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 purelystatistical. 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 ofpredictable 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 HML 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, ManchesterUniversity.

†Corresponding Author: Alex P. Taylor, Accounting and Finance Group, Manchester Business School, TheUniversity of Manchester, Booth Street West, Manchester, M15 6PB, England, e-mail: alex.taylor@mbs.ac.uk,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 orwhether it is a function of sentiment and waves of optimism and pessimism. The issue ofwhether 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 alleviatedby ‘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 returnsare 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 dataunder 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 seriesas predictor variables.

There is less prior evidence on whether the measured time variation in returns can beattributed 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 riskvariation 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, bytime-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 thattime-variation in the covariance of the market return with a pricing kernel that is spannedeither 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 returnsin 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 pastportfolio 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 returnsis 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 doesnot 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 rationalpricing, if we accept the tag de gustibus non est disputandum, since the pricing kernel mirrorsthe 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 combinesthis with the assumption of a factor structure of returns to identify shocks to the discount rate.

1e.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 newsabout 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 tothis 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 theshocks to discount rates. By assuming a stochastic process for the discount rate and aggregatingthese 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 Campbelland 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 changingmarket 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 returnsand assume the same AR(1) process for expected returns, they imply different sets of predictorvariables. 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 modelscasts 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 returnseries 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 riskpremium 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 twomodels 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 R2 obtainedfor 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 R2 for the parsimoniousversion 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 ourfindings 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 formarket 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 theFama-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)).

2Cf. Amihud (2002).3Fama and French (1995, 1996) argue that the value and size premia move closely with investment opportu-

nities.

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 Brennanet 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 theriskless 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 thetime-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 thatfor 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 ≈ 0.7-

0.8 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, SMB, HML, and HMZ (the spread between highand 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 SMB portfolio are similar to those for the market

portfolio in that there is no evidence that the expected returns are affected by time-varyingsentiment or liquidity. However for the arbitrage portfolio HMZ, 30% of the difference between

the expected return estimated from the unconstrained discount rate model and the expectedreturn from the risk-based model is attributable to time-variation in the Baker-Wurgler (2006)measure of sentiment. For HML, time variation in both sentiment and the Amihud (2002)

measure of illiquidity contribute to explaining the difference in the expected returns from thetwo 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 thetwo 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 thediscount rate model. Section 7 compares the time series of risk premium estimates from the twomodels. 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 pricingkernel 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 marketreturn. 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 positivebut 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 negativerelation depending on the method used.4 More recently, Ghysels et al. (2005) establish a

4For 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 returnrelation 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 betweena 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 notspecified.

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 thesecond 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 thebond 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 HML portfolio asa second factor, and find that the lagged market volatility and covariance of its return with

the return on the Fama-French (1993) HML portfolio predict market excess returns over theperiod 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 assetpricing 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 dependson the volatility of the pricing kernel, and tighter bounds that require further specificationof 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 levelsof 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 thepricing 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 orstochastic discount factor specification.

Our discount rate model builds on the distinction between between discount rate news andcash 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 isto 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 areconstructed 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 portfolioreturns is related to Pastor and Stambaugh (2009) who use prior beliefs on the correlation be-tween 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 re-turns, 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 portfolioreturns. 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 evidenceof 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 sampleperiod. 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 componentsto 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 ofportfolio 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 dividendyield 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 as13%. 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 timeseries 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 arisingfrom 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 accountof 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 spuriousregression for the case of highly persistent expected returns. The same concerns over data-

mining potentially arise in our empirical analysis. However, a major advantage of our approachis 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 ofno 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 dataunder 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 theeffects of data mining on levels of significance, in our approach for a candidate set of spanning

5The 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 bysumming 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 timevarying 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 thespace 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 mt+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(mt+1, RM,t+1) (1)

where we have imposed the normalization that Et(mt+1) = 1.

We can write the kernel, mt+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:

mt+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

αM,t = −covt(bm(t)RM,t+1, RM,t+1) ≡ −bm(t)σ2M,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 combinationof 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 portfoliop, p = 1, · · · , P as:

6Time 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=1δ

cpbp(t) for a set of constant portfolio

weights δcp, so that the sensitivity of the pricing kernel to market returns, bm(t), can be expressed

as a linear combination of the portfolio loadings on the market return, −ΣMp=1δ

cpbp(t). Equation

(3) shows that if the expected market return is to be a time-varying function of its variance,

then bm(t) must be time-varying, and the requirement that bm(t) = −ΣMp=1δ

cpbp(t) for a set

of constant portfolio weights δcp, 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 returnson a (possibly different) set of (factor) spanning portfolios, ΣM

p=1δdp(Rp,t+1 − Et[Rp,t+1]), for a

constant set of portfolio weights δd.

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 assetpricing 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 procedureidentifies 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 toprice the cross-section of asset returns.

The relation between cross-sectional asset pricing and time-variation in asset returns thatappears 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 thecovariance 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-sectionalasset 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, andwhich 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 thenegative 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 δc

p , p = 1, · · · , P , such that :

7Appendix A provides sufficient conditions for the existence of a set of loading-spanning portfolios.8Ross (2005), Zhou (2010) and Huang (2014) make the same point.9In 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

bm(t) = −ΣPp=1δ

cpbp(t) (5)

where bp(t) is the beta coefficient of beta-spanning portfolio p. Then

αM,t = covt(ΣPp=1δ

cpbp(t)RM,t+1, RM,t+1) = ΣP

p=1δcpcovt(Rp,t+1, RM,t+1) (6)

We assume that the risk premium, and therefore the covariance of the market return with thepricing kernel, follows an AR(1) process:

αM,t = a + ρα[αM,t−1 − a] + ξαt (7)

A sufficient condition for (7) is that the conditional covariances of the beta-spanning port-

folio 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 = a + ρα[

P∑

p=1

δcpcovt−1(Rp,tRM,t) − a] + ξαt (9)

where a =∑P

p=1δcpap, and ξα,t =

∑Pp=1

δcpRp,tRM,t.

Then the market risk premium can be written as an affine function of the geometricallyweighted average of past values of the weighted average of products of spanning portfolio and

market returns:

αM,t = a+∞∑

j=0

(ρα)jξα,t−j = a +∞∑

j=0

(ρα)j

P∑

p=1

δcpRp,t−jRM,t−j (10)

= a+

P∑

p=1

δcpx

cp,t(ρα) (11)

where xcp,t(ρα) ≡

∑∞j=0

(ρα)j[Rp,t−jRM,t−j].

Then the predictive system for the market excess return becomes:

RM,t+1 = a0 +

P∑

p=1

δcpx

cpt(β) + εt+1 (12)

xcpt(β) =

∞∑

s=0

βsRp,t−sRM,t−s (13)

9

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, δcp, 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 discountrate news.

Let µt be the expected log excess return on the market portfolio, so that the realized excessreturn can be written as:

rM,t+1 = µt + εt+1 (14)

We assume that µt follows an AR(1) process so that:

µt = µ + ρ[µt−1 − µ] + 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 thespace of innovations in cash flows and discount rate news, so that

rpt = βp0 + kpµt−1 +

M∑

j=1

βpjyjt + γpzt (16)

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 returnson 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

that∑P

p=1δdpβpj = 0, j = 1, · · · , M ;

∑Pp=1

δdpγp = 1. This ensures that the z-mimicking portfolio

10 See also Campbell and Shiller (1988), Campbell (1991), and Campbell and Ammer (1993).11We 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 isconsistent with shocks to the risk free rate playing only a minor role.

