GDP GROWTH AND STOCK MARKET VALUATION LIMITS:
FROM THE GORDON MODEL TO IBBOTSON S&P 500 RETURN DECOMPOSITIONS
David A. Brat, Department of Economics and Business
Randolph-Macon College, Ashland, VA 23005, 804-752-7353
dbrat@rmc.edu
Is the stock market over valued or under valued? In order
to assess these claims one must be able to indicate a true
value from which deviations may occur. Many economic
papers promise to explain valuation and many deliver by
providing insights into various relationships between
financial variables. However, very few papers have tried to
determine a baseline value for the stock market based on
supply side economic variables. This paper attempts to tie
GDP growth to earnings growth and ultimately to the S&P
500 index price. Several papers have supplied bits and
pieces of this puzzle and this paper is an attempt to put
them together into one framework.
decomposition which will clarify these relationships in
more detail. We have made some progress in this study
toward that end.
FIGURE 1
S&P and GDP
100
Logilinear Scale
ABSTRACT
Nominal
GDP
Nominal
S&P
Price
10
Nominal
S&P
Earnings
Nominal
S&P
Dividend
In the long run, we want to argue that stock market
performance cannot exceed the pace set by growth in
GDP. Exactly what the relationship between these two
variables is remains to be seen. Figure 1, shows GDP, S&P
500 nominal price, nominal earnings, and nominal
dividend normalized to a value of 1 on the left axis.
Clearly, all four variables increase at similar, but not
identical, rates. These series suggest several points
immediately. To simplify, let’s assume that nominal GDP
has grown at 6% over the long run. Then, assuming
capital’s share of income constant, Figure 1 shows
earnings growth at the same rate. Earnings paid out in
dividends have also grown at about 6% and capital
appreciation, nominal S&P Price, has also grown at 6%.
Both grow at 3% real rates. Thus, this simple model or
graph would suggest that if the total return to the S&P is
composed of the dividend payment plus capital
appreciation, we should expect a long run total real return
of about 6%, assuming zero inflation. According to
Diermeier [3] we are not far off as he concludes in his
study that the total real return is about 5.4%. However,
when one reviews the studies which include the recent run
up in stock prices, authors such as Lansing and Ibbotson
et. al. [6,4] conclude that the total return on equity is
10.7% over the past century.
In any case, we are still left with the question of why GDP
growth of 6% can generate total S&P returns of 6% or
10%. What is the upper limit and why is it a limit? We
intuitively know that a limit must exist but we are trying to
identify an accounting rule and some type of total return
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INTRODUCTION
47
1
Date
THE LITERATURE AND FRAMEWORK
One way of addressing the stock market value is presented
by Roger Ibbotson and Peng Chen. Ibbotson and Chen [4],
for example, discuss six ways to decompose historical
stock returns into their different components, and then
offer two methods to predict future returns using the
supply side of equity returns. They use inflation, earnings,
dividends, price to earnings ratio, dividend payout ratio,
book value per share, return on equity, and GDP per capita
as the components of the supply side of returns. They
conclude that earnings is in line with output increases.
Taking the average return over the previous 75 years,
Ibbotson and Chen argue that increases in the P/E ratio
account for only a small portion of stock returns, (1.25%).
GDP growth and earnings growth account for larger
portions of total return. Ibbotson and Chen state that the
supply side of equity returns are extremely important to
investors, as investors can only expect returns equal to
what companies can supply.
The six methods offered for deconstructing historical
returns are essentially identities, substituting terms for one
another in analyzing the supply side of equity returns. The
total nominal return on equities from 1926-2000 has been
10.7% according to work done by Ibbotson. This 10.7%
can be viewed in many ways. The first and most basic
decomposition is 10.7% = 3% inflation + 2% real risk free
rate on long-term U.S. bond + 5.24% equity risk premium,
the amount investors were compensated for investing in
common stocks rather than U.S. bonds.
Decomposition two states that 10.7% = 4.28% income or
dividend growth + 3 % real capital gain + 3% inflation.
Decomposition three indicates that 10.7%=4.28% dividend
income + 1.25 % growth in P/E + 1.75% growth in EPS +
3% inflation. The cap gain in two is simply dissected here.
Decomposition six indicates that 10.7% = 4.28% dividend
income + 1% growth in factor shares + 2 % growth in
GDP/POP + 3% inflation.
