Stock Returns are Predictable:
A Firm Level Analysis
DORON AVRAMOV AND TARUN CHORDIA*
First draft: November 10, 2002
This Revision: August 1, 2003
*Doron Avramov is from the University of Maryland, e.mail:davramov@rhsmith.umd.edu,
Tel: 301-405-0400, and Tarun Chordia is from the Goizueta Business School, Emory University, e.mail: Tarun Chordia@bus.emory.edu, Tel: 404-727-1620.
We thank Gurdip Bakshi, Devraj Basu, Jeff Busse, Wayne Ferson, Amit Goyal, Clifton
Green, Rick Green, Paul Irvine Narasimhan Jegadeesh, Avner Kalay, Shmuel Kandel, Dilip
Madan, Lubos Pastor, Gordon Phillips, Nagpurnanand Prabhala, Jay Shanken, Avi Wohl,
Goufu Zhou, seminar participants at Arizona State, Emory, Maryland, North Carolina, Tel
Aviv, Tulane, and especially an anonymous referee, for comments and suggestions. All
errors are out own.
Stock Returns are Predictable:
A Firm Level Analysis
Abstract
This paper studies return predictability at the firm level through the performance of
investment strategies that use conditioning information to build unconditional efficient
portfolios from individual stocks. The model that generates moments for portfolio
selection is a multivariate regression of excess returns on asset pricing factors with
alphas and betas that may be linear functions of information variables. Investment
strategies that use conditioning information outperform both passive benchmarks and
dynamic strategies that disregard predictability. The superior performance is attributed
to time varying alphas, betas, and risk premia, and is robust to inclusion of portfolio
constraints, estimation risk, model uncertainty, hedging demands, and transaction costs.
Introduction
Previous studies such as Keim and Stambaugh (1986) and Fama and French (1988,1989)
have documented that stock returns are predictable through time. Subsequently, incorporating predictability in empirical research has provided fresh insights into the behavior of
security prices and investment decisions. In asset pricing tests, Bossaerts and Green (1989),
Ferson and Harvey (1991), and Lettau and Ludvigson (2001) show that factor models with
predictability have been reasonably successful in explaining why average returns are different across stocks. In a portfolio evaluation context, Ferson and Schadt (1996) find that
fund managers perform better when evaluation measures account for predictability. In an
event study, Eckbo, Masulis, and Norli (2000) demonstrate that seasoned public offering
stocks do not underperform subsequent to the offering date when risk and return vary predictably with economic conditions. However, the extent to which stock return predictability
is economically important is controversial. Skepticism has emerged due to the possibility
of data mining (Foster et al. (1997)), statistical biases in predictive regressions (Stambaugh
(1999) and Ferson, Serkasian, and Simin (2003)), inferior performance of predictabilitybased investment strategies relative to passive benchmarks such as the market index (Allen
and Karjalainen (1999) and Coopers et al. (2001)), and weak out-of-sample performance of
predictive regressions (Bossaerts and Hillion (1999)). This paper studies predictability at
the stock level using an investment based metric that is potentially robust to these concerns.
The metric incorporates innovations in portfolio theory that extend the Markowitz
(1952, 1959) mean variance paradigm to account for predictability and multifactor demands.1 In particular, we study predictability through the step-ahead performance of the
Hansen-Richard (1987) unconditional strategy, formed using 2,832 NYSE-AMEX stocks,
that attains the smallest variance for a given mean among all possible portfolios that use
conditioning information. Ferson and Siegel (2001) provide an explicit solution for the
Hansen-Richard portfolio and illustrate some of its properties that are robust relative to
other utility maximizing strategies. Notably, this portfolio responds conservatively to extreme values of information variables. In contrast, conditionally efficient strategies, such as
that obtained by maximizing the exponential utility function under conditional normality,
are not robust to extreme signals. Predictability is also examined based on the performance
of a portfolio that is efficient in the Merton (1973) sense, formed by adding hedging demands
to the Hansen-Richard strategy. We derive an explicit solution for the Merton portfolio and
show that the hedging component is a minimum variance portfolio constructed such that the
covariance between that portfolio and state variables conforms to some desired hedging level.
Our metric is especially suitable for analyzing predictability at the individual stock level.
1
See Hansen and Richard (1987) and Ferson and Siegel (2001) for unconditional mean variance efficiency
with predictability and Fama’s (1996) treatment of Merton’s (1973) ICAPM for multifactor demands.
1
Indeed, cross-sectional predictability, such as the momentum effect of Jegadeesh (1990) and
the size and value premia documented by Basu (1977), Banz (1981), and Fama and French
(1992), have been studied at the stock level. However, empirical research on time series
predictability has focused on aggregates such as the market index and portfolios of stocks
and bonds. To our knowledge, a stock-level analysis is new in the context of time series
predictability. Such analysis is attractive for several reasons. First, it addresses critiques
about data mining and loss of information on portfolio-based procedures in empirical asset
pricing.2 Second, it is grounded in economic theory in that the equilibrium paradigms of
Bossaerts and Green (1989), Berk, Green, and Naik (1999), and Gomes, Kogan, and Zhang
(2002) have all suggested that expected return and risk change predictably at the individual
firm level. Third, it can shed light into potential sources of predictability. Indeed, Ferson
and Harvey (1991) find that changes in risk premia explain most of the predictable variation,
while changing beta is second order. However, they use portfolios and not individual stocks,
and they suggest that the importance of beta variation for portfolios may be muted, because
much of the variation of individual firm’s beta could be diversified out at the portfolio level.
We will examine whether their intuition holds for individual stocks, in an investment setting.
The statistical model that generates moments for portfolio selection is a multivariate
regression of excess returns on asset pricing factors, with alphas, betas, and market price of
beta risks that may be linear functions of lagged variables when predictability is suspected
and are constant over time otherwise. Portfolio strategies are derived under estimation risk
in a Bayesian framework, and are rebalanced on a monthly basis using a recursive scheme
that yields a time series of realized returns. Unconditional Sharpe ratios and alphas are
then calculated for evaluating portfolio performance. We also study a utility based metric
that indicates the maximal fee one would be willing to pay (or would request) so as to
switch from holding asset allocation that does not use conditioning information to dynamic
investment strategies that use information. Using such unconditional measures is advocated
by asymmetric information based paradigms such as those of Mayers and Rice (1979) and
Dybvig and Ross (1985). In Dybvig and Ross, for example, a conditionally efficient portfolio
need not appear efficient to one who does not observe the information used by the investor.
In contrast, the unconditionally efficient portfolio maximizes the measured performance.
All unconditional performance measures suggest that returns on individual stocks are
predictable through time. Hansen-Richard and ICAPM portfolio strategies that use conditioning information substantially outperform both passive benchmarks and dynamic mean
variance and multifactor efficient strategies that disregard predictability. For example, when
alpha, beta, and risk premia are all allowed to vary, regressing excess returns generated by
the step-ahead Hansen-Richard strategies on the Fama-French benchmarks yields statisti2
See Litzenberger and Ramaswamy (1979), Lo and MacKinlay (1990), and Berk (2000).
2
cally positive Jensen’s alphas to the tune of 1.07% per month when portfolio holdings are
unconstrained and 0.99% per month under no short selling. In general, when alpha, beta,
and risk premia are constant over time, Jensen’s alphas are significantly negative or zero.
The Sharpe ratio generated by the Hansen-Richard strategy is 0.19 when investments are
unconstrained, and is 0.22 under no short selling. The no predictability counterparts are
-0.09 and 0.11. In addition, the utility-based measure suggests that it takes an implausibly
large turnover to generate transaction costs so as to eliminate the gain from using conditioning information in building portfolios. We have also studied a case where a hypothetical
investor is unsure about which set of conditioning information is relevant in forecasting
future stock returns, thereby making portfolio decisions on the basis of a general model
that averages across single forecasting models using posterior probabilities as weights, as
in Avramov (2002). The economic value of exploiting conditioning information in building efficient portfolios from individual stocks is found to be robust to such model uncertainty.
Portfolio strategies with time-varying alphas substantially outperform portfolios with
asset-pricing restrictions, suggesting that variations in alpha are an important determinant
of predictability. This complements the evidence in Ferson and Harvey (1999) who test
conditional asset pricing models using portfolios as test assets. They show that the conditioning information enters significantly into the intercepts in regressions of excess returns
on asset pricing factors, thereby providing evidence against the model restrictions. In this
work, we document the economic significance of asset pricing misspecification when the
test assets are individual stocks. We also show that Hansen-Richard and ICAPM portfolio
strategies formed such that the market beta is time varying generate better performance
relative to their fixed beta counterparts. This lends support to the intuition of Ferson and
Harvey (1991) that beta variation could be meaningful at the stock-versus-aggregate level.
In fact, we have found that time variation in alpha, beta, and risk premia are, all, important
determinants of the superior performance of investments that use conditioning information.
Our finding that returns are predictable differs from those of Bossaerts and Hillion
(1999), Cooper et al. (2002), and many others who have called into question the extensive in-sample evidence on predictability. To explain the different outcomes, we note that
Bossaerts and Hillion (1999) examine predictability of market portfolios using statistical
model selection criteria. Our metric is economically motivated, and our analysis applies to
single stocks. In addition, we depart from the predictive regression framework of Bossaerts
and Hillion and others. Instead, we model excess returns using a factor model representation, similar to that of Ferson and Harvey (1999), where expected returns may be a
nonlinear function of lagged variables, and where time varying alpha, beta, and risk premia, all potential sources of predictability, can be studied simultaneously. Next, Cooper et
al. (2002) follow a strategy that updates the long-short stock portfolios every year based on
past data. For instance, the strategy may go long in the lowest size quintile or the highest
3
book-to-market ratio quintile of stocks. Our study forms portfolios based on mean-variance
and multifactor optimizations, thereby avoiding the sudden and extreme movement of a
stock in and out of the optimal investment portfolio. Also, Cooper et al. focus on firm-level
equity characteristics such as size and book-to-market as potential predictors. This paper studies predictability using the macroeconomic variables - term spread, default spread,
short-term interest rate, aggregate values of dividend yield, and idiosyncratic volatility. The
last instrument was entertained by Campbell et al. (2001) and was applied by Goyal and
Santa Clara (2002) in a predictability context. To summarize, much of the new evidence
about stock return predictability that this work documents is attributed to firm-versusaggregate level analysis and to the metric used to study predictability, which is based on
the performance of portfolios that are efficient in the sense of Hansen and Richard (1987)
as well as Merton (1973).
