if the application of rational

Peter Fortune

Professor of Economics at Tufts Uni-versity and Visiting Scholar at theFederal Reserve Bank of Boston. Theauthor is grateful to Richard Kopckeand Joe Peek for constructive com-ments, and to David Zanger for hisable research assistance.

T his is the second in a series of three papers assessing theperformance of the U.S. stock market. The first paper (Fortune1989) dispelled the myth of increasing stock market volatility: it

found that the monthly total rate of return on the Standard & Poor’s 500Composite Index has not been more volatile in the 1980s than inprevious periods. Indeed, the peak of stock market volatility was in the1930s. Others (for example, Schwert 1989) have reached the sameconclusion using data going as far back as 1859. These observationssuggest that investors do not face greater uncertainty about the returnson stocks than they have in the past, at least over periods of a month orlonger. They also suggest that firms need not be concerned that the costof equity capital has risen for risk-related reasons.

The present "paper addresses the question of the efficiency of thestock market--do stock prices correctly reflect available informationabout future fundamentals, such as dividends and interest rates? Statedin another way, is the volatility of stock prices due to variation infundamentals, or do other sources of volatility play a significant role?

The third paper will complete the trilogy by investigating the natureand consequences of very short-term (daily or intra-day) volatility.While the market’s volatility over a period of a month or longer has notbeen increasing, rare--but prominent--daily spikes in stock price vari-ation, which remain largely unexplained, have become the subject ofpublic policy debate. Hence, the next paper will address episodes suchas the October 19, 1987 crash and the break of October 13, 1989.

The present paper is structured as follows. Section I discusses themeaning of the Efficient Market Hypothesis (EMH) and draws out someof its implications for stock price behavior. The second section reviewsthe major stock market anomalies that cast clouds over the hypothesis ofmarket efficiency, while the third section assesses "modern" evidenceagainst the efficiency hypothesis. Section IV proposes an explanation formarket inefficiency that is consistent with much of the evidence mar-

shared in sections II and III. The paper concludeswith a brief summary.

The purpose of this paper is not to draw out thepolicy implications of market inefficiency--that is thetask of the next paper. However, the prominence ofinefficiencies suggests a role for public policies thatmight be counterproductive in an efficient market. Inshort, this paper suggests that recent proposals forchanges in margin requirements, introduction oftrading halts, and other reforms might be productive.

L The Efficient Market Hypothesis (EMH)Practitioners are interested in the stock market

because it is their bread and butter. Academic econ-omists are interested for a very different reason: forthem, the stock market provides an excellent labora-tory for the evaluation of microeconomic theory. Com-mon stocks are highly standardized products traded inan active auction market with very easy exit anal entryof both producers (firms issuing equity) and consumers(investors purchasing shares); as a result, the prices ofcommon stocks should conform to the implications ofthe theory of competitive markets.

The Efficient Market Hypothesis is the focus ofthe laboratory experiments, for it is the logical resultof the application of microeconomic theory to thedetermination of stock prices. As Marsh and Mertonpoint out (1986, p. 484):

To reject the Efficient Market Hypothesis for the wholestock market.., implies broadly that production deci-sions based on stock prices will lead to inefficient capitalallocations. More generally, if the application of rationalexpectations theory to the virtually "ideal" conditionsprovided by the stock market fails, then what confidencecan economists have in its application to other areas ofeconomics where there is not a large central market withcontinuously quoted prices, where entry to its use is notfree, and where short sales are not feasible transactions?

There were several reasons for the popularityenjoyed by the EMH in the 1960s and 1970s. First, itwas rooted in a very strong theoretical foundation.This foundation began with Samuelson’s work on thebehavior of speculative prices, in which he showedthat the prices of speculative assets should follow arandom walk (1965). It was further buttressed byHarry Markowitz’s theory of portfolio selection (1959)and by William Sharpe’s construction of the CapitalAsset Pricing Model (CAPM), which described theimplications of optimal portfolio construction for as-

set prices in security market equilibrium (1964); bothMarkowitz and Sharpe won the 1990 Nobel Prize inEconomics for their contributions. The final contribu-tion was Robert Lucas’s Rational Expectations Hy-pothesis (1978), which examined the implications ofoptimal forecasting for individual behavior and mac-roeconomic performance.

By the late 1970s, those who disputed the EMHfound themselves facing an avalanche of sharplypointed and well-argued opposing positions. Thetheorists had apparently won, in spite of the paucityof supportive evidence, and the prevailing yiew wasthat no systematic ways exist to make unusual re-turns on one’s portfolio. Practitioners were reducedto the position of ridiculing the EMH but could notmake effective arguments against it.1

A second reason for the popularity of the EMHwas the hubris of financial market practitioners dur-ing the 1960s. The "go-go" years had rested upon thenotion that opportunities for unusual profits were

"If the application of rationalexpectations theory to thevirtually "ideal’ conditions

provided by the stock marketfails, then what confidencecan economists have in itsapplication to other areas

of economics?"

abundant, and that it required only a reasonableperson and a bit of care to sort out the wheat from thechaff in financial markets. The EMH provided anantidote to this hubris, for it argued that opportuni-ties to make unusual profits were both rare andephemeral: by their very nature, they were the resultof temporary market disequilibria that are quicklyeliminated by the actions of informed traders. Thus,the EMH counseled healthy skepticism in investmentdecisions. This skepticism about "beat the market"strategies has led to the popularity of index funds,which allow investors to hold "the market" withoutworrying about individual stocks.

18 March/April 1991 New England Economic Review

The EMIt, Definition 1: Prices Are OptimalForecasts

The fundamental insight of the EMH is that assetprices reflect optimal use of all available information.A more formal statement is that the price of anactively traded asset is an optimal forecast of theasset’s "fundamental value." To understand this no-tion, suppose that market agents think of each pos-sible sequence of future events as a "state-of-the-world," and that there are N possible states of theworld, to each of which a number s (s = 1,2,3 .....N) is assigned. For example, state-of-the-world 1might be "dividends grow at 2 percent per yearindefinitely, and a constant discount rate of 5 percentshould be used," while s = 2 might be "dividendsgrow at 3 percent for two years, during which thediscount rate is 7 percent, but thereafter dividendsgrow at 1 percent and the discount rate is 4 percent."Suppose also that the set of all the available informa-tion at time t is denoted by fit and that vr(slf!t) is theprobability that state s will occur, conditional on theinformation set available at time t.

Then for each state-of-the-world we can calculatea fundamental value of the asset, which we denote asP*(s). Hence, P*(4) is the fundamental value if statenumber 4 occurs. Note that there is no single funda-mental value, rather there are N possible fundamen-tal values, one for each state. If Pt is the market priceand the expected fundamental value is E(P~lf~t) =~s P*(s)~’(sl~t), the EMH is embodied in the state-ment that the current price of the asset is equal to theexpected fundamental value, or, more concisely, thatthe price of an asset is the best estimate of itsfundamental value; that is,

(1) Pt = E(P~lf!t).

Consider the following simple example. Thereare three states-of-the-world: in state 1, the funda-mental value is $100, in state 2 it is $75, and in state 3the fundamental value is $40. Investors do not knowwhich state will materialize, but they have formed anassessment of the probability of occurrence of eachstate. Suppose that these probabilities are 0.25, 0.35and 0.40, respectively. The market price under theEMH will be the expected fundamental value: Pt =

0.25($100) + 0.35($75) + 0.40($40) = $67.25.The EMH carries a number of strong implications

about the behavior of asset prices. First, recall that ~-~t

contains all the relevant information available at timet; this includes historical information (for example,

past values of the asset price, the history of dividends,capital structure, operating costs, and the like), as wellas current publicly available information on the firm’spolicies and prospects. Because Pt already incorporatesall of the relevant information, the unanticipated compo-nent of the market price should be uncorrelated withany information available at the time the price is ob-served. A simple test of this proposition is to do aregression of Pt on a measure of the optimal forecastE(P~I fit) and upon any information that might be in(say, the history of the stock price). This crude form oftechnical analysis2 should result in a coefficient of 1.0 onE(P~If~t) and coefficients of zero on past stock prices,leading to the conclusion that all information in paststock prices has been embedded in the fundamentalvalue.3

The fundamental insight of theEMH is that asset prices reflect

optimal use of all availableinformation.

A second implication bears on the sequence overtime of prices under the EMH. Suppose we are attime t and we wish to forecast the price at time t + 1.If we knew the information that would be available attime t + 1, our forecast would be E(P~+llf!t+~). Butwe do not know, at time t, the information availableat time t + 1; we only know f!t. If r is the requiredrate of return on an asset with the risk level and othercharacteristics of the asset under consideration, thebest forecast of Pt+~ when we only know the ele-ments in ~t is E(P~+~ If!t).4 Now, any new informationarriving between time t and time t + 1 is, by defini-tion, random so its effect on price creates a randomdeviation from today’s best forecast.

From the optimal forecast definition of the EMHin (1), we see that the EMH implies the followingsequence of prices:

(2) Pt + ~ = (1 + r)Pt + ~t + 1, where E(~t + 1) = 0

Equation (2) says that the sequence of prices will be arandom walk with drift; price will vary randomlyaround a rising trend. Because new information israndom, having no predictable components, ~t+l hasa zero mean and is without serial correlation.

This is the basis of the "random walk" tests of

March/April 1991 New England Economic Review 19

the EMH which estimate equations like (2) and searchfor serial correlations in the residuals. Appendix 1uses time series analysis to determine whether dailychanges in the closing value of the S&P500 indexduring the 1980s are consistent with a random walk.The answer appears to be "almost, but not quite."The results show a five-day trading cycle that can beused to predict stock price movements, but this cycleis a very small source of the total variation in theS&P500. Hence, it might not be strong enough togenerate economic profits after transactions cost; thisis, of course, consistent with the EMH.

