Short-term interest rates and stock market anomalies

Paulo Maio1 Pedro Santa-Clara2

This version: October 20163

1Hanken School of Economics. E-mail: paulofmaio@gmail.com2Nova School of Business and Economics, NBER, and CEPR. E-mail: psc@novasbe.pt3We thank an anonymous referee and seminar participants at the Arne Ryde workshop, Finance

Down Under Conference, Rothschild Caesarea Center Conference, and the SGF Conference forhelpful comments on earlier drafts. We are grateful to Kenneth French, Amit Goyal, RobertShiller, Robert Stambaugh, and Lu Zhang for making available stock market data. Any remainingerrors are our own.

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 model

is consistent with Merton’s ICAPM framework and the key risk factor is the innovation on a

short-term interest rate—the Fed funds rate or the T-bill rate. This model explains a large

fraction of the dispersion in average returns of the joint market anomalies. Moreover, the

model compares favorably with alternative multifactor models widely used in the literature.

Hence, short-term interest rates seem to be relevant for explaining several dimensions of

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

is driven by an observable state variable. 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

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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

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);

equity 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). 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%. When the model is forced to price all 70

portfolios simultaneously, and thus the joint seven CAPM anomalies, we obtain a cross-

sectional R2 estimate of 58%. To account for the evidence showing that small caps represent

the biggest challenge for asset pricing models (see Fama and French (2012, 2015)), we also

use equal-weighted portfolios. The model does even better in pricing the equal-weighted

portfolios. Specifically, in the augmented test with all 70 portfolios the fit of the ICAPM is

around 67% for both versions of the model.

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. Following Maio

and Santa-Clara (2012), 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 investment

opportunities, measured by the excess market return and economic activity.

We compute an extensive sensitivity analysis to the performance of the ICAPM. Specif-

ically, we use alternative interest rate factors, estimate the model for a restricted sample,

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test the ICAPM with additional market anomalies, employ additional methods of statistical

inference for the risk price estimates, estimate the ICAPM by allowing for an unrestricted

zero-beta rate, employ double-sorted portfolios, use additional model evaluation metrics,

and estimate the ICAPM in the respective covariance and SDF representations. Overall, our

results are maintained, or even reinforced, under these alternative methods and empirical

designs.

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

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). 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 several equity-based sources of systematic risk (other than the market factor),

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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.1 In this sense, our simple 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

some of 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.2

Therefore, 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. 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

1There 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)).

2However, we do not claim that our simple model represents a new workhorse multifactor asset pricingmodel that explains nearly all the CAPM anomalies. For example, untabulated results suggest that themodel is not successful in pricing portfolios sorted on price momentum (Jegadeesh and Titman (1993)) orprofitability (Haugen and Baker (1996)). We claim instead that a rather simple model, based only on onemacro variable outside the equity market—the Fed funds rate or T-bill rate—makes a significant step forwardin explaining (in an economically consistent way) some of the most prominent market anomalies.

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), Campbell (1996), and more recently, Chen(2003), Brennan, Wang, and Xia (2004), Campbell and Vuolteenaho (2004), Guo (2006), Hahn and Lee(2006), Petkova (2006), Guo and Savickas (2008), Bali and Engle (2010), Botshekan, Kraeussl, and Lucas(2012), Garret and Priestley (2012), and Maio (2013a, 2013b).

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multifactor 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

significantly 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 asset growth, 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 present 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.

2 The model

2.1 A two-factor model

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

in discrete time to motivate our two-factor model. In the ICAPM, the additional factor(s)

(relative to the standard market factor from the baseline CAPM) represent the innovation(s)

in state variable(s) that forecast future changes in the investment opportunity set. The

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

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

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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that state variable (that is, under the CAPM framework). The reason is that the first asset

does not provide a hedge against future negative shocks in the expected return of aggregate

wealth since it offers high returns when expected wealth returns are also high.7 Therefore, a

rational risk-averse investor is willing to hold such an asset only if it offers a higher expected

return in excess of the risk-free rate (relative to the asset that is uncorrelated with the state

variable). This additional risk premium is captured by the term, λzβi,z, where λz stands

for the risk price associated with the hedging factor (the innovation in the state variable)

and βi,z is the respective factor loading for asset i. Thus, this additional risk premium stems

from aversion to intertemporal risk (unfavorable changes in future investment opportunities),

which a risk-averse investor wants to hedge.

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 return) within the ICAPM.8 There is previous evidence in the return predictability

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

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

Mercer, 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, (1)

E(Ri,t+1 −Rf,t+1) = λMβi,M + λTBβi,TB, (2)

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.

