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 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?
3.6.2 Volume, Volatility, and Tail Risk
Risk measures quantify the following three characteristics of a risk random variable.
1 Volume: a smaller risk is preferred. The mean measures volume.
2 Volatility: a risk exhibiting less volatility (or variability) is preferred. Variance and standard deviation measure volatility. We use the word ‘volatility’ in a sense parallel to stock price volatility. Volatility is two-sided.
3 Tail: a risk with a lower likelihood of extreme outcomes is preferred. The level of loss with a 1% probability of exceedance is a tail risk measure. Tail risk is one-sided.
These characteristics overlap.
A risk measure must reflect volume because we want it to mirror a risk preference satisfying the MONO axiom, smaller risks are preferred to larger ones—even if the small risk is volatile and the large risk is certain. A measure of variability or tail risk that ignores volume is called a measure of deviation. The eponymous standard deviation is the, well, standard example. Adding the mean to a measure of deviation creates a risk measure.
Risk-based capital (RBC) formulas are risk measures. Many of them are volume based. They compute target capital by applying factors to premium, reserve, or asset balances. The factors vary according to the risk of each element. Examples include NAIC RBC, the Solvency II Solvency Capital Requirement standard formula, and most rating agency capital adequacy models. Classification rating plans are also risk measures. They compute a premium as a function of risk characteristics, such as building value and location, construction, occupancy, protection, and use for property insurance.
A risk measure sensitive to volatility quantifies mild to moderately adverse outcomes: outcomes frequent enough that most actuaries see examples during their careers. Management is often concerned with quarterly results volatility and can suppose investors are similarly troubled. Standard deviation quantifies volatility risk very well, and a two deviations from the mean rule of thumb turns out to be a surprisingly accurate estimate of a 20-year event in many situations.
Tail risk represents something so extreme it may or may not be experienced during a career, nevertheless it must always be considered possible. A tail risk catastrophe event often doubles or triples the previous worst historical event.
Variability and tail risk are distinct. An outcome of winning ¤1 million or ¤3 million has variability, but much less tail risk for most people than the possibility of gaining or losing ¤1 million. The variability of the two is the same but the tail risk is different. Risk is always relative to a base.
In Section 5.2 we enumerate several mathematical properties that risk measures should have, providing another way to characterize them.
3.6.3 Applications of Risk Measures
Insurance company operations are governed by the interaction of two risk measures: a capital risk measure setting the needed amount of capital and a pricing risk measure determining its cost. These two roles should not be
