Assumption (7.1.(i) will he satisfied unless the stock price is expected to glow lot-ever al rale It or (aster. In Section 7.1.2 below, we discuss models of rational bubbles that relax (bis assumption.

Letting A. increase in (7.1.fi) and assuming (7.1.0), we obtain a formula expressing the stock price as the expected present value of future dividends out to lite infinite future, discounted at a constant rate, for future convenience we write tins expected present value as Рш:

(7.1.7)

An tinrcalisiie special case thai nevertheless provides some useful intuition occurs when dividends are expected to grow al a constant rale (, (which must be smaller than l{ to keep the stock price finite):

I-/I/m.I = (1 + ( ) l./ = [\+ С,У I),. (7.I.S)

Substituting (7.I ..8) into (7.1.7), we obtain the well-known Cordon growth model (Cordon l(.U>2) for the price of a slock with a constant discount rate II and dividend growth rate ( , where (1 < It:

>/, 11 О-гСШ,

Ihe Cordon growth model shows that the stock price is extremely sensitive to a permanent change in the discount rate R when It is close lo ( since the elasticity of the price with respect to the discount rate is (ill/d R)(Rf P) --/> /(/<- C).

It is important to avoid two common errors in interpreting these formulas. First, note that we have made no assumptions about equity repurchases by firms. F.qnity repurchases affect the time pattern of expected future dividends per share in (7.1.7), but they do not affect the validity of the formula itself. Problem 7.1 explores this point in more detail.

Second, the hypothesis that the expected stock return is constant tin (High lime is sometimes known as the martingale modeled.slock prices.- But a constant expected slock relui n does not imply a martingale for the slock price itself. Recall that a martingale for the price requires F.,[/,+ 1] = / whereas (7.I.-1) implies that

l.,/ il = (1 + R) li - F.,(0,+ ,. (7.1.10)

-.Sec Chapter lui- a cut-till ilist-ussiniHvl the itt.itlitt.tlt- hypothesis, l.tltov (1989) miivi-vs lln- martingale liicratun- li-nni S.iimielsim (! >[, >) on. Mori- general martingale results lor tisk-m-uiiali/i-il i>ii< г pi iii i-sM-t an- disi nsv-d in (hapicr 9.

= M Л + §±lV (7.1.11)

V +i /

The value of this portfolio at time t, discounted to time 0 at rate R, is

м, = -Ml-. (7.1.12)

(1+Л)

It is straightforward to show that M, is a martingale.

Even though the stock price Pt is not generally a martingale, it will follow a linear process with a unit root if the dividend D, follows a linear process with a unit root.4 In this case the expected present-value formula (7.1.7) relates two unit-root processes for P, and D,. It can be transformed to a relation between stationary variables, however, by subtracting a multiple of the dividend from both sides of the equation. We get

El its

(7.1.13)

Equation (7.1.13) relates the difference between the stock price and 1/7? times the dividend to the expectation of the discounted value of future changes in dividends, which is stationary if changes in dividends are stationary. In this case, even though the dividend process is nonstationary and the price process is nonstationary, there is a stationary linear combination of prices and dividends, so that prices and dividends are cointegrated,5

:,In iltt-special case where dividends are expected to grow at a constant rate C, this simplifies to E/,+i = (1 + (.)/,. The stock price is expected to grow at the same rate as the dividend, because the dividend-price ratio is constant in this case.

ijnosrly, a variable follows a stationary time-series process if shocks to the variable have temporary but not permanent effects. A variable follows a process with a unit root, also known as an integrated process, if shocks have permanent effects on the level of (he variable, but not onihe.change in (he variable. In this case the first difference of the variable is stationary, but the level is not. A martingale is a unit-root process where the immediate effect of a shock is the same as the permanent effect. See Chapter 2 or a textbook in time-series analysis such as Hamilton (1994) for precise definitions of these concepts.

Two variables with unit roots are coituegrated if some linear combination of the variables is stationary. See F.ugte and Granger (1987) or Hamilton (1994) for general discussion, or Campbell and Shiller (1987) for this application oftlie concept. Note that here the stationary linear combination of the variables involves the constant discount rale Я, which generally is not known a fnian.

The expected stock price next period does not equal the stock price today s would be required if the slock price were a martingale; raiher, the expected future slock price equals one plus the constant required return, (1 + R), times the current stock price, less an adjustment for dividend payments.3 To obtain a martingale, we must construct a portfolio for which all dividend payments are reinvested in the slock. At time t, this portfolio will have N, shares of the stock, where

MP* Ш

Although this formulation of the expected present-value model has been explored empirically by Campbell and Shiller (1987), West (1988b), and others, stock prices and dividends arc like many other macroeconomic time scries in that they appear to grow exponentially over lime rather than linearly. This means that a linear model, even one that allows for a unit root, is less appropriate than a loglincar model. Below we develop a present-value framework that is appropriate when dividends follow a loglincar process.