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

7In this reasoning, we are assuming that the state variable covaries positively with future investmentopportunities.

8Brennan 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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widely used in the literature. The first model is the baseline CAPM from Sharpe (1964)

and Lintner (1965), which is nested on our ICAPM. The second model is the Fama and

French (1993, 1996) three-factor model (FF3, henceforth), which adds a size factor (SMB)

and a value-growth factor (HML) to the standard market factor. The next two models

are the four-factor models of Carhart (1997) (C4) and Pastor and Stambaugh (2003) (PS4),

which augment FF3 with a momentum factor (UMD) and a stock liquidity factor (LIQ),

respectively. The fifth model is 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. 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.9

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, and is suitable for models containing factors that are not

traded.10 Specifically, in the version based on FFR the factor betas are estimated from the

9CMA is constructed in a different way than IA and the same occurs for RMW in relation with ROE(see Hou, Xue, and Zhang (2015) and Fama and French (2015) for details).

10Since the innovation in the short-term interest rate does not represent an (excess) return, we cannot usethe time-series regression approach, employed in Fama and French (1993, 1996, 2015) among others, to testthe ICAPM.

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time-series multiple regressions for each testing asset,

Ri,t+1 −Rf,t+1 = δi + βi,MRMt+1 + βi,FFRFFRt+1 + εi,t+1, (3)

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

rate.

The expected return-beta representation is estimated in a second step by the OLS cross-

sectional regression,

Ri −Rf = λMβi,M + λFFRβi,FFR + αi, (4)

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.11

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 and

matches the zero-beta rate, 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.12

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), (5)

where K denotes the number of factors (K = 2 in the ICAPM), and α is the (N × 1) vector

11If 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.

12Another 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).

8

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 (5) 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 small individual pricing errors. In both cases, the

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.13

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

represent a good approximation to the true finite sample distribution, we conduct a bootstrap

simulation to produce more robust p-values for the tests of individual significance of the factor

risk prices and also for the χ2-test. The bootstrap simulation consists of 5,000 replications

where the excess portfolio returns and risk factor realizations are simulated (with replace-

13Since 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 at 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.

9

ment from the original sample) independently and without imposing the ICAPM’s restric-

tions. Hence, the data-generating process is simulated under the assumption that the factors

are independent from the testing assets (“useless factors” as in Kan and Zhang (1999)).14

Moreover, the bootstrap accounts for the contemporaneous cross-correlation among the test-

ing assets, which often exhibit a small factor structure (see Lewellen, Nagel, and Shanken

(2010)).15 The full details of the bootstrap simulation are available in the online appendix.

As in Kan and Zhang (1999), Jagannathan and Wang (2007), Lewellen, Nagel, and

Shanken (2010), and Adrian, Etula, and Muir (2014), we evaluate the statistical significance

of the sample R2 estimates by computing empirical p-values based on the bootstrap simula-

tion described above. The empirical p-values correspond to the fractions of artificial samples

in which the pseudo explanatory ratio is higher than the sample estimate. By computing

the empirical p-values we account for the sampling error associated with the sample R2OLS

estimates. More specifically, under the assumption of independence between returns and

factors, 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 (2016) (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 ), (6)

which is relevant for 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

14Kan 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.

15Campbell and Vuolteenaho (2004) and Lioui and Maio (2014) also conduct bootstrap simulations inorder to obtain more “robust” standard errors for the risk price estimates in cross-sectional asset pricingtests.

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following equation,

Ri −Rf = RMβi,M + SMBβi,SMB +HMLβi,HML + αi,C , (7)

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.16

3.2 Data

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

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 St. Louis Fed. The data

on the risk factors associated with the CAPM, FF3, C4, and FF5 models described 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

16In other words, the risk price estimates for these factors can be different than the respective factormeans, which makes R2

OLS the correct metric to assess the explanatory power of our two-factor model.

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

period 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)).

Table 1 presents 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 presented in Panel B 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

(not tabulated) is not excessive (0.78). Moreover, 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.

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 low

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,

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1987)) refers to a pattern in which 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-growth 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 (Belo and Lin (2011)).

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

resent value-weighted 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 ra-

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

assets (PIA); and inventory growth (IVG). All the portfolio return 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 2 presents the descriptive statistics for high-minus-low spreads in returns between

the last and first deciles within each portfolio group. Most of these anomalies are economi-

cally 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 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

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deciles (0.81), while the two value-growth 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.

