Exponential Premium Principle¶

Definition¶

For risk-aversion parameter $\alpha>0$ and loss $X$ with $\mathbb{E}[\mathrm{e}^{\alpha X}]<\infty$ $\infty$ , the exponential premium is $\Pi_\alpha(X)=\alpha^{-1}\log\mathbb{E}[\mathrm{e}^{\alpha X}]=K_X(\alpha)/\alpha$ $K_{X} (α) / α$ .

Interpretation¶

This is the indifference price for exponential wealth utility. It gives extra weight to large losses and may be infinite for losses whose positive exponential moments do not exist.

Exponential premium of an exponential loss with rate λ=1 plotted against α: it starts at E[X]=1, follows 1+α/2 near zero and rises without bound as α approaches λ.

The figure draws the Examples formula with $\lambda=1$ $1$ , so $\mathbb{E}[X]=\operatorname{Var}(X)=1$ and $\Pi_\alpha(X)=\alpha^{-1}\log[1/(1-\alpha)]$ $α)]$ . The premium never falls below $\mathbb{E}[X]=1$ $1$ , as Jensen's inequality requires; near $\alpha=0$ it follows the dotted approximation $\mathbb{E}[X]+\alpha\operatorname{Var}(X)/2=1+\alpha/2$ $α /2$ ; and as $\alpha$ approaches $\lambda=1$ the moment $\mathbb{E}[\mathrm{e}^{\alpha X}]$ blows up, so the premium is infinite for $\alpha\ge\lambda$ $λ$ .

Move $\alpha$ towards $\lambda$ : $\Pi_\alpha(X)=\alpha^{-1}\log[\lambda/(\lambda-\alpha)]$ grows without bound and is infinite for $\alpha\ge\lambda$ $λ$ , while near $\alpha=0$ it follows $\mathbb E[X]+\tfrac12\alpha\operatorname{Var}(X)$ $\frac{1}{2} α Var (X)$ .

The cumulant generating function is the logarithm of the loss moment generating function (MGF), so $K_X(\alpha)/\alpha$ expresses the premium compactly. In insurance indifference premiums, substituting exponential wealth utility into the insurer's equation and cancelling the wealth factor gives precisely this formula.

Key Results¶

Jensen's inequality gives $\Pi_\alpha(X)\ge\mathbb{E}[X]$ $E [X]$ .
Translation consistency and independent additivity follow from exponential products.
Iterativity follows from the tower property: $\mathrm{e}^{\alpha\Pi_\alpha(X\mid Y)}=\mathbb{E}[\mathrm{e}^{\alpha X}\mid Y]$ $Y]$ .
It respects exponential order and hence stop-loss order.
For bounded $X$ , $\Pi_\alpha(X)\le\operatorname{ess\,sup}X$ $ess sup X$ .
Near $\alpha=0$ with sufficient moments and a finite MGF, $\Pi_\alpha(X)=\mathbb{E}[X]+\alpha\operatorname{Var}(X)/2+o(\alpha)$ $o (α)$ .

Examples¶

For exponential severity of rate $\lambda$ and $0<\alpha<\lambda$ $λ$ , premium is $\alpha^{-1}\log[\lambda/(\lambda-\alpha)]$ $α)]$ . At or above $\lambda$ it is infinite.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 12

Aliases: entropic premium, exponential utility premium, exponential principle, exponential premium Keywords: Π_α(X) = (1/α) log E[e^(αX)], risk-aversion parameter α, E[X] + α Var(X)/2, zero-utility premium with exponential utility, infinite premium for heavy tails Symbols: alpha, infinity, lambda, pi

Last updated: 16 October 2026 22:28