Download Cash-flow Risk, Discount Risk, and the Value Premium

Cash-flow Risk, Discount Risk, and the Value
Premium∗
Tano Santos
Pietro Veronesi
Columbia University and NBER
University of Chicago, CEPR and NBER
June 3, 2005
Abstract
We propose a general equilibrium model with multiple assets that explicitly ties the
time series properties of the market portfolio with the cross-sectional predictability of
returns on price sorted portfolios, the value premium. This link provides tight predictions
about the cash-flow characteristics of value and growth stocks, such as their fundamentals
cash-flow risk and their expected dividend growth. We show that (a) substantial crosssectional difference in fundamentals cash-flow risk is necessary to generate a plausible value
premium; (b) the time variation the aggregate expected return induces fluctuations in the
size of the value premium and that, in particular, the value premium is larger in “bad
times;” (c) the unconditional CAPM fails, and thus a value premium puzzle arises, because
of general equilibrium restrictions on the market portfolio; and (d ) an HML factor lines up
returns as it captures aggregate differences in cash-flow risk in the economy. Our model
also sheds light on the empirical performance of conditional asset pricing models that have
been recently proposed to address the value premium.
∗
VERY PRELIMINARY AND INCOMPLETE. PLEASE DO NOT CIRCULATE WITHOUT
PERMISSION.
I. INTRODUCTION
Historically, stocks with high book-to-market ratios, value stocks, have yielded higher
average returns than stocks with low book-to-market ratios, growth stocks. The CAPM’s major
failure is its inability to price book-to-market sorted portfolios. As a result of this failure, a
large collection of explanations have been proposed to address this value premium puzzle, from
the behavioral to the rational ones that argue that the value premium is simply a compensation
for the larger risk of value stocks.1 These explanations though are surprisingly detached from
the voluminous literature that focuses on the properties of the aggregate market portfolio, such
as the large equity premium and the high volatility and predictability of aggregate returns.
In this paper we argue that the time series behavior of the market portfolio imposes general
equilibrium restrictions on the behavior of the cross-section of average returns of price sorted
portfolios. These restrictions are important as they provide tight implications about the cashflow characteristics of value and growth stocks as well as about the variation over time of the
value premium itself. Our predictions are broadly consistent with empirical evidence.
More specifically, we adopt a representative agent economy where preferences are of the
external habit persistence type introduced by Campbell and Cochrane (1999). These preferences are successful in generating plausible quantitative implications for the market portfolio
through the time variation of the market price of consumption risk. We follow Menzly, Santos
and Veronesi (2004, MSV henceforth), and embed these preferences in a general equilibrium
setting with multiple risky assets.2 These assets have time varying expected dividend growth
and differ from each other in their cash-flow risk, that is, in the covariance of their cash-flow
with the aggregate economy. By generalizing some results of MSV, we show that (a) substantial cross-sectional differences in cash-flow risk are necessary to generate a plausible value
premium; (b) the time variation in the market price of consumption risk interacts with the
cross-sectional differences in cash-flow risk to induce fluctuations in the value premium and
that, in particular, the value premium is high whenever the market premium is also high;
(c) because of general equilibrium restrictions on the total wealth portfolio, the unconditional
1
For the value premium see Rosenberg, Reid, and Lanstein (1985) and Fama and French (1992). For be-
havioral explanations see for example Rosenberg, Reid, and Lanstein (1985), DeBondt and Thaler (1987) and
Lakonishok, Shleifer, and Vishny (1994). For the rational ones see Fama and French (1993), Lettau and Ludvigson (2001), Gomes, Kogan and Zhang (2003) among others.
2
To the best of our knowledge the first fully fledged general equilibrium model of the cross section of stock
returns is Gomes, Kogan, and Zhang (2003) who build on the partial equilibrium model of Berk, Green and
Naik (1999). See also Zhang (2005).
1
CAPM fails and thus a value premium puzzle obtains; and (d ) an HML factor, as in Fama
and French (1993), lines up returns as it captures aggregate differences in cash-flow risk in the
economy. In addition, the model also sheds light on the performance of the recently proposed
conditional CAPM models.
In order to understand the intuition of our results, consider first the case where all assets
have identical cash-flow risk.3 In this case, we show that whether an asset has a high or
low premium depends on the asset’s duration, by which we mean whether it has low or high
expected cash-flow growth. Assets with high expected cash-flow growth are relatively more
sensitive to shocks in the aggregate discount than otherwise identical assets with low expected
cash-flow growth.4 When only discount effects are present then, the risk-return trade-off is
only determined by the timing of cash-flows, that is by the asset’s duration. Can discount
effects generate the value premium? No. Assets with strong expected cash-flow growth have
high price-dividend ratios and, as just mentioned, a high sensitivity to changes in the aggregate
discount, and, as a consequence, command a higher premium. Thus, a counterfactual positive
relation obtains between price-dividend ratios and average excess returns. It follows that if the
value premium is to obtain, cross-sectional differences in cash-flow risk have to be sufficiently
strong to undo discount effects.
Indeed, consider now the case of a low duration asset whose cash-flow growth is strongly
positively correlated with the growth rate of the aggregate economy. In this case, and due
to its low expected dividend growth, the total value of this asset is mainly determined by
the current level of cash-flows, rather than by those in the future. The price of the asset is
then mostly driven by cash-flow shocks and the fundamental risk embedded in these cashflows drives also the risk of the asset. Thus, when cash-flows display substantial fundamental
risk, the asset’s premium is higher when the duration is lower. Can these cash-flow effects
generate the value premium? Yes. Assets with high cash-flow risk and low duration have low
price-dividend ratios. This is due to both the fact that they are risky, and thus prices have
to be low to compensate agents for the risk they take, and because they have low expected
dividend growth. Thus, potentially, the value premium can now arise, and whether it does
or not depends on how the tension between “discount effects” (high risk when the asset has
3
In a general equilibrium setting, all assets have identical cash-flow risk only if their covariance of dividends
with consumption equals the variance of consumption. That is, all assets have a cash-flow risk that equals the
one of the aggregate endowment itself.
4
The parallel with the standard intuition in fixed income is helpful here: the price of a zero discount bond
of longer maturity is more sensitive to shocks in the aggregate discount than one with shorter maturity.
2
a high duration) and “cash-flow effects” (high risk when the asset has low duration) resolves
quantitatively.
In addition to an unconditional value premium, our model provides predictions for its
dynamics over time. We show that discount risk effects interact with the cross-sectional dispersion in cash-flow risk to induce time series variation in the relative risk of value versus growth
stocks over the business cycle as measured by shocks to aggregate consumption. In particular,
consistently with the data, value stocks are particularly risky during “bad” times. This is
natural: Agents demand a relatively higher compensation for holding assets with cash-flows
that covary positively with consumption growth when faced with adverse shocks. Thus cashflow effects have a conditional effect as well. To put it differently, cross-sectional dispersion
in cash-flow risk results not only on cross-sectional dispersion in unconditional average excess
returns, but also on fluctuations of the value premium that reinforce the unconditional effects.
To evaluate the model’s ability to yield quantitatively plausible implications we simulate
an economy with 200 assets over 10,000 years of quarterly data. Throughout we mimic the
procedure employed in the literature of sorting assets into decile portfolios formed on the
basis of price-dividend ratios. We calibrate the discount parameters to match the time series
properties of the aggregate market portfolio and thus generate quantitatively relevant discount
effects. We then calibrate the cash-flow effects to obtain properties of the cross-section of
stock returns that match well those in the data. In order to do so, we assume that the crosssectional dispersion in cash-flow risk is symmetrically distributed around zero. In our model,
we find that in order to obtain the value premium some of the underlying assets need to have
cash-flow growth processes that are, in absolute value, strongly correlated with consumption
growth.5 That is, the strong discount effects that are needed to generate the standard battery
of moments for the aggregate portfolio imply that also very strong cash-flow effects are needed
to generate the standard battery of moments for the cross-section of price sorted portfolios.
Moreover, we show that the preferred parameterization also generates the fluctuations in the
value premium observed in the data.
We then proceed to interpret the success (and failure) of existing asset pricing models,
such as the CAPM, the conditional CAPM, and the Fama and French (1993) model in the
context of our framework. First, in our setup, the CAPM does not hold either conditionally or
unconditionally. The reason is that general equilibrium restrictions induce a mild predictability
5
In a general equilibrium setting, if some stocks are strongly correlated with consumption growth, others
must have a strong negative correlation with consumption growth.
3
in expected consumption growth and this breaks the perfect correlation between the stochastic
discount factor and the return of the market portfolio. We show that the CAPM performs
poorly in our simulations but that the conditional CAPM, which is a misspecified asset pricing
model in our setup, performs much better. The reason is well known: The conditional asset
pricing model captures the increase in the relative riskiness of value stocks in bad times, that
is, whenever the market premium is high. But this effect can only arise if value stocks are also
the assets that have high cash-flow risk unconditionally, as our simulations show. This provides
a rationale as to why the two different strands of rational explanations that have been recently
proposed to address the value premium, conditional asset pricing models such as Lettau and
Ludvigson (2001) and models based on cash-flow risk, such as Parker and Julliard (2005) and
Bansal, Dittmar, and Lundblad (2005), have both been shown to be relatively successful: They
are both sides of the same coin in the presence of discount effects.
We then turn our attention to the factor model of Fama and French (1993). In particular
we construct an HML factor which captures the cross-sectional dispersion in cash-flow risk.
Assets with a high cash-flow risk have, mechanically, a high loading on HML. The premium
on HML varies over time, as already mentioned, so the Fama and French model performs
well because it captures the sources of unconditional cross-sectional variation in average excess
returns through the loadings on HML as well as the sources of conditional variation through
the fluctuations in the premium on HML. Finally, when we run in simulated data the time
series regressions of the returns of the price sorted portfolios on the returns on HML, we obtain
loadings that are very similar to their empirical counterparts.
The paper proceeds as follows: Next section introduces the model. Section III contains
the model results for prices and expected returns. In particular, it discusses the source of the
value premium in our setting. Section IV contains a simulation of our model and comparison
to data and evaluates the model’s ability to match basic moments of the returns data, both
in the time series and the cross section. In particular we investigate the time series behavior
of the value premium as a function of the dividend yield of the market portfolio. Section V
evaluates and interprets the existing asset pricing models in light of our findings. Section VI
quantifies the magnitudes of the cash-flow risk effects that are needed to generate the value
premium. Section VII offers some extensions and additional discussions and VIII concludes.
All proofs are in Appendix.
4
II. THE MODEL
II.A Preferences
There is a representative investor who maximizes
Z ∞
E
u (Ct , Xt , t) dt ,
(1)
0
where the instantaneous utility function is give by
(
1−γ
t)
e−ρt (Ct −X
1−γ
u (Ct , Xt , t) =
e−ρt log (Ct − Xt )
if γ > 1
(2)
if γ = 1
In (2), the variable Xt denotes an external habit level and ρ denotes the subjective discount
rate.6 The exact specification of the external habit Xt is described below.
II.B Cash-flows
We consider an endowment economy with n financial assets. Each asset has an instantaneous dividend stream denoted by Dti , for i = 1, .., n. The aggregate endowment available
for consumption at any time t is then equal to the sum of dividends.7 The consumption good
is immediately perishable and non-storable, which yields the equilibrium restriction
Ct =
n
X
Dti
(3)
i=1
Equation (3) implies that specific assumptions made on the dividend processes immediately
translate into particular dynamics for aggregate consumption. Unfortunately, even when one
assumes relatively simple processes for Dti the resulting process for aggregate consumption is
difficult to work with and restrictive assumptions need to be made for tractability.8 In order
6
On habit persistence and asset pricing see Sundaresan (1989), Constantinides (1990), Abel (1990), Ferson
and Constantinides (1991), Detemple and Zapatero (1991), Daniel and Marshall (1997), Campbell and Cochrane
(1999), Li (2001), and Wachter (2000). These papers though only deal with the time series properties of the
market portfolio and have no implications for the risk and return properties of individual securities.
7
For consistency with the data, we should consider also other forms of income such as labor income. Although
doing so does not present any additional technical challenge, we assume nonetheless that total consumption
equals dividends as the results and the general equilibrium restrictions are easier to see in this case. In Santos
and Veronesi (2005) we provide a thorough discussion of the role of labor income in a framework similar to the
one presented here without habit formation.
8
Recently, Cochrane, Longstaff and Santa Clara (2004) managed to solve in closed form the case where n = 2,
dividends are log-normally distributed, and agents are endowed with log utility. Their modeling device seems
hard to generalize for n > 2 and different preferences.
5
to better understand the restrictions that have to hold in a general equilibrium setting and the
nature of our assumptions below, it is instructive to review the nature of the difficulty, as also
0
explained in Santos and Veronesi (2005). Let Dt = Dt1 , ..., Dtn be the vector of dividends,
and assume for instance that dividends are given by
dDti
= µiD (Dt ) dt + νi0 dBt
Dti
(4)
for some drifts µiD (Dt ), and where νi is a n × 1 constant vector, and dBt is a n × 1 vector of
Brownian motions. From equation (3) and Ito’s lemma, we find that the process for aggregate
consumption is then
dCt
= µc (st ) dt + σc (st )0 dBt
(5)
Ct
0
where st = s1t , ..., snt = Dt1 /Ct , , ..., Dtn /Ct are shares of consumption produced by dividends, and
µc (st ) =
n
X
sit µiD
and
σc (st ) =
i=1
n
X
sit νi
(6)
i=1
The main difficulty in obtaining tractable and interpretable formulas for asset prices lies
in the dependence of the drift and the volatility of the consumption process on the shares st =
s1t , ..., snt . Still, analytical formulas for asset prices can be obtained by making economically
plausible assumptions on the joint processes of consumption Ct and shares st , as advanced in
MSV and Santos and Veronesi (2005). Here we follow Santos and Veronesi (2005) and make
two assumptions:
Assumption 1: The aggregate consumption is given by
dCt
= µc (st ) dt + σc0 dBt
Ct
where
Above, θCF
µc (st ) = µc + µc,1 (st )
and
µc,1 (st ) = s0t θCF .
(7)
0
0
1 , ..., θ n
i
= θCF
CF , and σc = (σc , 0, ..., 0) . The specification of θCF is
explained below.