4 Main empirical results

In this section, we conduct the asset pricing cross-sectional tests 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 3. We can see that for all portfolio groups 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 pre-

mia among the deciles within each portfolio group. Moreover, only 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 indicated by the negative R2 estimates. Therefore, these

results 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 are displayed in Table 4. To save space, we only

present and discuss the results for the version based on FFR. The findings for the version

based on TB are qualitatively similar and are discussed in the online appendix. The results

for the test with the BM portfolios show that the ICAPM explains a large fraction of the

14

dispersion in average returns of the value-growth portfolios with an R2 estimate above 60%.

This sample R2 is statistically significant at the 5% level based on the empirical p-value

obtained from the bootstrap simulation. The point estimate for the “hedging” risk price,

λFFR, is negative and strongly statistically significant (1% level based on the empirical p-

value and 5% level based on the asymptotic p-value). When we use an alternative measure of

value-growth (EP), the model shows an increase in fit relative to the estimation with the BM

deciles (from 62% to 78%), with this sample R2 being strongly significant (1% level). This

suggests 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 from Table 2).

The estimate for λFFR is significant at the 5% level (based on both types of standard errors).

The results for the test with the equity duration deciles (DUR) show that the ICAPM

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

70% and this estimate is largely significant (5% level). The estimate for the hedging risk price

is negative and strongly significant (1% based on the empirical p-value). In the estimation

with the REV portfolios, we can see that the ICAPM offers a good explanatory power,

with a coefficient of determination of 52% (significant at the 5% level), which nevertheless

represents a lower fit than in the tests with either BM or DUR. The 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).

The rest of the table shows that the two-factor model has a large explanatory power

for investment-based anomalies. In the case of the IA portfolios the estimate for the cross-

sectional coefficient of determination is 65%, which is significant at the 1% level and repre-

sents a higher fit than in the test with the REV portfolios. In the estimation with the PIA

deciles, the explanatory power (59%) is only marginally lower than in the test with the IA

deciles (and is significant at the 5% level). The risk price estimates corresponding to the

interest rate factor are negative and strongly significant (at the 5% or or 1% level) when it

comes to price both of these portfolio groups.

15

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

for the ICAPM among all seven portfolio groups. The R2 estimate is 20%, and this point

estimate is not statistically significant at the 5% level (although there is significance at the

10% level). Despite this fact, the interest rate factor remain priced with a risk prices estimate

that is significant at the 5% level (based on both types of p-values). Across all the seven

cross-sectional tests, one fact remains robust: The ICAPM passes 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 market anomalies jointly. This is especially relevant

in our case since some of the risk price estimates differ significantly in magnitude across the

testing portfolios. Specifically, 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 for the augmented test indicate that the R2 estimate is 58%, which is way

above the corresponding explanatory ratio obtained for the baseline CAPM reported above

(-59%). This estimate of the explanatory ratio is strongly significant as indicated by the

empirical p-value around zero. Moreover, the model is not formally rejected by the χ2-

statistic as indicated by the asymptotic p-value quite close to one (while the empirical p-

value is clearly above 10%). Following the evidence for the individual anomaly tests, the

estimate for the interest rate risk price is negative and strongly significant (1% level) based

on both types of statistical inference. Hence, by using a larger cross-section of testing assets

we obtain higher statistical power in the risk price estimates in comparison to the single-

anomaly estimates.

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

16

separately and jointly the seven CAPM anomalies considered in the 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.17

4.2 Individual pricing errors

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

fit of the ICAPM it still 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 portfolios. We can see that the magnitudes of the pricing errors as-

sociated 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 pricing error is significantly below

the cross-section mean (among all the deciles in the 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 misspecification is

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

17Brennan, 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.

17

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 other six portfolio classes.

4.3 Which factors explain the anomalies?

The results above suggest that the innovation in the Fed funds 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 aggregate 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 are shown in Table 5.18 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 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 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 gap associated with the market factor has the wrong sign,

18The results for the version based on TB are similar and are available upon request.

18

which confirms why the baseline CAPM is not successful in pricing the value anomaly in our

sample. Hence, it is the innovation in the Fed funds 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 pricing

errors 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, the gap D1−D10 in average excess returns (low equity

duration stocks minus high duration stocks) is about 0.52% per month, which 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 at 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 spread: 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

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 corresponding to the DUR deciles.

For the IA portfolios 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 again 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 errors is only 0.15% per month. In

19

the case of the PIA portfolios, the gap (D1 −D10) in average 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 spread in risk premium of 0.34%. This

results in an average pricing error gap of 0.23%, less than half the raw spread in returns.

