Expected Returns and Risk in the Stock Market
M. J. Brennan∗
Anderson School, UCLA
Alex P. Taylor†
Manchester Business School
January 9, 2016
Abstract
In this paper we present new evidence on the predictability of stock returns, and examine
the extent to which time variation in expected returns is due to time variation in risk
exposure or due simply to mispricing or sentiment. In doing this we develop two new
models for the prediction of stock market returns, one risk-based, and the other purely
statistical. The pricing kernel model expresses the expected excess return as the covariance
of the market return with a pricing kernel that is a linear function of portfolio returns. The
discount rate model is based on the log-linear present value model of Campbell and Shiller
and predicts the expected excess return directly as a function of weighted past portfolio
returns. For aggregate market returns the two models provide independent evidence of
predictable variation in returns, with R2 of 6 − 8% for 1-quarter returns and 10-16% for
1-year returns. For value-based arbitrage portfolios such as HM L we do find evidence of
predictability from the discount rate model that is not captured by the risk-based model and
this additional predictability is related to measures of time-varying sentiment and liquidity.
Keywords: Predictability, Expected returns, Risk, Sentiment
JEL Classification Codes: G12, G14, G17
∗
Michael Brennan is Emeritus Professor at the Anderson School, UCLA, Professor of Finance, Manchester
University.
†
Corresponding Author: Alex P. Taylor, Accounting and Finance Group, Manchester Business School, The
University of Manchester, Booth Street West, Manchester, M15 6PB, England, e-mail: [email protected],
Tel: +44(0)161 275 0441, Fax: +44(0)161 275 4023.
1
1
Introduction
The two principal issues that arise in the literature on stock market predictability are whether
returns are predictable, and whether the predictability arises from time variation in risk or
whether it is a function of sentiment and waves of optimism and pessimism. The issue of
whether returns are in fact predictable is a vexed one in view of the highly persistent nature
of, and slight theoretical justification for, most of the predictor variables that have been used.
These purely statistical concerns, which are exacerbated by the data mining that is implicit in
the broad search for significant predictors by different researchers, are only partially alleviated
by ‘out of sample’ tests. Nevertheless, despite some prominent papers challenging the existence
of predictability,1 the professional consensus seems to be moving towards the view that returns
are predictable. We provide new and strong support for this view. Moreover, since our predictive
variables depend only on lagged portfolio returns, it is a simple matter to bootstrap the data
under the null hypothesis of no predictability to obtain powerful tests of the null hypothesis.
This is not generally possible for models that rely on macro-economic or accounting data series
as predictor variables.
There is less prior evidence on whether the measured time variation in returns can be
attributed to time variation in risk, simply because many of the tests of predictability are
motivated by either purely statistical models or simple present value models that exclude risk
variation from consideration. Significant exceptions include Merton (1980) and Ghysels et
al. (2005) who show that time-variation in market returns is driven, at least in part, by
time-variation in market volatility, and Scruggs (1998) and Guo et al. (2009) who consider
time-varying returns in relation to an ICAPM type pricing kernel. In this paper we show that
time-variation in the covariance of the market return with a pricing kernel that is spanned
either by the three Fama-French factors, or by the returns on the market portfolio and three
portfolios formed on the basis of lagged dividend yield, can explain 14-16% of 1-year returns
in sample, and 9-13% out of sample. We find no evidence that expected returns on the market
portfolio are influenced by time-varying sentiment or liquidity.
Our empirical analysis relies on two new models for the prediction of the market expected
(excess) return or discount rate. The first model expresses the expected excess return as the
covariance of the market return with a pricing kernel that is a linear function of portfolio returns.
The second model predicts the expected excess return directly as a function of weighted past
portfolio returns.
The first model, which we refer to as the pricing kernel model, constrains the predictors with
the discipline of an asset pricing model and assumes that the time variation in expected returns
is driven solely by time variation in risk, where the risk of the market return is measured by its
covariance with a portfolio which captures innovations in the pricing kernel. This model does
not seem to be consistent with irrationality except insofar as it can be shown that the pricing
kernel itself reflects ‘irrational’ concerns. This is a difficult task, even for the sceptic of rational
pricing, if we accept the tag de gustibus non est disputandum, since the pricing kernel mirrors
the marginal utility and therefore the tastes of the representative agent.
The second model, which we refer to as the discount rate model, exploits the accounting
identity of the log-linear present value model of Campbell and Shiller (1988), and combines
this with the assumption of a factor structure of returns to identify shocks to the discount rate.
1
e.g. Goyal and Welch (2008).
2
Our analysis rests on the intuition that past returns will be negatively related to future returns
insofar as realized returns reflect shocks to discount rates. However, since returns reflect news
about future cash flows as well as about discount rates, the past series of returns on a portfolio
of stocks provides only limited information about future expected returns. Our solution to
this problem of contamination of the discount rate signal by cash flow news is to form a linear
combination of portfolio returns to ‘soak up’ the cash flow news which enables us to identify the
shocks to discount rates. By assuming a stochastic process for the discount rate and aggregating
these shocks over time, we are able to arrive at an estimate of the current expected excess return.
This approach is a purely statistical one that makes use of the accounting identity of Campbell
and Shiller but that contains no economic assumptions. It is consistent with time variation in
expected returns being generated by cycles of excessive optimism and pessimism and changing
market liquidity,2 as well as being generated by time variation in risk and/or risk aversion.
The two models offer largely independent evidence on the existence of predictability since,
although they are similar in that they both extract information from past portfolio returns
and assume the same AR(1) process for expected returns, they imply different sets of predictor
variables. Therefore, evidence of a common predictable component from the two quite different
models is strong support for predictability. Moreover, comparing the results for the two models
casts light on the question on whether the predictability is rational or not. Under rational
pricing the discount rate model and the pricing kernel model imply the same expected return
series and, absent empirical problems concerning the ability of the selected portfolios to satisfy
the spanning requirements of the two models, any predictability that is captured by the discount
rate model but not by the pricing kernel model is in a sense outside the classical asset pricing
framework.
For the market portfolio, the expected return series generated by the two models of the
discount rate are related, with the highest correlation between the time series of quarterly risk
premium estimates from the two models being 0.62. Bootstrap simulations show that under
the null hypothesis the chance of observing the levels of predictability that we find for the two
models and a correlation between the two model predictions as high as we find is less than 1%.
The pricing kernel model estimates seem to be superior. The maximum in-sample R̄2 obtained
for this model is 8.3% for quarterly returns as compared with 5.6% for the discount rate model.
When the models are used to predict 1-year excess returns the in-sample R̄2 for the parsimonious
version of the pricing kernel model is 15.7% as compared with 9.9% for the discount rate model.
Out of sample, the discount rate model does not improve on a naive forecast, while the pricing
kernel model reduces the naive forecast error of one year market excess returns by 9-13%.
The simple CAPM predicts that the market return spans the pricing kernel. While our
findings are not consistent with this, we do find evidence that the component of the pricing
kernel that is associated with the market return itself contributes significantly (at the 1% level)
to time variation in expected market returns. We also find evidence that the projection of the
pricing kernel onto the three Fama-French (1993) factors provides strong predictive power for
market returns.3 This is consistent with these factors capturing important components of the
pricing kernel, and provides evidence against those who attribute the empirical success of the
Fama-French 3-factor model in pricing the cross-section of stock returns simply to data-mining
(Mackinlay (1995)) or market inefficiency (Lakonishok et al (1994)).
2
Cf. Amihud (2002).
Fama and French (1995, 1996) argue that the value and size premia move closely with investment opportunities.
3
3
We also examine the predictive properties of a restricted pricing kernel model in which the
variables that enter the pricing kernel are motivated by Merton’s (1973) ICAPM. As Brennan
et al. (2004) and Nielsen and Vassalou (2006) have shown, under reasonable assumptions the
pricing kernel can be shown to depend on only the market return, the Sharpe ratio, and the
riskless interest rate. When this restriction is imposed, the model predicts quarterly excess
returns with an R2 of 5.6% as compared with an R2 of 8.3% for the unrestricted model.
We have noted that the pricing kernel model attributes all time-variation in expected returns
to time-variation in risk, while the discount rate model takes no account of the source of the
time-variation in expected returns, so that differences in the expected returns from the two
models potentially reflect such factors as time-varying sentiment or liquidity. We find that
for returns on the market portfolio the only evidence of predictability that is not related to
changing risk is a small high frequency component associated with lead-lag effects in returns.
Otherwise the two models yield very similar components of variation with persistence ρ ≈ 0.70.8 for quarterly data. Neither the Amihud (2002) measure of illiquidity nor the Baker-Wurgler
(2006) measure of sentiment are significantly related to the expected market returns.
For returns on the spread portfolios, SM B, HM L, and HM Z (the spread between high
and zero dividend yield portfolios), the story is more complex. The discount rate model finds
considerable time series variation in the expected returns on these portfolios, explaining 12-16%
of the 1-year returns. The results for the SM B portfolio are similar to those for the market
portfolio in that there is no evidence that the expected returns are affected by time-varying
sentiment or liquidity. However for the arbitrage portfolio HM Z, 30% of the difference between
the expected return estimated from the unconstrained discount rate model and the expected
return from the risk-based model is attributable to time-variation in the Baker-Wurgler (2006)
measure of sentiment. For HM L, time variation in both sentiment and the Amihud (2002)
measure of illiquidity contribute to explaining the difference in the expected returns from the
two models.
The paper is organized as follows. In Section 2 we discuss how the paper is related to the
extensive existing literature that is concerned with return predictability. Section 3 develops the
two models of expected returns and Section 4 describes the data. Section 5 presents the main
empirical results for the pricing kernel model. Section 6 is concerned with the estimation of the
discount rate model. Section 7 compares the time series of risk premium estimates from the two
models. Section 8 reports further empirical findings, and Section 9 concludes.
2
Related Literature
The pricing kernel model originates with Merton (1980) who uses the simple CAPM pricing
kernel to forecast the expected return on the market portfolio: under the CAPM the covariance
of the pricing kernel with the market return is proportional to the variance of the market
return. Subsequent efforts to model the equity risk premium in terms of the volatility of
the market return have met with mixed success. Several authors have reported a positive
but insignificant relation between the variance of the market return and its expected value;
others find a significant but negative relation; and some find both a positive and a negative
relation depending on the method used.4 More recently, Ghysels et al. (2005) establish a
4
For references, see Ghysels et al. (2005).
4
significant positive relation between the monthly market risk premium and the variance of
returns estimated using daily data. We confirm the existence of a significant positive risk return
relation at the quarterly frequency, with the market variance being captured by a distributed
lag on past squared quarterly returns. Bandi et al. (2014) report a positive relation between
a low frequency component of market return variance and a similarly slow moving component
of the market excess return although the economic model that gives rise to this relation is not
specified.
Campbell and Vuolteenaho (2004), Brennan, Wang, and Xia (2004), and Petkova (2006)
all show that the value premium is correlated with innovations in their measures of investment
opportunities. Scruggs (1998) employs a two-factor ‘ICAPM-type’ pricing kernel in which the
second factor is the return on a bond portfolio to capture time-variation in the equity premium;
he finds that the equity premium is related to the covariance of the market return with the
bond return, although the results are sensitive to the assumption of a constant correlation
between bond and stock returns as pointed out by Scruggs and Glabadanidis (2003). Guo et
al. (2009) follow a similar approach, using the return on the Fama-French HM L portfolio as
a second factor, and find that the lagged market volatility and covariance of its return with
the return on the Fama-French (1993) HM L portfolio predict market excess returns over the
period 1963-2005, but not over earlier periods. Unlike these papers which allow the predictor
variable to be pre-determined by an ICAPM interpretation of their role in cross-sectional asset
pricing tests, our general pricing kernel model identifies directly the component of the pricing
kernel that is correlated with market returns.
Ross (2005) develops an upper bound on the predictability of stock returns which depends
on the volatility of the pricing kernel, and tighter bounds that require further specification
of the pricing kernel have been provided by Zhou (2010) and Huang (2013). These tighter
bounds ‘provide a new way to diagnose asset pricing models’ (Huang, 2013, p1). Our levels
of predictability fall well within the Ross bounds, and the level of predictability that we find
from the pricing kernel model is precisely that delivered by the (partial) specifications of the
pricing kernel that we propose. Our approach does not require the specification of the complete
stochastic discount factor and our goal is not to test any particular asset pricing model or
stochastic discount factor specification.
Our discount rate model builds on the distinction between between discount rate news and
cash flow news that was developed by Campbell (1991), and was used in a similar context
by Campbell and Ammer (1993) and Campbell and Vuolteenaho (2004). Their approach is
to extract the discount rate news from the coefficients of a VAR in which the state variables
are variables that are known to predict stock returns. In contrast, our state variables are
constructed as distributed lags of past returns on portfolios that are chosen to capture shocks
to the discount rate.
Our focus on the information about discount rate innovations that is contained in portfolio
returns is related to Pastor and Stambaugh (2009) who use prior beliefs on the correlation between discount rate shocks and portfolio returns to develop a Bayesian approach to predictive
regression systems. However, while we focus on the information contained in past portfolio returns, Pastor and Stambaugh are concerned primarily with the predictive power of the dividend
yield and the cay variable.
Our findings are also related to research on the predictive ability of lagged equity portfolio
returns. Hong, Torous and Valkanov (2007) regress market returns on previous period industry
5
and market returns and find evidence that some industries (when combined with the lagged
market return) lead the stock market at the monthly frequency; they interpret this as evidence
of slow diffusion of information. In this paper our concern is with more persistent variation in
the equity premium corresponding to business cycle or even lower frequencies. Our discount rate
model, which nests the simple approach of regressing the market return on the lagged portfolio
return, is also able to accommodate a stationary but persistent process for the expected return.
Eleswarapu and Reinganum (2004) find that 1-year stock market returns are negatively
related to the past return on glamour stocks. We find little evidence of this in our sample
period. Ludvigson and Ng (2007) estimate the principal components of a large number of equity
portfolio returns and other conditioning variables and examine the ability of these components
to predict market returns.
The discount rate model is also related to research by van Binsbergen and Koijen (2009)
that uses a Kalman filter to estimate the expected market return and dividend growth rate
from market returns and the price-dividend ratio. However, we use a linear combination of
portfolio returns instead of the dividend growth rate to filter out the cash flow news. Like van
Binsbergen and Koijen (2009), we assume that the expected excess return follows an AR(1)
process. Cochrane (2008) also provides statistically significant evidence on the predictive role
of the dividend yield by using the implication of the present value relation that the dividend
yield must predict either returns or dividend growth and showing that it does not predict the
latter.
