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, not 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.