Monograph on

Anomalies and Asset Allocation

S.P. Kothari Sloan School of Management, E52-325 Massachusetts Institute of Technology

50 Memorial Drive, Cambridge, MA 02142

(617) 253-0994 kothari@mit.edu

and

Jay Shanken

William E. Simon Graduate School of Business Administration University of Rochester, Rochester, NY 14627

(716) 275-4896 shanken@simon.rochester.edu

First draft: December 2000 Current version: June 2002

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Acknowledgements

We thank Michela Verardo for excellent research assistance. We are grateful to theResearch Foundation of the Institute of Chartered Financial Analysts and theAssociation for Investment Management and Research, the Bradley Policy ResearchCenter at the Simon School, and the John M. Olin Foundation for financial support.

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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 value-

weighted 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

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measuring the magnitude of risk-adjusted expected returns.1 In this monograph, the three

most prominent CAPM “anomalies” are considered: 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 an anomaly-based portfolio (or spread), in order to

exploit the gains to efficiency. The portfolio improvement depends, not only on the risk-

adjusted expected returns of these 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.

Traditional statistical tests of significance, while useful in many contexts, are not

particularly well suited to investment decision-making. In recent years, Bayesian

statistical methods have begun to achieve greater prominence in addressing asset allocation

problems.2 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.

1 Two notable exceptions are the recent work of Pastor (2000), which is closely related to our analysis, andHaugen and Baker (1996).

2 See Kandel and Stambaugh (1996).

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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. 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 of asset

allocation using the capital asset pricing model (CAPM) as the theoretical framework. The

chapter reviews portfolio theory, the CAPM, and the efficient markets hypothesis. Chapter

III reviews recent evidence challenging the efficient markets 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 results

of our analysis of historical data and optimal asset allocation by tilting the market index

portfolio toward value, momentum, or small stocks. This analysis also incorporates a

Bayesian perspective on optimal asset allocation. Chapter V considers the joint

optimization problem in which all three anomalies are considered simultaneously. Chapter

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VI summarizes the monograph and discusses its implications and directions for future

work.

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Chapter II

Asset allocation in a CAPM world

This chapter reviews the fundamental concepts of finance and their implications for

asset allocation. We review portfolio theory, the CAPM, and the efficient markets

hypothesis.3

Portfolio theory

In a mean-variance setting, Harry Markowitz in 1952 developed optimization

techniques to derive the efficient frontier of risky assets. An efficient frontier graphs the

portfolios with highest expected return for given levels of portfolio return variance.

Estimated values of expected return, standard deviation of return, and pairwise covariances

of returns for all available risky securities are the inputs to deriving the efficient frontier.

Portfolio theory suggests that an investor should choose a portfolio on the efficient frontier

to maximize expected utility. This assumes that the investor cares only about the mean

and variance properties of overall portfolio returns. 4

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 a unique risky “tangency” portfolio on the efficient

frontier and the risk-free asset. Relatively risk-averse investors will invest a larger fraction

of their assets in the risk-free asset whereas relatively more risk-tolerant investors will opt

3 For a detailed treatment of the concepts in this chapter, see Bodie, Kane, and Marcus (1999), chapters 6-9and 12, or Ross, Westerfield, and Jaffe (1996), chapters 9 and 10.

4 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

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 representing the best possible combinations of portfolio

expected return and standard deviation.

The CAPM

The Capital Asset Pricing Model (CAPM) builds on Markowitz’s portfolio theory

ideas and further simplifies investors’ asset allocation decision.5 The CAPM is derived

with an additional critical assumption that all 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 this demand for assets must equal the supply, in equilibrium, it follows that

the tangengy 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, it

shows that expected return is a increasing linear function of its covariance risk or beta.

Beta is defined as

b i = Cov(Ri, Rm)/Var(Rm)

5 The CAPM is credited to Sharpe (1964), Lintner (1965), and Mossin (1966).

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where Cov(Ri, Rm) is the covariance of security i’s return with the return on the market

portfolio and Var(Rm) 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(Ri) = Rf + b i (E(Rm) – R f),

where E(Ri) is security i’s expected rate of return, Rf is the risk free rate of return, and

(E(Rm) – R f) is the market risk premium. In addition to it’s importance in portfolio

analysis, beta is helpful in corporate valuation and investment (i.e., capital budgeting)

decisions.

Efficient markets hypothesis6

The efficient markets hypothesis states that security prices rapidly and accurately

reflect all information 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 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 non-existent.

6 For detailed reviews of the efficient markets hypothesis and empirical literature on market efficiency, seeFama (1970, 1991) and MacKinlay (1997).

7 This notion of informational financial market efficiency should not be confused with the earlier concept ofthe mean-variance efficiency of a portfolio.

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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 betting that the inefficiency will be corrected over

a given horizon exposes the arbitrageur to 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

strategy should be passive (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. Investors’ views about these issues can have important implications for

asset allocation by affecting their confidence that deviations observed in the past will

persist in the future. Therefore, we briefly review the relevant theory and evidence in the

next chapter.

8 See Shleifer and Vishny (2000).

9 The proportion of assets an investor invests in the market portfolio is a function of the investor’s risktolerance. Since investors’ risk tolerance changes endogenously with their wealth, the fraction of theirwealth invested in the market portfolio is also expected to vary with their wealth levels. This is beyond thescope of the monograph.

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Chapter III

Recent evidence challenging market efficiency and itsimplications 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 quickly 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.

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prices to new information is consistent with market efficiency, but efficiency also requires

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 new and that it may take several years before prices incorporate the

full impact of 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 over-

reaction to corporate news as a result of investors’ behavioral biases or limited capability

to process information.1 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 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).

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Since the patterns are not really descriptive of the properties of the underlying distribution,

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-to-

twelve 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 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.

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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 (2001)

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.

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Evidence on long horizon 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.12

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

because typically only a few firms experience an event each month. Fortunately, research

also shows that a few firm characteristics (e.g., firm size, value and growth stocks, 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 creates a potentially implementable

12 Evidence of long-horizon predictability following corporate events and past security price performanceappears in 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, andMandelker (1992) estimate bidder firms’ price performance; dividend initiations and omissions are examinedin Michaely, Thaler, and Womack (1995); performance following proxy fights is studied in Ikenberry andLakonishok (1993); Ikenberry, Lakonishok, and Vermaelen (1995) and Mitchell and Stafford (2000) examinereturns 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.

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investment opportunity to exploit characteristic-based return predictability without losing

too much diversification.

The firm characteristics most highly associated with future returns are the book-to-

market ratio, firm size, and past security price performance or momentum. Banz (1981)

and 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.13 Rosenberg, Lanstein, and Reid (1985) and

recently Fama and French (1992) show that value stocks significantly outperform growth

stocks. Average return of the highest decile of stocks ranked according to the book-to-

market ratio is almost one percent per month more than the lowest decile of book-to-

market ratio stocks. The Jensen alpha of value (growth) stocks is economically and

statistically significantly positive (negative).14 The high expected return on value stocks

might reflect compensation for some sort of distress factor risk. An alternative

interpretation is that growth stocks are overpriced glamour stocks that earn subsequent 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,15 price

momentum at intermediate intervals of six-to-twelve months (see Jegadeesh and Titman,

13 Handa, Kothari, and Wasley (1989) and Kothari, Shanken, and Sloan (1995) show that the size effect ismitigated when portfolios’ CAPM betas are estimated using annual returns.

14 Kothari, Shanken, and Sloan (1995) show that the book-to-market ratio effect documented in the literatureis exaggerated in part because of survival biases inherent in the Compustat database and that the effect isconsiderably attenuated among the larger stocks and in industry portfolios.

15 See Jegadeesh (1990), Lehmann (1990), and Ball, Kothari, and Wasley (1995).

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1993), and price reversal over longer horizons of three-to-five years.16 Only the

momentum effect appears to be robust (in the post-1940 period) and we examine the extent

to which an investor can improve the risk-return trade-off by titling asset allocation to

exploit price momentum.

Implications for asset allocation

Long-horizon market inefficiency evidence implies that some investment strategies

havesignificant non-zero CAPM alphas. The intuitive implication for asset allocation is to

tilt the investment portfolio away from the market portfolio and toward the positive-alpha

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 might be accompanied by residual risk that reflects return variation

unrelated to the market index returns. The greater the residual risk, the lesser the

recommended tilt toward the investment strategy. The optimal asset allocation decision

that accounts for the magnitude of potential abnormal return and the residual risk of the tilt

investment portfolio is formally derived in Treynor and Black (1973) and discussed in the

next chapter.

We extend the traditional optimal asset allocation analysis with Bayesian methods

of analysis that combines 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

16 See DeBondt and Thaler (1980 and 1985), Chan (1988), Ball and Kothari (1989), Chopra, Lakonishok, andRitter (1992), Lakonishok, Shleifer, and Vishny (1994), and Ball, Kothari, and Shanken (1995).

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return that might be earned in future from an investment strategy and the speed with which

capital markets might assimilate information into prices in future). In addition, there might

be a concern that the historical evidence on the magnitude of abnormal return from an

investment strategy exaggerates true performance because of researchers’ data-mining and

data-snooping biases that might have skewed the historical performance of an investment

strategy.

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.

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Chapter IV

Optimal portfolio tilts

Overview

This chapter presents empirical evidence on the historical performance of value-

versus-growth 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) Portfolio performance over two-year horizons and an evaluation of portfolioturnover entailed in rebalancing tilt portfolios after a one-year holdingperiod.

(ii) Results of a Bayesian predictive analysis that avoids over-fitting throughthe incorporation of priors and recognition of parameter uncertainty.

(iii) Results of optimum asset allocation when the market portfolio consists ofboth bond and equity securities.

(iv) A limited analysis of joint optimization over a market index and all threeanomalies.

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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 book-

to-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.

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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 delisting securities, 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 in small

stocks are less liquid and low prices typically result in 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 we assemble

for optimal asset allocation analysis. The total number of firm-year observations from

1963 to 1998 is 100,904, with an average of about 2,800 firms per year. If we had not

22

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 reported in the last row of Table 1 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.17 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 have low book values and thus 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, which raises the average for our sample. Since book value data on the

17 The market capitalization numbers are not adjusted for inflation through time, so both real and nominaleffects cause variation in market values across years.

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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 value-

weight market return is the abnormal performance of a portfolio or the Jensen alpha. The

CAPM regression is estimated using the time series of annual post-ranking quintile

portfolio returns from July 1963 to July 1998. The identity of the stocks in each quintile

portfolio changes annually as all available stocks are re-ranked each July 1 on the basis of

their market capitalization, book-to-market ratio, or past one-year performance. The

CAPM regressions are:

24

Rqt – R ft = a q + bq (Rmt – R ft ) + eqt (1)

where

Rqt – R ft is the buy-and-hold, value-weight excess return on quintile portfolio q for yeart, defined as the quintile portfolio return minus the annual risk-free rate;

Rmt – R ft is the excess return on the CRSP value-weight market return;

a q is the abnormal return (or Jensen alpha) for portfolio q over the entire estimationperiod;

bq is the CAPM beta risk of portfolio q over the entire estimation period, and

eqt is the residual risk.

Tables 3, 4, and 5 report performance for allocations tilted toward size, book-to-

market, and momentum portfolios. Specifically, using size portfolios as an example, we

report the performance of a portfolio consisting of X% of the smallest or largest 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). We report several performance measures for each tilt

portfolio. These include: the average annual excess return on a tilt portfolio from 1963 to

1998; 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).

