Taxation of a GMWB Variable Annuity in a Stochastic Interest Rate Model
Andrea Molent
TL;DR
This paper develops a numerical framework to evaluate GMWB variable annuities considering both stochastic interest rates and taxation, revealing their significant impact on policy valuation and withdrawal strategies.
Contribution
It introduces a novel numerical method to jointly model taxation and stochastic interest rates in GMWB valuation, enhancing accuracy over previous models.
Findings
Taxation and stochastic interest rates significantly influence GMWB valuation.
Accounting for these factors alters optimal withdrawal strategies.
The model explains the popularity of GMWB products for retirement planning.
Abstract
Modeling taxation of Variable Annuities has been frequently neglected but accounting for it can significantly improve the explanation of the withdrawal dynamics and lead to a better modeling of the financial cost of these insurance products. The importance of including a model for taxation has first been observed by Moenig and Bauer (2016) while considering a GMWB Variable Annuity. In particular, they consider the simple Black-Scholes dynamics to describe the underlying security. Nevertheless, GMWB are long term products and thus accounting for stochastic interest rate has relevant effects on both the financial evaluation and the policy holder behavior, as observed by Gouden\`ege et al. (2018). In this paper we investigate the outcomes of these two elements together on GMWB evaluation. To this aim, we develop a numerical framework which allows one to efficiently compute the fair value…
| Description | Parameter | Value |
|---|---|---|
| Age at inception | ||
| Premium | ||
| Years to maturity | ||
| Annual guaranteed amount | ||
| Excess withdrawal fee | ||
| Fee rate | to be determined | |
| Income tax rate | or | |
| Capital gain tax rate | or | |
| Early withdrawal penalty |
| Description | Parameter | Value |
|---|---|---|
| Initial fund value | ||
| Fund volatility | , | |
| Initial interest rate | , | |
| Interest rate mean reversion speed | ||
| Interest rate mean | ||
| Interest rate volatility | ||
| Correlation |
| Description | Parameter | Value |
|---|---|---|
| Time step per year | ||
| Points in | ||
| Points in | ||
| Points in | ||
| Withdrawal step |
| No Taxation | |||||
|---|---|---|---|---|---|
| With Taxation | |||||
| No Taxation | |||||
|---|---|---|---|---|---|
| With Taxation | |||||
| Transition to | Transition to | |
|---|---|---|
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Taxation of a GMWB Variable Annuity in a Stochastic Interest Rate Model
Andrea Molent Dipartimento di Scienze Economiche e Statistiche, Università degli Studi di Udine, Italy - [email protected]
Abstract
Modeling taxation of Variable Annuities has been frequently neglected but accounting for it can significantly improve the explanation of the withdrawal dynamics and lead to a better modeling of the financial cost of these insurance products. The importance of including a model for taxation has first been observed by Moenig and Bauer [19] while considering a GMWB Variable Annuity. In particular, they consider the simple Black-Scholes dynamics to describe the underlying security. Nevertheless, GMWB are long term products and thus accounting for stochastic interest rate has relevant effects on both the financial evaluation and the policy holder behavior, as observed by Goudenège et al. [12]. In this paper we investigate the outcomes of these two elements together on GMWB evaluation. To this aim, we develop a numerical framework which allows one to efficiently compute the fair value of a policy. Numerical results show that accounting for both taxation and stochastic interest rate has a determinant impact on the withdrawal strategy and on the cost of GMWB contracts. In addition, it can explain why these products are so popular with people looking for a protected form of investment for retirement.
Keywords: Variable Annuities, taxation, stochastic interest rate, optimal withdrawal, tree method.
1 Introduction
Variable Annuities are tax deferred investment contracts with insurance coverage. The market for such products has been steadily growing in the past years all around the world and 2019 has set best sales year since 2008 in Unites States. According to the Secure Retirement Institute [27], the Variable Annuity sales in 2019 amounted to over $100 billions, which represents almost half of the total annuity sales. In this paper we focus on a particular type of Variable Annuity, called Guaranteed Minimum Withdrawal Benefit (GMWB) which promises to return the entire initial investment by means of cash withdrawals during the policy life, plus a final payment amounting to the remaining account value at the contract maturity. Usually, the policy holder (hereinafter PH) pays the whole premium as a lumpsum and he is entitled to withdraw at each contract anniversary a variable amount, with a minimum guaranteed. Thanks to the guarantee included in the policy, the PH can withdraw money from his account even if it has run out. Moreover, if the PH death occurs before the contract maturity, then his heirs receive the remaining account value as a lumpsum payout. The premium paid at contract inception determines the risky account, which changes over time according to a financial index (usually a fund) but it is also reduced due to the fees applied by the insurer and by withdrawals made by the PH.
