Pareto Loss Model¶

Definition¶

The course's Pareto loss model has survivor function $\bar F_X(x)=[\beta/(\beta+x)]^\alpha$ for $x\ge0$ $0$ , shape $\alpha>0$ and scale $\beta>0$ $0$ . It is the Lomax or Pareto type II convention, whose support starts at zero; an unshifted Pareto type I model has a different support.

Interpretation¶

The polynomial tail allows occasional very large losses. The shape controls which moments are finite, so a valid loss distribution need not have finite variance.

Lomax survivor function (800/(800+x))² with the deductible 300 marked; the curve passes 0.529 at x = 300 and the red area beyond 300, about 581.82, decays slowly to the right.

With $\alpha=2$ and $\beta=800$ $800$ , the survivor function falls only like $x^{-2}$ , so a visible share of probability remains far to the right. The total area under $\bar F_X$ is $\mathbb{E}[X]=800$ $800$ . The red area beyond the deductible $d=300$ is the stop-loss transform $\pi_X(300)=800^2/1100\approx581.82$ $581.82$ , the mean excess per loss; dividing by $\bar F_X(300)=(8/11)^2\approx0.529$ gives the mean excess per payment, $1100$ . Because the tail is so heavy, part of the red area lies beyond the plotted range.

Change the shape $\alpha$ and the scale $\beta$ : the mean is infinite for $\alpha\le1$ and the variance for $\alpha\le2$ $2$ . Move $d$ to see the mean excess $(\beta+d)/(\alpha-1)$ grow linearly in $d$ , unlike the constant mean excess of the dashed exponential.

The ordinary deductible separates per-loss excess $(X-d)_+$ from excess conditional on a payment. The Pareto survivor ratio above $d$ gives a conditional excess law with scale $\beta+d$ $d$ , while integrating the tail gives the expected payment per ground-up loss.

Key Results¶

$f_X(x)=\alpha\beta^\alpha/(\beta+x)^{\alpha+1}$ for $x>0$ $0$ .
$\mathbb{E}[X]=\beta/(\alpha-1)$ for $\alpha>1$ $1$ ; variance is $\beta^2\alpha/[(\alpha-1)^2(\alpha-2)]$ for $\alpha>2$ $2$ .
For $\alpha>1$ $1$ , the stop-loss transform is $\pi_X(d)=\beta^\alpha(\beta+d)^{1-\alpha}/(\alpha-1)$ $1)$ .
Conditional excess over $d$ is Lomax with the same shape and scale $\beta+d$ $d$ .
Every positive exponential moment is infinite, so an exponential premium is infinite.

Examples¶

With $\alpha=2$ $2$ , $\beta=800$ and deductible 300, mean ground-up loss is 800 but variance is infinite. Mean excess per loss is $800^2/1100\approx581.82$ $581.82$ , and mean excess per payment is 1100.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 2

Aliases: Lomax distribution, Pareto type II, shifted Pareto distribution, Pareto distribution, Lomax, Pareto II Keywords: heavy tail, survivor function (β/(β + x))^α, mean excess (β + d)/(α - 1), infinite variance, infinite exponential premium Symbols: alpha, beta, pi

Last updated: 17 October 2026 21:53