Equity Valuation Cannot Outgrow the Economy Over the Long Run.
Over the very long term the total dividend and capitalization of an economy's equity market should grow at the same rate as the GDP. Using this relationship, the expected total return to the equity market is the sum of the expected GDP growth rate and current dividend yield. Long-term U.S. historical data show a remarkable consistency with this proposition. A risk premium parameter that takes into account GDP growth potential, the interest rate and the dividend yield is identified. An equation that relates the risk premium to the interest rate and dividend yield is derived using the familiar LM curve of macroeconomic theory. Using this equation, it is suggested that bond prices are the ultimate check that brings stocks and GDP growth in line over the long run.
The U.S. stock market has been putting up an astonishing performance for the past decade. Equity market capitalization as measured by the Wilshire 5000  index quadrupled from the end of 1989 through 1999, creating enormous wealth. The combined capitalization of stocks listed on the New York Stock Exchange and Nasdaq  totaled $17.4 trillion at the end of 1999. The (NYSE+Nasdaq capitalization)/GDP ratio changed from 0.80 in 1993 to 1.87 in 1999. In these six years, the New York Stock Exchange and Nasdaq have created wealth that equals 1.8 times 1993 GDP .
Many believe that the fast pace with which the equity market has been advancing in the past decade can be maintained and even be surpassed in the future. Some predictions of the level of the Dow-Jones Industrial averages in the near future were raised from 36,000 to 40,000 to 100,000 [3,4,5]. (The Dow was 11,497 at the end of 1999.) On the other hand, others are nervous about the high level of the stock market and fear that a crash of unprecedented scale is in the making. One author predicted that the Dow is likely to fall as much as ninety-eight percent in early 2000s, and the market will probably not bottom out until it reaches a level of ninety-five .
While this paper will not make another prediction of the future market level, it does provide a framework that takes into account key macroeconomic factors that determine the market level in the long run.  Within this framework, one can make a sensible judgment on the possible long-run future market level and can factor that into investment decisions.
Corporate Profits, Dividends and GDP
On the income-side of the GDP accounts, corporate after-tax earnings are the total amount available for distribution each year to the owners of corporate business. The total dividend is the amount that is actually distributed. Wages, taxes, interest, rent and depreciation are the other main categories of factor income comprising the income-side of GDP accounts. The share of dividends in the total GDP should be fairly stable in order of magnitude over the long term. This share is determined by the structure of the economy. Figure 1 shows the after tax earning and dividend as percentage of GDP in the U.S. for 1959-1999 . After-tax earnings averaged 5.5 percent and dividends 2.7 percent of GDP for this period. The maximum and minimum of dividend/GDP ratio were 4.1 percent and 2.0 percent over this period.
Over the long term the dividend/GDP ratio is expected to fluctuate, but the order of magnitude should remain the same, under normal conditions. It is unlikely for the dividend/GDP ratio to get to, say, fifteen percent without a complete structural change of the economy coupled with drastic wealth redistribution. In the limiting case of an infinite time horizon, the expected average dividend growth rate must converge toward the GDP growth rate. Otherwise, the dividend/GDP ratio will either go up unbounded (when dividend grows faster than the GDP) or go down to zero (when dividend grows slower than GDP). Both cases are not possible for a functioning economy. We summarize this notion as following proposition.
Proposition 1. Over an infinite time horizon, the expected average growth rate of an economy total corporate dividend is the same as that of GDP.
Valuation Model of the Equity Market
We now derive a valuation model for the total equity market of an economy. To proceed, let us assume that: (1) all the companies accounted for in the GDP accounts are listed on one of the nation's stock exchanges, (2) foreign companies make up a negligible part of the total domestic equity market and (3) foreign earnings of domestic companies can be substituted by domestic earnings of foreign companies. These assumptions ensure that the dividend reported in the GDP is the same as the total dividend of all companies in the concerned equity market. This is not strictly true in reality for the U.S., but the approximation will not change the conclusions or model substantially. Denote the expected average GDP annual growth rate into perpetuity as g, the current total corporate annual trailing dividend D, capitalization of the equity market P and an appropriate discount rate d. The market capitalization is the sum of all discounted future dividends into perpetuity. Proposition 1 tells us that the dividend is expec ted to grow at the same average rate, g, as GDP. Assume that the dividend is paid out annually. Discounting all future expected cash flows at the rate d, which is greater than g, we have
(1) P = D[(1 + g)/(1 + d) + [(1 + g).sup.2]/[(1 + d).sup.2] + [(1 + g).sup.3]/[(1 + d).sup.3] + ... + [(1 + g).sup.n]/[(1 + d).sup.n] + ...]
