Description
People rationalize for various reasons. Rationalization may differentiate the original deterministic explanation of the behavior or feeling in question.
Rationalizing Momentum Interactions
Doron Avramov and Satadru Hore
?
?
Doron Avramov is at the Robert H. Smith School of Business, University of Maryland,
email: [email protected]. Satadru Hore is at the University of Iowa, email:
[email protected]. We thank Ravi Bansal, Darrell Du?e, and Monika Piazzasi for
useful comments. All errors are our own responsibility.
Rationalizing Momentum Interactions
Momentum pro?tability concentrates in high information uncertainty and high credit risk
?rms and is virtually nonexistent otherwise. This paper rationalizes such momentum inter-
actions in equilibrium asset pricing. In our paradigm, dividend growth is mean reverting,
expected dividend growth is persistent, the representative agent is endowed with stochastic
di?erential utility of Du?e and Epstein (1992), and leverage, which proxies for credit risk,
is modeled based on the Abel’s (1999) formulation. Using reasonable risk aversion levels we
are able to produce the observational momentum e?ects. In particular, momentum prof-
itability is especially large in the interaction between high levered and risky cash ?ow ?rms.
It rapidly deteriorates and ultimately disappears as leverage or cash ?ow risk diminishes.
1 Introduction
Momentum e?ects in stock returns are robust. Fama and French (1996) show that momen-
tum is the only deviation from the CAPM unexplained by the Fama and French (1993)
model. Schwert (2003) demonstrates that pro?t opportunities, such as the size and value
e?ects as well as equity premium predictability, typically disappear, reverse, or attenuate
following their discovery. Momentum is an exception. Speci?cally, Jegadeesh and Tit-
man (2001, 2002) document momentum pro?tability in the period after its discovery in
Jegadeesh and Titman (1993). Korajczyk and Sadka (2004) ?nd that momentum survives
trading costs, whereas Avramov, Chordia, and Goyal (2006) show that the pro?tability
of the other past-return anomaly, namely reversal, disappears in the presence of trading
costs. Fama and French (2007) argue that momentum is among the few robust anomalies.
Momentum robustness has generated a plethora of behavioral and rational explanations.
1
Empirical work also uncovers momentum interactions. In particular, Zhang (2006) ?nds
that momentum concentrates in high information uncertainty stocks, i.e., stocks with high
return volatility, high cash ?ow volatility, small market capitalization, or high analysts’
earnings forecast dispersion, and points to behavioral interpretations. Avramov, Chordia,
Jostova, and Philipov (2007) document that momentum prevails only among high credit
risk stocks and is nonexistent otherwise and, moreover, the credit risk e?ect dominates the
information uncertainty e?ect. Indeed, the momentum-credit risk relation could point to
1
Behavioral: Barberis, Shleifer, and Vishny (1998), Daniel, Hirshleifer, and Subrahmanyam (1998), and
Hon, Lim, and Stein (2000). Rational: Berk, Green, and Naik (1999) and Johnson (2002). Moskowitz and
Grinblatt (1999) argue that industry momentum explains momentum in individual stocks. Avramov and
Chordia (2006) show that while momentum is unexplained by risk based asset pricing models, removing
business cycle e?ects from model mispricing completely eliminates momentum e?ects. These last two
studies indicate that momentum could be attributable to missing factors in rational asset pricing models.
1
rational interpretations of momentum. However, this is an empirical ?nding that has not
been formalized in an equilibrium model.
2
Indeed, thus far there has not been any attempt,
to our knowledge, to theoretically rationalize or even behavioralize momentum interactions.
This paper studies momentum interactions from an equilibrium perspective and ulti-
mately shows that the concentration of momentum in high information uncertainty as well
as high credit risk stocks is perfectly consistent with rational asset pricing. In particular,
we develop a representative agent general equilibrium paradigm extending the Lucas (1978)
economy. Here, dividend growth is mean reverting, expected dividend growth is persistent,
and the representative agent is endowed with the recursive utility form of Du?e and Ep-
stein (1992), which is the continuous time analog of the Epstein and Zin (1989) and Weil
(1989) preferences. Moreover, based on the novel formulation of Abel (1999), our equilib-
rium clearing condition requires that the stream of dividends serve for both consumption
and debt repayment. The leverage role in asset pricing goes back at least to Merton (1974).
Our model is fairly general from the perspective of both preferences and dynamics. The
utility speci?cation breaks the tight association between the elasticity of intertemporal sub-
stitution and the risk aversion measure, which are reciprocals of each other under power
preferences. Moreover, our consumption dynamics closely follows the high economic growth
along with small consumption growth documented in the US. Collectively, our model al-
lows one to match key asset pricing regularities based on reasonable risk aversion measures.
In essence, we extend the theoretical work of Johnson (2002) along two important di-
2
Another interaction that we do not explore here is the volume e?ect on momentum pro?tability docu-
mented by Lee and Swaminathan (2000).
2
mensions. First, whereas Johnson does not explicitly formulate investor preferences, here
the dynamics of the underlying economic fundamentals is tied down to stochastic di?erential
utility of the representative agent. Moreover, Johnson focuses on understanding momen-
tum e?ects, whereas we attempt to rationalize the concentration of momentum pro?tability
among stocks with some particular styles. That is, the question of interest is how momen-
tum payo?s vary in equilibrium with ?nancial leverage, which proxies for credit risk, as well
as information uncertainty measures such as stock return volatility and cash ?ow volatility.
3
Based on simulations, we are able to generate high enough equity premium using rea-
sonable risk aversion measures, consumption dynamics, and leverage. More importantly,
we are able to rationalize momentum interactions, as we generate strong momentum ef-
fects for the interaction between high leverage and risky cash ?ow ?rms. We demonstrate
that momentum e?ects deteriorate and ultimately disappear as leverage or cash ?ow risk
diminishes. More speci?cally, the correlation between observed and expected return is pos-
itive and monotonically increases with leverage. The monotonic relation holds at high,
medium, and low levels of expected growth rate volatility, at high and low autocorrelation
of expected growth rate, and at the entire range of risk aversion measures (5 to 10) analyzed.
Next, we show that the expected return spread between the highest and lowest past
year cumulative return portfolios increases with leverage. For example, when operating
cash ?ows are highly volatile, the expected return spread is 11.49% for high leverage and
only 0.27% for low leverage stocks. On the other hand, the overall spreads are small when
3
The leverage proxy for credit risk is sound. For one, there is a strong correlation (the time series
mean of the cross sectional Spearman Rank Correlation is 0.34) between S&P credit rating and leverage.
Moreover, leverage is a crucial determinant in modeling default risk.
3
either the volatility of expected growth in cash ?ows is small or the expected growth in
cash ?ows is not highly persistent. Indeed, while leverage is crucial, risk and persistence of
cash ?ow growth are also important determinants of momentum e?ects in the cross section
of returns. On the other hand, momentum is only mildly related to investor’s risk aversion.
To summarize, it has been documented that momentum interacts with ?rm-level in-
formation uncertainty measures and credit conditions. Our collective evidence shows that
equilibrium momentum indeed concentrates in the interaction between risky cash ?ows and
high credit risk ?rms. In the presence of recursive preferences, ?nancial leverage, and per-
sistent dividend growth, one can match equity premia, riskfree rate, return predictability,
and especially the observational momentum pro?tability at reasonable risk-aversion levels.
The paper proceeds as follows. Section 2 describes the economic setup including in-
vestor preferences and the dynamics of the underlying economic fundamentals. Section 3
derives the interaction of momentum with credit risk and information uncertainty. Sec-
tion 4 reports the simulation results coming up from the theoretical formulations. Section
5 concludes and provides suggestions for future work. Technical details are in the appendix.
2 The economic setup
2.1 Preferences and Dynamics
In formulating investor preferences, we depart from the regular power utility speci?cation.
Theoretically, power preferences put a heavy restriction on elasticity of inter-temporal sub-
4
stitution (EIS) and risk aversion – they are reciprocals of each other – even when risk
aversion and EIS are distinct economic quantities. EIS is about a deterministic consump-
tion path as it measures the willingness to exchange consumption today for consumption
tomorrow conditioned on a current riskfree interest rate, whereas risk aversion is about
preference over a random quantity. Empirically, the power utility restriction gives rise to
the equity premium and riskfree rate puzzles as well as the failure of the consumption based
asset pricing model to explain the cross section dispersion in average stock returns.
Instead, we employ stochastic di?erential utility (SDU) of Du?e and Epstein (1992).
The SDU is the continuous time analog of the Epstein-Zin (1989) and Weil’s (1990) recursive
preferences, which break the tight association between risk aversion and EIS. The SDU is
identi?ed by a pair of functions (f
?
, A(J)), called an aggregator, where A(J) is local risk-
aversion and f
?
represents the relative preference between immediate consumption and the
certainty equivalent of utility derived from future consumption. An ordinally equivalent
representation of the SDU is given by the normalized aggregator (f, 0), which is formulated
as
f(C, J) =
?(1 ??)J
1 ?
1
?
_
C
1?
1
?
((1 ??)J)
1
?
?1
1??
?1
_
. (1)
In equation (1), C denotes the current consumption, J is the continuation utility (or the
value function) attributable to future consumption streams, ? is the EIS, ? is the discount
rate standing for the time preference, and ? is the relative risk-aversion parameter. Under
5
this convention, the time t value function of an agent can be written as
J
t
= E
t
_
?
t
f(c
s
, J
s
)ds. (2)
We derive below an explicit solution for J
t
assuming that ? = 1. This assumption is
innocuous. In particular, Bansal and Yaron (2004) and Ai (2007) show that ? = 1 is the
point of equivalence between wealth and substitution e?ects, but preferences over risk are
still determined by risk-aversion. That is, if ? < 1 (? > 1), the income (substitution)
e?ect dominates. In other words, high growth rate can lead an agent to consume more
(income e?ect) or invest more (substitution e?ect). Which course of action is undertaken
depends upon the ? parameter. Indeed, EIS primarily deals with determining the riskfree
rate, which is not at the core of our study. The normalized aggregator based on ? = 1 is
the limit of (1) taking the form
f(C, J) = ?(1 ??)J
_
log C ?
log(1 ??)J
1 ??
_
. (3)
Next, as in Johnson (2002), we assume that the dividend growth follows a geometric
path with stochastic expected growth rate. Speci?cally, the joint system of the observed
dividend growth and the unobserved expected dividend growth are formulated as
dD
t
D
t
= X
t
dt +?
D
dW
1
, (4)
dX
t
= ?(
¯
X ?X
t
)dt +?
x
dW
2
, (5)
where X
t
is the expected dividend growth, ? stands for the speed of mean reversion,
¯
X
is the long run mean of X
t
, ?
D
is the volatility of dividend growth, ?
x
is the volatility of
6
expected dividend growth, and the correlation between the two Brownian motions is ?.
To account for leverage, we build on the novel formulation of Abel (1999). Abel is able
to generate, in a fairly simple setting, low variability of the riskfree rate along with a large
equity premium, both of which are patterns documented in the US economy in the post-war
period. In particular, we assume that the equilibrium consumption is a portion of dividend
C = D
?
, (6)
while the remainder of the dividend stream is distributed as adjustment cost or debt pay-
ment in the economy. The no-leverage case ? = 1 depicts an economy with no adjustment
cost wherein the agent consumes the full dividend streams. In the ? < 1 case, which is at
the core of our analysis, the rest of dividends go towards debt payment. Then stocks in
this economy are residual claims on the consumption stream net debt payments.
In the presence of leverage, the consumption growth dynamics takes the form
dC
C
= µ
C
(X
t
)dt +?
C
dW
1
, (7)
where
µ
C
(X
t
) = ?
_
X
t
+
1
2
(? ?1)?
2
D
_
(8)
?
C
= ??
D
, (9)
suggesting that expected consumption growth is slower than expected dividend growth as
7
long as ? < 1. This is consistent with the US economy which demonstrates slow consump-
tion growth along with relatively fast economic growth. Notice also that in a levered econ-
omy, the volatility of consumption ??
D
is smaller than the volatility of economic growth.
The utility process J satis?es the Bellman equation with respect to equilibrium con-
sumption
DJ(C, X, t) +f(C, J) = 0 (10)
where DJ is the di?erential operator applied to J with respect to {C, X, t} with the
boundary condition J(C, x, T) = 0. In the analysis that follows, we are interested in the
equilibrium as T ? ?. Thus, we drop the explicit time dependence assuming that the
agent is in?nitely long-lived and has reached the equilibrium over time. We can formulate
an exact solution to the value function which is stated in the following proposition.
Proposition 1. An exact solution to the di?erential equation in (10) is given by
J(C
t
, X
t
) =
C
1??
t
1 ??
exp (u
1
X
t
+u
2
) (11)
where
u
1
=
(1 ??)?
? +?
(12)
u
2
=
(1 ??)?
?
_
(? ?1 ???)?
2
D
2
+
?
¯
X
? +?
+
(1 ??)?
? +?
_
?
2
x
2(? +?)
+?
D
?
x
?
__
. (13)
Proof: see the appendix.
Unlike in power preferences, SDU incorporates the agent’s value function in the current
utility. Thus, expected growth rate enters into the agent’s utility through the value func-
8
tion J. Indeed, expected growth rate has important implications for future consumption
streams. Higher expected growth rate indicates higher expected future consumption and
hence higher future utility which is re?ected through higher current value function. Thus,
J
X
is positive. On the other hand, J
XX
is negative suggesting that J is concave in X or
J
X
increases in X in diminishing rates. We also run some simulations to learn the impact
of the value function on the current utility. The current utility is increasing (decreasing)
in growth rates under low (high) leverage and low (high) uncertainty of future growth rates.
Moreover, the autocorrelation of X is important for understanding the expected growth
rate e?ect on future utility. If expected growth rate is highly persistent (low ?) then high
X
t
implies high future expected growth rate, which, in turn, implies that the agent expects
to consume more in the future. Thus J
X
increases with expected growth rate persistence.