The remainder of the paper proceeds as follows. Section 1 develops a framework for analyzing predictability at the individual stock level using an investment-based metric. Section
2 describes the data, and Section 3 presents the results. Section 4 offers conclusions and
potential avenues for future research. Unless otherwise noted, all derivations are presented
in the appendix.
1
A framework for studying predictability at the stock level
This section sets forth the framework for studying return predictability at the individual
stock level. The section is organized as follows. Part 1.1 describes the statistical models for stock returns, the market portfolio, state variables believed to mimic investmentconsumption uncertainties, and conditioning information. The statistical structure generates inputs for deriving the demand function for stocks under mean variance and multifactor
efficiency. Part 1.2 describes the Hansen-Richard demand function, and part 1.3 describes
a portfolio that is efficient in the Merton’s ICAPM sense. The econometric approach for
deriving moments for portfolio selection with estimation risk is described in part 1.4.
1.1
The statistical structure
The underlying models for stock returns, factor-mimicking portfolios, and predictors are
rt = α(zt−1 ) + β(zt−1 )ft + vt ,
(1)
α(zt−1 ) = α0 + α1 zt−1 ,
(2)
β(zt−1 ) = β0 + β1 (IK ⊗ zt−1 ) ,
(3)
ft = af + Af zt−1 + vf t ,
(4)
zt = az + Az zt−1 + vzt ,
(5)
4
where rt is an N -vector of returns on investable stocks in excess of the riskfree rate, ft
contains excess returns on a market index and S = K − 1 zero-cost hedging portfolios
believed to mimic future consumption-investment uncertainties, and zt is an M -vector of
variables observed at time t that are potentially related to the probability distribution function of future asset returns. The intercepts α0 and α1 are N -vector and N × M matrix,
respectively, β(zt−1 ) is an N × K matrix of conditional betas, the symbol ⊗ denotes the
Kronecker product, and vt is an N -vector of zero mean security-specific events. Modeling α
and β as linear functions of instruments goes back to Rosenberg and Marathe (1979), later
generalized by Shanken (1990) and Ferson and Harvey (1998, 1999). The predictive regression characterization for mimicking portfolios captures potential changes in the market
price of beta risks. The predictors follow vector AR(1). The vector autoregression characterization has been adopted in a portfolio choice context by Kandel and Stambaugh (1996),
Stambaugh (1999), Barberis (2000), Avramov (2002, 2003), among others, as a means of
dealing with the fact that information variables, albeit predetermined, evolve stochastically.
Substituting (2) and (3) into the right hand side of (1) and then using the identity
ft ⊗ zt−1 = (IK ⊗ zt−1 ) ft yields the following representation for excess stock returns
rt = α0 + α1 zt−1 + β0 ft + β1 [ft ⊗ zt−1 ] + vt .
(6)
Under scenarios where α0 = 0 and α1 = 0, the regression (6) stands for an unconditional
representation of a conditional asset pricing model with K fundamental factors (ft ) and KM
scaled factors (ft ⊗ zt−1 ). Indeed, if returns are predictable one could expand the set of
factors to risk-adjust asset returns. For example, Ferson and Schadt (1996) use the CAPM
version of (6) for evaluating performance of mutual fund managers. Eckbo et al. (2000) follow Ferson and Schadt (1996) in risk-adjusting stock returns using a scaled factor model for
investigating the long run performance of stocks of firms that make seasoned public offerings. This work examines whether mean variance and ICAPM portfolio strategies based on
models with scaled factors outperform their fixed-beta and no-predictability counterparts.
The statistical structure implies three sources of time series predictably. These are predictable α (α1 = 0), β (β1 = 0), and risk premia (AF = 0). Indeed, Ferson and Harvey
(1991) find that most of the evidence on predictability is captured by a risk-based asset pricing model, especially due to changes in risk premia, whereas beta variation is second order.
However, they use portfolio returns and not individual stocks, and they suggest that the
importance of time-varying betas for portfolios may be muted relative to individual stocks.
We will examine whether that intuition holds for individual stocks, in a portfolio choice setting.3 In particular, we study three beliefs about the predictability structure. The first rules
3
Several studies address the implications of time varying beta at the stock level. For example, Chan (1988)
and Ball and Kothari (1989) argue that changing beta can explain the success of contrarian strategies. In
equilibrium paradigms, Bossaerts and Green (1989) show that shocks to the idiosyncratic component of the
5
out predictability, setting the parameters α1 , β1 , and Af equal to zero. The second allows
all these parameters to depart from zero. The third is a fixed-beta model (β1 = 0). Each
of the three specifications is examined under two polar beliefs about the validity of asset
pricing models. The first disregards pricing restrictions, thereby leaving the intercepts α0
and α1 unrestricted. The second imposes pricing restrictions. With unrestricted intercepts
factor models with predictability are merely conditional covariance models. With intercepts
restricted to be zero, a risk-based asset pricing model explains stock return predictability.
As noted earlier, for any specification of ft in (1), a total of 3×2 return generating models
are examined. The models differ with respect to beliefs about the predictability structure
and about the validity of asset pricing models. Each of the six specifications is first studied
when ft is believed to be the market factor, and then studied when the market factor is
augmented with some state variables perceived to mimic future consumption-investment
uncertainties. Under the former, the various models produce moments for unconditionally
efficient mean-variance portfolios. Under the later, the moments are used for deriving multifactor efficient portfolios, which combine mean variance portfolios with hedging demands as
in Merton (1973). Thus, beliefs about single versus multifactor models imply a different demand function for securities. Explicit solutions for the portfolio strategies are derived below.
Based on the underlying statistical models for excess returns, factors, and predictors,
the mean and variance that go into mean-variance and multifactor efficient portfolios are
µt = α(zt ) + β(zt )(af + Af zt ),
(7)
Σt = β(zt )Σf f β(zt ) + Σrr ,
(8)
where Σf f and Σrr are the covariance matrices of vf t and vt , respectively. Section 1.4
presents explicit expressions for the Bayesian versions of µt and Σt that account for estimation risk. The covariance matrix Σrr is assumed to be diagonal. Otherwise, the estimated
covariance matrix may be singular. To illustrate, consider an investment universe that
comprises 1,000 securities. (In this work, portfolios are often formed using more than 1,000
stocks.) Then, one has to estimate 1,000 variance and 499,500 covariance parameters. Our
sample contains, at most, 461 return observations per stock. Thus, there are, at most,
461,000 return observations, less than the number of parameters in the covariance matrix. Indeed, Jobson and Korkie (1980, 1981), Frost and Savarino (1986), Best and Grauer
(1991), and Black and Litterman (1992) note that the performance of mean-variance portfolio strategies formed on the basis of a covariance matrix estimated from historical returns
is poor even when the number of securities is considerably smaller than 1,000. Poor performance is commonly attributed to an imprecise estimation of a large number of parameters.
firm’s exogenous cash flow alter the risk and expected return. Berk, Green, and Naik (1999) and Gomes,
Kogan, and Zhang (2001) show that the market beta is related to firm level size and book-to-market.
6
Beyond estimation feasibility, imposing diagonal covariance matrix is tractable computationally. Applying the inverse matrix theorem (e.g., Seber (1984 p. 520)) to (8) yields
−1
−1
−1
−1
Σ−1
t = Σrr − Σrr β(zt ) β(zt ) Σrr β(zt ) + Σf f
−1
β(zt ) Σ−1
rr .
(9)
Hence, the inversion of the large-scaled (N × N ) matrix Σt involves the inversion of the
diagonal matrix Σrr and the low-dimensional (K × K) matrices Σf f and β(zt ) Σ−1
rr β(zt ) +
−1
Σf f . In addition, the structure of the covariance matrix allows one to readily approach
an investment universe comprising securities with different return histories. This aspect is
desirable in real time investment, as stocks enter and leave the sample periodically. Indeed,
different return histories are permissible in our framework. Earlier papers by Stambaugh
(1997) and Pastor and Stambaugh (2002) analyze investments whose histories differ in
length. Relative to them, we allow expected returns, variances, and covariances to vary
with the state of the economy. In particular, observe from (7) and (8) that alpha, beta, and
the residual variance are all based on a return history of a stock, which may be shorter than
that of the market factor, state variables, and predictors. Longer histories of the factors and
predictors are useful in estimating the parameters af , Af , and Σf f , thereby improving the
precision of estimated expected returns, variances, and covariances of short-history stocks.
1.2
Unconditional mean-variance efficiency with conditioning information
We evaluate the step-ahead performance of the Markowitz-mean-variance strategy that disregards predictability versus the unconditional efficient strategy of Hansen and Richard
(1987) that uses conditioning information. When predictability is ruled out, the Markowitz
portfolio is a natural choice. When predictability is accounted for, the Hansen-Richard
portfolio is attractive relative to other utility maximizing portfolios such as these of Kim
and Omberg (1996) and Campbell and Viceira (1999), as noted by Ferson and Siegel (2001).
First, it is conditionally and unconditionally efficient, whereas other conditionally efficient
portfolios need not be unconditionally efficient. For example, portfolios obtained by maximizing the conditional expected value of the exponential utility function under normality
are not unconditionally efficient. In addition, the Hansen-Richard portfolio is robust to
extreme values of the predictors. In general, the conditionally efficient strategy can result
in large long or short positions for extreme values of the signal while the unconditionally
efficient strategy displays a conservative response to extreme signals (see also Basu and
Stremme (2003) for a comparison). In this work, the Hansen-Richard portfolio is obtained
as an outcome of quadratic utility optimization. Ferson and Siegel (2001) show that maximizing the conditional expected value of a quadratic utility function is equivalent to finding
an unconditionally efficient portfolio. Thus, of all conditional mean variance strategies one
could have formed only the quadratic utility portfolio maximizes the measured performance.
7
Next, we describe the paradigm for asset allocation. The investment universe at time t
comprises Nt individual stocks and a riskless asset. The number of investable stocks varies
over time as stocks enter (via IPOs) and leave (bankruptcy, merger, acquisition, etc.) the
sample periodically. A hypothetical investor maximizes the conditional expected value of
U (Wt , Rp,t+1 , at , bt ) = at + Wt Rp,t+1 −
bt 2 2
W R
,
2 t p,t+1
(10)
where at is some constant, Wt denotes the time t wealth, bt stands for the absolute risk
aversion parameter, and Rp,t+1 is the portfolio return computed as Rp,t+1 = 1+rf t +wt rt+1 ,
with rf t being the riskless rate, rt+1 the Nt+1 -vector of the time t + 1 stock returns in excess
of the riskless rate, and wt the portfolio weights. Now, let µt and Σt denote the expected
value and variance of excess stock returns conditioned upon predictive variables observed
at the end of time t, and let Λt = [Σt + µt µt ]−1 . Taking conditional expectation from both
sides of (10) and manipulating yield the following formulation for the utility maximization
wt = arg max wt µt −
wt
1
w Λ−1 wt ,
2(1/γt − rf t ) t t
(11)
(bt wt )
is the relative risk aversion parameter. We keep the difference 1/γt − rf t
where γt = (1−b
t wt )
fixed over time. For notation clarity, we suppress the time subscript from γ and rf .
