Ordering of risks

Comparing risks is the very essence of the actuarial profession. This chapter offers mathematical concepts and tools to do this, and derives some important results of non-life actuarial science that can be derived. There are two reasons why a risk, representing a non-negative random financial loss, would be universally preferred to another. One is that the other risk is larger, see Section 7.2, the second is that it is thicker-tailed (riskier), see Section 7.3. Thicker-tailed means that the probability of extreme 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 preferring the risk with uniformly lower stop-loss premiums describes 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 order 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 about the attractiveness of risks about which there is no consensus in a group of decision makers all considered sensible.