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Convex Order and More Dangerous Risks

Definition

For integrable losses, convex order X≤cxYX\le_{\mathrm{cx}}Y means E[h(X)]≤E[h(Y)]\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 E[X]=E[Y]\mathbb{E}[X]=\mathbb{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 YY spreads its mass to 0 and 2. On the right, E[(X−d)+]=(1−d)+\mathbb{E}[(X-d)_+]=(1-d)_+ and E[(Y−d)+]=(2−d)+/2\mathbb{E}[(Y-d)_+]=(2-d)_+/2 start from the common mean at d=0d=0, and the second is never smaller, so XX precedes YY in stop-loss order with equal means, which is the characterization of X≤cxYX\le_{\mathrm{cx}}Y in the definition. The gap between the lines reflects the extra dispersion of YY, which also shows in the variances 0 and 1.

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)=xh(x)=x and h(x)=−xh(x)=-x are convex.
  • With finite second moments, Var⁡(X)≤Var⁡(Y)\operatorname{Var}(X)\le\operatorname{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}\{0,2\} in convex order. Their variances are 0 and 1.

Prerequisites

Sources

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