A conditionally efficient mean-variance portfolio strategy that solves (11) is given by
wt = (1/γ − rf )Λt µt ,
= (1/γ
(12)
Σ−1
t µt
− rf )
,
1 + µt Σ−1
t µt
(13)
where (13) follows by applying the inverse matrix theorem to Λt . Similarly, the Markowitzmean-variance portfolio that does not use conditioning information is w = (1/γ − rf )Λµ,
where Λ = (µµ + Σ)−1 , and µ and Σ are the no predictability analogs of µt and Σt .
The conditionally efficient portfolio (12) is equivalent to the unconditional portfolio of
Ferson and Siegel (2001) presented in equation (12) of their paper when γ takes the value
γ=
1
,
µp /ζ + rf
(14)
µ Σ−1 µ
where µp is the excess expected return target and ζ = E(µt Λt µt ) = E 1+µt Σt −1tµ , with
t
t
t
E denoting the expected value taken with respect to the unconditional distribution of the
predictors. In words, for a given mean there exists a quadratic utility function with a
relative risk aversion parameter given in (14) that yields a conditionally efficient portfolio
that is also unconditionally efficient in the sense of Hansen and Richard (1987). Note from
(14) that γ and µp are inversely related. Indeed, risk-averse investors will pick conservative
portfolios. The relation between γ and ζ is also intuitive. Observe that ζ is positively
8
associated with the Sharpe ratio. In particular, based on Ferson and Siegel, the maximal
ζ
. Other things being equal, higher γ is associated with
squared Sharpe ratio is equal to (1−ζ)
higher ζ because more conservative agents will require higher compensation for taking risk.
−1
µΣ µ
, which differs from that perceived
Note that for the Markowitz’s agent ζ = (1+µ
Σ−1 µ)
by an agent who acknowledges the possibility of predictability. In fact, agents who agree
that returns are predictable but disagree about the relevant predictors and/or about the
source of predictability perceive a different ζ. Hence, it follows from (14) that specifying the same expected return target across the Markowitz’s and Hansen and Richard’s
agents means that the risk aversion parameter would be different across the agents. In
our empirical examinations, we will hold the primitive parameters of the quadratic utility
fixed across agents. That is, the risk aversion parameter is γ0 under all scenarios. Then, the
Markowitz’s portfolio is (1/γ0 −rf )Λµ. The Hansen-Richard counterpart is (1/γ0 −rf )Λt µt .
Indeed, we have found it tractable to implement utility maximization in the empirical
analysis. Then, all investors have the same risk preferences regardless of beliefs in the
existence of predictability, sources of predictability, and relevant predictors. In addition,
using a utility function one can implement a utility based metric such as that in Fleming et
al. (2000). The average utility computed based on returns on step-ahead strategies indicates
the maximal fee an investor would be willing to pay (or would request) for switching from
the Markowitz to the Hansen-Richard strategy. The maximal fee is a useful performance
measure because it examines whether the added value of using conditioning information, if
there is one, survives reasonable transaction costs. There are some computational merits as
well. We have found that in the presence of non negativity constraints and a large universe
of investable stocks, it is easier to implement a quadratic programming code to maximize
(11) versus to minimize the portfolio variance subject to an expected payoff target.
1.3
Accounting for Merton’s (1973) hedging demands
Predictability is also studied through the performance of portfolios that are efficient in the
Merton’s (1973) ICAPM sense. In Merton, demands for risky securities are affected by
possible uncertain changes in future investment opportunities. Hence, the ICAPM portfolio combines the Markowitz-mean-variance portfolio with hedging portfolios that mimic
uncertainties about future consumption-investment opportunities. The demand function
for stocks in the presence of hedging is derived below. First, observe from equations (1)
and (4) that the unexpected value of stock returns is β(zt−1 )vf t + vt , where vt is orthogonal to the unexpected value of the state variables. Note also that the disturbance term
in (4) can be partitioned into vf t = [vmkt,t , v1,t , . . . , vS,t ] , where vmkt,t and vs,t are the
time t innovations in the market portfolio and the state variable s, respectively. Next,
let Vs,t−1 = β(zt−1 )covt−1 (vf t , vs,t ), which is an Nt−1 -vector of covariances between unex-
9
pected values of stock returns and state variable s, and let VS,t−1 = [V1,t−1 , . . . , VS,t−1 ] be
the Nt−1 × S matrix. In Fama (1996), portfolio optimization for an ICAPM investor is formulated as minω 21 ω Σω subject to (i) ω µ = µp , and (ii) ω VS = δS , where δS is an S-vector
of targets for the covariance between unexpected returns on the ICAPM portfolio and the
state variables. We derive an ICAPM portfolio by maximizing (11) subject to ω VS = δS .
An explicit solution is given by (time dependence is suppressed for notational convenience)
ω=
(1/γ − rf ) −1
Σ µ
(1 + µ Σ−1 µ)
+ Σ−1 VS (VS Σ−1 VS )−1 δS −
Hansen-Richard’s demand
(1/γ − rf )
V Σ−1 µ .
(1 + µ Σ−1 µ) S
Merton’s demand
(15)
The demand function (15) has two components. The first is the unconditionally efficient Hansen-Richard portfolio. The second reflects the Merton’s demand for risky assets
as a vehicle to hedge against unfavorable shifts in the investment opportunity set. The
explicit solution (15) is attractive since is facilitates a straightforward derivation of ICAPM
portfolios from lifetime consumption-portfolio decisions for a large universe of stocks. One
caveat is that the solution requires that δS be explicity specified in advance. Of course, an
ICAPM portfolio may be sensitive to that specification. In the absence of any theoretical
or empirical rationale for the desired hedge, we implement a sensitivity analysis.
The ICAPM portfolio has an intuitive interpretation. The reader may note that this
portfolio could be obtained as the sum of two portfolios solved in two sequential optimizations. The first optimization yields the unconditional portfolio of Hansen and Richard
(1987), ωhr . In particular, the term in parenthesis on the right-hand-side of (15) is δ̃S =
δS − VS ωhr , which is the difference between the target for the desired hedge, as set by the
investor, and the covariance between innovations in the Hansen-Richard portfolio and the
state variables. If that difference is zero then hedging demands are nonexistent since the
Hansen-Richard portfolio itself fully hedges against unanticipated shocks to investmentconsumption opportunities. If that difference is not zero, however, then the investor conducts a second optimization where the hedging component, ωh , minimizes ωh Σωh such that
ωh VS = δ̃S . The portfolio ωh that solves this optimization is the hedging demand on the
right hand side of (15). That is, the hedging demand is the minimum variance portfolio such
that the covariance between that portfolio and the state variables is equal to the component
in the target, δ̃S , left unhedged (or possibly over hedged) by the Hansen-Richard portfolio.
1.4
Asset allocation with estimation risk
Academics and practitioners often approach asset allocation by first specifying a model for
stock returns, which in the context of this work amounts to determining the collection of
information variables and factors, and then replacing the parameter values in the mean and
variance expressions by maximum likelihood estimates. This common practice implies that
10
the specified return model is undoubtfully correct and that the sample estimates are the
true parameter values. Asset allocation based on such practice often involves taking large
long positions in some assets and large short positions in others (see, e.g., Best and Grauer
(1991)). Extreme long and short positions could be due to imprecise estimation of the
moments.4 If so, incorporating estimation risk in a Bayesian framework could help improve
the performance of mean variance strategies. Indeed, Frost and Savarino (1986) and Jorion
(1986) use Bayes methods to reduce estimation errors and improve portfolio performance.
We construct efficient portfolios accounting for estimation risk. We make several assumptions about the data generating process. First, as in Barberis (2000) and many others, prior beliefs about model parameters are noninformative. In addition, the residuals in
(1), (4), and (5) are i.i.d. normally distributed. Indeed, the normality assumption is often
rejected as returns display fate tails. However, Tu and Zhou (2003) show that normality works well in evaluating portfolio performance for a mean-variance investor. Next, let
α̂(zT ), β̂(zT ), âf , Âf , Σ̂f f , Σ̂rr be the maximum likelihood estimators of the parameters
in (7) and (8). Maximum likelihood estimators for all parameters are derived analytically
in part A of the appendix. Under the most general case considered here where intercepts
are unrestricted and predictability is due to time varying alpha, beta, and the market price
of beta risk, the mean and variance that go into portfolio optimization, say at time T , are
µT
= α̂(zT ) + β̂(zT )(âf + Âf zT ),
(16)
ΣT
= P1 β̂(zT )Σ̂f f β̂(zT ) + P2 Σ̂rr .
(17)
The predictive mean is simply obtained by replacing the parameters in (7) by their maximum likelihood estimates. The predictive variance is larger than the maximum likelihood
analog of (8) as it incorporates two additional parameters P1 and P2 , each of which is greater
than one, that reflect the additional uncertainty about predicted returns due to estimation
T
(1 + δT ) where
risk. Exact expressions for these parameters are given by P1 = T −M −2K−3
δT is described in equation (24) in the appendix, P2 is an N × N diagonal matrix whose i-th
Ti
element is Ti −M −K−KM
−3 (1 + δiT ), where Ti is the number of monthly return observations
recorded for stock i and δiT is described in equation (25) in the appendix. In addition,
the Bayesian expression for VS , the covariance between unexpected stock returns and state
variables that goes into multifactor efficient portfolios, is equal to the maximum likelihood
estimate of VS multiplied by (1 + δT ). The term δT reflects the impact of estimation risk.
2
Data
The data includes monthly individual stock returns for an average of 1,268 large NYSE and
AMEX stocks over the sample period July 1962 through December 2000. A small fraction
4
Though, Green and Hollifield (1992) argue that the presence of a single dominant factor would result in
extreme weights even in the absence of estimation errors.
11
of stocks is present over the entire sample period and a number of stocks enter and exit the
sample. The overall number of stocks included is 2,832. We consider only the large stocks
by excluding the smallest quartile of stocks from the sample each month. We require at
least 61 months of returns for a stock to be included in the sample. Ten trading days in
each month are also required for each stock. The time-series, cross-sectional average market capitalization of the stocks in the sample is $1,716 million and the median is $373 million.
