Conditional Beta[1]

8/8/2019 Conditional Beta[1]

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The Conditional Beta and the Cross-Section of Expected Returns

Turan G. Bali

, Nusret Cakici, and Yi Tang

ABSTRACT

This paper examines the cross-sectional relation between conditional betas and expected stock returnsfor the sample period of July 1963 December 2004. The portfolio-level analyses and the firm-levelcross-sectional regressions indicate a positive and significant relation between conditional betas and thecross-section of expected returns. The average return difference between high- and low- portfolios is

in the range of 0.89% to 1.01% per month depending on the time-varying specification of conditionalbeta. After controlling for size, book-to-market, liquidity and momentum, the positive relation betweenmarket beta and expected returns remains economically and statistically significant. These results arerobust across different measures of conditional beta, alternative portfolio partitions, firm-level andportfolio-level cross-sectional regressions, and excluding the AMEX and NASDAQ stocks.

Key words: market beta, conditional beta, CAPM, conditional CAPM, expected stock returns

JEL classification: G10, G11, C13

Turan Bali is a professor of finance at the Department of Economics and Finance, Zicklin School of Business,

Baruch College, City University of New York, One Bernard Baruch Way, Box 10-225, New York, New York10010. Phone: (646) 312-3506, Fax: (646) 312-3451, E-mail: [email protected]; Nusret Cakici is aprofessor of finance at the Department of Economics, City College of New York, City University of New York,Convent Avenue at 138th Street, New York, New York 10031, Phone: (212) 650-6204, Fax: (212) 650-6341, E-mail: [email protected]; Yi Tang is a Ph.D. student at the Department of Economics and Finance, ZicklinSchool of Business, Baruch College, City University of New York, One Bernard Baruch Way, Box 10-225, NewYork, New York 10010, Phone: (646) 312-3396, Fax: (646) 312-3451, E-mail: [email protected]. Corresponding author.


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I. Introduction

The capital asset pricing model (CAPM) of Sharpe (1964), Lintner (1965), and Black (1972)

implies the mean-variance efficiency of the market portfolio in the sense of Markowitz (1959). The

primary implication of the CAPM is that there exists a positive linear relation between expected

returns on securities and their market betas, and variables other than beta should not capture the

cross-sectional variation in expected returns. However, over the last three decades, a large number

of studies have tested the empirical performance of the static (or unconditional) CAPM in explaining

the cross-section of realized average returns. The findings of earlier studies indicate that firm size,

book-to-market ratio, earnings-to-price ratio, liquidity, and momentum have significant explanatory

power for average stock returns, while market beta has little or no power.

Early tests of the CAPM are based on the cross-sectional regressions of average stock

returns on estimates of individual stock betas.1

Two obvious problems with these tests are errors-in- variables and residual correlations. First, beta estimates for individual stocks are imprecise,

generating a measurement error problem when they are used to explain average returns. To improve

the accuracy of estimated betas, Blume (1970), Friend and Blume (1970), and Black, Jensen, and

Scholes (1972) use portfolios instead of individual stocks in their cross-sectional tests. Since

estimates of betas for diversified portfolios are more precise than estimates for individual stocks,

using portfolios in the cross-section regressions of average returns on betas diminishes the errors-in-

variables problem.

Second, the regression residuals have common sources of variation. Positive correlation in

the residuals yields downward bias in the usual OLS estimates of the standard errors of the cross-

sectional regression slopes. Fama and MacBeth (1973) introduce a method for addressing the

inference problem caused by correlation of the residuals in cross-sectional regressions. Rather than

estimating a single cross-section regression of average monthly returns on betas, they estimate

month-by-month cross-section of regressions of monthly returns on betas. The time-series averages

of the monthly slopes and intercepts and their standard errors are used to test whether the average

market risk premium is positive and the average intercept is equal to the risk-free rate.In very early cross-sectional tests, Douglas (1969), Black, Jensen, and Scholes (1972), Miller

and Scholes (1972), Blume and Friend (1973), and Fama and MacBeth (1973) find that the average

slope coefficient on beta is less than the average excess market return and the intercept is greater

1 The CAPM predicts that the intercept in these cross-sectional regressions is the risk-free interest rate and thecoefficient on market beta is the expected market risk premium.


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than the average risk-free interest rate.2 In their widely cited study, Fama and French (1992)

examine the static version of the CAPM and find both at the firm and portfolio level that the cross-

sectional relation between market beta and average return is flat.3 They interpret this flat relation as

strong empirical evidence against the CAPM.

As indicated by Jagannathan and Wang (1996), while a flat relation between the

unconditional expected return and the unconditional market beta may be evidence against the static

CAPM, it is not necessarily evidence against the conditional CAPM. The CAPM was originally

developed within the framework of a hypothetical single-period model economy. The real world,

however, is dynamic and hence, expected returns and betas are likely to vary over time. Even when

expected returns are linear in betas for every time periods, based on the information available at the

time, the relation between the unconditional expected return and the unconditional beta could be

flat.4

There is substantial empirical evidence that conditional betas and expected returns depend

on the nature of the information available at any given point in time and vary over time.5 In this

paper, we investigate whether time-varying conditional betas can explain the cross-section of

expected returns at the firm and portfolio level. Earlier studies use either a single or rolling long

sample of monthly data in estimating beta. Rather than using a long sample of monthly data, we use

daily returns within a month to compute realized beta for each stock trading at the NYSE, AMEX,

and NASDAQ for the sample period of July 1963 to December 2004. We propose three alternative

specifications of expected future beta based on the past information on realized beta. Specifically,

we use autoregressive, moving average, and GARCH-in-mean models to obtain time-varying

conditional betas for each stock.

For each specification of conditional beta, we find that stocks with high (low) market betas

have, on average, high (low) average returns. The portfolio-level analyses and the firm-level cross-

sectional regressions indicate that the positive relation between the conditional betas and the cross-

section of average returns is economically and statistically significant. Average portfolio returns

increase almost monotonically when moving from low-beta to high-beta portfolios. The R

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values

2 The evidence that the relation between beta and average return is too flat is also confirmed in time-series tests (e.g.,Friend and Blume (1970), and Black, Jensen, and Scholes (1972)).3 Jegadeesh (1992) obtains results similar to Fama and French (1992).4 This is because an asset that is on the conditional mean-variance frontier need not be on the unconditional frontier, asDybvig and Ross (1985) and Hansen and Richard (1987) point out.5 An incomplete list includes Bollerslev, Engle, and Wooldridge (1988), Harvey (1989, 2001), Shanken (1990, 1992),Ferson and Harvey (1991, 1999), Fama and French (1997), Lettau and Ludvigson (2001), Campbell and Vuolteenaho(2004), Jostova and Philipov (2005), Petkova and Zhang (2005), Lewellen and Nagel (2006), and Ang and Chen (2005).


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from the regression of average portfolio returns on average portfolio betas are in the range of 82%

to 98% for 10, 20, 50, and 100 beta portfolios. When we form the equal-weighted decile portfolios

by sorting the NYSE/AMEX/NASDAQ stocks based on conditional beta, the average return

difference between decile 10 (high- ) and decile 1 (low- ) portfolios is in the range of 0.89% to

1.01% per month depending on the time-varying specification of conditional beta. For 20, 50, and

100 beta portfolios, the average return difference ranges from 1.01% to 1.23% per month. To check

whether our findings are driven by small, illiquid, and low-price stocks, we exclude the AMEX and

NASDAQ stocks and form the beta portfolios by sorting only the NYSE stocks based on the

conditional betas. The results indicate that excluding the AMEX and NASDAQ sample has almost

no effect on our original findings. We also control for the well-known cross-sectional effects

including size and book-to-market (Fama and French (1993, 1995, 1996)), liquidity (Amihud (2002)

and Pstor and Stambaugh (2003)), and past return characteristics (Jegadeesh and Titman (1993)).After controlling for these effects, we estimate the cross-sectional beta-premium to be in the range

of 0.86% to 1.46% per month.

The paper is organized as follows. Section II contains the data and variable definitions.

Section III discusses the average raw returns and the average risk-adjusted returns on beta

portfolios. Section IV presents the firm-level cross-sectional regression results. Section V provides a

battery of robustness checks including portfolio-level cross-sectional regressions, testing long-term

predictive power of conditional betas, and some additional tests after controlling for liquidity and

momentum, after excluding the AMEX and NASDAQ sample, and after controlling for

microstructure effects. Section VI presents cross-sectional implications of the conditional CAPM

approach. Section VII concludes the paper.

II. Data and Variable Definitions

The first data set includes all New York Stock Exchange (NYSE), American Stock Exchange

(AMEX), and NASDAQ financial and nonfinancial firms from the Center for Research in Security

Prices (CRSP) for the period from July 1963 through December 2004. We use the daily stock returnsto generate the conditional beta measures. The second data set is the COMPUSTAT, which is

primarily used to obtain the book values for individual stocks.