The first row of the column labeled “0% = VWRet” in Table 3 shows that the

average annual excess return on the CRSP value-weight portfolio from 1963 to 1998 is

7.4%. Values to its right are average excess returns for portfolios with increasing

allocation to the smallest size quintile portfolio, with the column labeled “100% = Rt”

25

invested entirely in the smallest quintile portfolio. The average excess return on the

smallest size quintile portfolio is 8.6%.

This portfolio’s a is -0.5% (standard error 2.9%) and its b 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 low-

priced, small market capitalization stocks together result in an insignificant a for the

smallest size quintile portfolio. Without our data screen, the small-firm a is 3.3%. The

row labeled “Alpha” reports Jensen alpha for the various allocations. Since the first a

value refers to the a 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

a 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,

with the last column showing the smallest quintile portfolio’s Sharpe ratio to be only

31.7%. The right-most column reports the ratio of the optimal portfolio, 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 1963-1998 period, the value-

weight portfolio has the optimal Sharpe ratio.

[Table 3]

Another measure of portfolio performance that is, perhaps, more intuitive than the

Sharpe ratio is M2. M2 is the excess return on a hypothetical portfolio, p*, consisting of

the portfolio in question and T-bills such that the return volatility of p* is the same as that

26

of the value-weight market portfolio.18 If the size-tilted portfolio’s volatility exceeds that

of the value-weight portfolio, then p* will be long in T-bills so as to lower the risk. The

return on the resulting portfolio, p*, is referred to as M2. The value-weight portfolio’s M2

is simply its excess return. Table 3 shows that tilting toward the smallest size quintile

portfolio results in a lower M2 than that of the value-weight portfolio. In the extreme, the

M2 of the smallest size quintile portfolio is 5.6% compared to 7.4% for the value-weight

portfolio.

Table 3 also reports two additional performance measures – c_Sharpe and c_M2.

These measures are derived by placing weights of c on zero and (1 - c) on the estimated a

and using this average as the true a 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 in the past performance of an

investment strategy, the larger will be the value of c. We report results under the

assumption that only half of the historical a 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. 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

18 M2 is named after Franco Modigliani and Leah Modigliani. They introduced the measure in their paperappearing the Journal of Portfolio Management in 1997.

27

quintile portfolio is 32.7% compared to 31.7% without the c adjustment and 41.5% for the

value- weight portfolio.

In addition to reporting the performance of a series of portfolios tilted toward the

smallest size quintile portfolio, we report performance for the optimal tiltin 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 ratio2] 1/2

This information ratio is defined as a /Std(e), where a is the Jensen alpha of the tilt

portfolio, i.e., 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 a , i.e.,

eqt from equation (1).19 The optimal amount of tilting increases in the magnitude of the a

and 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 a is the

reward for doing so.

Table 3 reports that the optimal portfolio has a Sharpe ratio of 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.

19 The optimal portfolio’s composition is determined using the following formula: Optimal allocation to thetilt portfolio = (X/Sharpe ratio of the VWRt)/(1 + (1 - b)X), where X = (1 – c)*Information ratio/Std(e).

28

Results in Table 3 for a strategy of titling toward the largest size quintile suggest

that, for the 1963-to-1998 period, investors would not have gained much 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 M2 of the resulting portfolio is

approximately the same as the value-weight portfolio, 7.4%. Since small stocks performed

poorly, an investor would have been slightly better off by excluding small stocks from the

value-weight portfolio or by tilting toward the largest stocks.

Although probably of lesser practical relevance, we also include results for a

strategy of tilting toward the spread between quintiles 5 and 1. The “asset” in this context

should be viewed as a position consisting of $1 in T-bills and $1 on each side of large-

small firm spread. With the proliferation of exchange-traded funds tied to a variety of

indexes, implementing such spreads may eventually become more realistic. The alpha for

this particular spread is quite small, however.

Optimal tilting toward book-to-market quintile portfolios

Table 4 reports the results of tilting portfolios toward extreme book-to-market

stocks. The format is similar to that of Table 3 for size portfolios. Table 4 shows that

investing in highest book-to-market (i.e., value) stocks yields a highly significant Jensen

alpha of 4.7% (standard error 1.5%) per annum. As the value-weight portfolio is tilted

toward the fifth quintile of book-to-market ranked stocks, the Sharpe ratio increases from

41.5% for the value-weight portfolio to 64% for the fifth quintile of book-to-market stocks.

The corresponding M2 performance measures increase from 7.4% to 11.3%. The c_Sharpe

and c_M2 measures also rise, but obviously not as spectacularly.

29

Interestingly, the optimal portfolio is fully invested in the highest book-to-market

ratio stocks, even after cutting the estimated alpha in half. One interpretation of the

superior performance of the high book-to-market ratio stocks is that these are distressed

stocks that ex post exhibited superior performance from 1963-1998. An investor’s

confidence in the persistence of such superior performance will determine the extent to

which one tilts the investment portfolio toward value stocks. Notwithstanding the

distressed nature of these stocks, we emphasize that we have applied the investment filter

rules such that we only include stocks priced greater than $2 and stocks in size deciles two

through ten. This enhances the practicality dimension of investing in these stocks.

[Table 4]

Table 4 also shows that growth stocks (i.e., low book-to-market stocks) did not

perform well, though the value-weight a of -0.7% is statistically indistinguishable from

zero. As with small firms, tilting toward growth stocks lowers the Sharpe ratio and M2

measure. However, the lack of statistical significance leaves us less confident about the

potential benefits from shorting the growth stocks based on this historical performance.

The spread results for book-to-market are notable in several respects and are similar

whether we impose our small/low price filter or not. First, we now have an interior

optimum with nearly forty percent of the optimal portfolio in the spread, dropping to about

one-third when alpha is cut in half. It might seem odd that average excess return declines

as the spread weight increases, despite the 5.4% alpha, but this is due to the fact that the

spread beta is negative; the high book-to-market quintile has a significantly lower beta than

the low book-to-market quintile. We also note that the optimal value spread tilt produces a

30

much lower M2 than the optimal value tilt due to the small (negative) alpha for growth and

its relatively high residual risk.

It is interesting that the information ratio for the spread is much lower than that for

the high book-to-market quintile, 33% vs. 56%, even though shorting the low book-to-

market stocks increases the alpha. The reason is that the spread is exposed to much more

residual risk - 16.5%, as compared to 8.4% for quintile 5. As a result, one is better off

investing in the high book-to-market stocks than in the spread. In fact, investing 50% or

more of the portfolio quintile dominates the optimal spread position. The optimal M2

values are 11.3% and 9.4%, for q5 and q5-q1, respectively, based on the full alpha. Note

that this spread should be highly correlated with the much-heralded Fama-French HML

book-to-market factor. Thus, an investment strategy that tries to mimic this factor by

forming an optimal tilt with the market index appears to be dominated by other simple tilt

strategies.

Optimal tilting toward momentum quintile portfolios

A momentum investment strategy is highly profitable historically (see table 5). We

rank stocks on the basis of their performance over one year ending on May 31 of each

calendar year and the investment strategy is implemented one month later from July 1.

Skipping a month avoids well-known bid-ask effects that bias performance downward.

The worst performance quintile of stocks earns an average excess return of 4.3%

compared to the value-weight portfolio’s 7.4 average annual excess return. This translates

into a Jensen alpha of -3.8% (standard error 1.9%). Without our data screen, the “loser”

quintile alpha is even lower at –5.7%. The best performance quintile portfolio earns 4.1%

31

abnormal return (standard error 2.0%). As with book-to-market, the optimal position is to

be fully invested in the quintile 5 stocks. When the alpha is cut in half , the optimal

position is about 80% invested in the quintile 5 stocks, though allocations from 60% to

100% yield similar performance measures.

In contrast to what we saw for the value spread, a strategy of shorting the “losers”

and going long in the “winner” quintile results in even higher optimal Sharpe ratios and

M2s, with the optimal weight about one-half and M2 of 11%.20 The improvement observed

here is a reflection of the fact that the loser alpha is almost as large in magnitude as the

winner alpha. At very high levels of investment in the momentum spread, the residual risk

effect dominates and the performance ratios quickly deteriorate.

[Table 5]

Summary

Our results on the benefits of tilting an investment portfolio toward extreme size

stocks, value and growth stocks, or momentum portfolios lead to several conclusions. The

risk-return trade-off is not improved much by titling portfolios toward extreme size stocks.

Combining the market portfolio with value (high book-to-market) stocks or past winners

(momentum) results in significant increases in the Sharpe ratio and M2. Even if an

investor believes that only half of the past good performance of the value and momentum

strategies is sustainable in future, such tilting strategies would be desirable.

20 The estimate of residual risk is higher than the standard deviation under “100% = rt.” This apparentcontradiction is due to the degrees-of-freedom adjustments, one for total variance and two for residualvariance.

32

The Bayesian Approach to Asset Allocation

Motivation for the Bayesian analysis. The preceding analysis uses historical data

to estimate the inputs to the asset allocation problem and provides results for a variety of

tilt strategies. However, even if we believe that the portfolio parameters are (relatively)

constant over time, it is important to consider the potential impact of estimation error on

our portfolio decisions. Unfortunately, traditional statistical analysis is not well suited to

this task. The standard errors reported earlier can be used to derive confidence intervals

for, say, alphas, but how should such observations be translated into an investment

decision?

Intuition for the Bayesian analysis. Intuitively, if an alpha is not estimated with

much precision, then it is more likely that the apparent abnormal return (positive alpha) is

due to chance and, therefore, may not be a good indication of what will be observed in the

future. In such a case, it would seem sensible to tilt less aggressively in the direction of the

given anomaly. The extent to which we should “discount” the historical evidence because

of this additional uncertainty or estimation risk is not so clear, however.

A related issue is that, even before looking at the data, we may have some prior

notion as to a plausible range of values for alpha. This prior belief could be based on

observations of returns in earlier periods or in other countries. Or, it might be based more

on economic theory and one’s general view about the efficiency of financial markets and

the relevance of simple theories like the CAPM.21 Recall that the CAPM implies alpha

should be zero when the index is the true market portfolio of all assets.

21 See Pastor (2000).

33

Whatever the source of one’s prior belief, suppose, for example, that an annualized

alpha bigger than 4% is judged implausibly large and yet an estimate of 4.7% is obtained.

In light of this prior belief, the expectation for future abnormal return is clearly less than

4.7%. The extent to which we will want to lower or shrink the estimated value naturally

depends on the confidence we have in our initial belief, as compared to the precision of the

statistical estimate.

Bayesian analysis provides an appealing framework in which to formalize these

ideas and incorporate them in an asset allocation decision. Academic work on portfolio

optimization has increasingly utilized Bayesian methods in recent years, the study by

Pastor (2000) being the most relevant for the issues considered here. In Bayesian analysis,

initial beliefs about return parameters are represented in terms of prior probability

distributions. For convenience, we assume normal distributions for priors as well as

returns. Using a basic law of conditional probability referred to as Bayes rule, the data are

combined with one’s initial beliefs to form an updated posterior probability distribution

that reflects the learning that has occurred.