The EMIt, Definition 2: Risk-Adjusted Returns AreEqualized

The EMH can be restated in a different manner,which focuses on the rate of return on individualassets rather than their prices. The Capital AssetPricing Model (CAPM), developed by Markowitz andSharpe, states that in an efficient market t~e risk-adjusted expected returns on all securities are equal;any differences across assets in expected rates ofreturn are due to "risk premia" arising from unavoid-able (or "systematic") uncertainty.

The CAPM distinguishes between two types ofrisk: systematic risk, which affects all securities, eachto a different degree, and unsystematic, risk, which isunique to individual securities. Unsystematic risk canbe avoided by an appropriate diversification of port-folios, based on the variances and covariances ofsecurity returns. Because unsystematic risk can beavoided without any sacrifice in the return expectedfrom the investor’s portfolio, it imposes no riskpremium.

However, systematic risk, which affects all secu-rities and is, therefore, unavoidable, will earn a riskpremium. The CAPM defines a simple measure of theamount of systematic risk a security contains: its"beta coefficient." A security’s beta coefficient mea-sures the marginal contribution of that security to themarket portfolio’s risk: if/3 = 0, adding the securityto the optimal portfolio does not affect portfolio risk;if /3 < 0, portfolio risk is reduced by adding thesecurity to the portfolio: and if /3 > 0, the securityadds to the portfolio’s risk. The returns on a securitywith a beta of 1.0 move with the market--when thereturn on the market portfolio changes by 1 percent,the return on a 13 -- 1.0 security moves in the samedirection by 1 percent. Securities with betas greaterthan 1.0 have above-average risks, while securitieswith betas below 1.0 have below-average risks.

According to the CAPM, the realized return on asecurity is described by the following "characteristicline":

(3)

where Ri is the realized return on the specific secur-ity, rf is the return on a risk-free asset (such as U.S.Treasury bills), Rm is the rate of return on the marketportfolio (such as the S&P500), and vi is a zero-meanrandom variable whose variance measures the unsys-tematic risk of that security. The slope of this charac-teristic line, /3i, is the security’s beta coefficient. Thebeta can be estimated by a bivariate regression of theexcess return on an asset, (Ri - rf), on the excessreturn on the market, (Rm - rf).

Equation (3) describes the relationship betweenthe realized return on an individual security and therealized return on the market. This provides the basisfor answering the question, "What is the normalreturn on a security?" From the characteristic line wecan see that because E(vi) = 0, the expected return onan asset will be a linear function of the asset’s risklevel, as measured by its beta coefficient. For everysecurity, the expected return will lie on the same

The Capital Asset Pricing Modelstates that in an efficient market

the risk-adjusted expected returnson all securities are equal; any

differences across assets inexpected rates of return are due to

"risk premia’" arising fromunavoidable uncertainty.

straight line, called the Security Market Line (SML),which relates expected return to risk (beta). The SMLis described by the following equation:

(3’) E(Ri) = r~ + (Em -

where E(Ri) is the expected return on the ith security,and Era is the expected return on the market portfolio.This relationship represents the optimal forecast of a


security’s rate of return (rather than its price); underthe EMH any deviations of the actual return, Ri, fromthis relationship must be random.

The term (Em - rf)~i is the "risk premium" forthe ith security; it is the product of the market rewardfor a unit of risk, defined as the expected excessreturn on the market portfolio, and of the security’srisk level, measured by its beta. The SML says thatthe optimal forecast of the return on a security is therisk-free interest rate plus a risk premium. Hence, ona risk-adjusted basis (when the risk premium isdeducted from the expected return), all securities areexpected to earn a return equal to the risk-free inter-est rate. Securities that have high betas, hence addingmore to the portfolio risk, will carry higher expectedreturns because risk-averse investors will requirehigher returns on average to compensate them for theadditional risks.

Each possible value of rf and Em will ha.v.e adiffei!ent SML describing security market equilibrium.Figure 1 shows the SML under the assumptions r~ =8 percent and Em = 12 percent. In this case theintercept will be 8 percent and the slope of the SMLwill be 4. The market portfolio would be at point"M," with a beta of 1.0 and an expected return of 12percent; to verify this, simply substitute ]3 = 1.0 intoequation 3’. Discovery of a security whose expectedreturn is above (or below) the SML is an indication ofmarket inefficiency, for that security is expected togive a risk-adjusted return above (or below) therequired level; that is, it is underpriced (or over-priced).

Thus, the SML can be used to describe theexpected returns that are consistent with an efficientmarket. For example, point "HDL" on Figure 1represents Handelman Corporation, a distributor ofhome entertainment media (records, video tapes,etc.). As of October 1990, Handelman’s beta coeffi-cient was 1.4, so the hypothetical SML predicts areturn on HDL of 13.6 percent. Handelman’s higherreturn is attributable solely to its above-average risk.

Some Caveats

Both forms of the EMH rest on a strong assump-tion: the market equilibrium of asset prices is inde-pendent of the distribution across investors of thetwo basic raw materials of investment: informationand wealth. In short, all those things that makedifferent investors evaluate assets differently aretreated as of negligible importance. Among these"’irrelevant" factors are differences in probability as-

Figure 1

Efficien t Markets:The Security Market Line

E(Ri)I

SML

13.612

( - ) o 1.o 1.4 (+)

The equation for the SML is E = rf + (Em - rf) I}i.

The SML drawn above assumes rf= 8% and Era= 12%.

sessments (more optimistic investors will invest alarger share in favored assets), differences in transac-tions costs (investors with low costs, such as financialinstitutions, will devote more resources to stocksthan will high-cost investors, such as individuals),and differences in tax rates paid by investors.

If these factors can be ignored, the prices orreturns on assets will be determined solely by funda-mentals. But if they are important, prices can deviate,perhaps persistently, from fundamental values. In-deed most explanations of inefficiency in securitymarkets rest on some form of heterogeneity amonginvestors.

II. Stock Market Anomalies and the EMIl

The theoretical victories of the EMH were notsupported by empirical evidence. True, some studiesdid support the EMH; for example, numerous studiesshowed that stock prices were random walks in thesense that past stock prices provided no useful infor-mation in predicting future stock prices. But thesestudies do not represent the preponderance of theevidence for two reasons. First, gross inefficienciescan coexist with random walks in stock prices, as in


the case of rational bubbles (which is discussedbelow). Second, and more important, by the 1980s avast literature on stock market anomalies had devel-oped. These anomalies, defined as departures fromefficient markets that allow economic agents to enjoyunusually high (risk-adjusted) returns, appeared tolead to rejection of the EMH.

This section reviews some of the major anoma-lies in stock price determination that have been thetraditional basis for rejecting the EMH. While a panelof coroners might not declare the EMH officiallydead, by the late 1980s the burden of proof hadshifted to the EMH adherents. The anomalies dis-cussed in this section are among the list of causes thatwould appear on the death certificate.

The Small-Firm Effect

Arguably the best-known anomaly in stockprices is the Small-Firm Effect: the common stocks ofsmall-capitalization companies have, on averdge, ex-hibited unusually high rates of return throughoutmost of this century. This is shown clearly in Figure2, which reports the accumulated values (assumingreinvestment of dividends) of an investment of onedollar in January of 1926 in two portfolios: the S&P500and a portfolio of small-firm stocks. While small firmssuffered more in the Great Depression, their growthsince that episode has been far more dramatic thanthe growth in a portfolio represented by the S&P500.

According to the EMH, the small-firm effectshould be due solely to higher beta coefficients forsmall stocks; in other words, the higher rate of returnis solely due to higher risks. Any unwarrantedgrowth should lead investors to restructure theirportfolios to include more small-cap firms, therebydriving the price of small-cap stocks up relative tohigh-capitalization stocks and restoring a "normal"relationship in which all firms enjoy the same rate ofreturn, after adjustment for risk. The evidence sug-gests, however, that the higher return on small-capstocks is not attributable to higher risk. While inrecent years the small-firm effect has disappeared,the puzzle is that it existed for so many years, in spiteof general awareness that it was there.

The Closed-End Mutual Fund Puzzle

Another well-known anomaly involving a spe-cific class of firms is the Closed-End Mutual FundPuzzle, reflected in the discounts (and occasionalpremia) on closed-end mutual funds (Lee, Shleifer

Figure 2

Accumulated Value of$1 Invested in 1926

Log Value8

6

4

2S&,P 500

0

-2

-41926 1932 1938 1944 |950 1956 1962 1968 1974 1980 1986

Source: Author’s calculations, using Ibbotson (1990).

and Thaler 1990b; Malkiel 1977). Closed-end mutualfunds differ from open-end mutual funds in thatopen-end funds keep the prices of their shares at thenet asset value (NAV) by promising to buy or sell anyamount of their shares at NAV. Closed-end funds, onthe other hand, issue a fixed number of shares atinception, and any trading in those shares is betweeninvestors; this allows the closed-end fund share priceto deviate from NAV, that is, closed-end funds cantrade at either a discount or a premium. If the EMH isvalid, then any sustained discount or premium onclosed-end fund shares must be due to unique char-acteristics of the fund’s assets or charter. In theabsence of such distinguishing characteristics, anydiscounts or premiums would induce investors toengage in arbitrage that would eliminate the discountor premium. For example, an unwarranted discountwould lead investors to buy the closed-end fundshares and sell short a portfolio of stocks identical tothat held by the fund, thereby capturing a risklessincrease in wealth equal to the discount. A premium,on the other hand, would induce investors to sellshort the closed-end fund and buy an equivalentportfolio of stocks.

But closed-end fund shares typically sell at dis-counts, and the discounts are often substantial. Figure


3 shows the average year-end discount in the period1970-89 for seven major diversified closed-end fundcompanies.5 It is clear that the discounts move inverselyto stock prices; periods of bull markets, such as 1968-70and 1982-86, are associated with low discounts, whilebear markets (the 1970s and 1987) are associated withhigh discounts. Thus, the price paid for a dollar ofclosed-end fund assets is procyclical.

