Momentum in an ICAPM World

December 30, 2019

ABSTRACT

I empirically test the ability of the ICAPM to explain the momentum anomaly in equity markets.

Using the Epstein and Zin formulation of Campbell and Vuolteenaho (2004) and Campbell et

al (2017) as inspiration, this paper derives an ICAPM representation of the Stochastic Discount

Factor (SDF) which implies that the momentum risk premium has three main components: cash

flow, discount rate and volatility risk.Winner stocks are those that have higher cash flow betas and

lower discount rate and volatility betas whereas loser stocks are those for whom the converse is

true. Thus the momentum strategy can be seen as a bet against a stocks cash flow, discount rate

and volatility betas. This no- arbitrage model of momentum fits the data reasonably well, though

the central prediction that winner stocks have higher cash flow betas and lower discount rate and

volatility betas is not empirically supported.

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1. Introduction

The momentum effect, the tendency of short run returns to predict future returns, has mystified

the asset pricing anomalies literature for over three decades. Traditional factor models such as the

CAPM or the Fama-French 3 factor model (FF) cannot explain the statistically significant alphas

attached to the momentum strategy (Fama and French, 1996). Introducing ad-hoc macroeconomic

variables into this multifactor framework adds little predictive power for these factor models as

momentum alphas are still largely unexplained (Chen, Roll, and Ross 1986; Li, Vassalou, and Xing

2006; Liu and Zhang 2008).

Whilst it is tempting to do so, one should not interpret the failure of traditional factor models

to price momentum as evidence against the efficient markets hypothesis. Rather, the failure can

largely be explained by a misspecification in the factor structure. Traditional reduced form factor

models such as the CAPM and FF assume that equity pricing is symmetric, that is asset prices

respond identically to positive and negative shocks of the same magnitude. Yet recent evidence

suggests that the time varying equity risk premium is largely asymmetric, with downside risk priced

more heavily by investors than upside risk. In fact, Lettau, Maggiori and Weber (2014) find that the

Downside Risk CAPM (DR CAPM), which allows time variation in market betas, has far greater

predictive power than the CAPM or FF for the cross-section of equity returns, especially in bearish

markets. This asymmetric element to equity pricing is especially important for momentum, given

the documented crash risk in momentum strategy returns across bad states of nature (Daniel and

Moskowitz, 2016).

This paper seeks to address this asymmetric pricing issue by considering an ICAPM SDF model

based on Epstein-Zin (EZ) recursive preferences which assumes that the utility function is non-

separable across states of nature. This allows news about future bad states to influence investor’s

utility today and consequently asset prices. Using EZ preferences, our stylised no-arbitrage model

is able to identify three main determinants of the SDF: cash flow, discount rate and volatility risk.

The SDF positively covaries with cash flow risk whereas discount rate and volatility risk attract a

negative risk premium.

This implies that momentum is a bet on the cash flow, discount rate and volatility betas

of a stock . ’Winner’ stocks are those with higher cash flow betas and lower discount rate and

volatility betas whereas ’loser’ stocks are those for whom the converse is true. I follow Campbell

and Vuolteenaho (2004) in making the assumption that cash flow and volatility risk attract greater

risk prices than discount rate risk. I make this assumption because discount rate risk is transitory:

higher discount rates today may reduce contemporaneous wealth but will improve investment op-

portunities in the future. On the other hand, shocks to cash flows and volatility have permanent

effects on investor wealth meaning that risk averse investors demand greater compensation for these

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components of the long run risk equation (Campbell and Vuolteenaho, 2004).

Intuitively, the ’winner’ portfolio loads heavily on cash flow risk because it is largely comprised

of small and value stocks which are highly sensitive to news about future cash flows (Campbell

and Vuolteenaho, 2004). This is largely because small and value stocks generate cash flows in the

immediate future and are thus less sensitive to discount rates, leaving cash flow risk as the main

source of variation. In contrast, the ’loser’ portfolio loads heavily on discount rate risk because

large and growth stocks generate more distant cash flows that are more sensitive to market-wide

discount rates (Campbell and Vuolteenaho, 2004). Thus momentum is long cash flow risk but short

discount rate risk.

