Asset allocation paper June 2002 - mit.edukothari/attach/Asset allocation paper June 2002.pdf ·...
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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 [email protected]
and
Jay Shanken
William E. Simon Graduate School of Business Administration University of Rochester, Rochester, NY 14627
(716) 275-4896 [email protected]
First draft: December 2000 Current version: June 2002
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Acknowledgements
We thank a reviewer for helpful suggestions and Michela Verardo for excellent research assistance. We are grateful to the Research Foundation of the Institute of Chartered Financial Analysts and the Association for Investment Management and Research, the Bradley Policy Research Center at the Simon School, and the John M. Olin Foundation for financial support.
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, we
consider the implications for asset allocation of the three most prominent CAPM
“anomalies”: expected return effects that are negatively related to firm size (market
capitalization), and positively related to firm book-to-market ratios and past-year
momentum. For each anomaly, we estimate the amount that investors should tilt their
portfolios away from the market index, toward the anomaly-based portfolio (or spread), in
order to exploit the gains to efficiency.2 However, the same principles of modern portfolio
theory can be applied to other investment strategies that are expected to generate positive
risk-adjusted returns (e.g., an earnings-based strategy, accruals strategy, or a trading-
volume-based anomaly).
The portfolio improvement obtained by tilting an index toward an anomaly-based
strategy depends, not only on the risk-adjusted expected returns of the three strategies, but
also on residual risk, i.e., that portion of risk that is not related to variation in the market
index returns. This risk measure has received little attention in the academic literature, but
it is important for asset allocation. We also follow up on the performance of each strategy
in the second year after portfolio formation to get a rough indication of the relevance of
portfolio rebalancing. Finally, we examine asset allocation across all three anomalies and
the market index.
Our focus on the three most prominent anomalies should not be interpreted as
suggesting that we believe these anomalies will persist in the future. Each investor will
1 Two notable exceptions are the recent work of Pastor (2000), which is closely related to our analysis, and Haugen and Baker (1996). 2 For a practical guide to implementing an active portfolio investment management strategy that is grounded in modern finance, see Waring, Whitney, Pirone, and Castille (2000).
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have his or her own beliefs about the likely performance of these and other strategies.
Traditional statistical tests of significance, while useful in many contexts, are not
particularly well suited to investment decision-making in this sort of context. In recent
years, Bayesian statistical methods have begun to achieve greater prominence in
addressing asset allocation problems.3 Part of the appeal of the Bayesian perspective is
that it provides the analyst or investor a rigorous framework in which to combine
somewhat qualitative judgments about future returns with the statistical evidence in
historical data. Such judgments or “prior beliefs” might be based on an analyst’s views
concerning the ability of financial markets to efficiently process information and the speed
with which this occurs. Related opinions about the extent to which expected returns are
compensation for risk or, instead, induced by mispricing and behavioral biases are also
relevant.
While academic literature in this area sometimes focuses on very technical
mathematical issues, the main ideas are fairly simple and very intuitive. We provide a
basic introduction to Bayesian methods, which will hopefully bring the reader close to the
state-of-the-art fairly quickly. These methods are then applied in our portfolio analysis of
expected return anomalies. Good quantitative money managers also recognize the
inevitable influence that repeated searches through the historical evidence (“data-mining”)
can have on one’s views and the need to adjust for this influence. They will typically be
inclined to try to exploit a pattern observed in past data if there is a good “story” to go with
it. We consider this issue as well.
Outline of the monograph. Chapter II reviews the finance theory on asset
allocation in the framework of the capital asset pricing model (CAPM). The chapter
3 See Kandel and Stambaugh (1996).
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reviews portfolio theory, the CAPM, and the efficient market hypothesis. Chapter III
reviews recent evidence challenging the efficient market hypothesis. We summarize
findings suggesting economically significant profitability of trading strategies that invest in
value, momentum, and small stocks. We also discuss the implications of the evidence
indicative of market inefficiency for optimal asset allocation. Chapter IV presents the
results of our analysis of historical data and its implications for improving asset allocation
by tilting the market index toward portfolios of value, momentum, or small stocks.
Chapter V presents the intuition for and an application of a Bayesian perspective on
optimal asset allocation. Chapter VI examines the tilt portfolios’ performance over a two-
year horizon. In Chapter VII we consider the joint optimization problem in which all three
anomalies are considered simultaneously. Chapter VIII summarizes the monograph and
discusses its implications and directions for future work.
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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 discuss portfolio theory, the CAPM, and the efficient market
hypothesis.4
Portfolio theory
In a mean-variance setting, it is assumed that an investor’s utility increases with the
mean and decreases in the variance of overall portfolio returns.5 The mean is the expected
return on the portfolio while the variance is the measure of the portfolio’s total risk. The
efficient frontier is a graph of the set of portfolios with highest expected return for each
given level of portfolio return variance. Thus, modern portfolio theory implies that, in
order to maximize expected utility, an investor should choose a portfolio on the efficient
frontier. In 1952, Harry Markowitz developed optimization techniques for deriving the
efficient frontier of risky assets. The inputs to this derivation are estimated values of
expected return, standard deviation of return, and pairwise covariances of returns for all
available risky securities.
An investor’s portfolio selection problem is simplified with the availability of a
risk-free asset. An opportunity to invest in risky and risk-free assets implies that all
efficient portfolios consist of combinations of the risk-free asset and a unique “tangency”
4 For a detailed treatment of the concepts in this chapter, see Bodie, Kane, and Marcus (1999), chapters 6-9 and 12, or Ross, Westerfield, and Jaffe (1996), chapters 9 and 10. 5 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.
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portfolio of the risky assets. Investors who are relatively more risk-averse will invest a
larger fraction of their assets in the risk-free asset, whereas relatively more risk-tolerant
investors will opt for a greater fraction of their investment in the tangency portfolio. All of
these combinations of the tangency portfolio and the risk-free asset lie on a straight line
when expected return is plotted against standard deviation of return. This line, called the
capital market line, is the efficient frontier and represents the best possible combinations of
portfolio expected return and standard deviation.
