Do Stock Prices Move Too Much to Be Justified by Subsequent Changes

Do Stock Prices Move Too Much to be Justifiedby SubsequentChanges in Dividends?

By ROBERTJ. SHILLER*

A simple model that is commonlyused to and RichardPorter on the stock market,and interpret movements in corporate common by myself on the bond market. stock. price indexes asserts that real stock To illustrategraphically why it seems that prices equal the present value of rationally stock pricesare too volatile,I have plotted in expected or optimally forecastedfuture real Figure 1 a stock price index p, with its ex dividendsdiscounted by a constant real dis- post rational counterpart p* (data set 1).' count rate. This valuation model (or varia- The stock price index pt is the real Standard tions on it in which the real discount rate is and Poor's Composite Stock Price Index (de- not constant but fairly stable) is often used trended by dividing by a factor proportional by economistsand marketanalysts alike as a to the long-run exponential growth path) and plausible model to describe the behaviorof p* is the present discounted value of the aggregatemarket indexes and is viewed as actual subsequentreal dividends (also as a providing a reasonable story to tell when proportionof the same long-rungrowth fac- people ask what accounts for a sudden tor).2 The analogous series for a modified movement in stock price indexes. Such Dow Jones IndustrialAverage appear in Fig- movementsare then attributedto "new in- ure 2 (data set 2). One is struck by the formation" about future dividends. I will smoothness and stability of the ex post ra- refer to this model as the "efficientmarkets tional price series p* when compared with model"although it shouldbe recognizedthat the actualprice series.This behaviorof p* is this name has also been applied to other due to the fact that the presentvalue relation models. relatesp* to a long-weightedmoving average It has often been claimed in popular dis- of dividends(with weights correspondingto cussions that stock price indexes seem too discount factors) and moving averagestend "volatile," that is, that the movements in to smooth the series averaged. Moreover, stock price indexes could not realisticallybe while real dividendsdid vary over this sam- attributedto any objectivenew information, ple period, they did not vary long enough or since movementsin the priceindexes seem to far enough to cause majormovements in p*. be "too big" relative to actual subsequent For example, while one normally thinks of events. Recently, the notion that financial the Great Depression as a time when busi- asset prices are too volatile to accord with ness was bad, real dividends were substan- efficient markets has received some econo- tially below their long- run exponential metric supportin papers by Stephen LeRoy growth path (i.e., 10-25 percent below the

*Associateprofessor, University of Pennsylvania,and 'The stock price index may look unfamiliarbecause researchassociate, National Bureauof Economic Re- it is deflatedby a priceindex, expressed as a proportion search.I am gratefulto ChristineAmsler for research of the long-rungrowth path and only Januaryfigures assistance,and to her as well as BenjaminFriedman, are shown.One mightnote, for example,that the stock Irwin Friend, Sanford Grossman, Stephen LeRoy, marketdecline of 1929-32 looks smallerthan the recent StephenRoss, and JeremySiegel for helpfulcomments. decline. In real terms,it was. The Januaryfigures also This researchwas supportedby the NationalBureau of miss both the 1929peak and 1932 trough. EconomicResearch as part of the ResearchProject on 2The price and dividendseries as a proportionof the the ChangingRoles of Debt and Equity in Financing long-rungrowth path are definedbelow at the beginning U.S. Capital Formation sponsoredby the American of Section I. Assumptionsabout public knowledgeor Councilof Life Insuranceand by the National Science lack of knowledgeof the long-run growth path are Foundation under grant SOC-7907561. The views important,as shall be discussedbelow. The seriesp* is expressedhere are solelymy own and do not necessarily computed subject to an assumptionabout dividends representthe views of the supportingagencies. after 1978.See text and Figure3 below.

