Extending the CAPM: the Reward Beta Approach

by

Graham Bornholt

Griffith University

Key words: CAPM, Asset pricing; Reward beta; Size effect; Book to market effect

JEL Classification: G12; G24; G31

Dr Graham Bornholt Department of Accounting, Finance and Economics Gold Coast Campus, Griffith University PMB 50 Gold Coast Mail Centre QLD 9726 Tel: (07) 555 28851 Email: g.bornholt@griffith.edu.au

The author thanks the participants at research seminars at Melbourne University, RMIT, Griffith University, and the 9th AIBF Banking and Finance Conference for their helpful comments on this paper. Thanks also go to two anonymous referees whose comments helped improve the paper. _____________

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Extending the CAPM: the Reward Beta Approach

ABSTRACT

This paper offers an alternative method for estimating expected returns. The

proposed reward beta approach performs well empirically and is based on asset

pricing theory. The empirical section compares this approach with the CAPM and the

Fama-French three-factor model. In out-of-sample testing, both the CAPM and the

three-factor model are rejected. In contrast, the reward beta approach easily passes

the same test. In robustness checks, the reward beta approach consistently

outperforms both the CAPM and the three-factor model.

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1. Introduction

Practitioners often use the Sharpe-Lintner capital asset pricing model (CAPM)

to produce estimates of a firm’s cost of capital. However, a number of empirical

studies (particularly Fama and French, 1992) have raised doubts about the validity of

the CAPM. After reviewing this evidence, Fama and French (2004) conclude that the

CAPM’s empirical problems invalidate most of its current applications. As doubts

about the reliability of the CAPM accumulate, multifactor models such as Fama and

French’s (1993) three-factor model are receiving more attention in empirical research.

Yet the three-factor model is no panacea. There are two main problems with this

model. Firstly, the method used by Fama and French to construct their size and book-

to-market factors is empirically driven and seems ad-hoc. As a result, the three-factor

model lacks a strong theoretical basis derived from asset pricing theory. Secondly, its

appeal in practice is limited by the need to find reliable forward-looking estimates of

the three factor-sensitivities and the three factor-premiums.

Given the deficiencies of both the CAPM and the three-factor model, finance

practitioners need a better method to estimate expected returns. This paper introduces

the reward beta approach. Reward beta estimates are used to replace CAPM beta

estimates in the security market line.

The theoretical justification for the reward beta approach comes from its

consistency with a wide range of plausible asset pricing models, including the CAPM.

These include mean-risk asset pricing models that differ from the CAPM to the extent

that their betas differ from the CAPM beta. Whichever of these models is correct,

however, the reward beta has the same value as the correct mean-risk beta. By basing

our approach on reward betas, we avoid using betas from the wrong model.

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Consideration needs to be given to the impact of size and book-to-market

effects on estimates of expected returns. Fama and French (1992) provide strong

evidence for size and book-to-market effects. In this paper, size and book-to-market

effects are incorporated directly into reward beta estimates through the use of

portfolios. This approach is tested against both the CAPM and the Fama-French three-

factor model using US stock data.

The test results overwhelmingly support the reward beta approach. In out-of-

sample testing, both the CAPM and the Fama-French three-factor models are rejected

as models of the expected returns for these portfolios. The CAPM results are

particularly poor. In robustness checks, the reward beta approach consistently

outperforms both the CAPM and the three-factor model.

This paper is organized as follows. All returns in the paper are discrete returns.

Ex-ante returns are random variables and are denoted by using uppercase R, and

realized returns by using lowercase r. Section 2 describes the reward beta approach,

while Section 3 specifies the compatible version of the market model to be used in

testing the competing models. Section 4 contains the empirical study comparing the

out-of-sample performance of the CAPM, the three-factor model and the reward beta

approach. Section 5 contains some final comments.

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2. The Reward Beta Approach

If there are N risky securities, then the Sharpe-Lintner capital asset pricing

model (CAPM) can be written

, , )][(β][ NiallforrRErRE fmifi ≤−+= (1)

where rf denotes the risk-free rate, Ri and Rm denote the random returns of security i

and the market, respectively, and where iβ = cov(Ri, Rm)/var(Rm) is the CAPM beta.

