Exponential Order¶

Definition¶

For nonnegative losses, exponential order $X\le_eY$ means $\mathbb{E}[\mathrm{e}^{\alpha X}]\le\mathbb{E}[\mathrm{e}^{\alpha Y}]$ for every $\alpha>0$ $0$ . Comparisons may use extended values, but a finite exponential premium at a given $\alpha$ requires the corresponding moment to be finite.

Interpretation¶

The comparison tests every exponential risk-aversion parameter. It is weaker than agreement among all increasing concave wealth utilities.

$Two moment generating functions for α>0: M_X(α)=e^α for the certain loss 1 and the dashed M_Y(α)=(1+e^{2α})/2 for the loss 0 or 2, which lies above it for every α>0.$

The figure plots the two moment generating functions of the Examples case. Both curves start at 1 when $\alpha=0$ with the same slope, because $X$ and $Y$ have the same mean 1; for every $\alpha>0$ the curve of $Y$ lies above that of $X$ , which is exactly $X\le_eY$ $Y$ . The gap widens as $\alpha$ grows, because a larger risk-aversion parameter weights the outcome 2 of $Y$ more heavily.

Move $\alpha$ : $M_X(\alpha)\le M_Y(\alpha)$ at every $\alpha>0$ $0$ , so $X\le_e Y$ $Y$ , and the exponential premium of $Y$ is the larger one. Shrink the spread $s$ to 0 and the two curves coincide.

The moment generating function evaluates $\mathbb{E}[\mathrm{e}^{\alpha X}]$ . Exponential order compares these values over positive arguments; finiteness at the chosen argument is needed when turning the comparison into a finite premium.

Key Results¶

Stop-loss order implies exponential order, since $\mathrm{e}^{\alpha x}$ is increasing and convex.
For $u(w)=-\mathrm{e}^{-\alpha w}$ $- e^{- α w}$ , exponential order gives the reverse order of expected utilities of $w-X$ and $w-Y$ $Y$ .
With equal means and MGFs finite near zero, a Taylor comparison gives $\operatorname{Var}(X)\le\operatorname{Var}(Y)$ $Var (Y)$ .
If both positive MGFs are infinite, the comparison can carry little information about finite premiums.

Examples¶

For $X=1$ and $Y$ equal 0 or 2 with probability $1/2$ , $M_X(\alpha)=\mathrm{e}^\alpha\le(1+\mathrm{e}^{2\alpha})/2=M_Y(\alpha)$ for every positive $\alpha$ .

Prerequisites¶

Sources¶

Risk Insurance · Lecture 10

Aliases: MGF order for positive arguments, exponential utility risk ordering, exponential ordering Keywords: X ≤_e Y, E[e^(αX)] ≤ E[e^(αY)], comparison of moment generating functions, stop-loss order implies exponential order Symbols: alpha

Last updated: 16 October 2026 22:28