Multiperiod mean-variance efficient portfolios with endogenous liabilities

We study the optimal policies and mean-variance frontiers (MVF) of a multiperiod mean-variance optimization of assets and liabilities (AL). This makes the analysis more challenging than for a setting based on purely exogenous liabilities, in which the optimization is only performed on the assets while keeping liabilities fixed. We show that, under general conditions for the joint AL dynamics, the optimal policies and the MVF can be decomposed into an orthogonal set of basis returns using exterior algebra. This formalism, novel to financial applications, allows us to study analytically the structure of optimal policies and MVF representations under endogenous liabilities in a multidimensional and multiperiod setting. Using a numerical example, we illustrate our methodology by analysing the impact of the rebalancing frequency on the MVF and by highlighting the main differences between exogenous and endogenous liabilities.