Several reasons are offered for closed-end funddiscounts. First, because of potential capital gainstaxes on unrealized appreciation, a new buyer ofclosed-end fund shares faces a tax liability if the fundshould sell appreciated securities; this potential taxliability justifies paying a lower price than the marketvalue of the underlying securities.6 Second, closed-end funds might have limited asset marketability ifthey buy letter stock or privately placed debt, whichcannot be sold to the public without incurring theexpense of obtaining Securities and Exchange Com-mission (SEC) approval or the restrictions on corpo-rate policy often required by public market investors.Third, agency costs, in the form of high managementfees or lower management performance, might ex-plain the discounts.

Figure 3

Average Premium (+) or Discount (-)on Seven Closed-Ended Funds

Percent10

-10

-20

-3O1970 1972 1974 1976 1978 1980 1982 1984 1986 1988

Note: The seven companies are listed in text footnote 5.Data from Barron’s.

Malkiel (1977) found that the discounts werelarger than could be accounted for by these factors,and other work has confirmed that this appears to bea true anomaly. To this should be added anotherpuzzle: at inception, the initial public offering (IPO)of closed-end fund shares must incur underwritingcosts and, as a result, the shares must be priced at apremium over NAV, after which the price of sea-soned shares typically moves to a discount within sixmonths. Why would informed investors buy the IPO,thereby paying the underwriting costs via capitallosses as discounts emerge? Clearly, something irra-tional is going on!

Weekend and Janumnd Effects

Another class of anomalies focuses on specifictime periods or seasonalities. Cross (1973) reportedevidence of a Weekend Effect, according to whichweekends tend to be bad for stocks; large marketdecreases tend to occur between the close on Fridayand the close on Monday. Later work showed thatthe weekend effect really occurs between the Fridayclose and the Monday opening. In Appendix 1 aweekend effect is added to the time series model ofstock prices for the 2,713 trading days in the 1980s.The result is resounding statistical support for aweekend effect. A plausible explanation of the week-end effect is that firms and governments release goodnews during market trading, when it is readily ab-sorbed, and store up bad news for after the close onFriday, when investors cannot react until the Mondayopening.

In recent years the January Effect has receivedconsiderable attention; the rate of return on commonstocks appears to be unusually high during January.The primary explanation is the existence of tax-lossselling at year end: investors sell their losing stocksbefore year end in order to obtain the tax savingsfrom deducting those losses from capital gains real-ized during the year. The selling pressure in lateDecember is then followed by buying pressure inJanuary as investors return to desired portfolio com-positions. However, this explanation is not consistentwith the EMH, according to which investors with nocapital gains taxes, such as pension funds, shouldidentify any tendency toward abnormally low pricesin December and should become buyers of stocksoversold in late December. This means that tax-lossselling should affect the ownership of shares but nottheir price.

The January effect has been thoroughly investi-


gated, and has been found to be more complicatedthan originally thought. Keim (1983) has shown thatthe January effect appears to be due largely to pricebehavior in the first five trading days of January; it isreally an Early-January Effect. Also, Reinganum(1983) found that the January effect and the small-firm effect are commingled: the January effect appearsto exist primarily for small firms and, in fact, much ofthe small-firm effect occurs in January.

In the time-series analysis reported in Appendix1, a test was added for the January effect and for anearly-January effect. The results do not support aJanuary effect of either type in the 1980s, at least forthe S&PS00. The fact that it does not appear for largefirms, which dominate the S&PS00 and are the firmsof primary interest to institutional investors, is con-sistent with the EMH. Arbitrage by well-informedinstitutional investors appears to prevent any late-December selling pressure from affecting the shareprices of large-capitalization firms. This does notprovide any conclusions about the January eitfect forsmall firms.

Figure 4

Annual Excess Return on Stocks,Classified by Value Line

Rank, 1965 to 1989

Total Return %

30

2O

10

0

-10

-20

-30

Rank 1lessRank 3

1965 1969 1973 1977 1981 1985 1989

Source: Value Line Investment Survey.

The Value Line Enigma

Yet another well-known anomaly is the ValueLine Enigma. The Value Line Investment Surveyproduces reports on 1700 publicly traded firms. Aspart of its service, Value Line ranksthe commonstocks of these firms in terms of their "timeliness," bywhich it means the desirability of purchasing thefirm’s shares. Value Line employs five timelinessranks, from most timely (Rank 1) to least timely (Rank5). Rank 3 is the designation for firms projected toincrease in line with the market.

Figure 4 reports the annual average excess returnsfor Rank 1 and Rank 5 stocks. These returns arecomputed as the difference between the mean returnson stocks in the stated rank and the overall mean(Rank 3) returns. The computation assumes that thestocks are bought at the beginning of the year andsold at the end of the year, and transaction costs arenot considered. It is clear that Rank I stocks generallyperform better than average. In only five of thetwenty-five years do Rank 1 stocks underperform theaverage; the probability of this happening by chanceis only 0.00046.7 Also, Rank 5 stocks tend to under-perform. In only three of the twenty-five years doRank 5 stocks perform better than average, and thenthe difference is small.

If the stock market is efficient, only one reasonexists for higher rank stocks to generate higher re-

turns: they have a higher level of market risk, that is,higher beta coefficients. Black (1971) found that themean beta coefficients were roughly the same forstocks in each rank, concluding that the rankingsystem did have predictive value. However, Lee(1987) found that a stock’s beta coefficient is inverselyrelated to its Value Line rank: stocks for whichpurchase is timely tend to have higher betas. Thissuggests that the better performance of stocks ranked1 and 2 is, at least in part, due to the higher averagereturns normally associated with higher risk.

Holloway (1981) examined the value of bothactive and passive trading strategies based on theValue Line Ranking System. An active trading policywas defined as purchasing Rank 1 stocks at thebeginning of a year and holding them until theearliest of either the end of the year or a downgradeof the stock’s Value Line rank, at which time the stockwould be replaced by another Rank 1 stock to be helduntil year end. A passive, or buy-and-hold, strategywas defined as purchasing Rank I stocks at the outsetof a year and selling them at year end. The activetrading strategy generated higher returns than didthe passive strategy when transaction costs were notconsidered, but was inferior to the buy-and-holdstrategy when reasonable transaction costs were as-


sessed. Hence, active trading using the Value Lineranking system is not a profitable strategy for inves-tors.

However, Holloway found that even after ad-justments for transactions costs and for risk, a passivestrategy using Rank 1 stocks outperformed a passivestrategy using Rank 3 stocks; the Value Line RankingSystem did provide profitable information for thosewho are willing to buy and hold. It is noteworthy thatthis advantage existed even when adjustments weremade for both transaction costs and risk (beta).

IlL "’Modern’" Evidence of Ineffic_iency

The previous section reports the results of "tra-ditional" approaches to assessing the EMH: examina-tion of specific examples of departures from theEMH, called anomalies. During the 1980s several"modhrn" approaches were developed. These are thetopic of this section.

Excess Volatility of Stock Prices

One of the more controversial "modern" tests ofthe EMH is based on the observed volatility of stockprices. Leroy and Porter (1981) and Shiller (1981)concluded that the observed amount of stock pricevolatility is too great to be consistent with the EMH.In order to understand this "excess volatility" argu-ment, refer back to the first definition of an efficientmarket: a market is efficient if the price of the asset isan optimal forecast of the fundamental value, that is,if Pt = E(P~l~t).

The logic of the excess volatility argument isbased upon a property of statistical theory: the opti-mal forecast of a random variable must,.on average,vary by no more than the amount of variation in therandom variable being forecasted. Thus, if the marketprice is an optimal forecast of the fundamental val-ue-as the EMH implies--it should vary less than(and certainly no more than) the fundamental value.

A formal statement of the excess volatility argu-ment is that the relationship between the fundamen-tal price under the actual state (s) and the optimalforecast of the fundamental price is

(4) Ps* = E(P*lf~t) + ~s

where ~5 is a random variable that measures thedeviation between the fundamental value for thestate which actually occurs, Ps*, and the optimal

forecast of the fundamental value, E(P*lf~t). If theforecast is optimal, these deviations must be randomand uncorrelated with the forecast itself. Now, theEMH implies that P = E(Ps*lf~t), which means

(4’) P2 = P + ~s.

In other words, the correct price (conditional onknowing the true state) is equal to the market priceplus a random term, denoted by ~s, which measuresthe surprise resulting when the true state is known.This random term must be uncorrelated with P,because P is the optimal forecast and, therefore,already reflects any systematic information.

This provides the basis for the variance boundstests of the EMH. Equation (4’) shows that thevariance of the fundamental price is equal to thevariance of the market price plus the variance of thesurprise. Turning this around produces the followingrelationship:

(4") VAR(Pt) = VAR(P*t) - VAR(~t).

Because variances must be non-negative, if theEMt:t is valid the variance of the market price must beno greater than the variance of the fundamentalvalue, or:

(4") VAR(Pt) -< VAR(P~.

Consider the following simple example, summa-rized in Table 1. Assume three states of the world, ineach of which the dividend-price ratio is 10. In state 1dividends paid at year end will be $10 and the

Table 1Example of Variance Bounds TestsThree States--Three Years

Fundamental Probability of State s in YearState Value

(s) (P~’) 1 2 31 $100 .25 .50 .102 75 .50 .30 .803 40 .25 .20 .10

Market Price $72.50 $ 80.50 $74.00Modal Fundamental

Value* $75.00 $100.00 $75.00"The modal fundamental value assumes that the most likely stateoccurs in each year.


fundamental value is $100, in state 2 dividends are$7.50 and the fundamental value is $75, and in state 3dividends are $4 with a fundamental value of $40.These fundamental values are shown in column 2 ofTable 1. At the beginning of each year the dividend tobe paid at the end of the year is not known becausethe state that actually occurs is not known, but theprobability of each state occurring is known. There-fore, at the beginning of each year investors knowonly the probability distribution of states and the

Under the null hypothesis of theEMH, the market price must varyby no more than the fundamentalprice. Any "excess" volatility is,therefore, a symptom of market

inefficiency.

dividend payment that each state entails. In thisexample, changes in the probability distributionacross states correspond to the notion that newinformation is received by investors at the beginningof each year.

