Stock market prices.

Lo, Andrew W.

Stock Market Prices

Since the catastrophic stock market crash of October 1929 and the resulting Great Depression, economists and policymakers have been extremely interested in the behavior of financial asset prices. The Securities Exchange Act of 1934 and the creation of the Securities and Exchange Commission were direct consequences of the turbulent markets of the 1920s, and much subsequent regulatory legislation has been designed to reduce the wild price swings generally associated with "speculative" investors. In the wake of the more recent "October Massacre," understanding how and why equity prices fluctuate has never been more important. This summary describes some of what my coauthors and I have learned recently about the random nature of stock price movements.

The Random Walk

One of the earliest characterizations of rationally determined stock prices is the random walk model, which says that future price changes cannot be predicted from past price changes. First developed from rudimentary economic considerations of "fair games," the random walk has received broad support from the many early empirical studies confirming the unpredictability of stock returns, generally using daily or monthly returns of individual securities.

However, one of my papers with A. Craig McKinlay shows that the random walk model does not fit aggregate weekly returns during 1962-87.(1) In fact, the weekly returns of a portfolio containing one share of each security traded on the New York and American Stock Exchanges (called an "equally weighted" portfolio) exhibit an autocorrelation of 30 percent, implying that about 10 percent of the variability of next week's return is explained by this week's return! An equally weighted portfolio containing only the stocks of "smaller" companies, companies with relative low market values, has an autocorrelation of 42 percent and is as high as 49 percent during 1975-87.

This fact surprises many economists because a violation of the random walk hypothesis necessarily implies that price changes can be forecast to some degree. The existence of these weekly correlations suggests that there are unexploited profit opportunities. Two other facts add to this puzzle: 1) weekly portfolio returns are strongly positively autocorrelated, but the returns to individual securities generally are not; in fact, the average autocorrelation across securities is negative (but insignificant); and 2) the predictability of returns is quite sensitive to the holding period: serial dependence is strong and positive for daily and weekly returns but is virtually zero for returns over a month, a quarter, or a year.

Lead-Lag Effects, Contrarian Profits, and Size

Since the autocorrelation of portfolio returns is the sum of the individual stocks' autocorrelations and their cross-autocorrelations (for example, the correlation of this week's return on stock A with next week's return on stock B), we look to the cross-autocorrelations to explain the fact that portfolio returns are forecastable and individual stock returns are not. MacKinlay and I find that these cross-autocorrelations are strongly positive and exhibit a distinct lead-lag pattern: the returns on "larger" stocks--stocks with larger market values--almost always lead those of "smaller" stocks.(2) That is, this week's returns of large stocks can forecast next week's returns of smaller stocks, but not vice-versa. Since individual stocks are weakly negatively correlated on average, the positive correlation of weekly portfolio returns is completely caused by these lead-lag effects.

Such effects are also an important source of the apparent profitability of contrarian investment strategies, strategies that buy "losers" and sell "winners." For example, suppose the market consists of only two stocks, A and B, with returns that are uncorrelated individually but positively cross-correlated. If A's return is higher than the market this week, the contrarian will sell it and buy B. But if A and B are positively cross-autocorrelated, a higher return for A today implies a higher return for B tomorrow (on average). Thus the contrarian investor profits (on average) from buying B. Although A's past returns cannot be used to forecast its future returns, they can be used to forecast B's future returns, and contrarian trading strategies inadvertently benefit from this.

Our results show that at least half of the expected profits from one particular contrarian strategy are the result of lead-lag effects. Economic models attempting to explain the 30 percent autocorrelation in portfolio returns now must do so in a very specific way: they must provide a mechanism by which the returns of smaller companies lag those of larger ones.

Other aspects of the behavior of stock returns also seem to be related to the company's market value or "size." For example, small stocks are largely responsible for the "January effect," an empirical regularity in which equity returns over the past 25 years have been consistently higher than usual between the last few trading days of December and the first few of January. Also, the returns of small stocks are generally more volatile than those of large stocks. Moreover, for the contrarian trading strategy that MacKinlay and I examine, small stocks tend to yield higher expected profits. These empirical observations probably signal substantial differences between the ecomonic structure of small and large corporations. But how these differences are manifested in the behavior of equity returns cannot be reliably determined through data analysis alone.

In a related context, MacKinlay and I have shown that when empirical facts motivate the search for additional empirical facts in the same data, this can lead to anomalous findings that are more apparent than real.(3) Moreover, the more we scrutinize a collection of data, the more likely we are to find interesting (spurious) patterns. Since stock market prices are perhaps the most studied economic quantities to date, financial economists must be particularly vigilant. The importance of size would be much more convincing if it were based on a model of economic equilibrium in which the relationship between size and the behavior of asset returns is well articulated. I hope to provide such a model in the near future.