The CAPM
The Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965),
builds on Markowitz’s portfolio theory ideas and further simplifies an investor’s asset
allocation decision. The CAPM is derived with an additional critical assumption that
investors have homogenous expectations, which means that all market participants have
identical beliefs about securities’ expected returns, standard deviations, and pairwise
covariances. With homogenous expectations and the same investment horizon, all
investors would arrive at the same efficient frontier. Therefore, they would hold
combinations of the same tangency portfolio and the risk-free asset. Since total investor
demand for assets must equal the supply, in equilibrium, it follows that the tangency
portfolio is the value-weighted portfolio of all risky assets in the economy, called the
market portfolio.
The CAPM gives rise to a mathematically elegant relation between the expected
rate of return on a security and its risk relative to the market portfolio. Specifically, the
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theory implies that expected return is an increasing linear function of its covariance risk or
beta. Beta is defined as
βi = Cov(Ri, Rm)/Var(Rm)
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 + βi (E(Rm) – Rf),
where E(Ri) is security i’s expected rate of return, Rf is the risk free rate of return, and
(E(Rm) – Rf) is the market risk premium. In addition to its importance in portfolio
analysis, beta is often used in corporate valuation and investment (i.e., capital budgeting)
decisions.
Efficient market hypothesis6
The efficient market hypothesis states that security prices rapidly and accurately
reflect all information that is available at a given point in time.7 Security markets tend
toward (informational) efficiency because a large number of market participants actively
compete among themselves to gather and process information and trade on that
information. Ideally, this process moves security prices until those prices reflect the
6 For detailed reviews of the efficient markets hypothesis and empirical literature on market efficiency, see Fama (1970, 1991) and MacKinlay (1997). 7 This notion of informational financial market efficiency should not be confused with the earlier concept of the mean-variance efficiency of a portfolio.
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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.
In the short-run, prices may not completely adjust to new information due to
various trading costs. More generally, markets may be inefficient because of behavioral
biases in investor beliefs (excessive-optimism or pessimism, overconfidence, etc.).
Deviations from efficiency can persist if, in betting that the inefficiency will be corrected
over a given horizon, the arbitrageur is exposed to substantial risk that the “mispricing”
will get worse before it gets better.8
Portfolio theory, the CAPM, and the efficient market hypothesis jointly have
remarkably simple implications for investors’ asset allocation decisions. Investors should
hold a combination of the risky market portfolio and the risk-free asset and the investment
approach should be a passive buy-and-hold strategy (i.e., invest in index funds).9 The
picture is less clear, however, if we believe that the CAPM does not hold and if we doubt
market efficiency. We explore the attendant complexities in the remaining chapters of this
monograph.
A large body of evidence suggests that security returns exhibit significant
predictable deviations from the CAPM and that the capital markets are inefficient in
certain respects. As discussed in detail later on, these CAPM deviations or risk-adjusted
returns are captured by a statistical parameter referred to as Jensen’s alpha. Investors’
views about these capital market issues can have important implications for asset
8 See Shleifer and Vishny (2000). 9 The proportion of assets invested in the market portfolio is a function of the investor’s risk tolerance which may change endogenously with their wealth.
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allocation by affecting their confidence that positive alphas observed in the past will persist
in the future. Therefore, we briefly review the relevant theory and evidence in the next
chapter.
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Chapter III
Recent evidence challenging market efficiency and its implications for asset allocation
This chapter summarizes recent evidence indicating informational inefficiency in
the U.S. and international capital markets. Some of the evidence suggests capital markets
take several years to reflect information about underlying economic fundamentals in stock
prices. This evidence of apparent market inefficiency has implications for an investor’s
asset allocation decisions. Informed investors should tilt their portfolios away from the
market portfolio and in a direction that exploits the inefficiency. The optimal extent of
such tilting will depend on risk and other factors that are considered later.
Return predictability in short-window event studies
There is overwhelming evidence that security prices rapidly adjust to reflect new
information reaching the market.10 Starting with Fama, Fisher, Jensen and Roll (1969),
short-window event-study research documents the market’s quick response to new
information. This research analyzes large samples of firms experiencing a wide range of
events like stock splits, merger announcements, management changes, dividend
announcements, earnings releases, etc. There is evidence that the market reacts within
minutes of public announcements of firm-specific information like earnings and dividends
or macroeconomic information like inflation data, or interest rates. Rapid adjustment of
10 For an excellent summary of this research, see Bodie, Kane, and Marcus (1999), chapters 12 and 13.
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prices to new information is consistent with market efficiency, but efficiency also requires
that this response is, in some sense, rational or unbiased. If both conditions hold, any
opportunity to benefit from the news is short lived and investors only earn a normal rate of
return thereafter.
Longer-horizon return predictability
In the past two decades, a large body of academic and practitioner research has
begun to challenge market efficiency.11 Mounting evidence suggests that revisions in
beliefs in response to new information do not always reflect unbiased forecasts of future
economic conditions now, and that it may take several years before prices incorporate the
full impact of the news. As the market seems to correct the initial mispricing over several
subsequent years, long-term abnormal expected returns may be possible for an informed
investor who tries to profit from this correction.
Behavioral models of investor behavior hypothesize systematic under- and over-
reaction to corporate news as a result of investors’ behavioral biases or limited capability
to process information. Barberis, Shleifer, and Vishny (1998), Daniel, Hirshleifer, and
Subramanyam (1998 and 2001), and Hong and Stein (1999) develop models to explain the
apparent predictability of stock returns at various horizons. These models draw upon
experimental evidence and theories of human judgment bias or limited information-
processing capabilities, as developed in cognitive psychology and related fields.
The representativeness bias (Kahneman and Tversky, 1982) causes people to
over-weight information patterns observed in past data, which might just be random.
11 The discussion in this chapter draws on Fama (1998).
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Since the patterns are not really descriptive of the true properties of the underlying process,
they are not likely to persist. For example, investors might extrapolate a firm’s past history
of high sales growth and thus overreact to sales news (see Lakonishok, Shleifer, and
Vishny, 1994, and DeBondt and Thaler, 1980 and 1985).
On the other hand, investors may be slow to update their beliefs in the face of new
evidence as a result of the conservatism bias (Edwards, 1968). This can contribute to
investor under-reaction to news and lead to short-term momentum in stock prices (e.g.,
Jegadeesh, 1990, and Jegadeesh and Titman, 1993). The post-earnings announcement
drift, i.e., the tendency of stock prices to drift in the direction of earnings news for three-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 to the initial private information that induces momentum.
