Now suppose that in addition to the five percent "normal" bond, the government introduced a second kind of perpetuity, on which annual payments increased every year at some fixed rate—say, two percent. Let's call this a growth bond. What interest rate should the growth bond pay today in order to pay exactly as much over its lifetime as the normal five percent bond would? Clearly, the initial rate on the growth bond should be lower than five percent—because growth would make payments higher in the future. But how much lower?

That seems like a difficult math problem, but actually the solution is simple—and it's treated in nearly every finance textbook. A normal bond and a growth bond are equivalent in present value if the sum of the growth bond's interest rate plus its growth rate is equal to the normal bond's interest rate. So if the normal bond is paying five percent and the growth bond's payments will rise by two percent a year, then the growth bond should start off paying three percent. It's that easy.

Of course, the timing of the payments is different. In the first year a $1,000 normal bond will pay $50 in interest, whereas the growth bond will pay $30. In the second year the growth bond will pay $30.60, and in the tenth year $35.85. And so on. If the sum of the growth bond's interest rate and growth rate is bigger than the interest rate of the normal bond, then the growth bond is paying its holders too much money. What does "paying too much" mean? Simply that the growth bond is underpriced.

Let's bring these mathematical calculations back to the world of stocks.

***** First, think of a stock as being the same thing as a growth bond.

***** Second, think of a stock's dividend yield as being the same thing as the growth bond's interest rate.

***** Third, think of the growth rate of the stock's annual dividend as the same thing as the growth rate of the growth bond's annual interest payment.

***** And remember that we want the stock to provide the same flow of cash as a normal long-term Treasury bond.

The stock will provide that cash flow if the sum of its dividend yield and the growth rate of its dividends (how much they increase, on average, per year) is equal to the interest rate on that normal Treasury bond. If the sum is greater than the Treasury rate, then the stock is paying too much.

Just like a growth bond, a stock that pays too much is underpriced. Again, think of bonds. If one bond that costs $1,000 is paying interest of $100 a year while all other bonds that cost $1,000 are paying interest of $50 a year, then, obviously, the bond that is paying $100 a year is too cheap at $1,000. Its price should rise to $2,000, making its return the same five percent.

Think of the stock market, as represented by the S&P 500. When the sum of the S&P's dividend yield plus the growth rate of its dividends exceeds the rate on a normal Treasury bond, then the market has not reached the perfectly reasonable price. The market is too cheap. It needs to rise some more.

Let's get down to the real numbers. We need look at only three things: the interest rate on long-term Treasury bonds, the dividend yield on stocks, and the expected long-term growth rate of stock dividends.

The first two numbers are easy to find in any newspaper. Earlier this year the rate on a thirty-year T-bond was roughly 5.5 percent, and the dividend yield for the typical stock in the Dow Jones Industrial Average was 1.5 percent—both rates low by traditional standards. The third rate—the growth of dividends per share—is not listed in newspapers, but its history is easy to discover, using statistics developed by the Yale economist Robert Shiller. (If you would like to see the data yourself, they are available on the Web at www.econ.yale.edu/~shiller/chapt26.html.)

The figures are compelling. From 1977 to 1997 the growth rate for dividends was 6.1 percent. From 1946 to 1997 the rate was 6.2 percent. So the past two decades have not really been much different from the rest of the postwar period. The consistency of these numbers is important. There are two possible explanations for the apparent undervaluation of stocks back in the late 1970s. Either dividends grew far more than could possibly have been expected or people were too cautious about the risks of stocks. Since dividend growth over the past twenty years is almost precisely the same as over the past fifty years, the growth should have come as no big surprise.

But back to our calculations.

To start, let's assume that dividends will grow in the future at the same fairly steady rate as they have over the past half century—by about 6.2 percent. You can see that stocks are paying too much when you add the yield of dividends (about 1.5 percent in 1998) to the growth rate (6.2 percent) and you get 7.7 percent, or 2.2 percentage points more than the T-bond rate (5.5 percent). Thus stocks put more money in your pocket than bonds, even though stocks are actually less risky than bonds over long periods.

Of course, applying the simple formula here is a problem, because the growth rate is bigger than the current interest rate. If firms grew that fast forever and the interest rate did not change, then the present value of future dividends would be infinite. One way to solve this problem is to rely on the simple model we used for Wells Fargo—breaking up a firm's life into adolescence and adulthood.

Suppose, for example, that all the companies in the market will grow at six percent a year for the next ten years and then grow at 0.5 percent below the GDP growth rate after that. In that case the present value of future dividends that you buy when you buy $100 worth of a portfolio representing the entire S&P 500 is $172. In other words, the market would have to rise immediately by 72 percent under these extremely modest assumptions to reach the PRP.

