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However, note that while the current stock price reflects information about all future dividends, all dividends are not equally important. Because of discounting, current dividends influence the current stock price more than dividends in the far future. The discounting works as follows. The current stock price depends on the discounted expected price and dividends next period, P(0) = [P(1) + D(1)]/(1 + r). We also know that next period's price equals the discounted value of expected prices and dividends in period 2. Therefore, in influencing today's stock price, current dividends are discounted only once, D(1)/(1 + r), but dividends expected in period 2 are discounted twice. That is because we have to discount P(1) when considering its influence on P(0) and because P(1) itself depends on discounted dividends in period 2. Therefore, the influence of period 2 dividends on the current stock price is twice discounted, D(2)/(1 + r)2. Because (1 + r)2 is greater than (1 + r), future dividends drive the current stock price less than current dividends do. In fact, the further ahead we look, the less influence dividends have on current prices.

We have therefore shown that the current share price depends on the discounted sum of future dividends. This is why we called this approach a means of valuing shares that depends on fundamental value. Under this model, how much you are prepared to pay for the stock depends on the underlying profitability of the company and the dividends it pays. It does not depend on whether the Dow Jones has fallen below 6000 on the third Tuesday of a month with an R in its name—the price you should pay depends only on the profitability of the underlying asset. However, we still need to tie up a loose end here. We have shown that the current share price depends on the discounted stream of dividends and the discounted expected stock price several periods in the future. In other words, the current share price, P(0), depends on all discounted dividends over the next, say, 40 years, D(1)/(1 + r) + D(2)/(1 + r)2 + ... all the way to D(40)/ (1 + r)40, but it also depends on the discounted value of the share price in 40 years' time: P(40)/(1 + r)40. We still have to solve the problem that if stock prices are forward-looking, then the current stock price must depend on both future dividends and how much investors think the stock will be worth in the future. Unless we can somehow get round this dependence on future prices, we will have a circularity that we cannot solve.

4 We have that P(1) = [P(2) + D(2)]/(1 + r) and that P(0) = [P(1) + D(1)]/(1 + r). Therefore, we can write P(0) = [[P(2) + d(2)]/(1 + r) + D(1)]/(1 + r) = P(2)/(1 + r)2 + D(2)/(1 + r)2 + D(1)/(1 + r). However, we also have that P(2) = [P(3) + D(3)]/(1 + r), so we can use this to write P(0) = P(3)/ (1+ r)3 + D(3)/(1 + r)3 + D(2)/(1 + r)2 + D(1)/(1 + r). If we continue in this manner, we would arrive at P(0) = P(40)/(1 + r)40 + D(40)/(1 + r)40 + ... + D(2)/(1 + r)2 + D(1)/(1 + r). We could, of course, carry on like this indefinitely and arrive at an expression p(0) = P(N)/(1 + r)N + D(N)/(1 + r)N + ... + D(2)/ (1 + r)2 + D(1)/(1 + r), where N is any large integer you want to specify.

However, we can probably ignore this discounted future price term—that is, we can expect that P(40)/(1 + r)40 is small. So long as the average annual growth in share prices over the next 40 years is less than r, then P(40) is less than (1 + r)40 times its value today; if that is true over any long horizon, P(n)/(1 + r)n eventually goes to zero.

But would the share price in the distant future be small relative to the discount factor? The answer is basically—yes. The reason is straightforward. The real rate of return on relatively safe assets (like inflation-proof government bonds) has been around 3-4% over the past 20-odd years (the only period when bonds guaranteeing a real rate of return have existed). This means that a typical share (which is riskier than these safe assets) is likely to have a required real rate of return in excess of 3-4%. In other words, the value of r in equation (1) is likely to be greater than .03, and probably significantly greater (note that we are talking in real terms). The sustainable, or long-run, rate of growth of real GDP in nearly all developed economies is under 3%. So now consider what would happen if the price of an IBM share were expected to consistently rise in real terms by more than r, when r itself is in excess of the growth of GDP. This would mean that the value of one share in IBM would, eventually, be larger than the whole GDP of nearly any advanced economy. This seems implausible, so it seems sensible to assume that share prices ultimately grow at a slower rate than r. If this is the case, then the discounted share price eventually tends to zero, and we can write the current share price as depending only on the discounted sum of future dividends.

This implies that share prices can, indeed should, move in response to anything that changes the expected value of dividends at any point in the future (such as new products or inventions, changes in regulatory rules, increases in corporate taxes) or anything that causes people to change their required rates of return over any period in the future (changes in interest rates, changes in how investors assess risk). Anything that causes the expectation of the dividend to be paid in the future to rise will cause stock prices to rise now. And anything that causes investors to require a higher rate of return causes share prices to fall now. Therefore, we can begin to see why stocks might be so volatile and dependent on rumor and information.