In an interesting recent paper, Kelly and Pruitt (2013) combine cross sectional information
on book-to-market ratios to forecast 1-year stock returns and obtain out of sample R2 as high as
13%. In this paper we use both quarterly and 1-year stock returns: for 1-year returns we obtain
in sample R2 of 14-16%, and out of sample R2 of 9-13%. Like ours, their estimates of the time
series of expected excess returns have low persistence relative to previous findings. However,
the predictability that we identify is not strongly related to that of the Kelly and Pruitt (2013)
model.5 The cay predictor of Lettau and Ludvigson (2001) also identifies a component of the
expected return that is essentially orthogonal to our model predictions.
Two important issues that arise in the extensive literature on predictability are the inference
problems caused by highly persistent predictor variables, and the effects of data-mining arising
from the collective search for predictor variables by the research community. Stambaugh (1999),
Torous et al. (2004), and Campbell and Yogo (2006) develop test procedures that take account
of persistence. Foster et al. (1997) analyze the effect of overfitting data in the context of
predictive regressions. Ferson et al (2003) examine the interaction of data-mining and spurious
regression for the case of highly persistent expected returns. The same concerns over datamining potentially arise in our empirical analysis. However, a major advantage of our approach
is that it allows us to assess whether the levels of predictability that we find can be explained
by overfitting of the data. First, since we use only portfolio returns as predictor variables,
it is straightforward to compute significance levels by simulation under the null hypothesis of
no predictability or serial independence of returns. In contrast, when macro-economic series
are used as predictor variables more extended assumptions are required to simulate the data
under the null hypothesis. Secondly, whereas the previous literature has involved search over
an undefined domain of potential predictors which does not lend itself to an assessment of the
effects of data mining on levels of significance, in our approach for a candidate set of spanning
5
The correlation between the predicted excess return series is less than 0.25.
6
portfolios the search is over a well-defined set of predictor variables characterized by a single
weighting parameter, β. This allows us to assess the effects of data-mining on significance levels.
Our analysis indicates that the level of predictability found cannot be explained by data-mining
or the persistence of the predictor variables.
3
Two Models of Expected Returns
In this section we derive two different models of expected returns. Both are constructed by
summing up past shocks: the first, the pricing kernel model, sums up past shocks to the
covariance of the pricing kernel with the market return; the second, the discount rate model,
sums up past shocks to the discount rate. Both models assume that the equity premium follows
an AR(1) process. The first model assumes that we can find a set of portfolios whose time
varying beta coefficients span the time-varying loading of the pricing kernel on the return on
the market portfolio. The second assumes that we can find a set of portfolios that spans the
space of aggregate cash flow and discount rate shocks.
To see the relation between the two models of the expected market excess return let RM,t+1
denote the excess return on the market portfolio from time t to t + 1, and let m̃t+1 denote the
pricing kernel at time t + 1. Then it follows from the definition of the pricing kernel that αM,t,
the expected excess return on the market portfolio at time t, is given by:
αM,t = −covt (m̃t+1 , RM,t+1)
(1)
where we have imposed the normalization that Et (m̃t+1 ) = 1.
We can write the kernel, m̃t+1 , as the sum of a component that is a time-varying linear
function of the market return, mt+1 , and a component that is orthogonal to the market return,
ηt+1 :
m̃t+1 = am (t) + bm (t)RM,t+1 + ηt+1
≡ mt+1 + ηt+1
(2)
where covt(ηt+1 , RM,t+1) = 0, and bm (t) captures time variation in the sensitivity of the pricing
kernel to the market return.6 Then
2
αM,t = −covt (bm(t)RM,t+1 , RM,t+1) ≡ −bm (t)σM,t
(3)
so that time variation in the equity premium is controlled by the component of the pricing
kernel that is correlated with market returns.
The pricing kernel model assumes that bm (t) can be written as a fixed linear combination
of the loadings of a set of portfolio returns on the market return. We call these portfolios ‘
(pricing kernel) beta-spanning portfolios’. Thus write Rp,t, the return on spanning portfolio
p, p = 1, · · ·, P as:
6
Time variation in am (t) and bm (t) implies that the pricing kernel (2) defines a ‘conditional factor model’(Cf.
Cochrane, 2002, Ch. 8.)
7
Rp,t+1 = ap(t) + bp(t)RM,t+1 + up (t + 1)
(4)
Then the pricing kernel model assumes that bm (t) = −ΣPp=1 δpc bp(t) for a set of constant portfolio
weights δpc , so that the sensitivity of the pricing kernel to market returns, bm(t), can be expressed
c
as a linear combination of the portfolio loadings on the market return, −ΣM
p=1 δp bp (t). Equation
(3) shows that if the expected market return is to be a time-varying function of its variance,
c
then bm (t) must be time-varying, and the requirement that bm (t) = −ΣM
p=1 δp bp (t) for a set
c
of constant portfolio weights δp , then means that the beta coefficients of the beta-spanning
portfolios also be time-varying.7
The discount rate model on the other hand assumes that innovations in αM,t, and therefore
in bm (t)vart(RM,t+1), can be expressed as a linear combination of the innovations in the returns
d
on a (possibly different) set of (factor) spanning portfolios, ΣM
p=1 δp (Rp,t+1 − Et [Rp,t+1 ]), for a
constant set of portfolio weights δ d .
The pricing kernel model illustrates the intimate nature of the relation between cross-section
asset pricing and asset return dynamics, since the pricing kernel is the basis of cross-section asset
pricing while the dynamics of the covariance between the market return and the pricing kernel
describe the dynamics of the equity premium.8 We should note however, that our procedure
identifies only mt+1 , the component of the pricing kernel whose covariance with the market
return exhibits time series variation, rather than the whole pricing kernel that is required to
price the cross-section of asset returns.
The relation between cross-sectional asset pricing and time-variation in asset returns that
appears in the ICAPM has led Campbell (1993, pp499-500) to note that ‘the intertemporal
model suggests that priced factors should be found not by running a factor analysis on the
covariance matrix of returns, nor by selecting important macro-economic variables. Instead,
variables that have been shown to forecast stock-market returns should be used in cross-sectional
asset pricing studies.’9 We shall proceed in part in the reverse direction, by testing whether
variables (portfolio returns) that have shown to be important in cross-section asset pricing, and
which therefore belong in the pricing kernel, also have important information for forecasting
market returns.
3.1
The Pricing Kernel and Expected Returns
Equation (3) expresses the arithmetic expected excess return on the market portfolio as the
negative of the conditional covariance of the market return with the pricing kernel:
αM,t = −covt (bm(t)RM,t+1 , RM,t+1)
where bm (t) is the conditional sensitivity of the innovation in the pricing kernel to the market
return, the pricing kernel ’beta’. Assume that there exists a set of beta-spanning portfolios,
p = 1, · · · , P , and constant portfolio weights δpc , p = 1, · · · , P , such that :
7
Appendix A provides sufficient conditions for the existence of a set of loading-spanning portfolios.
Ross (2005), Zhou (2010) and Huang (2014) make the same point.
9
In contrast, Fama writes of ‘the multi-factor models of Merton (1973) and Ross (1976) ‘that they are an
empiricist’s dream ... that can accommodate... any set of factors that are correlated with returns’. (Fama 1991,
p. 1594)
8
8
bm (t) = −ΣPp=1 δpc bp(t)
(5)
where bp(t) is the beta coefficient of beta-spanning portfolio p. Then
αM,t = covt(ΣPp=1 δpc bp(t)RM,t+1, RM,t+1) = ΣPp=1 δpc covt (Rp,t+1 , RM,t+1)
(6)
We assume that the risk premium, and therefore the covariance of the market return with the
pricing kernel, follows an AR(1) process:
αM,t = ā + ρα[αM,t−1 − ā] + ξαt
(7)
A sufficient condition for (7) is that the conditional covariances of the beta-spanning portfolio returns in equation (6) follow AR(1) processes with the same persistence parameter, ρα,
and with innovations which are equal up to a constant to the product of the spanning portfolio
and market returns:
covt(Rp,t+1 , RM,t+1) = ap + ρα[covt−1 (Rp,t, RM,t) − ap ] + Rp,tRM,t
(8)
Then
αM,t = ā + ρα[
P
X
δpc covt−1 (Rp,tRM,t) − ā] + ξαt
(9)
p=1
where ā =
PP
c
p=1 δp ap ,
and ξα,t =
PP
c
p=1 δp Rp,t RM,t .
Then the market risk premium can be written as an affine function of the geometrically
weighted average of past values of the weighted average of products of spanning portfolio and
market returns:
αM,t = ā+
∞
X
(ρα)j ξα,t−j = ā +
j=0
= ā+
P
X
∞
X
j=0
(ρα)j
P
X
δpc Rp,t−j RM,t−j
(10)
p=1
δpc xcp,t(ρα)
(11)
p=1
where xcp,t (ρα) ≡
P∞
j=0 (ρα )
j
[Rp,t−j RM,t−j ].
Then the predictive system for the market excess return becomes:
P
X
RM,t+1 = a0 +
δpc xcpt(β) + t+1
(12)
p=1
xcpt(β)
=
∞
X
β s Rp,t−s RM,t−s
s=0
9
(13)
where we have substituted β for ρα to ensure consistency of notation with the discount
rate model that follows. An attractive feature of the model is that the parameters a0 , δpc , β
can be estimated relatively easily since, given an estimate of β, the estimation reduces to
a standard predictive regression with predictor variables xcpt(β). We discuss the estimation
details in Section 5.
3.2
The Discount Rate Model
Our analysis for this model is motivated by the log-linear model of Campbell and Vuolteenaho
(2004)10 which decomposes the unexpected return on stocks into cash flow news and discount
rate news.
Let µt be the expected log excess return on the market portfolio, so that the realized excess
return can be written as:
rM,t+1 = µt + t+1
(14)
We assume that µt follows an AR(1) process so that:
µt = µ̄ + ρ[µt−1 − µ̄] + zt
(15)
We refer to zt , the innovation in the expected market return, as the discount rate news.
Our second assumption is that there exists a set of P well diversified (factor) spanning
portfolios whose excess returns, rpt, p = 1, · · ·, P follow an exact factor model and span the
space of innovations in cash flows and discount rate news, so that
rpt = βp0 + kp µt−1 +
M
X
βpj yjt + γpzt
(16)
j=1
where yjt (j=1,· · ·,M) denotes innovations in common cash flow factors, and zt is the discount
rate news.
We are implicitly assuming that shocks to the risk free rate are small and can be subsumed
in the cash flow news.11 The second term in (16) captures time variation in the expected returns
on the spanning portfolios which depend on variation in the systematic expected return factor,
µt−1 . The third term corresponds to cash flow news which we allow to have a factor structure,
and the fourth term is the effect of aggregate discount rate news. The number of spanning
portfolios, P , is equal to one plus the number of cash flow innovations: P = M + 1.
Consider a ‘z-mimicking’ portfolio whose weights on the P spanning portfolios, δp , are such
P
P
that Pp=1 δpd βpj = 0, j = 1, · · ·, M ; Pp=1 δpd γp = 1. This ensures that the z-mimicking portfolio
10
See also Campbell and Shiller (1988), Campbell (1991), and Campbell and Ammer (1993).
We have repeated the analysis using gross returns in place of excess returns: the proportion of the return
that is attributable to discount rate news is largely independent of which definition of returns is used, which is
consistent with shocks to the risk free rate playing only a minor role.
11
10
return loads only on the discount rate news, zt . Then it is easily shown that the discount rate
news can be written as a linear function of the return on the z-mimicking portfolio:
zt =
δ0d
+
P
X
δpd rpt − wµt−1 ,
(17)
p=1
where δ0d = −
PP
d
p=1 δp βp0 ,
w=
PP
d
p=1 δp kp .
Then combining equations (14) and (17), the process of the market expected log return is:
µt = (1 − ρ)µ̄ + (ρ − w)µt−1 + δ0d +
P
X
δpd rpt
(18)
p=1
Substituting recursively for µt−j , the expected log excess market return, µt , may be written
as a linear function of geometrically weighted past returns on the P spanning portfolios:
µ̄(1 − β − w) + δ0d X d d
+
δp xpt(β)
(1 − β)
P
µt =
(19)
p=1
where β ≡ ρ − w, and
xdpt (β) =
∞
X
β s rp,t−s
(20)
s=0
Combining (14) with (19), the log excess return on the market portfolio may be written as:
rM,t+1 = a0 +
P
X
δpd xdpt(β) + t+1
(21)
p=1
where the predictor variables, xpt (β), are weighted averages of past log returns on the
spanning portfolios as shown in (20).12 Note that, comparing equations (15) and (18), the
weighting parameter, β, may deviate from the true persistence of the expected return, ρ: we
report both parameters below.
Equation (21) is the basis of our discount rate model based predictive regression. However,
in most of our empirical analysis we shall substitute arithmetic returns for the logarithmic
returns that follow strictly from the Campbell log-linearization.
Then the predictive system becomes:
RM,t+1 = a0 +
P
X
δpd xdpt(β) + t+1
(22)
β s Rp,t−s
(23)
p=1
xdpt (β) =
∞
X
s=0
12
We shall henceforth use the term ‘predictor variable’ for xdpt , the variables formed as distributed lags on the
returns on the spanning portfolios, rpt .
11
where RM,t and Rp,t denote the (arithmetic) excess returns on the market portfolio and portfolio
p respectively.
Note first that the only difference between the predictive systems from the discount rate
based model, defined by equations (22) and (23), and the pricing kernel based model, defined
by (12) and (13), lies in the definition of the predictive variables which we have denoted by
xdpt (β) and xcpt (β). The former is a weighted average of past returns on portfolio p, while the
latter is a weighted average of past products of portfolio returns and market returns. Secondly,
while we have developed the two models for the prediction of the market excess return, it
is clear that the same approaches can be used, mutatis mutandis, to predict returns on any
portfolio, by regressing the portfolio returns on a set of predictor variables formed as geometric
weighted averages either of past returns on a set of spanning portfolios (xdpt(β)) or of the
products of past returns on a set of spanning portfolios with the return on the portfolio whose
return is to be predicted (xcpt (β)). In section 8.2 we shall apply the models to the prediction
of returns on certain arbitrage or spread portfolios. Thirdly, we have no a priori method
of identifying the spanning portfolios. Therefore we shall consider different candidate sets
of spanning portfolios whose choice is discussed in the following section. While we refer to
them as ‘spanning portfolios’, they are in fact only candidate sets of spanning portfolios whose
adequacy must be judged by the predictive performance of the models. Finally, we observe
that, in contrast to earlier studies that use financial ratios such as interest rates and dividend
yields as predictors, our predictor variables, xdpt(β) and xcpt(β), are constructed simply from
past returns on the portfolios. This will facilitate tests of the model.
