New light on the portfolio allocation problem

The basics of the mean-variance portfolio optimisation procedure have been well understood since the seminal work of Markowitz in the 1950's. A vector x of asset weights, restricted only by requiring its components to add to 1, is to be chosen so that the linear combination μ_p = x′μ of the expected asset returns μ (or, expected excess returns), which represents the expected return on a portfolio, is maximised for a specified level of "risk", σ_p, the standard deviation of the portfolio. The efficient frontier is the curve traced out in (μ_p, σ_p) space by portfolios whose return/risk tradeoff is optimal in this sense. The portfolio with the maximum Sharpe ratio is the portfolio with the highest return/risk tradeoff achievable from the assets, and under some conditions can be obtained as the point of tangency of a line from the origin to the efficient frontier, as is well known. But when the tangency approach fails, which it commonly can, the question arises as to the maximum Sharpe ratio achievable from the assets. This problem, which has not been dealt with before, is solved explicitly in this paper, and the corresponding optimal portfolio found. The suggested procedure is easily implemented when the usual inputs - estimates of mean excess returns and their covariance matrix - are available.