Alternative Investments. Black Keith H.

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      1.5.5 Expressing Utility Functions with Higher Moments

      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).

      1.5.6 Expressing Utility Functions with Value at Risk

      The preceding representation of preferences in terms of moments of the return distribution is the most common approach to modeling preferences involving uncertain choices. It is theoretically sound as well. However, the investment industry has developed a number of other measures of risk, most of which are not immediately comparable to the approach just presented. For instance, in the CAIA Level I book, we learned about value at risk (VaR) as a measure of downside risk. Is it possible to use this framework to model preferences in terms of VaR? It turns out that in a rather ad hoc way, one can use the preceding approach to model preferences on risk and return when risk is measured by VaR. That is, we can rank investment choices by calculating the following value:

(1.7)

      Here, λ can be interpreted as the degree of risk aversion toward VaR, and VaR_α is the value at risk of the portfolio with a confidence level of α.

We can further generalize Equation 1.7 and replace VaR with other measures of risk. For instance, one could use risk statistics, such as lower partial moments, beta with respect to a benchmark, or the expected maximum drawdown.

      1.5.7 Using Risk Aversion to Manage a Defined Benefit Pension Fund

      To complete our discussion of objectives, we now consider an application of the previous framework to present the objectives of a defined benefit (DB) pension fund. The following information is available:

      

Current value of the fund: €V billion

      

Number of asset classes considered: N

      

Return on asset class i: R_i

      

Weight of asset class i in the portfolio: w_i

      

Return on the portfolio:

Assuming that the preferences of the DB fund can be expressed as in Equation 1.5, the portfolio manager will select the weights, w_i, such that the expected utility is maximized. That is, Equation 1.8 expresses the objective function that is maximized by choosing the values of w_i. Of course, the portfolio manager must ensure that the weights will add up to one and some or all of the weights will need to be positive.

(1.8)

      1.5.8 Finding Investor Risk Aversion from the Asset Allocation Decision

      As mentioned previously, the value of the risk aversion has an intuitive interpretation. The expected excess rate of return on the optimal portfolio (E[R_P] − R_f) divided by its variance, σ²P, is equal to the degree of risk aversion, λ:

(1.9)

      The value of the parameter of risk aversion, λ, is chosen in close consultation with the plan sponsor. There are qualitative methods that can help the portfolio manager select the appropriate value of the risk aversion. The portfolio manager may select a range of values for the parameters and present asset owners with resulting portfolios so that they can see how their level of risk aversion affects the risk-return characteristics of the portfolio under current market conditions.

EXHIBIT 1.3 Hypothetical Risk Returns for Two Portfolios

Example: Consider the information for two well-diversified portfolios shown in Exhibit 1.3. The riskless rate is 2 % per year.

      Assuming that these are optimal portfolios for two asset owners, what are their degrees of risk aversion?

We know from Equation 1.9 that the expected excess return on each portfolio divided by its variance will be equal to the degree of the risk aversion of the investor who finds that portfolio optimal.

      Aggressive investor: (15 % − 2 %)/(16%²) = 5.1

      Moderate investor: (9 % − 2 %)/(8%²) = 10.9

      As expected, the aggressive portfolio represents the optimal portfolio for a more risk-tolerant investor, while the moderate portfolio represents the optimal portfolio for a more risk-averse investor.

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