Assumption 2: For each i, the share sit follows the mean reverting process
dsit = φ si − sit dt + sit σ i (st ) · dBt
where
σ i (st ) = νi0 −
n
X
j=1
6
sjt νj0
(8)
(9)
The cash-flow model (8) imposes a structure on the relative size of firms, where “size”
is measured as the fraction of total output produced by a given firm. In particular, it imposes
the economically plausible assumption that no firm will take over the economy, as sit > 0 for
P
all i. Finally, the volatility σ i (st ) in (9) ensures that ni=1 sit = 1 for all t. It is worth noting
that although the form of the volatility σ i (st ) in (9) seems an ad-hoc formulation, it actually
stems from the model (4) - (5), as it is possible to verify by Ito’s lemma.
II.C Cash-flow risk
Given Assumptions 1 and 2, an application of Ito’s Lemma shows that dividends, Dti =
sit Ct , evolve according to the following process:
dDti
i
= µiD,t dt + σD
(st ) dBt
Dti
(10)
where
µiD,t
= µc +
i
θCF
+φ
si
−1
sit
(11)
i
σD
(st ) = σc + σ i (st )
(12)
In these formulas,
i
θCF
= νi0 · σc
First, note that the expected dividend growth µiD,t = Et
h
dDti
Dti
i
depends on the relative
share si /sit . When this quantity is low, the asset’s relative contribution to total consumption is
below its long term average and the asset has a higher expected dividend growth. MSV test this
prediction in a set of industry portfolios and find strong support for it. In addition, note that
the long term dividend growth of this asset is given by µc , the unconditional expected return
i , which is asset specific and it depends on
of consumption growth, as well as a parameter θCF
the correlation of the stock shares with consumption growth.
Second, in this paper and because of the assumption of habit formation preferences, the
stochastic discount factor is only driven by shocks to consumption growth. Thus, cash-flow
risk is measured by the covariance of dividends with consumption growth, given by
i
dDt dCt
i
i
,
= σc σc0 + θCF
− s0t θCF
σCF,t ≡ Covt
Dti Ct
(13)
i
The quantity σCF,t
, the conditional cash-flow risk of asset i, will play a prominent role in this
i
paper. The term θCF
− s0t · θCF is parametrically indeterminate, that is, adding a constant
7
i
to all θCF
leaves this term unaffected, as
identifiability restriction
n
X
Pn
i
i=1 st
= 1. For this reason, we also impose the
j
sj θCF
= 0.
(14)
j=1
Thus, we obtain that the unconditional covariance between asset i’s cash-flow growth and
consumption growth is given by
σ iCF
i
i dDt dCt
i
= σc σc0 + θCF
.
,
= E σCF,t = E Covt
Dti Ct
(15)
i
In other words, the parameter θCF
regulates the relative cash-flow risk of individual assets,
as it is linear in the unconditional covariance of dividend growth with consumption growth
σ iCF . It is useful to emphasize that the benchmark level of risk of an asset is the riskiness of
aggregate consumption. An asset is deemed risky (safe) if its cash flows are more (less) risky
than aggregate consumption. This is a general equilibrium restriction, as by definition, the
variance of consumption growth must be a weighted average of its covariances with individual
dividend growth. In what follows we use the expression cash-flow risk when we refer to either
i
σ iCF or θCF
as there is a one to one mapping between one and the other.
As a final remark on cash-flows, note that the model is internally consistent: If we apply
the general equilibrium restriction on the drift of the consumption process, (6), to the dividend
process (10)
X
n
dCt
E
=
sit µiD,t = µc + s0t θCF ,
Ct
i=1
which equals (7) in Assumption 1. Note then that consumption growth is not i.i.d. but rather
has some predictable components which are linked to variation in the vector of shares, st . Still,
i
as we show below there is very little predictability in practice as the parameters θCF
are very
small.9
II.D Habit Dynamics
As advanced by Campbell and Cochrane (1999), the fundamental state variable driving
the attitudes towards risk in the habit model (1) − (2) is the surplus consumption ratio, St =
(Ct − Xt ) Ct−1 . In order to obtain closed form solutions for prices when there are multiple
9
Briefly, we show in simulations below that expected consumption growth fluctuates between a maximum of
2.22% and a minimum of 1.87%, a very mild variation compared to the 1.5% standard deviation of consumption
growth that we assume. Indeed, consistently with the empirical evidence, predictability regressions in our
simulated data produce a negligible level of predictability of consumption growth.
8
securities, MSV model the inverse surplus consumption ratio Yt = St−1 as a mean reverting
process. Unfortunately, their modelling device cannot be applied when γ > 1. In addition,
i
they only obtained approximate formulas for the case where θCF
6= 0. For these reasons we
opt here for a different strategy and model directly the process
γ
Ct
= St−γ
Gt =
Ct − Xt
(16)
To model the dynamics of Gt start by noticing that one important difference with respect
to the models of Campbell and Cochrane (1999) and MSV is the fact that in our model
consumption has some predictable components. Thus it is important to specify the process
for Gt in a way that is consistent with a time-varying expected consumption growth. To have
guidance on the type of process for Gt , it is instructive to consider the implications for Gt under
the standard assumption that Xt is an exponentially weighted average of past consumption
levels, as in Constantinides (1990) and Detemple and Zapatero (1991), that is,
Z t
Xt = λ
e−λ(t−τ ) Cτ dτ
−∞
Ito’s Lemma immediately shows that dXt = λ (Ct − Xt ) dt. Thus, an application of Ito’s
Lemma to (16) yields the process
dGt = [µG (Gt ) − σG (Gt ) µc,1 (st )] dt − σG (Gt ) σc dBt1 ,
(17)
where µG (Gt ) and σG (Gt ) > 0 are complicated functions of Gt , provided in equations (31)
and (32) in the Appendix. Equation (17) provides a strong intuition on how the time varying
component of the drift rate of consumption µc,1 (st ) should enter the process for Gt = St−γ .
Namely, a higher expected consumption growth µc,1 (st ) implies a lower drift rate of Gt . The
intuition is that an increase in the expected growth rate of consumption leads to a prediction of
higher future consumption compared to current habit Xt and thus a higher surplus consumption
ratio St in the future. Thus, given (16), this implies a lower expected Gt . In what follows, as in
MSV and Campbell and Cochrane, we simplify (17) to obtain a more manageable process, but
we retain the intuition that µc,1 (st ) enters the drift rate of Gt , in the same form as specified
in (17) and assume the following drift and diffusion processes:
µG (Gt ) = k G − Gt
and
σG (Gt ) = α (Gt − λ) .
(18)
Notice then that the drift of Gt has two components to it. The first one is a mean
reversion component and captures the basic idea of habit persistence models, namely that
9
the habit Xt eventually “catches up” with Ct . The second component links the drift rate of
Gt to the time varying component of the drift rate of consumption growth. Once again, in
order to retain the habit formation formulation, it is important to assume that the coefficient
that multiplies µc,1 (st ) equals the diffusion term in the process itself. As for the diffusion
component, as in MSV, λ ≥ 1 bounds Gt from below at λ and α > 0 transmits the innovations
in consumption growth, dBt1 , to the convexity of the utility function. These assumptions allow
us to obtain closed form formulas for asset prices. Note that the habit model of MSV is a
special case of (17) and (18): In fact, MSV assume that γ = 1, and that consumption growth
is i.i.d., an assumption that obtains here by setting µc,1 (st ) = 0.
III. EQUILIBRIUM ASSET PRICES AND RETURNS
In this section we characterize the prices and returns associated with the economy described in the previous section. Our strategy to do this is standard. Given (2) , the stochastic
discount factor is given by
mt = e−ρt (Ct − Xt )−γ = e−ρt Ct−γ Gt .
Then we can use Ito’s Lemma, together with our assumptions on the dynamics of Ct and
Gt = St−γ to show that
dmt
0
= −rtf dt + σm
dBt ,
mt
the first, and only non-zero, entry in the diffusion component vector, σm , is given by
1
σm
= − [γ + α (1 − λStγ )] σc .
Then we exploit our assumptions on the dynamics of Ct , Gt = St−γ and sit to solve for
Z ∞ Z ∞ mτ
mτ
i
i
i
Pt = E t
Dτ dτ = Et
sτ Cτ dτ
mt
mt
t
t
(19)
(20)
in closed form. We then use our expressions for prices to compute returns,
dRti =
dPti + Dti dt
− rdt
Pt
and calculate the expected excess returns
Et dRti = −cov
dmt
, dRi
mt
10
0 i
= −σm
σR ,
(21)
i is the diffusion component associated with the returns of asset i. Our purpose is
where σR
to obtain closed form expressions for (20) and (21) and relate them to the parameters in our
model.
III.A The total wealth portfolio
It is useful to start characterizing some basic properties of the total wealth portfolio as
the intuition for some of these results becomes useful later.
Proposition 1: The price-consumption ratio, the expected excess return and diffusion terms of the total wealth portfolio are, respectively:
PtT W
Ct
= α0T W (st ) + α1T W (st ) Stγ
(22)


n

 S γ α (1 − λS γ )
X
2
TW j
t
t
σ
+
w
σ
Et dRtT W = (γ + α (1 − λStγ ))
jt
CF,t  (23)
 f1T W (st ) + Stγ c
j=1
n
X
Stγ α (1 − λStγ )
TW j
=
wjt
σD (st ) ,
(24)
γ σc +
T
W
f1 (st ) + St
j=1
n
o
TW
where the functions α0T W (st ),α1T W (st ), f1T W (st ) and wjt
are given in ApTW
σR,t
pendix.
The results are similar to the ones found by Campbell and Cochrane (1999) and MSV,
and we refer to those papers for more detail. Briefly, the price-consumption ratio of the total
wealth portfolio is increasing in the surplus consumption ratio St . This is intuitive: a high
surplus consumption ratio implies a low local curvature of the utility function, a “less risk
averse” attitude by part of the representative agent, and this in turn translates into higher
price-dividend ratios.
Differently from Campbell and Cochrane (1999) and MSV, the total wealth portfolio
depends also on the entire vector of shares st . The reason is that in our setting, the general
equilibrium restriction (5) generates a mild predictability in consumption growth (see Assumption 1). The functions α0T W (st ) and α1T W (st ) are typically decreasing in expected consumption
growth, because in our set up the elasticity of intertemporal substitution is less than one. Thus,
this component implies that an increase in µc (st ) result in lower prices.10
10
To review the economic reasoning, a low elasticity of intertemporal substitution implies a desire for con-
sumption smoothing. Thus, an increase in expected consumption growth yields a higher desire of current
consumption, and thus lower savings. The consumer then sell stocks and bonds, resulting in a decrease of the
P/C ratio of the total wealth portfolio.
11
Turning to the expected excess returns, the first term in parenthesis reflects the curvature
of the utility function of the representative agent, and thus the degree of risk aversion: High
curvature parameter γ or low surplus St imply high expected returns. The first term of the
expression in brackets is instead linked to discount effects: As shown in the pricing function,
changes in St induce a volatility of stock returns which is perfectly correlated with the stochastic
discount factor, and thus it is priced. MSV discuss this effect more thoroughly.
The novel term is the second one in the bracket, which is the premium investors require
because of changes in expected consumption growth. This second term is typically negative.
The reason is that our modeling device induces a mild positive correlation between shocks to
consumption growth and shocks to expected consumption growth. Thus, since as explained
earlier, negative shocks to consumption growth are correlated with positive shocks to prices,
this component carries a negative premium.
III.B. Prices and returns for individual securities
Proposition 2: The price of asset i is given by
i
i
Pti
s
s
γ
i
i γ
i
i
= α0 + α1 St + α2 (st ) i + α3 (st ) St
i
Dt
st
sit
(25)
where α0i , α1i are positive constants and α2i (st ) and α3i (st ) are positive functions
of the share vector st given in appendix.
Proposition 2 shows that the price-dividend ratios of asset i is increasing in the surplus
consumption ratio St . As it was true for the total wealth portfolio, a high St implies a lower
risk aversion of the representative agent, and thus higher prices of assets. In addition, however,
the price-dividend ratio is increasing in the relative share, si /sit , which as shown in expression
(11) , determines the time varying component of expected dividend growth. A high expected
dividend growth results naturally in a higher price-dividend ratio. The last term shows that
shocks to expected dividend growth have a different effect depending on the level of the surplus consumption ratio: The higher the surplus consumption ratio the stronger the impact of
expected dividend growth shocks on the price-dividend ratio.
Finally, as it was true for the total wealth portfolio, the price of each individual asset
also depends on functions of the vectors of shares α2i (st ) and α3i (st ). As explained earlier,
these functions are typically decreasing in expected consumption growth because the elasticity
of intertemporal substitution is less than one in our setting. Thus, an increase in the expected
consumption growth decreases price-dividend ratios.
12
i
σR,t
Proposition 3: (a) The diffusion term of the return process for asset i is given by


γ
γ
X
1
S α (1 − λS )
i i
i j
i + ηit
= t i t σc + 
σD (st ) +
ηjt
σD (st )
(26)
γ
s
s
i
i
f1 si , st + St
1 + f2 (St , st ) si
j6=i
t
t
(b) The expected excess return of asset i is given by
Et dRti = µDISC
+ µCF
i,t
i,t
where


µDISC
i,t
= (γ + α (1 −
γ
γ
γ  αSt (1 − λSt )  2
σc
λSt ))
i
f1i ssi , st + Stγ
t

= (γ + α (1 − λStγ )) 
µCF
i,t
(27)

1
1 + f2i (St , st )
i i
i + ηit
σCF,t +
s
sit

X
i j
ηjt
σCF,t 
(28)
j6=i
with
f1i
i
s
/sit , st
α0i + α2i (st ) si /sit
>0
= i
α1 + α3i (st ) si /sit
and
f2i (St , st ) =
α2i (st ) + α3i (st ) Stγ
> 0,
α0i + α1i Stγ
i are given in Appendix.
and ηjt
Proposition 3 shows that the expected excess returns the effects are divided in two.
These two terms correspond to the two sources of shocks to returns that are shown in the
expression for the diffusion component of returns, (26) . The first correspond to shocks in the
discount factor and the second to cash-flow shocks, both its own, the second term in expression
(26) , and shocks to the cash-flows in the rest of the assets in the economy.
III.B.1 Discount risk effects
The source of this component of the risk premium, µDISC
, is the variation of the agi,t
gregate discount – proxied by Stγ . To interpret further this term it is useful to notice first
that
∂Pti /Pti
Stγ
.
γ
γ =
∂St /St
f1 si /sit , st + Stγ
(29)
Thus the volatility of an asset’s return is linked to the elasticity of prices to shocks in the
variable driving the aggregate discount, which is Stγ . The variance of these shocks is linked to
α (1 − λStγ ) σc ,
13
which is the diffusion component of dStγ /Stγ , the inverse of our state variable Gt , as it follows
from a basic application of Ito’s Lemma to (17) . Clearly, only the component of these shocks
that covaries with the shocks to the stochastic discount factor is priced which, given (19) is
[γ + α (1 − λStγ )] α (1 − λStγ ) σc2 .
