Dynamic Mean Variance Analysis

Abstract

We analyze the conditional versions of two closely connected mean-variance investment problems, the replication of a contingent claim on the one hand and the selection of an efficient portfolio on the other hand, in a general discrete time setting with incomplete markets. We exhibit a positive process h which summarizes two pieces of economically meaningful information. As a function of time, it describes the time dimension of the investment opportunity set through its link with the notion of dynamic Sharpe ratio. As a function of the states of the world, it can be used as a correction lens for myopic investors, and it reveals the gap between static and dynamic mean-variance investment strategies. A short sighted investor who corrects the probability distribution with the help of h acts optimally for long horizons. We describe the dynamic mean-variance efficient frontier conditioned on the information available at a future date in the form of a two fund separation theorem. The dynamic Sharpe ratio measures the distance from of an investment strategy to the efficient frontier. We explain how optimal dynamic Sharpe ratios aggregate through time and we study the time consistency rules which efficient portfolios must follow. We investigate the effect of a change of investment horizon, in particular we show that myopia is optimal as soon as the process h is deterministic.