk-neutral value of all future cashflows to the policy-holder at time t . At maturity t N = T , the policyholder obtains an liquidation cash flow V ( T − , X ( T − )). Both account balances becomes zero at T , that is, W ( T ) = A ( T ) = 0.For example, the liquidation cash flow may be given by V ( T − , X ( T − )) = A ( T − ) − β max( A ( T − ) − G N ,
0) + max( W ( T − ) − A ( T − ) , , (9)which assumes a nominal withdrawal γ N = A ( T − ) of remaining guarantee account bal-ance subject to a penalty implied by (4) with contract amount G N , plus the liquidationof the wealth account after this withdrawal. Different contracts may define differentliquidation cash flows. We emphasize here that the analysis presented in this articledoes not require this particular form (9). Note also that the liquidation cash flow eitherdoes not depend on γ N or the dependence is only formal, in that γ N is always chosento maximize the liquidation cash flow.
3. The Policyholder’s Value Function
Having introduced the modeling framework and contract specifications, we can nowcalculate the policyholder’s value function V ( t, X ( t )) as the risk-neutral expected valueof policyholder’s future cash flows at time t ∈ [0 , T ]. Valuing the future cash flows underthe risk-neutral pricing approach assumes the policyholder’s cash flow may be replicatedby self-financing portfolios of the same initial value. Although the policyholder maynot have the resources to carry out any hedging strategies in person, we assume themarket is liquid enough to trade such strategies without frictions, and the policyholderhas access to the market through independent agents. Thus the risk-neutral valuationof the policyholder’s future cash flows can be regarded as the value of the remainingterm of the VA contract from the poliyholer’s perspective.6ollowing Section 2, for a given withdrawal strategy Γ, the jump condition at thewithdrawal time t n for the policyholder’s value function V ( t, X ( t )) is given by V ( t − n , X ( t − n )) = C n ( γ n , X ( t − n )) + V ( t n , X ( t n )) , (10)i.e., the remaining policy value just before the withdrawal is the sum of the cash flowfrom the withdrawal and the remaining policy value immediately after the withdrawal.Here, the withdrawal γ n is given as (5) by applying the strategy of choice.The policy value at t ∈ ( t n − , t n ) is given by the discounted expected future policyvalue under the risk-neutral probability measure as V ( t, X ( t )) = E Q t (cid:104) e − (cid:82) tnt r ( s ) ds V ( t − n , X ( t − n )) (cid:105) , (11)where e − (cid:82) tnt r ( s ) ds is the discount factor. The initial policy value, given by V (0 , X (0)),can be calculated backward in time starting from the terminal condtion V ( T − , X ( T − )),using (10) and (11), as described in Algorithm 1.As an illustrative example, we assume r ( t ) ≡ r , σ ( t ) ≡ σ and α tot ( t ) ≡ α tot asconstants, and thus V ( t, X ( t )) = V ( t, W ( t )) for t ∈ ( t n − , t n ). We now derive thepartial differential equation (PDE) satisfied by the value function V through a hedgingargument. We consider a delta hedging portfolio that, at time t ∈ ( t n − , t n ), takes along position of the value function V and a short position of W ( t ) ∂ W V ( t,W ( t )) S ( t ) shares ofthe index S . Here ∂ W V ( t, W ( t )) ≡ ∂V ( t,W ) ∂W | W = W ( t ) is the partial derivative of V ( t, W )with respect to the second argument, evaluated at W ( t ). Denoting the total value ofthis portfolio at t ∈ ( t n − , t n ) as Π V ( t ), the value of the delta hedging portfolio is givenby Π V ( t ) = V ( t, W ( t )) − W ( t ) ∂ W V ( t, W ( t )) . (12)By Ito’s formula and (1), the SDE for Π V can be obtained as d Π V ( t ) = (cid:0) ∂ t V ( t, W ( t )) − α tot W ( t ) ∂ W V ( t, W ( t )) + σ W ( t ) ∂ W W V ( t, W ( t )) (cid:1) dt, (13)for t ∈ ( t n − , t n ). Since the hedging portfolio Π V is locally riskless, it must grow at therisk-free rate r , that is d Π V ( t ) = r Π V ( t ) dt . This along with (12) implies that the PDEsatisfied by the value function V ( t, W ) is given by ∂ t V − rV + ( r − α tot ) W ∂ W V + 12 σ W ∂ W W V = 0 , (14)for t ∈ ( t n − , t n ) and n = 1 , . . . , N . The boundary conditions at t n are specified by(9) and (10). The valuation formula (11) or the PDE (14) may be solved recursivelyby following Algorithm 1 to compute the initial policy value V (0 , X (0)). It should benoted that (11) is general, and does not depend on the simplifying assumptions madein the PDE derivation. 7 lgorithm 1 Recursive computation of V (0 , X (0)) choose a withdrawal strategy Γ initialize V ( T − , X ( T − )), e.g., as (9) set n = N while n > do compute V ( t n − , X ( t n − )) by solving (11) or (14) with terminal condition V ( t − n , X ( t − n )) compute the withdrawal amount γ n − by applying the strategy Γ as (5) compute V ( t − n − , X ( t − n − )) by applying jump condition (10) n = n − end while4. The Insurer’s Liability Function and the Fair Fee Rate The GMWB contract may be considered from the insurer’s perspective by examiningthe insurer’s liabilities, given by the risk-neutral value of the cash flows that must bepaid by the insurer in order to fulfill the GMWB contract.On any withdrawal date t n , the actual cash flow received by the policyholder isgiven by (4). This cash flow is first