Value, momentum, and short-term interest rates

Paulo Maio1 Pedro Santa-Clara2

First version: July 2011

This version: December 20113

1Hanken School of Economics. E-mail: paulofmaio@gmail.com.2Millennium Chair in Finance. Nova School of Business and Economics, NBER, and CEPR. E-mail:

psc@novasbe.pt.3We thank John Cochrane for comments on a preliminary version of this paper.

Abstract

This paper offers a simple asset pricing model that goes a long way forward in explaining the value

and momentum anomalies. We specify a three-factor conditional intertemporal CAPM, denoted as

(C)ICAPM, where the factors (other than the market return) are the market factor scaled by the lagged

state variable and the “hedging”, or intertemporal, risk factor. These two factors are based on the

same macroeconomic state variable: the short-term interest rate. We test our three-factor model with

25 portfolios sorted on size and book-to-market and 25 portfolios sorted on size and momentum. The

(C)ICAPM outperforms the Fama and French (1993) three-factor model in pricing both sets of portfolios,

and only marginally underperforms the Carhart (1997) four-factor model. The ICAPM hedging risk

factor explains the dispersion in risk premia across the BM portfolios, while the scaled factor prices the

dispersion in risk premia across the momentum portfolios. According to our model, value stocks enjoy

higher expected returns than growth stocks because they have higher interest rate risk; that is, they have

more negative loadings on the hedging factor. Past winners also enjoy higher average returns than past

losers, because they have greater conditional market risk; that is, past winners have higher market betas

in times of high short-term interest rates.

Keywords: Cross-section of stock returns; Asset pricing; Intertemporal CAPM; Conditional

CAPM; Conditioning information; State variables; Linear multifactor models; Predictability of

returns; Fama-French factors; Value premium; Momentum; Long-term reversal in returns

JEL classification: G12; G14; E44

1 Introduction

There is much evidence that the standard Sharpe (1964)-Lintner (1965) Capital Asset Pricing

Model (CAPM) cannot explain the cross-section of U.S. stock returns in the post-war period.

Value stocks (stocks with high book-to-market ratios, (BM)), for example, outperform growth

stocks (low BM), which is known as the value premium anomaly [Rosenberg, Reid, and Lanstein

(1985), Fama and French (1992)]. Also, stocks with high prior one-year returns outperform

stocks with low prior returns, which is the momentum anomaly [Jegadeesh and Titman (1993)].

We offer a simple asset pricing model that goes a long way forward in explaining these two

anomalies. We specify a three-factor conditional intertemporal CAPM, denoted as (C)ICAPM,

that merges the conditional CAPM (CCAPM) and the intertemporal CAPM (ICAPM) from

Merton (1973). The factors in the model are the market equity premium (as in both the CCAPM

and the ICAPM); the market factor scaled by the state variable (as in the CCAPM); and the

“hedging” or intertemporal factor (as in the ICAPM).

The first source of systematic risk other than the market factor (the scaled factor) arises from

time-varying betas. The second source of systematic risk (the innovation in the state variable)

arises because stocks that are more correlated with good future investment opportunities should

earn a higher risk premium as they do not provide a hedge for reinvestment risk (unfavorable

changes in aggregate wealth for future periods).

In the empirical applications of both the ICAPM and CCAPM, the ultimate source for the

additional risk factors (relative to the usual market factor) is the same, that is, a time-varying

market risk premium in the current (CCAPM) or future (ICAPM) periods, or time-varying

betas (CCAPM), where the time variation is driven by common state variables.1

In our three-factor model, we use short-term interest rates (proxied either by the Federal

funds rate, FFR, or the relative or stochastically detrended Treasury-bill rate, RREL) as the

single state variable that drives both future aggregate investment opportunities and conditional

market betas. There is evidence in the return predictability literature that short-term interest

rates forecast expected (excess) market returns, especially at short forecasting horizons [Camp-

bell (1991), Hodrick (1992), Jensen, Mercer, and Johnson (1996), Patelis (1997), Thorbecke

1State variables used to proxy for the expected market return or conditional betas are largely borrowed fromthe fast growing literature on equity premium predictability: the slope of the yield curve or term structure spread[Campbell (1987), Fama and French (1989)]; the spread between higher- and lower-rated corporate bond yields(default spread) [Keim and Stambaugh (1986), Fama and French (1989)]; short-term interest rates [Campbell(1991), Hodrick (1992)]; and aggregate valuation ratios like the dividend yield [Fama and French (1988, 1989)]or the earnings yield [Campbell and Shiller (1988)], among others.

1

(1997), Ang and Bekaert (2007), among others]. Thus, both the Fed funds rate and the relative

T-bill rate represent valid ICAPM state variables.

It is not surprising that a factor model based on the short-term interest rate would per-

form well in driving equity risk premia. Specifically, the Fed funds rate represents the major

instrument of monetary policy, so changes in it should reflect the privileged information of the

monetary authority about the future state of the economy.2

We test the two versions of our three-factor model with 25 portfolios sorted on size and

book-to-market and 25 portfolios sorted on size and momentum. The cross-sectional tests show

that the (C)ICAPM explains a large percentage of the dispersion in average equity premia of

the two portfolio groups, with explanatory ratios around 70%. The (C)ICAPM outperforms the

Fama and French (1993) three-factor model when it comes to pricing both sets of portfolios,

and is only marginally behind the Carhart (1997) four-factor model, which has explanatory

ratios around 80%. The ultimate sources of systematic risk in our model (other than the

market factor), however, are associated with a single variable from outside the equity market,

the short-term interest rate. In contrast, the four-factor model has three equity financial-based

sources of systematic risk (other than the market factor). Thus, our model is more parsimonious

in this sense.

Moreover, our model represents an application of the ICAPM using a macroeconomic vari-

able, while the foundation for the Carhart (1997) model is less clear.3 In this sense, our model

is a step in the direction of a fundamental model of asset pricing instead of simply explaining

equity portfolio returns with the returns of other equity portfolios. In other words, the Fed funds

rate or the relative interest rate are not a priori mechanically related to the test portfolios, as is

the case with some of the equity-based factors in Fama and French (1993) and Carhart (1997).

Interestingly, the (C)ICAPM outperforms both the Fama and French (1993) and Carhart (1997)

models in fitting the difficult-to-price small-growth portfolio.

The hedging risk factor explains the dispersion in risk premia across the size-BM portfolios,

while the scaled factor prices the dispersion in risk premia across the size-momentum portfolios.

According to our model, value stocks enjoy higher expected returns than growth stocks because

they have more exposure to changes in the state variable; that is, they have more negative

2Bernanke and Blinder (1992) and Bernanke and Mihov (1998) argue that the Fed funds rate is a good proxyfor Fed policy actions.

3There is some evidence that the Fama-French size and value factors proxy for future investment opportunities[Petkova (2006) and Maio and Santa-Clara (2011)] and future GDP growth [Vassalou (2003)]. The justificationfor the momentum factor that Carhart (1997) uses is more controversial.

2

loadings on the hedging factor. One possible explanation for these loadings is that many value

firms have a poor financial position and thus are more sensitive to rises in short-term interest

rates that further constrain their access to external finance.

As for explaining momentum, in our model past winners enjoy higher average returns than

past losers because they have greater conditional market risk; that is, they have higher market

betas in times of high short-term interest rates. The explanation for this is that winner and loser

stocks have different characteristics at different points of the business cycle. Specifically, during

economic expansions, which are associated with high short-term interest rates, winners tend to

be cyclical firms, which have high market betas. Conversely, in recessions, with low short-term

interest rates, winners tend to be non-cyclical firms, with low market betas [see Grundy and

Martin (2001) and Daniel (2011) for a related discussion].

The results of the (C)ICAPM hold under a battery of robustness checks: conducting a

bootstrap simulation; including bonds in the test assets; testing the model simultaneously on

value and momentum portfolios; using an alternative measure of the innovation in the state

variable; including the market equity premium in the test assets; estimating the model with an

alternative sample; using alternative standard errors for the factor risk prices; and estimating

the model in expected return-covariance form.

We also test the (C)ICAPM over an alternative group of portfolios, 25 portfolios sorted on

size and long-term prior returns (SLTR25), to assess whether the model explains the long-term

reversal anomaly [De Bondt and Thaler (1985, 1987)]. The results show that our three-factor

model can explain a significant fraction of the dispersion in equity premia of these portfolios,

with explanatory ratios above 50%. As in the test with the SBM25 portfolios, it is the hedging

factor that drives the explanatory power of the (C)ICAPM over the SLTR25 portfolios.

Our work is related to the growing empirical literature on the ICAPM, in which the factors

(other than the market return) proxy for future investment opportunities.4 It is also related

to the large conditional CAPM literature, which postulates that the CAPM should hold on a

period-by-period basis, i.e., conditionally rather than unconditionally.5

4An incomplete list of papers that have implemented empirically testable versions of the original ICAPM overthe cross section of stock returns includes Shanken (1990), Campbell (1996), and more recently, Chen (2003),Brennan, Wang, and Xia (2004), Campbell and Vuolteenaho (2004), Guo (2006), Hahn and Lee (2006), Petkova(2006), Guo and Savickas (2008), and Bali and Engle (2010).

5An incomplete list of references includes Ferson, Kandel, and Stambaugh (1987), Harvey (1989), Cochrane(1996, 2005), He, Kan, Ng, and Zhang (1996), Jagannathan and Wang (1996), Ghysels (1998), Ferson and Harvey(1999), Lewellen (1999), Lettau and Ludvigson (2001), Wang (2003), Petkova and Zhang (2005), Avramov andChordia (2006), and Ferson, Sarkissian, and Simin (2008).

3

Our paper is organized as follows. In Section 2, we derive our three-factor model. Section

3 describes the econometric methodology and the data. In Section 4, we present and analyze

the main results for the cross-sectional tests of the (C)ICAPM. Section 5 provides a number of

robustness checks. In Section 6, we analyze the long-term reversal anomaly.

2 A three-factor model

We use a simple version of the Merton (1973) intertemporal CAPM (ICAPM) in discrete time

(the full derivation is presented in Appendix A).6 The expected return-covariance equation is

given by

Et(Ri,t+1)−Rf,t+1 = γ Covt(Ri,t+1 −Rf,t+1, Rm,t+1) + γz Covt(Ri,t+1 −Rf,t+1,∆zt+1), (1)

where Ri,t+1 denotes the return on asset i; Rf,t+1 stands for the risk-free rate; γ denotes the

(constant) coefficient of relative risk aversion (RRA); Rm,t+1 is the market return; and γz

represents the (covariance) risk price associated with state-variable risk, which is given by

γz ≡ −JWz(Wt, zt)JW (Wt, zt)

.

In this expression, JW (·) denotes the marginal value of wealth (W ), and JWz(·) represents a

second-order cross-derivative relative to wealth and the state variable (z). γz can be interpreted

as a measure of aversion to state variable/intertemporal risk, with ∆zt+1 = zt+1−zt representing

the innovation in the state variable.

We can rewrite the pricing equation (1) in expected return-beta form:

Et(Ri,t+1)−Rf,t+1 = γVart(Rm,t+1)Covt(Ri,t+1 −Rf,t+1, Rm,t+1)

Vart(Rm,t+1)

+γz Vart(∆zt+1)Covt(Ri,t+1 −Rf,t+1,∆zt+1)

Vart(∆zt+1)

= λM,tβi,M,t + λz,tβi,z,t, (2)

where λM,t and λz,t represent the conditional (beta) risk prices associated with the market and

state variable factors, respectively, and βi,M,t and βi,z,t denote the corresponding conditional

betas for asset i.7 Thus, although the market price of covariance risk is constant over time, the6Cochrane (2005) presents a similar covariance pricing equation based on a continuous time pricing kernel.7We call the innovation to the state variable a risk factor.

4

market price of beta risk is time-varying.

We assume that the conditional beta associated with the state variable innovation is constant

through time, that is, βi,z,t = βi,z, but, following the conditional CAPM literature [Harvey

(1989), Ferson and Harvey (1999), Lettau and Ludvigson (2001), and Petkova and Zhang (2005),

among others], we let the conditional market beta for asset i be linear in the lagged state variable:

βi,M,t = βi,M + βi,Mzzt. (3)

We estimate βi,M and βi,Mz (and also βi,z) from the time-series multiple regression:

Ri,t+1 −Rf,t+1 = ai + βi,M,tRm,t+1 + βi,z∆zt+1 + εi,t+1

= ai + (βi,M + βi,Mzzt)Rm,t+1 + βi,z∆zt+1 + εi,t+1

= ai + βi,MRm,t+1 + βi,MzRm,t+1zt + βi,z∆zt+1 + εi,t+1, (4)

where βi,M and βi,Mz represent the unconditional betas associated with the market factor

(Rm,t+1) and scaled factor (Rm,t+1zt), respectively.8

By substituting equation (3) in (2), we obtain a three-factor model:

Et(Ri,t+1)−Rf,t+1 = λM,tβi,M + λM,tztβi,Mz + λz,tβi,z. (5)

By applying the law of iterated expectations, we define the model in unconditional form:

E(Ri,t+1 −Rf,t+1) = E(λM,t)βi,M + E(λM,tzt)βi,Mz + E(λz,t)βi,z

= λMβi,M + λMzβi,Mz + λzβi,z, (6)

where λM , λMz, and λz represent the unconditional risk prices for the market, scaled, and

“hedging” factors, respectively. This is a conditional intertemporal CAPM, (C)ICAPM.