In order to manage GMWB contracts, insurers usually employ hedging techniques which rely on the computation of the fair prices of the policies in a risk neutral probability framework. In addition, the hedging costs are offset by deducting a proportional fee from the risky asset account. Moreover, the mortality risk is hedged by using the law of large numbers (see Bernard and Kwak [2] and Lin et al. [17] for an explanation of move-based and semi-static hedging of Variable Annuities). Price and Greeks calculation usually relies on numerical computations, which are based on a convenient model of the product, of the financial market, and nonetheless of the behavior of the PH. In fact, since the PH can choose (within certain limits established by the contract) the amount to be withdrawn, he can decisively drive the total payoff of the contract. Anyway, ordinary techniques for pricing American and Bermudan options lead to prices which differ significantly from market observations (Moenig and Bauer [19]). A possible explanation to the theoretical-empirical price gap, can be found in a correct model for the dynamics of taxation that the customer must face. In this regard, Moenig and Bauer [19] propose to model taxation imposed to the PH and to consider a subjective valuation of the contract. Specifically, they show that when accounting for taxation, PH withdraws less frequently than without taxes and by employing ordinary pricing techniques, one can obtain prices which are in line with empirical observations. Moreover, Moenig and Zhu [20] observe that the preferential tax treatment has been one of the key factors that have made Variable Annuities such a popular instrument and thus correctly modeling taxation can improve the explanation of the still unclear mechanisms about these products. We stress out that the investigations in [19] and [20] have been performed by assuming the Black-Scholes model for the underlying fund.
Interest rates is another relevant factor in Variable Annuities evaluation. As observed by Goudenège et al. [12], since GMWB contracts have long maturities that could last almost 25 years, the Black-Scholes model seems to be unsuitable for such a long time interval as it assumes constant interest rate and volatility. Several authors have investigated the possibility of evaluating GMWB contracts while considering a stochastic interest rate. For example, Peng et al. [24] develop an analytic approximation of the fair value of the GMWB under the Vasicek stochastic interest rate model. Donnelly et al. [7] consider pricing and Greeks calculation through an Alternating Direction Implicit method in the advanced Heston-Hull-White model. Dai et al. [6] develop a tree based model to include both stochastic interest rate and mortality in their evaluation framework. Gudkov et al. [14] employ the operator splitting method to price GMWB products under stochastic interest rate, volatility and mortality. Shevchenko and Luo [28] employ high order Gauss-Hermite quadrature to evaluate the GMWB contract under the Vasicek interest rate model. Recently, Goudenège et al. [13] exploit a hybrid tree-PDE method together with Machine Learning techniques to efficiently evaluate the GMWB contract in a model that considers both stochastic interest rate and stochastic volatility. More generally, as far as pricing of Variable Annuities in a stochastic interest rate framework is considered, it is worth mentioning the work of Bacinello and Zoccolan [1] that develops a Monte Carlo flexible approach to study the impact of threshold fee on the optimal surrender strategy about a product including accumulation and death guaranteed benefits under a model which considers stochastic interest rate, volatility and mortality. We also mention Goudenège et al. [11], who employ the hybrid Tree-PDE method to evaluate a GLWB contract under stochastic interest rate.
In this paper we present an investigation about GMWB pricing and PH behavior when both tax treatment and stochastic interest rate are considered. In particular, following Moenig and Bauer [19] and Moenig and Zhu [20], we model taxation of GMWB through a constant marginal income tax rate on all policy earnings and a constant marginal tax rate on capital gains from investments outside of the policy. Moreover, we also include a premium based model for taxation of the insurer, which was neglected in previous researches. Because of taxation, the evaluation of the contract is not straightforward, so we exploit the same subjective risk-neutral valuation methodology employed in [19]. In particular, in this framework, the value of a given post-tax cash flow is the amount necessary to set up a pre-tax portfolio that replicates the considered cash flow. This causes the insurer and the PH to evaluate the policy differently and we investigate both the two perspectives. As far as the stochastic interest rate is concerned, we consider the Hull-White model (Hull [16]), which is often employed by both academics and practitioners for its easiness of calibration and simple probability distribution. This model has already been employed in other research works concerning GMWB Variable Annuities (e.g. [7], [6], [12] and [13]). We stress out that considering both taxation and stochastic interest rate is a challenging task because of the computational effort required to consider many factors together. In particular, evaluating a GMWB policy in the considered model is a four (plus time) dimensional problem, which means a high computational cost in terms of both computing time and working memory required. Moreover, the evaluation of a policy through the subjective risk-neutral valuation methodology requires the resolution of many fixed point problems, and this increases even more the computational cost. Finally, we assume the PH to employ an optimal withdrawal strategy, which implies the numerical resolution of a dynamic control problem. In order to manage such a computational effort, we use a backward dynamic approach that exploits a tree approach to compute the fair contract price. In particular, we employ a trinomial tree to approximate the stochastic interest rate process through a Markov chain, which represents an efficient numerical solution already used by Goudenège et al. [13]. It is worth noting that tree methods have already been used to study the GMWB contract. In this regard, we mention the works of Costabile [4] and of Costabile et al. [5] that employ a trinomial tree to evaluate a GMWB policy and to investigate the PH decisions while including exogenous factors in the model.
In order to test our approach, we perform some numerical experiments. Specifically, we study how the evaluation of the policy varies according to the insurer and to the PH perspectives and how the withdrawal strategy is modified, by including or not including taxation and by changing the parameters of the interest rate and the fund. Numerical results show many interesting findings. First of all, if taxation is considered, the fair value of the policy for the PH is higher than the fair value for the insurer. This means that the PH attributes a higher price to the policy than the insurer does, so buying and selling the contract can be a good deal for both of them. Secondly, we observe that taxation and interest rate modeling have a significant impact on the withdrawal strategies of the PH.
