Monograph on. Anomalies and Asset Allocation

Transcription

1 Monograph on Anomalies and Asset Allocation S.P. Kothari Sloan School of Management, E Massachusetts Institute of Technology 50 Memorial Drive, Cambridge, MA (617) and Jay Shanken William E. Simon Graduate School of Business Administration University of Rochester, Rochester, NY (716) First draft: December 2000 Current version: June 2002

2 2 Acknowledgements We thank a reviewer for helpful suggestions and Michela Verardo for excellent research assistance. We are grateful to the Research Foundation of the Institute of Chartered Financial Analysts and the Association for Investment Management and Research, the Bradley Policy Research Center at the Simon School, and the John M. Olin Foundation for financial support.

3 Chapter I Introduction The issue of how an investor should combine financial investments in an overall portfolio so as to maximize some objective is fundamental to both financial practice and to understanding the process that determines prices in a financial market. A key principle underlying modern portfolio theory is that there is no point in bearing portfolio risk unless it is compensated by a higher level of expected return. This is formalized in the concept of a mean-variance efficient portfolio, one that has as high a level of expected return as possible for the given level of risk, and incurs the minimum risk needed to achieve that expected return. Although efficiency is an appealing concept, it is far from obvious just what the composition of an efficient portfolio should be. The classic theory of risk and return called the capital asset pricing model (CAPM) provides a starting point. It implies that the valueweighted market portfolio of financial assets should be efficient. However, the accumulated empirical evidence of the past two decades or so indicates that stock indices like the S&P 500 are not (mean-variance) efficient. This literature has uncovered various firm characteristics that are significantly related to expected returns beyond what would be explained by their contributions to the risk of the market index. Whether this is due to limitations of the theory or the use of a stock market index in place of the true market portfolio, the practical implication is that one can construct portfolios that dominate the simple market index. Surprisingly, not much of the work exploring the empirical limitations of the CAPM has adopted an asset allocation perspective. Rather, the focus has been on

4 2 measuring the magnitude of risk-adjusted expected returns. 1 In this monograph, we consider the implications for asset allocation of the three most prominent CAPM anomalies : expected return effects that are negatively related to firm size (market capitalization), and positively related to firm book-to-market ratios and past-year momentum. For each anomaly, we estimate the amount that investors should tilt their portfolios away from the market index, toward the anomaly-based portfolio (or spread), in order to exploit the gains to efficiency. 2 However, the same principles of modern portfolio theory can be applied to other investment strategies that are expected to generate positive risk-adjusted returns (e.g., an earnings-based strategy, accruals strategy, or a tradingvolume-based anomaly). The portfolio improvement obtained by tilting an index toward an anomaly-based strategy depends, not only on the risk-adjusted expected returns of the three strategies, but also on residual risk, i.e., that portion of risk that is not related to variation in the market index returns. This risk measure has received little attention in the academic literature, but it is important for asset allocation. We also follow up on the performance of each strategy in the second year after portfolio formation to get a rough indication of the relevance of portfolio rebalancing. Finally, we examine asset allocation across all three anomalies and the market index. Our focus on the three most prominent anomalies should not be interpreted as suggesting that we believe these anomalies will persist in the future. Each investor will 1 Two notable exceptions are the recent work of Pastor (2000), which is closely related to our analysis, and Haugen and Baker (1996). 2 For a practical guide to implementing an active portfolio investment management strategy that is grounded in modern finance, see Waring, Whitney, Pirone, and Castille (2000).

5 3 have his or her own beliefs about the likely performance of these and other strategies. Traditional statistical tests of significance, while useful in many contexts, are not particularly well suited to investment decision-making in this sort of context. In recent years, Bayesian statistical methods have begun to achieve greater prominence in addressing asset allocation problems. 3 Part of the appeal of the Bayesian perspective is that it provides the analyst or investor a rigorous framework in which to combine somewhat qualitative judgments about future returns with the statistical evidence in historical data. Such judgments or prior beliefs might be based on an analyst s views concerning the ability of financial markets to efficiently process information and the speed with which this occurs. Related opinions about the extent to which expected returns are compensation for risk or, instead, induced by mispricing and behavioral biases are also relevant. While academic literature in this area sometimes focuses on very technical mathematical issues, the main ideas are fairly simple and very intuitive. We provide a basic introduction to Bayesian methods, which will hopefully bring the reader close to the state-of-the-art fairly quickly. These methods are then applied in our portfolio analysis of expected return anomalies. Good quantitative money managers also recognize the inevitable influence that repeated searches through the historical evidence ( data-mining ) can have on one s views and the need to adjust for this influence. They will typically be inclined to try to exploit a pattern observed in past data if there is a good story to go with it. We consider this issue as well. Outline of the monograph. Chapter II reviews the finance theory on asset allocation in the framework of the capital asset pricing model (CAPM). The chapter 3 See Kandel and Stambaugh (1996).

