Skewness

Skewness is a measure of how lopsided a probability distribution or a set of data is around its average. A symmetric distribution, such as the normal distribution, has a skewness of zero. A distribution with a long tail of high values has positive skewness, and one with a long tail of low values has negative skewness.

In finance, skewness describes the balance between large gains and large losses in a series of returns. Two investments can share the same average return and the same standard deviation and yet differ sharply in the risk of a severe loss. Skewness, together with kurtosis, captures that difference.

Skewness is the third standardized moment of a distribution. For a sample of $n$ observations it is commonly calculated as:

• $x_i$ is the $i$-th observation.
• $\bar{x}$ is the mean of the observations.
• $s$ is their standard deviation, here computed with $n$ in the denominator.

Cubing the deviations keeps their sign, so large negative deviations pull the result down and large positive ones push it up. Dividing by $s^3$ removes the units. Statistical software often applies a small-sample adjustment, so reported values can differ slightly.

Consider five monthly returns of -6, 1, 1, 2 and 2 percent. The mean is zero. The squared deviations sum to $36 + 1 + 1 + 4 + 4 = 46$, giving a variance of 9.2 and a standard deviation of about 3.03. The cubed deviations sum to $-216 + 1 + 1 + 8 + 8 = -198$, an average of -39.6. Skewness is $-39.6 / 3.03^3 = -39.6 / 27.9$, or about -1.42. The series is strongly negatively skewed: four small gains and one large loss.

In a positively skewed distribution most outcomes are modest and a few are very large, so the mean lies above the median. A lottery ticket is the extreme case. In a negatively skewed distribution most outcomes are small gains and a few are large losses, and the mean lies below the median.

Returns on broad equity indices, measured over days or months, have historically been negatively skewed: markets tend to fall faster than they rise. Returns on individual shares over long horizons are positively skewed, because a loss is limited to the amount invested while a gain is not. Research on United States shares has shown that a small minority of companies account for most of the market's long-run gain. Compounded returns also produce positive skew in prices, which is one reason the lognormal distribution is used to model them.

Strategies have characteristic skew, and it is a useful guide to how they fail.

• Negative skew. Selling volatility through options collects small premiums most of the time and occasionally suffers a large loss. Carry trades and many arbitrage strategies using leverage share this profile.
• Positive skew. Trend following takes many small losses when markets move sideways and occasional large gains when a sustained trend develops. Buying options has the same shape.
• Mixed cases. Equity momentum earns steady returns in most periods but is subject to a rare, sharp momentum crash.

A negatively skewed strategy can show a long, smooth record that conceals its real risk. Measures such as the Sharpe ratio, which use only the mean and standard deviation, flatter such strategies. The Sortino ratio and the maximum drawdown are often used alongside it for this reason.

Investors appear to like positive skew and dislike negative skew. Assets with lottery-like payoffs tend to be priced high and to earn low average returns, while assets that add negative skew to a portfolio must offer higher expected returns to be held. Asset pricing research has formalized this through coskewness, the contribution of an asset to the skewness of the market portfolio.

Option prices reveal the skewness that the market expects. Out-of-the-money puts on equity indices trade at higher implied volatilities than out-of-the-money calls, a pattern known as the volatility skew. It implies a distribution of future prices with a heavier left tail than the one assumed by the basic option pricing model.

Sample skewness is imprecise. Because deviations are cubed, a single extreme observation can dominate the result, and adding or removing one crisis month can change the sign. For normally distributed data the standard error of sample skewness is roughly the square root of $6 / n$, so even several years of monthly returns give a wide margin of uncertainty. A strategy that has not yet experienced its worst outcome will show a skewness that understates the danger. Skewness also depends on the measurement interval, and daily, monthly and annual returns on the same asset can have different values.