In the case of the IVG portfolios, 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. Overall, the results of this subsection confirm the evidence above that the driving

force of the ICAPM in pricing the seven portfolio groups is the interest rate factor.

4.4 Factor betas and intuition

The analysis above shows that the innovation in the Fed funds 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 corresponding dispersion in average returns.

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

displayed in Figure 3. 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 FFR (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-

20

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. 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.

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.19

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

19These results are consistent with the evidence in Maio (2014a) 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

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

interest rates, and less subjective to changes in long-term discount rates.20

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

described in Section 2. The results are presented in Table 6. To save space, we only report

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

20This 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

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 where all

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

OLS), in line with the evidence

presented in Maio (2016).

By comparing the R2OLS values associated with 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 (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 these 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%.

In sum, 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 (2016) 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.21

21In comparison, in the alternative multifactor models several factors drive the explanatory power (forexample, in the case of FF5 the HML factor drives value-based anomalies while the CMA factor helps

23

4.6 Equal-weighted portfolios

In this subsection, we estimate the ICAPM by using equal-weighted portfolios. Employing

equal-weighted portfolios enables us to address the evidence that small caps represent the

biggest challenge for asset pricing models (see Fama and French (2012, 2015)). The results

for the cross-sectional tests involving the equal-weighted portfolios are displayed 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, across all the

seven portfolio classes it turns out that the sample R2 are greater than the corresponding

values in the estimation with value-weighted deciles. Specifically, the explanatory ratios vary

between 61% (test with PIA) and 94% (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 ratio around 80% (compared to 20% in the benchmark case). 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 portfolios, 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 at the 5% or 1% level, the exception being the

tests with the BM deciles.22 Moreover, the model passes the specification test in all cases as

shown by the p-values clearly above 5%.

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

67%, which is significant at the 1% level. Moreover, the risk price estimates for the hedging

factor are negative and strongly significant (5% or 1% level) for all testing assets. The strong

performance of the ICAPM is remarkable given that these anomalies are more accentuated

among small stocks, thus imposing a bigger challenge on asset pricing models.

explaining the investment-based anomalies).22The existence of wide confidence intervals for the cross-sectional R2 in the tests with BM deciles confirms

the 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 these portfolios.

24

4.7 Sensitivity analysis

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

presented and discussed in detail in the internet appendix. Here, we only briefly summarize

the key findings.

First, we estimate both versions of the ICAPM by using alternative definitions of the

interest rate factors—the first-difference in interest rates. The results show that the ex-

planatory ratios and risk price estimates are very similar to the benchmark case, and this

holds for both versions of the model.

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. 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 performance of the model.

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.23

Fourth, we conduct alternative methods of statistical inference for the risk price estimates

associated with the two-factor model. Specifically, we compute the t-ratios employed in Fama

and MacBeth (1973) and Jagannathan and Wang (1998), and the misspecification-robust t-

ratios from Kan, Robotti, and Shanken (2013). The results show that the interest rate risk

23Unreported results show that our model is not successful in explaining the price momentum (e.g., Je-gadeesh and Titman (1993) and Fama and French (1996)) and profitability anomalies (e.g., Haugen andBaker (1996) and Novy-Marx (2013)). The reason is that there is not enough dispersion in the interest ratebetas with the right sign among those portfolios. Specifically, past short-term winners have more positiveinterest rate loadings than past losers, which interacted with the negative interest rate price of risk, generatesa spread in risk premia in the wrong direction to match the raw momentum profits. In any case, a singlefactor like the innovation in FFR or TB (the role of the market factor is only in matching the cross-sectionalmean risk premium) can’t have a large explanatory power for a large number of anomalies. This stems fromthe small 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).

25

prices are strongly significant (1% level) by using the three types of t-ratios.

Fifth, we estimate the ICAPM by specifying a second-pass OLS cross-sectional regression

with an unrestricted zero-beta rate as in Kan, Robotti, and Shanken (2013) and others. The

results indicate that the estimates for the excess zero-beta rate are largely insignificant in

both versions of the model, hence, the model is able to match the zero-beta rate. On the

other hand, the interest rate factors remain priced.

Sixth, we estimate the model by using double-sorted portfolios on size and other anoma-

lies. Specifically, we use 25 portfolios sorted on size and book-to-market ratio, 25 portfolios

sorted on size and asset growth, and 25 portfolios sorted on size and long-term return reversal.

The results suggest that the ICAPM offers a high explanatory power for these three groups

of double-sorted portfolios as indicated by the R2 estimate of 50% in the joint estimation

with the 75 portfolios.

