Write a report of 1,000 words discussing the title below. “What is meant by the term ‘Market Efficiency’ when used in the context of financial markets? Explain what

Write a report of 1,000 words discussing the title below. “What is meant by the term ‘Market Efficiency’ when used in the context of financial markets? Explain what

his term refers to and why this concept is important for investors.” Guidelines: A report should comprise facts, a discussion of the facts supported by evidence/examples or data, and finishing with a concluding paragraph. Supplementary charts +/or other data may be included. Possible elements of the report might include; *Summary and overview of what efficiency in financial markets – are there degrees of efficiency? * Ask the question – How do we know whether this concept actually exists or not? What tests for Market Efficiency exist, discuss – the articles provide examples of tests What examples of stock price movements that might demonstrate market efficiency can be provided; Source: google or yahoofinance.com for charts of share price movements which might illustrate market efficiency or inefficiency * Explain why is it important? Explain the implications for investors of financial market efficiency vs. financial market inefficiency.

Student Economic Review, Vol. 21, 2007 167 WHY MIGHT SHARE PRICES FOLLOW A RANDOM WALK? SAMUEL DUPERNEX Senior Sophister The Efficient Markets Hypothesis no longer holds the impervious position in finance it once did, Consequently the assumption that share prices follow a random walk is now uncertain. Samuel Dupernex defines and discusses the random walk model, outlining its relationship to the efficiency of markets. Empirical evidence is used to investigate the arguments for and against the model. Introduction As recent as 30 years ago, the efficient market hypothesis (EMH) was considered a central proposition in finance. By the mid-1970s there was such strong theoretical and empirical evidence supporting the EMH that it seemed untouchable. However, recently there has been an emergence of counter arguments refuting the EMH. The EMH is the underpinning of the theory that share prices could follow a random walk. Currently there is no real answer to whether stock prices follow a random walk, although there is increasing evidence they do not. In this paper a random walk will be defined and some of the literature on the topic will be discussed, including how the random walk model is associated with the idea of market efficiency. Then the arguments for and against the random walk model will be presented. It will be shown that, in some cases, there is empirical evidence on the same issue that could be used to support or challenge the theory. Random Walks and the Efficient Market Hypothesis As mentioned above, the idea of stock prices following a random walk is connected to that of the EMH. The premise is that investors react instantaneously to any informational advantages they have thereby eliminating profit opportunities. Thus, prices always fully reflect the WHY MIGHT SHARE PRICES FOLLOW A RANDOM WALK? 168 information available and no profit can be made from information based trading (Lo and MacKinley, 1999). This leads to a random walk where the more efficient the market, the more random the sequence of price changes. However, it should be noted that the EMH and random walks do not amount to the same thing. A random walk of stock prices does not imply that the stock market is efficient with rational investors. A random walk is defined by the fact that price changes are independent of each other (Brealey et al, 2005). For a more technical definition, Cuthbertson and Nitzsche (2004) define a random walk with a drift (δ) as an individual stochastic series Xt that behaves as: Xt = ∂+ Xt−1 + εt +1 ~ ,0( ) 2 1 σ ε ε iid t+ The drift is a simple idea. It is merely a weighted average of the probabilities of each price the stock price could possibly move to in the next period. For example, if we had €100 and this moved either 3.0% up or 2.5% down with P=0.5 for each case, then the drift would be 0.25%, calculated by (Brealey et al, 2005): 0.5(0.03) + 0.5(-0.025) = 0.0025 = 0.25% However, even though it is useful, the model is quite restrictive as it assumes that there is no probabilistic independence between consecutive price increments. Due to this, a more flexible model called the ‘martingale’ was devised. This improved on the random walk model as it can “be generated within a reasonably broad class of optimizing models” (LeRoy, 1989:1588). A martingale is a stochastic variable Xt which has the property that given the information set Ωt , there is no way an investor can use