Finally, J
X
decreases in ?. In particular, if the agent discounts the future more (higher
?) then the impact of expected growth rate on the value function (future utility) diminishes.
The next section derives the pricing kernel dynamics, the asset return dynamics, the
correlation between observed realized returns and expected returns, and especially the link
between leverage, information uncertainty, and stock return momentum.
3 Asset Pricing
Du?e and Epstein (1992) show that the pricing kernel for SDU is given by ?
t
= exp(
_
t
0
f
J
ds)f
C
,
where f
J
and f
C
are the derivatives of f(C, J) in (3) with respect to J and C. Hence we
can ?nd the explicit pricing kernel dynamics as stated in the proposition below.
9
Proposition 2. The pricing kernel dynamics is given by
d?
?
= ?r
f
t
dt ????
D
dW
1
+u
1
?
x
dW
2
(14)
where
r
f
t
= ?X
t
+u
1
???
D
?
x
? +?(u
2
+ 1) ?u
1
?
¯
X ?
1
2
???
2
D
(?? + 1) ?
1
2
?
2
x
u
2
1
= µ
C
(X
t
) +? ??
2
C
? +
(1 ??)?
? +?
?
C
?
x
?. (15)
Proof: see the appendix.
Observe from equation (14) that leverage is an important determinant of both the drift
and di?usion of the pricing kernel dynamics. Moreover, our equilibrium riskfree rate for-
mulated in (15) has two attractive features compared to its power utility counterpart. For
comparison, the corresponding riskfree rate for power utility is r
f
t
= ?µ
C
(X
t
)+??
?(?+1)?
2
C
2
.
First, the riskfree rate based on the power utility is highly sensitive to expected con-
sumption growth. In particular, a one-percent increase in the growth rate is followed by
?-percent increase in the riskfree rate - a nuisance at the heart of the riskfree rate puzzle.
Indeed, the riskfree rate determines the intertemporal substitution e?ect between deter-
ministic consumption streams, whereas risk-aversion determines the agent’s preference over
risky bets. Thus, risk aversion should not exert such a considerable in?uence on the con-
sumption growth riskfree rate sensitivity. In the SDU case, there is a much more realistic
one-to-one relationship between expected consumption growth and riskfree rate.
10
Second, for realistic values of µ
C
and ?
C
, the power utility riskfree rate is increasing
under believable values of risk-aversion - which again misconstrues the nature of the riskfree
rate. In other words, higher risk-aversion is required to match the equity-premia, it also
correspondingly increases the riskfree rate. In our case, as in Du?e and Epstein (1992),
the riskfree rate is strictly decreasing in risk-aversion. Thus, if indeed higher risk-aversion
is needed to match the equity premia, it does not pose any challenge to match low riskfree
rate. It should also be noted that the riskfree rate does vary through time with X
t
.
We next establish the equilibrium dividend price ratio and the return dynamics.
Proposition 3. The equilibrium price-dividend ratio
Pt
Dt
, denoted by G(X
t
), is
G(X
t
) =
_
?
0
exp(P
1
(?)X
t
+P
2
(?))d? (16)
where P
1
and P
2
are the solutions of a system of ODEs given in the appendix.
Furthermore, the excess return dynamics is given by
dR
t
= µ
R
t
dt +?
D
dW
1
+
G
X
G
?
x
dW
2
(17)
dµ
R
t
= (·)dt +
_
G
X
G
_
X
(???
D
?
x
? ?u
1
?
2
x
)?
x
dW
2
. (18)
It immediately follows that the instantaneous covariance between realized and expected
return is given by
E
t
__
dR
t
?µ
R
t
dt
_ _
dµ
R
t
?(·)dt
_¸
=
_
G
X
G
_
X
_
?
D
?
x
? +
G
X
G
?
2
X
_
(???
D
?
x
? ?u
1
?
2
x
). (19)
We can now go further and formulate the SDEs of realized cumulative excess returns
11
and expected excess returns for an investment horizon of l periods. In particular,
R
t,t+l
= R
t
+
_
t+l
t
µ
R
s
ds +
_
t+l
t
?
D
dW
1
+
_
t+l
t
G
X
G
?
x
dW
2
(20)
µ
R
t,t+l
= µ
R
t
+
_
t+l
t
(·)ds +
_
t+l
t
_
G
X
G
_
X
(???
D
?
x
? ?u
1
?
2
x
)?
x
dW
2
(21)
As a ?rst pass to understand momentum e?ects in stock returns we compute the cor-
relation between R
t,t+l
and µ
R
t,t+l
. The covariance between realized and expected return
is
Cov
t
(R
t,t+l
, µ
R
t,t+l
) = E
t
_
(R
t,t+l
?E
t
R
t,t+l
)(µ
R
t,t+l
?E
t
µ
R
t,t+l
)
¸
= ?
x
(???
D
?
x
? ?u
1
?
2
x
)
_
t+l
t
_
G
X
G
_
X
_
?
D
? +
G
X
G
?
x
_
ds. (22)
Then, the variances are computed as
V
t
(µ
R
t+l
) = (???
D
?
x
? ?u
1
?
2
x
)
2
?
2
x
_
t+l
t
__
G
X
G
_
X
_
2
ds (23)
V
t
(R
t+l
) =
_
t+l
t
_
?
2
D
+
_
G
X
G
?
x
_
2
+ 2
G
X
G
?
D
?
x
?
_
ds. (24)
Hence, the correlation is
?(l) =
_
t+l
t
_
G
X
G
_
X
(?
D
? +
G
X
G
?
x
)ds
_
_
t+l
t
__
G
X
G
_
X
_
2
ds ·
_
t+l
t
_
?
2
D
+
_
G
X
G
?
x
_
2
+ 2
G
X
G
?
D
?
x
?
_
ds
. (25)
The ?(l) function depicts the correlation between expected return and realized return
up to horizon l. As l ?0, ?(l) is the instantaneous correlation between realized return up
until now (time 0) and expected return. As l increases, it becomes the correlation between
12
realized return over the period 0 ? l with the expected return at the end of that period.
The leading term on the numerator of ?(l) can be written as
GG
XX
?(G
X
)
2
G
2
=
1
G
2
_
_
?
0
exp(·)d?
_
?
0
exp(·)P
2
1
(?)d? ?
__
?
0
exp(·)P
1
(?)d?
_
2
_
(26)
which is positive from direct application of Cauchy-Schwartz inequality to functions P
1
(?)
_
exp(·)
and
_
exp(·), both of which are integrable in the domain. As we show below, the term
G
X
G
is always positive for ? < 1. The autocorrelation is positive unless ? < 0 and |?
D
?| >
G
X
G
?
x
.
3.1 Interpretation and implications
3.1.1 Consumption Growth
The equilibrium condition produces consumption growth which follows the dynamics in (7).
If there is no leverage (? = 1), then consumption growth is the same as the dividend growth
and we are back to a Lucas (1978) economy setting. In the case of a slave economy (? = 0),
there is no consumption growth and consumption at every point in time is nonrandom. Of
course, one can also attribute the ? = 0 case to a riskfree bond. At intermediate level of
leverage (0 < ? < 1), which is the focus of our analysis below, we observe key features of
post-war US data. First, the volatility of consumption growth is much smaller than the
volatility of dividend growth. Furthermore, in the presence of high growth rate of dividends
the consumption growth rate can still be low especially if the economy is highly levered.
13
3.1.2 Price-Dividend ratio
The business cycle e?ect on the P/D ratio is instantly observable as
G
X
=
1 ??
?
_
?
0
exp(P
1
(?)X
t
+P
2
(?))
_
1 ?e
???
_
d?. (27)
Thus the P/D ratio is increasing in the growth rate X
t
as long as ? < 1. The e?ect is
most pronounced for low ? (high leverage), it deteriorates as ? grows (lower leverage), and
ultimately vanishes as ? approaches one (no leverage). It should be noted that in the power
utility case G
X
is positive only if ?? < 1. Essentially, this restriction puts an undue burden
on ? to be less than
1
?
. Here, the condition that the P/D ratio increases in the business
cycle variable does not depend on risk-aversion. This appealing feature is due to the use of
the stochastic di?erential utility. Notice also that the ? < 1 case points to a wealth e?ect
in the economy in the presence of increasing economic growth rate. That is, if X
t
increases,
the stock price rises relative to the dividend - and even more so relative to consumption.
The equilibrium D/P ratio is shown in Figure 1 for plausible values of the parameters
underlying the stock return dynamics. The e?ect of risk-aversion is straightforward. Higher
risk-aversion increases expected return thus reducing the current stock price or increasing
the D/P ratio. The e?ect of leverage on the D/P ratio, however, is not straightforward.
Figure 1 shows that leverage and the D/P ratio are nonlinearly related. The D/P ratio
?rst increases as ? grows and then decreases as ? approaches one. This pattern is attributed
to the interaction of leverage with the growth rate X
t
. In the no leverage (? = 1) case the
P
1
(?) in (16) is 0, suggesting that leverage and growth rate do not interact. At high growth
14
rates with small leverage (high ?), agents encounter high consumption growth and thus low
marginal utility of consumption. Agents would therefore invest more and consequently the
stock price would rise, triggering low D/P ratio. It is intuitive to think that the reverse
would take place at high leverage (small ?). However, at high leverage, P
1
(?) is high and
it interacts strongly with the growth rate. Thus, the stock price rises with high leverage
and high growth rate and the D/P ratio again falls. In essence, when leverage is high,
consumption reacts slowly in response to high growth rate and the agent invests more in
anticipation of good times in the future, thus raising prices and lowering the D/P ratio.
3.1.3 Expected return and leverage
The expected excess stock return is given by
µ
R
t
= (???
2
D
?u
1
?
D
?
x
?) +G
X
(???
D
?
x
? ?u
1
?
2
x
)
D
P
. (28)
Thus, G
X
– the response of the P/D ratio to the economic growth rate – guides the ex-
pected return dynamics. Notice that for stocks with high ? (low leverage) the ability of
the dividend yield to forecast future returns diminishes. In particular, for ? = 1, G
X
= 0
suggesting that all time series e?ects on expected stock return vanish completely. Likewise,
at ? = 0, µ
R
t
= 0 which implies that in a fully levered economy, there is no consumption
growth and the only security available is a bond with a constant stream of consumption.
As such, the risk-premia vanishes and the bond yields a rate of return equal to the discount
rate ?. Thus, return predictability is pronounced for stocks with intermediate values of ?.
Interestingly, observe from Figure 2 that even in the absence of time series e?ects on ex-
15
pected stock return, our model can still deliver the relatively high equity premium observed
in the US over the past century. In particular, the second term in the ?rst parenthesis in
(28), u
1
?
D
?
x
? =
(1??)??
D
?x?
?+?
, which is a contribution of the stochastic di?erential utility,
provides the explanation. Small values of ? and ? could greatly magnify this expected
return component, thus producing expected return terms that match the high equity pre-
mium at a relatively low risk-aversion.
Figure 2 also displays the non-linear dependence of expected return on leverage. Clearly,
at ? = 0, µ
R
t
= 0 and at ? = 1, excess return is constant. The time-series dependence and
its interaction with leverage is provided at intermediate levels of ?. In such intermediate
levels, whenever we observe high D/P ratio in Figure 1, we also observe high expected
return level since G
X
> 0. The e?ect is magni?ed as risk-aversion increases, as expected.
This positive relationship is empirically shown through the positive coe?cients in predictive
regressions as in Cochrane (2007).
4 Understanding momentum interactions
As noted earlier for ? = 1 or ? = 0 the excess return is zero or constant, respectively,
and the momentum e?ect is nonexistent. We focus on the more realistic 0 < ? < 1 do-
main where the full dynamics of the model can be interacted with leverage. Our goal is
to calibrate the correlation between observed return and expected return as well as the
overall momentum pro?tability for 42 distinct speci?cations of parameters underlying the
return dynamics with each speci?cation standing for one particular class of ?rms. Thus,
16
our model could ultimately deliver prominent predictions on the cross section of such ?rms,
and it does, as we show below.
The parameter settings are described in Table 1. In the simulation exercises the seven
settings described in Table 1 will be interacted with six leverage levels - thus overall we
consider 42 distinct cases. For all the seven [A-G] settings, we assume that the discount
rate for our in?nitely lived agent is small (? = 0.01) and the expected dividend growth is
highly autoregressive (? = 0.05) with the long run mean growth rate (
¯
X) at 5%. Based on
the relationship in (25) and (27), the claim going forward is that high variance in expected
dividend growth rate generates the momentum e?ect that is more pronounced for high
levered ?rms. High variance could emerge due to both high volatility in expected growth
rate innovation (high ?
x
) and highly persistent expected dividend growth (low ?).
In Table 1, settings A and B display high variance of dividend growth (high ?
D
and ?
x
),
C and D display low, and E exhibits medium. The distinction between A and B (C and
D) is higher risk-aversion parameter for B (D). Both E and F exhibit the same volatility
of dividend growth (?
D
) but F displays lower expected growth rate volatility (?
x
). Setting
G takes higher ?, which means less persistent expected dividend growth rate, and at the
same time holds ?
D
and ?
x
at a moderate level.
The next section describes the simulations made to assess the impact of leverage, volatil-
ity of dividend growth, volatility of expected dividend growth, risk aversion, and persistence
of expected dividend growth on momentum e?ects.
17
4.1 Correlation between realized and expected returns
We ?rst explore the correlation between realized and expected returns. The system of re-
alized and expected returns is simulated forward (starting from X
0
=
¯
X) for investment
horizons of 3, 6, 9, and 12 months and then the correlation between observed investment
return and expected return is computed. Table 2 shows the correlation for all eight set-
tings, each of which is interacted with six levels of leverage ranging between 0.15 (high
leverage) and 0.90 (low leverage). The ?gures represent the correlation formulated in (25)
for investment horizons of 3-12 months. The unchanging correlation pattern represents the
stability of correlation over time. Johnson (2002) depicts similar correlation structure.