6 4 reviews portfolio theory, the CAPM, and the efficient market hypothesis. Chapter III reviews recent evidence challenging the efficient market hypothesis. We summarize findings suggesting economically significant profitability of trading strategies that invest in value, momentum, and small stocks. We also discuss the implications of the evidence indicative of market inefficiency for optimal asset allocation. Chapter IV presents the results of our analysis of historical data and its implications for improving asset allocation by tilting the market index toward portfolios of value, momentum, or small stocks. Chapter V presents the intuition for and an application of a Bayesian perspective on optimal asset allocation. Chapter VI examines the tilt portfolios performance over a twoyear horizon. In Chapter VII we consider the joint optimization problem in which all three anomalies are considered simultaneously. Chapter VIII summarizes the monograph and discusses its implications and directions for future work.

7 5 Chapter II Asset allocation in a CAPM world This chapter reviews the fundamental concepts of finance and their implications for asset allocation. We discuss portfolio theory, the CAPM, and the efficient market hypothesis. 4 Portfolio theory In a mean-variance setting, it is assumed that an investor s utility increases with the mean and decreases in the variance of overall portfolio returns. 5 The mean is the expected return on the portfolio while the variance is the measure of the portfolio s total risk. The efficient frontier is a graph of the set of portfolios with highest expected return for each given level of portfolio return variance. Thus, modern portfolio theory implies that, in order to maximize expected utility, an investor should choose a portfolio on the efficient frontier. In 1952, Harry Markowitz developed optimization techniques for deriving the efficient frontier of risky assets. The inputs to this derivation are estimated values of expected return, standard deviation of return, and pairwise covariances of returns for all available risky securities. An investor s portfolio selection problem is simplified with the availability of a risk-free asset. An opportunity to invest in risky and risk-free assets implies that all efficient portfolios consist of combinations of the risk-free asset and a unique tangency 4 For a detailed treatment of the concepts in this chapter, see Bodie, Kane, and Marcus (1999), chapters 6-9 and 12, or Ross, Westerfield, and Jaffe (1996), chapters 9 and More sophisticated approaches take into account potential hedging demands for securities (e.g., Merton, 1973, and Long, 1974) when the characteristics of the investment opportunity set change over time. Consideration of these issues is beyond the scope of this monograph.

8 6 portfolio of the risky assets. Investors who are relatively more risk-averse will invest a larger fraction of their assets in the risk-free asset, whereas relatively more risk-tolerant investors will opt for a greater fraction of their investment in the tangency portfolio. All of these combinations of the tangency portfolio and the risk-free asset lie on a straight line when expected return is plotted against standard deviation of return. This line, called the capital market line, is the efficient frontier and represents the best possible combinations of portfolio expected return and standard deviation. The CAPM The Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965), builds on Markowitz s portfolio theory ideas and further simplifies an investor s asset allocation decision. The CAPM is derived with an additional critical assumption that investors have homogenous expectations, which means that all market participants have identical beliefs about securities expected returns, standard deviations, and pairwise covariances. With homogenous expectations and the same investment horizon, all investors would arrive at the same efficient frontier. Therefore, they would hold combinations of the same tangency portfolio and the risk-free asset. Since total investor demand for assets must equal the supply, in equilibrium, it follows that the tangency portfolio is the value-weighted portfolio of all risky assets in the economy, called the market portfolio. The CAPM gives rise to a mathematically elegant relation between the expected rate of return on a security and its risk relative to the market portfolio. Specifically, the

9 7 theory implies that expected return is an increasing linear function of its covariance risk or beta. Beta is defined as β i = Cov(R i, R m )/Var(R m ) where Cov(R i, R m ) is the covariance of security i s return with the return on the market portfolio and Var(R m ) is the variance of the return on the market portfolio. It is identical to the (true) slope coefficient in the regression of i s returns on those of the market and thus indicates the relative sensitivity of security i to aggregate market movements. The CAPM linear risk-return relation is E(R i ) = R f + β i (E(R m ) R f ), where E(R i ) is security i s expected rate of return, R f is the risk free rate of return, and (E(R m ) R f ) is the market risk premium. In addition to its importance in portfolio analysis, beta is often used in corporate valuation and investment (i.e., capital budgeting) decisions. Efficient market hypothesis 6 The efficient market hypothesis states that security prices rapidly and accurately reflect all information that is available at a given point in time. 7 Security markets tend toward (informational) efficiency because a large number of market participants actively compete among themselves to gather and process information and trade on that information. Ideally, this process moves security prices until those prices reflect the 6 For detailed reviews of the efficient markets hypothesis and empirical literature on market efficiency, see Fama (1970, 1991) and MacKinlay (1997). 7 This notion of informational financial market efficiency should not be confused with the earlier concept of the mean-variance efficiency of a portfolio.

