Policy Limit¶

Definition¶

A policy limit $u\ge0$ caps the insurer's payment. Without a deductible, payment is $Y=\min(X,u)$ $min (X, u)$ . With an ordinary deductible $d$ and maximum payment $u$ , it is $Y=\min((X-d)_+,u)$ $d)_{+}, u)$ . A maximum covered ground-up loss is a different convention and must be stated separately.

Interpretation¶

The cap creates a point mass at the maximum payment whenever losses can reach or exceed it. A deductible controls the bottom of the covered layer (the band of loss amounts from $d$ to $d+u$ that the insurer pays); a payment limit controls its width.

Payment against ground-up loss with deductible 100 and limit 500: zero up to 100, rising with slope 1 to 500 at loss 600, then flat; a dashed grey line shows a ground-up cap paying 400.

With $d=100$ and $u=500$ $500$ , the red payment $Y=\min((X-d)_+,u)$ is zero below the deductible, rises one-for-one through the covered layer from $d=100$ to $d+u=600$ $600$ , and stays at $u=500$ for every larger loss; all those losses form the point mass at the maximum payment. A loss of 800 is therefore paid 500. The dashed grey line shows the other convention from the example: capping the ground-up loss at 500 before applying the deductible pays $500-100=400$ $400$ .

Move $X$ , $d$ and $u$ : the payment $Y=\min((X-d)_+,u)$ rises with slope 1 between $d$ and $d+u$ and stays at $u$ beyond, which differs from capping the ground-up loss first. The readouts give $\mathbb E[Y]$ and the point mass $\mathbb P(Y=u)$ for an illustrative exponential loss.

The stop-loss transform and reinsurance values the unbounded layer above a retention $d$ , the amount the insurer keeps. When the transforms are finite, subtracting the transform at $d+u$ from the transform at $d$ removes payments beyond the cap. The finite-layer integral below remains valid even when the uncapped transforms are infinite.

Key Results¶

$\mathbb{E}[\min(X,u)]=\int_0^u\bar F_X(t)\,\mathrm{d}t$ for nonnegative $X$ .
$\displaystyle\mathbb{E}[\min((X-d)_+,u)]=\int_d^{d+u}\bar F_X(t)\,\mathrm{d}t$ for finite $d,u\ge0$ $0$ . If $\mathbb{E}[X]<\infty$ $\infty$ , this also equals $\pi_X(d)-\pi_X(d+u)$ $u)$ .
For $u>0$ $0$ , the capped excess has mass $\mathbb{P}(X\ge d+u)$ at $u$ .
For $u>0$ $0$ , its mass at zero is $F_X(d)$ ; at $u=0$ the entire payment is zero.

Examples¶

If deductible is 100 and maximum insurer payment is 500, a ground-up loss of 800 gives payment 500. A ground-up coverage cap of 500 instead would give payment 400.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 2

Aliases: maximum insurer payment, coverage limit, payment limit Keywords: limited expected value, min((X - d)+, u), covered layer, point mass at the limit, E[min(X, u)] Symbols: infinity, integral, pi

Last updated: 14 October 2026 08:21