Overconfidence about private information also causes investors to downplay the
importance of publicly disseminated information. Therefore, information releases like
earnings announcements generate incomplete price adjustments in this context.
Subsequent earnings outcomes eventually reveal the true implications of the earlier
evidence, however, resulting in predictable price reversals over long horizons.
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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 (2002)
demonstrates that anomalous-looking patterns in returns can also arise in a model in which
fully rational investors gradually learn about certain features of the economic environment.
These patterns would be observed in the data with hindsight, but could not be exploited by
investors in real time. Clearly, sorting out all these issues is a challenging task.
Next, we review the evidence indicating return predictability. However, we caution
the reader that, in addition to the unresolved theoretical issues, there is no consensus
among academics on the interpretation of the existing empirical evidence. In particular,
Fama (1998) argues that much of the evidence on abnormal long-run return performance is
questionable because of methodological limitations and the more general effect of data
mining.
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Evidence on return predictability
Research indicates long-horizon predictability of returns following a variety of
corporate events and past security price performance. The corporate events include stock
splits, share repurchases, extreme earnings performance announcements, bond rating
changes, dividend initiations and omissions, seasoned equity offerings, initial public
offerings, etc. Evidence of long-horizon predictability following corporate events and past
security price performance appears in the following studies (see Fama, 1998, for a detailed
discussion). Fama, Fisher, Jensen, and Roll (1969), and Ikenberry, Rankine, and Stice
(1995) examine price performance following stock splits; Ibbotson (1975) and Loughran
and Ritter (1995) study post-IPO price performance; Loughran and Ritter (1995) document
negative abnormal returns after seasoned equity offerings; Asquith (1983) and Agrawal,
Jaffe, and Mandelker (1992) estimate bidder firms’ price performance; dividend initiations
and omissions are examined in Michaely, Thaler, and Womack (1995); performance
following proxy fights is studied in Ikenberry and Lakonishok (1993); Ikenberry,
Lakonishok, and Vermaelen (1995) and Mitchell and Stafford (2000) examine returns
following open market share repurchases; and, Litzenberger, Joy, and Jones (1971), Foster,
Olsen, and Shevlin (1984) and Bernard and Thomas (1990) study post-earnings
announcement returns.
The main conclusion from these studies is that, in many cases, the magnitude of
abnormal returns is not only statistically highly significant, but economically large as well.
However, from the standpoint of asset allocation and investment strategy, predictable
returns following corporate events provide a limited opportunity to exploit the inefficiency
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because typically only a few firms experience an event each month. Fortunately, research
also shows that a small number of firm characteristics (e.g., firm size, value and growth
attributes, i.e., the book-to-market ratio, and past price performance, i.e., momentum) are
highly successful in predicting future returns. Moreover, a large number of securities share
the firm characteristics that are correlated with substantial magnitudes of future returns.
The availability of a large pool of securities to invest in reduces the loss of diversification
entailed in trying to exploit the characteristic-based return predictability. .
The firm characteristics most highly associated with future returns are the book-to-
market ratio, firm size, and past security price performance or momentum. Banz (1981)
and, more recently, Fama and French (1993) provide evidence that small size (low market
capitalization) firms earn positive CAPM-risk-adjusted returns. That is, small firm
portfolios exhibit a positive Jensen alpha.12 Rosenberg, Lanstein, and Reid (1985) and
Fama and French (1992) show that value stocks significantly outperform growth stocks
when value is defined as the level of a firm’s book-to-market ratio. The average return of
the highest decile of stocks ranked according to book-to-market is almost one percent per
month more than for the lowest decile of stocks. The Jensen alpha of value (growth)
stocks is significantly positive (negative), both economically and statistically.13 One
possibility is that the high expected return on value stocks reflects compensation for some
sort of distress-related factor risk. An alternative interpretation is that growth stocks are
12 Handa, Kothari, and Wasley (1989) and Kothari, Shanken, and Sloan (1995) show that the size effect is mitigated when portfolios’ CAPM betas are estimated using annual returns. 13 Kothari, Shanken, and Sloan (1995) show that the book-to-market ratio effect documented in the literature is exaggerated in part because of survival biases inherent in the Compustat database and that the effect is considerably attenuated among the larger stocks and in industry portfolios.
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overpriced glamour stocks that subsequently earn low returns (see Lakonishok, Shleifer,
and Vishny, 1994, and Haugen, 1995).
A large literature examines whether past price performance predicts future returns.
There is mixed evidence to suggest price reversal at short intervals up to a month14 and
over longer horizons of three-to-five years,15 with more compelling evidence of price
momentum at intermediate intervals of six-to-twelve months (see Jegadeesh and Titman,
1993). Only the momentum effect appears to be robust (in the post-1940 period) to the
form of risk-adjustment and other technical considerations, hence we examine the extent to
which an investor can improve the risk-return trade-off by tilting the asset allocation so as
to exploit such price momentum.
The preceding discussion identifies three characteristic-based investment strategies
that historically have produced positive abnormal returns. The next chapter presents mean-
variance optimization techniques that can be used to exploit the abnormal-return
generating ability of these anomaly-based investment strategies. However, the
optimization analysis is intended only to serve as a guiding tool for investment managers
by highlighting the potential impact of tilt strategies on portfolio risk and return. In
general, managers will also be guided by their own research, their beliefs about the
likelihood that historically successful strategies will continue to perform well in the future,
market conditions prevailing at the time of their investment decisions, and other factors
like transaction costs, international diversification, and taxes.
14 See Jegadeesh (1990), Lehmann (1990), and Ball, Kothari, and Wasley (1995). 15 See DeBondt and Thaler (1980 and 1985), Chan (1988), Ball and Kothari (1989), Chopra, Lakonishok, and Ritter (1992), Lakonishok, Shleifer, and Vishny (1994), and Ball, Kothari, and Shanken (1995).
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Implications for asset allocation
Evidence of market inefficiency often translates into investment strategies that have
significant non-zero CAPM alphas. The intuitive implication for asset allocation is to tilt
the investment portfolio away from the passive market portfolio and toward the 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 will typically expose the investor to residual risk that
reflects return variation unrelated to the market index returns. The greater the residual risk,
the lesser the recommended tilt. The optimal asset allocation decision that accounts for the
magnitude of potential abnormal return as well as the residual risk incurred is formally
derived in a classic paper by Treynor and Black (1973) and is discussed in the next
chapter.
