Loss Distributions: Modeling Insurance Claims with Probability

Choosing a Loss Distribution

Insurance losses are typically non-negative, right-skewed, and may have heavy tails. Common choices include the exponential, Pareto, lognormal, gamma, and Weibull distributions. The choice depends on the type of insurance and the tail behavior of losses. Light-tailed distributions (exponential, gamma) are used for routine claims; heavy-tailed distributions (Pareto, lognormal) model catastrophic or liability claims.

On Exam P, you do not need to justify a distributional choice, but you should be able to work with each of these distributions efficiently.

The Pareto Distribution

The single-parameter Pareto distribution has survival function S(x) = (theta / (theta + x))^alpha for x > 0. The mean is theta / (alpha - 1) (exists only when alpha > 1) and the variance is alpha * theta^2 / ((alpha - 1)^2 * (alpha - 2)) (exists only when alpha > 2). The Pareto is heavy-tailed, meaning its tail decreases polynomially rather than exponentially.

For deductibles, E[(X - d)+] = (theta + d) / (alpha - 1) * (theta / (theta + d))^alpha when alpha > 1. This formula appears frequently on Exam P.

The Lognormal Distribution

If ln(X) ~ Normal(mu, sigma^2), then X follows a lognormal distribution. E[X] = exp(mu + sigma^2/2) and Var(X) = exp(2*mu + sigma^2) * (exp(sigma^2) - 1). Probabilities are computed by converting to the normal: P(X <= x) = Phi((ln(x) - mu) / sigma). The lognormal is widely used for modeling insurance claim sizes because it is always positive, right-skewed, and has a heavier tail than the exponential.