4
Data
The market excess return, RM,t, is defined as the difference between the return on the Standard
and Poor’s 500 portfolio for quarter t and RF t, which is the risk free rate for the quarter, taken
as the return on a 3-month Treasury Bill. The S&P500 return is taken from CRSP, and the
Treasury Bill rate series is from the Federal Reserve Bank of St Louis. We use quarterly data
on portfolio returns from 1927.3 to 2010.4. The estimation period is 1946.1 to 2010.4, while
the earlier returns are used to calculate the value of the predictor variables at the beginning of
1946.1.
The predictor variables for the discount rate model, xdpt (β), are formed from the lagged
excess returns on the spanning portfolios using equation (23). The predictor variables for the
pricing kernel model, xcpt(β), are formed from lagged cross products of market excess returns
and excess returns on the spanning portfolios using equation (13).13
We consider the following proxies for the spanning portfolios in both the discount rate and
the pricing kernel models: (i) the market portfolio (M); (ii) the three Fama-French portfolios
(FF3); (iii) the market portfolio and and 3 portfolios formed on the basis of dividend yield
(3DP): the highest and lowest yielding quintiles of dividend paying stocks, and a portfolio of
non-dividend paying stocks. We consider in addition two expanded sets of spanning portfolios:
6BM − S is the market portfolio plus the 6 Fama-French size and book-to-market sorted
portfolios; and 6DP consists of the market portfolio and 5 quintiles of stocks ranked by dividend
13
Note that for the discount rate model the spanning portfolios are assumed to span the space of discount rate
and aggregate cash flow shocks, while for the pricing kernel model the betas of the loading-spanning portfolios
are assumed to span the beta of the pricing kernel.
12
yield plus the zero dividend yield portfolio. Data on these portfolio returns were taken from
the website of Ken French.
The market portfolio was included as a proxy for a spanning portfolio because of the mixed
prior evidence that the market variance predicts future market returns discussed above, and
because this portfolio spans the pricing kernel under the CAPM and so provides a natural
baseline for the pricing kernel model. The three Fama-French portfolios were included because
of the empirical success of the FF model in pricing the cross-section of asset returns, which
suggests that these portfolios belong in the pricing kernel. The dividend yield portfolios were
included because portfolios that differ in yield will have different durations and have different
sensitivities to discount rate shocks, including those caused by shocks to the covariance between
the pricing kernel and asset returns.14 The expanded sets of portfolios were included because in
the discount rate model the dimensionality of cash flow shocks seems likely to be greater than
the number of portfolios required to span the pricing kernel. In addition, expanding the set of
spanning portfolios helps us to assess the spanning adequacy of our original candidate sets of
spanning portfolios.
We also compare our predictor variables with variables that have been used earlier in the
literature. Following Goyal and Welch (2008), these include, the Dividend (Earnings) yield on
the market portfolio, which is defined as the log of the ratio of dividends (earnings) on the
S&P500 over the past 12 months to the lagged level of the index; the Book-to-market value
ratio for the Dow Jones Industrial Average; the Stock Variance which is the sum of squared
daily returns on the S&P500 index over the previous quarter; the 3 month Treasury Bill rate;
the Long Term Yield which is the yield on long term US government bonds; the Term Spread
which is the difference between the Long Term Yield and the Treasury Bill rate; Inflation which
is the one month lagged inflation rate; the Default Yield Spread which is the difference between
BAA and AAA-rated corporate bond yields; and cay which is the consumption, wealth, income
ratio of Lettau and Ludvigson (2001). Fuller descriptions of these variables are to be found in
Goyal and Welch (2008) and the actual data series were taken from the website of Amit Goyal.
The small-stock value spread is often used as a state variable in models of predictable stock
returns as in Brennan et al. (2001), Cohen et al. (2003), and Campbell and Vuolteenaho
(2004). It is constructed from the book-to-market values of portfolios formed by a 2 by 3 sort
on size and book-to-market ratio, available from the website of Ken French. It is defined as the
log(BE/ME) of the small high-book-to-market portfolio minus the log(BE/ME) of the small
low-book-to-market portfolio. The book-to-market values for these portfolios are defined on a
yearly basis and the method described in Campbell and Vuolteenaho (2004) is used to construct
monthly values of the value spread.
Eleswarapu and Reinganum (2004) find that yearly stock market returns are negatively
related to the lagged returns on glamour stocks over the prior 36 month period. Following
Eleswarapu and Reinganum (2004) we consider five portfolios formed by sorting on the bookto-market ratio. The glamour portfolio is defined as the quintile with the lowest book-to-market
ratio and its cumulative log return over the past 36 months is used as the predictor variable.
We utilize the portfolio data from Ken French’s website rather than construct quintiles in the
slightly different manner described in Eleswarapu and Reinganum (2004). We find qualitatively
similar results to theirs, obtaining an R2 ≈ 5% over their sample period, and R2 ≈ 3% over our
sample period in regressions on annual excess log returns.
14
See Brennan and Xia (2006) and Lettau and Wachter (2007).
13
We compare our model predictions with those of Kelly and Pruitt (2013). For this purpose
we use their 12 month in-sample forecasts constructed using 100 portfolios.15 Finally, we explore
the dependence of the expected return series from the two models on measures of sentiment
and stock market illiquidity. For the former we use the Baker-Wurgler (2006) sentiment index
taken from http://people.stern.nyu.edu/jwurgler. Illiquidity is measured by the Amihud (2002)
measure of illiquidity and we are grateful to Sahn-Wook Huh for calculating this measure for
us.
Empirical Tests and Estimates of the Pricing Kernel model
5
We start by estimating equations (12) and (13) for the pricing kernel model. A non-linear
maximum likelihood estimator is implemented by choosing values of β, forming the predictor
variables from equation (13), and then running an OLS regression of market excess returns
on the predictor variables to estimate the regression coefficients, δ. The MLE estimator of β
is the value that minimizes the sum of squared residuals (or equivalently maximizes the R2 )
in (12). However, a problem with this MLE estimator is that, as Stambaugh (1986) shows,
the small sample bias in the R2 is increasing in the persistence parameter of the predictor
portfolio, ρ, and therefore in the weighting parameter β which is being estimated.16 The higher
bias in R2 associated with higher values of β will tend to result in estimates of β that are too
high. Therefore we employ a bias-adjusted procedure which is described in Appendix B. Having
estimated the parameters (a0 , δc , β), we form time series estimates of the expected market excess
P
return, α̂M,t = â0 + Pp=1 δ̂pc xcpt(β̂), and use these estimates to compute the autocorrelation of
the expected market excess return, ρα.
We use quarterly returns on the different sets of beta-spanning portfolios to form the predictor variables, xc (β), and report estimation results for prediction horizons of both 1-quarter
and 1-year for the sample period from 1946.1 to 2010.4
Table 1 reports the results of tests of predictability for the pricing kernel model for the
1-quarter and 1-year horizon, using different sets of spanning portfolios, along with estimates of
the weighting parameter, β, and the persistence parameter, ρα. The primary sets of spanning
portfolios are (i) the market portfolio (M ); (ii) the three Fama-French portfolios (F F 3); and (iii)
the market portfolio and the three dividend yield portfolios (3DP ). Panel A reports the results
for prediction of the quarterly excess return on S&P500 portfolio, and Panel B for prediction
of the 1-year return. Wc is the Wald statistic corrected for bias from an estimation that seeks
to maximize this bias-adjusted statistic.
For the 1-quarter horizon the null hypothesis of no predictability is rejected at the 1% level
for all three sets of spanning portfolios except F F 3 where the significance level is only 5%
when the Wald criterion is used. After bias correction the fraction of the variance of returns
explained by the model (R2c ) ranges from 4.1% for M to 8.3% for 3DP . The significance of
the results for the single spanning portfolio, M , implies that time variation in the volatility
of the simple CAPM pricing kernel has predictive power for the equity premium and that the
market excess return is predicted by a weighted average of past squared market returns. This
result contrasts with the findings of Goyal and Welch (2008), but is consistent with the results
15
We thank the authors for making these forecasts available to us.
Bootstrap simulations show that for the BM-S model the 5% critical values of R2 and the Wald statistic
start to rise rapidly once β exceeds about 0.9.
16
14
of Ghysels et al. (2005), although these authors estimate the market variance from daily data.
The weighting parameter, β, is 0.67±0.02 for F F 3 and 3DP , and the estimated autocorrelation,
ρα, is 0.82 ± 0.02; the values for M are somewhat lower. The autocorrelation implies a half-life
for shocks to the discount rate of about 3.25 quarters.
When the forecast horizon is extended to 1 year Panel B shows that the single predictive
variable of the M model is no longer significant under the R2 criterion, although the F F 3 and
3DP sets of spanning portfolios yield R2c of 14% and 15.7% respectively which are significant
at the 1% level. Under the Wald criterion the predictions are significant at the 5% level for all
three sets of spanning portfolios. As with the quarterly estimates, the estimates of β and ρα
for the 3DP and F F 3 spanning portfolios are very close, and the expected return estimates for
the different sets of spanning portfolios are very closely related: Panel B of Table 2 shows that
the correlation between the 3DP and F F 3 estimates is 0.91, while the correlations of these
estimates with the M estimates is 0.62 and 0.73 respectively.
If the sets of spanning portfolios we have selected do not in fact span the innovation in the
pricing kernel, we should expect that increasing the number of portfolios in the spanning set
would increase the predictive power of the model. To assess this we also fit the model using two
expanded sets of spanning portfolios: 6BM − S is the market portfolio plus six size and bookto-market portfolios and nests the F F 3 set of portfolios; similarly 6DP is the market portfolio
plus an expanded set of dividend yield sorted portfolios and nests 3DP . For the dividend yield
sorted portfolios, there is no evidence that spanning is improved with the larger set of portfolios:
as we move from 3DP to 6DP the corrected R2c actually falls for both quarterly and 1-year
return predictions. On the other hand, expanding the set of spanning portfolios from F F 3 to
6BM − S does improve the corrected R2c from 0.063 (0.140) to 0.073 (0.171) for the quarterly
(1-year) forecasts, suggesting that F F 3 may be too parsimonious to fully capture the pricing
kernel. Nevertheless, in the interests of parsimony and to minimize the perils of data-mining,
we focus our attention on the three primary sets of spanning portfolios for the pricing kernel
model.
Panel A of Table 2 reports the estimates of (a0 , δ c, β, ρα) from the quarterly return predictions for the three sets of spanning portfolios. Significance levels for the coefficients are
computed from standard errors calculated from bootstrap simulations under the alternative
hypothesis.17
Components of the pricing kernel that are significant at the 5% level or better include the
market portfolio (except for 3DP ) and the two other F F 3 portfolios (SM B only at the 10%
level), as well as the zero dividend yield portfolio. Their signs are generally consistent with prior
knowledge of the pricing kernel. Thus the positive coefficients on RM and HM L are consistent
with a positive risk premium for the market portfolio and for the HM L portfolio, while the
insignificant coefficient on SM B offers no support for a small firm premium. The results for the
3DP spanning portfolios suggest that there is a positive premium associated with covariance
with returns on a portfolio of high yield (value) stocks, and a negative premium associated with
covariance with zero yield (growth) stocks.
In summary, the results are consistent with time-variation in the covariance of the market return with the pricing kernel leading to variation in expected market returns. The high
correlation between the 3DP and F F 3 risk premium estimates, as well as the limited improvement from increasing the set of beta spanning portfolios, suggests that the betas of both the
17
Note that β is undefined under the null.
15
3DP and F F 3 sets of spanning portfolios do a good job of spanning the pricing kernel beta:
bM (t) ≈ Σ3p=1 δpc (t).
5.1
Expected return series
Panel B of Table 2 shows that the average risk premium predicted for each of the three sets
of spanning portfolio is 1.78% per quarter: this is equal to the sample average excess return.
The 3DP estimates yield the most variability in the risk premium: 2.54%, which is 1.4 times
the mean premium. The M estimates are much less variable: their standard deviation is only
1.71% while the standard deviation of the F F 3 estimates is 2.21%. Figure 1 plots the expected
1-year return series obtained using the three different sets of spanning portfolios. It is apparent
that the series based on F F 3 and 3DP track each other well, reaching low points around the
dot-com bubble and rising steeply in the wake of the financial crisis.
There are three pronounced peaks in the series. We list them in decreasing order of importance and for each we report equity premium estimates from the F F 3 (3DP ) models which can
be compared with the mean 1-year equity premium of 7.7%. In 2009 following the collapse of
Lehman Brothers in September 2008 the estimated premium was 37.9% (38.5%); in 1975 during
the inflationary recession following the first oil crisis, 29.2% (33.8%); in 1956 following a period
of heightened tension over Formosa, 30% (26%). These are very pronounced fluctuations in
the estimated equity premium, and it is striking that the two sets of spanning portfolios yield
relatively similar estimates.
Interestingly, the estimated equity premium from both models was negative from the third
quarter of 2000 to the last quarter of 2001:18 it seems that at least part of the runup in equity
prices around the turn of the millennium can be attributed to a sharp decline in the equity
premium.19
5.2
Out of Sample Tests
Table 3 reports the out of sample forecasting power of the pricing kernel model for different
horizons, using different sets of spanning portfolios. Results are reported for both quarterly
forecasts that are derived by estimating the model on quarterly returns and for 1-year forecasts
that are based on parameter estimates from fitting the model to one year excess returns. The
models are estimated initially over the period 1946.1 to 1965.4 and the parameter estimates are
used to forecast the market excess return for 1966.1. Then the estimation period is extended by
one quarter for the next forecast, and so on. For the multi-quarter and multi-year forecasts we
compare the sum of the realized excess returns over the next k quarters (years) with forecasts of
the sum based on the forecast of αM,t+1 and the parameters of the AR1 process estimated over
the same period. The table reports the ratio of the mean square model forecast errors to the
mean square error of a forecast that is based on the out of sample historical mean also starting
in 1928.2: a ratio less than unity implies that the model outperforms the naive historical mean
forecast. The 3DP model shows generally the strongest out of sample forecast power. For the
18
Boudoukh et al. (1993) report reliable evidence that the ex ante equity market risk premium is negative in
some states of the world.
19
It was in September 1999 that James K. Glassman and Kevin A. Hassett published their article Dow 36,000,
which argued that future dividends on the market should be discounted at a rate below the Treasury bond rate.
16
quarterly returns forecasts it reduces the error variance relative to the naive model by 3%,10%,
12% and 16% for forecasts of 1,2,3 and 4 quarters, and for 1-year returns by 13%, 13% and
7% for forecasts of 1, 2 and 3 years. This compares with the error variance reduction reported
by Kelly and Pruitt (2013) of ‘up to 13%’. The F F 3 quarterly return forecasts improve on
the naive forecast by 4-6% and the 1-year forecast improvement is 9%. Even using the single
spanning portfolio, M , reduces the forecast error by 2-3% for 1 and 2 quarter forecasts.
