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Expected Dividend Growth, Valuation Ratios and Rational Optimism
Gary Santoni
Department of Economics
Ball State University
Muncie, IN, 47306
Email: gsantoni@gw.bsu.edu
Alex Tabarrok
George Mason University
Tabarrok@gmu.edu
Published as:
Santoni, G. and A. Tabarrok. 2002. Expected Dividend Growth, Valuation Ratios
and Rational Optimism. Journal of Financial and Economic Practice 1 (1): 110119.
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Stock market valuation ratios reached historically unprecedented levels in the late 1990s
and continue to be very high today.1 Chart 1 shows the behavior of the price earnings ratio for
stocks contained in the Standard and Poor’s Index. The data, which are updated from Shiller
(1996), are annual for the years 1871-2000. As indicated, the P/E ratio of 28 recorded in 2000
had never before been attained in the previous 130 years. The data in Chart 2 show that the
recent behavior of the dividend price ratio relative to its historical norm is even more stark.
As of this writing (late 2001), stock prices have fallen, especially in the once high-flying
Internet sector. The various ratios have fallen to more "reasonable" levels but are still high
relative to historical averages. While prices have fallen so have dividends and earnings and the
issue raised in the literature concerns the ratio of prices to various measures of the fundamentals.
As Chart 3 shows, these ratios are independent of the level of prices. The dividend price ratio,
for example, (which in late 2001 stands at 0.017), is still below its 1996 level of 0.024 when
Federal Reserve Chairman Alan Greenspan characterized market participants as “irrational,” and
well below its historical average of 0.048. In addition, it is also clear from Charts 1 and 2 that
there are times other than the late 1990s (for example, 1893-96, 1930-34, 1960-72, and 1991-93)
when the P/E ratio was unusually high and the dividend price ratio unusually low.2
This paper uses the Gordon growth model to explain variation in these valuation ratios.
In particular, the model is used to show that acceleration in the expected dividend growth rate
beginning in the late 1950's is consistent with the behavior of the price earnings and dividend
price ratios since that time. This explanation of the behavior of the ratios provides a more
optimistic outlook for the future behavior of stock prices than, for example, Campbell and Shiller
(1998) who suggest that the recent levels of “...these ratios are extraordinarily bearish for the
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U.S. stock market (p. 24)” or Fama and French (2000) who argue that, “...the low end-of-sample
dividend yield implies low expected (future) stock returns (p.15).”3
The Gordon Growth Model and Valuation Ratios
We use Gordon’s (1962) growth model to assess the behavior of the price earnings and
dividend price ratios. This model says that the current price of a stock, Pt , depends on three
factors: 1) the dividend the stock paid in the previous period, Dt 1 , the current period expectation
of the future growth rate in dividends, g t , and the current period discount rate relevant for stocks
in the firm’s risk category, rt , as related in equation (1).
Pt  Dt 1 (
1  gt
)
rt  g t
(1)
Notice that equation (1) can be rearranged to obtain an expression for the dividend price ratio as
in equation (2).
Dt 1 rt  g t

Pt
1 gt
(2)
Furthermore, if both sides of equation (1) are divided by trailing earnings, Et 1 , as in equation
(3), an expression for the price earnings ratio is obtained.
Pt
D
1 gt
 t 1 (
)
Et 1 Et 1 rt  g t
(3)
Equations (2) and (3) are useful. They indicate that both valuation ratios are driven by
the discount rate, rt , and the expected growth rate in dividends, g t . In particular, an increase in
the discount rate increases the dividend-price ratio while an increase in the expected growth rate
in dividends lowers it. These changes operate in reverse on the price-earnings ratio. In addition,
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the price-earnings ratio increases with an increase in the proportion of earnings paid out in
dividends, Dt 1 / Et 1 .
The dividend price and price earnings ratios are both very sensitive to changes in the
expected growth rate in dividends and discount rate. For example, consider the dividend price
ratio given by equation (2). The mean of this ratio is 0.048 over the 1871-2000 interval. Since
g t is small, typically around 2 percent (see further below), this implies that the mean difference
between rt and g t is about 0.05. Suppose the mean of g were one percentage point higher with
the mean of r unchanged. In this case, the dividend price ratio would fall from about 0.048 to
about 0.038 – almost a 25 percent reduction.
A similar result holds for the price earnings ratio. If the mean difference between rt and
g t is about 0.05, the term inside the parentheses in equation (3) is about 20. Since the average
payout ratio, Dt 1 / Et 1 , is about 0.65, equation (3) implies an average price earnings ratio of 13.
