Time-consistent Optimal Portfolio Strategy for Asset-liability Management under Mean-variance Criterion

This paper studies the time-consistent optimal portfolio strategy of an investor with an exogenous liability. Assume that the investor adopts the mean-variance criterion and trades continuously in a market consisting of one risk-free asset and one risky asset; and the price of the risky asset and the value of the exogenous liability are governed by geometric Brownian motions. An extended Hamilton-Jacobi-Bellman equation is derived, and the analytical expressions of the time-consistent optimal portfolio strategy and the mean-variance efficient frontier are obtained. A numerical example is provided to show the results. Our main findings are: (1) introducing an exogenous liability makes the time-consistent optimal portfolio strategy be a stochastic process; (2) the efficient frontier under the time-consistent optimal strategy for asset-liability management is below both the one under the time-consistent optimal strategy in the case of no liability and the one under the pre-commitment optimal strategy for asset-liability management