(30)
The component of the asset’s premium that is linked to discount effects is the product of (29)
and (30) . Cross-sectional variation in the discount effects can only be driven by differences in
the price elasticity (29), which is in turn driven by the behavior of the function f1 si /sit , st .
We have been unable to obtain a general characterization of this function, but for parameter
values that are empirically relevant we find that
∂f1 si /sit , st
< 0,
∂ si /sit
and thus assets with a higher expected dividend growth, as measured by the relative share
si /sit , display stronger discount effects. The intuition is straightforward: stocks with a high
expected dividend growth pay the bulk of its proceeds far in the future. Thus, minor variations
in the aggregate discount rate – through the risk aversion of the representative investor – result
in large percentage variations of the price of the asset, as also shown in the first term of the
i
diffusion function σR,t
in equation (26). This variation is naturally priced and thus the higher
required premia associated with assets with large relative shares.
III.B.2 Cash-flow risk effects
The source of premia related to cash-flow shocks, µCF
i,t , has two components to it. The
first is related to shocks in the asset’s dividends, which is the second term in expression (26),
and the second is related to shocks in the dividends of the rest of the assets in the economy,
which, as shown in (25) , affect the price of asset i as well. The logic for the sources of the
premia linked to cash-flow shocks is the same as in the discount effects case. First it can be
easily shown that the elasticity of the price with respect to shocks to its own dividends is,
∂Pti /Pti
1
i
i + ηit
=
.
i
i
s
i
∂Dt /Dt
1 + f2 (St , st ) si
t
i (s ) , and recall that its conditional
The diffusion term of the dividend process of asset i is σD
t
i
covariance with consumption growth is denoted by σCF,t
. The first component of µCF
i,t is then
the component of the dividend shocks that covaries with shocks to the stochastic discount
14
factor multiplied by the effect that these shocks have on the price of asset i, as measured by
the elasticity. A similar logic applies to the second term in µCF
i,t . Indeed it can be shown that
∂Pti /Pti
∂Dtj /Dtj
i
= ηjt
for j 6= i.
As before this component of the premium results from the product of this (cross) elasticity
j
and the priced component of the shock to asset j’s dividends, σCF,t
.
How does the current level expected dividend growth, as measured by si /sit , affect the
cash-flow risk component of expected stock returns? Given the conditional covariance of the
i
dividend of asset i with aggregate consumption, σCF,t
, the first term of (28) is unambiguous:
i
Since f2i (St , st ) > 0, if the asset is “risky”, that is, if σCF,t
> 0, then a high expected dividend
growth translates in a lower premium stemming from current dividend volatility. The intuition
is also clear: a stock that pays more in the future than today has a relatively low dividend
i
compared to the future. Thus, the risk embedded in current dividends, σCF,t
, has a relatively
low impact on the total risk of stock. In the limit, if the stocks does not pay any dividend
today, it cannot have any “cash-flow risk”, as there is zero current covariance of dividends
with consumption. If instead the asset’s dividends covary negatively with consumption growth
i
(σCF,t
< 0), then a high expected dividend growth increases the risk premium. The argument,
of course, is the converse of the previous one.
The effect that the current expected dividend growth of asset i has on the second term
of the cash-flow risk component of stock return return (28) is more difficult to interpret. To
quantify these effects, the top panel of Figure 1 plots the quantity µCF
i,t as a function of the
h
i
i
at the steady state, that is, for the case where
unconditional cash-flow risk σ iCF = E σCF,t
St = S and st = s. As it can be seen, the cash-flow component of expected return is increasing
in σ iCF . Note however, that there is a negative “bias” in this component of expected excess
return. Indeed the case σ iCF = 0 still implies a negative expected excess return stemming from
cash-flow risk effects. This is due to the second component in (28), which is related to the
time variation in the aggregate expected consumption growth. As we discussed in the case of
the total wealth portfolio, this component carries typically a negative risk premium. Finally,
i
the bottom panel of Figure 1 plots µCF
i,t as a function of σ CF for the case where St = S but for
i
a random draw of shares st . Although an increasing pattern in σCF
can be easily seen, cross-
sectional differences in si /sit may make the component µCF
i,t of an asset with high unconditional
cash-flow risk σ iCF temporarily lower than that of an asset with lower cash-flow risk σ iCF .
15
III.C The value premium
In order to gauge the source of the value premium in our model it is convenient to turn
i
to Figure 2. Panels A, B, and C plot µDISC
, µCF
i,t
i,t , and the total Et dRt respectively against
the relative share si /sit for various levels of the asset’s unconditional cash-flow risk σ iCF , which
i
correspond to different values of θCF
(see expression (15)). In all cases, the level of surplus St
is set to its steady state value S. The parameters used are those of the calibration exercise
discussed in detail in the next section.
Start with Panel A. As discussed in Section III.B.2, the discount risk component of
expected return is increasing in the relative share si /sit , that is, with expected dividend growth.
The reason is that assets with high relative shares are more sensitive to shocks in the stochastic
discount factor. These shocks are naturally priced and thus the higher required premia of assets
with high relative shares. In addition, the discount risk component of expected returns does
depend as well on the asset’s unconditional cash-flow risk σ iCF : Stocks with higher cash-flow
risk σ iCF have a larger discount risk component in expected returns. The intuition is that
stocks with a higher σ iCF are riskier and as a consequence have lower prices. It follows that
changes in the stochastic discount factor have a larger impact, in percentages, on the prices of
assets with higher levels of cash-flow risk.
Panel B of Figure 2 plots the cash-flow risk component of expected returns which, as
discussed in Section III.B.2, is decreasing in expected dividend growth for stocks with high
cash-flow risk. Finally, Panel C reports the total expected return for each asset that is obtained
by adding to the discount risk component the cash-flow risk component of stock returns.
III.C.1 Discount risk effects and the “growth premium”
What does this decomposition of expected excess returns imply for the value premium?
Notice first that, given expression (25), sorting assets according to their price-dividend ratio
is akin to sorting them on both cash-flow risk, σ iCF , and expected dividend growth si /sit . In
particular, value stocks (assets with low P/D ratios) are, on average, associated with high
σ iCF and low expected dividend growth si /sit . Consider now the case where cross-sectional
i
differences in cash-flow risk are “small” (e.g. θCF
≈ 0 for all i). Then, σ iCF are roughly
the same across all assets and thus the sorting procedure simply picks differences in expected
dividend growth as measured by si /sit . In this case, the total expected excess return are
as in Panel A as discount effects dominate. Since low price-dividend ratio stocks are those
corresponding to low relative shares si /sit , value stocks are found on the left-hand side of the
panel and thus have low expected excess returns. Similarly, high price-dividend ratio stocks
16
are those with high si /sit and thus growth stocks are found on the right-hand side of the panel
and have high expected excess returns. To summarize, if cross-sectional differences in cashflow risk are “small”, then growth stocks have higher expected excess returns than value stocks.
That is, a “growth premium” obtains.
III.C.2 Cash-flow risk effects
It follows from the discussion above that if a value premium is to obtain there must
be substantial cross-sectional differences in cash-flow risk. To see this, consider now Panel C,
which reports the total expected return when both discount effects (Panel A) and substantial
cash-flow effects (Panel B) are present. Value stocks (assets with low P/D ratio) have, on
i , and low expected dividend growth, that is, low si /si .
average, high risk, that is, high σCF
t
This combination corresponds to the area around the top-left corner of the plot, that is, to high
expected excess return. Conversely, growth stocks (assets with high P/D ratios) must have a
i
combination of low σCF
and high si /sit . This combination can be found on the bottom-right
corner of the plot. As it can then be seen then value stocks will command a high premium
and growth stocks a low (and even negative) premium. To summarize then if cross-sectional
differences in cash-flow risk are “large”, then value stocks have higher expected excess returns
than growth stocks. That is, a “value premium” obtains.
III.C.3 Cash-flow risk effects and the dynamics of the value premium
Campbell and Cochrane’s (1999) important contribution is to show how a strong variation in risk preferences is able to generate the main time series properties of the aggregate
market portfolio, including a high equity risk premium, large volatility of returns, a low and
constant interest rate, a sizable Sharpe ratio, and, importantly, a strong predictability in aggregate excess return.11 We have shown that in such an environment, if cross-sectional differences
are mainly driven by cross-sectional differences in expected dividend growth, then a “growth
premium” obtains rather than a “value premium.” Given that what is empirically observed is
the latter and not the former, it must be the case that cross-sectional differences in cash-flow
risk must be the main determinant of the cross-section of stock returns. Clearly, the “size”
of these cash-flow risk effects can only be assessed in a model where the strong discount risk
effects needed to generate the time series properties of the market portfolio are present, otherwise one would underestimate the strength of the cash-flow risk effects that are in turn needed
11
More generally, Melino and Yang (2003) have shown in the context of a simple and pedagogical example
that a very sensitive stochastic discount factor is needed to match the basic properties of the market portfolio.
In particular what is required is a high variation in the conditional variance of the stochastic discount factor.
17
to obtain the value premium.
The presence of discount risk effects which are associated with the time series variation
in risk preferences has additional implications in what concerns the dynamics of the value
premium. Essentially, discount risk effects interact with the cross-sectional dispersion in cashflow risk to induce fluctuations in the value premium, as shown in Figure 3. This figure plots the
expected excess returns of three assets against the surplus consumption ratio, St . The dotted
line shows the expected excess return for the market portfolio; the solid line corresponds to
the expected excess return on a representative value stock with high cash-flow risk and low
expected dividend growth; finally the dash line corresponds to the premium of a representative
growth stock with low cash-flow risk and high expected dividend growth. As it can be seen,
when the surplus consumption ratio is low (high), the value premium is high (low). This is
i
are particularly riskier when the
intuitive: In our framework assets with a high value of θCF
representative agent’s is highly risk averse which occurs whenever adverse consumption growth
shocks depress the surplus consumption ratio, increasing in turn the market premium and its
dividend yield. Thus in our model the value premium has a strong predictable component
being high (low) when the market premium is high (low).
IV. THE VALUE PREMIUM AND ITS DYNAMICS
In this section we conduct a simulation study to evaluate the extent to which the model
can match the moments of the return data that have become standard in the literature both
in the time series and the cross-section. These moments can be found in Table I. The data
set is standard and it is very briefly described in the Notes to Table I. Panel A shows mean
and standard deviation for the returns on the market portfolio and the risk free rate. Panel
B shows the predictability regressions of Fama and French (1988) and Campbell and Shiller
(1988), for two different sample periods, which are meant to emphasize that long horizon return
forecastability is sensitive to the particular period under consideration. Panels A and B are the
standard concern of the equity premium literature. Panel C shows the value premium and its
corresponding puzzle, the failure of the CAPM to generate the large cross-sectional dispersion
in average returns across book-to-market sorted portfolios. Panel C is the standard concern of
the cross-sectional literature.
We leave the study of the CAPM’s failure to explain the value premium for Section
V.A. The ability of a model to match the stylized patterns in Panels A and B is essentially
related to the properties of the stochastic discount factor that the model generates. As it is
18
well known, habit persistence models à la Campbell and Cochrane, such as the present one,
are relatively successful in reproducing these patterns in the time series of aggregate returns.
Panel C instead is related to both discount and cash-flow effects and it is the focus of our
investigation but without loosing sight of the model’s ability to stay close to the facts in Panel
A and B. The presence of discount risk effects also generates implications for the time series
behavior of the value premium and we study these effects in Section IV.C.
IV.A Details of the simulation
We simulate the model presented in Section II.B with 10,000 years of quarterly data for
200 firms. We sort these assets into ten portfolios according to their price-dividend ratio in
an effort to mimic the standard procedure used in the cross-sectional literature and focus our
analysis on these ten portfolios. Table II contains the parameter values that are going to be
used throughout. These values were chosen to generate moments for our simulated economy
that are close to their empirical counterparts in Table I. Panel A of Table II contains our choice
of parameters for the consumption process and preferences. Average consumption growth, µc ,
is set at 2% and the standard deviation is set at 1.5%. This latter value should be measured
against a standard deviation of consumption growth for the postwar sample of 1.22% and the
one for the longer sample starting in 1889, which is 3.32%.12 As for the preference parameters,
our choice of γ is between the values used by MSV, γ = 1, and the value used by Campbell and
Cochrane (1999), γ = 2. Although the process that is modeled is Gt = St−γ , it is more intuitive
and clear to think about our choices in terms of the local curvature of the utility function,
γSt−1 , as it is the economically relevant magnitude for the effects of interest. Our choices
imply a steady state value of the local curvature of the utility function of γS
−1
= 48, higher
than the already high value of Campbell and Cochrane (1999) which is 35. The minimum
value of this local curvature is 27.75. Finally the parameter k and α are similar to the values
chosen by MSV.
As for the share process, we assume that all of the 200 simulated assets have the same
steady state contribution to overall consumption, si = 1/200 = .005. Also the speed of mean
reversion is set at φ = .052, a value that is only slightly lower than the one estimated by
MSV for the overall market portfolio. The key parameter of interest in our model is the one
i . Our general equilibrium setting requires that
that controls differences in cash-flow risk, θCF
this parameter is symmetrically distributed around zero, as the aggregate market must have a
12
See Campbell and Cochrane (1999) Table 2.
19
variance equal to the one of consumption, that is the cash-flow risk parameters must be
i
θCF
∈ −θCF , θCF ,
i
where θCF > 0, so that for any asset with cash-flow risk θCF
there is a “mirror” asset with
i . Throughout, and with some abuse of terminology, we refer to
cash-flow risk equal to −θCF
θCF as the cash-flow risk parameter but the reader should keep in mind that it is the support
of the cash-flow risk parameters of individual assets.
In what follows, we start discussing a baseline case with θCF = .00325 for it generates
quantitatively plausible implications for the cross-section of stock returns. We investigate this
case in detail and then, in Section VI, we study the behavior of the model under different
assumptions on the size of the parameter θCF . We also postpone a discussion of the size of the
cash-flow risk effects until that section.
IV.B The time series properties of the market and the value premium
Table III is the analog to Table I but in simulated data. Panel A shows the implications
of our model for the aggregate data. The model generates a sizable, if slightly low, equity
premium and volatility of stock returns, and the risk free rate moments are reasonable. Panel
B of Table III shows the predictability regressions for all the standard horizons. As already
mentioned the model does well in this dimension: The coefficients all have positive signs and
increase with the forecasting horizon as do the t−statistics. The R2 s are relatively lower than
their empirical counterparts but not far off the mark for the case of the 1948-2001 sample.
