As we explained in the previous chapter, risk-management problems can be viewed as problems aimed at optimizing a variety of criteria.1 In this chapter, we examine how risk measures are defined and used to help select solutions to the risk-management problems of either individual or institutional investors. There are many different risk measures, but their use is based on the same considerations: Each measure attempts both to describe risks and to provide some information for solving risk-management problems. However, since any risk measure gives a summary description of lottery characteristics, it can only be used correctly in the context of particular optimization problems, as we show in this chapter.
A risk measure is a function, denoted by ρ, that maps lotteries X into the set of real numbers R:2
ρ: X → R
The function ρ is defined to have a positive range. For example, if a risk management problem is formulated as maximizing an expected utility function u, and if u has a positive range,3 then:
ρ(X) ≡ m − E[u(X)]
where m is an appropriately large positive constant, could serve as a risk measure. While this observation shows how risk measures are related to risk-management problems, it is not very informative because it simply restates the original problem. Fortunately, there are several other approaches to finding helpful measures, as we demonstrate in the balance of this chapter.