The most important method in terms of our paper is
decomposition six, the GDP method.
There are several important findings noted by Ibbotson and
Chen. [4] “First, the growth in corporate earnings is in line
with the growth of the overall economic productivity as
seen in Figure 1. Second, P/E increases account for only
1.25% of the 10.7% total equity returns. Most of returns
are attributable to dividend payments and nominal earnings
growth (including inflation and real earnings growth).
Third, the increase in relative factor share of the equity can
be fully attributed to the increase in the P/E ratio. Overall
economic productivity outgrew both corporate earnings
and dividends from 1926 through 2000. Fourth, despite the
record earnings growth in the 1990s, the dividend yield
and the payout ratio declined sharply, which renders
dividends alone a poor measure for corporate profitability
and future earnings growth.” [4]
RESEARCH QUESTION
We are getting warm enough to make some comments
relating to our thesis. First, we have now found a link
between GDP growth and stock market returns. However,
the surprise is that GDP growth only accounts for 2% of
the 10.7% nominal average return over our period. How
can this be? Part of our problem may be related to the
period examined and the fact that we are dealing with only
the mean values over a long period. For example, the
dividend income accounted for 4.3% real returns over this
period. However, Figure 3 in Ibbotson and Chen shows
that the dividend yield has fallen drastically from highs of
6% or so in the 1930s – 1950s to only 1.1% in 2000. In
addition, their Figure 4 shows that the dividend payout
ratio has fallen from 60% over the bulk of the sample
period to the current lows of 32%. In addition, the results
above are complicated by the fact that the nominal average
return is 10.7% from 1926 – 2000 but if one does not
include the period of the recent stock run-up, this number
falls. Diermeier [3] for example forecasts the real return to
be 5.4% from 1963-1982. An 8.4 % “nominal” return is
then reasonable and this is significantly lower than the
10.7% return used in the Ibbotson paper. Which result is
relevant in tracking the fundamental value of the stock
market? Or should we just expect on average numbers to
give us an on average conclusion?
In summary, in the Ibbotson paper we have found a gold
mine in terms of what we were out to find. They state the
exact part of total returns explained by GDP growth and it
is small. It appears that we are now faced with another
problem. Here, we seem to have the answers as they fall
out of accounting decompositions but we lack the theory to
explain the upper and lower bounds we seek for these
decompositions. In the first and most basic decomposition,
how large can the equity risk premium be and what would
set the bounds for this number? Historically it has been
about 5% but could it have been 10%? Why or why not?
Is this payment to investors in any way linked to real
output? By looking to the decomposition method alone, the
answer seems to be no. GDP growth accounts for only 2%
of the 10.7% according to method 6. We have also noted
that overall, productivity has outgrown both earnings and
dividends and this may bode well for “expected future
earnings.” It may also suggest an answer to why S&P
returns can exceed GDP growth, as they do.
So while we are getting warm and the story gets more
interesting, we hope, we need to go back to the theory with
the hope of unifying this story.
THEORY AND METHOD
The remainder of this paper will address the questions
posed immediately above in our prior work. [1] First, the
paper will show the mathematical links between the
Gordon model [5]and the Ibbotson decompositions, and
the assumptions necessary to make the decompositions.
Second, it will address the sensitivity of these
decompositions to changes in the mean values as the
sample period changes from 1926-2000 to 1963-1984 to
1984- present. Finally, the paper will review and assess
what the appropriate model might be for explaining market
behavior today.
In brief, the Gordon model gets us from:
P0 = ∑ (Dt/(1+k)t
to P0=∫Dte-ktdt to
k = D0/P0+ g,
where P is a share’s price, D is the Dividend and k is the
rate of profit. “Translated, the final equation above means
that the rate of profit at which a share of common stock is
selling is equal to the current dividend, divided by the
current price (the dividend yield), plus the rate at which the
dividend is expected to grow. Gordon refers to k as the
growth rate of profit.” [5]
Importantly, the difference between this model and the
dividend yield (D/P) is the assumption of growth. The
latter assumes that the dividend will remain constant. To
assume a constant rate of growth and estimate it (the
growth rate) to be equal to the current rate appears to be a
better alternative. Under this model, the dividend will
grow at the rate br, which is the product of the fraction of
income retained b, and the rate of return earned on net
worth r and so g = b*r. One can also arrive at g directly by
taking some average of the past rate of growth in a
corporation’s dividend. [5] The average value chosen is
one of the topics under investigation in the current paper.
