Part II

THE FOCUS OF THE FIVE CHAPTERS in Part I has been the Random Walk Hypothesis and the forecastability of asset returns through time. In the three chapters of Part II, we shift our focus to questions of the predictability of relative returns for a given time period. On average, is the return to one stock or portfolio higher than the return to another stock or portfolio? If the answer to this age-old question is yes, can we explain the difference, perhaps through differences in risk?

These questions are central to financial economics since they bear directly on the trade-off between risk and expected return, one of the founding pillars of modern financial theory. This theory suggests that lower risk investments such as bonds or utility stocks will yield lower returns on average than riskier investments such airline or technology stocks, which accords well with common business sense: investors require a greater incentive to bear more risk, and this incentive manifests itself in higher expected returns. The issue, then, is whether the profits of successful investment strategies can be attributed to the presence of higher risks—if so, then the profits are compensation for risk-bearing capacity and nothing unusual; if not, then further investigation is warranted.

Over the past three decades, a number of studies have reported so-called anomalies, strategies that, when applied to historical data, lead to return differences that are not easily explained by risk differences. For example, one of the most enduring anomalies is the “size effect,” the apparent excess expected returns that accrue to stocks of small-capitalization companies-in excess of their risks—which was first discovered by Banz (1981). Rozeff and Kinney (1976), Keim (1983), and Roll (1983) document a related anomaly: small capitalization stocks tend to outperform large capitalization stocks by a wide margin over the turn of the calendar year. Other well-known anomalies include: the Value Line enigma (Copeland and Mayers, 1982); the profitability of return-reversal strategies (DeBondt and Thaler, 1985; Rosenberg, Reid, and Lanstein, 1985; Lehmann, 1990; Chopra, Lakon-ishok, and Ritter, 1992); the underreaction to earnings announcements or “post-earnings announcement drift” (Ball and Brown, 1968; Bernard and Thomas, 1990); the relation between price/earnings ratios and expected returns (Basu, 1977); the relation between book-value/market-value ratios and expected returns (Fama and French, 1992); the volatility of orange juice futures prices (Roll, 1984b); and calendar effects such as holiday, weekend, and turn-of-the-month seasonalities (Lakonishok and Smidt, 1988).

In light of these anomalies, it is clear that we need a risk/reward benchmark to tell us how much risk is required for a given level of expected return. In the academic jargon, we require an equilibrium asset-pricing model. The workhorse asset-pricing model that virtually all anomaly studies use to make risk adjustments is the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965). In the CAPM framework, an asset's “beta” is the relevant measure of risk—stocks with higher betas should earn higher returns on average. And in many of the recent anomaly studies, the authors argue forcefully that differences in beta cannot fully explain the magnitudes of return differences, hence the term “anomaly.”

There are many possible explanations for these anomalies, but most of them fall neatly into two general categories: risk-based alternatives and non-risk-based alternatives. Risk-based alternatives include the CAPM and multifactor generalizations such as Ross's (1976) Arbitrage Pricing Theory and Merton's (1973) Intertemporal CAPM, all developed under the assumptions of investor rationality and well-functioning capital markets. Among these alternatives, the source of deviations from the CAPM is omitted risk factors, i.e., the CAPM is an incomplete model of the risk/reward relation, hence average return differences cannot be explained solely by the standard CAPM betas.

Non-risk-based alternatives include statistical biases that might afflict the empirical methods, the existence of market frictions and institutional rigidities not captured by standard asset-pricing models, and explanations based on investor irrationality and market inefficiency.

The heated debate among academics concerning the explanation for the anomalies motivated our studies in chapters 7 and 8. Chapter 7 provides a general statistical framework to distinguish between the risk-based and non-risk-based alternatives. The idea is to consider the overall risk/reward trade-off implied by the anomaly and ask if it can be plausibly explained by risk differences. We conclude that the magnitude of the expected return differences is too large to be explained purley by differences in risk.