Ωt to profit beyond the level which is consistent with the risk inherent in the security (Elton et al, 2002). The martingale is superior to the random walk because stock prices are known to go through periods of high and low turbulence. This behaviour could be represented by a model “in which successive conditional variances of stock prices (but not their successive levels) are positively autocorrelated” (LeRoy, 1989:1590). This could be done with a martingale, but not with a random walk.1 Fama (1970) stated that there are three versions of efficient markets: 1 Samuelson (1965) proved this result. SAMUEL DUPERNEX 169 1. Weak-form: Ω comprises of historical prices only, meaning that it is not possible to earn superior risk adjusted profits which are based on past prices (Shleifer, 2000). This leads to the random walk hypothesis. 2. Semi-strong form: Ω includes historical prices and all publicly available information as well. 3. Strong form: “Ω is broadened still further to include even insider information” (LeRoy, 1989:1592). Each of these forms has been tested and some of the results of these studies will be discussed later in the paper. As the strong form is considered somewhat extreme, analysis focuses on the weak and semi-strong forms. Arguments against the Random Walk Model There has been myriad of empirical research done into whether there is predictability in stock prices. Below, a summary of the main theories will be presented. Short-Run and Long-Run Serial Correlations and Mean Reversion Lo and MacKinley (1999) suggest that stock price short-run serial correlations are not zero. They also propose that in the short-run stock prices can gain momentum due to investors ‘jumping on the bandwagon’ as they see several consecutive periods of same direction price movement with a particular stock. Shiller (2000) believes it was this effect that led to the irrational exuberance of the dot-com boom. However, in the long-run this does not continue and in fact we see evidence of negative autocorrelation. This has been dubbed ‘mean reversion’ and although some studies (e.g. Fama and French (1988)) found evidence of it, its existence is controversial as evidence has not been found in all research. Chaudhuri and Wu (2003) used a Zivot-Andrews sequential test model to increase test power, thus decreasing the likelihood that previous results were a result of data-mining and obtained better results. To date, this method has not been widely adopted. Market Over- and Under-reaction Fama (1998) argues that investors initially over or under-react to the information and the serial correlation explained above is due to them fully WHY MIGHT SHARE PRICES FOLLOW A RANDOM WALK? 170 reacting to the information over time. The phenomenon has also been attributed to the ‘bandwagon effect’. Hirshleifer discusses ‘conservatism’ and argues that “under appropriate circumstances individuals do not change their beliefs as much as would a rational Bayesian in the face of new evidence” (Hirschleifer, 2001:1533). He asserts that this could lead to over-reaction or underreaction. Seasonal Trends Here, evidence is found of statistically significant differences in stock returns during particular months or days of the week. The ‘January effect’ is the most researched, but Bouman and Jacobsen (2002) also find evidence of lower market returns in the months between May and October compared with the rest of the year. One problem with finding patterns in stock market movements is that once found, they soon disappear. This seems to have been the case with the January effect, as traders quickly eliminated any profitable opportunities present because of the effect. Size Fama and French (1993) found evidence of correlation between the size of a firm and its return. It appears that smaller, perhaps more liquid firms, garner a greater return than larger firms. Figure 1 shows the results: Figure 1. Average monthly returns for portfolios formed on the basis of size (1963-1990) Source: Malkiel, 2003 SAMUEL DUPERNEX 171 However, it should be noted that the results may not accurately reflect reality, as this size trend has not been seen from the mid 1980’s onwards. In addition to this, the beta measure in the CAPM2 may be incorrect, as Fama and French (1993) point out. The market line was in fact flatter than the beta of the CAPM would have you believe. An illustration of this can be seen in Figure 2 below, where the market line should follow a fit of points 1-10. Figure 2. Average Premium Risk (1993-2002), % Source: Brealey, R. A., Myers, S. C. and Allen (2005:338) Dividend Yields Some research has been done on the ability of initial dividend yields to forecast future returns. As can be seen from the Figure.3, generally a higher rate of return is seen when investors purchase a market basket of equities with a higher initial dividend yield. It should be noted that this trend does not work dependably with individual stocks. 