These results simply reproduce the good performance of the model in Campbell and Cochrane
(1999) and MSV in what refers to the market portfolio.
Panel C of Table III contains the average excess returns for the ten portfolios sorted on
P/D. The value premium obtains nicely in our setup. Indeed the value premium is a healthy
5.88%, only slightly above the empirically observed one of 5.50%. Notice though that the
average excess returns for each portfolio are below their empirical counterparts. The reason
is that as mentioned the model misses the equity premium by about 3%. This low premium
can be seen as well in the Sharpe ratio but, importantly, they decrease with the price-dividend
ratio, an important feature of the data (see Table I.)
i
The line denoted Avge θCF
× 100 reports the average cash-flow risk parameter for each
of the decile portfolios. As the intuition developed in Section III.C suggested, the sorting
i
procedure picks cross-sectional variation in the cash-flow risk parameter, θCF
: Stocks in the
value portfolio, portfolio 10, have, on average, a high cash-flow risk parameter whereas the
20
opposite is true for the growth portfolio, portfolio 1. In our framework, and in line with much
of the recent empirical research on this issue,13 value stocks are indeed riskier in the cash-flow
sense and the strength of this effect is enough to undo the natural “discount riskiness” of
growth stocks. Indeed, as shown also in Section III.C and in particular in the top panel of
Figure 2, if discount risk effects dominate a growth premium obtains. Substantial discount risk
effects are needed to match the time series properties of the market portfolio though and we
show in the next section that these effects interact in turn with the cross-sectional dispersion
of cash-flow risk to generate interesting dynamics for the value premium.
IV.C The dynamics of the value premium
As it is well known, habit persistence models induce fluctuations in the representative
agent’s attitudes towards risk, which in turn translate into predictable and highly volatile
stock returns. In this section we show that these fluctuations also interact with the crosssectional dispersion in cash-flow risk to induce in turn a predictable and volatile value premium.
These fluctuations in the value premium are argued as a primary reason for the failure of the
unconditional CAPM, a topic to which we return below.
To ascertain the time series variation of the value premium Table IV Panel A shows the
average excess return of the first and tenth decile portfolio as a function of whether the marketto-book ratio of the market portfolio is above or below a certain percentile, denoted by c. For
instance, the first line shows that the average excess rate of return of the first decile (growth)
portfolio is 13.18% if the market-to-book of the market portfolio is below the 15th percentile
of its empirical distribution and that of the tenth decile (value) portfolio is 23.57%. The value
premium is then 10.38%. Instead when the market-to-book is above the 15th percentile the
first decile portfolio has an average excess return of 5.73% and the tenth portfolio has one of
10.35% for a total value premium of 4.62%, which is considerably lower than the previous one.
This pattern is consistent independently of the percentile that is chosen as a cut-off point. The
value premium is higher whenever the market-to-book of the market portfolio is low. Notice
that these are also periods where the average excess return of the market is high, as shown in
M
the columns headed by R .
How does the model perform in this dimension? Panel B of Table IV reports the same
calculations as in Panel A but with simulated data. The only difference is that, naturally,
instead of using the market-to-book we use the price-dividend ratio of the market portfolio to
13
See Bansal, Dittmar, and Lundblad (2005), Campbell and Vuolteenaho (2005), Parker and Julliard (2005),
and Hansen Heaton and Li (2005).
21
identify the state. The pattern is indeed very similar with the only exception of the level of
the premia which is, as already discussed, lower than in the data. The value premium is higher
when the price-dividend ratio of the market portfolio is low than when it is high.
In summary then, the discount risk effects needed to replicate the time series properties
of the market portfolio interact with the cross-sectional dispersion in cash-flow risk to generate variation in the value premium. Value stocks are particularly risky during bad times,
periods when the aggregate market premium and its dividend yield are high relative to their
unconditional mean, an effect that is present both in the data and the model.
V. THE CAPM AND OTHER ASSET PRICING MODELS
As shown in the previous section the model matches well the basic moments of return
data both in the time series and the cross-section. Whereas several general equilibrium models
have been put forth to address the time series properties of the market portfolio this has
been much less the case when it comes to the cross-section of stock returns.14 Still, several
empirically based asset pricing models have been proposed to explain the patterns in the crosssection. To what extent is our model consistent with these empirically based asset pricing
models? How can our model illuminate the apparent success (and failure) of these models?
In this section we address these questions. We focus our analysis on three important models:
The CAPM of Sharpe (1964) and Lintner (1965), the Fama and French (1993) model and the
conditional asset pricing models proposed of late of which Lettau and Ludvigson (2001) is the
foremost example. It is important to emphasize that in our framework, all these models are
misspecified but as we show below, with the exception of the CAPM they “work” rather well
as they capture complementary aspects of the data.
V.A The CAPM
V.A.1 The CAPM and the value premium puzzle
Any asset pricing model that generates the value premium has also to be consistent
with the value premium puzzle, the inability of the CAPM to price “market” value sorted
portfolios.15 The value premium puzzle can be seen in the last line of Table I Panel C (CAPM
14
15
See Gomes, Kogan and Zhang (2003) for a notable exception.
The inability of the CAPM to explain the cross section of average returns is pronounced in the postwar
sample used in this paper. Recently though Ang and Chen (2005) and Fama and French (2005) show that the
behavior of the CAPM in the earlier sample covering 1927-1963 is much better. Still Daniel and Titman (2005,
Table 3) and Fama and French (2005) perform triple sorts, on ME, BE/ME and (preformation) market beta to
find variation in average returns unrelated to beta thus rejecting the CAPM.
22
β). The beta of the sorted portfolios is flat if not slightly decreasing in the market-to-book,
at odds with the strong increasing pattern in average returns. The CAPM produces no crosssectional dispersion in its measure of risk when confronted with substantial variation in average
returns. To do this more formally we turn to Table V Panel A where we report the results of
time series regressions of returns on the excess returns on the market portfolio,
Rtp = α + β M RtM + pt
for
p = 1, 2, · · · , 10.
We do this for both empirical (Panel A-1) and simulated data (Panel A-2). The panel shows
the intercepts in the time series, α, and its corresponding t−statistic, t (α) . It also reports the
beta on the market portfolio, β M and its t−statistic, t β M . We have omitted the t−statistic
on the loading for the case of simulated data because, consistently with the empirical data,
they are all strongly significant (well above 100).
Start with the case of the empirical data, Panel A-1. The intercepts, “alphas” of the
CAPM time series regressions are large and statistically significant. Growth stocks have large
negative intercepts whereas value stocks have large positive ones. The poor performance of
the CAPM can also be seen in line 1 of Panel A in Table VI, where we report the standard
Fama-MacBeth cross-sectional regressions. The coefficient is not statistically significant, enters
with the wrong sign and the R2 is just 11%.
Turn next to the time series regressions in simulated data, Panel A-2 of Table V. Unlike
the case in the empirical data, the betas cross-sectionally correlate positively with average
excess returns, an important issue on which more below. Still the cross-sectional dispersion
in betas is not enough to match the cross-sectional dispersion in average returns generated by
the model. Indeed the pattern and statistical significance of the intercepts in simulated data
is strikingly similar to its counterpart in empirical data. A visual impression of this result can
be obtained in the bottom-right panel of Figure 4, which shows the average excess returns for
the ten decile simulated portfolios plotted against the CAPM fitted returns.
In summary, the cross-sectional dispersion of betas is not enough to generate a large
cross-sectional dispersion in CAPM fitted excess returns, at least large enough to match the
average excess returns on the simulated decile portfolios. While average returns range between
3.01% for high price-dividend ratio stocks and 8.88% for low price-dividend ratio stocks, the
“fitted” returns only range between 3.90% and 5.34% (for the fitted returns see Table III Panel
C). That is, the model not only generates the value premium but also the value premium puzzle.
23
V.A.2 Fama-MacBeth regressions in simulated data
In our simulated data, the CAPM betas are cross-sectionally positively correlated with
average excess returns, although their spread is not enough to generate the observed spread
in average returns. Still, given that the CAPM betas are ordered in the right direction, crosssectional regressions that impose no constraints on the level of estimated market premium
may immediately induce a good fit as measured for instance by the cross-sectional R2 . This
can be seen in line 5 of Panel B in Table VI, where we run the Fama-MacBeth regression
in artificial data: The CAPM produces a good fit with an R2 of 88%. Moreover the market
enters significantly and with the right sign. The estimated quarterly market premium though is
3.35%, which corresponds to an annualized value above 13%. This number should be compared
to the market premium in our model which is 4.41% (see Table III Panel A.) Thus the CAPM
“works” in our model at the expense of an unreasonable level in the market premium. This
message has recently been emphasized by Lewellen and Nagel (2005) and Daniel and Titman
(2005): A small but slightly positive cross-sectional covariation between betas and average
returns can result in the unwarranted support of asset pricing models that fail to impose
economically based restrictions on the size of the premia of the proposed factors.
V.B The Fama and French (1993) model
V.B.1 Cash-flow risk effects, discount risk effects, and HML
The Fama and French (1993) model has become a standard benchmark in asset pricing
tests. How well does it work in our set up? To answer this question we construct an HML factor
in artificial data that is long the three top decile portfolios and short the bottom three shown in
i
Table III Panel C. This panel also reports the average cash-flow risk parameter θCF
for each of
the decile portfolios. Recall that cross-sectional differences in the expected covariance between
consumption and dividend growth are exclusively driven by this parameter, see expression
i
(15) . As already discussed, there is a clear ordering of average θCF
across decile portfolios.
i
Value stocks, say portfolio 10, has a much larger value of θCF
than growth stocks, say portfolio
i
1. HML then captures cross-sectional variation in θCF
across price-dividend sorted portfolios.
Recall that in our framework, as we saw in Section III.C, the value premium is present only
when cash-flow effects are large enough and this is exactly what HML summarizes. Still, it
i
is important to emphasize that differences in cash-flow risk θCF
also yield differences in the
impact that discount effects have on expected returns: Assets with large cash-flow risk have
low prices and thus shocks to the aggregate discount have a larger percentage impact on prices.
24
HML then is not a factor capturing cash-flow risk solely, but also, partly, the size of discount
i
risk. Does the large spread in θCF
implied by our choice of θCF = .00325 translate into
unreasonable loadings on HML? We investigate this question next.
V.B.2. Time series and cross-sectional regressions evidence
Table V Panel B presents the results of time series regressions,
Rtp = α + βM RtM + βHM L RtHM L + pt
for
p = 1, 2, · · · , 10.
Panel B-1 shows the results in the case of the empirical data. The results are well know.
The intercepts go down considerably and only one of them is statistically significant; value
(growth) stocks have a large (small) loading on HML and the inclusion of HML in the time
series regression collapses the betas on the market portfolio around 1 (see Fama and French
(1993, page 21-26)).
Panel B-2 shows the time series regression in simulated data. Again we do not report
the t−statistic on the loadings on the market and HML as they are all above 100. Turning
first to the loadings on the market portfolio, notice that, as it was the case in the empirical
sample, adding HML to the time series regressions has the effect of reducing the spread in the
i and collapse them around 1. As Fama and French (1993) note this pattern is
estimates of βM
related to the negative correlation between the market and the returns on HML.
As for the loading on the HML portfolio notice that it has a strong cross-sectional
variation which reflects the cross-sectional variation in the underlying cash-flow risk of the
different portfolios. Indeed the loading on HML of the growth portfolio is −.29 whereas that
of the value portfolio is 1.01. That HML helps considerably in addressing the value premium
puzzle in simulated data can be seen in the size of the intercepts of the time series regressions,
which drop considerably relative to the size of the intercepts when only the market portfolio
is present. Moreover there is no longer any pattern in the variation of the intercept across
decile portfolios, which shows that HML is capturing the systematic pattern of misspricing
documented in Panel A. Notice that, with the exception of the growth portfolio, the intercepts
are all statistically significant but that again their t−statistics are remarkably low, given that
they are the result of a misspecified model tested in 10,000 years of quarterly data.
The evidence in the Fama-MacBeth regression confirms the time series evidence. Line 2
of Table VI Panel A shows that HML enters significantly and the estimated size of the premium
on HML is very close to the average excess return of the HML portfolio. This is also the case
in our simulated regression, which is shown in line 6 of Panel B in Table VI. The coefficient
25
on the loading on HML is almost identical to its empirical counterpart and, once annualized,
close to our estimated average excess return of the HML portfolio, which is 3.91%. The only
caveat is that the market portfolio is significant in our simulated Fama-MacBeth regressions
whereas it is not in the empirical data. Yet, this table shows that the inclusion of HML in the
cross-sectional regression aligns the portfolios correctly, as the intercept is now close to zero
(with t−statistics equal to −1.06 even with 40,000 observations) and the (quarterly) market
premium equals 1.32%, which annualized is 5.28%, still higher than the average market return
in simulation (4.40%), but much smaller than the 13.4% obtained for the CAPM case.
V.C Conditional asset pricing models
Conditional asset pricing models have been proposed recently to address the failure of
traditional models such as the CAPM in explaining the cross-sectional variation in average
returns in book-to-market sorted portfolios. The idea, as advanced by Hansen and Richard
(1987), is that the CAPM may fail unconditionally but may hold conditionally. Thus tests of
the CAPM that ignore the use of conditioning information are simply misspecified. Researchers
have reacted to this observations by using as a proxy for investors’ information set variables
that are known to forecast returns in the time series.16 Typically these models result in testing
a multifactor model where the additional factor, other than the market, is the market itself
interacted with the proposed conditioning variables.
Lines 3 and 4 in Panel A of Table VI shows that conditioning by the dividend yield of
the market portfolio and the cay variable of Lettau and Ludvigson (2001) results also in a
coefficient for the instrumented market that is strongly significant. In addition the R2 is an
impressive 83% and 81% respectively. Panel B, line 7 shows that our model does also well in this
dimension. When we interact the returns of the market portfolio with the simulated dividend
yield of the market portfolio we obtain a strongly significant coefficient and, once again, of
similar magnitude to its empirical counterpart. The intuition behind these conditional asset
pricing models is that they capture the fact that value stocks become relatively riskier in bad
times, as shown in Table IV. Recall that the conditional CAPM does not hold in our setup.
What this section shows is that any model that captures the conditional effects that arise out
i
of the interaction of discount effects with the cross-sectional dispersion in θCF
is bound to
mechanically improve on the dismal performance of the unconditional CAPM.
