Assessing Skewness and Kurtosis in the Return Distribution
When returns fall outside of a normal distribution, the distribution exhibits skewness or kurtosis. Skewness is known as the third "moment" of a return distribution and kurtosis is known as the fourth moment of the return distribution, with the mean and the variance being the first and second moments, respectively. (Variance is a statistic that is closely related to standard deviation; both measure the dispersion of an investment's historical returns.) Ideally, investors should consider all four moments or characteristics of an investment's return distribution.
Skewness: Skewness measures the degree of asymmetry of a distribution around its mean. Positive skewness indicates a distribution with an asymmetric tail extending toward more positive values. Negative skewness indicates a distribution with an asymmetric tail extending toward more negative values.
Kurtosis: Kurtosis measures the degree to which a distribution is more or less peaked than a normal distribution. Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis indicates a relatively flat distribution. A normal distribution has a kurtosis of 3. Therefore, an investment characterized by high kurtosis will have "fat tails" (higher frequencies of outcomes) at the extreme negative and positive ends of the distribution curve. A distribution of returns exhibiting high kurtosis tends to overestimate the probability of achieving the mean return.
Figure 5 illustrates both the skewness and kurtosis in the return distribution for the S&P 500 Index from Figure 4. The skewness is negative, which tells us that the returns are negatively biased. Because kurtosis measures the steepness of the curve, we can tell that there is a steep curve by reviewing the kurtosis number. A kurtosis less than zero indicates a relatively flat distribution.
Skewness and kurtosis are important because few investment returns are normally distributed. Investors often predict future returns based on standard deviation, but such predictions assume a normal distribution. An investment's skewness and kurtosis measure how its distribution differs from a normal distribution and therefore provide an indication of the reliability of predictions based on the standard deviation. As Figure 6 highlights, two investments with very different distribution profiles can have the same mean and standard deviation. Therefore, it is useful to consider other methods for predicting returns.
Source: "An Introduction to Omega, Con Keating and William Shadwick, The Finance Development Center, 2002
Table 1 summarizes the key characteristics of a return distribution.