To the best of our knowledge, this is the first analysis about GMWB pricing and withdrawal strategy which accounts for both taxation and stochastic interest rate. Our research could be useful both for the qualitative observations obtained and for the numerical solutions adopted.
The reminder of the paper is organized as follows. Section 2 introduces the stochastic model for the underlying and the interest rate processes. Section 3 describes the GMWB contract and the taxation model. Section 4 presents pricing assumptions. Section 5 describes the pricing method and the technical measures. Section 6 shows numerical results on various examples. Finally, Section 7 draws the conclusions.
2 The Stochastic Model
In order to define the notation used throughout the rest of the paper, let us introduce the Black-Scholes Hull-White model. The Hull-White model [16] is one of historically most important interest rate models, which is nowadays often used for option pricing purposes. In particular, the existence of closed formulas for the price of bonds, caplets and swaptions is one of the important advantages of this model. Furthermore, it is capable of generating negative interest rates, actually observed in the markets in recent years. We report the dynamics of the Black-Scholes Hull-White model, which combines the dynamics of the interest rate with the dynamics of the underlying:
[TABLE]
where and are Brownian motions with . Moreover and are positive values and the initial values and are given. Furthermore, is a deterministic function which is completely determined by the market values of the zero-coupon bonds by calibration (see Brigo and Mercurio [3]) so that the theoretical prices of the zero-coupon bonds match exactly the market prices.
Let denote the market price of the zero-coupon bonds at time [math] for the maturity . The market instantaneous forward interest rate is then defined by
[TABLE]
It is well known that the (short) interest rate process can be written as
[TABLE]
where is a stochastic process whose dynamics is given by
[TABLE]
and is a real valued function with
[TABLE]
Then, the Black-Scholes Hull-White model can be described by the following relations:
[TABLE]
The flat curve case is a particular case for the market price of a zero-coupon bonds: in this specific case, the price at time of a zero-coupon bond with maturity is given by
[TABLE]
and the function is given by
[TABLE]
We stress out that assuming a flat curve for the price of bonds is not essential for the development of our model, but it simplifies the numerical settings.
3 Modeling the contract
3.1 Modeling taxation
In order to model taxation, we follow the same approach proposed by Moenig and Bauer [19], which in turn is a simplified version of the model currently in force in the Unites States.
As far as the PH is concerned, taxes are due on future investment gains and not on the invested amount. In particular, we assume a constant marginal income tax rate to be applied on all policy earnings and a constant marginal tax rate to be applied to the capital gains from investments outside of the policy. This means that if the PH sets up a portfolio that replicates the after tax policy cash flows, then is the tax rate applied on gains of such a portfolio. On the contrary, is the tax rate applied on all gains caused by PH’s withdrawals form the policy. In particular, earnings are withdrawn before the initial premium, following last-in first-out approach.
In order to complete tax modeling we have to consider taxation concerning the insurer, which is usually of two types: premium taxation and net income taxation (see Skipper [29]). Determining life insurer profit is a challenge because of the difference in timing between premium payments and claim payments, so premium taxes are the most common. Furthermore, as far as Unites States life insurance system is concerned, the insurance companies can elect to be taxed based on either premiums or net income (see Nissim [23]). For for sake of simplicity, we assume premium based taxation, that is the insurer pays a certain percentage of the gross premium as taxes. So, the tax due by the insurer is thus , where is the premium tax rate. Such a rate usually varies between and (see Moran [21]). Obviously the insurer has to recover this tax cost, therefore we assume that such an amount is applied indirectly to the customer as an entry cost, which reduces the gross premium and determines the net premium , given by .
3.2 The GMWB contract
We study here a simple version of the GMWB contract which was first investigated by Moenig and Bauer [19]. We consider an -year old individual that purchased a GMWB policy with a finite integer maturity against the payment of a single gross premium . Then entry expenses are deducted from the gross premium and the net premium is credited to the policy’s account. There are three variables which determine the state of a policy at time , namely the account value , the benefit base and the tax base whose values at time are equal to the policy net premium, that is
[TABLE]
In particular, the account value represents the risky account of the policy, which changes as if it were invested in a market fund, aside from being reduced by withdrawals and management costs. The benefit base represents the guarantee inherent in the policy as it regulates the maximum withdrawal that the PH can make, while the tax base represents the amount that may still be withdrawn from the policy free of tax.
Let denote the time of the contract anniversary, i.e. . The variables and do not change during the time between two consecutive anniversaries, that is for , while varies according to an underlying investment fund changes. This fund is usually chosen by the customer from a list proposed by the insurer. Specifically, let us term the value of the underlying fund, which evolves according to (2.1). Then, for , follows the same dynamics of with the exception that fees are subtracted continuously, that is
[TABLE]
The variable in (3.2) is the (constant) fee rate and it controls the fees withdrawn by the account value.
