Ex Post Efficient Set Mathematics

This paper considers efficient set mathematics for the case where the covariance matrix of asset returns is assumed known but ex ante the vector of expected returns is replaced by an estimated or forecast value. It is shown that the ex post mean and variance differ from the standard results. Consequently the maximum Sharpe ratio portfolio also differs from the standard result. However, even with uncertainty about the vector of expected returns, subject to the assumptions made about the joint distribution of actual returns and estimated mean returns, ex post Sharpe ratio maximisers hold the ex post market portfolio. The properties of the zero beta portfolio are similar to the standard results leading to a capital market line. The ex post Capital Asset Pricing Model incorporates an intercept and the betas are not the same as those computed ex ante. The results are illustrated with an example.