Tail Weight

Study guide: Exam MAS-I

Tail Weight describes how much probability a severity distribution puts on very large losses — how slowly its survival function $S(x) = 1 - F(x)$ falls to 0. A heavy-tailed distribution such as the Pareto makes extreme claims far more likely than a light-tailed one such as the gamma, and distributions are compared by four related measures.

$\lim_{x \to \infty} \frac{S_X(x)}{S_Y(x)} = \lim_{x \to \infty} \frac{f_X(x)}{f_Y(x)}$

$e(d) = E[X - d \mid X > d]$

Limiting ratio. If the limit is $\infty$ , $X$ has the heavier right tail; if it is a positive constant, the tails are proportionally equivalent; if it is $0$ , $X$ is lighter. The density form follows from L'Hôpital's rule and helps when $S$ has no closed form.
Existence of moments. The fewer positive moments $E[X^k]$ are finite, the heavier the tail. Gamma and Weibull have all moments (light); the Pareto has $E[X^k] < \infty$ only for $k < \alpha$ (heavy, heavier as $\alpha$ falls). The lognormal has every moment but no moment generating function, so the moment test and the MGF test can disagree.
Hazard Rate. A decreasing hazard rate $h(x) = f(x)/S(x)$ signals a heavy tail and an increasing one a light tail. The Pareto's hazard decreases; the Weibull's decreases for $\tau < 1$ , is constant for $\tau = 1$ (the exponential) and increases for $\tau > 1$ .
Mean excess loss $e(d)$ , the average amount by which losses exceeding $d$ exceed it, is constant for the exponential — by memorylessness, the excess over any $d$ is exponential with the same mean — but rises linearly in $d$ for the Pareto, $e(d) = (d + \theta)/(\alpha - 1)$ when $\alpha > 1$ : the larger a Pareto loss already is, the larger its expected excess. It also prices an excess layer: the expected loss above a retention $R$ is $E[(X - R)_+] = S(R)\,e(R)$ .

Pareto Versus Exponential at the Extreme Losses are modelled either as exponential with mean 1,000 or as Pareto with $\alpha = 3$ , $\theta = 2{,}000$ (also mean 1,000). Compare $P(X > 10{,}000)$ under each model, decide which tail is heavier, and say which moments of the Pareto exist.
Answer
 $\begin{align*} S_{\text{Par}}(10{,}000) &= \left(\frac{2{,}000}{12{,}000}\right)^{3} \\ &= 0.00463 \\ S_{\text{Exp}}(10{,}000) &= e^{-10} \\ &= 0.0000454 \end{align*}$ 
With the same mean, the Pareto makes a $10,000 loss about 100 times as likely. In the limit, $S_{\text{Par}}(x)/S_{\text{Exp}}(x) = (2{,}000)^3 e^{x/1{,}000}/(x + 2{,}000)^3 \to \infty$ , because an exponential grows faster than any polynomial, so the Pareto's tail is heavier. Its moments exist only for $k < 3$ : the mean and variance are finite, $E[X^3]$ is not, while the exponential has all its moments.

Hazard Rate and Mean Excess Loss of a Pareto For the Pareto with $\alpha = 3$ , $\theta = 2{,}000$ , find the hazard rate and the mean excess loss at $d = 0$ , $1{,}000$ and $10{,}000$ .
Answer From $f(x) = \alpha\theta^{\alpha}/(x+\theta)^{\alpha+1}$ and $S(x) = \theta^{\alpha}/(x+\theta)^{\alpha}$ :
 $\begin{align*} h(x) &= \frac{f(x)}{S(x)} \\ &= \frac{\alpha}{x + \theta} \end{align*}$ 
Losses above $d$ are again Pareto, shifted to start at $d$ with scale $d + \theta$ , so
 $\begin{align*} e(d) &= \frac{d + \theta}{\alpha - 1} \\ &= \frac{d + 2{,}000}{2} \end{align*}$ 
$d$ $h(d)$ $e(d)$

0 0.00150 1,000

1,000 0.00100 1,500

10,000 0.00025 6,000

The hazard falls and the mean excess climbs linearly — both signatures of a heavy tail. An exponential with mean 1,000 would have constant $h = 0.001$ and constant $e(d) = 1{,}000$ .