Implementing the Bayesian analysis for asset allocation. For pedagogical

purposes, we initially suppose that alpha is the only unknown parameter. Let a 0 be the

prior expected value for alpha and s (a 0) the prior standard deviation. Say a 0 = 0, the

value implied if the market index is mean-variance efficient. If s (a 0) = 2%, then an alpha

of 4% or more is a two-standard deviation “event” with probability just 0.023. Of course,

the actual alpha is either greater than 4% or not, but this probability quantifies our

subjective judgment that values that large are implausible.

34

Now, let a denote the given estimate of alpha and se(a ) its standard error. Say a

is 4.7%, as above, and se(a ) = 1.5%, the values observed earlier for the high book-to-

market quintile in Table 4. In this context, Bayes rule implies that the posterior mean is a

precision-weighted average of the estimate and the prior mean:

a * = [ a0 . 1/var(a 0) + a . 1/var(a )] / [1/var( a 0) + 1/var(a )], (2)

where precision is technically defined as the reciprocal of variance. If the prior uncertainty

is large relative to the informativeness in the data, i.e., if var(a 0) is high in relation to

var(a ), Bayes rule places most of the weight on the estimate a . Alternatively, if there is

not much data or if the data are quite noisy, var(a ) is large and a * is closer to the prior

mean a 0.

In our example, 1/var(a 0) + 1/var(a ) = 1/0.022 + 1/0.0152 = 2,500 + 4,444.44 =

6,944.441, so a * = (2500/6,944.44) . 0 + (4,444/6,944.44) . 0.047 = 0.03 or 3%. Since the

estimate here is a bit more precise than the prior, greater weight is placed on the estimate,

as compared to the prior mean. As a result, the posterior mean of 3% is closer to 4.7%

than to 0.

Having discussed the idea of shrinking an estimate toward a prior mean, we now

turn to the other important consideration of estimation risk, namely, implication for asset

allocation. We observed earlier that the optimal amount of tilting toward a quintile or

spread portfolio depends on its residual risk as well as its alpha and the Sharpe ratio of the

market index. From a Bayesian perspective, uncertainty about the true value of alpha is

naturally recognized as an additional source of risk that confronts an investor.

Conventional risk measures ignore such uncertainty. To convey this point, we rewrite

equation (2) as a Bayesian predictive regression:

35

Rq – R f = a* + bq (Rm – R f) + [ eq + (aq - a*)], (3)

where a * is the posterior mean for alpha discussed above.

Looking forward, an investor’s uncertainty about the manner in which the true

alpha deviates from its posterior expected value is a form of residual or non-market risk.

Recall that the residual term eq reflects economic influences that affect the stocks in

portfolio q, but do not have a net impact on the market index. Likewise, whether our

expected value for alpha is too high or too low will have no bearing on whether the market

subsequently goes up or down. Therefore, a q-a* is uncorrelated with the market return,

and with eqt as well, by a similar argument.

In order to make an optimal portfolio decision, we need to know how much

additional residual variability is induced by the alpha “parameter uncertainty.” More

formally, we need to know the variance of the posterior probability distribution for alpha.

Fortunately, there is a simple and intuitive mathematical result that delivers this variance:

the posterior precision is just the sum of the precisions of the prior and the estimated alpha

and is given by the denominator of (2). Recalling our earlier computations, this is

6,944.44, so the posterior standard deviation, s *( a ), is 0.012 or 1.2%, as compared to the

prior standard deviation of 2%. The reduction from 2% to 1.2% is an indication of the

extent to which observing the historical data has narrowed our belief about the true value

of alpha.

Given the posterior standard deviation for alpha, the next step is to quantify the

overall predictive residual risk, s *(res), perceived by the investor. This is the standard

deviation of the quantity in brackets in (3). Since eqt is uncorrelated with a -a*, s *(res)

is (0.0842 + 0.0122).5 = 8.5%. Interestingly, the uncertainty about alpha only increases

36

residual risk by 0.1% from the regression estimate of 8.4%. Although one might be

inclined to attribute this to the fairly tight prior distribution assumed for alpha, that is not

the cause. To see this, suppose the prior is totally uninformative, i.e., let s (a 0) approach

infinity in (2). Now, the posterior moments are identical to the sample moments: a * = a

= 4.7% and s *( a ) = se(a ) = 1.5%. The implied value of s *( a ) increases only slightly,

however, and is still about 8.5%. The investors’ uncertainty is simply dominated by the

variability of the residual component of return in this case. Parameter uncertainty is a

second-order effect.

We make similar computations without the simplifying assumption that alpha is the

only unknown parameter in the decision problem. The relevant formulas appear in the

appendix. With uninformative priors for alpha and beta, but treating the sample residual

variance as the true value of var(eq), we have s *(res) = 8.6%. If, instead, we let the data

dominate our belief about var(eq) and use uninformative priors for all the regression

parameters, s *(res) increases to 8.9%.

To examine the impact of estimation risk on asset allocation, we combine the

original estimate, a = 4.7%, with our most conservative estimate of residual risk, s *(res)

= 8.9%. This risk measure now takes on the role played earlier by the sample estimate of

var(eq). For simplicity, we specify uninformative priors for the mean and variance of the

market index as well. By an argument similar to that for alpha and residual risk,

uncertainty about the market’s true mean return increases the (predictive) risk perceived by

the investor and lowers the perceived market Sharpe ratio. Other things equal, this market

effect tends to increase the optimal weight on the tilt portfolio.

37

Recall from our earlier analysis of the book-to-market anomaly, which ignored

parameter uncertainty, that it was optimal to be fully invested in the high book-to-market

quintile (assuming no-short-selling). The corresponding M2 value was 11.3%, as

compared to the market expected return of 7.4% (market standard deviation = 17.7%).

With parameter uncertainty, being fully invested is still the optimal strategy. Now, the

predictive market risk is 18.5% and the optimal M2 is perceived to be 11.2%. The main

point, however, is that the investor is not affected by ignoring parameter uncertainty in this

case.

As a more interesting illustration, suppose we shrink the estimate of alpha with

weights c = 0.7 on zero and 0.3 on a = 4.7%. The resulting value for alpha is 1.9%.

Without incorporating parameter uncertainty, the optimal weight on quintile 5 would be

74.6%, still quite high despite the more conservative assumption about alpha. The

Bayesian optimal weight is just a bit lower at 72.5%, as the increase in residual risk

apparently dominates the market risk effect. Using the “wrong” weight, 74.6%, would

reduce the perceived M2 by less than a basis point from the optimal predictive value of

7.9%. Even in this case of an unconstrained optimum, neglecting estimation risk has

virtually no effect on the investor. Only the desired degree of Bayesian shrinkage is

important. Similar conclusions hold for tilts involving the small firm and high momentum

quintiles.22

Summary of the Bayesian analysis. The Bayesian analysis is a simple and

intuitive approach to incorporating information about the imprecision or uncertainty in the

22 Shrinkage implies that the prior for alpha is informative, which means that our analysis with uninformativepriors overstates the impact of estimation risk in this case. Also, our conclusions are essentially unchanged ifparameter uncertainty regarding the market index parameters is ignored.

38

historical estimates of alpha or beta. This uncertainty increases the perceived (or

predictive) residual risk of the investment portfolio. If this effect is greater than the effect

of uncertainty about the market expected return, it should incline the investor to tilt the

portfolio less aggressively toward the anomaly. The preceding discussion formalizes these

concepts in the context of optimal asset allocation. We find that, under plausible

assumptions, giving consideration to parameter uncertainty changes the optimal asset

allocation to some extent, but not substantially.

Performance of tilt portfolios over two-year horizons

Previous analysis examines portfolio performance over one year immediately

following the formation of the size, book-to-market, and momentum tilt portfolios.

Performance measurement over one year implicitly assumes that the optimal portfolio

would be rebalanced every year. However, to minimize transaction costs and because of a

longer investment horizon, an investor might be reluctant to rebalance the portfolio

annually. Therefore, we also examine performance in the second year following the tilt

portfolio construction. The focus is on examining whether the expected gains from tilting

in the first year are sustained in the second year without rebalancing.

To implement a strategy of investing in the second year, we sort stocks on firm

characteristics at the end of June of year t, but measure tilt portfolio returns starting in July

of year t+1. The sample spans 35 years, and starts in July 1964. Tables 6-8 report tilt

portfolios’ performance for the second year after portfolio formation. These tables

correspond to Tables 3-5 that report tilt portfolios’ performance for the year immediately

following their formation.

39

Tables 6-8 show that tilting toward extreme size portfolios remains unbeneficial in

terms of improvement in the risk-return trade-off. The momentum strategy exhibits signs

of reversal, with an alpha of –4.1% in the second year (standard error = 2.5%). Reversal in

the second year is more consistent with momentum in the first year being a continuation of

overreaction to information that arrives during the portfolio formation year, rather than an

adjustment to an initial underreaction.

Value stocks (quintile 5) put forth a strong risk-adjusted performance even in the

second year after their formation (see Table 7). They earn an average excess return of 10%

as compared to the value-weight market return of 7.1%. With a beta of just 0.78, the

value-stock Jensen alpha is 4.5% (standard error = 1.1%), similar to the 4.7% alpha for the

first year.23 The information ratio actually increases in the second year, to 0.73 from 0.56

earlier, since residual risk declines substantially, from 8.4% to 6.2%. Since the c_Sharpe

and c_M2 are maximized at the 100% tilt, the optimal portfolio would invest entirely in

value stocks and short the value-weight market portfolio. More realistically, the implied

strategy is to invest primarily in value stocks.

[Tables 6-8]

Optimum asset allocation using the market portfolio of both bond and equity

securities

Financial planners typically recommend asset allocation that consists of substantial

investments in both equity and bond securities. It is common to encounter a mix of

approximately 60% equities and 40% bonds.24 Therefore, it is of interest to examine

23 This is consistent with results in La Porta, Lakonishok, Shleifer, and Vishny (1997).

24 For example, Jonathan Clements writes in the Wall Street Journal on July 31, 2001, page C1, “… youbuild a balanced portfolio of typically 60% stocks and 40% bonds…”

40

whether, and by how much, one should deviate from such a balanced market portfolio in

light of the size, book-to-market, and momentum anomalies discussed earlier. We

construct a time series of returns on a market portfolio by combining the CRSP value-

weight portfolio returns with 10-year U.S. Government bond returns. Except for using a

different market index, we then repeat our empirical analysis of the benefits of tilting

toward size, market-to-book, and momentum quintile portfolios. The results are reported

in Tables 9-11. .

The 60/40 portfolio has lower excess return as well as lower risk than the all-stock

index. The market Sharpe ratio declines slightly with the inclusion of bonds, from 0.42 to

0.38, and alphas relative to the 60/40 market are a bit higher. Given the lower volatility of

the 60/40 mix, the betas of the stock portfolios naturally increase. The changes in residual

risk are less consistent, with noteworthy increases for growth stocks (from 9.9% to 12.8%)

and past losers (from 10.3% to 12.3%). Despite these mostly minor changes, the

implications for tilting and optimal asset allocation are quite similar to those discussed

earlier with the all-stock market portfolio: heavy tilts toward value and high momentum

stocks, with less aggressive tilts toward the value and momentum spreads due to the

moderating effect of residual risk.

Summary

This chapter analyzes the optimal tilts and their associated risk/return

characteristics using information about the historical performance of the market portfolio

and anomaly portfolios based on size, value and growth, and momentum. We find that

value and momentum portfolios earn positive CAPM-risk adjusted returns and an investor

41

would substantially improve the risk-return trade-off by tilting the portfolio toward them.

