Maximum Claim in a Portfolio¶

Definition¶

For nonnegative i.i.d. severities $X_i$ independent of count $N$ , the maximum claim is $M=\max(X_1,\ldots,X_N)$ $max (X_{1}, \dots, X_{N})$ , with $M=0$ if $N=0$ $0$ . Let $F_X$ be the severity cumulative distribution function (CDF) and $G_N$ the count probability generating function (PGF).

Interpretation¶

Aggregate loss and maximum claim measure different exposures. Conditioning on the count reduces the maximum to the probability that every claim is at most a chosen threshold.

Poisson PGF exp(-5(1-p)) plotted against p = F_X(z), rising from e^-5 at p = 0 to 1 at p = 1, with the point at p = 0.9 marked at height 0.607.

The figure plots $\mathbb{P}(M\le z)=G_N(F_X(z))$ for $N\sim\operatorname{Pois}(5)$ as a function of $p=F_X(z)$ $F_{X} (z)$ , the probability that a single claim is at most $z$ . In the example a claim exceeds 1000 with probability $0.1$ , so $p=0.9$ and the probability that no claim exceeds 1000 is $G_N(0.9)=\mathrm{e}^{-0.5}\approx0.607$ $0.607$ . At $p=0$ the curve starts at $G_N(0)=\mathbb{P}(N=0)=\mathrm{e}^{-5}$ $e^{- 5}$ , the chance of no claims at all, and it reaches 1 at $p=1$ $1$ . The steep rise near $p=1$ shows how strongly the maximum depends on the single-claim tail when several claims are expected.

The collective risk model supplies a random claim count and independent common severities. To study the maximum rather than their sum, condition on the same count: all $n$ claims are at most $z$ with probability $F_X(z)^n$ .

Key Results¶

For $z\ge0$ $0$ , $\mathbb{P}(M\le z)=G_N(F_X(z))$ $G_{N} (F_{X} (z))$ .
If $N\sim\operatorname{Pois}(\lambda)$ $Pois (λ)$ , $\mathbb{P}(M\le z)=\exp[-\lambda(1-F_X(z))]$ $F_{X} (z))]$ .
With fixed count $n\ge1$ $1$ , this becomes $F_X(z)^n$ .
The no-claim convention matters at zero and when evaluating moments.

Examples¶

With Poisson mean 5 and probability $0.1$ that one claim exceeds 1000, the probability that no claim exceeds 1000 is $\mathrm{e}^{-0.5}\approx0.60653$ $0.60653$ .

Prerequisites¶

Sources¶

Risk Insurance · Lecture 4

Aliases: largest claim size, maximum of a random number of claims, maximum claim, largest claim Keywords: P(M ≤ z) = G_N(F_X(z)), exp(-λ(1 - F_X(z))), probability that no claim exceeds a threshold Symbols: lambda

Last updated: 27 October 2026 11:52