With regards to volatility risk, the momentum strategy can be seen as a strategy that is short

volatility because of the option-like behaviour of the losers. In bear markets, the loser portfolio

becomes highly levered and in accordance to Merton (1974) behaves like a deep out of the money

call option on the market portfolio. When markets correct, the steep convexity of the loser portfolio

results in the losers significantly outperforming the winners, initiating a ’momentum crash’ (Daniel

and Moskowitz, 2016). Thus, momentum is a bet against volatility with the volatility risk price

being negative in our framework.

I find that this long run risk formulation of momentum anomaly has large explanatory power for

the returns to ten momentum sorted portfolios. The adjusted R2 ranges between 60-90% and the

signs of the cash flow, discount rate and volatility risk factors are highly signiifcant at a 1% level

and conform to the basic predictions of our model, with cash flow risk attracting a positive risk pre-

mium and discount rate and volatility risk attracting a negative risk premium. However I find that

the magnitudes of the coefficients are not consistent with our model: the loser portfolio attracts

higher cash flow, discount rate and volatility betas relative to the winner portfolio. One possible

explanation that may explain the failure of our model predictions may be that higher moments

of cash flow, discount rate and volatility risk are important in pricing momentum returns. Kalev,

Saxena and Zolotoy (2017) find that this is certainly the case for the size and the value anomalies,

indicating that an avenue for future research would be to extend this framework to include these

higher moments. Given recent evidence that higher moments of the return distribution do matter

for momentum (Barroso and Santa Clara, 2014), there is great potential for such an avenue of

research to yield critical results for the pricing of the momentum anomaly.

The remainder of this paper is organised as follows. Sections 2 and 3 introduce my theoretical

model and discuss the key implications for the momentum risk premium. Section 4 discusses my

empirical methodology and estimation results. Section 5 concludes this paper.

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2. Model

My theoretical model assumes a complete markets framework where a unique stochastic discount

factor (SDF) mt+1 exists that prices all assets in the universe. Specifically, I use the SDF framework

of Campbell et al (2017) which assumes that the representative agent has Epstein and Zin (EZ)

preferences that takes the following form:

mt+1 = θ ln δ − θ

ψ∆ct+1 + (θ − 1)rt+1 (1)

∆ct+1 refers to the log value of reinvested wealth, rt+1 refers to the return on the aggregate wealth

portfolio Wt which is the market value of the consumption stream owned by the representative

agent. δ refers to the subjective time preference parameter. ψ refers to the elasticity of intertem-

poral substitution (EIS) and θ= 1−γ1− 1

ψ

where γ is the relative risk aversion parameter.

By introducing separate parameters for relative risk aversion and EIS, long run consumption risk is

allowed to influence investor’s contemporaneous utility as per EZ preferences through the parame-

ter rt+1. So long as γ< 1ψ , the investor is concerned about both short run and long run consumption

growth risk.

To avoid empirical issues associated with consumption data, Campbell (1993) demonstrates that

one can apply a log-linear transformation of equation 1 to substitute out consumption and redefine

the SDF in terms of news to future cash flows Nc,t+1, news to future discount rates Nd,t+1 and news

to future volatility Nv,t+1. Campbell and Vuolteenaho (2004) and Campbell et al (2017) use this

transformation alongside the Campbell-Shiller return decomposition to derive a process for shocks

to mt+1 as follows:

mt+1 − Etmt+1 = −γNc,t+1 +Nd,t+1 + φNv,t+1 (2)

where:

Nc,t+1 = (Et+1 − Et)

∞∑s=1

ρsct+1+s

Nd,t+1 = (Et+1 − Et)∞∑s=1

ρsrt+1+s

Nv,t+1 = (Et+1 − Et)

∞∑s=1

ρsvart+s(mt+1+srt+1+s) (3)

In this stylised framework, the risk price for Nc,t+1 is γ times greater than the risk price for Nd,t+1.