The "market’s" problem is to determine a marketprice that best reflects that information. Table 1assumes three years, with columns 3 to 5 showing theprobability distribution of states in each year. Therow marked "market price" shows the EMH marketprice, defined as the statistical expectation of funda-mental values in each year. Thus, as time passes, themarket price should increase from $72.50 to $80.50,then fall to $74.00; the sample standard deviation ofthe market price would be $4.25.

But the "correct" price, defined as the funda-mental value associated with the realized state,would exhibit even larger movements. For example,if in each year the modal (most likely) state occurs,then the sequence of states is 2, 1, 2 and the funda-mental values (row 5) would be $75 in year 1, $100 inyear 2, and $75 in year 3. The sample standarddeviation of these three "correct" prices would be$14.43, much greater than the sample standard devi-ation of the market price.8 This result is consistentwith the EMH.

Shiller’s excess volatility tests were conducted as

follows. He assumed a dividend valuation model inwhich the fundamental value is the present value ofthe perpetual stream of dividends resulting in eachstate-of-the-world. Using actual data on dividendspaid over a very long period of time, and an assump-tion about the terminal price of shares, he calculateda time series for the fundamental value of the S&P500index. He then compared the variance of that serieswith the variance of the observed values of theS&P500 and found that, if the discount rate wasassumed to be constant, the variance of the marketprice was about six times the variance of the funda-mental value--dramatic refutation of the EMH. How-ever, if the discount rate was allowed to vary withinterest rates (so that fundamental values exhibitedgreater variation), the market price had a varianceabout 1.5 times the variance of the fundamental price.In either case, the volatility of the stock market wasgreater than the upper bound implied by the EMH,leading Shiller to reject the EMH.

Under the null hypothesis of the EMH, themarket price must vary by no more than the funda-mental price. But Shiller’s discovery implies eitherthat the EMH is invalid or his test is invalid. This is acommon problem of statistical tests: one must makeassumptions about the world in order to constructany test, but one cannot know whether rejection ofthe null hypothesis is due to the invalidity of thehypothesis or to the invalidity of the assumptions.

The conclusion that excess volatility exists hasbeen criticized for a number of reasons, each of whichcan be seen as a criticism of the test. Marsh andMerton (1986) disputed one of the assumptions un-derlying Shiller’s test--that dividends are a stationarytime series--and showed that if the process by whichdividends are set is non-stationary, the EMH test isreversed: under the EMH, market prices should bemore volatile than fundamental values. Kleidon(1986) has criticized the excess volatility test on sta-tistical grounds, arguing that the Shiller test is anasymptotic test, assuming a very large sample ofobservations over time, and that the data availableare necessarily finite, hence small-sample biases canweaken the test. In addition, the power of the testagainst reasonable alternative hypotheses is quitelow, meaning that the test is not likely to reject theEMH when it should be rejected.

Whatever the validity of the excess volatilitytests, they do provide an additional reason--otherthan observed anomalies--to doubt the validity of theEMH, and they have had a significant effect on thestate of academic thinking about market efficiency.

26 MarchlApril 1991 Nezo England Economic Review

Speculative Bubbles

It has long been a common practice to look backon dramatic collapses in asset prices and assign themto the bursting of a bubble. For example, followingthe October 1987 crash, many observers pointed outthat stock prices had risen so rapidly in 1986 and 1987that a bubble surely existed.

The notion of a "bubble" is a familiar one: abubble reflects a difference between the fundamentalvalue of an asset and its market price. Unfortunately,while the notion of a bubble has rhetorical force, it isa far more slippery concept than it appears. Clearly, abubble is not merely a random deviation of price fromvalue, for the law of large numbers suggests thatpurely random deviations will wash out over timewithout any necessity of collapse.

The bubble concept has been powerful becauseof the notion of self-fulfillment: bubbles are self-fulfilling departures of prices from fundamental val-ues which continue until, for some reason, the con-ditions of self-fulfillment disappear. What do wemean by self-fulfilling bubbles? Recall that financialtheory states that the market value of an asset (in-cluding dividends received) at the end of one periodmust be the market price at the end of the previousperiod, adjusted for growth at the required rate ofreturn (r). That is, in equilibrium, where r is therequired rate of return associated with the asset’s risklevel and Et denotes an expectation conditional oninformation at time t:

(5) Et(Pt + 1 + Dt + 1) = (1 + r)Pt.

This difference equation, when solved recur-sively, gives the following stock price model, which isthe well-known present discounted value model:

(6) Pt = ~ [(1 + r) - kEtDt + k].k=l

One definition of a self-fulfilling speculative bub-ble is that its presence does not violate this descrip-tion of asset prices. In that case, called a "rationalbubble," market observers could not see the presenceof a bubble and would not behave in ways thateliminate it. But how would a bubble remain invisi-ble? To do this, it must be true that its existence doesnot violate the process shown in equation (5). If wedefine Bt as the size of the bubble, we can see that ifthe bubble is expected to grow at the required rate ofreturn, that is, if EtBt+1 = (1 + r)Bt, the bubble will be

viable. In this case investors do not care if they arepaying for a bubble because they expect to get therequired return on that investment.

This definition of a rational bubble implies somevery strong restrictions on bubbles. One is that bub-bles cannot be negative: in order to be self-fulfilling, anegative bubble must become more negative at thegeometric rate r, but the stock price will grow at a rateless than r because dividends are paid.9 From (5) wecan see that Et(Pt + 1) = (1 + r)Pt - Et(Dt + 1).Hence, a negative rational bubble must ultimatelyend in a zero price, a result that, once acknowledged,must lead to the elimination of the negative bubble.Thus, while the market price might be below thefundamental value at a specific point in time, itcannot be the result of a rational bubble.

Because there is no upward limit on prices, apositive bubble can exist, although with some im-plausible consequences. First, as time passes a posi-tive rational bubble must represent an increasingproportion of the asset’s price. This is because thebubble must grow at the rate r, while the price growsat a rate less than r because of dividend payments.

Bubbles are self-fulfillingdepartures of prices from

fundamental values, whichcontinue until, for some reason,the conditions of self-fulfillment

disappear.

But the idea that investors can project an indefiniteincrease in the relative size of the bubble underminesthe existence of the bubble. Surely, if investors un-derstand that a positive bubble means that the bubblemust be an increasingly important component ofprice, they will imagine that at some time the bubblemust burst. But as soon as they realize that it mustburst, it zoill burst!

For example, suppose investors believe that apositive bubble exists but that it will not burst untilthe year 2091. They must, then, realize that in theyear 2090 the market price must reflect only thefundamental value because the year 2090 investorswill not pay for a bubble knowing that it will disap-pear. But if the year 2090 price is the fundamental


value, no bubble can exist in 2089, and therefore theyear 2089 price must be equal to the fundamentalvalue. This chain of reasoning leads to the conclusionthat a bubble cannot exist now. This will be true evenif the collapse of a supposed bubble will not occuruntil after the passage of a very great (approachinginfinite) time: as long as the resale price of the assetplays a negligibly small role in price determination, abubble cannot exist!1°

If a rational bubble can never emerge, what is leftof the notion of bubbles? Remember that a crucialassumption of the "bUbbles cannot exist" paradigm isthat investors behave as if they have an infinite timehorizon. If investors have finite horizons, and plan tosell their shares before the present value of the salebecomes negligibly small, they will not project cashflows into the indefinite future, but will formjudgments about the price at which the asset can besold at the end of the horizon.1~ If, for example, thehorizon is five years, the market price of the assetnow will be described by the standard valuationequation Pt = ~[(1 + r)-kEtDt+k] + (l+r)-5EtPt+s. Ifthe expected resale price is simply the present valueof expected dividends beyond that point, we arereally back to the infinite-horizon model in which theultimate resale price is irrelevant and bubbles cannotexist. For example, if EtPt+5 = ~;2 (1 + r)-kEtDt+k wecan see that the correct price will be described byequation (6).

Thus, the presence of a resale price whose ex-pected value is not hinged to dividends beyond thatpoint is necessary to the existence of rational bubbles.While it might be "rational" to use the infinite-horizon valuation model, it is not "realistic." Inves-tors and traders do form judgments about the price atwhich they can sell assets, but they do not believethat the buyers are using an infinite-horizon model todecide the value of the asset.

Thus, rational bubbles are realistic descriptionsof stock price performance; if the "market’s" horizonis shorter than the time to the popping of a bubble,the bubble can continue. This is the essence of the"Greater Fool" explanation of speculative episodes:you will knowingly pay a price above fundamentalvalue because you believe that someone later on willpay an even greater premium over fundamentalvalue.

How should one go about testing the role ofrational bubbles? This question is difficult to answer,for a rational bubble will not affect the sequence ofprices until it breaks. The analysis of such low prob-ability events is called the "peso problem": market

prices will not reflect the effects of very low probabil-ity events even if they should have dramatic effectswhen they appear. Hence, it would be impossible touncover a rational bubble as long as it exists. How-ever, the disappearance of a bubble, such as a majordecline in stock prices, can be examined to determinewhether it was preceded by a speculative bubble inprice.

Using the Ibbotson (1990) data for monthly re-turns on common stocks (S&P500) and one-monthTreasury bills, a measure of stock price bubbles forthe period February 1926 to December .1988 wasconstructed. This was done by computing, for eachmonth, the difference between the total return oncommon stocks and the required return. The requiredreturn was computed as the one-month Treasury billrate plus a risk premium. Denoting the actual returnas Rt and the required return as (rt + 0), where rt isthe one-month Treasury bill rate and 0 is the riskpremium, the bubble at time t is:

(7) Bt= [l+Rt-(rt+ 0)]Bt-1.

This approach assigns any difference betweenthe observed return and the required return to

Figure 5

Estimated S&P 500Stock Market Bubble

Bubble: January 1926=12.0

1.8

1.6

1.4

1.2

1.0

.8

.6

.4

.2

0 I I I I I I I I ! I1926 1932 1938 1944 1950 1956 1962 1968 1974 1980 1986

Source: Author’s calculations, using Ibbotson (1990).

