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Ex Post Efficient Set Mathematics

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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 C*apital 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.

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*Journal of Mathematical Finance*, Vol. 3 No. 1A, 2013, pp. 201-210. doi: 10.4236/jmf.2013.31A019.

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