Compound Risk Distribution¶

Definition¶

The distribution of $\displaystyle S=\sum_{i=1}^NX_i$ is a compound distribution when $N$ is a nonnegative integer variable and the $X_i$ are i.i.d. severities independent of $N$. Let $\mu=\mathbb{E}[X_1]$ and $\sigma^2=\operatorname{Var}(X_1)$$\mathrm{Var}(X_1)$.

Interpretation¶

Compounding combines uncertainty in the number of claims with uncertainty in their sizes. It differs from mixing a parameter of one distribution.

The figure reads the formula for $F_S$ as a two-stage experiment. The count is drawn first, with probability $\mathbb{P}(N=n)$ for each branch; given $N=n$$n$, the loss is a sum of $n$ independent severities, whose law is the $n$-fold convolution $F_X^{*n}$ (and $S=0$ when $n=0$$0$). Averaging the branch laws with the branch probabilities gives $F_S$. In the Bernoulli Example only the branches $n=0$ and $n=1$ remain, with weights $1-q$ and $q$, which is why the result is a simple mixture.

The collective risk model supplies the random-sum assumptions. Given $N=n$$n$, convolution of independent random variables gives $F_X^{*n}$, and the probability generating function averages transform powers over the count. Adam’s law: total expectation applied to $\mathbb{E}[S\mid N]=N\mu$ gives the mean. The conditioning trick for variance combines $\operatorname{Var}(S\mid N)=N\sigma^2$ with variation in $N\mu$ to give the two variance terms.

Key Results¶

• $\mathbb{E}[S]=\mathbb{E}[N]\mu$$\mathbb{E}[N]\mu$.
• $\operatorname{Var}(S)=\mathbb{E}[N]\sigma^2+\operatorname{Var}(N)\mu^2$$\mathrm{Var}(N){\mu }^2$, with finite second moments.
• With moment generating function (MGF) $M$ and PGF $G$, $M_S(t)=G_N(M_X(t))$ where the expression is finite; for integer severities $G_S(z)=G_N(G_X(z))$$G_N(G_X(z))$.
• $\displaystyle F_S(s)=\sum_{n\ge0}\mathbb{P}(N=n)F_X^{*n}(s)$$n)F_X^{\ast n}(s)$, with $F_X^{*0}$ the cumulative distribution function of the constant zero.

Examples¶

If $N$ is Bernoulli with probability $q$, then $S$ is zero with probability $1-q$ and otherwise has the severity law. Thus a compound Bernoulli model is also a simple mixture.

Prerequisites¶

• Collective Risk Model
• Probability Generating Function
• Convolution of Independent Random Variables
• Adam’s Law: Total Expectation
• Conditioning Trick for Variance

Related Concepts¶

• Compound Poisson Distribution
• Mixture Distribution

Sources¶

• Risk Insurance · Lecture 4

Aliases: random sum distribution, compound frequency-severity distribution, compound distribution, random sum, compound sum, stopped-sum distribution Keywords: E[S] = E[N] μ, Var(S) = E[N] σ^2 + Var(N) μ^2, M_S(t) = G_N(M_X(t)), n-fold convolution, compound Bernoulli Symbols: mu, sigma, sum

Last updated: 17 October 2026 21:53