Speculative Return

The Dividend Discount Model appears frequently in the theory of the stock market. There are many variants. The variant that I shall address is that of Vanguard’s John Bogle.

To learn the details of the Dividend Discount Model and its variants, read one of Gummy’s tutorials. Gummy is a prominent poster and a retired mathematics professor. His real name is Peter Ponzo.
Gummy's (Peter Ponzo's) web site
Gummy's Dividend Discount Model Tutorial

The return from the stock market can be broken into two parts: the investment return and the speculative return.

The investment return equals the dividend yield plus the dividend growth rate. Assuming that dividends remain at a constant fraction of earnings, the investment return also equals the dividend yield plus the earnings growth rate. When looking at a broad-based index such as the S&P500, smoothed earnings (such as the average of the previous ten years of earnings) grow at a steady rate. You can calculate the investment return of a broad-based index accurately with information that is always available.

The equation for the investment return is accurate in the intermediate-term (typically ten years, but ranging from 5 years to 20 years). It is overcome by the speculative return in the short-term. In the long-term, the mathematical assumptions behind it (specifically, the desirability and availability of suitable reinvestments) break down. In the very long-term, the stock market has returned 6.5% to 7.0% (annualized, real, total return).

The speculative return is the effect of price changes relative to the same income stream. John Bogle prefers to estimate it by dividing the final price-to-earnings ratio by the initial price-to-earnings ratio. An alternative way to phrase the identical calculation is to divide the initial earnings yield by the final earnings yield. [The earnings yield is the inverse of the price-to-earnings ratio, usually expressed as a percentage.] Another calculation, which is closer to the Dividend Discount Model, is to divide the initial dividend yield by the final dividend yield. This ratio is the same, provided that dividends remain a constant fraction of earnings.

Here is a fine point. When we convert from changes in price-to-earnings ratios to percentage changes, we take the difference in the price-to-earnings ratios and divide by the initial price-to-earnings ratio. This is the proper, standard calculation. It is close to, but different from, similar calculations using earnings yield. Taking the difference and then dividing by the initial earnings yield or by the final earnings yield or by the average of the initial and final earnings yields: all produce different results.

Results

I made graphs of the 10-Year (annualized, real, total) stock market (S&P500) return versus the initial percentage earnings yield 100E10/P, the percentage earnings yield in year ten minus the percentage earnings yield in year one and the average of the percentage earnings yield in years ten and one.

Letting x = the percentage earnings yield 100E10/P component and y = the annualized, real, total stock market return, the equations using 1923-1970 data are:
1) x = 0.2828y + 5.304 with R-squared = 0.4136 when x is the initial percentage earnings yield,
2) x = -0.5529y + 3.7759 with R-squared = 0.9128 when x is the difference in earnings yields and
3) x = 0.0063y + 7.1919 with R-squared = 0.0004 when x is the average of the initial and final earnings yields.

[To put all three equations on one graph required me to place the annualized, real, total stock market return on the horizontal axis. I have retained my normal definitions of x as an earnings yield and y as the annualized, real, total return.]

Notice from the first equation that earning yield and the total return are positively correlated. Higher earnings yields go together with higher total returns. Lower earnings yields go together with lower total returns. This is consistent with theory and with earlier results.

Notice from the second equation that total returns are even more closely related to the difference of the percentage earnings yield at year ten and at year one. The value of R-squared has increase from 0.4136 to a very high 0.9128. The change in the sign of the slope appears because the percentage earnings yield at year one has a minus sign when taking the difference.

Finally, notice that the total return is essentially independent of the average of the earnings yields at years one and ten. R-squared is 0.0004.

I have taken similar data using 1921-1970. The relationships are distorted by the exceedingly high earnings yields in 1921 and 1922. The values of R-squared are 0.3804 for the initial earnings yield, 0.8142 for the difference and 0.0206 for the average.

I have plotted similar data using the difference in earnings yields divided by the average of the earnings yields in years one and ten. It increased the value of R-squared from 91% to 92%. It is better, as suggested by theory, but it is not worth the additional effort.

For a specified 10-Year annualized, real, total return, the initial earnings yield varied from minus 2.5% to plus 5%. [Divide minus 2.5% and plus 5% by the slope of 0.2828 to find the variation in the 10-Year real, annualized, total return versus the percentage earnings yield.]

Taking the difference of earnings yields reduces the amount of scatter.

For a specified 10-Year annualized, real, total return, the difference in the earnings yield varied from minus 2% to plus 1%. [Divide by the slope of -0.5529 to find the variation in the 10-Year real, annualized, total return versus the percentage earnings yield.] The relationship is a little bit tighter for 10-Year returns close to zero.

For 5-Year returns:
1) x = -0.3567y + 2.4248 with R-squared = 0.907 when x is the difference in earnings yields and
2) x = 0.0086y + 7.1172 with R-squared = 0.0021 when x is the average of the initial and final earnings yields.

The scatter about the difference in the percentage earnings yield was minus 1% to plus 1% when the 5-Year real, annualized, total return was below 10%. It was minus 3% to plus 2% when the total return was above 10%. [Divide by the slope of –0.3567 to find the variation in the 5-Year real, annualized, total return for a specified difference in the percentage earnings yield.]

Remarks

Obviously, we do not know the percentage earnings yield ten years into the future. But we can look at sensitivities.

Today’s earnings yield is 3.5%. If P/E10 were to fall to 20, which would still be high by historical standards, the earnings yield would be 5.0%, which is a (year ten minus year one) difference of plus 1.5%. The equation predicts a change in the real, annualized, total return equal to 1.5% divided by the slope of –0.5529, which is –2.7%. That is, overall return would be the investment return minus 2.7%.

If P/E10 were to fall to 14 to 15, which has been typical historically, the earnings yield would increase from today’s 3.5% to 7%. This is a difference of plus 3.5%. From the equation, the speculative return would be minus 6.3%. This would overwhelm any investment return.

If P/E10 were to fall to 8, which is still comfortably within the historical range, the earnings yield would rise to 12.5%. The difference would be plus 9.0%. The speculative return would be 9.0% / [minus 0.5529]. That is, minus 16.3%. The overall ten-year result would be a loss of 10% or, possibly, worse.

Have fun.

John Walter Russell
August 9, 2005