This follows from the fact that cash flow shocks permanently reduce wealth whereas positive dis-

count rate shocks are transitory and are partially offset by higher future returns (Campbell and

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Vuolteenaho, 2004). In a similar vein, the risk price for Nv,t+1 is φ times greater than the risk price

for Nd,t+1. The parameter ρ is a discount coefficient.

Substituting equation 2 into equation 1 yields a stochastic process for the expected log SDF Etmt+1 :

Etmt+1 = κ+ Et[−γ∞∑s=0

ρs∆ct+1+s +

∞∑s=0

ρsrt+1+s + φ

∞∑s=1

ρsvart+s(mt+1+srt+1+s)] (4)

Finally I arrive at my specification for mt+1 by substituting equation 4 into equation 2 which yields

a stochastic process that depends on both expectations and shocks to cash flows, discount rates

and volatility:

mt+1 = −γNc,t+1 +Nd,t+1 + φNv,t+1 + κ − γEc,t + Ed,t + φEv,t (5)

where:

Ec,t = Et[∞∑s=1

ρsct+1+s]

Ed,t = Et[

∞∑s=1

ρsrt+1+s]

Ev,t = Et[∞∑s=1

ρsvart+s(mt+1+srt+1+s)] (6)

Equation 5 represents our stylised theoretical model of the SDF. In a world of no-arbitrage, mt+1

defined by equation 5 is such that the Asset Pricing Euler Equation must hold for any asset i:

Et(mt+1ri,t+1) = 1 (7)

3. Model Implications for Risk Premia

Equation 5 implies that in a world where investors have EZ preferences, there are three main

drivers of the SDF mt+1: cash flow, discount rate and volatility risk. γ and φ represent the risk

loadings that the SDF attaches to cash flow and volatility risk, whereas the loading for the discount

rate component is normalised to 1. Thus, the SDF attaches greater risk loading to cash flow and

volatility risk by a factor of γ and φ respectively.

Cash flow risk components Nc,t+1 and Ec,t attract a negative coefficient -γ in our SDF model. This

is because when there future cash flows are expected to be low, consumption today decreases which

results in a lower intertemporal marginal rate of substitution (IMRS) and consequently a higher

SDF mt+1. This implies that assets that are positively correlated with Nc,t+1 and Ec,t are

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risky and hence demand a risk premium .

Conversely, volatility risk components Nv,t+1 and Ev,t attract a positive coefficient φ in the SDF

model. This represents the volatility risk premium: Risk averse agents have a preference for con-

sumption smoothing: that is they would like to moderate their consumption over multiple periods.

This implies that risk averse investors have a preference for assets that are expected to have lower

future volatility, enabling consumption smoothing. Thus the volatility terms attract a positive risk

price φ. This implies that assets that are negatively correlated with Nv,t+1 and Ev,t are

risky and hence demand a risk premium

Finally, the discount rate components Nd,t+1 and Ed,t also attracts a positive coefficient, albeit it

is smaller than the volatility components by a factor of φ. This is because discount rate risk is

transitory: whilst higher discount rates will reduce wealth today, they improve investment oppor-

tunities for the future allowing for a partial offset for that contemporaneous wealth loss. If long

run risk concerns dominate an investors risk aversion, as they should under an EZ framework, then

assets that positively covary with discount rate risk are valuable. In other words assets that are

negatively correlated with Nd,t+1 and Ed,t are risky and hence demand a risk premium .

These 3 observations of our stylised model lead us to Propositions 1-3:

Proposition 1: If a stock has a higher cash flow beta, that is the stock positively co-

varies with Nc,t+1 and Ec,t, they should have higher returns on average than stocks for

which the converse is true

Proposition 2: If a stock has a higher discount rate beta, that is the stock positively

covaries with Nd,t+1 and Ed,t, they should have lower returns on average than stocks

for which the converse is true

Proposition 3: If a stock has a higher volatility beta, that is the stock positively co-

varies with Nv,t+1 and Ev,t, they should have lower returns on average than stocks for

which the converse is true

I discuss the implications of this model for the momentum anomaly in the next section.