The CAPM assumes all investors choose mean-variance efficient portfolios. A

number of authors have presented alternative mean-risk asset pricing models by

replacing the mean-variance assumption with a specific mean-risk assumption (for

example, Kaplanski, 2004, and Bawa and Lindenberg, 1977).

Bornholt (2006) extends these cases by deriving a broad class of mean-risk asset

pricing models that includes the CAPM as a special case. The risk measures used in

the derivation of these models are those that are consistent with expected utility theory

and risk aversion. The risk measure that investors are assumed to use determines the

mean-risk beta (denoted giβ ) for that particular model. The risk measure that investors

actually use determines which of these models is correct. Although these competing

models have differing definitions for their mean-risk betas, the models all have

security market lines with the same form as the CAPM:

. , )][(β][ NiallforrRErRE fmgifi ≤−+= (2)

This common feature means that there is an alternative way to define the correct

value of beta. Rearranging (2) yields:

. , ][][

β NiallforrRErRE

fm

figi ≤

−−

=

6

We see that the correct mean-risk beta equals the ratio of the security’s risk premium

to the market risk premium. Since risk premiums are considered in finance to be the

reward for bearing risk, the latter ratio is called the reward beta (denoted riβ ). That is,

, , ][][

β NiallforrRErRE

fm

firi ≤

−−

= (3)

where the subscript r is used in order to differentiate this definition of beta from other

beta definitions. By basing beta estimation on reward betas, we avoid needing to

know which mean-risk asset pricing model is correct.

This equivalence between the reward beta and the correct mean-risk beta means

that using (2) with the reward beta riβ replacing giβ is justified by mean-risk asset

pricing theory. Accordingly, the reward beta approach to estimating expected returns

involves estimating the right-hand side of the following equation:

. , )][(β][ NiallforrRErRE fmrifi ≤−+= (4)

Comparing equation (4) with (2) indicates that the CAPM and this new approach will

differ in practice because their respective beta estimates differ. The remainder of this

section describes the preferred method for estimating reward betas.

Size and book-to-market effects are likely to pose problems for all mean-risk

models. For portfolios of stocks of the same size, for example, Tables 2 and 4 of Fama

and French (1993) typically show increasing average excess returns and decreasing

CAPM betas as book-to-market equity increases.1 Simply replacing variance with

another portfolio risk measure seems unlikely to lead to a model with betas that would

reverse this contradictory result.

1 This result can also be seen in Table 1. The limited Australian evidence does not display this

feature (see Gaunt, 2004; Faff, 2001).

7

Given these doubts, the reward beta approach uses forward-looking portfolio

reward beta estimates in (4) to estimate expected returns. For equities, this method

partitions the universe of stocks into portfolios periodically in such a way that stocks

in the same portfolio are those judged by the researcher to have had similar risk at the

time of portfolio formation, and then uses the reward beta estimate from the portfolio

in which the security currently belongs as the estimate of the individual security’s

reward beta.

Specifically, if fmj rrr and ,, denote the sample average returns of portfolio j, the

market proxy, and the risk-free rate, respectively, for a given estimation period then

the resulting estimate for the reward beta (denoted rjβ ) is

. )()(

βfm

fjrj rr

rr−−

= (5)

If a portfolio is composed of securities of similar risk, then its reward beta estimate

can be assigned to the securities currently in that portfolio. That is, if security i

currently belongs to portfolio j then security i’s current beta estimate is rjβ . This

process means that an individual security’s beta estimate is based on the post

portfolio-inclusion returns of securities that were in its current portfolio in the past. It

is this feature that makes these beta estimates forward-looking.

The reward beta estimator rjβ in (5) is a ratio estimator. Although it is a

consistent estimator of the portfolio’s reward beta, it may be biased in small samples.

However, since the composition of each portfolio is updated regularly, portfolio betas

can be estimated over much longer periods than the five years that is usually

considered appropriate for individual securities. As a result, sample sizes used in

portfolio reward beta estimation may be quite large. For example, sample size is 330

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(months) in the empirical study reported in this paper, and such a large sample will

ameliorate any potential bias.