At each anniversary time , the continuation of the policy is determined according to the survival of the PH during the last year of the contract. In order to describe the policy revaluation mechanisms, let us denote with and the account values just before and after any cash-flow at time (we use the same notation for and ). If the PH has passed away during the previous year, then his heirs receive the death benefit , which is paid at time and it is given by the residual account value net of taxation, that is
[TABLE]
where is the income tax rate and . After the payment of the death benefit, the contract ends and it has no residual value. On the contrary, if the PH has not passed away, then he is entitled to withdraw an amount within some limits. According to the contract, the withdrawal amount selected by the PH must satisfy the following relation:
[TABLE]
where is a positive constant value called the annual guaranteed amount and it is stated in the contract. In particular, if then the PH is entitled to withdraw at each contract anniversary exactly an amount equal to throughout the duration of the contract. After the withdrawal has been performed, the new account value is given by
[TABLE]
while the new benefit base and tax base are given by
[TABLE]
and
[TABLE]
respectively.
The PH does not receive the whole amount withdrawn because some fees and tax may be applied. Specifically, the PH receives the withdrawn amount reduced by the fees due to the insurer for withdrawing more than the guaranteed amount and also reduced by a penalty for early withdrawals and by the taxation. Specifically, the net amount he receives is given by
[TABLE]
being the cost for withdrawing an amount exceeding , an early withdrawal penalty for any withdrawal before the age of 59.5 years and the income taxes associated with the withdrawal. In particular,
[TABLE]
[TABLE]
and
[TABLE]
The coefficient in (3.9) is a non-negative coefficient called surrender charge, which usually decreases with time and it is zero within the term of the contract. Moreover, in (3.10) is another non-negative coefficient that determines the penalty for an early withdrawal. In particular, since these contracts are usually employed as a supplement to the retirement pension, we assume that when the contract maturity is achieved, the PH must be older than years, so penalty is not applied at last withdrawal at time .
Finally, after the last withdrawal has been made at time , the alive PH receives the remaining account value net of taxes, that is
[TABLE]
and the contract ends.
4 Pricing assumptions
In this Section we present the pricing framework. First of all, we observe that asset pricing under taxation is not direct since, as observed by Ross [26], taxation leads to the loss of uniqueness of prices. In fact, the valuation of a specific cash flow depends on the personal endowment and tax rates. Following the same approach of Moenig and Bauer [19], we define the value of a given post-tax cash flow as the amount of money that a specific agent needs to create a financial portfolio (made of stocks and bonds) that, after taxation, replicates the considered cash flow. Clearly, such a value depends on the specific taxation applied to the agent and this can be significantly different between the customer and the insurer. In addition, taxation may be different in relation to the financial instrument considered: a lighter taxation is usually applied to insurance products (such as VA policies) and a heavier taxation for financial products (such as the securities in a replicating portfolio).
First of all, we present how to evaluate the GMWB contract assuming PH’s subjective valuation and then we present the same while assuming insurer’s subjective valuation. The main differences in the two perspectives are due to taxation and to control on withdrawals. As far as taxation is considered, the PH has to pay taxes on both policy earnings and capital gains outside the policy. On the contrary, the taxation applied to the insurer is much simpler: a percentage of the gross premium. As far as withdrawals are concerned, the PH selects optimal withdrawals in order to maximize the expected value of its assets, net of taxation: if taxation is applied, such a value is not equal to insurer’s liability. Thus, the amount withdrawn by the PH is optimal for him, but it could be different from the worst amount computed considering the insurer’s point of view, that is the amount that maximizes insurer’s liability to the PH. This means that the PH withdraws money trying to maximize his economic return, rather than trying to maximize the outputs of the insurer: since these two strategies do not coincide, the costs for the insurer are lower than the worst withdrawing case.
Finally, we underline that the considered framework captures an interesting feature of insurance products. Taxation makes GMWB policies particularly attractive to customers: although taxes are applied on the earnings of the policy, the tax regime is particularly favorable for this type of product and therefore it is more convenient for the customer buying the policy rather than reproducing it through a replicating portfolio.
In the next Subsections, we show how to compute the initial contract value according to the PH and to the insurer’s subjective valuation. We stress out that in both cases we compute the cost of the replicating portfolio under the same risk neutral measure for the Black-Scholes Hull-White model (see Brigo and Mercurio [3]).
4.1 Policyholder’s subjective valuation
Following Moenig and Bauer [19], we consider the PH’s subjective valuation of the contract. This means we compute the amount of money that a PH needs to set a hypothetical replicating portfolio, which replicates the post-tax policy cash-flows. Specifically, let denote the fair value according to an alive PH of a GMWB contract at time , being the interest rate, the account value, the guarantee base and the tax base respectively. Specifically, following the same approach of Moenig and Bauer [19], represents the average option value across the many policies sold to the customers that are still alive at time .
Finally, in order to compute PH’s subjective value of the contract at time , we proceed backward in time, starting from contract’s maturity at time and by taking into account the changes that occur to the policy status parameters.
