Why Historical Returns Can Look Extreme for Volatile Stocks

Summary

The document investigates a chart showing sharply rising estimated average annual returns among stocks with high volatility. The estimates use historical daily prices, moving windows of 1,000 observations, and returns adjusted for the risk-free rate. The author considers survivorship and selection bias, skewed returns, and noisy averages as possible explanations, noting that stocks surviving for decades may be an unusually successful subset. The same pattern appears when volatility is measured using either a current estimate or a stock-specific historical value.

A simulation of zero-mean Brownian returns with occasional large bankruptcy drops reproduces the extreme averages. This suggests the pattern can arise from the sampling and selection process even when the true mean is zero, rather than indicating high expected returns for volatile stocks. The discussion is exploratory: it does not provide a formal derivation or quantify how each sampling choice contributes. The practical context is estimating historical mean returns for synthetic option prices, where the author had capped the mean prediction pending further investigation.

Key ideas

• Moving-window averages of historical returns can become unstable for high-volatility stocks.
• Stocks observed over long periods may be subject to survivorship and selection effects.
• A zero-mean return simulation with occasional bankruptcy losses reproduces the high estimated means.
• The observed pattern is not evidence by itself that volatile stocks have high expected returns.

Tags

# Insanely high mean annual returns for volatile stocks in historical data


# Insanely high mean annual returns for volatile stocks in historical data












Chart of 'Average Historical 1y Stock Returns' vs 'Current Volatility'.

The average calculated as moving window, stock returns adjusted for the current risk free rate.

The data 250 stocks, daily prices starting from 1972 each, 14_000 data points. Moving window size 1000 points.

What is going on here?

The first part of the chart up until vol=0.0005 looks reasonable, but after that the average stock returns just going into the sky.

Is it survivorship and selection biases and skewed, non normal and high noise? (mean of 1000 points looks quite jumpy and unstable).

Looks like selection bias, maybe, if stock is very volatile and yet it managed to not go bankrupt for 55 years, then on average, such stocks indeed are very profitable.

I published the data and code.

Also instead of the current volatility `EWMA[(log return)^2, span=365/3]` I plot against stock historical volatility (constant for each stock), again similar picture:

Why I need it and my solution

I need the 1y mean prediction to generate synthetic stock option prices for the past history.

So far I just forcefully limited the mean prediction at max value at 0.0005 volatility ignoring the high returns of high-vol stocks. I think it's some random artefact.

UPDATE:

It seems I found what causes it. I made simulation code, random brownian walks with mean zero and 0.3% bankruptcies drop, results are the same, very high mean returns for high volatility stocks (the real mean is 0):

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.