TABLE OF CONTENTS Comparing risks is the very essence of the actuarial profession. This chapter offers some mathematical concepts and tools to do this, and gives some important results of non-life actuarial science that can be derived. A risk, by which we mean a non-negative random variable, can be preferable to another for two reasons. One is that the other risk is larger, see Section 10.2, the second is that it is thicker-tailed ( riskier), see Section 10.3. Thicker-tailed means that the probability of large values is larger, making a risk with equal mean less attractive because it is more spread and therefore less predictable. We show that having thicker tails means having larger stop-loss premiums.
We also show that the latter is equivalent to the common preferences between risks of all risk averse decision makers. From the fact that a risk is smaller or less risky than another, one may deduce that it is also preferable in the mean-variance ordering that is used quite generally. In this ordering, one prefers the risk with the smaller mean, and the variance serves as a tie-breaker. This ordering concept, however, is inadequate for actuarial purposes, since it leads to decisions that many sensible decision makers would dispute. We give several invariance properties of the stop-loss order. The most important one for actuarial applications is that it is preserved under compounding, when either the number of claims or the claim size distribution is replaced by a riskier one.
In Section...