The correlation between expected excess return and cumulative excess return monoton-
ically increases with leverage. The monotonic relationship holds at high, medium, and low
levels of growth rate volatility, at high and low autocorrelation of expected growth rate,
and at the entire range of risk aversion measures considered here. While all cases examined
exhibit positive correlation between observed and expected returns, we show below that
momentum pro?tability prevails only in a few of the settings and is nonexistent in others.
4.2 Instantaneous expected return spreads
Table 3 exhibits expected excess return over a one-year period in which past cumulative
returns have been classi?ed into ten deciles, with column 1 (10) pertaining to the low-
est (highest) observed returns. There are several insights emerging about the role that
leverage, dividend growth volatility, expected dividend growth volatility, risk aversion, and
persistence play in generating momentum e?ects. Here, momentum pro?tability is de?ned
18
as the expected return spread between the highest and lowest observed return portfolios.
Momentum and leverage. Momentum pro?tability monotonically increases with
leverage regardless of the case considered, with settings A, B, and E displaying large and
economically meaningful expected return spreads. Focusing on A, the expected return
spread is 11.49% [26.51%-15.02%] per year for ? = 0.15 while it is only 0.27% for ? = 0.9.
Moving to B, the expected return spread is 13.16% for ? = 0.15 and is 1.71% for ? = 0.9.
The corresponding ?gures for E are 9.87% and 0.26%. On the other hand, the overall
spreads are small for the C, D, and G settings. This indicates that while leverage is crucial
there are some other important determinants of momentum e?ects.
Before we move on, it should be noted that the expected returns reported in Table 3
might seem quite large, especially the ones indicating strong momentum e?ects. However,
notice that the time discount parameter ? is small (0.01) at this stage. We reexamine some
of the sub-cases using higher time-discount parameter. The evidence is reported in Table
4. Indeed, increasing ? brings down expected return at every decile to more realistic levels.
The interesting evidence, however, is that the expected return spread between the highest
and lowest deciles is preserved. To illustrate, for case A and ? = 0.25 the spread is 7.56%,
for B and ? = 0.15 is 9.96%, for B and ? = 0.9 is 0.57%, and for E and ? = 0.2 is 5.87%.
Momentum and information uncertainty. Settings A, B, and E display the highest
expected return spreads across the ten deciles, as noted earlier. For the other settings the
spreads are much lower and are often even negligible. To illustrate, the highest spread for
C is 2.95% [15.01%-12.06%], 4.86% for D, 1.05% for F, and only 0.29% for G. Cases A, B,
19
and E are all characterized by high (0.07) to moderate (0.05) expected dividend growth
volatility, which, from the ?rm perspective, amounts to relatively high volatility cash ?ows.
To this point we are able to rationalize previously documented momentum interactions.
In particular Zhang (2006) ?nds that momentum concentrates in high information uncer-
tainty stocks and points to behavioral interpretations. Avramov, Chordia, Jostova, and
Philipov (2007) document that momentum prevails only among high credit risk stocks.
Whereas such momentum-credit risk interaction could point to rational interpretations,
this is purely an empirical ?nding thus far that has not been formalized in an equilibrium
model. The collective evidence here shows that equilibrium momentum e?ects should con-
centrate in the interaction of risky cash ?ows and highly levered ?rms. Interestingly, neither
leverage alone nor cash ?ow volatility alone are su?cient to generate momentum e?ects.
Focusing on information uncertainty measures, a valid point to make is that the volatil-
ity of expected dividend growth (?
x
) is the primary force of momentum e?ects, whereas the
volatility of the unexpected dividend growth (?
D
) plays a marginal role. Note in particular
that cases E and F are virtually identical with the only exception being ?
x
= 0.05 in E
versus ?
x
= 0.02 in F. Nevertheless, the expected return spreads in E are considerably
higher for all leverage levels.
Momentum and risk aversion. Setting G features the highest risk aversion but
nevertheless yields the lowest expected return spreads between the ten portfolios, ranging
between 0.02% and 0.29%. The immediate takeout is that the risk aversion measure is not
a key parameter in generating momentum e?ects. Let us also compare A versus B as well
20
as C versus D. For cases B and D, the higher risk-aversion simply increases expected re-
turn at every level of leverage and still preserves expected return spread across the deciles.
Indeed, the momentum pro?tability in B (D) is slightly higher than than of A (C), suggest-
ing that risk aversion has some e?ect, albeit relatively small, in explaining the return spread.
Momentum and expected growth rate persistence. Case G is di?erent from the
previous settings in that it features the lowest autocorrelation of expected growth rate,
which reduces the total variance of expected growth rate. In case G, the shocks to the sys-
tem are exactly the same as in case E (?
D
= 0.06 and ?
x
= 0.05). Therefore, with higher
risk-aversion (10 versus 5) we can only expect the momentum e?ect of case E to be exac-
erbated, just like case B compared to case A. However, with lower ? the expected return
spread across the high and low performing portfolios is minuscule, even at high levels of
leverage. Therefore, high ?, which reduces the variance of expected growth rates, exhibits
no expected return di?erential that characterizes the momentum e?ect in the data. Persis-
tence is indeed crucial in generating momentum e?ects and it overwhelms risk aversion.
The evidence emerging from Table 3 also suggests that even when there is strong corre-
lation between realized and expected return at high levels of leverage in high, medium, and
low levels of volatility (see Table 2), low cash ?ow volatility does not produce economically
signi?cant expected return spreads across the di?erent deciles even when leverage is high.
21
4.3 Holding period return spreads
What makes the momentum e?ect a conundrum is the holding period pro?t. The strategy
of buying winners and selling short losers produces 8-12% ex post payo?s according to
Jegadeesh and Titman (1993). We next simulate holding period returns based on one year
formation period and conventional holding periods of 3-12 months. Table 5 reports momen-
tum pro?tability which is the return spread between the top and bottom past return deciles.
Consistent with the evidence reported thus far it follows that the high volatility case A,
which produces high ex-ante expected excess returns, also generates high ex post holding
period returns. Focusing on the one year holding period, momentum pro?tability is 10.35%
for ? = 0.15 and is only 0.37% for ? = 0.9. The low volatility case D fails to generate high
momentum pro?tability. For the one year holding period the momentum payo? is 3.2%
for high leverage and 0.13% at low leverage. Furthermore, the moderate volatility case E
generates moderate levels of investment returns ranging between 0.26% and 8.98% for the
one year holding period. The evidence in Case G with the lowest autocorrelation con?rms
the earlier observation that holding period returns based on low autocorrelation and high
volatility are small and similar in magnitude to the high autocorrelation low volatility case
D (recall ?
x
= 0.05 in case G while ?
x
= 0.03 in case D). In summary, we con?rm that
momentum e?ects concentrate in ?rms with high leverage as well as highly volatile and
persistent expected cash ?ows growth.
22
5 Conclusion
Previous work shows that momentum e?ects in stock returns are robust, thus invoking a
plethora of behavioral and rational explanations. Previous work also uncovers momentum
interactions. In particular, momentum concentrates in stocks with high return volatility,
high cash ?ow volatility, small market capitalization, high analysts’ earnings forecast dis-
persion, as well as high credit risk. Thus far, there has not been any attempt, to our
knowledge, to theoretically rationalize or even behavioralize such momentum interactions.
This paper embraces this task. In particular, it analyzes momentum interactions from a
rational equilibrium perspective and ultimately shows that the concentration of momentum
in high information uncertainty as well as high credit risk stocks is perfectly consistent with
rational asset pricing. Our economic setup is fairly general from the perspectives of both
preferences and dynamics. The stochastic di?erential utility of Du?e and Epstein (1992)
employed here breaks the tight association between the elasticity of inter-temporal substi-
tution and the risk aversion measure. Moreover, our consumption dynamics, which is based
on the novel formulation of Abel (1999), closely follows the high economic growth along with
small consumption growth documented in the US post-war data. Collectively, our model
allows one to match key regularities in asset pricing using reasonable risk aversion measures.
We use simulations to ?nd out that our paradigm indeed predicts strong equilibrium
momentum e?ects for the interaction between high leverage and risky cash ?ow ?rms. Mo-
mentum pro?tability deteriorates and ultimately disappears as either leverage or cash ?ow
risk diminishes. More speci?cally, the correlation between observed and expected returns is
positive and monotonically increasing with leverage. The monotonic relationship holds at
23
high, medium, and low levels of expected growth rate volatility, at high and low autocor-
relation of expected growth rate, and at the entire range of risk aversion measures analyzed.
Moreover, the expected return spread between the highest and lowest past year cumu-
lative return portfolios increases with leverage. For example, when operating cash ?ows
are highly volatile, the expected return spread is 11.49% for high leverage and only 0.27%
for low leverage. On the other hand, the overall spreads are small when either the volatil-
ity of expected cash ?ows growth is small, or the expected growth in cash ?ows in not
highly persistent. This indicates that while leverage is crucial, risk and persistence in
cash ?ows growth are both important determinants of momentum e?ects. The collective
evidence thus shows that equilibrium momentum pro?tability concentrates in the interac-
tion between risky cash ?ows and high levered ?rms which is perfectly consistent with data.
Looking forward, there are several suggestions for future work. First, currently leverage
is exogenous. We ask: given a particular leverage level - what is the overall momentum
e?ect? It would be quite appealing to endogenize leverage and make it a ?rm-decision
variable. Next, our focus here has been on ?rm level interactions. It has also been shown
that momentum displays strong business cycle e?ects. Our setting can readily be extended
to analyze possible rational business cycle e?ects in momentum pro?tability. Finally, from
an empirical perspective, one could analyze the joint e?ect of leverage, expected dividend
growth risk, and dividend growth persistence on the cross section of average returns.
24
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28
0.2
0.4
0.6
0.8
1
3
4
5
6
0
0.05
0.1
0.15
0.2
?
Dividend?Price Ratio
?
Figure 1: Dividend-Price ratios implied by the model for ? = 0.01, ? = 0.1,
¯
X = 0.05,
?
D
= .05, ?
x
= .035, ? = .35. The state is set to
¯
X, i.e.X
0
=
¯
X
.
29
0.2
0.4
0.6
0.8
1
3
4
5
6
0
0.05
0.1
0.15
0.2
?
Expected Excess Return
?
Figure 2: Expected excess return implied by the model for ? = 0.01, ? = 0.1,
¯
X = 0.05,
?
D
= .05, ?
x
= .035, ? = .35. The state is set to
¯
X, i.e.X
0
=
¯
X
.
30
Table 1:This table lists the set of parameters we are considering. A and B are cases with high volatility
of dividend growth and expected dividend growth. Case B is the same as A with higher risk-aversion.
Cases C and D are with low volatilities with D having higher risk-aversion than C. E and F have moderate
volatilities and higher correlation where F has lower volatility of expected dividend growth. G is similar to
E, except with lower autocorrelation.
? ? ?
¯
X ?
D
?
x
?
A 0.01 5.0 0.05 0.05 0.08 0.07 0.2
B 0.01 8.0 0.05 0.05 0.08 0.07 0.2
C 0.01 5.0 0.05 0.05 0.04 0.03 0.1
D 0.01 8.0 0.05 0.05 0.04 0.03 0.1
E 0.01 5.0 0.05 0.05 0.06 0.05 0.2
F 0.01 5.0 0.05 0.05 0.06 0.02 0.2
G 0.01 10.0 0.20 0.05 0.06 0.05 0.2
31
Table 2:This table shows the correlation between expected excess return and cumulative excess return
based on investment horizons of 3,6,9 and 12 months. The system is simulated forward for 5000 paths
using Monte Carlo integration subsequent to which we compute the ?(l) function. Then the average of the
?(l) function is reported for each l. In all cases, X
0
=
¯
X.
? 3 6 9 12
A 0.15 0.97 0.97 0.97 0.97
0.25 0.95 0.95 0.95 0.95
0.30 0.91 0.91 0.91 0.91
0.40 0.87 0.87 0.87 0.87
0.70 0.71 0.71 0.71 0.71
0.90 0.51 0.51 0.51 0.51
D 0.15 0.98 0.98 0.98 0.98
0.20 0.97 0.97 0.97 0.97
0.30 0.95 0.95 0.95 0.95
0.40 0.92 0.92 0.92 0.92
0.70 0.77 0.77 0.77 0.77
0.90 0.52 0.52 0.52 0.52
E 0.15 0.99 0.99 0.99 0.99
0.20 0.97 0.97 0.97 0.97
0.30 0.94 0.94 0.94 0.94
0.40 0.91 0.91 0.91 0.91
0.70 0.75 0.75 0.75 0.75
0.90 0.57 0.57 0.57 0.57
G 0.15 0.96 0.96 0.96 0.96
0.20 0.95 0.95 0.95 0.95
0.30 0.92 0.92 0.92 0.92
0.40 0.89 0.89 0.89 0.89
0.70 0.73 0.73 0.73 0.73
0.90 0.47 0.47 0.47 0.47
32
Table 3: This table shows the average instantaneous expected excess return (annualized percentage)
under the di?erent parameter settings subsequent to one year in which the cumulative return has fallen
into 1 of the 10 deciles labeled 1-10. Column 1 is the expected excess return for the lowest decile and
Column 10 is for the highest. The system is simulated forward for 5000 di?erent paths for one year, and
each path here depicts one security over the year. For each path, we compute the observed return in (20)
and the corresponding expected return. At the end of one year, we sort the 5000 paths based on observed
return and assign them into the ten portfolios. The average of the expected return for each portfolio
(equally-weighted) is then reported. In all cases, X
0
=
¯
X.