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 = δd0 +

P∑

p=1

δdprpt − wµt−1, (17)

where δd0 = −

∑Pp=1

δdpβp0, w =

∑Pp=1

δdpkp.

Then combining equations (14) and (17), the process of the market expected log return is:

µt = (1 − ρ)µ + (ρ − w)µt−1 + δd0 +

P∑

p=1

δdprpt (18)

Substituting recursively for µt−j, the expected log excess market return, µt, may be writtenas a linear function of geometrically weighted past returns on the P spanning portfolios:

µt =µ(1 − β − w) + δd

0

(1 − β)+

P∑

p=1

δdpxd

pt(β) (19)

where β ≡ ρ − w, and

xdpt(β) =

∞∑

s=0

βsrp,t−s (20)

Combining (14) with (19), the log excess return on the market portfolio may be written as:

rM,t+1 = a0 +

P∑

p=1

δdpxd

pt(β) + εt+1 (21)

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), theweighting parameter, β, may deviate from the true persistence of the expected return, ρ: wereport 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 logarithmicreturns that follow strictly from the Campbell log-linearization.

Then the predictive system becomes:

RM,t+1 = a0 +

P∑

p=1

δdpxd

pt(β) + εt+1 (22)

xdpt(β) =

∞∑

s=0

βsRp,t−s (23)

12We 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 ratebased model, defined by equations (22) and (23), and the pricing kernel based model, definedby (12) and (13), lies in the definition of the predictive variables which we have denoted by

xdpt(β) and xc

pt(β). The former is a weighted average of past returns on portfolio p, while thelatter 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, itis 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 geometricweighted averages either of past returns on a set of spanning portfolios (xd

pt(β)) or of the

products of past returns on a set of spanning portfolios with the return on the portfolio whosereturn is to be predicted (xc

pt(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 methodof identifying the spanning portfolios. Therefore we shall consider different candidate setsof 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 whoseadequacy 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 dividendyields as predictors, our predictor variables, xd

pt(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 RFt, which is the risk free rate for the quarter, takenas the return on a 3-month Treasury Bill. The S&P500 return is taken from CRSP, and theTreasury 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, whilethe 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 andthe 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

13Note that for the discount rate model the spanning portfolios are assumed to span the space of discount rateand aggregate cash flow shocks, while for the pricing kernel model the betas of the loading-spanning portfoliosare 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 mixedprior evidence that the market variance predicts future market returns discussed above, andbecause 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 becauseof 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 wereincluded 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 betweenthe 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 thanthe 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 ofspanning portfolios.

We also compare our predictor variables with variables that have been used earlier in theliterature. 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 theS&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 squareddaily 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 Spreadwhich is the difference between the Long Term Yield and the Treasury Bill rate; Inflation whichis 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, incomeratio 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 stockreturns 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 sorton size and book-to-market ratio, available from the website of Ken French. It is defined as thelog(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 ayearly 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 negativelyrelated 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 book-to-market ratio. The glamour portfolio is defined as the quintile with the lowest book-to-marketratio 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 theslightly 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 oursample period in regressions on annual excess log returns.

14See 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 explorethe 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 indextaken 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 forus.

5 Empirical Tests and Estimates of the Pricing Kernel model

We start by estimating equations (12) and (13) for the pricing kernel model. A non-linearmaximum 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 returnson 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 higherbias 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. Havingestimated the parameters (a0, δ

c, β), we form time series estimates of the expected market excess

return, αM,t = a0 +∑P

p=1δcpx

cpt(β), 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 pre-dictor 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 the1-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 spanningportfolios are (i) the market portfolio (M); (ii) the three Fama-French portfolios (FF3); and (iii)

the market portfolio and the three dividend yield portfolios (3DP ). Panel A reports the resultsfor prediction of the quarterly excess return on S&P500 portfolio, and Panel B for predictionof 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% levelfor all three sets of spanning portfolios except FF3 where the significance level is only 5%

when the Wald criterion is used. After bias correction the fraction of the variance of returnsexplained by the model (R2

c) 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 volatilityof the simple CAPM pricing kernel has predictive power for the equity premium and that themarket 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

15We thank the authors for making these forecasts available to us.16Bootstrap 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.

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 FF3 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 FF3 and3DP sets of spanning portfolios yield R2

c 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 allthree sets of spanning portfolios. As with the quarterly estimates, the estimates of β and ρα

for the 3DP and FF3 spanning portfolios are very close, and the expected return estimates forthe different sets of spanning portfolios are very closely related: Panel B of Table 2 shows that

the correlation between the 3DP and FF3 estimates is 0.91, while the correlations of theseestimates 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 thepricing 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 twoexpanded sets of spanning portfolios: 6BM − S is the market portfolio plus six size and book-

to-market portfolios and nests the FF3 set of portfolios; similarly 6DP is the market portfolioplus 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 R2

c actually falls for both quarterly and 1-year

return predictions. On the other hand, expanding the set of spanning portfolios from FF3 to6BM − S does improve the corrected R2

c from 0.063 (0.140) to 0.073 (0.171) for the quarterly(1-year) forecasts, suggesting that FF3 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, δc, β, ρα) from the quarterly return pre-

dictions for the three sets of spanning portfolios. Significance levels for the coefficients are

computed from standard errors calculated from bootstrap simulations under the alternativehypothesis.17

Components of the pricing kernel that are significant at the 5% level or better include themarket portfolio (except for 3DP ) and the two other FF3 portfolios (SMB only at the 10%

level), as well as the zero dividend yield portfolio. Their signs are generally consistent with priorknowledge of the pricing kernel. Thus the positive coefficients on RM and HML are consistent

with a positive risk premium for the market portfolio and for the HML portfolio, while theinsignificant coefficient on SMB offers no support for a small firm premium. The results for the

3DP spanning portfolios suggest that there is a positive premium associated with covariancewith returns on a portfolio of high yield (value) stocks, and a negative premium associated withcovariance with zero yield (growth) stocks.

In summary, the results are consistent with time-variation in the covariance of the mar-

ket return with the pricing kernel leading to variation in expected market returns. The highcorrelation between the 3DP and FF3 risk premium estimates, as well as the limited improve-

ment from increasing the set of beta spanning portfolios, suggests that the betas of both the

17Note that β is undefined under the null.

15

3DP and FF3 sets of spanning portfolios do a good job of spanning the pricing kernel beta:

bM(t) ≈ Σ3p=1δ

cp(t).

5.1 Expected return series

Panel B of Table 2 shows that the average risk premium predicted for each of the three setsof 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 timesthe mean premium. The M estimates are much less variable: their standard deviation is only

1.71% while the standard deviation of the FF3 estimates is 2.21%. Figure 1 plots the expected1-year return series obtained using the three different sets of spanning portfolios. It is apparent

that the series based on FF3 and 3DP track each other well, reaching low points around thedot-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 impor-tance and for each we report equity premium estimates from the FF3 (3DP ) models which can

be compared with the mean 1-year equity premium of 7.7%. In 2009 following the collapse ofLehman 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 periodof 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 yieldrelatively similar estimates.

Interestingly, the estimated equity premium from both models was negative from the thirdquarter 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 equitypremium.19

5.2 Out of Sample Tests

Table 3 reports the out of sample forecasting power of the pricing kernel model for differenthorizons, using different sets of spanning portfolios. Results are reported for both quarterlyforecasts 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. Themodels 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 byone 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 ofthe 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 themean 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 meanforecast. The 3DP model shows generally the strongest out of sample forecast power. For the

18Boudoukh et al. (1993) report reliable evidence that the ex ante equity market risk premium is negative insome states of the world.