28 MarchlApril 1991 New England Economic Review

Table 2Probability Model for Stock Market CrashesMonthly Data 1926-88

Size of Crash Over Next 12 Months

CONSTANT - 1.5611(-10.81)

IogBUBBLE~_I +.9327(+3.99)

TIME n.a.

Number of Months 755Proportion Predicted .8808Mean Probability .7019 "Number of Months

Followed by Crashes 90

One Standard Deviationa Two Standard Deviations Three Standard Deviations-1.3795 -2.4792 -1.2698 -3.7940 -1.9569

(-6.34) (-12.38) (-4.50) (-10.49) (-3.92)

+.8240 +2.1689 + 1.3931 + 1.7837 +. 1897( + 3.24) ( + 4.52) ( + 2.99) ( + 2.23) ( + .22)

-.0007 n.a. -.0058 n.a. -.0139(- 1.12) (- 4.36) (-2.46)

755 755 755 755 755.8808 .9603 .9603 .9881 .9881.7024 .8611 .8754 .9411 .9532

90 30 30Note: Numbers in parentheses are t-statistics. The parameters are estimated using a Iogil model, according to whichProb(crash) = 1/1{1 + exp[-(a + bX)]}, where X is lhe list of explanatory variables (IogBUBBLE and TIME).’n.a. = not applicable.a In the 755 months in the period 1926:2 to 1988:12, the change in log bubble over the next 12 months had a mean ot -0.0208 and a standarddeviation of 0.2226. Hence, a "one-standard-deviation" crash is defined as a 12-month change in the logarithm of the bubble by an amount of-0.2434 or less. A two-standard-deviation crash is a change of -0.4660 or less.

changes in a bubble. Assigning any arbitrary positivevalue to the initial bubble makes it possible to traceout the path of the bubble using this differenceequation. Note that the value assigned to the initialbubble is irrelevant since our interest is in movementsin a bubble, not in its absolute size.

Figure 5 shows the path of our measure of thebubble over the period February 1926 to December1988, using the assumptions that the initial bubble is1.0 in January of 1926 and that 0 is the average riskpremium over the entire sample period.12 The timeseries shown in Figure 5 indicates a very large spec-ulative bubble in the late 1920s as the return on stockssharply exceeded the required return. This was fol-lowed by the crashes of 1929-32. The next bubbleemerged in the 1955-65 period, when the bubbleappeared to remain high for a considerable period oftime before a prolonged "crash" lasting through the1970s. The bubble fell to a low point in late 1982 thatmatched the lows of the 1930s.

The Crash of 1987 has often been attributed to aspeculative bubble emerging as prices rose dramati-cally in 1986 and the first nine months of 1987.However, our bubble measure does not support thisinterpretation. While stock returns were above therequired return, creating an expanding bubble, thesize of the bubble in September 1987 was so small that

it could not be used to predict the crash.Does our measure of a stock market bubble have

any predictive value? Clearly, the concept of a bubbleis intended to explain market crashes, not simplymild and temporary price declines. Therefore, wehave employed a probabilistic model to determinewhether the probability of a future crashis related tothe size of the bubble. In an efficient market norelationship exists but, if speculative bubbles do exist,the probability of a crash should be a direct functionof the size of the bubble.

Table 2 reports our results for several differentdefinitions of a crash. A "one standard deviationcrash" occurs if over the next twelve months thechange in (the logarithm of) the bubble is more thanone standard deviation below the mean change. 13 Wealso consider crashes of two and three standarddeviations. The probability of a crash is assumed tobe described by a logistic function, and the result is alogit14 model in which the probability of a crash overthe next twelve months is a function of the logarithmof the bubble at the end of the previous month. Inorder to correct for any trends in the relationship, wehave also added a linear trend variable (TIME).

The results for 1926-88 suggest that the bubbledoes have predictive value: for both one- and two-standard-deviation definitions of a crash, the variable

March/April 1991 Nezo England Economic Review 29

log BUBBLEt_1 is statistically significant and has apositive sign: the bigger the bubble, the greater theprobability of a crash in the next twelve months. Thebubble size loses its predictive value if a crash is athree-standard deviation fall, but only 8 months fitthis definition, and a model cannot predict eventsthat occur so rarely.

Mean Reversion h~ Stock Retinas

The phenomenon of mean reversion is the ten-dency for stocks that have enjoyed high (low) returnsto exhibit lower (higher) returns in the future; that is,returns appear to regress toward the mean. A semi-nal test of mean reversion was undertaken by Famaand French (1988), who regressed the rate of returnfor a holding period of N months upon the rate ofreturn during the previous N months. For example,they regressed the return over an 18-month periodupon the return over the previous 18 months; anegative slope coefficient indicates mean reversionover an 18-month horizon. Fama and French foundevidence of mean reversion for holding periodslonger than 18 months.

While we have some skepticism about the Fama-French tests (the results appeared to be due primarilyto the inclusion of the 1930s in the sample period),the phenomenon of mean reversion has been sup-ported by other tests. Poterba and Summers (1988)found that the variances of holding-period returns donot increase in proportion to the length of the holding

Mean reversion can be thought ofas another anomaly, but it isreally much more: it is the

common theme in most tests ofstock market efficiency.

period, an indication of mean reversion.15 Othervariants of mean reversion tests also confirm theexistence of mean reversion. For example, De Bondtand Thaler (1985) have found evidence of both awinner’s curse and a loser’s blessing in stock prices.Stocks that have experienced a recent reduction intheir P/E ratios tend to have higher rates of returnthan equivalent stocks that have not been "losers,"

while stocks that have experienced increases in theirP/Es tend to be losers in subsequent periods. Theloser’s blessing appears to be more dramatic than the

16winner’s curse.Mean reversion can be thought of as another

anomaly, but it is really much more: it is the commontheme in most tests of stock market efficiency. Forexample, Shiller’s excess volatility tests are an indi-rect test for mean reversion, and the analysis ofbursting speculative bubbles is really an examinationof sudden mean reversions.

IV. Why Is the Stock Market Inefficient?Abundant evidence casts doubt on the Efficient

Market Hypothesis. The natural next question is,"Why?" What aspects of investor behavior mightaccount for these departures from the predictions ofeconomic theory?

An important preliminary to answering thisquestion is the observation that many of the anoma-lies shown in the previous section are really manifes-tations of one fundamental anomaly: the small-firmeffect. For example, the January effect is primarily acharacteristic of small firms (Keim 1983; Reinganum1983), and the winner’s curse and loser’s blessing arealso most prominent among small firms (De Bondtand Thaler 1985). Furthermore, the closed-end fundpuzzle appears to be the result of the similarity of themarkets for closed-end funds and for small-firmstock: Lee, Shleifer and Thaler (1990a) show thatdiscounts on closed-end funds are highly correlatedwith performance of small-firm stocks, that institu-tions tend to shy away from both small-firm stocksand closed-end fund shares, and that the transactionsize of both small-firm stocks and closed-end fundshares tends to be much lower than the transactionsize for large-firm stocks.

This suggests a common denominator formany--but not all--of the departures from the EMH.They tend to be concentrated in stocks traded inrelatively narrow markets where the "smart money"is not as likely to play. In short, inefficiencies mightbe associated with a form of market segmentation inwhich the EMH applies to stocks of large firms whichare the province of financial institutions with accessto research on fundamentals, while inefficiencies,and their associated profitable opportunities, appearto be concentrated among those who invest in thestocks of smaller firms, traded in less active marketswith a lower quality of information.


Market hzeff!’ciency and Market Segmentation

The claim that market participants can be seg-mented into highly informed and less informed in-vestors, and that this fact is an important componentof stock price determination, will create a sense ofd~ja vu: presenting it to market practitioners is a caseof preaching to the choir. Indeed, the "smart money-dumb money" distinction has been around for aslong as markets have existed, and it was enshrined inthe work of early dissidents in the random walkdebate. For example, Cootner (1964) argued that theprofits of professional investors, who have low trans-actions costs, come from observing the random walkof stock prices produced by nonprofessionals andstepping in when prices wander sufficiently far fromthe efficient price.

Why has this view enjoyed a renaissance amongacademic economists? It is not merely because aca-demics get to put notches on their guns when theydisturb the conventional wisdom. Nor is it solely dueto the more important reason that anomalies havebecome too numerous and well-documented to ig-nore. Each of these has played a role, but the funda-mental reason is that only recently have economistsprovided a theoretical foundation for market segmen-tation.

EMH theorists rejected the market segmentationapproach for several reasons. The first was that itclearly assumes irrational behavior, with the unso-phisticated investors (henceforth called small inves-tors) somehow driving prices of stocks (primarily ofsmall firms) away from fundamental value. The sec-ond is that it assumes that the smart money allowsthis to happen and fails to step in when very smallopportunities arise; in short, arbitrage is incomplete.The third problem is survival of the small investors; ifsmall investors are buying high and selling low, asthey must if they are giving large investors an oppor-tunity for profits, then the population of small inves-tors should diminish over time and the inefficienciesshould disappear.

The possibility that some investors are "irra-tional" is not sufficient to induce inefficiency; nobodybelieves that all investors are rational, and so long asthese investors are infra-marginal they are merelygiving profits to large investors. True, theorists donot like irrationality, and might question why itshould exist; but that is a question of psychology, notof economics. Furthermore, the proposition that irra-tionality is self-correcting because the irrational play-ers incur losses and leave the game does not work

well for two reasons. First, the mortality of investors,and the difficulty of transmitting wisdom and expe-rience to the young, mean that a new crop of inves-tors is always emerging which, if given sufficientendowments, can become players. The 1980s were anexample of this, with young professionals havinglimited experience handling large amounts of money.Second, as we shall argue, it is possible that irra-

It is possible that irrationalinvestors do, in fact, get rewarded

and are not eliminated.

tional investors do, in fact, get rewarded and are noteliminated.