421 422 THE AMERICAN ECONOMIC REVIEW JUNE 1981

Index 300- Index 2000-

225! 1500 p

150- * 1000-

75- 500-

0 I year yeor 0 I I I I 1 1870 1890 1910 1930 1950 1970 1928 1938 1948 1958 1968 1978

FIGURE 1 FIGURE 2 Note: Real Standardand Poor'sComposite Stock Price Note:Real modified Dow JonesIndustrial Average (solid Index (solid line p) and ex post rationalprice (dotted line p) and ex post rational price (dotted line p*), line p*), 1871- 1979,both detrendedby dividinga long- 1928-1979,both detrendedby dividing by a long-run run exponentialgrowth factor. The variablep* is the exponentialgrowth factor. The variablep* is the present present value of actual subsequentreal detrendeddi- value of actual subsequentreal detrendeddividends, vidends, subject to an assumptionabout the present subject to an assumptionabout the present value in value in 1979 of dividends thereafter.Data are from 1979 of dividendsthereafter. Data are from Data Set 2, Data Set 1, Appendix. Appendix. growth path for the Standard and Poor's that p, =E,( p*), i.e., p, is the mathematical series, 16-38 percentbelow the growthpath expectation conditional on all information for the Dow Series)only for a few depression availableat time t of p*. In other words,p, is years: 1933, 1934, 1935, and 1938. The mov- the optimal forecast of p*. One can define ing averagewhich determinesp* will smooth the forecast error as u,= p* -pt. A funda- out such short-runfluctuations. Clearly the mental principleof optimal forecastsis that stock marketdecline beginningin 1929 and the forecast error u, must be uncorrelated ending in 1932 could not be rationalizedin with the forecast;that is, the covariancebe- terms of subsequentdividends! Nor could it tween p, and u, must be zero. If a forecast be rationalizedin terms of subsequentearn- error showed a consistent correlationwith ings, since earningsare relevantin this model the forecast itself, then that would in itself only as indicators of later dividends. Of imply that the forecast could be improved. course, the efficient marketsmodel does not Mathematically,it can be shown from the say p=p*. Might one still suppose that this theory of conditional expectations that u, kind of stock market crash was a rational must be uncorrelatedwith p,. mistake,a forecasterror that rationalpeople If one uses the principlefrom elementary mightmake? This paperwill explorehere the statisticsthat the varianceof the sum of two notion that the very volatility of p (i.e., the uncorrelatedvariables is the sum of their tendency of big movements in p to occur variances, one then has var(p*) var(u)+ again and again) implies that the answer is var(p). Since variancescannot be negative, no. this means var(p) ) ?var(p*) or, converting To give an idea of the kind of volatility to more easily interpretedstandard devia- comparisonsthat will be made here, let us tions, considerat this point the simplestinequality which puts limits on one measureof volatil- (1) (p or(P*) ity: the standarddeviation of p. The efficient marketsmodel can be describedas asserting This inequality (employed before in the VOL. 71 NO. 3 SHILLER: STOCK PRICES 423 papers by LeRoy and Porter and myself) is how restrictions implied by efficient markets violated dramatically by the data in Figures on the cross-covariance function of short- 1 and 2 as is immediately obvious in looking term and long-term interest rates imply in- at the figures.3 equality restrictions on the spectra of the This paper will develop the efficient long-term interest rate series which char- markets model in Section I to clarify some acterize the smoothness that the long rate theoretical questions that may arise in con- should display. In this paper, analogous im- nection with the inequality (1) and some plications are derived for the volatility of similar inequalities will be derived that put stock prices, although here a simpler and limits on the standard deviation of the in- more intuitively appealing discussion of the novation in price and the standard deviation model in terms of its innovation representa- of the change in price. The model is restated tion is used. This paper also has benefited in innovation form which allows better un- from the earlier discussion by LeRoy and derstanding of the limits on stock price Porter which independently derived some re- volatility imposed by the model. In particu- strictions on security price volatility implied lar, this will enable us to see (Section II) that by the efficient markets model and con- the standard deviation of tvp is highest when cluded that common stock prices are too information about dividends is revealed volatile to accord with the model. They ap- smoothly and that if information is revealed plied a methodology in some ways similar to in big lumps occasionally the price series that used here to study a stock price index may have higher kurtosis (fatter tails) but and individual stocks in a sample period will have lower variance. The notion ex- starting after World War II. pressed by some that earnings rather than It is somewhat inaccurate to say that this dividend data should be used is discussed in paper attempts to contradict the extensive Section III, and a way of assessing the im- literature of efficient markets (as, for exam- portance of time variation in real discount ple, Paul Cootner's volume on the random rates is shown in Section IV. The inequalities character of stock prices, or Eugene Fama's are compared with the data in Section V. survey).5 Most of this literature really ex- This paper takes as its starting point the amines different properties of security prices. approach I used earlier (1979) which showed Very little of the efficient