2 Capital Asset Pricing Model WHY MIGHT SHARE PRICES FOLLOW A RANDOM WALK? 172 Figure 3. The Future 10-Year Rates of Return When Stocks are purchased at Alternative Initial Dividend Yields (D/P) Source: Malkiel, B. G. (2003:66) However, Malkiel (2003) notes that as dividend yields are intrinsically linked with interest rates, this pattern could be due more to the general economic condition rather than just dividend yields. Also, dividends are becoming replaced by things such as share repurchase schemes, so this indicator may no longer be useful. Shiller looked at how dividend present value was related to stock prices. There seemed to be very little correlation. For example, during the bull market of the 1920s, the S&P Composite Index (in real terms) rose by 415.4%, while the dividend present value increased by only 16.4% (Shiller, 2000). The results are seen in Figure 4 below: SAMUEL DUPERNEX 173 Figure 4. Stock Price and Dividend Present Value: 1871–2000. Source: Shiller (2000:186) Value vs. Growth Firms It has been noted by many 3 that in the long-term, value (low price to earnings (P/E) and price to book-value (P/BV) ratios) firms tend to generate larger returns than growth (high P/E and P/BV ratios) firms. In addition, Fama and French (1993) found there to be good explanatory power when the size and P/BV were used concurrently. Fama and French (1995) then took this idea further and asserted that there are 3 main factors that affect a stock’s return4 : 1. The return on the market portfolio less the risk-free rate of interest. 2. The difference between the return on small and large firm stocks. 3. The difference between the return on stocks with high book-tomarket ratios and stocks with low book-to-market ratios (Brealey and Myers, 2005) These arguments are powerful and could lead people to doubt the EMH and random walks, assuming that the CAPM is correct. However, as Malkiel 3 Hirshleifer (2001), Malkiel (2003) and Fama and French (1993), among others 4 This is part of the arbitrage pricing theory, which does not assume that markets are efficient. Instead it assumes that stocks returns are linearly related to a set of factors, and the sensitivity to each factor depends on the stock in question. WHY MIGHT SHARE PRICES FOLLOW A RANDOM WALK? 174 (2003) points out, it may be that the CAPM fails to take into account all the appropriate aspects of risk. Arguments for the Random Walk Model Shleifer (2000) identified three main arguments for EMH: 1. Investors are rational and hence value securities rationally. 2. Some investors are irrational but their trades are random and cancel each other out. 3. Some investors are irrational but rational arbitrageurs eliminate their influence on prices. If all these exist, then both efficient markets and stock prices would be very unpredictable and thus would follow a random walk. Brealy and Myers (2005) employed a statistical test to assess the EMH by looking for patterns in the return in successive weeks of several stock market indices. Figure 5. Scatter diagrams showing the return in successive weeks on two stock market indices between May 1984 and May 2004 Source: Brealey, R. A., Myers, S. C. and Allen, F. (2006:338) SAMUEL DUPERNEX 175 Some of the results appear in Figure 5 and show almost no correlation in the returns. Event Studies Event studies help test the semi-strong form of the EMH. One such study examined how the release of news regarding possible takeover attempts affected abnormal returns. The results, illustrated below in Figure 6, showed that: • Share prices rose prior to announcement as information is leaked. • Share prices jump on the day of announcement. • Share prices steadied after the takeover, showing that news affects prices immediately. Figure 6. Cumulative abnormal returns of shareholders of targets of takeover attempts around the announcement date Source: Shleifer, A. (2000:8) In another study, Scholes (1972) observed how prices reacted to noninformation by seeing how share prices reacted to large share sales by large WHY MIGHT SHARE PRICES FOLLOW A RANDOM WALK? 176 investors. This study is important as it directly deals with the issue of the availability of close substitutes for individual securities5 . Scholes finds they lead to small price changes and that this could be due to negative news regarding the share sale. Thus, the results support the random walk theory. Predictability of Technical Trading Strategies Fama (1965) found evidence that there was no long-term profitability to be found in technical trading strategies. Malikiel (2003) supports this view and provides us with evidence, such as Figure 7, that more often than not traders find it difficult to perform better than the benchmark indices. When they do, their success is often not repeated in the long-run. Figure 7. Percentage of Various Actively Managed Funds Outperformed by Benchmark Index 10 Years to 12/31/01 Source: Malkiel, B. G. (2003:79) 5 This is central to the arguments of arbitrage in the EMH, as the theory states that ‘a security’s price is determined by its value relative to that of its close substitutes and not on market supply’ (Shleifer, 2000) SAMUEL DUPERNEX 177 On the other hand, why are there investors with sophisticated tools if their efforts are futile? This does seem to be the problem, as clearly rational investors would not invest if they could not ‘beat the market’. Indeed there is evidence to support this point of view. Lo, Mamaysky and Wang (2000) found that “through the use of sophisticated nonparametric statistical techniques… [analysts] may have some modest predictive power” (Malkiel, 2003:61) Mis-pricing There are many theories that assume mis-pricing. Mis-pricing does not affect our belief in the EMH or random walks so long as the profitable opportunities are small or they are the result of public information being misunderstood or misused by everyone. Conclusion As many of the results have contradictory evidence, it is very difficult to come to a conclusion. Data mining is certainly a problem, as one can manipulate data to support their findings. Also, many of the results could be due to chance. It has also been suggested by Conrad that the evidence on crosssectional predictability could be due to “missing risk factors in a multifactor model … [and conclude that the] pricing errors are persuasive evidence against linear multifactor model and therefore for other types of models, or they are evidence of data-snooping biases, significant market frictions, or market inefficiencies” (Conrad, 2000:516). However, evidence suggests that markets are to a certain extent predictable. This does not mean that there are opportunities for arbitrage though, because these would soon be exploited and then vanish. In the real world (with taxes, transaction costs etc.) you can have some predictability without there being profitable opportunities. It seems that stocks do approximately follow a random walk, but there are other factors, such as those discussed by Fama and French (1995), which appear to affect stock prices as well. Studies on random walks and the EMH are important, as they can give us some information on the relative efficiency of markets. The EMH can be used as a benchmark for measuring the efficiency of markets, and WHY MIGHT SHARE PRICES FOLLOW A RANDOM WALK? 178 from this we have at least a rough idea as to whether the stocks are likely to follow a random walk. Bibliography Brealey, R.A., Myers, S.C. and Allen, F. (2005) Corporate Finance: 8th Edition. New York: McGraw-Hill Irwin. Bouman, S. and Jacobsen, B. (2002) ‘The Halloween Indicator, “Sell in May and Go Away”: Another Puzzle’. American Economic Review 92:5:1618- 1635. Chaudhuri, K. and Wu, Y. (2003) “Random Walk versus Breaking Trend in Stock Prices: Evidence From Emerging Markets”. Journal of Banking and Finance 27:4:575-592. Conrad, J. (2000) “A Non-Random Walk Down Wall Street: Book Review”. Journal of Finance. 55:1:515-518. Cuthbertson, K. and Nitzche, D. (2004) Quantitative Financial Economics, 2 nd edition. Chichester: John Wiley and Sons. Elton, E.J., Gruber, M.J., Brown, S.J. and Goetzmann, W.N. (2002) Modern Portfolio Theory and Investment Analysis. New York: John Wiley and Sons. Fama, E.F. (1965) “The Behaviour of Stock Market Prices”. Journal of Business 38:34-106. Fama, E.F. (1970) “Efficient Capital Markets: A Review of Theory and Empirical Work”. Journal of Finance 383-417. Fama, E.F. (1998) “Market Efficiency, Long-Term Returns, and Behavioural Finance.” Journal of Financial Economics 49:3:283-306. Fama, E.F. and French, K.R. (1988) “Permanent and Temporary Components of Stock Prices”. Journal of Political Economy 96:246-273. Fama, E.F. and French, K.R. (1993) “Common Risk Factors in the Return on Stocks and Bonds”. Journal of Finance 33:3-56. SAMUEL DUPERNEX 179 Fama, E.F. and French, K.R. (1995) “Size and Book-to-Market Factors in Earnings and Returns”. Journal of Finance. 