Short Term Interest Rate and Stock Market

Electronic copy available at: http://ssrn.com/abstract=1986787

Short-term interest rates and stock market anomalies

Paulo Maio1 Pedro Santa-Clara2

This version: March 20163

1Hanken School of Economics. E-mail: [email protected] School of Business and Economics, NBER, and CEPR. E-mail: [email protected] thank Yakov Amihud, Kevin Aretz, Hendrik Bessembinder, John Cochrane, Mark Grin-

blatt, Bruce Grundy, Joachim Inkmann, Arne Klein, Shimon Kogan, Adriano Rampini, JesperRangvid, Christian Wagner, Hao Zhou, and seminar participants at the 2012 Arne Ryde workshop,2012 Finance Down Under Conference in Melbourne, 2012 Rothschild Caesarea Center Conferencein Herzliya, and the 2012 SGF Conference in Zurich for helpful comments. We are grateful to Ken-neth French, Amit Goyal, Robert Shiller, Robert Stambaugh, and Lu Zhang for making availablestock market data. Previous versions circulated with the titles “Value, momentum, and short-terminterest rates” and “The Fed and stock market anomalies.” Maio acknowledges financial supportfrom the Hanken Foundation. Santa-Clara is supported by a grant from the Fundacao para aCiencia e Tecnologia (PTDC/EGE-GES/101414/2008). Any remaining errors are our own.

Electronic copy available at: http://ssrn.com/abstract=1986787

Abstract

We present a simple two-factor model that helps explaining several CAPM anomalies—value

premium, return reversal, equity duration, asset growth, and inventory growth. The key risk

factor is the innovation on a short-term interest rate, either the Fed funds rate or the T-bill

rate. This model explains a large percentage of the dispersion in average returns of the

joint anomalies, with cross-sectional R2 estimates of 58% and 67% in the estimation with

value- and equal-weighted portfolios, respectively. Moreover, the model compares favorably

with alternative multifactor models. Hence, short-term interest rates seem to be relevant for

explaining cross-sectional equity risk premia.

Keywords: cross-section of stock returns; asset pricing; intertemporal CAPM; state vari-

ables; linear multifactor models; predictability of returns; value premium; long-term rever-

sal in returns; equity duration anomaly; corporate investment anomaly; inventory growth

anomaly

JEL classification: E44; G12; G14

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)). The long-term reversal in returns anomaly (De

Bondt and Thaler (1985, 1987)) refers to stocks with low returns in the long past having

higher average returns, while past winners have lower returns. Moreover, there is evidence

showing that stocks with high duration earn lower average returns than stocks with low

duration (Dechow, Sloan, and Soliman (2004)). On the other hand, stocks of firms that

invest more tend to have lower average returns than the stocks of firms that invest less

(Titman, Wei, and Xie (2004), Cooper, Gulen, and Schill (2008), Lyandres, Sun, and Zhang

(2008)), which represents the investment anomaly in broad terms.

We offer a simple asset pricing model that goes a long way in explaining several CAPM

anomalies, that is, patterns in cross-sectional equity risk premia that are not explained by

the CAPM. We specify a two-factor intertemporal CAPM (ICAPM, Merton (1973)) in which

the factors are the market equity premium and the “hedging” or intertemporal factor. The

second source of systematic risk (the innovation in the state variable) arises because stocks

that are more correlated with future investment opportunities should earn a higher risk

premium since they do not provide a hedge for reinvestment risk (unfavorable changes in

aggregate wealth in future periods). In the traditional empirical applications of the ICAPM,

the ultimate source for the additional risk factor (in addition to the usual market factor) is

related with a time-varying market risk premium in future periods, where the time variation

is driven by an observable state variable.1 In our simple model, we use a proxy for short-

term interest rates—either the the Federal funds rate (FFR) or the three-month T-bill rate

(TB)—as the single state variable that drives future aggregate investment opportunities.

There is evidence in the return predictability literature that short-term interest rates forecast

expected excess market returns, especially at short horizons (Campbell (1991), Hodrick

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

Bekaert (2007), and Maio (2014b), among others). Thus, either FFR or TB represents a

valid ICAPM state variable. Following the bulk of the ICAPM literature, the innovations in

1State variables used to proxy for the expected market return are largely borrowed from the large literatureon 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)); and aggregate valuation ratios like the dividendyield (Fama and French (1988, 1989)) or the earnings yield (Campbell and Shiller (1988), Campbell andVuolteenaho (2004), Maio (2013c)), among others.

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short-term interest rates are constructed from a first-order autoregressive process.

We test our two-factor model with decile portfolios sorted on the book-to-market ratio

(Rosenberg, Reid, and Lanstein (1985), BM); earnings-to-price ratio (Basu (1983), EP); eq-

uity duration (Dechow, Sloan, and Soliman (2004), DUR); long-term prior returns (De Bondt

and Thaler (1985), REV); firms’ investment-to-assets ratio (Cooper, Gulen, and Schill (2008),

IA); changes in property, plant, and equipment plus changes in inventory scaled by assets

(Lyandres, Sun, and Zhang (2008), PIA); and inventory growth (Belo and Lin (2011), IVG).

To account for the evidence showing that small caps represent the biggest challenge for asset

pricing models (see Fama and French (2012, 2015)), we use both value- and equal-weighted

portfolios. The cross-sectional tests show that the ICAPM explains a large percentage of the

dispersion in average equity premia of the seven portfolio groups, with explanatory ratios

that are in most cases around or above 40% in the tests based on value-weighted portfolios

and above 60% in the estimation with equal-weighted portfolios. When the model is forced

to price all 70 portfolios simultaneously, and thus the joint seven CAPM anomalies, we ob-

tain cross-sectional R2 estimates of 58% and 48% in the versions based on FFR and TB,

respectively. In the augmented test based on equal-weighted portfolios the fit of the ICAPM

is even larger as indicated by the explanatory ratios of 67% in both versions of the model.

Our results are maintained, or even reinforced, if we use alternative interest rate factors

(obtained from first-differences in either FFR or TB) or estimate the model on a subsample

that ends in 2006 (to assess the impact of the recent financial crisis). Moreover, the ICAPM

has also a large explanatory power for equal-weighted portfolios sorted on two additional

market anomalies—cash-flow-to-price ratio (Lakonishok, Shleifer, and Vishny (1994)) and

investment growth (Xing (2008)).

In all cross-sectional tests, the risk price estimates for the innovation in the short-term

interest rate are negative and strongly statistically significant in most cases. These estimates

are consistent with the ICAPM since both the Fed funds rate and the T-bill rate (in levels)

are negatively correlated with future excess market returns and economic activity.

Critically, the interest rate risk factor explains the dispersion in risk premia across the

seven portfolio classes enumerated above. Thus, according to our model, value stocks, past

long-term losers, stocks with low duration, stocks of firms that invest less, and firms that

build lower inventories enjoy higher expected returns than growth stocks, past long-term

winners, high-duration stocks, firms that invest more, and firms that build higher inventories,

respectively. The reason is that the former stocks have more exposure to changes in the

state variable; that is, they have more negative loadings on the interest rate factor. One

possible explanation for these loadings is that many of these value, past loser, low-duration,

and low-investment (low-inventory) firms, have a poor financial position and expectations

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of modest growth in future cash flows, and thus are more sensitive to rises in short-term

interest rates that further constrain their access to external finance and the investment in

profitable projects that could enhance the firm value.

The ICAPM compares favorably with alternative multifactor models widely used in the

literature like the three-factor model from Fama and French (1993), the four-factor models

proposed by Carhart (1997), Pastor and Stambaugh (2003), and Hou, Xue, and Zhang

(2015), or the recent five-factor model from Fama and French (2015) when it comes to explain

these seven market anomalies. Specifically, the ICAPM outperforms the models from Hou,

Xue, and Zhang (2015) and Fama and French (2015) in the estimation with value-weighted

portfolios and the models from Fama and French (1993) and Pastor and Stambaugh (2003)

when tested on the equal-weighted deciles. This is remarkable since the factors in our model

(other than the market factor) are associated with a single variable from outside the equity

market—the Fed funds rate or the T-bill rate. In contrast, all these alternative models have

equity-based sources of systematic risk (other than the market factor), thus our model is

more parsimonious. Perhaps more important, the ICAPM represents an application of the

ICAPM using a macroeconomic variable, while the foundation for the alternative models is

less clear.2 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, our state variable, the short-term interest rate, is not a priori

mechanically related to the test portfolios, as is the case with the equity-based factors in

the alternative models. The model also outperforms other factor models that rely on macro

variables (mainly factors retrieved from the equity premium predictability literature like the

term spread, default spread, or market dividend yield) and that can also be interpreted as

applications of the ICAPM.

However, we do not claim that our simple model represents a new workhorse multifactor

asset pricing model that explains nearly all the CAPM anomalies. For example, untabulated

results suggest that the model is not successful in pricing portfolios sorted on price momen-

tum (Jegadeesh and Titman (1993)) or profitability (Haugen and Baker (1996)). We claim

instead that a rather simple model, based only on one macro variable outside the equity

market—the Fed funds rate or T-bill rate—makes a significant step forward in explaining

(in an economically consistent way) some of the most prominent market anomalies. Hence,

the money market, and monetary policy actions in particular, seems to have a lot to say

about cross-sectional equity risk premia. After all, it is not totally surprising that a factor

model based on short-term interest rates would perform well in driving equity risk premia.

2There is some evidence that the Fama-French size and value factors proxy for future investment oppor-tunities (Petkova (2006) and Maio and Santa-Clara (2012)) and future GDP growth (Vassalou (2003)).

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The Fed funds rate represents one of the major instrument of monetary policy3, so changes

in it should reflect the privileged information of the monetary authority about the future

state of the economy.4

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.5 The mul-

tifactor model used in this paper is also related to alternative multifactor models that use an

interest rate risk factor to help explaining (some of) the cross-section of stock returns (see,

for example, Brennan, Wang, and Xia (2004), Petkova (2006), and Lioui and Maio (2014)).

The main innovation relative to these works is that we force our model to explain signifi-

cantly more market anomalies than these previous studies, which basically have focused on

explaining the value premium. Thus, we show that risk factors related to short-term interest

rates can also help explaining other CAPM anomalies like corporate investment, inventory

growth, long-term return reversal, and equity duration. Our paper is also related with a

broad literature focusing on the interaction between monetary policy actions (measured by

short-term interest rates) and the stock market.6

The paper is organized as follows. In Section 2, we derive our two-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 ICAPM. In Section 5, we evaluate the

consistency of our model with the ICAPM framework, while Section 6 shows a comparison

with alternative ICAPM specifications.

3Bernanke and Blinder (1992) and Bernanke and Mihov (1998) argue that the Fed funds rate is a goodproxy for Fed policy actions, while Fama (2013) shows that the Fed funds rate converges quickly to the Fedfunds target rate.

4For example, Romer and Romer (2000) and Peek, Rosengren, and Tootell (2003) provide evidence thatthe Federal Reserve has informational advantages about the economy and financial institutions.

5An incomplete list of papers that have implemented empirically testable versions of the original ICAPMover the cross section of stock returns includes Shanken (1990), Cochrane (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), Bali and Engle (2010), Botshekan, Kraeussl, and Lucas(2012), Garret and Priestley (2012), and Maio (2013a, 2013b).

6A list of recent papers includes Gilchrist and Leahy (2002), Rigobon and Sack (2003, 2004), Bernankeand Kuttner (2005), Chen (2007), Balvers and Huang (2009), Bjørnland and Leitemo (2009), Lioui and Maio(2014), and Maio (2014a, 2014b).

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2 The model

2.1 A two-factor model

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

time.7 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). (2)

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, which represents a risk factor.8

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

Et(Ri,t+1)−Rf,t+1 = γ Vart(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, (3)

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. Thus, although the market price of covariance risk is constant over time,

the market price of beta risk is potentially time-varying.

We assume that the conditional betas associated with both factors are (approximately)

constant through time, that is, βi,M,t = βi,M and βi,z,t = βi,z.9 Thus, by applying the law of

7The full derivation is available upon request. Cochrane (2005) presents a similar covariance pricingequation based on a continuous time pricing kernel.

8In the empirical applications conducted in the following sections, we construct the innovation in thestate variable from a first-order auto-regressive process.

9Assuming time-varying betas, which are affine in the lagged state variable, gives rise to an additionalscaled market factor that would increase the model’s fit (see Cochrane (2005) and Lettau and Ludvigson(2001), among others).

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iterated expectations, unconditional expected returns are given by

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

= λMβi,M + λzβi,z, (4)

where λM and λz represent the unconditional risk prices for the market and “hedging”

factors, respectively. This model represents a two-factor Intertemporal CAPM (ICAPM).

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

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

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

against future negative shocks in the expected returns of aggregate wealth, since it offers

high returns when expected returns are also high.10 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.

We use two short-term interest rates—the Fed funds rate (FFR) and the three-month

T-bill rate (TB)—as the state variables that drive investment opportunities (expected stock

market returns) within the ICAPM.11 There is previous evidence in the return predictabil-

ity literature that short-term interest rates forecast (with a negative sign) expected excess

market returns, especially at short horizons (Campbell (1991), Hodrick (1992), Jensen, Mer-

cer, and Johnson (1996), Patelis (1997), Ang and Bekaert (2007), and Maio (2014b), among

others).

Therefore, the two versions of our two-factor model taken to the data in the following

sections are given by

E(Ri,t+1 −Rf,t+1) = λMβi,M + λFFRβi,FFR, (5)

E(Ri,t+1 −Rf,t+1) = λMβi,M + λTBβi,TB, (6)

where λFFR and λTB represent the risk prices for the innovations in FFR and TB, respec-

tively, and βi,FFR and βi,TB denote the respective factor loadings for asset i.

2.2 Alternative factor models

We compare the performance of the two-factor ICAPM with alternative factor models widely

used in the literature. The first model is the baseline CAPM from Sharpe (1964) and Lintner

10In this reasoning, we are assuming that the state variable covaries positively with future investmentopportunities.

11Brennan and Xia (2006) and Nielsen and Vassalou (2006) show that the intercept of the capital marketline, which corresponds to the risk-free rate, represents one valid state variable in the ICAPM.

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(1965), which is nested on our ICAPM:

E(Ri,t+1 −Rf,t+1) = λMβi,M . (7)

The second model is the Fama and French (1993, 1996) three-factor model (FF3, hence-

forth), the most widely used model in the empirical asset pricing literature, which seeks to

offer a risk-based explanation for both the size and value anomalies. 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, (8)

where (λSMB, λHML) denote the 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 third model is the four-factor model from Carhart (1997) (C4), which incorporates

a momentum factor (UMD, up-minus-down short-term past returns) to the FF3 model:

E(Ri,t+1 −Rf,t+1) = λMβi,M + λSMBβi,SMB + λHMLβi,HML + λUMDβi,UMD. (9)

The fourth model is the four-factor model from Pastor and Stambaugh (2003) (PS4),

which adds a stock liquidity factor (LIQ, high-minus-low liquidity) to FF3:

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

Next, we estimate the four-factor model from Hou, Xue, and Zhang (2015) (HXZ4). This

model includes an investment factor (IA, low-minus-high investment-to-assets ratio) and a

profitability factor (ROE, high-minus-low return on equity) in addition to the market and

size (ME) factors:

E(Ri,t+1 −Rf,t+1) = λMβi,M + λMEβi,ME + λIAβi,IA + λROEβi,ROE. (11)

Finally, we use the five-factor model from Fama and French (2015, 2016, FF5), which

augments the FF3 model by an investment (CMA) and a profitability (RMW ) factor:

E(Ri,t+1 −Rf,t+1) = λMβi,M + λSMBβi,SMB + λHMLβi,HML + λCMAβi,CMA + λRMWβi,RMW .

