Multi-period mean–variance asset–liability management with uncontrolled cash flow and uncertain time-horizon

This paper considers an asset–liability management problem under a multi-period mean–variance model with uncontrolled cash flow and uncertain time-horizon. The difference from the existing literature is that the liability is assumed to be influenced not only by the stochastic return of the liability but also by some uncontrolled cash flows, which can be explained as, for example, stochastic expenditure of individual investors, or claim processes of insurers. Firstly, the original problem is translated into a standard multi-period stochastic optimal control problem by introducing a Lagrange multiplier, and the corresponding analytical solution is derived by adopting the dynamic programming approach. Secondly, according to Lagrange duality theorem, closed-form expressions for the efficient investment strategy and the mean–variance efficient frontier are obtained. Moreover, a multi-period version of two-fund separation theorem is proved, and some special cases are discussed. Finally, some numerical examples are presented to illustrate our results.