50:131-155. Hirshleifer, D. (2001) “Investor Psychology and Asset Pricing”. Journal of Finance 56:4:1533-97. LeRoy, S.F. (1989) “Efficient Capital Markets and Martingales”. Journal of Economic Literature 27:1583-1621. Lo, A.W. and MacKinley. A.C. (1999) A Non-Random Walk Down Wall Street. Princeton: Princeton University Press. Lo, A.W., Mamaysky, H. and Wang, J. (2000) “Foundations of Technical Analysis: Computational Algorithms, Statistical Inference, and Empirical Implementation”. Journal of Finance. 55:4:1705-1765. Malkiel, B.G. (2003) “The Efficient Market Hypothesis and Its Critics”. Journal of Economic Perspectives 17:1:59-82. Scholes, M. (1972) “The Market for Securities: Substitution versus Price Pressure and Effects of Information on Share Prices”. Journal of Business 45:179-211. Shiller, R.J. (2000) Irrational Exuberance. Princeton: Princeton University Press. Shleifer, A. (2000) Inefficient Markets: An Introduction to Behavioural Finance. Oxford: Oxford University Press.

Electronic copy available at: http://ssrn.com/abstract=991509 1 EFFICIENT MARKETS HYPOTHESIS Andrew W. Lo To appear in L. Blume and S. Durlauf, The New Palgrave: A Dictionary of Economics, Second Edition, 2007. New York: Palgrave McMillan. The efficient markets hypothesis (EMH) maintains that market prices fully reflect all available information. Developed independently by Paul A. Samuelson and Eugene F. Fama in the 1960s, this idea has been applied extensively to theoretical models and empirical studies of financial securities prices, generating considerable controversy as well as fundamental insights into the price-discovery process. The most enduring critique comes from psychologists and behavioural economists who argue that the EMH is based on counterfactual assumptions regarding human behaviour, that is, rationality. Recent advances in evolutionary psychology and the cognitive neurosciences may be able to reconcile the EMH with behavioural anomalies. There is an old joke, widely told among economists, about an economist strolling down the street with a companion. They come upon a $100 bill lying on the ground, and as the companion reaches down to pick it up, the economist says, ‘Don’t bother – if it were a genuine $100 bill, someone would have already picked it up’. This humorous example of economic logic gone awry is a fairly accurate rendition of the efficient markets hypothesis (EMH), one of the most hotly contested propositions in all the social sciences. It is disarmingly simple to state, has far-reaching consequences for academic theories and business practice, and yet is surprisingly resilient to empirical proof or refutation. Even after several decades of research and literally thousands of published studies, economists have not yet reached a consensus about whether markets – particularly financial markets – are, in fact, efficient. The origins of the EMH can be traced back to the work of two individuals in the 1960s: Eugene F. Fama and Paul A. Samuelson. Remarkably, they independently developed the same basic notion of market efficiency from two rather different research agendas. These differences would propel the them along two distinct trajectories leading to several other breakthroughs and milestones, all originating from their point of intersection, the EMH. Electronic copy available at: http://ssrn.com/abstract=991509 2 Like so many ideas of modern economics, the EMH was first given form by Paul Samuelson (1965), whose contribution is neatly summarized by the title of his article: ‘Proof that Properly Anticipated Prices Fluctuate Randomly’. In an informationally efficient market, price changes must be unforecastable if they are properly anticipated, that is, if they fully incorporate the information and expectations of all market participants. Having developed a series of linear-programming solutions to spatial pricing models with no uncertainty, Samuelson came upon the idea of efficient markets through his interest in temporal pricing models of storable commodities that are harvested and subject to decay. Samuelson’s abiding interest in the mechanics and kinematics of prices, with and without uncertainty, led him and his students to several fruitful research agendas including solutions for the dynamic assetallocation and consumption-savings problem, the fallacy of time diversification and logoptimal investment policies, warrant and option-pricing analysis and, ultimately, the Black and Scholes (1973) and Merton (1973) option-pricing models. In contrast to Samuelson’s path to the EMH, Fama’s (1963; 1965a; 1965b, 1970) seminal papers were based on his interest in measuring the statistical properties of stock prices, and in resolving the debate between technical analysis (the use of geometric patterns in price and volume charts to forecast future price movements of a security) and fundamental analysis (the use of accounting and