markets literature evidence suggesting that long-term bond bears directly on the characteristic feature of yields are too volatile to accord with simple the model considered here: that expected expectations models of the term structure of real returns for the aggregate stock market interest rates.4 In that paper, it was shown are constant through time (or approximately so). Much of the literature on efficient markets concerns the investigation of nomi- 3Some people will object to this derivation of (I) and say that one might as well have said that E,(p,) =p,* nal "profit opportunities" (variously defined) i.e., that forecasts are correct "on average," which would and whether transactions costs prohibit their lead to a reversal of the inequality (1). This objection exploitation. Of course, if real stock prices stems, however, from a misinterpretation of conditional are "too volatile" as it is defined here, then expectations. The subscript t on the expectations operator E means "taking as given (i.e., nonrandom) all there may well be a sort of real profit op- variables known at time t." Clearly, pt is known at time t portunity. Time variation in expected real and p* is not. In practical terms, if a forecaster gives as interest rates does not itself imply that any his forecast anything other than Et( p*), then high forecast is not optimal in the sense of expected squared forecast error. If he gives a forecast which equals E( p,*) 5It should not be inferred that the literature on only on average, then he is adding random noise to the efficient markets uniformly supports the notion of ef- optimal forecast. The amount of noise apparent in Fig- ficiency put forth there, for example, that no assets are ures I or 2 is extraordinary. Imagine what we would dominated or that no trading rule dominates a buy and think of our local weather forecaster if, say, actual local hold strategy, (for recent papers see S. Basu; Franco temperatures followed the dotted line and his forecasts Modigliani and Richard Cohn; William Brainard, John followed the solid line! Shoven and Lawrence Weiss; and the papers in the 4This analysis was extended to yields on preferred symposium on market efficiency edited by Michael stocks by Christine Amsler. Jensen). 424 THEAMERICAN ECONOMIC REVIEW JUNE 1981 trading rule dominates a buy and hold real interest rate r so that -y= 17/(1+4r). In- strategy, but really large variations in ex- formation at time t includes Pt and Dt and pected returns might seem to suggest that their lagged values, and will generally in- such a trading rule exists. This paper does clude other variablesas well. not investigatethis, or whether transactions The one-period holding return Ht costs prohibit its exploitation.This paper is (APt +I+Dt)/Pt is the return from buying concerned,however, instead with a more in- the stock at time t and selling it at time t+ 1. teresting (from an economic standpoint) The first term in the numeratoris the capital question: what accounts for movements in gain, the second term is the dividend re- real stock prices and can they be explained ceived at the end of time t. They are divided by new information about subsequentreal by P, to provide a rate of return.The model dividends?If the model fails due to excessive (2) has the propertythat Et(Ht) r. volatility,then we will have seen a new char- The model (2) can be restatedin terms of acterizationof how the simple model fails. series as a proportionof the long-rungrowth The characterizationis not equivalent to factor: pt = PtI/kA dt =Dt/Xt? T where other characterizationsof its failure,such as the growthfactor is - T =(l + g)- T9,g is the that one-period holding returns are fore- rate of growth,and T is the base year. Divid- castable, or that stocks have not been good ing (2) by At-T and substitutingone finds6 inflationhedges recently. 00 The volatility comparisons that will be 2 made here have the advantagethat they are (3) Pt= (Xy)k Etdt+k insensitive to misalignment of price and k=O dividend series, as may happen with earlier 00 k data when collection procedures were not = 'Etdt+k ideal. The tests are also not affected by the k=O practice, in the constructionof stock price and dividend indexes, of dropping certain The growth rate g must be less than the stocks from the sample occasionallyand re- discount rate r if (2) is to give a finite price, placing them with other stocks, so long as and hence y- AXy<1, and definingr by y the volatility of the series is not misstated. 7/( +r), the discount rate appropriatefor These comparisonsare thus well suited to the pt and dt series is r> 0. This discount existing long-termdata in stock price aver- rate i is, it turns out, just the mean divi- ages. The robustnessthat the volatility com- dend divided by the mean price, i.e, r= parisonshave, coupled with their simplicity, E(d)/E( p).7 may account for their popularityin casual discourse. 6No assumptionsare introducedin going from (2) to (3), since (3) is just an algebraictransformation of (2). I I. TheSimple Efficient Markets Model shall,however, introduce the assumptionthat d, is jointly stationarywith information,which means that the (unconditional)covariance between d, and zt-k,where zt is Accordingto the simple efficient markets any informationvariable (which might be d, itself orp,), model, the real price P, of a share at the dependsonly on k, not t. It follows that we can write beginningof the time period t is given by expressionslike var(p) without a time subscript.In contrast,a realizationof the randomvariable the conditional expectation E,(d1+k) is a function of time since it 00 depends on informationat time t. Some stationarity (2) Pt = Yk+ EtDt+k O