19It 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% and7% 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 FF3 quarterly return forecasts improve onthe 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 bootstrappedunder 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 bycomparing the sample statistic with the distribution of the bootstrapped statistics. The 3DP

and FF3 forecast improvements for the quarterly forecasts and for the 1 year forecast aresignificant 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 givento 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 quartermarket 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 smallsample 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 theregression 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 sampleperformance 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 tothe 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 ofREVOOS 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 meanand 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 variablesthe improvement is less than 1%. The two predictive variable model reported in Panel B isthe 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 predictorvariables and corresponding averages of the forecast equity premium from the pricing kernel

model using different sets of spanning portfolios. Considering only correlations that are greaterthan 0.3 in absolute value, we see that the predicted premium from the CAPM kernel, M , hascorrelations of -0.36 with Glamor, 0.55 with Stock Variance, and 0.46 with the Default Yield

Spread. The predictions from the FF3 kernel have correlations of 0.41 and -0.40 with Stock

20See 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 ofspanning portfolios are compared with the 4 most significant ‘other predictors’ by regressing themarket 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 estimatesfor 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 theirtheoretical 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 valueof 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 otherpredictor 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 port-folio 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 factorsas well as the discount rate news. Three or four portfolios may well be insufficient for this.Therefore, in addition to the 3DP and FF3 sets of spanning portfolios we include the two

expanded sets of spanning portfolios: 6BM −S is the market portfolio plus the 6 Fama-Frenchsize 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. Themodel 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 marketportfolio alone cannot span the innovations in both cash flows and the discount rate, even ifthere 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 asingle 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 FF3or 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 varyingstochastically: increases in expected dividend growth increase stock prices and are associated with increases indiscount 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 inducea negative association between market returns and expected future returns. The two effects are offsetting sothat 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 expecteddividend 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

FF3 is significant at the 5% level, the persistence parameter for the estimated predicted excessreturn, 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 thepredictions 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 aresignificant at the 5% level for both 3DP and 6DP sets of spanning portfolios. The persistenceparameter 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 FF3 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 ofparameter estimates for the 1-quarter implementation of the discount rate model. Virtuallynone of the individual parameter estimates except β and ρα for 3DP and 6DP are significant

at the 5% level, and the only δd parameter estimate that is significant is the coefficient of SMBwhen the set of spanning portfolios is FF3.

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 theout 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 ofportfolios necessary to span the space of cash flow and discount rate innovations, which is alsomanifest in the lack of significance of the δ parameter estimates in Table 8.

7 Comparison of Pricing Kernel and Discount Rate model esti-

mated risk premium series

Table 9 reports comparative statistics for the two model estimates of the risk premium: forthe discount rate model we use the estimates based on FF3 and 6DP spanning portfolios andfor the pricing kernel model the FF3 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 3DPspanning 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 theirclose 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 returnseries; 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 FF3 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 ofspanning 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 the6DP discount rate model is 0.59 (0.61), and between the FF3 pricing kernel model and the

22Note that the FF3 estimates for the discount rate model show very low persistence.

19

6DP discount rate model is 0.54 (0.62). Only the FF3 discount rate model estimates show

relatively low correlations with the other series and, as we have noted, the FF3 portfolios donot appear to be adequate to span the cash flow and discount rate news.

Simultaneous bootstrap simulations for the two models provide further confirmation thatthe 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 describedin Section 5. For each sample the FF3 pricing kernel model and 6DP discount rate model

predictors of quarterly market excess returns were estimated. The simulations imply that theprobability 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 estimatedrisk premium series is taken into account.

Panel C reports the R2 from univariate and bivariate regressions of the quarterly and 1-yearmarket excess return on the pricing kernel model predictions using FF3 and 3DP spanning

portfolios and the discount rate model predictions using the FF3 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 correspondingrow 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 oneof 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 withthe 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. Forexample the adjusted R2 for the 6DP quarterly forecasts from the discount rate model risesfrom 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 elementsof 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 FF3 or 6DP spanning portfolios are combined with the 6DP discount

rate model prediction indicates that the two models are identifying a common component ofpredictability 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 effectof 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 theexpected 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 andin 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 mar-

20

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 canbe tested in principle by regressing an estimate of the conditional covariance on the expected

return forecast, αM,t:

Ct+1 = a + bαM,t + εt+1 (24)

where Ct+1 is the estimated covariance of returns in period (t + 1), and αM,t is the estimate

formed at the end period t of the arithmetic excess market return for period (t+1). If log returnswithin 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 pricingkernel 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 logreturns, 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 thatcan be estimated using high frequency data and the covariance of quarterly or annual returnsuncertain. 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 thecorresponding 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 tounity 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 estimatingcovariances 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 usemore 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 frequencyof 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 thecovariance. The table is based on the pricing kernel model of predicted expected excess market

returns for 1 quarter and 1 year using the FF3 beta-spanning portfolios. The parameters forthe 1-quarter prediction model on shown in the central column of Table 2. αy

M,t and αqM,t are the

1-year and 1-quarter return predictions. The proxies for the 1-year and 1-quarter covariances,Cy

t+1and Cq

t+1are 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 usingquarterly 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 countervailingforces 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 theproxy 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

23Levhari and Levi (1977) showed that CAPM betas vary systematically with the return interval even if returnsare iid. Lo and MacKinlay (1990) show that the returns on large firms systematically lead the returns on smallfirms. Gilbert et al. (2014) show that the difference between daily and quarterly betas depends on the opacityof the firm accounts. Brennan and Zhang (2014) show that the ratio of long to short horizon betas depends onfirm 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 lowfrequency 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 areassumed 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 inthe 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 calculatedby dividing the market excess return for each quarter by a scaled version of the realized dailyvolatility 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 bespanned 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∑

p=1

δSRp xSR

pt (βSR) + εt+1 (25)

where

xSRpt (β) =

∞∑

s=0

βsrp,t−s

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 isassumed to be given by the realized return on the portfolio with weights proportional to δSR

p ,

which we denote by RSR,t.

Equation (25) was estimated over the period 1946.1 to 2010.4 and the parameter estimatesare 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 R2 is 0.083 when the dependentvariable is the reward to volatility ratio, as compared with 0.065 when the dependent variableis the raw excess return, and two of the δSR

p 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, uRFt , were projected onto the the 6DP portfolio returns. The estimates of the

parameters, δRFp , 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 toδRFp were used to calculate the returns on the RF mimicking portfolio, RRF,t.

24Briefly: in a continuous time diffusion setting the instantaneous investment opportunity set is completelydescribed by the interest rate and the Sharpe ratio. If these two variables follow a joint Markov process, thenthey are sufficient statistics for the entire investment opportunity set.

22

Then the predictive system for the market excess return is:

RM,t+1 = a0 − δRMxICRM,t − δSRxIC

SR,t − δRF xICRF,t (26)

where

xICRM,t(β) =

∞∑

s=0

βsRM,t−sRM,t−s

xICSR,t(β) =

∞∑

s=0

βsRSR,t−sRM,t−s

xICRF,t(β) =

∞∑

s=0

βsRF RRF,t−sRM,t−s

The weighting parameter for the RF variable, βRF , was set equal to the estimated autoregressiveparameter of the AR(1) process for RF , and the other parameters were estimated as describedabove, 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 3DPpricing 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 forthe two ICAPM state variables which are calculated using bootstrap standard errors are both inexcess 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 covariationwith 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 expectedreturns 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 inthe pricing kernel model. Other possible determinants of expected returns include investor

sentiment (Baker and Wurgler (2006)) and market illiquidity (Amihud (2002)). Moreover Bakerand 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 psychologicalfactors 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 independentof 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 suchnon-risk factors.