paid out of the policyholder’s real withdrawalfrom the wealth account, which is equal to min( W ( t − n ) , γ n ), the smaller of the nominalwithdrawal and the available wealth. If the wealth account has an insufficient balance,the rest must be paid by the insurer. If the real withdrawal exceeds the cash flowentitled to the policyholder, the insurer keeps the surplus. The payment made by theinsurer at t n is thus given by c n ( γ n , X ( t − n )) = C n ( γ n , X ( t − n )) − min( W ( t − n ) , γ n ) . (15)At any time t , we denote the net liability function as L ( t, X ( t )), which refers to therisk-neutral value of all future payments made to the policyholder by the insurer, lessthe present value of all future insurance fee incomes.The insurance fees, charged at the rate α ins ( t ) , t ∈ [0 , T ], is called fair if the totalfees exactly compensate for the insurer’s total liability, such that the net liability is zeroat time t = 0. That is, L (0 , X (0)) = 0 . (16)Note that L (0 , X (0)) depends on α ins ( t ), which was made implicit for notational sim-plicity. If α ins ( t ) ≡ α ins is a constant, its value can be found by solving (16). The fairinsurance fees represent the hedging cost for the insurer to deliver the GMWB guar-antee to the policyholder, which is often regarded as the value of the GMWB rider,at least from the insurer’s perspective. We emphasize here that this value may not beequal to the added value of the GMWB rider to the policyholder’s wealth account, aswe will show in Section 5.To compute L (0 , X (0)) we first note that at maturity T , the terminal condition on L is given by L ( T − , X ( T − )) = V ( T − , X ( T − )) − W ( T − ) , (17)8.e., the insurer must pay for any amount of the policyholder’s final value not coveredby the available wealth. Depending on the GMWB contract details, this amount canbe negative, in which case the insurer gets paid. This happens, for example, if theliquidation cash flow is given by (9), and there are more penalties applied to the fi-nal (compulsory) withdrawal due to forced liquidation of a high final balance of theguarantee account. The jump condition on L at t n is given by L ( t − n , X ( t − n )) = c n ( γ n , X ( t − n )) + L ( t n , X ( t n )) , (18)i.e., upon withdrawal, the net liability is reduced by the amount paid out.At t ∈ ( t n − , t n ), the net liability function is given by the risk-neutral value of theremaining liabilities at t n before any benefit is paid, less any insurance fee incomes overthe period ( t, t n ), discounted at the risk-free rate. Specifically, we have L ( t, X ( t )) = E Q t (cid:104) e − (cid:82) tnt r ( s ) ds L ( t − n , W ( t − n ) , A ( t − n )) (cid:105) − E Q t (cid:104) (cid:82) t n t e − (cid:82) st r ( u ) du α ins ( s ) W ( s ) ds (cid:105) . (19)Note that the net liability, viewed from time t forward, is reduced by expecting toreceive insurance fees over ( t, t n ). Since this reduction decreases with time, the netliability increases with time.To give an example, we again assume constant r ( t ) ≡ r , σ ( t ) ≡ σ , α ins ( t ) ≡ α ins , α tot ( t ) ≡ α tot . Under these simplifying assumptions we have L ( t, X ( t )) = L ( t, W ( t )),for t ∈ ( t n − , t n ). To derive the PDE satisfied by L ( t, W ), consider a delta hedgingportfolio that, at time t ∈ ( t n − , t n ), consists of a long position in the net liabilityfunction L and a short position of W ( t ) ∂ W L ( t,W ( t )) S ( t ) shares of the index S . The value ofthe delta hedging portfolio, denoted as Π L ( t ), is given byΠ L ( t ) = L ( t, W ( t )) − W ( t ) ∂ W L ( t, W ( t )) . (20)By Ito’s formula and (1), we obtain the SDE for Π L as d Π L ( t ) = (cid:0) ∂ t L ( t, W ( t )) − α tot W ( t ) ∂ W L ( t, W ( t )) + σ W ( t ) ∂ W W L ( t, W ( t )) (cid:1) dt, (21)where t ∈ ( t n − , t n ). Since the hedging portfolio Π L is locally riskless and must grow atthe risk-free rate r , as well as increase with the insurance fee income at rate α ins W ( t )(see remarks after (19)), we must also have d Π L ( t ) = (cid:0) r Π L ( t ) + α ins W ( t ) (cid:1) dt . Thisalong with (20) implies that the PDE satisfied by the value function L ( t, W ) is givenby ∂ t L − α ins W − rL + ( r − α tot ) W ∂ W L + 12 σ W ∂ W W L = 0 , (22)for t ∈ ( t n − , t n ). The initial net liability can thus be computed by recursively solving(19) or (22) from terminal and jump conditions (17) and (18), as described in Algorithm2. 9 lgorithm 2 Recursive computation of L (0 , X (0)) choose a withdrawal strategy Γ initialize L ( T − , X ( T − )) as (17) set n = N while n > do compute L ( t n − , X ( t n − )) by solving (19) or (22) with terminal condition L ( t − n , X ( t − n )) compute the withdrawal amount γ n − by applying the strategy Γ as (5) compute L ( t − n − , X ( t − n − )) by applying jump condition (18) n = n − end while5. Policy Value Maximization vs Liability Maximization In the previous sections, the withdrawal strategy γ n , n = 1 , . . . , n − We first formulate the policyholder’s value maximization problem, i.e., maximiz-ing the initial policy value V (0 , X (0)) by optimally choosing the sequence γ n for n =1 , . . . , N −
