Stop-Loss Separation Theorem¶

Theorem¶

For nonnegative integrable losses, if $X\le_{\mathrm{sl}}Z$ and $\mathbb{E}[X]<\mathbb{E}[Z]$$\mathbb{E}[Z]$, the separation theorem guarantees a loss law $Y$ such that $X\le_{\mathrm{st}}Y\le_{\mathrm{sl}}Z$ and $\mathbb{E}[Y]=\mathbb{E}[Z]$$\mathbb{E}[Z]$.

Interpretation¶

The comparison can be separated into an increase in loss size followed by an equal-mean convex-order comparison. The intermediate variable describes a distribution; the theorem does not specify a unique insurance contract.

The figure draws the Examples case. The first arrow moves all the mass of $X=0$ up to $Y=1$$1$, an increase in loss size ($X{\le }_{\mathrm{st}}$$X\le_{\mathrm{st}}Y$$Y$); the second spreads the mass of $Y$ out to 0 and 2 around the common mean 1 ($Y{\le }_{\mathrm{cx}}$$Y\le_{\mathrm{cx}}Z$$Z$). Together the two steps decompose the original comparison $X\le_{\mathrm{sl}}Z$$Z$, in which $\mathbb{E}[X]=0<\mathbb{E}[Z]=1$$1$.

Stochastic order forms the first comparison, increasing loss size from $X$ to $Y$. Stop-loss order forms the second, comparing expected excesses of $Y$ and $Z$ at every retention; matching their means turns this second step into a dispersion comparison.

Key Results¶

• The equal-mean second step is $Y\le_{\mathrm{cx}}Z$$Z$.
• The theorem gives existence, not uniqueness of $Y$.
• Outside the theorem's hypothesis: if $X$ and $Z$ already have equal means, then $X\le_{\mathrm{cx}}Z$ directly, so $Y=X$ gives the same decomposition.
• Matching the means alone is insufficient; both order requirements must also hold.

Examples¶

Let $X=0$ and $Z$ equal 0 or 2 with equal probabilities. Choosing $Y=1$ gives $X\le_{\mathrm{st}}Y$$Y$, $Y\le_{\mathrm{cx}}Z$ and matching means of 1 for $Y,Z$.

Prerequisites¶

• Stochastic Order
• Stop-Loss Order
• Convex Order and More Dangerous Risks

Related Concepts¶

• Risk Orders Under Aggregation

Sources¶

• Risk Insurance · Lecture 11

Aliases: equal-mean intermediate risk, separation of size and dispersion, separation theorem Keywords: X ≤_st Y ≤_cx Z, stochastic order followed by convex order, intermediate risk

Last updated: 17 October 2026 21:53