The economic intuition underlying the (C)ICAPM is that an asset that covaries positively

with changes in the state variable earns a higher risk premium than an asset that is uncorrelated

with the state variable. The reason is that the first asset does not provide a hedge against

future negative shocks in the returns of aggregate wealth, since it offers high returns when

8The interaction variable, Rm,t+1zt, is often interpreted as a managed return. See Hansen and Richard (1987),Cochrane (1996, 2005), Bekaert and Liu (2004), Brandt and Santa-Clara (2006), among others.

5

future aggregate returns are also high.9 Therefore, a rational investor is willing to hold such an

asset only if it offers a higher expected return in excess of the risk-free rate. This additional

risk premium is captured by the term λzβi,z.

The term λMzβi,Mz represents an additional risk premium that arises from the fact that the

market beta is time-varying and increases with the state variable. When z increases, asset i

becomes more correlated with the market return, making this asset riskier. This three-factor

model is parsimonious, since a single state variable drives the two sources of systematic risk

(other than the market factor).

In the empirical tests of the (C)ICAPM we use the Fed funds rate (FFR), and in an alterna-

tive version the relative Treasury bill rate (RREL), as the single state variable that drives future

aggregate investment opportunities (market returns), and that also drives conditional market

betas. There is strong evidence in the return predictability literature that short-term interest

rates forecast expected market returns, especially at short-term forecasting horizons [Campbell

(1991), Hodrick (1992), Jensen, Mercer, and Johnson (1996), Patelis (1997), Thorbecke (1997),

and Ang and Bekaert (2007), among others].10

3 Econometric methodology and data

In this section, we describe the econometric methodology and the data used in the asset pricing

tests conducted in the following sections.

3.1 Econometric methodology

The empirical methodology is the time-series/cross-sectional regressions approach (TSCS) pre-

sented in Cochrane (2005) (Chapter 12), which enables us to obtain direct estimates for factor

betas and (beta) prices of risk. This method has been employed by Brennan, Wang, and Xia

(2004), and Campbell and Vuolteenaho (2004), among others. The factor betas are estimated

from the time-series multiple regressions for each test asset:11

Ri,t+1 −Rf,t+1 = δi + βi,MRMt+1 + βi,M,zRMt+1zt + βi,z∆zt+1 + εi,t. (7)

9In this reasoning, we are assuming that the state variable covaries positively with future investment oppor-tunities.

10Under some assumptions, Brennan and Xia (2006) and Nielsen and Vassalou (2006) show that the intercept ofthe capital market line, which corresponds to the risk-free rate, represents one valid state variable in the ICAPM.

11The lagged conditioning variable is previously demeaned, which is a common practice in the conditionalCAPM literature [see, for example, Lettau and Ludvigson (2001) and Ferson, Sarkissian, and Simin (2003)].

6

We use the monthly excess market return (RM) to compute the betas, rather than the raw

market return, as in most applications of linear factor models in the empirical asset pricing

literature. RM is based on the value-weighted market return from CRSP and it is available

on Kenneth French’s website. RMt+1zt denotes the scaled factor (the interaction between the

equity premium and the lagged state variable), and ∆zt+1 ≡ zt+1− zt stands for the innovation

in the short-term interest rate, z = FFR or RREL.

The expected return-beta representation from equation (6) is estimated in a second step by

the OLS cross-sectional regression:

Ri −Rf = λMβi,M + λM,zβi,M,z + λzβi,z + αi, (8)

which produces estimates for factor risk prices (λ) and pricing errors (αi). In this cross-sectional

regression, Ri −Rf represents the average time-series excess return for asset i.12

We do not include an intercept in the cross-sectional regression since we want to impose

the economic restrictions associated with the model. If the model is correctly specified, the

intercept in the cross-sectional regression should be equal to zero; that is, assets with zero betas

with respect to all the factors should have a zero risk premium relative to the risk-free rate.13

Other studies use generalized least squares (GLS) or weighted least squares (WLS) cross-

sectional regressions to estimate factor risk prices in the cross-section of returns [e.g., Ferson

and Harvey (1999), Shanken and Zhou (2007), Lewellen, Nagel, and Shanken (2010)]. The OLS

cross-sectional regression is economically appealing and easy to interpret since it assigns equal

weight to all testing returns. Thus, we can assess if some economically interesting group of

portfolios (e.g., value or momentum portfolios) is properly priced by each model. Furthermore,

the GLS or WLS cross-sectional regressions are more difficult to interpret, since the testing

returns usually receive large positive and negative weights (the weights come from the inverse

of the covariance matrix of the residuals associated with the time-series regressions). Therefore,

it is harder to assess whether a particular model is able to explain the CAPM anomalies.

Moreover, use of OLS regressions allows us to directly compare different models, unlike either

12If the factor loadings are based on the whole sample, the risk price estimates from the TSCS approach arenumerically equal to the risk price estimates from Fama and MacBeth (1973) regressions. The standard errors ofthe risk price estimates in the Fama-MacBeth procedure, however, do not take into account the estimation errorin the factor loadings from the first-pass time-series regressions. In the TSCS approach, we use Shanken (1992)standard errors that correct for the error-in-variables bias, as discussed below.

13Another reason for not including the intercept in the cross-sectional regressions is that often the marketbetas for equity portfolios are very close to 1 (e.g., 25 size/book-to-market portfolios), creating a multicollinearityproblem [see Jagannathan and Wang (2007)].

7

GLS or WLS regressions, in which the weights are model-specific, and thus prevent us from

directly comparing the fit of two different models (e.g., (C)ICAPM versus the CAPM).

A test for the null hypothesis that the N pricing errors are jointly equal to zero (that is, the

model is perfectly specified) is given by

α′Var (α)−1 α ∼ χ2(N −K), (9)

where K denotes the number of factors (K = 3 in the (C)ICAPM), and α is the (N × 1) vector

of cross-sectional pricing errors.

Both the t-statistics for the factor risk prices and the computation of Var(α) are based on

Shanken (1992) standard errors, which introduce a correction for the estimation error in the

factor betas from the time-series regressions, thus accounting for the “error-in-variables” bias

in the cross-sectional regression [see Cochrane (2005), Chapter 12].

Although the statistic (9) represents a formal test of the validation of a given asset pricing

model, it is not particularly robust [Cochrane (1996, 2005), Hodrick and Zhang (2001)]. In

some cases, the near singularity of Var(α), and the inherent problems in inverting it, points to

rejection of a model with low pricing errors. In other cases, it is possible that the low values for

the statistic are a consequence of low values for Var(α)−1 (overestimation of Var(α)), rather

than the result of low individual pricing errors. In both cases, this asymptotic statistic provides

a misleading picture of the overall fit of the model.

A simpler and more robust measure of the global fit of a given model over the cross-section

of returns is the cross-sectional OLS coefficient of determination:

R2OLS = 1− VarN (αi)

VarN (Ri −Rf ),

where VarN (·) stands for the cross-sectional variance. R2OLS represents a proxy for the propor-

tion of the cross-sectional variance of average excess returns explained by the factors associated

with a given model.

A related measure is the mean absolute pricing error, computed as

MAE =1N

N∑i=1

|αi|,

which represents the average pricing error associated with a given model.

8

3.2 Data and variables

The data on the Federal funds rate and the three-month Treasury bill rate (TB) are from

the FRED database (St. Louis Fed). The relative Treasury-bill rate (RREL) represents the

difference between TB and its moving average over the previous twelve months, RRELt =

TBt − 112

∑12j=1 TBt−j . The portfolio return data, the one-month Treasury bill rate used to

construct portfolio excess returns, and the risk factors from alternative models are all obtained

from Kenneth French’s data library. The sample period we use is 1963:07–2009:12, where the

starting date coincides with most cross-sectional asset pricing tests in the literature.

Table 1 presents descriptive statistics for the factors in the (C)ICAPM,RMt+1, RMt+1FFRt,

RMt+1RRELt, ∆FFRt+1 and ∆RRELt+1. We also present descriptive statistics for the size

(SMB), value (HML), and momentum factors (UMD) from the Fama and French (1993) and

Carhart (1997) factor models.

We can see that the three (C)ICAPM factors are not persistent, with the innovation in the

Fed funds rate being the most persistent variable, with an autoregressive coefficient of 0.40.

Moreover, the three factors are not significantly correlated among themselves, with correla-

tion coefficients varying between -0.19 (RMt+1FFRt and ∆FFRt+1) and 0.15 (RMt+1 and

RMt+1FFRt), when the state variable is FFR. In the version with RREL the magnitudes of

the correlations among the three factors are smaller than 0.12. Hence the three factors from

the (C)ICAPM seem to proxy for different sources of systematic risk. As for the correlation

with the other risk factors, RMt+1FFRt is marginally negatively correlated with HML (-0.26)

and marginally positively correlated with UMD (0.22), while RMt+1RRELt is also slightly

positively correlated with the momentum factor (0.25).

Figure 1 depicts the time-series of the changes in both the Fed funds rate and RREL. We can

see that these two variables present an approximate pro-cyclical pattern, with sharp increases

during economic expansions, and some significant declines during recessions. The average Fed

funds rate change in expansions (as measured by the National Bureau of Economic Research

(NBER)) is 0.06% per month, and in recessions -0.38% per month. In the case of ∆RREL, we

have an average of 0.14% in expansions and -0.21% in recessions.

9

4 Main empirical results

4.1 Testing the (C)ICAPM

We assess whether the three-factor (C)ICAPM explains the value and momentum anomalies.

The value premium corresponds to the empirical evidence showing that value stocks (stocks

with a high book-to-market ratio) have higher average returns than growth stocks (stocks with

a low book-to-market) [see Rosenberg, Reid, and Lanstein (1985), and Fama and French (1992),

among others]. This spread in average returns is called an “anomaly” in the sense that the

baseline CAPM [Sharpe (1964) and Lintner (1965)] is not able to explain such a premium [see

Fama and French (1992, 1993, 2006)]. We use the standard 25 size/book-to-market portfolios

(SBM25) from Fama and French (1993) to test the value premium puzzle.

The momentum anomaly is that past winners (stocks with higher returns in the recent past)

continue to have subsequent higher returns, while past losers continue to underperform in the

near future [Jegadeesh and Titman (1993), and Chan, Jegadeesh, and Lakonishok (1996), among

others]. This return premium is not explained by either the baseline CAPM or the Fama and

French (1993) three-factor model [see Fama and French (1996)]. In fact, the momentum anomaly

represents one of the major challenges for most asset pricing models in the literature (Cochrane

(2007)). In order to assess the explanatory power of the (C)ICAPM for the momentum anomaly

we use 25 portfolios sorted on both size and prior one-year returns (SM25).14 The use of these

portfolios allows us to assess whether momentum is persistent across different size groups [see

Fama and French (2008)].15

The estimation results for the (C)ICAPM are displayed in Table 2. The results for the test

with the SBM25 portfolios (Panel A) show that the (C)ICAPM’s version with FFR explains a

significant fraction of the dispersion in average returns of these portfolios, with an R2 estimate

of 70% and an average pricing error of only 0.10% per month (which compares with a cross-

sectional average portfolio risk premium of 0.67% per month). Moreover, the model passes the

χ2 test with a p-value of 6%. The point estimate for the “hedging” risk price, λz, is negative

and strongly statistically significant (1% level), while the point estimate for the risk price of the

scaled factor, λM,z, is largely insignificant.

In the version with RREL, the model’s fit is somewhat worse, but still shows a good ex-

14Fama and French (1996), Bansal, Dittmar, and Lundblad (2005), Liu and Zhang (2008), He, Huh, and Lee(2010), and Maio (2011), among others, conduct asset pricing tests over portfolios sorted on momentum.

15Some authors argue that double-sort portfolios produce a greater dispersion in average returns [see, forexample, Lakonishok, Shleifer, and Vishny (1994)].

10

planatory power with an R2 estimate of 46% and an average pricing error of 0.14% per month.

This version also passes the χ2 test with a p-value of 9%. As in the version with FFR, the

estimate for λz is negative and strongly significant, while the estimate for λM,z is now positive,

although not significant at the 10% level.

Thus, the key factor that drives the fit of the model over the SBM25 portfolios seems to be

the innovation in the short-term interest rate, ∆FFRt+1 or ∆RRELt+1, rather than the scaled

factor, RMt+1FFRt or RMt+1RRELt.16

The results for the test with the SM25 portfolios (Panel B) indicate that the (C)ICAPM

based on FFR also explains a large fraction of the dispersion in average returns of these portfo-

lios, with an R2 estimate of 71%, which is very close to the explanatory ratio in the test with the

SBM25 portfolios. The average pricing error is 0.16% per month (compared to a cross-sectional

average portfolio risk premium of 0.60% per month), which is higher than the corresponding

mispricing in the test with SBM25, confirming that the size-momentum portfolios are harder

to price than the size-BM portfolios. The (C)ICAPM does not pass the χ2 test, although this

rejection is largely explained by a mismeasured inverse of the covariance matrix of the pric-

ing errors, Var(α), given the good fit associated with the model. The point estimate for the

risk price of the scaled factor is positive and strongly significant (1% level), but the risk price

estimate associated with the hedging factor is not statistically significant at the 10% level.

In the version based on RREL, the model’s fit is marginally above the first version, with an

explanatory ratio of 74% and an average pricing error of 0.15% per month. This shows that this

version of the model performs relatively better in pricing the SM25 portfolios than the SBM25

portfolios. The model is rejected by the χ2 statistic only marginally (p-value = 4%). As in the

case of FFR, the estimate for λM,z is positive and strongly significant, while the estimate for

λz is negative and significant at the 5% level.

