A reader commented and asked: “I frequently read that stock prices are not normally distributed, and that by assuming they are, an investor will tend to underestimate market risk. One paper I read says their distribution is leptokurtic, a distribution that has a more acute peak around the mean (that is, a higher probability than a normally distributed variable of values near the mean) and fatter tails (that is, a higher probability than a normally distributed variable of extreme values). My question is, given this fact, is there a practical way for retail investors who are not statisticians and who don’t have access to sophisticated tools, to better estimate risks than using functions that assume a normal distribution?”

You mean the distribution of returns, not prices.

To the degree that return distributions are actually leptokurtic, investors who treat them as normal overestimate future return predictability and thereby underestimate risk. For a return distribution that is sufficiently “wild” in this way, the mean (standard deviation) is not a good indicator for expected return (return volatility). In other words, a single additional observation may materially shift the mean and standard deviation even for a “large” sample. Accordingly, the sample size-confidence level (statistical significance) relationship derived for the normal distribution breaks down. This indeterminacy is fundamental (not just a matter of tool selection), theoretically limiting predictive power, as though nature wants to hide the future.

For further discussion, see:

“The Black Swan: The Impact of the Highly Improbable”

“The Fourth Quadrant: No Realm for the Normal”

“Different Paths to the Same (Disconcerting) Destination?”

“How Can You Avoid the Fat Left Tails?”