Index TABLE 1- DEFINITIONSOF PRINCIPALSYMBOLS 300- y =real discount factor for series before detrending; y= 1/(1 +r) y= real discountfactor for detrendedseries; _=Ay 225- D, = real dividendaccruing to stock index (before detrending) d, = real detrendeddividend; d, D,/Xt+ - T = 150 A first differenceoperator x, _x,-x, St= innovationoperator; 8,x,X+ k-E,X,+k E,t IX,+k; E= unconditionalmathematical expectations operator. mean of x. 75- E(x) is the true (population) Et = mathematicalexpectations operator conditional on informationat time t; E,x, _E(x,II,) where I, is the vector of informationvariables known at year 0-l time t. 1870 1890 1910 1930 1950 1970 A= trendfactor for priceand dividendseries; A- I +g where g is the long-rungrowth rate of price and FIGURE 3 dividends. Note: Alternative measures of the ex post rational price P, = real stock priceindex (beforedetrending) p*, obtained by alternative assumptions about the pre- r pi = real detrendedstock price index;p P/AtT sent value in 1979 of dividends thereafter. The middle = rationalstock index (expression4) are p, ex post price curve is the p* series plotted in Figure 1. The series r= one-periodreal discountrate for series before de- computed recursively from terminal conditions using trending dividend series d of Data Set 1. r= realdiscount rate for detrendedseries; r= (1 -y )y r2 = two-periodreal discountrate for detrendedseries; r2=(I +r_)2-I We may also write the model as noted t= time (year) above in terms of the ex post rationalprice T=base year for detrendingand for wholesaleprice series p1*(analogous to the ex post rational index; PT=PT -nominal stock price index at interest rate series that JeremySiegel and I time T used to study the Fishereffect, or that I used to study the expectationstheory of the term structure).That is, p1*is the presentvalue of actual subsequentdividends: tion, the result would be to add or subtract an exponentialtrend from the p* shown in (4) Pt =Et( Pt*) Figure 1. This is shown graphicallyin Figure 00 3, in which p* is shown computed from + where P k= ldtk alternative terminal values. Since the only k=O thing we need know to compute p* about Since the summationextends to infinity, we dividends after 1978 is p* for 1979, it does never observep* without some error. How- not matter whetherdividends are "smooth" ever, with a long enough dividend series we or not after 1978. Thus, Figure 3 represents may observe an approximate p . If we choose our uncertaintyabout p*. an arbitraryvalue for the terminalvalue of There is yet another way to write the p* (in Figures 1 and 2, p* for 1979was set at model, which will be useful in the analysis the average detrended real price over the which follows. For this purpose, it is con- sample) then we may determinep1* recur- venient to adopt notation for the innovation sively by p* =Y(p*?I +dt) working back- in a variable. Let us define the innovation ward from the terminal date. As we move operator t -Et -Et-1 where Et is the con- back from the terminaldate, the importance ditional expectationsoperator. Then for any of the terminalvalue chosendeclines. In data variableXt the term StXt+k equals EtXt+k - set (1) as shown in Figure 1, y is .954 and Et IXt+k which is the change in the condi- Y'08 =.0063 so that at the beginning of the tional expectation of Xt+k that is made in sample the terminal value chosen has a response to new information arriving be- negligibleweight in the determinationof pt*. tween t- 1 and t. The time subscript t may If we had chosen a differentterminal condi- be dropped so that SXk denotes StXt+k and 426 THE AMERICAN ECONOMIC REVIEW JUNE 1981

