Compound Poisson and Tweedie

Let N be a non-negative integer-valued claim count with PMF p_N, and let X_1, X_2, ... be independent and identically distributed claim severities, also independent of N. The aggregate loss is S = X_1 + X_2 + ... + X_N, with S = 0 when N = 0.

When N is Poisson, S is compound Poisson. When N is negative binomial, S is compound negative binomial. Klugman Loss Models develops both as the standard FAM and ASTAM aggregate-loss frameworks.

When N is in the (a, b, 0) class (Poisson, Binomial, Negative Binomial) and X is integer-valued with PMF f, the PMF g of S satisfies a recursion. The recursion converts what would be an infinite convolution into a step-by-step computation that is feasible by spreadsheet.

The starting value depends on the frequency family. For compound Poisson with mean λ, g(0) = exp(-λ(1 - f(0))). For compound negative binomial with parameters r and β, g(0) = (1 + β(1 - f(0)))^{-r}.

If N is Poisson with mean λ and each X_i is gamma with shape α and scale θ, then S is a member of the Tweedie family with power parameter p strictly between 1 and 2. The Tweedie has a positive probability mass at S = 0 (corresponding to N = 0) and a continuous density on S > 0.

The Tweedie variance function is Var(S) = φ μ^p where μ = E[S] and φ is a dispersion parameter. This power-law variance is what lets Tweedie sit inside a GLM as the natural pure-premium target: the same coefficients explain mean loss whether the predictor effect runs through frequency or severity.

Standard severity GLMs (gamma, lognormal) discard policy-years with no claim, which throws away information. Standard frequency GLMs ignore severity. Tweedie pure-premium GLMs model both at once on the entire portfolio, including zero-claim observations.

The actuarial cost is a more complex likelihood; the benefit is one model that is consistent with the underlying compound-Poisson story and that respects the unit-rating exposure structure. ASTAM treats Tweedie as a standard aggregate-loss tool and PA references it as a GLM family.

Claim frequency is Poisson with mean 4 per year; severity is exponential with mean 1,000. Compound Poisson aggregate S has mean λ × E[X] = 4 × 1,000 = 4,000 and variance λ × E[X^2] = 4 × (2 × 1,000^2) = 8,000,000. Standard deviation is about 2,828.

The CV of S is 2,828 / 4,000 = 0.707. Compare to the CV of the severity X alone, which is 1.0. Aggregating across 4 expected claims tightens the relative spread by a factor of √4, exactly as the law of large numbers predicts.

Frequency is NegBin with r = 2 and β = 0.5, so E[N] = 1 and Var(N) = 1.5. Severity is gamma with mean 500 and variance 250,000 (so CV = 1).

Aggregate mean is E[N] × E[X] = 500. Aggregate variance is E[N] × Var(X) + Var(N) × E[X]^2 = 1 × 250,000 + 1.5 × 250,000 = 625,000. Standard deviation is 790.6, and the aggregate CV is 1.58 — wider than the severity CV because the frequency is overdispersed.

When N is Poisson and X is Inverse Gaussian, the compound S still has a closed-form mean and variance. The Inverse Gaussian is itself a Tweedie at p = 3 and is the natural severity model when fitted gamma is too light-tailed and lognormal is too heavy-tailed for the available data.

Inverse Gaussian gets less coverage on ASTAM than Lognormal or Gamma but is on the table and appears occasionally as a model-comparison option. Its MLE for the mean parameter is the sample mean; its MLE for the shape parameter is closed-form and worth knowing.