How effective these estimates will be depends on whether or not stocks in the

same portfolio do have similar risk. Fama and French (1992, 1993, 1995, 1996) have

argued that if stocks are priced rationally then size and book-to-market equity must

proxy for underlying risk factors.2 If we accept their risk-based explanation of these

two effects then portfolios formed on size and book-to-market equity are composed of

stocks with similar risk, and so are amenable to the portfolio method of beta

estimation. These are the portfolios that are used to test the reward beta approach in

the empirical study in Section 4.

3. A Version of the Market Model

Before this testing can be undertaken, a version of the market model is needed

that is compatible with the reward beta approach. The appropriate version is:

, ])[(β)][(β jmmjfmrjfj RERrRErR ε+−+−=− (6)

where j denotes portfolio j, and jε is a random error term with 0][ =jE ε and

0][ =jmRE ε . This model is used in the cross-section regression tests reported in the

next section, and is called the reward beta model.

In this model, portfolio j’s expected return is determined by its reward beta rjβ ,

the risk-free rate and the market risk premium. The model’s error specification

implies that jβ in (6) equals the portfolio’s CAPM beta. The coefficient jβ in the

reward beta model contributes to the volatility of portfolio j’s return and controls the

2 To date, this risk-based explanation remains controversial. Lakonishok et. al. (1994) argue

for a mis-pricing explanation of these effects. See also Daniel et. al. (2001).

9

covariance between the portfolio’s and the market’s returns, but has no effect on its

expected value (unless rjj ββ = ). This means that while the value of the CAPM beta

may be useful ex-post for fitting the model to data, it is not relevant ex-ante for

estimating expected returns. If the CAPM were to hold then rjj ββ = , and the reward

beta model would reduce to the standard CAPM version of the market model applied

at the portfolio level.

4. Empirical Evaluation

The current study compares the CAPM, the Fama-French three-factor model

and the reward beta approach. Fama and French (1993) constructed three factors to

explain the cross-section of stock expected returns: (i) the excess return on a market

portfolio; (ii) the difference between returns on small and large stocks (“small minus

big,” denoted SMB); and (iii) the difference between the returns on high and low

book-to-market equity (“high minus low,” denoted HML). The three-factor model for

portfolio j is,

, ][][][][ HMLEhSMBEsrREbrRE jjfmjfj ++−=− (7)

where the factor sensitivities, bj , sj , and hj , are the coefficients from the time-series

regression,

. )()()( jjjfmjjfj HMLhSMBsrRbarR ε+++−+=− (8)

4.1 Size and Book-to-Market Effects

Fama and French (1993) use the three-factor model to explain the

cross-sectional variation in the returns of 25 portfolios of stocks sorted on size and

book-to-market equity comprising NYSE, AMEX and NASDAQ stocks. Fama and

French (1992, 1993, 1995, 1996) argue that size and book-to-market equity must

proxy for two underlying risk factors if stocks are priced rationally. Although their

10

risk-based explanation of these two effects remains controversial, if it is true then

portfolios of stocks sorted on size and book-to-market equity are composed of stocks

with similar risk. Accordingly, this study uses updated versions of the same 25 Fama-

French value-weighted portfolios formed on size and book-to-market equity that were

used by Fama and French (1993) to test the three-factor model.

4.2 Data

All monthly portfolio returns, market returns, Fama-French factor returns, and

risk-free returns used in this study were sourced from Kenneth French’s website, and

covered the period July 1963 to December 2003.3 The one-month Treasury bill rate is

used for the risk-free rate, while the proxy for the market return is the return of the

CRSP value-weighted index of all NYSE, AMEX and NASDAQ stocks.