Nonsynchronous Trading

Perhaps the simplest explanation of the predictability in returns is a kind of measurement error to which financial data are particularly susceptible, often called the "infrequent trading" or "nonsynchronous trading" problem. This arises when prices recorded at different times are treated as if they were sampled simultaneously. For example, the daily prices of financial securities quoted in the Wall Street Journal are usually "closing" prices, prices at which the last transaction in each of those securities occurred on the previous business day. If the last transaction in stock A occurs at 2 p.m. and the last transaction in stock B occurs at 4 p.m., then included in B's closing price is information not available when A's closing price was set.

This can create spurious predictability in asset returns since economywide shocks will be reflected first in the prices of the most frequently traded securities, with less frequently traded stocks responding with the lag. Even when there is no statistical relationship between stocks A and B, their measured returns will seem cross-autocorrelated simply because we have mistakenly assumed that they are measured simultaneously.

MacKinlay and I have constructed an explicit model of this phenomenon that is capable of generating size-determined lead-lag patterns (since small stocks trade less frequently than large stocks do), and positive portfolio correlation in weekly returns.(4) Using this frame-work, we can estimate the degree of nonsynchronous trading implicit in the observed means, variances, and autocorrelations of the data. With weekly returns, the infrequent trading necessary to produce an autocorrelation of 30 percent is empirically implausible, requiring securities to go for several days without trading on average. Therefore, infrequent trading may be responsible for a portion of the observed autocorrelation, but it cannot explain all of it.

Long-Term Memory

In contrast to the positive autocorrelation MacKinlay and I find in short-horizon stock returns, others have reported negative serial correlation in longer-horizon (three-to-five-year) returns for the longer 1926-87 sample period. This may be a sympton of "long-range dependence" or long memory in asset returns, a kind of dependence often found in natural phenomena. Unlike conventional models of economic time series in which shocks of the remote past have little influence on the distant future, the serial dependence of long-memory time series decays far more slowly. This has profound economic and econometric implications: long-range dependence can change the optimal portfolio mix drastically for any individual and also affects the statistical procedures that we use to learn about asset returns.

To test for long memory in stock returns, I develop a statistic based on Benoit Mandelbrot's "rescaled range," which is robust to the short-horizon serial correlation discussed above.(5) In contrast to an earlier study that claims to have uncovered long memory in the stock market using Mandelbrot's procedure, I show that there is no evidence of long-range dependence in daily, weekly, monthly, or annual returns over various sample periods once short-term correlations are properly taken into account. Joseph G. Haubrich and I have also looked for long-term memory in aggregate output, with much the same results.(6) Although we show that long-range dependence can arise naturally in an equilibrium model of real business cycles, the current empirical evidence is not supportive.

Directions for Future Research

Perhaps the most pressing fact in need of a theory is that the predictability of stock returns is strongest for weekly holding periods. Several equilibrium models of time-varying expected rates of return already have been proposed as explanations of long-horizon return predictability, but they require unrealistic parameter values to capture weekly variations in stock price changes. Moreover, none of the models is yet able to generate the kind of lead-lag structure exhibited by the data. This suggests the need for innovation in asset pricing paradigms, perhaps by a more explicit modeling of learning behavior, the transmission of information, and the microstructure of financial markets. Although traditionally considered inappropriate for academic scrutiny, subjects such as technical analysis and market psychology may play an important role in future models of rational economic equilibrium in asset markets.

(1)A. W. Lo and A. C. MacKinlay, "Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test," NBER Reprint No. 1180, May 1989. (2)A. W. Lo and A. C. MacKinlay, "When Are Contrarian Profits Due to Stock Market Overreaction?" NBER Working Paper No. 2977, May 1989. (3)A. W. Lo and A. C. MacKinlay, "Data-Snooping Biases in Tests of Financial Asset Pricing Models," NBER Working Paper No. 3001, June 1989. (4)A. W. Lo and A. C. MacKinlay, "An Econometric Analysis of Non-synchronous Trading," NBER Working Paper No. 2960, May 1989. (5)A. W. Lo, "Long-Term Memory in Stock Market Prices," NBER Working Paper No. 2984, May 1989. (6)J. G. Haubrich and A. W. Lo, "The Sources and Nature of Long-Term Memory in the Business Cycle," NBER Working Paper No. 2951, April 1989.