1. Following the principle of dynamic programming, this is accomplished bychoosing the withdrawal γ n as γ n = Γ V ( t n , X ( t − n )) = arg max γ ∈A (cid:8) C n ( γ, X ( t − n )) + V (cid:0) t n , X ( t n | X ( t − n ) , γ ) (cid:1)(cid:9) (23)in the admissible set A = { γ : γ ≥ , A ( t n | X ( t − n ) , γ ) ≥ } . Here, we used X ( t n | X ( t − n ) , γ )and A ( t n | X ( t − n ) , γ ) to denote the state variables X ( t n ) and A ( t n ) at t n after withdrawal γ is made, given the value of the state variables X ( t − n ) before the withdrawal. At anywithdrawal time t n the policyholder chooses the withdrawal γ ∈ A to maximize thesum of the cash flow he receives and the present value of the remaining term of thepolicy. The strategy Γ V given by (23) is called the value maximization strategy .On the other hand, the optimization problem from the insurer’s perspective con-siders the most unfavourable situation for the insurer. That is, by making suitablechoices of γ n ’s, the policyholder attempts to maximize the net initial liability function L (0 , X (0)). Even though a policyholder has little reason to pursue such a strategy, thefair fee rate under this strategy is guaranteed to cover the hedging cost of the GMWBrider regardless of the withdrawal strategy of the policyholder (assuming the insurer10an perfectly hedge the market risks). The withdrawal γ n for this strategy is given by γ n = Γ L ( t n , X ( t − n )) = arg max γ ∈A (cid:8) c n ( γ, X ( t − n )) + L (cid:0) t n , X ( t n | X ( t − n ) , γ ) (cid:1)(cid:9) , (24)i.e., the sum of the cash flow paid by the insurer and the net liability of the remainingterm of the contract is maximized. The strategy Γ L given by (24) is called the liabilitymaximization strategy . To clarify the different implications on the fair insurance fees between the valueand the liability maximization strategies, we now establish the relationship betweenthe policy value V and the net liability L by defining the process M ( t, X ( t )) := L ( t, X ( t )) + W ( t ) − V ( t, X ( t )) , (25)for t ∈ [0 , T ]. From (17) we obtain M ( T − , X ( T − )) = 0 , (26)as the terminal condition for M . From (11) and (19) we find the recursive relation for M as, M ( t, X ( t )) = E Q t (cid:104) e − (cid:82) tnt r ( s ) ds M ( t − n , X ( t − n )) (cid:105) + W ( t ) − E Q t [ W ( t − n )] − E Q t (cid:104)(cid:82) t n t e − (cid:82) st r ( u ) du α ins ( s ) W ( s ) ds (cid:105) . (27)Note that the second and third lines in (27) can be identified with the time- t risk-neutralvalue of management fees over ( t, t n ). To see this, we first note that the difference of thefirst two terms is the time- t risk-neutral value of the total fees charged on the wealthaccount over ( t, t n ), and the expectation in the third term is the time- t risk-neutralvalue of the insurance fees over the same period.In lights of (26) and (27), the quantity M ( t, X ( t )) defined by (25) is precisely thetime- t risk-neutral value of future management fees. From (25), the policy value maybe written as V ( t, X ( t )) = W ( t ) + L ( t, X ( t )) − M ( t, X ( t )) , (28)i.e., the sum of the wealth and the value of the GMWB rider, less the value of futuremanagement fees. At t = 0, this gives V (0 , X (0)) + M (0 , X (0)) = W (0) + L (0 , X (0)) . (29)Therefore, maximizing L (0 , X (0)) in general is not the same as maximizing V (0 , X (0)),since the total management fee M (0 , X (0)) depends on the withdrawal strategy. Thefair fee condition (16) becomes V (0 , X (0)) + M (0 , X (0)) = W (0) , (30)11s in contrast to the V (0) = W (0) condition often seen in the literature, when no man-agement fees are charged, in which case the two strategies Γ V and Γ L coincide. Whenthe management fee rate is positive, the liability maximization strategy Γ L by definitionleads to the maximal initial net liability, thus by (16) the maximal fair insurance feerate. We now consider the two strategies in an idealized world where the policy valueand the net liability processes can be perfectly replicated using self-financing tradingstrategies. As is mentioned above, assuming that the fair insurance fee follows from theliability maximization strategy, the insurer is guaranteed a nonnegative profit, regard-less of the actual withdrawal strategies of the policyholder. If the policyholder behavesdifferently from this strategy, in particular, if he follows the value maximization strat-egy, the insurer generally makes a positive profit.On the other hand, consider the situation where the policyholder purchases his policyfrom a middle agent, who in turn purchases the same policy for the same price from aninsurer in the name of the policyholder, and handles withdrawals at his own choice, butfulfills any withdrawal requests from the policyholder according to the GMWB contract.In other words, the middle agent provides the GMWB guarantee to the policyholder,and follows the value maximization strategy when making his own withdrawals fromthe insurer. Then regardless of the policyholder’s withdrawal behavior or the fees, themiddle agent always makes a nonnegative profit. If the policyholder behaves differentlyfrom the policy value maximization strategy, she receives a value less than the maximalpolicy value received by the agent, who therefore makes a positive profit out of thesuboptimal behavior of the policyholder. Given that the middle agent will carry outwithdrawals that would maximize the value rather than the cost to the insurer, theinsurer in turn can afford to charge a less expensive fee than those implied by theliability maximization strategy, leading to more value for the middle agent and thepolicyholder. The seemingly win-win situation come at the loss of the wealth manager,who now expect to receive less management fees. In this case the middle agent in effectreduces the market frictions represented by the management fees by maximizing hisown value, and at the same time help improving the policyholder’s value.