The 25 Fama-French portfolios were formed on size and book-to-market equity

as follows:

“The portfolios, which are constructed at the end of each June, are the

intersections of 5 portfolios formed on size (market equity, ME) and 5 portfolios

formed on the ratio of book equity to market equity (BE/ME). The size

breakpoints for year t are the NYSE market equity quintiles at the end of June of

t. BE/ME for June of year t is the book equity for the last fiscal year end in t-1

divided by ME for December of t-1. The BE/ME breakpoints are NYSE

quintiles. …. The portfolios for July of year t to June of t+1 include all NYSE,

AMEX, and NASDAQ stocks for which we have market equity data for

3 Thanks go to Kenneth French for making his portfolio data widely available at

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

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December of t-1 and June of t, and (positive) book equity data for t-1.”

(Kenneth French’s website, webpage Data_Library/tw_5_ports.html)

4.3 Design

To evaluate the three competing approaches, the overall sample is split into two

parts: within-sample and out-of-sample. The within-sample period is used to calculate

time-series estimates of each model’s parameters. These estimates are then used in

cross-section regressions to test the competing models’ explanations of portfolio

out-of-sample average excess returns.

Fama and French (1992) used data covering the period July 1963 to December

1990. After that paper was published, there was considerable discussion about

whether their results were due to data dredging and also whether the CAPM

anomalies they outlined would persist or would the anomalies disappear subsequently.

As a result, the end of 1990 seems a natural point to end the within-sample period

because this allows these issues to be addressed during the current investigation.

Thus the (initial) within-sample estimation period is from July 1963 to

December 1990 (330 months). The out-of-sample period covers the remaining

months: January 1991 to December 2003 (156 months). All 330 months of the within-

sample period were used to estimate portfolio reward betas, CAPM betas, and three-

factor sensitivities for each of the 25 portfolios.

4.4 Within-Sample Estimates of Betas and Factor Sensitivities

Panel A of Table 1 contains the average monthly percentage excess returns on

the 25 Fama-French size-BE/ME value-weighted portfolios for the within-sample

period July 1963 to December 1990. Panel B contains the traditional CAPM beta

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estimates.4 A portfolio’s CAPM beta is the slope from the regression of its monthly

excess returns on the market proxy’s monthly excess returns. Panel C contains the

reward betas calculated according to (5). Similarly, Table 2 reports within-sample

estimates of the factor sensitivities from the three-factor time-series regression (8).

The regressions that follow use the coefficients in these tables as explanatory

variables.

[TABLES 1 AND 2 ABOUT HERE]

4.5 Out-of-Sample Tests

To set the stage, Figure 1 plots observed risk premiums (the 25 out-of-sample

average excess returns) against within-sample CAPM beta estimates, and against

within-sample reward beta estimates. Given that these betas are only estimates, and

that the asset pricing models are only one-period models that model expectations not

sample averages, we should not expect estimates to fall perfectly on a straight line.

Rather, if a model is the correct one then we should expect its estimates to fall

approximately on a straight line (with positive slope) when observed risk premiums

are plotted against that model’s beta estimates.

Figure 1 shows that the CAPM fails this informal test, at least relative to the

excellent performance of the reward beta approach. The three portfolios with the

largest CAPM betas have the smallest average excess returns. The better performance

of the reward beta approach is verified in the following regressions tests.

4 The problem identified by Fama and French (1992) for both the standard CAPM and the

zero-beta version is evident here. For the Small quintile, for example, average monthly

excess returns in Panel A increase monotonically from left to right whereas the

corresponding CAPM betas in Panel B decrease almost monotonically from left to right.

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[FIGURE 1 ABOUT HERE]

Cross-section regressions are used to test the three competing explanations for

portfolio average excess returns. These explanations are based on (6) (with the CAPM

as a special case), (7) and (8). For portfolio j’s ex-ante average excess returns

(denoted fj rR − ), the alternative models are:

CAPM: , )(β jfmjfj rRrR ε+−=− (9)

Three-factor: , )( jjjfmjfj LMHhBMSsrRbrR ε+++−=− (10)

Reward beta: , ])[(β)][(β jmmjfmrjfj RERrRErR ε+−+−=− (11)

where the averages are calculated over the out-of-sample period.

The standard cross-section regression methodology is employed. Estimates of

betas and factor sensitivities are the explanatory variables in these regressions.