The idea that the value of a share should equal the discounted value of expected future dividends has a powerful logic behind it. But is it right? After all, some companies pay no dividends, show no inclination to do so, and yet have high share prices. Microsoft has been one of the star performers in the U.S. stock market over the past 20 years; yet until the end of that period, it did not pay one cent in dividends. But the dividend/discount account of stock prices is not, at least in principle, inconsistent with how Microsoft was valued for all those years when it paid no dividends. The dividend discount model says that the expectation of a dividend being paid at some point in the future generates value today. That point could be 20 years off; as long as the eventual payoff is big, so is the stock price today. In fact, the payoff need not actually come as a dividend. Share repurchases by the company, or purchases of stock by another company that launches a successful takeover bid, are other means of distributing cash to shareholders. (We can think of these as forms of dividend payment; in a takeover, the purchase of stock is really a final dividend payment to the holders of shares in the company that is taken over.)

So, in principle, the massive valuations of some dot.com (Internet) companies that were seen at the end of the 1990s and the notion that stock values reflect the present discounted value of future cash distributions by the company are not inconsistent. In practice, however, most stock valuations in the Internet frenzy at the end of the 1990s were hard to reconcile with rational valuation, and their stock values subsequently slumped. Consider what happened to the Internet stock At Home:

At Home went public at $17 per share, and its stock valuation went through the roof. At the time of its IPO [initial public offering of shares] in July 1997, At Home's revenue was only $750,000 but its market capitalization was $2.6 billion. By the summer of1998 its market capitalization was up to $5.6 billion, even though in the previous 12 months its revenue was only $12 million. And by the time At Home announced the acquisition of Excite in January 1999, At Home capitalization was all the way up to $12.4 billion, even though in its most recent quarter the company posted a loss of $7.6 million. "You'd think you'd do forward earnings estimates based on the year 2000," quips one venture capitalist. "I guess I was wrong—it's actually based on the year 3000."5

Startup companies certainly do have uneven profiles for dividends. This is less true for mature companies or for the market as a whole. Figure 17.5 showed that an index of real dividends paid on all industrial stocks since 1871 is fairly steady; certainly it appears less volatile than the index of stock prices themselves. If dividends—either on an individual stock or the aggregate of all companies—grow at a roughly constant rate, we can use simple algebra to come up with a useful way of expressing share prices.

First, recall what the dividend discount model predicts for the price today of a stock, denoted P(0)

P(0) = D(1)/(1 + r) + D(2)/(1 + r)2 + D(3)/(1 + r)3 + - + D(N)/(1 + r)N

where for any value of j, D(j) is the expected dividend to be paid in period j. We assume here that the required rate of return does not change from period to period. Suppose dividends grow at a constant rate g, so that

D(2) = (1 + g)D(1) D(3) = (1 + g)D(2) = (1 + g)2 D(1) D(4) = (1 + g)D(3) = (1 + g)3 D(1) and so forth We can then write the stock price

P(0) = [D(1)/(1 + r)]* {1 + (1 + g)/(1 + r) + [(1 + g)/(1 + r)]2 + [(1 + g)/(1 + r)]3-+ [(1 + g)/(1 + r)]N}

This messy looking formula is just an infinite geometric progression. Each term in the final set of braces is a constant multiple of (1 + g)/(1 + r) times the previous term, and there are an infinite number of such terms to be added. As long as g is less than r (and we argued above that this is plausible), such a sum has a finite limit of (1 + r)/(r — g), which gives a value for P(0) of

5 Perkins and Perkins, The Internet Bubble (New York: Harper Collins, 1999) p. 169.

If we rearrange this equation we get D(1)/P(0) = r - g

This equation says that the ratio of the next dividend paid to the current stock price, which is called the (prospective) dividend yield, is the difference between the required rate of return and the long-run growth of dividends. Because we can measure the dividend yield on a stock, or on an index of the stocks of many companies, accurately, we now have a way of judging what the gap between the required rate of return and the anticipated growth of dividends is. We just take the latest stock price and dividends, and by forming the dividend yield, we can measure the implied expected excess return over and above the anticipated growth of dividends. This simple and useful relation says that if the dividend yield is low, either the required rate of return on equities is itself low or the anticipated growth of dividends is high.

We can try applying this relation to real-life data. Figure 17.10 shows the dividend yield on the Shiller index of U.S. stock prices from 1871. In 1929, on the eve of the massive fall in U.S. stock prices, the dividend yield on U.S. stocks was around 4%. This was significantly below the average from the previous 50 years. Recall that our relation says that if the dividend yield is low, either the required rate of return on equities is itself low or the anticipated growth of dividends is high. In 1929, short-term nominal interest rates were about 4%, and inflation had been low. So real interest rates might have been just under 4%. If we add an equity risk premium of about 3% to a real interest rate of just under 4%, we generate a required real return on stocks of about 7%. Now if we suppose that people expected dividends to grow in real terms by about 3%, we can explain a dividend yield of around 4%. Because the U.S. economy had been growing steadily for much of the 1920s, an expectation of 3% real growth of dividends might not have seemed unreasonable. Certainly the great U.S. economist Irving Fisher believed on the eve of the stock market crash that U.S. equity prices were sustainable. In fact,

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