In order to assess the statistical significance of the relative error variance statistics for the out
of sample forecasts, the 10,000 samples of the market and portfolio returns were bootstrapped
under the assumption of no predictability as described in Appendix B, and the out of sample
forecast procedure was applied to the generated data. Significance is then determined by
comparing the sample statistic with the distribution of the bootstrapped statistics. The 3DP
and F F 3 forecast improvements for the quarterly forecasts and for the 1 year forecast are
significant at the 1% or 5% levels. The 3DP 1-year forecast for 2 years is also significant at the
5% level.
5.3
Comparison with other predictor variables
The extensive prior literature on stock return predictability demands that attention be given
to the performance of the pricing kernel model relative to that of earlier predictors that have
been proposed. Panel A of Table 4 reports the results of quarterly regressions of the 1 quarter
market excess returns for the period 1946.1-2009.4 on 13 different predictor variables that have
widely used.20 With the exception of the Lettau-Ludvigson (2001) cay and the Kelly-Pruitt
(2013) prediction (kp), the in sample R2 are less than 1%, despite the fact that there is a small
sample bias in the R2 for many of the predictor variables due to their high autocorrelations,
rho. To account for the small sample bias we report bootstrapped significance levels for the
regression coefficient which are indicated by stars in the table. For cay (kp) the in-sample R2
is 4.2% (2.3%) and the autocorrelation of the predictor is 0.925 (0.931). The out of sample
performance of the different predictors is represented by REVOOS , which is the ratio of the
variance of the prediction error yielded by rolling out of sample regressions starting in 1966.1 to
the error variance of a simple historical mean predictor. The historical mean and the predictive
models are estimated over the period from 1928.2 to the year before the forecast. A value of
REVOOS less than unity implies that the predictive model improves on the naive historical
mean forecast. Excluding cay and kp, only 4 of the predictors improve on the historical mean
and then only by modest amounts. cay reduces the forecast error variance by 2.1%, kp by 1.5%,
the Default Yield Spread by 1.3%, and the Glamor variable by 1.1%. For the other variables
the improvement is less than 1%. The two predictive variable model reported in Panel B is
the ICAPM motivated model of Guo and Savickas (2009) which performs poorly in our sample
period (which is longer than theirs).
Table 5 reports correlations between 4 quarter moving averages of these other predictor
variables and corresponding averages of the forecast equity premium from the pricing kernel
model using different sets of spanning portfolios. Considering only correlations that are greater
than 0.3 in absolute value, we see that the predicted premium from the CAPM kernel, M , has
correlations of -0.36 with Glamor, 0.55 with Stock Variance, and 0.46 with the Default Yield
Spread. The predictions from the F F 3 kernel have correlations of 0.41 and -0.40 with Stock
20
See Goyal and Welch (2008).
17
Variance and the T-bill yield and 0.33 with the Term Spread; the predictions of the 3DP kernel
have correlations of 0.36 and -0.34 with the Stock Variance and the Term Spread.
In Table 6 the risk premium estimates from the pricing kernel model for different sets of
spanning portfolios are compared with the 4 most significant ‘other predictors’ by regressing the
market excess returns on the competing predictors. Regressions 1-4 include the Dividend-price
ratio, Glamor, and the Kelly-Pruitt prediction along with the model risk premium estimates
for the three sets of spanning portfolios. None of the other predictors is significant while
the coefficients of the pricing kernel model predictors are highly significant and close to their
theoretical value of unity. Regressions 5-8 repeat the analysis when cay is added to the list of
other predictors. cay is highly significant with t-statistic in excess of 3.6 in all the regressions.
The model predictors remain highly significant in the presence of the cay variable and the
coefficients of the model predictions remain within one standard error of their theoretical value
of unity. It appears that the pricing kernel model predictors are capturing a component of time
variation in expected returns that is largely orthogonal to that captured by cay and the other
predictor variables.
6
Empirical Test and Estimates of the Discount Rate model
Tables 7 and 8 report the results of estimating equations (22) and (23) for the discount rate
model. The estimation procedure parallels that for the pricing kernel model described above
except that the predictor variables, xd (β), are geometrically weighted averages of simple portfolio returns rather than the products of the returns with the return on the market portfolio.
For the discount rate model the spanning portfolios are required to span the cash flow factors
as well as the discount rate news. Three or four portfolios may well be insufficient for this.
Therefore, in addition to the 3DP and F F 3 sets of spanning portfolios we include the two
expanded sets of spanning portfolios: 6BM − S is the market portfolio plus the 6 Fama-French
size and book-to-market sorted portfolios, while 6DP is the market portfolio plus 6 portfolios
formed on the basis of dividend yield.
We consider first the market portfolio, M , as the candidate single spanning portfolio. The
model is then unable to predict market returns and the estimation does not even converge
for 1-year return predictions. This is what we should expect since the returns on the market
portfolio alone cannot span the innovations in both cash flows and the discount rate, even if
there is only a single cash flow factor.21 This candidate spanning portfolio is therefore omitted
from the subsequent tables and we concentrate on sets of spanning portfolios with more than a
single member.
Considering next the quarterly return predictions for the other sets of spanning portfolios
reported in Panel A of Table 7, we find that when the spanning portfolios are either F F 3
or 6BM − S, both of which are based on size and book-to-market sorts, the model identifies
21
Menzly et al. (2004) present a model in which risk aversion and expected dividend growth are varying
stochastically: increases in expected dividend growth increase stock prices and are associated with increases in
discount rates; this induces a positive association between lagged market returns and expected future returns.
On the other hand, increases in risk aversion increase discount rates and reduce stock prices and therefore induce
a negative association between market returns and expected future returns. The two effects are offsetting so
that the net relation between market movements and future returns becomes insignificant. Lettau and Ludvigson
(2003) and van Binsbergen and Koijen (2009) provide empirical evidence of positive covariation between expected
dividend growth rates and discount rates which is consistent with Menzly et al.
18
a high frequency component of the variation in the discount rate. While the prediction for
F F 3 is significant at the 5% level, the persistence parameter for the estimated predicted excess
return, ρα , is only 0.268, implying a half-life of only about one half of a quarter or six weeks;
the persistence parameter obtained using the 6BM − S portfolios is a little higher but the
predictions are statistically insignificant. On the other hand, expanding the set of dividend
yield based spanning portfolios from 3DP to 6DP almost doubles the R2 ; the predictions are
significant at the 5% level for both 3DP and 6DP sets of spanning portfolios. The persistence
parameter for the predicted return series obtained using these dividend yield based sets of
spanning portfolios is in excess of 0.8 and is comparable to that reported for the pricing kernel
model in Table 1. Panel B shows that when the model is used to predict 1-year returns, only
the predictions based on the 6DP set of spanning portfolios are significant - at the 5% or 10%
level depending on the criterion. However, all sets of spanning portfolios except F F 3 identify
a component with a persistence around 0.8.
Table 8, which corresponds to Table 2 for the pricing kernel model, reports the full set of
parameter estimates for the 1-quarter implementation of the discount rate model. Virtually
none of the individual parameter estimates except β and ρα for 3DP and 6DP are significant
at the 5% level, and the only δ d parameter estimate that is significant is the coefficient of SM B
when the set of spanning portfolios is F F 3.
The out of sample predictive power of the model was estimated following the procedure
described in Section 5.2. However, in contrast to the results for the pricing kernel model the
out of sample discount rate model forecasts failed to outperform the naive forecast. This is
probably due to the larger number of parameter estimates required by the large number of
portfolios necessary to span the space of cash flow and discount rate innovations, which is also
manifest in the lack of significance of the δ parameter estimates in Table 8.
Comparison of Pricing Kernel and Discount Rate model estimated risk premium series
7
Table 9 reports comparative statistics for the two model estimates of the risk premium: for
the discount rate model we use the estimates based on F F 3 and 6DP spanning portfolios and
for the pricing kernel model the F F 3 and 3DP estimates.22 Panel A shows that the standard
deviation of the risk premia is highest for the pricing kernel model estimates based on the 3DP
spanning portfolios (2.54%), and then for the 6DP discount rate model estimates (2.39%). Panel
B shows that the correlation between these quarterly (1-year) series is 0.59 (0.61), and their
close relation is shown in Figure 2. Despite their different conceptual and empirical bases it
is apparent that the two models are identifying a common component of the expected return
series; we have seen that their persistence parameters, ρα, are 0.83 and 0.88.
Panel B shows that the correlation between the pricing kernel model quarterly (1-year)
estimates based on the F F 3 and 3DP spanning portfolios is 0.92 (0.85), confirming the visual
impression from Figure 1. This high correlation gives us some assurance that the two sets of
spanning portfolios are spanning the same component of the pricing kernel. The cross-model
correlation between the quarterly (1-year) estimates for the 3DP pricing kernel model and the
6DP discount rate model is 0.59 (0.61), and between the F F 3 pricing kernel model and the
22
Note that the F F 3 estimates for the discount rate model show very low persistence.
19
6DP discount rate model is 0.54 (0.62). Only the F F 3 discount rate model estimates show
relatively low correlations with the other series and, as we have noted, the F F 3 portfolios do
not appear to be adequate to span the cash flow and discount rate news.
Simultaneous bootstrap simulations for the two models provide further confirmation that
the predictability that they identify is not spurious. 10,000 bootstrap data samples of the
portfolio and market returns were generated under the null hypothesis in the manner described
in Section 5. For each sample the F F 3 pricing kernel model and 6DP discount rate model
predictors of quarterly market excess returns were estimated. The simulations imply that the
probability of obtaining the joint levels of predictability reported for the two models in Tables
1 and 7 is 0.14%; and this declines to 0.03% when the correlation of 0.54 between the estimated
risk premium series is taken into account.
Panel C reports the R2 from univariate and bivariate regressions of the quarterly and 1-year
market excess return on the pricing kernel model predictions using F F 3 and 3DP spanning
portfolios and the discount rate model predictions using the F F 3 and 6DP spanning portfolios.
The numbers on the diagonal are the R2 from the univariate regressions while the off-diagonal
numbers are the R2 from the bivariate regressions with predictors given by the corresponding
row and column headings. Comparing the diagonal and off-diagonal terms, we see that for both
pricing kernel models the prediction can be improved by combining it with the prediction of one
of the discount rate models. For example, the adjusted R2 for 1-year predictions of the pricing
kernel models using 3DP spanning portfolios increases from 19.4% to 25.7% when combined with
the 6DP discount rate model prediction. Similarly, for both discount rate models the prediction
can be improved by combining it with the prediction of one of the pricing kernel models. For
example the adjusted R2 for the 6DP quarterly forecasts from the discount rate model rises
from 8.7% to 11-12% when combined with one of the pricing kernel model forecasts. Thus
the forecasts of neither model dominate those of the other. Rather, while there are elements
of predictability that are not captured by the pricing kernel model but are captured by the
discount rate model and vice versa, the modest increase in the R2 when the discount rate model
predictions using the F F 3 or 6DP spanning portfolios are combined with the 6DP discount
rate model prediction indicates that the two models are identifying a common component of
predictability which is consistent with Figure 2 and the correlations reported in Panel B.
8
Additional empirical findings
In this section we report first some results on forecasts of the covariance of the pricing kernel
with the market return derived from the pricing kernel model. Secondly, we analyze the effect
of restricting the pricing kernel specification to a particular version of the ICAPM. Then we
consider the ability of the two models of the discount rate to capture time-variation in the
expected returns on some additional portfolios. Finally, we relate the time-variation in the
expected return series from the discount rate model to time-variation in market liquidity and
in sentiment.
8.1
Predicting the covariance of the pricing kernel with the market return
The pricing kernel model rests on the familiar result that the expected excess return on the
market portfolio is determined by the conditional covariance of the pricing kernel with the mar20
ket return as shown in Equation (1). It follows that if the model forecasts the expected excess
return it should also forecast the corresponding conditional covariance. Such an implication can
be tested in principle by regressing an estimate of the conditional covariance on the expected
return forecast, α̂M,t:
Ĉt+1 = a + bα̂M,t + t+1
(24)
where Ĉt+1 is the estimated covariance of returns in period (t + 1), and α̂M,t is the estimate
formed at the end period t of the arithmetic excess market return for period (t+1). If log returns
within a quarter or a year were approximately iid then it would be possible to estimate the
quarterly or 1-year covariance, Cq , Cy of the log of the market return with the log of the pricing
kernel return by appropriately scaling estimates of the covariance derived from high frequency
data. However, the covariance that we require is the covariance of the arithmetic not the log
returns, and there is also extensive evidence of lagged cross auto-correlations between security
and portfolio returns,23 both factors making the relation between the short run covariance that
can be estimated using high frequency data and the covariance of quarterly or annual returns
uncertain. To illustrate this, Table 10 reports scaled ratios of the covariance of the market
return with the pricing kernel return calculated using daily, weekly, and monthly returns to the
corresponding covariance calculated using quarterly returns. If long run returns were simply
sums of short run returns and the returns were iid, we would expect these ratios to be equal to
unity apart from sampling error. In fact, the ratios are considerably in excess of unity and even
in excess of two for daily returns. This implies that we have to be very cautious in estimating
covariances of long interval returns by simply scaling up estimates of the covariance obtained
from short interval or high frequency returns.
Therefore we face a quandary: we cannot estimate the conditional covariance of returns
over the next quarter (year) from the observed one quarter or one year return, and yet if we use
more high frequency data we are uncertain of the relation between the covariance of the one
quarter (year) return and the covariance of higher frequency returns; the higher is the frequency
of returns used the more efficient will be the estimator but also the more biased. Therefore
in Table 11 we report the results of estimating equation (24) using different proxies for the
covariance. The table is based on the pricing kernel model of predicted expected excess market
returns for 1 quarter and 1 year using the F F 3 beta-spanning portfolios. The parameters for
the 1-quarter prediction model on shown in the central column of Table 2. α̂yM,t and α̂qM,t are the
1-year and 1-quarter return predictions. The proxies for the 1-year and 1-quarter covariances,
y
q
Ĉt+1
and Ĉt+1
are appropriately scaled sample covariances using daily, weekly, monthly and
(for the 1-year covariance) quarterly returns. The proxies that use relatively low frequency
data are very noisy: for example the proxy for the 1-year covariance that is estimated using
quarterly data is based on only four observations; on the other hand, as we have mentioned, the
proxies obtained from high frequency data are likely to be highly biased. These countervailing
forces are apparent in the table: in Panel A the R2 for the prediction of the 1-year covariance
rises from 0.003, when we use the proxy estimated from daily returns, to 0.097 when we use the
proxy obtained from 12 monthly returns, but then declines to 0.063 for the proxy based on 4
quarterly returns. For the prediction of the 1-quarter covariance the R2 rises monotonically as
23
Levhari and Levi (1977) showed that CAPM betas vary systematically with the return interval even if returns
are iid. Lo and MacKinlay (1990) show that the returns on large firms systematically lead the returns on small
firms. Gilbert et al. (2014) show that the difference between daily and quarterly betas depends on the opacity
of the firm accounts. Brennan and Zhang (2014) show that the ratio of long to short horizon betas depends on
firm size, book-to-market ratio, number of analysts etc.
