Severity, Frequency, and Aggregate Models

You are given empirical or descriptive information about losses and asked to fit or recognize a severity distribution and compute moments.

Match the shape (skew, tail heaviness) to a distribution family — exponential and gamma for moderate tails, Pareto for very heavy tails, lognormal for multiplicative loss processes. Compute E[X], E[X^2], Var(X) using the standard tables, watching for which parameterization the formula uses. For policy modifications, compute E[X ∧ u] (limited expected value) directly from definitions.

E[X ∧ u] = ∫₀^u (1 - F(x)) dx; differentiate to recover the density.

A frequency distribution is given, possibly mixed (e.g., gamma mixing distribution), and you must find the mean, variance, or PMF.

Identify the (a, b, 0) or (a, b, 1) class — Poisson, binomial, negative binomial fall into (a, b, 0). For mixed Poisson, the mixing distribution's mean is the unconditional E[N] and the variance follows from the law of total variance: Var(N) = E[Var(N|Λ)] + Var(E[N|Λ]). Negative binomial arises naturally from Poisson-gamma mixing.

Mixed Poisson with mixing dist M: E[N] = E[M], Var(N) = E[M] + Var(M).

A compound model has frequency N and i.i.d. severities X_i, and you must compute E[S], Var(S), or P(S ≤ s) for aggregate S = X_1 + ... + X_N.

Use compound moments: E[S] = E[N] E[X], Var(S) = E[N] Var(X) + Var(N) (E[X])^2. For exact distributions, use convolutions of severity or the moment generating function approach. For approximate computation, use the normal approximation or the recursive formula (next bullet).

Compound formula: E[S] = E[N] E[X]; Var(S) = E[N] Var(X) + Var(N) (E[X])^2.

A discrete severity distribution is paired with an (a, b, 0) or (a, b, 1) frequency distribution and you need P(S = s) for several values of s.

Use Panjer's recursion: f_S(s) = (1 / (1 - a · f_X(0))) · Σ_{x=1}^{s} (a + b · x/s) · f_X(x) · f_S(s - x). Start from f_S(0) = M_N(log f_X(0)) and iterate. Be careful with f_X(0) — if severity has a point mass at 0, it changes the boundary.

Panjer's recursion: f_S(s) = (1 / (1 - a f_X(0))) Σ (a + b x/s) f_X(x) f_S(s - x).