Our investigation of optimal asset allocation also incorporates Bayesian methods of
analysis that combine investors’ qualitative judgments about future returns with the
evidence in historical data. The qualitative judgments might be based on an investor’s
subjective assessment of the extent of market inefficiency (i.e., the magnitude of abnormal
return that might be earned in the future from an investment strategy and the speed with
which capital markets might assimilate information into future prices). In addition, there
might be a concern that the historical evidence on the magnitude of abnormal returns
exaggerates the true performance of an investment strategy because of data-mining (data-
snooping) biases inherent in the research process that might have skewed the historical
performance of an investment strategy.
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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 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 portfolio turnover entailed in rebalancing tilt portfolios after a one-year holding period.
(ii) Results of a Bayesian predictive analysis that avoids over-fitting through
the incorporation of priors and recognition of parameter uncertainty.
(iii) Optimal asset allocation results when the market portfolio consists of both bond and equity securities.
(iv) A limited analysis of joint optimization over a market index and all three
anomaly strategies.
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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 delisted securities, when available, we include
their delisting return as reported on the CRSP tapes. This prevents survival bias from
exaggerating an investment strategy’s performance.
The empirical analysis gives consideration to the practical feasibility of mutual
funds implementing the asset allocation recommendations in this monograph. Toward this
end, we therefore exclude stocks with impracticably small market capitalization and low
prices from our analysis. Investments of an economically meaningful magnitude at current
market prices can be difficult in small stocks, as they are less liquid, and low prices
typically are associated with high transaction costs. Therefore, we report results of optimal
asset allocation by restricting the universe of stocks analyzed to those with market
capitalization in excess of the smallest decile of stocks listed on NYSE and stock price
greater than $2.
Descriptive statistics
Table 1 reports descriptive statistics for the sample of equity securities that we
assemble for optimal asset allocation analysis. The total number of firm-year observations
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from 1963 to 1998 is 100,904, with an average of about 2,800 firms per year. If we had
not excluded stocks priced lower than $2 or stocks in the lowest decile of the market
capitalization of NYSE stocks, the number of securities each year would have been
approximately 4,800. The average annual buy-and-hold return on these securities is 14%,
with a cross-sectional standard deviation of 42%. Because of some spectacular winners,
the median annual return is considerably lower at 9%.
The average return for year t-1, the year prior to investment, is reported in the last
row of Table 1 and is much higher at 22.8% compared to the average return of 14.3% for
year t. The large difference is attributable to the exclusion of low-priced and small market
capitalization stocks. Stocks experiencing negative returns decline in price and market
value by the end of year t-1. We eliminate many of those stocks as rather impractical for
investment purposes. Thus, the stocks retained for investment at the beginning of year t
have typically performed relatively well in the prior year, which naturally boosts the
average return for year t-1 of the stocks retained. Of course, all of our portfolio analysis is
forward-looking and, therefore, not subject survivor bias.
The average market capitalization of the sample securities is $723 million, but the
median stock’s market value is only $143 million.16 The mean book-to-market ratio is
1.04, which is a result of two contributing factors. First, we exclude small market
capitalization and low-priced stocks, many of which also have low book values because of
asset write-offs, restructuring charges etc., and therefore low book-to-market ratios.
Second, although book-to-market ratios in the 1990s have been at the low end of the
distribution of book-to-market ratios, book-to-market ratios in the 1970s were quite high,
16 The market capitalization numbers are not adjusted for inflation through time, so both real and nominal effects cause variation in market values across years.
23
which raises the average for our sample. Since book value data on the Compustat is not
available as frequently as return data on CRSP, there are only 75,272 firm-year
observations in the analysis using the book-to-market ratio.
Optimal tilting toward size quintile portfolios
We present evidence on large-sample historical performance of small-versus-large
market capitalization stocks, value-versus-growth stocks, and the momentum effect. To
assess the performance of each strategy, we form quintile portfolios on July 1 of each year
by ranking all available stocks on their book-to-market ratios, market capitalization, or past
one-year performance (momentum). We estimate each portfolio’s risk-adjusted
performance for the following year. We then measure the performance of a portfolio
formed by tilting the value-weight market portfolio toward the quintile portfolios, with the
weight of a quintile portfolio ranging from zero to 100% and that of the value-weight
portfolio declining from 100% to zero. That is, the value-weight portfolio is gradually
tilted all the way toward a quintile portfolio. Optimal tilt is when the Sharpe ratio of the
tilt portfolio attains the maximum.
We estimate a portfolio’s risk-adjusted performance using the CAPM regression.
The estimated intercept from a regression of portfolio excess returns on the excess value-
weight market return is the abnormal performance of the portfolio, also referred to as the
Jensen alpha. The CAPM regression is estimated using the time series of annual 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
24
on the basis of their market capitalization, book-to-market ratio, or past one-year
performance. The CAPM regressions are:
Rqt – Rft = αq + βq (Rmt – Rft) + εqt (1)
where
Rqt – Rft is the buy-and-hold, value-weight excess return on quintile portfolio q for year t, defined as the quintile portfolio return minus the annual risk-free rate; Rmt – Rft is the excess return on the CRSP value-weight market return;
αq is the abnormal return (or Jensen alpha) for portfolio q over the entire estimation period; βq is the CAPM beta risk of portfolio q over the entire estimation period, and
εqt 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 (Q1) or largest (Q5)
market capitalization quintile portfolio and (100 – X)% of the CRSP value-weight
portfolio. X varies from 0 (i.e., no tilt toward a size quintile portfolio) to 100% (i.e., all the
investment in a size quintile portfolio).
Although probably of lesser practical relevance, we also include results for a
strategy of tilting toward the spread between quintiles 5 and 1. With the proliferation of
exchange-traded funds tied to a variety of indexes, implementing such spreads may
eventually become more realistic. The conventional size-based strategy of emphasizing
small firms would correspond to a negative position in this spread, but the risk-adjusted
performance of this small-large strategy is slightly negative for our sample. Our
25
presentation of results for the Q5-Q1 size spread will serve as an introduction to the
notation and concepts of the monograph. The more interesting findings for the Q5-Q1
value and momentum strategies will then follow.