10 8 market participants consensus beliefs based on all of the information available to them. In an efficient market, rewards to technical analysis and fundamental analysis are nonexistent. In the short-run, prices may not completely adjust to new information due to various trading costs. More generally, markets may be inefficient because of behavioral biases in investor beliefs (excessive-optimism or pessimism, overconfidence, etc.). Deviations from efficiency can persist if, in betting that the inefficiency will be corrected over a given horizon, the arbitrageur is exposed to substantial risk that the mispricing will get worse before it gets better. 8 Portfolio theory, the CAPM, and the efficient market hypothesis jointly have remarkably simple implications for investors asset allocation decisions. Investors should hold a combination of the risky market portfolio and the risk-free asset and the investment approach should be a passive buy-and-hold strategy (i.e., invest in index funds). 9 The picture is less clear, however, if we believe that the CAPM does not hold and if we doubt market efficiency. We explore the attendant complexities in the remaining chapters of this monograph. A large body of evidence suggests that security returns exhibit significant predictable deviations from the CAPM and that the capital markets are inefficient in certain respects. As discussed in detail later on, these CAPM deviations or risk-adjusted returns are captured by a statistical parameter referred to as Jensen s alpha. Investors views about these capital market issues can have important implications for asset 8 See Shleifer and Vishny (2000). 9 The proportion of assets invested in the market portfolio is a function of the investor s risk tolerance which may change endogenously with their wealth.

11 9 allocation by affecting their confidence that positive alphas observed in the past will persist in the future. Therefore, we briefly review the relevant theory and evidence in the next chapter.

12 10 Chapter III Recent evidence challenging market efficiency and its implications for asset allocation This chapter summarizes recent evidence indicating informational inefficiency in the U.S. and international capital markets. Some of the evidence suggests capital markets take several years to reflect information about underlying economic fundamentals in stock prices. This evidence of apparent market inefficiency has implications for an investor s asset allocation decisions. Informed investors should tilt their portfolios away from the market portfolio and in a direction that exploits the inefficiency. The optimal extent of such tilting will depend on risk and other factors that are considered later. Return predictability in short-window event studies There is overwhelming evidence that security prices rapidly adjust to reflect new information reaching the market. 10 Starting with Fama, Fisher, Jensen and Roll (1969), short-window event-study research documents the market s quick response to new information. This research analyzes large samples of firms experiencing a wide range of events like stock splits, merger announcements, management changes, dividend announcements, earnings releases, etc. There is evidence that the market reacts within minutes of public announcements of firm-specific information like earnings and dividends or macroeconomic information like inflation data, or interest rates. Rapid adjustment of 10 For an excellent summary of this research, see Bodie, Kane, and Marcus (1999), chapters 12 and 13.

13 11 prices to new information is consistent with market efficiency, but efficiency also requires that this response is, in some sense, rational or unbiased. If both conditions hold, any opportunity to benefit from the news is short lived and investors only earn a normal rate of return thereafter. Longer-horizon return predictability In the past two decades, a large body of academic and practitioner research has begun to challenge market efficiency. 11 Mounting evidence suggests that revisions in beliefs in response to new information do not always reflect unbiased forecasts of future economic conditions now, and that it may take several years before prices incorporate the full impact of the news. As the market seems to correct the initial mispricing over several subsequent years, long-term abnormal expected returns may be possible for an informed investor who tries to profit from this correction. Behavioral models of investor behavior hypothesize systematic under- and overreaction to corporate news as a result of investors behavioral biases or limited capability to process information. Barberis, Shleifer, and Vishny (1998), Daniel, Hirshleifer, and Subramanyam (1998 and 2001), and Hong and Stein (1999) develop models to explain the apparent predictability of stock returns at various horizons. These models draw upon experimental evidence and theories of human judgment bias or limited informationprocessing capabilities, as developed in cognitive psychology and related fields. The representativeness bias (Kahneman and Tversky, 1982) causes people to over-weight information patterns observed in past data, which might just be random. 11 The discussion in this chapter draws on Fama (1998).

14 12 Since the patterns are not really descriptive of the true properties of the underlying process, they are not likely to persist. For example, investors might extrapolate a firm s past history of high sales growth and thus overreact to sales news (see Lakonishok, Shleifer, and Vishny, 1994, and DeBondt and Thaler, 1980 and 1985). On the other hand, investors may be slow to update their beliefs in the face of new evidence as a result of the conservatism bias (Edwards, 1968). This can contribute to investor under-reaction to news and lead to short-term momentum in stock prices (e.g., Jegadeesh, 1990, and Jegadeesh and Titman, 1993). The post-earnings announcement drift, i.e., the tendency of stock prices to drift in the direction of earnings news for three-totwelve months following an earnings announcement (e.g., Ball and Brown, 1968, Litzenberger, Joy, and Jones, 1971, and others) could also be a consequence of the conservatism bias. Stock price over- and under-reaction can also be an outcome of investor overconfidence and biased self-attribution, two more human-judgment biases. Overconfident investors place too much faith in their private information about the company s prospects and thus over-react to it. In the short run, overconfidence and attribution bias (contradictory evidence is viewed as due to chance) together result in a continuing overreaction to the initial private information that induces momentum. Overconfidence about private information also causes investors to downplay the importance of publicly disseminated information. Therefore, information releases like earnings announcements generate incomplete price adjustments in this context. Subsequent earnings outcomes eventually reveal the true implications of the earlier evidence, however, resulting in predictable price reversals over long horizons.

