A share of stock, you will recall, is a certificate of ownership in the business firm named on that certificate. I you own 1 share of ABC Co. stock and the ABC Co. has sold a total of 1 million of such shares, then you are entitled to the following: (1) one millionth of the annual cash dividend the firm's board of director might approve, and (2) in case the company is terminated, one millionth of its "residual" or "liquidation" value, defined as the net cash proceeds from the sale of all of its assets minus the repayment of all of the debt owed by the firm. Furthermore, if the stock is so-called "common stock," you typically also will have (3) one of a million votes on certain matters that must be approved by shareholders, among them the approval of a board of directors proposed by management or an alternative board proposed by certain groups of shareholders seeking to install its preferred slate of directors.
Taken together, what is the so-called "intrinsic value" of these rights inherent in a stock certificate? Modern finance theory holds that this value is almost wholly determined by the future cash flow the share of stock promises its holder. According to the theory, that value is determined by a two-step process: (1) the projection of the future stream of annual cash dividends likely to be yielded by the stock and (2) the conversion of that projected future cash flow into a present-value equivalent by means of an appropriate discount rate1.
To see how that theory might help us think about Chairman Greenspan's query, let us develop a very simple model of stock valuation based on that theory and then apply it to the famous Standard and Poor's Composite Index of 500 Stocks (the S&P 500). That index is a composite of the stocks issued by 500 individual business firms. It can be viewed as something akin to a large mutual fund, although one so large that it is often taken as in index of "the market" as a whole. We shall explore whether, in our view, the current level of the S&P 500 is above the level that a market composed of rational investors would accord the stocks represented in that index.
A. A Very Simple Model of Stock Valuation
Imagine a corporation that has traditionally paid out to shareholders a dividend equal to an average of 50% of its annual earnings. We call this the dividend-payout ratio, hereafter referred to by the symbol d. That ratio need not be exactly 50% every single year, of course, because annual earnings do fluctuate about a smoother longer-run trend. But let us assume that over the longer run d = 0.50.
1. Assumed Future Growth in Earnings Per Share (EPS): Denote the firms's total annual earnings, divided by the number of shares of common stock outstanding, as EPS, the earnings per share. Assume that, over the longer run, as far as the eye can see, EPS is expected to growth at an annual compound growth rate equal to g. If we let subscript o denote the present (time t=0), then we assume that
[1]EPSt = EPSt-1 (1+g)
or EPSt= EPS0(1 + g)t for all t in the foreseeable future.
2. Assumed Future Growth in Dividends per Share: If we assume that a constant dividend-payout ratio of d will be maintained in the future, then the dividend per share now expected to be received at some future time t will be DIV t = d EPSt. Plugging in for EPSt from equation [1] we can then express DIVt as:
[2]DIVt = [d EPS0](1 + g)t for all t in the foreseeable future.
3. The Present Value of this Steadily Growing Dividend Stream: Now it can be shown and should be taken for granted here, that the present value of this perpetually growing stream of dividends, discounted at the compound rate ke, will be
[3] P0 = [d EPS0](1 + g)/(ke - g) ,
where ke is the so-called "cost of equity financing." P0 would be the price someone who projects the dividend stream defined by equation [2] would be willing to pay for that stream now if (s)he wished to earn an annual compound rate of return of ke on that investment.
4. The Warranted PE-Ratio: If we divide both sides of equation [3] by EPS0, that is, the current and known earnings per share, we obtain the famous Price-to-Earnings Ratio the investor would be willing to pay, that is, the price per dollar of current earnings per share. This ratio is widely known in the trade as the PE-Ratio. It is a staple among financial analysts. Here that ratio reduces to the very simple expression:
[4]P/EPS0 = d(1 + g)/(ke -g) .
5. Justifiable PE Ratios: Table 1 (below) shows, for this model, the corresponding justifiable PE-ratios for different combinations of projected growth rates in earnings per share (g) and discount rates (ke), at a dividend-payout ratio of d = 50%. The first insight to be gained from this table is just how sensitive the justifiable PE ratios are to our assumptions about (1) the future growth in corporate earnings and dividends--i.e., growth rate g; (2) to the proper discount rate ke at which future cash flows from stocks are to be converted to present values and (3) the future dividend-payout ratio. That, of course, is one reason why the "experts" on TV can spin whatever yarn they wish to spin about the current state of the stock market.