Seventh, we compute two additional metrics proposed by Kan, Robotti, and Shanken

(2013) to evaluate the performance of the ICAPM—an alternative cross-sectional OLS R2

(ρ2 and associated specification tests) and the Qc-statistic, which tests the null hypothesis

that the pricing errors are jointly equal to zero. Overall, these two additional evaluation

metrics, and associated model specification tests, lend further support to the ICAPM.24

Eighth, we define and estimate the ICAPM in expected return-covariance representation

by using first-stage GMM as in Cochrane (2005). The results show that the covariance risk

prices for the interest rate factors are negative and strongly significant, in line with the

results for the benchmark beta representation.

Finally, we estimate the ICAPM in the stochastic discount factor (SDF) representation by

using first-stage GMM. The results suggest that our ICAPM is correctly specified based on

the Hansen and Jagannathan (1997) distance metric. Moreover, by employing the sequential

procedure proposed in Gospodinov, Kan, and Robotti (2014) we find that the interest rate

factor is priced.

24We thank the referee for suggesting this analysis.

26

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 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 hence should earn a lower

total risk premium than an asset that is uncorrelated with the state variable. This means a

negative risk premium associated with the hedging factor, which in turn implies a negative

risk price for that factor (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 estimates for the interest rate

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

out that the corresponding state variable (Fed funds rate) should forecast a decline in future

aggregate wealth. To test whether FFR forecasts 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 + bqFFRt + ut+1,t+q, (8)

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.25 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,

25We only report the results associated with FFR. The results for TB are quite similar and availableupon request.

27

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 incorporate 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 are 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.26

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

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 + bqFFRt + ut+1,t+q, (9)

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 8. We can see that FFR

forecasts a decline in the equity premium at all horizons. Yet, the associated slopes are not

statistically significant at any horizon, and the R2 assumes 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

26In related work, Boons (2016) 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.

28

and Bekaert (2007)), suggesting that the forecasting power of these variables has declined in

recent years.27

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. The largest forecasting power is achieved at q = 12 and q = 24, with R2 estimates

of 8%.

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

the coefficients associated with FFR 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%.28

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

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

27By 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.

28The slopes associated with short-term interest rates remain significant in most cases when we add thecurrent values of IPG or ∆e as predictors.

29

(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. (2016), 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 aggregate 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 9. 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. 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

30

significantly lower than our benchmark ICAPM as indicated by the sample R2 around 30%.

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

ICAPM specifications, with a cross-sectional R2 of 66% and this estimate is significant at

the 1% level. This represents a marginally larger fit than the benchmark ICAPM, yet, we

must stress that the explanatory power of the value spread for some of the portfolios (BM

and EP) is somewhat mechanical, exactly in the same way as the role played by HML.29

We conduct some robustness checks to the analysis of the alternative ICAPM models.

The results are presented and discussed in detail in the internet appendix. First, we estimate

an augmented ICAPM specification that includes all alternative state variables. The results

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. We also

conclude 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 specification.

Second, we use the two additional performance metrics, ρ2 and Qc, to evaluate the

performance of the alternative ICAPM models. The results show that only two of the

alternative ICAPM models (those based on TERM and vs) are not rejected by the R2-

based test (ρ2 = 1) at the 5% level.

Third, we compute the asymptotic pairwise tests of equality of ρ2, proposed by Kan,

Robotti, and Shanken (2013), among all ICAPM models (including our two-factor model).

The results indicate that the ICAPM based on FFR and TB dominate (in statistical terms)

most of the other ICAPM models when we use standard errors computed under the as-

sumption of correctly specified models. When we employ standard errors computed under

29The 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)).

31

the assumption of a misspecified model there are no statistical significant differences in ρ2

among any two models. This should arise from the large standard errors associated with ρ2

for some of the alternative models.

Fourth, we estimate the alternative ICAPM models in the SDF representation by using

the Hansen-Jagannathan (HJ) method. The results indicate that most alternative ICAPM

versions are rejected by the HJ-specification test at the 5% level. The only exception is the

version based on the value spread. This suggests that nearly all alternative ICAPM models

are misspecified. 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.

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%. When the model is forced

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

obtain a cross-sectional R2 estimate of 58%. The model does even better in pricing the

32

equal-weighted portfolios. Specifically, in the augmented test with all 70 portfolios the fit

of the ICAPM is around 67%. 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. Following Maio and Santa-Clara (2012), 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 investment opportunities, measured by the excess market return and economic

activity.