(2) P = D[(1 + g)/(d - g)]
The total capitalization of the equity market is proportional to the total dividend D and increases as g, the expected future average GDP growth rate, increases and decreases as d, the required discount rate, increases. Let us take a closer look at the meaning of the rate d. Multiplying both sides of equation (1) by (1+d) and use the definition of P, we have
(3) (1 + d)P = D(1 + g) + P(1 + g)
D(1+g) is the expected dividend received at the end of next year. P(1+g) is the expected value, at the end of next year, of the sum of all discounted future cash flows beyond next year. By definition, P(1+g) is the expected capitalization for next year. Therefore, d can be interpreted as the expected total return rate of the equity market. The total return is composed of a dividend of D(1+g) and a capital appreciation of gP. Denoting the current dividend yield D/P as [Y.sub.D], we can, following Graham and Dodd  and others, write the total return rate,
(4) d = g + [Y.sub.D](l + g)
Equation (4) is the traditional result that says that the expected total return rate of the equity market has two parts: one from the capital appreciation rate, which is the same as the GDP growth rate over the long term, and the other from the current dividend yield.
Capital Appreciation Over the Long Term
We can generalize equation (3) for an arbitrary year n. Rewrite equation (1) as,
(5) (1 + D).sup.n] P= D(1 + g)[((l + d).sup.n] -[(1 + g).sup.n])/d - g + [(1 + g).sup.n]P
The first term on the RHS of equation (5) is the dividend income up to year n, and the second term, [(1 + g).sup.n]P, by definition is the market capitalization at end of year n. This implies that the market capitalization grows at the annual rate of g.
Proposition 2. Over an infinite time horizon, the expected total capitalization of the equity market grows at the same rate as that of the economy's GDP.
From Propositions 1 and 2, we can infer that the expected dividend yield over an infinite time horizon is a constant. For a finite time horizon, the Dividend/GDP ratio fluctuates; the economy has expansions and recessions; and the stock market has bubbles and crashes. How long do we have to wait to see the "long term" effect kick in? Let us now turn to U.S. historical data in this century to see how these conclusions hold up in reality. Table 1 shows for selected periods the total return to equity, the dividend yield, the equity capital market appreciation rate, and the historical U.S. GDP growth rate. The U.S. equity market data are taken from the celebrated book by Siegel . The GDP (GNP) data are from BEA  and the U.S. Census Bureau .
It is seen from the table that over the periods 1926-97, 1946-97 and 1966-97, the capital appreciation rates were not very different from the GDP growth rates. Therefore, over the long term, the capital appreciation rate has tended to track the GDP growth rate, as expected. Over shorter periods, however, the discrepancy can be substantial. Notice that over the period 1966-81, the equity market grew much slower (2.6 percent) than the nominal GDP rate (9.6 percent). On the other hand, during the recent bull market of 1981-97, the equity capitalization increased at a very high rate (12.9 percent), while the GDP grew at a more moderate rate (6.4 percent). By combining the two periods, the difference between the capital appreciation rate and GDP growth rate for 1966-97 is only 0.3 percent. Thus, the explosive capital gain of the stocks during 1981-97 could be interpreted as merely making up for the time lost during the previous one-and-one-half decades. For the longer period of 1926-1997, despite all the ups and downs in all aspects of American life, equity market capitalization and GDP have been on average growing at a similar rate. Again, for all the recent seemingly incredible returns on stocks, they have not outgrown the economy over the long period of 1926-97. 
From Table 1, we see that dividend yield has been on a slightly declining trend. But the order of magnitude remains the same for periods longer than thirty years. It appears that a period of about thirty years may be the time scale for the equity capital appreciation rate and the GDP growth rate to converge and therefore can be interpreted as "long term." This is the same time scale for "long term" as implied in Siegel . This is also about the length of time that discount rates converge to perpetuities. Data in Table 1, however, are not extensive enough to be conclusive. More research should be conducted to establish a more precise time scale for GDP growth and the stock appreciation rate convergence.
Even though for the infinite time horizon the total equity capitalization has to grow at the same rate as GDP, over an arbitrarily long finite time horizon there is no guarantee that the two growth rates will converge. What market mechanism exists that tends to bring the two growth rates in line over time? We will provide one plausible mechanism in the next section.
