Stop-Loss Transform and Reinsurance¶

Definition¶

For a nonnegative integrable loss $X$ , its stop-loss transform at retention $d\ge0$ is $\pi_X(d)=\mathbb{E}[(X-d)_+]$ $d)_{+}]$ . In reinsurance, an insurer passes (cedes) part of its losses to a reinsurer under a contract called a treaty and keeps the rest, its retention. A stop-loss treaty pays the excess of an aggregate loss over its retention; the same positive-part function can describe excess-of-loss cover on an individual claim.

Interpretation¶

The transform is the expected covered tail beyond a deductible. It distinguishes risks that have the same mean but different upper-tail exposure.

$Left: survivor function of an exponential loss with rate 1/2, shaded beyond d = 2; right: its stop-loss transform 2e^{-d/2}, starting at the mean 2 and equal to 0.74 at d = 2.$

The figure applies the Examples formula with $\lambda=1/2$ $1/2$ , so $\bar F_X(t)=\mathrm{e}^{-t/2}$ and $\pi_X(d)=2\mathrm{e}^{-d/2}$ $2 e^{- d /2}$ . The shaded area under the survivor function beyond $d=2$ equals $\pi_X(2)=2\mathrm{e}^{-1}\approx0.74$ $0.74$ , the height of the marked point on the right. As the retention grows, the transform falls convexly from $\pi_X(0)=\mathbb{E}[X]=2$ towards zero, and its slope at $d$ is $-\bar F_X(d)$ .

Move the retention $d$ : the red area under $\bar F_X$ to the right of $d$ is $\pi_X(d)$ , drawn as the point on the right-hand curve, and the tangent there has slope $-\bar F_X(d)$ . Change $\lambda$ to rescale the curve, which starts at $\pi_X(0)=\mathbb E[X]=1/\lambda$ $1/ λ$ .

An ordinary deductible produces the payment $(X-d)_+$ $d)_{+}$ . Its expectation is therefore the stop-loss transform: the function records the expected ceded payment as the retention changes.

Key Results¶

$\pi_X(d)=\int_d^\infty\bar F_X(t)\,\mathrm{d}t$ $\int_{d}^{\infty} \overset{ˉ}{F}_{X} (t) d t$ .
It is nonincreasing and convex in $d$ , with $\pi_X(0)=\mathbb{E}[X]$ and $\pi_X(d)\to0$ $0$ .
Its right derivative is $-\mathbb{P}(X>d)$ $d)$ ; its left derivative is $-\mathbb{P}(X\ge d)$ $d)$ .
The retained loss $\min(X,d)$ plus the ceded excess $(X-d)_+$ equals $X$ .

Examples¶

For exponential loss of rate $\lambda$ , $\pi_X(d)=\mathrm{e}^{-\lambda d}/\lambda$ $e^{- λ d} / λ$ . This follows by integrating its survivor function from $d$ onwards.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 2

Aliases: stop-loss premium, excess-of-loss payment, stop-loss transform, stop-loss reinsurance, stop-loss treaty Keywords: π_X(d) = E[(X - d)+], retention, ceded loss, reinsurance treaty, integral of the survivor function Symbols: infinity, integral, lambda, pi

Last updated: 17 October 2026 21:53