The conditioning variables are selected based on studies such as Keim and Stambaugh
(1986), Campbell and Shiller (1988), and Fama and French (1989), who examine predictability of returns on portfolios of stocks and bonds. We study dividend yield, term spread, default spread, the three-month t-bill yield, and aggregate idiosyncratic volatility. Dividend
yield is the total dividend payments on the value weighted CRSP index over the previous
12 months divided by the current level of the index. Dividend yield is included as a proxy
for time variation in the unobservable risk premium. The default spread is the difference
between the average yield of bonds rated BAA by Moodys and the average yield of bonds
with a Moodys rating of AAA, and is included to capture the effect of default premiums.
The term spread is measured as the difference between the average yield of Treasury bonds
with more than ten years to maturity and the average yield of T-bills that mature in three
months. The yield on the three-month T-bill serves as a proxy for expectations of future economic activity. The idiosyncratic volatility has been introduced by Campbell et al. (2001).
The average daily idiosyncratic volatility each month for each stock is estimated as follows. Daily idiosyncratic variance is defined as the stock return less the equally weighted
market return squared. Monthly volatility or the standard deviation is square root of the
average daily variance each month, calculated for stocks that have at least ten days of return
data each month. We use the average daily volatility instead of the monthly volatility (obtained by summing up the daily variances each month as in Goyal and Santa-Clara (2002))
because the monthly variances of less frequently traded stocks are likely to be biased downwards as compared to monthly variances of more frequently traded stocks. The market
idiosyncratic volatility is the cross-sectional average of the individual firm volatilities.
3
Results
Portfolio strategies with estimation risk are obtained based on a recursive scheme. That is,
portfolios are derived using the first P monthly observations, are then rebalanced using the
first P + 1 monthly observations, and so on, . . ., and are finally rebalanced using the first
T − 1 monthly observations, with T denoting the sample size. In the empirical experiments,
P ranges between 120 and T − 120 in a one-month interval. The month t realized excess
return on a step-ahead strategy is obtained by multiplying the month t − 1 portfolio weights
12
by the month t realized excess returns. The recursive scheme produces T − P such realized
excess returns on 3 × 2 × 2 dynamic trading strategies that differ with respect to (i) beliefs
about the predictability structure, (ii) beliefs about the validity of asset pricing models,
and (iii) whether or not portfolio constraints are incorporated. Portfolio decisions are made
when the difference γ1 − rf in (13) and (15) is kept constant over time. The difference is
0.089, which corresponds to a risk aversion parameter of γ = 7, within the range of estimates
summarized by Mehra and Prescott (1985), and an annual riskfree rate of 6%. Examining
some other risk aversion and riskfree rate values have yielded similar performance rating.
3.1
The ex post performance of unconditional mean-variance strategies
Predictability is first studied through the step-ahead performance of the Hansen-Richard
versus the Markowitz portfolios. The Bayesian mean and variance that go into the HansenRichard portfolios are described in (16) and (17). The no predictability moments are derived
in the appendix. At this stage, ft in (1) is the market factor only. Table I reports measures for evaluating the ex post performance of three strategies. Strategy M1 disregards
predictability (the Markowitz portfolio), M2 allows predictable alphas (unless pricing restrictions are imposed), betas, and the market price of beta risks, and M3 is the fixed-beta
analog of M2 . Each of the three strategies is studied under 2 × 2 scenarios where optimal
portfolios are first unconstrained and are then constrained by no short selling, and where
CAPM restrictions are first discarded (unrestricted intercepts) and are then imposed. In
Table I, µ and σ are the average and standard deviation of excess realized returns. SR is the
Sharpe ratio obtained by dividing µ by σ. The parameters α1 and β are the Jensen’s (1969)
alpha and slope in a regression of excess realized returns on contemporaneous values of the
market factor, and α2 is the Jensen’s alpha in a similar regression where Fama-French (1993)
factors replace the market factor. The t statistic tests the equality of alphas to zero. Table
I reports quantities derived using 341 monthly excess realized returns, which corresponds
to P = 120. Figure 1, analyzed below, plots performance measures where P takes on all
the range of values from 120 to T −120. The measures µ, σ, SR, α1 , and α2 are scaled by 100.
Note that although the FF-factors do not play any role in deriving mean variance portfolios they are used to estimate the Jensen’s alpha. Our attempt here is to examine whether
predictability based strategies outperform an investment in the Fama-French benchmarks,
which, in turn, is known to produce high Sharpe ratios. Note also that we impose portfolio
constraints, which appears to restrict the ability to optimize the portfolios. That restriction
would be valid, however, if expected returns, variances, and covariances were known. These
quantities are obviously unknown. In practice, many institutional investors are prohibited
from taking short positions either through explicit rules or via the implicit threat of lawsuit
for violating fiduciary standards. Other investors often voluntarily impose constraints on
portfolio holdings since they find short sale to be costly, some are unable to borrow the stock
13
(see Gezcy, Musto, and Reed (2002)). Empirically, Frost and Savarino (1988) suggest that
imposing constraints improves performance of mean variance portfolios since it can reduce
sampling errors. On the other hand, Green and Hollifield (1992) imply that imposing constraints introduces specification errors since such constraints do not appear in population.
Thus, imposing constraints involves a tradeoff between estimation and specification errors.
Table I presents figures that support firm-level return predictability. The HansenRichard strategies consistently outperform the Markowitz portfolios and they outperform
passive benchmarks. Let us first analyze the case where intercepts are unrestricted. With
short selling permitted the mean return obtained by the Markowitz strategies is -1.76% per
month, though this figure is statistically insignificant in that the t-value for testing µ = 0 is
√
-1.64 ( −1.76
19.80 × 341). The Jensen’s alpha obtained by regressing realized returns, produced
by Markowitz strategies, on both the market index and the Fama-French benchmarks is
negative but statistically indistinguishable from zero under conventional significance levels.
Incorporating predictability considerably improves performance. First, the average value
of realized excess returns is significantly positive given by 1.18% per month under M2 and
0.97% per month under M3 . Second, the Hansen-Richard strategies are less volatile. The
standard deviation is 6.2% for both M2 and M3 , while it is 19.8% for M1 . Not only do
Hansen-Richard strategies outperform the Markowitz portfolios, but they outperform passive investments in the market index and Fama-French benchmarks as well. Focusing on
M3 , alphas are significantly positive given by α1 = 0.98% and α2 = 1.06% per month.
Imposing portfolio constraints seems to substantially improve the performance of the
Markowitz portfolios. For example, the Sharpe ratio rises from -0.089 to 0.110, and alphas
become positive, given by α1 = 0.14% and α2 = 0.03%, though insignificantly so. The
apparent improvement in performance is consistent with Frost and Savarino (1988) who
emphasize the role that portfolio constraints play in reducing estimation errors. Indeed,
this study derives constrained and unconstrained Markowitz strategies under estimation
risk. This, presumably, reduces estimation errors. The substantially improved performance
in the presence of constraints could indicate model misspecification. Put differently, modeling stock returns so that alpha, beta, and risk premia are all time invariant does not appear
to be a robust characterization for return dynamics even when estimation risk is included.
When predictability is recognized and portfolio strategies are constrained, the expected
return on the step-ahead strategies rises relative to unconstrained scenarios, but so do
volatility and systematic risk. Thus, the overall change in performance is unclear. A larger
Sharpe ratio indicates that imposing constraints improves performance. A smaller α2 provides evidence to the contrary. Nevertheless, two observations are unambiguous. First,
strategies M2 and M3 outperform M1 , generating much larger Sharpe ratios and alphas.
Second, these predictability-based strategies outperform passive benchmarks, in that α1 and
14
α2 are both significantly positive. These findings emerge also when portfolio holdings are
unconstrained, demonstrating the robust economic value of exploiting return predictability.
Incorporating CAPM restrictions hurts performance of Hansen-Richard strategies. With
short selling permitted α1 declines from 1.07% to 0.70% (M2 ) and from 0.98% to 0.86%
(M3 ), and α2 declines from 1.07% to 0.67% (M2 ) and from 1.06% to 0.76% (M3 ). The reduction is even sharper when short selling is not allowed. Nevertheless, alphas are, in most
cases, significantly positive. A potential explanation for deteriorating performance is that
predictable asset mispricing is an important determinant of firm-level return predictability.
This finding complements the evidence in Ferson and Harvey (1999) who test asset pricing
models where test assets are portfolios sorted on size and book-to-market and on industry
groups. They show that intercepts in Fama-French regressions vary predictably, thereby
rejecting the conditional version of Fama-French. Here, we map their findings into the
case where test assets are individual stocks. Stock-level analysis addresses critiques such as
data mining and loss of information noted by Litzenberger and Ramaswamy (1979), Lo and
MacKinlay (1990), and Berk (2000). While our portfolio strategies are formed using the
market factor in (1), we have also obtained (results not reported) economically significant
alphas when portfolios are formed using the Fama-French benchmarks in (1).
Thus far, we have not considered transaction costs in the analysis. From a practical
investment perspective, however, their impact could be important. Berkowitz, Logue, and
Noser (1988) estimate one-way transaction costs of 23 basis points for institutional investors
in NYSE stocks. That figure could be smaller in the context of our work since the smallest
quartile of stocks is eliminated from the investment universe. To gauge the economic gain of
firm-level predictability under reasonable transaction costs, we compare the 23 basis point
figure to the maximal fee an investor would be willing to pay so as to switch from the
Markowitz-mean-variance portfolio to the Hansen-Richard strategy. A utility-based metric could be especially useful for measuring the maximal fee. In particular, observe from
1
(µ2 + σ 2 ), where µ and σ are those
(11) that the realized average utility is µ − 2(1/γ−r
f)
reported in Table I. Now, let ∆U 1 and ∆U 2, respectively, be the monthly differences in
realized utility between strategies M2 and M1 and between strategies M3 and M1 . We
find that under unrestricted intercepts, ∆U 1 and ∆U 2 are 3.18% and 2.97% when portfolios are unconstrained and are 1.97% and 0.77% when short selling is precluded. Under
CAPM, the corresponding quantities are smaller but are still much larger than the 23 basis
point figure. This suggests that it takes an implausibly large turnover to call into question the use of conditioning information in forming unconditional mean variance portfolios.
Table I also indicates that predictability survives transaction costs when the maximal fee
is computed as the difference between alphas of strategies with versus without predictability.
