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

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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 National Bureauof 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).
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