As a preliminary to exploring this, we estimate 1-year expected returns for 3 spread portfo-

lios using the two models: the spread portfolios are the Fama-French SMB and HML 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 HMZ (Highminus Zero). The models are estimated by simply replacing the market excess return in equa-

tions (12) and (22) with the spread portfolio return; the spanning portfolios for both modelsare the FF3 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 reducingthe error in the 1 year forecast of returns by 9%, 6% and 7% relative to the naive model for

the SMB, HML and HMZ portfolios respectively. This out of sample forecast performancefor 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 thepricing kernel model for the HML and HMZ spread portfolios: thus there is little evidencethat 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 bytime variation in sentiment or liquidity. However, for SMB the pricing kernel model yields

predictability that is not only significant, but is greater than that of the discount rate model:the R2

c rises from 12.5% to 16.8%. Moreover, the correlation between the two model estimates

for SMB 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 portfolioscalculated from the discount rate model using the FF3 spanning portfolios. The expectedreturns on HML and HMZ have a correlation of 0.77 but, while the expected returns on

HML and SMB have a positive correlation (0.20), the expected returns on HMZ and SMBare almost uncorrelated. Figure 3, which plots the three expected return series, shows that the

expected return on HMZ was strongly negative for most of the 1970’s: high dividend yieldstocks 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 betweenthe quarterly FF3 discount rate model estimates of the 1-year expected returns and the corre-

sponding 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, thecorresponding 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 , wasestimated 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 HML 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 isno evidence that Illiquidity affects the expected returns on the market portfolio or the other

25We align the BW measure for the end of year t with the expected return for year t because most of thevariation in the BW measure is associated with variables from year (t−1) (new issue returns, the relative pricingof 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 SMB. However, BW sentimentis significantly positively associated with the non-risk component of the expected returns onHML (t = 2.65) and HMZ (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 expectedreturns 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 sentimenteffect for the book-to-market ratio. Standardized time series of the HMZ portfolio 1-year

expected return and the Baker-Wurgler sentiment series are plotted in Figure 4. It is clear thatthey 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 com-ponent of the expected returns on HMZ and HML is associated with time varying sentiment

and illiquidity (HML).

Stambaugh et al. (2012) have argued that anomalies in stock returns are due in largepart 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 whichgives rise to overpricing; and that the returns on the short leg of a strategy are more negativewhen 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 threearbitrage 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 returnson 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 ascalculated 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 marketneutral.

Rp,t = alowDlow,t−1 + ahighDhigh,t−1 + βRmt + εp,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 arbi-trage portfolio from the discount rate model with FF3 as spanning portfolios. The regression

was estimated using overlapping quarterly data and t-statistics were calculated using Newey-West 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 measuresthe 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 ofindividual portfolio to the time variation in the spread portfolio returns. Contrary to the expec-

tations raised by the Stambaugh et al. findings, the time-variation in the returns on SMB andHML are mainly due to the variation in the long side returns- Small firms for SMB, and both

26Amihud (2002) does find an association between expected Illiquidity and the market return although thesignificance of the results becomes marginal when the standard deviation of the market return is included in theregression.

25

big and small high book-to-market returns for HML. The variation in the short side returns

is only approximately one eighth that of the long side returns. Moreover, for the constituentportfolios of HML 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 wedo not have an equilibrium model against which to assess returns there is not much evidence

that the time variation in returns on HML and SMB is due to periodic overpricing. Ratherthe highest absolute market adjusted returns are for the long portfolios in the high state - 6.7%for the Small firm component of SMB and 9.5% for the small high book-to-market portfolio of

HML. In contrast, returns for the supposedly overpriced short side in the high state are only1.8% for Big firms in SMB and 1-2% for the low book-to-market portfolios in HML. Thus,

even though we have seen that the returns on HML are strongly related to sentiment, thereis 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, HMZ, the evidence favoring the Stambaugh et

al. hypothesis is much stronger. Here more of the time variation comes from the short sideportfolio 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 evidenceon 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, wherethe 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 aweighted average of past cross-products of the returns on the beta-spanning portfolios and themarket 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 discountrates 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 returnson 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-yearreturns with a corrected R2 of 14-16%. Out of sample, it reduces the mean square forecasterror 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 isalso 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 Lettau-Ludvigson (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 forexpected 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 identifyexpected returns with persistence of approximately 0.8

The three Fama-French factors, FF3, 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%) to7.3% (17.1%) when the spanning portfolios are increased from FF3 to a full set of 6 size

and book-to-market sorted portfolios, which indicates that the FF3 spanning is not perfect.Nevertheless, the major role of the FF3 factors in capturing the time-variation of expected

returns is important new evidence in support of their role in rational cross-sectional assetpricing.

We also examine the predictive properties of a restricted pricing kernel model in which thevariables 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 isimposed, 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 onthe 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. Sincethis 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 differentpredictor 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 someconfidence 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 identifypredictability with a persistence of around 0.8 in quarterly data, the FF3 (and 6BM − S)

versions of the discount rate model pick up a high frequency component with a persistence ofaround 0.3 in quarterly data.

The pricing kernel model attributes all the time variation in expected returns to time vari-ation 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 in-vestigate the role of non-risk factors in determining expected returns we also fitted the FF3

discount rate model to the 1-year returns on the arbitrage or spread portfolios, HML, SMBand HMZ, where HMZ 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 componentwith a persistence above 0.85 in quarterly data. Only a portion of the time-variation in the

expected returns on the HML and HMZ 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 HML

(HMZ) that is not captured by the risk-based model is explained by the Baker-Wurgler sen-timent variable and illiquidity. We find no evidence that the expected returns on either themarket portfolio or SMB 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 arbitrageportfolios (HML and HMZ) 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 HML and HMZ are related

27

to the Baker-Wurgler measure of sentiment.

An issue that we have not explored is that, while the evidence from the FF3 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 HML isdue 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 leavethis 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 onthe 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=1δ

cpbp(t), where δc

p is a set of constantportfolio 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-elementvector of state variables, xt, so that:

bp(t) = bp0 + b′p1xt

where bp0 is an (Px1) 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 asan affine function of the same state variables:

bm(t) = bm0 + b′m1xt

where bm0 is a scalar and bm1 is a Kx1 vector. Then a sufficient condition for the pricingkernel model spanning assumption, bm(t) = ΣP

p=1δcpbp(t), is that the (1xP ) vector δc satisfies:

δcbp0 = bm0

δcb′p1 = b′

m1

These (K+1) equations in the P unknowns, δcp, 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 theestimated 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 β = argmaxβ[R2c] where R2

c =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 approachwhich 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 (TxP )matrix of the fitted volatilities and the (TxP ) matrix of standardized return innovations. Toconstruct each bootstrap data sample we randomly select T (Px1)-vectors of standardized

innovations and construct the period t vector of returns by multiplying the fitted volatilitiesfor 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 theportfolios 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 spanningportfolio returns we generate the predictor variables for the pricing kernel model, xc

pt(β) =∑∞s=0

βsRp,t−sRM,t−s for different values of β and calculate the R2 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 10,000times 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 de-

termining the proportion of the 10,000 bootstrap samples in which the calculated value of thecorrected R2

c ≡ 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 bootstrapestimation.

