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Short-term interest rates and stock market anomalies
Paulo Maio1 Pedro Santa-Clara2
This version: October 20163
1Hanken School of Economics. E-mail: [email protected] School of Business and Economics, NBER, and CEPR. E-mail: [email protected] 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
1
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,
2
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),
3
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).
4
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).
5
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.
6
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.
7
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.
10
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.
11
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,
12
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
13
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