Buy-and-Hold Projections
I have constructed tables from the equations in You Can't Count on 7% to assist buy-and-hold investors.
You Can't Count on 7%
Buy-and-Hold Equations
Here are the equations.
Here is the regression equation for the 10-year stock return and the percentage earnings yield 100E10/P (using 1923-1972 data): y = 1.5247x-4.5509 where y is the annualized real return in percent and x is 100E10/P or 100/[P/E10]. The confidence limits are plus and minus 6%.
Here is the regression equation for the 20-year stock return and the percentage earnings yield 100E10/P (using 1923-1972 data): y = 1.0849x-1.4488 where y is the annualized real return in percent and x is 100E10/P or 100/[P/E10]. The confidence limits are plus and minus 4%.
Here is the regression equation for the 30-year stock return and the percentage earnings yield 100E10/P (using 1923-1972 data): y = 0.4159x+3.764 where y is the annualized real return in percent and x is 100E10/P or 100/[P/E10]. The confidence limits are plus and minus 2%.
Determining P/E10
The tables require you to know P/E10 (or 100E10/P).
P/E10 is the value (price) of the S&P500 index divided by the average of the most recent ten years of earnings. Professor Robert Shiller maintains an S&P500 database, which includes P/E10, at his web site.
Professor Shiller’s Web Site
Professor Shiller’s Data
The smoothed earnings E10 vary slowly. Assuming that E10 has not changed enough to make a significant difference: The latest value of P/E10 equals [the old P/E10 value]*[the latest index value (price) / the old index value].
The index value in July 2005 was 25.95 (rounded). The S&P500 index was 1178.28 or 1178 (rounded).
When the S&P500 index is 1160, P/E10 = 25.6. When the S&P500 index is 1180, P/E10 = 26.0. When the S&P500 index is 1200, P/E10 = 26.4. When the S&P500 index is 1220, P/E10 = 26.9. When the S&P500 index is 1240, P/E10 = 27.3.
When the S&P500 index is 1100, P/E10 = 24.2.
When the S&P500 index is 1000, P/E10 = 22.0.
When the S&P500 index is 900, P/E10 = 19.8.
When the S&P500 index is 800, P/E10 = 17.6.
When the S&P500 index is 700, P/E10 = 15.4.
When the S&P500 index is 600, P/E10 = 13.2.
The percentage earnings yield is 100%/[P/E10]. For example, when P/E10 = 25, the percentage earnings yield is 4%. When P/E10 = 20, the percentage earnings yield is 5%.
Confidence Limits
The tables show intermediate probabilities.
I have used eyeball estimates for the confidence limits. I assign them a confidence level of 90% (two-sided).
A normal (Gaussian, bell shaped) distribution has a 90% confidence interval that extends plus and minus 1.64 standard deviations about the mean. Five percent of the outcomes are above the upper confidence limit and five percent of the outcomes are below the lower confidence limit.
Half way between one of these confidence limits and the Calculated Rate is 0.82 standard deviations from the mean. The confidence level of plus and minus 0.82 standard deviations about a normal (Gaussian, bell shaped) distribution is 60% (actually, 58.8%). Twenty percent of the outcomes are above the upper intermediate level and twenty percent of the outcomes are below the lower intermediate level.
I have identified five levels of probability. An outcome has a 50%-50% of being above or below the Calculated Rate. It has a 20% chance of being below the lower intermediate confidence level. It has a 5% chance of being below the lower confidence limit (which I have determined from an eyeball estimate). It has an 80% chance of being below (and a 20% chance of being above) the upper intermediate level. It has a 95% chance of being below (and a 5% chance of being above) the upper confidence limit.
These intermediate confidence levels are very easy to calculate. The confidence limits for the 10-year equations are plus and minus 6%. The intermediate levels are plus and minus 3%.
The confidence limits for the 20-year equations are plus and minus 4%. The intermediate levels are plus and minus 2%.
The confidence limits for the 30-year equations are plus and minus 2%. The intermediate levels are plus and minus 1%.
These estimates are coarse. I have limited my confidence interval to 90% about the mean because the details of the probability distribution are unknown. I am unwilling to assign a higher level of confidence to any projection. A normal (Gaussian, bell shaped) distribution is adequate under such conditions. Finally, because of the high amount of randomness in year-to-year returns (single years), the data have an adequate number of degrees of freedom.
Tables
Calculated Rates
P/E10 100E10/P 10-Year 20-Year 30-Year
25 4% 1.55% 2.89% 5.43%
20 5% 3.07% 3.98% 5.84%
15 6.67% 5.61% 5.78% 6.54%
10 10% 10.70% 9.40% 7.92%
Confidence Levels for 10-Year Projections
P/E10 100E10/P 5% 20% 50% 80% 95%
25 4% (4.4) (1.4) 1.55 4.6 7.6
20 5% (2.9) 0.1 3.07 6.1 9.1
15 6.67% (0.4) 2.6 5.61 8.6 11.6
10 10% 4.7 7.7 10.70 13.7 16.7
Confidence Levels for 20-Year Projections
P/E10 100E10/P 5% 20% 50% 80% 95%
25 4% (1.1) 0.9 2.89 4.9 6.9
20 5% 0.0 2.0 3.98 6.0 8.0
15 6.67% 1.8 3.8 5.78 7.8 9.8
10 10% 5.4 7.4 9.40 11.4 13.4
Confidence Levels for 30-Year Projections
P/E10 100E10/P 5% 20% 50% 80% 95%
25 4% 3.4 4.4 5.43 6.4 7.4
20 5% 3.8 4.8 5.84 6.8 7.8
15 6.67% 4.5 5.5 6.54 7.5 8.5
10 10% 5.9 6.9 7.92 8.9 9.9
Applying these Projections
These equations are best used in conjunction with our other design equations.
Equations for Design
These equations are not meant to help you grit your teeth and ride out the bad times when prices fall sharply. Rather, they are to help you invest sensibly and knowledgeably.
You may decide to delay a stock investment until you can get reasonable prices. Varying portfolio allocations slowly in accordance with changes in stock valuations is in the tradition of Benjamin Graham and other great investors. The notion of buying stocks regardless of prices is not.
Have fun.
John Walter Russell
September 7, 2005