Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 54

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

(3.14)

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 Based on your own views of risk, write down a few properties you believe a consistent risk preference should exhibit. For example, if you prefer X to Y what can you say about X + W vs. Y + W for another risk W?