These conclusions also apply when we use a market index consisting of 60% equities and

40% bonds.

We also explore optimal asset allocation based on the Bayesian approach of

incorporating estimation risk due touncertainty about the true ex ante performance

measures and asset allocation taking into account potential biases in historical performance

due to data snooping and survival. Adjustments to reflect an investor’s beliefs about the

effect of data snooping biases on historical performance (reducing alphas by 50%) still

result in aggressive tilts toward value and momentum.. The effect of incorporating

estimation risk in asset allocation decisions is modest in the contexts we examine.

42

Chapter V

Optimal asset allocation with all three anomalies

In this chapter, we consider the optimal asset allocation for two sets of risky assets:

i) the stock market index, large firms, value firms, and high momentum firms, and ii) the

stock market index, the size spread (large-small), the value spread (high-low book-to

market), and the momentum spread (winners-losers). The individual risk and return

characteristics of these assets were examined in Chapter 4. Table 12 (not included yet)

contains the residual correlations for each set of assets. The correlation between large

firms and value firms is -0.30. The other correlations in Panel A are also negative but

closer to zero. The correlation in Panel B between the book-to-market and momentum

(size) spreads is -0.27 (-0.20).

Following Treynor and Black (1973), we structure the asset allocation decision in

terms of an optimal active portfolio of the anomaly-based investments and an optimal

combination of the market index and the active portfolio.25 Based on the historical

estimates, the optimal strategy for the first set of assets entails an “unreasonably” large

(-563%) short position in the market index. In fact, even if we let c = 0.9 and reduce all

the alphas by 90%, the optimal strategy still shorts the market. When short-selling is ruled

out, the active portfolio (c = 0) consists of 77% in value firms and 23% in winner firms and

the optimal portfolio has no (direct) investment in the stock market index. The optimal M2

is 11.5%, much higher than the 7.4% excess return on the market. Note that the greater

25 In the unrestricted short-selling case, we use the formulas in Gibbons, Ross, and Shanken (1989) whichgeneralize those in Treynor and Black (1973) to accommodate nonzero residual correlation.

43

investment in value stocks, as compared to winners, is consistent with the higher

information ratio for those firms, as observed earlier.

Letting c = 0.5 does not change any of the weights, but lowers the M2 to 9.1%.

Finally, if we let c = 0.75, then the active portfolio consists of 8% in large firms, 60% in

value firms, and 32% in winner firms. Still, there is no investment in the market index and

the optimal M2 is now 8.0%. Not much changes if we assume that the large firm alpha is

zero. The small optimal investment in large firms is driven by the diversification benefit

of the negative residual correlations with value, and to a lesser extent, winner firms. This

diversification benefit becomes relevant when the cost, in terms of lost expected return

from investment in value and winner stock is reduced sufficiently. The relative robustness

of optimal portfolio weights to substantial reductions in forward-looking alphas is, we

think, interesting and somewhat surprising.

Now consider investment in our second set of assets, the stock index and spread

portfolios. Recall that the “asset” in the case of a spread is a position consisting of $1 in

T-bills and $1 on each side of spread. In this case, the unrestricted optimal allocation

looks more conventional. The active portfolio invests 12% in the size spread, 42% in the

value spread, and 46% in the momentum spread. The optimal allocation puts 35% in the

stock index and 65% in the active portfolio, resulting in an M2 of 13.9%, higher than the

11.5% for the first set of assets. The high residual risks of the individual spread positions

are substantially reduced by diversifying across spreads, making it possible to exploit the

high alphas more efficiently than with the single spread tilts examined in Chapter 4.

In general, when short-selling is not restricted, it can be shown algebraically that

increasing c leaves the active portfolio unchanged. Naturally, the weight on the active

44

portfolio is lowered, however. With c = 0.5 (0.75), that weight is 54% (40%) and the

corresponding M2 drops to 9.4% (7.9%).

45

Chapter VI

Summary, conclusions, and directions for future research

Our main findings are as follows. First, there is essentially no size effect in our

data over the period 1963-98. This is due, in part, to our exclusion of very small, low-

priced stocks in an attempt to approximate realistic investment strategies. As in other

papers, the book-to-market and momentum effects are large. When we rank stocks from

low to high and form quintiles, the spreads in alpha between quintiles 5 and 1 are 5.4% and

8.0%, respectively. Considering the anomalies separately and examining feasible

strategies involving either high book-to-market (value) or strong momentumstocks, the

optimal allocation is to be fully invested in quintile 5. Moreover, this is true for value even

if we inject a healthy dose of conservatism and reduce the alphas by half! The optimal tilt

toward strong momentum stocks is about 80% with the reduced alpha.

Less extreme optimal tilts are obtained when the quintile 5 – 1 spreads (i.e., long in

quintile 5 and short in quintile 1) are considered, although these strategies would seem less

relevant from a practical investment perspective. Interestingly, the residual risk of the

value spread portfolio is so high that an investor would be better off with an aggressive

position in high book-to-market stocks as compared to an optimal spread position. Thus,

despite the higher alpha of the spread portfolio, i.e., our version of the Fama-French HML

factor portfolio, its risk-return characteristics are not as attractive as those of the high

book-to-market portfolio when considered solely in combination with the value-weight

portfolio.

The tenor of our results is unchanged when a 60/40 market index of stocks and

bonds is used. We also track the performance of each strategy in the second year after

46

portfolio formation so as to provide some indication of the extent to which portfolio

rebalancing is warranted. The value strategy of investing in high book-to-market stocks

continues to deliver strong abnormal performance in the second year, while the high

momentum stock alpha turns negative suggesting the possibility of continuing investor

overreaction as the source of momentum profits.

When we optimize with the market index, large-cap stocks, value stocks, and

strong momentum stocks, the optimal asset allocation is about three-quarters in value and a

quarter in momentum, even if we reduce the alphas by half. There is no direct investment

in the market index. Using the size, value, and momentum spreads, instead,,produces the

highest performance measure over all the scenarios we consider, an M2 of 13.9% as

compared to the market excess return of 7.4%. The optimal allocation is about one-third in

the value-weight index, about 30% each for the value and momentum spreads, and the rest

in size. With alphas cut in half, the M2 drops to 9.4%. Almost half of the optimal portfolio

is now invested in he market index, with about a quarter each in value and momentum.

That value and momentum should play an important role in asset allocation was to

be expected given the literature on CAPM anomalies. The extent to which aggressive

investment in these anomalies seems to be called for, even with substantial reductions in

alphas and the incorporation of Bayesian estimation risk, is more surprising. We hope to

have provided insights concerning the risk and return characteristics of anomaly-based

investment strategies that will be useful to investors in making future asset allocation

decisions.

It would be of interest to expand the analysis by including international investment

opportunities as well as tax considerations. There is also a large literature on stock return

47

predictability that considers changes in risk and expected return over time. Incorporating

this sort of information might lead to improved asset allocation decisions.

48

References

Agrawal, A., Jaffe, J., Mandelker, G., 1992, The post-merger performance of acquiringfirms - a reexamination of an anomaly, Journal of Finance 47, 1605-1621.

Asquith, P., 1983, Merger bids, uncertainty, and stockholder returns, Journal ofFinancial Economics 11, 51-83.

Ball, R., Brown, P., 1968, An empirical evaluation of accounting income numbers,Journal of Accounting Research 6, 159-177.

Ball, R., Kothari, S., 1989, Nonstationary expected returns: Implications for tests ofmarket efficiency and serial correlation in returns, Journal of FinancialEconomics 25, 51-74.

Ball, R., Kothari, S., Shanken, J., 1995, Problems in measuring portfolio performance:An application to contrarian investment strategies, Journal of FinancialEconomics 38, 79-107.

Ball, R., Kothari, S., Wasley, C., 1995, Can we implement research on stock-tradingrules? Journal of Portfolio Management 21, 54-63.

Banz, R., 1981, The relationship between return and market value of common stocks,Journal of Financial Economics 9, 3-18.

Barberis, N., Shleifer, A., Vishny, R., 1998, A model of investor sentiment, Journal ofFinancial Economics 49, 307-343.

Bernard, V., Thomas, J., 1990, Evidence that stock prices do not fully reflect theimplications of current earnings for future earnings, Journal of Accounting &Economics 13, 305-340.

Bodie, Z., Kane, A., Marcus, A., 1999, Investments (New York, NY: Irwin McGraw-Hill).

Chan, K., 1988, On the contrarian investment strategy, Journal of Business 61, 147-163.

Chopra, N., Lakonishok, J., Ritter, J., 1992, Measuring abnormal performance: Doesthe market overreact? Journal of Financial Economics 32, 235-268.

Daniel, K, Grinblatt, M., Titman, S., Wermers, R., 1997, Measuring mutual fundperformance with characteristic-based benchmarks, Journal of Finance 52,1035-1058.

Daniel, K., Hirshleifer, D., Subramanyam, A., 1998, Investor psychology and securitymarket under- and overreactions, Journal of Finance 53, 1839-1885.

49

Daniel, K., Hirshleifer, D., Subramanyam, A., 2001, Overconfidence, arbitrage, andequilibrium asset pricing, Journal of Finance 56, 921-965.

DeBondt, W., Thaler, R., 1985, Does the stock market overreact? Journal of Finance40, 793-805.

DeBondt, W., Thaler, R., 1987, Further evidence of investor overreaction and stockmarket seasonality, Journal of Finance 42, 557-581.

Edwards, W., 1968, Conservatism in human information processing, In: Kleinmutz, B.(Ed.), Formal Representation of Human Judgement (New York, NY: Wiley).

Fama, E., 1965, The behavior of stock market prices, Journal of Business 38, 34-105.

Fama, E., 1970, Efficient capital markets: A review of theory and empirical work,Journal of Finance 25, 383-417.

Fama, E., 1990, Stock returns, expected returns, and real activity, Journal of Finance45,1089-1108.

Fama, E., 1991, Efficient capital markets: II, Journal of Finance 46, 1575-1617.

Fama, E., 1998, Market efficiency, long-term returns, and behavioral finance, Journalof Financial Economics 49, 283-306.

Fama, E., Fisher, L., Jensen, M., Roll, R., 1969, The adjustment of stock prices to newinformation, International Economic Review 10, 1-21.

Fama, E., French, K., 1988, Permanent and temporary components of stock prices,Journal of Political Economy 96, 246-273.

Fama, E., French, K., 1989, Business conditions and expected returns on stocks andbonds, Journal of Financial Economics 25, 23-49.

Fama, E., French, K., 1993, Common risk factors in the returns on stocks and bonds,Journal of Financial Economics 33, 3-56.

Fama, E., French, K., 1996, Multifactor explanations of asset pricing anomalies,Journal of Finance 51, 55-84.

Fama, E., French, K., 1992, The cross-section of expected returns, Journal of Finance47, 427-465.

Fama, E., MacBeth, J., 1973, Risk, return, and equilibrium: Empirical tests, Journal ofPolitical Economy 81, 607-636.

Foster, G., Olsen, C., Shevlin, T., 1984, Earnings releases, anomalies and the behaviorof security returns, The Accounting Review 59, 574-603.

50

Handa, P., Kothari, S., Wasley, C., 1989, The relation between the return interval andbetas: Implications for the size effect, Journal of Financial Economics 23, 79-100.

Haugen, R., 1995, The New Finance: The Case Against Efficient Markets (EnglewoodCliffs, NJ: Prentice Hall).