21
we move to proxies derived from lower return frequencies. Despite the difficulties of proxying
the time-varying 1-year or 1-quarter covariances of the market return with the pricing kernel,
the table provides strong evidence that in forecasting the excess return on the market portfolio
with the pricing kernel model we are also forecasting the conditional covariance of the low
frequency market return with the pricing kernel.
8.2
An ICAPM pricing kernel
The pricing kernel that we have specified so far depends on undefined state variables that are
assumed to be spanned by the returns on the different sets of portfolios we have introduced.
In this section we restrict the pricing kernel to depend on the market return, the innovation in
the interest rate and the innovation in the Sharpe ratio. These three variables are motivated
by the discussion of the ICAPM in Brennan et al. (2004) and Nielsen and Vassalou (2006).24
First, a time series of realized reward to volatility ratios, RM.t+1 /σM,t+1 , was calculated
by dividing the market excess return for each quarter by a scaled version of the realized daily
volatility during the quarter. The Sharpe ratio, which is the predicted value of the reward to
volatility ratio, is assumed to follow an AR(1) process, and its innovations are assumed to be
spanned by the market return and the returns on the 6DP portfolios. Then, following the logic
of Section 3.1, the realized reward to volatility ratio is given by:
Rt+1 /σM,t+1 = a0 +
P
X
SR
δpSR xSR
) + t+1
pt (β
(25)
p=1
where
xSR
pt (β) =
∞
X
β s rp,t−s
s=0
Equation (25) is essentially the discount rate model of expected returns, except that the market
excess return is standardized by the estimated volatility. The innovation in the Sharpe ratio is
assumed to be given by the realized return on the portfolio with weights proportional to δpSR ,
which we denote by RSR,t.
Equation (25) was estimated over the period 1946.1 to 2010.4 and the parameter estimates
are reported in the first column of Table 12. Comparing these estimates with those for the
6DP discount rate model in Tables 7 and 8, we note that the R̄2 is 0.083 when the dependent
variable is the reward to volatility ratio, as compared with 0.065 when the dependent variable
is the raw excess return, and two of the δpSR are significant at the 5% level.
Secondly, an AR(1) model was fitted to the quarterly risk free rate series and the innovations
from the model, uRF
t , were projected onto the the 6DP portfolio returns. The estimates of the
parameters, δpRF , are reported in the second column of Table 12. We see that the portfolio
returns capture about 63% of the variation in the residual. Portfolio weights proportional to
δpRF were used to calculate the returns on the RF mimicking portfolio, RRF,t.
24
Briefly: in a continuous time diffusion setting the instantaneous investment opportunity set is completely
described by the interest rate and the Sharpe ratio. If these two variables follow a joint Markov process, then
they are sufficient statistics for the entire investment opportunity set.
22
Then the predictive system for the market excess return is:
IC
IC
RM,t+1 = a0 − δRM xIC
RM,t − δSR xSR,t − δRF xRF,t
(26)
where
xIC
RM,t (β) =
∞
X
xIC
SR,t (β) =
∞
X
xIC
RF,t (β) =
∞
X
β s RM,t−s RM,t−s
s=0
β s RSR,t−s RM,t−s
s=0
s
βRF
RRF,t−s RM,t−s
s=0
The weighting parameter for the RF variable, βRF , was set equal to the estimated autoregressive
parameter of the AR(1) process for RF , and the other parameters were estimated as described
above, using market and portfolio returns for the period 1946.1 to 2010.4.
The results for the restricted pricing kernel model are shown in the third column of Table
12. The bias-adjusted R2 of 0.052 compares with the values of 0.083 for the unrestricted 3DP
pricing kernel model, and of 0.041 for the CAPM pricing kernel model (M ),shown in Table 1,
and the value of 0.056 for the 6DP discount rate model shown in Table 7. The t-statistics for
the two ICAPM state variables which are calculated using bootstrap standard errors are both in
excess of two. The parameter estimates imply that there is a positive risk premium associated
with covariation with the Sharpe ratio, but a negative premium associated with covariation
with the interest rate.
Overall, the results show that time variation in the covariance of a simple ICAPM pricing
kernel with the market return captures a significant fraction of the time variation in expected
returns that is captured by the more general pricing kernels that we have considered.
8.3
Sentiment, illiquidity and predicted returns
Thus far we have either left the determinants of expected returns unspecified as in the discount
rate model or assumed that expected returns are determined solely by changing risk as in
the pricing kernel model. Other possible determinants of expected returns include investor
sentiment (Baker and Wurgler (2006)) and market illiquidity (Amihud (2002)). Moreover Baker
and Wurgler have found that their measure of investor sentiment is an important determinant of
the expected returns on spread portfolios that are long portfolios of small, high book-to-market,
or high dividend yield stocks and short the corresponding portfolios of big, low book-to-market
and low dividend yield stocks. To the extent that investor sentiment reflects only psychological
factors and is independent of risk as captured by the pricing kernel, we should expect that
the pricing kernel model estimates of expected return would be unrelated to investor sentiment.
Similarly, we also expect that the pricing kernel estimates of expected return will be independent
of illiquidity to the extent that market illiquidity is independent of aggregate risk. On the other
hand, the expected returns estimated using the discount rate model may well be related to such
non-risk factors.
As a preliminary to exploring this, we estimate 1-year expected returns for 3 spread portfolios using the two models: the spread portfolios are the Fama-French SM B and HM L factor
23
returns and the returns on a portfolio that is long in the top 20% of firms ranked by dividend
yield and short in a portfolio of non-dividend paying stocks, which we denote by HM Z (High
minus Zero). The models are estimated by simply replacing the market excess return in equations (12) and (22) with the spread portfolio return; the spanning portfolios for both models
are the F F 3 portfolios. Selected model parameter estimates are reported in Table 13. Panel A
shows that there is significant predictability for the 3 spread portfolio returns using the discount
rate model, with the 1-year R2 ranging from 12 to 16% after bias correction. The (quarterly)
persistence parameter for all three portfolios is around 0.86, implying a half-life for shocks of 4.6
quarters. Out of sample tests for the discount rate model show the model predictions reducing
the error in the 1 year forecast of returns by 9%, 6% and 7% relative to the naive model for
the SM B, HM L and HM Z portfolios respectively. This out of sample forecast performance
for the spread portfolios is in marked contrast to the poor out of sample performance of the
discount rate model when applied to market excess returns.
On the other hand, Panel B shows little evidence of significant predictability from the
pricing kernel model for the HM L and HM Z spread portfolios: thus there is little evidence
that time variation in the returns on these portfolios is driven by time-varying risk which raises
the question of whether the time-variation in expected returns on these portfolios is driven by
time variation in sentiment or liquidity. However, for SM B the pricing kernel model yields
predictability that is not only significant, but is greater than that of the discount rate model:
the R2c rises from 12.5% to 16.8%. Moreover, the correlation between the two model estimates
for SM B is 0.45 while it is less than 0.3 for the other two spread portfolios.
Table 14 shows the correlations between the 1-year expected returns on the spread portfolios
calculated from the discount rate model using the F F 3 spanning portfolios. The expected
returns on HM L and HM Z have a correlation of 0.77 but, while the expected returns on
HM L and SM B have a positive correlation (0.20), the expected returns on HM Z and SM B
are almost uncorrelated. Figure 3, which plots the three expected return series, shows that the
expected return on HM Z was strongly negative for most of the 1970’s: high dividend yield
stocks had very low expected returns relative to those on zero yield ‘growth’ stocks during this
period.
To determine whether the difference between the discount rate model and the pricing kernel
model estimates of expected returns are related to sentiment or liquidity the difference between
the quarterly F F 3 discount rate model estimates of the 1-year expected returns and the corresponding pricing kernel model estimates were regressed on the Baker and Wurgler (2006) (BW)
measure of investor sentiment and the average value of the Amihud (2002) measure of market
illiquidity for the previous year.25 The results are reported in Table 15. For completeness, the
corresponding difference for the market portfolio was also included. In this case the discount
rate model estimate of the 1-year expected excess return for the market portfolio, RM , was
estimated using the 6DP set of predictor portfolios. The market illiquidity variable contributes
to the explanation of the difference between the expected return series for the HM L portfolio:
the positive regression coefficient (t = 2.33) implies that value stocks have a higher expected
return than growth stocks after adjusting for risk when Illiquidity is high. However there is
no evidence that Illiquidity affects the expected returns on the market portfolio or the other
25
We align the BW measure for the end of year t with the expected return for year t because most of the
variation in the BW measure is associated with variables from year (t − 1) (new issue returns, the relative pricing
of dividend and non-dividend paying stocks, and share turnover).
24
two spread portfolios after adjusting for risk.26
Consistent with BW, we find no evidence that their measure of sentiment is associated with
(the non-risk based component of the) expected returns on SM B. However, BW sentiment
is significantly positively associated with the non-risk component of the expected returns on
HM L (t = 2.65) and HM Z (t = 5.19). When BW sentiment is high, value firms have high
expected returns relative to growth firms, and high dividend yield stocks have high expected
returns relative to non-dividend paying stocks. While the dividend-yield spread portfolio results
are consistent with BW, those authors were unable to find a statistically significant sentiment
effect for the book-to-market ratio. Standardized time series of the HM Z portfolio 1-year
expected return and the Baker-Wurgler sentiment series are plotted in Figure 4. It is clear that
they track each other closely.
The difference between the expected return series from the unconstrained discount rate
model and that from the risk based pricing kernel model is an estimate of the component of the
expected return that is not explained by risk. We have shown that this non-risk based component of the expected returns on HM Z and HM L is associated with time varying sentiment
and illiquidity (HM L).
Stambaugh et al. (2012) have argued that anomalies in stock returns are due in large
part to impediments to short sales, and that mispricing is generally over-pricing that cannot be
arbitraged away. They find that anomalies are highest following periods of high sentiment which
gives rise to overpricing; and that the returns on the short leg of a strategy are more negative
when sentiment is high, while the returns on the long leg are largely invariant to sentiment. This
argument and the related findings suggest that the time-variation in the returns on the three
arbitrage portfolios will be mainly due to time-variation in the returns on the short leg whose
profits will come from periodic overpricing. To determine whether this is the case, the returns
on the long and short portfolios underlying each of the arbitrage portfolios were regressed
on dummy variables that capture whether the expected return on the arbitrage portfolio as
calculated using the discount rate model is above or below its median value (Dlow , Dhigh). The
market return, Rm,t was included as a control since the long and short portfolios are not market
neutral.
Rp,t = alow Dlow,t−1 + ahigh Dhigh,t−1 + βRmt + p,t
The dependent variable is the one year return on the long or short portfolio, and the dummy
variables were determined using estimates of expected 1-year return on the corresponding arbitrage portfolio from the discount rate model with F F 3 as spanning portfolios. The regression
was estimated using overlapping quarterly data and t-statistics were calculated using NeweyWest standard errors with 4 lags. The results are shown in Table 16. In the table the long
component of each spread portfolio is shown above the short component. ahigh − alow measures
the difference between the market adjusted returns in states when the expected return on the
spread portfolio is high and low and it therefore provides a simple measure of the contribution of
individual portfolio to the time variation in the spread portfolio returns. Contrary to the expectations raised by the Stambaugh et al. findings, the time-variation in the returns on SM B and
HM L are mainly due to the variation in the long side returns- Small firms for SM B, and both
26
Amihud (2002) does find an association between expected Illiquidity and the market return although the
significance of the results becomes marginal when the standard deviation of the market return is included in the
regression.
25
big and small high book-to-market returns for HM L. The variation in the short side returns
is only approximately one eighth that of the long side returns. Moreover, for the constituent
portfolios of HM L there is no significant difference between the variation in returns for large
and small firms where impediments to short sales are likely to be larger. Finally, although we
do not have an equilibrium model against which to assess returns there is not much evidence
that the time variation in returns on HM L and SM B is due to periodic overpricing. Rather
the highest absolute market adjusted returns are for the long portfolios in the high state - 6.7%
for the Small firm component of SM B and 9.5% for the small high book-to-market portfolio of
HM L. In contrast, returns for the supposedly overpriced short side in the high state are only
1.8% for Big firms in SM B and 1-2% for the low book-to-market portfolios in HM L. Thus,
even though we have seen that the returns on HM L are strongly related to sentiment, there
is not much evidence of the effects of short sales impediments which would imply that most of
the time-variation in returns would come from the short-side portfolios.
However, for the dividend spread portfolio, HM Z, the evidence favoring the Stambaugh et
al. hypothesis is much stronger. Here more of the time variation comes from the short side
portfolio of zero yield stocks and there is a strong suggestion of overpricing in the -6.1% average
risk adjusted return on this portfolio in the high state.
9
Conclusion
In this paper we have shown that it is possible to extract the expected returns on the market
portfolio from the lagged returns on a set of spanning portfolios, and have provided new evidence
on the predictability of (excess) stock returns. Our first model, the pricing kernel model,
assumes that time variation in expected returns is driven solely by time variation in risk, where
the risk of the market return is measured by its covariance with a portfolio whose returns capture
innovations in the pricing kernel. This model expresses the expected excess market return as a
weighted average of past cross-products of the returns on the beta-spanning portfolios and the
market portfolio. The second model, the discount rate model relies on the fact that in a world
of time varying discount rates, returns on common stock portfolios reflect shocks to discount
rates as well as to cash flow expectations. By assuming that the expected return follows an
AR1 process we are able to express the expected return as a weighted average of past returns
on a portfolio whose weights are chosen so that its return has exposure only to discount rate
innovations.
The pricing kernel model predicts quarterly returns with a corrected R2 of 6-8% and 1-year
returns with a corrected R2 of 14-16%. Out of sample, it reduces the mean square forecast
error by 3-4% for quarterly forecasts and by 9-13% for 1-year forecasts. These predictions and
forecast improvements are highly statistically significant. The predictive power of the model is
also robust to the inclusion of other predictor variables that have been examined previously,
and the component of predictability that we identify is essentially orthogonal to the LettauLudvigson (2001) cay variable. The persistence of the estimated expected return series is
considerably less than the persistence of variables that have been commonly used to proxies for
expected returns such as the dividend-price ratio, the earnings-price ratio and the T-bill rate.