Technically, the tilt “asset” in this context should be viewed as a position
consisting of $1 in T-bills and $1 on each side of the large-small firm spread. In other
words, the investor in this asset is implicitly assumed to receive interest on the proceeds
from the short sale of the small-firm quintile. This combined position has a net investment
of $1, unlike the spread itself, which is a zero-investment portfolio with the rate of return
undefined. Since we focus on excess returns, the return on the $1 investment in T-bills is
netted out and the performance measures are determined completely by the spread in
returns between large and small quintile stocks. The return calculations ignore the impact
of margin requirements that may be associated with either long or short positions. If the
spread portfolio generates a positive alpha, then an investor can improve performance by
tilting toward the spread. In this case, an X% “tilt toward the spread” entails a $(100 – X)
investment in the value-weight portfolio and $X in the spread asset.
We report a variety of statistics for each tilt portfolio. These include: the average
annual excess return on a tilt portfolio from 1963 to 1999; standard deviation of the excess
return; the Sharpe ratio (i.e., the ratio of the average excess return to the standard deviation
of excess return); the Jensen alpha and CAPM beta, which are estimated using regression
eq. (1). In addition to Table 3, we include figure 1 which plots the behavior of three
portfolio performance metrics (excess returns, M2, and C_M2, described below) for
portfolio allocations tilted toward the size quintile portfolios 1 and 5, and the Q5-Q1
(large-small) spread. The graphical presentation of the information is helpful in
26
visualizing the costs and benefits of tilting toward various investment strategies. The
graphs also aid in gaining an understanding of the performance metric’s sensitivity to small
deviations from the optimal portfolio. If the sensitivity is low, then the potential loss in
performance for moderate deviations from the optimum would not be great. Given the
inherent limitations of any analysis of this sort, our confidence in the relevance of the
results would be substantially reduced if too much sensitivity were observed..
The first row of the column labeled “0% = vwrt” in Table 3 shows that the average
annual excess return on the CRSP value-weight portfolio from 1963 to 1999 is 7.4%.
Values to its right are average excess returns for portfolios with increasing allocations to
the smallest size quintile portfolio, with the column labeled “100% = rt” invested entirely
in the smallest quintile portfolio. The average excess return on the smallest size quintile
portfolio is 8.6%.
This portfolio’s α is -0.5% (standard error 2.9%) and its β is 1.24 (standard error
0.15). Thus, the size effect (Banz, 1981) is not observed in this sample. The poor
performance of small stocks in the 1980s and our decision to exclude extremely low-
priced, low market capitalization stocks together result in an insignificant α for the
smallest size quintile portfolio. Without our data screen, the small firm α is 3.3%. The
row labeled “Alpha” reports Jensen alpha for the various allocations. Since the first α
value refers to the α of the value-weight portfolio, it must be zero. As the portfolio is
increasingly tilted towards the smallest quintile portfolio, the reported values approach the
α of the smallest quintile portfolio, i.e., -0.5%.
Below the portfolios’ alphas, we report their Sharpe ratios. The market portfolio’s
Sharpe ratio is 41.5%. Tilting toward small stocks dramatically lowers the Sharpe ratio,
27
with the last column showing the smallest quintile portfolio’s Sharpe ratio to be only
31.7%. The right-most column reports the ratio for the optimal portfolio with unrestricted
short-selling, i.e., the portfolio with the highest Sharpe ratio. Since tilting toward the
smallest quintile portfolio increases volatility faster than the increase in average returns,
for the 1963-1999 period, the value-weight portfolio has the optimal Sharpe ratio.
[Table 3 & Figure 1]
An equivalent measure of portfolio performance that some analysts prefer to report
is M2, which is the excess return on the portfolio after an adjustment to make its volatility
equal to that of the market index. It can be shown that M2 is a positive linear
transformation of the Sharpe ratio – hence the two performance measures provide identical
rankings of portfolios. More formally, M2 is the excess return on a hypothetical portfolio,
p*, which takes positions in the given portfolio, as well as T-bills, such that the return
volatility of p* is the same as that of the value-weight market portfolio.17 If the size-tilted
portfolio’s volatility (return standard deviation) exceeds that of the value-weight portfolio,
then p* will include a long position in T-bills so as to lower the risk.18 The return on the
resulting portfolio, p*, is referred to as M2. The value-weight portfolio’s M2 is simply its
excess return which serves as the benchmark that we hope to beat by exploiting active
positions in the anomaly-based portfolios. Since the unit of measurement for M2 is
percentage excess return, it may be a more intuitive measure than the Sharpe ratio. Table 3
shows that tilting toward the smallest size quintile portfolio results in a lower M2 than that
17 M2 is named after Franco Modigliani and Leah Modigliani. They introduced the measure in their paper appearing the Journal of Portfolio Management in 1997. 18 In the opposite situation, leverage (shorting the riskless asset) would be used to raise the volatility of p* to that of the market index. In this case, performance would be overstated insofar as the borrowing rate exceeds the T-bill rate used in the computation.
28
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 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 α and
using this average as the true α for the quintile portfolio. In this way, we capture an
investor’s confidence (or lack of confidence) in the historical performance of an
investment strategy. The lower an investor’s confidence that the past performance of an
investment strategy will persist, the larger will be the value of c. We report results under
the assumption that only half of the historical α of a portfolio can be expected in the future,
i.e., c = 0.5.
An investor might believe that historical performance is exaggerated because of
data snooping, survivor biases, luck, or because the investment opportunity will be
arbitraged away in the future as a result of public knowledge of the opportunity. The
results in Table 3 for a strategy of tilting towards the smallest quintile portfolio with c =
0.5 show, not surprisingly, that tilting remains unattractive. The c_Sharpe ratio of the
smallest size quintile portfolio is 32.7% compared to 31.7% without the c adjustment and
41.5% for the value- weight portfolio. The slight improvement is due to the fact that the
estimated small-firm alpha is negative.
