Maximize the sharpe ratio and minimize a VaR

In addition to its role as the optimal ex ante combination of risky assets for a risk-averse investor, possessing the highest potential return-for-risk tradeoff, the tangency or Maximum Sharpe Ratio portfolio in the Markowitz (1952, 1991) procedure plays an important role in asset management, as it minimizes the probability that a future portfolio return falls below the risk-free or reference rate. This is a kind of Value at Risk (VaR) property of the portfolio. In this paper we demonstrate the way this VaR, and related quantities, vary along the efficient frontier, emphasizing the special role played by the tangency portfolio. The results are illustrated with an analysis of the market crash of October 1987, as an episode of extreme negative market movements, where the tangency portfolio performs best (loses least!) among a variety of portfolios