3.1 Model Implications for Momentum

Since propositions 1-3 imply that stocks with higher cash flow betas and lower discount rate and

lower volatility betas outperform on average, we can motivate the following propositions about the

momentum portfolio:

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Proposition 4: If Proposition 1 holds, momentum returns are increasing in cash

flow beta differentials between winner and loser stocks.

Proposition 5: If Proposition 2 holds, momentum returns are decreasing in dis-

count rate beta differentials between winner and loser stocks.

Proposition 6: If Proposition 3 holds, momentum returns are decreasing in volatil-

ity beta differentials between winner and loser stocks.

Propositions 4-6 imply that the momentum strategy invests in stocks that have higher cash flow

betas and lower discount rate and volatility betas and short stocks for whom the converse is true.

Thus momentum is a bet on a stock’s cash flow, discount rate and volatility betas. I will empirically

test our theory of momentum using 10 US momentum sorted portfolios. I outline my empirical

methodology next.

3.2 Empirical Framework

To test the prediction of our model that winner stocks have higher cash flow betas and lower dis-

count rate and volatility betas than loser stocks, it is necessary to construct estimates of these betas

for both winner and loser stocks. I do this using three main measurements. Firstly, I estimate the

betas using first differences multivariate regressions for the 10 momentum sorted portfolios. Sec-

ondly, I estimate a lag adjusted beta estimate that accounts for stale prices. Finally, I estimate the

betas by GMM system, using moment conditions from the Asset Pricing Euler Equation derived

under conditional lognormality.

Multivariate Regression Betas

The multivariate regression approach is the most basic approach to estimating our betas of interest.

I run the follow regression for all 10 momentum sorted portfolios:

∆ri,t+1 = α+βi,NC∆Nc,t+1+βi,ND∆Nd,t+1+βi,NV ∆Nv,t+1+βi,EC∆Ec,t+βi,ED∆Ed,t+βi,EV ∆Ev,t

(8)

Here ri,t+1 is the monthly return on momentum portfolio i at month t+ 1. ∆ is the first difference

operator. Using the coefficients as our estimates of the beta, it is easy to ascertain whether or not

the winner portfolio betas are larger than the loser portfolios.

I use a first differences approach for regression to control for persistence in the news terms Nc,t+1,

Nd,t+1 and Nv,t+1. As we will detail shortly, we estimate these terms using a VAR system com-

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prising of highly persistent state variables. Thus these news terms are also likely to be persistent.

Hence I take first differences to ensure an estimation model that has valid statistical inference. I

then scale the betas by a factor of 100 to account for the scale differences between Nc,t+1, Nd,t+1

and Nv,t+1 and the dependent variable ri,t+1.

Lag Adjusted Beta

I define our lag adjusted betas in accordance with Campbell and Vuolteenaho (2004). The shock

betas are defined as follows:

βi,ND =ˆCov(ri,t, NDt)

ˆV ar(NCt, NDt)+

ˆCov(ri,t, NDt−1)

ˆV ar(NCt, NDt)

βi,NC =ˆCov(ri,t, NCt)

ˆV ar(NCt, NDt)+

ˆCov(ri,t, NCt−1)

ˆV ar(NCt, NDt)

βi,NV =ˆCov(ri,t, NV t)

ˆV ar(NCt, NDt)+

ˆCov(ri,t, NV t−1)

ˆV ar(NCt, NDt)

(9)

Similarly I define the expectation betas as:

βi,ED =ˆCov(ri,t, EDt−1)

ˆV ar(NCt, NDt)

βi,EC =ˆCov(ri,t, ECt−1)

ˆV ar(NCt, NDt)

βi,EV =ˆCov(ri,t, EV t−1)

ˆV ar(NCt, NDt)

(10)

These beta measurements differ from standard definitions of beta because of very special features

of the data. We are using data from the 10 US momentum sorted portfolios available from the

Kenneth French data library spans from January 1929 to December 2010. Since some of the stocks

in these momentum portfolios were not traded frequently in the early parts of the sample, it is

possible that the presence of stale prices may give the misleading perception that our news terms

are apriori causing portfolio returns (Scholes and Williams 1977; Dimson 1979). To remove this

concern we follow Campbell and Vuolteenaho (2004) in adding a lagged component to the beta

measurement of βi,ND, βi,NC and βi,NV . Since our expectation terms for time t are already condi-

tional on time t− 1, we need not make a similar adjustment for βi,ED, βi,EC and βi,EV .