In addition to reporting performance for a series of portfolios tilted toward the
smallest size quintile portfolio, we report performance for the optimal tilt in the absence of
short-selling constraints. It can be shown [see Treynor and Black (1973)] that the Sharpe
ratio of the optimal portfolio, which appears in the right-most column of Table 3, is
Sharpe(Optimal) = [(Sharpe(VWRt)2 +Information ratio2]1/2
29
The information ratio in this equation is defined as α/Std(e); α is the Jensen alpha of the
portfolio strategy that will be added to the simple index position (the smallest size quintile
portfolio in the example here) and Std(e) is the standard deviation of the residuals from the
CAPM regression used to estimate the α, i.e., the standard deviation estimate for εqt in
equation (1).19 The optimal amount of tilting increases with the magnitude of the α, and it
decreases with the residual uncertainty. This is logical since we must bear residual risk by
tilting away from a simple diversified position in the market index and α is the reward for
doing so.
The last column in Table 3 reports that the optimal portfolio’s Sharpe ratio is
41.6%. Since the value-weight market portfolio’s Sharpe ratio is 41.5%, and since tilting
toward the smallest size portfolio reduces the Sharpe ratio, an investor must short the
smallest size-quintile portfolio to reach optimality. However, the Sharpe ratio improves
only marginally so, essentially, the optimal strategy would be to simply invest in the
market portfolio.
Results in Table 3 for a strategy of tilting toward the largest size quintile suggest
that, for the period from 1963 to 1999, investors would have gained only slightly by
investing in large stocks. Even though tilting the value-weight portfolio toward the largest
size quintile portfolio by about 90% maximizes the Sharpe ratio, the M2 of the resulting
portfolio is approximately the same as that for the value-weight portfolio, 7.4%. Since
small stocks performed poorly, an investor would have been a bit better off by excluding
small stocks from the value-weight portfolio or by tilting toward the largest stocks. This
19 The optimal portfolio’s composition is determined using the following formula: Optimal allocation to the tilt portfolio = (X/Sharpe ratio of the VWRt)/(1 + (1 - β)X), where X = (1 – c)*Information ratio/Std(e).
30
can be seen by comparing the Sharpe ratios or M2 values of the portfolios tilted toward
large stocks, Q5, with those for the portfolios tilted toward the small stocks, Q1.
Finally, we find that the Q5 – Q1 size spread portfolio has a small alpha of 0.8%
(standard error = 3.3%) that is not statistically significant. Not surprisingly, tilting toward
this spread portfolio barely improves the Sharpe ratio or M2. The optimal Sharpe ratio is
41.6%, which is close to the Sharpe ratio of the value-weight portfolio. Thus, it is optimal
to invest almost the entire portfolio in the value-weight index.
Optimal tilting toward book-to-market quintile portfolios
Table 4 and Figure 2 report the results of tilting portfolios toward extreme book-to-
market stocks. The formats are the same as those for the size portfolios in Table 3 and
Figure 1. Table 4 shows that investing in the 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 64.0% for the fifth
quintile. The corresponding M2 performance measures increase from 7.4% to 11.3%. The
c_Sharpe and c_M2 measures, computed with alphas cut in half, also rise, but obviously
not as spectacularly.
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
31
which one tilts the investment portfolio toward value stocks. Notwithstanding the potential
distressed nature of these stocks, we emphasize that we have applied the investment filter
rules that 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 and Figure 2]
Table 4 also shows that growth stocks (i.e., low book-to-market stocks) did not
perform well, though the value-weight α 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. The spread has a large alpha of 5.4%, while the spread
beta is negative: the high book-to-market quintile has a significantly lower beta than the
low book-to-market quintile. It might seem odd that average excess return declines as the
spread weight increases, despite the 5.4% alpha. This is mechanically driven by the fact
that the spread return is lower than the market return. The benefit of exploiting the small
(negative) alpha for growth, by shorting quintile 1, is dwarfed by the impact of the
relatively high residual risk for growth.
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 (negative alpha) stocks increases the alpha a bit. The reason is that the spread is
32
exposed to much more residual risk -- 16.5%, as compared to 8.4% for quintile 5. The
residual risk of the spread would be dampened if the residual returns for value and growth
were positively correlated (they would be partially hedged in the spread), but in fact the
correlation is slightly negative. As a result, one is much better off investing in a value
strategy that emphasizes high book-to-market stocks, as compared to one that tries to
exploit the spread. In fact, investing 50% or more of the portfolio in quintile 5 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 and
Figure 3). We rank stocks on the basis of their performance over one year ending on May
31 of each calendar year and implement the investment strategy one month later starting on
July 1. Skipping a month avoids well-known bid-ask effects that bias momentum
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 a
4.1% abnormal return (standard error 2.0%). As with book-to-market, the optimal position
33
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 equal to 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. This more than offsets the increased residual risk from investing in
the spread, as reflected in the higher information ratio: 45.3% for the spread and 38.3% for
the winner quintile stocks. At very high levels of investment in the momentum spread, the
residual risk effect dominates and the performance ratios quickly deteriorate.
[Table 5 and Figure 3]
Summary
Our results on the benefits of tilting an investment portfolio toward extreme size
stocks, value and growth stocks, or momentum stocks lead to several conclusions. The
risk-return trade-off is not improved much by tilting 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 positive performance of the value and
momentum strategies is sustainable in the future, such tilting strategies would be desirable.
20 The estimate of residual risk is higher than the standard deviation under “100% = rt.” This apparent contradiction is due to the degrees-of-freedom adjustments, one for total variance and two for residual variance.
34
Chapter V
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 we may have some prior notion as to a plausible range of
values for alpha, even before looking at the data. 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
35
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.
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%. Naturally, the extent to which we will want to lower or shrink the estimated value
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 from observing the data.
Implementing the Bayesian analysis for asset allocation.
For pedagogical purposes, we initially suppose that alpha is the only unknown
parameter. Let α0 be the prior expected value for alpha and σ(α0) the prior standard
deviation. Say α0 = 0, the value implied if the market index is mean-variance efficient. If
21 See Pastor (2000) for an interesting analysis of the role of a pricing model in beliefs.
36
σ(α0) = 2%, then an alpha of 4% or more is a two-standard deviation “event” with
probability less than 0.023. Of course, the actual alpha is either greater than 4% or not, but
this probability quantifies our subjective judgment that such large values are implausible.
Now, let α̂ denote the given estimate of alpha and se(α̂ ) its standard error. Say α̂
is 4.7%, as above, and se(α̂ ) = 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:
α* = [α0 . 1/var(α0) + α̂ . 1/var(α̂ )] / [1/var(α0) + 1/var(α̂ )], (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(α0) is high in relation to
var(α̂ ), Bayes rule places most of the weight on the estimate α̂ . Alternatively, if there is
not much data or if the data are quite noisy, var(α̂ ) is large and α* is closer to the prior
mean α0.