(12)

CMA is constructed in a different way than IA and the same occurs for RMW in relation

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with ROE (see Hou, Xue, and Zhang (2015) and Fama and French (2015) for details).

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

We use the time-series/cross-sectional regression approach presented in Cochrane (2005)

(Chapter 12), which enables us to obtain direct estimates for factor betas and prices of risk.

This method has been employed by Brennan, Wang, and Xia (2004) and Campbell and

Vuolteenaho (2004), among others. Specifically, in the version based on FFR the factor

betas are estimated from the time-series multiple regressions for each test asset,

Ri,t+1 −Rf,t+1 = δi + βi,MRMt+1 + βi,FFRFFRt+1 + εi,t+1, (13)

where RM is the excess market return and FFR stands for the innovation in the Fed funds

rate.

The expected return-beta representation from equation (4) is estimated in a second step

by the OLS cross-sectional regression,

Ri −Rf = λMβi,M + λFFRβi,FFR + αi, (14)

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

12If the factor loadings are based on the whole sample, the risk price estimates from the two-pass regressionapproach are numerically equal to the risk price estimates from Fama and MacBeth (1973) regressions. Thestandard errors of the risk price estimates in the Fama-MacBeth procedure, however, do not take into accountthe estimation error in the factor loadings from the first-step time-series regressions.

13Another reason for not including the intercept in the cross-sectional regressions is that often the marketbetas for equity portfolios are very close to one, creating a multicollinearity problem in the cross-sectionalregression (see Jagannathan and Wang (2007)). Estimating the cross-sectional regression without intercept iscommon in the literature (see Campbell and Vuolteenaho (2004), Cochrane (2005), Yogo (2006), Jagannathanand Wang (2007), Lioui and Maio (2014), among others).

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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), (15)

where K denotes the number of factors (K = 2 in the 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 (15) 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, leads to rejection of a model even 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, which is widely

used in the literature, 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

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

associated with a given model.14

Since the asymptotic theory implicit in the Shanken (1992) standard errors might not

represent a good approximation to the true finite sample distribution in some cases, we

conduct a bootstrap simulation to produce more robust p-values for the tests of individ-

ual significance of the factor risk prices and also for the χ2-test. The bootstrap simulation

consists of 5,000 replications in which the excess portfolio returns and risk factor realiza-

tions are simulated (with replacement from the original sample) independently and without

imposing the ICAPM’s restrictions. Hence, the data-generating process is simulated under

14Since we do not include an intercept in the cross-sectional regression, this metric can assume negativevalues. This means that the factor betas underperform the cross-sectional average risk premium in explainingcross-sectional variation in risk premia. Similar R2 measures are used in Campbell and Vuolteenaho (2004),Yogo (2006), Maio (2013a, 2013b), Lioui and Maio (2014), among others.

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the assumption that the factors are independent from the testing assets (“useless factors”

as in Kan and Zhang (1999)).15 Moreover, the bootstrap accounts for the contemporane-

ous cross-correlation among the testing assets, which often exhibit a small factor structure

(see Lewellen, Nagel, and Shanken (2010)).16 The full details of the bootstrap simulation

algorithm are available in Appendix A.17

Following Lewellen, Nagel, and Shanken (2010) and Adrian, Etula, and Muir (2014), we

evaluate the statistical significance of the point R2 estimates by computing 95% and 90%

empirical confidence intervals for the coefficient of determination in the cross-sectional re-

gressions based on the bootstrap simulation.18 We want to investigate the following question:

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

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

Following Maio (2015) (see also Cochrane (2005) and Lewellen, Nagel, and Shanken

(2010)), we also compute the “constrained” cross-sectional R2,

R2C = 1− VarN(αi,C)

VarN(Ri −Rf ), (16)

which is relevant the alternative multifactor models where all the factors represent excess

stock returns. This metric is similar to R2OLS, but relies on the pricing errors (αi,C) from

a constrained regression that restricts the risk price estimates to be equal to the respective

factor means. For example, in the case of FF3, these pricing errors are obtained from the

following equation,

Ri −Rf = RMβi,M + SMBβi,SMB +HMLβi,HML + αi,C , (17)

where RM , SMB, and HML denote the sample means of the market, size, and value factors,

respectively. We should note that this restriction does not apply to the ICAPM since the

hedging factors do not represent holding-period returns on traded portfolios. In other words,

the risk price estimates for these factors can be different than the respective factor means,

which makes R2OLS the correct metric to assess the explanatory power of our two-factor

15Kan and Zhang (1999), Kleibergen (2009), and Gospodinov, Kan, and Robotti (2014) show that theusual t-ratios tend to overstate the statistical significance of risk price estimates when the factors are useless.

16Campbell and Vuolteenaho (2004) and Lioui and Maio (2014) also conduct a bootstrap simulation inorder to obtain more “robust” standard errors for the risk price estimates in cross-sectional asset pricingtests.

17In unreported results, we compute the t-ratios based on the Fama and MacBeth (1973) method. TheFama-MacBeth t-ratios indicate greater statistical significance for both interest rate risk prices than withthe Shanken’s t-ratios.

18In related work, Kan, Robotti, and Shanken (2013) derive an asymptotical distribution for the cross-sectional R2.

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model.

3.2 Data

Following the bulk of the empirical ICAPM literature (e.g., Cochrane (1996), Campbell

and Vuolteenaho (2004), Petkova (2006), Botshekan, Kraeussl, and Lucas (2012), and Maio

(2013a, 2013b)), the innovations in both ICAPM state variables are obtained from the fol-

lowing AR(1) process,

FFRt+1 = 0.000 + 0.991FFRt, R2 = 0.98,

(0.99)(147.26),

TBt+1 = 0.000 + 0.992TBt, R2 = 0.98,

(0.89)(153.18),

with OLS t-ratios presented in parentheses. The innovation in the Fed funds rate is given by

FFRt+1 ≡ FFRt+1−0.000−0.991FFRt and similarly for TB. The data on the Federal funds

rate and the three-month Treasury-bill rate are from the FRED database (St. Louis Fed).

The data on the risk factors associated with the CAPM, FF3, C4, and FF5 models presented

in the previous section (RM , SMB, HML, UMD, RMW , and CMA) are retrieved from

Kenneth French’s data library. LIQ is retrieved from Robert Stambaugh’s webpage, while

the data on the remaining factors (ME, IA, and ROE) are obtained from Lu Zhang. The

sample period we use is 1972:01 to 2013:12, where the starting date is restricted by the data

availability on some of the factors (e.g., the factors used in Hou, Xue, and Zhang (2015)).

Tables 1 and 2 present descriptive statistics for the factors in the ICAPM, RM , FFR,

and TB. We also present descriptive statistics for the factors associated with the alternative

multifactor models. We can see that the two “hedging” factors are not persistent as indicated

by the autoregressive coefficients of 0.40 and 0.33 for FFR and TB, respectively. Thus, the

innovations in short-term interest rates are significantly less serially correlated than the

original interest rates. Moreover, the interest rate factors are significantly less volatile than

the equity premium. The correlations displayed in Table 2 show that the market factor is

nearly uncorrelated with both interest rate factors, with correlations around -0.14. On the

other hand, the two “hedging” factors are highly correlated, but the degree of comovement

is not excessive (0.78). We can also see that both interest rate factors are uncorrelated with

the alternative risk factors as the correlation coefficients are below 0.10 (in absolute value)

in all cases.

11

In the following sections, we study whether the two-factor ICAPM explains a variety of

CAPM anomalies—value premium, equity duration, long-term reversal in stock returns, cor-

porate investment, and inventory growth. The value premium corresponds to the empirical

evidence showing that value stocks (stocks with a high book-to-market or earnings-to-price

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

earnings-to-price ratio) (see Basu (1983), Rosenberg, Reid, and Lanstein (1985), and Fama

and French (1992), among others). There is strong evidence showing that this spread in

average returns is not explained by the baseline CAPM from Sharpe (1964) and Lintner

(1965) (see Fama and French (1992, 1993, 2006)).

The equity duration anomaly follows the evidence showing that stocks exhibiting lower

duration have higher average returns than high-duration stocks (see Dechow, Sloan, and

Soliman (2004)). The long-term reversal in returns anomaly (De Bondt and Thaler (1985,

1987)) is that stocks with low returns over the last five years have higher subsequent returns,

while past long-term winners have lower future returns. This anomaly should be related to

the value anomaly, as long-term underperformers end up with high book-to-market ratios.

Broadly speaking the investment-based anomalies refer to the evidence showing that

stocks of firms that invest more have lower average returns than the stocks of firms that

invest less (Cooper, Gulen, and Schill (2008), Lyandres, Sun, and Zhang (2008), Fama and

French (2008)). We analyze three investment related anomalies, which refer to different

components of corporate investment: investment-to-assets ratio (e.g., Cooper, Gulen, and

Schill (2008) and Hou, Xue, and Zhang (2015)); changes in property, plant, and equipment

plus changes in inventory scaled by assets (Lyandres, Sun, and Zhang (2008)); and inventory

growth (see Belo and Lin (2011)).

Therefore, the portfolio return data used in the cross-sectional asset pricing tests repre-

sent deciles sorted on the book-to-market ratio (BM); earnings-to-price ratio (EP); equity

duration (DUR); long-term prior returns (REV); investment-to-assets ratio (IA); changes in

property, plant, and equipment plus changes in inventory scaled by assets (PIA); and inven-

tory growth (IVG). We use both value- and equal-weighted portfolios. Using equal-weighted

portfolios enables us to address the evidence showing that small caps represent the biggest

challenge for asset pricing models (see Fama and French (2012, 2015)). All the portfolio re-

turn data are obtained from Lu Zhang. The one-month Treasury bill rate used to construct

portfolio excess returns is obtained from Kenneth French’s data library.

Table 3 presents the descriptive statistics for high-minus-low spreads in returns between

the last and first decile within each value-weighted portfolio group. Most of these anomalies

are economically significant as the average spreads in returns are around or above 0.40% per

month in absolute value. The anomaly showing the largest spread in average returns is BM

12

with an average gap of 0.69%, followed by EP (0.58%). The anomaly with lower average

return is IVG with an average gaps in returns of 0.36% (in magnitude). The pairwise

correlations in Panel B indicate that there is no excessive overlapping among the different

anomalies. The largest correlation (in magnitude) occurs for the spreads associated with

the EP and DUR deciles (0.81), while the two value spreads (based on BM and EP) show

a correlation of 0.67. On the other hand, the correlations among the investment-based

anomalies (IA, PIA, and IVG) are all around 0.50, which suggests that these three anomalies

represent to a large extent different dimensions of cross-sectional equity risk premia.

Table 4 shows the descriptive statistics associated with the spreads high-minus-low based

on the equal-weighted portfolios. With the exception of IVG (with an average gap of -47

basis points), all anomalies have spreads in average returns above 0.50% in magnitude. In

fact, all the seven spreads are clearly above the corresponding magnitudes based on value-

weighted portfolios. The most significant anomalies seem to be BM and IA with average

absolute gaps in returns of 80 and 73 basis points, respectively. These results suggest that

the equal-weighted portfolios represent a bigger challenge to asset pricing models than the

value-weighted portfolios since the spreads in average returns among the extreme deciles are

wider. The pairwise correlations in Panel B indicate that there is no excessive overlapping

within the seven anomalies, similarly to the case of value-weighted portfolios. The largest

correlations occur for the spreads associated with the BM, EP, and DUR deciles, with abso-

lute correlations marginally above 0.80. On the other hand, the spreads corresponding to the

investment anomalies (IA, PIA, and IVG) show correlations around 0.70, which represents

a higher degree of comovement than in the case of the value-weighted portfolios.

4 Main empirical results

In this section, we conduct the cross-sectional test of the two-factor ICAPM and alternative

factor models.

4.1 Testing the ICAPM

As a reference point for the benchmark results associated with the ICAPM, we present the

results for the the cross-sectional tests of the baseline CAPM. The results are presented in

Table 5. We can see that for all value-weighted portfolio groups (Panel A) the OLS R2

estimates assume negative values, varying between -18% (test with the REV deciles) and

-118% (test with IA deciles). This means that the CAPM performs worse than a model that

predicts constant risk premia among the deciles within each portfolio group. Moreover, only

13

in the tests with the BM and REV deciles does the CAPM pass the specification test (at

the 5% level) based on the asymptotic inference. Yet, this formal statistical validation of the

model does not imply any economic significance as suggested by the negative R2 estimates.

In the cross-sectional tests involving the equal-weighted portfolios (Panel B) the fit of the

CAPM is even worse than in the estimation with value-weighted portfolios: the OLS R2

estimates are even more negative, varying between -72% (test with PIA deciles) and values

below -130% (tests with DUR and EP deciles). Therefore, the results from Table 5 confirm

why the characteristics associated with the seven portfolio classes analyzed in this paper are

called market or CAPM anomalies, that is, the CAPM cannot explain the dispersion in risk

premia among each of these portfolio sorts.

The estimation results for the ICAPM applied to the value-weighted portfolios are dis-

played in Table 6. The results for the test with the BM portfolios (Panel A) show that the

ICAPM explains a large fraction of the dispersion in average returns of the value-growth

portfolios, with R2 estimates above 60% in the two versions of the model. In both cases,

these R2 estimates are statistically significant at the 10% level as they lie above the upper

limit of the corresponding 10% bootstrapped confidence interval. The point estimates for

the “hedging” risk prices, λFFR and λTB, are negative and strongly statistically significant

(1% level based on the empirical p-values and 5% level based on the asymptotic p-values).

When we use an alternative measure of value-growth (EP, Panel C), the model’s version

based on FFR shows an increases in fit relative to the test with the BM deciles (from 62%

to 78%), with this R2 estimate being strongly significant (5% level). On the other hand, the

fit of the version based on TB decreases to 40%, thus shoing that the portfolio risk premia

associated with the two measures of value-growth (BM and EP) are far from being strongly

correlated (as already suggested in Table 3). That result also shows that the performance of

the two model’s versions exhibits some discrepancies despite the large correlation of the two

interest rate factors, as shown in the last section. The estimate for λFFR is significant at the

5% level (based on both types of standard errors) while for λTB we have stronger statistical

significance (1% level).