Table 3 shows results for the regressions of out-of-sample average excess returns on

within-sample estimates of reward beta, CAPM beta and factor sensitivities. The

coefficient entries for a particular model in the table are estimates of the relevant

out-of-sample averages for that model as described by (9), (10) or (11).

Each coefficient entry is also the average slope on the same explanatory

variable from the corresponding month-by-month Fama-Macbeth regressions, and

each t-statistic is the average slope divided by its time-series standard error from the

month-by-month Fama-Macbeth regressions. [TABLE 3 ABOUT HERE]

As (9), (10) and (11) do not have intercepts when considered cross-sectionally,

the first three regressions test the three models by including an intercept. The intercept

in the CAPM regression is not significant, but neither is the coefficient on its beta, and

its adjusted R-squared value is very low. While the three-factor and reward beta

models have very similar adjusted R-squared values, the three-factor model is rejected

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because its intercept is highly significant (t = 4.11), whereas the intercept in the

reward beta model’s regression is not significantly different from zero (t = 0.30).

The next three regressions in the table make the situation clearer. Without the

added intercept, although the coefficient in the CAPM regression is highly significant

when there is no intercept, its goodness of fit is considerably worse (adjusted

R-squared = – 0.37). In the three-factor regression, only the coefficient on the market

beta is significant. Moreover, its adjusted R-squared value is now only 0.31 without

the intercept (down from 0.57). Contrast these results with the reward beta regression

results. In the latter regression, the coefficient on reward beta is highly significant

(t = 2.80) while the large adjusted R-squared value (0.58) is unchanged by the

removal of the intercept.

The value (0.40) of the coefficient on reward beta is the model’s estimate of the

(ex-ante) monthly market risk premium, and is very close to the observed market risk

premium of 0.46 percent per month calculated over for the whole period July 1963 to

December 2003.5 The lack of statistical significance for the coefficient on the CAPM

beta in this regression is not surprising since this coefficient is just an estimate of the

difference between the market’s average return in the out-of-sample period and its

expected value [see (11)].

For completeness, the final regression in the table tests the reward beta model

augmented with the size and book-to-market factor sensitivities. As might be expected

from the results of the earlier regressions, there is nothing to be gained by augmenting

the reward beta model with these factor sensitivities. The coefficient on reward beta

5 This degree of closeness is largely accidental. In the robustness investigation reported in

Table 4, the values of this coefficient range between 0.259 and 0.516.

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retains its significance (t-value = 3.14) whereas the factor sensitivities’ coefficients

are not significant. Moreover, the overall fit of this regression is slightly less (adjusted

R-squared = 0.54) than the fit of the reward beta model.

The above regression tests do not support the doubts that were raised about

Fama and French’s (1992) results mentioned earlier. The portfolios in the current

study are formed on size and book-to-market equity, and the out-of-sample period is

after the data period used in their study. If their results were due to data dredging, or if

the size and book-to-market effects they reported disappeared after 1990, then

within-sample reward betas would not have worked so well in the out-of-sample

regression tests.

4.6 Robustness

The regressions in Table 3 confirm the importance of reward beta in explaining the

cross-section of average returns. The reward beta approach strongly dominates both

the CAPM and the three-factor model in these regression tests. To see if the success

of the reward beta approach is dependent upon the research design employed,

robustness checks were undertaken.

The CAPM beta estimates used as explanatory variables in Table 3 are

estimated using all of the within-sample period, whereas the usual recommendation is

to use only the last five years of data when estimating betas. However, when only the

last five years of the within-sample period (1986-1990) are used to estimate CAPM

betas, the CAPM regression test results are even worse than those reported in Table 3

and are not shown here.

The second robustness check varies the point at which the whole sample is split

into the within-sample and the out-of sample periods. All the results up to now are

based on the within-sample period ending at the end of December 1990, giving a

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13 year out-of-sample period. To test the sensitivity of the results to moving the split

point, the calculations in Tables 1, 2 and 3 are repeated with the within-sample period

ending at the end of December in each year from 1987 to 1993, inclusive. As a result,

the out-of-sample periods vary from 17 years down to 10 years.