16
See, among others, the conditional asset pricing models of Jagannathan and Wang (1996), Ferson and
Harvey (1999), Lettau and Ludvigson (2001), and Santos and Veronesi (2005).
26
VI. THE SIZE OF THE CASH-FLOW RISK
VI.A Do value stocks have larger cash-flow risk?
Many of our results depend on the extent to which value stocks are also those with high
cash-flow risk. Is this the case? Recently there has been a flurry of research that is remarkably
consistent on answering this question affirmatively. For instance, Cohen, Polk and Vuolteenaho
(2003) obtain cash-flow betas by regressing different measures of firm’s profitability on the
corresponding measures of market profitability. Panel A of Table VIII reports some of these
regressions, which we discuss in more detail below in Section VI.C, and it is taken directly
from Cohen, Polk, and Vuolteenaho (2003, Table II Panel B).17 Notice that independently of
the definition of cash-flow used, value stocks have always larger regressions coefficients on the
corresponding aggregate cash-flow measure than growth stocks, though the magnitudes are
very different depending on the definition used.
Also Bansal, Dittmar, and Lundblad (2005) regress dividend growth on a moving average
of consumption growth rates, an approach similar to the one of Parker and Julliard (2005),
and find that indeed cash-flow betas are larger for value sorted portfolios (see their Table 1.)
Recently as well, and building on these authors, Hansen, Heaton, and Li (2005) have focused
on the valuation of cash-flows that are cointegrated with consumption,
dt = λct + ζt + φ (xt ) ,
where xt is a variable driving changes in expected consumption growth. Here an increase in
the value of λ also increases the long run covariation of consumption growth. They find that
the low (high) BE/ME portfolios have indeed low (high) values of λ. That is, value stocks have
larger (long run) covariation with consumption growth.18 In summary then there is by now
substantial evidence that indeed value stocks have “more” cash-flow risk than growth stocks.19
The issue of course is whether they have “enough of it.” We turn to this questions next by
evaluating the model under different assumptions on the value of the parameter that controls
the cross-sectional dispersion of risk θCF .
17
18
The remaining regressions are very similar and are omitted in the interest of space.
These authors emphasize that whether the parameter λ is well identified depends on the presence of the
time trend. In their absence λ is only very poorly identified. As they point out, the economic interpretation of
these time trends is difficult.
19
Campbell and Vuolteenaho (2005) show in the context of an ICAPM model that value stocks are the ones
with a high cash-flow beta and they show that the premium associated with this loading is relatively higher than
the premium associated with the discount beta. Growth stocks have a high discount beta and a low cash-flow,
or bad, beta and thus command a lower premium.
27
VI.B The value premium and the market portfolio under different values of θCF
VI.B.1 Discount risk only
Table VII shows several of the moments of interest in simulated data for the different
values of the cash-flow risk parameter θCF considered. Start with the case θCF = 0, that is,
i
the case where there are no cross-sectional differences in cash-flow risk (θCF
= 0 for all i,)
and only discount effects matter. Section III.C.1 argues that in this instance a counterfactual
positive relation obtains between price-dividend ratios and average excess returns. In this
simulation, this effect is shown in the top left panel of Figure 4, which plots the average log of
the price-dividend ratio of the ten sorted portfolios versus their corresponding average excess
returns. This result is reported also in Table VII in the column marked 10 − 1. This column
denotes the difference in average excess returns between portfolio 10, the value portfolio, and
portfolio 1, the growth portfolio, which in this case is −3.65%, that is, as expected, a “growth”
premium obtains.
The results regarding the moments of the market portfolio when θCF = 0 are also
off target, but in the opposite direction.20 The model generates an excessively high equity
M
premium, R , of 9.90% and a high annualized standard deviation of returns, vol RM , of
24.16%, which is higher than its empirical counterpart of 16.25%. The level of the risk free rate
is reasonable but its volatility is slightly high. Finally, Table VII also reports the coefficients and
2
2
R2 s for the three b12 and R12
and four year b16 and R16
quarterly predictability regressions
of market returns on the log dividend-price ratio. The t−statistics are omitted throughout
because they are all well above statistical significance. The model generates predictability
magnitudes that are too high relative to the ones observed in the 1948-2001 sample, but that
accord well those of the sample that excludes the extraordinary market of the late 90s.
VI.B.2 Discount and cash-flow risk
The next rows of Table VII show the behavior of the model as we increase θCF . First,
notice that, as expected from the discussion in Section III.C, when we allow for a larger
i , the value premium increases. As already seen when
θCF , and thus a bigger spread in θCF
θCF = .00325, the value premium is 5.88%, only slightly above the observed one of 5.50%
20
This effect, however, is not due to any shortcoming of the model – as Campbell and Cochrane (1999) and
MSV show, the model can replicate the main empirical predictions of the aggregate portfolio – but to our specific
choice of parameters for the representative agent. Below we show that the presence of cash-flow risk puts all
of these moments back in line, and Table VII aims at showing differences in performance by varying only the
parameter θCF , while keeping the preference parameters constant.
28
(Table I Panel C) and the negative relation between price-dividend ratios and average excess
returns obtains. This negative relation is also clearly shown in the top right panel of Figure 4.
Second, the implications for the market portfolio improve, becoming closer to their empirical
counterparts, as we increase θCF . When, for instance, θCF = .003, the equity premium is
6.63%, the volatility of returns is 16.88% and the volatility of the risk free rate has decreased
to 4.65%. That the value premium improves as the cash-flow risk parameter increases is
consistent with the intuition provided in Section III.C: As θCF increases those assets with high
i , become particularly risky; this depresses its price-dividend
cash-flow risk parameters, θCF
ratio and increases its required expected excess return. More puzzling at first is the behavior
of the moments of the market portfolio as we increase θCF .
To understand these effects, recall first that our framework implies a mild predictability
of consumption growth. Moreover, it can be shown that consumption growth and expected
consumption growth are positively correlated. In addition, in our model the representative
agent has a low intertemporal elasticity of substitution and thus a preference for consumption
smoothing. These two ingredients combine to yield an effect on prices that is absent from
the habit persistence models of Campbell and Cochrane (1999) and MSV, which assume that
consumption growth is i.i.d. Assume for instance that a negative shock to consumption growth
occurs. The intuition of habit persistence models is that the corresponding drop in prices will
be sharper than in models where habits are not present because the increase in the representative agent’s risk aversion that accompanies a negative consumption shock gives an additional
negative push to prices. This effect is mitigated in our present model because a negative consumption shock is associated, on average, with a drop in expected consumption growth. Strong
preferences for intertemporal consumption smoothing implies that the representative agent will
attempt to save more when expected consumption growth decreases, increasing his demand for
stocks and bonds. This additional demand for assets thus reduces the drop in prices, its corresponding volatility and effectively reduces both the equity premium and the predictability.
The size of this counterbalancing effect depends on the “size” of the term µc,1 (st ) = s0t θCF ,
i
which governs the variation in expected consumption growth, see Assumption 1. If all θCF
are close to zero, as it is the case when θCF is low, then this effect is negligible, but if they
are large – as it is necessary to obtain substantial cash-flow effects – then the intertemporal
substitution effect will be large.
VI.C The magnitude of the cash-flow risk
As seen in Table VII, the value premium obtains in our set up if the cash-flow risk
29
parameter θCF is set to be equal to .00325. To evaluate how “big” this value is it is useful
to compute the range of correlation coefficients between dividend growth and consumption
growth implied by this choice of θCF , which is
i
dDt dCt
∈ [−.37, .42]
corr
,
Dti Ct
These bounds are large. In this model, in order to generate a sizable value premium we
need to assume that some stocks’ dividends have a very high correlation with consumption
growth, while others must have a strong negative correlation with consumption growth. To
put these number in perspective, Campbell and Cochrane (1999) report a correlation between
aggregate dividends and consumption of about .20.21
The need for large parameters can be seen in slightly different way in Table VIII. In Panel
A we reproduce the results of Cohen, Polk, and Vuolteenaho (2003) who compute measures of
cash-flow risks for the average stock in decile portfolios. To illustrate the type of calculations,
consider their use of individual stocks dividend growth, which is closest to the case discussed
in this paper. For each time t, each stock i in portfolio p, p = 1, ..., 10, they compute the
i
regression coefficient βt,t+R
R
X
ρjCP V ∆dit+j−1,t+j
=
i
αt,t+R
+
i
βt,t+R
j=1
R
X
i
ρjCP V ∆dmkt
t+j−1,t+j + εt+j−1,t+j
j=1
where ∆dit,t+1 and ∆dm
t,t+1 are the dividend growth of asset i and the market, respectively,
ρjCP V = .95 is a discount, and R is the regression time horizon. They call the regression
i
coefficient βt,t+R
the cash-flow beta.
Notice first that, irrespective of the cash-flow measure used, the regression coefficients
are indeed decreasing in the market-to-book: Value stocks have higher cash-flow betas than
growth stocks.
Magnitudes are different depending on which cash-flow measure is used.
P4
j
If either (accumulated) return on equity,
j=0 ρ ROE`,t+j,j+1 , or (accumulated) dividend
P4
j
growth,
j=0 ρ ∆d`,t+j,j+1 , is used as a measure of cash-flow growth, the regression coefficients roughly double when we go from growth stocks to value stocks. If instead we use
(accumulated) earnings growth relative to market value, (X`,t+4,4 − X`,t−1,0 ) /M E`,t−1,0 , the
coefficients increase by a factor of 10. Finally if (accumulated) earnings relative to market,
P4
j
j=0 ρ (X`,t+j,j+1 /M E`,t+j−1,j ) , is used they increase almost by a factor of 20.
21
As a caveat to this number, recall that in our paper we impose the general equilibrium restriction that
Pn
Dtj = Ct , so that, by construction, the correlation between aggregate dividends and consumption has to
j=1
be 1.
30
We also report in Panel B of Table VIII the corresponding regressions in simulated data
for several values of θCF . Clearly we do not have the counterparts to many of the variables
used by Cohen, Polk, and Vuolteenaho (2003) and we can only regress dividend growth for
each of the individual portfolios on the aggregate market dividend growth. The increase in
the regression coefficients is also a clear pattern in simulated data with the exception of the
case where θCF = 0. For values of θCF > 0, the growth portfolio has a much lower cash-flow
beta as defined by Cohen, Polk and Vuolteenaho (2003). For θCF = .001, the magnitudes
match well their empirical counterparts when either accumulated return on equity or dividend
growth, which is the measure closest to the one used in simulated data, are used and are
lower when earnings growth to market is used. Recall though that a value θCF = .00325 was
needed to generate the value premium. The spread in the regressions coefficients in this case
is large. Moreover the sign for growth stocks is negative, at odds with the empirical evidence
irrespective of the measure used and the magnitudes for the value portfolio stocks are high if
any of the first three measures discussed above is used.
To summarize then, there is supportive evidence that the cross-sectional dispersion in
cash-flow risk goes in the right direction: Value stocks do indeed have a higher cash-flow risk
estimates than growth stocks. Still, the magnitudes that are needed in this model to generate
the observed pricing effects are too big given the empirical evidence.
VII. DISCUSSION
VII.A Is consumption growth predictable by the stock market?
We saw above that the presence of predictable components in consumption growth has
a strong implications for the properties of the market portfolio in the presence of strong cashflow effects. Given this, a reasonable concern is whether the model implies a counterfactual
predictability of consumption growth by financial variables. This is not the case. Table IX
shows the results of long term quarterly regressions of consumption growth on the log dividendprice ratio. D/P is indeed a statistically significant predictor of consumption growth at all
horizons and it enters with the right positive sign but the t−statistics are surprisingly very
low considering that we are using 10,000 years of quarterly data. Moreover fluctuations in the
dividend yield explain, say, a puny .6% at the two year horizon and even less for the rest of
the horizons.
31
VII.B On the definition of firms
Our model is very stylized with respect to the definition of firms. In our setting, a firm
is infinitely lived with a cash-flow risk parameter that is chosen at time “0” and kept constant
throughout. Both assumptions are unrealistic: Firms do not live forever and, importantly,
it is likely that the riskiness of a firm cash-flows change with the life cycle of the firm. For
instance, Fama and French (2004) report that new lists have cash-flow characteristics that
are quite different from old firms. Similarly, Pastor and Veronesi (2003) show that young
firms have higher M E/BE ratios than old firms, everything else equal. Although it would
be valuable to incorporate such assumptions in our model, we do not believe the conclusions
would change. The reason is that our model’s predictions are about “price sorted” portfolios
and their characteristics depend, to a first degree, on expected dividend growth and the level
of cash-flow risk. For instance, suppose we assume that every period new firms are born, and,
consistently with a life cycle model, new firms have higher expected dividend growth (they are
i . Assume also that
born with a low relative share) and, say, a low cash-flow risk parameter θCF
i
increases, as fixed costs and transformation costs increase, thereby
during the life cycle, θCF
increasing the sensitivity of cash flows to the business cycle. In equilibrium, when there are
many assets, we will still have a cross-section of stocks that are characterized by either high
or low cash-flow risk, as well as either high and low expected dividend growth. The sorting
procedure would then pick the same type of stocks that the current sorting procedure does.
That is, growth stocks would be those with high expected dividend growth and low cash-flow
risk, while value stocks would be those with low expected dividend growth and high cash-flow
risk. The implications for returns would then be the same as in the current setting.
VII.C Relation to the literature
An important message of this paper is that discount effects interact with the crosssectional dispersion in cash-flow risk to generate fluctuations in the value premium that are
countercyclical: The value premium is high whenever the market premium is high as well.
Thus discount effects magnify the effects of a given cross sectional dispersion in cash-flow risk.
It is important to recall that in the absence this cross-sectional dispersion, a growth premium
obtains and it is also countercyclical. The data, as we now know indicates otherwise. This
observation illuminates why “cash-flow beta based” models such as Bansal et al. (2005) and
conditional asset pricing models such as Lettau and Ludvigson (2001) do both well in explaining
the value premium. They are different sides of the same coin: Cross-sectional differences in
32
cash-flow risk, which is what these cash-flow beta based models capture, induce time series
variation in the value premium in the presence of strong discount effects, which is what the
conditional CAPM models capture. Thus both approaches are complementary in addressing
the value premium puzzle.