The procedure was then repeated, replacing the quarterly excess return as dependent vari-able 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 calculatestandard errors and significance.

27Estimations are performed by a grid search over 0 ≤ β ≤ 0.95.

30

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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. Weestimate the model with three different sets of spanning portfolios: the S&P500 portfolio (M), the 3 Fama-Frenchportfolios (FF3), 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-Frenchportfolios (FF3), and for the discount rate model the expanded spanning portfolios based on dividend yield areused, which are the S&P500 portfolio and six portfolios sorted on dividend yield (6DP ). The plot shows theexpected 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 fromthe discount rate model. SMB and HML are the Fama-French size and book-to-market factors. HMZ is aportfolio formed on dividend yields that is long a portfolio of the 20% of stocks with highest dividend yields andshort a portfolio of non-dividend paying stocks. The expected returns for the spread portfolios are derived usingthe discount rate model with the 3 Fama-French portfolios (FF3) as spanning portfolios. The plot shows theexpected 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-yearreturn of the HMZ portfolio, HMZ(t) ≡ Et[RHMZ,t+1]. The expected return is derived using the discount rate

model where the spanning portfolios are the 3 Fama-French portfolios (FF3). HMZ is a portfolio formed ondividend yields that is long a portfolio of the 20% of stocks with highest dividend yields and short a portfolio ofnon-dividend paying stocks. BW(t+1) denotes the sentiment series advanced by one year in order to make itsprincipal components more timely. The plot shows the expected 1-year returns and sentiment measure for eachquarter over the period 1965Q3-2010Q4. Both series are standardized to zero mean and unit volatility.

37

Panel A. Predicting quarterly excess returns

Spanning β ρα w R2 W R2c Wc

Portfolios

M 0.522 0.631 0.109 0.043 21.558 0.041 ∗∗∗ 20.166 ∗∗∗

FF3 0.655 0.801 0.146 0.066 28.756 0.063 ∗∗∗ 23.809 ∗∗

3DP 0.694 0.829 0.135 0.088 46.775 0.083 ∗∗∗ 54.803 ∗∗∗

Expanded sets of spanning portfolios6BM − S 0.650 0.778 0.128 0.081 59.991 0.073 ∗∗ 46.989 ∗∗

6DP 0.675 0.819 0.144 0.079 60.659 0.071 ∗∗ 60.151 ∗∗

Panel B. Predicting 1-year excess returns

Spanning β ρα w R2 W R2c Wc

Portfolios

M 0.400 0.505 0.105 0.038 11.969 0.033 10.857 ∗∗

FF3 0.562 0.719 0.157 0.161 33.846 0.140 ∗∗∗ 28.614 ∗∗

3DP 0.547 0.727 0.180 0.184 36.503 0.157 ∗∗∗ 37.971 ∗∗

Expanded sets of spanning portfolios6BM − S 0.486 0.653 0.167 0.212 95.181 0.171 ∗∗∗ 75.079 ∗∗

6DP 0.501 0.693 0.192 0.173 73.281 0.130 ∗∗ 53.447 ∗∗

Table 1: Pricing kernel model: tests of predictability and persistence parameter

estimates

The table reports the results of tests of the null hypothesis of no predictability of the market excess return forthe pricing kernel model: RM,t+1 = a0 +

PP

p=1 δpxcpt(β) + εt+1 where xc

pt(β) =P∞

s=0 βsRp,t−sRM,t−s. Panel Apredicts quarterly and panel B 1-year returns. RM,t+1 is the quarterly (or 1-year) excess return on the S&P500index; Rp,t, p = 1, · · · , P , are the quarterly excess returns on a set of predictor portfolios. We estimate the modelwith three different sets of spanning portfolios: the S&P500 portfolio (M), the 3 Fama-French portfolios (FF3),and the S&P500 portfolio and three portfolios sorted on the basis of dividend yield (3DP ). The expanded sets ofspanning portfolios are the S&P500 portfolio and the six portfolios sorted on the basis of size and book-to-marketratio (6BM − S), and the S&P500 portfolio and six portfolios sorted on the basis of dividend yield (6DP ). Thesample period is 1946.1 to 2010.4 (the prediction period). Estimations are performed by a grid search over0 ≤ β ≤ 0.95. The parameters are chosen to maximize the R2

c 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 bya 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 Waldstatistic calculated using the Newey-West (1987) correction with 4 lags. Wc denotes the bias corrected value forW which is calculated by maximizing the bias-adjusted Wald statistic. Significance levels for Wc are calculatedby the bootstrap procedure, and indicated by stars.

38

Panel A: parameter estimates

Spanning portfolios: M FF3 3DP

β 0.522∗∗∗ 0.655∗∗∗ 0.694∗∗∗

(0.18) (0.10) (0.08)ρα 0.631∗∗∗ 0.801∗∗∗ 0.829∗∗∗

(0.18) (0.09) (0.08)a0 -0.000 -0.004 -0.003

(0.01) (0.02) (0.02)RM 1.316∗∗∗ 1.531∗∗∗ 1.529

(0.43) (0.39) (1.45)Fama-French portfolios

SMB -1.165∗

(0.67)HML 1.268∗∗

(0.59)Dividend yield sorted portfolios

zero -1.658∗∗∗

(0.53)Lo20 0.787

(0.97)Hi20 1.175

(0.73)

R2 0.043∗∗∗ 0.066∗∗∗ 0.088∗∗∗

Panel B. Descriptive statistics of risk premium estimates

M FF3 3DP

Mean 1.78% 1.78% 1.78%Std.Dev. 1.71 2.21 2.54Min. 0.09 -2.64 -5.17Max. 10.76 11.96 12.67

Correlation matrixM 1.00FF3 0.73 1.003DP 0.62 0.91 1.00

Table 2: Pricing kernel model: parameter estimates for quarterly forecastPanel A reports parameter estimates from the regression:

RM,t+1 = a0 +

PX

p=1

δpxcpt(β) + εt+1

with different spanning portfolio quarterly returns, Rp,t−s used to form the predictors, xcpt(β) =

P∞

s=0 βsRp,t−sRM,t−s. The dependent variable is the quarterly excess return on the S&P500 portfolio, RM,t. Weestimate the model with three different sets of spanning portfolios: the S&P500 portfolio (M), the 3 Fama-Frenchportfolios (FF3), and the S&P500 portfolio and three portfolios sorted on the basis of dividend yield (3DP ). Thesample 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. PanelB reports means and standard deviations for the risk premium estimates obtained using the different sets ofspanning portfolios, as well as the correlations between the estimates.

