The Intrinsic Value of a Share of Stock
Modern finance theory holds that the "intrinsic value" of a share of stock is determined by the future cash flow that a well-informed, "rational" investor-one who has realistic notions about the stock's future performance-expects to receive from it. Our model rational investor (perhaps a research analyst) is assumed to project the cash dividends the share is expected to yield as long as it is held and the price at which the share will eventually be resold. That future resale price will, of course, depend on what cash dividends potential future buyers of the stock are likely to project.

Discount Rates, Risk Premiums, and PE Ratios
In a world in which money can be invested at interest, a dollar of dividends receivable in some future year is not worth a dollar today. Consequently, the intrinsic value of a share of stock is not simply the sum of the future cash inflows expected from that stock. It is the sum of the so-called "present-value equivalents" of these projected cash flows.
To illustrate: if an investor could earn 5 percent per year on money invested in a bank account today, then $100 receivable one year hence would be worth only about $95.24 today. Why? Because at 5 percent interest, $95.24 deposited today would earn $4.76 in interest over the next 12 months, so that a year from now there would be exactly $100 in the account. In the jargon of economics, the interest rate used to calculate this present-value equivalent of $95.24 is called the discount rate. One can think of it as an interest rate applied in reverse.
Thanks to the compounding of interest-which means that, at the end of every year, interest earned during that year is added to the amount that earns interest in subsequent years-$100 receivable 10 years hence would be worth only $61.39 today. If one used a higher discount rate of, say, 7 percent in this conversion, then $100 receivable 10 years hence would have a present value of $50.83. In general, the higher the discount rate, the lower the present value of a given future cash flow. It follows that any event that makes investors apply a higher discount rate to the expected cash flows from stocks-for example, Chairman Greenspan's statement about "irrational exuberance"-will send the stock market plummeting.
A bank account is a neat model to explain the principle of discounting, but it does not furnish a proper benchmark for valuing stocks. A better candidate is the rate of return investors can earn on long-term U.S. Treasury bonds-currently close to 7 percent. Because investments in stocks are riskier than investments in government bonds, however, rational investors add a risk premium to that benchmark to arrive at the discount rate they use in the valuation of stocks. This risk premium tends to be low in the optimistic atmosphere of bull markets and high in the nervous atmosphere of bear markets.
Finally, it is customary in the stock market to express the intrinsic value of stock as a multiple of the current reported earnings per share. For example, if a company has 100 million shares outstanding at a price of $12 a share, and if it has after-tax profits of $75 million, then that firm's price-earnings ratio would be 16. In bull markets, PE ratios tend to be relatively high. They are low when bears reign.

The Intrinsic Value of a Broad Market Index
Let us now turn our attention to a broad market index such as the Standard and Poor's 500 Composite. The S&P 500 represents a selected number of stocks traded on the two major national exchanges and in the over-the-counter market. Figures 1 through 3 present data for the S&P 500 from 1965 to 1996.
Figures 1 and 2 show that dividends per share and earnings per share of the S&P 500 have grown at average annual rates of 6.3 and 6.4, respectively. Figure 3 tracks the S&P 500's PE ratio. It's important to note that historically the PE ratio of the S&P 500 has rarely stayed above 20 or below 10 for long. It currently hovers near 20.
Figure 4 illustrates how the PE ratio that rational investors ought to be willing to pay for the S&P 500 varies according to two assumptions: (1) the future growth in dividends per share and (2) the discount rate at which these future dividends are converted into present-value equivalents, to arrive at the S&P 500's intrinsic value. The graph assumes that, on average, a constant fraction of 50 percent of the S&P 500's earnings per share will be paid out in dividends. The S&P 500's actual dividend-payout ratio has fluctuated closely about that percentage for the last three decades.
If rational investors expected future dividends per share to grow at an annual compound rate of 7 percent indefinitely, and if they added a risk premium of 3 percentage points to the current 7-percent yield on long-term Treasury bonds (to arrive at a 10-percent discount rate for the conversion of future dividends into present values), then these investors ought to be willing to pay a PE ratio of 17.8 for the S&P 500. On the other hand, a PE ratio of 20 at a projected 7-percent growth in earnings and dividends implies a discount rate of only 9.7 percent-7 percent plus a risk premium of 2.7 percent.
I pose two questions:
First, given the historical trends of earnings and the average annual growth rate of the gross domestic product (5.5 percent since 1990), can one realistically project that the S&P 500's earnings and dividends per share will grow at 7 percent? As seen in Figure 2, since about 1992, the S&P 500's earnings per share did grow much more steeply than its long-run historic trend of 6.4 percent. It remains to be seen whether that departure from the historical trend is more or less permanent.
Second, is a risk premium between 2.5 and 3 percent above the yield on long-term Treasury bonds adequate compensation for the higher risk inherent in stocks? It may be, if one assumes that our currently low inflation rate of less than 3 percent continues indefinitely, and especially if one agrees with a recent study suggesting that the actual inflation rate is 1 to 1.5 percent lower than the officially reported rate. A discount rate of 9 to 10 percent, then, would translate into an expected real (inflation-adjusted) rate of return on stocks of roughly 8 percent. Perhaps investors will remain satisfied with that average real rate of return on stocks.

In conclusion, there is no clear answer to Chairman Greenspan's query. If there were, the market undoubtedly would adjust to it overnight. As Figure 4 illustrates, the intrinsic value of stocks is highly sensitive to prevailing assumptions about future company earnings and dividends and about the discount rate at which future dividends should be converted to present values. In my view, a PE ratio of 20 for the S&P 500 represents highly optimistic assumptions. At the same time, I can respect investors who consider these assumptions defensible, as long as they make them explicit.