Читать книгу Alternative Investments - Black Keith H. - Страница 17

When the expected utility is presented as in Equation 1.5, we are assuming that risk can be measured using variance or standard deviation of returns. This assumption is reasonable if investment returns are approximately normal. While the normal distribution might be a reasonable approximation to returns for equities, empirical evidence suggests that most alternative investments have return distributions that significantly depart from the normal distribution. In addition, return distributions from structured products tend to deviate from normality in significant ways. In these cases, Equation 1.5 will not be appropriate for evaluating investment choices that display significant skewness or excess kurtosis. It turns out that Equation 1.5 can be expanded to accommodate asset owners' preferences for higher moments (i.e., skewness and kurtosis) of return distributions. For example, one may present expected utility in the following form:

(1.6)

Here, S is the skewness of the portfolio value; K is the kurtosis of the portfolio; and λ₁, λ₂, and λ₃ represent preferences for variance, skewness, and kurtosis, respectively. It is typically assumed that most investors dislike variance (λ₁ > 0), like positive skewness (λ₂ > 0), and dislike kurtosis (λ₃ > 0). Note that the signs of coefficients change.

Example: Consider the information about two hedge fund indices in Exhibit 1.2.

If we set λ₁ = 10 and ignore higher moments, the investor would select the HFRI Fund Weighted Composite as the better investment, as it would have the higher expected utility (0.075 to 0.055). However, if we expand the objective function to include preference for positive skewness and set λ₂ = 1, then the investor would select the HFRI Fund of Fund Defensive as the better choice, because it would have a higher expected utility (0.29 to –0.54).