This is close to the actual average of 13.9. Again, a one percentage point increase in the mean of
g results in an increase in the price earnings ratio from about 13 to 16.25 – an increase of 25
percent. The large swings in the dividend price and price earnings ratios shown in Charts 1 and
2 can therefore be explained by relatively mild variation in the expected growth rate of
dividends.4
Some Potential Explanations of Current Levels of the Valuation Ratios
The most common explanation for high valuation ratios is that these are signs of an
unsustainable price bubble brought about by irrational exuberance. Shiller (1989, 2000) and
Campbell and Shiller (1998, 1988) are proponents of this explanation. A few economists,
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however, have tried to explain recent valuation ratios on the basis of a permanent fall in the
discount rate – a fundamental factor.
For example, Fama and French (2000) argue, in the context of the Gordon model, that a
decline in the discount rate, rt , explains the appreciation in stock prices in recent years as well as
higher price earnings and lower dividend price ratios.5 Siegel (1999) presents a similar argument
and suggests that the high current level of stock prices relative to the fundamentals imply,
“Either 1) future stock returns (discount rates) are going to be lower than historical averages, or
2) earnings (and hence other fundamentals such as dividends or book value) are going to rise at a
more rapid rate in the future (p.14).” Like Fama and French, Siegel (1999, p. 15) concludes that
the bulk of the explanation for the recent behavior of the valuation ratios shown in Charts 1 and 2
is a decline in the discount rate. However, Siegel indicates that this cannot be the whole story
because relying solely on this explanation requires a discount rate below the current yield on
inflation-protected government bonds. Since these bonds are safer than equities, their expected
returns should be lower not higher than expected equity returns. As a result, Siegel suggests that
part of the explanation “must be due to the expectation that future earnings (and dividend)
growth will be significantly above the historical average (p. 14).” A similar suggestion is made
in Carlson (2001, p. 3), Carlson and Pelz (2001, p. 2) and Balke and Wohar (2001). We pursue
this alternative explanation.
Of course, expectations regarding future growth in earnings and dividends are not
directly observable but we provide a new method for “backing out” these expectations from
Gordon’s model. Using the model, we ask what investors would have had to expect about
dividend growth rates to generate the price series that actually occurred. These derived
expectations can then be compared to the actual series of dividend growth to see whether the
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derived series is plausibly rational (Is the derived series consistent with actual dividend
growth?).6 Doing this necessarily requires us to drop the assumption used in constructing the
Gordon model that the dividend growth rate has a mean that is constant over time. Instead, we
follow Barsky and DeLong (1993, p. 293) who “...see investors as having to estimate period by
period a growth rate that is nonstationary.”7 The conflict between this assumption and the model
is why we regard our procedure as a back-of-the-envelope calculation. We use the Gordon
model despite this problem because it produces a startling result – the sharp swings in the price
earnings and dividend price ratios shown in Charts 1 and 2 are consistent with a simple model
based solely on fundamentals so long as that model allows for changes in the expected growth
rate of dividends.8
Expected Growth in Real Dividends
Chart 4 plots the expected annual growth rate in real dividends that we derive from the
model. To obtain the series, we solve equation (1) for g t . Our calculations use the actual values
observed for Pi and Dt 1 . In addition, we assume that the real discount rate for stocks, rt , is
constant and equal to 8.0 percent over the entire period.
The expected growth rate in dividends implied by the Gordon model oscillated around a
mean value of 2.6 percent with a standard deviation of 1.4 percentage points prior to 1956. Since
then the growth rate accelerated sharply rising to a mean level of 4.3 percent. The standard
deviation in the later period is 1.1 percentage points. Because we have worked backwards by
solving the model for the expected growth rate in dividends implied by the actual time series of
stock prices and our assumption regarding the discount rate, it is very important to ask whether
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the solution for g implied by the model is plausibly rational. Is the series that we derive from
the model a reasonable representation of the expected value of the realized (observed) series?
