The Cauchy Distribution in Simple Terms

Most financial models assume that returns follow a normal distribution but what if this assumption is flawed? What if extreme events happen far more often than expected?

In 1827, Augustin-Louis Cauchy introduced a probability function while studying wave propagation and resonance:

Cauchy Probability Density Function (PDF):

\[ f(x; x_0, \gamma) = \frac{1}{\pi \gamma} \cdot \frac{\gamma^2}{(x - x_0)^2 + \gamma^2} \]

• \( x_0 \) (location): The peak and median of the distribution.
• \( \gamma \) (scale): Controls spread and tail thickness.

Unlike the normal distribution, the mean of a Cauchy distribution does not exist.

For most probability distributions, we compute the mean (expected value) using the integral:

\[ E[X] = \int x \cdot f(x) \ dx \]

where \( f(x) \) is the probability density function (PDF) of the Cauchy distribution.

If this integral converges to a finite value, the mean is well-defined. This works for distributions like the normal distribution, where extreme values decay fast enough to ensure convergence.

When we attempt to compute \( E[X] \), the integral diverges because the Cauchy distribution has tails that decay too slowly. This means the mean is not defined, adding more data does not stabilize the average.

In many distributions, the location parameter coincides with the mean because the mean is well-defined. However, in the Cauchy distribution:

• The mean does not exist.
• The location parameter \( x_0 \) still exists and represents the center of the distribution.

The median is the value that divides the probability distribution into two equal halves. Unlike the mean, it is always well-defined, even in heavy-tailed distributions.

Since extreme values strongly influence the mean (when it exists), but do not affect the median, the median is a more robust measure of central tendency when dealing with outliers.

Portfolio theory assumes averaging returns across assets reduces risk. In a Cauchy-like world, increasing sample size does not stabilize returns, diversification does not always reduce risk.

Standard risk measures assume finite variance, but the Cauchy distribution has infinite variance, rendering traditional metrics (like standard deviation) to measure risk useless.

In a Cauchy world, volatility is infinite, outliers dominate risk.

• Use the Median Instead of the Mean.
• Use Interquartile Range (IQR) for Risk.
• Use Conditional VaR.

Cauchy’s work challenges traditional finance by showing that risk can be fundamentally unstable.

Markets are not (often) Gaussian…