For each month from July 1963 to December 2004, the following variables are computed for

each firm in the sample:


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SIZE:Following the existing literature, the firm size (ME) is measured by the natural logarithm of

the market value of equity (a stocks price times shares outstanding in millions of dollars) for each

stock.

BOOK-TO-MARKET: Following Fama and French (1992), we compute a firms book-to-market

ratio (BE/ME) using its market equity at the end of December of year t1 and the book value of

common equity plus balance-sheet deferred taxes for the firms latest fiscal year ending in calendar

year t1.6

REALIZED BETA:

To estimate monthly beta for an individual stock, we assume a single factor return generating

process in the form of a market model:

tditdmtititdi RR ,,,,,,,, ++= , (1)

where tdiR ,, is the daily return on stockion daydof montht, tdmR ,, is the daily market return on

daydof montht, and tdi ,, is the residual term.7 ti , is the intercept and ti, is the realized beta of

stocki in month tdefined as

=

=

==n

d

tmtdm

n

d

tmtdmtitdi

tdm

tdmtdi

ti

RR

RRRR

RVar

RRCov

1

2,,,

1

,,,,,,

,,

,,,,

,

)(

))((

)(

),( , (2)

where tiR , is the average daily return on stockiin montht, tmR , is the average daily market return

in montht, andnis the number of daily return observations in month t. In our empirical analysis,

tdmR ,, is measured by the CRSP daily value-weighted index, i.e., the daily value-weighted average

returns of all stocks trading at the NYSE, AMEX, and NASDAQ.

Earlier studies generally use monthly returns in estimating beta and testing the CAPM and

other factor models. We use daily returns and in principle, betas will be estimated more precisely

with higher-frequency data, just as Merton (1980) observed for variances. In practice, using daily

6 To avoid giving extreme observations heavy weight in our analysis, following Fama and French (1992), the smallest andlargest 0.5% of the observations on book-to-market ratio are set equal to the next largest and smallest values of the ratio(the 0.005 and 0.995 fractiles).7 French, Schwert, and Stambaugh (1987), Campbell, Lettau, Malkiel and Xu (2001), Goyal and Santa-Clara (2003), andBali, Cakici, Yan and Zhang (2005) use within-month daily returns to estimate the monthly market variance or themonthly idiosyncratic or total volatility of each stock trading at the NYSE, AMEX, and NASDAQ.


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returns creates microstructure issues caused by nonsynchronous trading. Nonsynchronous prices

can have a big impact on short-horizon betas. Lo and MacKinlay (1990) show that small stocks react

with a significant (week or more) delay to common news, so daily beta will miss much of the small

stock covariance with market returns. To mitigate the problem, in the robustness check section, we

exclude the AMEX and NASDAQ stocks and control for the size effect using two-dimensional

size/beta portfolios based on the NYSE sample. Also, following Dimson (1979), we include both

current and lagged market returns in the regressions and estimate realized beta as the sum of the

slopes ( 1,

ti and2,

ti ) in equation (3):

tditdmtitdmtititdi RRR ,,,1,2,,,

1,,,, +++= , (3)

where the sum of the slopes, 2,1,,

tititi += , adjusts for nonsynchronous trading (see Scholes and

Williams (1977) and Dimson (1979)).

Time-varying conditional betas are estimated based on the following autoregressive of order

one AR(1), moving average of order one MA(1), and GARCH(1,1)-in-mean models:

AR(1): ttt aa ++= 110 , 1101|1 )|( +== tARtttt aaE ,

21

2 )|( = ttE , (4)

MA(1): ttt bb ++= 110 , 1101|1)|(

+== t

MA

tttt bbE ,2

12 )|( = ttE , (5)

GARCH-in-mean: tttt cc ++= 2

1|10 ,2

1|101|1)|( +== tt

GARCHtttt ccE , (6)

2

12

2

110

2

1|1

2

)|( ++== ttttttE ,

where we drop the i subscript in equations (4)-(6) in order to save space. )|( 1 ttE denotes the

current conditional mean of realized beta estimated with the information set at time t1, 1 t . In

AR(1) and MA(1) models, the conditional variance of realized beta, denoted by )|( 12

ttE , is

assumed to be constant.8 In GARCH-in-mean model, the conditional variance of realized beta is

assumed to follow the GARCH(1,1) model of Bollerslev (1986).9 In this paper, we compare the

conditional betas (AR

tt 1| ,

MA

tt 1| ,

GARCH

tt 1| ) with the lagged realized betarealized

t 1 in terms of their

power to predict the cross-section of one-month-ahead average stock returns.

8 At an earlier stage of the study, we used AR(2), MA(2), ARMA(1,1), and ARMA(2,2) models in estimating theconditional betas. Since the predictive power of these alternative models turns out to be similar, we choose to presentresults from the most parsimonious AR(1) and MA(1) specifications.9 The GARCH-in-mean model is originally introduced by Engle, Lilien and Robins (1987) and then used by manyresearchers to model the conditional mean of asset returns as a function of the conditional volatility.


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Earlier studies find significant persistence in the conditional beta estimates for industry, size

or book-to-market portfolios (e.g., Braun, Nelson, Sunier (1995), and Ang and Chen (2005)).

However, these studies do not estimate conditional betas at the firm-level. Ang, Chen, and Xing

(2006) compute realized beta at the firm-level using daily returns over the past 12 months, and

propose alternative measures of downside risk based on the unconditional realized betas. According

to their descriptive statistics, the average AR(1) coefficient of realized betas are in the range of 0.077

to 0.675 depending on their specification of downside risk. Motivated by these studies, we generate

conditional beta estimates for each stock using AR(1) and MA(1) models given in equations (4)-(5).

Jagannathan and Wang (1996) examine the relation between unconditional betas and the

cross-section of unconditional expected returns by assuming that the conditional CAPM holds

period-by-period. As described in Jagannathan and Wang (1996), when the conditional CAPM is

assumed to hold for each period, cross-sectionally, the unconditional expected return on any asset isa linear function of its expected beta and its beta-premium sensitivity. In other words, the standard

static (or unconditional) CAPM leads to a two-factor unconditional asset pricing model, where the

first factor is the unconditional market beta that measuresaverage market riskandthe second factor

is the unconditional premium beta which measures beta-instability risk. According to this model,

stocks with higher expected betas should have higher unconditional expected returns. Similarly,

stocks with betas that are correlated with the market risk premium and hence are less stable over the

business cycle should also have higher unconditional expected returns. Jagannathan and Wang

(1996) indicate that the beta-premium sensitivity of an asset measures the instability of the assets

beta over the business cycle.

Motivated by the conditional CAPM literature, we model the conditional mean of market

beta as a function of its conditional variance as in the GARCH-in-mean specification. Equation (6)

models the current conditional mean and conditional variance of realized betas as a function of the

information set at time t1.

To provide an alternative justification for our use of the GARCH-in-mean model, we

compute the correlations between (i) the realized beta ( t ) and the conditional standard deviation

of realized beta ( t ); (ii) the realized beta ( t ) and the conditional variance of realized beta ( t );

(iii) the conditional mean of realized beta ( )|( 1 ttE ) and the conditional standard deviation of

realized beta ( t ); and(iv) the conditional mean of realized beta ( )|( 1 ttE ) and the conditional

variance of realized beta ( t ).


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The panel below presents the percentiles of the correlation measures for all stocks trading at the

NYSE, AMEX, and NASDAQ. The correlation statistics indicate a strong relation between the

monthly realized betas and their conditional volatility as well as a strong relation between the

conditional mean of monthly realized betas and their conditional volatility.

Correlation t , t t ,2t )|( 1 ttE , t )|( 1 ttE ,

2t

1% -0.625 -0.650 -0.988 -0.989

5% -0.390 -0.422 -0.971 -0.973

10% -0.272 -0.301 -0.944 -0.946

20% -0.150 -0.168 -0.838 -0.826

30% -0.069 -0.080 -0.505 -0.474

40% 0.002 -0.004 0.073 0.066

50% 0.067 0.067 0.583 0.549

60% 0.130 0.135 0.791 0.773

70% 0.196 0.204 0.882 0.877

80% 0.273 0.287 0.929 0.931

90% 0.390 0.412 0.963 0.967

95% 0.493 0.520 0.979 0.981

99% 0.696 0.711 0.992 0.993

Similarly, the estimated slope coefficients ( 1c ) in tttt cc ++= 2

1|10 are found to be statistically

significant which provides further justification of our use of the GARCH-in-mean model.