8X denotes SX0 or ,X,. Since conditional forecasterror ut =P* -Pt is uncorrelatedwith expectations operators satisfy EjEk = Pt. However, the forecast error ut is not Emin(j k) it follows that E-m,aX,+k =Et-m serially uncorrelated.It is uncorrelatedwith (Et Xt+ k Et- IXt+ k) = Et-m Xt+ k - all information known at time t, but the Et-mXt+k 0, m > 0. This means that StXt+k lagged forecast error ut_1 is not known at must be uncorrelatedfor all k with all infor- time t since P'*I is not discoveredat time t. mation known at time t- 1 and must, since In fact, ut= lz3k= +kpt+k as can be seen lagged innovationsare informationat time t, by substitutingthe expressionsfor pt and pt' be uncorrelatedwith St,Xt+j t'Paul Samuelson emphasized sion test. We might thus regress the trans- should, in contrast to AtXpt-Pt-p, by ef- formed variable ut -Yu+ I on variables ficient markets,be unforecastable.In prac- known at time t. Since ut - yuti tice, with our data, 6tpt so measured will y , this amounts to testing whether approximatelyequal Apt. the innovation in price can be forecasted.I The model also impliesthat the innovation will performand discusssuch regressiontests in price is related to the innovationsin di- in SectionV below. videndsby To find a limit on the standarddeviation 00 of Sp for a given standarddeviation of dt, (5) stPt = Yyk Stdt+k first note that d, equals its unconditional k=O expectationplus the sum of its innovations: This expressionis identicalto (3) except that 00 St replaces Et. Unfortunately,while 6tpt is observable in this model, the terms (6) dt=E(d)+ 2 St-kdt Stdt+k k=0 are not directlyobservable, that is, we do not know when the public gets informationabout a particulardividend. Thus, in deriving in- If we regard E(d) as E-0(dt), then this equalitiesbelow, one is obliged to assumethe expression is just a tautology. It tells us, "worst possible" pattern of informationac- though,that d t=0,t 1,2,.... arejust different crual. linear combinationsof the same innovations Expressions(2)-(5) constitute four differ- in dividendsthat enter into the linearcombi- ent representationsof the same efficient nation in (5) which determine 8tpt t= marketsmodel. Expressions(4) and (5) are 0, 1, 2,.... We can thus ask how large particularlyuseful for derivingour inequali- var (8p) might be for given var(d). Since ties on measuresof volatility. We have al- innovations are serially uncorrelated, we ready used (4) to derive the limit (1) on the know from (6) that the varianceof the sum is standarddeviation of p given the standard deviation of p*, and we will use (5) to derive 81t follows that var(u)=var(8p)/(l y2) as LeRoy a limit on the standarddeviation of Sp given and Porternoted. They base theirvolatility tests on our the standarddeviation of d. inequality(1) (whichthey call theorem2) and an equality restriction a2(p) +a2(8p)/(l I-2)=a2(p*) (their One issue that relates to the derivationof theorem3). They found that,with postwarStandard and (1) can now be clarified.The inequality(1) Poor earnings data, both relations were violated by was derived using the assumption that the samplestatistics. VOL. 71 NO. 3 SHILLER: STOCK PRICES 427 the sum of the variances: where v is the Lagrangeanmultiplier. The first-order conditions for j0= , . .. .0 are 00 00 aj, (7) var(d)= 2 var(dk)= 2 k=O k=O 0 (10) a- =2 2 ok )7 2paj 0 Our assumptionof stationarityfor d, implies that var(8t_kd) -var(Sdk) u.2 is indepen- which in turn means that a. is proportional dent of t. to j. The second-orderconditions for a In expression(5) we have no information maximum are satisfied, and the maximum that the varianceof the sum is the sum of the can be viewed as a tangencyof an isoquant variancessince all the innovationsare time t for var(op), which is a hyperplane in innovations,which may be correlated.In fact, (Jo, 91, 'g2'... space, with the hypersphererep- for given a6,a, the maximumvariance resentedby the constraint.At the maximum of the sum in (5) occurswhen the elementsin (u2 = (1-y2)var(d )y2k and var(Sp) the sum are perfectly positively correlated. y2var(d)/(1-y2) and so, converting to This meansthen that so long as var(Sd)#O0, standard deviations for ease of interpreta- td,+k =akS(d(, where ak =Gk/GO- Substitut- tion, we have ing this into (6) implies

