Why Fat Tails Break Averages

For bell-curve returns, huge losses are so rare that the average loss is finite. Some markets have fat tails that shrink only like 1/x². Comparing integrals shows the probabilities still add up, but the average loss doesn't: x/(1 + x²) is at least 1/(2x), and ∫ 1/x diverges.

How it works

Comparison test for improper integrals

In one sentence

If $0 \le f(x) \le g(x)$ for $x \ge a$ , then $\int_a^\infty g$ converging implies $\int_a^\infty f$ converges, and $\int_a^\infty f$ diverging implies $\int_a^\infty g$ diverges.

General formula

\frac{1}{1+{\htmlData{sym=0}{x}}^2} \le \frac{1}{{\htmlData{sym=0}{x}}^2} \;\Rightarrow\; \int_1^{\htmlData{sym=1}{\infty}} \frac{d{\htmlData{sym=0}{x}}}{1+{\htmlData{sym=0}{x}}^2} < {\htmlData{sym=1}{\infty}}, \qquad \frac{\htmlData{sym=0}{{x}}}{1+{\htmlData{sym=0}{x}}^2} \ge \frac{1}{2{\htmlData{sym=0}{x}}} \;\Rightarrow\; \int_1^{\htmlData{sym=1}{\infty}} \frac{{\htmlData{sym=0}{x}}\,d{\htmlData{sym=0}{x}}}{1+{\htmlData{sym=0}{x}}^2} = {\htmlData{sym=1}{\infty}}

If a curve stays under one with finite area, its area is finite too; if it stays above one with infinite area, so is its own.

What the symbols mean

$x$
Loss size
How big a loss is, in units of the typical day's move.
$\infty$
No upper limit
Every loss size, however large, counts.

A loss distribution's density must have total area 1, and its average loss is $\int x f(x)\,dx$ .
For a fat-tailed (Cauchy) density, $\frac{1}{1+x^2} \le \frac{1}{x^2}$ and $\int_1^\infty \frac{dx}{x^2}$ converges, so the probabilities add up.
But $\frac{x}{1+x^2} \ge \frac{1}{2x}$ for $x \ge 1$ , and $\int_1^\infty \frac{dx}{2x}$ diverges, so the average loss is infinite.
For bell-curve returns, $e^{-x^2/2}$ shrinks so fast that both integrals converge.

Learn the math

Improper IntegralsCalc II · Lesson 8, Integration Techniques Comparison Test for Improper IntegralsCalc II · Lesson 9, Integration Techniques