Table I presents percentiles of the time-series mean and standard deviation of realized and

expected conditional betas. The statistics presented in Panel A of Table I are based on realized betas

computed using daily returns over the previous month without lagged market return. The statistics

shown in Panel B of Table I are based on realized betas computed using daily returns over the

previous month with the lagged market return. In both panels, we only report the sample mean of

the realized betas because theoretically the mean of conditional betas is the same as the mean of

realized beta. Theoretically, the means should be the same, but the discrepancies are caused by the

filtration of conditional beta. As expected, the standard deviation of realized betas is greater than thestandard deviation of expected betas. In both panels, the standard deviation of conditional betas

obtained from the GARCH-in-mean model is somewhat greater than the standard deviation of

conditional betas obtained from the AR(1) and MA(1) models.


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We compare the conditional betas ( ARtt 1| ,MAtt 1| ,

GARCHtt 1| ) with the lagged realized beta

realizedt 1 in terms of their power to predict the one-month-ahead realized beta,

realizedt . Table II

shows the percentiles of R2 values from the regression of one-month-ahead realized betas on the

lagged realized beta and conditional betas. The performance of conditional betas in predicting theone-month-ahead realized beta is much higher than the lagged realized beta. The one percentile of

R2 is 0.01% for realizedt 1 and the 99 percentile of R2 is 26.82%. The corresponding figures are 1.14%

and 33.87% for GARCHtt 1| .

These results provide some explanation for why the earlier studies that use lagged realized

beta or unconditional beta could not identify a positive and significant relation between market beta

and expected stock returns. We think that to generate more accurate measures of expected futures

betas and to explain the cross-sectional variation in stock returns, one needs to use conditionalbetas.

III. Average Returns and FF-3 Alphas on Beta Portfolios

A. Univariate Portfolio-Level Analysis

Table III presents the equal-weighted average returns of decile portfolios that are formed by

sorting the NYSE/AMEX/NASDAQ stocks based on the lagged realized beta, and the conditional

AR(1), MA(1), and GARCH-in-mean betas. The results in Panel A of Table III are based on the

realized betas computed using daily returns over the previous month without lagged market return.

When portfolios are sorted based on the lagged realized beta, realizedt 1 , the average return

difference between decile 10 (high- ) and decile 1 (low- ) is about 0.49% per month with the

Newey-West (1987) t-statistic of 2.53. Although this result provides evidence against the empirical

validity of CAPM, it is not conclusive because as discussed earlier realizedt 1 is not a precise estimator

of realizedt . The static CAPM predicts a contemporaneous positive relation between expected stock

returns and market betas. However, we cannot use the current realized beta in empirical tests

because of the statistical problems indicated by Miller and Scholes (1972) and Fama and MacBeth

(1973).

We now examine the cross-sectional predictive power of the conditional beta measures

which are shown to be more accurate estimators of realizedt . When decile portfolios are sorted based


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on ARtt 1| ,MAtt 1| , and

GARCHtt 1| , the average return difference between decile 10 (high-) and decile 1

(low-) is in the range of 0.74% to 0.92% per month and highly significant.

In addition to the average raw returns, Panel A of Table III also presents the magnitude and

statistical significance of the intercepts (FF-3 alphas) from the regression of the equal-weighted

portfolio returns on a constant, excess market return, SMB and HML factors. As shown in Panel A

of Table III, the 10-1 difference in the FF-3 alphas is negative for realizedt 1 , but it is positive and

highly significant for the AR(1), MA(1), and GARCH-in-mean betas.

The results shown in Panel B of Table III are based on the realized betas computed using

daily returns over the previous month with the lagged market return. When portfolios are sorted

based on realizedt 1 , the average return difference between high- and low- portfolios is negative but

marginally significant. When decile portfolios are sorted based on

AR

tt 1|

,

MA

tt 1|

, and

GARCH

tt 1|

, theaverage return difference between high- and low- portfolios is positive, in the range of 0.89% to

1.01% per month, and highly significant. Panel B of Table III also shows that the 10-1 difference in

the FF-3 alphas is negative for realizedt 1 , but it is positive and highly significant for the AR(1), MA(1),

and GARCH-in-mean betas. Overall, the results in Table III indicate that the strong positive relation

between the conditional betas and expected returns is robust to the measurement of realized betas.

To save space, in the following sections, we report results only from the realized beta

measures estimated with the lagged market return.

B. Controlling for Size and Book-to-Market

We test whether there is a positive relation between conditional beta and expected returns

after controlling for size and book-to-market. We control for size by first forming decile portfolios

ranked based on market capitalization. Then, within each size decile, we sort stocks into decile

portfolios ranked based on GARCH-in-mean beta so that decile 1 (10) contains stocks with the

lowest (highest) market beta. Panel A of Table IV shows that in each size decile, the highest (lowest)

beta decile has a higher (lower) average returns. The column labeled Average Returns averagesacross the 10 size deciles to produce decile portfolios with dispersion in market beta, but which

contain all sizes of firms. This procedure creates a set of decile beta portfolios with near-identical

levels of firm size and thus these decile beta portfolios control for differences in size. After

controlling for size, the average return difference between high- and low- portfolios is 1.41% per


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month with the Newey-West t-statistic of 3.48. Thus, market capitalization does not explain the high

(low) returns to high (low) beta stocks.

We also control for book-to-market (BM) by first forming decile portfolios ranked based on

the ratio of book value of equity to market value of equity. Then, within each BM decile, we sort

stocks into decile portfolios ranked based on GARCH-in-mean beta so that decile 1 (10) contains

stocks with the lowest (highest) market beta. Panel B of Table IV shows that in each BM decile, the

highest (lowest) beta decile has a higher (lower) average returns. The last two columns report the

average returns and Newey-West t-statistics of 10 beta portfolios after controlling for BM. The

average return difference between high- and low- portfolios is 1.16% per month with the Newey-

West t-statistic of 3.69. Thus, book-to-market ratio does not explain the high (low) returns to high

(low) beta stocks.10

Table V shows the average return differences and FF-3 alphas on high- minus low-portfolios within each size and book-to-market decile. As shown in Panel A of Tabel V, for all

specifications of conditional beta, the average return differences and FF-3 alphas are positive and

economically significant within each size decile. Except for the two biggest size portfolios (Size-9,

Size-10), the average return differences and FF-3 alphas are also statistically significant at the 5%

level or better. For example, for the smallest size decile, the average return difference between decile

10 (high- ) and decile 1 (low- ) is 2.49% per month for ARtt 1| , 2.57% per month forMAtt 1| , and

2.50% per month for

GARCH

tt 1| . The corresponding FF-3 alphas are 2.02% per month for

AR

tt 1| ,

2.06% per month for MAtt 1| , and 2.02% per month forGARCHtt 1| . This strong positive relation between

market beta and expected return is present for Size-1 to Size-9 portfolios, and the relation becomes

somewhat weaker for the largest size portfolio (Size 10).

Panel B of Table V shows that for all specifications of conditional beta, the average return

differences and FF-3 alphas are positive and economically significant within each book-to-market

decile. The average return differences and FF-3 alphas are also statistically significant at the 5% level

or better. For example, for the lowest BM decile, the average return difference between decile 10(high-) and decile 1 (low-) is 1.50% per month for ARtt 1| , 1.49% per month for

MAtt 1| , and 1.29%

per month for GARCHtt 1| . The corresponding FF-3 alphas are 1.60% per month forARtt 1| , 1.54% per

10 In Table IV, decile portfolios are formed based on the GARCH-in-mean beta estimates. The results from the AR(1)and MA(1) models are very similar to those in Table IV. They are available from the authors upon request.


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month for MAtt 1| , and 1.36% per month forGARCHtt 1| . Although there is no obvious pattern, the strong

positive relation between market beta and expected return is more pronounced for BM-1 to BM-9

portfolios, and the relation becomes weaker for the highest BM portfolio.

To check whether the results in Tables IV and V are driven by small and illiquid stocks, in

the robustness check section, we report average returns from the double-sort size/beta and

BM/beta portfolios after excluding the AMEX and NASDAQ stocks. As will be discussed, after

controlling for size and book-to-market, we find a positive and highly significant relation between

conditional betas and the cross-section of expected returns for the NYSE sample as well.

IV. Firm-Level Cross-Sectional Regressions

We have so far tested the significance of conditional beta measures at the portfolio level. We

will now examine the cross-sectional relation between market beta and expected returns at the firm

level using Fama and MacBeth (1973) regressions.