If the growth rate that has prevailed since 1946 continues for another fifty years before tailing off, then buying $100 worth of stock today will get you dividends worth $270 in present value. But dividends have grown faster than GDP for some time. Perhaps we should be more aggressive. If dividends just keep up with GDP (rather than falling behind by half a point) after ten more years of six percent growth, then their present value climbs to $329. If the six percent growth lasts twenty years, the present value climbs to $360; if fifty years, to $460.

After weighing the historical evidence, we support an estimate based on one of the last two numbers. The PRP for the market overall should be 260 to 360 percent higher than it is now—three and a half to four and a half times as high. Since the dividend yield for the Dow was around 1.5 percent earlier this year, the market should rise until the yield is about 0.4 percent.

It is easy to pull the same answer out of the growth-bond relationship. If the dividend yield is 0.4 percent and the Treasury bond yield is 5.5 percent, then the equation (cash returns from bonds = cash returns from stocks, or cash returns from bonds = dividend yield plus the growth rate of dividends) balances if that growth rate equals 5.1 percent: 5.5 = 0.4 + 5.1. This number is more than one percentage point below the average growth rate of dividends since 1946, so it seems to us perfectly reasonable.

But wait. These calculations have been based on numbers that don't allow for inflation. Inflation makes saving for tomorrow less attractive, because one dollar tomorrow can't buy as much as one dollar today. Although your dividends will be higher ten years from now, a new car or a trip to Europe will cost more, so shouldn't we take inflation into account?

Fine. Let's correct for inflation. Since the interest payment on a T-bond is the same every year, the bond's future payments are worth less and less as inflation erodes the value of the dollar. To account for this degradation, economists talk about the "real yield" of a bond, which is the nominal, or stated, interest rate minus the inflation rate.

So let's look at some numbers that correct for inflation. In its forecast at the beginning of this year the Congressional Budget Office predicted that inflation will rise at an average of about 2.6 percent a year through 2009. That means that the real yield on long-term Treasuries paying 5.5 percent is about 2.9 percent. For stocks the dividend yield is 1.5 percent. We don't need to adjust it for inflation, so long as we adjust the growth rate for inflation.

And how do we get the real growth rate for dividends? One source is the data developed by Robert Shiller. From 1946 to 1997 dividends per share grew at a real annual rate of 2.2 percent. From 1977 to 1997 the rate was 2.3 percent. From 1987 to 1997 it was 3.0 percent.

Add the middle real-growth rate to the dividend yield of 1.5 percent and you get a total of 3.8 percent, or 0.9 percent more than the real interest rate on bonds. Use the more recent figure for real growth and the difference is even larger.

If we assume that the real growth of dividends will continue at two to three percent, then we find, again, that stocks are paying too much. The only way this imbalance can be corrected is for stocks to rise in price.

But there is a serious problem with these numbers. The dividend payouts are far, far too low. Why?

Two reasons. First, when a firm pays out dividends to its shareholders, the shareholders are forced to pay tax immediately on the dividends. When the firm retains its earnings, the shareholders pay no tax. As we pointed out above, firms have been gradually learning that shareholders prefer not to pay taxes, and the fraction of earnings that is paid out in dividends has dropped dramatically—from above 70 percent in the 1930s to less than 40 percent today. This downward trend absolutely does not reflect a decline in firms' ability to pay dividends.

Second, the data we have used so far are based on the dividends of the S&P 500. One intriguing characteristic of the recent bull market is that many of the firms that have soared are computer and Internet companies that pay no dividends. This change in the composition of the S&P 500 means that dividend statistics are currently biased downward. (Our assumption is that a firm like Microsoft, with $20 billion in cash, will put money in shareholders' pockets in the future.) A better measure might correct for this big-firm low-dividend bias by looking at the market as a whole.

To do this, we constructed an aggregate measure of dividend yield and growth rate of dividends for all companies from the Federal Reserve's Flow of Funds tables. These numbers make the market look even better. The dividend yield for U.S. companies last year, according to calculations from the Fed data, was 2.0 percent, and the growth rate of dividends averaged 9.4 percent over the preceding twenty years. Adding those two numbers, we get 11.4 percent, as against the six percent we have been using in our conservative calculations.

How much will prices have to rise? Until they reach our PRP, or perfectly reasonable price. And what, precisely, is that?

Let's step back. If a stock's dividend payout in dollars stays the same but the stock rises in price, its yield will decline. Take AT&T. Suppose it pays a dividend of $1.50 a share while shares are trading at $100. Its yield is 1.5 percent. Now assume that AT&T triples in price but its dividend stays the same; its yield becomes 0.5 percent.

Let's suppose that the entire market is represented by that single share of AT&T. After all, at the start of this year the stocks in the Dow Jones Industrial Average were offering a dividend yield of about 1.5 percent. If the entire market triples in price and the market's dividend payout in dollars stays the same, the yield will drop to 0.5 percent.

Add that yield to our conservative real growth rate of dividends (2.3 percent) and you get 2.8 percent—approximately equal to the real T-bond interest rate. The equation balances.