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

literature. Specifically, the ICAPM outperforms the models from Hou, Xue, and Zhang

(2015) and Fama and French (2015). 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 some of the equity-based

factors in the alternative models. Our model also outperforms other factor models 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.

33

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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, respec-

tively. RMW and CMA denote the Fama-French profitability and investment factors. The

sample is 1972:01–2013:12. φ designates the first-order autocorrelation coefficient. Panel B

contains the correlations between the interest rate factors and each of the alternative factors.

Panel A

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

Panel B

FFR TB

RM −0.14 −0.14SMB −0.05 −0.04HML −0.05 −0.08UMD 0.04 0.08LIQ 0.01 −0.02ME −0.04 −0.01IA 0.01 −0.00ROE 0.06 0.06RMW 0.04 0.06CMA −0.05 −0.06

42

Table 2: Descriptive statistics for spreads in returnsThis table reports descriptive statistics for the “high-minus-low” spreads in returns associ-

ated with different portfolio classes. The portfolios are deciles sorted on book-to-market ra-

tio (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 as-

sets (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

43

Table 3: 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. λMdenotes 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) and empirical p-values

(in brackets) obtained from a bootstrap simulation. The column labeled χ2 presents the statistic

(first row) and associated asymptotic (in parenthesis) and empirical (in brackets) p-values for the

test on the joint significance of the pricing errors. The column labeled R2OLS denotes the cross-

sectional OLS R2 with the corresponding empirical p-value shown in brackets. R2C 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.λM χ2 R2

OLS R2C

BM 0.70 16.00 −0.41 −0.29(3.33) (0.067)[0.000] [0.063] [0.289]

DUR 0.68 22.31 −0.85 −0.62(3.23) (0.008)[0.000] [0.045] [0.342]

EP 0.67 23.12 −0.74 −0.54(3.21) (0.006)[0.000] [0.049] [0.344]

REV 0.71 12.74 −0.18 −0.05(3.37) (0.175)[0.000] [0.093] [0.156]

IA 0.60 21.56 −1.18 −0.98(2.92) (0.010)[0.000] [0.069] [0.334]

PIA 0.57 26.36 −0.43 −0.38(2.78) (0.002)[0.000] [0.056] [0.220]

IVG 0.60 18.10 −0.55 −0.45(2.91) (0.034)[0.000] [0.073] [0.204]

All 0.65 105.58 −0.59 −0.45(3.12) (0.003)[0.001] [0.092] [0.218]

44

Table 4: Factor risk premia for ICAPMThis 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 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. 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) and empirical p-values (in brackets) obtained from a bootstrap

simulation. The interest rate factor is the innovation on the Fed funds rate. The column labeled χ2 presents

the statistic (first row) and associated asymptotic (in parenthesis) and empirical (in brackets) p-values for

the test on the joint significance of the pricing errors. The column labeled R2OLS denotes the cross-sectional

OLS R2 with the corresponding empirical p-value shown in brackets. 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 R2OLS

BM 0.61 −0.67 5.25 0.62(2.89) (−2.25) (0.731)[0.000] [0.006] [0.190] [0.043]

DUR 0.61 −0.78 4.48 0.70(2.89) (−2.49) (0.811)[0.000] [0.003] [0.343] [0.013]

EP 0.59 −0.80 4.87 0.78(2.73) (−2.19) (0.771)[0.000] [0.011] [0.324] [0.004]

REV 0.64 −0.58 5.29 0.52(3.01) (−1 .94 ) (0.726)[0.000] [0.019] [0.208] [0.043]

IA 0.57 −0.82 3.28 0.65(2.69) (−2.18) (0.916)[0.001] [0.009] [0.587] [0.005]

PIA 0.56 −0.80 4.46 0.59(2.68) (−2.47) (0.814)[0.001] [0.003] [0.545] [0.010]

IVG 0.61 −0.44 7.37 0.20(2.91) (−2.01) (0.498)[0.000] [0.017] [0.216] [0.065]

All 0.60 −0.71 36.09 0.58(2.86) (−2.85) (0.999)[0.003] [0.002] [0.422] [0.000]

45

Table 5: 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). 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

Panel A (BM)

D1 0.38 0.66 −0.20 −0.08D10 1.07 0.58 0.39 0.11Dif. −0.69 0.08 −0.59 −0.18

Panel B (DUR)

D1 0.83 0.64 0.17 0.02D10 0.31 0.75 −0.30 −0.14Dif. 0.52 −0.11 0.48 0.16

Panel C (EP)

