Stop-Loss Order¶

Definition¶

For integrable nonnegative losses, stop-loss order $X\le_{\mathrm{sl}}Y$ means $\mathbb{E}[(X-d)_+]\le\mathbb{E}[(Y-d)_+]$ for every $d\ge0$$0$. The course states it for $d>0$$0$; continuity as $d\downarrow0$ supplies the mean comparison at zero.

Interpretation¶

Every excess-loss layer has no larger expected payment for $X$. This order combines size and dispersion, so it does not require equal means.

The figure takes the Examples case at retention $d=1$$1$. The payment $(x-1)_+$ is convex, so the expected payment of $Y$, which is 0 or 2 with equal probabilities, is the midpoint of the chord between $(0,0)$ and $(2,1)$, namely 0.5; the constant $X=1$ sits at the kink and pays 0. The same chord argument works at every retention, which gives $\mathbb{E}[(X-d)_+]\le\mathbb{E}[(Y-d)_+]$ for all $d\ge0$$0$, that is, $X\le_{\mathrm{sl}}Y$$Y$.

The stop-loss transform and reinsurance is the expected payment above each retention. Ordering those transforms yields the definition. Jensen's inequality explains the constant-loss example: $x\mapsto(x-d)_+$ is convex, so a constant at the mean has no larger expected excess than the original loss.

Key Results¶

• It is equivalent to increasing convex order: $\mathbb{E}[h(X)]\le\mathbb{E}[h(Y)]$ for increasing convex $h$ whenever the expectations are defined.
• It implies $\mathbb{E}[X]\le\mathbb{E}[Y]$ and is transitive.
• Stochastic order and the equal-mean thicker-tailed comparison each imply stop-loss order.
• Without equal means, stop-loss order need not imply a variance comparison.
• Increasing convex transformations preserve the order when the transformed quantities meet the needed integrability conditions.

Examples¶

Let $X=1$ and $Y$ equal 0 or 2 with equal probabilities. At $d=1$$1$, expected excesses are 0 and 0.5; their stop-loss transforms are ordered at every retention.

Prerequisites¶

• Stop-Loss Transform and Reinsurance
• Jensen Inequality

Related Concepts¶

• Convex Order and More Dangerous Risks
• Exponential Order
• Risk Orders Under Aggregation
• Thicker-Tailed Comparison
• Second-Order Stochastic Dominance
• Crossing Criteria for Risk Orders
• Stochastic Order

Sources¶

• Risk Insurance · Lecture 9

Aliases: increasing convex order, ordering of stop-loss premiums, stop-loss ordering Keywords: X ≤_sl Y, E[(X - d)+] ≤ E[(Y - d)+], increasing convex functions, every retention d

Last updated: 14 October 2026 08:21