Pareto Distribution

The Pareto distribution on the SOA Loss Models tables has shape α > 0 and scale θ > 0. The support is x > 0, not x > θ. This is the Pareto Type II (Lomax) form, not the Pareto Type I that some textbooks introduce. Confusing the two changes both the support and the moment formulas.

Below x = 0 the PDF is zero. The density is maximized at x = 0 and decays as a power of (x + θ) — that polynomial decay is what makes the tail heavy.

The Pareto mean exists only when α > 1; the variance only when α > 2; the k-th moment only when α > k. For α close to 1 the mean is large; for α between 1 and 2 the mean exists but the variance does not. This is the actuarial signature of heavy tails: large losses are not rare enough for higher moments to converge.

On excess-of-loss reinsurance pricing, fitted α values below 2 are common for catastrophe layers, which is why standard parametric variance-based capital formulas cannot be used in those layers without modification.

Pareto has a useful conditional distribution. Given that X exceeds a threshold d, the excess X − d is again Pareto with the same shape α but with a shifted scale θ + d. This is why Pareto is the natural model for excess-of-loss reinsurance: data above any retention is still Pareto.

If X given a rate Λ is exponential with rate Λ, and Λ itself is gamma-distributed with shape α and rate θ (equivalently, an inverse-scale parameter), then the unconditional X is Pareto with shape α and scale θ. This is the continuous analogue of the gamma-mixed-Poisson identity that produces the negative binomial.

The narrative is the same: heterogeneity in an underlying exponential rate, across policies or across loss types, produces an unconditional heavy-tailed distribution even though every conditional component is light-tailed. Klugman Loss Models presents this in the mixture-distributions chapter.

With scale θ known, the Pareto shape MLE has a clean closed form. Given observations x_1, ..., x_n, the MLE is the inverse of the average log-ratio:

With both α and θ unknown, the likelihood equations are coupled and have no closed-form solution. Numerical methods are used in practice. ASTAM examples often hold θ fixed at a small value (or at a deductible) and ask for α̂.

Severities are Pareto with α = 3 and θ = 1,000. The probability a loss exceeds 5,000 is (1,000 / 6,000)^3 ≈ 0.00463. The mean loss is 1,000 / 2 = 500.

The expected excess over a 5,000 retention, given that the loss exceeds 5,000, is the mean of the conditional Pareto with shape 3 and scale 6,000, which is 6,000 / 2 = 3,000. The unconditional pure premium for the excess layer is 0.00463 × 3,000 ≈ 13.9.

Observed losses with known scale θ = 500: 800, 1,500, 2,000, 4,000, 6,000. Log-ratios ln((x_i + 500)/500) are ln(2.6) = 0.956, ln(4) = 1.386, ln(5) = 1.609, ln(9) = 2.197, ln(13) = 2.565. Sum is 8.713.

MLE for shape is α̂ = 5 / 8.713 ≈ 0.574. With α̂ < 1, the fitted Pareto has infinite mean. The fit is signaling that this small sample is consistent with a very heavy tail; a larger sample, a goodness-of-fit check, and an alternative model (Lognormal, Weibull) should all be considered before any premium decision.