D1 0.38 0.70 −0.30 −0.02D10 0.96 0.57 0.28 0.11Dif. −0.58 0.14 −0.58 −0.13

Panel D (REV)

D1 0.96 0.78 0.07 0.11D10 0.55 0.79 −0.18 −0.06Dif. 0.41 −0.01 0.25 0.17

Panel E (IA)

D1 0.76 0.61 0.12 0.03D10 0.34 0.72 −0.26 −0.12Dif. 0.42 −0.11 0.38 0.15

Panel F (PIA)

D1 0.85 0.59 0.16 0.10D10 0.36 0.67 −0.17 −0.14Dif. 0.49 −0.08 0.34 0.23

Panel G (IVG)

D1 0.76 0.62 −0.05 0.19D10 0.40 0.71 −0.17 −0.14Dif. 0.36 −0.09 0.11 0.33

46

Tab

le6:

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).λM

,λSM

B,λH

ML

,λUM

D,

andλLIQ

den

ote

the

risk

pri

cees

tim

ate

s(i

n%

)fo

rth

em

ark

et,

size

,va

lue,

mom

entu

m,

an

dli

qu

idit

yfa

ctors

,

resp

ecti

vely

.λM

E,λIA

,an

dλROE

rep

rese

nt

the

risk

pri

ces

ass

oci

ate

dw

ith

the

Hou

-Xu

e-Z

han

gsi

ze,

inve

stm

ent,

an

dp

rofi

tab

ilit

yfa

ctors

,re

spec

tive

ly.

λRM

Wan

dλCM

Ad

enot

eth

eri

skp

rice

esti

mat

esfo

rth

eF

am

a-F

ren

chp

rofi

tab

ilit

yan

din

vest

men

tfa

ctors

.B

elow

the

risk

pri

cees

tim

ate

sare

dis

pla

yed

t-st

atis

tics

bas

edon

Sh

anke

n’s

stan

dar

der

rors

(in

pare

nth

esis

)an

dem

pir

icalp-v

alu

es(i

nb

rack

ets)

ob

tain

edfr

om

ab

oots

trap

sim

ula

tion

.T

he

colu

mn

lab

eled

χ2

pre

sents

the

stat

isti

c(fi

rst

row

)an

dass

oci

ate

dasy

mp

toti

c(i

np

are

nth

esis

)an

dem

pir

ical

(in

bra

cket

s)p-v

alu

esfo

rth

ete

ston

the

join

t

sign

ifica

nce

ofth

ep

rici

ng

erro

rs.

Th

eco

lum

nla

bel

edR

2 OLS

den

ote

sth

ecr

oss

-sec

tional

OL

SR

2w

ith

the

corr

esp

on

din

gem

pir

icalp-v

alu

esh

own

in

bra

cket

s.R

2 Cre

pre

sents

the

con

stra

ined

cros

s-se

ctio

nalR

2.

Th

esa

mp

leis

1972:0

1–2013:1

2.

Itali

c,u

nd

erli

ned

,an

db

oldt-

rati

os

den

ote

stati

stic

al

sign

ifica

nce

atth

e10

%,

5%,

and

1%le

vel

s,re

spec

tive

ly.

λM

λSM

BλHM

LλUM

DλLIQ

λM

EλIA

λROE

λRM

WλCM

Aχ

2R

2 OLS

R2 C

10.

60−

0.0

10.

4687.0

40.

70

0.67

(2.91

)(−

0.0

2)(2.99

)(0.0

51)

[0.0

02]

[0.9

94]

[0.0

01]

[0.0

55]

[0.0

00]

20.

600.1

20.4

20.5

974.

23

0.74

0.75

(2.89

)(0.6

6)(2.76

)(1.6

1)(0.2

28)

[0.0

02]

[0.7

25]

[0.0

03]

[0.2

08]

[0.1

06]

[0.0

00]

30.

61−

0.00

0.46

−0.1

386.

60

0.71

0.65

(2.92

)(−

0.00

)(2.98

)(−

0.2

7)(0.0

45)

[0.0

02]

[0.9

99]

[0.0

01]

[0.8

90]

[0.0

51]

[0.0

00]

40.

610.0

90.3

2−

0.21

86.5

50.

68

0.30

(2.94

)(0.4

7)(2.99

)(−

1.15

)(0.0

46)

[0.0

01]

[0.8

05]

[0.0

01]

[0.4

38]

[0.0

50]

[0.0

00]

50.

61−

0.03

0.43

−0.1

30.

2483.

