3.6 Risk Measures
An ice cream manufacturer wants to introduce a product that customers will prefer over existing ones. It would be very helpful to have a way of predicting customer ice cream preferences. As far as a customer is concerned their preference may be very simple and intuitive: “this brand tastes better!” That preference is not expressed in a way that the manufacturer can use to predict customer responses to a new product. However, through taste tests the manufacturer is able to determine general principles which do predict the preferences of most customers. For example, most customers prefer ice cream with a higher fat content and natural rather than artificial ingredients. This information is a good start but ideally the ice cream manufacturer can find a way to represent it numerically—an ice cream measure, if you will—which will enable them to analyze it much more easily.
Risk preferences have many parallels with ice cream preferences. Both are somewhat idiosyncratic and personalized—but some general principles about them can be determined, although it is harder for risk since there is no simple taste test to elicit risk preferences. Like ice cream manufacturers, risk management professionals would benefit from having a risk measure that quantifies a true risk preference and allows them to predict how individuals act. In this section, we try to find such risk measures. It is important to note that risk preferences are opposite to ice cream preferences in the sense that better ice cream is preferable, whereas more risk is not.
Formally, a risk measure is a real-valued functional on a set of random variables that quantifies a risk preference—the way an individual or group of individuals decides risk questions. The random variables represent risks, and the risk measure conducts a taste test; given two, it predicts which one is preferred, i.e. has lower risk. A risk capital formula, such as NAIC RBC or Solvency II SCR, and a classification rating plan are archetypal risk measures. Section 6.5 provides a compendium of other standard risk measures.
3.6.1 Risk Preferences and Risk Measures
A risk preference models the way we compare risks and how we decide between them. It captures our intuitive notions of riskiness and converts them into a form we can use to predict future preferences. Using the ice cream analogy, the manufacturer needs to convert “this tastes better” into a series of preferences about ice cream ingredients, which can be used to predict the desirability of a new product.
Risk preferences are defined on a set of loss random variables S. We write X⪰Y if the risk X is preferred to Y. If X⪰Y and Y⪰X we are neutral between X and Y.
A risk preference for insurance loss outcomes needs to have the following three properties.
1 Complete (COM) for any pair of prospects X and Y either X⪰Y or Y⪰X or both, that is, we can compare any two prospects.
2 Transitive (TR) if X⪰Y and Y⪰Z then X⪰Z.
3 Monotonic (MONO) if X≤Y in all outcomes then X⪰Y.
The second property ensures the risk preference is logically consistent. The third reflects the reality that large positive outcomes for losses are less desirable than small ones. If X⪰Y then X is generally smaller or tamer than Y. The third property also ensures the risk preference takes into account the volume or size of loss, even when there is no variability. For example a uniform random loss between 0 and ¤1 million is preferred to a certain loss of ¤1 million, even though the former is variable and the latter is fixed.
Example 32 X⪰Y iff E[X]≤E[Y] defines a risk neutral preference. X⪰Y iff E[X]+SD(X)≤E[Y]+SD(Y) defines a mean-variance risk preference. Notice the order of the inequalities in both cases.
A risk measure is a numerical representations of risk preferences. If S and the preference ⪰ have certain additional properties then it is possible to find a risk measure ρ:S→R that represents it, in the sense that
The reversed inequality arises because ρ measures risk, and less risk is preferred to more.
The risk measure collapses a risk preference into a single number. It facilitates simple and consistent decision making. We consider risk measures and risk preferences in more detail in Section 5.
Exercise 33