Thus, the scaled factor seems to be the key factor that drives the explanatory power of the

(C)ICAPM over the SM25 portfolios. Therefore, the two key factors in the (C)ICAPM seem

to measure two different and complementary sources of systematic risk. The hedging factor is

able to capture the value anomaly, and the scaled factor prices momentum.

16Brennan, Wang, and Xia (2004) and Petkova (2006) also price the SBM25 portfolios with multifactor modelsthat contain the innovation in short-term interest rates as one of the factors. However, it is not clear in theirmodels what is the contribution of the interest rate factor to drive the explanatory power over the size/BMportfolios.

11

4.2 Comparison with alternative factor models

We compare the performance of the (C)ICAPM with three alternative linear factor models, the

baseline unconditional CAPM; the Fama and French (1993, 1996) three-factor model (FF3);

and the Carhart (1997) four-factor model (C4). FF3, the most widely used model in the

empirical asset pricing literature, seeks to offer a risk-based explanation for both the size and

value premiums. To the excess market return, Fama and French add two factors – SMB (small

minus big), and HML (high minus low) – to account for the size and value premiums.

The FF3 model can be represented in expected return-beta form as

E (Ri,t+1 −Rf,t+1) = λMβi,M + λSMBβi,SMB + λHMLβi,HML, (10)

where (λSMB, λHML) denote the (beta) risk prices associated with the SMB and HML factors,

respectively, and (βi,SMB, βi,HML) stand for the corresponding factor betas for asset i.

The four-factor model is represented as

E (Ri,t+1 −Rf,t+1) = λMβi,M + λSMBβi,SMB + λHMLβi,HML + λUMDβi,UMD, (11)

where λUMD denotes the risk price associated with the momentum factor, and βi,UMD represents

the corresponding beta for asset i. The novelty relative to the FF3 model is the risk premium

associated with the momentum (UMD) factor. UMD (up minus down or winner minus loser)

refers to the return of a self-financing portfolio (like SMB and HML), representing the spread

in average returns between past short-term winner stocks and past short-term loser stocks.

The results for these two factor models are displayed in Table 3. We can see that the baseline

CAPM cannot price both sets of equity portfolios, with explanatory ratios of -42% and -18% in

the tests with SBM25 and SM25, respectively. These negative estimates indicate that the model

performs more poorly than a model that predicts constant risk premia in the cross section.

The FF3 model, however, explains a significant proportion of the dispersion in average

returns of the SBM25 portfolios, with an R2 estimate of 67% and an average mispricing of

0.10% per month. These results are consistent with the evidence in Fama and French (1993,

1996). The risk price estimate associated with HML is statistically significant at the 1% level,

while the risk price for SMB is not significant. Yet the FF3 model cannot price the SM25

portfolios, as illustrated by the nearly zero estimate of the coefficient of determination (3%),

12

which is in line with previous evidence [Fama and French (1996)].

When we compare the performance of FF3 and the (C)ICAPM (version with FFR), we see

that both models have similar performance in pricing the SBM25 portfolios, but the (C)ICAPM

(both versions) clearly outperforms in pricing the size-momentum portfolios. In other words,

the (C)ICAPM can explain the two anomalies, while the FF3 can price only the value premium.

The C4 model explains a large fraction of the cross-sectional dispersion in average returns,

with explanatory ratios of 78% and 85% in the tests with SBM25 and SM25, respectively, and

the UMD factor is priced in both cases (1% level). Thus, the four-factor model outperforms the

(C)ICAPM (version with FFR) by about 8% and 14% (in the explanatory ratios) in explaining

the SBM25 and SM25 portfolios, respectively. When we compare against the version based

on RREL, the C4 model outperforms by about 0.32% and 0.11% in pricing the SBM25 and

SM25 portfolios, respectively. Note that the C4 model includes three independent sources of

systematic risk (in addition to the market return), while in the (C)ICAPM the two key sources

of systematic risk (scaled factor and innovation in the state variable) are associated with the

same state variable, the short-term interest rate.

4.3 Individual pricing errors

Although both R2OLS and MAE represent measures of the overall fit of the (C)ICAPM, it is

important to assess the relative explanatory power of the model over the different portfolios

within a certain group (e.g., value versus growth portfolios, or past winners versus past losers

portfolios).

Figure 2 plots the pricing errors (and respective t-statistics) associated with the SBM25

portfolios in the version with FFR. For the (C)ICAPM, the biggest negative outlier is the

extreme large-value portfolio (S5BM5) with a pricing error of -0.21% per month, and the main

positive outliers are the small-value portfolios (S1BM4 and S1BM5), with pricing errors of

0.26% per month. In terms of statistical significance, only the pricing error for portfolio S1BM5

is significant at the 5% level. In comparison, in the test with the FF3 model there are seven

portfolios with significant pricing errors. It is interesting to see that the (C)ICAPM outperforms

both the FF3 and C4 models in pricing the extreme small-growth portfolio (S1BM1) with a

pricing error of -0.15% per month, compared to mispricing of -0.41% and -0.31% for FF3 and C4,

respectively. This portfolio is particularly hard to price for most models in the empirical asset

pricing literature [see, for example, Fama and French (1993) and Campbell and Vuolteenaho

13

(2004)].

In untabulated results for the version with RREL, there are four portfolios with significant

pricing errors at the 5% level (S1BM1, S1BM4, S1BM5 and S5BM1), which is consistent

with the lower explanatory power of this version relative to the version with FFR in pricing

the SBM25 portfolios.

Figure 3, which is similar to Figure 2, provides a visual representation of the model’s fit

(version with FFR) in a cross-sectional test with the SM25 portfolios. The main negative out-

liers associated with the (C)ICAPM are the big-intermediate (S5M3) and big-winner (S5M4)

portfolios, with pricing errors of -0.37% and -0.40% per month, respectively. The main positive

outlier is the small-winner portfolio (S1M5), with a pricing error of 0.43% per month. We can

see that for most portfolios the (C)ICAPM produces significantly lower pricing errors than the

FF3 model and similar errors to the C4 model.

Regarding the statistical significance, there are five portfolios with significant mispricing

(S1BM5, S3BM5, S5BM3, S5BM4 and S5BM5). However, in the case of the alternative

factor models, the number of significant pricing errors is greater (nine and eighteen for C4 and

FF3, respectively). Untabulated results show that in the version based on RREL, there are

only three portfolios with significant mispricing (S1BM3, S1BM5 and S2BM1).

4.4 Which factors explain the value and momentum premiums?

The results in Table 2 suggest that the innovation in the state variable drives the fit of the

(C)ICAPM for pricing the SBM25 portfolios, while the scaled factor seems to drive the ex-

planatory power of the model for the SM25 portfolios.

To see more clearly which factors drive the explanatory power of the (C)ICAPM in pricing

each set of portfolios, we conduct an “accounting analysis” of the contribution of each factor to

the overall fit of the model. Specifically, we compute the average factor risk premium (average

beta times risk price) for each factor and across every book-to-market(BM)/momentum quintile.

For example, the average market risk premium associated with the BM/momentum quintile j

is given by

λMβj,M ,

where the average beta for BM/momentum quintile j = 1, ..., 5 is computed as the simple

14

average of the market betas for portfolio j across the 5 size quintiles within SBM25 or SM25:

βj,M =15

5∑i=1

βi,M , i = 1j, 2j, 3j, 4j, 5j,

where the first number refers to the size quintile, and j refers to the BM/momentum quintile.

The results for this accounting decomposition are shown in Table 4. The spread in average

excess returns between the first (Q1, growth) and the fifth BM quintile (Q5, value) is -0.53%

per month, which corresponds to the (symmetric of the) value premium in our sample. This gap

has to be (partially) matched by the risk premium associated with one or more of the factors

in the (C)ICAPM, as shown in the respective beta pricing equation (6), if this model is able to

price the value premium.

In the version with FFR, the spread Q1 − Q5 in the market risk premium is 0.13% per

month, which moves the model farther from explaining the value premium, and confirms why

the baseline CAPM does not price the value anomaly. The spread associated with the scaled

factor has the right sign, but the magnitude is quite low (-0.03% per month). Thus, it is

the innovation in the Fed funds rate that accounts for the value premium, with a spread in

the respective risk premium of -0.58% per month, which more than explains the original value

premium of -0.53%. Only -0.04%, of the original gap of -0.53%, is left unexplained by the three-

factor ICAPM; this is another way to gauge the success of the model in explaining the value

anomaly. Thus, value stocks covary negatively with innovations to the Fed funds rate, which

has a negative risk price.

In the version based on RREL, the accounting decomposition is qualitatively similar. The

gaps in risk premiums associated with the market, scaled and hedging factors are 0.15%, 0.17%

and -0.69%, respectively, producing a mispricing Q1−Q5 of -0.16% per month. Thus, it is the

hedging factor that drives the value premium.

With regard to the momentum spread, the gap Q1 − Q5 (loser minus winner) in average

excess returns is about -1% per month, nearly twice the size of the value premium in our sample.

As in the case of the value anomaly, the CAPM cannot explain momentum, as the gap (Q1−Q5)

in the market risk premium is positive (0.14% per month).

A similar positive spread in risk premium (0.19% per month) is generated by the innovation

in the Fed funds rate, showing that this factor does not help to price momentum. Thus, it is

the scaled factor that is key in pricing momentum, generating a gap Q1−Q5 in risk premia of

15

about -1.16% per month that more than matches the original return spread of -1.01%. Only

-0.18% of this last spread is left unexplained by the three-factor ICAPM, thus justifying the

large fit of the model in pricing the SM25 portfolios, as documented above.

In the case of the version based on RREL, once again we have similar results. The spreads

in risk premiums for the market, scaled and hedging factors are 0.08%, -1.43% and 0.49%,

respectively, leading to a mispricing of only -0.14% per month.

These results confirm that the two non-market factors in the (C)ICAPM drive two different

sources of systematic risk, one related to the value premium, and another related to momentum.

We conduct a similar decomposition for the FF3 and C4 models to assess the factors that

drive the explanatory power of these models over the value and momentum quintiles, which is

presented in Table 5. In the case of the FF3 model, the gap Q1−Q5 associated with the HML

factor is -0.56% per month, which is about the same as the risk premium gap associated with

the hedging factor in the (C)ICAPM, and nearly matches the original value spread of -0.53%.

When it comes to pricing the momentum quintiles, the spread associated with HML is only

-0.29% per month, about one-third of (the size of) the original momentum spread of -1.01%,

thus leading to a gap Q1 − Q5 in mispricing of -0.84% per month. In other words, the FF3

cannot price the momentum spread.

In the case of the C4 model and BM quintiles, the risk premium gap associated with HML

is very similar to the corresponding spread of HML in the FF3 model (-0.55% per month).

For the momentum quintiles, the risk premium spread associated with UMD is -0.94%, which

almost matches the original momentum spread. Thus, the key factor that prices the value

premium in both models is HML (similarly to the hedging factor in the (C)ICAPM), while the

UMD factor drives momentum in the C4 model (just like the scaled factor in the (C)ICAPM).

These results also suggest that the innovation in our state variable is correlated (condi-

tional on the other factors of the (C)ICAPM) with the HML factor, while the scaled factor

is correlated with the UMD factor. To assess this conjecture, we conduct time-series multiple

16

regressions:17,18

HMLt+1 = ρ0 + ρ1RMt+1 + ρ2RMt+1zt + ρ3∆zt+1, (12)

UMDt+1 = ρ0 + ρ1RMt+1 + ρ2RMt+1zt + ρ3∆zt+1. (13)

To measure the individual statistical significance of the regressors, we compute heteroskedasticity-

robust GMM standard errors [White (1980)].

The results displayed in Table 6 show that in the regression for HML (Panel A), conditional

on the market return, both the hedging factor, ∆FFRt+1, and the scaled factor, RMt+1FFRt,

are negatively correlated with HMLt+1, and the slopes are strongly significant (1% level). In

the regression for UMD, the scaled factor is positively correlated with UMDt+1, and this

effect is strongly significant (at the 1% level), while the slope associated with ∆FFRt+1 is not

significant at the 10% level. The correlations are far from perfect, however, as indicated by the

R2 estimates of 16% and 8% in the regressions for HML and UMD, respectively.

When the state variable is RREL, the results are qualitatively similar. Both RMt+1RRELt

and ∆RRELt+1 are negatively correlated with HMLt+1, although the slope associated with

the scaled factor is not significant at the 10% level. In the regression of UMD, both factors are

positively correlated with UMDt+1, and both coefficients are statistically significant.

Hence, ∆FFRt+1 and ∆RRELt+1 both measure some of the risks captured by HML, and

the same happens to the scaled factor in relation to UMD, but these effects are only partial.

This result for UMD is consistent with other evidence showing that the payoffs of momentum

strategies can be, at least partially, accounted for by lagged macroeconomic variables linked to

the business cycle as is the case of the Fed funds rate or relative Treasury bill rate [see Chordia

and Shivakumar (2002) and Ahn, Conrad, and Dittmar (2003)].

4.5 Factor betas and intuition

Our analysis shows that the innovation in the Fed funds (or in the relative T-bill rate) is the

factor in the (C)ICAPM responsible for pricing the value spread, and the scaled factor accounts

for the momentum anomaly. Put differently, there is a dispersion in the betas associated with

17Other evidence shows that some of the risk factors in the ICAPM or conditional CAPM measure approxi-mately the same types of risks associated with HML [e.g., Lettau and Ludvigson (2001), Vassalou (2003), Hahnand Lee (2006), Petkova (2006), among others].