AT A DIVIDEND-PAYOUT RATIO OF 50% Annual Growth Rate (g) Across |
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| k | 2% | 3% | 4% | 5% | 6% | 7% | 8% | 9% | 10% |
| 8.0 | 8.5 | 10.3 | 13.0 | 17.5 | 26.5 | 53.5 | na | na | na |
| 8.5 | 7.8 | 9.4 | 11.6 | 15.0 | 21.2 | 35.7 | 108.0 | na | na |
| 9.0 | 7.3 | 8.6 | 10.4 | 13.1 | 17.7 | 26.8 | 54.0 | na | na |
| 9.5 | 6.8 | 7.9 | 9.5 | 11.7 | 15.1 | 21.4 | 36.0 | 109.0 | na |
| 10.0 | 6.4 | 7.4 | 8.7 | 10.5 | 13.3 | 17.8 | 27.0 | 54.5 | na |
| 10.5 | 6.0 | 6.9 | 8.0 | 9.5 | 11.8 | 15.3 | 21.6 | 36.3 | 110.0 |
| 11.0 | 5.7 | 6.4 | 7.4 | 8.8 | 10.6 | 13.4 | 18.0 | 27.3 | 55.0 |
| 11.5 | 5.4 | 6.1 | 6.9 | 8.1 | 9.6 | 11.9 | 15.4 | 0.0 | 36.7 |
| 12.0 | 5.1 | 5.7 | 6.5 | 7.5 | 8.8 | 10.7 | 13.5 | 18.2 | 27.5 |
| 12.5 | 4.9 | 5.4 | 6.1 | 7.0 | 8.2 | 9.7 | 12.0 | 15.6 | 22.0 |
| 13.0 | 4.6 | 5.2 | 5.8 | 6.6 | 7.6 | 8.9 | 10.8 | 13.6 | 18.3 |
| 13.5 | 4.4 | 4.9 | 5.5 | 6.2 | 7.1 | 8.2 | 9.8 | 12.1 | 15.7 |
| 14.0 | 4.3 | 4.7 | 5.2 | 5.8 | 6.6 | 7.6 | 9.0 | 10.9 | 13.8 |
| 14.5 | 4.1 | 4.5 | 5.0 | 5.5 | 6.2 | 7.1 | 8.3 | 9.9 | 12.2 |
| 15.0 | 3.9 | 4.3 | 4.7 | 5.3 | 5.9 | 6.7 | 7.7 | 9.1 | 11.0 |
| 15.5 | 3.8 | 4.1 | 4.5 | 5.0 | 5.6 | 6.3 | 7.2 | 8.4 | 10.0 |
B. Putting a Value on the S&P 500
Now, you may justly wonder how useful such a model could be in the real world. Would the earnings per share of any firm ever grow in perpetuity at a constant growth rate g? First, note that at any discount rate ke above, say, 10%, 50 years and eternity are pretty much the same as far as present values go, because $ 1 due 50 years hence and discounted at ke = 10% is worth only about $.0085 now--less than a penny. But even 50 years is a long horizon, of course. Second, however, one may wish to use this model not so much for single firms, but for a broad market index, such as the S&P 500. There exists for the S&P 500 the analogue of a share of stock that has a "price" and for which one can calculate the "earnings per share" and the "dividend payout ratio," and these numbers are widely available in the press and on the Internet.
In the past decade or so, the average annual growth rate of gross domestic product (GDP) in nominal terms (that is, not adjusted for inflation) has fluctuated between 3% and close to 8%. The average annual growth rate since 1990 has been closer to 5 percent. (After adjustment for inflation, the comparable real growth rate has been about 2.2%). If all goes well, perhaps we can assume that the long-run average growth rate of nominal GDP in the foreseeable future will be somewhere between 5% and 7%, although year-to-year fluctuations out the year-to-year growth rate outside this range. Finally, it may not be a bad approximation to assume that the earnings per share of as broad a stock market index as the S&P 500 will tend to grow over the long run at roughly the same rate of growth as does nominal GDP, because together the firms in the S&P 500 form a substantial part of the general economy. Furthermore, it might be reasonable to believe that investors would apply to such a broad index a discount rate ke that is, say, 200-400 basis points (2-4 percentage points) above the yield on 20- to 30-year US Treasury bonds. For example, if the market yield to maturity on 30 year US Treasury bonds is 7%, then proper ke for the S&P 500 might be around 9%-11%. Finally, we have already noted that the dividend-payout ratio of the S&P 500 has tended to hover about 50%. It is likely to do so in the future. With these assumptions, we can use Table 1 to explore whether one may now call the S&P 500 overvalued.
C. Is the S&P 500 Now "Overvalued"?
Let us assume that the future longrun growth rate g for the earnings and dividend per share of the S&P 500 index will fall somewhere between 5% to 7%. The yield on longterm U.S.Treasury bonds recently has been between 6.5% and 7%. Adding a reasonable risk premium for stocks on top of these bond yields suggests a proper discount rate for the S&P 500 (our ke) somewhere between 9% and 11%. Now go find where in Table 1 these numbers put you. You will conclude that, within the context of the United States economy now and in the foreseeable future, PE-ratios above 15 for the S&P 500 reflect a certain degree of optimism that may or may not be warranted.