In our example, 1/var(α0) + 1/var(α̂ ) = 1/0.022 + 1/0.0152 = 2,500 + 4,444.44 =
6,944.441, so α* = (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 the impact of parameter uncertainty on risk.
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
37
recognized as an additional source of risk that confronts an investor. Conventional risk
measures ignore this estimation risk. To convey this point, we rewrite equation (2) as a
Bayesian predictive regression:
Rq – Rf = α∗ + βq (Rm – Rf) + [εq + (αq − α∗)], (3)
where α* 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 εq 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, αq−α∗ is uncorrelated with the market return,
and with εqt 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 “parameter uncertainty” associated with
alpha. More formally, we require 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 alpha
estimate, as given by the denominator of (2). Recalling our earlier computations, this is
6,944.44, so the posterior standard deviation, σ*(α), 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.
38
Given the posterior standard deviation for alpha, the next step is to quantify the
overall predictive residual risk, σ*(res), perceived by the investor.22 This is the standard
deviation of the quantity in brackets in (3). Since εqt is uncorrelated with α −α∗, σ*(res)
is (0.0842 + 0.0122).5 = 8.5%. Interestingly, the uncertainty about alpha only increases the
perceived 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 σ(α0)
approach infinity in (2). Now, the posterior moments are identical to the sample moments:
α* = α̂ = 4.7% and σ*(α) = se(α̂ ) = 1.5%. The implied value of σ*(α) increases only
slightly, however, and is still about 8.5%. The investors’ uncertainty is just 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(εq), we have σ*(res) = 8.6%. If, instead, we let the data
dominate our belief about var(εq) and use uninformative priors for all the regression
parameters, σ*(res) increases to 8.9%.
To examine the impact of estimation risk on asset allocation, we combine the
original estimate, α̂ = 4.7%, with our most conservative estimate of residual risk, σ*(res)
= 8.9%. This risk measure now takes on the role played earlier by the regression estimate
22 In Bayesian analysis, one refers to “posterior” uncertainty when talking about parameter values and “predictive” uncertainty when the future value of a random variable like residual return, whose distribution depends on the parameters, is considered. Both concepts involve beliefs formed after observing the past data.
39
of residual standard deviation. 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.
Recall from our earlier analysis of the book-to-market anomaly, that it is optimal to
be fully invested in the high book-to-market quintile (assuming no-short-selling) when
parameter uncertainty is ignored. 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 barely 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 α̂ = 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
40
important. Similar conclusions hold for tilts involving the small firm and high momentum
quintiles.23
Summary of the Bayesian analysis.
The Bayesian analysis is a simple and intuitive approach to incorporating
information about the imprecision or uncertainty in the 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 strategy. 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.
23 Shrinkage implies that the prior for alpha is informative, which means that our analysis with uninformative priors overstates the impact of estimation risk in this case. Also, our conclusions are essentially unchanged if parameter uncertainty regarding the market index parameters is ignored.
41
Chapter VI
Additional Tilt-Portfolio Results
The previous analysis examined portfolio performance over the one year
immediately following the formation of the size, book-to-market, and momentum tilt
portfolios. For an investor with a longer horizon, performance measurement over one year
implicitly assumes that the optimal portfolio will be rebalanced every year. This may
entail considerable transaction costs and is only warranted if the performance of the
portfolio is likely to decay substantially over time. However, if the relevant characteristics
of stocks do not change much over the one-year holding period, then similar performance
results might be anticipated if the stock is held for an additional year. This seems plausible
for stocks ranked by size and book-to-market, but not for momentum which, by its nature,
is relatively short-lived. We now turn to some empirical evidence on this issue.
Tilt portfolio characteristics over two-year horizons
Table 6 reports transition probabilities for stocks moving from one quintile
portfolio to another in one year. The underlying data are the same as that used in optimal
portfolio construction. The second row of Table 6 shows that a stock in the smallest size
quintile has a 43.9% probability of being in the same size quintile in the following year.
The corresponding probability for the lowest book-to-market quintile is 54.4%, and it is
20.7% for the lowest quintile of momentum stocks. The transition to the “missing” cell is
quite high for stocks in all quintiles and especially so for the stocks in the first quintile.
Stocks end up in the “missing” category in year t+1 because of mergers, acquisitions,
delistings, and bankruptcies, as well as because they do not meet the investment criteria we
42
have employed (i.e., stocks priced above $2 and stocks that are not in the lowest decile of
market capitalization for NYSE stocks).
Generally, the transition probabilities for stocks ranked on momentum are roughly
the same, regardless of the initial quintile. Momentum is an indication of persistence, in
the sense that high (low) past-year quintile returns are associated with high (low) average
or expected returns for the following year. However, the realized returns in any holding
period will, like returns generally, be dominated by surprises that represent deviations from
the expected returns. Thus, conditioning on the past year’s return provides limited
information about next year’s return. Thus, a given stock is about as likely to be in one
future return (momentum) quintile as any other. In other words, the relatively uniform
transition probability distribution for momentum rankings is not surprising.
[Table 6]
The transition probabilities in Table 6 indicate that a non-trivial fraction of the
stocks in all the extreme portfolios except the large-stock portfolio end up in another
quintile portfolio after one year. Next, we present evidence on the degree to which firm
characteristics differ from their initial characteristics as a function of the investment
horizon. Table 7 reports simple averages of firm size, book-to-market ratio, and
momentum for each quintile portfolio in the formation (ranking) year t and the five
following years. The average market capitalization of the stocks in size quintile 1 is $43.3
million in the formation year t. This rises steadily from $57.3 million in year t+1 to $114.1
million in year t+5. In calculating the average market values we include all the stocks in
year t that survive in the successive future years. That is, we do not drop stocks that fail to
43
continue to meet our initial investment criteria - minimum stock price of $2 and market
capitalization in the lowest decile for NYSE stocks.
The largest size quintile stocks’ average market value increases modestly from
almost $3 billion in year t to $3.2 billion in year t+1. Value stocks, i.e., stocks in the
highest book-to-market quintile, also remain at approximately the same level in year t+1,
with the average book-to-market ratio declining from 3.0 to 2.9. Although the first-year
momentum and second-year reversal are apparent for the (simple) average returns in the
last panel of Table 7, mean-reversion after the quintile formation year is dramatic. This is
to be expected since the stocks are, by construction, ranked on their ex post realized returns
for year t, while the averages for years t+1 through t+5 provide an indication of ex ante
expected returns.