While many of these variables have a persistence in quarterly data of above 0.9 we identify
expected returns with persistence of approximately 0.8
The three Fama-French factors, F F 3, are found to span a significant component of the
26
pricing kernel that gives rise to the time variation in expected market returns, although the
predictive power of the model for quarterly (1-year) returns increases from 6.3% (14%) to
7.3% (17.1%) when the spanning portfolios are increased from F F 3 to a full set of 6 size
and book-to-market sorted portfolios, which indicates that the F F 3 spanning is not perfect.
Nevertheless, the major role of the F F 3 factors in capturing the time-variation of expected
returns is important new evidence in support of their role in rational cross-sectional asset
pricing.
We also examine the predictive properties of a restricted pricing kernel model in which the
variables that enter the pricing kernel are motivated by Merton’s (1973) ICAPM and depend on
only the market return, the Sharpe ratio, and the riskless interest rate. When this restriction is
imposed, the model predicts quarterly excess returns with a corrected R2 of 5.6% as compared
with 8.3% for the unrestricted model.
The discount rate model predicts quarterly (1-year) market returns with a corrected R2 of 6%
(10%) when the set of spanning portfolios is the market portfolio and six portfolios formed on
the basis of dividend yield (6DP ). The larger set of spanning portfolios required by this model
is due to the need to span the cash flow as well as discount rate shocks on the portfolios. Since
this requires the estimation of a larger set of coefficients, the individual coefficient estimates are
not significant, although the quarterly predictions themselves are significant at the 5% level;
and out of sample the model does not improve on the performance of a naive forecast.
Although the two model forecasts have quite different conceptual bases and employ different
predictor variables, the forecasts themselves are quite closely related; the highest correlation
between the time series of forecasts from the two models is around 0.6. This gives us some
confidence that the models are identifying a common component of the expected return series.
But, while the pricing kernel model and and the 6DP version of the discount rate model identify
predictability with a persistence of around 0.8 in quarterly data, the F F 3 (and 6BM − S)
versions of the discount rate model pick up a high frequency component with a persistence of
around 0.3 in quarterly data.
The pricing kernel model attributes all the time variation in expected returns to time variation in risk, while the discount rate model imposes no such restrictions and potentially allows
for expected returns to be affected by factors such as sentiment or liquidity. Therefore to investigate the role of non-risk factors in determining expected returns we also fitted the F F 3
discount rate model to the 1-year returns on the arbitrage or spread portfolios, HM L, SM B
and HM Z, where HM Z is a portfolio that is long high dividend yield stocks and short zero
yield stocks. The expected return series for all three portfolios have a predictable component
with a persistence above 0.85 in quarterly data. Only a portion of the time-variation in the
expected returns on the HM L and HM Z spread portfolios is captured by the risk based pricing
kernel model, and we find that 7% (30%) of the variation in the expected returns on HM L
(HM Z) that is not captured by the risk-based model is explained by the Baker-Wurgler sentiment variable and illiquidity. We find no evidence that the expected returns on either the
market portfolio or SM B are afected by sentiment or illiquidity.
Thus we have shown that a significant component of the variation in expected returns
on stock portfolios is attributable to time-variation in risk. For the value-based arbitrage
portfolios (HM L and HM Z) there is also evidence of ‘mispricing’ which is associated with
waves of optimism and pessimism in financial markets and changing illiqidity. In support of the
‘sentiment’ hypothesis, we have shown that expected returns on HM L and HM Z are related
27
to the Baker-Wurgler measure of sentiment.
An issue that we have not explored is that, while the evidence from the F F 3 pricing kernel
model suggests that the Fama-French factors capture important aspects of the pricing kernel,
we have found that a significant component of the variation in the expected return on HM L is
due to mispricing relative to the risk-based model. Is it possible that this portfolio can capture
an important element of the pricing kernel while itself being subject to mispricing? We leave
this intriguing issue to future research.
28
10
Appendix A
Pricing Kernel Spanning Assumption
In the pricing kernel model we assume that the time-varying loading of the pricing kernel on
the return on the market portfolio, bm(t), is spanned by the time-varying betas of the (pricing
kernel) beta-spanning portfolios, bp(t): bm(t) = ΣPp=1 δpc bp(t), where δpc is a set of constant
portfolio weights and bp(t) is the beta of portfolio p. To motivate this assumption assume that
the vector of spanning portfolio betas, bp (t), can be written as an affine function of a K-element
vector of state variables, xt, so that:
bp (t) = bp0 + b0p1 xt
where bp0 is an (P x1) vector of constants and bp1 is a (KxP ) matrix of constants. Assume
further that the pricing kernel sensitivity to the market return, bm(t), can also be written as
an affine function of the same state variables:
bm(t) = bm0 + b0m1 xt
where bm0 is a scalar and bm1 is a Kx1 vector. Then a sufficient condition for the pricing
kernel model spanning assumption, bm (t) = ΣPp=1 δpc bp (t), is that the (1xP ) vector δ c satisfies:
δ c bp0 = bm0
δ c b0p1 = b0m1
These (K+1) equations in the P unknowns, δpc , will in general have a solution if P ≥ K + 1.
11
Appendix
Estimation Procedure To adjust for the small sample bias arising from persistence in the
predictor variables we proceed as follows. First we estimate the small sample bias in the
estimated R2 as a function of the estimated β under the null hypothesis of no predictability,
BR2 (β). Then our bias-corrected estimator of β is given by β̂ = argmaxβ [R2c ] where R2c =
R2 (β) − BR2 (β). In a similar fashion we calculate a bias corrected Wald statistic, W c . To
calculate the bias in the estimated R2 under the null, BR2 (β), we adopt a bootstrap approach
which reflects the null hypothesis of no predictability. Specifically, we fit a GARCH(1,1) to the
returns on the market portfolio and each of the spanning portfolios and save both the (T xP )
matrix of the fitted volatilities and the (T xP ) matrix of standardized return innovations. To
construct each bootstrap data sample we randomly select T (P x1)-vectors of standardized
innovations and construct the period t vector of returns by multiplying the fitted volatilities
for period t by the randomly selected standardized innovations and adding the intercepts from
the GARCH estimation. In this way we preserve the exact time series of volatilities for the
portfolios and the vector of mean returns, as well as the cross-sectional correlation structure,
while ensuring that the returns are serially independent. Then from the simulated spanning
portfolio returns we generate the predictor variables for the pricing kernel model, xcpt (β) =
P∞ s
2
s=0 β Rp,t−sRM,t−s for different values of β and calculate the R from regressing the market
29
excess return on the generated predictor variables. For each generated sample we estimate the
parameters (a0 , δ, β) in equation (12) and calculate the resulting R2 . Repeating this 10,000
times we calculate BR2 (β) as the average value of R2 in the bootstrapped samples.27 The
predictor variables, xcp (β), are formed by truncating the summation in equations (13) at 1927.3.
The predictive regressions and the return vectors which are sampled for the bootstrap start in
1946.1.
Given the parameter estimates, we assess the statistical significance of our results by determining the proportion of the 10,000 bootstrap samples in which the calculated value of the
corrected R2c ≡ R2 − BR2 (β) (or corrected Wald statistic) exceeds that calculated using the
actual data. Standard errors of the parameter estimates are also obtained from the bootstrap
estimation.
The procedure was then repeated, replacing the quarterly excess return as dependent variable with the 1-year excess return starting in the same quarter. Although this induces overlap
in the dependent variable this is accounted for in the bootstrap simulations used to calculate
standard errors and significance.
27
Estimations are performed by a grid search over 0 ≤ β ≤ 0.95.
30
12
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33
Figure 1: Market expected returns estimated from the pricing kernel model
The figure shows the expected 1-year excess returns for S&P500 estimated from the pricing kernel model. We
estimate the model with three different sets of spanning portfolios: the S&P500 portfolio (M ), the 3 Fama-French
portfolios (F F 3), and the S&P500 portfolio and three portfolios sorted on the basis of dividend yield (3DP ).
The plot shows the expected 1-year excess returns for each quarter over the period 1946Q1-2010Q4.
34
Figure 2: Comparison of expected returns from pricing kernel and discount rate models
The figure compares the expected 1-year excess returns for S&P500 estimated from the pricing kernel model
with that of the discount rate model. The spanning portfolios for the pricing kernel model are the 3 Fama-French
portfolios (F F 3), and for the discount rate model the expanded spanning portfolios based on dividend yield are
used, which are the S&P500 portfolio and six portfolios sorted on dividend yield (6DP ). The plot shows the
expected 1-year excess returns for each quarter over the period 1946Q1-2010Q4.
35
Figure 3: Expected 1-year returns for spread portfolios
The figure shows the expected 1-year returns for the spread portfolio returns (SMB, HML, HMZ) estimated from
the discount rate model. SM B and HM L are the Fama-French size and book-to-market factors. HM Z is a
portfolio formed on dividend yields that is long a portfolio of the 20% of stocks with highest dividend yields and
short a portfolio of non-dividend paying stocks. The expected returns for the spread portfolios are derived using
the discount rate model with the 3 Fama-French portfolios (F F 3) as spanning portfolios. The plot shows the
expected 1-year returns for each quarter over the period 1946Q1-2010Q4.
36
Figure 4: The Baker-Wurgler sentiment measure and expected returns on the HMZ portfolio
The figure shows the relation between the Baker-Wurgler (BW) measure of sentiment and the estimated 1-year
return of the HMZ portfolio, HM Z(t) ≡ Et [RHM Z,t+1 ]. The expected return is derived using the discount rate
model where the spanning portfolios are the 3 Fama-French portfolios (F F 3). HM Z is a portfolio formed on
dividend yields that is long a portfolio of the 20% of stocks with highest dividend yields and short a portfolio of
non-dividend paying stocks. BW(t+1) denotes the sentiment series advanced by one year in order to make its
principal components more timely. The plot shows the expected 1-year returns and sentiment measure for each
quarter over the period 1965Q3-2010Q4. Both series are standardized to zero mean and unit volatility.
37
Panel A. Predicting quarterly excess returns
Spanning
Portfolios
β
ρα
w
R2
W
R2c
M
FF3
3DP
0.522
0.655
0.694
0.631
0.801
0.829
0.109
0.146
0.135
0.043
0.066
0.088
21.558
28.756
46.775
0.041
0.063
0.083
0.081
0.079
59.991
60.659
0.073
0.071
Expanded sets of spanning portfolios
6BM − S 0.650 0.778 0.128
6DP
0.675 0.819 0.144
Wc
∗∗∗
∗∗∗
∗∗∗
∗∗
∗∗
20.166
23.809
54.803
46.989
60.151
∗∗∗
∗∗
∗∗∗
∗∗
∗∗
Panel B. Predicting 1-year excess returns
Spanning
Portfolios
β
ρα
w
R2
W
R2c
M
FF3
3DP
0.400
0.562
0.547
0.505
0.719
0.727
0.105
0.157
0.180
0.038
0.161
0.184
11.969
33.846
36.503
0.033
0.140
0.157
0.212
0.173
95.181
73.281
0.171
0.130
Expanded sets of spanning portfolios
6BM − S 0.486 0.653 0.167
6DP
0.501 0.693 0.192
Wc
∗∗∗
∗∗∗
∗∗∗
∗∗
10.857
28.614
37.971
75.079
53.447
∗∗
∗∗
∗∗
∗∗
∗∗
Table 1: Pricing kernel model: tests of predictability and persistence parameter
estimates
The table reports the results of tests of the
of no predictability P
of the market excess return for
P null hypothesis
∞
c
c
s
the pricing kernel model: RM,t+1 = a0 + P
δ
x
(β)
+
where
x
(β)
=
p
t+1
pt
pt
p=1
s=0 β Rp,t−s RM,t−s . Panel A
predicts quarterly and panel B 1-year returns. RM,t+1 is the quarterly (or 1-year) excess return on the S&P500
index; Rp,t , p = 1, · · · , P , are the quarterly excess returns on a set of predictor portfolios. We estimate the model
with three different sets of spanning portfolios: the S&P500 portfolio (M ), the 3 Fama-French portfolios (F F 3),
and the S&P500 portfolio and three portfolios sorted on the basis of dividend yield (3DP ). The expanded sets of
spanning portfolios are the S&P500 portfolio and the six portfolios sorted on the basis of size and book-to-market
ratio (6BM − S), and the S&P500 portfolio and six portfolios sorted on the basis of dividend yield (6DP ). The
sample period is 1946.1 to 2010.4 (the prediction period). Estimations are performed by a grid search over
0 ≤ β ≤ 0.95. The parameters are chosen to maximize the R2c of the predictive regression, where R2c denotes the
R2 of the predictive regression adjusted to correct for small sample bias. Levels of significance are determined by
a bootstrap procedure in which returns over the period 1946.1 to 2010.4 are sampled under the null hypothesis:
∗ ∗∗ ∗∗∗
, ,
denote significance at the 10%, 5%, and 1% levels.
ρα is the first order autocorrelation of the estimated market risk premium, and w ≡ ρα − β. W is the Wald
statistic calculated using the Newey-West (1987) correction with 4 lags. Wc denotes the bias corrected value for
W which is calculated by maximizing the bias-adjusted Wald statistic. Significance levels for Wc are calculated
by the bootstrap procedure, and indicated by stars.
38
Panel A: parameter estimates
Spanning portfolios:
M
FF3
3DP
β
0.522∗∗∗
(0.18)
0.631∗∗∗
(0.18)
-0.000
(0.01)
1.316∗∗∗
(0.43)
0.655∗∗∗
(0.10)
0.801∗∗∗
(0.09)
-0.004
(0.02)
1.531∗∗∗
(0.39)
0.694∗∗∗
(0.08)
0.829∗∗∗
(0.08)
-0.003
(0.02)
1.529
(1.45)
ρα
a0
RM
Fama-French portfolios
-1.165∗
(0.67)
1.268∗∗
(0.59)
SM B
HM L
Dividend yield sorted portfolios
-1.658∗∗∗
(0.53)
0.787
(0.97)
1.175
(0.73)
zero
Lo20
Hi20
R̄2
0.043∗∗∗
0.066∗∗∗
0.088∗∗∗
Panel B. Descriptive statistics of risk premium estimates
M
FF3
3DP
Mean
Std.Dev.
Min.
Max.
1.78%
1.71
0.09
10.76
1.78%
2.21
-2.64
11.96
1.78%
2.54
-5.17
12.67
Correlation matrix
M
FF3
3DP
1.00
0.73
0.62
1.00
0.91
1.00
Table 2: Pricing kernel model: parameter estimates for quarterly forecast
Panel A reports parameter estimates from the regression:
RM,t+1 = a0 +
P
X
δp xcpt (β) + t+1
p=1
with
spanning portfolio quarterly returns, Rp,t−s used to form the predictors, xcpt (β) =
P∞ different
s
s=0 β Rp,t−s RM,t−s . The dependent variable is the quarterly excess return on the S&P500 portfolio, RM,t . We
estimate the model with three different sets of spanning portfolios: the S&P500 portfolio (M ), the 3 Fama-French
portfolios (F F 3), and the S&P500 portfolio and three portfolios sorted on the basis of dividend yield (3DP ). The
sample period is 1946.1 to 2010.4 (the prediction period). The parameters are estimated by Supβ (R2 − bias).