18Ferguson and Shockley (2003) and Hahn and Lee (2006) conduct similar time-series regressions for SMBand HML.

17

the hedging factor within the size-BM portfolios that fits the value premium, and there is a

similar dispersion in the betas associated with the scaled factor within the size-momentum

portfolios that fits the momentum premium.

The multiple-regression betas associated with both factors in the case of the SBM25 port-

folios are displayed in Figure 4. We can see that value stocks have negative betas associated

with ∆FFRt+1, while growth stocks have positive betas for this same factor. This dispersion

in betas scaled by the negative risk price for ∆FFRt+1 generates a spread in risk premia. A

similar pattern holds for the factor loadings associated with ∆RRELt+1.

Why are value stocks more (negatively) sensitive to unexpected rises in short-term interest

rates? One possible explanation is that many of these firms are near financial distress as a

result of a sequence of negative shocks to their cash flows [Fama and French (1992)], and are

thus more sensitive to rises in short-term interest rates. According to the credit channel theory

of monetary policy [Bernanke and Gertler (1995)], a monetary tightening (increase in the Fed

funds rate) represents an increase in financial costs and restricts access to external financing.

This effect should be stronger for firms in poorer financial position, as typically those firms

have higher costs of external financing, and the value of their assets (which act as collateral

for new loans) is relatively depressed. Increases in interest rates would thus constrain access

to financial markets and prevent those firms from investing in profitable investment projects.

This mechanism is consistent with the analysis of Lettau and Wachter (2007) who show that

the prices (and realized returns) of value stocks are more sensitive to shocks in near-term cash

flows, while the prices of growth stocks are more related to shocks to discount rates (long-term

expected returns).

The analysis of the betas for the SM25 portfolios in Figure 5 shows that past winners have

slightly positive betas with the scaled factor, while past losers have large negative betas. This

dispersion in betas multiplied by the positive risk price of the scaled factor, RMt+1FFRt, gener-

ates the risk premium necessary to explain the momentum spread. In the case of RMt+1RRELt

the dispersion of betas (negative versus positive) between past losers and winners is even more

clear. Thus, past winners are riskier not because they have higher market betas in average times,

but because they are more correlated with the market in periods of high short-term interest

rates.

18

We can assess this in greater detail by computing the average conditional market betas:19

βi,M,t = βi,M + βi,M,zzt,

where zt represents the average of the scaling variable calculated over periods with high and low

interest rates. A period with high interest rates occurs when the Fed funds rate or RREL is

1.5 standard deviations above its mean; similarly, a period with low interest rates occurs when

FFR (RREL) is 1.5 standard deviations below its mean. When we consider all the periods, the

average conditional market beta corresponds to the unconditional (multiple-regression) market

beta since the scaling variable has unconditional zero mean, E(FFRt) = E(RRELt) = 0.

Figure 6 plots the average conditional market betas in the test with the SM25 portfolios

for both versions of the model.20 In Panels A and B, we can see that past winners have lower

unconditional market betas than past losers across all size quintiles. That is, past losers are

unconditionally riskier than past winners. This shows the inability of the simple CAPM to

price the momentum portfolios. In Panels C and D, however, we can see that in periods of

high interest rates, past winners have higher market betas than past losers, an effect that is

robust across all size deciles. On the other hand, in periods with low interest rates, past losers

have higher market betas than past winners, as shown in Panels E and F. Thus, past winners

are riskier than past losers because they have greater market risk in times of high short-term

interest rates.

Why are past winners riskier than past losers in periods with high interest rates? A possible

explanation relies on the different characteristics of winner and loser stocks at different points of

the business cycle. That is, during economic expansions (which are associated with high short-

term interest rates) winners tend to be cyclical firms, which have high market betas. Conversely,

during recessions (periods with low short-term interest rates) winners tend to be non-cyclical

firms, with low market betas. The changing composition of the momentum portfolios leads to

the time variation in its market betas.

This reasoning is consistent with evidence in the momentum literature that momentum

profits are pro-cyclical.21 The mean of the momentum factor (UMD) in economic expansions

19Lettau and Ludvigson (2001) perform a similar analysis.20The analysis is conducted only for the SM25 portfolios, since the scaled factor is not relevant to price the

SBM25 portfolios.21See Johnson (2002), Chordia and Shivakumar (2002), Cooper, Gutierrez, and Hameed (2004), Sagi and

Seasholes (2007), and Stivers and Sun (2010). Specifically, the theoretical analysis in Johnson suggests thatmomentum profits might be the result of episodic but persistent shocks in cash flows, which can be related with

19

(as classified by the NBER) is 0.85% per month compared to only 0.01% per month in recessions.

Thus, the time variation in market betas matches the time variation in momentum returns. This

variation in risk premium is justified because under positive business conditions and high short-

term interest rates, the market future risk premium is low. Risk-averse investors are willing to

invest in winner stocks, which are cyclical at this point of the cycle and have high betas, only

if these stocks sell at a greater discount, that is, offer a higher expected return.

Our results are also consistent with the evidence provided in Grundy and Martin (2001)

and Daniel (2011) that momentum profits are associated with time-varying market betas of

winner and loser portfolios. They find that after a bear equity market, the market beta of the

momentum factor is low since past winners have low betas (defensive stocks that performed

relatively better in the bear market) and past losers have high betas (aggressive or cyclical

stocks that underperformed more in the bear market). At the same time, in a bear market

interest rates are usually at low levels, and so it follows that past losers have high betas when

interest rates are low while past winners have low betas. On the other hand, in a bull market

interest rates are at high levels, and thus past winners (those that have outperformed in the

bull market) have high market betas while past losers exhibit low betas. Thus, interest rates

represent an instrument that signals time variation in market betas of the winner and loser

portfolios as a result of changing market conditions and hence of the changing composition of

the momentum portfolios and of their market betas.

Figure 7 shows that there is some correlation over time between the momentum factor and

the conditional market beta of UMD, computed as

βUMD,M,t = βUMD,M + βUMD,M,zzt.

Specifically, the momentum crashes that occurred in 2001 and 2009 [as documented by Daniel

(2011)] are roughly associated with a sharp decline in the current and lagged market betas of

the UMD factor.

5 Additional results

In this section, we apply a battery of robustness checks to our main results.

short-term business conditions.

20

5.1 Bootstrap simulation

Following Lewellen, Nagel, and Shanken (2010), we estimate empirical confidence intervals for

the coefficient of determination and average pricing error in the cross-sectional regressions. We

use a bootstrap simulation with 5,000 replications in which the excess portfolio returns and risk

factor realizations are simulated (with replacement from the original sample) independently and

without imposing the (C)ICAPM’s restrictions. Thus, the data generating process is derived

under the assumption that the model is not true. We want to investigate the following question:

under the assumption that the (C)ICAPM does not hold, how likely is it that we obtain the fit

found in the data. In other words, are our results in the cross-sectional tests spurious?

In untabulated results and in the test with SBM25, the 95% confidence intervals for R2

are [−1.09, 0.19] and [−1.08, 0.19] when the state variables are FFR and RREL, respectively.

The 95% confidence intervals for the average pricing error are [0.48, 0.72] and [0.48, 0.72] for

the versions with FFR and RREL, respectively. When we compare these intervals with the

actual estimates, it follows that for both versions of the model the estimated coefficients of

determination (70% and 46%) are well above the upper bounds on the intervals. Simultaneously,

the sample MAE estimates (0.10% and 0.14%) are significantly below the lower bounds on the

corresponding empirical intervals.

In the test with SM25, the confidence intervals for MAE are quite similar to those in the

test with SBM25, which implies that the MAE estimates from the original sample (0.16% and

0.15%) are statistically significant. On the other hand, the 95% interval for R2 is [−0.46, 0.27]

for both versions of the model, implying that also in this case, the actual R2 estimates of 71%

and 74% are well above the upper limit.

Overall, these results suggest that the fit of the model in pricing the BM and momentum

portfolios is not spurious.

5.2 Pricing bond returns

Adding bond returns to the empirical tests of the (C)ICAPM enables us to assess whether the

model can jointly price stocks and bonds.22 We add to each equity portfolio group the excess

returns on seven Treasury bonds with maturities of 1, 2, 5, 7, 10, 20, and 30 years. The data

are available from CRSP. This involves a total of 32 test assets in each estimation (SBM25 or

22Fama and French (1993) and Koijen, Lustig, and Van Nieuwerburgh (2010) also estimate factor models overthe joint cross-section of stock and bond returns.

21

SM25).

The results are presented in Table 7. In the test with SBM25 and version based on FFR, the

explanatory ratio increases to 85% from 70% in the benchmark test, while the average pricing

error is 0.09% per month. The point estimate for λFFR is slightly lower (-0.48) than in the

benchmark test, but remains strongly significant (1% level).

In the test with SM25, the R2 and average pricing error estimates are the same as in the

tests for the equity portfolios. The risk price for the scaled factor, λM,FFR, is close to the

corresponding estimate in the benchmark test and is significant at the 1% level. In the tests

with either portfolio group, the model is rejected by the χ2 test, likely mainly because of a poor

inversion of Var(α) when the number of test returns is relatively large.

In the version based on RREL, the explanatory ratio in the test with SBM25 is 57% (up from

46% in the benchmark test), and the average pricing error is 0.16% per month. The estimate

for the hedging risk price is significantly lower in magnitude than in the baseline case (-0.18%)

but is still significant at the 1% level. In the test containing the SM25 portfolios, the fit of the

model is basically the same as in the benchmark case, with a coefficient of determination of

76%. As before, the estimates for λM,RREL and λRREL are positive and negative, respectively,

but only the scaled factor is priced.

We also estimate the alternative linear factor models by including bond risk premiums in

the menu of test assets. Untabulated results show explanatory ratios of 31% and 12% for the

baseline CAPM in the tests with SBM25 and SM25, respectively. This shows that the CAPM

has some explanatory power over bond risk premia.

The FF3 model has a fit very similar to the (C)ICAPM (version with FFR) in the test with

SBM25 (R2 = 82%), but it underperforms significantly in the test with SM25 (R2 = 26%). The

explanatory ratio for the C4 model is quite similar to the (C)ICAPM in the test with SBM25

(88%), while it outperforms in the test with SM25, with an explanatory ratio of 89%.

Overall, when we price equity and bond risk premia jointly, the results for the (C)ICAPM

are quite similar to the benchmark results.

5.3 Pricing alternative equity portfolios

We estimate the (C)ICAPM with alternative equity portfolios – 10 portfolios sorted on size,

10 portfolios sorted on BM and 10 momentum portfolios, for a total of 30 portfolios. This

cross-sectional test enables us to check whether our three-factor model prices simultaneously

22

the BM and momentum portfolios.

The results are displayed in Table 8. We can see that in the version based on FFR the

model’s fit is smaller than in the tests with either SBM25 or SM25, with an explanatory ratio

of 29% and an average pricing error of 0.15% per month. The risk price estimates for the

non-market factors have the same signs than in the test with SM25, and both estimates are

significant at the 1% level.

In the version with RREL, the explanatory power is significantly greater than in the first

version, with an R2 of 60% and an average mispricing of 0.12% per month. This fit is halfway

the one obtained for the tests with the SBM25 and SM25 portfolios. The risk price estimates

for the scaled and hedging factors have the same signs as in the version with FFR, and both

estimates are statistically significant.

In untabulated results, the explanatory ratios for the FF3 and C4 models are -9% and

85%, respectively. These results show that overall, the (C)ICAPM does a good job in pricing

simultaneously the size, BM and momentum portfolios.

5.4 Pricing the market return

We next reestimate the (C)ICAPM by including the market equity premium (RM) in the set of

test assets.23 This enables us to assess whether the model can jointly price the equity portfolios

(SMB25 or SM25) and the market return.

Results not tabulated show that the (C)ICAPM fit is very close to that of the benchmark

test including only equity portfolios, with R2 estimates of 71% and 69% in the tests with SBM25

and SM25, respectively, when the state variable is FFR. In the version based on RREL, the

explanatory ratios are 48% and 74% in the tests with SBM25 and SM25, respectively. For both

versions of the model, the risk price estimates are also nearly the same as in the benchmark test

of the (C)ICAPM.

Thus, forcing the model to price the aggregate equity premium does not have an impact on

the fit of the (C)ICAPM.

23Lewellen, Nagel, and Shanken (2010) advocate that when the factors are returns, they should be included inthe set of test assets.

23

5.5 Alternative ICAPM specification

In an alternative ICAPM specification, the innovation in the state variable represents the resid-

ual from an AR(1) model:

∆zt+1 ≡ εt+1 = zt+1 − φz − ρzzt. (14)

By using this new proxy for ∆zt+1, we want to assess whether the results for the (C)ICAPM

are sensitive to the measurement of the innovation in the state variable.

Untabulated results are very similar to the benchmark test using the first difference in either

FFR or RREL. In the version based on FFR, the explanatory ratios are 72% and 71% in the

tests with SBM25 and SM25, respectively, while average pricing errors are the same as in the

benchmark test. In the version with RREL, the R2 estimates are 54% and 76% in the tests

with SBM25 and SM25, respectively. The corresponding MAE estimates are 0.13% and 0.14%

per month, which are very similar to the corresponding values in the benchmark test. The point

estimates for the factor risk prices are also very close to the estimates in the benchmark test,

for both versions of the model.

Thus, the results of the (C)ICAPM are robust to the way we measure the innovation in the

state variable, the hedging risk factor.