We present the time-series averages of the slope coefficients from the cross-section of

average stock returns on the lagged realized beta, conditional beta, size, and book-to-market. The

average slopes provide standard Fama-MacBeth tests for determining which explanatory variables

on average have non-zero expected premiums. Monthly cross-sectional regressions are run for the

following econometric specifications:

tititittit

realized

tittti MEBEMER ,1,1,,31,,21,,1,0, )/log(log ++++= , (7)

tititittitAR

ttittti MEBEMER ,1,1,,31,,21|,,1,0, )/log(log ++++= , (8)

tititittitMA


tititittitGARCH


where tiR , is the realized return on stocki in month t, 1,log tiME is the natural logarithm of market

equity for firm i in month t1, )/log( 1,1, titi MEBE is the natural logarithm of the ratio of book

value of equity to market value of equity for firm i in month t1,realizedti 1, isthe lagged realized beta

of stocki in month t1, and ARtti 1|, ,MA

tti 1|, , andGARCH

tti 1|, are the conditional expected beta of stocki

in month testimated with the information set at month t1.

Table VI reports the time series averages of the slope coefficients i,t (i = 1, 2, 3) over the

498 months from July 1963 to December 2004. The Newey-West adjusted t-statistics are given in


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parentheses. The results show a negative but insignificant relation between the lagged realized beta

and the cross-section of average stock returns since the average slope, 1,t, from the monthly

regressions of realized returns on realizedti 1, alone is about 0.07% with a t-statistic of 1.19. The

univariate regression results indicate a significantly positive relation between average stock returnsand conditional betas. The average slope, 1,t, from the monthly regressions of realized returns on

ARtti 1|, ,

MAtti 1|, , or

GARCHtti 1|, is in the range of 0.44% to 0.50% and statistically significant at the 5% or

better. These values imply a reasonable expected market risk premium of 5.33% to 5.87% per

annum.

The univariate regression results also indicate a significantly negative relation between

average stock returns and firm size. The average slope, 2,t, from the monthly regressions of realized

returns on 1,log tiME alone is about 0.24% with a t-statistic of 4.75. The parameter estimatesshow a significantly positive relation between average stock returns and book-to-market ratio. The

average slope, 3,t, from the monthly regressions of realized returns on )/log( 1,1, titi MEBE alone is

about 0.42% with a t-statistic of 5.98. The findings of negative size and positive book-to-market

effect in Fama-MacBeth regressions are consistent with Fama and French (1992) and related studies.

The strong positive relation between conditional beta and expected stock returns is found to

be robust across different econometric specifications. When we include size to the univariate

regressions the average slope coefficient on

AR

tti 1|, ,

MA

tti 1|, , orGARCH

tti 1|, is about 0.80% and statisticallysignificant at the 1% level. When book-to-market is included to the univariate regressions, the

average slope coefficient on conditional betas is in the range of 0.65% to 0.68% and statistically

significant at the 1% level. When both size and book-to-market are included to the univariate

regressions, the average slope coefficient on the conditional betas is about 0.90% and statistically

significant at the 1% level.

The R2 values from the univariate regressions of realized returns on conditional beta are in

the range of 2.02% to 2.13%. When size and book-to-market are included to these univariate

regressions, the R2 values increase to 4.70% to 4.87%. Although the R2 values from univariate and

multivariate cross-sectional regressions are small, they are consistent with the earlier studies that

report R2 for the firm-level cross-sectional regressions.

Brennan, Chordia, and Subrahmanyam (1998) show that the standard Fama-MacBeth (1973)

approach presents problems if the independent variables used in the cross-sectional regressions (i.e.,


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factor loadings such as beta) are measured with error. One method of dealing with this measurement

error problem is to use the information from the first-stage regressions (in which the factor loadings

are estimated) to correct the coefficient estimates in the second stage regressions. To deal with this

errors-in-variables problem, Brennan et al. (1998) first obtain monthly slope coefficients from the

standard Fama-MacBeth regressions, and then run a time-series regression of the slope coefficients

on the factor portfolio returns. The intercept and its standard error from the second-stage time-

series regression are used to test the null hypothesis. The authors call the intercept purged estimator

because it purges the monthly estimates of the factor dependent component.

In our context, to correct for measurement error in the average slope coefficients on realized

and conditional betas, we first obtain the monthly slope coefficients ( t,1 ) from the cross-sectional

regressions:ti

realized

titttiR ,1,,1,0, ++= , tiAR

ttitttiR ,1|,,1,0, ++= , tiMA

ttitttiR ,1|,,1,0, ++= ,

and tiGARCH

ttitttiR ,1|,,1,0, ++= . Then, we construct two alternative beta-mimicking portfolios. At

the beginning of each month from July 1963 to December 2004, all NYSE/AMEX/NASDAQ

stocks are sorted into 10 beta portfolios with the NYSE and CRSP breakpoints. The value-weighted

average return difference between decile 10 and decile 1 is used as a proxy for the realization of the

beta-related risk factor. Once we generate two alternative factor portfolio returns based on the

CRSP and NYSE breakpoints, we regress t,1 on a constant and the factor portfolio returns that

mimic beta-related risk factor (Ft):

ttt uF ++= 10,1 ,

where the intercept 0)

is the purged estimator of the average slope coefficient on realizedti 1, ,AR

tti 1|, ,

MAtti 1|, , and

GARCHtti 1|, .

The results in Panel B of Table VI show that the intercept is economically and statistically

insignificant for the lagged realized beta, whereas the intercepts for AR(1), MA(1), and GARCH-in-

mean beta estimates are positive and highly significant. The magnitude and statistical significance of

the intercepts are very similar for the two beta-mimicking portfolios generated with the CRSP andNYSE breakpoints. In fact, the t-statistics of the intercepts presented in Panel B are not different

from the t-statistics of the average slope coefficients reported in Panel A of Table VI. Hence, we

conclude that the potential errors-in-variables problem in our conditional beta estimates is not

significant enough to alter our main conclusions.


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In addition to the slope coefficients from the univariate Fama-MacBeth regressions, we also

correct for measurement error in the slope coefficients from the multivariate regressions with size

and book-to-market. However, as shown in Panel A of Table VI, the magnitude and statistical

significance of the average slope coefficients on the conditional betas are much higher in the

multivariate regressions than in the univariate regressions. Hence, after correcting for the errors-in-

variables problem, the Newey-West t-statistics of the intercepts indicate 1% level of significance.

V. Robustness Check

A. Alternative Portfolio Partitions

Table VII presents the equal-weighted average returns of 20, 50, and 100 portfolios that are

formed by sorting the NYSE/AMEX/NASDAQ stocks based on the conditional AR(1), MA(1),

and GARCH-in-mean betas. The average return difference between high- and low- portfolios is

in the range of 0.83% to 1% per month for ARtt 1| , 0.89% to 1.11% per month forMAtt 1| , and 1.06%

to 1.31% per month GARCHtt 1| . All these average return differences are statistically significant at the

5% level or better. In addition to the average raw returns, Table VII also reports the magnitude and

statistical significance of the FF-3 alphas. The 10-1 difference in the FF-3 alphas is positive and

highly significant for the AR(1), MA(1), and GARCH-in-mean betas.

In addition to the firm-level Fama-MacBeth regressions, we examine the cross-sectional

relation between conditional beta and expected returns at the portfolio level. We present the time-

series averages of the slope coefficients from the cross-section of average portfolio returns on the

conditional portfolio beta:

tiAR

ttptttpR ,1|,,1,0, ++= , (11)

tiMA

ttptttpR ,1|,,1,0, ++= , (12)

tiGARCH

ttptttpR ,1|,,1,0, ++= , (13)

where tpR , is the realized return on portfoliop in month tcalculated as the equal-weighted average

returns of all stocks in portfolio p, and ARttp 1|, ,MA

ttp 1|, , andGARCH

ttp 1|, are the conditional expected

beta of portfolio p obtained from the equal-weighted average conditional beta of all stocks in

portfoliop in month testimated with the information set at month t1.


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First, we form 10, 20, 50, and 100 portfolios by sorting the NYSE/AMEX/NASDAQ

stocks based on the conditional AR(1), MA(1), and GARCH-in-mean betas. Then, for each month

from July 1963 to December 2004, we compute each portfolios return as the equal-weighted

average return of all stocks in the portfolio. Similarly, we calculate the portfolios conditional beta as

the equal-weighted average conditional beta of all stocks in the portfolio. Finally, we run the

univariate regressions of average portfolio returns on the average conditional portfolio beta for each

month from July 1963 to December 2004.

Table VIII presents the time series averages of the slope coefficients and the Newey-West

adjusted t-statistics in parentheses. The univariate regression results indicate a significantly positive

relation between average portfolio returns and average portfolio betas. For 10-beta portfolios, the

average slope from the monthly regressions of average portfolio returns on ARttp 1|, ,MA

ttp 1|, , and

GARCHttp 1|, is about 0.44%, 0.47%, and 0.53%, respectively. These average slope coefficients have a

Newey-West t-statistic of 2.25, 2.37, and 2.61, respectively. A notable point in Table VIII is that for

20, 50, and 100 beta-portfolios, the average slope coefficients are very similar. In other words, the

results are very robust across different portfolio formations. These slope coefficients, which imply

an expected market risk premium of 5.28% to 6.36% per annum, are also very similar to our earlier

findings from the firm-level cross-sectional regressions. The difference between the results reported

in Table VI and VIII is the magnitude of R2 values. As expected, the R2 values are much higher for

the portfolio-level regressions. More specifically, the R2

measures are about 58%-59% for 10-betaportfolios, 48%-49% for 20-beta portfolios, 35%-36% for 50-beta portfolios, and 25%-26% for 100-

beta portfolios.