Haugen, R., Baker, N., 1996, Commonality in the Determinants of Expected Returns,Journal of Financial Economics.

Hong, H., Stein, J., 1999, A unified theory of underreaction, momentum trading, andoverreaction in asset markets, Journal of Finance 54, 2143-2184.

Ibbotson, R., 1975, Price performance of common stock new issues, Journal ofFinancial Economics 2, 235-272.

Ikenberry, D., Lakonishok, J., 1993, Corporate governance through the proxy contest:evidence and implications, Journal of Business 66, 405-435.

Ikenberry, D., Lakonishok, J., Vermaelen, T., 1995, Market underreaction to openmarket share repurchases, Journal of Financial Economics 39, 181-208.

Ikenberry, D., Rankine, G., Stice, E., 1995, What do stock splits really signal? Journalof Financial and Quantitative Analysis 31, 357-375.

Jegadeesh, N., 1990, Evidence of predictable behavior of security returns, Journal ofFinance 45, 881-898.

Jegadeesh, N., Titman, S., 1993, Returns to buying winners and selling losers:Implications for stock market efficiency, Journal of Finance 48, 65-91.

Kahneman, D., Tversky, A., 1982,Intuitive predictions: Biases and correctiveprocedures, Reprinted in Kahneman, Slovic, and Tversky, Judgement underUncertainty: Heuristics and Biases, (Cambridge, England: CambridgeUniversity Press).

Kothari, S., Shanken, J., Sloan, R., 1995, Another look at the cross-section of expectedreturns, Journal of Finance 50, 185-224.

Lakonishok, J., Shleifer, A., Vishny, R., 1994, Contrarian investment, extrapolation,and risk, Journal of Finance 49, 1541-1578.

La Porta, R., Lakonishok, J., Shleifer, A., Vishny, R., 1997, Good news for valuestocks: further evidence on market efficiency, Journal of Finance 52, 859-874.

Lehmann, B., 1990, Fads, martingales, and market efficiency, Quarterly Journal ofEconomics 105, 1-28.

51

Lewellen, J., Shanken, J., 2001, Market efficiency, rational expectations, andestimation risk, forthcoming in the Journal of Finance.

Lintner, J., 1965, The valuation of risky assets and the selection of risky investments instock portfolios and capital budgets, Review of Economics and Statistics 47,13-37.

Litzenberger, R., Joy, M., Jones, C., 1971, Ordinal predictions and the selection ofcommon stocks, Journal of Financial and Quantitative Analysis 6, 1059-1068.

Long, J., 1974, Stock prices, inflation, and the term structure of interest rates, Journalof Financial Economics 1, 131-170.

Loughran, T., Ritter, J., 1995,The new issues puzzle, Journal of Finance 50, 23-51.

MacKinlay, A., 1997, Event studies in economics and finance, Journal of EconomicLiterature 35, 13-39.

Markowitz, H., 1952, Portfolio selection, Journal of Finance 7, 77-91.

Merton, R., 1973, Intertemporal capital asset pricing model, Econometrica 41, 867-887.

Michaely, R., Thaler, R., Womack, K., 1995, Price reactions to dividend initiations andomissions: overreaction or drift, Journal of Finance 50, 997-997.

Mitchell, M., Stafford, E., 2000, Managerial decisions and long-term stock priceperformance, Journal of Business 73, 287-329.

Modigliani, F., Modigliani, L., 1997, Risk-adjusted performance, Journal of PortfolioManagement, 45-54.

Mossin, J., 1966, Equilibrium in a capital asset market, Econometrica 34, 768-783.

Pastor, L., 2000, Portfolio slection and asset pricing models, Journal of Finance 55,179-223.

Rosenberg, B., Reid, K., Lanstein, R., 1985, Persuasive evidence of marketinefficiency, Journal of Portfolio Management 11, 9-17.

Ross, S., Westerfield, R., Jaffee, J., 1996, Corporate Finance, 4th edition, Mc-Graw-Hill, New York, NY.

Sharpe, W., 1964, Capital asset prices: A theory of market equilibrium underconditions of risk, Journal of Finance 19, 425-442.

Shleifer, A., Vishny, R., 1997, The limits of arbitrage, Journal of Finance 52, 35-55.

52

Treynor, J., Black, F., 1973, How to use security analysis to improve portfolioselection? Journal of Business 46, 66-86.

53

Table 1

Descriptive statistics for the entire sample

The sample includes firms with price greater than or equal to $2 and market capitalizationgreater than or equal to the 10th percentile of NYSE stocks.

Variable N (avg per year) Mean Median Std Dev

Return year t in % 2802 14.3 9.3 42.0

Market Value in $MM 2802 723.5 143.5 2,893.7

Price in $ 2802 27.59 21.43 122.87

Book-to-Market 2090 1.04 0.66 4.79

Return year t-1in % 2560 22.8 13.9 51.6_________________________________________________________________

Note: These descriptive statistics are the time-series average of the cross-sectional statistics (number ofobservations, mean, median, standard deviation) obtained every year.

The total number of firm-years is 100,904. All are used for size portfolios.

For BM portfolios, the total number of firm-years is 75,272.

For momentum portfolios using return for year t-1, the total number of firm-years is 92,182.

In some of the analysis, we also form portfolios on the basis of term and default spreads, both measured atthe end of June of year t. Term spread is the difference between the yield on AAA bonds with maturity over10 years and the yield on one-month Treasury Bills. Default spread is the difference between the yield onBAA bonds with maturity over 10 years and the yield on AAA bonds with comparable maturity.

54

Appendix

First, we evaluate the predictive residual variance assuming that the priors for alpha

and beta are uninformative, the residual standard deviation is known, and returns are

jointly normally distributed. In this case, it is well known that the posterior distribution ofthe regression parameters mirrors the usual sampling distribution for the regression

estimates; i.e., (a , b) is jointly normally distributed with mean (a , b ) and variance matrix

s 2(e).(X ’X) -1, where X is the Tx2 matrix of independent variables including a constantvector (Zellner, 1971).

Consider the regression equation of quintile excess return on the market index

excess return. We use the simpler notation, yt = a + bxt + et . For out-of-sample return

observations, x and y, the corresponding predictive regression is

y = a + b x + [ e + (a- a ) + (b- b )x] , (A1)

where a and b are regression estimates based on data from t=1 to T. From the Bayesian

perspective, these estimates are the predictive regression coefficients (a * and b * in ourearlier notation) and the expression in brackets is the predictive regression residual. As

before, the residual consists of the regression disturbance plus additional terms that reflect

uncertainty about the parameters a and b.

The predictive residual can be viewed as the difference between y and the

regression-based prediction of y conditional on a known value of x. This is referred to asthe prediction error in standard regression analysis. Its conditional variance is well known

and given by the expression:

var(e){1 + [1 + (x - x )2/sx2]/T}, (A2)

where x is the sample mean and sx2 the variance of the market returns (maximum

likelihood estimates).

Whereas x is known in the classical prediction problem, the future market return isyet to be realized when making the asset allocation decision. Therefore, the relevant

predictive residual variance for the quintile return y is the average value of the variance in

55

(A2) over all possible values of x. In this context, the regression estimates and the sample

moments, x and sx2, are known and hence treated as nonrandom. With an uninformative

prior for the market return parameters, the predictive mean is xand the predictive variancefor x is sx2.(T+1)/(T-3), a bit larger than the sample variance (Zellner, 1971). Taking the

expectation of (A2) with respect to this distribution for x, the predictive residual varianceis

var* (res) = var(e) . œßø

ŒºØ

-

-+

)3(

)1(21

TT

T. (A2)

Interestingly, the influence of x has dropped out, simplifying the final result. With

an uninformative prior for var(e), it can be shown that var(e) is replaced in (A2) by the

usual unbiased residual variance estimate multiplied by (T-2)/(T-4). These formulaspermit a quick evaluation of the potential impact of parameter uncertainty on residual risk

without requiring a careful formulation of prior beliefs.

Term premium Yield on AAA bonds with maturity over 10 years - yield on one month treasury bills.

Default premium Yield on BAA bonds - yield on aaa bonds (both with maturity over 10 years).

Rt Tilt portfolio, for exaple, size or book-to-market quintile portfolio.

ExRt Time series average of annual excess return on a tilt portfolio (over 36 years, 1964-1999). Excess return is raw return on the portfolio minus annualized yield on one-month Treasury bill.

10%, 20% .. Percentage of rt in each tilted portfolio return. 0% indicates the value-weighted market return, 100% indicates rt (the portfolio return).

Std(ExRt) Standard deviation of excess return

Alpha Intercept of a CAPM regression, where the factor is the excess market return (vwret - risk free rate). The alpha ofeach tilt portfolio is a fraction of the alpha of the total return portfolio, i.e. alpha for 10% tilt is 10%* (alpha of rt).

Beta Slope coefficient from a CAPM regression, where the factor is the excess market return (vwret - risk free rate)

Sharpe Sharpe ratio = ExRt/Std(ExRt)

c Measure of an investor's degree of confidence in historical performance. c = 0.5 is used, which means prospective asset allocations are based on 50% of the historical alpha estimates.

c_Sharpe (exrt - c*alpha)/std(exrt) -- with alpha varying for each tilt portfolio and for each characteristic quintile.

Table 2Legend for all the tables in the Monograph

M2 M-square measure of performance = sharpe * Std(exrt of the value weight market portfolio) = Excess return of a combination of the portfolio and the risk free rate with the same standard deviation of the market

c_M2 c_sharpe * exrt (of market) = Excess return of a combination of the portfolio and the risk free rate with the same std of the market

Optimal Sh Sharpe ratio of the optimal portfolio = sqrt(ShM2 + ((1-c) * (alpha/std(eA)))2)

Optimal M2 optimal Sh * Std(ExRt+C29)

Standard errors Standard errors are below each estimate.For alpha and beta, the standard error comes from the regression.For sharpe ratios, an approximate standard error is calculated as sqrt(var mean ret/T)/std dev ret = 1/sqrt(T),where T = 18 for low spreads and high spreads. For the information ratio, an approximate standard error is calculated as std error of alpha/std error of the regression.

Differences high-low Differences are calculated between estimates from the high-spread sample and the low-spread sample.Standard errors of the differences are calculated assuming that the two series are independent, as sqrt(var1/T1 + var2/T2)

Variable N (avg per year) Mean Median Std Dev

Return year t in % 2802 14.3 9.3 42

Market Value in $MM 2802 723.5 143.5 2,893.70

Price in $ 2802 27.59 21.43 122.87

Book-to-Market 2090 1.04 0.66 4.79

Return year t-1 in % 2560 22.8 13.9 51.6

obtained every year.

For momentum portfolios using return for year t-1 , the total number of firm-years is 92,182.

Table 1Descriptive statistics for the entire sample

The descriptive statistics are the time-series average of the cross-sectional statistics (number of observations, mean, median, standard deviation)

Deafult spread is the difference between the yield on BAA bonds with maturity over 10 years and the yield on AAA bonds with comparable maturity.

The sample includes firms with price greater than or equal to $2 and market capitalization greater than or equal to the 10th percentile of NYSE stocks.

The total number of firm-years is 100,904. All are used for size portfolios. For BM portfolios, the total number of firm-years is 75,272.