5.6 Alternative standard errors

We use alternative standard errors for the factor risk prices and pricing errors. These GMM-

based standard errors can be interpreted as a generalization of the Shanken (1992) standard

errors to the extent that they relax the implicit assumption of independence between the factors

and the residuals from the time-series regressions [see Cochrane (2005) (Chapter 12)]. The full

details are provided in Appendix B.

Untabulated results show that the t-statistics for the risk price estimates based on the new

standard errors lead to the same qualitative decisions as the Shanken t-statistics. Specifically,

λz in the test with SBM25 and λM,z in the test with SM25 are both significant at the 1%

level. The main difference occurs with the χ2 statistic in the tests with SBM25, which now

has p-values of 2% and 4% in the versions with FFR and RREL, respectively. This values are

related to a poor inversion of the covariance matrix of the pricing errors, given their lowness

(0.10% or 0.14% per month).

24

5.7 Alternative sample

We estimate out three-factor model in the test with the SM25 portfolios for the 1963:07-2008:12

period. We want to assess whether the fit of the model in pricing the momentum portfolios is

robust to removing the momentum crash occurred in 2009, as documented by Daniel (2011).

Untabulated results show that the explanatory power of the version based on FFR is only

marginally lower than in the test for the full sample, with an R2 estimate of 60% and an average

pricing error of 0.22% per month. On the other hand, in the version based on RREL, the fit of

the (C)ICAPM is basically the same as in the benchmark test (R2 = 73%,MAE = 0.17%). In

both versions, the point estimates of the risk price for the scaled factor are strongly significant

(1% level).

Overall, these results show that the 2009 momentum crash does not have a meaningful

impact on the capacity of the model in pricing the momentum portfolios.

5.8 Estimating the (C)ICAPM in expected return-covariance representation

We define and test the (C)ICAPM in expected return-covariance representation:

E(Ri,t+1 −Rf,t+1) = γM Cov(Ri,t+1 −Rf,t+1, RMt+1)

+γM,z Cov(Ri,t+1 −Rf,t+1, RMt+1zt) + γz Cov(Ri,t+1 −Rf,t+1,∆zt+1), (15)

where (γM , γM,z, γz) denote the covariance risk prices associated with the market return, the

scaled factor, and the innovation in the state variable, respectively.

This version of the model is equivalent to an expected return-single beta pricing equation.

Thus, the model should fit as well as the version with multiple-regression betas, although the

risk prices might have different signs, given possible correlation among the factors. Though, as

the factors in the (C)ICAPM are not significantly correlated, as shown in Table 1, the factor risk

prices should have the same signs in either multiple- or single-regression betas (or equivalently,

covariances).

We estimate specification (15) by first-stage GMM [Hansen (1982) and Cochrane (2005)].

This method uses equally weighted moments, which is conceptually equivalent to running an

OLS cross-sectional regression of average excess returns on factor covariances (right-hand side

variables). One advantage of using the GMM procedure is that we do not need to have previous

estimates of the individual covariances, since these are implied in the GMM moment conditions.

25

The GMM system has N + 3 moment conditions, where the first N sample moments corre-

spond to the pricing errors for each of the N testing returns:

gT (b) ≡ 1T

T−1∑t=0

(Ri,t+1 −Rf,t+1)− γM (Ri,t+1 −Rf,t+1) (RMt+1 − µM )

−γM,z(Ri,t+1 −Rf,t+1) (RMt+1zt − µM,z)

−γz(Ri,t+1 −Rf,t+1) (∆zt+1 − µz)

RMt+1 − µM

RMt+1zt − µM,z

∆zt+1 − µz

= 0.

i = 1, ..., N, (16)

In this system, the last three moment conditions enable us to estimate the factor means.

Thus, the estimated covariance risk prices from the first N moment conditions correct for the

estimation error in the factor means, as in Cochrane (2005) (Chapter 13) and Yogo (2006).

There are N − 3 overidentifying conditions (N + 3 moments and 2× 3 parameters to estimate).

The standard errors for the parameter estimates and the remaining GMM formulas are presented

in Appendix C. By defining the first N residuals from the GMM system as the pricing errors

associated with the N test assets, αi, i = 1, ..., N , the χ2, R2OLS , and MAE measures are defined

analogously to the formulas presented in Section 3.

The GMM estimation results are displayed in Table 9. As expected, the R2 and MAE

estimates are the same as in the benchmark test of the beta pricing equation. Now, however,

the (C)ICAPM version based on FFR is rejected by the χ2 statistic in the estimation with

the SBM25 portfolios (p-value = 1%), which again should be the result of a poor inversion of

Var(α). The point estimate for γFFR is negative and statistically significant (at the 5% level)

in the test with SBM25, while the point estimate for γM,FFR is positive and strongly significant

(1% level) in the test with SM25. Thus, the signs of the covariance risk prices of the non-market

factors are the same as in the test of the beta pricing equation. The market covariance risk

prices, γM , are negative, but these point estimates are largely insignificant.

In the version with RREL, the (C)ICAPM continues to pass the χ2 test when the test

portfolios are SBM25 (p-value = 6%). The estimates for γM,RREL and γRREL have the same

signs as λM,RREL and λRREL, in the benchmark test. The hedging risk price in the test with

SBM25 and the scaled factor risk price in the test with SM25 are statistically significant at the

26

1% and 5% levels, respectively. In contrast with the version based on FFR, the estimates for

the market risk price are now positive, although largely insignificant.

Overall, the estimation results for the covariance pricing equation are consistent with those

in the benchmark test.

We also estimate the expected return-covariance equation by including an intercept that

represents a proxy for the excess zero-beta rate:

E(Ri,t+1 −Rf,t+1) = γ0 + γM Cov(Ri,t+1 −Rf,t+1, RMt+1)

+γM,z Cov(Ri,t+1 −Rf,t+1, RMt+1zt) + γz Cov(Ri,t+1 −Rf,t+1,∆zt+1). (17)

As we note in Section 3, if the (C)ICAPM is correctly specified, the estimate for γ0 should not

be statistically different from zero.

Results not tabulated show that the point estimates for γ0 are nearly zero and largely

insignificant in the tests with both the SBM25 and SM25 portfolios, and for both versions of

the model. Moreover, the MAE and R2 estimates are nearly the same as in the benchmark

restricted pricing equation without intercept, thus showing that the constant factor plays no

relevant role.

These results seem to suggest that the (C)ICAPM is not misspecified. That is, there are no

relevant missing risk factors, at least when it comes to price the value and momentum portfolios.

5.9 Nested models

The (C)ICAPM consists of two important nested models. The standard ICAPM can be obtained

as a special case of the (C)ICAPM by imposing βi,Mz = 0, i.e., that the conditional market beta

is constant over time:

E(Ri,t+1 −Rf,t+1) = λMβi,M + λzβi,z. (18)

Similarly, the conditional CAPM in unconditional form can be obtained from (6) by imposing

λz = 0; that is, investment opportunities are constant through time:

E(Ri,t+1 −Rf,t+1) = λMβi,M + λMzβi,Mz. (19)

We estimate the two nested models of the (C)ICAPM: the two-factor conditional CAPM

in equation (19), and the two-factor (unconditional) ICAPM in equation (18). This analysis

27

allows us to evaluate the incremental explanatory power of the benchmark three-factor model

against each nested model in pricing both sets of equity portfolios.

The estimation results are displayed in Table 10. In the test with SBM25, the conditional

CAPM based on FFR has some explanatory power over the size-BM portfolios with an R2

estimate of 34% and an average mispricing of 0.15% per month. The fit of the two-factor

ICAPM is significantly better, with an explanatory ratio of 67% and an average pricing error

of 0.11% per month, for almost the same explanatory power as in the benchmark (C)ICAPM.

Moreover, the point estimate for λFFR is negative and strongly significant (1% level).

When the state variable is RREL, the CCAPM cannot price the size/BM portfolios, with

a coefficient of determination of -18% and a MAE estimate of 0.22% per month. On the other

hand, the fit of the two-factor ICAPM (R2 = 40%) is almost the same as in the (C)ICAPM,

and the estimate for λz is significant at the 1% level.

In the test with SM25, the two-factor ICAPM cannot price these portfolios, with a negative

R2 estimate (-18%) and an average pricing error as high as 0.32% per month. On the other

hand, the fit of the conditional CAPM is nearly the same as that of the (C)ICAPM, with a

coefficient of determination of 69% and an MAE estimate of 0.16% per month. Moreover, the

risk price of the scaled factor is positive and highly significant.

The results for the version based on RREL are qualitatively similar. The two-factor ICAPM

performs poorly with an explanatory ratio of just 7%. In contrast, the fit of the CCAPM (66%)

is close to that of the benchmark three-factor model and the risk price estimate for the scaled

factor is significant at the 1% level.

Thus, these results are consistent with the analysis so far. The (C)ICAPM provides the

“best of both worlds,” that is, the best characteristics of the two nested models. It includes

the hedging risk factor that prices the BM portfolios (as in the baseline ICAPM), and also the

scaled factor that prices the momentum portfolios (as in the conditional CAPM).

6 Long-term reversal

Can the (C)ICAPM explain the long-term reversal in returns anomaly [De Bondt and Thaler

(1985, 1987)]? The anomaly is that stocks with low returns over the long term (three to five

years) have higher subsequent future returns, while past long-term winners have lower future

returns. This long-term mean reversion in stock returns is not explained by the CAPM. This

28

anomaly should be closely related to the value anomaly, as long-term underperformers end up

with high book-to-market ratios.

To test the explanatory power of the (C)ICAPM for this anomaly, we use 25 portfolios sorted

on both size and long-term past returns (SLTR25). The portfolios come from the intersection

of five portfolios formed on size (market equity) and five portfolios formed on past returns (13

to 60 months before the portfolio formation date). The portfolios are obtained from Kenneth

French’s data library.24

The results for the (C)ICAPM pricing equation in the test with the SLTR25 portfolios are

shown in Table 11. We can see that the (C)ICAPM based on FFR has considerable explanatory

power, with a coefficient of determination of 63%, while the corresponding average pricing error

is 0.09% per month. This is a relatively similar fit to the test with the SBM25 portfolios.

Moreover, the point estimate for λFFR is negative and strongly significant (1% level), while the

risk price of the scaled factor is largely insignificant.

In the version based on RREL, the fit is only marginally lower with an explanatory ratio

of 51% and an average mispricing of 0.11% per month. Moreover, the hedging risk factor is

strongly priced (1% level). Thus, as in the test with SBM25, most of the explanatory power of

the model over the SLTR25 portfolios seems to be driven by the hedging factor.

We also estimate the alternative factor models with these SLTR25 portfolios. Results not

tabulated show that the baseline CAPM cannot price these portfolios, with an R2 of -9%, and

an average pricing error of 0.17% per month. The FF3 model significantly outperforms the

CAPM, with an explanatory ratio of 75%, marginally better than the fit of the (C)ICAPM.

Moreover, the risk price for HML is strongly priced. The C4 model has the best overall fit,

with an R2 estimate of 92%, indicating that UMD, in addition to HML, helps to price these

portfolios.

The plot of the individual pricing errors, presented in Figure 8, shows that the main outlier in

the test with the (C)ICAPM’s version with FFR is the small/past winner portfolio (S1LTR5),

with a pricing error of -0.36% per month; the corresponding mispricing in the case of the FF3

model is -0.33% per month. The pricing errors for portfolios S1LTR3 and S1LTR5 are statis-

tically significant at the 5% level, while in the case of the FF3 model there are three portfolios

with significant errors. Untabulated results show that, similarly to the version with FFR,

24Fama and French (1996), Da (2009), and Da and Warachka (2009), among others, also conduct asset pricingtests over portfolios sorted on prior long-term returns.

29

when the state variable is RREL only the pricing errors associated with portfolios S1LTR3

and S1LTR5 are statistically significant. Thus, as in the test over the size/BM portfolios, the

(C)ICAPM seems to behave much like the FF3 model.

We also conduct an “accounting decomposition” of the long-term reversal spread, similar to

the analysis made for the value and momentum spreads in Section 4. In untabulated results,

the gap Q1−Q5 in average excess returns (past long-term loser minus past long-term winner)

is about 0.45% per month, which corresponds to the long-term reversal spread in our sample.

This premium is comparable to the size of the value premium reported above (0.53%). The

risk premium (beta times risk price) gap (Q1−Q5) associated with the market factor is -0.03%

per month, thus confirming that the baseline CAPM cannot price the long-term reversal (LTR)

quintiles.

The spreads in risk premium associated with the hedging and scaled factors are 0.26% and

0.09% per month, respectively. Of the original 0.45% spread in returns, 0.14% is not explained

by the model, which represents about one-third of the original gap. Thus, the key factor

responsible for the explanatory power of the (C)ICAPM over the long-term reversal portfolios

is the hedging factor, similar to the results obtained for the value premium.

When the state variable is RREL, the results are qualitatively similar: the risk premium

gaps for the market, scaled and hedging factors are -0.04%, 0.05% and 0.19%, respectively,

producing a spread in pricing errors of 0.25% per month. Thus, as in the case with FFR, the

hedging factor drives most of the explanatory power of the model over the long-term reversal

spread.

An analogous decomposition for the FF3 model shows that the HML factor is the key driver

of the LTR spread, with a gap in risk premium of 0.34% per month, while the SMB makes a

marginal contribution (gap of 0.02%) leading to a gap in mispricing of 0.11% per month. In

the case of the C4 model, the gaps in risk premium associated with the SMB, HML, and

UMD factors are 0.12%, 0.26%, and 0.07% per month, respectively, producing a gap Q1−Q5

in average pricing error of only 0.02% per month, consistent with the high explanatory ratio.

