Читать книгу Alternative Investments - Black Keith H. - Страница 34

A problem that is especially acute with alternative investments is that the mean-variance optimization approach considers only the mean and variance of returns. This means that the optimization model does not explicitly account for skewness and kurtosis. Investors' expected utility can be expressed in terms of mean and variance alone if returns are normally distributed. However, when making allocations to alternative investments and other investments with nonzero skewness and nonzero excess kurtosis, portfolio optimizers tend to suggest portfolios with desirable combinations of mean and variance but with highly undesirable skewness and kurtosis. In other words, although mean-variance optimizers can identify the efficient frontier and help create portfolios with the highest Sharpe ratios, they may be adding large and unfavorable levels of skewness and kurtosis to the portfolio.

For example, two assets with returns that have the same variance may have very different skews. In a competitive market, the expected return of the asset with the large negative skew might be substantially higher than that of the asset with the positive skew to compensate investors willing to bear the higher downside risk. A mean-variance optimizer typically places a much higher portfolio weight on the negatively skewed asset because it offers a higher mean return with the same level of variance as the other asset. The mean-variance optimizer ignores the unattractiveness of an asset's large negative skew and, in so doing, maximizes the error.

There are three common ways to address this complication. First, as we saw earlier, it is possible to expand our optimization method to account for skewness and kurtosis of asset returns. Second, we can continue with our mean-variance optimization but add the desired levels of the skewness and kurtosis as explicit constraints on the allowed solutions to the mean-variance optimizer, such as when the excess kurtosis of the portfolio returns is not allowed to exceed 3, or when the skewness must be greater than –0.5. A problem with incorporating higher moments in portfolio optimization is that these moments are extremely difficult to predict, as they are highly influenced by a few large negative or positive observations. In addition, in the second approach, a portfolio with a desired level of skewness or kurtosis may not be feasible at all. Finally, the analyst may choose to explicitly constrain the weight of those investments that have undesirable skew or kurtosis. For example, the allocation to a hedge fund strategy that is known to have large tail risk (e.g., negative skew) might be restricted to some maximum weight.