? 1 2 3 4 5 6 7 8 9 10
A 0.15 15.02 17.13 18.25 18.83 19.48 20.04 20.67 21.39 22.99 26.51
0.25 18.75 20.32 21.08 21.50 21.91 22.26 22.68 23.12 24.09 26.13
0.30 20.37 21.79 22.43 22.85 23.22 23.52 23.86 24.28 25.11 26.88
0.40 22.94 24.11 24.65 24.98 25.26 25.50 25.76 26.13 26.73 28.12
0.70 25.77 26.21 26.43 26.55 26.65 26.74 26.83 26.97 27.22 27.71
0.90 21.61 21.68 21.74 21.76 21.78 21.79 21.81 21.84 21.90 21.98
B 0.15 20.13 21.91 22.81 23.28 23.76 24.15 24.66 25.18 26.32 33.29
0.20 23.45 24.98 25.73 26.21 26.65 26.99 27.37 27.88 28.83 30.89
0.30 28.56 29.92 30.56 30.97 31.25 31.54 31.88 32.26 33.00 34.64
0.40 32.43 33.55 34.05 34.43 34.61 34.81 35.14 35.41 35.99 37.27
0.70 34.25 34.78 35.12 35.77 36.12 36.88 37.22 37.45 37.89 38.14
0.90 37.30 37.66 37.89 37.98 38.08 38.15 38.24 38.36 38.62 39.01
C 0.16 12.06 13.07 13.53 13.71 13.87 14.01 14.16 14.30 14.58 15.01
0.25 7.88 8.51 8.85 9.00 9.14 9.30 9.44 9.61 9.95 10.62
0.30 8.17 8.72 9.01 9.14 9.27 9.40 9.53 9.69 9.97 10.57
0.40 8.71 9.15 9.38 9.48 9.58 9.69 9.79 9.90 10.14 10.62
0.70 8.57 8.76 8.84 8.88 8.93 8.96 8.99 9.04 9.13 9.30
0.90 5.96 5.98 6.00 6.01 6.02 6.02 6.02 6.03 6.05 6.07
D 0.15 10.56 11.94 12.63 12.93 13.20 13.46 13.72 13.99 14.52 15.42
0.20 9.69 10.32 10.67 10.81 10.97 11.11 11.27 11.43 11.79 12.52
0.30 10.83 11.38 11.66 11.79 11.92 12.06 12.19 12.33 12.63 13.24
0.40 11.83 12.31 12.52 12.64 12.76 12.85 12.96 13.07 13.32 13.83
0.70 12.05 12.47 12.33 12.39 12.42 12.46 12.50 12.55 12.64 12.82
0.90 8.77 8.79 8.81 8.82 8.82 8.83 8.83 8.84 8.86 8.88
33
Table 3 (Continued):
? 1 2 3 4 5 6 7 8 9 10
E 0.15 11.54 13.39 14.47 14.98 15.50 16.03 16.57 17.21 18.57 21.41
0.20 13.15 14.28 14.84 15.13 15.43 15.68 15.97 16.27 16.96 18.38
0.30 14.11 15.13 15.64 15.89 16.15 16.37 16.62 16.90 17.48 18.72
0.40 15.68 16.51 16.89 17.11 17.33 17.49 17.69 17.91 18.37 19.30
0.70 16.94 17.27 17.42 17.53 17.58 17.65 17.74 17.82 17.99 18.37
0.90 13.49 13.55 13.59 13.60 13.61 13.62 13.64 13.66 13.69 13.75
F 0.20 7.16 7.47 7.62 7.68 7.74 7.79 7.85 7.90 8.01 8.21
0.25 6.30 6.64 6.79 6.86 6.94 7.00 7.07 7.14 7.28 7.57
0.30 6.34 6.63 6.78 6.84 6.91 6.96 7.03 7.09 7.23 7.50
0.40 6.72 6.95 7.05 7.11 7.17 7.21 7.25 7.31 7.42 7.63
0.70 7.09 7.17 7.20 7.22 7.24 7.25 7.27 7.28 7.32 7.40
0.90 5.74 5.75 5.75 5.76 5.76 5.77 5.77 5.78 5.78 5.79
G 0.15 7.25 7.34 7.38 7.39 7.41 7.42 7.44 7.45 7.48 7.54
0.20 8.33 8.48 8.54 8.57 8.60 8.63 8.66 8.68 8.74 8.85
0.30 10.02 10.21 10.30 10.34 10.34 10.41 10.45 10.49 10.57 10.71
0.40 11.22 11.34 11.48 11.52 11.56 11.59 11.62 11.69 11.74 11.90
0.70 12.07 12.14 12.17 12.19 12.20 12.21 12.22 12.24 12.27 12.33
0.90 9.54 9.54 9.55 9.55 9.55 9.55 9.55 9.55 9.55 9.56
34
Table 4: This table repeats the exercise in Table 3 with di?erent time preference param-
eter ?. Certain sub-cases are taken and they are replicated with a higher ?. The ?rst line
for each sub-case is copied from the corresponding line on Table 3, and the following line is
the repeat of the same simulation with higher beta, such that the transversality condition
is still satis?ed. This table shows that increasing ? lowers expected return at every decile
but still maintains a healthy di?erence between the highest and lowest deciles.
? 1 2 3 4 5 6 7 8 9 10
A(? = 0.25) 0.01 18.75 20.32 21.08 21.50 21.91 22.26 22.68 23.12 24.09 26.13
0.08 10.88 12.23 12.96 13.34 13.76 14.11 14.53 14.99 16.05 18.42
B(? = 0.15) 0.01 20.13 21.91 22.81 23.28 23.76 24.15 24.66 25.18 26.32 33.29
0.08 11.80 13.44 14.33 14.82 15.37 15.81 16.36 16.98 18.40 21.76
B(? = 0.90) 0.01 37.30 37.66 37.89 37.98 38.08 38.15 38.24 38.36 38.62 39.01
0.08 18.86 18.91 18.96 18.97 18.99 18.99 19.01 19.02 19.07 19.13
E(? = 0.20) 0.01 13.15 14.28 14.84 15.13 15.43 15.68 15.97 16.27 16.96 18.38
0.055 7.64 8.69 9.29 9.59 9.89 10.20 10.54 10.89 11.71 13.51
35
Table 5: This table shows the holding period return di?erential based on investment horizons of 3-12
months. The formation period is one year. We use 5000 di?erent paths where each path denotes one stock.
At the end we sort the observed return into ten equally weighted portfolios. The average di?erence between
the top and bottom decile is reported in this table. In all cases, X
0
=
¯
X.
? 3 6 9 12
A 0.15 2.78 5.52 7.87 10.35
0.25 1.78 3.49 5.05 6.65
0.30 1.58 3.11 4.46 5.87
0.40 1.24 2.46 3.54 4.66
0.70 0.46 0.89 1.29 1.82
0.90 0.09 0.18 0.26 0.37
D 0.15 3.21 3.25 3.28 3.20
0.20 2.63 2.65 2.71 2.63
0.30 2.25 2.27 2.32 2.23
0.40 1.85 1.85 1.91 1.85
0.70 0.72 0.67 0.73 0.72
0.90 0.11 0.12 0.14 0.13
E 0.15 2.38 4.75 6.75 8.98
0.20 1.48 2.96 4.23 5.62
0.30 1.10 2.18 3.13 4.14
0.40 0.87 1.72 2.48 3.27
0.70 0.33 0.65 0.95 1.30
0.90 0.06 0.13 0.18 0.26
G 0.15 1.41 2.78 4.00 3.96
0.20 1.26 2.47 3.57 3.49
0.30 1.04 2.03 2.97 2.83
0.40 0.83 1.63 2.36 2.27
0.70 0.31 0.61 0.86 0.87
0.90 0.06 0.12 0.19 0.18
36
6 Appendix
Preferences
In continuous time, the recursive utility function takes the form of stochastic di?erential
utility. The stochastic di?erential utility U : L
2
?R is a mapping from a square integrable
space to the real line and is de?ned by two primitive functions: (f, A) where f : R
+
×R ? R
and A : R ? R. For any consumption process C ? L
2
, the utility process J is the unique
SDE
dJ
t
=
_
?f(C
t
, J
t
) ?
1
2
A(J
t
)?
v
?
v
_
dt +?
v
dB
t
with boundary condition J
T
= 0. The di?erent components are - J
t
, a continuation utility
for the agent given consumption C
t
, f(C
t
, J
t
) is an ordinal map of date t
s consumption
and continuation utility, and A(J
t
) is a measure of local risk-aversion. If given an initial
consumption C
t
and as long as the solution of the above SDE is well-de?ned, the utility at
time t is de?ned as U(C
t
) = J
t
. Under certain conditions, the above SDE is well-de?ned
and hence the utility exists. The function U is monotonic and risk-averse for A ? 0. Given
an f and two functions A
?
and A, let U
?
and U be the two utilities corresponding to the
aggregators (f, A
?
) and (f, A). If A
?
? A, then U
?
is more risk-averse than U, i.e. any con-
sumption stream rejected by a deterministic consumption path by one will also be rejected
by another. A convenient normalization that produces an ordinally equivalent utility func-
tion is achieved by setting A = 0, which means the above SDE solves E
t
[dJ
t
] +f(C, J) = 0
for normalized aggregator (f, 0). The normalization is useful because it produces a much
simpler Bellman equation to be solved than if A = 0. Fortunately, there exists a transfor-
mation from (
¯
f, A) to (f, 0) such that the utilities generated from both will be ordinally
equivalent. Further discussion of the aggregators and the normalization that leads to an
37
ordinally equivalent representation of the aggregators is given in Du?e and Epstein (1992).
Proof of Proposition 1: The Bellman equation in (10) can be written as
J
C
C?
_
X
t
+
1
2
(? ?1)?
2
D
_
+J
X
?(
¯
X ?X
t
)+
1
2
J
CC
C
2
?
2
?
2
D
+
1
2
J
XX
?
2
X
+J
XC
C??
D
?
x
? +f(C, J) = 0
The continuation utility J has a solution of the form
(1 ??)J = exp(u
0
ln C
t
+u
1
X
t
+u
2
)
Substituting it in and collecting terms, reduces the above equation to a system of ODEs
that can be solved recursively
u
0
= (1 ??)
u
1
=
(1 ??)?
? +?
u
2
=
(1 ??)?
?
_
(? ?1 ???)?
2
D
2
+
?
¯
X
? +?
+
(1 ??)?
? +?
_
?
2
x
2(? +?)
+?
D
?
x
?
__
Thus, the continuation utility function reduces to J(C
t
, X
t
) =
C
1??
t
1??
exp(u
1
X
t
+u
2
).
Proof of Proposition 2: The pricing kernel for stochastic di?erential utility can be
38
written as
d?
?
=
df
C
f
C
+f
J
dt
Using the above utility function, de?ne g = f
C
=
?(1??)J
C
and f
J
= ??(1 +u
1
X +u
2
). Use
Ito’s Lemma on g and (5) and (7) one can rewrite the pricing kernel as
d?
?
= ?r
f
t
dt ????
D
dW
1
+u
1
?
x
dW
2
r
f
t
= ?X
t
+u
1
???
D
?
x
? +?(u
2
+ 1) ?u
1
?
¯
X ?
1
2
???
2
D
(?? + 1) ?
1
2
?
2
x
u
2
1
Proof of Proposition 3: The ?rm stock price is given by
P
t
=
1
?
t
E
t
_
?
t
?
s
D
s
ds
=
1
?
t
_
?
t
E
t
?
s
D
s
ds
Applying Feynman-Kac, we know E
t
[?
s
D
s
] = f(?
t
D
t
, X
t
, s ?t). Applying Ito’s Lemma,
P
t
= D
t
G(X
t
)
where G(X
t
) =
_
?
t
exp(P
1
(s ?t)X
t
+P
2
(s ?t))ds. Making a change of variable ? = s ?t,
G(X
t
) =
_
?
0
exp(P
1
(?)X
t
+ P
2
(?))d?. P
1
(?) and P
2
(?) satisfy a set of ODEs that can be
solved recursively with initial conditions P
1
(0) = P
2
(0) = 0
P
1
(?) = ?(? ?1) ??P
1
(?)
39
P
2
(?) = u
1
?
¯
X +
1
2
?
2
D
(?? ?1) ??(u
2
+ 1) +P
1
(?)
_
?
¯
X +u
1
?
2
x
?(?? ?1)?
D
?
x
?
¸
+
1
2
_
(?? ?1)
2
?
2
D
+u
2
1
?
2
x
?2u
1
(?? ?1)?
D
?
x
? +P
2
1
(?)?
2
x
¸
The solution of P
1
(?) is given by P
1
(?) =
1??
?
(1 ?e
???
) and then that can be used to solve
for P
2
(?). Plugging in P
1
(?), P
2
(?) becomes
P
2
(?) = a? +b(e
???
?1) +c(1 ?e
?2??
)
where
a =
_
u
1
+
1 ??
?
__
¯
X? ?(?? ?1)?
D
?
x
? +
1
2
?
2
x
_
u
1
+
1 ??
?
__
+
1
2
?
2
D
??(?? ?1)
??(u
2
+ 1)
b =
1 ??
?
2
_
?
2
x
(1 ??)
?
+?
¯
X +u
1
?
2
x
?(?? ?1)?
D
?
x
?
_
c =
?
2
x
4
(1 ??)
2
?
3
The transversality condition holds for a < 0, which holds for believable parameter values.
Applying Ito’s lemma to P
t
= D
t
G(X
t
), we derive the process for cumulative excess return
dR
t
=
Dt+dP
Pt
?r
f
t
dR
t
= µ
R
t
dt +?
D
dW
1
+
G
X
G
?
x
dW
2
dµ
R
t
= (·)dt +
_
G
X
G
_
X
(???
D
?
x
? ?u
1
?
2
x
)?
x
dW
2
where µ
R
t
= (???
2
D
? u
1
?
D
?
x
?) +
G
X
G
(???
D
?
x
? ? u
1
?
2
x
). The latter is derived from the
equilibrium argument that the expected excess return is given by µ
R
t
= ?Cov
t
_
d?t
?t
, dP
t
_
.