Analysis of the factor loadings in Figure 9 sheds light on the way the (C)ICAPM, more

precisely, the hedging factor, prices the long-term reversal anomaly. We can see that, across

all size quintiles, past long-term losers have relatively high negative betas associated with the

innovation in the Fed funds rate, while past long-term winners have positive loadings (within

the first size quintile, negative betas but with lower magnitudes). This spread in betas scaled

30

by the corresponding risk price generates the risk premium necessary to partially explain the

long-term reversal return spread. When the state variable is RREL, with the exception of the

first size quintile, we also have negative factor loadings for past long-term losers and positive

betas for past winners.

Why are past long-term losers have greater interest risk than past long-term winners? Past

long-term losers are likely to have a long sequence of negative shocks in their cash flows, and

hence become more financially constrained through time. Hence, these firms will be more

sensitive to additional negative shocks in their earnings, specifically further rises in short-term

interest rates. Hence, past long-term losers act much like value stocks, while past-winners

behave more like growth stocks.

7 Conclusion

We offer a simple asset pricing model that goes a long way forward in explaining the value and

momentum anomalies. We specify a three-factor conditional intertemporal CAPM, denoted as

(C)ICAPM. The factors are the market equity premium, the market factor scaled by the state

variable (arising from time-varying market betas), and the “hedging” or intertemporal factor.

These last two factors are based on the same macroeconomic state variable, the Federal funds

rate or the relative T-bill rate.

We test our three-factor model with 25 portfolios sorted on size and book-to-market and 25

portfolios sorted on size and momentum. The cross-sectional tests show that the (C)ICAPM

explains a large faction of the dispersion in average equity premia of the two portfolio groups,

with explanatory ratios around 70%. The (C)ICAPM outperforms the Fama and French (1993)

three-factor model when it comes to pricing both sets of portfolios, and only marginally under-

performs the Carhart (1997) four-factor model.

The ultimate non-market sources of systematic risk in our model are associated with one

single variable, a proxy for short-term interest rates; in the four-factor model, there are three

unrelated non-market sources of systematic risk. Moreover, the factors in Carhart (1997) are self-

financing portfolios related to the test portfolios, while we use a macroeconomic state variable

that a priori is not mechanically related to the test portfolios.

The ICAPM hedging risk factor explains the dispersion in risk premia across the book-to-

market portfolios, and the scaled factor prices the dispersion in risk premia across the momentum

31

portfolios. According to our model, the reason that value stocks enjoy higher expected returns

than growth stocks is because they have higher interest rate risk; that is, they have more

negative factor loadings on the hedging factor. Furthermore, in our model past winners enjoy

higher average returns than past losers because they have greater conditional market risk; that

is, past winners have higher market risk in times of high short-term interest rates.

32

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39

A Derivation of the ICAPM in discrete time

The problem for the representative investor in the economy is stated as

J (Wt, zt) ≡ max{Ct+j}∞j=0,{ωi,t+j}∞j=0

Et

∞∑j=0

δjU (Ct+j)

s.t.

Wt+1 = Rp,t+1(Wt − Ct)

Rp,t+1 = g (zt) + εt+1

Rp,t+1 =∑N

i=1 ωi,tRi,t+1

,

and can be represented in a dynamic programming framework, as follows:

J (Wt, zt) ≡ maxCt,ωi,t

{U (Ct) + δ Et [J (Wt+1, zt+1)]}

s.t.

Wt+1 = Rp,t+1(Wt − Ct)

Rp,t+1 = g (zt) + εt+1

Rp,t+1 =∑N

i=1 ωi,tRi,t+1

, (A.1)

where J (Wt, zt) denotes the time t value function; U (Ct) denotes the utility over consumption;

Rp,t+1 is the gross return on the aggregate portfolio; zt is the state variable that forecasts Rp,t+1;

ωi,t is the weight for asset i in the representative investor’s portfolio; and δ is a time-subjective

discount factor.25 εt+1 represents a forecasting error, and g (zt) denotes a function of the state

variable that represents the component of the market return that is predictable by the state

variable.

The first-order condition (f.o.c.) with respect to Ct is equal to

UC(Ct) = δ Et [JW (Wt+1, zt+1)Rp,t+1] , (A.2)

where UC(·) and JW (·) denote the first-order partial derivatives of U(·) relative to Ct and J (·)

with respect to Wt+1, respectively.

The return on aggregate wealth can be rewritten as

Rp,t+1 =N−1∑i=1

ωi,t (Ri,t+1 −Rf,t+1) +Rf,t+1, (A.3)

25For notational convenience we assume there is only one state variable, i.e., zt is a scalar.

40

where Rf,t+1 denotes a benchmark return (for example, the risk-free rate), and where we impose

the constraint that the portfolio weights must sum up to 1,∑N

i=1 ωi,t = 1.26 Then, the f.o.c.

with respect to ωi,t is given by

Et [JW (Wt+1, zt+1) (Wt − Ct) (Ri,t+1 −Rf,t+1)] = 0. (A.4)

By applying the envelope theorem to (A.1), JW (·) can be represented as

JW (Wt, zt) =∂Ct

∂Wt{UC(Ct)− δ Et [JW (Wt+1, zt+1)Rp,t+1]}+ δ Et [JW (Wt+1, zt+1)Rp,t+1]

+∂ωi,t

∂WtEt [JW (Wt+1, zt+1) (Wt − Ct) (Ri,t+1 −Rf,t+1)] . (A.5)

By using Equations (A.2) and (A.4), Equation (A.5) simplifies to

JW (Wt, zt) = δ Et [JW (Wt+1, zt+1)Rp,t+1] , (A.6)

and by combining with Equation (A.2), this leads to the usual envelope condition:

JW (Wt, zt) = UC(Ct). (A.7)

By updating (A.7), substituting the result in (A.2), and rearranging, we obtain the Euler

equation:

1 = Et

[δUC(Ct+1)UC(Ct)

Rp,t+1

]= Et

[δJW (Wt+1, zt+1)JW (Wt, zt)

Rp,t+1

]. (A.8)

Given (A.8), we can substitute consumption out of the model, and the resulting stochastic

discount factor (SDF) is equal to

Mt+1 = δJW (Wt+1, zt+1)JW (Wt, zt)

. (A.9)

To derive the Euler equation for an arbitrary individual risky return, Ri,t+1, by using the

law of iterated expectations, the f.o.c. with respect to ωi,t can be rewritten as

Et (Mt+1Ri,t+1) = Et (Mt+1Rf,t+1) . (A.10)

26The normalization that the benchmark return is the Nth asset does not play any role in the derivation.

41

By substituting (A.3) in (A.8), and rearranging, we obtain,

1 =N−1∑i=1

ωi,t Et [Mt+1 (Ri,t+1 −Rf,t+1)] + Et (Mt+1Rf,t+1) . (A.11)

By using (A.10), we derive the pricing equation for asset i:

1 = Et (Mt+1Rf,t+1) = Et (Mt+1Ri,t+1) . (A.12)

To linearize the model, we use the general expected return-covariance representation:

Et(Ri,t+1)−Rf,t+1 = −Covt(Ri,t+1 −Rf,t+1,Mt+1)

Et(Mt+1). (A.13)

By using Stein’s lemma, we can rewrite the covariance term Covt(Ri,t+1 −Rf,t+1,Mt+1) as:27

Covt(Ri,t+1 −Rf,t+1,Mt+1) = Covt

[Ri,t+1 −Rf,t+1, δ

JW (Wt+1, zt+1)JW (Wt, zt)

]=

δ

JW (Wt, zt){Et[JWW (Wt+1, zt+1)] Covt(Ri,t+1 −Rf,t+1,Wt+1)

+ Et[JWz(Wt+1, zt+1)] Covt(Ri,t+1 −Rf,t+1, zt+1)}

=δ

JW (Wt, zt)

{Wt Et[JWW (Wt+1, zt+1)] Covt

(Ri,t+1 −Rf,t+1,

Wt+1

Wt

)+ Et[JWz(Wt+1, zt+1)] Covt(Ri,t+1 −Rf,t+1, zt+1)} . (A.14)

The conditional mean SDF is given by

Et(Mt+1) =δ

JW (Wt, zt)Et[JW (Wt+1, zt+1)]. (A.15)

By substituting Equations (A.14) and (A.15) into (A.13), we obtain:

Et(Ri,t+1)−Rf,t+1 = −Wt Et[JWW (Wt+1, zt+1)]Et[JW (Wt+1, zt+1)]

Covt

(Ri,t+1 −Rf,t+1,

Wt+1

Wt

)−Et[JWz(Wt+1, zt+1)]

Et[JW (Wt+1, zt+1)]Covt(Ri,t+1 −Rf,t+1, zt+1). (A.16)

27For applications of the Stein (1981) lemma to asset pricing, see, for example, Brandt and Wang (2003),Cochrane (2005), and Balvers and Huang (2009).

42

Finally, by assuming the approximations,

Et[JW (Wt+1, zt+1)] = JW (Wt, zt),

Et[JWW (Wt+1, zt+1)] = JWW (Wt, zt),

Et[JWz(Wt+1, zt+1)] = JWz(Wt, zt),

we obtain the ICAPM pricing equation:

Et(Ri,t+1)−Rf,t+1 = γ Covt(Ri,t+1 −Rf,t+1, Rm,t+1)− JWz(Wt, zt)JW (Wt, zt)

Covt(Ri,t+1 −Rf,t+1, zt+1),

(A.17)

where γ ≡ −WtJWW (Wt,zt)JW (Wt,zt)

denotes the parameter of relative risk aversion (assumed to be con-

stant), and we use the result from the intertemporal budget constraint that the return on

aggregate wealth is approximately equal to the change in wealth, Wt+1

Wt' Rm,t+1.28

Since Covt(Ri,t+1 − Rf,t+1, zt) = 0, we use the innovation in the state variable, which is

measured by the first difference in zt+1:29

∆zt+1 = zt+1 − zt. (A.18)

The resulting pricing equation is given by

Et(Ri,t+1)−Rf,t+1 = γ Covt(Ri,t+1−Rf,t+1, Rm,t+1)− JWz(Wt, zt)JW (Wt, zt)

Covt(Ri,t+1−Rf,t+1,∆zt+1).

(A.19)

This specification is also consistent with the original ICAPM in continuous time, which is based

on the innovations in the state variables.

28This is true if consumption is low relative to wealth, Ct �Wt.29The simple change corresponds to the innovation if the state variable follows a random-walk process.

43

B Cross-sectional regressions with GMM robust standard er-

rors

The GMM system equivalent to the time series/cross-sectional regressions approach has a set

of moment conditions given by

gT (Θ) =1T

∑T

t=1(rt −Rf,t1N − δ − βft)∑Tt=1(rt −Rf,t1N − δ − βft)⊗ ft∑T

t=1(rt −Rf,t1N − βλ)

=

0(N×1)

0(NK×1)

0(N×1)

, (B.20)

where rt(N ×1) is a vector of simple returns; 1N (N ×1) is a vector of ones; δ(N ×1) is a vector

of constants for the time series regressions; β(N ×K) is a matrix of K factor loadings for the N

test assets; ft(K × 1) is a vector of common factors used to price assets; λ(K × 1) is a vector of

beta risk prices; ⊗ denotes the Kronecker product; and 0 denotes conformable vectors of zeros.

The first two sets of moment conditions identify the factor loadings (including the constants

or Jensen alphas), and thus are equivalent to the time-series regressions. These moment condi-

tions are exactly identified with N +NK orthogonality conditions and N +NK parameters to

estimate. The third set of moments corresponds to the cross-sectional regression, and identifies

the beta risk prices, λ. Hence, the third set of moments has N moment conditions and K

parameters to estimate, leading to N −K overidentifying restrictions, which also corresponds

to the number of overidentifying conditions in the entire system.

System (B.20) represents a straightforward generalization of the system presented in Cochrane

(2005) (Chapter 12), for the case of K > 1 risk factors affecting the cross-section of returns.

The vector of parameters to estimate in this GMM system is given by

Θ′ =[δ′ β∗ λ′

], (B.21)

where β∗ ≡ vec(β′)′, and vec is the operator that enables us to stack the factor loadings for the

N assets into a column vector.

The matrix that chooses which moment conditions are set to zero in the GMM first-order

44

condition, agT (Θ) = 0, is given by

a =

IN ⊗ IK+1 0(N(K+1)×N)

0(K×N(K+1)) β′

, (B.22)

where Im denotes an identity matrix of order m.

The matrix of sensitivities of the moment conditions to the parameters is given by

d ≡ ∂gT (Θ)∂Θ′

= −

IN IN ⊗

(1T

∑Tt=1 f ′t

)0(N×K)

IN ⊗(

1T

∑Tt=1 ft

)IN ⊗

(1T

∑Tt=1 ftf ′t

)0(NK×K)

0(N×N) IN ⊗ λ β

. (B.23)

The variance-covariance matrix of the moments, S, has the form:

S =∞∑

j=−∞E

rt −Rf,t1N − δ − βft

(rt −Rf,t1N − δ − βft)⊗ ft

rt −Rf,t1N − βλ

rt−j −Rf,t−j1N − δ − βft−j

(rt−j −Rf,t−j1N − δ − βft−j)⊗ ft−j

rt−j −Rf,t−j1N − βλ

′

=∞∑

j=−∞E

εt

εt ⊗ ft

β(ft − E(ft)) + εt

εt−j

εt−j ⊗ ft−j

β(ft−j − E(ft)) + εt−j

′ , (B.24)

where εt ≡ rt − Rf,t1N − δ − βft, represents the vector of time-series residuals. In the last

equality, we impose the null that the asset pricing model relation is true, E (rt −Rf,t1N ) = βλ:

rt −Rf,t1N − βλ = rt −Rf,t1N − E (rt −Rf,t1N )

= rt −Rf,t1N − δ − β E(ft) = β(ft − E(ft)) + εt. (B.25)

By using the general GMM formula for the variance-covariance matrix of the parameter

estimates,

Var(Θ) =1T

(ad)−1aSa′(ad)−1′, (B.26)

the last K elements of the main diagonal give the variances of the estimated factor risk prices,

used to calculate the t-statistics.

