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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.