The results of these additional calculations for the CAPM are uniformly bad

(without an intercept, the CAPM regression adjusted R-squared values are all

negative) and are not shown here. When intercepts are added to the three-factor and

reward beta models, the intercept in the three-factor regression is significant at the 1%

level in every case whereas the intercept in the reward beta regression is not

significant in any case. These results strongly reinforce the earlier rejection of the

three-factor model.6

Table 4 reports the effect of changing the split point on regression tests of the

reward beta and three-factor models (both without intercepts). While the regression

coefficients, t-statistics and adjusted R-squared values in Table 4 are quite sensitive to

the endpoint of the within-sample period, the reward beta model strongly dominates

the three-factor model in every case. The adjusted R-squared values from the reward

beta regressions are all substantially larger than the corresponding three-factor

adjusted R-squared values. [TABLE 4 ABOUT HERE]

4.7 Interpretation of Results

The poor performance of the CAPM in explaining the cross-section of average

returns just adds to the already strong empirical evidence against the CAPM (see, for

example, Fama and French, 2004). The empirical advantage the reward beta approach

has over the three-factor model may be partly due to the way the two approaches deal

6 These results are available from the author on request.

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with interactions between size and book-to-market effects. Interactions between size

and book-to-market effects are automatically incorporated into the portfolio beta

estimates in the reward beta approach. In contrast, the three-factor model uses

separate size and book-to-market factors, and so is less able to take account of such

interactions.

There is another reason why the reward beta approach may give better results

than those provided by estimating more-specialized models of expected returns. As

long as the ratios of portfolio risk premiums to the market risk premium are

reasonably stable through time, the reward beta approach ought to produce acceptable

results. Such stability could have a rational or an irrational basis.

5. Conclusion

Both the CAPM and the Fama-French three-factor model are known to have

deficiencies. The empirical evidence does not support the CAPM, while the

three-factor model lacks theoretical asset pricing justification and its appeal is limited

in practice by estimation problems.

The reward beta approach, on the other hand, is based on asset pricing theory

and is strongly supported by the empirical evidence reported in this paper. These

advantages make this approach a better choice across a range of applications.

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REFERENCES

Bawa, V. S., and E. B. Lindenberg, 1977, Capital market equilibrium in a mean-lower

partial moment framework, Journal of Financial Economics 5, 189-200.

Bornholt, G. N., 2006, Expected utility and mean-risk asset pricing models, Working

paper (Griffith University, Gold Coast, QLD).

Daniel, K. D., D. Hirshleifer, and A. Subrahmanyam, 2001, Covariance risk,

mispricing, and the cross-section of security returns, Journal of Finance 56,

921-965.

Faff, R., 2001, An examination of the Fama and French three-factor model using

commercially available factors, Australian Journal of Management, 26, 1-17.

Fama, E. F., and K. R. French, 1992, The cross-section of expected stock returns,

Journal of Finance 47, 427-465.

Fama, E. F., and K. R. French, 1993, Common risk factors in the returns on stocks

and bonds, Journal of Financial Economics 33, 3-56.

Fama, E. F., and K. R. French, 1995, Size and book-to-market factors in earnings and

returns, Journal of Finance 50, 131-155.

Fama, E. F., and K. R. French, 1996, Multifactor explanations of asset pricing

anomalies, Journal of Finance 51, 55-84.

Fama, E. F., and K. R. French, 1997, Industry costs of equity, Journal of Financial

Economics 43, 143-193.

Fama, E. F., and K. R. French, 2004, The capital asset pricing model: Theory and

evidence, Journal of Economic Perspectives 18, 25-46.

19

Gaunt, C., 2004, Size and book to market effects and the Fama French three factor

asset pricing model: evidence from the Australian stock market, Accounting and

Finance, 44, 27-44.

Kaplanski, G., 2004, Traditional beta, downside risk beta and market risk premiums,

The Quarterly Review of Economics and Finance, 44, 636-653.

Lakonishok, J., A. Shleifer, and R. W. Vishny, 1994, Contrarian investment,

extrapolation, and risk, Journal of Finance 49, 1541-1578.