Our model is also related to Lettau and Wachter (2005). These authors have proposed
a model where firms solely differ in the timing of their cash-flows, that is in their duration,
and where the economy is subject to discount shocks. Their model allows for the possibility of
investor sentiment shocks that are totally uncorrelated to shocks in the aggregate endowment
process. Growth stocks, which pay far in the future, are more sensitive to shocks in investor
sentiment but investors do not fear these shocks and thus the lower premium growth stocks
command. A value premium automatically obtains as discount risk goes unpriced.
VIII. CONCLUSIONS
Two effects combine to determine the cross-section of average returns: discount and
cash-flow effects. We have shown that if only the former are present then assets with high
valuation ratios as measured by the price-dividend ratio, growth stocks, command a high
premium and assets with low price-dividend ratios command a low premium. The reason is
that if only discount effects are present, then an asset’s conditional premium only depends on
how sensitive is its price to shocks in the stochastic discount factor. This sensitivity is directly
linked to the asset’s duration, that is, it is linked to whether the asset is delivering the bulk of
its contribution to consumption far in the future or not. Assets with longer duration are more
sensitive to shocks in the stochastic discount factor than otherwise identical assets with lower
duration. These assets are also those with stronger dividend growth and, as a consequence,
have higher price-dividend ratios. The result is then a positive relation between price dividend
ratios and average excess returns.
This last implication is counterfactual. The value premium is the observation that
“value” stocks, stocks with low prices to fundamentals have, on average, higher excess returns than “growth” stocks. This implication can only be obtained in the presence of strong
cash-flow effects. The intuition is simple enough: assets with high cash-flow risk have lower
prices and higher expected excess returns in order to compensate investors for the positive
covariation between the assets’ cash-flow and consumption growth. Thus cash-flow effects are
the key ingredient to deliver the value premium.
The magnitudes of these cash-flow effects can only be assessed in the presence of realistic
33
discount effects. We know that strong discount effects are needed to deliver plausible quantitative implications for the return properties of the market portfolio. We have proposed a
framework, which is a version of the model used by Campbell and Cochrane (1999) and Menzly, Santos, and Veronesi (2004), where the stochastic discount effects are such as to deliver
plausible implications for the market portfolio. A key ingredient is that the representative
investor’s attitudes towards risk fluctuate depending on the realization of shocks to aggregate
consumption. We have shown that this effect results in making value stocks relative riskier than
growth stocks in downturns, as measured by negative shocks to aggregate consumption. This
is natural: Agents demand a relatively higher compensation for holding assets with cash-flows
that covary positively with consumption growth when faced with adverse shocks. Thus cashflow effects have a conditional effect as well. To put it differently, cross-sectional dispersion
in cash-flow risk results not only on cross-sectional dispersion in unconditional average excess
returns, but also on variation in the conditional premia that further reinforces the unconditional effect. Indeed we show that the value premium increases considerably during bad times.
This may explain why the two different strands of rational explanations proposed to address
the value premium, conditional asset pricing models such as Lettau and Ludvigson (2001) and
models based on cash-flow risk, such as Parker and Julliard (2005) and Bansal Dittmar and
Lundblad (2005), have been relatively successful.
We have used the model to interpret the factor model of Fama and French (1993).
In particular we have constructed an HML factor and found that it captures cross-sectional
dispersion in cash-flow risk. Assets with a high cash-flow risk have, mechanically, a high
loading on HML. The premium on HML varies over time, as already mentioned, so the Fama
and French model performs well because it captures the sources of unconditional cross-sectional
variation in average excess returns through the load on HML as well as conditional variation
through the fluctuations in the premium on HML.
The model then provides a coherent view of the time series and the cross-section of average returns but it is to the latter that the present paper makes a contribution by quantitatively
assessing what is needed to deliver the value premium. Much remains to be done, of course.
In particular it would be interesting to investigate the small sample properties of our model
to address one question where our paper is slightly at odds with the empirical results, namely,
whether a rational asset pricing model such as Fama and French (1993) can be distinguished
in finite sample from a characteristic model. That is, to what extent can these cash-flow effects
that so key a role play in the cross-section be uncovered with the sample sizes at our disposal?
34
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38
APPENDIX
Rt
The habit dynamics: If Xt = λ −∞ e−λ(τ −t) Cτ dτ we have dXt = λ (Ct − Xt ) dt. Define then Gt =
f (Ct , Xt ) = (Ct / (Ct − Xt ))γ . We then have
1
fC = −γGt Gtγ − 1 Ct−1
(
1
2
1
1 )
+1
γ
γ
fCC =
γ (γ − 1) G Gt − 1 + 2γ Gt − 1 Gtγ
Ct−2
fX
=
γGt
1
(Ct − Xt )
1
1
where we used Gtγ = Ct / (Ct − Xt ) and Gtγ − 1 = Xt / (Ct − Xt ). Ito’s Lemma then yields
dGt = µG (Gt ) − σG (Gt ) µ{c,1} (st ) dt − σG (Gt ) σc dBt1
where
µG (Gt )
=
σG (Gt )
=
1
2
1
1
+1
1
γλGt + γ (γ − 1) G Gtγ − 1 σc2 + γ Gtγ − 1 Gtγ σc2 − σG (Gt ) µc
2
1
γGt Gtγ − 1
(31)
(32)
Proof of Proposition 1. This is a corrollary to Proposition 2, and it is proved below.
Proof of Proposition 2. Part (a). Pricing Formula. The pricing formula is
Z ∞
uc (Cτ , Xτ ) i
Dτ dτ
Pti = Et
e−ρ(τ −t)
uc (Ct , Xt )
t
Z ∞
γ −1
−ρ(τ −t) 1−γ
= Ct Gt Et
e
Cτ Gτ siτ dτ
t
We divide the proof in two parts: First, we obtain a general pricing formula which depends on the state variables.
Second, we obtain analytical solutions for the coefficients of these state variables.
Part a.1: A pricing formula. For this proof, it is convenient to rewrite the share processes in its general
form as
n
X
dsit =
sjt λji dt + sit ν i −s0t ν dBt
j=1
i
P
where λji = φs , for i 6= j, and λii = − j6=i λij = −φ
quantities
qti = Ct1−γ Gt sit and
P
j6=i
sj = −φ 1 − si = φsi − φ. Define the two
pit = Ct1−γ sit
and the 2n × 1 vector yt = [qt , pt ]. An application of Ito’s Lemma and tedious algebra shows
b y yt dt + Σy,t dBt
dyt = Λ
where
"
by =
Λ
bq
Λ0 + Θ
0
39
b qp
Θ
bp
Λ0 + Θ
#
,
b i for i = q, p, qp are diagonal matrices with ii element given by
Λ =φ (s × 10n ) , Θ
1
γ (1 − γ) σc2 − k − (1 − γ) σc2 α + (1 − γ) θi − αθi
2
i
θbqp
= kG + (1 − γ) σc2 αλ + αλθi
1
θbpi = (1 − γ) µc − γ (1 − γ) σc2 + (1 − γ) θi
2
and Σy,t is an appropriate matrix. Assuming existence of the expectation in the pricing function, we can apply
Fubini’s theorem
Z ∞
Z ∞
h
i
−ρ(τ −t) i
i
γ −1
Et e−ρ(τ −t) yτi dτ
e
yτ dτ = Ctγ G−1
Pt = Ct Gt Et
t
θbqi
=
(1 − γ) µc −
t
t
h
i
b y , eω(τ −t)
The expectation in the integral can be computed as follows: Let ω be the vector of eigenvalues of Λ
the diagonal matrix with ii element given by eωi (τ −t) and U the matrix of associated eigenvectors. Then, we
can write
2n X
2n
i
h
i
h
X
uik e(ωk −ρ)(τ −t) u−1
Et e−ρ(τ −t) yτi = ιi · U· eω(τ −t) · U−1 · yt e−ρ(τ −t) =
jk yjt
k=1 j=1
h
i
where u−1
is the jk element of U −1 . Substituting into the expectation, and taking the integral, we find
jk
h
i
Z ∞
2n X
2n uik u−1
2n
i
h
X
X
jk
Et e−ρ(τ −t) yτi dτ =
bij yjt
yjt =
ρ
−
ω
k
t
j=1
j=1
k=1
where
bij =
h
i
2n uik u−1
X
kj
k=1
ρ − ωk
Below, we obtain these coefficients in closed form. Note, however, that by substituting yjt = qjt for j = 1, ..., n
and yjt = pj−n,t for j = n + 1, .., n we obtain
!
Z ∞
n
n
X
X
i
−ρ(τ −t) i
γ −1
i
i
γ −1
b2j pj,t
e
yτ dτ = Ct Gt
b1j qj,t +
Pt = Ct Gt Et
t
j=1
n
X
=
Ctγ G−1
t
=
n X
bi1j + bi2j Stγ sjt
Ct
Ct1−γ Gt
n
X
1−γ
bi1j sjt + Ct
j=1
!
bi2j sjt
j=1
j=1
j=1
Part a.2: Analytical formulas for bi1,j and bi2,j . We finally obtain a closed form formula for bij ’s, and
thus, of bi1j and bi2j . First, note that we can write
bij = ιi · U· Ω−1 · U−1 ιj
b y on the principal diagonal. But then, since U· Ω−1 ·U−1 =
where Ω is the matrix with the eigevalues of Iρ− Λ
−1
by
Iρ − Λ
we have that for i = 1, ..., n and j = 1, ..., 2n
−1
by
bij = ιi · Iρ − Λ
· ιj
−1
by
We now explicitly compute these quantities. Define B = Iρ − Λ
, so that
by = I
B Iρ−Λ
40
Making this explicit, for every i = 1, .., n (row) we have
2n
X
b y = ιi
bij Iρ−Λ
j
j=1
where
by
b y and ιi is a (1 × 2n) row vector with 1 in ith position, and zero
Iρ−Λ
is the jth row of Iρ−Λ
j
elsewhere. For every i, we have a system of equations that pins down bij for all j = 1, .., 2n. We now solve
this system of equation. To limit the number of indices involved, we do this exercise for i = 1. Of course, the
methodology works for every i. For i = 1 we have then the following two systems of equations. The first holds
for j = 1, .., n and the second for the remaining n rows:
n
X
b11 ρ − φs1 + φ − θbq1 −
b1j φsj
=
1
(row 1)
=
0
(row 2)
j=2
n
X
−b11 φs1 + b12 ρ − φs2 + φ − θbq2 −
b1j φsj
j=3
..
.
−
n−1
X
b1j φsj + b1n ρ − φsn + φ − θbqn
=
0
(row j)
j=1
n
X
1
−b11 θqp
+ b1n+1 ρ − φs1 + φ − θbp1 −
b1n+j φsj
=
0
(row n + 1)
=
0
(row n + 2)
j=2
n
X
2
−b12 θqp
− b1n+1 φs1 + b1n+2 ρ − φs2 + φ − θbp2 −
b1n+j φsj
j=3
..
.
n
−b1n θqp
−
n−1
X
b1n+j φsj + b12n ρ − φsn + φ − θbpn
=
0
j=1
The first set of equation is readily solved. In fact, we can write
b11
=
αq1 + αq1 × φ
n
X
b1j sj
j=1
b1k
=
αqk × φ
n
X
b1j sj
for
k = 2, .., n
j=1
where
1
αqi = ρ + φ − θbqi
Multiply both sides of each row k = 1, ..., n by sk and sum across rows to obtain
!
n
n
n
X
X
X
1 j
1 1
j j
1 j
bj s = s αq +
st αq φ
bj s
j=1
Define the constants
Hq =
n
X
j=1
j=1
sj αqj and Kq =
1
1 − φHq
j=1
41
(row 2n)
Solving for
Pn
j=1
b1j sj we obtain the quantity
n
X
b1j sj = s1 αq1 Kq
j=1
Thus
b11
b1k
=
αq1 + αq1 × φs1 αq1 Kq
=
αqk
1
× φs
αq1 Kq
(33)
for
k = 2, .., n
Hence, the first term in the P/C ratio obtained earlier, i.e.
n
n
X 1 j X 1 j γ
Pt1
=
b1j st +
b2j st St
Ct
j=1
j=1
is given by
n
X
b11j sjt = αq1 s1t + φs1 αq1 Kq
j=1
n
X
αqk sjt
j=1
b1j .
b11j
=
where recall that for j = 1, ..., n we defined earlier
We now turn to the second system of equations, which for k = 1, ..., n can be rewritten as
b1n+k = αpk φ
n
X
k
b1n+j sj + b1k αpk θbqp
j=1
with
1
αpk = ρ + φ − θbpk
and b1k given in (33) - (34). Substitute b1k first, to obtain
b1n+1
=
αp1 φ
n
X
1
1
b1n+j sj + αpq
+ αpq
× φs1 αq1 Kq
j=1
b1n+k
=
αpk φ
n
X
k
b1n+j sj + αpq
φs1 αq1 Kq
j=1
where
k
k
αpq
= αpk θbqp
αqk
As before, for k = 1, .., n multiply both sides by sk and sum across k’s to obtain
! n
!
n
n
n
X
X 1
X
X
k 1
1 1
k k
j
k k
s bn+k = αpq s +
s αp φ
bn+j s +
s αpq φs1 αq1 Kq
j=1
k=1
k=1
Let
Hp =
n
X
k=1
!
k
s
αpk
k=1
and solve for
Pn
k=1
sk b1n+k to find
n
X
sk b1n+k
=
1 1
αpq
s Kp
+
k=1
n
X
!
k
s
k
αpq
k=1
where
Kp =
1
(1 − φHp )
42
φs1 αq1 Kq Kp
(34)
Substitute back into b1n+1 and b1n+k and find
b1n+1
=
1
αpq
+ s1 g11
b1n+k
=
s1 gk1
where for k = 1, ..., n
(
gk1
=
αq1 φ
αpk
1
αp1 θpq
Kp
+
n
X
!
j
s
j
αpq
!