6. Numerical Examples
To demonstrate the impact of management fees on the fair fees of GMWB contracts,we carry out in this section several numerical experiments. We investigate how thepresence of management fees will lead to different fair fees for the two withdrawalstrategies studied in previous sections under different market conditions and contractparameters.
For illustration purposes, we assume a simple GMWB contract as specified by (4),(7), (9) as well as constant r , σ , α m and α ins so that the PDEs (14) and (22) are valid.12e consider different contractual scenarios and calculate the fair fees implied by (16)under the withdrawal strategies given in Section 5.It is assumed that the wealth and the guarantee accounts start at W (0) = A (0) = 1.The maturities of the contracts range from 5 to 20 years, with annual contractualwithdrawals evenly distributed over the lifetime of the contracts. The first withdrawaloccurs at the end of the first year and the last at the maturity. The management feerate ranges from 0% up to 2%.We consider several investment environments with the risk-free rate r at levels 1%and 5%, and the volatility of the index σ at 10% and 30%, to represent different marketconditions such as low/high growth and low/high volatility scenarios. In addition, thepenalty rate β may take values at 10% or 20%.We compute the initial policy value V (0 , X (0)) as well as the initial net liability L (0 , X (0)) at time 0 numerically by following Algorithms 1 and 2 simultaneously. Thewithdrawal strategies Γ L and Γ V are considered separately. The PDEs (14) and (22) aresolved using Crank-Nicholson finite difference method (Crank and Nicolson [7], Hirsa[12]) with appropriate terminal and jump conditions for both functions under bothstrategies. This leads to the initial values and liabilities V (0 , X (0); Γ L ), L (0 , X (0); Γ L ), V (0 , X (0); Γ V ) and L (0 , X (0); Γ V ) under both strategies. Here, we made the depen-dence of these functions on the strategies explicit.The fair fee rates under both strategies were obtained by solving (16) using a stan-dard root-finding numerical scheme. Note that in the existing literature, Forsyth andVetzal [10] considered only the strategy Γ L and computed the “total value”, equivalentto V (0 , X (0); Γ L ) + M (0 , X (0); Γ L ), and obtained the fair fee rate by requiring this totalvalue to be equal to W (0), the initial investment, which is the same as requiring theinitial liability to be equal to 0 as indicated by (16). The fair fees and corresponding total policy values are shown in Figures 1 and 2 fortwo market conditions: a low return market with high volatility ( r = 1% , σ = 30%) anda high return market with low volatility ( r = 5% , σ = 10%), respectively. Fair fee ratesobtained for all market conditions and contract parameters can be found in Tables 1and 2. The corresponding policy values are listed in Tables 3 and 4.The first observation to note from these numerical results is that the fair fee rateimplied by the liability maximization strategy is always higher, and the correspondingpolicyholder’s total value always lower, than those implied by the value maximizationstrategy, unless management fees are absent, in which case these quantities are equal.These are to be expected from the definitions of the two strategies.We also observe from these numerical results that, under the market condition oflow return with high volatility, a much higher insurance fee rate is required than underthe market condition of high return with low volatility, for the obvious reason thatunder adverse market conditions, the guarantee is more valuable. Moreover, a higherpenalty rate results in a lower insurance fee since a higher penalty rate discourages13he policyholder from making more desirable withdrawals that exceed the contractedvalues.Furthermore, the results show that under most market conditions or contract spec-ifications, the fair insurance fee rate obtained is highly sensitive to the management feerate regardless of the withdrawal strategies, as seen from Figures 1 and 2. In partic-ular, the fair fee rate implied by the liability maximization strategy always increaseswith the management fee rate, since the management fees cause the wealth account todecrease, leading to higher liability for the insurer to fulfill. On the other hand, the fairfee rate implied by the value maximization strategy first increases then decreases withthe management fee rate, since at high management fee rates, a rational policyholdertends to withdraw more and early to avoid the management fees, which in turn reducesthe liability and generates more penalty incomes for the insurer.A major insight from the numerical results is that with increasing management fees,the value maximization withdrawals of a rational policyholder deviates more from theliability