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

We study a portfolio optimization problem involving the loss averse policyholder of a variable annuity with a guaranteed minimum maturity benefit. This financial guarantee is financed via a fee withdrawn directly from the investment account, which impacts the net investment return. A fair pricing constraint is defined in terms of the risk-neutral value of the final contract payout. We propose a new fee structure that adjusts to the investment mix maximizing policyholder’s utility while keeping the contract fairly priced. We seek the investment strategy that maximizes the policyholder’s expected utility of terminal wealth after the application of a financial guarantee and subject to the fair pricing constraint. We assume that the policyholder’s risk attitude is relative to a reference level, risk-seeking towards losses and risk-averse towards gains. 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 find an optimal portfolio for a wide range of fees, while keeping the contract fairly priced.