15 13 In summary, behavioral finance theory shows how investor judgment biases can contribute to security price over- and under-reaction to news events. The existing evidence suggests that it can take up to several years for the market to correct the initial error in its response to news events. These conclusions should be viewed with some skepticism, however. The behavioral theories have, for the most part, been created to fit the facts. Initially, overreaction was advanced as the main behavioral bias relevant to financial markets. Only after the strong evidence of momentum at shorter horizons became widely acknowledged were the more sophisticated theories developed. As just discussed, current explanations for momentum range from underreaction to short-term continuing overreaction. Thus, it is difficult to identify a particular behavioral paradigm at this point. Moreover, recent work by Lewellen and Shanken (2002) demonstrates that anomalous-looking patterns in returns can also arise in a model in which fully rational investors gradually learn about certain features of the economic environment. These patterns would be observed in the data with hindsight, but could not be exploited by investors in real time. Clearly, sorting out all these issues is a challenging task. Next, we review the evidence indicating return predictability. However, we caution the reader that, in addition to the unresolved theoretical issues, there is no consensus among academics on the interpretation of the existing empirical evidence. In particular, Fama (1998) argues that much of the evidence on abnormal long-run return performance is questionable because of methodological limitations and the more general effect of data mining.

16 14 Evidence on return predictability Research indicates long-horizon predictability of returns following a variety of corporate events and past security price performance. The corporate events include stock splits, share repurchases, extreme earnings performance announcements, bond rating changes, dividend initiations and omissions, seasoned equity offerings, initial public offerings, etc. Evidence of long-horizon predictability following corporate events and past security price performance appears in the following studies (see Fama, 1998, for a detailed discussion). Fama, Fisher, Jensen, and Roll (1969), and Ikenberry, Rankine, and Stice (1995) examine price performance following stock splits; Ibbotson (1975) and Loughran and Ritter (1995) study post-ipo price performance; Loughran and Ritter (1995) document negative abnormal returns after seasoned equity offerings; Asquith (1983) and Agrawal, Jaffe, and Mandelker (1992) estimate bidder firms price performance; dividend initiations and omissions are examined in Michaely, Thaler, and Womack (1995); performance following proxy fights is studied in Ikenberry and Lakonishok (1993); Ikenberry, Lakonishok, and Vermaelen (1995) and Mitchell and Stafford (2000) examine returns following open market share repurchases; and, Litzenberger, Joy, and Jones (1971), Foster, Olsen, and Shevlin (1984) and Bernard and Thomas (1990) study post-earnings announcement returns. The main conclusion from these studies is that, in many cases, the magnitude of abnormal returns is not only statistically highly significant, but economically large as well. However, from the standpoint of asset allocation and investment strategy, predictable returns following corporate events provide a limited opportunity to exploit the inefficiency

17 15 because typically only a few firms experience an event each month. Fortunately, research also shows that a small number of firm characteristics (e.g., firm size, value and growth attributes, i.e., the book-to-market ratio, and past price performance, i.e., momentum) are highly successful in predicting future returns. Moreover, a large number of securities share the firm characteristics that are correlated with substantial magnitudes of future returns. The availability of a large pool of securities to invest in reduces the loss of diversification entailed in trying to exploit the characteristic-based return predictability.. The firm characteristics most highly associated with future returns are the book-tomarket ratio, firm size, and past security price performance or momentum. Banz (1981) and, more recently, Fama and French (1993) provide evidence that small size (low market capitalization) firms earn positive CAPM-risk-adjusted returns. That is, small firm portfolios exhibit a positive Jensen alpha. 12 Rosenberg, Lanstein, and Reid (1985) and Fama and French (1992) show that value stocks significantly outperform growth stocks when value is defined as the level of a firm s book-to-market ratio. The average return of the highest decile of stocks ranked according to book-to-market is almost one percent per month more than for the lowest decile of stocks. The Jensen alpha of value (growth) stocks is significantly positive (negative), both economically and statistically. 13 One possibility is that the high expected return on value stocks reflects compensation for some sort of distress-related factor risk. An alternative interpretation is that growth stocks are 12 Handa, Kothari, and Wasley (1989) and Kothari, Shanken, and Sloan (1995) show that the size effect is mitigated when portfolios CAPM betas are estimated using annual returns. 13 Kothari, Shanken, and Sloan (1995) show that the book-to-market ratio effect documented in the literature is exaggerated in part because of survival biases inherent in the Compustat database and that the effect is considerably attenuated among the larger stocks and in industry portfolios.