maximization withdrawals assumed by the insurer. In particular, it is seen byexamining Figures 1 and 2 that the fair fee rates implied by the two strategies differmore significantly under the following conditions: • longer maturity T , • lower penalty rate β , • higher index return r , and • higher management fee rate α m .Moreover, careful examination of results listed in Tables 1 and 2 reveals that the indexvolatility σ does not seem to contribute significantly to this discrepancy. These obser-vations are intuitively reasonable: The contributors listed above all imply that the totalmanagement fees M (0) will be higher. There are more incentives to withdraw early toachieve more values in the form of reduced management fees. The corresponding differ-ences between the policyholder’s values follow similar patterns. Of particular interestis that in some cases, as shown in Figure 2, the fair fee rate implied by maximizingpolicyholder’s value can become negative. This implies that the policyholder wouldwant to withdraw more and early due to high management fees to such an extent, thatthe penalties incurred exceed the total value of the GMWB rider. On the other hand,the fair fee rate implied by maximizing the liability is always positive.
7. Conclusions
Determining accurate hedging costs of VA guarantees is a significant issue for VAproviders. While the presence of management fees is typically ignored in the VA liter-ature, it was demonstrated in this article that the impact of management fees on thepricing of GMWB contract is significant. As a form of market friction similar to tax14onsideration, management fees can affect policyholders’ withdrawal behaviors, caus-ing large deviations from the “optimal” (liability maximization) withdrawal behaviorsoften assumed in the literature.Two different policyholder’s withdrawal strategies were considered: liability maxi-mization and value maximization when management fees are present. We demonstratedthat these two withdrawal strategies imply different fair insurance fee rates, where max-imizing policy value implies lower fair fees than those implied by maximizing liability,or equivalently, maximizing the “total value” of the contract, which represents themaximal hedging costs from the insurer’s perspective.More importantly, we quantitatively demonstrated that the difference between theinitial investment and the total value of the policyholder is precisely the total valueof the management fees, which is also the cause of the discrepancy between the twowithdrawal strategies considered in this article. The two strategies coincide when man-agement fees are absent. We identified a number of factors that contribute to thisdiscrepancy through a series of illustrating numerical experiments. Our findings iden-tify the management fees as a potential cause of discrepancy between the fair fee ratesimplied by the liability maximization strategy, often assumed from the insurer’s per-spective for VA pricing, and the prevailing market rates for VA contracts with GMWBor similar riders.
Acknowledgement
This research was supported by the CSIRO-Monash Superannuation Research Clus-ter, a collaboration among CSIRO, Monash University, Griffith University, the Univer-sity of Western Australia, the University of Warwick, and stakeholders of the retirementsystem in the interest of better outcomes for all. This research was also partially sup-ported under the Australian Research Council’s Discovery Projects funding scheme(project number: DP160103489). We would like to thank Eckhard Platen and XiaolinLuo for useful discussions and comments. 15 gmt fee rate (%) F a i r f ee r a t e ( % ) - = 10%, T = 5 Yrs P o li c y v a l u e fair fee ( L max )fair fee ( V max) V (0) ( L max) V (0) ( V max) Mgmt fee rate (%) F a i r f ee r a t e ( % ) - = 20%, T = 5 Yrs P o li c y v a l u e Mgmt fee rate (%) F a i r f ee r a t e ( % ) - = 10%, T = 10 Yrs P o li c y v a l u e Mgmt fee rate (%) F a i r f ee r a t e ( % ) - = 20%, T = 10 Yrs P o li c y v a l u e Mgmt fee rate (%) F a i r f ee r a t e ( % ) - = 10%, T = 20 Yrs P o li c y v a l u e Mgmt fee rate (%) F a i r f ee r a t e ( % ) - = 20%, T = 20 Yrs P o li c y v a l u e Figure 1: Fair insurance fee rates and policy values as a function of management fee rates α m for risk-free rate r = 1% and volatility σ = 30%, for penalty rates β = 10% ,
20% and maturities T = 5 , , gmt fee rate (%) F a i r f ee r a t e ( % ) - = 10%, T = 5 Yrs P o li c y v a l u e fair fee (L max) fair fee (V max)V (0) (L max)V (0) (V max) Mgmt fee rate (%) F a i r f ee r a t e ( % ) - = 20%, T = 5 Yrs P o li c y v a l u e Mgmt fee rate (%) F a i r f ee r a t e ( % ) -0.500.5 - = 10%, T = 10 Yrs P o li c y v a l u e Mgmt fee rate (%) F a i r f ee r a t e ( % ) - = 20%, T = 10 Yrs P o li c y v a l u e Mgmt fee rate (%) F a i r f ee r a t e ( % ) -101 - = 10%, T = 20 Yrs P o li c y v a l u e Mgmt fee rate (%) F a i r f ee r a t e ( % ) -0.8-0.6-0.4-0.200.2 - = 20%, T = 20 Yrs P o li c y v a l u e Figure 2: Fair insurance fee rates and policy values as a function of management fee rates α m for risk-free rate r = 5% and volatility σ = 10%, for penalty rates β = 10% ,
20% and maturities T = 5 , , able 1: Fair fee rate α ins (%) based on the liability maximization strategy Γ L . Parameters α m r (%) σ (%) β (%) T
0% 0 .