39

Panel A. Quarterly forecasts

Spanning Portfolios: M FF3 3DPHorizon

1 quarter 0.97 ∗∗ 0.96 ∗∗∗ 0.97 ∗∗∗

2 0.98 ∗ 0.94 ∗∗ 0.90 ∗∗∗

3 1.00 0.95 ∗∗ 0.88 ∗∗∗

4 1.02 0.95 ∗∗ 0.84 ∗∗∗

Panel B. 1-year forecasts

Spanning Portfolios: M FF3 3DPHorizon

1 year 1.02 0.91∗∗ 0.87∗∗∗

2 1.06 0.99 0.87∗∗

3 1.15 1.07 0.93∗

Table 3: Relative Error Variance of Out of Sample Return forecasts for the Pricing

Kernel ModelThe 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 valuebelow one indicates that the model outperforms the naive forecast which is the sample mean calculated fromdata up to the quarter of the forecast. The quarterly forecasts are made each quarter and are extended out to 4quarters using the estimated parameters of the AR(1) process. The 1-year forecasts are also made quarterly andare extended out to 3 years using the estimated parameters of the AR(1) process. The model and the historicalmean are estimated using data starting in 1926.2 and ending in the quarter before the forecast is made. Thespanning portfolios are the the S&P500 portfolio (M), the 3 Fama-French portfolios (FF3), and the S&P500portfolio and three portfolios sorted on the basis of dividend yield 3DP . Levels of significance are determined bya 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 α β R2adj ρ REVOOS

Variable

Panel A: univariate regressions

dp 0.110 (2.66) 0.027 (2.20) 0.018 0.984 1.015ep 0.060 (1.48) 0.015 (1.03) 0.004 0.950 1.058b/m 0.003 (0.19) 0.027 (1.17) 0.003 0.980 1.177VS 0.105 (2.00) -0.059 (-1.65) 0.007 0.816 1.000glam 0.025 (3.24) -0.024 (-1.31) 0.003 0.900 0.989svar 0.018 (3.23) -0.100 (-0.15) -0.004 0.456 1.008tbl 0.030 (3.22) -0.280 (-1.59) 0.007 0.949 1.001lty 0.028 (2.33) -0.171 (-0.94) 0.000 0.981 1.019tms 0.008 (1.04) 0.625 (1.63) 0.008 0.834 0.994infl 0.022 (3.34) -0.478 (-1.10) 0.001 0.474 0.996dfy 0.012 (0.88) 0.619 (0.42) -0.003 0.885 0.987kp -0.020 (-1.20) 0.313** (2.50) 0.023 0.931 0.985cay 0.017 (3.11) 0.897*** (3.64) 0.042 0.925 0.979

Panel B: bivariate regression

svar 0.007 (0.81) 1.117 (0.68) -0.005 0.620 1.240scov -0.552 (-0.17) 0.712

Table 4: Predictive regressions using other predictor variables

The table reports estimates of the equation:

RM,t+1 = α + βXt + εt+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 laggedvalue of the predictor variable. ρ is the auto-correlation of the variable. t-statistics are in parentheses and areadjusted 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 thelagged level of the index; the Value Spread V S, is the log book-to-market ratio of the small high-book-to-marketportfolio minus the log book-to-market ratio of the small low-book-to-market portfolio; Glamour, glam, is thecumulative 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 ofsquared daily returns on the S&P500 index over the previous quarter; tbl is the 3 month Treasury Bill rate; theLong Term Yield, lty is the yield on long term US government bonds; the Term Spread, tms, is the differencebetween the Long Term Yield and the Treasury Bill rate; Inflation, infl, is the one month lagged inflation rate;the Default Yield Spread, dfy, is the difference between BAA and AAA-rated corporate bond yields; and cayis the consumption, wealth, income ratio of Lettau and Ludvigson (2001) which is available over the period1952.3-2010.4. Fuller descriptions of these variables are to be found in Goyal and Welch (2008) and the actualdata series were taken from the website of Amit Goyal. kp is the in-sample predicted one year excess returnfrom Kelly and Pruitt (2012). Panel B reports the results from a multivariate predictive regression in which thepredictors are svar and scov, the sum of the daily cross-products of the market excess return with the HMLreturn. Levels of significance for β, indicated by stars, are determined by a bootstrap procedure in which returnsover 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 quarterlyreturn 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 beforethe forecast year.

41

Spanningportfolios: M FF3 3DP

PredictorVariable

dp -0.05 0.12 0.12ep -0.21 -0.12 -0.13b/m 0.05 0.06 0.04VS 0.09 -0.11 -0.10glam -0.36 -0.27 -0.23svar 0.55 0.41 0.36tbl -0.04 -0.40 -0.29lty 0.09 -0.27 -0.15tms 0.30 0.33 0.34infl 0.03 -0.17 -0.21dfy 0.46 0.19 0.25kp -0.06 0.21 0.24cay 0.01 0.04 0.07

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 whichare 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 threeFama-French portfoios (FF3) 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 0.010 0.015 0.018 -0.009 -0.010 -0.002 -0.003(0.61) (0.56) (0.86) (1.06) (-0.41) (-0.45) (-0.10) (-0.12)

glam -0.020 -0.001 -0.003 -0.002 -0.033 -0.012 -0.013 -0.013(-1.06) (-0.03) (-0.15) (-0.08) (-1.68) (-0.58) (-0.66) (-0.67)

kp 0.211 0.236 0.095 0.034 0.322 0.349 0.206 0.153(1.09) (1.27) (0.51) (0.18) (1.59) (1.77) (1.03) (0.74)

cay 1.002 0.914 0.878 0.848(3.87) (3.65) (3.75) (3.74)

Pricing kernel predictions:

M 1.008 0.944(4.03) (3.52)

FF3 0.923 0.874(5.04) (4.73)

3DP 0.949 0.881(5.78) (5.61)

R2 0.022 0.061 0.078 0.101 0.066 0.100 0.116 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 variablesand forecasts from the pricing kernel model for different sets of spanning portfolios. The spanning portfolios arethe S&P500 portfolio (M), the three Fama-French portfoios (FF3) and the S&P500 portfolio and three portfoliosformed 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 β ρα w R2 W R2c Wc

Portfolios

M -0.252 -0.145 0.107 0.009 3.498 0.008 2.418FF3 0.153 0.268 0.115 0.046 17.452 0.044 ∗∗ 14.055 ∗∗

3DP 0.884 0.877 -0.007 0.034 12.365 0.030 ∗ 13.130 ∗∗

Expanded sets of spanning portfolios6BM − S 0.248 0.334 0.086 0.042 21.713 0.036 11.8556DP 0.923 0.856 -0.067 0.065 33.553 0.056 ∗∗ 22.920 ∗∗

Panel B. Predicting 1-year excess returns

Spanning β ρα w R2 W R2c Wc

Portfolios

M dncFF3 0.014 0.066 0.052 0.023 8.115 0.021 4.4073DP 0.825 0.830 0.005 0.094 17.132 0.055 10.099

Expanded sets of spanning portfolios6BM − S 0.887 0.767 -0.120 0.121 23.739 0.042 8.8156DP 0.902 0.847 -0.055 0.181 52.138 0.099 ∗ 36.368 ∗∗

Table 7: Discount rate model: tests of predictability and persistence parameter

estimates

The table reports the results of tests of the null hypothesis of no predictability of the market excess return forthe discount rate model : RM,t+1 = a0 +

PP

p=1 δpxdpt(β) + εt+1 where xd

pt(β) =P∞

s=0 βsRp,t−s. Panel A predictsquarterly 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 withfive different sets of spanning portfolios: the market portfolio (M), the 3 Fama-French portfolios (FF3), theS&P500 portfolio and three portfolios sorted on the basis of dividend yield (3DP ); six portfolios formed on thebasis of firm size and book to market ratio (6BM -S), and the S&P500 portfolio and six portfolios formed on thebasis of dividend yield (6DP ). The sample period is 1946.1 to 2010.4 (the prediction period). Estimations areperformed by a grid search over 0 ≤ β ≤ 0.95. The parameters are chosen to maximize the R2

c of the predictiveregression, where R2

c denotes the R2 of the predictive regression adjusted to correct for small sample bias. Levelsof significance are determined by a bootstrap procedure in which returns over the period 1946.1 to 2010.4 aresampled 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 Waldstatistic calculated using the Newey-West (1987) correction with 4 lags. Wc denotes the bias corrected value forW which is calculated by maximizing the bias-adjusted Wald statistic. Significance levels for Wc are calculatedby the bootstrap procedure, and indicated by stars.

