Mathematical Finance (q-fin.MF)

Abstract: Being a long-term investor has become an argument by itself to sustain larger allocations to risky assets, but - although intuitively appealing - it is rarely stated exactly why capital markets would provide a better opportunity set to investors with long investment horizons than to other investors. In this paper, it is shown that if in fact the equity risk-premium is slowly mean-reverting then an investor committing to a long-term deterministic investment strategy would realize a better risk-return trade-off in a mean-variance optimization than investors with shorter investment horizons. It is well known that the problem of mean-variance optimization cannot be solved by dynamic programming. Instead, the principle of Calculus of Variations is applied to derive an Euler-Lagrange equation characterizing the optimal investment strategy. It is a main result that the optimization problem is equivalent to a spectral problem by which explicit solutions to the optimal investment strategy can be derived for an equilibrium market of bonds and equity. In this setting, the paper contributes to portfolio choice in continuous time in the tradition of Markowitz.