2% 0 .
4% 0 .
6% 0 .
8% 1% 1 .
2% 1 .
4% 1 .
6% 1 .
8% 2%1 10 10 5 3.08 3.48 3.96 4.59 5.41 6.65 8.66 12.18 17.67 23.53 29.9210 1.66 1.92 2.25 2.66 3.21 3.97 5.08 6.77 9.26 12.17 15.3120 0.82 0.97 1.17 1.43 1.77 2.22 2.83 3.68 4.81 6.19 7.7120 5 3.08 3.47 3.95 4.55 5.34 6.48 8.39 12.01 17.66 23.55 29.9210 1.66 1.91 2.23 2.62 3.13 3.83 4.88 6.60 9.20 12.17 15.3120 0.81 0.96 1.16 1.40 1.71 2.13 2.72 3.57 4.74 6.17 7.7130 10 5 15.05 15.92 16.92 18.02 19.27 20.70 22.37 24.43 26.88 29.91 33.6910 8.38 8.93 9.55 10.22 10.98 11.85 12.84 13.98 15.34 16.93 18.8120 4.32 4.64 5.00 5.41 5.87 6.39 6.98 7.66 8.45 9.35 10.3720 5 13.99 14.82 15.73 16.80 18.01 19.40 21.06 23.11 25.64 28.86 32.9510 7.85 8.38 8.97 9.63 10.36 11.22 12.22 13.38 14.74 16.43 18.4120 4.00 4.33 4.69 5.10 5.55 6.07 6.66 7.34 8.11 9.01 10.025 10 10 5 0.43 0.47 0.50 0.55 0.59 0.64 0.69 0.75 0.81 0.88 0.9610 0.18 0.20 0.21 0.23 0.25 0.28 0.30 0.33 0.36 0.39 0.4320 0.08 0.09 0.10 0.11 0.12 0.13 0.15 0.16 0.18 0.20 0.2220 5 0.40 0.44 0.48 0.52 0.57 0.62 0.68 0.74 0.81 0.88 0.9610 0.11 0.12 0.14 0.16 0.18 0.20 0.22 0.25 0.28 0.31 0.3520 0.03 0.04 0.04 0.05 0.05 0.06 0.07 0.08 0.09 0.10 0.1230 10 5 5.33 5.48 5.65 5.81 5.99 6.17 6.35 6.55 6.75 6.96 7.1710 2.91 3.02 3.12 3.23 3.35 3.47 3.60 3.73 3.86 4.01 4.1620 1.58 1.65 1.74 1.82 1.91 2.00 2.10 2.21 2.32 2.43 2.5520 5 4.97 5.13 5.28 5.44 5.61 5.79 5.97 6.15 6.35 6.55 6.7610 2.27 2.35 2.43 2.52 2.61 2.71 2.81 2.91 3.02 3.13 3.2520 1.08 1.13 1.19 1.24 1.31 1.37 1.43 1.50 1.58 1.65 1.7318 able 2: Fair fee rate α ins (%) based on the policy value maximization strategy Γ V . Parameters α m r (%) σ (%) β (%) T
0% 0 .
2% 0 .
4% 0 .
6% 0 .
8% 1% 1 .
2% 1 .
4% 1 .
6% 1 .
