The Loss Distribution Approach: Frequency, Severity, and Aggregate Losses

For most choices of severity distribution $G$ (lognormal, GPD, Pareto) and frequency distribution $p_N$, the aggregate distribution $F_{S_N}$ has no closed form. Three tools handle this:

1. Moments (Proposition 13.8): $\mathbb{E}[S_N] = \mathbb{E}[N]\mathbb{E}[X_1]$ and $\text{Var}(S_N) = \text{Var}(N)\mathbb{E}[X_1]^2 + \mathbb{E}[N]\text{Var}(X_1)$ — closed-form regardless of the distributions, via conditional expectation.
2. Panjer recursion (Theorem 13.16): For $N$ in the binomial/Poisson/negative binomial Panjer class and discrete severities, a one-pass recursion computes the aggregate PMF exactly in $O(m^2)$ time.
3. Normal and translated-gamma approximations: Moment-matching approximations for the bulk of $F_{S_N}$; the GPD is added for the upper tail.

A fourth tool — Poisson mixture models — explains the heavy-tailed loss frequency observed empirically: if the Poisson rate $\lambda$ is itself random (e.g. gamma-distributed, capturing uncertainty about the true loss rate), the marginal count distribution is negative binomial, with heavier tails than a plain Poisson.

• define a compound distribution $S_N$ (Definition 13.1) and state its two key assumptions;
• derive $\mathbb{E}[S_N]$ and $\text{Var}(S_N)$ via the law of total expectation and variance;
• state the Panjer class and apply the Panjer recursion for $N \sim \text{Poi}(\lambda)$ with discrete severity;
• explain why the normal approximation is inadequate for heavy-tailed severities and describe the GPD alternative;
• define a Poisson mixture model and show the Poisson-gamma mixture is a negative binomial.

1. 1. The compound distribution
2. 1. Moments of compound distributions
3. 1. The Panjer recursion
4. 1. Moment approximations
5. 1. Poisson mixture models