The results for the test with the equity duration deciles (DUR, Panel B) show that the

ICAPM performs marginally better than in the test with the BM deciles. The explanatory

ratios are around 70% and these estimates are largely significant (5 % level). Thus, the

estimated fit of the ICAPM is more statistically significant when the model is tested on

the DUR portfolios than when tested on the BM and EP deciles. The estimates for the

two hedging risk prices are negative and strongly significant (1% based on the empirical

p-values). In the case of the REV portfolios (Panel D) we can see that the ICAPM offers a

good explanatory power, with a coefficient of determination of 52% in both versions of the

14

model (statistically significant in the version based on FFR), which nevertheless represents

a lower fit than in the tests with either BM or DUR. The point estimate for λFFR is negative

and marginally significant based on the Shanken’s t-statistic, although the empirical p-value

points to stronger statistical significance (5% level). On the other hand, the estimate of

λTB is strongly significant (5% or 1% level) according to both the asymptotic and empirical

p-values.

In Panels E and F, we can see that the two-factor model has a high explanatory power for

investment-based portfolios. In the case of the IA portfolios (Panel E), the estimates for the

cross-sectional coefficient of determination are 65% and 56% in the versions corresponding

to FFR and TB, respectively, which represents a higher fit than in the test with the REV

portfolios. Moreover, both of these estimates are statistically significant at the 5% level. In

the estimation with the PIA deciles (Panel F), the explanatory power of the version based on

FFR (59%) is only marginally lower than in the test with the IA deciles. On the other hand,

the specification associated with TB registers a larger drop in fit relative to the test with

IA (from 56% to 39%). Yet, in both versions, the R2 estimates are statistically significant.

The risk price estimates corresponding to both FFR and TB are negative and strongly

significant (at the 5% or or 1% level) when it comes to price both of these portfolio groups.

The cross-sectional test including the IVG deciles as test assets registers the lowest fit

for the ICAPM among all seven portfolio groups. The R2 estimates are 20% and 13% in the

versions based on FFR and TB, respectively, and these point estimates are not statistically

significant at the 10% level. Despite this fact, the interest rate factors remain priced, with

risk prices estimates that are significant at the 5% level based on the empirical p-values.

Furthermore, across all the seven cross-sectional tests, one fact remains robust: both versions

of the model pass the χ2 test with both asymptotic and empirical p-values clearly above 5%,

that is, the model is formally validated in statistical terms.

We conduct an additional cross-sectional test of the ICAPM that includes all 70 equity

portfolios simultaneously. This test is significantly more demanding than the previous tests

since we force the model to explain all anomalies jointly. This is especially relevant in the

case here since some of the interest rate factor risk price estimates differ significantly in

magnitude across the testing portfolios. For example, the estimates for λFFR vary between

-0.44% (test with IVG deciles) and -0.82% (test with IA). The different risk price estimates

across different portfolio groups arises from the fact that several of these market anomalies

are not significantly correlated as shown in the last section.

The results in Panel H indicate that the R2 estimate in the version based on FFR is 58%,

which is well above the corresponding explanatory ratio obtained for the baseline CAPM, as

shown in Table 5 (-59%). The fit of the version based on TB is only slightly lower (48%),

15

and in both versions of the ICAPM the point estimates of the explanatory ratio are strongly

significant since they lie above the upper limit of the corresponding 95% empirical confidence

interval. Moreover, the model is not formally rejected by the χ2 statistic as indicated by the

asymptotic p-values quite close to one. Following the evidence for the individual anomaly

tests, the estimates for both interest rate risk prices are negative and strongly significant

(1% level). Thus, by using a larger cross-section of testing assets we obtain higher statistical

power in the estimates obtained from the cross-sectional regressions.

The results for the cross-sectional tests involving the equal-weighted portfolios are dis-

played in Table 7. We can see that when it comes to price the equal-weighted portfolios

the fit of the ICAPM is larger than in the case of the value-weighted portfolios. Indeed, for

both versions of the model and across all the seven portfolio classes it turns out that the R2

estimates are greater than the corresponding values in the estimation with value-weighted

deciles. Specifically, in the version based on FFR the explanatory ratios vary between 61%

(test with PIA) and 94% (DUR), whereas the range of these estimates for the version based

on TB is between 66% (test with REV) to 86% (DUR). The most notable improvement

against the test with value-weighted portfolios shows up in the case of the IVG deciles as

indicated by the explanatory ratios around 80%, compared to values around or below 20% in

the benchmark test. This shows that the behavior of market anomalies, and the performance

of factor models in explaining them, can vary widely among value- and equal-weighted port-

folios, that is, size can play an important role within these anomalies (see Fama and French

(2008)). In nearly all cases, the R2 estimates are statistically significant, the exception being

the tests with the BM deciles.19 Moreover, both versions of the model pass the specification

test as shown by the p-values clearly above 5% in all cases.

In the more challenging test including all the seven anomalies, we obtain a fit as large as

67% in both versions of the model, which is significant at the 5% level. This confirms that the

performance of the two versions of the model is relatively similar when it comes to explain

the equal-weighted portfolios, thus signaling a relevant improvement in fit for the version

based on the T-bill rate relative to the benchmark test with value-weighted deciles. The risk

price estimates for the hedging factor are negative and strongly significant (1% level) in both

versions, similarly to the benchmark test. Overall, the evidence from Table 7 indicates that

the fit of the ICAPM for the cross-section of equal-weighted portfolios is even stronger than

in the benchmark tests with value-weighted portfolios. This result is remarkable given that

these anomalies are more accentuated among small stocks, thus imposing a bigger challenge

19The existence of wide confidence intervals for the cross-sectional R2 in the tests with BM deciles confirmsthe evidence in Lewellen, Nagel, and Shanken (2010) and Kan, Robotti, and Shanken (2013) that there isconsiderable sampling error associated with this statistic for cross-sectional tests that rely on BM portfolios.

16

on asset pricing models.

In sum, the results from this subsection show that the ICAPM is successful in pricing the

seven CAPM anomalies considered in this paper. Second, the key factor that drives the fit

of the model seems to be the innovation in short-term interest rates, rather than the market

factor.20

4.2 Individual pricing errors

Although the cross-sectional coefficient of determination represents a measure of the overall

fit of the ICAPM it remains important to assess the relative explanatory power of the

model over the different portfolios within a certain group (e.g., value versus growth, or

low-investment versus high-investment stocks).

Figure 1 plots the pricing errors (and respective t-statistics) associated with the BM,

DUR, EP, and REV value-weighted portfolios for the version based on FFR.21 We can see

that the magnitudes of the pricing errors associated with these four groups are quite small,

and none of these errors are statistically significant at the 10% level. Most pricing errors

fall below 0.15% in magnitude, the few exceptions being the third (pricing error of -0.17%)

and sixth (0.16%) BM deciles and the eighth return reversal (-0.18%) decile. Yet, this level

of mispricing is significantly below the cross-section mean (among all the deciles in a group)

risk premium of 0.67% and 0.70% per month for the BM and REV groups, respectively.

Figure 2, which is similar to Figure 1, provides a visual representation of the model’s

fit in cross-sectional tests with the IA, PIA, and IVG portfolios. Similarly to the first four

anomalies, most of the deciles associated with these three sorts have pricing errors that are

both economically and statistically insignificant. Only for the first decile within the IVG

group do we have statistical significance at the 10% level (t-ratio of 1.68). Most magnitudes

of the pricing errors are below 0.15%, the exceptions being the first two IVG deciles, with

pricing errors in the range of 15 to 19 basis points. Yet, this level of mispricing is way below

the average risk premium within the IVG deciles (0.59%). Both Figures 1 and 2 also show

that the pricing errors associated with most portfolio groups have a non-monotonic pattern,

in contrast with the raw average returns, thus confirming the large fit of the ICAPM. The

exception are the IVG deciles with positive (negative) pricing errors for the first (last) deciles,

which is in line with the evidence above showing that the fit of the ICAPM for these portfolios

is significantly lower than for the remaining six portfolio classes.

20Brennan, Wang, and Xia (2004) and Petkova (2006) price 25 size-BM portfolios with multifactor modelsthat contain the innovation in short-term interest rates as one of the factors. However, it is not clear intheir models what is the contribution of the interest rate factor to drive the explanatory power for theseportfolios.

21The results for the version based on TB are qualitatively similar and are available upon request.

17

Figures 3 and 4 plot the pricing errors associated with the equal-weighted portfolios.

The pattern is approximately similar to the benchmark case with value-weighted deciles.

Among the 70 portfolios only two portfolios have statistically significant pricing errors at

the 10% level—the fourth BM decile and the last REV decile with t-ratios of -1.70 and -1.74,

respectively. There are five portfolios with absolute errors above 0.15%—fourth (-0.20%) and

last (0.21%) BM deciles, first IA decile (0.20%), and third (-0.19%) and last (-0.19%) PIA

deciles. Yet, with the exception of the fourth BM decile already mentioned, none of these

estimates are significant at the 10% level. Within all seven portfolio groups, and specifically

in the IVG group, there is no monotonic pattern in the pricing errors with respect to rank,

which is in line with the large fit of the model in terms of explaining the equal-weighted

portfolios compared to the corresponding tests based on value-weighted deciles.

4.3 Which factors explain the anomalies?

The results above suggest that the innovation in either the Fed funds rate or the T-bill

rate drives the fit of the ICAPM for pricing each of the seven market anomalies. To see

more clearly which factors drive the explanatory power of the 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 factor risk premium (beta times risk

price) for each factor and for both the first and last deciles within each portfolio group. For

example, the market risk premium associated with the first BM decile is given by

λMβ1,M ,

and similarly for the other interest rate factors.

The results for this accounting decomposition for the version based on FFR are shown in

Table 8.22 In Panel A, the spread in average excess returns between the first (D1, growth) and

the last BM decile (D10, value) is -0.69% per month, which corresponds to the (symmetric of

the) value premium in our sample. The corresponding spread associated with the EP deciles

(Panel C) is slightly smaller (-0.58%). Each of these gaps must be (partially) matched by

the risk premium associated with one or more of the factors in the ICAPM, as shown in the

respective beta pricing equation (4), if this model is able to price the value premium. The

spread D1−D10 in the market risk premium is 0.08% and 0.14% in the tests with BM and

EP, respectively. In other words, the spread associated with the market factor has the wrong

sign to drive the value premium, which confirms why the baseline CAPM is not successful

in pricing the value anomaly in our sample. Thus, it is the innovation in the Fed funds

22The results for the version based on TB are similar and are available upon request.

18

rate, with a spread in the respective risk premium of -0.59% per month, that accounts for

the BM premium of -0.69%. Only -0.18%, of the original gap of -0.69%, is left unexplained

by the two-factor ICAPM. In the test with the EP deciles, the fit is even higher as the

risk premium associated with FFR exactly matches the original return spread of -0.58%,

originating an average gap in mispricing of only -0.13%. Thus, value stocks covary negatively

with innovations in the Fed funds rate, which has a negative risk price.

In the case of the DUR portfolios (Panel B), the gap D1−D10 in average excess returns

(low equity duration stocks minus high duration stocks) is about 0.52% per month, which

corresponds to the size of the equity duration anomaly in our sample. This premium is

somewhat lower than the BM premium reported above (0.69%). The risk premium gap

(D1 − D10) associated with the market factor goes again in the wrong direction (-0.11%

per month), thus justifying why the CAPM fails in explaining these portfolios. The hedging

factor is the key factor that prices the duration anomaly with a gap in risk premia of 0.48%

per month that nearly matches the original return premium. Of the original 0.52% spread

in returns, only 0.16% is not explained by the model. The behavior of the ICAPM in pricing

the REV deciles (Panel D) is similar to the duration portfolios. Of the original spread in

average excess returns (past long-term losers minus past winners) of 0.41%, it turns out that

0.25% is matched by the risk premium of the interest rate factor while the gap in the market

risk premium is around zero (-0.01%). The resulting spread in average pricing error is 0.17%,

which is about the same magnitude of the error spread associated with the DUR deciles.

For the IA portfolios (Panel E) we have a return spread (D1−D10) of 0.42% per month,

which represents the same size of the long-term reversal return spread. The spreads in risk

premia associated with the market factor has the wrong sign (-0.11%), which justifies the

poor performance of the CAPM for these portfolios. In contrast, the gap in risk premium

for the innovation in the Fed funds rate is 0.38%, which almost equals the original spread in

risk premia. Consequently, the gap (D1 − D10) in pricing error is only 0.15% per month.

In the case of the PIA portfolios (Panel F), the gap (D1−D10) in average excess returns is

marginally higher than in the case of the IA deciles, at 0.49% per month. As in the case of IA

the hedging factor is the key driving force in the model with a gap in risk premium of 0.34%.

This results in an average mispricing of 0.23%, less than half the raw spread in returns.

In the case of the IVG portfolios (Panel G), the contribution of the interest rate factor is

the smallest among all the seven anomalies, with a spread in risk premia corresponding to

this factor of only 0.11%. This justifies the lower explanatory power of the model for these

deciles: out of the original spread in risk premia of 0.36% per month, 0.33% is still left

unexplained by the ICAPM.

The results for the equal-weighted deciles (Panels H to N) are qualitatively similar to

19

those based on the value-weighted portfolios. As before, it turns out that the market factor

leads the two-factor model further away in explaining any of these seven anomalies as the gaps

(D1−D10) in the market risk premium always have the wrong sign. Thus, the innovation

in FFR is the key factor that drives the explanatory power of the model with gaps in risk

premium that are close to the original spreads in returns, thus producing gaps (D1−D10) in

pricing errors with low magnitudes. Specifically, in the test with the IVG deciles the spread in

risk premium for the hedging factor is 0.41%. Such estimate almost matches the raw spread

in returns of 0.47%, thus leading to a spread in pricing errors of only 0.14% per month. This

explains why the ICAPM performs much better in pricing the equal-weighted IVG deciles

compared to the value-weighted counterparts. Overall, the results of this subsection confirm

the evidence above that the driving force of the ICAPM is the interest rate factor.

4.4 Factor betas and intuition

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

factor in the ICAPM responsible for pricing the seven anomalies considered in this paper.

Put differently, there is a dispersion in the betas associated with the hedging factor within

each of the seven portfolio groups that fits the related anomalies.

The multiple-regression betas associated with the innovation in the Fed funds rate for the

value-weighted portfolios are displayed in Figure 5. In Panels A and C, we can see that value

stocks (stocks with a high book-to-market or earnings-to-price ratio) have negative interest

rate betas while growth stocks have positive betas for this same factor. This dispersion in

betas scaled by the negative risk price for FFR generates a positive spread in risk premia

between value and growth stocks.

In the case of the equity duration portfolios (Panel B), it turns out that low duration

stocks have negative betas associated with the innovation in the Fed funds rate (similarly to

value stocks), while high duration stocks have positive loadings. This spread in betas scaled

by the interest rate risk price generates the risk premium necessary to explain the equity

duration return spread. A similar pattern holds for the return reversal deciles (Panel D),

with past long-term losers having negative interest rate betas and past long-term winners

exhibiting positive loadings.

Regarding the IA and PIA portfolios (Panels E and F), we have a similar pattern to the

duration and return reversal anomalies: the low deciles have negative betas associated with

the hedging factor while the top deciles have positive betas. This spread in betas scaled by

the negative price of risk for the interest rate factor explains why stocks of firms that invest

less (low IA or PIA ratios) earn higher average returns than stocks of firms that invest more.

