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BAM Intelligence

## On the Distribution of Long-Run Stock Returns

It is well-known that the distributions of daily and monthly equity returns are leptokurtic (fat-tailed) relative to the normal distribution. In other words, the shape of their return distribution is more peaked than you’d find in a normal, or bell curve, distribution. In such a distribution, small changes are less frequent than in a normal distribution, and extreme events (a large price move) are more likely to happen.

Additionally, extreme events are potentially much larger (the central peak is narrower, but the tails are significantly longer and fatter) than in a normal distribution. However, as Eugene Fama and Kenneth French note in their May 2017 paper, “Long-Horizon Returns,” because we have fewer observations, we know less about the distributions of long-horizon returns.

To study distributions of long-horizon (up to 30 years) U.S. stock returns, Fama and French used bootstrap simulations. A bootstrap simulation uses actual monthly returns from history, but rearranges them to get different “lifetimes” of return streams.

This is a common technique used when the author of a study doesn’t want to make any assumption about the underlying distribution, but believes each return reasonably represents an independent observation from the same distribution.

The Fama and French study covered the period July 1926 to December 2016, and they sought the answers to two questions. First, how does the distribution of investment payoffs change as we extend the horizon? Second, how does uncertainty about expected returns affect the distribution of long-horizon payoffs?

Study Findings

Following is a brief summary of their findings:

• Bootstrap distributions of continuously compounded returns approach normal distributions for longer return horizons. If continuously compounded returns approach a normal distribution (i.e., the returns are symmetric on both sides of the mean), we would expect the natural log of the same variable would have a log-normal distribution (a distribution that is skewed right with a fat right tail).
• However, as Fama and French extended return horizons, there was little evidence that payoff distributions converge to the log-normal distributions suggested by convergence to normal of continuously compound returns—30-year returns are as close to (or far from) log-normal as monthly returns.
• Skewness and kurtosis increase at longer horizons. This means that the tails are fatter even though the end results are more concentrated around the long-term average.
• The tails of bootstrap payoff distributions are further to the left (more extreme for the left tail, less extreme for the right) than the log-normal distribution predicts, and the middle of the distribution is shifted to the right.

For investors, the important implication of these findings is that the longer the horizon, the greater the dispersion of returns. In other words, just as real-world options prices suggest, the longer the horizon, the greater the risk of equity investing.

This commentary originally appeared September 6 on ETF.com

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