S&P500 Regression Equations

The value of x is the percentage earnings yield 100E10/P (or 100/[P/E10]), where P/E10 is Professor Robert Shiller's measure of valuation. The value of y is the Calculated Return.

Professor Shiller maintains an S&P500 database at his web site. It includes monthly values of P/E10.
Professor Shiller's web site

Sequences starting from 1881-1962.

Equations of real, annualized total returns for 20, 30 and 40-Year Sequences

1881-1962:

20 years:
y = 0.5316x + 2.362
plus 4.2 and minus 4.2
R-squared = 0.3172

30 years:
y = 0.2591x + 4.2433
plus 2.8 and minus 2.6
R-squared = 0.2198

40 years:
y = 0.3411x + 3.8111
plus 2.0 and minus 2.0
R-squared = 0.5746

Calculated Returns at different valuations

Estimates at today's valuations:
100E10/P = 3.5%
20-year return: 4.22%
30-year return: 5.15%
40-year return: 5.00%

Estimates for January 2000:
P/E10 = 43.77
20-year return: 3.58%
30-year return: 4.84%
40-year return: 4.59%

Estimates at typical valuations:
P/E10 = 14
20-year return: 6.16%
30-year return: 6.09%
40-year return: 6.25%

Sequences starting from 1921-1962.

Equations of real, annualized total returns for 20, 30 and 40-Year Sequences

1921-1962:

20 years:
y = 0.5209x + 2.9554
plus 3.8 and minus 4.0
R-squared = 0.2521

30 years:
y = 0.2624x + 4.8916
plus 2.2 and minus 2.2
R-squared = 0.2215

40 years:
y = 0.2896x + 4.6044
plus 1.4 and minus 1.4
R-squared = 0.5555

Calculated Returns at different valuations

Estimates at today's valuations:
100E10/P = 3.5%
20-year return: 4.78%
30-year return: 5.81%
40-year return: 5.62%

Estimates for January 2000:
P/E10 = 43.77
20-year return: 4.15%
30-year return: 5.49%
40-year return: 5.27%

Estimates at typical valuations:
P/E10 = 14
20-year return: 6.68%
30-year return: 6.77%
40-year return: 6.67%

Remarks

I suspect that the equations derived from the 1921-1962 sequences are better in spite of using less data. I suspect, but I do not know for sure, that there is an anomaly similar to what we have found with Historical Surviving Withdrawal Rates. That is, the effects of P/E10 differ in the earlier period (especially, during 1900-1910) from those in later years.

Total return calculations versus valuations from the 1921-1962 equations look better than those from 1882-1962. They are closer to what we have observed at typical valuations (i.e., P/E10 = 14).

Today's valuation drag down the 30 and 40-year returns by about 1% when compared with normal valuations. They drag the 20-year calculated returns down even more.

Don't overlook the confidence limits. They are wide, about plus and minus 4% at 20 years, 2.2% at 30 years (using the 1921-1962 data) and 1.4% at 40 years. These are from eyeball estimates. They are at a high level of confidence. Actual results are likely to be closer to the calculated returns.

Have fun.

John Walter Russell
October 6, 2005