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

Abstract

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 an 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 available in marketed products 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 writers losses by only ever performing nonwithdrawal, 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 partial differential equation 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 an...

A Unified Pricing of Variable Annuity Guarantees under the Optimal Stochastic Control Framework

“…Also, there is no death benefit; it is assumed that the beneficiary will maintain the contract in the case of a policyholder death. This contract has only basic features facilitating comparison of results from different academic studies, such as [4,5,7,14].…”

“…It is natural to form a uniform grid in A so that optimal withdrawal strategies can be tested on a constant increment δA = A j+1 − A j , as has been done successfully in [7] for pricing of a basic GMWB specified by (14) and (15). However, extensive numerical tests show that if a uniform grid in A is used for pricing GMAB with ratchets and optimal withdrawals (our numerical example in Section 7), then neither linear interpolation nor cubic interpolation in A can achieve an efficient convergence in pricing results.…”

“…Also, there is no death benefit; it is assumed that beneficiary will maintain the contract if the case of policyholder death. This contract has only basic features facilitating comparison of results from different academic studies, such as , Dai et al (2008), Luo and Shevchenko (2015c), Luo and Shevchenko (2015a). Specifications common in the industry include cases where the contractual amount G n is specified to be different from G n = W (0)(t n − t n−1 )/T and a penalty is applied to both the withdrawn amount and the benefit base.…”

“…Huang and Kwok (2015) developed a regression-based MC method for pricing GLWB under the bang-bang strategy in the case of stochastic volatility. GMWB pricing under the bang-bang strategy was studied in Luo and Shevchenko (2015c). The difficulty with applying the well known Least-Squares MC introduced in Longstaff and Schwartz (2001) for pricing VA riders under the optimal strategy is due to the fact that the paths of the underlying VA wealth account are affected by the withdrawals.…”

Valuation of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Stochastic Interest Rate

“…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. GMWB pricing under bang-bang strategy was studied in Luo and Shevchenko (2015c), and Huang and Kwok (2015) have developed a regression-based MC method for pricing GLWB. For GMWB under the optimal withdrawal strategy, the numerical evaluations have been developed by Dai et al (2008) and Chen and Forsyth (2008) using finite difference PDE methods and by Luo and Shevchenko (2015a) using direct integration method.…”