4.1.1 Value function at a contract anniversary
First of all, let us denote with the policy value at maturity, after the last withdrawal is performed. Such an amount is given by the final payoff, that is
[TABLE]
Now, let us focus on the -th contract anniversary, at time . Since we are assuming that the PH is alive, then he is entitled to perform a withdrawal from his account. Let and represent the values of the policy just before and after the PH has withdraw money respectively. In particular, are the state parameters before withdrawing at time , while are the state parameters after withdrawing at time . Please, observe that there is no need to distinguish between the value of the interest rate before and after the withdrawal, because such a value is not modified by the withdrawal, so we simply write in both cases. We can write the relation between the two policy values in the general form
[TABLE]
where we underline the dependence of many variables on the withdrawal by denoting them as a function of . In particular, equations (3.5), (3.6), (3.7), (3.9), (3.10) and (3.11) express the dependence of and on respectively. The PH might adopt a static withdrawal strategy, which means he withdraw an amount equal to , regardless of the value taken from the policy state parameters. Such a strategy is easy to be implemented, but may be not the optimal one for him. We rather assume that the PH selects the amount in order to maximize the expected value of his assets – contract plus net withdrawal–, that is
[TABLE]
where
[TABLE]
is the maximum withdrawal allowed by the contract. We observe that, at maturity, the optimization problem (4.3) can be easily solved as the continuation value after the payment is given by the final payoff, which has a closed formulation. Specifically, one can prove that the optimal withdrawal in this particular case is given by
[TABLE]
Moreover, by using equations (4.1), (4.2) and (4.5), one can obtain the following expression:
[TABLE]
In general, when considering the optimal withdrawal at time , there is no closed formula as for the last anniversary . In the general case, the optimal withdrawal must be approximated by a numerical procedure.
4.1.2 Dynamics of the value function between two anniversaries
During the time between two contract anniversaries and , the variables and do not change. Changes of the policy value are solely due to the passage of time and to the changes of the account value and of the interest rate . Following Moenig and Bauer [19], the subjective risk-neutral value at time of is given via a nonlinear implicit equation:
[TABLE]
where stands for and stands for . Furthermore, is the death benefit that may be paid at time in case of death and it is computed according to (3.3). Moreover, is the probability that the alive PH, aged exactly at time , will die in one year, while is the probability that he will survive at least one more year. We stress out that the use of death and survival probabilities is possible if a large number of contract holders is assumed: in this case, mortality risk is diversifiable.
4.2 Insurer’s subjective valuation
Let denote the fair value of the GMWB contract but according to insurer’s subjective value, that is the amount of money that the insurer needs to set a replicating portfolio. The valuation according to the insurer differs from the valuation according to the PH for some reasons. First of all, the taxation applied to the insurer only concerns the initial gross premium and it is not applied to the replicating portfolio. Secondly, the insurer must shell out an amount gross of taxes, while the PH receives the net amount. Finally, the insurer has no decision-making power and undergoes the PH’s choices regarding the amount to be withdrawn. Just as done for the PH’s subjective valuation, in order to compute insurer’s subjective value at contract inception, we proceed backward in time.
4.2.1 Value function at a contract anniversary
Let be the policy value at maturity according to the insurer, after the last withdrawal is performed. Such an amount is given by the final payoff before tax, that is the residual account value:
[TABLE]
Moreover, since the optimal withdrawal at time is given by (4.5), one can prove the following relation:
[TABLE]
Now, let us focus on the -th contract anniversary at time . The functions and represent the value of the contract just before and after the PH has withdrawn the amount , which is the solution of problem (4.3). The following relation holds,
[TABLE]
Equation (4.10) is similar to equation (4.2) but taxes are not subtracted because the insurer has to pay the amount before taxation.
4.2.2 Dynamics of the value function between two anniversaries
As opposed to the PH, the insurer pays no taxes on the replicating portfolio. The subjective risk-neutral value at time of , is given by the discounted expected future value of the death benefit plus the value of the policy, that is
[TABLE]
where stands for and stands for .
5 Pricing method
The fair value of the GMWB contract at time according to the PH’s subjective perspective, denoted by , can be computed by moving backward in time. The terminal condition is expressed by (4.1). In order to proceed backward, we have to solve the nonlinear implicit equation (4.7) in for , and apply relations (4.2) and (4.3) to handle the jumps due to withdrawals at each contract anniversary.
With a similar approach, the initial fair value of the contract according to the insurer’s perspective, denoted by , can be computed by starting from the terminal condition , by solving backward equation (4.11) and by applying relation (4.10). We observe that computing requires the knowledge of the optimal withdrawals, which can be achieved through the parallel computation of .
We stress out that the evaluation problems of and are four dimensional problems (plus the time variable) and this represents a non-trivial challenge which requires an efficient numerical method to be solved.
5.1 Problem discretisation
The variables that determine the state of the policy at any time are the and . To tackle the problem numerically, we prefer to replace with , since the dynamics of is simpler and one can easily compute from through (2.3). We consider a set of discrete values for , for , for and for , and we define a 4 dimensional grid .
First of all, since the benefit base and the tax base are non-negative values that do not exceed , it is worth exploiting an uniform partition of the interval to define and . In particular, we set
[TABLE]
and
[TABLE]
where and are two positive integers.
As opposed to and , the account value assumes non-negative unbounded values. Anyway, because of withdrawals and fees applied by the insurer, such a value should not grow too much during the life of the policy. In fact, as observed by MacKay et al. [18] and by Bacinello and Zoccolan [1] in a similar context, when the account value is very high there is a great incentive for the PH to surrender the contract by withdrawing all the money. So, following same principle of the spatial grid employed by Haentjens and In’t Hout [15], we consider as a non-uniform distribution of points which is more dense where the process is more likely to be. Specifically, we consider two sets of points: the first set
[TABLE]
is made of uniformly distributed points between [math] and and the second one
[TABLE]
is made of points which are distributed uniformly in log between and . Then, and we term the -th point of . Moreover, for seek of simplicity, we consider and we term the number of elements of . We stress out that the coefficient and are determined empirically in order to give accurate results and their small variations do not produce impacts on the numerical results.