40
doc_681110601.pdf
People rationalize for various reasons. Rationalization may differentiate the original deterministic explanation of the behavior or feeling in question.
Rationalizing Momentum Interactions
Doron Avramov and Satadru Hore
?
?
Doron Avramov is at the Robert H. Smith School of Business, University of Maryland,
email: [email protected]. Satadru Hore is at the University of Iowa, email:
[email protected]. We thank Ravi Bansal, Darrell Du?e, and Monika Piazzasi for
useful comments. All errors are our own responsibility.
Rationalizing Momentum Interactions
Momentum pro?tability concentrates in high information uncertainty and high credit risk
?rms and is virtually nonexistent otherwise. This paper rationalizes such momentum inter-
actions in equilibrium asset pricing. In our paradigm, dividend growth is mean reverting,
expected dividend growth is persistent, the representative agent is endowed with stochastic
di?erential utility of Du?e and Epstein (1992), and leverage, which proxies for credit risk,
is modeled based on the Abel’s (1999) formulation. Using reasonable risk aversion levels we
are able to produce the observational momentum e?ects. In particular, momentum prof-
itability is especially large in the interaction between high levered and risky cash ?ow ?rms.
It rapidly deteriorates and ultimately disappears as leverage or cash ?ow risk diminishes.
1 Introduction
Momentum e?ects in stock returns are robust. Fama and French (1996) show that momen-
tum is the only deviation from the CAPM unexplained by the Fama and French (1993)
model. Schwert (2003) demonstrates that pro?t opportunities, such as the size and value
e?ects as well as equity premium predictability, typically disappear, reverse, or attenuate
following their discovery. Momentum is an exception. Speci?cally, Jegadeesh and Tit-
man (2001, 2002) document momentum pro?tability in the period after its discovery in
Jegadeesh and Titman (1993). Korajczyk and Sadka (2004) ?nd that momentum survives
trading costs, whereas Avramov, Chordia, and Goyal (2006) show that the pro?tability
of the other past-return anomaly, namely reversal, disappears in the presence of trading
costs. Fama and French (2007) argue that momentum is among the few robust anomalies.
Momentum robustness has generated a plethora of behavioral and rational explanations.
1
Empirical work also uncovers momentum interactions. In particular, Zhang (2006) ?nds
that momentum concentrates in high information uncertainty stocks, i.e., stocks with high
return volatility, high cash ?ow volatility, small market capitalization, or high analysts’
earnings forecast dispersion, and points to behavioral interpretations. Avramov, Chordia,
Jostova, and Philipov (2007) document that momentum prevails only among high credit
risk stocks and is nonexistent otherwise and, moreover, the credit risk e?ect dominates the
information uncertainty e?ect. Indeed, the momentum-credit risk relation could point to
1
Behavioral: Barberis, Shleifer, and Vishny (1998), Daniel, Hirshleifer, and Subrahmanyam (1998), and
Hon, Lim, and Stein (2000). Rational: Berk, Green, and Naik (1999) and Johnson (2002). Moskowitz and
Grinblatt (1999) argue that industry momentum explains momentum in individual stocks. Avramov and
Chordia (2006) show that while momentum is unexplained by risk based asset pricing models, removing
business cycle e?ects from model mispricing completely eliminates momentum e?ects. These last two
studies indicate that momentum could be attributable to missing factors in rational asset pricing models.
1
rational interpretations of momentum. However, this is an empirical ?nding that has not
been formalized in an equilibrium model.
2
Indeed, thus far there has not been any attempt,
to our knowledge, to theoretically rationalize or even behavioralize momentum interactions.
This paper studies momentum interactions from an equilibrium perspective and ulti-
mately shows that the concentration of momentum in high information uncertainty as well
as high credit risk stocks is perfectly consistent with rational asset pricing. In particular,
we develop a representative agent general equilibrium paradigm extending the Lucas (1978)
economy. Here, dividend growth is mean reverting, expected dividend growth is persistent,
and the representative agent is endowed with the recursive utility form of Du?e and Ep-
stein (1992), which is the continuous time analog of the Epstein and Zin (1989) and Weil
(1989) preferences. Moreover, based on the novel formulation of Abel (1999), our equilib-
rium clearing condition requires that the stream of dividends serve for both consumption
and debt repayment. The leverage role in asset pricing goes back at least to Merton (1974).
Our model is fairly general from the perspective of both preferences and dynamics. The
utility speci?cation breaks the tight association between the elasticity of intertemporal sub-
stitution and the risk aversion measure, which are reciprocals of each other under power
preferences. Moreover, our consumption dynamics closely follows the high economic growth
along with small consumption growth documented in the US. Collectively, our model al-
lows one to match key asset pricing regularities based on reasonable risk aversion measures.
In essence, we extend the theoretical work of Johnson (2002) along two important di-
2
Another interaction that we do not explore here is the volume e?ect on momentum pro?tability docu-
mented by Lee and Swaminathan (2000).
2
mensions. First, whereas Johnson does not explicitly formulate investor preferences, here
the dynamics of the underlying economic fundamentals is tied down to stochastic di?erential
utility of the representative agent. Moreover, Johnson focuses on understanding momen-
tum e?ects, whereas we attempt to rationalize the concentration of momentum pro?tability
among stocks with some particular styles. That is, the question of interest is how momen-
tum payo?s vary in equilibrium with ?nancial leverage, which proxies for credit risk, as well
as information uncertainty measures such as stock return volatility and cash ?ow volatility.
3
Based on simulations, we are able to generate high enough equity premium using rea-
sonable risk aversion measures, consumption dynamics, and leverage. More importantly,
we are able to rationalize momentum interactions, as we generate strong momentum ef-
fects for the interaction between high leverage and risky cash ?ow ?rms. We demonstrate
that momentum e?ects deteriorate and ultimately disappear as leverage or cash ?ow risk
diminishes. More speci?cally, the correlation between observed and expected return is pos-
itive and monotonically increases with leverage. The monotonic relation holds at high,
medium, and low levels of expected growth rate volatility, at high and low autocorrelation
of expected growth rate, and at the entire range of risk aversion measures (5 to 10) analyzed.
Next, we show that the expected return spread between the highest and lowest past
year cumulative return portfolios increases with leverage. For example, when operating
cash ?ows are highly volatile, the expected return spread is 11.49% for high leverage and
only 0.27% for low leverage stocks. On the other hand, the overall spreads are small when
3
The leverage proxy for credit risk is sound. For one, there is a strong correlation (the time series
mean of the cross sectional Spearman Rank Correlation is 0.34) between S&P credit rating and leverage.
Moreover, leverage is a crucial determinant in modeling default risk.
3
either the volatility of expected growth in cash ?ows is small or the expected growth in
cash ?ows is not highly persistent. Indeed, while leverage is crucial, risk and persistence of
cash ?ow growth are also important determinants of momentum e?ects in the cross section
of returns. On the other hand, momentum is only mildly related to investor’s risk aversion.
To summarize, it has been documented that momentum interacts with ?rm-level in-
formation uncertainty measures and credit conditions. Our collective evidence shows that
equilibrium momentum indeed concentrates in the interaction between risky cash ?ows and
high credit risk ?rms. In the presence of recursive preferences, ?nancial leverage, and per-
sistent dividend growth, one can match equity premia, riskfree rate, return predictability,
and especially the observational momentum pro?tability at reasonable risk-aversion levels.
The paper proceeds as follows. Section 2 describes the economic setup including in-
vestor preferences and the dynamics of the underlying economic fundamentals. Section 3
derives the interaction of momentum with credit risk and information uncertainty. Sec-
tion 4 reports the simulation results coming up from the theoretical formulations. Section
5 concludes and provides suggestions for future work. Technical details are in the appendix.
2 The economic setup
2.1 Preferences and Dynamics
In formulating investor preferences, we depart from the regular power utility speci?cation.
Theoretically, power preferences put a heavy restriction on elasticity of inter-temporal sub-
4
stitution (EIS) and risk aversion – they are reciprocals of each other – even when risk
aversion and EIS are distinct economic quantities. EIS is about a deterministic consump-
tion path as it measures the willingness to exchange consumption today for consumption
tomorrow conditioned on a current riskfree interest rate, whereas risk aversion is about
preference over a random quantity. Empirically, the power utility restriction gives rise to
the equity premium and riskfree rate puzzles as well as the failure of the consumption based
asset pricing model to explain the cross section dispersion in average stock returns.
Instead, we employ stochastic di?erential utility (SDU) of Du?e and Epstein (1992).
The SDU is the continuous time analog of the Epstein-Zin (1989) and Weil’s (1990) recursive
preferences, which break the tight association between risk aversion and EIS. The SDU is
identi?ed by a pair of functions (f
?
, A(J)), called an aggregator, where A(J) is local risk-
aversion and f
?
represents the relative preference between immediate consumption and the
certainty equivalent of utility derived from future consumption. An ordinally equivalent
representation of the SDU is given by the normalized aggregator (f, 0), which is formulated
as
f(C, J) =
?(1 ??)J
1 ?
1
?
_
C
1?
1
?
((1 ??)J)
1
?
?1
1??
?1
_
. (1)
In equation (1), C denotes the current consumption, J is the continuation utility (or the
value function) attributable to future consumption streams, ? is the EIS, ? is the discount
rate standing for the time preference, and ? is the relative risk-aversion parameter. Under
5
this convention, the time t value function of an agent can be written as
J
t
= E
t
_
?
t
f(c
s
, J
s
)ds. (2)
We derive below an explicit solution for J
t
assuming that ? = 1. This assumption is
innocuous. In particular, Bansal and Yaron (2004) and Ai (2007) show that ? = 1 is the
point of equivalence between wealth and substitution e?ects, but preferences over risk are
still determined by risk-aversion. That is, if ? < 1 (? > 1), the income (substitution)
e?ect dominates. In other words, high growth rate can lead an agent to consume more
(income e?ect) or invest more (substitution e?ect). Which course of action is undertaken
depends upon the ? parameter. Indeed, EIS primarily deals with determining the riskfree
rate, which is not at the core of our study. The normalized aggregator based on ? = 1 is
the limit of (1) taking the form
f(C, J) = ?(1 ??)J
_
log C ?
log(1 ??)J
1 ??
_
. (3)
Next, as in Johnson (2002), we assume that the dividend growth follows a geometric
path with stochastic expected growth rate. Speci?cally, the joint system of the observed
dividend growth and the unobserved expected dividend growth are formulated as
dD
t
D
t
= X
t
dt +?
D
dW
1
, (4)
dX
t
= ?(
¯
X ?X
t
)dt +?
x
dW
2
, (5)
where X
t
is the expected dividend growth, ? stands for the speed of mean reversion,
¯
X
is the long run mean of X
t
, ?
D
is the volatility of dividend growth, ?
x
is the volatility of
6
expected dividend growth, and the correlation between the two Brownian motions is ?.
To account for leverage, we build on the novel formulation of Abel (1999). Abel is able
to generate, in a fairly simple setting, low variability of the riskfree rate along with a large
equity premium, both of which are patterns documented in the US economy in the post-war
period. In particular, we assume that the equilibrium consumption is a portion of dividend
C = D
?
, (6)
while the remainder of the dividend stream is distributed as adjustment cost or debt pay-
ment in the economy. The no-leverage case ? = 1 depicts an economy with no adjustment
cost wherein the agent consumes the full dividend streams. In the ? < 1 case, which is at
the core of our analysis, the rest of dividends go towards debt payment. Then stocks in
this economy are residual claims on the consumption stream net debt payments.
In the presence of leverage, the consumption growth dynamics takes the form
dC
C
= µ
C
(X
t
)dt +?
C
dW
1
, (7)
where
µ
C
(X
t
) = ?
_
X
t
+
1
2
(? ?1)?
2
D
_
(8)
?
C
= ??
D
, (9)
suggesting that expected consumption growth is slower than expected dividend growth as
7
long as ? < 1. This is consistent with the US economy which demonstrates slow consump-
tion growth along with relatively fast economic growth. Notice also that in a levered econ-
omy, the volatility of consumption ??
D
is smaller than the volatility of economic growth.
The utility process J satis?es the Bellman equation with respect to equilibrium con-
sumption
DJ(C, X, t) +f(C, J) = 0 (10)
where DJ is the di?erential operator applied to J with respect to {C, X, t} with the
boundary condition J(C, x, T) = 0. In the analysis that follows, we are interested in the
equilibrium as T ? ?. Thus, we drop the explicit time dependence assuming that the
agent is in?nitely long-lived and has reached the equilibrium over time. We can formulate
an exact solution to the value function which is stated in the following proposition.
Proposition 1. An exact solution to the di?erential equation in (10) is given by
J(C
t
, X
t
) =
C
1??
t
1 ??
exp (u
1
X
t
+u
2
) (11)
where
u
1
=
(1 ??)?
? +?
(12)
u
2
=
(1 ??)?
?
_
(? ?1 ???)?
2
D
2
+
?
¯
X
? +?
+
(1 ??)?
? +?
_
?
2
x
2(? +?)
+?
D
?
x
?
__
. (13)
Proof: see the appendix.
Unlike in power preferences, SDU incorporates the agent’s value function in the current
utility. Thus, expected growth rate enters into the agent’s utility through the value func-
8
tion J. Indeed, expected growth rate has important implications for future consumption
streams. Higher expected growth rate indicates higher expected future consumption and
hence higher future utility which is re?ected through higher current value function. Thus,
J
X
is positive. On the other hand, J
XX
is negative suggesting that J is concave in X or
J
X
increases in X in diminishing rates. We also run some simulations to learn the impact
of the value function on the current utility. The current utility is increasing (decreasing)
in growth rates under low (high) leverage and low (high) uncertainty of future growth rates.
Moreover, the autocorrelation of X is important for understanding the expected growth
rate e?ect on future utility. If expected growth rate is highly persistent (low ?) then high
X
t
implies high future expected growth rate, which, in turn, implies that the agent expects
to consume more in the future. Thus J
X
increases with expected growth rate persistence.