20

Table 1

Within-Sample Average Monthly Percent Excess Returns, CAPM betas and

Reward Betas for 25 Fama and French portfolios formed on Size and

BE/ME: 7/63-12/90, 330 Months The within-sample period is from July 1963 until December 1990. Calculations are based on monthly

excess returns (from French) for 25 Fama and French portfolios formed on size and book-to-market

equity from NYSE, AMEX and NASDAQ stocks. A portfolio’s average monthly percent excess return

is the average of its monthly excess returns over the within-sample period. A portfolio’s CAPM beta is

the slope from the regression of its monthly excess returns on the market’s monthly excess returns,

where the market’s excess return is the difference between the return of the CRSP value-weighted

index of all NYSE, AMEX and NASDAQ stocks and the one-month Treasury bill rate (from French).

A portfolio’s reward beta is the ratio of its average monthly percent excess return divided by the

average (0.337) of the market’s monthly percent excess returns for the within-sample period.

Size Book-to-Market Equity (BE/ME) Quintiles Quintile Low 2 3 4 High

Panel A: average monthly percent excess returns Small 0.174 0.573 0.624 0.811 0.931

2 0.282 0.574 0.780 0.845 0.941 3 0.313 0.602 0.590 0.761 0.825 4 0.354 0.288 0.552 0.725 0.837

Big 0.276 0.288 0.351 0.460 0.499 Panel B: CAPM betas

Small 1.44 1.26 1.16 1.08 1.11 2 1.44 1.24 1.13 1.05 1.12 3 1.36 1.16 1.04 0.98 1.07 4 1.22 1.14 1.04 0.97 1.08

Big 1.00 0.98 0.88 0.84 0.86 Panel C: Reward betas

Small 0.52 1.70 1.85 2.41 2.76 2 0.84 1.70 2.31 2.51 2.79 3 0.93 1.79 1.75 2.26 2.45 4 1.05 0.85 1.64 2.15 2.48

Big 0.82 0.85 1.04 1.36 1.48

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Table 2

Within-Sample Three-Factor Sensitivities from Time-Series Regressions for

25 Fama and French portfolios formed on Size and BE/ME: 7/63-12/90, 330

Months

jjjfmjjfj HMLhSMBsrRbarR ε+++−+=− )(

In the time-series regressions, a portfolio’s monthly excess return is regressed against Rm – rf ,

SMB and HML for the within-sample period from July 1963 to December 1990. Rm – rf is the

difference between the return of the CRSP value-weighted index of all NYSE, AMEX and

NASDAQ stocks and the one-month Treasury bill rate, and SMB and HML are the Fama and

French factor returns (from French): small minus big and high minus low.

Size Book-to-Market Equity (BE/ME) Quintiles Quintile Low 2 3 4 High

b

Small 1.05 0.98 0.94 0.90 0.95 2 1.10 1.02 0.98 0.97 1.06 3 1.10 1.02 0.97 0.98 1.05 4 1.06 1.08 1.05 1.03 1.15

Big 0.95 1.03 0.98 1.01 1.03 s

Small 1.39 1.27 1.14 1.09 1.18 2 1.00 0.94 0.83 0.71 0.84 3 0.70 0.62 0.55 0.44 0.65 4 0.30 0.25 0.24 0.21 0.37

Big – 0.20 – 0.21 – 0.26 – 0.22 – 0.03 h

Small – 0.28 0.08 0.27 0.38 0.62 2 – 0.49 0.02 0.25 0.46 0.70 3 – 0.43 0.04 0.31 0.50 0.70 4 – 0.45 0.02 0.32 0.55 0.74

Big – 0.44 – 0.01 0.19 0.57 0.76

22

Table 3

Out-of-Sample Cross-Section Regressions of Portfolio Average Monthly

Excess Returns on within-sample Reward betas, CAPM betas and 3-factor

Model Factor Sensitivities for 25 Fama and French portfolios formed on Size

and BE/ME: Out-of-Sample Period 1/91-12/03 (156 Months) The Tables 1 and 2 within-sample Reward betas, CAPM betas and 3-factor model factor sensitivities

for the 25 Fama and French portfolios formed on size and book-to-market equity are used as

explanatory variables in seven out-of-sample cross-section regressions. In these regressions, the

dependent variable is portfolio time-series average monthly excess return for the period from January

1991 to December 2003. Each reported coefficient is also the average slope from the corresponding

month-by-month cross-section regressions, and each reported t-statistic (in parentheses) is the average

slope divided by its time-series standard error from the month-by-month regressions.

