The "cash yield" from these numbers is much higher than the dividend yield for the market as a whole, confirming the intuition that our dividends-only calculations were extremely cautious.

For the market as a whole, the nominal growth rate for cash flow has been about 10 percent annually since the Second World War, as against a 7.3 percent growth in earnings (using the Shiller data) over this period; for the past twenty years cash flow has grown at 12.5 percent, and earnings have grown at 6.7 percent.

More impressive, for all corporations the average cash-flow yield last year was 3.3 percent, as compared with the 1.5 percent dividend yield we found above.

The cash flow from owning stocks, then, may be more than double the lowly dividends of today, and it has grown faster than dividends as well. If we use this alternative measure to evaluate how high the market needs to go to reach the PRP, then our conclusions become even more optimistic. When we used dividends, it looked like the market would have to triple. By this measure, which gives firms credit for money earned on behalf of shareholders but not paid out in dividends, the market would have to grow by a factor of six, even if we assume that cash flow will grow at about the same rate as dividends have grown in the past. If we assume that the recent spurt in cash flow will continue, the PRP is even higher.

Our circumspect assumptions show that the PRP is much higher than the market valuation is today, but it cannot be calculated precisely. For that reason we like to think of the perfectly reasonable price as defining the upper boundary of a "comfort zone." As long as you are a long-term investor with a diversified portfolio, you should not be concerned about warnings of overvaluation or manias or bubbles—provided that P/Es are under 100 and the Dow is below 36,000.

One common warning is that stocks will fall as interest rates rise. But unless the rate increase is dramatic and long-lasting, we are not greatly concerned. The reason is that our theory depends on the level of real interest rates, which have been steady throughout modern U.S. history. For example, if the nominal interest rate on Treasury bonds shoots up to eight percent, the reason for the rise will almost certainly be that investors are worried about higher inflation and are demanding extra compensation for lending their money to the government. But higher inflation would also push up revenues, earnings, and dividends for corporations, so the increase would not change our bullish story. If, however, inflation stays in check but interest rates climb to, say, 10 percent, then real rates will have reached a point where stocks are no longer cheap compared with bonds. That is very much a long shot.

We can't know whether the PRP for the Dow is really 27,000 or 54,000. However, we are sure of two things: thinking about stocks in the way we have described forces investors to ask the right questions, and anyone who claims that the market is too high today is viewing matters from an outdated and flawed perspective.

The Decline of the Risk Premium

INVESTORS aren't familiar with our PRP calculations, and they've been warned constantly that they are overvaluing the market, yet they have bid stock prices up dramatically. Why?

Earlier we presented evidence that stocks are less risky than bonds in the long run. We decided to make the cautious assumption that stocks and bonds are equally risky; then the PRP of a stock is the one that produces a flow of cash over time that equals the flow of cash from a Treasury bond.

But we have also shown that in the past investors who have owned stocks have put much more cash in their pockets than those who have owned bonds.

Remember that to find out the returns you can expect from a stock, you simply add the dividend yield to the anticipated growth rate of dividends. That number has typically been much higher than the yield on a Treasury bond, whose interest payments don't rise over time.

For example, early this year the stock of AlliedSignal, Inc., a diversified manufacturer that makes aerospace and automotive products, was paying a dividend of 68 cents a year and trading at $45 a share, for a yield of about 1.5 percent. The growth rate of the company's dividends was projected at 11.5 percent. Add 1.5 percent and 11.5 percent and you get the expected cash return from the stock—13 percent. Meanwhile, a long-term Treasury bond was yielding 5.5 percent.

Think of the 13 percent cash return on AlliedSignal stock as exactly comparable to the payments from a risky bond paying a fixed rate of 13 percent. For a risky bond, that extra 7.5 points over the Treasury rate is called the risk premium, and so it is for the risky stock. It is the extra cash flow that investors demand to compensate them for the extra risk of owning stocks instead of Treasury bonds.

But wait. In truth there is no extra risk in stocks. On average, stocks are actually less risky than bonds over long horizons. But investors have historically not believed this. They have perceived that stocks were riskier, so they have demanded higher returns. How much higher?

No official risk-premium figure is reported every year, but by making some assumptions we can construct a series—a historical record. Once again using the raw data compiled by Robert Shiller, we can go all the way back to the 1870s. The risk premium is the number of percentage points you have to add to the Treasury-bond rate in order to make the total equal to the dividend yield for stocks plus the growth rate of dividends. It is easy to find the bond rate and the dividend yield, but we have to guess what people expected that the growth rate of dividends would be each year. For simplicity we made the assumption that the expected growth rate was a consistent six percent—about the average for the entire postwar period.

Take a year when the dividend yield was three percent and the bond rate was five percent. Yield (three percent) plus the growth rate of dividends (six percent) equals nine percent. Subtract the bond rate (five percent) from nine percent and you get an estimate of the risk premium: four percent.

