Valuation of Variable Annuities with Guaranteed Minimum Withdrawal and Death Benefits via Stochastic Control Optimization

TL;DR

This paper develops a numerical method to value variable annuities with combined guaranteed benefits, accounting for optimal policyholder behavior, financial and mortality risks, revealing challenges in fee structures for long-term contracts.

Contribution

It introduces an efficient Gauss-Hermite quadrature method to solve the stochastic control problem for complex annuity products with death benefits and analyzes fee implications.

Findings

Adding death benefits complicates fee structures for long maturities.

Optimal policyholder behavior affects the feasibility of continuous fee models.

Upfront or fixed installment fees are more practical than proportional fees for long-term contracts.

Abstract

In this paper we present a numerical valuation of variable annuities with combined Guaranteed Minimum Withdrawal Benefit (GMWB) and Guaranteed Minimum Death Benefit (GMDB) under optimal policyholder behaviour solved as an optimal stochastic control problem. This product simultaneously deals with financial risk, mortality risk and human behaviour. We assume that market is complete in financial risk and mortality risk is completely diversified by selling enough policies and thus the annuity price can be expressed as appropriate expectation. The computing engine employed to solve the optimal stochastic control problem is based on a robust and efficient Gauss-Hermite quadrature method with cubic spline. We present results for three different types of death benefit and show that, under the optimal policyholder behaviour, adding the premium for the death benefit on top of the GMWB can be problematic for contracts with long maturities if the continuous fee structure is kept, which is ordinarily assumed for a GMWB contract. In fact for some long maturities it can be shown that the fee cannot be charged as any proportion of the account value -- there is no solution to match the initial premium with the fair annuity price. On the other hand, the extra fee due to adding the death benefit can be charged upfront or in periodic instalment of fixed amount, and it is cheaper than buying a separate life insurance.