Finally Campbell and Vuolteenaho (2004) show that under mild assumptions, ˆV ar(NCF,t,−NDR,t)

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is equivalent to ˆV ar(reM,t − Et−1reM,t), which is the denominator in standard definitions of beta.

Thus the denominator of the beta estimates in equations 9 and 10 are identical to the standard

CAPM formulation.

GMM

The derivation of the empirical GMM moment condition comes from Campbell et al (2017). I start

from the Asset Pricing Euler Equation which is defined as follows:

Et(Mt+1Ri,t+1) = 1 (11)

Here Mt+1 is the Stochastic Discount Factor (SDF) and Ri,t+1 is the return on asset i at time

t+1. Assuming that Mt+1 and Ri,t+1 are jointly conditionally lognormal, we have that Mt+1Ri,t+1

is also conditionally lognormal. Using the property that for any lognormal random variable X

log(E(X))=E(logX) + 12var(logX), we have that:

Et(mt+1 + ri,t+1) +1

2V art(mt+1 + ri,t+1) = 0 (12)

mt+1 and rei,t+1 are the log SDF and log returns respectively. We also know by Jensen’s inequality:

Et(ri,t+1) +1

2V art(ri,t+1) ≈ (EtRi,t+1 − 1) (13)

Substituting equation 15 into 14, we arrive at the moment condition for any asset i:

Et(mt+1) + Et(Ri,t+1) − 1 + Et(mt+1ri,t+1) − Et(mt+1)Et(ri,t+1) (14)

If this moment condition holds for asset i, it must also hold for a reference asset j:

Et(mt+1) + Et(Rj,t+1) − 1 + Et(mt+1rj,t+1) − Et(mt+1)Et(rj,t+1) (15)

Subtracting 17 from 16 leads us to our general GMM moment condition:

Et[Ri,t+1 −Rj,t+1 − (ri,t+1 − rj,t+1)(mt+1 − Etmt+1)] (16)

Plugging equation 2 into equation 17 yields our testable GMM moment condition:

Et[Ri,t+1 −Rj,t+1 − (ri,t+1 − rj,t+1)(−γNc,t+1 +Nd,t+1 + φNv,t+1)] (17)

Again, the risk price for Nd,t+1 is normalised to 1. Using equation 17 as our final moment condition,

I use GMM to estimate our beta estimates γ, φ and τ . I follow Campbell et al (2017) in using the

CRSP value weighted index as the reference asset Rr,t+1. The 10 momentum sorted portfolios are

used as the test assets in the GMM estimation.

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4. News Estimation

In order to estimate the betas using the approaches outlined in the previous section, the news terms

Nc,t+1, Nd,t+1, Nv,t+1, Ec,t, Ed,t and Ev,t must be estimated. This is done via a VAR system that

I detail below.

4.1. VAR Methodology

I follow Campbell et al (2014) in using the following VAR system to estimate the news terms:

zt+1 = α + Γ(zt − z) + σtut+1 (18)

zt+1 is an m x 1 vector of state variables with excess market returns rt+1 as the first element. α and

ut+1 are m x 1 vectors of constant parameters and residuals respectively. z and Γ are m x 1 the m

x m vector of constant parameter estimates. Finally σt is the conditional variance of market returns.