Diermeier [3] has specified the final model we will use in
our study. It is derived directly from the Gordon model
above. In Diermeier, if r= the total return on financial
assets, d = the income return, a= the capital appreciation
and n = financial net new issues, the two equations relate
these variables to each other and to g, the growth rate of
financial assets.
If r = d +a and n = g – a, where g = b*r from above, then
r = d + (g – n) and this is the supply model of expected
return we will use to investigate current rates of S&P
returns.
DATA AND SIMPLE FORECASTS
Due to space constraints, I will put forward three simple
forecasts using this Diermeier framework in algebraic
form. In each case, I will use the same algebraic model,
but will fill it in with data from three different sources.
First the data from the Diermeier study itself was from the
period 1963- 1982, not the most representative S&P trend.
r = d + (g - n)
5.4 = 7.5 + (2.6 – 4.7)
where
a = g -n
-2.1= (2.6-4.7)
Alternatively, when one uses Ibbotson average data over
the period 1926-2000, one obtains the following return:
r = d + b * r - (g – a)
r = 4.3 + (40%* 4) - (1.6% - 1) = 4.3 + 1.6 – 0.6 = 5.3 %
Finally, when one uses current data proxies, one obtains:
r = d + b * r - (g – a)
r = 1.1 + (60%*1.1) - (.66% - 1) = 1.1 + .66 + .34 = 2.0 %
These are all real rates of return. By contrast, the Ibbotson
average rate of return over the long run was a nominal
figure of 10.7%. A real rate of 7.7% is implied as they use
a long-run average of 3% CPI.
Ibbotson decomposition two above states that long-run
nominal returns of 10.7% = 4.28% income or dividend
growth + 3 % real capital gain + 3% inflation. If we plug
in the current data proxies used in the Diermeier format
above, we find a much lower return in the Ibbotson model.
r = 1.1% income or dividend growth + 3% real capital gain
+ 1% inflation + 0.5% reinvestment return = 5.6%
The real return would be 4.6%.
Ibbotson reports two forecasts himself. [4] First, the longrun supply of U.S. equity returns based on the earnings
model is 9.37%, yielding a 6.37 real return.
Second, the forward-looking dividends model is also
referred to as the constant-dividend-growth model (or the
Gordon model). Using current data, Ibbotson finds an
estimate of equity returns = 5.44%. However, they use the
long-run CPI average of 3% in forecasting this figure.
With a CPI of only 1%, the real equity return forecast
would be only 3.44%.
CONCLUSIONS
The goal of this paper was to find and examine hard links
between stock market returns and GDP growth. We found
the link we were looking for but it was unexpectedly small,
and the link itself was really only between GDP growth
and real capital gains. The GDP link to dividend income
was not found. We have found that productivity growth
outpaced earnings and dividend growth and this may
partially explain why equity returns can exceed present
GDP growth figures. We then presented several simple
but highly influential models forecasting equity returns.
We have the upper and lower bounds of forecasts and we
continue to seek a more thorough understanding of the link
between real economic activity and these equity returns.
REFERENCES
[1] Armstrong, JC. and Brat, David. “Supply Side
Determinants of Stock Market Valuation” Proceedings
of SeInforms, 2003.
[2] Asness, Clifford. “Stocks versus Bonds: Explaining
the Equity Risk Premium.” Financial Analysts Journal
56(2), 2000, pp. 96-113.
[3] Diermeier, Jeffrey and Ibbotson, Roger and Siegel,
Laurence. “The Supply of Capital Market Returns.”
The Financial Analysts Journal, 40(2), 1984, pp.74-80.
[4] Ibbotson, Roger G., and Peng Chen. "Stock Market
Returns in the Long Run: Participating in the Real
Economy." Yale University, International Center for
Finance, Working Paper No. 00-44, March, 2002.
[5] Gordon, Myron and Shapiro, Eli. “Capital Equipment
Analysis: The Required Rate of Profit.” Management
Science, Octorber 1956, pp. 102-110.
[6] Lansing, Kevin. “Searching for Value in the U.S. Stock
Market.” San Francisco Federal Reserve: Economic
Letter, 2002.