The Determination of Stock Prices

A simple, intuitive model of stock prices posits that the expected value of the current share price is the discounted present value of tomorrow's expected share price plus any expected dividend payout (cash flows), that is,

This equation is the basis of the dividend discount model and has a forward solution by successive substitution, that is, substituting in for for i = 1, . . ., ∞. For example, we first substitute the following for on the right-hand side (i = 1):

This yields the following:

In the limit, successive substitution generates the following forward solution:

We assume that the terminal value (as t approaches ∞) is equal to zero, that is:

If we assume that the expected dividend is constant, it is now easy to show from what we know about power functions, namely that:

Thus, our expression for price is equivalent to:

The constant dividend assumption is probably too restrictive but it does serve to make clear that the price of equity is the solution to a simple dividend discounting process. The constant dividend and infinite life of the firm together imply that price is like an annuity, that is, a constant dividend payment in perpetuity. Rearranging terms, we can see that acts like the capitalization rate (or cap rate); it is the yield on the asset:

To put this in perspective, take the March 2009 S&P 500 level, once again, of 757 but with the long run average p/e ratio equal to 16 and assume a dividend payout ratio of 0.4. The dividend is 0.4 times earnings. The p/d ratio is therefore 37.5, implying a cap rate equal to 2.67 percent, which was very close to the 2.82 percent yield on the 10-year Treasury at the time. Annual earnings were very depressed, however. If these earnings were to rebound to their pre-crisis levels, say, $80, then the cap rate of 2.67 percent would indicate an index level equal to , a level the S&P 500 did finally reach by November of 2010. The index achieved that level, however, through exceptional earnings growth over that one-and-a-half year period, in which annual real earnings rose from $7.2 to $77.1! This growth was due in large part to massive monetary and fiscal stimulus coupled with cost-cutting efficiency at the corporate level (including rising unemployment) and historically low capital costs. One important point to be made here is that despite the cap rate of 2.67 percent, this index managed to grow, not by 2.67 percent annually, but by 36 percent annually, all due to abnormal earnings growth as the economic recovery began in earnest in the summer of 2009.

The dividend discount model follows from the efficient markets hypothesis (EMH)—the notion that prices reflect firm fundamentals. The world of the EMH is dominated by rational agents, whose subjective models describing how markets work coincide with the objective, or true, market dynamic. These agents therefore have rational expectations derived from complete information sets, which effectively eliminate any chance that prices could systematically deviate from fundamentals. Consistent with the EMH, therefore, is the premise that prices are somehow right, suggesting that when they deviate from fundamentals, rational agents will exploit arbitrage opportunities that force prices back to fundamentals.

Shiller (1981) showed that actual stock prices are too volatile for the dividend discount model (he substitutes the average historical dividend and uses an average discount rate on equity). In reality, investors probably do not use an average of historical dividend; rather, they more likely try to estimate future dividends, in which case the model would predict more volatile prices consistent with Barsky and DeLong (1993).

In any case, this pricing model is a function of our notion of expected dividends and the interest rate, r. While this may appear to be a simple pricing model, understand that the determination of both future expected dividends and the discount rate are econometrically challenging pursuits. Moreover, while the theoretical price depends on both an econometric model of future expected dividends and a theoretical interest rate, the observed price is the market's notion of present value and, clearly, these two numbers can be quite different. Thus, as we see further on, though models may be intuitively appealing, they often produce forecasts that are wildly inconsistent with actual outcomes. We will return to the topic of modeling error later.