The fundamental objection to market segmenta-tion is the arbitrage objection. Large investors (it isargued) will ensure that prices must be nearly effi-cier~t. Because they care only about intrinsic value, ifprices diverge from the efficient price, they willengage in arbitrage to restore the equality. The arbi-trage objection can be dealt with by forgoing twoimplicit assumptions: that investors never plan toliquidate their stock positions, and that riskless arbi-trage is possible. Each of these assumptions is ad-dressed in turn.

As in the discussion of speculative bubbles, ifinvestors have finite horizons, they will be concernedabout the resale price of the security and form judg-ments about what that will be at the end of theirhorizon. One way of forming those judgments--theway proposed by traditional finance theory--is toestimate the future price as a present value of divi-dends received from that point on; that simply bringsin the infinite life assumption through the back door.Another approach--which seems more plausible--isto recognize that investors are concerned with resaleprice and that theh" forecasts of resale price may wellnot reflect solely their judgments about future divi-dends; perhaps even more important will be theirestimates of what other investors will be willing topay. This was stated clearly by John Maynard Keynes(1964, pp. 155-56), who said of professional investorsand speculators:

They are concerned, not with what an investment isreally ~vorth to a man who buys it "for keeps", but with

March/April 1991 Nezo England Economic Review 31

what the market will value it at, under the influence ofmass psychology, three months or a year hence ...professional investment may be likened to those news-paper competitions in which the competitors have topick out the six prettiest faces from a hundred photo-graphs, the prize being awarded to the competitorwhose choice most nearly corresponds to the averagepreferences of the competitors as a whole... We havereached the degree where we devote our intelligences toanticipating what average opinion expects average opin-ion to be.

But (an efficient market adherent might respond)even if many investors are forming judgments aboutresale prices that differ from fundamental value, awell-financed body of highly informed investors canprevent that from affecting market prices by engagingin riskless arbitrage. The response to this is that fewopportunities for riskless arbitrage exist, and to theextent that arbitrage involves some risk, risk-averseinvestors will require a positive expected return(above opportunity costs), allowing inefficient pricingto continue.

An example of this is discounts on closed-endfunds. Clearly, the market for closed-end fund sharesmust be dominated by investors who adopt a highprobability that resale prices will deviate from funda-mental values. If this were not true, investors wouldestimate that resale prices would be equal to futurefundamental value, and riskless arbitrage would en-sure that the current price reflects current fundamen-tal value. For example, if current investors think thatthe future resale price will be above fundamentalvalue, they will buy the closed-end fund shares andsell short a bundle of shares that replicate the closed-end fund portfolio. By doing this, they will enjoyprofits, and their profit-seeking activities will ensurethat current price is equal to current fundamentalvalue.

Noise Trading

We have argued that prices do diverge systemat-ically from fundamental values because prices candiverge systematically from fundamental values, be-cause even well-informed investors are risk averseand will not engage in sufficient arbitrage activity toprevent this. This view has been formalized recentlyby a model of irrationality called "noise trading" byits proponents (Black 1986; Shleifer and Summers1990). The noise trading model proposes that animportant segment of the market consists of investorswho bid prices away from fundamentals, thus intro-

ducing "noise" into stock prices. This noise, or "in-vestor sentiment," is sufficiently broad in its impact,affecting many stocks, that investors cannot avoid itby diversification and must accept it as a source ofsystematic risk. Because it is systematic and undiver-sifiable, the noise affects the rate of return investorsrequire on stocks and, therefore, market prices. Notall stocks are affected equally by noise risk--stocks ofwell-established firms, which are traded among in-formed investors, might not carry much noise risk,while stocks of small firms are more likely to bear thisrisk.

A simple model of noise trading is presented byDeLong, Shleifer, Summers and Waldmann (1990).This model, discussed in more detail in Appendix 2,assumes that young investors buy stocks and oldinvestors sell stocks (to the young) to live on in theirdotage. Sophisticated investors form optimal fore-casts of the future price, but unsophisticated inves-tors, called "noise traders," develop biased forecasts.Because sophisticated investors are risk averse andarbitrage is risky due to the possibility that the extentof price misperception by noise traders might change,sophisticated investors will not fully arbitrage awaythe influence of noise trading. Thus, noise traders candrive the market price away from the fundamentalvalue.

The noise trading model proposesthat an important segment of themarket consists of investors who

bid prices away fromfundamentals, thus introducing

"noise" into stock prices.

In the DeLong-Shleifer-Summers-Waldmannmodel, the degree of price misperception exhibitedby noise traders--the difference between their fore-casts and optimal forecasts--is assumed to be arandom variable (denoted as p), which follows anormal probability distribution with mean p* andvariance cr2. If p* = 0, noise traders agree with so-phisticated traders "on average," but noise traderforecasts will temporarily differ from sophisticatedforecasts at any moment. In this case the equilibrium


price of an asset will normally be below the funda-mental value, the discount being necessary to com-pensate sophisticated traders for the risk that stockprices will deviate from fundamental value evenmore in the future. If noise traders are pessimistic(/)* < 0), the normal discount from fundamental valuewill be higher, while if noise traders are optimistic(p* > 0), the normal discount will be lower or apremium might emerge. In addition to the normaldiscount arising from the average price mispercep-tion of noise traders, there is a temporary randomdiscount due to temporary variations in optimismand pessimism (/~ - p*).

The EMH adherent would ask why sophisticatedinvestors do not dominate the market and, througharbitrage, force the market price to equal the funda-mental value. If this did occur, the perceptions ofnoise traders would alter the ownership of stocks(sophisticated traders holding more when noise trad-ers are pessimistic and less when noise traders areoptimistic), but price misperceptions would not affectthe market price of an asset.

The answer is, as noted above, that arbitrage isnot riskless: no individual sophisticated trader canknow that all sophisticated traders together will forceequality of the market price with the fundamentalvalue. There is always the possibility that noisetraders will influence stock prices and that the sophis-ticated trader, when he arrives at the end of hishorizon, will be forced to sell at a price even furtherbelow fundamental value than his cost.

This model is overly simple, designed for expos-itory purposes and not as a strict representation ofreality. But it does explain a number of importantphenomena. For example, it explains the excess vol-atility of common stock prices found by Shiller: in theabsence of noise trading, stock prices would alwaysequal fundamental values, but with noise trading,stock prices will be more volatile than fundamentalvalues because of the changing perceptions of noisetraders. It also can explain the small-firm effect, andthe related anomalies (for example, the January ef-fect, the loser’s blessing, the closed-end fund puzzle):the small-firm effect exists because the noise risk ishigher among small firm.s, which are not as favoredby sophisticated traders and in which noise tradersplay a larger role.

Furthermore, this simpl6 model explains howthe phenomenon of noise trading can persist. Fried-man (1953) argued that, in the long run, prices mustconform to fundamentals because speculators whopaid incorrect prices would either go broke if they

tended to buy high and sell low, or would force pricesto equal fundamental value if they were sharpenough to buy low and sell high. In either case, whatwe now call noise trading would be a temporaryphenomenon. In this model, however, noise traderscreate a more risky environment, but because theireffects are pervasive and not idiosyncratic to individ-ual stocks, the risk is not diversifiable and must earna reward. Indeed, not only is noise trading consistentwith an average return above the riskless rate, but itcan be consistent with a higher average return fornoise traders than for sophisticated investors if, asseems likely, noise traders tend to invest more heav-ily in noise-laden stocks, hence earning more of therisk premium associated with small stocks.

Noise Trading with Fads: A Si~nulation

The noise trading model can easily generatemodels of stock price movements that mimic thesharp breaks and apparent patterns visible in the realdata. The crux of this is the possibility of "fads" thataffect investor sentiment, measured by p. Smallchanges in these opinions can translate into verylarge changes in stock prices. For example, usingequation (A2.1) in Appendix 2, we can calculate theeffect of a change in investor sentiment of noisetraders.

As noted above, noise trader price perceptionscan be decomposed into two types: the normal per-ception is the average value of p, denoted by p*, andis the degree of optimism that prevails over fairlylengthy periods; temporary price perceptions, de-noted by/9 - p*, prevail at any moment of time but donot affect the trend of stock prices.

Changes in the normal price perception, or p*,can result in very large changes in stock prices.According to equation (A2.1), a change in p* inducesa change in stock price by the amount (/x/r)/9", where/x is the proportion of traders who are noise tradersand r is the real interest rate. Assuming/x = 0.05 andr = 0.0042 per month (5 percent per year), wecalculate (/x/r) = 11.90: a change in p* by 0.01 (or 1percent of fundamental value) will alter the stockprice by 0.1190, or about 12 per cent of fundamentalvalue.

Changes in stock prices due to variations ininvestor sentiment will be random so long as thenormal price perception of noise traders is constant.However, if p* is serially correlated--its current valuedepends on previous values--stock price changes canexhibit sharp breaks that do not conform to a model


Figure 6

Simulated 50-Year Stock Price History

Stock Price (Actual/Fundamental)1.05

1.0

.95

.90

.85

.80

54 107 160 213 266 319 372 425 478 531 584Month

Source: See Appendix 2.

of simple random variation about an equilibriumlevel. This is likely to happen when there are fads inperceptions, as when optimism is reinforced by ear-lier phases of optimism.

To illustrate this, we have simulated monthlybehavior of a hypothetical stock market using equa-tion (A2.1). The settings for the crucial parameters arediscussed in Appendix 2. In order to introduce thepossibility of realistic results from a noise tradingmodel we need to add the notion of "fads andfashions," in which investor interactions createwaves of investor sentiment. Our way of introducingthe possibility of contagious behavior is to assumethat "opinion" follows a random walk shown by theautoregressive process:

where ~t is white noise with mean E(~t)= 0 and2 In this simple case, investorsconstant variance ~.

standing at a moment of time will forecast a constantvalue of p* because zero is their optimal forecast of ~for every period. But the actual value of p* will followa path determined by equation (8). Because stockprices are very sensitive to p*, this can result inrealistic stock market cycles.