43

FF3 6BM-S 3DP 6DP

β 0.153 0.248 0.884∗∗∗ 0.923∗∗∗

(0.24) (0.24) (0.18) (0.13)ρα 0.268 0.334 0.877∗∗∗ 0.856∗∗∗

(0.24) (0.23) (0.19) (0.14)a0 0.018∗∗ 0.021∗ 0.026 0.030

(0.01) (0.01) (0.02) (0.02)RM 0.145∗ 0.764 0.312 0.272

(0.08) (0.53) (0.19) (0.25)Fama-French portfolios

smb -0.321∗∗∗

(0.11)hml -0.102

(0.10)Size and book-to-market sorted portfolios

sl 0.042(0.15)

sn 0.000(0.31)

sh -0.341(0.21)

bl -0.406(0.36)

bn 0.062(0.25)

bh 0.010(0.18)

Dividend yield sorted portfolios

zero -0.030 -0.040(0.06) (0.06)

Lo20 -0.188 -0.103(0.13) (0.13)

Qnt2 -0.251(0.16)

Qnt3 0.034(0.16)

Qnt4 0.209(0.16)

Hi20 -0.120 -0.169∗

(0.09) (0.10)R2 0.046∗∗ 0.042 0.034∗ 0.065∗∗

Table 8: Discount rate model: parameter estimates for quarterly forecastsThis table reports parameter estimates from the regression:

RM,t+1 = a0 +

PX

p=1

δpxdpt(β) + εt+1

with different spanning portfolio quarterly returns, Rp,t−s used to form the predictors, xdpt(β) =

P∞

s=0 βsRp,t−s.The dependent variable is the quarterly excess return on the S&P500 portfolio, RM,t. We estimate the modelwith four different sets of spanning portfolios: the 3 Fama-French portfolios (FF3), the S&P500 portfolio andthree portfolios sorted on the basis of dividend yield (3DP ); six portfolios formed on the basis of firm size andbook to market ratio (6BM -S), and the S&P500 portfolio and six portfolios formed on the basis of dividendyield (6DP ). The sample period is 1946.1 to 2010.4 (the prediction period). The parameters are estimatedby Supβ (R2 − bias). Standard errors (in parentheses) are calculated from a bootstrap simulation using 10000realizations.

44

Panel A. Quarterly expected returns

Model Pricing Kernel Discount Rate

Spanning portfolios: FF3 3DP FF3 6DP

Mean 1.78% 1.78% 1.78% 1.78%Std. Dev. 2.21 2.54 1.90 2.39

Panel B. Correlations between risk premium estimates

Model Pricing Kernel Discount Rate

Spanning portfolios: FF3 3DP FF3 6DP

QuarterlyPricing Kernel - FF3 1.00Pricing Kernel - 3DP 0.92 1.00

Discount Rate - FF3 0.22 0.28 1.00Discount Rate - 6DP 0.54 0.59 0.35 1.00

1-yearPricing Kernel - FF3 1.00Pricing Kernel - 3DP 0.85 1.00

Discount Rate - FF3 0.33 0.27 1.00Discount Rate - 6DP 0.62 0.61 0.30 1.00

Panel C. R2 from univariate and bivariate regressions

Model Pricing Kernel Discount Rate

Spanning portfolios: FF3 3DP FF3 6DP

QuarterlyPricing Kernel - FF3 0.074Pricing Kernel - 3DP 0.096 0.099

Discount Rate - FF3 0.108 0.126 0.054Discount Rate - 6DP 0.110 0.120 0.113 0.087

1-yearPricing Kernel - FF3 0.167Pricing Kernel - 3DP 0.197 0.194

Discount Rate - FF3 0.241 0.200 0.030Discount Rate - 6DP 0.172 0.257 0.205 0.200

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 3Fama-French portfolios (FF3), and the S&P500 portfolio and in turn, the three (six) portfolios sorted on the basisof dividend yield (3DP, 6DP ). Panel B reports correlations between the different estimates of the market riskpremium. The expected market excess returns are calculated quarterly. A rolling four-quarter average is takento reduce noise before calculating the correlations in Panel B. Panel C reports the R2 values from (i) univariatepredictive 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 Weekly Monthly Quarterly

covd(Rm,pk)Nd

covq(Rm,pk)covw(Rm,pk)Nw

covq(Rm,pk)covm(Rm,pk)Nm

covq(Rm,pk)covq(Rm,pk)covq(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 quarterlymarket excess returns using the pricing kernel model with FF3 spanning portfolios, reported in Table 8. Rpk isthe return on this portfolio. Covariances are calculated using daily, weekly, monthly and quarterly returns andthe sample period is 1946.1-2010.4. The covariances are then converted to a quarterly basis by multiplying bythe number of days (weeks, months), Nd etc., in a quarter and divided by the covariance from quarterly data,covq(Rm, Rpk).

46

Panel A: Pricing kernel estimated from prediction of 1-year return

Cyt+1 = a + bαy

M,t + εt+1

a b R2

Covarianceestimated:

Quarterly 0.025 0.433 0.063(1.86) (2.78)

Monthly 0.052 0.328 0.097(5.78) (3.56)

Weekly 0.098 0.176 0.007(6.19) (1.13)

Daily 0.117 0.184 0.003(5.71) (1.15)

Panel B: Pricing kernel estimated from prediction of 1-quarter return

Cqt+1 = a + bαq

M,t + εt+1

a b R2

Covarianceestimated:

Monthly 0.015 0.455 0.066(5.14) (3.28)

Weekly 0.020 0.520 0.043(5.90) (2.50)

Daily 0.023 0.497 0.026(5.08) (2.63)

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,Cy

t+1 (Cqt+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:

C ≡ covt(R, δc′Rp) ≡ n

nX

i=1

[(Rt+i− < R >)(δc′Rp,t+i− < δ′Rp >)]

where δc 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 covarianceestimates are regressed on the 1-year (1 quarter) predicted market excess return, αy

M,t (αq

M,t).

Cyt+1 = a + bαy

M,t + εt+1

Cqt+1 = a + bαq

M,t + εt+1

where αy

M,t =P

s=0 βsδc′Rp,t−sRt−s is the predicted 1-year market excess return using the Fama-French portfo-lios, and δc is the vector of portfolio weights from the 1-year return prediction and αq

M,t is defined analogously.t-statistics are in parentheses and are adjusted for serial correlation in the residuals using the Newey-Westcorrection (1987) with 4 lags. The sample period is 1946.1-2010.4.

