Skewness, Kurtosis, and Strategies That Lie

This book invokes fat tails constantly. The moments that make a tail fat are worth defining once.

Variance is the second moment about the mean. The third and fourth, standardized, are:

Skewness measures asymmetry. Negative skew means the left tail is longer — occasional large losses among many small gains. Kurtosis measures tail weight; a normal distribution has kurtosis 3, and the excess over 3 is what people mean by “fat-tailed.” Equity index returns are reliably negatively skewed with substantial excess kurtosis: crashes gap, rallies grind.

Why this exposes a whole category of strategy. Consider a strategy that earns a small premium almost every month and occasionally loses catastrophically — selling out-of-the-money puts, carry trades, most yield-enhancement products, and anything described as “consistent income” (section “Option Greeks: Impact of Volatility on Options”). Such a strategy displays, for years:

• a high Sharpe ratio, because realized volatility is genuinely low between events;
• a small maximum drawdown, because none has happened yet;
• strong negative skew and enormous kurtosis — the only two statistics that reveal the problem.

The Sharpe ratio is not merely uninformative here; it is actively misleading, because the strategy is manufacturing the appearance of low risk by relocating the risk into a tail the measure ignores. Before accepting any track record with an unusually smooth return series, look at the third and fourth moments — and remember that a strategy which has not yet met its tail will show excellent statistics right up until the day it does.

The private-asset version of the same trick. Illiquid holdings — private equity, venture funds, non-traded REITs, private credit — are marked by appraisal rather than by market. Appraisals lag, smooth, and cluster around the last mark, which mechanically suppresses reported volatility and reported correlation with public markets. The result is a flattering Sharpe ratio and a small apparent drawdown, neither of which describes the economics of the position; it describes the reporting convention. This is the “volatility laundering” discussed in section “How the Fund Makes Money on Its LPs”. The tell is high first-order autocorrelation in the reported return series — genuine market returns are close to serially uncorrelated, while smoothed appraisals are not. When a private fund reports a Sharpe ratio above its public-market equivalent, assume the difference is measurement, not skill, until shown otherwise.