20

In the case of the IVG deciles (Panel G), the pattern in betas is not as clear as for the IA and

PIA deciles since both the extreme first and last deciles have positive interest rate loadings.

Yet, the beta estimate for the last decile is three times as large as for the first decile, thus

generating a spread in risk premium in the right direction.

Figure 6, which is similar to Figure 5, displays the loadings for the innovation in the

Fed funds rate in the case of the equal-weighted portfolios. The results are even more clear

than in the case of the value-weighted portfolios as the pattern in betas when we move from

the first to the last deciles exhibits a more approximate monotonic behavior among each of

the seven classes. In particular, in the case of the IVG deciles (Panel G) we have negative

loadings for the first deciles and a positive beta estimate for the last decile. This generates

a higher spread in the risk premium attached to the hedging factor than in the benchmark

case, thus justifying why the fit of the ICAPM for these portfolios is significantly larger than

for the corresponding value-weighted deciles.

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 (1989, 1990, 1995), Bernanke, Gertler,

and Gilchrist (1994), among others), 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.23

Why are stocks with higher duration less sensitive to rises in short-term interest rates?

One possible explanation is that these high duration stocks act to some degree like growth

stocks, which have significant growth options and few assets in place. Hence, their discount

rates are more sensitive to fluctuation in long-run risk premia that reflect changes in the

riskiness of their distant cash flows, and thus their current prices (returns) are less sensitive

to short-term interest rates. On the other hand, many of the the low duration stocks are

“cash cows” with stable earnings streams but few growth opportunities. This makes them

acting more like value stocks, whose current prices are more sensitive to rises in short-term

23These results are consistent with the evidence in Maio (2013a) showing that the more negative interestrate betas of value stocks relative to growth stocks are a result of a more negative effect (of a Fed fundsrate rate increase) into the cash flows of value stocks compared to growth stocks, while the impact in futurediscount rates is less important.

21

interest rates, and less subjective to changes in long-term discount rates.24

Why do past long-term losers have greater interest rate 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, caused specifically by 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. Regarding the investment anomaly, firms that

face higher financing constraints are likely to invest less, much like past long-term losers, and

thus are more sensitive to changes in short-term interest rates. Therefore, these firms earn

higher interest rate risk (and hence, larger risk premia) than firms with higher investment

growth.

4.5 Alternative multifactor models

We compare the performance of the ICAPM against the alternative multifactor models

presented in Section 2. The results are presented in Table 9. To save space, we only report

the results for the cross-sectional test including all 70 portfolios simultaneously. Starting with

the estimation containing the value-weighted portfolios (Panel A), we can see that all five

models seem to deliver a large explanatory power for the seven CAPM anomalies as judged

by the R2OLS estimates around 70% in all cases. Yet, for the HXZ4 and FF5 models this

large fit is partially spurious as it comes at the cost of implausible risk price estimates, that

is, estimated risk prices that are significantly different than the corresponding factor means

reported in Table 1. In fact, both λSMB and λRMW within FF5 and λROE in HXZ4 have

negative estimates, which are far away from the correct estimates between 0.20% (SMB)

and 0.57% (ROE) presented in Table 1. Consequently, the R2C estimates of 30% and 52%

(for HXZ4 and FF5, respectively) are fairly below the corresponding OLS estimates of 68%

and 74%. This shows that the correct metric to evaluate multifactor models in which all

the factors represent excess stock returns is R2C , instead of R2

OLS, in line with the evidence

presented in Maio (2015).

By comparing the R2OLS values corresponding to the ICAPM against the constrained

R2 of the alternative factor models it turns out that the ICAPM version based on FFR

compares quite favorably with both HXZ4 and FF5 models, the new workhorses in the

empirical asset pricing literature (see Fama and French (2015) and Hou, Xue, and Zhang

24This mechanism is consistent with the analysis of Lettau and Wachter (2007) who show that the prices(and realized returns) of value and low duration stocks stocks are more sensitive to shocks in near-termcash flows, while the prices of growth and high duration stocks are more related to shocks to discount rates(long-term expected returns).

22

(2015)). On the other hand, the fit of the ICAPM is only marginally lower than both FF3

and PS4 models (58% versus 67% and 65% for such models), yet, the liquidity model is

rejected by the specification test (based on the asymptotic inference). Overall, the model

with the largest explanatory power for the joint 70 portfolios is the C4 model with an R2C of

75%.

The results for the estimation with the equal-weighted portfolios (Panel B) indicate that

all five models are clearly rejected by the χ2 statistic, as indicated by the asymptotic p-values

of zero. The fit of FF3, C4, and PS4 is lower than in the estimation with value-weighted

portfolios (especially for FF3 and PS4) with R2C estimates that vary between 41% (FF3)

and 69% (C4). This is in part explained by the fact that HML—the factor that drives

the explanatory power of these models for some of the testing assets (e.g., BM and EP)—is

constructed from value-weighted BM portfolios, and thus the fit for equal-weighted portfolios

is not as large as for the value-weighted counterparts. On the other hand, the performance

of both HXZ4 (80%) and FF5 (77%) rises significantly in the tests with equal-weighted

portfolios in comparison to the benchmark case. When we compare the performance of our

two-factor model against the alternative models it follows that both versions of the ICAPM

clearly outperform the FF3 and PS4 models (67% versus values marginally above 40%) and

have basically the same performance as the Carhart’s model. The ICAPM only looses against

both HXZ4 and FF5, but the difference in the explanatory ratios is relatively limited (around

10% in both cases).

Overall, the results of this subsection show that the performance of the two-factor ICAPM

is quite satisfactory in comparison with the alternative multifactor models widely used in the

literature. We should note that some of the factors in the alternative models are designed

in such a way to price (almost) mechanically the testing portfolios (see Nagel (2013) and

Maio (2015) for a related discussion). The reason is that these factors are constructed from

portfolios sorted on the same characteristics as the testing assets. This is the case of HML

with regards to the BM portfolios and the cases of both IA and CMA with regards to

the IA deciles. Thus, the fact that our simple model can outperform multifactor models

containing these factors in terms of explaining these CAPM anomalies seems remarkable.

Additionally, our model is more parsimonious since only one factor—the innovation in a

short-term interest rate—helps explaining several different anomalies. In comparison, in the

alternative multifactor models several factors drive the explanatory power (for example, in

the case of FF5 the HML factor drives value-based anomalies while the CMA factor helps

explaining the investment-based anomalies).

23

4.6 Sensitivity analysis

We conduct some robustness checks to the main results discussed above. The results are

displayed, and discussed in more detail, in the internet appendix. First, we estimate both

versions of the ICAPM by using new interest rate factors. Following Hahn and Lee (2006)

and Maio and Santa-Clara (2012), the innovation in the Fed funds rate corresponds to the

first-difference on this variable, FFRt+1 ≡ FFRt+1 − FFRt, and similarly for TB. The

results show that both the explanatory ratios and risk price estimates are very similar to the

corresponding values in the benchmark specification, and this holds for both versions of the

model and for the tests with both value- and equal-weighted portfolios.

Second, we estimate the ICAPM for a subsample that ends in 2006:12. The goal is to

evaluate the impact of the recent financial crisis on the fit of the ICAPM given the large

spike in stock market volatility observed during the 2007–2009 period. Overall, the fit of the

ICAPM is larger in the restricted sample than in the full sample, which suggests that the

financial crisis has had a negative effect on the fit of the ICAPM.

Third, we estimate the ICAPM with portfolios related with two additional anomalies.

We employ deciles sorted on cash-flow-to-price ratio (CFP, Lakonishok, Shleifer, and Vishny

(1994)) and investment growth (IG, Xing (2008)). The results indicate that the ICAPM has

strong explanatory power for the equal-weighted portfolios associated with the CFP and IG

anomalies. However, our model is not successful in explaining the price momentum (e.g.,

Jegadeesh and Titman (1993) and Fama and French (1996)) and profitability anomalies (e.g.,

Haugen and Baker (1996) and Novy-Marx (2013)). The reason is that there is not enough

dispersion in the interest rate betas with the right sign among those portfolios. Specifically,

past short-term winners have more positive interest rate loadings than past losers, which

interacted with the negative interest rate price of risk, generates a spread in risk premia in

the wrong direction to match the raw momentum profits. In any case, a single factor like

the innovation in FFR or TB (the role of the market factor for explaining cross-sectional

dispersion in equity risk premia is residual, as shown above) can’t have a large explanatory

power for a large number of anomalies. This stems from the very low correlation among

many of these patterns in cross-sectional average returns (see Fama and French (2015) and

Hou, Xue, and Zhang (2015) and also the evidence in the previous section).

5 Consistency with the ICAPM

In this section, we assess more formally the consistency of our two-factor model with the

the ICAPM framework of Merton (1973). Following Maio and Santa-Clara (2012), if a

24

state variable forecasts a decline in future aggregate financial wealth, the asset’s covariance

with its innovation should earn a negative risk premium in the cross-section of stocks. The

intuition is that if a given asset is positively correlated (without any loss of generality) with

a state variable that forecasts a decline in the expected stock market return, it pays well

when the expected aggregate wealth is lower. Therefore, this asset provides a hedge against

negative changes in future wealth for a multi-period risk-averse investor and thus should earn

a negative risk premium (which in turn implies a negative risk price given the assumption

of a positive covariance with the innovation in the state variable).

The results in the last section show that the risk price for both interest rate factors

are consistently negative. Thus, to achieve consistency with the ICAPM it turns out that

the corresponding state variable (Fed funds rate or T-bill rate) should forecast a decline in

future aggregate wealth. To test whether FFR and TB forecast excess market returns at

multiple horizons, we conduct monthly long-horizon single predictive regressions (Keim and

Stambaugh (1986), Campbell (1987), Fama and French (1988, 1989)),

ret+1,t+q = aq + bqzt + ut+1,t+q, (18)

where ret+1,t+q ≡ ret+1 + ...+ ret+q is the continuously compounded excess market return over q

periods into the future (from t+ 1 to t+ q), and re is the excess log market return. z stands

for the relevant state variable (FFR or TB) that forecasts the equity premium. The proxy

for the market return is the value-weighted CRSP return, and to compute excess log returns

we subtract the log of the one-month T-bill rate. We use forecasting horizons of 1, 3, 6,

9, 12, 24, and 36 months ahead. The statistical significance of the regression coefficients is

assessed by using Newey and West (1987) asymptotic t-statistics with q lags to account for

the serial correlation in the regression residuals that stems from using overlapping returns.

As a robustness check, we also report Hodrick (1992) t-ratios, which introduce a correction

for the overlapping pattern in the residuals.

Given the Roll’s critique (Roll (1977)), we also investigate whether short-term interest

rates forecast a decline in future economic activity. Since the stock index is an imperfect

proxy for aggregate wealth, it is likely that changes in the future return on the unobservable

wealth portfolio might be related with future economic activity. Specifically, several forms

of non-financial wealth, like labor income, houses, or small businesses, are related with the

business cycle, and hence, economic activity.25

We use the log growth in the industrial production index (IPG) and the log growth

25In related work, Boons (2014) evaluates the consistency of an alternative ICAPM specification (includingthe term spread, default spread, and dividend yield) with the ICAPM, where investment opportunities aremeasured by economic activity.

25

in aggregate earnings (∆e) as the proxies for economic activity. The data on industrial

production are obtained from the St. Louis FED, whereas the level of earnings associated

with the S&P index are retrieved from Robert Shiller’s webpage. We run the following

univariate regressions to forecast economic activity

yt+1,t+q = aq + bqzt + ut+1,t+q, (19)

where y ≡ IPG,∆e and yt+1,t+q ≡ yt+1 + ... + yt+q denotes the forward cumulative sum in

either IPG or ∆e.

The results for the forecasting regressions are presented in Table 10. We can see that

both short-term interest rates forecast a decline in the equity premium at all horizons. Yet,

the associated slopes are not statistically significant at any horizon, and the R2 assume tiny

values. These results are partially at odds with previous evidence showing that the level of

short-term interest rates is a significant predictor of the equity premium at short horizons

(e.g., Patelis (1997) and Ang and Bekaert (2007)), suggesting that the forecasting power of

these variables has declined in recent years.26

The results for the predictive regressions associated with industrial production growth

(Panel B) indicate significantly stronger forecasting power in comparison with the equity

premium. As in the case of the market return the slopes are negative at all horizons. Yet,

in this case we find strong statistical significance as the coefficients associated with FFR

are significant at the 5% or 1% level (based on both types of t-ratios) at all horizons, except

q = 36. On the other hand, the slopes associated with TB are significant at the 10% or 5%

level, for horizons between three and 24 months. The largest forecasting power is achieved at

q = 24, with R2 estimates of 8% and 6% in the regressions with FFR and TB, respectively.

In the regressions associated with future earnings growth (Panel C), we can see that the

coefficients associated with both FFR and TB are also negative at all forecasting horizons.

These estimates tend to be significant (based on the Hodrick t-ratios) for horizons beyond

six months, and specifically at longer horizons (q > 12) we have strong significance based on

both types of standard errors. The largest fit is achieved at longer horizons with explanatory

ratios above 10% in the regressions with either FFR or TB.27

In sum, the results of this section show that the negative risk price estimates associated

with both interest rate factors are consistent with the ICAPM, when future investment

26By conducting the predictive regressions for the 1972:01–2000:12 period, we find that the slopes as-sociated with FFR and TB at short-horizons (one and three months) are statistically significant. Maio(2014b) shows that the change in FFR (instead of its level) is a robust and significant predictor of theequity premium at short horizons.

27The slopes associated with short-term interest rates remain significant in most cases when we add thecurrent values of IPG or ∆e as predictors.

26

opportunities are measured by economic activity in addition to the return on the equity

index.

6 Comparison with alternative ICAPM specifications

We compare the performance of the ICAPM with alternative two-factor models that can also

be interpreted as empirical applications of the Merton’s ICAPM. That is, the risk factors

(other than the market factor) represent variables that are frequently used to forecast stock

market returns in the predictability literature.

The alternative factors are the innovations on the term spread (TERM), default spread

(DEF ), log market dividend yield (dp), log aggregate price-earnings ratio (pe), value spread

(vs), and stock market variance (SV AR). Several ICAPM applications have used innovations

in these state variables as risk factors to price cross-sectional risk premia (e.g., Campbell and

Vuolteenaho (2004), Hahn and Lee (2006), Petkova (2006), Maio (2013a, 2013b), Campbell

et al. (2014), among others).

TERM represents the yield spread between the ten-year and the one-year Treasury

bonds, and DEF is the yield spread between BAA and AAA corporate bonds from Moody’s.

The bond yield data are available from the St. Louis Fed Web page. dp is computed as the

log ratio of annual dividends to the level of the S&P 500 index. pe denotes the log price-

earnings ratio associated with the same index, where the earnings measure is based on a

10-year moving average of annual earnings. The data on the price, dividends, and earnings

are retrieved from Robert Shiller’s website. As in Campbell and Vuolteenaho (2004), vs

represents the difference in the log book-to-market ratios of small-value and small-growth

portfolios, where the book-to-market data are from French’s data library. SV AR is the

realized stock market volatility, which is retrieved from Amit Goyal’s webpage. As in our

benchmark ICAPM, the innovations in the alternative state variables are constructed from

an AR(1) process.

