Optimal mean-variance efficiency of a family with life insurance under inflation risk

We study an optimization problem of a family under mean-variance efficiency. The market consists of cash, a zero-coupon bond, an inflation-indexed zero-coupon bond, a stock, life insurance and income-replacement insurance. The instantaneous interest rate is modeled as the Cox-Ingersoll-Ross (CIR) model, and we use a generalized Black-Scholes model to characterize the stock and labor income. We also take into account the inflation risk and consider our problem in the real market. The goal of the family is to maximize the mean of the surplus wealth at the retirement or death of the breadwinner and minimize its variance by finding a portfolio selection. The efficient frontier and optimal strategies are derived through the dynamic programming method and the technique of solving associated nonlinear HJB equations. We also present a numerical illustration to explore the impact of economical parameters on the efficient frontier.