The distribution of stock returns

There is ample evidence that stock returns are not normally distributed. They tend to have leptokurtic distributions. What does that mean?

Leptokurtic distributions, compared to normal distributions, are more peaked in the center and have fatter tails. To give you an example, suppose that a stock has an annual mean return μ of 5% with a standard deviation σ of 10%. If the stock's returns were normally distributed, 68.26% of the return observations would fall between -5% and 15% (μ plus/minus 1σ). But, if the distribution is leptokurtic, more than 68.26% of the return observations will fall in this range.

You might say, "So what?" But, wait till you hear the whole story. Coming back to the stock that has μ=5% and σ=10%, if the stock's returns were normally distributed, less than 0.00006% of the return observations would be smaller than -45% or larger than 55% (μ plus/minus 5σ). Now, this is the crucial bit. If the distribution is leptokurtic, the probability of observing an extreme return (i.e., less than -45% or more than 55%) can be much higher. And, this matters a lot for investors. The fact that stock return distributions are leptokurtic tells investors that extreme returns occur more often than they would under a normal distribution.

Professor Eugene Fama

Professor Eugene Fama's research was influential in spreading the message that stock return distributions are fat tailed. That is, the extreme parts (tails) of stock distributions carry higher probability density than normal distributions.

To understand this concept better consider the following example. We have mentioned above that the probability of observing a daily return that is more than 5σ away from μ is 0.00006% under the normal distribution. Imagine that you could collect daily returns on a stock over 2 million trading days, which is 8,000 years. Multiplying 2 million trading days with 0.00006% yields 1.2. Therefore, under a normal distribution, we would expect that the stock yields a daily return that is 5σ away from μ (approximately) only once in 8,000 years.

Fama investigated daily returns of a bunch of stocks over 5 years. What he found was extraordinary. The stocks were yielding extreme daily returns much more often than the normal distribution predicts. For instance, AT&T had 6 observations of daily returns that are more than 5σ away from μ in just 5 years! Well, this is little bit more frequent than 1 such observation in 8,000 years!!!

The lesson we learn from Fama's research is that days with extreme returns are not rare in stock markets. They occur way more often than we might want to believe...

Bibliography

Fama, Eugene F., 1965, Behavior of Stock-Market Prices, Journal of Business

• Tweet