In addition, if we use the formula for the variance-covariance matrix of the GMM moment

45

conditions (errors),

Var(gT (Θ)) =1T

(IN(K+2)−d(ad)−1a

)S(IN(K+2)−d(ad)−1a

)′, (B.27)

we obtain the covariance matrix of the cross-sectional pricing errors (α) from the bottom-right

(N × N) block of Var(gT (Θ)), which is used to conduct the test that the pricing errors are

jointly equal to zero:

α′Var(α)−1α ∼ χ2(N −K). (B.28)

The Shanken (1992) standard errors can be derived as a special case of the GMM “robust”

standard errors derived above, as noted by Cochrane (2005) (Chapter 12). If we assume that

εt is jointly i.i.d.; εt and ft are independent; and finally ft has no serial correlation, then the

spectral density matrix S in (B.24) specializes to

S = E

εt

εt ⊗ ft

β(ft − E(ft)) + εt

εt

εt ⊗ ft

β(ft − E(ft)) + εt

′

=

Σ Σ⊗ E(f ′t) Σ

Σ⊗ E(ft) Σ⊗ E(ftf ′t) Σ⊗ E(ft)

Σ Σ⊗ E(f ′t) βΣfβ′ + Σ

, (B.29)

where Σf ≡ E [(f t − E(ft))(f t − E(ft))′] represents the variance-covariance matrix associated

with the factors, and Σ ≡ E(εtε′t) denotes the variance-covariance matrix associated with the

residuals from the time-series regressions. By replacing (B.29) in (B.26) we obtain the Shanken

variances for the estimated factor risk premia:

Var(λ) =1T

[(β′β

)−1β′Σβ

(β′β

)−1(

1 + λ′Σ−1f λ

)+ Σf

]. (B.30)

Similarly, the Shanken variances for pricing errors are given by

Var(α) =1T

(IN − β

(β′β

)−1β′)

Σ(IN − β

(β′β

)−1β′)(

1 + λ′Σ−1f λ

). (B.31)

46

C GMM formulas

Following Cochrane (2005), the weighting matrix associated with the GMM system (16) is given

by

W =

W∗ 0

0 IK

, (C.32)

where W∗ = IN is an N -dimensional identity matrix; 0 denotes a conformable matrix of zeros;

and IK denotes a K-dimensional identity matrix, with K representing the number of factors in

the model. In this specification, W∗ is the weighting matrix for the first N moment conditions

(corresponding to the N pricing errors), while IK is the weighting matrix associated with the

last K orthogonality conditions that identify the factor means.

The risk price estimates b have variance formulas given by

Var(b) =1T

(d′Wd)−1d′WSWd(d′Wd)−1, (C.33)

where d ≡ ∂gT (b)∂b′ represents the matrix of moments’ sensitivities to the parameters; and S is

an estimator for the spectral density matrix S derived under the heteroskedasticity-robust or

White (1980) standard errors (that is, no lags of the moment functions are considered in the

computation of S).

The variance–covariance matrix for the moments from first-stage GMM is given by

Var(gT (b)

)=

1T

(IN+K−d(d′Wd)−1d′W

)S(IN+K−Wd(d′Wd)−1d′

), (C.34)

where the first (N,N) block of (C.34) represents the covariance matrix of the N pricing errors.

47

Table 1: Descriptive statistics for (C)ICAPM factorsThis table reports descriptive statistics for the risk factors from the (C)ICAPM and alternative fac-tor models. RMt+1, RMt+1zt and ∆zt+1 denote the market, scaled and intertemporal risk fac-tors from the (C)ICAPM. The conditioning variables (z) are the Fed funds rate (FFR) and therelative T-bill rate (RREL). SMBt+1, HMLt+1, and UMDt+1 denote the size, value, and mo-mentum factors, respectively. The sample is 1963:07-2009:12. φ designates the first order au-tocorrelation coefficient. The correlations between the state variables are presented in Panel B.

Panel AMean (%) Stdev. (%) Min. (%) Max. (%) φ

RMt+1 0.42 4.51 −23.14 16.05 0.095SMBt+1 0.25 3.19 −16.67 22.19 0.059HMLt+1 0.41 2.95 −12.78 13.84 0.151UMDt+1 0.72 4.37 −34.69 18.35 0.065

RMt+1FFRt −0.01 0.17 −1.06 0.84 0.134∆FFRt+1 −0.01 0.57 −6.63 3.06 0.403

RMt+1RRELt 0.00 0.05 −0.47 0.22 0.147∆RRELt+1 0.00 0.50 −4.93 2.57 0.314

Panel BSMBt+1 HMLt+1 UMDt+1 RMt+1FFRt ∆FFRt+1 RMt+1RRELt ∆RRELt+1

RMt+1 0.30 −0.32 −0.14 0.15 −0.14 −0.06 −0.12SMBt+1 1.00 −0.24 −0.01 0.04 −0.05 0.06 −0.00HMLt+1 1.00 −0.16 −0.26 −0.05 −0.04 −0.11UMDt+1 1.00 0.22 0.05 0.25 0.08

RMt+1FFRt 1.00 −0.19 0.38 −0.20∆FFRt+1 1.00 −0.01 0.69

RMt+1RRELt 1.00 −0.09∆RRELt+1 1.00

48

Table 2: Factor risk premia for (C)ICAPMThis table reports the estimation and evaluation results for the three-factor Conditional IntertemporalCAPM ((C)ICAPM). The estimation procedure is the time-series/cross-sectional regressions approach.The test portfolios are the 25 size/book-to-market portfolios (SBM25, Panel A) and 25 size/momentumportfolios (SM25, Panel B). λM denotes the beta risk price estimate for the market factor; λM,z denotesthe risk price associated with the scaled factor; and λz represents the risk price associated with theintertemporal risk factor. The conditioning variables (z) are the Fed funds rate (FFR) and the relativeT-bill rate (RREL). Below the risk price estimates (in %) are displayed t-statistics based on Shankenstandard errors (in parenthesis). The column labeled MAE(%) presents the mean absolute pricing error(in %), and the column labeled R2

OLS denotes the cross-sectional OLS R2. The column labeled χ2

presents the χ2 statistic (first line), and associated asymptotic p-values (in parenthesis) for the test onthe joint significance of the pricing errors. The sample is 1963:07-2009:12. Italic, underlined and boldnumbers denote statistical significance at the 10%, 5% and 1% levels, respectively.

λM λM,z λz χ2 MAE(%) R2OLS

Panel A (SBM25)FFR 0.49 −0.01 −0.61 33.36 0.10 0.70

(2.22) (−0.20) (−2.83) (0.06)RREL 0.53 0.03 −0.64 31.10 0.14 0.46

(2.57) (1.05) (−2.90) (0.09)Panel B (SM25)

FFR 0.66 0.14 −0.27 55.76 0.16 0.71(3.22) (4.50) (−1.54) (0.00)

RREL 0.53 0.06 −0.53 35.18 0.15 0.74(2.32) (3.23) (−2.08) (0.04)

49

Table 3: Factor risk premia for alternative factor modelsThis table reports the estimation and evaluation results for alternative models – the CAPM (Row 1),the Fama-French three-factor model (Row 2) and the Carhart four-factor model (Row 3). The esti-mation procedure is the time-series/cross-sectional regressions approach. The test portfolios are the 25size/book-to-market portfolios (SBM25, Panel A) and 25 size/momentum portfolios (SM25, Panel B).λM , λSMB , λHML, λUMD denote the beta risk price estimates for the market, size, value and momentumfactors, respectively. Below the risk price estimates (in %) are displayed t-statistics based on Shankenstandard errors (in parenthesis). The column labeled MAE(%) presents the mean absolute pricing er-ror (in %), and the column labeled R2

OLS denotes the cross-sectional OLS R2. The column labeled χ2

presents the χ2 statistic (first line), and associated asymptotic p-values (in parenthesis) for the test onthe joint significance of the pricing errors. The sample is 1963:07-2009:12. Italic, underlined and boldnumbers denote statistical significance at the 10%, 5% and 1% levels, respectively.

Row λM λSMB λHML λUMD χ2 MAE(%) R2OLS

Panel A (SBM25)1 0.60 109.20 0.22 −0.42

(2.94) (0.00)2 0.40 0.22 0.49 83.86 0.10 0.67

(2.07) (1.62) (3.81) (0.00)3 0.47 0.22 0.48 3.46 32.69 0.09 0.78

(2.43) (1.59) (3.67) (3.72) (0.05)Panel B (SM25)

1 0.52 118.30 0.32 −0.18(2.56) (0.00)

2 0.49 0.46 −0.73 106.17 0.29 0.03(2.53) (3.05) (−2.82) (0.00)

3 0.51 0.21 0.45 0.82 75.40 0.11 0.85(2.64) (1.40) (2.13) (4.37) (0.00)

50

Table 4: Average risk premia across book-to-market and momentum quintilesThis table reports the average risk premium (average beta times (beta) risk price) for each factor, acrossquintiles for book-to-market (BM) and prior short-term returns (momentum, M). The model is thethree-factor (C)ICAPM when the state variables (z) are the Fed funds rate (FFR) and the relativeT-bill rate (RREL). E(R) denotes the average excess return for each BM and M quintile, and α

represents the average pricing error per quintile. RMt+1, RMt+1zt and ∆zt+1 denote the market, scaledand intertemporal risk factors from the (C)ICAPM. All the values are presented in percentage points.BM1 and M1 denote the lowest BM and M quintile, respectively, and Dif. denotes the difference acrossextreme quintiles. The sample is 1963:07-2009:12.

E(R) RMt+1 RMt+1zt ∆zt+1 αPanel A (FFR)

BM1 0.36 0.62 −0.01 −0.22 −0.04BM5 0.89 0.50 0.02 0.36 0.01Dif. −0.53 0.13 −0.03 −0.58 −0.04M1 0.07 0.92 −0.90 0.12 −0.07M5 1.07 0.78 0.26 −0.08 0.11Dif. −1.01 0.14 −1.16 0.19 −0.18

Panel B (RREL)BM1 0.36 0.68 0.14 −0.41 −0.05BM5 0.89 0.53 −0.02 0.27 0.11Dif. −0.53 0.15 0.17 −0.69 −0.16M1 0.07 0.71 −0.72 0.17 −0.09M5 1.07 0.63 0.71 −0.32 0.05Dif. −1.01 0.08 −1.43 0.49 −0.14

Table 5: Average risk premia across BM and momentum quintiles: Alternative modelsThis table reports the average risk premium (average beta times (beta) risk price) for each factor, acrossquintiles for book-to-market (BM) and prior short-term returns (momentum, M). The models are theFama-French model (FF3, Panel A), and the Carhart model (C4, Panel B). E(R) denotes the averageexcess return for each BM and M quintile, and α represents the average pricing error per quintile.RM , SMB, HML, and UMD denote the market, size, value, and momentum factors, respectively. Allthe values are presented in percentage points. BM1 and M1 denote the lowest BM and M quintile,respectively, and Dif. denotes the difference across extreme quintiles. The sample is 1963:07-2009:12.

Panel A: FF3E(R) RMt+1 SMBt+1 HMLt+1 α

BM1 0.36 0.43 0.14 −0.19 −0.02BM5 0.89 0.43 0.12 0.37 −0.02Dif. −0.53 0.00 0.03 −0.56 0.01M1 0.07 0.64 0.27 −0.22 −0.63M5 1.07 0.50 0.29 0.07 0.21Dif. −1.01 0.14 −0.02 −0.29 −0.84

Panel B: C4E(R) RMt+1 SMBt+1 HMLt+1 UMDt+1 α

BM1 0.36 0.50 0.14 −0.19 −0.09 −0.00BM5 0.89 0.50 0.12 0.36 −0.10 0.01Dif. −0.53 0.00 0.03 −0.55 0.01 −0.02M1 0.07 0.59 0.12 0.02 −0.61 −0.05M5 1.07 0.57 0.13 0.02 0.33 0.03Dif. −1.01 0.02 −0.01 −0.00 −0.94 −0.08

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Table 6: Time-series regressions for HML and UMD

This table reports the results from time-series regressions of HML (Panel A) and UMD (Panel B)on the (C)ICAPM factors, RMt+1, RMt+1zt and ∆zt+1. The conditioning variables (z) are the Fedfunds rate (FFR) and the relative T-bill rate (RREL). Below the coefficient estimates are displayedheteroskedasticity-robust t-statistics (in parenthesis). The column labeled R

2denotes the adjusted co-

efficient of determination. The sample is 1963:07-2009:12. Italic, underlined and bold numbers denotestatistical significance at the 10%, 5% and 1% levels, respectively.

const. RMt+1 RMt+1zt ∆zt+1 R2

Panel A (HMLt+1)FFR 0.00 −0.20 −4.16 −0.73 0.16

(3.88) (−5.54) (−4.56) (−3.80)RREL 0.00 −0.22 −4.24 −0.91 0.13

(4.07) (−6.06) (−1.48) (−3.77)Panel B (UMDt+1)

FFR 0.01 −0.16 6.63 0.58 0.08(5.04) (−2.48) (3.45) (1.47)

RREL 0.01 −0.11 19.99 0.82 0.08(5.09) (−1 .81 ) (4.01) (2.15)

Table 7: Factor risk premia for (C)ICAPM: Bond returnsThis table reports the estimation and evaluation results for the three-factor Conditional IntertemporalCAPM ((C)ICAPM). The estimation procedure is the time-series/cross-sectional regressions approach.The test portfolios are 7 Treasury bond returns plus 25 equity portfolios. The equity portfolios are the25 size/book-to-market portfolios (SBM25, Panel A) and 25 size/momentum portfolios (SM25, PanelB). λM denotes the beta risk price estimate for the market factor; λM,z denotes the risk price associatedwith the scaled factor; and λz represents the risk price associated with the intertemporal risk factor. Theconditioning variables (z) are the Fed funds rate (FFR) and the relative T-bill rate (RREL). Below therisk price estimates (in %) are displayed t-statistics based on Shanken standard errors (in parenthesis).The column labeled MAE(%) presents the mean absolute pricing error (in %), and the column labeledR2

OLS denotes the cross-sectional OLS R2. The column labeled χ2 presents the χ2 statistic (first line),and associated asymptotic p-values (in parenthesis) for the test on the joint significance of the pricingerrors. The sample is 1963:07-2009:12. Italic, underlined and bold numbers denote statistical significanceat the 10%, 5% and 1% levels, respectively.