The results for the alternative ICAPM specifications are displayed in Table 11. To save

space, we only report the results for the augmented cross-sectional test including all 70

portfolios simultaneously. We can see that the performance of the alternative two-factor

models is rather weak as the OLS R2 estimates are negative for most portfolio groups, and

this result holds for both value- and equal-weighted portfolios. Thus, the alternative factor

models do not outperform the baseline CAPM when it comes to explain the seven joint

anomalies. The few exceptions are the models based on TERM and vs, in which cases the

explanatory ratios are positive. Yet, the fit of the model based on TERM is significantly

lower than our benchmark ICAPM as indicated by the R2 estimates around 30% (for both

27

value- and equal-weighted deciles), and only in the test with value-weighted deciles are these

estimates statistically significant.

The ICAPM based on vs is by far the best performing model among the alternative

ICAPM specifications, with R2 of 66% and 69% in the tests with value- and equal-weighted

portfolios, respectively, and both of these estimates are significant. This represents basically

the same fit associated with our benchmark ICAPM in the estimation with equal-weighted

portfolios, while representing a marginally larger fit in the test with value-weighted portfolios.

Yet, we must stress that the explanatory power of the value spread for some of the portfolios

(BM and EP) is nearly mechanical, exactly in the same way as the role played by HML.28

We also estimate the following augmented ICAPM specification,

E(Ri,t+1 −Rf,t+1) = λMβi,M + λzβi,z + λTERMβi,TERM + λDEFβi,DEF

+λdpβi,dp + λpeβi,pe + λvsβi,vs + λSV ARβi,SV AR, (20)

where z ≡ FFR, TB, and λTERM , λDEF , λdp, λpe, λvs, and λSV AR denote the risk price esti-

mates for the innovations on the term spread, default spread, log dividend yield, smoothed

log price-to-earnings ratio, value spread, and stock market variance, respectively. The objec-

tive of testing this large-scale model is to check whether the interest rate risk prices remain

significant in the presence of all the alternative ICAPM factors. That is, we want to assess

whether interest rate risk is subsumed by the alternative risk factors frequently employed in

the ICAPM alternative.

We estimate the model above for the joint seven market anomalies by using both value-

and equal-weighted portfolios. Results reported in the internet appendix show that the risk

price estimates for either FFR or TB remain strongly significant (at the 5% or 1% levels)

when we add the alternative ICAPM factors. Moreover, the risk price estimates for the

alternative factors are not significant at the 5% level in most cases. The exceptions are

the risk prices associated with TERM (estimation with value-weighted portfolios), DEF

(estimation with equal-weighted portfolios), and vs (for both value- and equal-weighted

portfolios). However, as referred above, the explanatory power of vs for several of these

portfolios is somewhat mechanical.

We can also see that the explanatory ratios of the augmented model are not dramatically

higher than the corresponding estimates for our benchmark two-factor ICAPM, particularly

the version based on FFR. This means that we don’t loose much by excluding these other

factors from our model, while enjoying the benefits of a much more parsimonious specifica-

28The reason is that this spread represents the difference between the log BM ratio of small value andsmall growth stocks, which is highly correlated with the corresponding spread in average returns due to adynamic accounting decomposition (see Cohen, Polk, and Vuolteenaho (2003)).

28

tion. Overall, the results of this section show that our two-factor ICAPM tends to outperform

the alternative ICAPM specifications when it comes to price the seven market anomalies.

Moreover, the interest rate factors are not subsumed by the alternative ICAPM risk factors.

7 Conclusion

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

CAPM anomalies—the value premium, long-term reversal in returns, equity duration, the

corporate investment anomaly, and the inventory growth anomaly. We specify a two-factor

ICAPM containing the market equity premium and the “hedging” or intertemporal factor,

which represents the innovation in a macroeconomic state variable—the Federal funds rate

(FFR) or T-bill rate (TB).

We test our two-factor model with decile portfolios sorted on the book-to-market ratio;

earnings-to-price ratio; equity duration; long-term prior returns; firms’ investment-to-assets

ratio; changes in property, plant, and equipment plus changes in inventory scaled by assets;

and inventory growth. The cross-sectional tests show that the ICAPM explains a large

percentage of the dispersion in average equity premia of the seven portfolio groups, with

explanatory ratios that are in most cases around or above 40% in the tests based on value-

weighted portfolios and above 60% in the estimation with equal-weighted portfolios. When

the model is forced to price all 70 portfolios simultaneously, and thus the joint seven CAPM

anomalies, we obtain cross-sectional R2 estimates of 58% and 48% in the versions based on

FFR and TB, respectively. In the augmented test based on equal-weighted portfolios the fit

of the ICAPM is even larger as indicated by the explanatory ratios of 67% in both versions

of the model.

The ICAPM compares favorably with alternative multifactor models widely used in the

literature like the three-factor model from Fama and French (1993), the four-factor models

proposed by Carhart (1997), Pastor and Stambaugh (2003), and Hou, Xue, and Zhang

(2015), or the recent five-factor model from Fama and French (2015) when it comes to explain

these seven market anomalies. Specifically, the ICAPM outperforms the models from Hou,

Xue, and Zhang (2015) and Fama and French (2015) in the estimation with value-weighted

portfolios and the models from Fama and French (1993) and Pastor and Stambaugh (2003)

when tested on the equal-weighted deciles. This is remarkable since the factors in our model

(other than the market factor) are associated with a single variable from outside the equity

market—the Fed funds rate or the T-bill rate. Thus, our state variable is not a priori

mechanically related to the test portfolios, as is the case with the equity-based factors in

the alternative models. Our model also outperforms other factor models that rely on macro

29

variables (mainly factors retrieved from the equity premium predictability literature like the

term spread, default spread, or market dividend yield) and that can also be interpreted as

applications of the ICAPM.

The interest rate risk factor explains the dispersion in risk premia across the seven port-

folio classes enumerated above. Thus, according to our model, value stocks, past long-term

losers, stocks with low duration, stocks of firms that invest less, and firms that build lower

inventories enjoy higher expected returns than growth stocks, past long-term winners, high-

duration stocks, firms that invest more, and firms that build higher inventories, respectively.

The reason is that the former stocks have more exposure to changes in the state variable; that

is, they have more negative loadings on the interest rate factor. One possible explanation

for these loadings is that many of these value, past loser, low-duration, and low-investment

(low-inventory) firms, have a poor financial position and expectations of modest growth in

future cash flows, and thus are more sensitive to rises in short-term interest rates that further

constrain their access to external finance and the investment in profitable projects that could

enhance the firm value.

30

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36

A Bootstrap simulation

The bootstrap algorithm associated with the cross-sectional regression consists of the follow-ing steps:

1. For each empirical test, we estimate the time-series regressions to obtain the factorloadings,

Ri,t+1 −Rf,t+1 = δi + βi,MRMt+1 + βi,FFRFFRt+1 + εi,t+1,

and in a second step, the expected return-beta representation is estimated by an OLScross-sectional regression,

Ri −Rf = λMβi,M + λFFRβi,FFR + αi.

We compute and save both the t-statistics associated with the risk price estimates and

also the χ2 statistic, both based on Shanken (1992) standard errors,[t(λM), t(λFFR),χ2

].

2. In each replication b = 1, ..., 5000, we construct a pseudo-sample of excess returns foreach testing asset (of size T ) by drawing with replacement:

{(Ri,t+1 −Rf,t+1)b, t = sb1, sb2, ..., s

bT}, i = 1, ..., N,

where the time indices sb1, sb2, ..., s

bT are created randomly from the original time se-

quence 1, ..., T . Notice that all excess returns have the same time sequence in order topreserve the contemporaneous cross-correlation between asset returns.

3. For each replication b = 1, ..., 5000, we construct an independent pseudo-sample of therisk factors:

{RM bt+1, FFR

b

t+1, t = rb1, rb2, ..., r

bT},

where the time sequence (rb1, rb2, ..., r

bT ) is independent from sb1, s

b2, ..., s

bT . The time

sequence is the same for all factors to preserve their cross-correlations.

4. In each replication, we estimate the ICAPM by the two-step procedure, but using theartificial data rather than the original data:

(Ri,t+1 −Rf,t+1)b = δbi + βbi,MRM

bt+1 + βb

i,FFRFFRb

t+1 + εbi,t+1,

(Ri −Rf )b = λbMβbi,M + λbFFRβ

bi,FFR + αb

i .

We compute and save both the t-statistics for the factor risk prices and the χ2 statistic,[t(λbM), t(λbFFR),χ2,b

], leading to an empirical distribution of the statistics. We also

compute the cross-sectional OLS R2 for each pseudo sample, R2,b.

5. The empirical p-value associated with the risk price for FFRt+1 (for a two-sided test)

37

is computed as

p(λFFR) =

[#{t(λbFFR) ≥ t(λFFR)

}+ #

{t(λbFFR) < −t(λFFR)

}]/5000, if λFFR ≥ 0[

#{t(λbFFR) ≤ t(λFFR)

}+ #

{t(λbFFR) > −t(λFFR)

}]/5000, if λFFR < 0

,

and similarly for the other factor risk prices. In the above expression, #{t(λbFFR) ≥ t(λFFR)

}denotes the number of replications in which the pseudo t-stats are greater than or equalto the t-ratio from the original sample.

The p-value for the χ2 statistic is computed as

p(χ2) = #{χ2,b ≥ χ2

}/5000.

Finally, we order the pseudo values of the cross-sectional coefficient of determination,R2,b, and compute a 90% confidence interval by using the 5% and 95% percentilesof the bootstrapped distribution. The construction of the 95% confidence interval isanalogous.

38

Table 1: Descriptive statistics for risk factorsThis table reports descriptive statistics for the risk factors associated with the ICAPM and

alternative factor models. FFR and TB denote the “hedging factors” when the state vari-

ables are the Fed funds rate and T-bill rate, respectively. RM , SMB, HML, UMD, and

LIQ denote the market, size, value, momentum, and liquidity factors, respectively. ME,

IA, and ROE represent the Hou-Xue-Zhang size, investment, and profitability factors, re-

spectively. RMW and CMA denote the Fama-French profitability and investment fac-

tors. The sample is 1972:01–2013:12. φ designates the first-order autocorrelation coefficient.

Mean (%) Stdev. (%) Min. (%) Max. (%) φ

RM 0.53 4.61 −23.24 16.10 0.08

FFR 0.00 0.59 −6.51 3.15 0.40

TB 0.00 0.49 −4.54 2.69 0.33SMB 0.20 3.13 −16.39 22.02 0.01HML 0.39 3.01 −12.68 13.83 0.15UMD 0.71 4.46 −34.72 18.39 0.07LIQ 0.43 3.57 −10.14 21.01 0.09ME 0.31 3.14 −14.45 22.41 0.03IA 0.44 1.87 −7.13 9.41 0.06ROE 0.57 2.62 −13.85 10.39 0.10RMW 0.29 2.25 −17.60 12.24 0.18CMA 0.37 1.96 −6.76 8.93 0.14

39

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40

Table 3: Descriptive statistics for spreads in returns: VW portfoliosThis table reports descriptive statistics for the “high-minus-low” spreads in returns associated

with different (value-weighted) portfolio classes. The portfolios are deciles sorted on book-

to-market ratio (BM), equity duration (DUR), earnings-to-price ratio (EP), long-term reversal

in returns (REV), investment-to-assets (IA), changes in property, plant, and equipment scaled

by assets (PIA), and inventory growth (IVG). The sample is 1972:01–2013:12. φ designates

the first-order autocorrelation coefficient. The pairwise correlations are presented in Panel B.

Panel AMean (%) Stdev. (%) Min. (%) Max. (%) φ

BM 0.69 4.86 −14.18 20.45 0.11IA −0.42 3.62 −14.39 11.83 0.04

PIA −0.49 3.00 −10.37 8.60 0.08DUR −0.52 4.34 −21.38 15.77 0.09EP 0.58 4.83 −15.47 22.53 0.02

REV −0.41 5.21 −32.99 18.08 0.06IVG −0.36 3.15 −9.69 12.04 0.07

Panel BBM IA PIA DUR EP REV IVG

BM 1.00 −0.50 −0.31 −0.71 0.67 −0.56 −0.32IA 1.00 0.55 0.36 −0.41 0.45 0.50

PIA 1.00 0.20 −0.19 0.32 0.44DUR 1.00 −0.81 0.34 0.25EP 1.00 −0.35 −0.25

REV 1.00 0.17IVG 1.00

41

Table 4: Descriptive statistics for spreads in returns: EW portfoliosThis table reports descriptive statistics for the “high-minus-low” spreads in returns associated

with different (equal-weighted) portfolio classes. The portfolios are deciles sorted on book-

to-market ratio (BM), equity duration (DUR), earnings-to-price ratio (EP), long-term reversal

in returns (REV), investment-to-assets (IA), changes in property, plant, and equipment scaled

by assets (PIA), and inventory growth (IVG). The sample is 1972:01–2013:12. φ designates

the first-order autocorrelation coefficient. The pairwise correlations are presented in Panel B.

Panel AMean (%) Stdev. (%) Min. (%) Max. (%) φ

BM 0.80 5.38 −28.19 30.47 0.09IA −0.73 3.26 −12.23 15.48 0.14

PIA −0.64 2.86 −11.92 12.46 0.10DUR −0.65 5.01 −19.03 26.18 0.13EP 0.65 4.54 −19.36 22.63 0.09

REV −0.59 4.11 −26.19 13.32 0.08IVG −0.47 2.61 −8.51 9.78 0.12

Panel BBM IA PIA DUR EP REV IVG

BM 1.00 −0.68 −0.41 −0.82 0.85 −0.53 −0.44IA 1.00 0.69 0.54 −0.56 0.58 0.68

PIA 1.00 0.36 −0.32 0.47 0.72DUR 1.00 −0.82 0.32 0.40EP 1.00 −0.40 −0.36

REV 1.00 0.39IVG 1.00

42

Table 5: Factor risk premia for CAPMThis table reports the estimation and evaluation results for the standard CAPM. The estimation

procedure is the two-pass regression approach. The test portfolios are decile portfolios sorted on

book-to-market ratio (BM), equity duration (DUR), earnings-to-price ratio (EP), long-term reversal

in returns (REV), investment-to-assets (IA), changes in property, plant, and equipment scaled by

assets (PIA), and inventory growth (IVG). “All” refers to a test including all portfolio groups.