Table I reports figures based on P = 120. This assumes that a hypothetical agent in15
vests in stocks after observing ten years of data and then rebalances the investment on a
monthly basis. Performance could be sensitive to the timing of initial investment. Thus,
we analyze performance using 221 cut off points, where P ranges from 120 to T − 120 in
a one-month interval. For example, under P = 220, the investor makes the first portfolio
selection based on the first 220 monthly observations and then rebalances the portfolio in
each of the following 241 months. Figure 1 displays performance measures using all these
cut off values. August of 1972 corresponds to P = 120. This case is described in Table
I. December of 1990 corresponds to P = T −120, which generates 120 realized excess returns.
The left (right) plots in Figure 1 display performance measures under unrestricted intercepts when portfolio holdings are unconstrained (no short selling). Plots A1 and A2 describe
α1 . Plots B1 and B2 exhibit Sharpe ratios. Plots C1 and C2 show the evolution of $1 invested in July 1972 in dynamic and passive strategies where cash dividends are reinvested.
In plots A1, A2, B1, and B2, the dashed-dotted lines correspond to Markowitz portfolios.
The solid and dashed lines reflect the performance of Hansen-Richard portfolio strategies,
where alpha, beta, and risk premia are all allowed to vary (solid lines), and where beta is
constant (dashed lines). In plots C1 and C2, the dashed lines describe a buy-and-hold investment in the value weighted CRSP index. The solid lines describe a dynamic investment
in the Hansen-Richard strategy, where alpha, beta, and risk premia are allowed to vary. The
Markowitz portfolio (dashed-dotted liens) appears to be omitted from Plots C1 and C2 because the value of $1 invested in the no predictability portfolio rapidly declines towards zero.
Overall, the evidence from Figure 1 supports the notion of gains from utilizing firmlevel predictability in forming efficient portfolios. In brief, unconditional Sharpe ratios and
alphas are consistently larger for Hansen-Richard strategies. Strikingly, observe from plots
C1 and C2 that the December 2000 value of the $1 invested in July 1972 in the HansenRichard strategy with dynamic rebalancing exceeds $200 under unconstrained portfolios,
and exceeds $1,000 under no short selling. For comparison, the December 2000 value of
the $1 is only $27.77 when invested passively in the market index, and is almost zero when
invested dynamically in the Markowitz portfolio strategy. Indeed, when portfolio weights
are unconstrained, the gain from predictability appears to decline when the initial investment is made in 1987 or after. Even then, alphas are still positive, though not statistically
significant, and Hansen-Richard portfolios outperform the Markowitz strategy. When portfolios are constrained, predictability gains are robust to the timing of initial investments.
In practice, as noted earlier, investors do not take unlimited short positions in building
portfolios. As we demonstrate, under constrained scenarios, investors substantially improve
performance using conditioning information. Improved performance is present also under
restricted intercepts (not plotted).
16
3.2
The ex post performance of ICAPM portfolio strategies
In this part, we analyze performance of ICAPM portfolio strategies. We have to make
decisions about (i) the state variables, (ii) the appropriate performance measures, and (iii)
the desired hedge level.
State variables: Economic theory has remained silent about which state variables have
pervasive effects on the prices of all assets. Merton (1973) suggests that changing interest rates are likely to be correlated with adverse shifts in the investment opportunity set.
Chen, Roll, and Ross (1986), Shanken (1990), and Scruggs (1998), have used state variables
related to interest rate risk to examine the empirical performance of asset pricing models
(Chen, Roll, and Ross and Shanken) and to understand the nature of the intertemporal relation between risk and return (Scruggs). Chen, Roll, and Ross (1986) and Shanken (1990)
show that exposures to interest rate risk are priced in the equity market. Scruggs finds that
when the market factor is augmented with excess returns on long-term bonds, the relation
between the market risk premium and market variance is positive. We follow Scruggs in
choosing excess returns on long-term bonds as a state variable. Bond returns are inversely
related to interest rate changes, thus providing a hedge against interest rate variation. For
robustness checks, we have run the analysis using default and term premia, variables studied
by Chen, Roll, and Ross (1986). Qualitatively, the results (not reported) remain unchanged.
Performance measures: By construction, a multifactor efficient portfolio does not generate (ex ante) the highest admissible Sharpe ratio. Hence, evaluating an ICAPM strategy
using unconditional mean variance measures is infeasible. Instead, we propose the following mechanism. First, regress excess realized returns generated by the step-ahead ICAPM
strategies on the contemporaneous values of the state variables. Then, construct orthogonal
returns, or returns that are uncorrelated with the state variables, as the sum of the regression intercepts and residuals. For the desired expected return and hedge, if the risk of the
step-ahead strategies is the value implied by the ICAPM, i.e., if ICAPM restrictions hold,
then regressing orthogonal returns generated by the step-ahead strategies on the orthogonal
component of the market portfolio would yield a zero intercept. That is, the orthogonal
returns reflect a pure wealth bet (see also the discussion in Polk (2000)). This suggests that
the unconditional mean variance evaluation measures could be implemented on the basis
of the orthogonal returns. Specifically, an ICAPM portfolio is the best performer if the orthogonal returns derived based on that portfolio generate the highest Sharpe ratio. Instead,
one could regress the orthogonal returns on the component of the market portfolio that is
uncorrelated with the state variables. An outperforming portfolio is the one that yields the
highest alpha. The same alpha could also be obtained by regressing realized returns generated by the step-ahead strategies on the market portfolio combined with the state variables.
17
How much hedging? : We are unaware of any theoretically or empirically motivated value
for δS in (15). Thus, we have run the analysis using the three values δS = VS ωmv , δS =
2VS ωmv , and δS = 12 VS ωmv , where ωmv is the unconditional mean variance portfolio. Under the first, the mean-variance portfolio fully hedges against adverse shifts in investment
opportunities, in that the Merton’s hedging demand disappears. In the second (third), the
mean-variance portfolio underhedges (overhedges) with respect to a target set by the agent.
Table II reports unconditional measures for evaluating the step-ahead performance of
ICAPM strategies where the state variable is excess returns on long-term bonds. Panels A,
B, and C present evaluation measures where δS = VS ωmv , δS = 2VS ωmv , and δS = 12 VS ωmv ,
respectively. Under the first, the ICAPM portfolio is efficient in the Hansen and Richard
sense as well, in the absence of hedging demands. Panel A exhibits evaluation measures
based on both constrained and unconstrained portfolios. Panels B and C present measures
where portfolios are unconstrained using both realized and orthogonal returns. Table II
indicates that, as in Ait-Sahalia and Brandt (2001), hedging demands are negligible relative to the mean variance demand. In particular, based on realized returns on step-ahead
strategies, the unconditional expected return and risk are almost identical across the three
hedging levels. We have run the analysis using other values for δS and using default and
term premia as state variables. We have found that the mean variance demand dominates.
We demonstrate that ICAPM dynamic strategies with conditioning information consistently outperform both passive benchmarks and dynamic multifactor efficient strategies
that do not use conditioning information. In brief, relative to M1 , specifications M2 and
M3 deliver higher Sharpe ratios and alphas whether ICAPM restrictions are imposed or
ignored, under constrained and unconstrained portfolio holdings, and using both realized
and orthogonal returns. Focusing on M1 , M2 , and M3 in panel B, for example, based on
orthogonal returns, Sharpe ratios are -0.101, 0.112, and 0.099, α1 -s are -1.96%, 0.58%, and
0.53% per month, and α2 -s are 0.36%, 0.61%, and 0.72% per month, respectively. In general,
efficient strategies that do not exploit predictability generate either significantly negative or
zero alphas. In contrast, predictability based strategies yield significantly positive alphas in
most cases examined. The difference in alphas corresponding to predictability versus no predictability strategies overwhelmingly exceeds transaction costs. The overall evidence on the
superior performance of trading strategies that use conditioning information is reinforced
in Figure 2. The plotted unconditional portfolio measures, based on orthogonal returns
and δS = 2VS ωmv , are consistently larger when strategies use conditioning information. In
addition, investing $1 (August 72) using conditioning information generates substantially
higher wealth relative to a passive investment in the market portfolio and relative to a dynamic investment in a multifactor efficient strategy that does not use information variables.
18
3.3
Asset allocation with information: Time-varying versus fixed-beta
strategies
Thus far, we have shown that models based on the statistical structure with predictability
generate better performance, and that time varying alpha contribute much to performance.
Whether risk is time varying is an open question. In particular, in the presence of time
varying market beta, Chan (1988) and Ball and Kothari (1989) have explained the success of contrarian strategies, and Chan and Chen (1988) have explained the size premium.
However, these studies do not model the potential variation in the market price of beta
risk. Ferson and Harvey (1991) and Evans (1994) find that when risk premia could vary,
beta variation becomes second order in explaining the evidence on predictability. Ferson
and Harvey (1991) also suggest that beta variation could be diversified out in the portfolio level. Ferson and Harvey (1999) however provide evidence of time varying betas. This
study explicitly examines the importance of beta variation at the stock level using the investment based metric by comparing performance of time varying versus fixed beta strategies.
Performance is studied when intercepts are unrestricted, given the already documented role
of time varying alpha, and when risk premia could vary predictability with lagged variables.
Figures 1 and 2 display measures that are useful for studying the performance of time
varying versus constant beta dynamic strategies. It emerges from Figure 1 that when
Hansen-Richard portfolio strategies are unconstrained, time varying beta strategies generate
larger Sharpe ratios. The difference in Sharpe ratios ranges between 0.012 and 0.060 per
month. Under no short selling, time varying beta strategies generate larger Jensen’s alphas.
The difference in α1 ranges between 24 and 107 basis points per month. The difference in
α2 is of the same magnitude. Performance measures from Figure 2 support time varying
beta as well in that Sharpe ratios and alphas are generally higher when risk is time varying.
Overall, our investment-based metric confirms the intuition of Ferson and Harvey (1991)
that firm-level beta variation is important even when the market price of beta risk varies.
3.4
Incorporating model uncertainty
We study the performance of the Hansen-Richard and Merton portfolios under uncertainty
about which set of predictors should be retained in the statistical model. When M variables
are suspected relevant in forecasting returns on N stocks there are 2M N competing specifications. Each of these keeps, for any stock, between 0 and M variables and drops the rest.
We address such model uncertainty using a Bayesian model averaging (BMA) approach.
At the heart of BMA, all competing models are nested in a ‘weighted’ model using posterior probabilities as weights. Portfolio decisions are then made based on the weighted model.
Studying the economic gain of using conditioning information in building portfolios
under model uncertainty potentially addresses concerns about model misspecification and
19
weak out-of-sample performance of predictive regressions. In particular, Avramov (2002)
notes that the step-ahead prediction errors generated by the weighted model satisfy certain
desirable properties including zero mean, zero serial correlation, and zero correlation with
the predicted returns. In contrast, the performance of prediction errors generated by model
selection criteria, such as those studied by Bossaerts and Hillion (1999), is unsatisfactory.