8% 2%1 10 10 5 3.08 3.47 3.96 4.57 5.36 6.57 8.55 12.12 17.66 23.55 29.9310 1.66 1.92 2.23 2.61 3.13 3.81 4.86 6.44 8.99 12.11 15.3020 0.82 0.96 1.09 1.21 1.33 1.44 1.50 1.59 1.65 1.70 1.8120 5 3.08 3.47 3.95 4.55 5.34 6.48 8.38 12.01 17.66 23.55 29.9210 1.66 1.91 2.23 2.62 3.12 3.80 4.84 6.56 9.19 12.17 15.3120 0.81 0.96 1.16 1.39 1.67 2.05 2.60 3.43 4.65 6.14 7.7030 10 5 15.05 15.92 16.91 18.00 19.25 20.66 22.32 24.34 26.79 29.80 33.5810 8.38 8.93 9.53 10.19 10.93 11.77 12.72 13.81 15.11 16.65 18.5120 4.32 4.63 4.94 5.27 5.59 5.90 6.20 6.49 6.76 6.99 7.1620 5 13.99 14.82 15.73 16.80 18.00 19.39 21.04 23.09 25.62 28.84 32.9310 7.85 8.38 8.96 9.62 10.35 11.20 12.19 13.34 14.70 16.38 18.3720 4.00 4.32 4.68 5.08 5.53 6.04 6.61 7.27 8.04 8.92 9.955 10 10 5 0.43 0.47 0.50 0.54 0.58 0.62 0.67 0.71 0.76 0.81 0.8010 0.18 0.19 0.21 0.22 0.23 0.22 0.17 0.07 -0.02 -0.14 -0.2620 0.08 0.09 0.08 0.04 -0.09 -0.23 -0.37 -0.51 -0.64 -0.79 -0.9320 5 0.40 0.44 0.48 0.52 0.57 0.62 0.68 0.74 0.81 0.88 0.9610 0.11 0.12 0.14 0.15 0.17 0.19 0.20 0.22 0.23 0.25 0.2620 0.03 0.04 0.04 0.04 0.03 0.02 -0.00 -0.10 -0.24 -0.38 -0.5330 10 5 5.33 5.48 5.64 5.79 5.95 6.11 6.27 6.43 6.60 6.75 6.9410 2.91 3.01 3.10 3.18 3.26 3.34 3.41 3.47 3.53 3.58 3.6020 1.58 1.64 1.68 1.72 1.74 1.75 1.75 1.74 1.74 1.73 1.7120 5 4.97 5.13 5.28 5.44 5.61 5.78 5.96 6.14 6.32 6.51 6.7010 2.27 2.35 2.43 2.51 2.59 2.67 2.74 2.82 2.88 2.94 3.0020 1.08 1.13 1.18 1.22 1.23 1.24 1.23 1.21 1.18 1.14 1.0919 able 3: Total policy value V (0 , X (0); Γ L ) based on the liability maximization strategy Γ L . Parameters α m r (%) σ (%) β (%) T
0% 0 .
2% 0 .
4% 0 .
6% 0 .
8% 1% 1 .
2% 1 .
4% 1 .
6% 1 .
8% 2%1 10 10 5 1.00 0.99 0.99 0.98 0.98 0.98 0.97 0.97 0.97 0.97 0.9710 1.00 0.99 0.98 0.97 0.97 0.96 0.95 0.95 0.95 0.95 0.9520 1.00 0.98 0.96 0.95 0.94 0.92 0.92 0.91 0.90 0.90 0.9020 5 1.00 0.99 0.99 0.98 0.98 0.98 0.97 0.97 0.97 0.97 0.9710 1.00 0.99 0.98 0.97 0.96 0.96 0.95 0.95 0.95 0.95 0.9520 1.00 0.98 0.96 0.95 0.93 0.92 0.91 0.91 0.90 0.90 0.9030 10 5 1.00 1.00 0.99 0.99 0.99 0.98 0.98 0.98 0.97 0.97 0.9710 1.00 0.99 0.99 0.98 0.97 0.97 0.96 0.96 0.96 0.95 0.9520 1.00 0.99 0.97 0.96 0.95 0.94 0.93 0.93 0.92 0.91 0.9120 5 1.00 1.00 0.99 0.99 0.98 0.98 0.98 0.98 0.97 0.97 0.9710 1.00 0.99 0.99 0.98 0.97 0.97 0.96 0.96 0.95 0.95 0.9520 1.00 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.92 0.91 0.915 10 10 5 1.00 0.99 0.99 0.98 0.98 0.97 0.96 0.96 0.95 0.95 0.9410 1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.9020 1.00 0.98 0.95 0.93 0.91 0.89 0.88 0.86 0.84 0.83 0.8220 5 1.00 0.99 0.99 0.98 0.98 0.97 0.96 0.96 0.95 0.95 0.9410 1.00 0.99 0.98 0.96 0.95 0.94 0.93 0.92 0.91 0.90 0.8920 1.00 0.97 0.95 0.93 0.91 0.89 0.87 0.85 0.83 0.81 0.8030 10 5 1.00 1.00 0.99 0.99 0.98 0.98 0.97 0.97 0.96 0.96 0.9610 1.00 0.99 0.98 0.98 0.97 0.96 0.95 0.95 0.94 0.93 0.9320 1.00 0.98 0.97 0.96 0.94 0.93 0.92 0.91 0.90 0.89 0.8820 5 1.00 0.99 0.99 0.98 0.98 0.97 0.97 0.96 0.96 0.95 0.9510 1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.9120 1.00 0.98 0.96 0.94 0.92 0.90 0.89 0.87 0.86 0.84 0.8320 able 4: Total policy value V (0 , X (0); Γ V ) based on the policy value maximization strategy Γ V . Parameters α m r (%) σ (%) β (%) T
0% 0 .
2% 0 .
4% 0 .
6% 0 .
8% 1% 1 .
2% 1 .
4% 1 .
6% 1 .