18 16 overpriced glamour stocks that subsequently earn low returns (see Lakonishok, Shleifer, and Vishny, 1994, and Haugen, 1995). A large literature examines whether past price performance predicts future returns. There is mixed evidence to suggest price reversal at short intervals up to a month 14 and over longer horizons of three-to-five years, 15 with more compelling evidence of price momentum at intermediate intervals of six-to-twelve months (see Jegadeesh and Titman, 1993). Only the momentum effect appears to be robust (in the post-1940 period) to the form of risk-adjustment and other technical considerations, hence we examine the extent to which an investor can improve the risk-return trade-off by tilting the asset allocation so as to exploit such price momentum. The preceding discussion identifies three characteristic-based investment strategies that historically have produced positive abnormal returns. The next chapter presents meanvariance optimization techniques that can be used to exploit the abnormal-return generating ability of these anomaly-based investment strategies. However, the optimization analysis is intended only to serve as a guiding tool for investment managers by highlighting the potential impact of tilt strategies on portfolio risk and return. In general, managers will also be guided by their own research, their beliefs about the likelihood that historically successful strategies will continue to perform well in the future, market conditions prevailing at the time of their investment decisions, and other factors like transaction costs, international diversification, and taxes. 14 See Jegadeesh (1990), Lehmann (1990), and Ball, Kothari, and Wasley (1995). 15 See DeBondt and Thaler (1980 and 1985), Chan (1988), Ball and Kothari (1989), Chopra, Lakonishok, and Ritter (1992), Lakonishok, Shleifer, and Vishny (1994), and Ball, Kothari, and Shanken (1995).

19 17 Implications for asset allocation Evidence of market inefficiency often translates into investment strategies that have significant non-zero CAPM alphas. The intuitive implication for asset allocation is to tilt the investment portfolio away from the passive market portfolio and toward the positivealpha investment strategy. The amount that we tilt the portfolio toward a particular investment strategy would increase in the magnitude of the abnormal return from the strategy. However, such a tilt will typically expose the investor to residual risk that reflects return variation unrelated to the market index returns. The greater the residual risk, the lesser the recommended tilt. The optimal asset allocation decision that accounts for the magnitude of potential abnormal return as well as the residual risk incurred is formally derived in a classic paper by Treynor and Black (1973) and is discussed in the next chapter. Our investigation of optimal asset allocation also incorporates Bayesian methods of analysis that combine investors qualitative judgments about future returns with the evidence in historical data. The qualitative judgments might be based on an investor s subjective assessment of the extent of market inefficiency (i.e., the magnitude of abnormal return that might be earned in the future from an investment strategy and the speed with which capital markets might assimilate information into future prices). In addition, there might be a concern that the historical evidence on the magnitude of abnormal returns exaggerates the true performance of an investment strategy because of data-mining (datasnooping) biases inherent in the research process that might have skewed the historical performance of an investment strategy.

20 18 While an analysis of the sort presented here can provide useful guidance about asset allocation decisions, quantitative optimization techniques should not be viewed as black boxes that produce uniquely correct answers. There are simply too many assumptions that go into any such optimization and so there will always be an important role for judgment in the allocation decision. Modern portfolio techniques can be an important tool for enhancing that judgment, however. From this perspective, we believe it is important to first explore the optimal tilt problems in detail for each of the strategies considered here. Going through this process will give the reader a good feel for the basic historical risk/return characteristics of these strategies in conjunction with a simple index strategy. At the end of the monograph, we provide some additional results on optimal portfolios based on simultaneous optimization across several strategies. This will take into account the correlations between the various returns in addition to their individual risk/return attributes.

21 19 Chapter IV Optimal portfolio tilts Overview This chapter presents evidence on the historical performance of value-versusgrowth stocks, small-versus-large market capitalization stocks, and the momentum effect, all for the past four decades. Consistent with prior research, we find that value and positive-momentum stocks outperform growth, large-cap, and low past return stocks using the CAPM risk-adjusted returns as the benchmark. We then examine the extent to which tilting an investor s portfolio in favor of value, small market capitalization, and momentum stocks improves an investor s risk-return trade-off. We estimate optimal asset allocation under a variety of assumptions about the investor s prior beliefs concerning the efficiency of a market index and the profitability of investing in value, small, or momentum stocks. The investor might believe that the historical alpha of these stocks overstates their forward-looking alpha because of a combination of factors, including data snooping, survival biases, and chance. We end this section summarizing the results of a sensitivity analysis that includes the following. (i) (ii) (iii) (iv) Portfolio performance over two-year horizons and an evaluation of portfolio turnover entailed in rebalancing tilt portfolios after a one-year holding period. Results of a Bayesian predictive analysis that avoids over-fitting through the incorporation of priors and recognition of parameter uncertainty. Optimal asset allocation results when the market portfolio consists of both bond and equity securities. A limited analysis of joint optimization over a market index and all three anomaly strategies.