Next, we describe investment opportunities under model uncertainty. Let p(m) denote
m
the posterior probability of model m, and let µm
t and Σt be the model-specific mean and
variance with estimation risk as in (16) and (17). In the presence of model uncertainty and
within-model estimation risk, the mean and variance of predicted excess stock returns are
p(m)µm
t ,
µt =
(18)
m
p(m)Σm
t +
Σt =
m
m
m
p(m)(µt − µm
t )(µt − µt ) .
(19)
The mean is a weighted average of means. The variance is a sum of a mixture of variances
and a model-uncertainty component, which emerges due to cross-model variation in means.
Model uncertainty carries implications for hedging demands as well. In particular, let
Vstm be the model-specific covariance between unexpected excess returns on investable stocks
and the state variables. The version of Vstm that integrates out the model dependence is
p(m)Vstm +
Vst =
m
m
m
p(m)(µt − µm
t )(µst − µst ) ,
(20)
m
where µm
st is an S-vector of model-specific mean of the state variables and µst =
m p(m)µst .
The first term on the right hand side of (20) is a mixture of the within-model covariance.
The second term reflects the impact of model uncertainty that arises due to cross-model covariation between conditional expected returns on the investable stocks and state variables.
Given that our investment universe is large, addressing 2M N models is computationally
prohibitive. Indeed, one could restrict attention to 2M specifications only, assuming that
the collection of retained predictors is identical across stocks. For example, if the dividend
yield is the only predictor suspected relevant then there are two competing models. In one
(i.i.d.), the dividend yield is discarded, while in the second, the dividend yield is believed to
predict returns on all stocks. This is, however, too restrictive if alpha and beta are modeled
as time varying. In particular, it is known that the posterior probability comprises both
an in-sample goodness-of-fit measure and a penalty term for model complexity (Kass and
Raftery (1995)). Retaining dividend yield in the second model can improve the goodness of
fit. However, any improvement is likely to be overwhelmed by an extremely large penalty
factor since alphas and betas of all stocks are modeled as linear functions of the dividend
yield, which may be an incorrect specification. Indeed, we have run the analysis focusing on
20
2M models where alpha, beta, and risk premia could vary and found that the i.i.d. model
dominates all other models due to an extremely large penalty factor. On the other hand,
if time varying risk premia is the only source of predictability, model uncertainty could be
implemented focusing on 2M models. Then, predictability, if it exists, is common across all
stocks. As a result, we study model uncertainty under restricted intercepts and fixed betas.
When alpha and beta are constant, the regression of the factor portfolios on lagged
variables in equation (4) is model specific, while the regression of excess returns on factors
in equation (1) is model invariant. Thus, we can implement the explicit solution for the
posterior probabilities derived by Avramov (2002). Performance measures of the step-ahead
strategies under model uncertainty are described in Table III. Panel A displays portfolio
measures corresponding to Markowitz versus Hansen-Richard strategies with and without
portfolio constraints. Panel B describes measures corresponding to ICAPM strategies using
both realized and orthogonal returns. The M1 models are Markowitz-mean-variance and
Merton-multifactor efficient portfolio strategies that do not use conditioning information,
and the M4 models stand for the Hansen-Richard and ICAPM portfolios formed on the
basis of a model that averages across individual return forecasting models using posterior
probabilities as weights. In Panel A, the superscripts mkt and ltb denote, respectively, that
the asset pricing factors in (1) are the market portfolio and the market portfolio augmented
with excess returns on long term bonds. In Panel B, we study three hedging levels given
by δS = VS ωmv , δS = 2VS ωmv , and δS = 12 VS ωmv . These are denoted by ltb1 , ltb2 and ltb3 ,
respectively. As noted earlier, under the first specification, the ICAPM portfolio is efficient
in the Hansen-Richard sense as well. Performance measures are reported when the initial
investment is made on the basis of the three cut off points P = 120, P = 180, and P = 240.
Panel A of Table III shows that exploiting risk premia variation leads to substantial
improvement in portfolio performance. Under P = 120, means, Sharpe ratios, and alphas
are larger for M4 -versus-M1 strategies. For example, focusing on CAPM, the Markowitz
portfolio yields α1 = 0.02% whether or not portfolio constraints are imposed, whereas
the Hansen-Richard strategy generates α1 = 0.83% when portfolios are unconstrained and
0.66% under no short selling. Similar performance rating is obtained when (i) the market
factor is augmented with returns on long term bonds, (ii) under the three hedging levels,
and (iii) when the cut off point is P = 180. Under P = 240, the unconditional measures also
support time varying risk premia. Focusing on Panel A, the largest Sharpe ratio, 0.255, and
the largest alphas, α1 = 1.14% and α2 = 1.03%, are obtained based on the weighted model.
In addition, all alphas are significantly positive. Moving to Panel B, using realized returns
portfolio strategies with predictability appear to dominate their no predictability counterparts, generating larger means, Sharpe ratios, and alphas. The evidence is less strong,
however, based on orthogonal returns when P=240, in that alphas are not significantly positive, though they are larger under predictability. For example, the difference in α1 is 42 basis
21
points per month under δS = 2VS ωmv and is 44 basis points per month under δS = 12 VS ωmv .
Interestingly, Table III shows that performance measures are substantially different
across ICAPM specifications in the presence of model uncertainty. For example, observe
ltb3
2
from Panel B that the average realized return is 1.03% under Mltb
4 , and 2.46% under M4 .
This suggests that hedging demands are important determinants of the demand function for
stocks when model uncertainty is accounted for. In particular, the cross model covariation
between expected returns on stocks and the state variables in (20), attributed to model
uncertainty, is meaningful in forming portfolios. In contrast, as noted earlier, when model
uncertainty is ignored hedging demands are negligible relative to the mean variance demand.
The analysis with model uncertainty shows that risk premia is important in explaining
the superior performance of predictability based trading strategies. This complements the
previous findings about the importance of variation in both alpha and beta. Thus, we
conclude that time varying alpha, beta, and risk premia are all important in explaining why
the Hansen-Richard strategies and their ICAPM extensions considerably outperform both
passive benchmarks and dynamic trading strategies that disregard conditioning information.
4
Conclusions
A long-standing question in financial economics has been whether returns are predictable
through time. Indeed, financial economists have identified publicly observed economic variables that explain substantial amounts of future return variation. However, predictability
has remained an open question because of (i) concerns about data mining, (ii) concerns
that the statistical methods implemented to study predictability could indicate spurious
results, (iii) the large gap between the evidence of predictability and the real time investment performance, and (iv) the poor out-of-sample performance of predictive regressions.
This paper examines predictability at the stock level, implementing an investment-based
metric that builds on recent innovations in portfolio theory. The idea is to compare the
portfolio performance of two groups of hypothetical fund managers. Managers belonging
to the first group form portfolios relying on factor models where alpha, beta, and risk premia are constant over time. Fund mangers belonging to the second group use factor models
with alpha, beta, and risk premia that may be linear functions of lagged variables. Dynamic
portfolio strategies are efficient in the sense of Hansen and Richard (1987) or Merton (1973).
The evidence strongly suggests that excess returns on individual stocks are predictable.
Hansen-Richard and ICAPM portfolio strategies that use conditioning information generate
substantially better performance relative to mean variance and multifactor efficient strategies that do not use conditioning information. The superior performance is attributed to
time varying alpha, beta, and risk premia. Moreover, this superior performance is robust
22
to inclusion of portfolio constraints, estimation risk, model uncertainty, and trading costs.
Our work suggests several avenues for future research. The proposed framework is
especially suitable for studying the impact of stock specific characteristics such as size,
book-to-market ratio, turnover, and momentum on firm-level return predictability. Alternatively, it is also possible to study investments in equity mutual funds when returns
may be predictable. Using fund specific or manager specific attributes may provide useful information that could help investors improve their allocation to actively managed funds.
23
Appendix
A. The likelihood function
The sample contains Ti monthly excess return observations of stock i (there are N = 2, 832
NYSE and AMEX traded stocks over the 8/1962-12/2000 period), T excess returns on K
benchmark assets, and T realizations of M forecasting variables. Stock i enters the sample
at time ti (IPO) and leaves at time ti + Ti − 1 (merger, acquisition, bankruptcy, etc.) or
remains till the end-of-sample period. Let us fix some notation. Let ri denote the Ti -vector
of excess returns on stock i, let Gi = [Gti , . . . , Gti +Ti −1 ] , where Gt = [1, zt−1 , ft , ft ⊗ zt−1 ] ,
let Γi = [αi0 , αi1 , βi0 , βi1 ] , where, for example, βi0 is the i-th security version of β0 , let
Z = [z1 , . . . , zT ] , let F = [f1 , . . . , fT ] , let X = [x0 , . . . , xT −1 ] where x0 = [1, z0 ] with z0
being the first observation of the forecasting variables, let Vf = [vf 1 , . . . , vf T ]’, let Vz =
[vz1 , . . . , vzT ]’, let Vrz be a T × N matrix whose i-th column contains Ti values of vit , the
security-specific residual in regression (1), when returns on security i are recorded and T −Ti
zeros when such returns are missing, let AZ = [az , Az ] , and let AF = [af , Af ] . In addition,
let QGi = ITi − Gi (Gi Gi )−1 Gi , let QX = IT − X(X X)−1 X, let WZ = [X, Vf , Vrz ], and
let QZ = IT − WZ (WZ WZ )−1 WZ . The stochastic processes for excess returns, factors, and
predictors can be rewritten as ri = Gi Γi + vi , F = XAF +vf , and Z = XAZ + vz . Denoting
the set of parameters by Θ, the likelihood function can be factored as
N
L(ri , . . . , rN , Z, F |Θ, z0 ) =
i=1
p(ri | F, Z, Γi , Σirr , z0 )
(21)
× p(F | Z, AF , Σf f , z0 )p(Z | ξ, Σzz.r.f , z0 ),
−1
where Σirr is the i-the diagonal element of Σrr , ξ = [AZ , Σzf Σ−1
f f , Σzr Σrr ] , Σzz.r.f = Σzz −
−1
Σzr Σ−1
rr Σrz − Σzf Σf f Σf z , Σf z is the conditional covariance between vf t and vzt , Σzz is the
conditional covariance of vzt , and Σrz is an N × M matrix whose i th row contains the
covariance between vit and vzt . Analytic expression for the likelihood is given by
N
L ∝
i=1
Ti
1
(Σirr )− 2 exp − Σirr ri QGi ri + (Γi − Γ̂i ) Gi Gi (Γi − Γ̂i )
2
T
1
× |Σf f |− 2 exp − tr Σ−1
f f F QX F + (AF − ÂF ) X X(AF − ÂF
2
T
1
ˆ
ˆ
× |Σzz.r.f |− 2 exp − tr Σ−1
zz.r.f Z QZ Z + (ξ − ξ) WZ WZ (ξ − ξ
2
(22)
,
where Γ̂i = (Gi Gi )−1 Gi ri , ÂF = (X X)−1 X F, and ξˆ = (WZ WZ )−1 WZ Z. Next, let Σ̂rr
rQ
r
be a diagonal matrix whose (i, i) element is i TGi i i , let Σ̂f f = F QTX F , and let Γ̂ be the
N security version of Γ̂i . The maximum likelihood estimates of α0 , α1 , β0 , and β1 are the
first column, the next M columns, the next K columns, and the last KM columns of Γ̂,
respectively. When estimation and model risks are disregarded in asset allocation decisions,
the above-derived maximum likelihood estimates replace the true parameters in equations
(7) and (8) to yield the mean vector and the covariance matrix of stock returns. To describe
the maximum likelihood estimates in the i.i.d. case, let the process for excess returns and
factors be rt = α + βft + ut and ft = µf + uf t . The corresponding unconditional moments
are µ = α+βµf and Σ = βVf f β +Vrr , where Vf f and Vrr are the covariance matrices of uf t
24
and ut , respectively. (Vrr is assumed diagonal.) Now, let Hi = [Hti , . . . , Hti +Ti −1 ] , where
i are estimated
Ht = [1, ft ] , and let Γi = [αi , βi ] . The parameters Γi , µf , Vf f , and Vrr
r Q
r
by (Hi Hi )−1 Hi ri , F TιT , F QTY F , and i THi i i , respectively, where QY = IT −
ITi − Hi (Hi Hi )−1 Hi , and ιT is a T -vector of ones.
