Why is the gamma distribution so popular for modeling severities?

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AcadiFi · Accepted Answer

The gamma distribution is a two-parameter family with PDF f(x) = (x^(k-1) * e^(-x/theta)) / (theta^k * Gamma(k)), where k is shape and theta is scale. Mean k*theta, variance k*theta^2. It is strictly positive, right-skewed, and flexible: k=1 gives exponential, large k approaches normal. This range fits operational loss severity beautifully.

Thundertree Insurance models fire-claim severity with k=2.5, theta=$40,000. Mean loss is $100,000, variance is 10 billion dollars squared, skewness is 2/sqrt(2.5) = 1.26. Probability of loss exceeding $300,000 is computed from the incomplete gamma function: P(X > 300,000) = 1 - P(5; 2.5) = 2.9%. Gamma also appears as the conjugate prior for Poisson rate in Bayesian credit models and as the distribution of sums of independent exponentials.

Fitting via MLE or method of moments is straightforward. Compared to lognormal, gamma has a lighter right tail, so underestimates extreme losses in fat-tailed exposures; analysts often pair gamma with a GPD tail beyond a threshold. Quantile-quantile plots against observed losses reveal fit quality; systematic deviation above the 95th percentile suggests moving to heavier-tailed alternatives.

Why is the gamma distribution so popular for modeling severities?

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