Finally, the construction of the set relies on the trinomial tree proposed by Goudenège et al. in [13]. Such a tree defines a discrete Markov chain that matched the first two moments of the process . We set
[TABLE]
where is a positive coefficient that depends on the standard deviation of the process and is a suitable integer value, thus is made of points uniformly distributed in . Appendix A presents technical details about the process , the coefficient and the integer .
5.2 Backward evaluation of
Once the grid has been build, we can start the computation of the numerical approximation of defined on at any time . In particular, for every policy anniversary , we compute a function such that for any point of , approximates . Moreover, we also compute a function such that for any point of , approximates . According to (4.6), the terminal condition at each point of is given by:
[TABLE]
where .
Suppose now the function to be known on . Let us fix and let us focus on the computation of by solving equation (4.7). In particular, following the same approach employed by Moenig and Bauer [19] under the Black-Scholes model, one can verify that the solution of equation (4.7) exists and is unique. Furthermore, according to equation (4.7), can be interpreted as the solution of a fixed point problem:
[TABLE]
with
[TABLE]
where stands for
[TABLE]
Such a problem can be faced by fixed point iterations. The key point consists in calculating the expected values that appears in (5.8). In order to tackle such a problem, we employ a tree approach. Technical details are explained in Appendix B.
Once the function is known, we can compute by solving the optimal withdrawal problem related to equation (4.3). So, let us fix again and let us focus on solving the following problem:
[TABLE]
with
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
The resolution of (5.10) is not trivial as the function has no smoothness properties. In particular has singular points, due to the presence of the positive part function, as well as a discontinuity point, due to the function at . Therefore, we approach the solution of the maximization problem (5.10) through a very simple approach: we evaluate the target function in a set of points and record the maximum value achieved on these points. In particular we consider as the union of two sets, and . The first set, is a set of uniformly distributed values, while the second set is a set of critical values that might not be included in . In particular, and are considered in order to handle the discontinuity at .
Evaluating the function requires the calculation of the function at for any . These points may not belong the grid as the values may not belong to and respectively. So, in order to compute , interpolation on of is required. To this aim, we employ trilinear interpolation (Gomes et al. [10]).
5.3 Backward evaluation of
Once an approximation of is available, we can tackle the policy evaluation according to the insurer’s perspective, that is computing . Approximating is easier than approximating because of two reasons. First of all, the implicit nonlinear equation (4.7) is replaced by an explicit equation (4.11). Secondly, the problem of computing the best withdrawal has been already solved while approximating , so we have just to recover the optimal withdrawals already computed.
Similarly to what we have done for , we consider a function such that for any point of , approximates and a function such that for any point of , approximates . According to (4.9), for any point of , the terminal condition is given by:
[TABLE]
Suppose now the function to be known on . Let us fix and let us focus on the computation of by computing the following expression:
[TABLE]
We compute such an expression by using the same tree approach employed to compute (5.8). Please observe that in this case, no fix point iterations are required because (5.20) gives through an explicit equation.
Suppose now the function to be known on . Let us fix and let us focus on the computation of . Let be the maximum point for the problem (5.10) for the point . The following relation holds:
[TABLE]
where , , , and are defined as in (5.13)-(5.17). Also in this case, interpolation is required and we employ again trilinear interpolation.
5.4 Sketch of the algorithm
We present the sketch of the algorithm to approximate the initial fair contract values and .
Set the terminal values and according to equations (5.6) and (5.19) for every point in . 2. 2.
For all
- (a)
Compute and by solving equations (5.8) and in (5.20) for every point in ; 2. (b)
Compute by solving equation (5.10) for every point in ; 3. (c)
Compute by solving equation (5.21) for every point in ; 3. 3.
Compute and by solving equations (5.8) and (5.20).
Values and approximate and respectively. We point out that the algorithm is fully parallelizable: in fact the computations for every point in are independent of each other.
A common practice in VAs context (see for example Forsyth and Vetzal [9]) consists in computing the fair policy cost , that is the particular value of that makes the insurer’s initial value of the policy equal to the net premium . To this aim, the algorithm can be plugged into the secant method to solve the equation .
6 Numerical Results
In this Section we report the results of some numerical tests. Tables 1, 2, 3 report the parameters used in the analysis. Moreover, in order to estimate the mortality and survival probabilities and , we employ the 2007 Period Life Table for the Social Security Area Population for the USA [30]. Finally, we underline that these parameters, with the exception of those for the interest rate process, are the same employed by Moenig and Bauer [19].
6.1 Computing the fair fee rate
We start by computing the fair fee rate according to the insurer’s subjective valuation. In particular, we consider some test cases with different values of the initial interest rate , the volatility of the interest rate and the volatility of the underlying fund . Numerical results are reported in Table 4.