Now look at the actual results. From 1872 to 1929 the risk premium averaged 6.5 percent a year. The 1929 crash scared investors so badly that they boosted the premium even higher. From 1933 to 1950 it hovered around 11 percent. As the Second World War ended and the economy picked up, fears began to diminish, and during the 1950s the average premium dropped to about 9.5 percent. In the 1960s it continued to fall. The rate then fluctuated wildly as Paul Volcker, the chairman of the Federal Reserve Board, launched a war against inflation by hiking interest rates and creating enormous uncertainties. But the premium fell all the way down to about 2.5 percent in the late 1990s.

What happened as investors wised up to reality? They gradually bid down the irrationally high risk premium.

What does this do to stock prices? Say you own a house in northern California in a neighborhood that everyone believes is an earthquake zone. You rent out the house for $500 a month, or $6,000 a year—the going rate in that part of the country. The market value of the house is $60,000—a relatively low figure, because of the threat of quakes and the risk of losing the entire investment. Therefore the annual return on your investment, if the investment is the value of the house, is 10 percent ($6,000 divided by $60,000).

Suddenly a new seismological study is released showing that in fact the neighborhood is not in an earthquake zone, so the risk of losing your house in a catastrophe plummets to practically zero. The value of the house soars to $100,000. The income from the tenant, however, remains the same: it was set by the market, because the tenant could always move to another neighborhood. Therefore your annual return is now six percent ($6,000 divided by $100,000).

This is a good illustration of what happens to the risk premium after a shift of perception. The risk-free rate of return on the house turned out to be six percent, but you were getting a return of 10 percent when it appeared that you were in an earthquake zone. So you were receiving a risk premium of four percentage points. As the risk was unmasked as illusion, the premium vanished. The value (or price) of the house rose, and the return fell.

This is exactly what has been happening in the stock market. On average the risk premium has historically been about seven percent, and its sharp decline is what has been propelling stocks higher in the bull market of the late twentieth century. Changes in other elements in the equation have been slight. Although many analysts talk about the growth in corporate profits, the truth is that dividend increases have been remarkably steady. Earnings growth has been fairly regular as well, and though nominal bond rates have gone up and down, the real rate has been notably consistent. No, it is the declining risk premium that explains the market's boom.

The historically high risk premium has posed a major intellectual challenge to financial scholars for a long time. Hundreds of academic papers have offered explanations, but none has received wide acceptance. Why has the risk premium now dropped? Scholars are again at a loss.

Without a theory, the experts have to fall back on the argument that the 458 percent return that was produced by the market from June of 1989 to June of this year reflected a temporary euphoria over stocks, and that the historical risk premium accurately reflects people's preferences, as irrational as they may appear. Investors are naturally scared to death of stocks, according to this view, and a huge correction is coming, in which the risk premium will revert to its historical norm.

If the risk premium did return to normal, the carnage would be devastating. Add a seven percent risk premium to a 5.5 percent Treasury yield and you get a 12.5 percent target for the sum of the dividend yield and the growth rate of dividends. Assume that growth is 6.5 percent—the yield would have to quadruple, from 1.5 percent to six percent. For that to happen, stock prices would have to fall by 75 percent. A $100 stock paying a $1.50 dividend would drop to $25 a share.

We aren't worried about a return to an absurdly high risk premium, however, because we have a different explanation for what has happened over the past few decades. We see the decline in the risk premium as reasonable and long-lasting, not as insane and transitory. Investors have gradually learned about stocks and how their prices move over time. The prices that held in the past reflected an irrationally high aversion to risk as we measure it today, but in the past our understanding of risk and its calculation was in its infancy.

As stock ownership expands, so does education by mutual funds, banks, brokerage firms, journalists, and scholars. Research is far better today than it was in the past, and it is easily disseminated on the Internet. Seventy years ago few investors understood that excessive trading undermines profits, that stock-price fluctuations tend to cancel themselves out over time, making stocks less risky than they might appear at first glance, and that it is extremely difficult to outperform the market averages. American investors have learned to buy and hold.

Over the past few decades investors have entered the stock market the way a cautious child enters cool water. First he puts in a toe and pulls back. Then he tentatively submerges his foot and leaves it there. Then he wades in up to his knees, to his waist. At last he dives in. Americans are now diving into the stock market, having found that the water's fine. Having survived a 508-point drop in the Dow in a single day in 1987, a 554-point drop in a single day in 1997, and an 1,800-point drop in just six weeks in 1998, they feel that the risks are not nearly so great as they feared. They are using their resources and energy to learn about stocks and the best strategies for owning them. It makes good sense that such efforts would push the price of stocks toward the PRP. As the information has arrived, investors have brought their beliefs up to date and become more willing to hold on to stocks.

To believe that the market is overvalued, you have to believe that the risk premium, once irrationally so large and getting rationally small, will become irrationally large again. It is our strong belief that the risk premium will continue to shrink, and for good reasons. The best reason is the one that has prevailed for the past 200 years: stocks are actually less risky, in the aggregate and over the long term, than bonds.

Every time an analyst says that P/E ratios are too high today in the light of historical experience, she is implicitly saying that the risk premium is too low. In other words, she expects investors to go back to the days of being irrationally risk-averse. Maybe they will—but we strongly doubt that the profitable lessons of the contemporary stock market, once learned, will quickly be unlearned.