Equity Market Risk Premium
The total return to the equity market comes from both capital gains and dividends. A low dividend yield may not necessarily mean less return. Also, a very low dividend yield may not necessarily signal an over-valued market. Other factors must be considered. We now develop a parameter that can properly measure the risk and reward of the equity market. We can decompose the total return rate d into a risk free rate, i, and a risk premium, rp. Thus, equation (4) can be written as,
(6) rp = g - [Y.sub.D](l + g)
where rp is the risk premium that the equity investors demand for holding stocks. It is the rate of return above the risk free interest rate, i. Obviously, with higher expected GDP growth rates, higher dividend yields and lower interest rates, the equity holders are better rewarded for the risk they are taking. With higher values of rp, investors are more cautious and demand a high premium for holding equity. With a low rp value, stocks are aggressively valued, and the investors are content with a low risk premium. The risk premium, which combines the effects of growth potential, interest rates and dividend yields, is a useful gauge for the level of market valuation.
Let us now further develop equation (6) into a more useful working formula. The IS-LM model of macroeconomic theory says that when factors of production are fully employed by the economy, total output and interest rates have to be related in such a way that supply and demand will be matched in both the goods and asset markets. The standard LM equation is M/Pr = L(Y,I), where M is the aggregate money supply, Pr the price level, Y the real output and I the interest rate. A variation of the (logarithmic) LM equation can be written in the form,
(7) g = b + mi
where the b and m parameters should be, according to the IS-LM model, primarily dependent on the expected growth rate of real aggregate money supply (M/Pr). Neglecting the g term in [Y.sub.D](1 + g), equation (6) becomes,
(8) rp = b + (m - 1)i + [Y.sub.D]
If we now define risk premium ratio as the ratio between the risk premium and the risk free rate, rpr = (rp/i), equation (8) can be written as
(9) YD = (rpr + l - m)i - b
For a given risk tolerance rpr, equation (9) gives an iso-rpr line that relates [Y.sub.D] with i. Instead of comparing the dividend yield (or E/P ratio) directly with the interest rate, equation (9)--having accounted for stock growth potential--gives an alternative that can be used just as easily.
Now we use historical data to give a sample estimate of b and m in equation (7). In valuing equation (7), i should be the U.S. government perpetual bond yield representing the risk free rate for an infinite time horizon. Since the longest maturity for the traded U.S. government bond is thirty years, we will use the 30-year TCMS (Treasury Constant Maturity Series) rate reported monthly by the Federal Reserve Board  as a proxy. To illustrate with a plausible estimate of g, the expected average future GDP growth rate at a particular time, we use a weighted average for the past five years of GDP, with more weight given to the most recent years. Specifically,
This simple model is an example of the so-called "adaptive expectations model". It in effect predicts the average
(10)g = [0.5.sub.[g.sub.0]] + [O.25.sub.[g-1]] + [O.125.sub.[g-2]] + [0.0625.sub.[g-3]] + [0.0625.sub.[g-4]]
future GDP growth by extrapolating the weighted moving average of the past five years' realized GDP growth rates . Other models to predict future GDP growth rates might have been used, but that should not change the general results.
Figure 2 illustrates the historical relationship between g, given by equation (10), and i, the 30-year TCMS rate, for 1980-99. Results of linear regression on these two quantities are reported in Table 2. R-squared for the regression is 0.817. The two-tail p-value for the null hypothesis, H (null): m=0, is 6x[10.sup.-8]. Substituting values of b and m in equation (8), we have,
(11) rp = 1.046 - 0.306i + [Y.sub.D]
If the values of b and m are relatively stable, equation (11) can be used to give a nile-of-thumb estimate for the equity market risk premium, knowing the readily available interest rate and dividend yield. Alternatively, we can use rpr and plug b and m into equation (9) to get
(12) [Y.sub.D] = (rpr + 0.306)i - 1.046
Figure 3 shows equation (12) for rpr=0.225 and rp=0 together with the historical ([Y.sub.D], i) pairs for 1980-99. For [Y.sub.D] we used the dividend yield of the S&P 500 index  as an approximation to that of the whole U.S. equity market. The rpr=0.225 line is the average line of these historical points, and rpr=0 represents the critical line. Any ([Y.sub.D], i) point falling below this ipr=0 line represents an extremely over-valued stock market because the risk premium for such a point is less than zero. It can be seen that the ([Y.sub.D], i) point for 1999 was close to a twenty-year low.