22 20 Data We construct a comprehensive database of NYSE, AMEX, and NASDAQ equity securities for our analysis. All firm-year observations with valid data available on the CRSP and Compustat tapes from 1963 to 1999 are included. We measure buy-and-hold (i.e., compounded) annual returns from July of year t to June of year t+1, starting in July 1963 (for a total of 36 years). Each year we include all firms with Compustat data available for calculating the book-to-market ratio and CRSP data available for calculating market capitalization and past one-year return (to assign stocks to momentum portfolios). We require that included securities have the book-to-market ratio, size, and momentum information prior to calculating their annual return starting on July 1. Specifically, we measure market capitalization at the end of June of year t (e.g., size is measured at the end of June 1963, and returns are computed for the period July 1963 June 1964). Book value is measured at the end of the previous fiscal year (typically, December of the previous year, i.e., December 1962 for returns computed in July 1963 June 1964). The December/July gap ensures that the book value number was publicly available at the time of portfolio formation. Following Fama-French 1993, book value is the Compustat book value of stockholders equity, plus balance sheet deferred taxes and investment tax credit (if available), plus post-retirement benefit liability (if available), minus the book value of preferred stock. The book value of preferred stock is the redemption, liquidation, or par value (in this order), depending on availability. The bookto-market ratio (BM) is calculated as the book value of equity for the fiscal year ending in calendar year t-1 divided by the market value of equity obtained at the end of June of year t.

23 21 We analyze the performance of value-weighted quintile portfolios each year. We construct these portfolios at the end of June of year t, based on size, BM, and momentum (returns during the previous year, i.e. July t-1 to June t). Portfolios based on BM do not include firms with negative or zero BM values. Portfolios based on momentum do not include firms that lack return data for the 12 months preceding portfolio formation. Some securities do not remain active for the 12-month period beginning on July 1. Firms delist as a result of mergers, acquisitions, financial distress, and violation of exchange listing requirements. In case of delisted securities, when available, we include their delisting return as reported on the CRSP tapes. This prevents survival bias from exaggerating an investment strategy s performance. The empirical analysis gives consideration to the practical feasibility of mutual funds implementing the asset allocation recommendations in this monograph. Toward this end, we therefore exclude stocks with impracticably small market capitalization and low prices from our analysis. Investments of an economically meaningful magnitude at current market prices can be difficult in small stocks, as they are less liquid, and low prices typically are associated with high transaction costs. Therefore, we report results of optimal asset allocation by restricting the universe of stocks analyzed to those with market capitalization in excess of the smallest decile of stocks listed on NYSE and stock price greater than $2. Descriptive statistics Table 1 reports descriptive statistics for the sample of equity securities that we assemble for optimal asset allocation analysis. The total number of firm-year observations

24 22 from 1963 to 1998 is 100,904, with an average of about 2,800 firms per year. If we had not excluded stocks priced lower than $2 or stocks in the lowest decile of the market capitalization of NYSE stocks, the number of securities each year would have been approximately 4,800. The average annual buy-and-hold return on these securities is 14%, with a cross-sectional standard deviation of 42%. Because of some spectacular winners, the median annual return is considerably lower at 9%. The average return for year t-1, the year prior to investment, is reported in the last row of Table 1 and is much higher at 22.8% compared to the average return of 14.3% for year t. The large difference is attributable to the exclusion of low-priced and small market capitalization stocks. Stocks experiencing negative returns decline in price and market value by the end of year t-1. We eliminate many of those stocks as rather impractical for investment purposes. Thus, the stocks retained for investment at the beginning of year t have typically performed relatively well in the prior year, which naturally boosts the average return for year t-1 of the stocks retained. Of course, all of our portfolio analysis is forward-looking and, therefore, not subject survivor bias. The average market capitalization of the sample securities is $723 million, but the median stock s market value is only $143 million. 16 The mean book-to-market ratio is 1.04, which is a result of two contributing factors. First, we exclude small market capitalization and low-priced stocks, many of which also have low book values because of asset write-offs, restructuring charges etc., and therefore low book-to-market ratios. Second, although book-to-market ratios in the 1990s have been at the low end of the distribution of book-to-market ratios, book-to-market ratios in the 1970s were quite high, 16 The market capitalization numbers are not adjusted for inflation through time, so both real and nominal effects cause variation in market values across years.

25 23 which raises the average for our sample. Since book value data on the Compustat is not available as frequently as return data on CRSP, there are only 75,272 firm-year observations in the analysis using the book-to-market ratio. Optimal tilting toward size quintile portfolios We present evidence on large-sample historical performance of small-versus-large market capitalization stocks, value-versus-growth stocks, and the momentum effect. To assess the performance of each strategy, we form quintile portfolios on July 1 of each year by ranking all available stocks on their book-to-market ratios, market capitalization, or past one-year performance (momentum). We estimate each portfolio s risk-adjusted performance for the following year. We then measure the performance of a portfolio formed by tilting the value-weight market portfolio toward the quintile portfolios, with the weight of a quintile portfolio ranging from zero to 100% and that of the value-weight portfolio declining from 100% to zero. That is, the value-weight portfolio is gradually tilted all the way toward a quintile portfolio. Optimal tilt is when the Sharpe ratio of the tilt portfolio attains the maximum. We estimate a portfolio s risk-adjusted performance using the CAPM regression. The estimated intercept from a regression of portfolio excess returns on the excess valueweight market return is the abnormal performance of the portfolio, also referred to as the Jensen alpha. The CAPM regression is estimated using the time series of annual postranking quintile portfolio returns from July 1963 to July The identity of the stocks in each quintile portfolio changes annually as all available stocks are re-ranked each July 1