By comparing the two rows of Table 4, we observe that, in all the considered cases, including taxation reduces the fees that reduce the account value, that is the policy cost. We can speculate that, the reason for this lies in the withdrawal strategy: if taxation is applied, the optimal withdrawal strategy from PH’s perspective (i.e. the strategy that maximizes value according to his subjective view) does not coincide with the worst strategy according to insurer’s perspective (i.e. the strategy that maximizes his liability). Moreover, we observe that the higher the interest rate volatility, the greater the policy cost. This is probably due to the fact that, by increasing the volatility of the interest rate, it is easier to observe very low (or negative) interest rates which make replicating the policy very expensive. Thus, both taxation and interest rate modeling have a sensitive impact on policy evaluation.
6.2 Comparing policy initial values
We compute now the PH’s initial subjective policy value with equal to , that is the break-even fee, as in Table 4: this is the amount of money the PH needs to replicate the policy on its own. Numerical results are reported in Table 5. We observe that if no taxation is applied, the subjective valuation of the PH equates the subjective valuation of the insurer and the contract is fair for both the two agents. Instead, when taxation is applied, the contract value according to PH’s subjective valuation increases because of the tax regime applied to the policy, which is advantageous compared to the tax regime applied to investment outside the policy, and thus in the replicating portfolio.
Now let us consider as the premium tax rate. With such a premium tax rate, the gross premium that the insurer requires to cover all the costs is , so that the net premium is and the contract is fair for the insurer. According to Table 5, in all the considered cases, the customer will be willing to pay much more than to buy the policy: for example, if , and then the PH’s subjective valuation of the policy is . Therefore, if the insurer sets a sale price between and , then the sale will be advantageous for both the PH and the insurer.
The model considered allows us to recreate a framework that makes VAs particularly interesting to customers: although taxes are applied on the earnings of the policy, the tax regime is particularly favorable for this type of product. Therefore the GMWB policy is attractive for the customer and profitable for the insurer.
6.3 Comparing the PH withdrawals with and without taxation
The last numerical test we propose consists in comparing the optimal withdrawals performed by the PH while considering or not taxation. In particular, we consider the case with , and (results for other parameters combinations are similar). Moreover, the value of is set as the break-even fee with taxes, that is basis points. Optimal withdrawal amounts at different anniversaries are reported in Figures 6.1, 6.2 and 6.3. The first column represents the optimal amount as a function of and , by considering different values for and and a zero tax rate, while in the second column by considering both a positive income tax rate and a positive capital gain tax rate. The area where the color is darker identifies higher withdrawals. Finally, in the third column, the difference between the optimal amount without taxation and with taxation. Here, green areas indicate that withdrawals without tax are higher, while red areas (not visible) indicate that withdrawals with tax are higher.
We can observe that the optimal amount depends on all the considered parameters. In particular, the withdrawal strategy may significantly change according to the actual interest rate, as shown in Figure 6.3 for . As far as the impact of taxation on withdrawal strategy is concerned, we find the same effect observed by Moenig and Bauer [19]: when taxation is applied, the PH withdraws less than when taxation is not considered. In fact, for all the numerical cases considered, the last columns in Figures 6.1, 6.2 and 6.3 show only positive values, which means the amount withdrawn with no taxation is higher. As observed in [19], in the absence of taxes, as the account value increases more and more, the PH is motivated to withdraw money instead of leaving it in the policy where it is reduced by fees. Conversely, if taxation is applied, withdrawals are taxed as ordinary income and they are subject to capital gain tax if invested in other products. Therefore it is more convenient for the PH not to withdraw the money, letting it grow within the policy. Such a difference in the withdrawal strategy is particularly clear in Figure 6.3: if , then the PH withdraws large amount of money when is high if taxation is neglected, whereas no money if taxation is applied.
7 Conclusions
In this paper we have investigate the impact of taxation on a GMWB Variable Annuity when stochastic interest rate is considered. We modeled taxation following the approach of Moenig and Bauer [19]: we have considered a subjective risk-neutral valuation methodology that considers differences in the taxation for both different products and market agents. Moreover, we have modeled stochastic interest rate through the Hull-White model. This analysis combines the effects of taxation and of the variable interest rate which, as already shown separately in other research work, can have a significant impact on the withdrawal choices and thus on the hedging costs. This analysis has been possible thanks to the use of an efficient numerical method based on a tree approach (Goudenège et al. [13]). Numerical results show many interesting facts. First of all both, taxation and interest rate modeling can have a relevant impact on policy evaluation: the break-even fee can change of several basis points when the parameters of these two factors change. Then, applying different taxation to insurer and to policy holder can lead to different policy evaluation: in particular policy holder’s valuation is higher than insurer’s valuation and this makes buying and selling the policy convenient for both the two agents. Moreover, numerical tests show that taxation clearly impacts on withdrawal strategy: it discourages the policy holder to perform withdrawals. This is useful to match theoretical prices to those actually observed on the real market. In conclusion, the model presented here represents an important extension in the evaluation of GMWB type policies.
Appendix A Markov chain to approximate
In this Appendix we explain how to design a discrete time Markov chain that approximates the process , based on the trinomial tree introduced in [13]. First of all, we consider a partition of the time interval in sub-intervals, that is per year. We define as the time increment and we term the -th time step for . Please observe that policy anniversaries are included in the time steps and in particular . We consider the set given by
[TABLE]
where the coefficient is the standard deviation of the random variable (which is the same for all values) and it is given by
[TABLE]
We define now a discrete time Markov chain whose state space is an opportune subset of and that matches the first two moments of the process . The process is designed so it weakly converges to the process : in particular converges to .
