Convex Order and More Dangerous Risks¶

Definition¶

For integrable losses, convex order $X\le_{\mathrm{cx}}Y$ means $\mathbb{E}[h(X)]\le\mathbb{E}[h(Y)]$ for every convex function for which comparison is defined. For nonnegative losses this is equivalent to stop-loss order together with $\mathbb{E}[X]=\mathbb{E}[Y]$ $E [Y]$ . The course calls the larger risk more dangerous in this equal-mean setting.

Interpretation¶

Convex order compares dispersion after removing a difference in mean. All risk-averse agents prefer the less dispersed loss when utility expectations are defined.

Left: the loss X equal to 1 surely and the loss Y equal to 0 or 2 with probability 1/2 each. Right: their stop-loss transforms, equal at d = 0 and with Y's line above X's.

The figure uses the Example's risks. Both have mean 1, but $Y$ spreads its mass to 0 and 2. On the right, $\mathbb{E}[(X-d)_+]=(1-d)_+$ and $\mathbb{E}[(Y-d)_+]=(2-d)_+/2$ start from the common mean at $d=0$ $0$ , and the second is never smaller, so $X$ precedes $Y$ in stop-loss order with equal means, which is the characterization of $X\le_{\mathrm{cx}}Y$ in the definition. The gap between the lines reflects the extra dispersion of $Y$ , which also shows in the variances 0 and 1.

Spread $X$ and $Y$ around their common mean 1: both stop-loss transforms start at $\mathbb E[X]=\mathbb E[Y]=1$ $1$ , and the more dispersed risk lies above at every $d$ , so $X\le_{\mathrm{cx}}Y$ when $Y$ has the larger spread.

Convexity and concavity determines which test functions enter the comparison. Convex functions penalize dispersion; their negatives are concave, explaining the preference of risk-averse wealth utilities for the less dispersed loss.

Key Results¶

The order forces equal means because both $h(x)=x$ and $h(x)=-x$ are convex.
With finite second moments, $\operatorname{Var}(X)\le\operatorname{Var}(Y)$ $Var (Y)$ ; if the distributions differ, this inequality is strict.
Variance ordering alone does not imply convex order.
Convex order is not preserved by every convex transformation, because equal means can be lost.

Examples¶

A constant loss of 1 is below the equal-probability loss on $\{0,2\}$ in convex order. Their variances are 0 and 1.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 10

Aliases: equal-mean stop-loss order, more dangerous order, convex order, more dangerous risk, convex ordering Keywords: X ≤_cx Y, convex test functions, equal means, variance comparison, constant versus two-point loss

Last updated: 17 October 2026 21:53