Strategic mean-variance investing under mean-reverting stock returns

Abstract

In this report we derive the strategic (deterministic) allocation to bonds and stocks resulting in the optimal mean-variance trade-off on a given investment horizon. The underlying capital market features a mean-reverting process for equity returns, and the primary question of interest is how mean-reversion effects the optimal strategy and the resulting portfolio value at the horizon. In particular, we are interested in knowing under which assumptions and on which horizons, the risk-reward trade-off is so favourable that the value of the portfolio is effectively bounded from below on the horizon. In this case, we might think of the portfolio as providing a stochastic excess return on top of a "guarantee" (the lower bound). Deriving optimal strategies is a well-known discipline in mathematical finance. The modern approach is to derive and solve the Hamilton-Jacobi-Bellman (HJB) differential equation characterizing the strategy leading to highest expected utility, for given utility function. However, for two reasons we approach the problem differently in this work. First, we wish to find the optimal strategy depending on time only, i.e., we do not allow for dependencies on capital market state variables, nor the value of the portfolio itself. This constraint characterizes the strategic allocation of long-term investors. Second, to gain insights on the role of mean-reversion, we wish to identify the entire family of extremal strategies, not only the optimal strategies. To derive the strategies we employ methods from calculus of variations, rather than the usual HJB approach.