00 (11) u(Sp)

ple, we can put an upper bound to the value. This phenomenonis commonly attri- standard deviation of the change in price buted to a tendency for new informationto (rather than the innovation in price) for given come in big lumps infrequently.There seems standard deviation in dividend. The only dif- to be a commonpresumption that this infor- ference induced in the above procedure is mation lumping might cause stock price that /p, is a different linear combination of changes to have high or infinite variance, innovations in- dividends. Using the fact that which would seem to contradictthe conclu- Apt =8tpt + t-I -dt- , we find sion in the precedingsection that the variance of price is limited and is maximizedif 00 forecastshave a simple (12) = autoregressivestruc- Apt I ykaitdt+k ture. k=O High sample kurtosis does not indicate 00 00 00 infinite varianceif we do not assume,as did +r ',b_ k dt+k- I 2 at-idt-I Fama (1965) and others, that price changes 1=1I k=O j=1l are drawn from the stable Paretianclass of distributions.'0The model does not suggest As above, the maximization of the variance that price changeshave a distributionin this of Sp for given variance of d requires that the class. The model instead suggests that the time t innovations in d be perfectly cor- existence of moments for the price series is related (innovations at different times are implied by the existenceof momentsfor the necessarily uncorrelated) so that again the dividendsseries. dividend process must be forecasted as an As long as d is jointly stationary with ARIMA process. However, the parameters informationand has a finite variance,then p, of the ARIMA process for d which maximize p*, Sp, and Ap will be stationaryand have a the variance of lp will be different. One finite variance."If d is normallydistributed, finds, after maximizing the Lagrangean ex- however, it does not follow that the price pression (analogous to (9)) an inequality variableswill be normallydistributed. In fact, slightly different from (1 1), they may yet show high kurtosis. To see this possibility, suppose the dividends are serially independentand identi- (13) a(/\p)