In addition to the month-by-month Fama-MacBeth regressions, we take the time-series

average of the monthly portfolio returns and the monthly portfolio betas and compute overall

average portfolio return and overall average portfolio beta for 10, 20, 50, and 100 portfolios. Figures

1-4 plot the average portfolio return against the average portfolio beta and provide a strong positive

relation between market beta and expected returns for all portfolio partitions. The R2 value is

97.94% for 10-beta portfolios, 95.97% for 20-beta portfolios, 88.47% for 50-beta portfolios, and

81.50% for 100-beta portfolios. We also report the slope coefficients for each portfolio partition in

Figures 1-4. The results are very similar to our earlier findings from the month-by-month firm-level

and portfolio-level Fama-MacBeth regressions: As shown in Figures 1-4, the slope on average


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portfolio beta is almost identical for different portfolio partitions: 0.53% for 10, 50, and 100-beta

portfolios, and 0.52% for 20-beta portfolios.

B. Long Term Predictive Power of Conditional Betas

We have so far tested the significance of conditional beta measures for one-month ahead

predictability of stock and portfolio returns. That is, the conditional betas are estimated using the

information set at time t1 and their pattern is compared with the cross-section of average stock and

portfolio returns at time t. We will now investigate whether the conditional betas can predict the 2-

month, 3-month, and up to 12-month ahead returns.

Table IX presents the equal-weighted returns of decile portfolios that are formed by sorting

the NYSE/AMEX/NASDAQ stocks based on the conditional GARCH-in-mean betas. 11 The first

column repeats our earlier result for 1-month ahead predictability: When decile portfolios are sorted

based on GARCHtt 1| , the average return difference between decile 10 (high- ) and decile 1 (low- ) is

1.01% per month with the Newey-West t-statistic of 2.83. To test 3-month ahead predictability, we

form decile portfolios by sorting stocks based on their conditional betas at time t2 obtained from

the information set at time t3, GARCHtt 3|2 . The average return difference between high- and low-

portfolios is 0.93% per month with the t-statistic of 2.60. As shown in Table IX, the conditional

beta can predict up to 12-month ahead because the average return difference between high- and

low- portfolios is 0.75% per month with the t-statistic of 2.09.

In addition to the average raw returns, Table IX also presents the magnitude and statistical

significance of the FF-3 alphas from the regression of the equal-weighted portfolio returns on a

constant, excess market return, SMB and HML factors. The 10-1 difference in the FF-3 alphas is

positive and significant at the 5% level or better up to 9-month ahead predictability. However, the

economic and statistical significance of FF-3 alpha gradually reduce to 0.43% per month with the t-

statistic of 1.88 for 12-month ahead returns.

C. Controlling for Liquidity and Momentum

This section investigates whether the positive relation between conditional beta and the

cross-section of expected returns holds after controlling for liquidity and momentum.

11 To save space, we do not present the results from ARtti 1|, andMA

tti 1|, , which are very similar to those in Table IX.

They are available upon request.


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Liquidity generally implies the ability to trade large quantities quickly, at low cost, and

without inducing a large change in the price level. Following Amihud (2002), we measure stock

illiquidity as the ratio of absolute stock return to its dollar volume:

tititi VOLDRILLIQ ,,, /||= , (14)

where Ri,t is the return on stock i in month t, and VOLDi,t is the respective monthly volume in

dollars. This ratio gives the absolute percentage price change per dollar of monthly trading volume.

As discussed in Amihud (2002),ILLIQi,t follows the Kyles (1985) concept of illiquidity, i.e., the

response of price to the associated order flow or trading volume. The measure of stock illiquidity

given in equation (14) can be interpreted as the price response associated with one dollar of trading

volume, thus serving as a rough measure of price impact.

We control for liquidity by first forming decile portfolios ranked based on the illiquidity

measure of Amihud (2002). Then, within each illiquidity decile, we sort stocks into decile portfolios

ranked based on the GARCH-in-mean betas so that decile 1 (10) contains stocks with the lowest

(highest) market beta. Although not presented in the paper, in each illiquidity decile, the highest

(lowest) beta decile has a higher (lower) average returns. The column labeled Illiquidity in Panel A

of Table X presents the average returns across the 10 illiquidity deciles to produce decile portfolios

with dispersion in market beta. This procedure creates a set of decile beta portfolios with near-

identical levels of illiquidity. Thus, these decile beta portfolios control for differences in illiquidity.

After controlling for illiquidity, the average return difference between high- and low- portfolios is

1.20% per month with the Newey-West t-statistic of 3.05. Thus, liquidity does not explain the high

(low) returns to high (low) beta stocks.

Alternatively, we measure liquidity of individual stocks using dollar trading volume and

obtain similar results. The column labeled Volume in Panel A of Table X presents the average

returns across the 10 volume deciles to produce decile portfolios with dispersion in market beta.

After controlling for dollar trading volume, the average return increase monotonically from 0.92%

to 2.38% when moving from low- to high- portfolios. The average return difference between

high- and low- portfolios is 1.46% per month with the Newey-West t-statistic of 3.68. Thus,trading volume does not explain the high (low) returns to high (low) beta stocks either.

We control momentum by first forming decile loser-winner portfolios ranked based on the

past 6-month average returns of individual stocks. Then, within each 6-month momentum portfolio,

we sort stocks into decile portfolios ranked based on GARCH-in-mean beta so that decile 1 (10)

contains stocks with the lowest (highest) market beta. The column labeled MOM6 in Panel A of


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Table X presents the average returns across the 10 momentum deciles to produce decile portfolios

with dispersion in market beta. This procedure creates a set of decile beta portfolios with near-

identical levels of past average 6-month returns. Thus, these decile beta portfolios control for

differences in momentum. After controlling for momentum, the average return difference between

high- and low- portfolios is 0.99% per month with the Newey-West t-statistic of 2.89. Thus,

momentum does not explain the high (low) returns to high (low) beta stocks. Similar results are

obtained when we form loser-winner portfolios based on the past 12-month average returns

(MOM12). The average return difference between high- and low- portfolios is 0.86% per month

with the t-statistic of 2.74.

Finally, we investigate whether the positive relation between conditional beta and the cross-

section of expected returns holds in the firm-level Fama-MacBeth regressions after controlling for

liquidity, momentum, size, and book-to-market. Table XI presents the time series averages of the slope coefficients and the Newey-West

adjusted t-statistics in parentheses. The regression results indicate a significantly positive relation

between average stock returns and the conditional GARCH-in-mean betas after controlling for

illiquidity, trading volume, past average 6-month and 12-month returns with and without size and

book-to-market. The average slope coefficient on GARCHtt 1| is about0.51% with 1tILLIQ and 0.52%

with 1tVOL , and both coefficients are highly significant. The average slope coefficient onGARCHtt 1| is

about 0.41% and significant at the 5% level when 16 tMOM or 112 tMOM is included along withGARCHtt 1| in the cross-sectional regressions.

When alternative measures of liquidity and momentum are included along with GARCHtt 1| , size

and book-to-market, the average slope coefficient on GARCHtt 1| becomes very stable in the range of

0.62% to 0.66% for different specifications. As shown in Table XI, for all these multivariate

regressions with liquidity, momentum, size, and book-to-market, the average slope coefficient on

GARCHtt 1| is statistically significant at the 1% level.

At an earlier stage of the study, we replicated our results presented in Tables X and XI using

the conditional beta estimates obtained from the AR(1) and MA(1) specifications. The results turn

out to be very similar to those from GARCHtt 1| . To save space, we do not present our findings from

ARtti 1|, and

MAtti 1|, . They are available upon request.


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D. Results from the NYSE Sample

We have so far presented results from the NYSE/AMEX/NASDAQ sample which includes

small, illiquid, and low-price stocks. To check the robustness of our findings, we exclude the AMEX

and NASDAQ stocks and form the beta portfolios by sorting only the NYSE stocks based on the

conditional GARCH-in-mean betas. Table XII shows that for the univariate sort of NYSE stocks,

the average return difference between high- and low- portfolios is about 0.86% with the Newey-

West t-statistic of 2.79. The 10-1 difference in the FF-3 alphas is 0.37% with the t-statistic of 2.44.