7.1.2 Rational Bubbles

In the previous section wc obtained an cxpeclational difference equation, (7r1.4), and solved it forward lo show thai the slock price musl equal Id the expected present value of future dividends. The argument relied on the assumption (7.1.5) that the expected discounted stock price, К periods inlthe future, converges to zero as the horizon A increases. In this section wcj discuss models that relax this assumption.

The convergence assumption (7.1.5) is essential for obtaining a unique so ution Id, to (7.1.4). Once wc drop ihe assumption, there is an infinite in mber of solutions to (7.1.4). Any solution can be written in the form

P, = Pm + ft.

where

. Г

-[тт.

(7.1.14)

(7.1.15)

The additional term B, in (7.1.14) appears in the price only because it is expected to be present next period, with an expected value (1 4- R) times its current value.

The term Pi>, is sometimes called fundamental value, and the term Bt is often called a rational bubble. The word bubble recalls some of the famous episodes in financial history in which asset prices rose far higher than could easily be explained by fundamentals, and in which investors appeared to be betting that other investors would drive prices even higher in the future.6 The adjective rational is used because the presence of II, in (7:1.14) is entirely consistent with rational expectations and constant expected returns.

Mackay (1852) is a classic reference on early episodes such as die Dutch tulipniania in the 17th Century and the bitulou South Sea Bubble and Paris Mississippi Bubble in the 18th Century. Kindlebergcr (1989) describes these and other more recent episodes, while Carber (1989) argues that Dutch tulip prices were more closely related to fundamentals than is commonly realized.

Il is easiest to illustrate the idea of a rational bubble with an example. Blanchard and Watson (1982) suggest a bubble of the form

I ( 4r) /{ + with probability я;

4+1 = (7.1.Hi)

[<s,+ i, with probability 1-я.

This obeys the restriction (7.1.15), provided that the shock satisfies Eis*i+i = 0. The Blanchard and Watson bubble has a constant probability, 1 - тг, of bursting in any period. If it does not burst, it grows al a rale -~ - 1, faster than R, in order to compensate for ihe probability of bursting. Many other bubble examples can be constructed; Problem 7.2 explores an example suggested by Frool and Obslfeld (1991), in which ihe bubble is a nonlinear function оГ the stocks dividend.

Although rational bubbles have attracted considerable attention, there are both theoretical and empirical arguments that can be used to rule out bubble solutions to the difference equation (7.1.4). Theoretical arguments may be divided into partial-equilibrium arguments and genet aUcquilibrium arguments.

In partial equilibrium, the first point to note is thai there can never be a negative bubble on an asset with limited liability. If a negative bubble were to exist, il would imply a negative expected asset price al some dale in ihe future, and this would be inconsistent with limited liability. A second important point follows from this: A bubble on a limited-liability asset cannot start within an asset pricing model. It must have existed since asset trading began if it exists today. The reason is that if the bubble ever has a zero value, its expected future value is also zero by condition (7.1.15). But since the bubble can never be negative, it can only have a zero expectation if it is zero in the future with probability one (I)iba and Grossman (1988)).

Third, a bubble cannot exist if there is any upper limit on the price of an asset. Thus a commodity-price bubble is ruled out by the existence of some high-priced substitute in infinitely elastic supply (for example, solar energy in the case of oil). Stock-price bubbles may be ruled out if firms impose an upper limit on slock prices by issuing slock in response to price increases. Finally, bubbles cannot exist on assets such as bonds which have a fixed value on a let initial date.

General-equilibrium considerations also limit the possibilities for rational bubbles. Tirole (1982) has shown that bubbles cannot exist in a model with a finite number of infinite-lived rational agents. The argument is easiest to see when short sales arc allowed, although il does not in fact depend on the possibility of short sales. If a positive bubble existed in an asset infinite-lived agents could sell the asset short, invest some of the proceeds to pay the dividend stream, and have positive wealth left over. This arbitrage opportunity rules out bubbles.

