Special Note about Mean Reversion

Rob Bennett has made an important discovery. Mean Reversion occurs faster when you adjust for valuations.

Procedure

I tabulated the real, annualized, total return of the S&P500 (without any expenses) at years 10, 20 and 30. I used Excel’s LINEST function to calculate regression statistics versus 100E10/P (the percentage earnings yield, 100/[P/E10]).

When data are independent, the spread of the data decreases with the square root of the number of years. The ratio of a standard error at year 10 and the standard error at year 20 would normally be the square root of 2. The ratio of the square of the standard error at year 10 and the square of the standard error at year 20 would normally be 2 (assuming statistical independence). Similarly, the ratio of the square of the standard error at year 10 and the square of the standard error at year 30 would normally be 3.

The spread in stock returns decreases faster. Such behavior defines MEAN REVERSION (or REVERSION TO THE MEAN). It is not simply that a mean exists. It is that the randomness snaps down much faster than independence would suggest.

Using LINEST, you can calculate the ratio of the square of the standard error of y at year 20 and the square of the standard error of y at year 10 by squaring the relevant sey terms or, more easily, taking the ratio of ssrid terms. This calculation shows the effect of MEAN REVERSION when you take valuations into account.

Using LINEST, you can calculate the effect of MEAN REVERSION when you ignore the effect of valuations, You take the ratio of the relevant sum of the squares totals: sstotal = ssreg + ssresid.

Rob Bennett's Mean Reversion Discovery

Data

Here are the standard errors of y (includes the effect of valuations):
At year 10: 4.203.
At year 20: 2.375.
At year 30: 1.397.

Here are the standard errors using the totals (excludes the effect of valuations):
At year 10: 5.361.
At year 20: 3.360.
At year 30: 1.690.

Using 1923-1975 returns, based on the spread at year 10:

When you ignore valuations, at year 20, the spread of the data falls to what would normally be expected at year 25.5.

When you ignore valuations, at year 30, the spread of the data falls to what would normally be expected at year 100.6.

Using 1923-1975 returns, based on the spread at year 10:

When you include the effect of valuations, at year 20, the spread of the data falls to what would normally be expected at year 31.3.

When you include the effect of valuations, at year 30, the spread of the data falls to what would normally be expected at year 90.6.

These relationships are not strong statistically. From this data set, I calculate the confidence level in favor of the existence of MEAN REVERSION as being close to 90%. The effect of including valuations has a confidence level in the neighborhood of 75% to 80%.

Conclusions

The spread of real, annualized, total return of the S&P500 decreases when you take valuations into account.

The spread of real, annualized, total return of the S&P500 decreases more rapidly at year 20 when you take valuations into account.

Whether or not you take valuations into account, the real, annualized, total return of the S&P500 exhibits MEAN REVERSION.

You should use this data set only as evidence to support what makes sense theoretically. Never rely on numbers alone.

Have fun.

John Walter Russell
June 27, 2006