Severity Distributions: Exponential, Pareto, Lognormal, Weibull for STAM
Learn key severity distributions for modeling insurance claim amounts on Exam STAM.
The Exponential Distribution
The exponential distribution with parameter theta has PDF f(x) = (1/theta)*e^(-x/theta) for x > 0, CDF F(x) = 1 - e^(-x/theta), mean theta, and variance theta^2. Its coefficient of variation is always 1. The exponential is the simplest continuous severity distribution and serves as the building block for more complex models.
The exponential has the memoryless property: P(X > s + t | X > s) = P(X > t). This means the conditional distribution of the remaining loss, given that the loss has already exceeded some amount s, is the same as the original distribution. In insurance terms, the expected excess loss above any threshold is always theta, regardless of the threshold. This is a useful property but is often unrealistic for actual claim data, where large claims tend to remain large.
The limited expected value is E[X ^ d] = E[min(X, d)] = theta * (1 - e^(-d/theta)). The mean excess function is e(d) = E[X - d | X > d] = theta, constant for all d. This constant mean excess function is the signature of the exponential distribution.
The Pareto Distribution
The two-parameter Pareto (Pareto Type II or Lomax) with parameters alpha > 0 and theta > 0 has PDF f(x) = alpha*theta^alpha / (x + theta)^(alpha+1) for x > 0. Its survival function S(x) = (theta/(x + theta))^alpha makes tail probability calculations straightforward. The mean is theta/(alpha - 1) for alpha > 1, and the variance is alpha*theta^2 / ((alpha-1)^2*(alpha-2)) for alpha > 2.
The Pareto is heavy-tailed: its k-th moment exists only when alpha > k. This means for alpha between 1 and 2, the mean exists but the variance does not. Heavy-tailed distributions are essential for modeling large liability claims, reinsurance losses, and catastrophic events where extreme values occur more frequently than lighter-tailed models predict.
The limited expected value is E[X ^ d] = theta/(alpha-1) * (1 - (theta/(d+theta))^(alpha-1)) for alpha not equal to 1. The mean excess function is e(d) = (d + theta)/(alpha - 1), which is increasing in d. This increasing mean excess function indicates that the larger a claim gets, the larger the expected additional amount becomes, making the Pareto appropriate for loss distributions with very long tails.
The Lognormal Distribution
If ln(X) ~ Normal(mu, sigma^2), then X follows a lognormal distribution. The PDF is f(x) = (1/(x*sigma*sqrt(2*pi))) * exp(-(ln(x) - mu)^2 / (2*sigma^2)) for x > 0. The mean is E[X] = exp(mu + sigma^2/2) and the variance is (exp(sigma^2) - 1) * exp(2*mu + sigma^2).
The lognormal is moderately heavy-tailed (heavier than the exponential but lighter than the Pareto) and is the most commonly used severity distribution for property and casualty insurance losses. Its popularity comes from the observation that loss amounts are often well-modeled by a process of multiplicative random effects, which produces a lognormal by the central limit theorem applied to log(X).
The limited expected value is E[X ^ d] = exp(mu + sigma^2/2) * Phi((ln(d) - mu - sigma^2)/sigma) + d * (1 - Phi((ln(d) - mu)/sigma)), where Phi is the standard normal CDF. While this formula appears complex, it relies only on standard normal CDF values that can be looked up in tables.
The Weibull Distribution
The Weibull with parameters tau > 0 (shape) and theta > 0 (scale) has survival function S(x) = exp(-(x/theta)^tau). The CDF is F(x) = 1 - exp(-(x/theta)^tau) and the PDF is f(x) = (tau/theta) * (x/theta)^(tau-1) * exp(-(x/theta)^tau). The mean is theta * Gamma(1 + 1/tau).
The shape parameter tau determines the hazard rate behavior. When tau > 1, the hazard rate h(x) = (tau/theta)*(x/theta)^(tau-1) is increasing, suggesting "aging" or wear-out failures. When tau < 1, the hazard rate decreases, suggesting "burn-in" failures. When tau = 1, the Weibull reduces to the exponential with constant hazard rate.
The Weibull is used for warranty claims, mechanical failure times, and any loss process where the hazard rate depends on the magnitude of the loss. It is lighter-tailed than the Pareto and lognormal for tau >= 1, but can be heavy-tailed when tau < 1.
The Gamma Distribution
The Gamma with parameters alpha > 0 (shape) and theta > 0 (scale) has PDF f(x) = x^(alpha-1) * e^(-x/theta) / (theta^alpha * Gamma(alpha)) for x > 0. The mean is alpha*theta and the variance is alpha*theta^2. When alpha = 1, the Gamma reduces to the exponential. When alpha is a positive integer, the Gamma is the distribution of the sum of alpha independent exponential(theta) variables.
The Gamma is a flexible distribution that can model a wide range of shapes. For alpha < 1, the PDF is decreasing and approaches infinity at x = 0. For alpha = 1, the PDF is exponential. For alpha > 1, the PDF is unimodal with a mode at (alpha-1)*theta. In actuarial applications, the Gamma is often used as a conjugate prior for the Poisson rate parameter in Bayesian credibility models, and as a severity distribution for moderate-sized claims.
Choosing and Comparing Severity Distributions
Selection depends on the tail behavior, the availability of data, and the intended use. Several diagnostic tools help compare distributions. The mean excess plot: plot the empirical mean excess function e(d) against d. A constant plot suggests exponential; an increasing plot suggests Pareto or another heavy-tailed distribution; a decreasing plot suggests light tails.
The log-log plot of the empirical survival function: plot ln(S_hat(x)) against ln(x). A linear relationship suggests a Pareto distribution, with the slope giving -alpha. Formal goodness-of-fit tests (chi-squared, Kolmogorov-Smirnov, Anderson-Darling) provide statistical evidence for or against a candidate distribution.
For Exam STAM, know the PDFs, CDFs, means, variances, and limited expected values E[min(X, d)] for each distribution. Limited expected values are essential for computing expected payments under deductibles and policy limits. Be able to compute E[(X - d)+] = E[X] - E[X ^ d] for excess loss calculations.
Study Tips and Common Pitfalls
The most common mistake is confusing the parameterization across distributions. The Pareto on Exam STAM uses the (alpha, theta) parameterization where alpha is the tail parameter and theta is the scale parameter. Different sources may use different conventions, so always verify which parameterization is being used.
Another frequent error is forgetting the conditions for moment existence. The Pareto mean requires alpha > 1, and the variance requires alpha > 2. Attempting to compute a moment that does not exist produces a meaningless answer. Practice computing limited expected values and mean excess functions for each distribution, as these calculations are heavily tested. Work through at least 6 problems covering different distributions with deductibles and limits.