In some of the analysis, we also form portfolios on the basis of term and default spreads, both measured at the end of June of year t . Term spread is the difference between the yield on AAA bonds with maturity over 10 years and the yield on one-month Treasury Bills.

rt Anomaly-based active portfolio: size, book-to-market, or momentum extreme quintiles/spreads.

exrt Time series average of annual excess returns on a tilt portfolio (over 36 years, 1964-1999). Excess return is the raw portfolio return minus the one-year riskless rate.

10%, 20% .. Percentage of rt in each tilt-portfolio return. 0% corresponds to the value-weighted market return, 100% to rt.

std(exrt) Standard deviation of excess return.

Alpha Intercept from a CAPM regression of excess active portfolio return on the excess market return. The alpha ofeach tilt portfolio is a fraction of the alpha of the active portfolio, e.g., alpha for 10% tilt is 10%* (alpha of rt).

Sharpe Sharpe ratio = exrt/std(exrt).

c Measure of an investor's lack of confidence in historical performance. c = 0.5 is used, which means that prospective asset allocations are based on 50% of the historical alpha estimates.

c_Sharpe Sharpe ratio based on reduced alpha = (exrt - c*alpha)/std(exrt) .

M2 M-square measure of performance = Sharpe * std(market excess return) = excess return on a combination of the active portfolio and the riskless asset that has the same standard deviation as the market portfolio.

c_M2 M-square measure based on reduced alpha = c_Sharpe * std(market excess return).

Table 2Legend for all tables in the Monograph

Beta Slope coefficient from a CAPM regression of excess active portfolio return on the excess market return.

std(eA) Standard deviation of residuals from the CAPM regression for active portfolio A.

Shp mkt Sharpe ratio of the value-weight market portfolio.

Info ratio Information ratio = alpha/std(eA).

Shp opt Sharpe ratio of the optimal portfolio with unrestricted short-selling and confidence level c = sqrt(Shp_mkt^2 + [(1-c) * (alpha/std(eA))]^2).

Standard errors Standard errors are given below each estimate.For alpha and beta, the standard errors come from the regression.For Sharpe ratios and information ratios, approximate standard errors are given.

SIZE std(vw) 1/sqrt(T)QUINTILE 17.4% 0.167

% of rt 0% = vwret 20% 40% 50% 60% 80% 100% = rt

exrt 7.4% 7.6% 7.8% 8.0% 8.1% 8.3% 8.6% Alpha Beta std(eA)std(exrt) 17.7% 18.8% 20.4% 21.3% 22.3% 24.6% 27.0% -0.5% 1.24 16.1%

Q1 Alpha 0.0% -0.1% -0.2% -0.3% -0.3% -0.4% -0.5% 2.9% 0.15sharpe 41.5% 40.4% 38.5% 37.3% 36.2% 33.9% 31.7%

c_sharpe 41.5% 40.7% 39.0% 37.9% 36.9% 34.7% 32.7% Shp mkt Info ratio Shp optM2 7.4% 7.1% 6.8% 6.6% 6.4% 6.0% 5.6% 41.5% -3.2% 41.6%

c_M2 7.4% 7.2% 6.9% 6.7% 6.5% 6.1% 5.8% 16.7% 18.1% 16.7%

exrt 7.4% 7.3% 7.3% 7.3% 7.3% 7.3% 7.3% Alpha Beta std(eA)std(exrt) 17.7% 17.5% 17.4% 17.3% 17.2% 17.1% 17.0% 0.3% 0.95 2.3%

Q5 Alpha 0.0% 0.1% 0.1% 0.1% 0.2% 0.2% 0.3% 0.4% 0.02sharpe 41.5% 41.8% 42.1% 42.2% 42.3% 42.5% 42.7%

c_sharpe 41.5% 41.7% 41.8% 41.8% 41.9% 41.9% 41.9% Shp mkt Info ratio Shp optM2 7.4% 7.4% 7.5% 7.5% 7.5% 7.5% 7.6% 41.5% 11.5% 41.9%

c_M2 7.4% 7.4% 7.4% 7.4% 7.4% 7.4% 7.4% 16.7% 18.1% 16.7%

exrt 7.4% 5.6% 3.9% 3.0% 2.2% 0.4% -1.3% Alpha Beta std(eA)std(exrt) 17.7% 13.6% 11.2% 11.0% 11.6% 14.5% 18.8% 0.8% -0.29 18.3%

Q5 - Q1 Alpha 0.0% 0.2% 0.3% 0.4% 0.5% 0.6% 0.8% 3.3% 0.18sharpe 41.5% 41.2% 34.6% 27.3% 18.6% 2.9% -7.0%

c_sharpe 41.5% 40.6% 33.2% 25.6% 16.6% 0.7% -9.1% Shp mkt Info ratio Shp optM2 7.4% 7.3% 6.1% 4.8% 3.3% 0.5% -1.2% 41.5% 4.2% 41.6%

c_M2 7.4% 7.2% 5.9% 4.5% 2.9% 0.1% -1.6% 16.7% 18.1% 16.7%

Performance of Portfolios Tilted Towards Size Quintile PortfoliosTable 3

Figure 1: Size portfolio tilts

Excess Returns

-2%0%2%4%6%8%

10%

0% =vwret

20% 40% 60% 80% 100%= Rt

Q1Q5Q5-Q1

M2

-2%0%2%4%6%8%

10%

0% =vwret

20% 40% 60% 80% 100%= Rt

Q1Q5Q5-Q1

C_M2

-4%-2%0%2%4%6%8%

0% =vwret

20% 40% 60% 80% 100%= Rt

Q1Q5Q5-Q1

BM std(vw) 1/sqrt(T)QUINTILE 17.4% 0.167

% of rt 0% = vwret 20% 40% 50% 60% 80% 100% = rt

exrt 7.4% 7.5% 7.7% 7.8% 7.9% 8.0% 8.2% Alpha Beta std(eA)std(exrt) 17.7% 18.5% 19.6% 20.2% 20.8% 22.1% 23.5% -0.7% 1.21 9.9%

Q1 Alpha 0.0% -0.1% -0.3% -0.4% -0.4% -0.6% -0.7% 1.8% 0.09sharpe 41.5% 40.5% 39.3% 38.5% 37.8% 36.3% 34.8%

c_sharpe 41.5% 40.9% 40.0% 39.4% 38.8% 37.6% 36.3% Shp mkt Info ratio Shp optM2 7.4% 7.2% 6.9% 6.8% 6.7% 6.4% 6.2% 41.5% -7.1% 41.7%

c_M2 7.4% 7.2% 7.1% 7.0% 6.9% 6.6% 6.4% 16.7% 18.1% 16.7%

exrt 7.4% 8.0% 8.7% 9.1% 9.4% 10.1% 10.8% Alpha Beta std(eA)std(exrt) 17.7% 17.2% 16.8% 16.7% 16.7% 16.7% 16.9% 4.7% 0.83 8.4%

Q5 Alpha 0.0% 0.9% 1.9% 2.3% 2.8% 3.7% 4.7% 1.5% 0.08sharpe 41.5% 46.8% 51.8% 54.2% 56.5% 60.6% 64.0%

c_sharpe 41.5% 44.1% 46.3% 47.2% 48.1% 49.4% 50.1% Shp mkt Info ratio Shp optM2 7.4% 8.3% 9.2% 9.6% 10.0% 10.7% 11.3% 41.5% 55.9% 50.1%

c_M2 7.4% 7.8% 8.2% 8.4% 8.5% 8.7% 8.9% 16.7% 18.1% 16.7%

exrt 7.4% 6.4% 5.5% 5.0% 4.5% 3.6% 2.6% Alpha Beta std(eA)std(exrt) 17.7% 13.2% 10.3% 9.8% 10.2% 13.1% 17.6% 5.4% -0.38 16.5%

Q5 - Q1 Alpha 0.0% 1.1% 2.2% 2.7% 3.2% 4.3% 5.4% 3.0% 0.16sharpe 41.5% 48.4% 53.1% 50.7% 44.1% 27.1% 14.9%

c_sharpe 41.5% 44.3% 42.6% 37.0% 28.3% 10.7% -0.5% Shp mkt Info ratio Shp optM2 7.4% 8.6% 9.4% 9.0% 7.8% 4.8% 2.6% 41.5% 32.7% 44.6%

c_M2 7.4% 7.8% 7.5% 6.6% 5.0% 1.9% -0.1% 16.7% 18.1% 16.7%

Performance of Portfolios Tilted Towards Book-to-Market Quintile PortfoliosTable 4

Book-to-market portfolio tiltsFigure 2

Excess Returns

0%2%4%6%8%

10%12%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

M2

0%2%4%6%8%

10%12%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1Q5Q5-Q1

C_M2

-2%0%2%4%6%8%

10%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1Q5Q5-Q1

0.5MOMENTUM std(vw) 1/sqrt(T)QUINTILE 17.4% 0.167

% of rt 0% = vwret 20% 40% 50% 60% 80% 100% = rt

exrt 7.4% 6.7% 6.1% 5.8% 5.5% 4.9% 4.3% Alpha Beta std(eA)std(exrt) 17.7% 18.2% 18.9% 19.3% 19.8% 20.9% 22.1% -3.8% 1.11 10.3%

Q1 alpha 0.0% -0.8% -1.5% -1.9% -2.3% -3.1% -3.8% 1.9% 0.10sharpe 41.5% 37.1% 32.4% 30.1% 27.9% 23.5% 19.5%

c_sharpe 41.5% 39.2% 36.5% 35.1% 33.7% 30.9% 28.2% Shp mkt Info ratio Shp optM2 7.4% 6.6% 5.7% 5.3% 4.9% 4.2% 3.4% 41.5% -37.2% 45.5%

c_M2 7.4% 6.9% 6.5% 6.2% 6.0% 5.5% 5.0% 16.7% 18.1% 16.7%

exrt 7.4% 8.3% 9.2% 9.6% 10.1% 11.0% 11.9% Alpha Beta std(eA)std(exrt) 17.7% 18.0% 18.5% 18.9% 19.3% 20.3% 21.4% 4.1% 1.05 10.8%

Q5 alpha 0.0% 0.8% 1.7% 2.1% 2.5% 3.3% 4.1% 2.0% 0.10sharpe 41.5% 45.8% 49.4% 50.8% 52.1% 54.0% 55.4%

c_sharpe 41.5% 43.5% 44.9% 45.3% 45.6% 45.8% 45.7% Shp mkt Info ratio Shp optM2 7.4% 8.1% 8.7% 9.0% 9.2% 9.6% 9.8% 41.5% 38.3% 45.7%

c_M2 7.4% 7.7% 7.9% 8.0% 8.1% 8.1% 8.1% 16.7% 18.1% 16.7%

exrt 7.4% 7.4% 7.4% 7.5% 7.5% 7.5% 7.6% Alpha Beta std(eA)std(exrt) 17.7% 14.4% 12.4% 12.0% 12.3% 14.2% 17.4% 8.0% -0.06 17.6%