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Dependent variable: SR RF RM

β 0.926∗∗∗ 0.973∗∗∗ 0.658∗∗∗

(0.08) (0.02) (0.11)a0 0.607∗∗ 0.011∗∗∗ 0.000

(0.24) (0.00) (0.01)RM 3.743 -0.026 1.580∗∗∗

(3.16) (0.02) (0.44)

Dividend yield sorted portfolios:

Zero -1.104 0.002∗∗∗

(0.73) (0.00)Lo20 -0.777 0.017∗

(1.61) (0.01)Qnt2 -4.16∗∗ 0.016∗

(2.02) (0.01)Qnt3 1.461 -0.047∗∗∗

(2.01) (0.01)Qnt4 2.859 0.058∗∗∗

(2.02) (0.01)Hi20 -2.386∗∗ -0.028∗∗

(1.27) (0.01)

SR 0.576∗∗

(0.24)RF -44.823∗∗

(22.40)

R2 0.083 0.626 0.056

R2c 0.052

Table 12: Restricted pricing kernel model: parameter estimates for quarterly fore-

cast

The first column reports parameter estimates for RM,t+1/σM,t+1 = a0 +PP

p=1 δSRp xSR

pt (β) + εt+1, where RM,t

is market excess return for quarter t, σM,t is an appropriately scaled estimate of the daily return volatility,and the 6DP predictor portfolio quarterly returns, Rp,t−s are used to form the predictor variables, xSR

pt (β) =P∞

s=0 βsRp,t−s.The second column reports estimates of δRF

p , 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

RM,t+1 = a0 − δRMxICRM,t − δSRxIC

SR,t − δRF xICRF,t

where xICRM,t(β) =

P∞

s=0 βsRM,t−sRM,t−s, xICRF,t(β) =

P∞

s=0 βsRF RRF,t−sRM,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 tothe autoregressive coefficient for the riskless interest rate.R2

c is the bias corrected estimate of R2. Standard errors (in parentheses) are calculated using a bootstrapprocedure 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

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Panel A. Discount rate model

Spread β ρα w R2 W R2c Wc

Portfolio

SMB 0.738 0.853 0.115 0.151 21.804 0.125 ∗∗ 17.451 ∗

HML 0.988 0.879 -0.109 0.217 28.536 0.161 ∗∗ 20.588 ∗∗

HMZ 0.981 0.879 -0.102 0.206 22.566 0.159 ∗∗∗ 16.324 ∗

Panel B. Pricing kernel model

Spread β ρα w R2 W R2c Wc

Portfolio

SMB 0.976 0.943 -0.033 0.202 30.172 0.168 ∗∗∗ 27.397 ∗∗

HML 0.464 0.618 0.154 0.084 14.204 0.063 12.384HMZ 0.510 0.690 0.180 0.097 9.655 0.078 ∗ 5.273

Table 13: Tests of predictability and persistence parameter estimates for 1-year

returns on spread portfoliosThis table reports selected parameter estimates for the predictability of 1-year spread portfolio returns fromthe discount rate and pricing kernel models using the 3 Fama-French portfolios (FF3) as spanning portfolios.The sample period is 1946.1 to 2010.4 (the prediction period). SMB and HML are the Fama-French size andbook-to-market factors. HMZ is a portfolio formed on dividend yields that is long a portfolio of the 20% ofstocks with highest dividend yields and short a portfolio of non-dividend paying stocks. Levels of significanceare determined by a bootstrap procedure in which returns over the period 1946.1 to 2010.4 are sampled underthe null hypothesis: ∗,∗∗ ,∗∗∗ denote significance at the 10%, 5%, and 1% levels. See notes to Table 1 for furtherdetails.

SMB HML HMZ

SMB 1.000 0.199 -0.031HML 0.199 1.000 0.774HMZ -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. SMB and HMLare the Fama-French size and book-to-market factors. HMZ is a portfolio formed on dividend yields that islong a portfolio of the 20% of stocks with highest dividend yields and short a portfolio of non-dividend payingstocks. Expected returns are calculated from the discount rate model using the three Fama-French factors (FF3)as spanning portfolios. The sample period is 1946.1 to 2010.4.

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Const. Sentiment Illiquidity R2

RM 0.01 -0.01 -2.06 0.00(0.88) (-0.83) (-1.09)

SMB 0.01 -0.01 -0.65 0.00(0.44) (-0.87) (-0.27)

HML -0.02 0.02 3.80 0.07(-1.55) (2.65) (2.33)

HMZ -0.04 0.06 1.23 0.30(-2.13) (5.19) (0.51)

Table 15: Expected return estimates, sentiment and illiquidityThis 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 + β1Sentimentt+1 + β2Illiquidityt + εt

The regression is estimated using expected 1-year returns on the market portfolio RM , and on each of the spreadportfolios (SMB, HML, HMZ). 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 (FF3) as spanningportfolios. The discount rate model estimates for the market portfolio use the 6DP portfolios as spanningportfolios. The sentiment variable is advanced by one year in order in order to make its principal componentsmore timely. SMB and HML are the Fama-French size and book-to-market factors. HMZ is a portfolioformed on dividend yields that is long a portfolio of the 20% of stocks with highest dividend yields and short aportfolio of non-dividend paying stocks. The sample period is from 1965.1 to 2010.4. The regression is estimatedon a time series of quarterly estimates of 1-year expected returns and t− statistics in parentheses are calculatedusing Newey-West standard errors with 4 lags. ∗,∗∗ ,∗∗∗ denote significance at the 10%, 5%, and 1% levels.

50

alow ahigh ahigh − alow β R2

SMB spread

Small -1.33 6.68 8.00 1.04 0.69(-0.79) (3.16)∗∗∗ (3.36)∗∗∗ (16.24)∗∗∗

Big 0.52 1.77 1.24 0.95 0.95(0.91) (2.60)∗∗∗ (1.82)∗ (31.86)∗∗∗

SMB -1.85 4.91 6.76 0.10 0.09(-1.32) (2.61)∗∗∗ (3.17)∗∗∗ (1.70)∗

HML spread

sh 2.60 9.48 6.88 0.99 0.62(1.08) (4.17)∗∗∗ (2.46)∗∗ (11.92)∗∗∗

bh -0.14 5.91 6.06 0.98 0.77(-0.14) (3.58)∗∗∗ (3.69)∗∗∗ (13.91)∗∗∗

sl -1.40 -1.79 -0.38 1.12 0.61(-0.57) (-0.82) (-0.12) (13.25)∗∗∗

bl -0.05 -1.16 -1.11 0.98 0.95(-0.09) (-1.67)∗ (-1.51) (45.40)∗∗∗

HML 1.95 9.17 7.21 -0.07 0.09(1.55) (4.59)∗∗∗ (3.97)∗∗∗ (-0.89)

HMZ spread

Hi20 0.07 4.76 4.69 0.84 0.68(0.05) (2.52)∗∗ (2.51)∗∗∗ (10.97)∗∗∗

zero 2.17 -6.07 -8.23 1.26 0.70(0.88) (-3.40)∗∗∗ (-3.15)∗∗∗ (18.64)∗∗∗

HMZ -2.09 10.83 12.92 -0.42 0.20(-0.68) (4.04)∗∗∗ (4.06)∗∗∗ (-3.84)∗∗∗

Table 16: Sources of time variation in spread portfolio returns

This table reports the results of estimating the regression:

Rp,t = alowDlow,t−1 + ahighDhigh,t−1 + βRmt + εp,t

where Rp is a portfolio 1-year return. Dlow (Dlow) is a dummy variable which is equal to one when the expected1-year return, Et−1[Rp,t], on the corresponding spread portfolio (SMB,HML, HMZ) is above (below) its medianvalue. The expected return on the spread portfolios are calculated from the discount rate model using the threeFama-French factors (FF3) as spanning portfolios. The regression is estimated using a quarterly time seriesof data over the period from 1946.1 to 2010.4. t − statistics in parentheses are calculated using Newey-Weststandard errors with 4 lags. ∗,∗∗ ,∗∗∗ denote significance at the 10%, 5%, and 1% levels.

51