The portfolios are value- and equal-weighted in Panels A and B, respectively. λM denotes the risk

price estimate (in %) for the market factor. Below the risk price estimates are displayed t-statistics

based on Shanken’s standard errors (in parenthesis). The column labeled χ2 presents the statistic

(first line) and associated asymptotic p-value (in parenthesis) for the test on the joint significance

of the pricing errors. The column labeled R2OLS denotes the cross-sectional OLS R2, while R2

C

represents the constrained cross-sectional R2. The sample is 1972:01–2013:12. Italic, underlined,

and bold t-ratios denote statistical significance at the 10%, 5%, and 1% levels, respectively. Risk

price estimates market with ***, **, * represent statistical significance at the 1%, 5%, and 10%

levels, respectively, based on the empirical p-values from a bootstrap simulation. Underlined values

of the χ2 statistic mean that the model is not rejected at the 5% level based on the p-values from

the bootstrap. R2OLS values market with ** and * indicate statistical significance (based on the

bootstrap) at the 5% and 10% levels, respectively.λM χ2 R2

OLS R2C λM χ2 R2

OLS R2C

Panel A (VW) Panel B (EW)

BM 0.703∗∗∗ 16.00 −0.41 −0.29 BM 0.567∗∗∗ 27.27 −0.82 −0.76(3.33) (0.067) (2.59) (0.001)

DUR 0.676∗∗∗ 22.31 −0.85 −0.62 DUR 0.606∗∗∗ 29.91 −1.35 −1.14(3.23) (0.008) (2.77) (0.000)

EP 0.674∗∗∗ 23.12 −0.74 −0.54 EP 0.651∗∗∗ 26.10 −1.36 −1.05(3.21) (0.006) (2.97) (0.002)

REV 0.707∗∗∗ 12.74 −0.18 −0.05 REV 0.694∗∗∗ 28.71 −0.89 −0.61(3.37) (0.175) (3.18) (0.001)

IA 0.603∗∗∗ 21.56 −1.18 −0.98 IA 0.610∗∗∗ 61.25 −0.90 −0.76(2.92) (0.010) (2.78) (0.000)

PIA 0.573∗∗∗ 26.36 −0.43 −0.38 PIA 0.627∗∗∗ 51.90 −0.72 −0.59(2.78) (0.002) (2.85) (0.000)

IVG 0.602∗∗∗ 18.10 −0.55 −0.45 IVG 0.642∗∗∗ 58.05 −0.92 −0.73(2.91) (0.034) (2.92) (0.000)

All 0.647∗∗∗ 105.58 −0.59 −0.45 All 0.627∗∗∗ 159.10 −1.01 −0.82(3.12) (0.003) (2.86) (0.000)

43

Table 6: Factor risk premia for ICAPM: VW portfoliosThis table reports the estimation and evaluation results for the two-factor ICAPM. The estimation

procedure is the two-pass regression approach. The test portfolios are value-weighted decile port-

folios sorted on book-to-market ratio (BM), equity duration (DUR), earnings-to-price ratio (EP),

long-term reversal in returns (REV), investment-to-assets (IA), changes in property, plant, and

equipment scaled by assets (PIA), and inventory growth (IVG). “All” refers to a test including all

portfolio groups. λM and λz denotes the risk price estimates (in %) for the market and interest rate

factors, respectively. Below the risk price estimates are displayed t-statistics based on Shanken’s

standard errors (in parenthesis). FFR and TB stand for the version of the model in which the

factor is the innovation on the Fed funds rate and T-bill rate, respectively. The column labeled χ2

presents the statistic (first line) and associated asymptotic p-value (in parenthesis) for the test on

the joint significance of the pricing errors. The column labeled R2OLS denotes the cross-sectional

OLS R2. The sample is 1972:01–2013:12. Italic, underlined, and bold t-ratios denote statistical

significance at the 10%, 5%, and 1% levels, respectively. Risk price estimates market with ***,

**, * represent statistical significance at the 1%, 5%, and 10% levels, respectively, based on the

empirical p-values from a bootstrap simulation. Underlined values of the χ2 statistic mean that

the model is not rejected at the 5% level based on the p-values from the bootstrap. R2OLS values

market with ** and * indicate statistical significance (based on the bootstrap) at the 5% and 10%

levels, respectively.λM λz χ2 R2

OLS λM λz χ2 R2OLS

Panel A (BM) Panel E (IA)

FFR 0.612∗∗∗ −0.673∗∗∗ 5.25 0.62∗ FFR 0.565∗∗∗ −0.816∗∗∗ 3.28 0.65∗∗

(2.89) (−2.25) (0.731) (2.69) (−2.18) (0.916)

TB 0.619∗∗∗ −0.527∗∗∗ 4.84 0.69∗ TB 0.590∗∗∗ −0.587∗∗∗ 4.46 0.56∗∗

(2.94) (−2.33) (0.775) (2.82) (−2.41) (0.813)

Panel B (DUR) Panel F (PIA)

FFR 0.613∗∗∗ −0.781∗∗∗ 4.48 0.70∗∗ FFR 0.561∗∗∗ −0.798∗∗∗ 4.46 0.59∗∗

(2.89) (−2.49) (0.811) (2.68) (−2.47) (0.814)

TB 0.633∗∗∗ −0.567∗∗∗ 3.97 0.74∗∗ TB 0.572∗∗∗ −0.725∗∗ 6.09 0.39∗

(2.99) (−2.84) (0.860) (2.72) (−2.08) (0.637)

Panel C (EP) Panel G (IVG)

FFR 0.585∗∗∗ −0.798∗∗ 4.87 0.78∗∗ FFR 0.605∗∗∗ −0.436∗∗ 7.37 0.20(2.73) (−2.19) (0.771) (2.91) (−2.01) (0.498)

TB 0.630∗∗∗ −0.429∗∗∗ 9.72 0.40 TB 0.606∗∗∗ −0.350∗∗ 7.62 0.13(2.98) (−2.66) (0.285) (2.92) (−1 .89 ) (0.472)

Panel D (REV) Panel H (All)

FFR 0.640∗∗∗ −0.576∗∗ 5.29 0.52∗ FFR 0.597∗∗∗ −0.709∗∗∗ 36.09 0.58∗∗

(3.01) (−1 .94 ) (0.726) (2.86) (−2.85) (0.999)

TB 0.682∗∗∗ −0.370∗∗∗ 6.37 0.52 TB 0.618∗∗∗ −0.491∗∗∗ 47.92 0.48∗∗

(3.22) (−2.18) (0.606) (2.96) (−3.00) (0.969)

44

Table 7: Factor risk premia for ICAPM: EW portfoliosThis table reports the estimation and evaluation results for the two-factor ICAPM. The estimation

procedure is the two-pass regression approach. The test portfolios are equal-weighted decile port-











significance at the 10%, 5%, and 1% levels, respectively. Risk price estimates market with ***,

**, * represent statistical significance at the 1%, 5%, and 10% levels, respectively, based on the

empirical p-values from a bootstrap simulation. Underlined values of the χ2 statistic mean that

the model is not rejected at the 5% level based on the p-values from the bootstrap. R2OLS values

market with ** and * indicate statistical significance (based on the bootstrap) at the 5% and 10%

levels, respectively.λM λz χ2 R2



FFR 0.469∗∗ −1.079∗∗∗ 5.97 0.75 FFR 0.415 −1.643∗∗∗ 2.57 0.85∗∗

(1 .77 ) (−2.17) (0.650) (1.26) (−2.45) (0.958)

TB 0.547∗∗∗ −0.755∗∗∗ 6.86 0.80 TB 0.544∗∗ −1.009∗∗∗ 5.63 0.83∗∗

(2.19) (−2.50) (0.551) (1.98) (−3.19) (0.689)


FFR 0.510∗∗ −1.011∗∗∗ 1.96 0.94∗∗ FFR 0.455 −1.440∗∗∗ 4.54 0.61∗

(1.97) (−2.51) (0.982) (1.49) (−2.65) (0.805)

TB 0.578∗∗∗ −0.722∗∗∗ 3.13 0.86∗∗ TB 0.558∗∗ −0.991∗∗∗ 4.80 0.72∗∗

(2.34) (−2.81) (0.926) (2.03) (−3.12) (0.779)


FFR 0.526∗∗∗ −0.852∗∗∗ 5.08 0.83∗∗ FFR 0.410 −1.590∗∗ 3.04 0.81∗∗

(2.15) (−2.66) (0.749) (1.25) (−2.24) (0.932)

TB 0.603∗∗∗ −0.574∗∗∗ 6.39 0.77∗ TB 0.534∗∗ −1.042∗∗∗ 4.96 0.76∗∗

(2.56) (−3.07) (0.603) (1 .90 ) (−2.69) (0.762)


FFR 0.583∗∗∗ −0.830∗∗∗ 6.14 0.75∗∗ FFR 0.498∗ −1.069∗∗∗ 34.57 0.67∗∗

(2.42) (−2.78) (0.631) (1 .91 ) (−2.83) (1.000)

TB 0.644∗∗∗ −0.535∗∗∗ 8.59 0.66∗ TB 0.577∗∗ −0.729∗∗∗ 46.99 0.67∗∗

(2.78) (−3.25) (0.378) (2.34) (−3.32) (0.976)

45

Table 8: Accounting of risk premia

This table reports the risk premium (beta times risk price) for each factor from the ICAPM for the

first and last decile portfolios. The portfolios are decile portfolios sorted on book-to-market ratio

(BM), equity duration (DUR), earnings-to-price ratio (EP), long-term reversal in returns (REV),

investment-to-assets (IA), changes in property, plant, and equipment scaled by assets (PIA), and

inventory growth (IVG). In Panels A to G the portfolios are value-weighted while in Panels H to

N the portfolios are equal-weighted. E(R) denotes the average excess return for the first and last

deciles, and α represents the average pricing error per decile. RM and FFR denote the market and

intertemporal risk factors from the ICAPM, respectively. All the values are presented in percentage

points. D1 and D10 denote the lowest and last deciles, respectively, and Dif. denotes the difference

across extreme deciles. The sample is 1972:01–2013:12.

E(R) RM FFR α E(R) RM FFR α

Panel A (BM, VW) Panel H (BM, EW)

D1 0.38 0.66 −0.20 −0.08 D1 0.22 0.69 −0.37 −0.09D10 1.07 0.58 0.39 0.11 D10 1.02 0.48 0.33 0.21Dif. −0.69 0.08 −0.59 −0.18 Dif. −0.80 0.21 −0.71 −0.30

Panel B (DUR, VW) Panel I (DUR, EW)

D1 0.83 0.64 0.17 0.02 D1 0.97 0.55 0.32 0.10D10 0.31 0.75 −0.30 −0.14 D10 0.32 0.79 −0.46 −0.01Dif. 0.52 −0.11 0.48 0.16 Dif. 0.65 −0.24 0.78 0.11

Panel C (EP, VW) Panel J (EP, EW)

D1 0.38 0.70 −0.30 −0.02 D1 0.37 0.76 −0.26 −0.12D10 0.96 0.57 0.28 0.11 D10 1.02 0.55 0.53 −0.06Dif. −0.58 0.14 −0.58 −0.13 Dif. −0.65 0.21 −0.80 −0.06

Panel D (REV, VW) Panel K (REV, EW)

D1 0.96 0.78 0.07 0.11 D1 1.00 0.75 0.14 0.11D10 0.55 0.79 −0.18 −0.06 D10 0.41 0.81 −0.29 −0.11Dif. 0.41 −0.01 0.25 0.17 Dif. 0.59 −0.06 0.43 0.22

Panel E (IA, VW) Panel L (IA, EW)

D1 0.76 0.61 0.12 0.03 D1 0.87 0.52 0.16 0.20D10 0.34 0.72 −0.26 −0.12 D10 0.14 0.61 −0.35 −0.12Dif. 0.42 −0.11 0.38 0.15 Dif. 0.73 −0.10 0.51 0.32

Panel F (PIA, VW) Panel M (PIA, EW)

D1 0.85 0.59 0.16 0.10 D1 0.94 0.54 0.29 0.11D10 0.36 0.67 −0.17 −0.14 D10 0.30 0.64 −0.14 −0.19Dif. 0.49 −0.08 0.34 0.23 Dif. 0.64 −0.09 0.44 0.30

Panel G (IVG, VW) Panel N (IVG, EW)

D1 0.76 0.62 −0.05 0.19 D1 0.80 0.50 0.25 0.05D10 0.40 0.71 −0.17 −0.14 D10 0.33 0.57 −0.16 −0.08Dif. 0.36 −0.09 0.11 0.33 Dif. 0.47 −0.07 0.41 0.14

46

Tab

le9:

Fac

tor

risk

pre

mia

for

alte

rnat

ive

mult

ifac

tor

model

sT

his

tab

lere

por

tsth

ees

tim

atio

nan

dev

alu

atio

nre

sult

sfo

ralt

ern

ati

vem

ult

ifact

or

mod

els.

Th

ees

tim

ati

on

pro

ced

ure

isth

etw

o-p

ass

regre

ssio

n

app

roac

h.

Th

ete

stp

ortf

olio

sar

ed

ecil

ep

ortf

olio

sso

rted

on

book-t

o-m

ark

etra

tio

(BM

),eq

uit

yd

ura

tion

(DU

R),

earn

ings-

to-p

rice

rati

o(E

P),

lon

g-

term

reve

rsal

inre

turn

s(R

EV

),in

vest

men

t-to

-ass

ets

(IA

),ch

an

ges

inp

rop

erty

,p

lant,

an

deq

uip

men

tsc

ale

dby

ass

ets

(PIA

),an

din

vento

rygro

wth

(IV

G).

Th

ep

ortf

olio

sar

eva

lue-

and

equ

al-w

eighte

din

Pan

els

Aan

dB

,re

spec

tive

ly.λM

,λSM

B,λH

ML

,λUM

D,

an

dλLIQ

den

ote

the

risk

pri

ce

esti

mat

es(i

n%

)fo

rth

em

arke

t,si

ze,

valu

e,m

omen

tum

,an

dli

qu

idit

yfa

ctors

,re

spec

tive

ly.λM

E,λIA

,an

dλROE

rep

rese

nt

the

risk

pri

ces

ass

oci

ate

d

wit

hth

eH

ou-X

ue-

Zh

ang

size

,in

vest

men

t,an

dp

rofi

tab

ilit

yfa

ctors

,re

spec

tive

ly.λRM

Wan

dλCM

Ad

enote

the

risk

pri

cees

tim

ate

sfo

rth

eF

am

a-F

ren

ch

pro

fita

bil

ity

and

inves

tmen

tfa

ctor

s.B

elow

the

risk

pri

cees

tim

ate

sare

dis

pla

yedt-

stati

stic

sb

ase

don

Sh

an

ken

’sst

an

dard

erro

rs(i

np

are

nth

esis

).

Th

eco

lum

nla

bel

edχ

2p

rese

nts

the

stat

isti

c(fi

rst

lin

e)an

dass

oci

ate

dasy

mpto

ticp-v

alu

e(i

np

are

nth

esis

)fo

rth

ete

ston

the

join

tsi

gn

ifica

nce

of

the

pri

cin

ger

rors

.T

he

colu

mn

lab

eled

R2 OLS

den

otes

the

cross

-sec

tional

OL

SR

2,

wh

ileR

2 Cre

pre

sents

the

con

stra

ined

cross

-sec

tion

alR

2.

Th

esa

mple

is

1972

:01–

2013

:12.