φKq Kp
)
+
k
αpq
Kq
j=1
Thus, the second part in the price-consumption ratio is given by
n
X
1 1
b12j sjt = αpq
st + s1
j=1
n
X
gk1 skt
k=1
Generalizing the above derivations for every i = 1, ...,n, we can finally write
i
i
Pti
s
s
i
i γ
i
i
=
α
+
α
S
+
α
(s
)
+
α
(s
)
Stγ
t
t
0
1
2
3
t
Dti
sit
sit
where
where
1
αqi = ρ + φ − θbqi
α0i
=
α1i
=
α2i (st )
=
θbi
i
pq
αpq
= ρ + φ − θbpi
ρ + φ − θbqi
φαqi Kq s0t αq
α3i
=
s0t gi
(st )
n o
i
k
gki = αqi φ αpk αpi θbpq
Kp + s0 αpq φKq Kp + αpq
Kq
and
1
γ + α σc2 − k + (1 − γ − α) θi
2
θbqi
=
(1 − γ) µc − (1 − γ)
i
θbqp
=
θbpi
=
kG + (1 − γ) σc2 αλ + αλθi
1
(1 − γ) µc − γ (1 − γ) σc2 + (1 − γ) θi
2
Part (a) of Proposition 3: The Diffusion Component of Stock Returns
As shown, we can write
Pti = Ct α0i sit + α2i (st ) si + α1i sit + α3i (st ) si Stγ
e
Define by Set = Stγ = G−1
t . Using Ito’s Lemma, it is immediate to see that the diffusion of dS is given by
σS (S γ ) = Stγ α (1 − λStγ ) σc
43
Thus, an application of Ito’s Lemma shows that the diffusion term of Pti is given by
α1i sit + α3i (st ) si Stγ α (1 − λStγ )
i
σc
σR,t = σc + i i
α0 st + α2i (st ) si + α1i sit + α3i (st ) si Stγ
(
)
n
X
α0i + α1i Stγ 1{i} + φαqi Kq αqk + gki
γ skt σ k (st )
+
i
i
i i
i i
i
i
s
+
α
s
s
α
s
+
α
(s
)
St
+
α
(s
)
t
t
t
t
1
0
3
2
k=1
i
where 1{i} is the indicator function for k = i. Since σD
(st ) = σc + σ i (st ), and since by construction
)
(
n
X
α0i + α1i Stγ 1{k=i} + φαqi Kq αqk + gki
γ skt = 1
i
i
i i
i i
i
i
α
s
s
+
α
s
s
St
+
α
(s
)
+
α
(s
)
t
t
t
t
0
1
2
3
k=1
we can rewrite
i
σR,t
=
=
)
(
n
X
α0i + α1i Stγ 1{k=1} + φαqi Kq αqk + gki
Stγ α (1 − λStγ )
k
γ skt σD
σc +
(st )
i
i
i
i i
i
i i
(s
)
s
+
α
f1i si /sit ; st + Stγ
(s
)
s
+
α
s
S
s
+
α
α
t
t
t
t
t
3
1
2
0
k=1




X i k
Stγ α (1 − λStγ )
1
i
i
+ ηi,t σD
(st ) +
ηk,t σD (st )
γ σc +
i
i
i
i
 1 + f (S; s ) s

f1 s /st ; st + St
2
t
sit
k6=i
where
f1i
and
i
ηk,t
si /sit ; st
=
α0i + α2i (st ) si /sit
α1i + α3i (st ) si /sit
f2 (S; st )
=
α2i (st ) + α3i (st ) Stγ
α0i + α1i Stγ
φαqi Kq αqk + gki skt
=
α0i sit + α2i (st ) si + α1i sit + α3i (st ) si Stγ
Note that also that
f10 < 0 if and only if
α2i (s)
αi
1
< 0i =
i
α3 (s)
α1
αpi θbpq
Q.E.D.
Part (b) of Proposition 3: The Expected Return
The expected return is obtained immediately from σR,t by using the formula
h
i
dm
Et dRtT W = −Covt dRti ,
mt
Q.E.D.
Proof of Proposition 1: (a) The price consumption ratio of the total wealth portfolio can be obtained
by simply adding the prices of individual securities. In particular, we find
α0T W (st )
=
n
X
i=1
α1T W (st )
=
n
X
i=1
αqi sit +
n
X
φsi αqi Kq
i=1
i
αpq
sit +
n
X
i=1
n
X
αqk sjt = 1 + φKq s0 αq αq0 st
j=1
n n
o
X
i
k
φsi
αpk αpq
Kp + sαpq φαqi Kq Kp + αpq
αqi Kq skt
k=1
44
Algebra shows
1
αq0 st
1 − φHq
1
α1T W (st ) =
((αp st ) Kp φsαpq + αpq st )
1 − φHq
Part (b). An application of Ito’s Lemma to PtT W = Ct α0T W (st ) + α1T W (st ) Stγ implies that the diffusion part
of the TW portfolios is given by
α0T W (st )
TW
σP,t
=
=
=
=
α1T W (st )
S γ α (1 − λStγ ) σc
(st ) + Stγ × α1T W (st ) t
αq + Stγ (Kp φsαpq αp + αpq )
+ 0
I (st ) σ (st )
αq st + Stγ × ((αp st ) Kp φsαpq + αpq st )
j
j
n
j
X
st
αq + Stγ Kp φsαpq αpj + αpq
S γ α (1 − λStγ )
k ν j − s0 · ν
P
σ
+
σc + Tt W
γ c
n
γ
k
k
k
f1 (st ) + St
st
k=1 αq + St × Kp φsαpq αp + αpq
j=1
σc +
α0T W
n
X
Stγ α (1 − λStγ )
TW
σ
+
wjt
σD (st )
c
f1T W (st ) + Stγ
j=1
with
f1T W (st ) =
and where
TW
wjt
α0T W (st )
α1T W (st )
j
j
j
αq + Stγ Kp φsαpq αpj + αpq
st
k
= Pn k
γ
k + αk
st
α
α
+
S
×
K
φsα
pq p
p
pq
q
t
k=1
TW
wjt
= 1. Given the form of the stochastic discount factor, we obtain
i
h
dmt
= −Covt dRtT W ,
Et dRtT W
mt
)
(
n
X
Stγ α (1 − λStγ )
T W jc
γ
σc +
wjt σCF,t
= (γ + α (1 − λS ))
f1T W (st ) + Stγ
j=1
are weights such that
P
j
Q.E.D.
45
Table I
Basic moments in empirical data: 1948-2001
Panel A: Summary statistics for the market portfolio
R
M
7.71%
vol(RM )
rf
vol(rf )
16.25%
1.44%
3.08%
Panel B: Predictability regressions
Panel B-1: Sample 1948-2001
Horizon
ln D
P
t−stat.
R2
4
.13
(2.13)
.09
8
.2
(1.65)
.10
12
.26
(1.34)
.11
Panel B-2: Sample 1948-1995
16
.35
(1.29)
.14
4
.28
(4.04)
.19
8
.48
(4.00)
.32
12
.63
(4.49)
.43
16
.78
(5.41)
.54
Panel C: The value premium
Portf.
R (%)
M E/BE
P/D
Sharpe Ratio
CAPM β
Growth
1
6.86
5.05
43.47
.352
1.13
2
7.77
2.68
31.38
.450
1.02
3
7.67
2.00
26.87
.452
1.01
4
7.63
1.63
24.65
.461
.95
5
8.53
1.38
22.65
.555
.88
6
9.96
1.18
21.62
.640
.89
7
8.39
1.01
20.64
.522
.88
8
11.00
.86
19.95
.657
.91
M
9
11.39
.70
20.00
.644
.92
Value
10
12.36
.45
21.77
.600
.98
Notes to Table I. Panel A: Summary statistics for the market portfolio. R is the annualized
average excess returns of the market portfolio over the three month Treasury Bill. vol RM is
the annualized standard deviation of the returns on the market portfolio. rf is the average risk
free rate, as measured by three-month Treasury Bill rate, and vol(rf ) is its annualized standard
deviation. Panel B: Predictability quarterly regressions of excess returns at the 1, 2, 3, and
4 year horizon on the log of the price dividend ratio of the market portfolio. t−stat denotes
the Newey-West t− statistic where the number of lags is the double of the forecasting horizon.
Panel C: R is the annualized average excess returns of each of the decile portfolios, M E/BE
is the average market-to-book and P/D the average price dividend ratio. CAPM β is obtined
by running time series regressions of excess return on each of the ten decile portfolios sorted
on M E/BE on the market excess return, where M E is the market equity and BE is the book
value. Quarterly dividends, returns, market equity and other financial series are obtained from
the CRSP-COMPUSTAT database. The sample period is 1948-2001. The construction of the
BE/ME sorted portfolios follows the standard procedure of Fama and French (1992): Each year t
portfolios are sorted into 10 BE/ME sorted portfolios using book-to-market ratios for year t − 1.
Returns on each of these portfolios are calculated from July of year t to June of year t + 1.
46
Table II
Model parameters used in the simulation
Panel A: Consumption and preference parameters
µc
σc
γ
ρ
γ/S
min{γ/St }
α
k
.02
.015
1.5
.072
48
27.75
77
.13
Panel B: Share process parameter in the base line model
n
θCF
si
φ
νi
200
.00325
.005
.052
0.47
Notes to Table II. Panel A: µc is the annual average growth rate of the consumption process, σc
is the standard deviation of consumption growth, γ is the coefficient controlling the local curvature
of the utility function, ρ is the subjective discount rate, G, λ, α and k are the parameters controlling
the dynamics of the process Gt = St−γ , where St = (Ct − Xt )Ct−1 is the surplus consumption ratio
and the process for Gt is given by
dGt = k G − Gt − α (Gt − λ) µc,1 (st ) dt − α (Gt − λ) σc dBt1 .
Panel B: The share process for i = 1, 2 · · · , n is
dsit = φ si − sit + sit σ i (st )dB0 t
i
n = 200 is the number of assets in our artificial economy. θCF
is the parameter controlling the
i
, which are distributed uniformly in the range
cash-flow risk. Each assets is assigned a value of θCF
above. si is the fraction that each assets constributes to consumption in the steady state and φ is
the speed of mean reversion of the share process. Finally, σ i (st ) = ν i − s0t ν where ν i are vectors
i
i
/σc , νi+1
= ν i , and the remaining entries equal to zero. The simulation consists of
with ν1i = θCF
10,000 years of daily data.
47
Table III
Basic moments in simulated data
Panel A: Summary statistics for the aggregate portfolio
E(RM )
vol(RM )
rf
vol(rf )
4.41%
13.74%
.75%
4.53%
Panel B: Predictability regressions
Horizon
ln D
P
t−stat.
R2 (%)
4
.21
(26.98)
5.40
8
.32
(33.05)
8.09
12
.39
(36.92)
8.55
16
.43
(39.17)
8.54
Panel C: The value premium
Portf.
R (%)
ln (P/D)
i
) × 100
Avge(θCF
Sharpe Ratio
CAPM β
CAPM fitt. ret. (%)
Growth
1
3.01
6.53
−.2705
.227
.88
3.90
2
3.70
5.05
−.1501
.260
.90
3.97
3
4.47
4.61
−.0628
.290
.98
4.30
4
4.95
4.32
−.0088
.307
1.03
4.53
5
5.41
4.09
.0275
.318
1.07
4.73
6
5.76
3.87
.0531
.326
1.11
4.90
7
5.98
3.65
.0727
.324
1.14
5.01
8
6.27
3.42
.0907
.324
1.16
5.13
9
6.84
3.14
.1066
.331
1.19
5.24
Notes to Table III. Panel A: Summary statistics for the market portfolio. E(RM ) is the annu-
alized average excess returns of the market portfolio over the three month Treasury Bill. vol RM
is the annualized standard deviation of the returns on the market portfolio. E(rf ) is the average
risk free rate and vol(rf ) is its annualized standard deviation. Panel B: Predictability quarterly
regressions of excess returns at the 1, 2, 3, and 4 year horizon on the log of the price dividend
ratio of the market portfolio. t−stat denotes the Newey-West t−statistic where the number of
lags is the double of the forecasting horizon. Panel C: Annualized average returns R, average log
price-dividend ratio, ln (P/D), and CAPM β. CAPM fitted returns are the returns resulting from
multiplying the CAPM betas from the previous line by the average excess return of the market
i
i
portfolio reported in Panel A. Avge(θCF
) × 100 refers to the average θCF
(multiplied by 100) for
the assets in the corresponding decile portfolio.
48
Value
10
8.88
2.70
.1391
.369
1.21
5.34
Table IV
The dynamics of the value premium
Panel A: Annualized average excess returns (%) in empirical data
.
Market-to-book of market portfolio > c
Market-to-book of market portfolio < c
c
15%
20%
25%
30%
35%
1
13.18
10.57
5.51
6.97
8.19
10
23.57
21.70
19.16
19.49
18.65
10-1
10.38
11.14
13.64
12.51
10.45
M
R
15.40
13.41
9.89
10.50
11.14
c
15%
20%
25%
30%
35%
1
5.73
5.95
7.31
6.82
6.15
10
10.35
10.06
10.11
9.32
8.98
10-1
4.62
4.11
2.80
2.50
2.83
M
R
6.34
6.31
6.99
6.62
5.87
Panel B: Annualized average excess returns (%) in simulated data
Price-dividend of market portfolio < c
c
15%
20%
25%
30%
35%
1
7.06
6.34
5.72
5.28
5.01
10
18.18
16.35
14.90
13.55
12.80
10-1
11.12
10.01
9.17
8.27
7.80
Price-dividend of market portfolio > c
M
R
10.11
9.07
8.17
7.49
7.08
c
15%
20%
25%
30%
35%
1
2.29
2.17
2.10
2.03
1.93
10
7.24
7.01
6.87
6.88
6.77
10-1
4.95
4.84
4.77
4.85
4.84
M
R
3.40
3.24
3.15
3.09
2.97
Notes to Table IV. Panel A: Annualized average excess returns in empirical data of the growth
(portfolio 1) and value (portfolio 10) portfolios depending on whether the market-to-book of the
market portfolio is below or above the c percentile of its empirical distribution. Panel B: Annualized average excess returns in simulated data of the growth (portfolio 1) and value (portfolio
10) portfolios depending on whether the simulated price-dividend ratio of the market portfolio is
M
below or above the c percentile of its distribution in simulated data. R is the average excess
return on the market portfolio in empirical data (Panel A) and simulated data (Panel B).
49
Table V
Asset pricing models: Time series regressions (quarterly)
Panel A: Time series regression Rtp = α + β M RtM + pt for p = 1, 2, · · · , 10
.
Panel A-2: Empirical data
Portf.
α
t(α)
Growth
1
−.46
(−2.00)
2
−.03
(−.18)
3
−.02
(−.14)
4
.07
(.32)
5
.44
(2.07)
6
.78
(3.73)
7
.40
(1.51)
8
.99
(3.73)
9
1.07
(3.32)
Value
10
1.20
(2.65)
βM t βM
1.13
(39.80)
1.02
(43.68)
1.01
(42.56)
.95
(30.32)
.88
(27.24)
.89
(27.27)
.88
(21.38)
.91
(21.33)
.92
(17.56)
.98
(14.16)
Portf.
