Guaranteed Minimum Withdrawal Benefit Research Articles

Semi-Static Hedging for GMWB in Variable Annuities

The Guaranteed Minimum Withdrawal Benefit (GMWB) is an option embedded in a variable annuity that guarantees the policyholder to get the initial investment back by making periodic withdrawals regardless of the impact of poor market performance. In the paper we discuss methods of pricing and hedging of some versions of GMWBs. In particular we develop a method of constructing semi-static hedging strategies that offer several advantages over dynamic hedging. The idea is to first find the closest path-independent option to the guarantee, or its liability part, and then to construct a portfolio of traded European options that approximates the optimal option. This strategy requires fewer portfolio adjustments than delta hedging and outperforms the latter when there are random jumps in the underlying price. We illustrate the proposed method with numerical examples.

Analysis of a penalty method for pricing a guaranteed minimum withdrawal benefit (GMWB)

Analysis of a penalty method for pricing a guaranteed minimum withdrawal benefit (GMWB)

Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates

We consider the pricing of variable annuities with the Guaranteed Minimum Withdrawal Benefit (GMWB) under the Vasicek stochastic interest rates framework. The holder of the variable annuity contract pays an initial purchase payment to the insurance company, which is then invested in a portfolio of risky assets. Under the GMWB, the holder can withdraw a specified amount periodically over the term of the contract such that the return of the entire initial investment is guaranteed, regardless of the market performance of the underlying asset portfolio. The investors have the equity participation in the reference investment portfolio with protection on the downside risk.The guarantee is financed by paying annual proportional fees. Under the assumption of deterministic withdrawal rates, we develop the pricing formulation of the value function of a variable annuity with the GMWB. In particular, we derive analytic approximation solutions to the fair value of the GMWB under both equity and interest rate risks, obtaining both the lower and upper bound on the price functions. The pricing behavior of the embedded GMWB under various model parameter values is also examined.

A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB)

In this paper, we outline an impulse stochastic control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB) assuming the policyholder is allowed to withdraw funds continuously. We develop a numerical scheme for solving the Hamilton–Jacobi–Bellman (HJB) variational inequality corresponding to the impulse control problem. We prove the convergence of our scheme to the viscosity solution of the continuous withdrawal problem, provided a strong comparison result holds. The scheme can be easily generalized to price discrete withdrawal contracts. Numerical experiments are conducted, which show a region where the optimal control appears to be non-unique.

The effect of modelling parameters on the value of GMWB guarantees

In this article, an extensive study of the no-arbitrage fee for Guaranteed Minimum Withdrawal Benefit (GMWB) variable annuity riders is carried out. The value of the GMWB guarantee increases substantially when taking into account the separation of mutual fund fees and the fees earmarked for hedging the guarantee. In addition, the fee is also adversely affected if the underlying risky asset follows a jump diffusion process. We also explore the effects of various modelling assumptions on the optimal withdrawal strategy of the contract holder, as well as the impact on the guarantee value of sub-optimal withdrawal behaviour. Our general conclusions are that only if several unrealistic modelling assumptions are made is it possible to obtain GMWB fees in the same range as is normally charged. In all other cases, it would appear that typical fees are not enough to cover the cost of hedging these guarantees.

Guaranteed Minimum Withdrawal Benefit in Variable Annuities

We develop a singular stochastic control model for pricing variable annuities with the guaranteed minimum withdrawal benefit. This benefit promises to return the entire initial investment, with withdrawals spread over the term of the contract, irrespective of the market performance of the underlying asset portfolio. A contractual withdrawal rate is set and no penalty is imposed when the policyholder chooses to withdraw at or below this rate. Subject to a penalty fee, the policyholder is allowed to withdraw at a rate higher than the contractual withdrawal rate or surrender the policy instantaneously. We explore the optimal withdrawal strategy adopted by the rational policyholder that maximizes the expected discounted value of the cash flows generated from holding this variable annuity policy. An effcient finite difference algorithm using the penalty approximation approach is proposed for solving the singular stochastic control model. Optimal withdrawal policies of the holders of the variable annuities with the guaranteed minimum withdrawal benefit are explored. We also construct discrete pricing formulation that models withdrawals on discrete dates. Our numerical tests show that the solution values from the discrete model converge to those of the continuous model.

Financial valuation of guaranteed minimum withdrawal benefits

Financial valuation OF GMWBs: We develop a variety of methods for assessing the cost and value of a very popular ‘rider’ available to North American investors on variable annuity (VA) policies called a Guaranteed Minimum Withdrawal Benefit (GMWB). The GMWB promises to return the entire initial investment, albeit spread over an extended period of time, regardless of subsequent market performance. First, we take a static approach that assumes individuals behave passively and holds the product to maturity. We show how the product can be decomposed into a Quanto Asian Put plus a generic term-certain annuity. At the other extreme of consumer behavior, the dynamic approach leads to an optimal stopping problem akin to pricing an American put option, albeit complicated by the non-traditional payment structure. Our main result is that the No Arbitrage hedging cost of a GMWB ranges from 73 to 160 basis points of assets. In contrast, most products in the market only charge 30–45 basis points. Although we suggest a number of behavioral reasons for the apparent under-pricing of this feature in a typically overpriced VA market, we conclude by arguing that current pricing is not sustainable and that GMWB fees will eventually have to increase or product design will have to change in order to avoid blatant arbitrage opportunities.