On the White Board - September 2009

Sep 15, 2009

Stock Market Skewness

We all know stock market returns aren’t normally distributed. Much attention has been given to the “fat-tailedness” of returns, especially in recent months. Another, less publicized example of non-normality arises when we consider the skewness of logarithmic returns. In some sense this example is more basic, as skewness is measured with the third moment about the mean whereas fat-tailedness is measured with the fourth moment about the mean (the first two moments are the mean and the standard deviation, which can be exactly fitted to a normal distribution).

This skewness persists whether we consider daily, weekly, monthly, or annual logarithmic returns. This is the opposite of the expected behavior of logarithmic returns, because we expect the Central Limit Theorem to apply to logarithmic returns since they are additive over time (the annual log return is the sum of 252 daily log returns). Robert Shiller’s monthly stock market data, going back to 1871 (available at http://www.econ.yale.edu/~shiller/data.htm), is used to estimate annual log return skewness.

Specifically, all 1650 contiguous 12-month periods are used in our computation, since that is the best estimate of skewness, but when computing the errorbar it is assumed that there are only 137 independent data points. The skewness of (overlapping) annual logarithmic returns is computed to be -0.83, whereas the errorbar for the skewness estimator assuming a normal distribution with 137 independent data points is 0.21. In other words, the annual returns exhibit a skewness which is four standard deviations away from normal skewness (which is zero). This is statistically significant, and interesting; most of the models now in use for such things as long-term modeling do not take this statistically significant negative skewness into account, which may result in overly-rosy long-term forecasts. The plot below is a histogram of the annual returns with a best-fit normal approximation.

What could explain this negative skewness? One possible explanation is that the market by its very nature tends to have bubbles - to make many smallish gains and then balance that with largish losses, resulting in negative skew.