In addition to the univariate sorts, we further examine the cross-sectional relation by

forming the beta portfolios within each size and book-to-market deciles. As presented in Table XII,

the average return difference between high- and low- portfolios is 0.84% after controlling for size

and 0.78% after controlling for book-to-market. Both return differences are statistically significant at

the 1% level. The 10-1 differences in the FF-3 alphas are also positive and highly significant. Theseresults indicate that excluding the AMEX and NASDAQ sample has almost no effect on our

previous findings. These results remain the same for alternative specifications of conditional beta

( ARtti 1|, andMA

tti 1|, ).

E. Controlling for Microstructure Effects and NYSE Breakpoint

In the previous section, we excluded the AMEX and NASDAQ stocks and presented the

return/beta estimates from the portfolios of NYSE stocks formed based on the NYSE breakpoints.

However, these results may be contaminated with microstructure effects because there is only one

month gap between the conditional beta estimates and portfolio returns. In this section, we skip the

month following portfolio formation to avoid microstructure effects and use the NYSE breakpoints

following Fama and French (1992) to generate beta portfolios of NYSE/AMEX/NASDAQ stocks

with a relatively more balanced average market share.12

Table XIII presents the average returns on the beta portfolios of NYSE/AMEX/NASDAQ

stocks with NYSE breakpoints after skipping the month following portfolio formation. When

portfolios are sorted based on the lagged realized beta, the average return difference between high-

and low- portfolios is economically and statistically insignificant. When portfolios are formed based

on the AR(1), MA(1), and GARCH-in-mean beta estimates, the average return difference between

12 Since there are so many small NASDAQ stocks in terms of market capitalization, portfolio breakdowns aredetermined using only NYSE stocks to avoid the beta portfolios which contain small stocks from being too small interms of average market share.


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decile 10 and 1 is about 0.70%, 0.72%, and 0.92% per month, respectively. Similar to our earlier

findings, for all conditional beta estimates, the 10-1 differences in average returns and FF-3 alphas

are positive and highly significant. Overall, the results in Table XIII indicate that forming portfolios

with CRSP or NYSE breakpoints and skipping the month following portfolio formation do not

affect our main conclusions.

F. Controlling for Size and Book-to-Market Simultaneously

When constructing beta portfolios, we control for size or book-to-market ratio, but not

both. In this section, we test whether the significantly positive relation between conditional beta and

expected returns remains intact after controlling for size and book-to-market simultaneously.

Table XIV presents average returns and FF-3 alphas on the quintile portfolios of realized

and conditional betas after controlling for size and book-to-market. At the beginning of each monthtfrom July 1963 to December 2004, all NYSE/AMEX/NASDAQ stocks are first sorted into 5 size

(market equity) portfolios. Then within each size portfolio, stocks are sorted into 5 BM (book-to-

market equity ratio) portfolios. Finally, within each portfolio formed based on the intersections of 5

size and 5 BM portfolios, stocks are sorted into 5 beta portfolios based on their realized and

conditional betas in month t-1.

As shown in Table XIV, when stocks in the 55 size/BM portfolios are sorted into five

realized beta ( realizedt 1 ) portfolios, the average return difference between high- and low- portfolios

is about 0.04% per month with a t-statistic of 0.32. Similar to our earlier findings from the univariate

and bivariate sorts, there is no significant relation between lagged realized beta and the cross-section

of expected returns from trivariate sorts either. When stocks in the 55 size/BM portfolios are

sorted into five AR(1), MA(1), and GARCH-in-mean beta portfolios, the average return difference

between high- and low- portfolios is about 1.09%, 1.10%, and 1.13% per month, respectively, and

these return differences are statistically significant at the 1% level. Moreover, for all conditional beta

estimates, the 5-1 differences in FF-3 alphas are positive and highly significant. Overall, the results in

Table XIV indicate that the significantly positive relation between conditional beta and the cross-

section of expected returns remains the same after controlling for both size and book-to-market.

As discussed in Section IV, we control for both size and book-to-market when we run the

firm-level cross-sectional regressions, and as shown in Panel A of Table VI, the average slope

coefficients on the conditional beta estimates remain to be positive and highly significant after


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controlling for size and book-to-market (see the last 3 rows in Panel A of Table VI). Given that we

have a significantly positive relation between conditional beta and expected returns from the firm-

level Fama-MacBeth regressions, the results from the 555 size/BM/beta portfolios shown in

Table XIV are not surprising.

VI. Cross-Sectional Implications of the Conditional CAPM

The static (or unconditional) capital asset pricing model (CAPM) of Sharpe (1964), Lintner

(1965), and Black (1972) indicates that there exists a positive linear relation between expected

returns on securities and their market betas:

)()( ,, tmiti RERE = , (15)

where )( ,tiRE is the unconditional expected excess return of asset i, )( ,tmRE is the unconditional

expected excess return of the market portfolio, and )(/),( ,,, tmtmtii RVarRRCov= is the

unconditional beta of asset i.

Fama and French (1992) and related studies find that the unconditional market beta cannot

explain the cross-sectional variation in expected stock returns. The unconditional CAPM was

derived by examining the behavior of investors in a hypothetical model in which they live for only

one period. In the real world investors live for many periods. Hence, in the empirical examination of

the CAPM, using data from the real world, it is necessary to make certain assumptions. One of the

most commonly made assumptions in the static CAPM framework is that that the betas of the assets

remain constant over time. However, this is not a reasonable assumption because the relative risk of

a firms cash flow is likely to vary over the business cycle. As indicated by Harvey (1989), Shanken

(1990), Jagannathan and Wang (1996), Ferson and Harvey (1991, 1999), and Lettau and Ludvigson

(2001), betas and expected returns will in general depend on the nature of the information available

at any given point in time and vary over time.

The conditional version of the CAPM imposes restriction that conditionally expected returns

on assets are linearly related to the conditionally expected return on the market portfolio in excess of

the risk-free rate. The coefficient in the linear relation is the assets conditional beta or the ratio of

the conditional covariance of the assets return with the market to the conditional variance of the

market:

)|()|(

)|,()|( 1,

1,

1,1,

1, ttm

ttm

ttmti

tti RERVar

RRCovRE

=

+

+

++

+(16)


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where )|( 1, ttiRE + is the conditional expected excess return of asset i, )|( 1, ttmRE + is the

conditional expected excess return of the market portfolio,)|(

)|,(

1,

1,1,

1,

ttm

ttmti

tiRVar

RRCov

=

+

++

+ is the

conditional market beta of asset i, and t denotes the information set at time t.

We now rewrite equation (16) to simplify the follow-up expressions:

( ) 1,1,1, | +++ = titmtti ARE , (17)

where ( )ttmtm REA = ++ |1,1, is the time t+1 conditional expected market risk premium.

Taking the unconditional expectation of both sides of equation (17), we obtain the

unconditional implication of the conditional CAPM:

[ ] ( )1,1,1, , +++ += titmimti ACovARE (18)

where ( )1,1, , ++ titmACov denotes the unconditional covariance, [ ] mtm AAE =+1, and [ ] itiE =+1, are

the unconditional means of the corresponding conditional estimates.

Notice that the last term in equation (18) depends only on the part of the conditional beta

that is in the linear span of the market risk premium. This motivates Jagannathan and Wang (1996)

to decompose the conditional beta of any asset i into two orthogonal components by regressing the

conditional beta on the market risk premium. For each asset i, we run the following regression:

1,1,1, )( +++ ++= timtmiiti uAA , (19)

where )(/),( 1,1,1, +++= tmtitmi AVarACov is the unconditional market beta-premium sensitivity that

measures the sensitivity of conditional beta to the market risk premium.

Substituting (19) into (18) gives:

[ ] ( )1,1, ++ += tmiimti AVarARE (20)

Hence, cross-sectionally, the unconditional expected excess return on any asset iis a linear function

of the unconditional average of its conditional market beta ( i) and its unconditional market beta-

premium sensitivity ( i). Equation (20) implies that stocks with higher expected betas have higher

unconditional expected returns. Likewise, stocks with betas that are prone to vary with the market

risk premium and hence less stable over the business cycle also have higher unconditional expected

returns. Hence, the one-factor conditional CAPM leads to a two-factor model for unconditional

expected returns.


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A complete test of the conditional CAPM specification requires estimation of expected beta

( i) and beta-premium sensitivity ( i). In this paper, we use the average conditional beta estimates

obtained from AR(1), MA(1), and GARCH-in-mean specifications as a proxy for i . We estimate

beta-premium sensitivity i using the lagged market return as a proxy for the expected market risk

premium, i.e., )(/),( ,1,, tmtitmi RVarRCov += , where the lagged market return, tmR , , is used as a

proxy for the time t+1 conditional expected market risk premium, ( ) tmttmtm RREA ,1,1, | == ++ .