26 24 on the basis of their market capitalization, book-to-market ratio, or past one-year performance. The CAPM regressions are: where R qt R ft = α q + β q (R mt R ft ) + ε qt (1) R qt R ft is the buy-and-hold, value-weight excess return on quintile portfolio q for year t, defined as the quintile portfolio return minus the annual risk-free rate; R mt R ft is the excess return on the CRSP value-weight market return; α q is the abnormal return (or Jensen alpha) for portfolio q over the entire estimation period; β q is the CAPM beta risk of portfolio q over the entire estimation period, and ε qt is the residual risk. Tables 3, 4, and 5 report performance for allocations tilted toward size, book-tomarket, and momentum portfolios. Specifically, using size portfolios as an example, we report the performance of a portfolio consisting of X% of the smallest (Q1) or largest (Q5) market capitalization quintile portfolio and (100 X)% of the CRSP value-weight portfolio. X varies from 0 (i.e., no tilt toward a size quintile portfolio) to 100% (i.e., all the investment in a size quintile portfolio). Although probably of lesser practical relevance, we also include results for a strategy of tilting toward the spread between quintiles 5 and 1. With the proliferation of exchange-traded funds tied to a variety of indexes, implementing such spreads may eventually become more realistic. The conventional size-based strategy of emphasizing small firms would correspond to a negative position in this spread, but the risk-adjusted performance of this small-large strategy is slightly negative for our sample. Our

27 25 presentation of results for the Q5-Q1 size spread will serve as an introduction to the notation and concepts of the monograph. The more interesting findings for the Q5-Q1 value and momentum strategies will then follow. Technically, the tilt asset in this context should be viewed as a position consisting of $1 in T-bills and $1 on each side of the large-small firm spread. In other words, the investor in this asset is implicitly assumed to receive interest on the proceeds from the short sale of the small-firm quintile. This combined position has a net investment of $1, unlike the spread itself, which is a zero-investment portfolio with the rate of return undefined. Since we focus on excess returns, the return on the $1 investment in T-bills is netted out and the performance measures are determined completely by the spread in returns between large and small quintile stocks. The return calculations ignore the impact of margin requirements that may be associated with either long or short positions. If the spread portfolio generates a positive alpha, then an investor can improve performance by tilting toward the spread. In this case, an X% tilt toward the spread entails a $(100 X) investment in the value-weight portfolio and $X in the spread asset. We report a variety of statistics for each tilt portfolio. These include: the average annual excess return on a tilt portfolio from 1963 to 1999; standard deviation of the excess return; the Sharpe ratio (i.e., the ratio of the average excess return to the standard deviation of excess return); the Jensen alpha and CAPM beta, which are estimated using regression eq. (1). In addition to Table 3, we include figure 1 which plots the behavior of three portfolio performance metrics (excess returns, M 2, and C_M 2, described below) for portfolio allocations tilted toward the size quintile portfolios 1 and 5, and the Q5-Q1 (large-small) spread. The graphical presentation of the information is helpful in

28 26 visualizing the costs and benefits of tilting toward various investment strategies. The graphs also aid in gaining an understanding of the performance metric s sensitivity to small deviations from the optimal portfolio. If the sensitivity is low, then the potential loss in performance for moderate deviations from the optimum would not be great. Given the inherent limitations of any analysis of this sort, our confidence in the relevance of the results would be substantially reduced if too much sensitivity were observed.. The first row of the column labeled 0% = vwrt in Table 3 shows that the average annual excess return on the CRSP value-weight portfolio from 1963 to 1999 is 7.4%. Values to its right are average excess returns for portfolios with increasing allocations to the smallest size quintile portfolio, with the column labeled 100% = rt invested entirely in the smallest quintile portfolio. The average excess return on the smallest size quintile portfolio is 8.6%. This portfolio s α is -0.5% (standard error 2.9%) and its β is 1.24 (standard error 0.15). Thus, the size effect (Banz, 1981) is not observed in this sample. The poor performance of small stocks in the 1980s and our decision to exclude extremely lowpriced, low market capitalization stocks together result in an insignificant α for the smallest size quintile portfolio. Without our data screen, the small firm α is 3.3%. The row labeled Alpha reports Jensen alpha for the various allocations. Since the first α value refers to the α of the value-weight portfolio, it must be zero. As the portfolio is increasingly tilted towards the smallest quintile portfolio, the reported values approach the α of the smallest quintile portfolio, i.e., -0.5%. Below the portfolios alphas, we report their Sharpe ratios. The market portfolio s Sharpe ratio is 41.5%. Tilting toward small stocks dramatically lowers the Sharpe ratio,