Finally, J
X
decreases in ?. In particular, if the agent discounts the future more (higher
?) then the impact of expected growth rate on the value function (future utility) diminishes.
The next section derives the pricing kernel dynamics, the asset return dynamics, the
correlation between observed realized returns and expected returns, and especially the link
between leverage, information uncertainty, and stock return momentum.
3 Asset Pricing
Du?e and Epstein (1992) show that the pricing kernel for SDU is given by ?
t
= exp(
_
t
0
f
J
ds)f
C
,
where f
J
and f
C
are the derivatives of f(C, J) in (3) with respect to J and C. Hence we
can ?nd the explicit pricing kernel dynamics as stated in the proposition below.
9
Proposition 2. The pricing kernel dynamics is given by
d?
?
= ?r
f
t
dt ????
D
dW
1
+u
1
?
x
dW
2
(14)
where
r
f
t
= ?X
t
+u
1
???
D
?
x
? +?(u
2
+ 1) ?u
1
?
¯
X ?
1
2
???
2
D
(?? + 1) ?
1
2
?
2
x
u
2
1
= µ
C
(X
t
) +? ??
2
C
? +
(1 ??)?
? +?
?
C
?
x
?. (15)
Proof: see the appendix.
Observe from equation (14) that leverage is an important determinant of both the drift
and di?usion of the pricing kernel dynamics. Moreover, our equilibrium riskfree rate for-
mulated in (15) has two attractive features compared to its power utility counterpart. For
comparison, the corresponding riskfree rate for power utility is r
f
t
= ?µ
C
(X
t
)+??
?(?+1)?
2
C
2
.
First, the riskfree rate based on the power utility is highly sensitive to expected con-
sumption growth. In particular, a one-percent increase in the growth rate is followed by
?-percent increase in the riskfree rate - a nuisance at the heart of the riskfree rate puzzle.
Indeed, the riskfree rate determines the intertemporal substitution e?ect between deter-
ministic consumption streams, whereas risk-aversion determines the agent’s preference over
risky bets. Thus, risk aversion should not exert such a considerable in?uence on the con-
sumption growth riskfree rate sensitivity. In the SDU case, there is a much more realistic
one-to-one relationship between expected consumption growth and riskfree rate.
10
Second, for realistic values of µ
C
and ?
C
, the power utility riskfree rate is increasing
under believable values of risk-aversion - which again misconstrues the nature of the riskfree
rate. In other words, higher risk-aversion is required to match the equity-premia, it also
correspondingly increases the riskfree rate. In our case, as in Du?e and Epstein (1992),
the riskfree rate is strictly decreasing in risk-aversion. Thus, if indeed higher risk-aversion
is needed to match the equity premia, it does not pose any challenge to match low riskfree
rate. It should also be noted that the riskfree rate does vary through time with X
t
.
We next establish the equilibrium dividend price ratio and the return dynamics.
Proposition 3. The equilibrium price-dividend ratio
Pt
Dt
, denoted by G(X
t
), is
G(X
t
) =
_
?
0
exp(P
1
(?)X
t
+P
2
(?))d? (16)
where P
1
and P
2
are the solutions of a system of ODEs given in the appendix.
Furthermore, the excess return dynamics is given by
dR
t
= µ
R
t
dt +?
D
dW
1
+
G
X
G
?
x
dW
2
(17)
dµ
R
t
= (·)dt +
_
G
X
G
_
X
(???
D
?
x
? ?u
1
?
2
x
)?
x
dW
2
. (18)
It immediately follows that the instantaneous covariance between realized and expected
return is given by
E
t
__
dR
t
?µ
R
t
dt
_ _
dµ
R
t
?(·)dt
_¸
=
_
G
X
G
_
X
_
?
D
?
x
? +
G
X
G
?
2
X
_
(???
D
?
x
? ?u
1
?
2
x
). (19)
We can now go further and formulate the SDEs of realized cumulative excess returns
11
and expected excess returns for an investment horizon of l periods. In particular,
R
t,t+l
= R
t
+
_
t+l
t
µ
R
s
ds +
_
t+l
t
?
D
dW
1
+
_
t+l
t
G
X
G
?
x
dW
2
(20)
µ
R
t,t+l
= µ
R
t
+
_
t+l
t
(·)ds +
_
t+l
t
_
G
X
G
_
X
(???
D
?
x
? ?u
1
?
2
x
)?
x
dW
2
(21)
As a ?rst pass to understand momentum e?ects in stock returns we compute the cor-
relation between R
t,t+l
and µ
R
t,t+l
. The covariance between realized and expected return
is
Cov
t
(R
t,t+l
, µ
R
t,t+l
) = E
t
_
(R
t,t+l
?E
t
R
t,t+l
)(µ
R
t,t+l
?E
t
µ
R
t,t+l
)
¸
= ?
x
(???
D
?
x
? ?u
1
?
2
x
)
_
t+l
t
_
G
X
G
_
X
_
?
D
? +
G
X
G
?
x
_
ds. (22)
Then, the variances are computed as
V
t
(µ
R
t+l
) = (???
D
?
x
? ?u
1
?
2
x
)
2
?
2
x
_
t+l
t
__
G
X
G
_
X
_
2
ds (23)
V
t
(R
t+l
) =
_
t+l
t
_
?
2
D
+
_
G
X
G
?
x
_
2
+ 2
G
X
G
?
D
?
x
?
_
ds. (24)
Hence, the correlation is
?(l) =
_
t+l
t
_
G
X
G
_
X
(?
D
? +
G
X
G
?
x
)ds
_
_
t+l
t
__
G
X
G
_
X
_
2
ds ·
_
t+l
t
_
?
2
D
+
_
G
X
G
?
x
_
2
+ 2
G
X
G
?
D
?
x
?
_
ds
. (25)
The ?(l) function depicts the correlation between expected return and realized return
up to horizon l. As l ?0, ?(l) is the instantaneous correlation between realized return up
until now (time 0) and expected return. As l increases, it becomes the correlation between
12
realized return over the period 0 ? l with the expected return at the end of that period.
The leading term on the numerator of ?(l) can be written as
GG
XX
?(G
X
)
2
G
2
=
1
G
2
_
_
?
0
exp(·)d?
_
?
0
exp(·)P
2
1
(?)d? ?
__
?
0
exp(·)P
1
(?)d?
_
2
_
(26)
which is positive from direct application of Cauchy-Schwartz inequality to functions P
1
(?)
_
exp(·)
and
_
exp(·), both of which are integrable in the domain. As we show below, the term
G
X
G
is always positive for ? < 1. The autocorrelation is positive unless ? < 0 and |?
D
?| >
G
X
G
?
x
.
3.1 Interpretation and implications
3.1.1 Consumption Growth
The equilibrium condition produces consumption growth which follows the dynamics in (7).
If there is no leverage (? = 1), then consumption growth is the same as the dividend growth
and we are back to a Lucas (1978) economy setting. In the case of a slave economy (? = 0),
there is no consumption growth and consumption at every point in time is nonrandom. Of
course, one can also attribute the ? = 0 case to a riskfree bond. At intermediate level of
leverage (0 < ? < 1), which is the focus of our analysis below, we observe key features of
post-war US data. First, the volatility of consumption growth is much smaller than the
volatility of dividend growth. Furthermore, in the presence of high growth rate of dividends
the consumption growth rate can still be low especially if the economy is highly levered.
13
3.1.2 Price-Dividend ratio
The business cycle e?ect on the P/D ratio is instantly observable as
G
X
=
1 ??
?
_
?
0
exp(P
1
(?)X
t
+P
2
(?))
_
1 ?e
???
_
d?. (27)
Thus the P/D ratio is increasing in the growth rate X
t
as long as ? < 1. The e?ect is
most pronounced for low ? (high leverage), it deteriorates as ? grows (lower leverage), and
ultimately vanishes as ? approaches one (no leverage). It should be noted that in the power
utility case G
X
is positive only if ?? < 1. Essentially, this restriction puts an undue burden
on ? to be less than
1
?
. Here, the condition that the P/D ratio increases in the business
cycle variable does not depend on risk-aversion. This appealing feature is due to the use of
the stochastic di?erential utility. Notice also that the ? < 1 case points to a wealth e?ect
in the economy in the presence of increasing economic growth rate. That is, if X
t
increases,
the stock price rises relative to the dividend - and even more so relative to consumption.
The equilibrium D/P ratio is shown in Figure 1 for plausible values of the parameters
underlying the stock return dynamics. The e?ect of risk-aversion is straightforward. Higher
risk-aversion increases expected return thus reducing the current stock price or increasing
the D/P ratio. The e?ect of leverage on the D/P ratio, however, is not straightforward.
Figure 1 shows that leverage and the D/P ratio are nonlinearly related. The D/P ratio
?rst increases as ? grows and then decreases as ? approaches one. This pattern is attributed
to the interaction of leverage with the growth rate X
t
. In the no leverage (? = 1) case the
P
1
(?) in (16) is 0, suggesting that leverage and growth rate do not interact. At high growth
14
rates with small leverage (high ?), agents encounter high consumption growth and thus low
marginal utility of consumption. Agents would therefore invest more and consequently the
stock price would rise, triggering low D/P ratio. It is intuitive to think that the reverse
would take place at high leverage (small ?). However, at high leverage, P
1
(?) is high and
it interacts strongly with the growth rate. Thus, the stock price rises with high leverage
and high growth rate and the D/P ratio again falls. In essence, when leverage is high,
consumption reacts slowly in response to high growth rate and the agent invests more in
anticipation of good times in the future, thus raising prices and lowering the D/P ratio.
3.1.3 Expected return and leverage
The expected excess stock return is given by
µ
R
t
= (???
2
D
?u
1
?
D
?
x
?) +G
X
(???
D
?
x
? ?u
1
?
2
x
)
D
P
. (28)
Thus, G
X
– the response of the P/D ratio to the economic growth rate – guides the ex-
pected return dynamics. Notice that for stocks with high ? (low leverage) the ability of
the dividend yield to forecast future returns diminishes. In particular, for ? = 1, G
X
= 0
suggesting that all time series e?ects on expected stock return vanish completely. Likewise,
at ? = 0, µ
R
t
= 0 which implies that in a fully levered economy, there is no consumption
growth and the only security available is a bond with a constant stream of consumption.
As such, the risk-premia vanishes and the bond yields a rate of return equal to the discount
rate ?. Thus, return predictability is pronounced for stocks with intermediate values of ?.
Interestingly, observe from Figure 2 that even in the absence of time series e?ects on ex-
15
pected stock return, our model can still deliver the relatively high equity premium observed
in the US over the past century. In particular, the second term in the ?rst parenthesis in
(28), u
1
?
D
?
x
? =
(1??)??
D
?x?
?+?
, which is a contribution of the stochastic di?erential utility,
provides the explanation. Small values of ? and ? could greatly magnify this expected
return component, thus producing expected return terms that match the high equity pre-
mium at a relatively low risk-aversion.
Figure 2 also displays the non-linear dependence of expected return on leverage. Clearly,
at ? = 0, µ
R
t
= 0 and at ? = 1, excess return is constant. The time-series dependence and
its interaction with leverage is provided at intermediate levels of ?. In such intermediate
levels, whenever we observe high D/P ratio in Figure 1, we also observe high expected
return level since G
X
> 0. The e?ect is magni?ed as risk-aversion increases, as expected.
This positive relationship is empirically shown through the positive coe?cients in predictive
regressions as in Cochrane (2007).
4 Understanding momentum interactions
As noted earlier for ? = 1 or ? = 0 the excess return is zero or constant, respectively,
and the momentum e?ect is nonexistent. We focus on the more realistic 0 < ? < 1 do-
main where the full dynamics of the model can be interacted with leverage. Our goal is
to calibrate the correlation between observed return and expected return as well as the
overall momentum pro?tability for 42 distinct speci?cations of parameters underlying the
return dynamics with each speci?cation standing for one particular class of ?rms. Thus,
16
our model could ultimately deliver prominent predictions on the cross section of such ?rms,
and it does, as we show below.
The parameter settings are described in Table 1. In the simulation exercises the seven
settings described in Table 1 will be interacted with six leverage levels - thus overall we
consider 42 distinct cases. For all the seven [A-G] settings, we assume that the discount
rate for our in?nitely lived agent is small (? = 0.01) and the expected dividend growth is
highly autoregressive (? = 0.05) with the long run mean growth rate (
¯
X) at 5%. Based on
the relationship in (25) and (27), the claim going forward is that high variance in expected
dividend growth rate generates the momentum e?ect that is more pronounced for high
levered ?rms. High variance could emerge due to both high volatility in expected growth
rate innovation (high ?
x
) and highly persistent expected dividend growth (low ?).
In Table 1, settings A and B display high variance of dividend growth (high ?
D
and ?
x
),
C and D display low, and E exhibits medium. The distinction between A and B (C and
D) is higher risk-aversion parameter for B (D). Both E and F exhibit the same volatility
of dividend growth (?
D
) but F displays lower expected growth rate volatility (?
x
). Setting
G takes higher ?, which means less persistent expected dividend growth rate, and at the
same time holds ?
D
and ?
x
at a moderate level.
The next section describes the simulations made to assess the impact of leverage, volatil-
ity of dividend growth, volatility of expected dividend growth, risk aversion, and persistence
of expected dividend growth on momentum e?ects.
17
4.1 Correlation between realized and expected returns
We ?rst explore the correlation between realized and expected returns. The system of re-
alized and expected returns is simulated forward (starting from X
0
=
¯
X) for investment
horizons of 3, 6, 9, and 12 months and then the correlation between observed investment
return and expected return is computed. Table 2 shows the correlation for all eight set-
tings, each of which is interacted with six levels of leverage ranging between 0.15 (high
leverage) and 0.90 (low leverage). The ?gures represent the correlation formulated in (25)
for investment horizons of 3-12 months. The unchanging correlation pattern represents the
stability of correlation over time. Johnson (2002) depicts similar correlation structure.