Ital

ic,

un

der

lin

ed,

and

bol

dt-

rati

os

den

ote

stati

stic

al

sign

ifica

nce

at

the

10%

,5%

,an

d1%

leve

ls,

resp

ecti

vely

.R

isk

pri

cees

tim

ate

s

mar

ket

wit

h**

*,**

,*

rep

rese

nt

stat

isti

cal

sign

ifica

nce

at

the

1%

,5%

,an

d10%

level

s,re

spec

tive

ly,

base

don

the

empir

icalp-v

alu

esfr

om

ab

oots

trap

sim

ula

tion

.U

nd

erli

ned

valu

esof

theχ2

stat

isti

cm

ean

that

the

mod

elis

not

reje

cted

at

the

5%

level

base

don

thep-v

alu

esfr

om

the

boots

trap

.R

2 OLS

valu

esm

arke

tw

ith

**an

d*

ind

icat

est

atis

tica

lsi

gn

ifica

nce

(base

don

the

boots

trap

)at

the

5%

an

d10%

leve

ls,

resp

ecti

vely

.

λM

λSM

BλHM

LλUM

DλLIQ

λM

EλIA

λROE

λRM

WλCM

Aχ

2R

2 OLS

R2 C

PanelA

(value-w

eighted

10.

603∗∗∗−

0.00

50.

456∗∗∗

87.0

40.

70∗∗

0.6

7(2.91

)(−

0.02

)(2.99

)(0.0

51)

20.

598∗∗∗

0.1

230.

419∗∗∗

0.58

774.

23

0.74∗∗

0.7

5(2.89

)(0.6

6)(2.76

)(1.6

1)(0.2

28)

30.

606∗∗∗−

0.00

10.

462∗∗∗

−0.1

3186.6

00.

71∗∗

0.6

5(2.92

)(−

0.00

)(2.98

)(−

0.27

)(0.0

45)

40.

607∗∗∗

0.0

910.

323∗∗∗−

0.20

586.

55

0.68∗∗

0.3

0(2.94

)(0.4

7)(2.99

)(−

1.1

5)(0.0

46)

50.

605∗∗∗−

0.02

50.

427∗∗∗

−0.

129

0.238∗∗

83.4

90.

74∗∗

0.5

2(2.93

)(−

0.13

)(2.79

)(−

0.9

4)(2.3

1)(0.0

61)

PanelB

(equal-weighted)

10.

754∗∗∗−

0.39

90.

668∗∗∗

126.2

10.

71∗∗

0.4

1(3.29

)(−

1.71

)(4.35

)(0.0

00)

20.

865∗∗∗−

0.11

60.

528∗∗∗

1.2

07∗∗∗

110.5

90.

81∗∗

0.6

9(3.73

)(−

0.50

)(3.52

)(3.27

)(0.0

00)

30.

818∗∗∗−

0.4

65∗

0.71

5∗∗∗

−1.0

29114.9

00.

73∗∗

0.4

3(3.58

)(−

1.97

)(4.50

)(−

1.76

)(0.0

00)

40.

748∗∗∗

−0.

063

0.52

2∗∗∗

0.2

64112.

82

0.84∗∗

0.8

0(3.33

)(−

0.27

)(4.67

)(1.3

9)(0.0

00)

50.

751∗∗∗−

0.27

30.

466∗∗∗

0.1

350.

517∗∗∗

112.0

40.

84∗∗

0.7

7(3.32

)(−

1.24

)(3.07

)(0.9

2)(4.37

)(0.0

00)

47

Table 10: Predictive regressionsThis table reports the results associated with single long-horizon predictive regressions for the excess market

return (Panel A), growth in industrial production (Panel B), and aggregate earnings growth (Panel C),

at horizons of 1, 3, 6, 9, 12, 24, and 36 months ahead. The forecasting variables are the Fed funds rate

(FFR) and the T-bill rate (TB). The original sample is 1972:01–2013:12, and q observations are lost in

each of the respective q-horizon regressions. For each regression, the first line shows the slope estimates,

whereas the second and third lines present Newey-West (in parentheses) and Hodrick (in brackets) t-ratios,

respectively. T-ratios marked with *, **, and *** denote statistical significance at the 10%, 5%, and 1%

levels, respectively. R2 denotes the coefficient of determination.

q = 1 q = 3 q = 6 q = 9 q = 12 q = 24 q = 36

Panel A (re)

FFR −0.08 −0.20 −0.31 −0.42 −0.52 −0.49 −0.52(−1.50) (−1.38) (−1.00) (−0.92) (−0.94) (−0.83) (−0.62)[−1.47] [−1.18] [−0.90] [−0.84] [−0.79] [−0.40] [−0.30]

R2 0.00 0.01 0.01 0.01 0.01 0.01 0.01TB −0.08 −0.19 −0.30 −0.40 −0.48 −0.51 −0.56

(−1.29) (−1.16) (−0.87) (−0.75) (−0.75) (−0.71) (−0.55)[−1.27] [−1.00] [−0.79] [−0.70] [−0.64] [−0.35] [−0.26]

R2 0.00 0.01 0.01 0.01 0.01 0.00 0.00

Panel B (IPG)

FFR −0.02 −0.08 −0.18 −0.27 −0.36 −0.51 −0.36(−2.31∗∗) (−2.33∗∗) (−2.31∗∗) (−2.35∗∗) (−2.53∗∗) (−2.51∗∗) (−1.34)[−2.20∗∗] [−2.72∗∗∗] [−2.87∗∗∗] [−2.86∗∗∗] [−2.87∗∗∗] [−2.23∗∗] [−1.15]

R2 0.01 0.04 0.06 0.07 0.08 0.08 0.03TB −0.02 −0.07 −0.16 −0.24 −0.32 −0.49 −0.34

(−1.44) (−1.68∗) (−1.73∗) (−1.76∗) (−1.87∗) (−1.99∗∗) (−1.08)[−1.41] [−2.03∗∗] [−2.23∗∗] [−2.26∗∗] [−2.31∗∗] [−1.89∗] [−0.92]

R2 0.01 0.02 0.03 0.04 0.05 0.06 0.02

Panel C (∆e)

FFR −0.09 −0.33 −0.86 −1.52 −2.29 −4.80 −5.31(−0.96) (−0.86) (−0.99) (−1.24) (−1.59) (−2.58∗∗∗) (−2.31∗∗)[−0.96] [−1.25] [−1.65∗] [−1.96∗∗] [−2.27∗∗] [−2.90∗∗∗] [−2.52∗∗]

R2 0.00 0.01 0.01 0.03 0.04 0.11 0.12TB −0.09 −0.34 −0.90 −1.62 −2.46 −5.74 −6.57

(−0.79) (−0.71) (−0.85) (−1.05) (−1.34) (−2.63∗∗∗) (−2.47∗∗)[−0.79] [−1.04] [−1.41] [−1.68∗] [−1.93∗] [−2.66∗∗∗] [−2.42∗∗]

R2 0.00 0.01 0.01 0.02 0.04 0.12 0.14

48

Table 11: Factor risk premia for alternative ICAPMThis table reports the estimation and evaluation results for alternative two-factor ICAPM mod-

els. The estimation procedure is the two-pass regression approach. The test portfolios are decile

portfolios sorted on book-to-market ratio (BM), equity duration (DUR), earnings-to-price ratio

(EP), long-term reversal in returns (REV), investment-to-assets (IA), changes in property, plant,

and equipment scaled by assets (PIA), and inventory growth (IVG). The portfolios are value- and

equal-weighted in Panels A and B, respectively. λM and λz denotes the risk price estimates (in

%) for the market and “hedging” factors, respectively. Below the risk price estimates are dis-

played t-statistics based on Shanken’s standard errors (in parenthesis). TERM , DEF , dp, pe,

vs, and SV AR stand for the ICAPM in which the factors are the innovation on the term spread,

default spread, log dividend yield, smoothed log price-to-earnings ratio, value spread, and stock

market variance, respectively. The column labeled χ2 presents the statistic (first line) and associ-

ated asymptotic p-value (in parenthesis) for the test on the joint significance of the pricing errors.

The column labeled R2OLS denotes the cross-sectional OLS R2. The sample is 1972:01–2013:12.

Italic, underlined, and bold t-ratios denote statistical significance at the 10%, 5%, and 1% levels,

respectively. Risk price estimates market with ***, **, * represent statistical significance at the

1%, 5%, and 10% levels, respectively, based on the empirical p-values from a bootstrap simulation.

Underlined values of the χ2 statistic mean that the model is not rejected at the 5% level based on

the p-values from the bootstrap. R2OLS values market with ** and * indicate statistical significance

(based on the bootstrap) at the 5% and 10% levels, respectively.λM λz χ2 R2


Panel A (VW) Panel B (EW)

TERM 0.612∗∗∗ 0.330∗∗∗ 45.42 0.32∗∗ TERM 0.398 0.610∗∗ 30.55 0.29(2.93) (2.88) (0.984) (1.44) (2.53) (1.000)

DEF 0.653∗∗∗ 0.040 95.24 −0.52 DEF 0.715∗∗∗ 0.215∗∗∗ 40.13 −0.32(3.14) (1.63) (0.016) (2.83) (3.22) (0.997)

dp 0.595∗∗∗ −1.925∗∗∗ 79.81 −0.20 dp 0.318 −2.822∗∗∗ 85.07 −0.61(2.87) (−2.96) (0.155) (1.32) (−3.17) (0.079)

pe 0.595∗∗∗ 2.046∗∗∗ 76.18 −0.16 pe 0.316 3.040∗∗∗ 77.46 −0.57(2.87) (3.01) (0.232) (1.29) (3.08) (0.203)

vs 0.597∗∗∗ −2.119∗∗∗ 78.08 0.66∗∗ vs 0.670∗∗∗ −2.737∗∗∗ 109.47 0.69∗∗

(2.88) (−3.47) (0.189) (3.00) (−4.69) (0.001)SV AR 0.604∗∗∗ −0.278∗∗∗ 76.38 −0.21 SV AR 0.328 −0.573∗∗∗ 56.65 −0.11

(2.91) (−2.71) (0.227) (1.29) (−2.92) (0.835)

49

Panel A (BM) Panel B (BM, t-stats)

Panel C (DUR) Panel D (DUR, t-stats)

Panel E (EP) Panel F (EP, t-stats)

Panel G (REV) Panel H (REV, t-stats)

Figure 1: Individual pricing errors (BM, DUR, EP, and REV): value-weighted portfoliosThis figure plots the pricing errors (in % per month, Panels A, C, E, and G), and respective t-statistics (Panels B,

D, F, and H) of different decile portfolios associated with the ICAPM based on the Fed funds rate. The portfo-

lios are value-weighted deciles sorted on book-to-market (BM), equity duration (DUR), earnings-to-price (EP), and long-

term reversal in returns (REV). The pricing errors are obtained from an OLS cross-sectional regression of average ex-

cess returns on factor betas. i = 1, ..., 10 designates a portfolio associated with the ith decile within each class.

50

Panel A (IA) Panel B (IA, t-stats)

Panel C (PIA) Panel D (PIA, t-stats)

Panel E (IVG) Panel F (IVG, t-stats)

Figure 2: Individual pricing errors (IA, PIA, and IVG): value-weighted portfoliosThis figure plots the pricing errors (in % per month, Panels A, C, and E), and respective t-

statistics (Panels B, D, and F) of different decile portfolios associated with the ICAPM based on

the Fed funds rate. The portfolios are value-weighted deciles sorted on investment-to-assets (IA),

changes in property, plant, and equipment scaled by assets (PIA), and inventory growth (IVG). The

pricing errors are obtained from an OLS cross-sectional regression of average excess returns on fac-

tor betas. i = 1, ..., 10 designates a portfolio associated with the ith decile within each class.51

Panel A (BM) Panel B (BM, t-stats)

Panel C (DUR) Panel D (DUR, t-stats)

Panel E (EP) Panel F (EP, t-stats)

Panel G (REV) Panel H (REV, t-stats)

Figure 3: Individual pricing errors (BM, DUR, EP, and REV): equal-weighted portfoliosThis figure plots the pricing errors (in % per month, Panels A, C, E, and G), and respective t-statistics (Panels B,

D, F, and H) of different decile portfolios associated with the ICAPM based on the Fed funds rate. The portfo-

lios are equal-weighted deciles sorted on book-to-market (BM), equity duration (DUR), earnings-to-price (EP), and long-

term reversal in returns (REV). The pricing errors are obtained from an OLS cross-sectional regression of average ex-

cess returns on factor betas. i = 1, ..., 10 designates a portfolio associated with the ith decile within each class.

52

Panel A (IA) Panel B (IA, t-stats)

Panel C (PIA) Panel D (PIA, t-stats)

Panel E (IVG) Panel F (IVG, t-stats)

Figure 4: Individual pricing errors (IA, PIA, and IVG): equal-weighted portfoliosThis figure plots the pricing errors (in % per month, Panels A, C, and E), and respective t-statistics

(Panels B, D, and F) of different decile portfolios associated with the ICAPM based on the Fed

funds rate. The portfolios are equal-weighted deciles sorted on investment-to-assets (IA), changes

in property, plant, and equipment scaled by assets (PIA), and inventory growth (IVG). The pric-

ing errors are obtained from an OLS cross-sectional regression of average excess returns on fac-

tor betas. i = 1, ..., 10 designates a portfolio associated with the ith decile within each class.53

Panel A: BM Panel B: DUR

Panel C: EP Panel D: REV

Panel E: IA Panel F: PIA

Panel G: IVG

Figure 5: Regression betas for FFR: value-weighted portfoliosThis figure plots the beta estimates associated with the innovation in the Fed funds rate, FFR. The portfolios are

value-weighted deciles sorted on book-to-market (BM), equity duration (DUR), earnings-to-price (EP), and long-term

reversal in returns (REV), on investment-to-assets (IA), changes in property, plant, and equipment scaled by assets

(PIA), and inventory growth (IVG). i = 1, ..., 10 designates a portfolio associated with the ith decile within each class.54

Panel A: BM Panel B: DUR

Panel C: EP Panel D: REV

Panel E: IA Panel F: PIA

Panel G: IVG

Figure 6: Regression betas for FFR: equal-weighted portfoliosThis figure plots the beta estimates associated with the innovation in the Fed funds rate, FFR. The portfolios are

equal-weighted deciles sorted on book-to-market (BM), equity duration (DUR), earnings-to-price (EP), and long-term

reversal in returns (REV), on investment-to-assets (IA), changes in property, plant, and equipment scaled by assets

(PIA), and inventory growth (IVG). i = 1, ..., 10 designates a portfolio associated with the ith decile within each class.55

B Internet appendix: not for publication

B.1 Sensitivity analysis

We conduct some robustness checks to the main results discussed in Section 4 in the mainpaper. First, we estimate both versions of the ICAPM by using new interest rate factors.Following Hahn and Lee (2006) and Maio and Santa-Clara (2012), the innovation in the Fed

funds rate corresponds to the first-difference on this variable, FFRt+1 ≡ FFRt+1 − FFRt,

and similarly for TB. The objective is to assess whether the fit of the ICAPM is driven bythe way the hedging factors are constructed. The results are displayed in Tables A.1 and A.2below. We can see that both the explanatory ratios and risk price estimates are very similarto the corresponding values in the benchmark specification, and this holds for both versionsof the model and for the tests with both value- and equal-weighted portfolios. These resultsare not surprising since both short-term interest rates are quite persistent as shown by theestimates of the AR(1) coefficients presented in Section 3, thus the innovations constructedfrom the two methods are highly correlated.