discount rate and (2) is the solution to the The natural extension of (2) to the case of difference equation."5With our Standard and time varying real discount rates is Poor data, the growth rate of real price is only about 1.5 percent, while the discount rate is about 4.8%+1.5%=6.3%. At these (14) Pt =Et (Dt+ll lk+r,,) rates, the value of the firm a few decades hence is so heavily discounted relative to its size that it contributes very little to the value which has the property that E,((1 +H1)/ of the stock today; by far the most of the (1 + r)) -1. If we set 1 + r = (aU/aCt)/ value comes from the intervening dividends. (aU/aC+ l), i.e., to the marginal rate of sub- Hence (2) and the implied p* ought to be stitution between present and future con- useful characterizations of the value of the sumption where U is the additively separable firm. utility of consumption, then this property is The crucial thing to recognize in this con- the first-order condition for a maximum of text is that once we know the terminal price expected utility subject to a stock market and intervening dividends, we have specified budget constraint, and equation (14) is con- all that investors care about. It would not sistent with such expected utility maximiza- make sense to define an ex post rational price tion at all times. Note that while r, is a sort from a terminal condition on price, using the of ex post real interest rate not necessarily same formula with earnings in place of known until time t+ 1, only the conditional dividends. distribution at time t or earlier influences price in the formula (14). IV. Time-VaryingReal DiscountRates As before, we can rewrite the model in terms of detrended series: If we modify the model (2) to allow real discount rates to vary without restriction (15) Pt -Et(Pt*) through time, then the model becomes un- testable. We do not observe real discount 00 k rates directly. Regardless of the behavior of where p 2 d j + 1 P1 and D1, there will always be a discount kP O t+k j0 1+ rate series which makes (2) hold identically. We might ask, though, whether the move- 1 A-#+j _(1 +rt)/X ments in the real discount rate that would be required aren't larger than we might have This model then implies that u(Pt) ?f(p') expected. Or is it possible that small move- as before. Since the model is nonlinear, how- ments in the current one-period discount rate ever, it does not allow us to derive inequali- coupled with new information about such ties like (11) or (13). On the other hand, if movements in future discount rates could movements in real interest rates are not too account for high stock price volatility?16 large, then we can use the linearization of p* (i.e., Taylor expansion truncated after the 15To understand this point, it helps to consider a linear term) around d=E(d) and r-=E(r-); traditional continuous time growth model, so instead of i.e., (2) we have PO=1?D,e -r'dt. In such a model, a firm has a constant earnings stream I. If it pays out all earnings, then D= I and = Ie -rfdt= I/r. If it pays 00 00 PO fj' (16) Pt - -Y dt+k ~E(d)(F)r Y rt+k out only s of its earnings, then the firm grows at rate k=-O ) ~k=O (I -s)r, Dt s=e(' -s)rt which is less than I at t=O, but higher than I later on. Then Po=0fosIe(' s)rte- rdt- fO'sle -srtdt=sI/(rs). If s#O (so that we're not divid- where y=1/(1+E(rF)), and a hat over a ing by zero) PO= J/r. variable denotes the variable minus its mean. '6James Pesando has discussed the analogous question: how large must the variance in liquidity premia be The first term in the above expression is just in order to justify the volatility of long-term interest the expression for p* in (4) (demeaned). The rates? second term represents the effect on p* of VOL. 71 NO. 3 SHILLER: STOCK PRICES 431 movements in real discount rates. This sec- TABLE 2- SAMPLESTATISTICS FOR PRICE ond term is identical to the expression for p* AND DIVIDEND SERIES in (4) except that dt+k is replaced by rt+k and the expression is premultiplied by Data Set 1: Data Set 2: Standard Modified -E(d)/E(r)- and Dow It is possible to offer a simple intuitive Poor's Industrial interpretation for this linearization. First note Sample Period: 1871-1979 1928-1979 that the derivative of 1(1 + rt+k), with re- spect to r evaluated at E(r) is -y Thus, a 1) E(p) 145.5 982.6 one percentage point increase in -t+k causes E(d) 6.989 44.76 2) r .0480 0.456 17(1 to drop by y2 times 1 percent, or +r,+k) r2 .0984 .0932 slightly less than 1 percent. Note that all 3) b=lnX .0148 .0188 terms in (15) dated t+k or higher are pre- o(b) (.0011) (1.0035) multiplied by 17(1+'+k). Thus, if rt+k is 4) cor(p, p*) .3918 .1626 increased by one percentage point, all else a(d) 1.481 9.828 Elements of Inequalities: constant, then all of these terms will be Inequality (1) reduced by about y2 times 1 percent. We can 5) a(p) 50.12 355.9 approximate the sum of all these terms as 6) a(p*) 8.968 26.80 yk-lE(d)/E(r), where E(d )/E(F) is the Inequality (11) value at the beginning of time t + k of a 7) a(Ap+d1-ip 1) 25.57 242.1 min(a) 23.01 209.0 constant dividend stream E(d) discounted 8) 4.721 32.20 by E(F), and yk- 1 discounts it to the pres- a(d)/Irj Inequality (13) ent. So, we see that a one percentage point 9) a(Lp) 25.24 239.5 increase in -t+k, all else constant, decreases min(a) 22.71 206.4 p' by about yk+ 'E(d)/E(rF), which corre- 10) a(d)/1/r 4.777 32.56 sponds to the kth term in expression (16). There are two sources of inaccuracy with this Note: In this table, E denotes sample mean, a denotes linearization. First, the present value of all standard deviation and 6 denotes standard error. Min (a) future dividends starting with time t+k is is the lower bound on a computed as a one-sided x2 95 percent confidence interval. The symbols p, d, r, F2, b, not exactly yk- 'E(d )/E(rF). Second, increas- and p* are defined in the text. Data sets are described in ing ?k by one percentage point does not the Appendix. Inequality (1) in the text asserts that the cause 1/(1 +rt+k) to fall by exactly y2 times standard deviation in row 5 should be less than or equal 1 percent. To some extent, however, these to that in row 6, inequality (11) that a in row 7 should be less than or equal to that in row 8, and inequality errors in the effects on p* of i-, r-+ 't+2' (13) that a in row 9 should be less than that in row 10. should average out, and one can use (16) to get an idea of the effects of changes in discount rates. expected real rate. The commercial paper To give an impression as to the accuracy rate ranges, in this sample, from 0.53 to 9.87 of the linearization (16), I computed p* for percent. It stayed below 1 percent for over a data set 2 in two ways: first using (15) and decade (1935-46) and, at the end of the then using (16), with the same terminal con- sample, stayed generally well above 5 per- dition p1*979IIn placelr of the unobserved cent for over a decade. In spite of this erratic series, I used the actual four- six-month behavior, the correlation coefficient between prime commercial paper rate plus a constant p* computed from (15) and p* computed to give it the mean r of Table 2. The com- from (16) was .996, and a(p *) was 250.5 and mercial paper rate is a nominal interest rate, 268.0 by (15) and (16), respectively. Thus the and thus one would expect its fluctuations linearization (16) can be quite accurate. Note represent changes in inflationary expecta- also that while these large movements in i- tions as well as real interest rate movements. cause p* to move much more than was I chose it nonetheless, rather arbitrarily, as a observed in Figure 2, a( p*) is still less than series which shows much more fluctuation half of a( p). This suggests that the variabil- than one would normally expect to see in an ity i- that is needed to save the efficient 432 THE AMERICAN ECONOMIC REVIEW JUNE 1981 marketsmodel is muchlarger yet, as we shall Cowles who said that the index is see. intended to represent,ignoring the ele- To put a formal lower bound on a(r) ments of brokeragecharges and taxes, given the variabilityof Ap, note that (16) what would have happenedto an inves- makes fl* the present value of zt, z tor's funds if he had bought, at the where zt-d - PE(d)/E(Q). We thus know beginningof 1871, all stocks quoted on from (13) that 2E(F)var(A p)

