Premium Principles and Their Properties¶

Definition¶

A premium principle is a rule $\Pi$ assigning a price to a loss law $X$ . The course considers nonnegative loading $\Pi(X)\ge\mathbb{E}[X]$ $E [X]$ , no rip-off $\Pi(X)\le\operatorname{ess\,sup}X$ (the essential supremum, the smallest $m$ with $\mathbb{P}(X\le m)=1$ $1$ , i.e. the largest possible loss), translation consistency $\Pi(X+c)=\Pi(X)+c$ $c$ , additivity for independent losses, proportionality $\Pi(aX)=a\Pi(X)$ and iterativity $\Pi(X)=\Pi(\Pi(X\mid Y))$ $Y))$ .

Interpretation¶

The properties describe how pricing reacts to certain cash flows, rescaling, pooling and conditioning. A plausible-looking formula need not satisfy all of them.

Two number lines: a general loss with the band from E[X] to ess sup X shaded as admissible premiums, and a sure loss X = 10 where the premium 12 lies outside that band.

Nonnegative loading and no rip-off together confine a premium to the green band $\mathbb{E}[X]\le\Pi(X)\le\operatorname{ess\,sup}X$ $ess sup X$ . For the deterministic loss $X=10$ in the example, both ends equal 10, so the band shrinks to a single point. The expected-value premium $1.2\,\mathbb{E}[X]=12$ satisfies the loading condition but lies above the largest possible loss, so it violates no rip-off.

Pick a principle and a loss, then move $\theta$ or $\alpha$ : each readout checks one property on this loss, and a single cross is a counterexample. The default $\Pi(X)=1.2\mathbb E[X]$ on $X=10$ charges 12 and violates no rip-off.

Stop-loss order is one way to check whether a pricing rule respects a risk comparison. For example, it orders finite means, so an expected-value premium with fixed nonnegative loading preserves the order. This does not imply that every premium principle has that property.

Key Results¶

Iterativity applies the same rule first conditionally and then to the random conditional premium.
A universal property needs a proof; one counterexample disproves it.
An expected-value loading with positive coefficient fails translation consistency for positive constants.
Risk-order compatibility can be checked by expressing a premium in terms of increasing or convex functions.
No rip-off is only a finite upper bound when the loss is essentially bounded, that is, $\operatorname{ess\,sup}X<\infty$ $\infty$ .

Examples¶

For $\Pi(X)=1.2\mathbb{E}[X]$ and deterministic $X=10$ $10$ , the premium is 12. It violates no rip-off, since the largest possible loss is only 10.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 11

Aliases: premium functional, premium principle, premium calculation principle, properties of premium principles Keywords: nonnegative loading, no rip-off, translation consistency, additivity, proportionality, iterativity, premium loading Symbols: alpha, infinity, pi, theta

Last updated: 17 October 2026 21:53