Variable Annuity with GMWB: Surrender Or Not, That Is the Question

Abstract

A variable annuity contract with Guaranteed Minimum Withdrawal Benefit (GMWB) promises to return the entire initial investment through cash withdrawals during the policy life plus the remaining account balance at maturity, regardless of the portfolio performance. We assume that market is complete in financial risk and also there is no mortality risk (in the event of policyholder death, the contract is maintained by beneficiary), thus the annuity price can be expressed as appropriate expectation. Under the optimal withdrawal strategy of a policyholder, the pricing of variable annuities with GMWB is an optimal stochastic control problem. The surrender feature allows termination of the contract before maturity, making it also an optimal stopping problem.Although the surrender feature is quite common in variable annuity contracts, there appears to be no published analysis and results for this feature in GMWB under optimal policyholder behavior - results found in the literature so far are consistent with the absence of such a feature. Recently, Azimzadeh and Forsyth (2014) prove the existence of an optimal bang-bang control for a Guaranteed Lifelong Withdrawal Benefits (GLWB) contract. In particular, they find that the holder of a GLWB can maximize a writer’s losses by only ever performing non-withdrawal, withdrawal at exactly the contract rate, or full surrender. This dramatically reduces the optimal strategy space. However, they also demonstrate that the related GMWB contract is not convexity preserving, and hence does not satisfy the bang-bang principle other than in certain degenerate cases. For GMWB under optimal withdrawal assumption, the numerical algorithms developed by Dai et al. (2008), Chen and Forsyth (2008) and Luo and Shevchenko (2015a) appear to be the only ones found in the literature, but none of them actually performed calculations with surrender option on top of optimal withdrawal strategy. Also, it is of practical interest to see how the much simpler bang-bang strategy, although not optimal for GMWB, compares with optimal GMWB strategy with surrender option.Recently, in Luo and Shevchenko (2015a), we have developed a new efficient numerical algorithm for pricing GMWB contracts in the case when transition density of the underlying asset between withdrawal dates or its moments are known. This algorithm relies on computing the expected contract value through a high order Gauss-Hermite quadrature applied on a cubic spline interpolation and much faster than the standard PDE methods. In this paper we extend our algorithm to include surrender option in GMWB and compare prices under different policyholder strategies: optimal, static and bang-bang. Results indicate that following a simple but sub-optimal bang-bang strategy does not lead to significant reduction in the price or equivalently in the fee, in comparison with the optimal strategy. We also observed that the extra value added by the surrender option strongly depends on volatility and the penalty charge; among other factors are contractual rate, maturity, etc. At high volatility or at low penalty charge, the surrender feature adds very significant value to the GMWB contract - the required fair fee is more than doubled in some cases. We also performed calculations for static withdrawal with surrender option, which is the same as bang-bang minus the "no-withdrawal" choice. We find that the fee for such contract is only less than 1% smaller when compared to the case of bang-bang strategy, meaning that the "no-withdrawal" option adds little value to the contract.