economic data to determine a security’s fair value). Among the first to employ modern digital computers to conduct empirical research in finance, and the first to use the term ‘efficient markets’ (Fama, 1965b), Fama operationalized the EMH hypothesis – summarized compactly in the epigram ‘prices fully reflect all available information’ – by placing structure on various information sets available to market participants. Fama’s fascination with empirical analysis led him and his students down a very different path from Samuelson’s, yielding significant methodological and empirical contributions such as the event study, numerous econometric tests of single- and multi-factor linear asset-pricing models, and a host of empirical regularities and anomalies in stock, bond, currency and commodity markets. The EMH’s concept of informational efficiency has a Zen-like, counter-intuitive flavour to it: the more efficient the market, the more random the sequence of price changes generated by such a market, and the most efficient market of all is one in which price changes are completely random and unpredictable. This is not an accident of nature, but is in fact the direct result of many active market participants attempting to profit from their information. Driven by profit opportunities, an army of investors pounce on even the smallest informational advantages at their disposal, and in doing so they incorporate their information Electronic copy available at: http://ssrn.com/abstract=991509 3 into market prices and quickly eliminate the profit opportunities that first motivated their trades. If this occurs instantaneously, which it must in an idealized world of ‘frictionless’ markets and costless trading, then prices must always fully reflect all available information. Therefore, no profits can be garnered from information-based trading because such profits must have already been captured (recall the $100 bill on the ground). In mathematical terms, prices follow martingales. Such compelling motivation for randomness is unique among the social sciences and is reminiscent of the role that uncertainty plays in quantum mechanics. Just as Heisenberg’s uncertainty principle places a limit on what we can know about an electron’s position and momentum if quantum mechanics holds, this version of the EMH places a limit on what we can know about future price changes if the forces of economic self-interest hold. A decade after Samuelson’s (1965) and Fama’s (1965a; 1965b; 1970) landmark papers, many others extended their framework to allow for risk-averse investors, yielding a ‘neoclassical’ version of the EMH where price changes, properly weighted by aggregate marginal utilities, must be unforecastable (see, for example, LeRoy, 1973; M. Rubinstein, 1976; and Lucas, 1978). In markets where, according to Lucas (1978), all investors have ‘rational expectations’, prices do fully reflect all available information and marginal-utilityweighted prices follow martingales. The EMH has been extended in many other directions, including the incorporation of non-traded assets such as human capital, state-dependent preferences, heterogeneous investors, asymmetric information, and transactions costs. But the general thrust is the same: individual investors form expectations rationally, markets aggregate information efficiently, and equilibrium prices incorporate all available information instantaneously. The random walk hypothesis The importance of the EMH stems primarily from its sharp empirical implications many of which have been tested over the years. Much of the EMH literature before LeRoy (1973) and Lucas (1978) revolved around the random walk hypothesis (RWH) and the martingale model, two statistical descriptions of unforecastable price changes that were initially taken to be implications of the EMH. One of the first tests of the RWH was developed by Cowles and Jones (1937), who compared the frequency of sequences and reversals in historical stock returns, where the former are pairs of consecutive returns with the same sign, and the latter are pairs of consecutive returns with opposite signs. Cootner (1962; 1964), Fama (1963; 1965a), Fama and Blume (1966), and Osborne (1959) perform related tests of the RWH and, 4 with the exception of Cowles and Jones (who subsequently acknowledged an error in their analysis – Cowles, 1960), all of these articles indicate support for the RWH using historical stock price data. More recently, Lo and MacKinlay (1988) exploit the fact that return variances scale linearly under the RWH – the variance of a two-week return is twice the variance of a oneweek return if the RWH holds – and construct a variance ratio test which rejects