8% 2%1 10 10 5 1.00 0.99 0.99 0.98 0.98 0.98 0.97 0.97 0.97 0.97 0.9710 1.00 0.99 0.98 0.97 0.97 0.96 0.95 0.95 0.95 0.95 0.9520 1.00 0.98 0.97 0.96 0.95 0.95 0.94 0.94 0.94 0.93 0.9320 5 1.00 0.99 0.99 0.98 0.98 0.98 0.97 0.97 0.97 0.97 0.9710 1.00 0.99 0.98 0.97 0.96 0.96 0.95 0.95 0.95 0.95 0.9520 1.00 0.98 0.96 0.95 0.93 0.92 0.91 0.91 0.90 0.90 0.9030 10 5 1.00 1.00 0.99 0.99 0.99 0.98 0.98 0.98 0.97 0.97 0.9710 1.00 0.99 0.99 0.98 0.97 0.97 0.97 0.96 0.96 0.95 0.9520 1.00 0.99 0.98 0.97 0.96 0.95 0.95 0.94 0.94 0.94 0.9320 5 1.00 1.00 0.99 0.99 0.98 0.98 0.98 0.98 0.97 0.97 0.9710 1.00 0.99 0.99 0.98 0.97 0.97 0.96 0.96 0.95 0.95 0.9520 1.00 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.92 0.91 0.915 10 10 5 1.00 0.99 0.99 0.98 0.98 0.97 0.97 0.96 0.96 0.95 0.9510 1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.94 0.94 0.93 0.9320 1.00 0.98 0.96 0.94 0.94 0.93 0.93 0.93 0.93 0.92 0.9220 5 1.00 0.99 0.99 0.98 0.98 0.97 0.96 0.96 0.95 0.95 0.9410 1.00 0.99 0.98 0.96 0.95 0.94 0.93 0.92 0.91 0.91 0.9020 1.00 0.97 0.95 0.93 0.91 0.89 0.88 0.87 0.86 0.86 0.8630 10 5 1.00 1.00 0.99 0.99 0.98 0.98 0.97 0.97 0.97 0.96 0.9610 1.00 0.99 0.98 0.98 0.97 0.97 0.96 0.96 0.95 0.95 0.9520 1.00 0.99 0.97 0.97 0.96 0.95 0.94 0.94 0.94 0.93 0.9320 5 1.00 0.99 0.99 0.98 0.98 0.97 0.97 0.96 0.96 0.95 0.9510 1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.94 0.93 0.92 0.9220 1.00 0.98 0.96 0.94 0.93 0.92 0.91 0.90 0.89 0.88 0.8821 eferencesReferences [1] Bacinello, A. R., Ortu, F., 1996. Fixed income linked life insurance policies withminimum guarantees: Pricing models and numerical results. European Journal ofOperational Research 91 (2), 235–249.[2] Bauer, D., Kling, A., Russ, J., 2008. A universal pricing framework for guaranteedminimum benefits in variable annuities. ASTIN Bulletin 38 (2), 621–651.[3] B´elanger, A. C., Forsyth, P. A., Labahn, G., 2009. Valuing guaranteed mini-mum death benefit clause with partial withdrawals. Applied Mathematical Finance16 (6), 451–496.[4] Chen, Z., Forsyth, P. A., 2008. A numerical scheme for the impulse control for-mulation for pricing variable annuities with a guaranteed minimum withdrawalbenefit (gmwb). Numerische Mathematik 109 (4), 535–569.[5] Chen, Z., Vetzal, K., Forsyth, P. A., 2008. The effect of modelling parameterson the value of GMWB guarantees. Insurance: Mathematics and Economics 43,165–173.[6] Cramer, E., Matson, P., Rubin, L., 2007. Common practices relating to fasb state-ment 133, accounting for derivative instruments and hedging activities as it re-lates to variable annuities with guaranteed benefits. In: Practice Note. AmericanAcademy of Actuaries.[7] Crank, J., Nicolson, P., 1947. A practical method for numerical evaluation of so-lutions of partial differential equations of the heat-conduction type. MathematicalProceedings of the Cambridge Philosophical Society 43 (1), 50–67.[8] Dai, M., Kuen Kwok, Y., Zong, J., 2008. Guaranteed minimum withdrawal benefitin variable annuities. Mathematical Finance 18 (4), 595–611.[9] Delbaen, F., Schachermayer, W., 2006. The Mathematics of Arbitrage. Springer.[10] Forsyth, P., Vetzal, K., 2014. An optimal stochastic control framework for deter-mining the cost of hedging of variable annuities. Journal of Economic Dynamicsand Control 44, 29–53.[11] Fung, M. C., Ignatieva, K., Sherris, M., 2014. Systematic mortality risk: An anal-ysis of guaranteed lifetime withdrawal benefits in variable annuities. Insurance:Mathematics and Economics 58 (1), 103–115.[12] Hirsa, A., 2012. Computational Methods in Finance. Chapman and Hall/CRCFinancial Mathematics Series. 2213] Ho, T. S. Y., Lee, S. B., Choi, Y. S., 2005. Practical considerations in managingvariable annuities. doi:10.1.1.114.7023.
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