Q5 - Q1 alpha 0.0% 1.6% 3.2% 4.0% 4.8% 6.4% 8.0% 3.2% 0.17sharpe 41.5% 51.4% 60.2% 61.9% 60.9% 53.0% 43.4%

c_sharpe 41.5% 45.8% 47.3% 45.3% 41.4% 30.5% 20.5% Shp mkt Info ratio Shp optM2 7.4% 9.1% 10.7% 11.0% 10.8% 9.4% 7.7% 41.5% 45.3% 47.3%

c_M2 7.4% 8.1% 8.4% 8.0% 7.3% 5.4% 3.6% 16.7% 18.1% 16.7%

Performance of Portfolios Tilted Towards Momentum Quintile PortfoliosTable 5

Momentum portfolio tiltsFigure 3

Excess Returns

0%2%4%6%8%

10%12%14%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

M2

0%2%4%6%8%

10%12%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

C_M2

0%

2%

4%

6%

8%

10%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

From quintile in yeat t

To quintile in year t+1 Size Book-to-Market Momentum

1 missing 34.6% 18.3% 19.4%1 1 43.9% 54.4% 20.7%1 2 18.7% 20.5% 16.4%1 3 2.6% 4.7% 14.5%1 4 0.2% 1.6% 14.7%1 5 0.0% 0.5% 14.2%

2 missing 14.9% 14.9% 13.5%2 1 17.3% 12.3% 16.1%2 2 48.3% 43.2% 20.9%2 3 17.9% 22.2% 20.3%2 4 1.6% 5.9% 17.2%2 5 0.1% 1.5% 11.9%

3 missing 10.8% 14.4% 12.3%3 1 1.9% 1.9% 14.7%3 2 15.0% 16.6% 20.3%3 3 56.6% 39.5% 21.8%3 4 15.5% 22.9% 19.0%3 5 0.2% 4.7% 11.9%

4 missing 9.3% 13.9% 12.8%4 1 0.2% 0.5% 16.1%4 2 0.8% 3.7% 18.4%4 3 11.6% 18.8% 19.7%4 4 68.6% 42.7% 19.3%4 5 9.5% 20.4% 13.8%

5 missing 7.4% 16.0% 15.7%5 1 0.0% 0.3% 20.6%5 2 0.0% 0.8% 14.9%5 3 0.1% 3.5% 13.6%5 4 6.2% 16.8% 16.6%5 5 86.2% 62.6% 18.7%

The percentages refer to the probability that a stock belonging to a given quintile portfolioof stocks ranked according to firm size, book-to-market, or momentum (I.e., past year's return) in year t would be missing or belong to another quintile portfolio in year t+1. A stock is missing in year t+1 if it is delisted or if it does not meet the investment criteria (I.e., price should be greater than $2 and the market capitalization should exceed that of the lowest decile of market capitalization for NYSE stocks).

Table 6One-year-ahead transition probabilities for stocks in a given quintile

Quintile in year t

Quintile formation Year t Year t+1 Year t+2 Year t+3 Year t+4 Year t+5

1 43 57 69 85 102 114 2 77 89 103 118 133 152 3 148 164 182 205 227 257 4 356 390 425 470 519 571 5 2,995 3,167 3,325 3,503 3,692 3,885

1 0.20 0.29 0.36 0.41 0.47 0.502 0.44 0.51 0.57 0.61 0.66 0.703 0.66 0.72 0.76 0.80 0.82 0.854 0.92 0.95 0.97 0.98 1.00 1.005 3.00 2.90 2.80 2.75 2.74 2.65

1 -24.2% 15.7% 20.2% 20.7% 20.5% 19.8%2 -0.6% 15.1% 17.0% 16.5% 17.7% 17.8%3 14.0% 16.2% 16.4% 16.9% 16.4% 16.9%4 32.4% 17.4% 16.6% 17.6% 17.1% 16.6%5 92.3% 21.8% 16.9% 17.5% 17.0% 17.3%

Future market values of stocks in a given size quintile in year t, in $ millions

Future book-to-market values of stocks in a given book-to-market quintile in year t

Future momentum (i.e., past year's return) on stocks in a given momentum quintile in year t

Table 7Firm characteristics of stocks through event time

SIZE std(vw) 1/sqrt(T)QUINTILE 17.6% 0.169

% of rt 0% = vwret 20% 40% 50% 60% 80% 100% = rt

exrt 7.1% 7.6% 8.2% 8.5% 8.7% 9.3% 9.8% Alpha Beta std(eA)std(exrt) 17.9% 18.8% 20.1% 20.9% 21.8% 23.7% 25.9% 1.4% 1.18 15.3%

Q1 Alpha 0.0% 0.3% 0.6% 0.7% 0.9% 1.1% 1.4% 2.8% 0.15sharpe 39.7% 40.7% 40.7% 40.5% 40.1% 39.0% 37.9%

c_sharpe 39.7% 39.9% 39.3% 38.7% 38.1% 36.6% 35.1% Shp mkt Info ratio Shp optM2 7.1% 7.3% 7.3% 7.2% 7.2% 7.0% 6.8% 39.7% 9.4% 40.0%

c_M2 7.1% 7.1% 7.0% 6.9% 6.8% 6.6% 6.3% 16.9% 18.2% 16.9%

exrt 7.1% 7.0% 6.9% 6.9% 6.8% 6.7% 6.7% Alpha Beta std(eA)std(exrt) 17.9% 17.7% 17.5% 17.5% 17.4% 17.3% 17.2% -0.1% 0.95 3.1%

Q5 Alpha 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% -0.1% 0.6% 0.03sharpe 39.7% 39.6% 39.5% 39.4% 39.3% 39.0% 38.7%

c_sharpe 39.7% 39.6% 39.5% 39.4% 39.4% 39.1% 38.9% Shp mkt Info ratio Shp optM2 7.1% 7.1% 7.1% 7.0% 7.0% 7.0% 6.9% 39.7% -1.9% 39.7%

c_M2 7.1% 7.1% 7.1% 7.1% 7.0% 7.0% 7.0% 16.9% 18.2% 16.9%

exrt 7.1% 5.0% 3.0% 2.0% 0.9% -1.1% -3.2% Alpha Beta std(eA)std(exrt) 17.9% 13.9% 11.5% 11.2% 11.6% 14.2% 18.2% -1.5% -0.23 18.0%

Q5 - Q1 alpha 0.0% -0.3% -0.6% -0.7% -0.9% -1.2% -1.5% 3.3% 0.17sharpe 39.7% 36.2% 26.1% 17.6% 8.1% -7.8% -17.4%

c_sharpe 39.7% 37.3% 28.7% 20.9% 12.0% -3.6% -13.3% Shp mkt Info ratio Shp optM2 7.1% 6.5% 4.7% 3.2% 1.5% -1.4% -3.1% 39.7% -8.3% 39.9%

c_M2 7.1% 6.7% 5.1% 3.7% 2.1% -0.6% -2.4% 16.9% 18.2% 16.9%

Table 8One Year Later Performance of Portfolios Tilted Towards Size Quintile Portfolios

Size tilt portfolio performance one year laterFigure 4

Excess Returns

-4%-2%0%2%4%6%8%

10%12%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

M2

-4%

-2%

0%

2%

4%

6%

8%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

C_M2

-4%

-2%

0%

2%

4%

6%

8%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

BM std(vw) 1/sqrt(T)QUINTILE 17.6% 0.169

% of rt 0% = vwret 20% 40% 50% 60% 80% 100% = rt

exrt 7.1% 6.7% 6.3% 6.2% 6.0% 5.6% 5.2% Alpha Beta std(eA)std(exrt) 17.9% 18.2% 18.7% 19.0% 19.3% 20.0% 20.9% -2.4% 1.08 8.1%

Q1 Alpha 0.0% -0.5% -1.0% -1.2% -1.5% -1.9% -2.4% 1.5% 0.08sharpe 39.7% 36.9% 33.9% 32.4% 30.9% 27.9% 25.0%

c_sharpe 39.7% 38.2% 36.5% 35.6% 34.7% 32.8% 30.8% Shp mkt Info ratio Shp optM2 7.1% 6.6% 6.1% 5.8% 5.5% 5.0% 4.5% 39.7% -29.8% 42.4%

c_M2 7.1% 6.8% 6.5% 6.4% 6.2% 5.9% 5.5% 16.9% 18.2% 16.9%

exrt 7.1% 7.7% 8.3% 8.6% 8.8% 9.4% 10.0% Alpha Beta std(eA)std(exrt) 17.9% 17.1% 16.5% 16.2% 15.9% 15.5% 15.2% 4.5% 0.78 6.2%

Q5 Alpha 0.0% 0.9% 1.8% 2.2% 2.7% 3.6% 4.5% 1.1% 0.06sharpe 39.7% 44.8% 50.1% 52.8% 55.5% 60.8% 65.9%

c_sharpe 39.7% 42.2% 44.7% 45.9% 47.1% 49.3% 51.1% Shp mkt Info ratio Shp optM2 7.1% 8.0% 9.0% 9.5% 9.9% 10.9% 11.8% 39.7% 72.7% 53.8%

c_M2 7.1% 7.6% 8.0% 8.2% 8.4% 8.8% 9.2% 16.9% 18.2% 16.9%

exrt 7.1% 6.6% 6.2% 5.9% 5.7% 5.2% 4.8% Alpha Beta std(eA)std(exrt) 17.9% 13.5% 9.9% 8.8% 8.4% 10.0% 13.5% 6.9% -0.30 12.6%

Q5 - Q1 alpha 0.0% 1.4% 2.8% 3.5% 4.1% 5.5% 6.9% 2.3% 0.12sharpe 39.7% 49.2% 62.1% 67.2% 67.6% 52.6% 35.3%

c_sharpe 39.7% 44.1% 48.2% 47.7% 43.1% 24.9% 9.8% Shp mkt Info ratio Shp optM2 7.1% 8.8% 11.1% 12.0% 12.1% 9.4% 6.3% 39.7% 54.7% 48.2%

c_M2 7.1% 7.9% 8.6% 8.5% 7.7% 4.5% 1.8% 16.9% 18.2% 16.9%

One Year Later Performance of Portfolios Tilted Towards Book-to-Market Quintile PortfoliosTable 9

Book-to-market tilt portfolio performance one year laterFigure 5

Excess Returns

-0.05

0

0.05

0.1

0.15

0%=v

wre

t

20%

40%

60%

80%

100% =R

t

1

5

Q5-Q1

M2

-0.04-0.02

00.020.040.060.08

0%=v

wre

t

20%

40%

60%

80%

100% =R

t15Q5-Q1

C_M2

-0.04-0.02

00.020.040.060.08

0%=v

wre

t

20%

40%

60%

80%

100% =R

t

15Q5-Q1

Excess Returns

0%2%4%6%8%

10%12%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

M2

0%2%4%6%8%

10%12%14%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

C_M2

0%

2%

4%

6%

8%

10%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

Momentum std(vw) 1/sqrt(T)Quintile 17.6% 0.169

% of rt 0% = vwret 20% 40% 50% 60% 80% 100% = rt

exrt 7.1% 7.5% 8.0% 8.2% 8.4% 8.8% 9.2% Alpha Beta std(eA)std(exrt) 17.9% 18.1% 18.5% 18.8% 19.1% 19.8% 20.7% 1.9% 1.03 9.6%

Q1 Alpha 0.0% 0.4% 0.8% 1.0% 1.1% 1.5% 1.9% 1.7% 0.09sharpe 39.7% 41.6% 43.0% 43.5% 43.9% 44.4% 44.6%

c_sharpe 39.7% 40.5% 40.9% 41.0% 40.9% 40.6% 40.0% Shp mkt Info ratio Shp optM2 7.1% 7.4% 7.7% 7.8% 7.9% 7.9% 8.0% 39.7% 19.9% 40.9%

c_M2 7.1% 7.3% 7.3% 7.3% 7.3% 7.3% 7.1% 16.9% 18.2% 16.9%

exrt 7.1% 6.9% 6.7% 6.6% 6.5% 6.3% 6.1% Alpha Beta std(eA)std(exrt) 17.9% 18.6% 19.4% 19.8% 20.2% 21.2% 22.2% -2.2% 1.17 7.5%