ιT ιT
T
, QHi =
B. Asset allocation in the presence of estimation risk
Conditioned on the return generating specification, the Bayesian expected value of future excess returns is equal to its classical counterpart obtained on the basis of maximum likelihood estimates. The model-specific predictive variance is larger than its classiT
cal counterpart due to estimation risk. In particular, let Σ̃f f = T −M −2K−3
Σ̂f f , and let
Ti
i
i
Σ̃rr = Ti −M −K−KM −3 Σ̂rr . Some algebra shows that the Bayesian predictive variance is
ΣT = AT + (1 + δT )β̂(zT )Σ̃f f β̂(zT ) ,
(23)
where AT is an NT × NT diagonal matrix whose i-the entry is Σ̃irr (1 + δiT ), and
1
1 + (z̄ − zT ) V̂z−1 (z̄ − zT ) ,
T
(24)
Â∗F xT xT Â∗F (Gi Gi )−1
(25)
δT
=
δiT
= tr
23
32
33
+ (1 + δT ) tr Σ̃f f Ω22
i + Ωi (IK ⊗ zT ) + (IK ⊗ zT )Ωi + (IK ⊗ zT )Ωi (IK ⊗ zT )
being
Â∗F = [IM +1 , ÂF , ÂF (IK ⊗ zT )], with Ωmn
i
 11
Ωi
−1

Ω21
(Gi Gi ) =
i
Ω31
i
,
the following partitions

Ω12
Ω13
i
i
.
Ω22
Ω23
i
i
32
33
Ωi Ωi
(26)
1
T
(27)
The two additional variance components δT and δiT are attributed to estimation risk. Next,
Ti
T
let Ṽf f = T −2K−3
V̂f f , and let Ṽrri = Ti −K−3
V̂rri . Some algebra shows that the Bayesian
predictive variance in the no predictability case is equal to
ΣT = BT + 1 +
β̂ Ṽf f β̂ ,
where BT is an NT × NT diagonal matrix whose i-the entry is Ṽrri (1 + δ̃iT ), and
δ̃iT
= tr
µ̂∗f µ̂∗f (Hi Hi )−1 + 1 +
1
T
tr Ṽf f Ω22
i
,
(28)
µ̂∗F = [1, µ̂f ] , with Ω22
i being the following partition
Ω11
Ω12
i
i
22
Ω21
Ω
i
i
(Hi Hi )−1 =
25
.
(29)
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Table I
Unconditional performance measures for mean variance efficient strategies
The table reports various measures for evaluating the ex post performance of three unconditionally efficient mean variance strategies. Strategy M1 is the Markowitz-mean-variance
portfolio that disregards predictability, M2 allows predictable alphas (unless the pricing
restrictions are imposed), betas, and the market price of beta risks, and strategy M3 is the
fixed-beta analog of M2 . Each of the three strategies is studied under 2 × 2 scenarios where
optimal portfolios are first unconstrained and are then constrained by no short selling, and
where the CAPM restrictions are first disregarded (denoted unrestricted intercepts) and are
then imposed. A recursive scheme is implemented to derive optimal portfolios, which yields
341 monthly excess realized returns on the twelve portfolio strategies. The evaluation measures are based on the monthly realized returns and are as follows: µ is the average of excess
realized returns, σ is the standard deviation, SR is the Sharpe ratio obtained by dividing µ
by σ, the parameters α1 and β are the intercept and slope in a regression of excess realized
returns on the market factor, α2 is the intercept in a similar regression where FF factors
replace the market index, and t(•) is a t-statistic that tests the significance of alphas. The
measures µ, σ, SR, α1 , and α2 are scaled by 100.
M1
M2
M3
Unconstrained
M1
M2
M3
No short selling
Unrestricted intercepts
µ
σ
SR
β
α1
t(α1 )
α2
t(α2 )
-1.76
19.80
-8.87
0.44
-2.00
-1.86
0.39
0.43
1.18
6.20
19.01
0.20
1.07
3.19
1.07
3.20
0.97
6.20
15.60
-0.02
0.98
2.88
1.06
3.14
1.00
9.08
11.02
1.55
0.14
0.46
0.03
0.11
3.03
13.83
21.93
2.47
1.66
3.89
0.99
2.46
1.75
7.85
22.32
1.21
1.08
3.59
0.80
2.69
0.53
4.96
10.71
0.70
0.14
0.69
-0.12
-0.61
0.66
2.90
22.64
0.30
0.49
3.52
0.29
2.20
Under CAPM
µ
σ
SR
β
α1
t(α1 )
α2
t(α2 )
0.14
1.17
11.62
0.21
0.02
0.57
-0.05
-1.35
0.58
3.61
16.18
-0.21
0.70
3.69
0.67
3.43
0.76
4.41
17.23
-0.19
0.86
3.66
0.76
3.17
30
0.14
1.17
11.69
0.21
0.02
0.58
-0.05
-1.34
Table II
Unconditional performance measures for ICAPM efficient strategies
The table reports unconditional measures for evaluating ex post performance of ICAPM
strategies where the state variable is excess returns on long-term bonds. Panels A, B,
and C present evaluation measures where the desired hedging level δS is given by δS =
VS ωmv , δS = 2VS ωmv , and δS = 12 VS ωmv , respectively, where wmv is unconditional efficient
mean variance portfolio and VS is the covariance between unexpected returns and unexpected values of the state variables. Panel A exhibits evaluation measures based on both
constrained and unconstrained portfolios. Panels B and C present measures where portfolios are unconstrained using both realized and orthogonal returns. Orthogonal returns are
obtained as the sum of intercepts and residuals in a regression of returns generated by the
ICAPM strategies on the state variable.
Panel A: δS = VS ωmv
M1
M2
M3
Unconstrained
M1
M2
M3
No short selling
Unrestricted intercepts
µ
σ
SR
β
α1
t(α1 )
α2
t(α2 )
-1.88
18.65
-10.06
0.42
-2.11
-2.08
0.32
0.39
0.95
5.64
16.89
0.24
0.82
2.72
0.79
2.57
0.69
5.36
12.96
0.02
0.68
2.33
0.84
2.88
1.03
8.94
11.57
1.55
0.17
0.59
0.07
0.23
2.95
13.04
22.62
2.37
1.63
4.23
1.06
2.90
1.75
7.81
22.45
1.22
1.08
3.64
0.78
2.67
1.03
5.73
17.91
0.92
0.52
2.46
0.23
1.13
1.41
4.77
29.62
0.19
1.31
5.10
1.08
4.23
Under ICAPM
µ
σ
SR
β
α1
t(α1 )
α2
t(α2 )
0.07
1.66
4.03
0.21
-0.05
-0.69
-0.06
-0.80
0.75
3.89
19.37
-0.09
0.81
3.81
0.81
3.78
1.65
6.58
25.02
-0.06
1.68
4.68
1.59
4.35
31
0.14
1.37
9.97
0.24
0.00
0.04
-0.07
-1.85
Panel B: δS = 2VS ωmv
M1
M2
M3
Realized
M1
M2
M3
Orthogonal
Unrestricted intercepts
µ
σ
SR
β
α1
t(α1 )
α2
t(α2 )
-1.89
18.61
-10.13
0.42
-2.12
-2.10
0.31
0.38
0.95
5.63
16.84
0.23
0.82
2.71
0.78
2.57
0.68
5.33
12.73
0.02
0.67
2.30
0.83
2.86
-1.88
18.61
-10.09
0.47
-1.96
-1.89
0.36
0.43
0.61
5.47
11.17
0.16
0.58
1.91
0.61
1.98
0.52
5.29
9.86
-0.03
0.53
1.77
0.72
2.44
0.56
3.11
18.02
-0.18
0.59
3.52
0.61
3.55
1.05
4.53
23.14
-0.20
1.08
4.34
1.04
4.10
Under ICAPM
µ
σ
SR
β
α1
t(α1 )
α2
t(α2 )
0.07
1.44
4.64
0.22
-0.06
-1.02
-0.07
-1.31
0.57
3.11
18.21
-0.16
0.66
3.99
0.65
3.89
1.12
4.54
24.67
-0.16
1.21
4.94
1.12
4.50
32
0.10
1.43
7.13
0.26
0.06
1.11
0.02
0.45
Panel C: δS = 12 VS ωmv
M1
M2
M3
Realized
M1
M2
M3
Orthogonal
Unrestricted intercepts
µ
σ
SR
β
α1
t(α1 )
α2
t(α2 )
-1.86
18.72
-9.91
0.42
-2.09
-2.06
0.33
0.40
0.96
5.66
17.00
0.24
0.83
2.74
0.79
2.59
0.73
5.43
13.40
0.03
0.71
2.40
0.87
2.93
-1.85
18.72
-9.88
0.47
-1.94
-1.85
0.39
0.46
0.62
5.51
11.27
0.17
0.59
1.92
0.62
2.00
0.54
5.38
10.11
-0.02
0.55
1.82
0.74
2.47
0.89
6.04
14.76
-0.02
0.89
2.64
0.97
2.85
2.33
11.49
20.28
0.04
2.32
3.60
2.32
3.54
Under ICAPM
µ
σ
SR
β
α1
t(α1 )
α2
t(α2 )
0.07
2.31
2.89
0.19
-0.04
-0.33
-0.03
-0.29
1.13
6.11
18.51
0.05
1.10
3.31
1.13
3.34
2.70
11.58
23.30
0.13
2.62
4.15
2.52
3.92
33
0.21
2.25
9.41
0.25
0.17
1.50
0.15
1.36
Table III: Unconditional performance measures under model uncertainty
Panel A: Unconditional mean variance portfolios
M1 stands for no predictability portfolios, and M4 stands for portfolios derived based on a
weighted model that averages across individual forecasting models using posterior probability as weight. The superscript mkt (ltb1 ) denotes that the market portfolio (market portfolio
plus excess return on long term bonds) is (are) the asset pricing factors. Performance measures are computed where the first portfolio is based upon the first P observations, and is
then rebalanced on a monthly basis using the next P + 1 observations, and so on, and it is
finally rebalanced using the first T − 1 observations, where T is the sample size.