The result8 of one such experiment are shown inFigure 6. The results show what appear to be system-atic patterns in the stock price imposed upon a bearmarket, in which the stock price falls from its funda-mental value of 1.0 to about 0.88 at the end of the 50years (600 months). Recall that the results are stockprices relative to fundamental value, so the figuredoes not mean that stock prices fall to 88 percent ofthe original value; an upward trend in the fundamen-tal value could allow the stock price to rise eventhough, because of noise trading, it is not rising asfast as it should.

Repeated experiments will show essentially thesame patterns of prolonged departures from funda-mental value, punctuated by sharp breaks in price,though the sequence of bull and bear markets will bedifferent in each simulation. It is clear that even withmild fads in noise trader misperceptions, there can bedramatic and apparently systematic cycles in stockprices. While this does not prove that noise trading isthe source of the kind of stock market cycles weobserve, it does show that noise trading, supple-mented with contagious investor interactions or fads,is a plausible way of explaining observed stock pricebehavior.

V. Summamd and ConclusionsThis paper assesses the current state of the

efficient market hypothesis, which was the conven-tional wisdom among academic economists in the1970s and most of the 1980s. It reviews the empiricalevidence and concludes that it provides an over-whelming case against the efficient market hypothe-sis. This evidence exists in the form of a number ofwell-established anomalies--the small firm effect, theclosed-end fund puzzle, the Value Line enigma, theloser’s blessing and winner’s curse, and a variety ofanomalies surrounding seasonality, such as the Jan-uary effect and the weekend effect. Many of theseanomalies are more pronounced among small-firmstocks, suggesting that the efficient market hypothe-sis might be more appropriate for stock of large firms,but analysis of the S&P500, which is dominated bylarge firms, also finds important anomalies such as aweekend effect, slow mean reversions in returns, andstock price volatility in excess of the amount pre-dicted by fundamentals.

These anomalies can be explained by resorting toa model of "noise trading," in which markets aresegmented with the "smart money" enforcing a high


degree of efficiency in the pricing of stocks of largefirms while less informed traders dominate the mar-ket for small firms. This model can explain many ofthe anomalies, and it can generate cycles in stockprices that are very similar to those observed in thereal world.

Our fundamental conclusion is that the efficientmarkets hypothesis is having a near-death experienceand is very likely to succumb unless new technology,as yet unknown, can revive it. This conclusion has anumber of policy implications. The fundamental im-plication is that security market inefficiency provides

Our fundamental conclusion isthat the efficient market

hypothesis is having a near-deathexperience and is very likely to

succumb unless new technology,as yet unknown, can revive it.

an economic foundation for public policy interven-tions in security markets. Clearly, if markets areefficient, hence conforming to the paradigm of purecompetition, there is little reason for a security mar-ket policy: the market works to correct imbalancesand to efficiently disseminate information.

However, if inefficiencies do abound, reflectingbarriers to entry in transactions or inefficient collec-

tion, processing, and dissemination of information,there might be a role for public policy. For example,the existence of the Securities and Exchange Commis-sion, whose primary function is to ensure equalaccess to relevant information, would be questionablein a world with an efficient market for information; inthat case market prices would more accurately reflectall relevant information. As another example, in aworld of efficient markets, sharp changes in prices,such as the October 1987 break, would reflect dra-matic changes in fundamentals and should not elicitpublic policy responses. But in an inefficient market,in which investor sentiment clouds the influence offundamentals, policies designed to mitigate pricechanges (daily price limits, market closings undercertain conditions) might be appropriate.

The objective of this paper is not an examinationof sharp and maintained price breaks such as Octoberof 1987. But the noise trading model does suggest onereason why that break appeared to be lasting in itseffect on stock prices. To the extent that the pricebreak was not associated with changes in fundamen-tals (and it is widely agreed that it was not), it couldhave adversely affected investor sentiment, inducingprices to remain below fundamental values for pro-longed periods. A useful analogy is the prevalence ofdiscounts on closed-end mutual funds--these dis-counts appear to exist because investors recognizethat they might get larger.

The purpose of the third paper in this trilogy willbe to pursue the question of dramatic breaks in stockprices, and to investigate the wisdom and efficacy ofpolicies designed to address potential problems ofshort-run stock price volatility that arise from stockmarket inefficiency.


Appendix 1: Time Series Analysis of Daily StockPrices in the 1980s

In this appendix we report some results of tests forrandom walks and for specific anomalies using daily stockprice data for the 1980s. Figure A-1 shows the data: thedaily closing price of the S&P500 for each of the 2,713trading days from January 2, 1980 to September 21, 1990.The chart reports the logarithm of the price index, ratherthan the index itself, for two reasons. First, analysis of stockprices is usually done on the logarithm of the price becausedesirable statistical properties are the result; in particular,the changes in log stock prices are close to normallydistributed, a property that allows a broad range of statis-tical tools to be brought to bear. Second, a graph of the logprice has the property that the slope of the line measuresthe percentage rate of change of the price; for example, thechart shows that the rate of increase in stock prices wasparticularly high from mid-1982 through mid-1983, thenslowed somewhat until the October 19, 1987 break, andafter that break, the rate of increase was lower than it hadbeen in the earlier bull market periods.

Are Daily Stock Prices a Random Walk?

An attempt to fit this model to the daily S&P500 closingprices for 1980-90 failed to support the random walkhypothesis: while the intercept and slope coefficients wereconsistent with the EMH, the residuals were not whitenoise, but showed significant autocorrelation. Further ex-perimentation using time series methods led us to conclude

Figure A-1

Logarithm of Daily S&P 500Closing Price

Log Price6.1

5.9

5.7

5.5

5.3

5.1

’1.7 ~

4.8

1980 1982 1984 1986 1988 1990

that the log of the daily closing value of the S&P500corresponded to an Integrated Moving Average (IMA)model; we found that an IMA(1,5) model, using firstdifferences and five daily moving average terms, wassufficient to eliminate autocorrelation. This equation, inwhich movements in the first difference of logP are de-scribed by a five-day moving average of white noise terms,is reported as equation 1 in Table A-1.

The moving average coefficients are statistically signif-icant, and reveal the following pattern. If there is a down-ward shock ("crash") in the price change, the followingadjustments will occur: on the following day the pricechange will be slightly more than the normal amount, afterwhich it will increase at a slightly below-normal rate forthree days, ending with a slightly above-normal increase onthe fifth day. After five days, if no other shocks haveoccurred, the abnormal behavior is over. Given the shortperiod of fluctuation, it is no surprise that longer-term data(such as monthly data) do not reveal departures from theEMH.

Thus, our data lead us to reject the random walkimplications of the EMH for such short intervals of time asone day. However, the departure from a random walk isnot of major economic significance; in short, it is not"bankable." First, the low coefficient of determination(~2 = 0.01) tells us that while the moving average terms arestatistically significant, they explain only about I percent ofthe variation in the change in log price--there is a highprobability that any potential profits from trading strategiesbased on the knowledge of the time series structure will beswamped by random variations. Second, even at its mostprofitable, the optimal trading strategy might not cover itscosts: the optimal strategy would be to buy (sell) the dayafter a major fall (rise) in prices, then sell (buy back) theportfolio after it is held for four days. If we calculate theprofits from doing this after a major price decline (definedas a decline greater than all but only 10 percent of pricedeclines) the profits are only about 1.4 percent of the initialcost; this would not cover retail transactions costs, thoughit could cover institutional transaction costs.

Is There a Weekend Effect in the 1980s?

In order to examine the Weekend Effect during the1980s we have re-estimated equation 1 of Table A-1 byadding two dummy variables: WKEND, which has a valueof 1 if the trading day is a Monday, zero otherwise, andHOLIDAY, which has a value of I if the current trading daywas preceded by a one-day holiday. The five-day movingaverage behavior reported in Table A-l, equation 1, isreproduced in equation 2. In addition, the WKEND dummyvariable has a coefficient that is both negative and statisti-cally significant. Thus, the daily data for the 1980s docontain a significant Weekend Effect. The HOLIDAYdummy is not statistically significant, indicating that one-day closings are not associated with systematic differencesin price behavior.17

Source: Data Resources, Inc. The Januamd Effect in the 1980s

Equations 3 and 4 of Table A-1 incorporate dummyvariables for the January Effect. Equation 3 includes JAN-


Table A-1Tests of Random Walk HypothesisIMA(1.5) Model, Dependent Variable = & IogP

Independent EquationVariable 1 2 3 4

Constant +.0004 +.0007 +.0007 +.0007(1.85) (3.12) (2.80) (3.07)

MA(1) +.0526 +.0546 +.0546 +.0546(2.74) (2.84) (2.84) (2.84)

MA(2) -.0370 -.0369 -.0369 -.0369(1.92) (1.92) (1.92) (1.92)

MR(3) -.0200 -.0193 -.0192 -.0193(1.04) (1.00) (.99) (1.00)

MR(4) -.0548 -.0544 -.0541 -.0544(2.85) (2.83) (2.81) (2.83)

MA(5) +.0575 +.0518 +.0518 +.0517(2.85) (2.69) (2.69) (2.69)

WKEND n.a. -.0017 -.0017 -.0017(3.19) (3.19) (3.19)

HOLIDAY n.a. +.0007 +.0096 +.0006(.23) (.19) (.21)

JANUARY n.a. n.a. +.0006 n.a.(.82)

EARLY JAN n.a. n.a. n.a. +.0007(.19)

~2 .0089 .0120 .0119 .0117SEE .0109 .0109 .0108 .0108Q(156) 169.7670 169.3580 169.1590 169.2850

p .2179 .2248 .2281 .2260

Note: The sample period is January 2, 1980~Seplember 21, 1990Numbers in parentheses are absolute values ol t-statistics, p is Iheprobability level of lhe Q-statistic (156dl).n.a. = not applicable

UARY, defined as 1 for trading days in January and zerootherwise, and equation 4 includes EARLYJAN, a dummyvariable defined as I in the first five trading days of Januaryand zero otherwise. In neither case is the evidence consis-tent with a January Effect in the 1980s; neither coefficient isstatistically significant. We cannot, however, conclude thatthe January Effect has disappeared; it is possible that it stillexists for small firms, but that it does not exist for the largefirms that are in the S&P500.