Q5 Alpha 0.0% -0.4% -0.9% -1.1% -1.3% -1.8% -2.2% 1.4% 0.07sharpe 39.7% 37.2% 34.6% 33.4% 32.1% 29.7% 27.4%

c_sharpe 39.7% 38.4% 36.9% 36.2% 35.4% 33.9% 32.4% Shp mkt Info ratio Shp optM2 7.1% 6.7% 6.2% 6.0% 5.8% 5.3% 4.9% 39.7% -29.5% 42.3%

c_M2 7.1% 6.9% 6.6% 6.5% 6.3% 6.1% 5.8% 16.9% 18.2% 16.9%

exrt 7.1% 5.1% 3.0% 2.0% 1.0% -1.1% -3.1% Alpha Beta std(eA)std(exrt) 17.9% 15.1% 13.0% 12.3% 11.9% 12.3% 13.9% -4.1% 0.14 13.9%

Q5 - Q1 alpha 0.0% -0.8% -1.6% -2.1% -2.5% -3.3% -4.1% 2.5% 0.13sharpe 39.7% 33.5% 23.2% 16.2% 8.1% -8.8% -22.4%

c_sharpe 39.7% 36.3% 29.6% 24.6% 18.4% 4.6% -7.6% Shp mkt Info ratio Shp optM2 7.1% 6.0% 4.2% 2.9% 1.4% -1.6% -4.0% 39.7% -29.6% 42.4%

c_M2 7.1% 6.5% 5.3% 4.4% 3.3% 0.8% -1.4% 16.9% 18.2% 16.9%

One Year Later Performance of Portfolios Tilted Towards Momentum Quintile PortfoliosTable 10

Momentum tilt portfolio performance one year laterFigure 6

Excess Returns

-4%-2%0%2%4%6%8%

10%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

M2

-6%-4%-2%0%2%4%6%8%

10%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

C_M2

-2%

0%

2%

4%

6%

8%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

Q5-Q1

SIZE std(vw) 1/sqrt(T)QUINTILE 13.2% 0.167

% of rt 0% = vwret 20% 40% 50% 60% 80% 100% = rt

exrt 5.1% 5.8% 6.5% 6.8% 7.2% 7.9% 8.6% Alpha Beta std(eA)std(exrt) 13.5% 15.3% 17.8% 19.2% 20.6% 23.8% 27.0% 0.8% 1.54 17.6%

Q1 alpha 0.0% 0.2% 0.3% 0.4% 0.5% 0.6% 0.8% 3.1% 0.22sharpe 37.5% 37.6% 36.4% 35.5% 34.7% 33.1% 31.7%

c_sharpe 37.5% 37.1% 35.5% 34.5% 33.6% 31.8% 30.2% Shp mkt Info ratio Shp optM2 5.1% 5.1% 4.9% 4.8% 4.7% 4.5% 4.3% 37.5% 4.5% 37.6%

c_M2 5.1% 5.0% 4.8% 4.7% 4.5% 4.3% 4.1% 16.7% 17.8% 16.7%

optimal TB weights wM optimal M2 std(eA) std(vw) w095.5% 5.1% 17.6% 13.2% 4.4%

exrt 5.1% 5.5% 5.9% 6.2% 6.4% 6.8% 7.3% Alpha Beta std(eA)std(exrt) 13.5% 14.1% 14.7% 15.1% 15.5% 16.2% 17.0% 1.1% 1.21 4.6%

Q5 alpha 0.0% 0.2% 0.4% 0.6% 0.7% 0.9% 1.1% 0.8% 0.06sharpe 37.5% 39.0% 40.3% 40.8% 41.3% 42.1% 42.7%

c_sharpe 37.5% 38.3% 38.8% 39.0% 39.1% 39.3% 39.5% Shp mkt Info ratio Shp optM2 5.1% 5.3% 5.4% 5.5% 5.6% 5.7% 5.8% 37.5% 24.2% 39.4%

c_M2 5.1% 5.2% 5.2% 5.3% 5.3% 5.3% 5.3% 16.7% 17.8% 16.7%

optimal TB weights wM optimal M2 std(eA) std(vw) w0-11.9% 5.3% 4.6% 13.2% 90.3%

Table 11Performance of Portfolios Tilted Towards Size Quintile Portfolios: Index is 60% Equity and 40% Bonds

Size stock portfolio and bond IndexFigure 7

Excess Returns

0%2%4%6%8%

10%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

M2

0%1%2%3%4%5%6%7%

0% = vwret 20% 40% 60% 80% 100% = rt

Q1

Q5

C_M2

0%1%2%3%4%5%6%

0% = vwret 20% 40% 60% 80% 100% = rt

Q1

Q5

BM std(vw) 1/sqrt(T)QUINTILE 13.2% 0.167

% of rt 0% = vwret 20% 40% 50% 60% 80% 100% = rt

exrt 5.1% 5.7% 6.3% 6.6% 6.9% 7.6% 8.2% Alpha Beta std(eA)std(exrt) 13.5% 15.0% 16.8% 17.8% 18.9% 21.2% 23.5% 0.7% 1.48 12.8%

Q1 alpha 0.0% 0.1% 0.3% 0.4% 0.4% 0.6% 0.7% 2.3% 0.16sharpe 37.5% 38.0% 37.5% 37.1% 36.7% 35.7% 34.8%

c_sharpe 37.5% 37.5% 36.7% 36.1% 35.6% 34.4% 33.3% Shp mkt Info ratio Shp optM2 5.1% 5.1% 5.1% 5.0% 4.9% 4.8% 4.7% 37.5% 5.6% 37.6%

c_M2 5.1% 5.1% 4.9% 4.9% 4.8% 4.6% 4.5% 16.7% 17.8% 16.7%

optimal TB weights wM optimal M2 std(eA) std(vw) w092.1% 5.1% 12.8% 13.2% 7.6%

exrt 5.1% 6.2% 7.4% 7.9% 8.5% 9.6% 10.8% Alpha Beta std(eA)std(exrt) 13.5% 13.8% 14.4% 14.7% 15.1% 15.9% 16.9% 5.2% 1.10 8.3%

Q5 alpha 0.0% 1.0% 2.1% 2.6% 3.1% 4.2% 5.2% 1.5% 0.10sharpe 37.5% 44.9% 51.2% 53.9% 56.4% 60.6% 64.0%

c_sharpe 37.5% 41.1% 43.9% 45.0% 46.0% 47.4% 48.4% Shp mkt Info ratio Shp optM2 5.1% 6.0% 6.9% 7.3% 7.6% 8.2% 8.6% 37.5% 63.5% 49.2%

c_M2 5.1% 5.5% 5.9% 6.1% 6.2% 6.4% 6.5% 16.7% 17.8% 16.7%

optimal TB weights wM optimal M2 std(eA) std(vw) w0-51.2% 6.6% 8.3% 13.2% 131.9%

Table 12Performance of Portfolios Tilted Towards Book-to-Market Quintile Portfolios: Index is 60% Equity and 40% Bonds

Book-to-market stock portfolio and bond IndexFigure 8

Excess Returns

0%2%4%6%8%

10%12%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

M2

0%

2%

4%

6%

8%

10%

0% = vwret 20% 40% 60% 80% 100% = rt

Q1

Q5

C_M2

0%1%2%3%4%5%6%7%

0% = vwret 20% 40% 60% 80% 100% = rt

Q1

Q5

22

Momentum std(vw) 1/sqrt(T)QUINTILE 13.2% 0.166667

% of rt 0% = vwret 20% 40% 50% 60% 80% 100% = rt

exrt 5.1% 4.9% 4.8% 4.7% 4.6% 4.5% 4.3% Alpha Beta std(eA)std(exrt) 13.5% 14.7% 16.2% 17.1% 18.0% 20.0% 22.1% -2.6% 1.37 12.3%

Q1 alpha 0.0% -0.5% -1.1% -1.3% -1.6% -2.1% -2.6% 2.2% 0.15sharpe 37.5% 33.4% 29.3% 27.4% 25.6% 22.3% 19.5%

c_sharpe 37.5% 35.2% 32.6% 31.3% 30.0% 27.6% 25.4% Shp mkt Info ratio Shp optM2 5.1% 4.5% 4.0% 3.7% 3.4% 3.0% 2.6% 37.5% -21.4% 39.0%

c_M2 5.1% 4.8% 4.4% 4.2% 4.0% 3.7% 3.4% 16.7% 17.8% 16.7%

optimal TB weights wM optimal M2 std(eA) std(vw) w0127.0% 5.3% 12.3% 13.2% -30.0%

exrt 5.1% 6.4% 7.8% 8.5% 9.1% 10.5% 11.9% Alpha Beta std(eA)std(exrt) 13.5% 14.6% 16.0% 16.8% 17.7% 19.5% 21.4% 5.0% 1.35 11.5%

Q5 alpha 0.0% 1.0% 2.0% 2.5% 3.0% 4.0% 5.0% 2.1% 0.14sharpe 37.5% 44.0% 48.6% 50.3% 51.8% 53.9% 55.4%

c_sharpe 37.5% 40.5% 42.3% 42.8% 43.2% 43.6% 43.6% Shp mkt Info ratio Shp optM2 5.1% 5.9% 6.6% 6.8% 7.0% 7.3% 7.5% 37.5% 43.8% 43.5%

c_M2 5.1% 5.5% 5.7% 5.8% 5.8% 5.9% 5.9% 16.7% 17.8% 16.7%

optimal TB weights wM optimal M2 std(eA) std(vw) w015.6% 5.9% 11.5% 13.2% 65.3%

Table 13Performance of Portfolios Tilted Towards Momentum Quintile Portfolios: Index is 60% Equity and 40% Bonds

Momentum stock portfolio and bond Index

Figure 9

Excess Returns

0%2%4%6%8%

10%12%14%

0% =vwret

20% 40% 60% 80% 100%= rt

Q1

Q5

M2

0%1%2%3%4%5%6%7%8%

0% = vwret 20% 40% 60% 80% 100% = rt

Q1

Q5

C_M2

0%1%2%3%4%5%6%7%

0% = vwret 20% 40% 60% 80% 100% = rt

Q1

Q5

No Short Selling No Short SellingC=0 C=0.5Large Stocks 0% 0%Value Stocks 77% 77%Winner Stocks 23% 23%Market Index 0% 0%

Unrestricted Optimal Allocation c = 0Size Spread Portfolio 8%Value Spread Portfolio 27%Momentum Spread Portfolio 30%Market Index 35%

Unrestricted Optimal Allocation c = 0.5Size Spread Portfolio 5%Value Spread Portfolio 20%Momentum Spread Portfolio 22%Market Index 41%

Figure 10: Optimal allocation with three anomalies

No Short Selling (C = 0 or C = 0.5)

0%

77%

23%

0%

Large StocksValue StocksWinner StocksMarket Index

Unrestricted Optimal Allocation, C = 08%

27%

30%

35%Size Spread Portfolio

Value SpreadPortfolioMomentum SpreadPortfolioMarket Index

Unrestricted Optimal Allocation, C = 0.56%

23%

25%

46%

Size Spread Portfolio

Value SpreadPortfolioMomentum SpreadPortfolioMarket Index