In Chapter 8 we turn to non-risk-based alternatives and focus specifically on statistical biases that arise from searching through the data until something “interesting” is discovered. Anomalies obtained in this way are hardly anomalous—they are merely manifestations of data-snooping biases, spurious statistical artifacts that are due to random chance. For example, a typical adult individual in the United States is unlikely to be taller than 6'5”, hence the average height in a random sample of individuals is probably less than 6'5”. However, if we construct our sample by searching for the tallest individuals in a given population—by focusing our attention only on professional basketball players, for example—then the average height in this biased sample may well exceed 6'5”. This comes as no surprise because we searched over the population for “tallness,” and the fact that the average height in this sample is greater than 6'5” does not imply that heights are increasing in the general population.

In much the same way, searching through historical data for superior investment performance may well yield superior performance, but this need not be evidence of genuine performance ability. Even in the absence of superior performance abilities, with a large enough dataset and a sufficiently diligent search process, spurious superior performance can almost always be found. While such biases are unavoidable in non-experimental disciplines such as financial economics, acknowledging this possibility and understanding the statistical properties of these biases can go a long way towards reducing their effects.

In Chapter 8, we provide a formal analysis of this phenomenon in the context of linear factor models, and show that even small amounts of data-snooping can lead to very large spurious differences in return performance. Because with hindsight one can almost always “discover” return differences that appear large, it is critical to account for this effect by modifying the standard statistical procedures appropriately and we propose one method for doing so.

Having explored the “downside” of data-snooping—finding superior performance when none exists—we consider the “upside” in Chapter 9, in which we maximize predictability directly by constructing portfolios of stocks and bonds in a very particular manner. Many financial economists and investment professionals have undertaken the search for predictability in earnest and with great vigor. But as important as it is, predictability is rarely maximized systematically in empirical investigations, even though it may dictate the course of the investigation at many critical junctures and, as a consequence, is maximized implicitly over time and over sequences of investigations.

In Chapter 9, we maximize the predictability in asset returns explicitly by constructing portfolios of assets that are the most predictable in a time-series context. Such explicit maximization can add several new insights to findings based on less formal methods. Perhaps the most obvious is that it yields an upper bound to what even the most industrious investigator will achieve in his search for predictability among portfolios (how large can data-snooping biases be?). As such, it provides an informal yardstick against which other findings may be measured. For example, the results in Part I imply that approximately 10% of the variation in the CRSP equal-weighted weekly return index from 1962 to 1985 can be explained by the previous week's returns—is this large or small? The answer will depend on whether the maximum predictability for weekly portfolio returns is 15% or 75%.

The maximization of predictability can also direct us towards more dis-aggregated sources of persistence and time-variation in asset returns, in the form of portfolio weights of the most predictable portfolio, and sensitivities of those weights to specific predictors, e.g., industrial production, dividend yield, etc. A primitive example of this kind of disaggregation is the lead/lag relation among size-sorted portfolios we documented in Chapter 5, in which the predictability of weekly stock index returns is traced to the tendency for the returns of larger capitalization stocks to lead those of smaller stocks. The more general framework of Chapter 9 includes lead/lag effects as a special case, but captures predictability explicitly as a function of time-varying economic risk premia rather than as a function of past returns only.

More importantly, maximizing predictability may be a better alternative than the current two-step procedure for exploiting predictabilities in asset returns: (1) construct a linear factor model of returns based on cross-sectional explanatory power, e.g., factor analysis, principal components decomposition, etc.; and (2) analyze the predictability of these factors. This two-step approach is motivated by the risk-based alternatives literature we alluded to earlier: the CAPM and ICAPM, the APT, and their many variants in which expected returns are linearly related to contemporaneous “systematic” risk factors.

While the two-step approach can shed considerable light on the nature of asset-return predictability—especially when the risk factors are known—it may not be as informative when the factors are unknown. For example, it is possible that the set of factors which best explain the cross-sectional variation in expected returns are relatively unpredictable through time, whereas other factors that can be used to predict expected returns are not nearly as useful contemporaneously in capturing the cross-sectional variation of expected returns. Therefore, focusing on the predictability of factors which are important contemporaneously may yield a very misleading picture of the true nature of predictability in asset returns. The approach in Chapter 9 offers an alternative in which predictability is maximized directly.

Taken together, the chapters in Part II present a more detailed analysis of the sources and nature of predictability in the stock and bond markets, providing the statistical machinery to differentiate between risk-based and non-risk-based explanations of asset-pricing anomalies, to quantify and control for the impact of data-snooping biases, and to exploit more fully the genuine predictabilities that might be present in the data.