The initial value is , so . Now, let us fix a value and suppose for a certain integer . Let
[TABLE]
be the expected value of the random variable . We term
[TABLE]
the index of the first element of whose value is bigger than the expected value of the process . Moreover, we also consider these three indices:
[TABLE]
In particular, if we define the variables
[TABLE]
and
[TABLE]
then and .
There are two alternatives for the future states of the process : it can move from either to , , , or to , , . Transition probabilities for both these two alternatives are stated in Table 6. In particular, it is possible to prove that if then while if then . Since , at least one of the two sets of probabilities is well defined. Moreover, transition probabilities in Table 6 have been computed in order to match the first two moments of the process : this means that the random vectors and , given , have the same mean and variance.
The choice between the two alternatives - , , or , , - is made in order to reduce the number of points connected with , which is the starting point. Since reverts to [math], it is sufficient, when possible, to choose the set with the points closest to . Specifically, if then can only move to , , (in fact at least one among and is not in ). If then can only move to , , (in fact at least one among and is not in ). Finally, if both choices are admissible: if , then moves to otherwise to .
The state space of is the connected component of , that is the set of points that the process can reach. Taking advantage of the symmetry and mean reversion properties of the process , and thus of one can prove that where is an integer. Moreover, by exploiting the definition of , one can prove that
[TABLE]
thus as .
Finally, we stress out that matches the first two moments of the process , thus weak convergence for is guaranteed ant it can be proved as done by Nelson and Ramaswamy [22].
To conclude, we observe that the set defined in (5.5) is equal to : the only difference concerns the indexing of the elements and in particular for .
Appendix B Computing expected value (5.8)
B.1 The binomial tree approach
In this Appendix we explain how to efficiently compute the expectation in (5.8). Such a computation can be seen as a particular case of a more general problem: computing
[TABLE]
where is a given function. Moreover and are two consecutive policy anniversaries times and thus . Furthermore, and .
First of all, let us consider the Gaussian vector given by
[TABLE]
It is possible to prove that the mean vector of is given by
[TABLE]
where
[TABLE]
Moreover, the covariance matrix of is given by
[TABLE]
with
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Let be the lower triangular Cholesky decomposition of , and suppose to be a Gaussian standard vector. So, the random vector has the same law of . Following the same approach of Ekvall [8] for multidimensional simulation, we can develop a binomial tree method to compute (B.1). Such a method exploits 3 independent binomial approximations of , and . In particular, we consider the binomial random variable
[TABLE]
and define
[TABLE]
It is well known that converges in distribution to a standard normal distribution so, if are i.i.d. random variables that have the same law of , then the vector given by
[TABLE]
converges in distribution to . Let be the support of and let
[TABLE]
for be the associated probabilities. Let be three integers in and let be the vector defined by
[TABLE]
Please observe that, since is lower triangular, does not depend on and , while does not depend on . Moreover
[TABLE]
In order to approximate (B.1), we replace and with and respectively. We obtain
[TABLE]
Such an expression converges to thanks to the properties of convergence in distribution for expected values (see for example Pollard [25]). Please observe that, by leaving the random variable as the third component in , the two variables and do not depend on . So, in order to evaluate (B.1), the function needs to be evaluated only times in place of . This is a relevant improvement, because evaluating the function many times can be time demanding. Moreover, if the function is known only on the grid – this is what happens for the functions and – then a two-dimensional interpolation is required.
B.2 Improving computational efficiency
The Markov chain introduced in Appendix A does not only provide a way to define the set but it can be used to improve the evaluation of (B.1). Suppose now in (B.1) to be equal to for a particular integer . Let so that and . In order to improve the discretisation of the random variable we replace in (B.17) with . The transition probabilities
[TABLE]
for can be obtained by computing the -power of transition matrix of , whose elements are determined according to Table 6. Finally, we conclude by observing that the support of the random variable is a subset of for every value in . We stress out that the support of the random variable has elements, while the support of has at most elements. Numerical tests show that is usually smaller than , so computational efficiency is improved: for example, with respect to our tests in Section 6, we have and .
The Markov chain helps us to discretize the process but in order to compute (B.19) we also have to simulate the whole random vector in (B.2). To do so, we have to compute the normal Gaussian increments associated to the transitions of . Let us define the discrete random variable as the standard score of , that is
[TABLE]
Since matches the first two moment of the random Gaussian variable , then matches the first two moments of a standard Gaussian variable and so it can be employed in place of . Moreover, and . Then, we define the vector given by
[TABLE]
which converges to and in particular . Let be three integers such that is in and are in . Let be the vector defined by
[TABLE]
where is the support of . Please, note that is equal to which is in . Thus, we obtain the following approximation of , based on the Markov chain :
[TABLE]
We conclude by observing that equation (B.24) has one important advantage over equation (B.19), that improves computational efficiency when the function is known only at the points of . Specifically, the computation of (B.19) requires a two-dimensional interpolation to evaluate the function outside while (B.24) requires only a one-dimensional interpolation because, as opposed to , is an element of . To this aim, we employ one-dimensional cubic spline interpolation, which is very fast and accurate.
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