Standard errors (in parentheses) are calculated from a bootstrap simulation using 10000 realizations. Panel
B reports means and standard deviations for the risk premium estimates obtained using the different sets of
spanning portfolios, as well as the correlations between the estimates.
39
Panel A. Quarterly forecasts
Spanning Portfolios:
Horizon
M
1 quarter
2
3
4
0.97
0.98
1.00
1.02
FF3
∗∗
∗
0.96
0.94
0.95
0.95
3DP
∗∗∗
∗∗
∗∗
∗∗
0.97
0.90
0.88
0.84
∗∗∗
∗∗∗
∗∗∗
∗∗∗
Panel B. 1-year forecasts
Spanning Portfolios:
Horizon
M
FF3
3DP
1 year
2
3
1.02
1.06
1.15
0.91∗∗
0.99
1.07
0.87∗∗∗
0.87∗∗
0.93∗
Table 3: Relative Error Variance of Out of Sample Return forecasts for the Pricing
Kernel Model
The table reports the ratio of the variance of the error in forecasting the market excess return using the pricing
kernel model with different sets of spanning portfolios to the variance of the error of a naive forecast. A value
below one indicates that the model outperforms the naive forecast which is the sample mean calculated from
data up to the quarter of the forecast. The quarterly forecasts are made each quarter and are extended out to 4
quarters using the estimated parameters of the AR(1) process. The 1-year forecasts are also made quarterly and
are extended out to 3 years using the estimated parameters of the AR(1) process. The model and the historical
mean are estimated using data starting in 1926.2 and ending in the quarter before the forecast is made. The
spanning portfolios are the the S&P500 portfolio (M ), the 3 Fama-French portfolios (F F 3), and the S&P500
portfolio and three portfolios sorted on the basis of dividend yield 3DP . Levels of significance are determined by
a bootstrap procedure in which returns over the period 1946.1 to 2010.4 are sampled under the null hypothesis:
∗ ∗∗ ∗∗∗
, ,
denote significance at the 10%, 5%, and 1% levels. The 1st training sample if from 1946.1 to 1965.12.
The first forecast made in 1966.1 and the last forecast is for 2010.4.
40
In sample regression
Predictor
Variable
α
R2adj
β
ρ
REVOOS
Panel A: univariate regressions
dp
ep
b/m
VS
glam
svar
tbl
lty
tms
infl
dfy
kp
cay
0.110
0.060
0.003
0.105
0.025
0.018
0.030
0.028
0.008
0.022
0.012
-0.020
0.017
(2.66)
(1.48)
(0.19)
(2.00)
(3.24)
(3.23)
(3.22)
(2.33)
(1.04)
(3.34)
(0.88)
(-1.20)
(3.11)
0.027
0.015
0.027
-0.059
-0.024
-0.100
-0.280
-0.171
0.625
-0.478
0.619
0.313**
0.897***
(2.20)
(1.03)
(1.17)
(-1.65)
(-1.31)
(-0.15)
(-1.59)
(-0.94)
(1.63)
(-1.10)
(0.42)
(2.50)
(3.64)
0.018
0.004
0.003
0.007
0.003
-0.004
0.007
0.000
0.008
0.001
-0.003
0.023
0.042
0.984
0.950
0.980
0.816
0.900
0.456
0.949
0.981
0.834
0.474
0.885
0.931
0.925
1.015
1.058
1.177
1.000
0.989
1.008
1.001
1.019
0.994
0.996
0.987
0.985
0.979
1.117
-0.552
(0.68)
(-0.17)
-0.005
0.620
0.712
1.240
Panel B: bivariate regression
svar
scov
0.007
(0.81)
Table 4: Predictive regressions using other predictor variables
The table reports estimates of the equation:
RM,t+1 = α + βXt + t+1
for the period 1946.1-2010.4. RM,t+1 is the 1 quarter S&P500 excess return in quarter t + 1 and Xt is the lagged
value of the predictor variable. ρ is the auto-correlation of the variable. t-statistics are in parentheses and are
adjusted for serial correlation in the residuals using the Newey-West correction (1987) with 4 lags.
Panel A reports univariate regression results. The predictor variables are: the Dividend (Earnings) yield, dp
(ep), defined as the log of the ratio of dividends (earnings) on the S&P500 over the past 12 months to the
lagged level of the index; the Value Spread V S, is the log book-to-market ratio of the small high-book-to-market
portfolio minus the log book-to-market ratio of the small low-book-to-market portfolio; Glamour, glam, is the
cumulative log return over the past 36 months of the quintile of stocks with the lowest book-to-market ratio;
b/m is the book-to-market ratio for the Dow Jones Industrial Average; the Stock Variance, svar, is the sum of
squared daily returns on the S&P500 index over the previous quarter; tbl is the 3 month Treasury Bill rate; the
Long Term Yield, lty is the yield on long term US government bonds; the Term Spread, tms, is the difference
between the Long Term Yield and the Treasury Bill rate; Inflation, inf l, is the one month lagged inflation rate;
the Default Yield Spread, df y, is the difference between BAA and AAA-rated corporate bond yields; and cay
is the consumption, wealth, income ratio of Lettau and Ludvigson (2001) which is available over the period
1952.3-2010.4. Fuller descriptions of these variables are to be found in Goyal and Welch (2008) and the actual
data series were taken from the website of Amit Goyal. kp is the in-sample predicted one year excess return
from Kelly and Pruitt (2012). Panel B reports the results from a multivariate predictive regression in which the
predictors are svar and scov, the sum of the daily cross-products of the market excess return with the HM L
return. Levels of significance for β, indicated by stars, are determined by a bootstrap procedure in which returns
over the period 1946.1 to 2010.4 are sampled under the null hypothesis: ∗ ,∗∗ ,∗∗∗ denote significance at the 10%,
5%, and 1% levels. REVOOS is the ratio of the out of sample variance of the error of a forecast of the quarterly
return using the predictor variable to the variance of the error of a naive forecast equal to the historical mean.
Out-of-sample forecasts start in 1966 and the model is estimated using data starting 1928.06 to one year before
the forecast year.
41
Spanning
portfolios:
M
FF3
3DP
-0.05
-0.21
0.05
0.09
-0.36
0.55
-0.04
0.09
0.30
0.03
0.46
-0.06
0.01
0.12
-0.12
0.06
-0.11
-0.27
0.41
-0.40
-0.27
0.33
-0.17
0.19
0.21
0.04
0.12
-0.13
0.04
-0.10
-0.23
0.36
-0.29
-0.15
0.34
-0.21
0.25
0.24
0.07
Predictor
Variable
dp
ep
b/m
VS
glam
svar
tbl
lty
tms
infl
dfy
kp
cay
Table 5: Correlations of Pricing Kernel Model risk premium estimates with other
predictor variables
The table reports the correlations between 4-quarter moving averages of the other predictor variables which
are defined in Table 4 and 4-quarter moving averages of the expected return estimates from the pricing kernel
model for different sets of spanning portfolios. The spanning portfolios are the S&P500 portfolio (M ), the three
Fama-French portfoios (F F 3) and the S&P500 portfolio and three portfolios formed on dividend yield (3DP ).
The sample period is from 1946.1 to 2010.4, except for cay which is from 1952.3 to 2010.4.
Regressor
1
2
3
4
5
6
7
8
dp
0.011
(0.61)
-0.020
(-1.06)
0.211
(1.09)
0.010
(0.56)
-0.001
(-0.03)
0.236
(1.27)
0.015
(0.86)
-0.003
(-0.15)
0.095
(0.51)
0.018
(1.06)
-0.002
(-0.08)
0.034
(0.18)
-0.009
(-0.41)
-0.033
(-1.68)
0.322
(1.59)
1.002
(3.87)
-0.010
(-0.45)
-0.012
(-0.58)
0.349
(1.77)
0.914
(3.65)
-0.002
(-0.10)
-0.013
(-0.66)
0.206
(1.03)
0.878
(3.75)
-0.003
(-0.12)
-0.013
(-0.67)
0.153
(0.74)
0.848
(3.74)
glam
kp
cay
Pricing kernel predictions:
M
1.008
(4.03)
FF3
3DP
2
R̄
0.022
0.061
0.923
(5.04)
0.078
0.944
(3.52)
0.949
(5.78)
0.101
0.066
0.100
0.874
(4.73)
0.116
0.881
(5.61)
0.137
Table 6: Regression of quarterly excess returns on Pricing kernel model predictions
and other predictors
The table reports the results of regressions of quarterly market excess returns on selected other predictor variables
and forecasts from the pricing kernel model for different sets of spanning portfolios. The spanning portfolios are
the S&P500 portfolio (M ), the three Fama-French portfoios (F F 3) and the S&P500 portfolio and three portfolios
formed on dividend yield (3DP ). t-statistics are in parentheses. The sample period is from 1946.1 to 2010.4,
except for the regressions involving cay which are from 1952.3 to 2010.4.
42
Panel A. Predicting quarterly excess returns
Spanning
Portfolios
β
ρα
w
R2
W
R2c
M
FF3
3DP
-0.252
0.153
0.884
-0.145
0.268
0.877
0.107
0.115
-0.007
0.009
0.046
0.034
3.498
17.452
12.365
0.008
0.044
0.030
Expanded sets of spanning portfolios
6BM − S 0.248
0.334
0.086
6DP
0.923
0.856
-0.067
0.042
0.065
21.713
33.553
0.036
0.056
Wc
∗∗
∗
∗∗
2.418
14.055
13.130
11.855
22.920
∗∗
∗∗
∗∗
Panel B. Predicting 1-year excess returns
Spanning
Portfolios
β
ρα
w
R2
W
R2c
Wc
M
FF3
3DP
dnc
0.014
0.825
0.066
0.830
0.052
0.005
0.023
0.094
8.115
17.132
0.021
0.055
4.407
10.099
0.121
0.181
23.739
52.138
0.042
0.099
Expanded sets of spanning portfolios
6BM − S 0.887 0.767 -0.120
6DP
0.902 0.847 -0.055
∗
8.815
36.368
∗∗
Table 7: Discount rate model: tests of predictability and persistence parameter
estimates
The table reports the results of tests of P
the null hypothesis of no predictability
the market excess return for
Pof
∞
d
d
s
the discount rate model : RM,t+1 = a0 + P
p=1 δp xpt (β) + t+1 where xpt (β) =
s=0 β Rp,t−s . Panel A predicts
quarterly and panel B 1-year returns. RM,t+1 is the quarterly (or 1-year) excess return on the S&P500 index;
Rp,t , p = 1, · · · , P , are the quarterly excess returns on a set of spanning portfolios. We estimate the model with
five different sets of spanning portfolios: the market portfolio (M ), the 3 Fama-French portfolios (F F 3), the
S&P500 portfolio and three portfolios sorted on the basis of dividend yield (3DP ); six portfolios formed on the
basis of firm size and book to market ratio (6BM -S), and the S&P500 portfolio and six portfolios formed on the
basis of dividend yield (6DP ). The sample period is 1946.1 to 2010.4 (the prediction period). Estimations are
performed by a grid search over 0 ≤ β ≤ 0.95. The parameters are chosen to maximize the R2c of the predictive
regression, where R2c denotes the R2 of the predictive regression adjusted to correct for small sample bias. Levels
of significance are determined by a bootstrap procedure in which returns over the period 1946.1 to 2010.4 are
sampled under the null hypothesis: ∗ ,∗∗ ,∗∗∗ denote significance at the 10%, 5%, and 1% levels. dnc denotes did
not converge.
ρα is the first order autocorrelation of the estimated market risk premium, and w ≡ ρα − β. W is the Wald
statistic calculated using the Newey-West (1987) correction with 4 lags. Wc denotes the bias corrected value for
W which is calculated by maximizing the bias-adjusted Wald statistic. Significance levels for Wc are calculated
by the bootstrap procedure, and indicated by stars.
43
β
ρα
a0
RM
Fama-French portfolios
smb
hml
Size and book-to-market sorted portfolios
FF3
6BM-S
3DP
6DP
0.153
(0.24)
0.268
(0.24)
0.018∗∗
(0.01)
0.145∗
(0.08)
0.248
(0.24)
0.334
(0.23)
0.021∗
(0.01)
0.764
(0.53)
0.884∗∗∗
(0.18)
0.877∗∗∗
(0.19)
0.026
(0.02)
0.312
(0.19)
0.923∗∗∗
(0.13)
0.856∗∗∗
(0.14)
0.030
(0.02)
0.272
(0.25)
-0.030
(0.06)
-0.188
(0.13)
-0.040
(0.06)
-0.103
(0.13)
-0.251
(0.16)
0.034
(0.16)
0.209
(0.16)
-0.169∗
(0.10)
0.065∗∗
-0.321∗∗∗
(0.11)
-0.102
(0.10)
sl
0.042
(0.15)
0.000
(0.31)
-0.341
(0.21)
-0.406
(0.36)
0.062
(0.25)
0.010
(0.18)
sn
sh
bl
bn
bh
Dividend yield sorted portfolios
zero
Lo20
Qnt2
Qnt3
Qnt4
Hi20
R̄2
0.046∗∗
0.042
-0.120
(0.09)
0.034∗
Table 8: Discount rate model: parameter estimates for quarterly forecasts
This table reports parameter estimates from the regression:
RM,t+1 = a0 +
P
X
δp xdpt (β) + t+1
p=1
P
s
with different spanning portfolio quarterly returns, Rp,t−s used to form the predictors, xdpt (β) = ∞
s=0 β Rp,t−s .
The dependent variable is the quarterly excess return on the S&P500 portfolio, RM,t . We estimate the model
with four different sets of spanning portfolios: the 3 Fama-French portfolios (F F 3), the S&P500 portfolio and
three portfolios sorted on the basis of dividend yield (3DP ); six portfolios formed on the basis of firm size and
book to market ratio (6BM -S), and the S&P500 portfolio and six portfolios formed on the basis of dividend
yield (6DP ). The sample period is 1946.1 to 2010.4 (the prediction period). The parameters are estimated
by Supβ (R2 − bias). Standard errors (in parentheses) are calculated from a bootstrap simulation using 10000
realizations.
44
Panel A. Quarterly expected returns
Model
Pricing Kernel
Spanning portfolios:
FF3
3DP
FF3
6DP
Mean
Std. Dev.