Chart 5 plots the expected growth rate in real dividends (shown in Chart 4) against the
realized annual growth rate. The realized growth rate is clearly much more volatile (standard
deviation = 0.13) than the expected growth rate (standard deviation = 0.015) as is typical of
realized versus expected values. Furthermore, the means of the two series do not differ from one
another in a statistical sense. The average annual difference between the realized and expected
growth rates is 0.015 which is statistically indistinguishable from zero (t-score=1.41). In
addition, the time series of the differences between the realized and expected growth rates is
white noise as expected if the growth rate derived from the Gordon model is the expected value
of the realized rate. 9
It is important to note that the data fails to reject the model even though we allow for only
one degree of freedom – changes in the expected dividend growth rate. Of course, if we were to
allow for variation in the discount rate, the same conclusion would hold a fortiori. We are not so
much arguing for a pure dividend growth explanation as saying that such an explanation is not
inconsistent with the data.
The above suggests that the expected growth rate in real dividends is arguably rational.
This together with the sensitivity of the dividend price and price earnings ratios to variation in
the expected growth in dividends implies that increases in expected dividend growth can account
for the current levels of the ratios. For example, expected dividend growth averaged 0.066 in the
last three years of the data. This, together with equation (2) and our assumption of an 8.0 percent
discount rate implies a dividend price ratio of 0.013. The average of the realized series over the
same period is 0.014.
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The rationality of the expected dividend growth rate can also be seen in the context of
changes in the macro-economy. Chart 4 suggests that with the exception of a break in the 1970s
the expected growth in real dividends has been on a long upward trend since about 1950.
Consistent with this, output volatility has been on a long-downward trend since the 1950s, again
with the exception of a break in the 1970s. Blanchard and Simon (2001), for example, estimate
that quarterly output volatility has declined theefold since the early 1950s, from about 1.5% to
less than 0.5% currently. As a result, output expansions have become considerably longer over
time and recessions have become shorter. It is exactly these sorts of changes that should be
reflected in expectations of real dividend growth, as indeed appears to be the case.
Conclusion
Large changes in valuation ratios can be explained by relatively small changes in the
expected dividend growth rate. We use the Gordon growth model to back out an "expected"
dividend growth rate and we compare this rate with the actual rate of dividend growth. We
cannot reject the hypothesis that our estimated rate is a rational expectation of the actual. As a
result, a model of valuation ratios based solely on a handful of fundamentals can easily explain
the variation in the data. In particular, the historically high ratios of the late 1990s and today are
consistent with rational expectations about dividend growth. Such expectations, moreover,
appear reasonable in the context of slowly accumlating, long-term changes in the economy that
are reducing output volatility, increasing the duration of expansions and reducing the duration of
recessions. Contrary to a number of recent analyses, optimism about the future is not ruled out
by the data.
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Endnotes
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1. Papers discussing valuation ratios include Balke and Wohar (2001), Campbell and Shiller
(1998), Siegel (1999), Carlson (2001), Carlson and Pelz (2001), Fama and French (2000) and
Shiller (1996).
2.Of course, prices can be unusually high or low relative to measures of the fundamentals. The
papers referenced above are more concerned with episodes when prices are high relative to the
fundamentals.
3.See Siegel (1999), pp. 10 and 15 for a similar argument.
4.See also Barsky and DeLong (1990, p. 269).
5.See Fama and French (2000) who conclude that, “The unexpected capital gains for 1950-1999
are largely due to a decline in the discount rate (p. 15).”
6.Using a much more complicated procedure, Donaldson and Kamstra (1996) and Barsky and
DeLong (1993) derive series for the expected growth rate in dividends. Both series cover
periods shorter than the period examined here and neither includes the more recent period during
which the valuation ratios shown in Charts 1 and 2 reached their record levels.
7.A nonstationary growth rate is one for which the mean varies over time. Dropping this
assumption is problematic since derivation of the Gordon Model makes use of it. (See Kamstra
and Donaldson 1996, p. 345).
8.We use the model in the same spirit as Fama and French (2000) who argue that their “...
conclusion does not rest on the validity of the Gordon Model. All valuation models say the
dividend yield is driven by expected future returns (discount rates) and expectations about future
dividends (p. 15).”
9.Our hypothesis is that the realized growth rate in real dividends, Gt , represents a single draw
from a distribution of possible outcomes at time t . The mean of the distribution is the expected
growth rate in real dividends, g t , that we derive from Gordon’s Model. Hence, Gt will differ
from g t by a white noise error, E t , as in the following equation.
Gt  g t  Et
The autocorrelations and partial autocorrelations of E t at all lags through 36 are statistically
indistinguishable from zero and the Ljung-Box Q statistic at 36 lags is 44.7. Hence, the test fails
to reject the null that E t is white noise.