Finally, let us try to answer the question posed in the last section: What market mechanism is available to ensure that equity capitalization and GDP grow at the same rate for the long run? It is seen clearly from equation (8) or (11) that the interest rates and the risk premiums act as opposing poles to attract investors. Equations (6) and (7) determine respectively the equity and bond valuation for a given expected GDP growth rate. Over a period of time, for a given series of realized GDP growth rates, the interest rate--and thus the bond value--are roughly determined by equation (7). The GDP growth rate influences the stock market through the interest rate via equation (8).
If equity capitalization grows faster than the GDP for an extended period of time, the dividend yield will decrease; and the risk premium will diminish. At a certain point, the risk premium would be so low that investors would switch out of stocks into bonds. This would keep the stocks in line with GDP growth potential markets and eventually restore the proper level of risk premium.
A similar argument holds true for the opposite case in which equity capitalization grows slower than GDP for an extended period of time. Therefore, bonds--a competing asset to stocks--whose value is determined on a macroeconomic level by GDP growth (equation (7))--is the ultimate check to ensure that equity capitalization and GDP growth rates converge for the long run. It is well known that the level of the stock market is negatively correlated with the interest rate level (positively correlated with bond price). Equation (8) or
(11.) fits the phenomenon in a firm theoretical framework.
In summary, the analysis above has the following salient features:
* Even though many authors have discussed the relation between GDP and stock valuation, this paper provides one systematic and unambiguous approach in a coherent, albeit simplified, theoretical framework. Many have wondered how far stocks can go given the economy as a constraint. This paper provides some additional insight to the "How far?" question.
* A risk premium equation is derived with a sound macroeconomic footing. This equation can potentially become an effective tool in valuing stocks.
The author wants to thank the editor and the anonymous referees. Their comments greatly improved the paper. He also thanks Water Cheung of CIBC World Markets and Jing Chen of National University of Singapore for their comments.
AVAILABILITY OF DATA AND CALCULATIONS
All data and calculations used in this article are available on www.eurodynamics.com.
Jian-Chiu Han has been a derivatives trader for the past seven years after receiving a Ph.D. from MIT. He has worked for Chase, AIG and CIBC World Markets.
(1.) For recent examples of macroeconometric approaches relating to stock market valuation in this journal, see Fair (2000) and Prakken (2000) [8,9]
(2.) BEA'S GDP data only go back to 1929. For the period of 1926-29, GNP data from  are used.
(3.) If the original 1926-97 stocks return data in  are extended to 1999 using the Wilshire 5000 index, the stocks' capital appreciation rate for 1926-99 is 6.41 percent. GDP growth rate is 6.43 percent for the same period.
(1.) Bloomberg Historical Data Series. Bloomberg LP (Available to Bloomberg data service subscribers. Details can be found at www.bloomberg.com/products.)
(2.) U.S. Bureau of Economic Analysis data, www.bea.doc.gov.
(3.) Glassman, J. and K. Hassett, Dow 36,000. New York: Random House, 1999.
(4.) Elias, D., Dow 40,000. New York: McGraw-Hill, 1999.
(5.) Kadlec, C., Dow 100,000. Upper Saddle River, NJ: Prentice Hall Press, 1998.
(6.) Prechter, R., At the Crest of Tidal Wave. New York: John Wiley & Sons, 1997.
(7.) Graham, Benjamin and David Dodd, Security Analysis. New York: McGraw Hill, 1951.
(8.) Fair, Ray C., "Fed Policy and the Effects of a Stock Market Crash on the Economy," Business Economics, April 2000, pp. 7-14.
(9.) Prakken, Joel L., "Potential Productivity and the Stock Market in the 'New' U.S. Economy," Business Economics, April 2000, pp. 15-19.
(10.) Siegel, J., Stocks for the Long Run. New York: McGraw-Hill, 1998.
(11.) U.S. Bureau of the Census, The Statistical History of the United States. New York: Basic Books, Inc., 1976.
(12.) Federal Reserve Board data, www.federalreserve.gov/release/.
U.S. EQUITY MARKET RETURN AND GDP GROWTH RATE (IN PERCENT) Period Equity total Dividend Yield: Equity capital Average GDP return: d [Y.sub.D] (1+g) appreciation growth rate 1926-97 10.6 4.6 6.0 6.5 1946-97 12.2 4.3 7.9 7.4 1966-97 11.5 3.9 7.6 7.9 1966-81 6.6 4.1 2.6 9.6 1981-97 16.7 3.7 12.9 6.4 STATISTICS OF LINEAR REGRESSION BETWEEN g AND i Variable value standard error t-statistics p-value (2 tail) b 1.046 0.6955 1.503 0.150 m 0.694 0.0788 8.805 [less than]0.001