Til olc (1985) has studied die possibility of bubbles wilhin the Diamond (НК)Г>) ovcrlapping-gcnctations model. In this model there is an infinite number of liniie-lived agents, but Tirole shows that even here a bubble cannot arise when the interest rate exceeds the growth tale of ihe economy, because the bubble would eventually become infinitely large relative lo the wealth of the economy. This would violate some agents budget constraint. Thus bubbles can only exisl in dynamically fi/VfrVfi ovcrlappiug-gcncrations economies that have ovei ai cumulated private capital, driving the interest rate down below the growth rate of the economy. Many economists feel that dynamic inefficiency is unlikely to occur in practice, and Abel, Mankiw, Summers, and Xcckhauser (1989) present empirical evidence that il does not describe the US economy.

There are also some empirical arguments against the existence of bul>-bles. The most important point is dial bubbles imply explosive behavior of various series. In the absence of bubbles, if the dividend I), follows a linear process with a unit root then the slock price P, has л unit root while the change in the price Д/, and the spread between price and a multiple of dividends - D,/l{ are stationary. With bubbles, these variables all have an explosive conditional expectation: )iiuyc-,[x,( l/( 1 + Ж/1 Х,+ д] ф 0 for X, = / Л/ or /, - D,j II. Empirically, there is little evidence ol explosive behavior in these series. Л caveat is that stochastic bubbles are nonlinear, so standard linear methods may fail lo delect the explosive behavior of the conditional expectation in these models.

Finally, wc note lli.it rational bubbles cannot explain the observed predictability of slock returns. Bubbles create volatility in prices withou: creating predictability in returns. To ihe extent that price volatility can be explained by return predictability, the bubble hypothesis is superfluous.

Although rational bubbles may be implausible, there is much to be learned from studying them. An important theme of this chapter is that small movements in expected returns can have large effects on prices if they are persistent. Conversely, large persistent swings in prices can have small elfins on expected returns in any one period. A rational bubble can be seen as the extreme case where price movements are so persistent-indeed, explosive-that thev have no effects on expected returns at all.

7.1.1 Л n Approximate Present-Value Relation with Time-Varying Expected Returns

So far we have assumed thai expected slock returns are constant. Ibis assumption is analytically convenient, but il contradicts the evidence in (Ihaplei 2 and in Section 7.2 that slock returns arc predictable.

It is much more difficult to work with present-value relations when expected slock returns are liinc-varying, for then the relation between prices and returns becomes nonlinear. One approach is to use a logliuear ap-

proximation, as suggested by Campbell and Shiller (1988a,b). The logliri-ear relation between prices, dividends, and returns provides an accounting framework: High prices must eventually be followed by high future dividends, low future returns, or some combination of the two, and investors expectations must be consistent with this, so high prices must be associated with high expected future dividends, low expected future returns, or some combination of the two. Similarly, high returns must be associated with upward revisions in expected future dividends, downward revisions in expectec: future returns, or some combination of the two (Campbell [1991]).

Thus the loglinear framework enables us to calculate asset price behavior under any model of expected returns, rather than just the model with constant expected returns. The loglinear framework has the additional advantage that it is tractable under the empirically plausible assumption that dividends and returns follow loglinear driving processes. Later in this chap-S ter we use the loglinear framework to interpret the large empirical literature! on predictability in long-horizon stock returns. 1

The loglinear approximation starts with the definition of the log stock 1 return r,+. Using (7.1.1), (7.1.2), and the convention that logs of variables are denoted by lowercase letters, we have

r,+ , = log(P(+, +A+i)-log(/°,)

= />,+, -p, + log(l + exp(rf,+, - />,+i)). (7.1.17)

The last term on the right-hand side of (7.1.17) is a nonlinear function of the log dividend-price ratio, /(d,+i -p,+\). Like any nonlinear function /(x,+i), it can be approximated around the mean of x,+\, x, using a first-order Taylor expansion:

/(хн-i) /(x)+/(3e)(*+i-x). (7.1.18)

Substituting this approximation into (7.1.17), we obtain

r,+i k + ppl+x + (1 -p)rf,+,-p (7.1.19)

where p and к are parameters of linearization defined by p = 1/(1 -f-exp(c/ - p)), where (d - p) is the average log dividend-price ratio, and к = - log(p) - (1 - p) \og{\/p - 1). When the dividend-price ratio is constant, then p = 1/(1 + D/P), the reciprocal of one plus the dividend-price ratio. Empirically, in US data over the period 1926 to 1994 the average dividend-price ratio has been about 4% annually, implying that p should be about 0.96 in annual dala, or about 0.997 in monthly data. The Taylor approximation (7.1.18) replaces the log of the sum of the stock price and the dividend in (7.1.17) with a weighted average of the log stock price and the log dividend in (7.1.19). The log stock price gets a weight p close to one, while the log dividend gets a weight 1 - p close to zero because the dividend is on average