At the start of this year, the Dow Jones Industrial Average was about 9,000. If the Dow, representing the entire market, tripled, then dividend yields would decline to their "perfectly reasonable" level—the level at which stocks put the same amount into your pocket as bonds. If we use the dividend yield from the Fed data as our starting point, the market needs to quadruple to reach the PRP.

Recognize that our assumptions are modest. First, we are looking just at dividends. Second, we are using a conservative estimate for real dividend growth. It could easily be three percent or higher.

Give this powerful idea some time to sink in: By our simple, logical calculation, stocks may be undervalued by as much as three quarters. They need to triple or quadruple to get to where they should be: the PRP.

But firms earn far more than they pay out in dividends, and those earnings, too, count in figuring out how much money ends up in an investor's pocket. How much do they count? A lot.

IF you own a restaurant or a dry-cleaning shop, you aren't picky about what the profits are called. What's important is that your cash flow in is greater than your cash flow out, and that you have actual money to put into your bank account. This positive flow of cash is the reason for investments.

But so far we have been using just one kind of cash flow—dividends—as the measure for all cash flow. This is a very conservative approach, because in recent years the total profits (official after-tax earnings, as reported to the Securities and Exchange Commission) of the average company have been nearly three times the dividends it pays to shareholders.

Earnings are much higher than dividends because many firms keep part (or all) of that money to make investments, which they expect will deliver more profits in the future. In addition, businesses recognize that there's a tax advantage to retaining earnings, because the tax rates that shareholders pay on dividends are higher than the rates they pay on capital gains, or increases in the value of their stock—in the case of rich shareholders, roughly twice as high.

Certainly earnings are real money, but it would be an egregious error to apply the analysis we just used for dividends to all earnings for all companies. In the case of most companies, however, some of the earnings beyond dividends will flow to shareholders. Now we will calculate how much—and, from that, make a less cautious but more reasonable estimate of how stocks should be priced than we did in our dividend-based estimate. We will also explain why a P/E ratio as high as 100 is justified for the market as a whole, and P/Es even higher are justified for some stocks.

But first listen to Warren Buffett, America's most successful investor, explaining to shareholders at the 1998 annual meeting of Berkshire Hathaway why not all earnings are created equal. He defined what he called a "wonderful business": "The business is wonderful if it gives you more and more money every year without [your] putting up anything—or [with your putting up] very little. And we have some businesses like that.... The worst business of all is the one that grows a lot, where you're forced to grow just to stay in the game at all, and where you're reinvesting the capital at a very low rate of return. And sometimes people are in those businesses without knowing it."

In other words, the fact that a company's earnings are rising every year doesn't mean that its long-run prospects are improving. Those rising earnings may have to go straight back into the business to pay for vital capital investments just to stay on an even keel.

Consider, for example, a limousine company that has to use all its profits each year to buy new cars to keep up with the competition. If it does not buy new cars and its rivals do, the company will lose customers and eventually go bankrupt. So even if its profits are rising at 10 percent a year, they never go into the pockets of the owners. They are sunk into new cars. In the end the limo company is an asset with no value, because it has no cash flow.

Now consider a company that owns a seaside hotel that requires almost no upkeep. As people get richer, demand for rooms increases, so the hotel raises its rates. But very little money has to go back into the hotel. If the hotel distributes the money that's left over to its shareholders, then we can do our usual calculations—dividend rate plus growth rate of dividends should equal or exceed the T-bond rate—and everything is fine. But if the hotel decides to retain all those earnings and stash the cash in, say, a savings account paying five percent interest, we have a problem. Earnings will be inflated in subsequent years by the interest from the bank—interest that the shareholders could have earned if the hotel had paid the dividends to them.

These are the two reasons that cash flow to shareholders is rarely equal to reported earnings: 1) huge reinvestments may be necessary just to maintain a firm's competitive position in the future, and 2) there's the potential for double-counting.

The reported earnings of most companies, very clearly, are greater than the cash that flows into your pocket over the time you own a stock. But just as clearly, quarterly dividends are *less *than that cash flow. After all, many of today's greatest success stories—Microsoft, Amgen, Dell Computer, to name a few—do not pay conventional dividends at all. Yet their stock prices have risen sharply—a sign that investors believe that they will see cash in the future.

Is there any way we can value a company when all we have is reported earnings?

Actually, there are several. One way is to use the two stages we introduced earlier—adolescence and adulthood—but with a subtle change. Assume that during adolescence, as a company pours its earnings back into the firm rather than handing them out to shareholders as dividends, the company will increase its earnings at a high rate. At maturity the firm will start paying dividends, and will increase them at a low rate. Back before tax considerations became an important determinant of payouts, mature firms paid out about 70 percent of earnings as dividends, so we will use that figure. Let's see how much cash can be expected to go into your pocket if you put your money in a popular firm, Cisco Systems, that had very high P/Es last spring but that doesn't pay dividends.

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