For each month, we run the following cross-sectional Fama-MacBeth regressions:

ti

AR

it

AR

itttiR ,,2,1,0, +++= , (21)

ti

MA

it

MA

itttiR ,,2,1,0, +++= (22)

ti

GARCH

it

GARCH

itttiR ,,2,1,0, +++= (23)

where ARi ,MA

i , andGARCH

i are the time-series average ofAR

tti 1|, ,MA

tti 1|, , andGARCH

tti 1|, ,

respectively. ARi ,MA

i , andGARCH

i are obtained from the regression ofAR

tti 1|, ,MA

tti 1|, , andGARCH

tti 1|,

on tmR , , respectively. tmR , is proxied by the lagged return on the CRSP value-weighted index.

Table XV presents the time-series averages of the slope coefficients and their Newey-West

adjusted t-statistics (in parentheses) from the monthly cross-sectional Fama-MacBeth regressions of

stock returns on their average conditional beta and beta premium sensitivity. The average slope on

AR

i ,MA

i , andGARCH

i is about 0.6220, 0.6189, and 0.5705 with the Newey-West t-statistic of

2.89, 2.85, and 2.67, respectively. But, the average slope coefficients on beta premium sensitivity are

economically and statistically insignificant for all specifications of the conditional beta measures. The

results indicate significantly positive relation between average conditional beta and the cross-section

of expected returns within the conditional CAPM framework.

VII. Conclusion

This paper investigates the cross-sectional relation between conditional betas and expected

stock returns for the sample period of July 1963 December 2004. First, we use daily returns within

a month to compute realized beta for each stock trading at the NYSE, AMEX, and NASDAQ and

then propose three alternative specifications of expected future beta based on the past information


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on realized beta. Specifically, we use autoregressive, moving average, and GARCH-in-mean models

to obtain time-varying conditional betas for each stock.

For each specification of conditional beta, we find that average portfolio returns increase

almost monotonically when moving from low-beta to high-beta portfolios. The portfolio-level

analyses and the firm-level cross-sectional regressions indicate that the positive relation between the

conditional betas and the cross-section of average returns is economically and statistically significant.

For the NYSE/AMEX/NASDAQ sample, the average return difference between high- and low-

portfolios is in the range of 0.89% to 1.01% per month depending on the time-varying specification

of conditional beta. To check whether our findings are driven by small, illiquid, and low-price stocks,

we exclude the AMEX and NASDAQ stocks and form the beta portfolios by sorting only the

NYSE stocks based on the conditional betas. The results indicate that excluding the AMEX and

NASDAQ sample has almost no effect on our original findings. We also control for the well-knowncross-sectional effects including size, book-to-market, liquidity, and momentum. After controlling

for these effects, we estimate the cross-sectional beta-premium to be in the range of 0.86% to 1.46%

per month. These results are robust across different measures of conditional beta.


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25

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Table I. Time-Series Mean and Standard Deviation of Realized and Condition

This table presents the percentiles of the time-series mean and standard deviation of realized and conditional betas for the2004. In Panel A, realized beta for each stock trading at NYSE/AMEX/NASDAQ is computed using daily returns ovmarket return. In Panel B, realized beta is computed using daily returns over the previous month with the lagged market ris used as a proxy for the market portfolio. Conditional betas are estimated based on the AR(1), MA(1), and GARCH-in-m


22

)( =tE

MA(1): ttt bb ++= 110 , 1101|1)|(

+== t

MA

tttt bbE ,22 )( =tE


1|10 ,2

1|101|1)|( +== tt

GARCHtttt ccE ,

21|1

2 )|( = ttttE

Panel A. Realized Beta is Estimated without the Lagged Market Return

MEAN 1% 5% 10% 20% 30% 40% 50% 60% 70% 80%realizedt -0.1426 0.0347 0.1112 0.2325 0.3503 0.4610 0.5642 0.6862 0.8163 0.97

STD DEV 1% 5% 10% 20% 30% 40% 50% 60% 70% 80%realizedt 0.3929 0.5348 0.6360 0.7841 0.9236 1.0641 1.2175 1.3879 1.5970 1.86

ARtt 1| 0.0019 0.0111 0.0228 0.0463 0.0701 0.0965 0.1255 0.1590 0.2009 0.26

MAtt 1| 0.0020 0.0109 0.0220 0.0442 0.0657 0.0891 0.1154 0.1439 0.1830 0.23

GARCHtt 1| 0.0014 0.0101 0.0164 0.0454 0.0747 0.1071 0.1431 0.1880 0.2435 0.32

Panel B. Realized Beta is Estimated with the Lagged Market Return

MEAN 1% 5% 10% 20% 30% 40% 50% 60% 70% 80%realizedt -0.1450 0.0692 0.1744 0.3217 0.4587 0.5885 0.7111 0.8392 0.9726 1.13

STD DEV 1% 5% 10% 20% 30% 40% 50% 60% 70% 80%realizedt 0.5428 0.7126 0.8439 1.0454 1.2325 1.4229 1.6222 1.8551 2.1360 2.48

ARtt 1| 0.0165 0.0395 0.0596 0.0920 0.1225 0.1539 0.1885 0.2312 0.2845 0.36

MAtt 1| 0.0026 0.0117 0.0234 0.0472 0.0723 0.0976 0.1260 0.1609 0.2071 0.27

GARCHtt 1| 0.0059 0.0101 0.0221 0.0461 0.0802 0.1189 0.1635 0.2201 0.2877 0.38


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Table II. Performance of Lagged Realized and Conditional Betas in Predicting Futur

This table presents the percentiles of the R2 values from the regression of one-month-ahead realized betas on the laggedsample period of July 1963 to December 2004. In Panel A, realized beta for each stock trading at NYSE/AMEX/NASDAthe previous month without lagged market return. In Panel B, realized beta is computed using daily returns over the preturn. The CRSP value-weighted index is used as a proxy for the market portfolio. Conditional betas are estimated based mean models. The following OLS regressions are run to obtain the R2 values:

trealizedt

realizedt dd ++= 110

tARtt

realizedt dd ++= 1|10

tMAtt


tGARCHtt



R2 1% 5% 10% 20% 30% 40% 50% 60% 70% 80%

realized

t 1 0.01% 0.11% 0.22% 0.69% 1.18% 1.94% 3.22% 4.66% 6.89% 9.94

AR

tt 1| 0.74% 2.04% 2.87% 4.58% 6.16% 7.88% 10.31% 13.12% 15.77% 19.2

MA

tt 1| 0.63% 1.94% 2.67% 4.48% 5.95% 7.65% 9.98% 12.54% 15.39% 18.7

GARCH

tt 1| 1.14% 2.46% 3.33% 4.93% 6.75% 8.90% 11.01% 13.80% 16.70% 20.6


R2 1% 5% 10% 20% 30% 40% 50% 60% 70% 80%

realized

t 1 0.01% 0.03% 0.08% 0.22% 0.46% 0.81% 1.21% 1.73% 2.65% 4.12

AR

tt 1| 1.29% 2.28% 3.12% 4.22% 5.29% 6.10% 7.13% 8.38% 10.20% 12.6

MA

tt 1| 1.29% 2.35% 3.14% 4.15% 5.19% 6.04% 7.13% 8.40% 10.19% 12.5

GARCH

tt 1| 1.17% 2.12% 3.09% 4.23% 5.04% 6.02% 7.10% 8.31% 10.50% 12.5


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Table III. Equal-Weighted Returns on Portfolios Sorted by Realized and Condit

Equal-weighted decile portfolios are formed every month from July 1963 to December 2004 by sorting the NYSE/Arealized and conditional beta. In Panel A, realized beta for each stock is computed using daily returns over the previreturn. In Panel B, realized beta is computed using daily returns over the previous month with the lagged market rindex is used as a proxy for the market portfolio. Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) row High-Low refers to the difference in monthly returns between portfolios 10 and 1. The row Alpha reports

Fama-French (1993) model. Newey-West (1987) adjusted t-statistics are reported in parentheses.


22 )( =tE

MA(1): ttt bb ++= 110 , 1101|1)|(

+== t

MA

tttt bbE ,22 )( =tE


1|10 ,2

1|101|1)|( +== tt

GARCHtttt ccE ,

21|1

2 )|( = ttttE


realized

t 1

AR

tt 1| MA

tt 1|

DecileAverage

Return

Average

Beta

Average

Return

Average

Beta

Average

Return

Average

Beta

Ave

Re

1 Low 1.56 -1.65 1.09 0.08 1.07 0.02 1

2 1.36 -1.37 1.21 0.24 1.21 0.25 1

3 1.38 -0.01 1.30 0.37 1.31 0.37 1

4 1.38 0.23 1.41 0.48 1.36 0.48 1.

5 1.35 0.47 1.37 0.58 1.41 0.59 1.