the RWH for weekly US stock returns indexes from 1962 to 1985. In particular, they find that variances grow faster than linearly as the holding period increases, implying positive serial correlation in weekly returns. Oddly enough, Lo and MacKinlay also show that individual stocks generally do satisfy the RWH, a fact that we shall return to below. French and Roll (1986) document a related phenomenon: stock return variances over weekends and exchange holidays are considerably lower than return variances over the same number of days when markets are open. This difference suggests that the very act of trading creates volatility, which may well be a symptom of Black’s (1986) noise traders. For holding periods much longer than one week – fcor example, three to five years – Fama and French (1988) and Poterba and Summers (1988) find negative serial correlation in US stock returns indexes using data from 1926 to 1986. Although their estimates of serial correlation coefficients seem large in magnitude, there is insufficient data to reject the RWH at the usual levels of significance. Moreover, a number of statistical artifacts documented by Kim, Nelson and Startz (1991) and Richardson (1993) cast serious doubt on the reliability of these longer-horizon inferences. Finally, Lo (1991) considers another aspect of stock market prices long thought to have been a departure from the RWH: long-term memory. Time series with long-term memory exhibit an unusually high degree of persistence, so that observations in the remote past are non-trivially correlated with observations in the distant future, even as the time span between the two observations increases. Nature’s predilection towards long-term memory has been well-documented in the natural sciences such as hydrology, meteorology, and geophysics, and some have argued that economic time series must therefore also have this property. However, using recently developed statistical techniques, Lo (1991) constructs a test for long-term memory that is robust to short-term correlations of the sort uncovered by Lo and MacKinlay (1988), and concludes that, despite earlier evidence to the contrary, there is little support for long-term memory in stock market prices. Departures from the RWH can be fully explained by conventional models of short-term dependence. 5 Variance bounds tests Another set of empirical tests of the EMH starts with the observation that in a world without uncertainty the market price of a share of common stock must equal the present value of all future dividends, discounted at the appropriate cost of capital. In an uncertain world, one can generalize this dividend-discount model or present-value relation in the natural way: the market price equals the conditional expectation of the present value of all future dividends, discounted at the appropriate risk-adjusted cost of capital, and conditional on all available information. This generalization is explicitly developed by Grossman and Shiller (1981). LeRoy and Porter (1981) and Shiller (1981) take this as their starting point in comparing the variance of stock market prices to the variance of ex post present values of future dividends. If the market price is the conditional expectation of present values, then the difference between the two, that is, the forecast error, must be uncorrelated with the conditional expectation by construction. But this implies that the variance of the ex post present value is the sum of the variance of the market price (the conditional expectation) and the variance of the forecast error. Since volatilities are always non-negative, this variance decomposition implies that the variance of stock prices cannot exceed the variance of ex post present values. Using annual US stock market data from various sample periods, LeRoy and Porter (1981) and Shiller (1981) find that the variance bound is violated dramatically. Although LeRoy and Porter are more circumspect about the implications of such violations, Shiller concludes that stock market prices are too volatile and the EMH must be false. These two papers ignited a flurry of responses which challenged Shiller’s controversial conclusion on a number of fronts. For example, Flavin (1983), Kleidon (1986), and Marsh and Merton (1986) show that statistical inference is rather delicate for these variance bounds, and that, even if they hold in theory, for the kind of sample sizes Shiller uses and under plausible data-generating processes the sample variance bound is often violated purely due to sampling variation. These issues are well summarized in Gilles and LeRoy (1991) and Merton (1987). More importantly, on purely theoretical grounds Marsh and Merton (1986) and