Guaranteed Minimum Withdrawal Benefit in Variable Annuities

Abstract

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 efficient 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.

Pricing guaranteed minimum withdrawal benefits under stochastic interest rates

“…They analyze the fair proportional fees that should be charged on the provision of the guarantee. Dai, Kwok and Zong (2008) develop a singular stochastic control model for pricing GMWB under dynamic withdrawal. An efficient finite difference algorithm using the penalty approximation approach is also proposed for solving the singular stochastic control model.…”

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

“…We will also confirm this from some numerical experiments. Note that as discussed in (Dai et al, 2008), no boundary condition is required at A = ω 0 due 156 to hyperbolic nature of the variable A. Since equations (3.7), (3.10) can be solved without any…”

Policyholder Exercise Behavior in Life Insurance: The State of Affairs

“…This is not to say that actually determining the optimal strategies within a risk-neutral valuation framework is trivial. It may require the solution of optimal control problems akin to the valuation of American or Bermudan options, and a great number of contributions in actuarial science have taken this approach to evaluate various types of contracts (Milevsky and Salisbury, 2006;Ulm, 2006;Bauer et al, 2008;Dai et al, 2008;Bauer et al, 2010;Bacinello et al, 2011, among many others).…”

“…However, as already indicated, several studies have found discrepancies with corresponding results and market observations when following the value-maximizing approach. For instance, with regards to GMWBs Milevsky and Salisbury (2006) report an "underpricing of this feature [GMWBs] in a typically overpriced VA market" and posit that "only if several unrealistic modeling assumptions are made it is possible to obtain GMWB fees in the same range as is normally charged" (for similar assertions, see Dai et al (2008); Blamont and Sagoo (2009)). Piscopo (2010) states that under a no-arbitrage valuation, "GLWB issued on the USA market are underpriced" and that "market fees are not sufficient to cover the market hedging cost of the guarantee."…”