6 1.41 0.72 1.44 0.69 1.46 0.70 1.

7 1.34 1.01 1.51 0.81 1.46 0.82 1.

8 1.27 1.37 1.53 0.96 1.54 0.96 1.9 1.25 1.91 1.58 1.16 1.60 1.15 1.

10 High 1.07 3.40 1.83 1.55 1.85 1.62 2.

HighLow-0.49

(-2.53)

0.74

(2.33)

0.78

(2.47)

0.

(2

Alpha-0.48

(-2.85)

0.50

(2.17)

0.53

(2.33)

0.

(2


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realized

t 1 AR

tt 1| MA

tt 1|

DecileAverage

Return

Average

Beta

Average

Return

Average

Beta

Average

Return

Average

Beta

Ave

Re

1 Low 1.44 -2.25 1.13 0.03 1.13 0.05 1

2 1.32 -0.55 1.15 0.33 1.15 0.33 1

3 1.33 -0.08 1.27 0.47 1.25 0.48 1

4 1.39 0.24 1.38 0.60 1.38 0.60 1.

5 1.42 0.53 1.36 0.71 1.37 0.72 1.

6 1.41 0.85 1.41 0.83 1.42 0.83 1.

7 1.36 1.21 1.49 0.95 1.49 0.95 1.

8 1.38 1.65 1.57 1.08 1.58 1.08 1.

9 1.25 2.32 1.62 1.27 1.64 1.26 1.

10 High 1.08 4.25 2.02 1.74 2.00 1.70 2.

HighLow-0.33

(-1.92)

0.89

(2.66)

0.87

(2.54)

1.

(2

Alpha-0.35

(-2.36)

0.63

(2.76)

0.60

(2.62)

0.

(3


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Table IV. Equal-Weighted Returns on Portfolios Sorted by GARCH-in-Mean Beta After Cont

In Panel A, we first form decile portfolios of NYSE/AMEX/NASDAQ stocks ranked based on their market capitalizationstocks into decile portfolios ranked based on GARCH-in-mean beta so that decile 1 (10) contains stocks with the lowest (hAverage Returns averages across the 10 size deciles to produce decile portfolios with dispersion in market beta and with nthese decile beta portfolios control for differences in size. In Panel B, we first form decile portfolios of NYSE/AMEX/Nbook-to-market ratios (BM). Then, within each BM decile, we sort stocks into decile portfolios ranked based on GARCH-in

stocks with the lowest (highest) market beta. The column labeled Average Returns averages across the 10 BM deciles to pin market beta and with near-identical levels of BM and thus these decile beta portfolios control for differences in book-to-m

Panel A. Equal-Weighted Returns on Beta-Portfolios After Controlling for

Small Size Size-2 Size-3 Size-4 Size-5 Size-6 Size-7 Size-8 Size-9

1 Low 2.85 0.61 0.48 0.63 0.68 0.78 0.88 0.83 0.84

2 2.12 0.98 1.03 0.83 0.88 0.86 1.11 1.00 1.05

3 2.63 0.98 1.06 0.86 0.81 0.84 0.99 1.10 0.96

4 2.40 1.31 1.13 1.05 0.96 0.98 1.12 1.07 1.03

5 2.93 1.29 1.20 1.19 1.13 1.12 1.24 1.24 1.01

6 3.18 1.51 1.17 1.18 1.15 1.18 1.16 1.13 1.06

7 3.39 1.79 1.55 1.29 1.17 1.17 1.19 1.11 1.28

8 3.49 1.83 1.62 1.65 1.53 1.40 1.27 1.34 1.11

9 4.46 2.18 1.88 1.63 1.72 1.69 1.28 1.26 1.28

10 High 5.36 3.04 2.31 2.23 2.05 2.13 2.03 1.75 1.51

High


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Panel B. Equal-Weighted Returns on Beta-Portfolios After Controlling for Book-

Low BM BM-2 BM-3 BM-4 BM-5 BM-6 BM-7 BM-8 BM-9

1 Low 0.35 0.40 0.77 0.86 0.82 1.15 1.21 1.42 1.35

2 0.19 0.65 0.72 0.90 1.03 1.05 1.22 1.49 1.31

3 0.51 0.79 0.93 1.04 1.02 1.16 1.19 1.32 1.60

4 0.59 0.70 1.11 1.10 1.05 1.22 1.47 1.49 1.90

5 0.69 1.01 1.19 1.11 1.13 1.27 1.45 1.57 1.84

6 0.45 0.92 0.90 1.20 1.20 1.36 1.57 1.64 1.71

7 0.85 1.09 1.39 1.31 1.27 1.37 1.52 1.68 1.79

8 1.00 1.19 1.34 1.40 1.17 1.67 1.54 1.70 1.91

9 1.03 1.39 1.34 1.58 1.63 1.73 1.89 2.05 2.43

10 High 1.64 1.62 1.68 1.90 2.04 2.07 2.44 2.31 2.94

High


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34

Table V. Average Return Differences and FF-3 Alphas within each Size and BM Deciles

In Panel A, we first form decile portfolios of NYSE/AMEX/NASDAQ stocks ranked based on their marketcapitalizations. Then, within each size decile, we sort stocks into decile portfolios ranked based on conditional beta so thatdecile 1 (10) contains stocks with the (lowest) highest market beta. Average return differences and alphas along with theirNewey-West (1987) adjusted t-statistics in parentheses are reported for each size decile. In Panel B, we first form decile

portfolios of NYSE/AMEX/NASDAQ stocks ranked based on their book-to-market ratios (BM). Then, within each BMdecile, we sort stocks into decile portfolios ranked based on GARCH-in-mean beta so that decile 1 (10) contains stockswith the lowest (highest) market beta. Average return differences and alphas along with their Newey-West (1987) adjustedt-statistics in parentheses are reported for each BM decile.

Panel A. Average Return Differences and FF-3 Alphas within Size Deciles

AR

tt 1| MA

tt 1| GARCH

tt 1|

Decile High Low Alpha High Low Alpha High Low Alpha

Small Size

2.49

(5.79)

2.02

(5.54)

2.57

(6.09)

2.06

(6.05)

2.50

(5.89)

2.02

(5.35)

Size-22.17

(4.94)

1.65

(4.55)

2.22

(5.01)

1.69

(4.67)

2.43

(5.34)

1.83

(5.14)

Size-31.92

(4.00)

1.40

(3.55)

1.88

(3.88)

1.33

(3.36)

1.83

(3.82)

1.27

(3.40)

Size-41.67

(3.65)

1.25

(3.48)

1.68

(3.57)

1.23

(3.44)

1.60

(3.46)

1.05

(3.44)

Size-51.54

(3.46)

1.14

(3.56)

1.52

(3.40)

1.13

(3.57)

1.37

(3.06)

0.94

(3.05)

Size-61.49

(3.25)

1.17

(3.59)

1.51

(3.21)

1.16

(3.57)

1.35

(2.83)

0.93

(2.69)

Size-71.19

(2.68)

1.01

(3.26)

1.08

(2.39)

0.88

(2.87)

1.15

(2.56)

0.88

(2.91)

Size-80.72

(1.67)

0.64

(2.34)

0.82

(1.87)

0.73

(2.64)

0.92

(2.09)

0.87

(3.05)

Size-90.60

(1.33)

0.63

(2.09)

0.65

(1.39)

0.67

(2.13)

0.67

(1.45)

0.65

(2.13)

Big Size0.24

(0.60)

0.42

(1.58)

0.24

(0.60)

0.43

(1.64)

0.28

(0.68)

0.50

(1.91)


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35

Panel B. Average Return Differences and FF-3 Alphas within Book-to-Market Deciles

AR

tt 1| MA

tt 1| GARCH

tt 1|

Decile High Low Alpha High Low Alpha High Low Alpha

Low BM 1.50(3.75)

1.60(4.59)

1.49(3.74)

1.54(4.50)

1.29(3.27)

1.36(4.21)

BM-21.27

(3.44)

1.27

(3.83)

1.31

(3.66)

1.33

(4.12)

1.22

(3.28)

1.19

(3.57)

BM-30.89

(2.58)

0.80

(2.71)

0.82

(2.37)

0.70

(2.37)

0.91

(2.61)

0.73

(2.67)

BM-40.98

(2.90)

0.68

(2.59)

0.93

(2.76)

0.65

(2.41)

1.03

(3.06)

0.67

(2.52)

BM-51.18

(3.48)

0.71

(2.71)

1.22

(3.48)

0.75

(2.73)

1.22

(3.22)

0.79

(2.65)

BM-6

0.90

(2.90)

0.44

(1.73)

0.84

(2.66)

0.39

(1.56)

0.92

(2.83)

0.37

(1.53)

BM-70.90

(2.46

Conditional Beta[1]

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