49

0.74

0.52

(2.93

)(−

0.13

)(2.79

)(−

0.9

4)(2.3

1)(0.0

61)

[0.0

02]

[0.9

44]

[0.0

03]

[0.5

61]

[0.0

29]

[0.0

58]

[0.0

00]

47

Table 7: Factor risk premia for ICAPM: EW portfoliosThis table reports the estimation and evaluation results for the two-factor ICAPM. The estima-

tion procedure is the two-pass regression approach. The test portfolios are equal-weighted 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 includ-

ing 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) and empirical p-values (in brackets) obtained from a

bootstrap simulation. The interest rate factor is the innovation on the Fed funds rate. The column

labeled χ2 presents the statistic (first row) and associated asymptotic (in parenthesis) and empir-

ical (in brackets) p-values for the test on the joint significance of the pricing errors. The column

labeled R2OLS denotes the cross-sectional OLS R2 with the corresponding empirical p-value shown

in brackets. 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

BM 0.47 −1.08 5.97 0.75(1 .77 ) (−2.17) (0.650)[0.028] [0.005] [0.103] [0.112]

DUR 0.51 −1.01 1.96 0.94(1.97) (−2.51) (0.982)[0.017] [0.002] [0.586] [0.001]

EP 0.53 −0.85 5.08 0.83(2.15) (−2.66) (0.749)[0.008] [0.001] [0.155] [0.020]

REV 0.58 −0.83 6.14 0.75(2.42) (−2.78) (0.631)[0.003] [0.000] [0.196] [0.023]

IA 0.42 −1.64 2.57 0.85(1.26) (−2.45) (0.958)[0.226] [0.007] [0.647] [0.006]

PIA 0.46 −1.44 4.54 0.61(1.49) (−2.65) (0.805)[0.164] [0.006] [0.432] [0.043]

IVG 0.41 −1.59 3.04 0.81(1.25) (−2.24) (0.932)[0.247] [0.014] [0.665] [0.004]

All 0.50 −1.07 34.57 0.67(1 .91 ) (−2.83) (1.000)[0.080] [0.005] [0.332] [0.003]

48

Table 8: 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 variable is the Fed funds rate (FFR).

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)

bq −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 0.01

Panel B (IPG)

bq −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 0.03

Panel C (∆e)

bq −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 0.12

49

Table 9: Factor risk premia for alternative ICAPMThis table reports the estimation and evaluation results for alternative two-factor ICAPM models.

The estimation procedure is the two-pass regression approach. The test portfolios are 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). λM and λz denotes the risk price

estimates (in %) for the market and “hedging” factors, respectively. Below the risk price estimates

are displayed t-statistics based on Shanken’s standard errors (in parenthesis) and empirical p-values

(in brackets) obtained from a bootstrap simulation. ˜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 row) and associated asymptotic (in parenthesis)

and empirical (in brackets) p-values for the test on the joint significance of the pricing errors. The

column labeled R2OLS denotes the cross-sectional OLS R2 with the corresponding empirical p-value

shown in brackets. 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

˜TERM 0.61 0.33 45.42 0.32(2.93) (2.88) (0.984)[0.002] [0.002] [0.302] [0.011]

DEF 0.65 0.04 95.24 −0.52(3.14) (1.63) (0.016)[0.000] [0.148] [0.054] [0.265]

dp 0.60 −1.93 79.81 −0.20(2.87) (−2.96) (0.155)[0.002] [0.001] [0.099] [0.152]

pe 0.60 2.05 76.18 −0.16(2.87) (3.01) (0.232)[0.002] [0.001] [0.104] [0.130]

vs 0.60 −2.12 78.08 0.66(2.88) (−3.47) (0.189)[0.003] [0.000] [0.098] [0.000]

SV AR 0.60 −0.28 76.38 −0.21(2.91) (−2.71) (0.227)[0.001] [0.003] [0.100] [0.152]

50

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)This figure plots the pricing errors (in % per month, Panels A, C, E, and G), and respective t-statistics (Pan-

els B, D, F, and H) of different decile portfolios associated with the ICAPM based on the Fed funds rate. The

portfolios are 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.

51

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)This 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 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.52

Panel A: BM Panel B: DUR

Panel C: EP Panel D: REV

Panel E: IA Panel F: PIA

Panel G: IVG

Figure 3: Regression betas for FFRThis figure plots the beta estimates associated with the innovation in the Fed funds rate, FFR. The portfo-

lios are deciles sorted on book-to-market (BM), equity duration (DUR), earnings-to-price (EP), and long-term rever-

sal 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.53