α
t(α)
Growth
1
−.22
(−16.87)
2
−.07
(−3.78)
3
.04
(2.21)
4
.10
(5.27)
5
.17
(7.89)
6
.22
(9.50)
7
.24
(9.70)
8
.28
(10.26)
9
.40
(12.48)
Value
10
.89
(20.07)
βM
.88
.90
.98
1.03
1.07
1.11
1.14
1.16
1.19
1.21
Panel A-2: Simulated data
Panel B: Time series regression Rtp = α + β M RtM + β HM L RtHM L + pt for p = 1, 2, · · · , 10
Panel B-1: Empirical data
Portf.
α
t(α)
Growth
1
.20
(1.13)
2
.17
(1.05)
3
.02
(.14)
4
−.12
(−.61)
5
.19
(.87)
6
.28
(1.58)
7
−.40
(−2.15)
8
.01
(.09)
9
−.08
(−.43)
Value
10
−.36
(−1.23)
βM t βM
1.04
(43.68)
.99
(51.25)
1.00
(46.13)
.98
(35.28)
.91
(30.25)
.96
(38.66)
.99
(39.90)
1.05
(48.04)
1.09
(39.61)
1.20
(29.85)
β HM L t β HM L
−.42
(−12.13)
−.12
(−2.37)
−.03
(−.68)
.12
(1.88)
.16
(3.62)
.31
(8.85)
.50
(10.35)
.61
(15.52)
.72
(21.04)
.97
(14.14)
Portf.
α
t(α)
Growth
1
−.01
(−1.24)
2
.05
(2.69)
3
.08
(4.36)
4
.08
(4.07)
5
.11
(5.26)
6
.14
(6.11)
7
.10
(4.23)
8
.05
(1.96)
9
.10
(3.46)
Value
10
.14
(5.67)
βM
.96
.94
.99
1.02
1.05
1.09
1.09
1.09
1.09
.96
β HM L
−.29
−.15
−.06
.03
.08
.11
.19
.32
.41
1.01
Panel B-2: Simulated data
Notes to Table V. Panel A: Time series regressions in empirical (Panel A-1) and simulated
(Panel A-2) data of returns on each of the book-to-market sorted portfolios on the market excess
return. Simulation parameters are contained in Table II. α denotes the intercept of the time
series regression and β M the regression coefficient. t(α) and t(β M ) denote the heteroskedasticity
corrected t−statistic. Panel B: Time series regressions in empirical (Panel B-1) and simulated
(Panel B-2) data of returns on each of the book-to-market sorted portfolios on the market excess
return and the returns on HML, where β HM L is the regression coeffcient on HML. The t−statistics
in simulated data have been omitted as they are all well above 100 for the case of the regression
coefficients, β M and β HM L .
50
Table VI
Asset pricing models: Fama-MacBeth regressions (quarterly)
Panel A: Empirical data
1.
2.
3.
4.
Const.
4.69
(3.21)
.36
(.23)
2.72
(2.24)
3.06
(2.48)
Mkt.
−2.52
(−1.65)
1.63
(.99)
−.87
(−.65)
−1.37
(−1.01)
SMB
HML
−.31
(−.31)
1.05
(2.16)
Mkt×log(D/P)
Mkt×cay
Adj. R2
11%
80%
1.71
(2.46)
83%
.06
(2.34)
81%
Panel B: Simulated data
5.
6.
7.
Const.
−2.19
(−21.77)
−.16
(−1.06)
1.91
(6.57)
Mkt.
3.35
(31.48)
1.32
(8.71)
−.95
(−3.06)
HML
Mkt×log(D/P)
1.08
(26.71)
Adj. R2
88%
99%
1.94
(12.60)
98%
Notes to Table VI. Panel A: Fama-MacBeth regressions in empirical data. Line 1, CAPM
regressions where Mkt. represents the average excess return of the market portfolio. Line 2, Fama
and French (1993) model, where SMB is the return on “small minus big” and HML is the return
on “high minus low”. Line 3, conditional CAPM regression where the dividend yield, log(D/P),
of the market portfolio is used as a conditioning variable. Line 4 conitional CAPM regression
where the variable cay of Lettau and Ludvigson (2001) is used as a conditioning variable. Panel
B: Fama-MacBeth regressions in simulated data. t−statistic in parenthesis and Adj. R2 is the
adjusted R2 .
51
Table VII
Cash-flow risk, the market portfolio and the value premium
Cash-flow risk
θCF × 100
.0
.1
.2
.3
.325
M
R
9.90
9.68
8.89
6.63
4.41
Market portfolio
vol(RM )
rf
vol rf
24.16
1.16
5.44
23.62
1.13
5.35
21.77
1.02
5.07
16.88
.81
4.65
13.74
.75
4.53
b12
.76
.74
.69
.52
.39
Predictability
2
2
R12
b16 R16
23.1 .78 22.4
21.7 .76 21.0
17.5 .71 16.6
9.3
.54
8.5
8.6
.43
8.5
Value premium
10 − 1 CAPM 10 − 1
−3.65
−3.76
−3.14
−3.27
−.73
−.98
4.71
3.33
5.88
1.44
Notes to Table VII. This table reports basic moments of the returns in simulated data as a
function of θCF ≥ 0, which determines the support on which the cash-flow risk parameters of inM
i
dividual firms are uniformly distributed, θCF
average excess
∈ [−θCF , θCF ]. R is the annualized
returns of the market portfolio over the three month Treasury Bill. vol RM is the annualized
standard deviation of the returns on the market portfolio. rf is the average risk free rate and
vol(rf ) is its annualized standard deviation. b12 and b16 are the regressions coefficients of the
quarterly predictability regressions of excess returns on the log of the price dividend ratio of the
2
2
market portfolio for the three and four year horizons. R12
and R16
are the corresponding R2 s.
The t−stats are omitted but they are all well above standard significance levels. 10-1 denotes
the value premium, defined as the difference between the average return on the value portfolio,
portfolio 10, and the growth portfolio, portfolio 1. CAPM 10-1 is the fitted CAPM value premium,
where the betas are calculated the standard way in simulated data and the market premium is the
M
corresponding R in each line.
52
Table VIII
The size of the cash-flow risk effects
Panel A: Empirical data
Cash-flow definition
P4 . j
p
j=0 ρ ROEt+j,j+1
std. err.
X
P4
j=0
3
.94
(.12)
4
.95
(.25)
5
.96
(.14)
6
.97
(.12)
7
.98
(.14)
8
1.12
(.25)
9
1.28
(.32)
Value
10
1.51
(.29)
.35
.65
.92
1.17
1.26
1.63
1.93
2.97
4.15
11.26
(.31)
(.31)
(.17)
(.16)
(.28)
(.69)
(1.01)
(2.26)
(3.26)
(10.76)
t+j−1,j
p
Xt+4,j+4 −Xt−1,0
.21
.66
1.46
1.61
.24
1.83
2.74
5.50
2.38
2.64
std. err.
(.19)
(.08)
(.52)
(.28)
(.61)
(.60)
(1.24)
(2.69)
(.60)
(1.65)
ρj ∆dpt+j,j+1
std. err.
.79
(.19)
.90
(.13)
.96
(.10)
1.03
(.13)
1.34
(.28)
1.44
(.46)
1.14
(.31)
1.44
(.88)
1.39
(.77)
1.28
(.91)
6
1.14
1.01
.35
.20
.63
7
1.21
.99
.40
.77
1.27
8
1.07
.89
.51
1.30
1.62
9
.91
.81
.63
1.98
2.46
Value
10
.98
1.31
2.72
4.68
4.71
p
M Et−1,0
P4
2
.91
(.31)
p
ρj M Et+j,j+1
p
std. err.
p
Growth
1
.72
(.52)
j=0
Panel B: Simulated data
θCF × 100
.0
.1
.2
.3
.325
Growth
1
1.20
.65
−1.32
−5.89
−7.36
2
1.13
.77
−.67
−3.93
−4.91
3
1.23
1.00
.01
−2.34
−2.70
4
1.20
.93
.05
−1.19
−1.21
5
1.05
.87
.26
−.52
−.37
Notes to Table VIII. Panel A reports the results of Cohen, Polk, and Vuolteenaho (2003, Table
II Panel B) for the following regressions with annual data for each the ten decile portfolios sorted
on market-to-book,
4
X
p
ρj ROEt+j,j+1
=
p
p
βCF,0
+ βCF,1
j=0
4
X
4
X
M
ρj ROEt+j
+ p4
j=0
ρj
p
Xt+j,j+1
p
M Et+j−1,j
j=0
p
Xt+4,j+4
p
− Xt−1,0
p
M Et−1,0
4
X
ρj ∆dpt+j,j+1
=
=
=
4
X
M
Xt+j,j+1
+ p4
M
M Et+j−1
j=0
M
M Xt+4 − Xt−1
p
p
+ p4
βCF,0 + βCF,1
M
M Et−1
p
p
βCF,0
+ βCF,1
p
p
βCF,0
+ βCF,1
j=0
4
X
ρj
p
ρj ∆dM
t+j + 4 .
(35)
j=0
ROE denotes the ratio of clean surplus earning (Xt = BEt − BEt−1 + Dt where BEt−1 is the
beginning of the period book equity and Dt are the dividends from CRSP) to BEt−1 . M Et−1
denotes the market value at the beginning of the period and ∆dpt+j,j+1 is the log of dividend
growth of decile portfolio p. The first subscript refers to the year of observation and the second to
the number of years after the sort. Similar quantities are defined for the market portfolio. GMM
standard errors computed using the Newey-West formula with four lags and leads are reported in
parenthesis. ρ is a constant, linked to one minus the dividend yield, set at .95. Panel B reports
the results of the regression (35) above for simulated data where the simulations are done using
the parameters contained in Table II, except the cash-flow risk parameter θCF , that determines
the support on which the cash-flow risk parameters of individual firms are uniformly distributed,
i
θCF
∈ [−θCF , θCF ]. t−statistics are omitted as they are well above statistical significance.
53
Table IX
Consumption growth predictability in simulated data
Horizon (quarters)
log(D/P)
t-stats
R2
4
.0071
(3.42)
.4%
8
.0136
(3.37)
.6%
12
.0146
(2.57)
.5%
16
.0159
(2.18)
.4%
Notes to Table IX. Time series predictability regression, in simulated data, of consumption
growth on the lagged dividend yield, log(D/P),
∆ct+k = α + γ log(D/P ) + t+k
for
k = 4, 8, 12,
and
16.
t−statistics are Newey-West t−statisticss where the number of lags is double the the predictability
horizon period.
54
Figure 1: The Cash Flow component of Expected Return
(A) The cash flow risk component of expected returns − steady state
0.03
0.02
μ
CF
0.01
0
−0.01
−0.02
−0.03
−4
−3
−2
−1
0
1
2
3
i
4
−3
Unconditional Cash Flow Risk σCF
x 10
(B) The cash flow risk component of expected returns − random relative share
0.04
0.03
0.02
μ
CF
0.01
0
−0.01
−0.02
−0.03
−0.04
−4
−3
−2
−1
0
1
2
i
Unconditional Cash Flow Risk σCF
3
4
−3
x 10
The top panel plots the steady-state cash flow component of individual assets’ expected
i
= E cov dD i /Di, dC/C . For
return against the unconditional cash flow risk parameter σCF
each asset, relative share is assumed equal to one, si /sit = 1, and suplus consumption ratio is
assumed equal to its steady state value St = S. The bottom panel reports the same quantities,
but under a random selecetion for relative shares si /sit .
Figure 2: The Expected Return and Expected Dividend Growth
(A) The discount risk component of expected returns
0.065
High cash flow risk
0.06
0.055
0.045
μ
DISC
0.05
0.04
0.035
Low cash flow risk
0.03
0.025
0
0.5
1
1.5
2
2.5
3
3.5
Expected dividend growth (sbari / s i)
4
4.5
5
4
4.5
5
4
4.5
5
(B) The cash flow risk component of expected returns
0.06
0.05
High cash flow risk
0.04
0.03
μ
CF
0.02
0.01
0
−0.01
−0.02
Low cash flow risk
−0.03
−0.04
0
0.5
1
1.5
2
2.5
3
3.5
Expected dividend growth (sbari / s i)
(C) Expected returns
0.12
High cash flow risk
0.1
0.08
E[ R ]
0.06
0.04
0.02
0
Low cash flow risk
−0.02
0
0.5
1
1.5
2
2.5
3
3.5
Expected dividend growth (sbari / s i)
The top panel plots the theoretical discount component of individual stock returns
plotted against the relative share si /sit , which proxies for expected dividend growth. This
i
=
quantity is computed for various levels of the asset unconditional cash flow risk σCF
i i
E cov dD /D , dC/C . The middle panel plots the cash flow risk component of stock returns, plotted against the relative share si /sit , again for various levels of unconditional cash
flow risk. The bottom panel reports the total conditional expected return for individual assets.
Figure 3: The Expected Return and Surplus Consumption Ratio
Expected returns
Market
Value
Growth
0.5
E[ R ]
0.4
0.3
0.2
0.1
0
0.02
0.025
0.03
0.035
0.04
Surplus Consumption Ratio S
0.045
0.05
0.055
This figure shows the theoretical expected return for the market portfolio (dotted line),
a representative value stock (solid line), and a representative growth stock (dash dotted line),
plotted against values of the surplus consumption ration St . The vertical dotted line is the
median value of the surplus consumption ratio St. The representative value (growth) stock is
chosen with low (high) expected dividend growth and high (low) cash flow risk θCF .
Figure 4: The Cross-Section of Stock Returns in Simulated Data
P/D ratios with Discount Effects
P/D ratios with Discount and CF Effects
4.5
8
7
4
log(P/D)
log(P/D)
6
3.5
5
4
3
3
2.5
8
9
10
11
mean return (%)
12
2
13
3
9
12
8
11
10
9
8
5
6
7
mean return (%)
8
9
Returns with Discount and CF Effects
13
fitted return (%)
fitted return (%)
Returns with Discount Effects
4
7
6
5
4
8
9
10
11
mean return (%)
12
13
3
3
4
5
6
7
mean return (%)
8
9
The top left panel plots the average price-dividend ratio (y-axis) of P/D sorted portfolios
versus their unconditional average return (x-axis) in artificial data, under the assumption that
j
i
= σCF = σc2 . Under the same
assets have no cross-sectional differences in cash flow risk, σCF
assumptions, the bottom left panel plots the “fitted” average return according to the CAPM,
i
mkt
], on the y-axis against the average return on the
i.e. E[ Returni ] = βCAP
M E[ Return
x-axis. The two right-hand panels plots the same quantities as the left hand panels, but under
the assumption that individual assets have cross-sectional differences in cash flow risk.