λM λM,z λz χ2 MAE(%) R2OLS

Panel A (SBM25)FFR 0.49 −0.04 −0.48 50.25 0.09 0.85

(2.29) (−1.34) (−3.19) (0.01)RREL 0.62 −0.02 −0.18 90.46 0.16 0.57

(3.05) (−1.15) (−2.66) (0.00)Panel B (SM25)

FFR 0.68 0.11 0.07 84.98 0.16 0.71(3.33) (4.31) (0.51) (0.00)

RREL 0.54 0.04 −0.11 79.55 0.14 0.76(2.57) (3.86) (−1.33) (0.00)

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Table 8: Factor risk premia for (C)ICAPM (30 portfolios)This table reports the estimation and evaluation results for the three-factor Conditional IntertemporalCAPM ((C)ICAPM). The estimation procedure is the time-series/cross-sectional regressions approach.The test portfolios are 10 portfolios sorted on size, 10 portfolios sorted on book-to-market and 10 mo-mentum portfolios. λM denotes the beta risk price estimate for the market factor; λM,z denotes the riskprice associated with the scaled factor; and λz represents the risk price associated with the intertemporalrisk factor. The conditioning variables (z) are the Fed funds rate (FFR) and the relative T-bill rate(RREL). Below the risk price estimates (in %) are displayed t-statistics based on Shanken standarderrors (in parenthesis). The column labeled MAE(%) presents the mean absolute pricing error (in %),and the column labeled R2

OLS denotes the cross-sectional OLS R2. The column labeled χ2 presentsthe χ2 statistic (first line), and associated asymptotic p-values (in parenthesis) for the test on the jointsignificance of the pricing errors. The sample is 1963:07-2009:12. Italic, underlined and bold numbersdenote statistical significance at the 10%, 5% and 1% levels, respectively.

λM λM,z λz χ2 MAE(%) R2OLS

FFR 0.51 0.10 −0.57 35.26 0.15 0.29(2.59) (3.14) (−2.90) (0.13)

RREL 0.49 0.05 −0.40 33.16 0.12 0.60(2.45) (3.46) (−2.31) (0.19)

Table 9: Factor risk premia for (C)ICAPM: Estimation by GMMThis table reports the estimation and evaluation results for the three-factor Conditional IntertemporalCAPM ((C)ICAPM). The estimation procedure is first-stage GMM with equally weighted errors. Thetest portfolios are the 25 size/book-to-market portfolios (SBM25, Panel A) and 25 size/momentumportfolios (SM25, Panel B). γM denotes the (covariance) risk price estimate for the market factor; γM,z

denotes the risk price associated with the scaled factor; and γz represents the risk price associated withthe intertemporal risk factor. The conditioning variables (z) are the Fed funds rate (FFR) and therelative T-bill rate (RREL). The first line associated with each row presents the covariance risk priceestimates, and the second line reports the asymptotic GMM robust t-statistics (in parenthesis). Thecolumn labeled MAE(%) presents the mean absolute pricing error (in %), and the column labeled R2

OLS

denotes the cross-sectional OLS R2. The column labeled χ2 presents the χ2 statistic (first line), andassociated asymptotic p-values (in parenthesis) for the test on the joint significance of the pricing errors.The sample is 1963:07-2009:12. Italic, underlined and bold numbers denote statistical significance at the10%, 5% and 1% levels, respectively.

γM γM,z γz χ2 MAE(%) R2OLS

Panel A (SBM25)FFR −0.26 −146.70 −193.96 42.07 0.10 0.70

(−0.12) (−1.23) (−2.17) (0.01)RREL 0.07 981.75 −249.40 33.18 0.14 0.46

(0.03) (1.13) (−3.19) (0.06)Panel B (SM25)

FFR −0.25 456.74 −56.46 45.25 0.16 0.71(−0.13) (2.87) (−0.92) (0.00)

RREL 1.54 1960.65 −196.00 37.07 0.15 0.74(0.57) (2.51) (−1 .85 ) (0.02)

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Table 10: Factor risk premia for nested modelsThis table reports the estimation and evaluation results for nested models of the (C)ICAPM. The twonested models are the two-factor conditional CAPM (Row 1) and the two-factor ICAPM (Row 2). Theestimation procedure is the time-series/cross-sectional regressions approach. The test portfolios are the25 size/book-to-market portfolios (SBM25, Panels A and C) and 25 size/momentum portfolios (SM25,Panels B and D). λM denotes the beta risk price estimate for the market factor; λM,z denotes the riskprice associated with the scaled factor; and λz represents the risk price associated with the intertemporalrisk factor. The conditioning variables (z) are the Fed funds rate (FFR) and the relative T-bill rate(RREL). Below the risk price estimates (in %) are displayed t-statistics based on Shanken standarderrors (in parenthesis). The column labeled MAE(%) presents the mean absolute pricing error (in %),and the column labeled R2

OLS denotes the cross-sectional OLS R2. The column labeled χ2 presentsthe χ2 statistic (first line), and associated asymptotic p-values (in parenthesis) for the test on the jointsignificance of the pricing errors. The sample is 1963:07-2009:12. Italic, underlined and bold numbersdenote statistical significance at the 10%, 5% and 1% levels, respectively.

Row λM λM,z λz χ2 MAE(%) R2OLS

Panel A (SBM25, FFR)1 0.52 −0.13 64.60 0.15 0.34

(2.48) (−3.43) (0.00)2 0.49 −0.78 27.21 0.11 0.67

(2.17) (−2.96) (0.25)Panel B (SM25, FFR)

1 0.68 0.13 63.86 0.16 0.69(3.31) (4.20) (0.00)

2 0.53 0.08 112.03 0.32 −0.18(2.66) (0.54) (0.00)

Panel C (SBM25, RREL)1 0.66 −0.04 69.65 0.22 −0.18

(3.21) (−1 .68 ) (0.00)2 0.58 −0.48 47.61 0.15 0.40

(2.68) (−2.81) (0.00)Panel D (SM25, RREL)

1 0.55 0.04 73.08 0.17 0.66(2.60) (3.79) (0.00)

2 0.55 0.53 53.24 0.30 0.07(2.54) (2.19) (0.00)

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Table 11: Factor risk premia for (C)ICAPM (size/long-term reversal portfolios)This table reports the estimation and evaluation results for the three-factor Conditional IntertemporalCAPM ((C)ICAPM). The estimation procedure is the time-series/cross-sectional regressions approach.The test portfolios are the 25 size/long-term reversal portfolios (SLTR25). λM denotes the beta riskprice estimate for the market factor; λM,z denotes the risk price associated with the scaled factor; andλz represents the risk price associated with the intertemporal risk factor. The conditioning variables(z) are the Fed funds rate (FFR) and the relative T-bill rate (RREL). Below the risk price estimates(in %) are displayed t-statistics based on Shanken standard errors (in parenthesis). The column labeledMAE(%) presents the mean absolute pricing error (in %), and the column labeled R2

OLS denotes thecross-sectional OLS R2. The column labeled χ2 presents the χ2 statistic (first line), and associatedasymptotic p-values (in parenthesis) for the test on the joint significance of the pricing errors. Thesample is 1963:07-2009:12. Italic, underlined and bold numbers denote statistical significance at the10%, 5% and 1% levels, respectively.

λM λM,z λz χ2 MAE(%) R2OLS

FFR 0.52 −0.03 −0.47 37.71 0.09 0.63(2.54) (−0.61) (−2.81) (0.02)

RREL 0.54 0.06 −0.49 19.25 0.11 0.51(2.47) (1.99) (−2.63) (0.63)

55

Panel A: FFR

Panel B: RREL

Figure 1: Short-term interest ratesThis figure plots the time-series for the monthly changes in the Fed fundsrate (∆FFR) and the relative T-bill rate (∆RREL). The sample is1963:07-2009:12. The vertical lines indicate the NBER recession periods.

56

Panel A: pricing errors

Panel B: t-statistics

Figure 2: Individual pricing errors (FFR): SBM25This figure plots the pricing errors (in %, Panel A), and respective t-statistics (Panel B) ofthe 25 size/book-to-market portfolios (SBM25) from the (C)ICAPM (version based on FFR);Fama-French model (FF3); and the Carhart model (C4). The pricing errors are obtainedfrom an OLS cross-sectional regression of average excess returns on factor betas. ij des-ignates a portfolio associated with the ith size quintile and jth book-to-market quintile.

57

Panel A: pricing errors

Panel B: t-statistics

Figure 3: Individual pricing errors (FFR): SM25This figure plots the pricing errors (in %, Panel A), and respective t-statistics (Panel B)of the 25 size/momentum portfolios (SM25) from the (C)ICAPM (version based on FFR);Fama-French model (FF3); and the Carhart model (C4). The pricing errors are ob-tained from an OLS cross-sectional regression of average excess returns on factor betas. ij

designates a portfolio associated with the ith size quintile and jth prior return quintile.

58

Panel A: RMt+1FFRt Panel B: RMt+1RRELt

Panel C: ∆FFRt+1 Panel D: ∆RRELt+1

Figure 4: Regression betas for SBM25This figure plots the multiple regression beta estimates associated with the SBM25 portfoliosfrom (C)ICAPM. The factors are the scaled factor (RMt+1FFRt, RMt+1RRELt) and the in-novations in the state variable (∆FFRt+1,∆RRELt+1). ij designates a portfolio associatedwith the ith size quintile and jth book-to-market quintile. The sample is 1963:07-2009:12.

59

Panel A: RMt+1FFRt Panel B: RMt+1RRELt

Panel C: ∆FFRt+1 Panel D: ∆RRELt+1

Figure 5: Regression betas for SM25This figure plots the multiple regression beta estimates associated with the SM25 portfoliosfrom (C)ICAPM. The factors are the scaled factor (RMt+1FFRt, RMt+1RRELt) and the in-novations in the state variable (∆FFRt+1,∆RRELt+1). ij designates a portfolio associatedwith the ith size quintile and jth prior return quintile. The sample is 1963:07-2009:12.

60

Panel A: All periods, FFR Panel B: All periods, RREL

Panel C: Periods with high FFR Panel D: Periods with high RREL

Panel E: Periods with low FFR Panel F: Periods with low RREL

Figure 6: Average conditional market betas for SM25This figure plots the average conditional market beta estimates associated with the SM25portfolios from (C)ICAPM, βi,M + βi,M,z E(zt). In Panels A and B all the periods areused, whereas in Panels C,D (E,F) only the periods in which FFR,RREL are 1.5 stan-dard deviations above (below) the mean are used. ij designates a portfolio associatedwith the ith size quintile and jth prior return quintile. The sample is 1963:07-2009:12.

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Panel A: FFR

Panel B: RREL

Figure 7: Short-term interest ratesThis figure plots the monthly time-series for the conditional market beta of the win-ner minus loser portfolio (W-L) and the UMD factor. The sample is 1963:07-2009:12.

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Panel A: pricing errors

Panel B: t-statistics

Figure 8: Individual pricing errors (FFR): SLTR25This figure plots the pricing errors (in %, Panel A), and respective t-statistics (Panel B)of the 25 size/long-term reversal portfolios (SLTR25) from the (C)ICAPM (version based onFFR); Fama-French model (FF3); and the Carhart model (C4). The pricing errors are ob-tained from an OLS cross-sectional regression of average excess returns on factor betas. ij

designates a portfolio associated with the ith size quintile and jth prior return quintile.

63

Panel A: RMt+1FFRt Panel B: RMt+1RRELt

Panel C: ∆FFRt+1 Panel D: ∆RRELt+1

Figure 9: Regression betas for SLTR25This figure plots the multiple regression beta estimates associated with the SLTR25 portfo-lios from (C)ICAPM. The factors are the scaled factor (RMt+1FFRt, RMt+1RRELt) and theinnovations in the state variable (∆FFRt+1,∆RRELt+1). ij designates a portfolio associ-ated with the ith size quintile and jth prior return quintile. The sample is 1963:07-2009:12.

64