high- to be attributedto new information is "academic,"in that it relies fundamentally about future real dividends if uncertainty on unobservablesand cannot be evaluated about future dividends is measuredby the statistically. samplestandard deviations of real dividends around their long-run exponential growth APPENDIX path. The lower bound of a 95 percent one- sided x2 confidenceinterval for the standard A. Data Set 1: Standard and Poor Series deviation of annual changes in real stock prices is over five times higher than the Annual 1871- 1979. The price series P, is upper bound allowed by our measureof the Standard and Poor's Monthly Composite observed variability of real dividends. The Stock Priceindex for Januarydivided by the failureof the efficient marketsmodel is thus Bureau of Labor Statistics wholesale price so dramaticthat it would seem impossibleto index (JanuaryWPI startingin 1900, annual attribute the failure to such things as data average WPI before 1900 scaled to 1.00 in errors, price index problems, or changes in the base year 1979). Standard and Poor's tax laws. Monthly Composite Stock Price index is a One way of saving the general notion of continuation of the Cowles Commission efficient markets would be to attribute the Common Stock index developed by Alfred movements in stock prices to changes in Cowlesand Associatesand currentlyis based expected real interest rates. Since expected on 500 stocks. real interest rates are not directly observed, The Dividend Series D, is total dividends such a theory can not be evaluatedstatisti- for the calendaryear accruingto the portfolio cally unless some otherindicator of real rates representedby the stocks in the index di- is found. I have shown, however, that the vided by the average wholesale price index movements in expected real interest rates for the year (annual average WPI scaled to that would justify the variability in stock 1.00 in the base year 1979). Startingin 1926 prices are very large- much larger than the these total dividends are the series "Div- movementsin nominalinterest rates over the idends per share... 12 months moving total sampleperiod. adjustedto index"from Standardand Poor's Another way of saving the generalnotion statistical service. For 1871 to 1925, total of efficient marketsis to say that our mea- dividends are Cowles series Da-1 multiplied sure of the uncertaintyregarding future di- by .1264 to correctfor change in base year. vidends- the sample standard deviation of the movementsof real dividendsaround their B. Data Set 2: Modified Dow Jones long- run exponential growth path-un- IndustrialAverage derstates the true uncertaintyabout future dividends.Perhaps the marketwas rightfully Annual 1928-1979. HereP, and D, referto fearfulof much largermovements than actu- real price and dividends of the portfolio of ally materialized.One is led to doubt this, if 30 stocks comprisingthe samplefor the Dow after a centuryof observationsnothing hap- Jones IndustrialAverage when it was created pened which could remotelyjustify the stock in 1928. Dow Jones averages before 1928 price movements. The movements in real exist, but the 30 industrialsseries was begun dividendsthe marketfeared must have been in that year. The published Dow Jones In- many times largerthan those observedin the dustrial Average, however, is not ideal in Great Depressionof the 1930's,as was noted that stocks are droppedand replacedand in above. Since the market did not know in that the weightinggiven an individualstock advancewith certaintythe growthpath and is affectedby splits. Of the original30 stocks, distributionof dividendsthat was ultimately only 17 were still includedin the Dow Jones observed,however, one cannot be sure that IndustrialAverage at the end of our sample. they were wrong to considerpossible major The publishedDow Jones IndustrialAverage events whichdid not occur.Such an explana- is the simple sum of the price per share of tion of the volatilityof stock prices,however, the 30 companiesdivided by a divisorwhich VOL. 71 NO. 3 SHILLER: STOCK PRICES 435 changes throughtime. Thus, if a stock splits G. E. P. Box and G. M. Jenkins, Time Series two for one, then Dow Jones continues to Analysis for Forecasting and Control, San include only one share but changes the di- Francisco: Holden-Day 1970. visor to prevent a sudden drop in the Dow W. C. Brainard,J. B. Shoven,and L. Weiss, "The Jones average. Financial Valuation of the Return to To produce the series used in this paper, Capital," Brookings Papers, Washington the Capital Changes Reporterwas used to 1980, 2, 453-502. trace changesin the companiesfrom 1928 to Paul H. Cootner, The Random Character of 1979. Of the original 30 companies of the Stock Market Prices, Cambridge: MIT Dow Jones IndustrialAverage, at the end of Press 1964. our sample(1979), 9 had the identicalnames, Alfred Cowles and Associates, Common Stock 12 had changedonly their names, and 9 had Indexes, 1871-1937, Cowles Commission been acquired,merged or consolidated.For for Research in Economics, Monograph these latter 9, the price and dividend series No. 3, Bloomington: Principia Press 1938. are continued as the price and div- E. F. Fama, "Efficient Capital Markets: A idend of the sharesexchanged by the acquir- Review of Theory and Empirical Work," ing corporation.In only one case was a cash J. Finance, May 1970, 25, 383-420. payment, along with shares of the acquiring , '"The Behavior of Stock Market corporation,exchanged for the sharesof the Prices," J. Bus., Univ. Chicago, Jan. 1965, acquiredcorporation. In this case, the price 38, 34-105. and dividend series were continued as the C. W. J. Granger,"Some Consequences of the price and dividend of the shares exchanged Valuation Model when Expectations are by the acquiringcorporation. In four cases, Taken to be Optimum Forecasts," J. Fi- preferredshares of the acquiringcorporation nance, Mar. 1975, 30, 135-45. were among shares exchanged. Common M. C. Jensen et al., "Symposium on Some shares of equal value were substituted for Anomalous Evidence Regarding Market these in our series. The numberof sharesof Efficiency," J. Financ. Econ., June/Sept. each firm includedin the total is determined 1978, 6, 93-330. by the splits, and effective splits effected by S. LeRoy and R. Porter, "The Present Value stock dividendsand merger.The price series Relation: Tests Based on Implied Vari- is the value of all these shares on the last ance Bounds," Econometrica, forthcoming. tradingday of the precedingyear, as shown M. H. Miller and F. 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Samuelson, "Proof that Properly Dis- amination of the ExpectationsTheory of counted Present Values of Assets Vibrate the Term Structure of Interest Rates," Randomly," in Hiroaki Nagatani and Kate unpublished manuscript, Michigan State Crowley, eds., Collected Scientific Papers Univ. 1980. of Paul A. Samuelson, Vol. IV, Cambridge: S. Basu, "The Investment Performanceof MIT Press 1977. CommonStocks in Relationto their Price- R. J. Shiller, "The Volatility of Long-Term Earnings Ratios: A Test of the Efficient Interest Rates and Expectations Models of Markets Hypothesis," J. Finance, June the Term Structure," J. Polit. Econ., Dec. 1977, 32, 663-82. 1979, 87, 1190-219. 436 THE AMERICAN ECONOMIC REVIEW JUNE 1981

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