1.78%
2.21
1.78%
2.54
1.78%
1.90
1.78%
2.39
Discount Rate
Panel B. Correlations between risk premium estimates
Model
Pricing Kernel
Spanning portfolios:
FF3
Pricing Kernel - F F 3
Pricing Kernel - 3DP
1.00
0.92
1.00
Discount Rate - F F 3
Discount Rate - 6DP
0.22
0.54
0.28
0.59
Pricing Kernel - F F 3
Pricing Kernel - 3DP
1.00
0.85
1.00
Discount Rate - F F 3
Discount Rate - 6DP
0.33
0.62
0.27
0.61
Discount Rate
3DP
FF3
6DP
Quarterly
1.00
0.35
1.00
1.00
0.30
1.00
1-year
Panel C. R̄2 from univariate and bivariate regressions
Model
Pricing Kernel
Discount Rate
Spanning portfolios:
FF3
FF3
Pricing Kernel - F F 3
Pricing Kernel - 3DP
0.074
0.096
0.099
Discount Rate - F F 3
Discount Rate - 6DP
0.108
0.110
0.126
0.120
Pricing Kernel - F F 3
Pricing Kernel - 3DP
0.167
0.197
0.194
Discount Rate - F F 3
Discount Rate - 6DP
0.241
0.172
0.200
0.257
3DP
6DP
Quarterly
0.054
0.113
0.087
0.030
0.205
0.200
1-year
Table 9: Comparison of the discount rate and pricing kernel models
This table reports statistics for the quarterly and 1-year excess return forecasts from the discount rate and pricing
kernel models for the period 1946.1 to 2010.4. The spanning portfolios are the the S&P500 portfolio (M ), the 3
Fama-French portfolios (F F 3), and the S&P500 portfolio and in turn, the three (six) portfolios sorted on the basis
of dividend yield (3DP, 6DP ). Panel B reports correlations between the different estimates of the market risk
premium. The expected market excess returns are calculated quarterly. A rolling four-quarter average is taken
to reduce noise before calculating the correlations in Panel B. Panel C reports the R2 values from (i) univariate
predictive regressions of the market returns on the fitted expected returns for each model (on the diagonal); and
(ii) for bivariate regressions on the fitted expected returns for two models (the off diagonal elements).
45
Daily
d
Weekly
Monthly
w
m
Quarterly
cov (Rm ,pk)Nd
cov q (Rm ,pk)
cov (Rm ,pk)Nw
cov q (Rm ,pk)
cov (Rm ,pk)Nm
cov q (Rm ,pk)
cov q (Rm ,pk)
cov q (Rm ,pk)
2.267
1.884
1.207
1.000
Table 10: Covariance ratios
This table reports covariance ratios for different return intervals. The covariance is between the market return,
Rm , and the portfolio with weights δc . The portfolio weights δc are those estimated for the prediction of quarterly
market excess returns using the pricing kernel model with F F 3 spanning portfolios, reported in Table 8. Rpk is
the return on this portfolio. Covariances are calculated using daily, weekly, monthly and quarterly returns and
the sample period is 1946.1-2010.4. The covariances are then converted to a quarterly basis by multiplying by
the number of days (weeks, months), Nd etc., in a quarter and divided by the covariance from quarterly data,
cov q (Rm , Rpk ).
46
Panel A: Pricing kernel estimated from prediction of 1-year return
y
Ĉt+1
= a + bα̂yM,t + t+1
a
b
R2
Covariance
estimated:
Quarterly
Monthly
Weekly
Daily
0.025
(1.86)
0.052
(5.78)
0.098
(6.19)
0.117
(5.71)
0.433
(2.78)
0.328
(3.56)
0.176
(1.13)
0.184
(1.15)
0.063
0.097
0.007
0.003
Panel B: Pricing kernel estimated from prediction of 1-quarter return
q
Ĉt+1
= a + bα̂qM,t + t+1
a
b
R2
Covariance
estimated:
Monthly
0.015
(5.14)
0.020
(5.90)
0.023
(5.08)
Weekly
Daily
0.455
(3.28)
0.520
(2.50)
0.497
(2.63)
0.066
0.043
0.026
Table 11: Predicting the covariance of the market return with the estimated pricing
kernel
The realized covariances between market returns and the pricing kernels are estimated for year (quarter) t + 1,
y
q
Ĉt+1
(Ĉt+1
) using quarterly, monthly, weekly and daily returns. Thus for the Panel A (B), if n daily (weekly,
monthly, quarterly) observations are used to estimate the realized covariance over the following year (quarter),
the realized covariance is defined by:
Ĉ ≡ covt(R, δc0 Rp ) ≡ n
n
X
[(Rt+i − < R >)(δc0 Rp,t+i − < δ0 Rp >)]
i=1
c
where δ is the kernel weight vector estimated by using the Fama-French portfolios to forecast 1-year (1-quarter)
returns. < R > is the mean return per period (day week etc.) during the year (quarter). The covariance
estimates are regressed on the 1-year (1 quarter) predicted market excess return, α̂yM,t (α̂qM,t ).
y
Ĉt+1
= a + bαyM,t + t+1
q
Ĉt+1
= a + bαqM,t + t+1
where αyM,t = s=0 β s δc0 Rp,t−s Rt−s is the predicted 1-year market excess return using the Fama-French portfolios, and δc is the vector of portfolio weights from the 1-year return prediction and αqM,t is defined analogously.
t-statistics are in parentheses and are adjusted for serial correlation in the residuals using the Newey-West
correction (1987) with 4 lags. The sample period is 1946.1-2010.4.
P
47
Dependent variable:
SR
RF
RM
β
0.926∗∗∗
(0.08)
0.607∗∗
(0.24)
3.743
(3.16)
0.973∗∗∗
(0.02)
0.011∗∗∗
(0.00)
-0.026
(0.02)
0.658∗∗∗
(0.11)
0.000
(0.01)
1.580∗∗∗
(0.44)
-1.104
(0.73)
-0.777
(1.61)
-4.16∗∗
(2.02)
1.461
(2.01)
2.859
(2.02)
-2.386∗∗
(1.27)
0.002∗∗∗
(0.00)
0.017∗
(0.01)
0.016∗
(0.01)
-0.047∗∗∗
(0.01)
0.058∗∗∗
(0.01)
-0.028∗∗
(0.01)
a0
RM
Dividend yield sorted portfolios:
Zero
Lo20
Qnt2
Qnt3
Qnt4
Hi20
0.576∗∗
(0.24)
-44.823∗∗
(22.40)
SR
RF
R̄2
0.083
R̄2c
0.626
0.056
0.052
Table 12: Restricted pricing kernel model: parameter estimates for quarterly forecast
P
SR SR
The first column reports parameter estimates for RM,t+1 /σM,t+1 = a0 + P
p=1 δp xpt (β) + t+1 , where RM,t
is market excess return for quarter t, σM,t is an appropriately scaled estimate of the daily return volatility,
SR
and
P∞ thes 6DP predictor portfolio quarterly returns, Rp,t−s are used to form the predictor variables, xpt (β) =
s=0 β Rp,t−s .
The second column reports estimates of δpRF , the coefficients from the regression of the residual from an AR(1)
model of the riskless interest rate, RF , on the portfolio returns.
The third column reports estimates of
IC
IC
RM,t+1 = a0 − δRM xIC
RM,t − δSR xSR,t − δRF xRF,t
P∞ s
P∞ s
IC
where xIC
RM,t (β) =
s=0 β RM,t−s RM,t−s , xRF,t (β) =
s=0 βRF RRF,t−s RM,t−s , and the portfolio returns RRF,t
and RSR,t are formed using weights proportional to the regression coefficients δRF and δSR . βRF is set equal to
the autoregressive coefficient for the riskless interest rate.
R̄2c is the bias corrected estimate of R̄2 . Standard errors (in parentheses) are calculated using a bootstrap
procedure in which returns over the period 1946.1 to 2010.4 are sampled under the null hypothesis: ∗ ,∗∗ ,∗∗∗
denote significance at the 10%, 5%, and 1% levels. are in parentheses. The sample period is 1946.1 to 2010.4
48
Panel A. Discount rate model
R2
W
R2c
Spread
Portfolio
β
ρα
w
SM B
HM L
HM Z
0.738
0.988
0.981
0.853
0.879
0.879
0.115
-0.109
-0.102
Spread
Portfolio
β
ρα
w
SM B
HM L
HM Z
0.976
0.464
0.510
0.943
0.618
0.690
-0.033
0.154
0.180
0.151
0.217
0.206
21.804
28.536
22.566
0.125
0.161
0.159
Wc
∗∗
∗∗
∗∗∗
Panel B. Pricing kernel model
R2
W
R2c
0.202
0.084
0.097
30.172
14.204
9.655
0.168
0.063
0.078
17.451
20.588
16.324
∗
∗∗
∗
Wc
∗∗∗
∗
27.397
12.384
5.273
∗∗
Table 13: Tests of predictability and persistence parameter estimates for 1-year
returns on spread portfolios
This table reports selected parameter estimates for the predictability of 1-year spread portfolio returns from
the discount rate and pricing kernel models using the 3 Fama-French portfolios (F F 3) as spanning portfolios.
The sample period is 1946.1 to 2010.4 (the prediction period). SM B and HM L are the Fama-French size and
book-to-market factors. HM Z is a portfolio formed on dividend yields that is long a portfolio of the 20% of
stocks with highest dividend yields and short a portfolio of non-dividend paying stocks. Levels of significance
are determined by a bootstrap procedure in which returns over the period 1946.1 to 2010.4 are sampled under
the null hypothesis: ∗ ,∗∗ ,∗∗∗ denote significance at the 10%, 5%, and 1% levels. See notes to Table 1 for further
details.
SM B
HM L
HM Z
SM B
HM L
HM Z
1.000
0.199
-0.031
0.199
1.000
0.774
-0.031
0.774
1.000
Table 14: Correlations of expected 1-year returns on spread portfolios
This table reports correlation of the 1 year expected expected returns on the spread portfolios. SM B and HM L
are the Fama-French size and book-to-market factors. HM Z is a portfolio formed on dividend yields that is
long a portfolio of the 20% of stocks with highest dividend yields and short a portfolio of non-dividend paying
stocks. Expected returns are calculated from the discount rate model using the three Fama-French factors (F F 3)
as spanning portfolios. The sample period is 1946.1 to 2010.4.
49
RM
SM B
HM L
HM Z
Const.
Sentiment
Illiquidity
R2
0.01
(0.88)
0.01
(0.44)
-0.02
(-1.55)
-0.04
(-2.13)
-0.01
(-0.83)
-0.01
(-0.87)
0.02
(2.65)
0.06
(5.19)
-2.06
(-1.09)
-0.65
(-0.27)
3.80
(2.33)
1.23
(0.51)
0.00
0.00
0.07
0.30
Table 15: Expected return estimates, sentiment and illiquidity
This table reports the results of regressing the difference in expected 1-year returns from the discount rate
model, µdrm,t , and the pricing kernel model, µpk,t , on the Baker-Wurgler (2006) measure of investor sentiment
(Sentiment) and the average value of the Amihud (2002) measure of market illiquidity for the previous year
(Illiquidity):
µdrm,t − µpk,t = c + β1 Sentimentt+1 + β2 Illiquidityt + t
The regression is estimated using expected 1-year returns on the market portfolio RM , and on each of the spread
portfolios (SM B, HM L, HM Z). The expected 1-year returns for the spread portfolios and for the pricing
kernel estimates for the market portfolio are derived using the 3 Fama-French portfolio (F F 3) as spanning
portfolios. The discount rate model estimates for the market portfolio use the 6DP portfolios as spanning
portfolios. The sentiment variable is advanced by one year in order in order to make its principal components
more timely. SM B and HM L are the Fama-French size and book-to-market factors. HM Z is a portfolio
formed on dividend yields that is long a portfolio of the 20% of stocks with highest dividend yields and short a
portfolio of non-dividend paying stocks. The sample period is from 1965.1 to 2010.4. The regression is estimated
on a time series of quarterly estimates of 1-year expected returns and t − statistics in parentheses are calculated
using Newey-West standard errors with 4 lags. ∗ ,∗∗ ,∗∗∗ denote significance at the 10%, 5%, and 1% levels.
50
alow
ahigh
ahigh − alow
β
R2
1.04
(16.24)∗∗∗
0.95
(31.86)∗∗∗
0.10
(1.70)∗
0.69
0.99
(11.92)∗∗∗
0.98
(13.91)∗∗∗
1.12
(13.25)∗∗∗
0.98
(45.40)∗∗∗
-0.07
(-0.89)
0.62
0.84
(10.97)∗∗∗
1.26
(18.64)∗∗∗
-0.42
(-3.84)∗∗∗
0.68
SM B spread
Small
Big
SMB
-1.33
(-0.79)
0.52
(0.91)
-1.85
(-1.32)
6.68
(3.16)∗∗∗
1.77
(2.60)∗∗∗
4.91
(2.61)∗∗∗
8.00
(3.36)∗∗∗
1.24
(1.82)∗
6.76
(3.17)∗∗∗
0.95
0.09
HM L spread
sh
bh
sl
bl
HML
2.60
(1.08)
-0.14
(-0.14)
-1.40
(-0.57)
-0.05
(-0.09)
1.95
(1.55)
9.48
(4.17)∗∗∗
5.91
(3.58)∗∗∗
-1.79
(-0.82)
-1.16
(-1.67)∗
9.17
(4.59)∗∗∗
6.88
(2.46)∗∗
6.06
(3.69)∗∗∗
-0.38
(-0.12)
-1.11
(-1.51)
7.21
(3.97)∗∗∗
0.77
0.61
0.95
0.09
HM Z spread
Hi20
zero
HMZ
0.07
(0.05)
2.17
(0.88)
-2.09
(-0.68)
4.76
(2.52)∗∗
-6.07
(-3.40)∗∗∗
10.83
(4.04)∗∗∗
4.69
(2.51)∗∗∗
-8.23
(-3.15)∗∗∗
12.92
(4.06)∗∗∗
0.70
0.20
Table 16: Sources of time variation in spread portfolio returns
This table reports the results of estimating the regression:
Rp,t = alow Dlow,t−1 + ahigh Dhigh,t−1 + βRmt + p,t
where Rp is a portfolio 1-year return. Dlow (Dlow ) is a dummy variable which is equal to one when the expected
1-year return, Et−1 [Rp,t ], on the corresponding spread portfolio (SM B,HML, HM Z) is above (below) its median
value. The expected return on the spread portfolios are calculated from the discount rate model using the three
Fama-French factors (F F 3) as spanning portfolios. The regression is estimated using a quarterly time series
of data over the period from 1946.1 to 2010.4. t − statistics in parentheses are calculated using Newey-West
standard errors with 4 lags. ∗ ,∗∗ ,∗∗∗ denote significance at the 10%, 5%, and 1% levels.
51