You can find actual PE-ratios for the major U.S. stock-market indexes tucked away somewhere in Barron's, a weekly publication by the Dow Jones Company. Currently (December, 1996), the PE-ratio for the S&P 500 is between 20 and 21. You will conclude from Table 1 that PE-ratios for the S&P 500 above 20 require truly optimistic assumptions about our economy. Indeed, as is shown in Figure 4, historically that ratio has rarely stayed above 20 for very long. Thus, it seems fair to say that we seem to be moving now into the more heroic corner of the defensible forecasts for the main factors that drive the intrinsic value of the S&P 500 (the area framed by the double line). By the way, as you can see in Figure 4, during the pessimistic 1970s, the PE ratio of the S&P 500 spent a good bit of time in the single digits.
Some people argue that the past longterm pattern of the growth of dividends and earnings per share of the S&P 500 may not be quite relevant for the future, and that the future value of g in our model is likely to exceed the long run growth of the US GDP for decades to come, for at least two reasons.
First, there has been some redistribution of our GDP away from a payout to labor (in the form of salaries and wages) and towards the owners of capital (in the form of profits). That shift was accomplished by the massive restructuring of American business during the past decade or so. The restructuring has helped to lower labor costs as a percent of total revenue and has increased the so-called "profit margin" (profits as a percentage of revenues) earned by business firms. The question is, of course, how long into the future a mere restructuring of business can yield these added profits. There must be a lower limit to cost cutting, and it may well have been reached in many firms. Future growth in earnings will therefore will have to come from more vigorous growth in revenues, that is, from growth in sales volume. This means that more units of output must be sold or the price received per unit of output must increase, or both. It is not clear whether that type of growth can for long exceed the growth of our GDP.
Second, some people argue that the American GDP is no longer a good guide to the growth path of corporate profits, because so much of American corporate profit has been earned from investments by American business firms in other countries with stronger economic growth. There may be something to that argument. On the other hand, to the extent that profits earned abroad drive the bottom line of American business firms, that bottom line is vulnerable to changes in foreign-exchange rates. Mere fluctuations in exchange rates therefore may make the earnings per share of American stocks more volatile and, thereby, add another element of risk to investment in American stocks that may, in the longer run, drive up the discount rate ke. Other things being equal, an increase in that discount rate would lower the PE-ratios of the associated stocks.
D. A Popular Short-Cut Method by Which to Assess the S&P 500
Some commentators on the stock market prefer to use a short cut approach to assessing the value of the S&P. They simply invert the P/E-ratio of the S&P 500 index and then compare that ratio with the prevailing yield to maturity on 30-year US Treasury bonds. (See, for example, Roger Lowenstein, "Market revisited: Why S&P Looks Dear," The Wall Street Journal, December 26, 1996.)
For example, if the PE-ratio of the S&P 500 stands at 20, its inverse is .05 or 5%. That number is then viewed by these analysts as the "earnings yield" on the S&P 500. In principle, argue these analysts, that yield ought to be above the yield on long term US Treasury bonds, because investments in stocks (even as broad-based a portfolio as the S&P 500) are more risky than investments in US Treasury bonds. These analysts would then argue that, if the S&P 500's PE ratio is 20 (so that its its inverse is 5%) at a time when the yield on long-term US Treasury bonds is, say, 6 to 7%, then clearly the market seems to use too low a discount rate ke for putting a value on stocks. In other words, the market is judged to be overvalued by these analysts.
This line of reasoning is fundamentally flawed. To illsutrate, both according to economic theory and in practice the PE-ratio of high-growth stocks tend to be much higher than those for low-growth stocks. It follows that what these analysts would call these stocks' "earnings yield" (the inverse of their PE ratios) tends to be much higher for low-growth stocks than it is for high-growth stocks. Does that mean anything? Would you therefore prefer low-growth stocks to high-growth stocks? Surely not. But what applies in a comparison between two stocks also applies when one compares the inverse PE ratio of the S&P 500 (a growth stock) with the yield on a fixed income (no growth) debt instrument such as a US Treasury bonds.
You can easily convince yourself with the aid of equation (4) for the PE-ratio that the inverse PE-ratio would be equal to the true yield implicit in the price of the S&P 500--the required rate of return we have called ke above--only if the dividend payout ratio d of the S&P 500 were equal to 1 and the future growth rate in its earnings and dividends per share g were exactly zero. Those are simply unrealistic assumptions.
Frankly, I never cease to be amazed by the reasoning of people in the so-called "real world"! But keep the faith. Sooner or later the "real world" always catches up with economic theory. It happens when management consultants dress up standard economic theory in some fancy new concept and sell it to the "real world." Rest assured that, eventually, the inverse PE ratio, too, will find its way to the trash heap.