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Sharpe Ratio

Under CAPM, the market portfolio is the portfolio on the efficient frontier with the highest Sharpe ratio. The slope of the capital market line equals the market (i.e. index) Sharpe ratio.

The Sharpe Ratio makes implicit assumptions which stem from CAPM. It assumes either 1) normally distributed returns or 2) mean-variance preferences.

Both assumptions are suspect:
1) The returns generated by most hedge funds exhibit negative skewness.
2) In addition to the mean and variance, people also care about skewness (they like it positive) and kurtosis (they don't like it), and higher moments matter too.

Because the Sharpe ratio is oblivious of all moments higher than the variance, it is prone to manipulation. Goetzmann, et al. (2004) proved that an optimal (high) Sharpe ratio strategy would produce a distribution with a truncated right tail and a fat left tail (see chart below). That is, generate regular modest profits punctuated by occasional crashes, i.e. negative skewness. As mentioned above, most investors prefer positive skewness, therefore, although a high Sharpe ratio is good thing, a high Sharpe ratio strategy is a bad thing. An optimal Sharpe ratio strategy would be to sell high-priced, low probability payoffs, for example short out-of-the-money puts or short out-of-the-money calls.

maximal Sharpe ratio

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