29 27 with the last column showing the smallest quintile portfolio s Sharpe ratio to be only 31.7%. The right-most column reports the ratio for the optimal portfolio with unrestricted short-selling, i.e., the portfolio with the highest Sharpe ratio. Since tilting toward the smallest quintile portfolio increases volatility faster than the increase in average returns, for the period, the value-weight portfolio has the optimal Sharpe ratio. [Table 3 & Figure 1] An equivalent measure of portfolio performance that some analysts prefer to report is M 2, which is the excess return on the portfolio after an adjustment to make its volatility equal to that of the market index. It can be shown that M 2 is a positive linear transformation of the Sharpe ratio hence the two performance measures provide identical rankings of portfolios. More formally, M 2 is the excess return on a hypothetical portfolio, p*, which takes positions in the given portfolio, as well as T-bills, such that the return volatility of p* is the same as that of the value-weight market portfolio. 17 If the size-tilted portfolio s volatility (return standard deviation) exceeds that of the value-weight portfolio, then p* will include a long position in T-bills so as to lower the risk. 18 The return on the resulting portfolio, p*, is referred to as M 2. The value-weight portfolio s M 2 is simply its excess return which serves as the benchmark that we hope to beat by exploiting active positions in the anomaly-based portfolios. Since the unit of measurement for M 2 is percentage excess return, it may be a more intuitive measure than the Sharpe ratio. Table 3 shows that tilting toward the smallest size quintile portfolio results in a lower M 2 than that 17 M 2 is named after Franco Modigliani and Leah Modigliani. They introduced the measure in their paper appearing the Journal of Portfolio Management in In the opposite situation, leverage (shorting the riskless asset) would be used to raise the volatility of p* to that of the market index. In this case, performance would be overstated insofar as the borrowing rate exceeds the T-bill rate used in the computation.

30 28 of the value-weight portfolio. In the extreme, the M 2 of the smallest size quintile portfolio is 5.6% compared to 7.4% for the value-weight portfolio. Table 3 reports two additional performance measures, c_sharpe and c_m 2. These measures are derived by placing weights of c on zero and (1 - c) on the estimated α and using this average as the true α for the quintile portfolio. In this way, we capture an investor s confidence (or lack of confidence) in the historical performance of an investment strategy. The lower an investor s confidence that the past performance of an investment strategy will persist, the larger will be the value of c. We report results under the assumption that only half of the historical α of a portfolio can be expected in the future, i.e., c = 0.5. An investor might believe that historical performance is exaggerated because of data snooping, survivor biases, luck, or because the investment opportunity will be arbitraged away in the future as a result of public knowledge of the opportunity. The results in Table 3 for a strategy of tilting towards the smallest quintile portfolio with c = 0.5 show, not surprisingly, that tilting remains unattractive. The c_sharpe ratio of the smallest size quintile portfolio is 32.7% compared to 31.7% without the c adjustment and 41.5% for the value- weight portfolio. The slight improvement is due to the fact that the estimated small-firm alpha is negative. In addition to reporting performance for a series of portfolios tilted toward the smallest size quintile portfolio, we report performance for the optimal tilt in the absence of short-selling constraints. It can be shown [see Treynor and Black (1973)] that the Sharpe ratio of the optimal portfolio, which appears in the right-most column of Table 3, is Sharpe(Optimal) = [(Sharpe(VWRt) 2 +Information ratio 2 ] 1/2

31 29 The information ratio in this equation is defined as α/std(e); α is the Jensen alpha of the portfolio strategy that will be added to the simple index position (the smallest size quintile portfolio in the example here) and Std(e) is the standard deviation of the residuals from the CAPM regression used to estimate the α, i.e., the standard deviation estimate for ε qt in equation (1). 19 The optimal amount of tilting increases with the magnitude of the α, and it decreases with the residual uncertainty. This is logical since we must bear residual risk by tilting away from a simple diversified position in the market index and α is the reward for doing so. The last column in Table 3 reports that the optimal portfolio s Sharpe ratio is 41.6%. Since the value-weight market portfolio s Sharpe ratio is 41.5%, and since tilting toward the smallest size portfolio reduces the Sharpe ratio, an investor must short the smallest size-quintile portfolio to reach optimality. However, the Sharpe ratio improves only marginally so, essentially, the optimal strategy would be to simply invest in the market portfolio. Results in Table 3 for a strategy of tilting toward the largest size quintile suggest that, for the period from 1963 to 1999, investors would have gained only slightly by investing in large stocks. Even though tilting the value-weight portfolio toward the largest size quintile portfolio by about 90% maximizes the Sharpe ratio, the M 2 of the resulting portfolio is approximately the same as that for the value-weight portfolio, 7.4%. Since small stocks performed poorly, an investor would have been a bit better off by excluding small stocks from the value-weight portfolio or by tilting toward the largest stocks. This 19 The optimal portfolio s composition is determined using the following formula: Optimal allocation to the tilt portfolio = (X/Sharpe ratio of the VWRt)/(1 + (1 - β)x), where X = (1 c)*information ratio/std(e).