1
1
Mmkt
Mmkt
Mltb
Mltb
1
4
1
4
1
1
Mmkt
Mmkt
Mltb
Mltb
1
4
1
4
Unconstrained
No Short Selling
P = 120
µ
0.14 0.72
σ
1.17 4.31
SR
11.62 16.76
β
0.21 -0.20
0.02 0.83
α1
t(α1 ) 0.57 3.61
α2
-0.05 0.75
t(α2 ) -1.35 3.18
0.07 1.51
1.66 6.39
4.03 23.61
0.21 0.00
-0.05 1.51
-0.69 4.33
-0.06 1.40
-0.80 3.95
0.14 0.88
1.17 3.85
11.69 22.79
0.21 0.38
0.02 0.66
0.58 3.56
-0.05 0.44
-1.34 2.42
0.14 2.02
1.37 8.34
9.97 24.18
0.24 0.77
0.00 1.59
0.04 3.86
-0.07 1.01
-1.85 2.54
P = 180
µ
0.19 0.42
σ
1.15 3.47
SR
16.17 12.19
β
0.22 -0.19
0.03 0.56
α1
t(α1 ) 0.79 2.75
α2
-0.04 0.66
t(α2 ) -1.46 3.23
0.10 1.07
1.61 6.11
6.41 17.46
0.24 -0.03
-0.07 1.09
-0.99 2.96
-0.10 1.18
-1.42 3.18
0.19 0.67
1.15 2.97
16.18 22.43
0.22 0.28
0.03 0.46
0.79 2.86
-0.04 0.43
-1.45 2.58
0.17 1.54
1.30 6.40
13.42 24.04
0.24 0.58
0.00 1.12
0.03 3.18
-0.08 0.91
-2.04 2.59
P = 240
µ
0.23 0.46
σ
1.22 2.95
SR
18.67 15.77
β
0.24 -0.02
0.01 0.48
α1
t(α1 ) 0.30 2.39
α2
-0.08 0.51
t(α2 ) -2.37 2.44
0.12 1.15
1.38 5.82
8.91 19.71
0.23 0.08
-0.09 1.07
-1.35 2.69
-0.16 1.03
-2.54 2.52
34
0.23 0.78
1.23 3.21
18.68 24.23
0.24 0.33
0.01 0.48
0.30 2.44
-0.08 0.41
-2.36 2.03
0.19 1.75
1.25 6.87
14.97 25.52
0.23 0.68
-0.02 1.14
-0.48 2.69
-0.12 0.82
-2.85 1.93
Panel B: ICAPM portfolios
M1 stands for the no predictability ICAPM portfolios, and M4 stands for ICAPM portfolios derived based on a weighted model that averages across individual forecasting models
using posterior probability as weight. The superscripts ltb2 and ltb3 stand for hedging level
given by δS = 2VS ωmv and δS = 12 VS ωmv , respectively. Performance measures are based on
realized excess returns on the step-ahead strategies (the first four columns) and on orthogonal returns (the next four columns) obtained as the sum of the intercepts and residuals in
a regression of excess realized returns on the state variable.
2
2
Mltb
Mltb
1
4
3
3
Mltb
Mltb
1
4
2
2
Mltb
Mltb
1
4
Realized
3
3
Mltb
Mltb
1
4
Orthogonal
P = 120
µ
σ
SR
β
α1
t(α1 )
α2
t(α2 )
0.07 1.03
1.44 4.36
4.64 23.64
0.22 -0.11
-0.06 1.09
-1.02 4.61
-0.07 1.00
-1.31 4.16
0.07 2.46
2.31 11.31
2.89 21.77
0.19 0.21
-0.04 2.35
-0.33 3.81
-0.03 2.20
-0.29 3.51
0.10 0.96
1.43 4.35
7.13 22.15
-0.17 0.15
0.06 0.99
1.11 4.09
0.02 0.94
0.46 3.84
0.21 2.08
2.25 11.21
9.41 18.52
-0.30 0.43
0.17 2.06
1.51 3.27
0.15 2.01
1.36 3.15
P = 180
µ
σ
SR
β
α1
t(α1 )
α2
t(α2 )
0.10 0.73
1.41 4.05
7.34 18.10
0.24 -0.13
-0.07 0.83
-1.29 3.42
-0.10 0.91
-1.86 3.73
0.10 1.74
2.20 11.01
4.67 15.77
0.25 0.16
-0.07 1.62
-0.63 2.44
-0.11 1.72
-0.92 2.58
0.14 0.65
1.40 4.04
9.97 16.04
-0.16 0.17
0.04 0.71
0.91 2.88
0.00 0.81
0.02 3.31
0.23 1.33
2.14 10.91
10.74 12.20
-0.28 0.46
0.12 1.31
1.17 1.94
0.08 1.50
0.79 2.21
P = 240
µ
σ
SR
β
α1
t(α1 )
α2
t(α2 )
0.12 0.79
1.31 3.74
8.92 21.12
0.24 -0.02
-0.10 0.81
-1.75 3.15
-0.15 0.80
-2.75 3.03
0.14 1.86
1.71 10.56
7.98 17.66
0.23 0.29
-0.07 1.61
-0.71 2.23
-0.18 1.49
-1.98 2.03
35
0.13 0.39
1.31 3.57
9.74 10.81
-0.13 0.44
0.01 0.43
0.23 1.70
-0.04 0.46
-0.89 1.79
0.12
1.71
6.80
-0.09
0.01
0.08
-0.08
-0.86
0.46
9.81
4.69
1.35
0.45
0.63
0.49
0.68
Plot A1: alpha
Plot A2: alpha
2
% per month
% per month
2
1
0
−1
−2
7208
7612
8012
1
0
−1
−2
7208
8412 8712 9012
7612
Plot B1: Sharpe ratio
% per month
30
20
10
0
−10
7208
7612
8012
Plot C1: The evolution of $1
200
150
100
50
0
7208
7812
8412
20
10
0
−10
7208
8412 8712 9012
Portfolio value in $
% per month
8412 8712 9012
Plot B2: Sharpe ratio
30
Portfolio value in $
8012
9012 9512 0012
7612
8012
8412 8712 9012
Plot C2: The evolution of $1
1000
800
600
400
200
0
7208
7812
8412
9012 9512 0012
Figure 1 - Ex post performance of unconditionally efficient mean-variance portfolios
The left (right) plots display performance measures under unrestricted intercepts when portfolio holdings
are unconstrained (no short selling). Plots A1 and A2 describe the intercepts in a regression of excess returns
generated by the step-ahead strategies on the market portfolio. Plots B1 and B2 exhibit Sharpe ratios. Plots
C1 and C2 show the evolution of $1 invested in July 1972 in dynamic and passive strategies where cash
dividends are reinvested. In plots A1, A2, B1, and B2, the dashed-dotted lines reflect the performance of
Markowitz portfolios that disregard predictability. The solid and dashed lines reflect the performance of
Hansen-Richard portfolios, where alpha, beta, and risk premia are time varying (solid lines), and where beta
is constant (dashed lines). In plots C1 and C2, the dashed lines describe a buy-and-hold investment in the
value weighted CRSP index. The solid lines describe a dynamic investment in the Hansen-Richard strategy,
where alpha, beta, and risk premia are time varying. The Markowitz portfolio (dashed-dotted liens) appears
to be omitted from the plots because the value of $1 invested in that portfolio rapidly declines to zero.
36
Plot A1: alpha
Plot A2: alpha
2
% per month
% per month
2
1
0
−1
−2
7208
7612
8012
1
0
−1
−2
7208
8412 8712 9012
7612
Plot B1: Sharpe ratio
% per month
30
20
10
0
−10
7208
7612
8012
Plot C1: The evolution of $1
200
150
100
50
0
7208
7812
8412
20
10
0
−10
7208
8412 8712 9012
Portfolio value in $
% per month
8412 8712 9012
Plot B2: Sharpe ratio
30
Portfolio value in $
8012
9012 9512 0012
7612
8012
8412 8712 9012
Plot C2: The evolution of $1
1000
800
600
400
200
0
7208
7812
8412
9012 9512 0012
Figure 2 - Ex post performance of ICAPM efficient portfolios
The left (right) plots display performance measures under unrestricted intercepts under unconstrained portfolios (no short selling). Plots A1 and A2 describe the intercepts in a regression of orthogonal returns
generated by ICAPM strategies on the orthogonal component of market portfolio. Plots B1 and B2 exhibit
Sharpe ratios. Plots C1 and C2 show the evolution of $1 invested in July 1972 in dynamic and passive strategies where cash dividends are reinvested. In plots A1, A2, B1, and B2, the dashed-dotted lines reflect the
performance of ICAPM portfolios under no predictability. The solid and dashed lines reflect the performance
of ICAPM portfolios, where alpha, beta, and risk premia are time varying (solid lines), and where beta is
constant (dashed lines). In plots C1 and C2, the dashed lines describe a buy-and-hold investment in the
value weighted CRSP index. The solid lines describe a dynamic investment in an ICAPM strategy, where
alpha, beta, and risk premia are time varying. The no predictability portfolio (dashed-dotted liens) appears
to be omitted from the plots because the value of $1 invested in that portfolio rapidly declines to zero.
37