RegressionModel Intercept Reward

Beta CAPM Beta b s h Adjusted

R2 CAPM + 1.43 – 0.377 – 0.01 intercept (1.24) (– 0.322)

3-factor + 2.96 – 2.12 0.237 0.415 0.57 intercept (4.11)** (– 2.51)* (0.73) (1.26)

Reward + intercept 0.354 0.366 0.038 0.58

(0.30) (3.03)** (0.03)

CAPM 0.894 – 0.37 (2.69)**

3-factor 0.717 0.309 0.499 0.31 (2.13)* (0.95) (1.50)

Reward 0.400 0.303 0.58 (2.80)** (0.69)

Reward 0.427 0.269 0.005 -0.060 0.54 augmented (3.14)** (0.71) (0.01) (-0.19)

* and ** indicate a (two-sided) significant difference from zero at the 5% and 1% levels, respectively.

23

Table 4

Robustness of Cross-Section Regression Results For various endpoints of within sample periods from the end of 1987 until the end of 1993, the within-

sample Reward betas, CAPM betas and 3-factor model factor sensitivities for the 25 Fama and French

portfolios formed on size and book-to-market equity are re-calculated and used as explanatory variables

in the out-of-sample cross-section regressions reported here. In these regressions, the dependent

variable is portfolio time-series average monthly excess return for the out-of-sample period from the

boundary to December 2003. Each reported coefficient is also the average slope from the

corresponding month-by-month cross-section regression, and each reported t-statistic (in parentheses)

is the average slope divided by its time-series standard error from the month-by-month regressions.

End of Within Period

Reward Beta

CAPM Beta b s h Adjusted

R2

12/1987 0.674 0.131 0.401 (2.22)* (0.48) (1.45) 0.18 0.271 0.286 0.39 (2.84) ** (0.79) 12/1988 0.669 0.115 0.351 (2.09)* (0.40) (1.20) 0.07 0.259 0.295 0.29 (2.44)* (0.76) 12/1989 0.607 0.189 0.384 (1.81) (0.62) (1.23) 0.14 0.334 0.214 0.45 (2.43)* (0.50) 12/1990 0.717 0.309 0.499 (2.13)* (0.95) (1.50) 0.31 0.400 0.303 0.58 (2.80)** (0.69) 12/1991 0.601 0.257 0.598 (1.73) (0.74) (1.67) 0.35 0.516 0.048 0.53 (2.90)** (0.10) 12/1992 0.604 0.235 0.501 (1.61) (0.64) (1.31) 0.22 0.432 0.129 0.44 (2.39)* (0.25) 12/1993 0.597 0.235 0.415 (1.46) (0.59) (1.01) 0.10 0.391 0.168 0.35 (1.99)* (0.30) * and ** indicate a (two-sided) significant difference from zero at the 5% and 1% levels, respectively.

24

Figure 1. Comparison of the out-of-sample performances of CAPM and Reward betas. (A)

Scatter plot of out-of-sample risk premium percent against within-sample CAPM beta for the 25 Fama

and French portfolios formed on size and book-to-market equity, and (B) scatter plot of out-of-sample

risk premium percent against within-sample Reward beta for the same 25 Fama and French portfolios.

The within-sample period is from July 1963 until December 1990, and the out-of-sample period is from

January 1991 until December 2003. Risk premium percent for a portfolio is the portfolio’s time-series

average monthly percent excess return over the out-of-sample period. Betas are estimated using the

whole within-sample period, and are listed in Table I.

0.00

0.50

1.00

1.50

2.00

0.00 0.50 1.00 1.50 2.00

(A) CAPM beta

risk

pre

miu

m %

0.00

0.50

1.00

1.50

2.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

(B) Reward beta

risk

pre

miu

m %