Risk Aversion and the Shape of the Utility Function
We are now prepared to introduce a more precise definition of risk aversion. An investor is said to be risk averse if his utility function is concave, which in turn means that the investor requires higher expected return to bear risk. Exhibit 1.1 displays the log function for various values of wealth. We can see that the level of utility increases but at a decreasing rate.
Alternatively, a risk-averse investor avoids taking risks with zero expected payoffs. That is, for risk-averse investors,
1.5.4 Expressing Utility Functions in Terms of Expected Return and Variance
The principle of selecting investment strategies and allocations to maximize expected utility provides a very flexible way of representing the asset owner's preferences for risk and return. The representation of expected utility can be made more operational by presenting it in terms of the parameters of the probability distribution functions of investment choices. The most common form among institutional investors is to present the expected utility of an investment in terms of the mean and variance of the investment returns. That is,
Here, μ is the expected rate of return on the investment, σ2 is the variance of the rate of return, and λ is a constant that represents the asset owner's degree of risk aversion. It can be seen that the higher the value of λ, the higher the negative effect of variance on the expected value. For example, if λ is equal to zero, then the investor is said to be risk neutral and the investment is evaluated only on the basis of its expected return. A negative value of λ would indicate that the investor is a risk seeker and actually prefers more risk to less risk.
The degree of risk aversion indicates the trade-off between risk and return for a particular investor and is often indicated by a particular parameter within a utility function, such as λ in Equation 1.5. The fact that the degree of risk aversion is divided by 2 will make its interpretation much easier. It turns out that if Equation 1.5 is used to select an optimal portfolio for an investor, then the ratio of the expected rate of return on the optimal portfolio in excess of the riskless rate divided by the portfolio's variance will be equal to the degree of risk aversion.
Example: Suppose λ = 5. Calculate the expected utility of investments C and D.
In this case, the expected utility of investment C is higher than that of investment D; therefore, it is the preferred choice. It can be verified that if λ = 1, then the expected utility of investments C and D will be 0.0059 and 0.00949, respectively, meaning that D will be preferred to C.
APPLICATION 1.5.4
Suppose that an investor's expected utility, E[U(W)], from an investment can be expressed as:
where W is wealth, μ is the expected rate of return on the investment, σ2 is the variance of the rate of return, and λ is a constant that represents the asset owner's degree of risk aversion.
Use the expected utility of an investor with λ = 0.8 to determine which of the following investments is more attractive:
Investment A: μ = 0.10 and σ2 = 0.04
Investment B: μ = 0.13 and σ2 = 0.09
The expected utility of A and B are found as:
Investment A:
Investment B:
Because the investor's expected utility of holding B is higher, investment B is more attractive.
EXHIBIT 1.2 Properties of Two Hedge Fund Indices
Source: HFR and authors' calculations.
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:
Here, λ can be interpreted as the degree of risk aversion toward VaR, and VaRα 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