Second, we estimate the ICAPM for a subsample that ends in 2006:12. The goal is toevaluate the impact of the recent financial crisis on the fit of the ICAPM given the large spikein stock market volatility observed during the 2007–2009 period. The results are presentedin Tables A.3 and A.4. Overall, the fit of the ICAPM is larger in the restricted sample thanin the full sample. In the tests with all 70 value-weighted portfolios the explanatory ratiosare 64% and 60% in the versions based on FFR and TB, respectively, which representsa significant increase in explanatory power for the later case. In the estimation with the70 equal-weighted deciles the R2

OLS estimates are 78% (version with FFR) and 76% (TB),which represents an increase of more than 10% relative to the corresponding values in the fullsample. Moreover, the risk price estimates for the interest rate factors are strongly significant(5% or 1% levls) in nearly all cases, the exception being the version based on FFR whentested on the value-weighted return reversal deciles. These results suggest that the financialcrisis has had a negative effect on the fit of the ICAPM, which might be related to the factthat during this period the Fed funds rate hit the zero lower bound.29

Third, we estimate the ICAPM with portfolios related with two additional anomalies.We employ deciles sorted on cash-flow-to-price ratio (CFP, Lakonishok, Shleifer, and Vishny(1994)) and investment growth (IG, Xing (2008)). We employ both value- and equal-weightedportfolios and these data are available from Lu Zhang. The results are presented in TableA.5. We can see that the performance of the ICAPM for the value-weighted portfoliosassociated with these two anomalies is modest. The largest fit in the estimation with theCFP deciles is 24% (version with FFR), while in the test with the IG deciles we obtain anexplanatory ratio of 31% (version based on TB). However, in the estimation with equal-weighted portfolios the performance of the ICAPM for these anomalies improves sharply. Inthe tests with CFP, the R2

OLS estimates are 87% and 80% in the versions with FFR and TB,respectively, while in the estimation with the IG deciles these estimates are 82% and 76%,respectively. In all cases (both value- and equal-weighted portfolios) the two-factor model

29This lead to a change in monetary policy, which was known by “quantitative easing” (see Wright (2012)and Swanson and Williams (2014) for a discussion of the effect on asset markets).

56

passes the specification test with p-values largely above 5%. In most cases, the risk priceestimates for the interest rate factor are strongly significant (5% or 1% level), the exceptionsbeing the version based on FFR when tested on the IG deciles with significance at the 10%level. In sum, these results indicate that the ICAPM has strong explanatory power for theequal-weighted portfolios associated with the CFP and IG anomalies.

57

Table A.1: Factor risk premia for ICAPM: VW portfolios (alternative specification)This table reports the estimation and evaluation results for the two-factor ICAPM. The estimation












significance at the 10%, 5%, and 1% levels, respectively.λM λz χ2 R2



FFR 0.615 −0.663 5.40 0.61 FFR 0.567 −0.814 3.24 0.65(2.91) (−2.27) (0.714) (2.70) (−2.18) (0.918)

TB 0.622 −0.519 5.00 0.68 TB 0.592 −0.585 4.42 0.57(2.95) (−2.35) (0.757) (2.83) (−2.41) (0.817)


FFR 0.616 −0.772 4.41 0.72 FFR 0.561 −0.785 4.85 0.56(2.90) (−2.52) (0.818) (2.68) (−2.45) (0.773)

TB 0.636 −0.556 4.04 0.74 TB 0.571 −0.711 6.47 0.36(3.00) (−2.87) (0.853) (2.73) (−2.07) (0.595)


FFR 0.589 −0.786 5.04 0.77 FFR 0.606 −0.440 7.15 0.24(2.75) (−2.22) (0.753) (2.92) (−2.06) (0.520)

TB 0.632 −0.420 9.96 0.39 TB 0.607 −0.355 7.34 0.17(2.99) (−2.69) (0.268) (2.92) (−1 .93 ) (0.500)


FFR 0.645 −0.559 5.60 0.49 FFR 0.600 −0.700 37.25 0.58(3.04) (−1 .96 ) (0.692) (2.87) (−2.87) (0.999)

TB 0.685 −0.360 6.67 0.49 TB 0.620 −0.483 49.30 0.47(3.24) (−2.18) (0.573) (2.97) (−3.02) (0.957)

58

Table A.2: Factor risk premia for ICAPM: EW portfolios (alternative specification)This table reports the estimation and evaluation results for the two-factor ICAPM. The estimation















FFR 0.475 −1.069 6.08 0.75 FFR 0.420 −1.655 2.54 0.85(1 .80 ) (−2.19) (0.638) (1.28) (−2.43) (0.960)

TB 0.553 −0.748 6.99 0.80 TB 0.551 −1.014 5.52 0.83(2.22) (−2.52) (0.538) (2.00) (−3.18) (0.701)


FFR 0.514 −1.007 1.97 0.94 FFR 0.461 −1.429 4.67 0.61(2.00) (−2.52) (0.982) (1.52) (−2.67) (0.792)

TB 0.583 −0.719 3.06 0.87 TB 0.565 −0.987 4.87 0.72(2.37) (−2.82) (0.931) (2.07) (−3.13) (0.771)


FFR 0.532 −0.844 5.28 0.82 FFR 0.416 −1.607 2.91 0.83(2.18) (−2.68) (0.727) (1.27) (−2.24) (0.940)

TB 0.608 −0.570 6.54 0.77 TB 0.541 −1.051 4.71 0.78(2.59) (−3.09) (0.587) (1 .93 ) (−2.69) (0.788)


FFR 0.583 −0.830 6.14 0.75 FFR 0.503 −1.063 34.94 0.67(2.42) (−2.78) (0.631) (1 .94 ) (−2.84) (1.000)

TB 0.644 −0.535 8.59 0.66 TB 0.583 −0.725 47.39 0.67(2.78) (−3.25) (0.378) (2.37) (−3.34) (0.973)

59

Table A.3: Factor risk premia for ICAPM: VW portfolios (alternative sample)This table reports the estimation and evaluation results for the two-factor ICAPM. The estimation















FFR 0.578 −0.682 4.99 0.70 FFR 0.545 −0.748 5.54 0.59(2.54) (−2.48) (0.759) (2.43) (−2.49) (0.699)

TB 0.573 −0.530 3.81 0.83 FFR 0.555 −0.575 5.71 0.59(2.53) (−2.60) (0.874) (2.47) (−2.63) (0.680)


FFR 0.575 −0.844 5.43 0.70 FFR 0.543 −0.736 5.02 0.58(2.52) (−2.64) (0.711) (2.42) (−2.62) (0.756)

TB 0.578 −0.651 4.27 0.84 FFR 0.541 −0.667 6.25 0.42(2.54) (−2.88) (0.832) (2.41) (−2.25) (0.619)


FFR 0.557 −0.834 4.49 0.81 FFR 0.578 −0.578 8.44 0.29(2.41) (−2.29) (0.811) (2.57) (−2.22) (0.392)

TB 0.586 −0.507 8.32 0.56 FFR 0.571 −0.478 8.26 0.25(2.57) (−2.66) (0.402) (2.54) (−2.15) (0.408)


FFR 0.605 −0.639 4.04 0.66 FFR 0.569 −0.742 35.04 0.64(2.65) (−1 .89 ) (0.854) (2.53) (−2.96) (1.000)

TB 0.640 −0.436 5.38 0.50 FFR 0.576 −0.551 41.50 0.60(2.80) (−2.11) (0.716) (2.56) (−3.05) (0.995)

60

Table A.4: Factor risk premia for ICAPM: EW portfolios (alternative sample)This table reports the estimation and evaluation results for the two-factor ICAPM. The estimation















FFR 0.416 −0.976 7.28 0.88 FFR 0.369 −1.366 3.90 0.88(1.51) (−2.76) (0.507) (1.16) (−3.11) (0.866)

TB 0.472 −0.694 7.50 0.89 FFR 0.458 −0.906 7.11 0.86(1 .78 ) (−3.08) (0.484) (1.59) (−3.73) (0.525)


FFR 0.448 −0.963 2.69 0.96 FFR 0.402 −1.259 5.79 0.65(1.64) (−2.86) (0.952) (1.31) (−3.16) (0.671)

TB 0.497 −0.709 4.41 0.92 FFR 0.469 −0.924 6.34 0.71(1 .88 ) (−3.08) (0.819) (1.62) (−3.49) (0.609)


FFR 0.477 −0.799 4.98 0.88 FFR 0.349 −1.360 3.71 0.85(1 .88 ) (−3.08) (0.760) (1.08) (−2.68) (0.882)

TB 0.535 −0.566 6.72 0.85 FFR 0.435 −0.953 6.40 0.78(2.11) (−3.34) (0.567) (1.47) (−2.99) (0.602)


FFR 0.532 −0.818 5.84 0.82 FFR 0.442 −0.987 44.38 0.78(2.06) (−2.85) (0.665) (1.62) (−3.26) (0.988)

TB 0.580 −0.560 8.18 0.68 FFR 0.500 −0.702 56.46 0.76(2.30) (−3.20) (0.416) (1 .90 ) (−3.62) (0.840)

61

Table A.5: Factor risk premia for ICAPM: additional anomaliesThis table reports the estimation and evaluation results for the two-factor ICAPM. The estimation

procedure is the two-pass regression approach. The test portfolios are value-weighted (Panels A and

B) and equal-weighted (Panels C and D) decile portfolios sorted on cash-flow-to-price ratio (CFP)

and investment growth (IG). λM and λz denotes the risk price estimates (in %) for the market and

interest rate factors, respectively. Below the risk price estimates are displayed t-statistics based on

Shanken’s standard errors (in parenthesis). FFR and TB stand for the version of the model in

which the factor is the innovation on the Fed funds rate and T-bill rate, respectively. The column

labeled χ2 presents the statistic (first line) and associated asymptotic p-value (in parenthesis) for

the test on the joint significance of the pricing errors. The column labeled R2OLS denotes the cross-

sectional OLS R2. The sample is 1972:01–2013:12. Italic, underlined, and bold t-ratios denote

statistical significance at the 10%, 5%, and 1% levels, respectively.λM λz χ2 R2

OLS

Panel A (CFP, VW)

FFR 0.613 −0.631 5.76 0.24(2.88) (−2.12) (0.674)

TB 0.630 −0.325 8.49 −0.00(2.99) (−2.30) (0.387)

Panel B (IG, VW)

FFR 0.566 −0.473 13.60 −0.22(2.72) (−1 .93 ) (0.093)

TB 0.579 −0.537 6.75 0.31(2.77) (−2.44) (0.564)

Panel C (CFP, EW)

FFR 0.513 −0.989 4.74 0.87(2.02) (−2.50) (0.785)

TB 0.598 −0.687 5.00 0.80(2.46) (−2.84) (0.757)

Panel D (IG, EW)

FFR 0.426 −1.696 1.72 0.82(1.19) (−1 .75 ) (0.988)

TB 0.559 −0.962 4.04 0.76(2.04) (−2.44) (0.854)

62

Tab

leA

.6:

Fac

tor

risk

pre

mia

for

augm

ente

dIC

AP

MT

his

tab

lere

por

tsth

ees

tim

atio

nan

dev

alu

atio

nre

sult

sfo

rth

eau

gmen

ted

ICA

PM

.T

he

esti

mat

ion

pro

ced

ure

isth

etw

o-p

ass

regre

ssio

n

app

roac

h.

Th

ete

stp

ortf

olio

sar

eva

lue-

wei

ghte

d(P

anel

A)

and

equ

al-w

eigh

ted

(Pan

elB

)d

ecil

ep

ortf

olio

sso

rted

on

book-t

o-m

ark

etra

tio

(BM

),eq

uit

yd

ura

tion

(DU

R),

earn

ings

-to-

pri

cera

tio

(EP

),lo

ng-

term

reve

rsal

inre

turn

s(R

EV

),in

vest

men

t-to

-ass

ets

(IA

),ch

an

ges

in

pro

per

ty,

pla

nt,

and

equ

ipm

ent

scal

edby

asse

ts(P

IA),

and

inve

nto

rygr

owth

(IV

G).λM

den

otes

the

mark

etri

skp

rice

esti

mate

s(i

n%

).

λFFR

,λTB

,λTERM

,λDEF

,λdp,λpe,λvs,

andλSVAR

den

ote

the

risk

pri

cees

tim

ates

for

the

inn

ovat

ion

son

the

Fed

fun

ds

rate

,T

-bil

l

rate

,te

rmsp

read

,d

efau

ltsp

read

,lo

gd

ivid

end

yie

ld,

smoot

hed

log

pri

ce-t

o-ea

rnin

gsra

tio,

valu

esp

read

,an

dst

ock

mark

etva

rian

ce,

resp

ecti

vely

.B

elow

the

risk

pri

cees

tim

ates

are

dis

pla

yedt-

stat

isti

csb

ased

onS

han

ken

’sst

and

ard

erro

rs(i

np

are

nth

esis

).T

he

colu

mn

lab

eled

χ2

pre

sents

the

stat

isti

c(fi

rst

lin

e)an

das

soci

ated

asym

pto

ticp-v

alu

e(i

np

aren

thes

is)

for

the

test

on

the

join

tsi

gnifi

can

ceof

the

pri

cin

ger

rors

.T

he

colu

mn

lab

eled

R2 OLS

den

otes

the

cros

s-se

ctio

nal

OL

SR

2.

Th

esa

mp

leis

1972

:01–2013:1

2.

Itali

c,un

der

lin

ed,

an

d

bol

dt-

rati

osd

enot

est

atis

tica

lsi

gnifi

can

ceat

the

10%

,5%

,an

d1%

level

s,re

spec

tive

ly.

λM

λFFR

λTB

λTERM

λDEF

λdp

λpe

λvs

λSVAR

χ2

R2 OLS

PanelA

(VW

)

10.5

89−

0.32

10.1

36−

0.00

6−

0.43

80.

441

−1.

480−

0.01

757.

67

0.7

3(2.84

)(−

2.84

)(2.3

5)

(−0.2

4)(−

0.90

)(0.9

2)(−

2.29

)(−

0.3

0)

(0.6

32)

20.5

94−

0.19

50.1

31−

0.00

6−

0.48

50.

504

−1.

637−

0.00

665.

75

0.7

1(2.86

)(−

2.2

5)(2.3

9)

(−0.2

9)(−

1.01

)(1.0

7)(−

2.62

)(−

0.1

0)

(0.3

48)

PanelB

(EW

)

10.7

34−

0.35

80.1

630.0

920.7

95−

1.11

1−

2.03

50.1

87

47.

19

0.8

8(3.12

)(−

2.12

)(1.68

)(2.5

6)

(1.3

0)(−

1.62

)(−

2.67

)(1.81

)(0.9

18)

20.7

61−

0.31

80.1

570.0

920.7

43−

1.06

7−

2.00

10.1

97

49.

37

0.8

7(3.25

)(−

2.5

3)(1.73

)(2.75

)(1.2

3)(−

1.58

)(−

2.64

)(1.9

7)(0.8

77)

63

Short Term Interest Rate and Stock Market

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