[Table 7]
The results in table 7 indicates that, if a tilt portfolio were not rebalanced at the end
of one year, its average market capitalization and book-to-market ratio would not be very
different from those of a portfolio reconstructed every year. On the other hand, the
evidence on transition probabilities in table 6 suggest that there may be some increase in
the dispersion of these characteristics over time. Ultimately, the performance of the
anomaly-based strategies over longer holding periods is an empirical question that we will
now address.
Performance of tilt portfolios over two-year horizons
The previous analysis examines portfolio performance over one year immediately
following the formation of the size, book-to-market, and momentum tilt portfolios. We
now examine performance in the second year following the tilt portfolio construction. The
44
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 (May, for momentum) of year t, but measure tilt portfolio
returns (using the firms that survive) starting in July of year t+1. The sample spans 35
years, and starts in July 1964. Tables 8-10 and Figures 4-6 report tilt portfolios’
performance for the second year after portfolio formation. These tables correspond to
Tables 3-5 and Figures 1-3 that report tilt portfolios’ performance for the year immediately
following their formation.
Tables 8-10 and Figures 4-6 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 to information.
Value stocks (quintile 5) put forth a strong risk-adjusted performance even in the
second year after their formation (see Table 9 and Figure 5). 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.24 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
24 This is consistent with results in La Porta, Lakonishok, Shleifer, and Vishny (1997).
45
entirely in value stocks and short the value-weight market portfolio. More realistically, the
implied strategy is to invest primarily in value stocks.
[Tables 8-10 and Figures 4-6]
Optimum asset allocation using the market portfolio of both bonds and stocks
Financial planners typically recommend asset allocations that consist of substantial
investments in both equity and bond securities. It is common to encounter a mix of
approximately 60% equities and 40% bonds. Therefore, it is of interest to examine
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 stock 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, book-to-market, and momentum quintile portfolios. The results are
reported in Tables 11-13 and Figures 7-9.
The 60/40 portfolio has a lower average 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 index are a bit higher. Given the
lower volatility of the 60/40 mix, the betas of the stock portfolios naturally increase.25 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
25 To get some intuition for this, suppose the bond return were riskless. Let βi = cov(Ri, Rm)/var(Rm) be the beta of security i with respect to the equity index return, Rm. The beta of this security with respect to the 60/40 blend of Rm and the riskless bond is cov(Ri, 0.6Rm + 0.4Rf)/var(0.6Rm + 0.4Rf) = 0.6Cov(Ri, Rm)/0.36Var(Rm) = 1.67βi because Rf does not covary with stock returns and its variance is assumed to be zero. The increase observed in the data reflects the relatively low variability of bond returns as compared to stocks.
46
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.
[Tables 11-13 and Figures 7-9]
47
Chapter VII
Optimal asset allocation with all-three-anomaly strategies
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. The correlations between
portfolio residual returns are also relevant to the joint optimization problem. The
estimated residual correlation between large firms and value firms is -0.30. The residual
correlation between the book-to-market and momentum (size) spreads is -0.27 (-0.20).
The other correlations are also negative but closer to zero.
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.26 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. This is depicted
graphically in Figure 10. The optimal M2 is 11.5%, much higher than the 7.4% excess
26 In the unrestricted short-selling case, we use the formulas in Gibbons, Ross, and Shanken (1989), which generalize those in Treynor and Black (1973) to accommodate nonzero residual correlation.
48
return on the market. Note that the greater investment in value stocks, as compared to
winners, is consistent with the higher information ratio for those firms, as observed earlier.
[Figure 10]
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 stocks 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 the 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 the spread. In this case, the unrestricted optimal allocation
with c = 0 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. As seen in Figure 10, this
translates into an overall portfolio composition of 8% in the size spread, 27% in the value
spread, 30% in the momentum spread, and the remaining 35% in the market index. The
optimal portfolio has 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
49
diversifying across spreads, making it possible to exploit the high alphas more efficiently
than with the single-spread tilts examined in Chapter IV.
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
portfolio is lowered, however. With c = 0.5 (0.75), that weight is 54% (40%), down from
65%, and the corresponding M2 drops to 9.4% (7.9%) from the 13.9% value when c = 0.
While reduced, we still see a strong role for the anomaly-based strategies even after
substantial reductions in the historical alphas.
50
Chapter VIII
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 earlier
work, 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 momentum stocks, 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 Q5 – Q1 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
market 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
51
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 the 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 a consideration of tax effects. There is also a large literature on
52
stock return predictability that documents changes in risk and expected return over time.
Incorporating this sort of information might lead to improved asset allocation decisions.
53
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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 of
the regression parameters mirrors the usual sampling distribution for the regression
estimates; i.e., (α, β) is jointly normally distributed with mean (α̂ , β̂ ) and variance matrix
σ2(ε).(X'X)-1, where X is the Tx2 matrix of independent variables including a constant
vector (Zellner, 1971).
Consider the regression equation of quintile excess return on the market index
excess return. We use the simpler notation, yt = α + βxt + εt . For out-of-sample return
observations, x and y, the corresponding predictive regression is
y = α̂ + β̂ x + [ε + (α−α̂ ) + (β− β̂ )x], (Α1)
where α̂ and β̂ are regression estimates based on data from t=1 to T. From the Bayesian
perspective, these estimates are the predictive regression coefficients (α* and β* in our
earlier 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 α and β.
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 as
the prediction error in standard regression analysis. Its conditional variance is well known
and given by the expression:
var(ε){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 is
yet 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
59
(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 x and the predictive variance
for 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 variance
is
var*(res) = var(ε) .
−
−+
)3(
)1(21
TT
T. (A2)
Interestingly, the influence of x has dropped out, simplifying the final result. With
an uninformative prior for var(ε), it can be shown that var(ε) is replaced in (A2) by the
usual unbiased residual variance estimate multiplied by (T-2)/(T-4). These formulas
permit a quick evaluation of the potential impact of parameter uncertainty on residual risk
without requiring a careful formulation of prior beliefs.
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