The correlation between expected excess return and cumulative excess return monoton-
ically increases with leverage. The monotonic relationship holds at high, medium, and low
levels of growth rate volatility, at high and low autocorrelation of expected growth rate,
and at the entire range of risk aversion measures considered here. While all cases examined
exhibit positive correlation between observed and expected returns, we show below that
momentum pro?tability prevails only in a few of the settings and is nonexistent in others.
4.2 Instantaneous expected return spreads
Table 3 exhibits expected excess return over a one-year period in which past cumulative
returns have been classi?ed into ten deciles, with column 1 (10) pertaining to the low-
est (highest) observed returns. There are several insights emerging about the role that
leverage, dividend growth volatility, expected dividend growth volatility, risk aversion, and
persistence play in generating momentum e?ects. Here, momentum pro?tability is de?ned
18
as the expected return spread between the highest and lowest observed return portfolios.
Momentum and leverage. Momentum pro?tability monotonically increases with
leverage regardless of the case considered, with settings A, B, and E displaying large and
economically meaningful expected return spreads. Focusing on A, the expected return
spread is 11.49% [26.51%-15.02%] per year for ? = 0.15 while it is only 0.27% for ? = 0.9.
Moving to B, the expected return spread is 13.16% for ? = 0.15 and is 1.71% for ? = 0.9.
The corresponding ?gures for E are 9.87% and 0.26%. On the other hand, the overall
spreads are small for the C, D, and G settings. This indicates that while leverage is crucial
there are some other important determinants of momentum e?ects.
Before we move on, it should be noted that the expected returns reported in Table 3
might seem quite large, especially the ones indicating strong momentum e?ects. However,
notice that the time discount parameter ? is small (0.01) at this stage. We reexamine some
of the sub-cases using higher time-discount parameter. The evidence is reported in Table
4. Indeed, increasing ? brings down expected return at every decile to more realistic levels.
The interesting evidence, however, is that the expected return spread between the highest
and lowest deciles is preserved. To illustrate, for case A and ? = 0.25 the spread is 7.56%,
for B and ? = 0.15 is 9.96%, for B and ? = 0.9 is 0.57%, and for E and ? = 0.2 is 5.87%.
Momentum and information uncertainty. Settings A, B, and E display the highest
expected return spreads across the ten deciles, as noted earlier. For the other settings the
spreads are much lower and are often even negligible. To illustrate, the highest spread for
C is 2.95% [15.01%-12.06%], 4.86% for D, 1.05% for F, and only 0.29% for G. Cases A, B,
19
and E are all characterized by high (0.07) to moderate (0.05) expected dividend growth
volatility, which, from the ?rm perspective, amounts to relatively high volatility cash ?ows.
To this point we are able to rationalize previously documented momentum interactions.
In particular Zhang (2006) ?nds that momentum concentrates in high information uncer-
tainty stocks and points to behavioral interpretations. Avramov, Chordia, Jostova, and
Philipov (2007) document that momentum prevails only among high credit risk stocks.
Whereas such momentum-credit risk interaction could point to rational interpretations,
this is purely an empirical ?nding thus far that has not been formalized in an equilibrium
model. The collective evidence here shows that equilibrium momentum e?ects should con-
centrate in the interaction of risky cash ?ows and highly levered ?rms. Interestingly, neither
leverage alone nor cash ?ow volatility alone are su?cient to generate momentum e?ects.
Focusing on information uncertainty measures, a valid point to make is that the volatil-
ity of expected dividend growth (?
x
) is the primary force of momentum e?ects, whereas the
volatility of the unexpected dividend growth (?
D
) plays a marginal role. Note in particular
that cases E and F are virtually identical with the only exception being ?
x
= 0.05 in E
versus ?
x
= 0.02 in F. Nevertheless, the expected return spreads in E are considerably
higher for all leverage levels.
Momentum and risk aversion. Setting G features the highest risk aversion but
nevertheless yields the lowest expected return spreads between the ten portfolios, ranging
between 0.02% and 0.29%. The immediate takeout is that the risk aversion measure is not
a key parameter in generating momentum e?ects. Let us also compare A versus B as well
20
as C versus D. For cases B and D, the higher risk-aversion simply increases expected re-
turn at every level of leverage and still preserves expected return spread across the deciles.
Indeed, the momentum pro?tability in B (D) is slightly higher than than of A (C), suggest-
ing that risk aversion has some e?ect, albeit relatively small, in explaining the return spread.
Momentum and expected growth rate persistence. Case G is di?erent from the
previous settings in that it features the lowest autocorrelation of expected growth rate,
which reduces the total variance of expected growth rate. In case G, the shocks to the sys-
tem are exactly the same as in case E (?
D
= 0.06 and ?
x
= 0.05). Therefore, with higher
risk-aversion (10 versus 5) we can only expect the momentum e?ect of case E to be exac-
erbated, just like case B compared to case A. However, with lower ? the expected return
spread across the high and low performing portfolios is minuscule, even at high levels of
leverage. Therefore, high ?, which reduces the variance of expected growth rates, exhibits
no expected return di?erential that characterizes the momentum e?ect in the data. Persis-
tence is indeed crucial in generating momentum e?ects and it overwhelms risk aversion.
The evidence emerging from Table 3 also suggests that even when there is strong corre-
lation between realized and expected return at high levels of leverage in high, medium, and
low levels of volatility (see Table 2), low cash ?ow volatility does not produce economically
signi?cant expected return spreads across the di?erent deciles even when leverage is high.
21
4.3 Holding period return spreads
What makes the momentum e?ect a conundrum is the holding period pro?t. The strategy
of buying winners and selling short losers produces 8-12% ex post payo?s according to
Jegadeesh and Titman (1993). We next simulate holding period returns based on one year
formation period and conventional holding periods of 3-12 months. Table 5 reports momen-
tum pro?tability which is the return spread between the top and bottom past return deciles.
Consistent with the evidence reported thus far it follows that the high volatility case A,
which produces high ex-ante expected excess returns, also generates high ex post holding
period returns. Focusing on the one year holding period, momentum pro?tability is 10.35%
for ? = 0.15 and is only 0.37% for ? = 0.9. The low volatility case D fails to generate high
momentum pro?tability. For the one year holding period the momentum payo? is 3.2%
for high leverage and 0.13% at low leverage. Furthermore, the moderate volatility case E
generates moderate levels of investment returns ranging between 0.26% and 8.98% for the
one year holding period. The evidence in Case G with the lowest autocorrelation con?rms
the earlier observation that holding period returns based on low autocorrelation and high
volatility are small and similar in magnitude to the high autocorrelation low volatility case
D (recall ?
x
= 0.05 in case G while ?
x
= 0.03 in case D). In summary, we con?rm that
momentum e?ects concentrate in ?rms with high leverage as well as highly volatile and
persistent expected cash ?ows growth.
22
5 Conclusion
Previous work shows that momentum e?ects in stock returns are robust, thus invoking a
plethora of behavioral and rational explanations. Previous work also uncovers momentum
interactions. In particular, momentum concentrates in stocks with high return volatility,
high cash ?ow volatility, small market capitalization, high analysts’ earnings forecast dis-
persion, as well as high credit risk. Thus far, there has not been any attempt, to our
knowledge, to theoretically rationalize or even behavioralize such momentum interactions.
This paper embraces this task. In particular, it analyzes momentum interactions from a
rational equilibrium perspective and ultimately shows that the concentration of momentum
in high information uncertainty as well as high credit risk stocks is perfectly consistent with
rational asset pricing. Our economic setup is fairly general from the perspectives of both
preferences and dynamics. The stochastic di?erential utility of Du?e and Epstein (1992)
employed here breaks the tight association between the elasticity of inter-temporal substi-
tution and the risk aversion measure. Moreover, our consumption dynamics, which is based
on the novel formulation of Abel (1999), closely follows the high economic growth along with
small consumption growth documented in the US post-war data. Collectively, our model
allows one to match key regularities in asset pricing using reasonable risk aversion measures.
We use simulations to ?nd out that our paradigm indeed predicts strong equilibrium
momentum e?ects for the interaction between high leverage and risky cash ?ow ?rms. Mo-
mentum pro?tability deteriorates and ultimately disappears as either leverage or cash ?ow
risk diminishes. More speci?cally, the correlation between observed and expected returns is
positive and monotonically increasing with leverage. The monotonic relationship holds at
23
high, medium, and low levels of expected growth rate volatility, at high and low autocor-
relation of expected growth rate, and at the entire range of risk aversion measures analyzed.
Moreover, the expected return spread between the highest and lowest past year cumu-
lative return portfolios increases with leverage. For example, when operating cash ?ows
are highly volatile, the expected return spread is 11.49% for high leverage and only 0.27%
for low leverage. On the other hand, the overall spreads are small when either the volatil-
ity of expected cash ?ows growth is small, or the expected growth in cash ?ows in not
highly persistent. This indicates that while leverage is crucial, risk and persistence in
cash ?ows growth are both important determinants of momentum e?ects. The collective
evidence thus shows that equilibrium momentum pro?tability concentrates in the interac-
tion between risky cash ?ows and high levered ?rms which is perfectly consistent with data.
Looking forward, there are several suggestions for future work. First, currently leverage
is exogenous. We ask: given a particular leverage level - what is the overall momentum
e?ect? It would be quite appealing to endogenize leverage and make it a ?rm-decision
variable. Next, our focus here has been on ?rm level interactions. It has also been shown
that momentum displays strong business cycle e?ects. Our setting can readily be extended
to analyze possible rational business cycle e?ects in momentum pro?tability. Finally, from
an empirical perspective, one could analyze the joint e?ect of leverage, expected dividend
growth risk, and dividend growth persistence on the cross section of average returns.
24
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28
0.2
0.4
0.6
0.8
1
3
4
5
6
0
0.05
0.1
0.15
0.2
?
Dividend?Price Ratio
?
Figure 1: Dividend-Price ratios implied by the model for ? = 0.01, ? = 0.1,
¯
X = 0.05,
?
D
= .05, ?
x
= .035, ? = .35. The state is set to
¯
X, i.e.X
0
=
¯
X
.
29
0.2
0.4
0.6
0.8
1
3
4
5
6
0
0.05
0.1
0.15
0.2
?
Expected Excess Return
?
Figure 2: Expected excess return implied by the model for ? = 0.01, ? = 0.1,
¯
X = 0.05,
?
D
= .05, ?
x
= .035, ? = .35. The state is set to
¯
X, i.e.X
0
=
¯
X
.
30
Table 1:This table lists the set of parameters we are considering. A and B are cases with high volatility
of dividend growth and expected dividend growth. Case B is the same as A with higher risk-aversion.
Cases C and D are with low volatilities with D having higher risk-aversion than C. E and F have moderate
volatilities and higher correlation where F has lower volatility of expected dividend growth. G is similar to
E, except with lower autocorrelation.
? ? ?
¯
X ?
D
?
x
?
A 0.01 5.0 0.05 0.05 0.08 0.07 0.2
B 0.01 8.0 0.05 0.05 0.08 0.07 0.2
C 0.01 5.0 0.05 0.05 0.04 0.03 0.1
D 0.01 8.0 0.05 0.05 0.04 0.03 0.1
E 0.01 5.0 0.05 0.05 0.06 0.05 0.2
F 0.01 5.0 0.05 0.05 0.06 0.02 0.2
G 0.01 10.0 0.20 0.05 0.06 0.05 0.2
31
Table 2:This table shows the correlation between expected excess return and cumulative excess return
based on investment horizons of 3,6,9 and 12 months. The system is simulated forward for 5000 paths
using Monte Carlo integration subsequent to which we compute the ?(l) function. Then the average of the
?(l) function is reported for each l. In all cases, X
0
=
¯
X.
? 3 6 9 12
A 0.15 0.97 0.97 0.97 0.97
0.25 0.95 0.95 0.95 0.95
0.30 0.91 0.91 0.91 0.91
0.40 0.87 0.87 0.87 0.87
0.70 0.71 0.71 0.71 0.71
0.90 0.51 0.51 0.51 0.51
D 0.15 0.98 0.98 0.98 0.98
0.20 0.97 0.97 0.97 0.97
0.30 0.95 0.95 0.95 0.95
0.40 0.92 0.92 0.92 0.92
0.70 0.77 0.77 0.77 0.77
0.90 0.52 0.52 0.52 0.52
E 0.15 0.99 0.99 0.99 0.99
0.20 0.97 0.97 0.97 0.97
0.30 0.94 0.94 0.94 0.94
0.40 0.91 0.91 0.91 0.91
0.70 0.75 0.75 0.75 0.75
0.90 0.57 0.57 0.57 0.57
G 0.15 0.96 0.96 0.96 0.96
0.20 0.95 0.95 0.95 0.95
0.30 0.92 0.92 0.92 0.92
0.40 0.89 0.89 0.89 0.89
0.70 0.73 0.73 0.73 0.73
0.90 0.47 0.47 0.47 0.47
32
Table 3: This table shows the average instantaneous expected excess return (annualized percentage)
under the di?erent parameter settings subsequent to one year in which the cumulative return has fallen
into 1 of the 10 deciles labeled 1-10. Column 1 is the expected excess return for the lowest decile and
Column 10 is for the highest. The system is simulated forward for 5000 di?erent paths for one year, and
each path here depicts one security over the year. For each path, we compute the observed return in (20)
and the corresponding expected return. At the end of one year, we sort the 5000 paths based on observed
return and assign them into the ten portfolios. The average of the expected return for each portfolio
(equally-weighted) is then reported. In all cases, X
0
=
¯
X.
