Portfolio Optimization With a Guaranteed Minimum Maturity Benefit and Risk-Adjusted Fees

Abstract

We study a portfolio optimization problem involving the rational policyholder of a variable annuity with a guaranteed minimum maturity benefit. This financial guarantee is financed via a fee withdrawn directly from the investment account, which impacts the net investment return. We propose a new fee structure that adjusts to the investment mix. A fair pricing constraint is defined in terms of the risk-neutral value of the final contract payout. We seek the investment strategy that maximizes the policyholder's expected utility of terminal wealth after the application of a financial guarantee and subject to the fair pricing constraint. We solve the associated constrained stochastic control problem using a martingale approach and analyze the impact of the fee structure on the optimal investment strategies and payoff. Numerical results show that it is possible to nd an optimal portfolio for a wide range of fees, while keeping the contract fairly priced.

Keywords: Variable annuity, portfolio optimization, expected utility, optimal control, non-concave utility, guaranteed minimum maturity benefit

JEL Classification: C61, G22