Our regressions do, however, allow us to concludethat the January Effect can no longer be used as a profitablestrategy for a broad range of large firms listed on the majorexchanges. The fact that it does not appear for large firms,which are the firms of primary interest to institutionalinvestors, is consistent with the Efficient Markets Hypoth-esis: arbitrage by institutional investors prevents late-De-cember selling pressure from affecting the share prices of

large-capitalization firms, while small-cap stocks that haveperformed poorly do not have the attention of institutionalinvestors and are oversold at year end.

Appendix 2: A Model of Noise Trading

The De Long-Shleifer-Summers-Waldmann model ofnoise trading results in a market price of common stockdescribed by the following equation:

(A2.1) P = 1 +/xp*/r +/x(p - p*)/(1 + r) - 23,/x2o2/[r(1 + r)2]

where /x is the proportion of investors who are noisetraders, r is the real interest rate on riskless securities, and3’ is the degree of absolute risk aversion, assumed to be thesame for all investors. It is assumed that the degree of pricemisperception by noise traders, denoted b~y f~, is normallydistributed ~vith mean /)* and variance ~rL. The price de-scribed by (A2.1) is the market price relative to the funda-mental value, so P = 1.0 means that the price is equal tofundamental value.

The last three terms reflect the influence of noisetrading; if/x = 0, there are no noise traders and Pt = 1, thatis, stock prices are determined by fundamentals alone. Thesecond term reflects the influence of the mean amount ofmispricing. If p* > 0 (p* < 0), noise traders are normallybullish (bearish), and to the extent that they are importantin the market (measured by/x), this raises (reduces) stockprices; sophisticated traders will take opposing positions(such as reducing their stock holdings when prices areabove fundamentals). The third term reflects the effect ofunusual bullishness or bearishness of noise traders, mea-sured by the random variable (f~ - p*); once again, the effectthis has on stock prices depends on the relative numbers ofnoise ffaders.

Finally, the last term reflects the effect of uncertaintyabout the future degree of price misperception. The neteffect of this is negative because the uncertainty imposed bynoise traders will discourage investment by both sophisti-cated traders and noise traders, both of whom realize thatprice reversals can occur. This effect will be smaller, the lessrisk averse investors are (if 3’ = 0 nobody cares about riskso it will not affect prices) and the less important are noisetraders (if/x = 0 noise traders do not exist so they cannotaffect prices). Given the values of 3" and/x, the stock pricewill be negatively related to the size of the variance of noisetrader misperceptions (~2).

If we measure the volatility of stock prices by thestandard deviation conditional on information available inthe previous period, for example, by s = Et_~(P~ - Et_~Pt)2,we see that:

(A2.2) s = ~x~r/(1 + r)

which says that volatility will be greater the larger therepresentation by noise traders, the larger the variability intheir price misperceptions, and the lower the rate of inter-est.

If this were the end of the story, our noise tradingmodel would predict that stock prices will deviate ran-domly around a normal value that is determined by thefollowing equation:


(A2.3) Pt = 1 + ixp*/r - 27/x2(z2/[r(1 + r)2].

If p* is zero, stocks will be chronically undervalued becauseall traders recognize that noise exists and that they mighthave to sell their assets at prices below their fundamentalvalues. This "noise risk" cannot be eliminated by diversi-fication, and results in a market price less than the funda-mental value. Any deviations from this constant price levelwould be purely random, arising from temporary devia-tions of p from p*. In short, the model would simulatevolatility but not replicate the patterns of bull and bearmarkets we observe in the real world.

In order to introduce the possibility of realistic resultsfrom a noise trading model we need to add the notion of"fads and fashions," in which investor interactions createwaves of investor sentiment. One way of introducing thepossibility of contagious behavior is to assume that opin-ions follow a random walk shown by the autoregressiveprocess:

where ~t is white noise with mean E(~t) = 0 and constant2 In this simple case, investors standing at avariance ~.

moment of time will forecast a constant value of p* becausezero is their optimal forecast of ~ for every period. But theactual value of p* will follow a path determined by equation(A2.4). Because stock prices are very sensitive to p*, this canresult in realistic stock market cycles.

In order to complete the simulation model, we assume/x = 0.05, that is, noise traders represent 5 percent of themarket. We also assume ~r~ = 0.005, so that opinion (p*)follows a simple random walk with the standard deviationof 0.005 in the steps.1~ Finally, we assume ~r = 0.01, so thatin 84 per cent of the trials (months) the random componentin the degree of mispricing (for example, p - p*) will bewithin +1 percent of the fundamental value of the stock.

Any movements in the simulated stock price must bedue to either e or (p - p*). We complete the ~ simulationmodel by assuming that both ~ and (p - p*) are normallydistributed with zero means and the standard deviationsassigned above. Using a random number generator to pickvalues of ~ and (p - p*) for each of our "months," we cantrack the stock price. This experiment ~vas done 600 timesto simulate the stock price over a period of 50 years (600months).

The results of one simulation are shown in Figure 6.

38 March/April 1991 New England Economic Reviezo

1 One popular joke intended to discredit the EMH was abouttwo economists walking along a street in Chicago (the bastion ofthe EMH): one observes a $20 bill lying on the sidewalk and beginsto bend down to get it, but the other tells him not to bother, for ifthe bill were really there it would already have been picked up!

2 Technical analysis is the use of historical information onstock prices to forecast future stock prices. Perhaps the bestexample of technical analysis is the Dow Theory, which identifiesspecific patterns in stock prices and uses them to form forecasts.

3 For example, a simple dividend valuation model with aconstant growth rate for dividends (g) and a constant real discountrate (r) implies E(P~ = D/(r - g); knowing current dividends pershare (D) and using estimates of r and g allows one to test the EMHby regressing Pt on a variable defined as D/(r - g) and any othervariable in the information set f~. Adjustment of the regressionmethod for measurement error will, of course, be necessary.

4 For expositional convenience we are ignoring the paymentof dividends. If dividends are included, the valuation equationmust be modified to read

Thus, dividends are assumed to be reinvested.s The companies are Adams Express (NYSE), Baker Fentress

(OTC), General American Investors (NYSE), Lehman Corporation(NYSE), Source Capital (NYSE), Tri-Continental (NYSE), and Ni-agara Shares (NYSE).

6 Of course, an open-end fund with unrealized appreciationexposes the investor to the same tax liability. But the liability cannotaffect the open-end fund’s share price because it is always equal to thenet asset value; the only effect is to induce investors with high taxrates to prefer closed-end funds over open-end funds.

7 If the Value Line ranking system is of no use, the probabiloity that Rank I stocks will outperform Rank 3 stocks is 0.50. Underthis null hypothesis the probability of 5 or fewer "failures" in 25years is only 0.00046. This provides a strong reason to believe thatthe ranking system does have merit.

8 The keen-eyed reader will observe that one can construct anexample from Table 1 that shows the fundamental value varyingless than the market price. For example, if the same state occurredin each year, the fundamental price would not vary at all! This is anartifact of the example: with a large number of possible states anda large number of time periods, the optimal forecast must vary byno more than the fundamental price.

9 From (5) we can see that Et(Pt+l) = (1 + OPt - E~(Dt+I). Ifno dividends are expected, the price will be expected to grow at therate r. If dividends are paid, the rate of increase in price will be less

than r, but the rational bubble must grow at the rate r.10 Thus, the imposition of the transversality condition

limt~ [(1 + r)-tPd = 0 is sufficient to crush a nascent bubble.n Hence a necessary condition for existence of a speculative

bubble is a finite time horizon. This is not, however, sufficient.Tirole (1982) has shown that even with a finite time horizon, aspeculative bubble cannot exist if expectations are rational, that is,if investors’ forecasts are optimal. Hence, bubbles require bothfinite horizons and non-optimal forecasting. Stated differently,rational bubbles require inefficient markets.

12 Attempts to identify time-variation in the risk premiumwere not successful, so the value of 0 was set at the sample averagefor (Rt - r0; this was 0.0070 per month or 0.0838 per year.

13 For the 755 months in the period 1926:2 to 1988:12, thechange in log bubble over the next 12 months had a mean of-0.0208 and a standard deviation of 0.2226. Hence, a "one-standard-deviation crash" is defined as a 12-month change in thelogarithm of the bubble by an amount of -0.2434 or less. Atwo-standard-deviation crash is a change of -0.4660 or less.

14 The logit model assumes that the probability of an event (rr)is a logistic function of the following form:~rt = 1/{1 + exp[-(~ + /3xt)]}.

is The logic of the Poterba-Summers test is simple. Supposethat the one-period rate of return on stocks is approximated bythe change in the logarithm of the price. Suppose further that--as many financial studies assume~the change in the logarithmcan be represented by a constant plus a random error term, solog Pt+l - logPt =/z + ~t+l. Then the average return over N periodsis approximately logPt+N - logPt = N/z + (~+1 + ~t+2 ¯ ¯ ¯ + ~t+N).

If the ElVlH is correct, the gs are identically and independentlydistributed. Denoting the variance of ~ as o"2, the variance of theN period return is VAR(logP~+N - logPt) = No-a: the variance ofreturns is proportional to the period over wl-dch the returns areexperienced. If, as Summers and Poterba conclude, the varianceincreases less than in proportion to the period, the return on stocks ismean-reverting.

16 The term "winner’s curse" is used here in a different waythan it is used in discussing the effects of mergers and acquisitions.In that context, the winner’s curse is the tendency of those whooutbid others to pay too high a price for the acquired firm.

17 We also found that three-day weekends were no differentfrom two-day weekends; the extra day makes no difference, just asa one-day holiday makes no difference.18 The other parameter in the model (T) really plays no

important role in the simulation. We set it at T = 0.10.


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if the application of rational

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