2.6 Heavy tails, power laws and the Hill estimator

Motivation

The tail determines the probability of the day that ends a trading career. Normal tails decay like $e^{-x^2/2}$ ; equity tails decay like $x^{-3}$ . At five standard deviations, the ratio between these is a factor of thousands.

Power-law tails and moments#

Setup

A random variable $X > 0$ has a Pareto tail with index $\alpha > 0$ if $P(X > x) = (x/u)^{-\alpha}$ for $x \ge u$ . More generally, $X$ is regularly varying with tail index $\alpha$ if $P(X > x) = L(x)\,x^{-\alpha}$ with $L$ slowly varying ( $L(tx)/L(x) \to 1$ as $x \to \infty$ for every $t > 0$ ). The Student- $t$ with $\nu$ degrees of freedom is regularly varying with $\alpha = \nu$ ; the normal is not regularly varying (its tail index is effectively infinite).

Proposition 2.11moments of a power-law tail

If $P(|X| > x) \sim C x^{-\alpha}$ , then $E|X|^p < \infty$ iff $p < \alpha$ .

Proof

$E|X|^p = \int_0^\infty p x^{p-1} P(|X| > x)\,dx$ . The integral over $[0, x_0]$ is finite. For large $x$ the integrand behaves like $pC x^{p-1-\alpha}$ , which is integrable at infinity iff $p - 1 - \alpha < -1$ , i.e. $p < \alpha$ .∎

So with $\alpha \approx 3$ : the mean and variance exist, the fourth moment does not, and the population kurtosis is infinite. The sample kurtosis then grows without bound as $n$ grows (more precisely, it is dominated by the largest observation), which is what the Nifty table showed.

The Hill estimator#

Proposition 2.12Hill estimator as conditional MLE

Let $X_{(1)} \ge X_{(2)} \ge \dots \ge X_{(n)}$ be the order statistics of $|r_t|$ . Treat the $k$ largest as draws from a Pareto tail above the threshold $u = X_{(k+1)}$ . The MLE of $\alpha$ is

\hat\alpha_k = \Big[\frac1k\sum_{i=1}^{k}\ln X_{(i)} - \ln X_{(k+1)}\Big]^{-1}. \tag{2.16}

If the tail is exactly Pareto above $u$ , then $k/\hat\alpha_k \sim \text{Gamma}(k, \alpha)$ , $E[\hat\alpha_k] = \frac{k}{k-1}\alpha$ and $\sqrt{k}(\hat\alpha_k - \alpha) \Rightarrow N(0, \alpha^2)$ .

Proof

Conditional on exceeding $u$ , the Pareto density is $f(x) = \alpha u^\alpha x^{-\alpha-1}$ , $x \ge u$ . The log-likelihood of $k$ exceedances $x_1, \dots, x_k$ is

\ell(\alpha) = k\ln\alpha + k\alpha\ln u - (\alpha+1)\sum_{i=1}^{k}\ln x_i.

$\ell'(\alpha) = k/\alpha + k\ln u - \sum\ln x_i = 0$ gives $\hat\alpha = k/\sum_i\ln(x_i/u)$ , and $\ell''(\alpha) = -k/\alpha^2 < 0$ confirms a maximum. Substituting $u = X_{(k+1)}$ gives (2.16). For the distribution: if $X$ is Pareto( $\alpha$ , $u$ ), then $Y = \ln(X/u)$ has $P(Y > y) = e^{-\alpha y}$ , i.e. $Y \sim \text{Exp}(\alpha)$ . By the Rényi representation, the $k$ log-excesses over the $(k+1)$ -th order statistic are distributed as $k$ iid Exp( $\alpha$ ) variables, so $\sum Y_i \sim \text{Gamma}(k, \alpha)$ with mean $k/\alpha$ . Then $E[\hat\alpha] = k E[1/\sum Y_i] = k\alpha/(k-1)$ (the mean of an inverse-gamma), and the delta method applied to $\bar Y$ , which has mean $1/\alpha$ and variance $1/(k\alpha^2)$ , gives $\operatorname{Var}(\hat\alpha) \approx \alpha^4/(k\alpha^2) = \alpha^2/k$ .∎

For distributions that are only regularly varying, $\hat\alpha_k$ is consistent if $k \to \infty$ and $k/n \to 0$ (Mason 1982), but it has a bias that depends on how fast $L(x)$ settles down. The choice of $k$ trades variance ( $\alpha/\sqrt k$ ) against bias (which grows with $k$ ). Plot $\hat\alpha_k$ against $k$ (a Hill plot) and look for a stable region.

Tail estimates: simulation and Nifty#

Worked example 2.6.1simulation, with hill_estimator from m02_tails.py

For samples of 200,000 draws from a Student- $t$ with $\nu = 3$ , the average of $\hat\alpha_k$ over 40 independent samples is $3.08, 2.98, 2.95, 2.89, 2.79$ at $k = 200, 500, 1000, 2000, 5000$ (sampling s.d. across samples $0.24, 0.10, 0.08, 0.06, 0.04$ ). The true value is 3. The estimate drifts down as $k$ grows because the Student- $t$ density departs from a pure power law away from the extreme tail; a single sample need not show the drift at small $k$ , where the sampling error is larger than the bias. For normal data, $\hat\alpha_{1000} \approx 10.2$ (average over 20 samples) with $n = 200{,}000$ , and it keeps rising as $n$ grows.

Worked example 2.6.2Nifty sample

Using $|r_t|$ : $\hat\alpha = 3.59, 3.05, 2.76, 2.47$ at $k = 50, 100, 200, 400$ , with standard errors $\alpha/\sqrt{k}$ of about 0.51, 0.30, 0.20, 0.12. The left tail alone gives $\hat\alpha_{100} = 2.61$ and the right tail $2.86$ . These are in the range reported for international equity indices and stocks: Gopikrishnan, Meyer, Amaral and Stanley (1998) found $\alpha \approx 3$ for US stock returns (the "inverse cubic law"). With $\alpha \approx 3$ for Nifty,

\frac{P(|r| > 10\sigma)}{P(|r| > 5\sigma)} \approx 2^{-3} = \frac18 \qquad\text{versus}\qquad \frac{2\bar\Phi(10)}{2\bar\Phi(5)} \approx 2.7 \times 10^{-17} \text{ under normality.}

Practitioner notes

With $\alpha \approx 3$ , variance exists, so the CLT applies to sums and monthly returns are closer to normal than daily returns (aggregational Gaussianity). Convergence is slow, and quarterly returns still show excess kurtosis.
A tail index between 2 and 4 makes moment-based risk measures (kurtosis, and the standard errors of variance estimates) unreliable. Quantile-based measures and the tail index itself are more stable.
Short-option books (M09) are short the tail. The premium is set by the market's estimate of tail probability; the seller's survival depends on its own estimate of $\alpha$ and on position size (M03).