Michener (1982) provide two explanations for violations of variance bounds that are perfectly consistent with the EMH. Marsh and Merton (1986) show that if managers smooth dividends – a well-known empirical phenomenon documented in several studies of dividend policy – and if earnings follow a geometric random walk, then the variance bound is violated in theory, in which case the empirical violations may be interpreted as support for this version of the EMH. 6 Alternatively, Michener constructs a simple dynamic equilibrium model along the lines of Lucas (1978) in which prices do fully reflect all available information at all times but where individuals are risk averse, and this risk aversion is enough to cause the variance bound to be violated in theory as well. These findings highlight an important aspect of the EMH that had not been emphasized in earlier studies: tests of the EMH are always tests of joint hypotheses. In particular, the phrase ‘prices fully reflect all available information’ is a statement about two distinct aspects of prices: the information content and the price formation mechanism. Therefore, any test of this proposition must concern the kind of information reflected in prices, and how this information comes to be reflected in prices. Apart from issues regarding statistical inference, the empirical violation of variance bounds may be interpreted in many ways. It may be a violation of EMH, or a sign that investors are risk averse, or a symptom of dividend smoothing. To choose among these alternatives, more evidence is required. Overreaction and underreaction A common explanation for departures from the EMH is that investors do not always react in proper proportion to new information. For example, in some cases investors may overreact to performance, selling stocks that have experienced recent losses or buying stocks that have enjoyed recent gains. Such overreaction tends to push prices beyond their ‘fair’ or ‘rational’ market value, only to have rational investors take the other side of the trades and bring prices back in line eventually. An implication of this phenomenon is price reversals: what goes up must come down, and vice versa. Another implication is that contrarian investment strategies – strategies in which ‘losers’ are purchased and ‘winners’ are sold – will earn superior returns. Both of these implications were tested and confirmed using recent US stock market data. For example, using monthly returns of New York Stock Exchange (NYSE) stocks from 1926 to 1982, DeBondt and Thaler (1985) document the fact that the winners and losers in one 36-month period tend to reverse their performance over the next 36-month period. Curiously, many of these reversals occur in January (see the discussion below on the ‘January effect’). Chopra, Lakonishok and Ritter (1992) reconfirm these findings after correcting for market risk and the size effect. And Lehmann (1990) shows that a zero-net-investment strategy in which long positions in losers are financed by short positions in winners almost always yields positive returns for monthly NYSE/AMEX stock returns data from 1962 to 7 1985. However, Chan (1988) argues that the profitability of contrarian investment strategies cannot be taken as conclusive evidence against the EMH because there is typically no accounting for risk in these profitability calculations (although Chopra, Lakonishok and Ritter, 1992 do provide risk adjustments, their focus was not on specific trading strategies). By risk-adjusting the returns of a contrarian trading strategy according to the capital asset pricing model, Chan (1988) shows that the expected returns are consistent with the EMH. Moreover, Lo and MacKinlay (1990c) show that at least half of the profits reported by Lehmann (1990) are not due to overreaction but rather the result of positive crossautocorrelations between stocks. For example, suppose the returns of two stocks A and B are both serially uncorrelated but are positively cross-autocorrelated. The lack of serial correlation implies no overreaction (which is characterized by negative serial correlation), but positive cross-autocorrelations yields positive expected returns to contrarian trading strategies. The existence of several economic rationales for positive cross-autocorrelation that are consistent with EMH suggests that the profitability of contrarian trading strategies is not sufficient evidence to conclude that investors overreact. The reaction of market participants to information contained in earnings announcements also has implications for the EMH. In one of the earliest studies of the information content

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