? 1 2 3 4 5 6 7 8 9 10
A 0.15 15.02 17.13 18.25 18.83 19.48 20.04 20.67 21.39 22.99 26.51
0.25 18.75 20.32 21.08 21.50 21.91 22.26 22.68 23.12 24.09 26.13
0.30 20.37 21.79 22.43 22.85 23.22 23.52 23.86 24.28 25.11 26.88
0.40 22.94 24.11 24.65 24.98 25.26 25.50 25.76 26.13 26.73 28.12
0.70 25.77 26.21 26.43 26.55 26.65 26.74 26.83 26.97 27.22 27.71
0.90 21.61 21.68 21.74 21.76 21.78 21.79 21.81 21.84 21.90 21.98
B 0.15 20.13 21.91 22.81 23.28 23.76 24.15 24.66 25.18 26.32 33.29
0.20 23.45 24.98 25.73 26.21 26.65 26.99 27.37 27.88 28.83 30.89
0.30 28.56 29.92 30.56 30.97 31.25 31.54 31.88 32.26 33.00 34.64
0.40 32.43 33.55 34.05 34.43 34.61 34.81 35.14 35.41 35.99 37.27
0.70 34.25 34.78 35.12 35.77 36.12 36.88 37.22 37.45 37.89 38.14
0.90 37.30 37.66 37.89 37.98 38.08 38.15 38.24 38.36 38.62 39.01
C 0.16 12.06 13.07 13.53 13.71 13.87 14.01 14.16 14.30 14.58 15.01
0.25 7.88 8.51 8.85 9.00 9.14 9.30 9.44 9.61 9.95 10.62
0.30 8.17 8.72 9.01 9.14 9.27 9.40 9.53 9.69 9.97 10.57
0.40 8.71 9.15 9.38 9.48 9.58 9.69 9.79 9.90 10.14 10.62
0.70 8.57 8.76 8.84 8.88 8.93 8.96 8.99 9.04 9.13 9.30
0.90 5.96 5.98 6.00 6.01 6.02 6.02 6.02 6.03 6.05 6.07
D 0.15 10.56 11.94 12.63 12.93 13.20 13.46 13.72 13.99 14.52 15.42
0.20 9.69 10.32 10.67 10.81 10.97 11.11 11.27 11.43 11.79 12.52
0.30 10.83 11.38 11.66 11.79 11.92 12.06 12.19 12.33 12.63 13.24
0.40 11.83 12.31 12.52 12.64 12.76 12.85 12.96 13.07 13.32 13.83
0.70 12.05 12.47 12.33 12.39 12.42 12.46 12.50 12.55 12.64 12.82
0.90 8.77 8.79 8.81 8.82 8.82 8.83 8.83 8.84 8.86 8.88
33
Table 3 (Continued):
? 1 2 3 4 5 6 7 8 9 10
E 0.15 11.54 13.39 14.47 14.98 15.50 16.03 16.57 17.21 18.57 21.41
0.20 13.15 14.28 14.84 15.13 15.43 15.68 15.97 16.27 16.96 18.38
0.30 14.11 15.13 15.64 15.89 16.15 16.37 16.62 16.90 17.48 18.72
0.40 15.68 16.51 16.89 17.11 17.33 17.49 17.69 17.91 18.37 19.30
0.70 16.94 17.27 17.42 17.53 17.58 17.65 17.74 17.82 17.99 18.37
0.90 13.49 13.55 13.59 13.60 13.61 13.62 13.64 13.66 13.69 13.75
F 0.20 7.16 7.47 7.62 7.68 7.74 7.79 7.85 7.90 8.01 8.21
0.25 6.30 6.64 6.79 6.86 6.94 7.00 7.07 7.14 7.28 7.57
0.30 6.34 6.63 6.78 6.84 6.91 6.96 7.03 7.09 7.23 7.50
0.40 6.72 6.95 7.05 7.11 7.17 7.21 7.25 7.31 7.42 7.63
0.70 7.09 7.17 7.20 7.22 7.24 7.25 7.27 7.28 7.32 7.40
0.90 5.74 5.75 5.75 5.76 5.76 5.77 5.77 5.78 5.78 5.79
G 0.15 7.25 7.34 7.38 7.39 7.41 7.42 7.44 7.45 7.48 7.54
0.20 8.33 8.48 8.54 8.57 8.60 8.63 8.66 8.68 8.74 8.85
0.30 10.02 10.21 10.30 10.34 10.34 10.41 10.45 10.49 10.57 10.71
0.40 11.22 11.34 11.48 11.52 11.56 11.59 11.62 11.69 11.74 11.90
0.70 12.07 12.14 12.17 12.19 12.20 12.21 12.22 12.24 12.27 12.33
0.90 9.54 9.54 9.55 9.55 9.55 9.55 9.55 9.55 9.55 9.56
34
Table 4: This table repeats the exercise in Table 3 with di?erent time preference param-
eter ?. Certain sub-cases are taken and they are replicated with a higher ?. The ?rst line
for each sub-case is copied from the corresponding line on Table 3, and the following line is
the repeat of the same simulation with higher beta, such that the transversality condition
is still satis?ed. This table shows that increasing ? lowers expected return at every decile
but still maintains a healthy di?erence between the highest and lowest deciles.
? 1 2 3 4 5 6 7 8 9 10
A(? = 0.25) 0.01 18.75 20.32 21.08 21.50 21.91 22.26 22.68 23.12 24.09 26.13
0.08 10.88 12.23 12.96 13.34 13.76 14.11 14.53 14.99 16.05 18.42
B(? = 0.15) 0.01 20.13 21.91 22.81 23.28 23.76 24.15 24.66 25.18 26.32 33.29
0.08 11.80 13.44 14.33 14.82 15.37 15.81 16.36 16.98 18.40 21.76
B(? = 0.90) 0.01 37.30 37.66 37.89 37.98 38.08 38.15 38.24 38.36 38.62 39.01
0.08 18.86 18.91 18.96 18.97 18.99 18.99 19.01 19.02 19.07 19.13
E(? = 0.20) 0.01 13.15 14.28 14.84 15.13 15.43 15.68 15.97 16.27 16.96 18.38
0.055 7.64 8.69 9.29 9.59 9.89 10.20 10.54 10.89 11.71 13.51
35
Table 5: This table shows the holding period return di?erential based on investment horizons of 3-12
months. The formation period is one year. We use 5000 di?erent paths where each path denotes one stock.
At the end we sort the observed return into ten equally weighted portfolios. The average di?erence between
the top and bottom decile is reported in this table. In all cases, X
0
=
¯
X.
? 3 6 9 12
A 0.15 2.78 5.52 7.87 10.35
0.25 1.78 3.49 5.05 6.65
0.30 1.58 3.11 4.46 5.87
0.40 1.24 2.46 3.54 4.66
0.70 0.46 0.89 1.29 1.82
0.90 0.09 0.18 0.26 0.37
D 0.15 3.21 3.25 3.28 3.20
0.20 2.63 2.65 2.71 2.63
0.30 2.25 2.27 2.32 2.23
0.40 1.85 1.85 1.91 1.85
0.70 0.72 0.67 0.73 0.72
0.90 0.11 0.12 0.14 0.13
E 0.15 2.38 4.75 6.75 8.98
0.20 1.48 2.96 4.23 5.62
0.30 1.10 2.18 3.13 4.14
0.40 0.87 1.72 2.48 3.27
0.70 0.33 0.65 0.95 1.30
0.90 0.06 0.13 0.18 0.26
G 0.15 1.41 2.78 4.00 3.96
0.20 1.26 2.47 3.57 3.49
0.30 1.04 2.03 2.97 2.83
0.40 0.83 1.63 2.36 2.27
0.70 0.31 0.61 0.86 0.87
0.90 0.06 0.12 0.19 0.18
36
6 Appendix
Preferences
In continuous time, the recursive utility function takes the form of stochastic di?erential
utility. The stochastic di?erential utility U : L
2
?R is a mapping from a square integrable
space to the real line and is de?ned by two primitive functions: (f, A) where f : R
+
×R ? R
and A : R ? R. For any consumption process C ? L
2
, the utility process J is the unique
SDE
dJ
t
=
_
?f(C
t
, J
t
) ?
1
2
A(J
t
)?
v
?
v
_
dt +?
v
dB
t
with boundary condition J
T
= 0. The di?erent components are - J
t
, a continuation utility
for the agent given consumption C
t
, f(C
t
, J
t
) is an ordinal map of date t
s consumption
and continuation utility, and A(J
t
) is a measure of local risk-aversion. If given an initial
consumption C
t
and as long as the solution of the above SDE is well-de?ned, the utility at
time t is de?ned as U(C
t
) = J
t
. Under certain conditions, the above SDE is well-de?ned
and hence the utility exists. The function U is monotonic and risk-averse for A ? 0. Given
an f and two functions A
?
and A, let U
?
and U be the two utilities corresponding to the
aggregators (f, A
?
) and (f, A). If A
?
? A, then U
?
is more risk-averse than U, i.e. any con-
sumption stream rejected by a deterministic consumption path by one will also be rejected
by another. A convenient normalization that produces an ordinally equivalent utility func-
tion is achieved by setting A = 0, which means the above SDE solves E
t
[dJ
t
] +f(C, J) = 0
for normalized aggregator (f, 0). The normalization is useful because it produces a much
simpler Bellman equation to be solved than if A = 0. Fortunately, there exists a transfor-
mation from (
¯
f, A) to (f, 0) such that the utilities generated from both will be ordinally
equivalent. Further discussion of the aggregators and the normalization that leads to an
37
ordinally equivalent representation of the aggregators is given in Du?e and Epstein (1992).
Proof of Proposition 1: The Bellman equation in (10) can be written as
J
C
C?
_
X
t
+
1
2
(? ?1)?
2
D
_
+J
X
?(
¯
X ?X
t
)+
1
2
J
CC
C
2
?
2
?
2
D
+
1
2
J
XX
?
2
X
+J
XC
C??
D
?
x
? +f(C, J) = 0
The continuation utility J has a solution of the form
(1 ??)J = exp(u
0
ln C
t
+u
1
X
t
+u
2
)
Substituting it in and collecting terms, reduces the above equation to a system of ODEs
that can be solved recursively
u
0
= (1 ??)
u
1
=
(1 ??)?
? +?
u
2
=
(1 ??)?
?
_
(? ?1 ???)?
2
D
2
+
?
¯
X
? +?
+
(1 ??)?
? +?
_
?
2
x
2(? +?)
+?
D
?
x
?
__
Thus, the continuation utility function reduces to J(C
t
, X
t
) =
C
1??
t
1??
exp(u
1
X
t
+u
2
).
Proof of Proposition 2: The pricing kernel for stochastic di?erential utility can be
38
written as
d?
?
=
df
C
f
C
+f
J
dt
Using the above utility function, de?ne g = f
C
=
?(1??)J
C
and f
J
= ??(1 +u
1
X +u
2
). Use
Ito’s Lemma on g and (5) and (7) one can rewrite the pricing kernel as
d?
?
= ?r
f
t
dt ????
D
dW
1
+u
1
?
x
dW
2
r
f
t
= ?X
t
+u
1
???
D
?
x
? +?(u
2
+ 1) ?u
1
?
¯
X ?
1
2
???
2
D
(?? + 1) ?
1
2
?
2
x
u
2
1
Proof of Proposition 3: The ?rm stock price is given by
P
t
=
1
?
t
E
t
_
?
t
?
s
D
s
ds
=
1
?
t
_
?
t
E
t
?
s
D
s
ds
Applying Feynman-Kac, we know E
t
[?
s
D
s
] = f(?
t
D
t
, X
t
, s ?t). Applying Ito’s Lemma,
P
t
= D
t
G(X
t
)
where G(X
t
) =
_
?
t
exp(P
1
(s ?t)X
t
+P
2
(s ?t))ds. Making a change of variable ? = s ?t,
G(X
t
) =
_
?
0
exp(P
1
(?)X
t
+ P
2
(?))d?. P
1
(?) and P
2
(?) satisfy a set of ODEs that can be
solved recursively with initial conditions P
1
(0) = P
2
(0) = 0
P
1
(?) = ?(? ?1) ??P
1
(?)
39
P
2
(?) = u
1
?
¯
X +
1
2
?
2
D
(?? ?1) ??(u
2
+ 1) +P
1
(?)
_
?
¯
X +u
1
?
2
x
?(?? ?1)?
D
?
x
?
¸
+
1
2
_
(?? ?1)
2
?
2
D
+u
2
1
?
2
x
?2u
1
(?? ?1)?
D
?
x
? +P
2
1
(?)?
2
x
¸
The solution of P
1
(?) is given by P
1
(?) =
1??
?
(1 ?e
???
) and then that can be used to solve
for P
2
(?). Plugging in P
1
(?), P
2
(?) becomes
P
2
(?) = a? +b(e
???
?1) +c(1 ?e
?2??
)
where
a =
_
u
1
+
1 ??
?
__
¯
X? ?(?? ?1)?
D
?
x
? +
1
2
?
2
x
_
u
1
+
1 ??
?
__
+
1
2
?
2
D
??(?? ?1)
??(u
2
+ 1)
b =
1 ??
?
2
_
?
2
x
(1 ??)
?
+?
¯
X +u
1
?
2
x
?(?? ?1)?
D
?
x
?
_
c =
?
2
x
4
(1 ??)
2
?
3
The transversality condition holds for a < 0, which holds for believable parameter values.
Applying Ito’s lemma to P
t
= D
t
G(X
t
), we derive the process for cumulative excess return
dR
t
=
Dt+dP
Pt
?r
f
t
dR
t
= µ
R
t
dt +?
D
dW
1
+
G
X
G
?
x
dW
2
dµ
R
t
= (·)dt +
_
G
X
G
_
X
(???
D
?
x
? ?u
1
?
2
x
)?
x
dW
2
where µ
R
t
= (???
2
D
? u
1
?
D
?
x
?) +
G
X
G
(???
D
?
x
? ? u
1
?
2
x
). The latter is derived from the
equilibrium argument that the expected excess return is given by µ
R
t
= ?Cov
t
_
d?t
?t
, dP
t
_
.
40
doc_681110601.pdf