After estimating the parameters of the VAR system, I linearly map Nc,t+1, Nd,t+1 and Nv,t+1 to

the shock vector ut. Our linear mapping process is captured by the expressions below:

Nc,t+1 = (e1T + e1Tλσt)ut+1

Nd,t+1 = e1Tλσtut+1

Nv,t+1 = e2Tλσtut+1 (19)

e1 is an m×m vector with 1 as the first element and 0 for all other elements. e2 is an m x 1 vector

with 1 tied to the element linked with volatility and 0 for all other elements. The term e1T + e1Tλ

represents the contribution that shocks to the state variables play in driving Nc,t+1. e1Tλ measures

the contribution of state variable shocks to Nd,t+1. Finally, e2λ measures the contribution of state

variable shocks to Nv,t+1. We multiply each of the news terms by the scalar term σt to account for

different error variances across the state variables in the VAR system. I outline how σt is estimated

later in this section.

λ is defined as:

λ = (ρΓ(I − ρΓ)−1 (20)

where ρ = 0.95112 and I is the identity matrix. Following Campbell and Vuolteenaho (2004) I use

this default value for ρ on the assumption that the annual consumption-wealth ratio is 5.2% for

the representative investor.

I adopt a similar linear mapping process to estimate Ec,t, Ed,t and Ev,t, mapping them to the lagged

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state variables:

Ec,t = (e1T + e1Tλσt)Γzt

Ed,t = (e1Tλσt)Γzt

Ev,t = (e2T + e2Tλσt)Γzt (21)

4.2. VAR State Variables

To operationalise our VAR approach we must choose specific state variables to enter our state

vector. We follow Campbell et al (2017) in choosing the following six state variables: excess mar-

ket returns (ReM,t+1), equity volatility (σt), term spread (TSt+1), risk free rate (RFt+1), price to

earnings (PEt+1) and the small stock value spread (V St+1). I define these variables in accordance

with Campbell et al (2017). Sampling information about these variables are detailed below in table

1:

Table 1: State Variable Sample Information

State Variables Source Sample Period Interval

ReM,t+1 CRSP Database January 1929 - Dec 2010 Monthly

σt Kenneth French Data Library January 1929 - Dec 2010 MonthlyTSt+1 CRSP Database January 1929 - Dec 2010 MonthlyRFt+1 Kenneth French Data Library January 1929 - Dec 2010 MonthlyPEt+1 Robert Shiller Website January 1929 - Dec 2010 MonthlyVSt+1 Kenneth French Data Library January 1929 - Dec 2010 Monthly

In terms of the first state variable, ReM,t+1 is defined as the difference between the log returns on the

CRSP value weighted index and the log risk free rate. The second variable is the predicted market

variance conditional on all state variables at time t. The third variable is the US term spread which

is defined as the difference between the 10 year US government yield and the 3 month US treasury

bill rate. The fourth variable is the 3 month US Treasury Bill Rate. The fifth variable is the

logarithm of the smoothed SP500 price-earnings ratio available via Professor Bob Shiller’s website.

Finally the small stock value spread is the return differential between small value and small growth

stocks. This spread is computed using the return differential between the small growth and small

value portfolios available on the Kenneth French Data Library.

Our choice of these state variables is motivated by the need for variables that predict future cash

flows, discount rates and volatility. Firstly, financial variables such as the price-earnings ratio, small

stock value spread and market excess returns drive the cross-section of equity returns, indicating

that they hold important information for predicting future discount rates. Secondly, macroeco-

nomic variables such as the term spread and the risk free rate capture broad shifts in the economy

and hence are likely to be good predictors for future cash flows. Finally, historical volatility is

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needed to predict future patterns in expected volatility.

4.3. Predicting Conditional Market Variance

In order to calculate the second state variable-conditional market variance σt, I follow the approach

of Campbell et al (2017). First I construct realized monthly market variance RV ARt+1 from daily

market return data using the standard definition of sample variance:

RV ARt+1 =1

N − 1[

N∑t=1

rt − r] (22)

Define rt+1 as the log daily return on the CRSP value weighted index and r as the average monthly

return for month t+ 1. I then run the following predictive regression:

RV ARt+1 = α+ φ1RV ARt + φ2MRt + φ3TSt + φ4RFt + φ5PEt + φ6V St (23)

To control for the heteroskedasticity of innovations to our state variables, Equation 16 is estimated

as a Weighted Least Squares (WLS) regression where each observation is weighted by the inverse

of the realized variance RV AR−1t . The predictive regression results are presented below:

Table 2: Variance Predictive Regression

Variance Prediction ParametersConstant RVARt MRt TSt RFt PEt VSt R2

(1) (2) (3) (4) (5) (6) (7) (8)0.001 0.70 -0.001 0.001 0.203 0.001 0.002 29.83%(0.88) (16.86) (-0.27) (3.13) (3.37) (-0.19) (1.53)

Table 2 clearly demonstrates that volatility is highly predictable, a result that has been well doc-

umented in the literature (Glosten, Jagannathan and Runke 1993; Ang et al 2006 and Bekaert,

Hodrick, Zhang 2011). This predictability largely arises from the persistence of lagged volatility

but the term spread and the risk free rate are also informative about expected volatility.

Using the above estimates of α and φ1 − φ6, I estimate σt as:

σt+1 = α+ φ1RV ARt + φ2MRt + φ3TSt + φ4RFt + φ5PEt + φ6V St (24)

I use σt+1 as the second state variable in the VAR system.

4.4. State Variable Summary Statistics

I document the mean, standard deviations and correlation matrix of all six state variables on the

11

next page. The early sample is from January 1929 to December 1969 and the later sample is from

January 1970 to December 2010. Note that Market Returns, Conditional Variance, Term Spread

and the Risk Free rate are denoted in percentage units whereas the PE Ratio and the Value Spread

is quoted in log real units.

Table 3. State Variable Summary Statistics

This table shows the summary statistics for the six state variables in the VAR system. The sample period

is from January 1929 to December 2010.

Full Sample Early Sample Later Sample

Variables µ σ µ σ µ σ

ReM 0.41% 5.49% 0.49% 6.17% 0.32% 4.74%σt+1 0.90% 2.07% 0.72% 2.16% 1.26% 1.84%TS 7.55% 8.09% 6.72% 5.85% 8.38% 9.74%RF 0.29% 0.26% 0.14% 0.14% 0.45% 0.25%PE 2.90 0.37 2.77 0.32 3.02 0.37VS 1.64 0.36 1.80 0.43 1.47 0.15

Correlations ReM,t+1 σt+1 TSt+1 RFt+1 PEt+1 VSt+1

ReM,t+1 1 -0.074 0.049 0.057 -0.006 -0.031

σt+1 -0.074 1 0.145 0.1263 -0.1929 0.2116TSt+1 0.049 0.145 1 -0.482 -0.103 0.2827RFt+1 0.057 0.1263 -0.482 1 0.1319 -0.4874PEt+1 -0.006 -0.1929 -0.103 0.1319 1 -0.3529VSt+1 -0.031 0.2116 0.2827 -0.4874 -0.3529 1

ReM,t 0.1147 -0.088 0.042 -0.063 0.070 -0.033

σt -0.066 0.685 0.148 0.118 -0.199 0.222TSt 0.050 0.1407 0.942 -0.489 -0.100 0.278RFt -0.445 0.140 -0.462 0.975 0.129 -0.485PEt -0.091 -0.184 -0.109 0.137 0.992 -0.350VSt -0.023 0.207 0.289 -0.488 -0.354 0.991

The correlation matrix outlines a few noteworthy trends. Firstly, many of the state variables ex-

hibit a strong autocorrelation structure. This is especially true of the term spread, risk free rate,

price-earning ratios and the small value stock spread, whose autocorrelations exceed 90%. Market

returns are far less persistent, though autocorrelations are still significant at 11.45%.

Secondly, the correlation matrix highlights clear economic relationships between the state variables.

The positive correlation between the term spread and the small stock value spread indicates that

long term bonds underperform exactly when growth stock prices are high. One possible explana-

tion, As outlined by Campbell and Vuolteenaho (2004), is that long term bonds are highly sensitive

to changes in inflationary expectations, inducing negative correlation with high duration growth

assets. Another interesting result is that the small stock value spread and the price-earning ratios

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are negatively correlated. If we take the PE ratio as a proxy for the cost of capital, this indi-

cates that growth stocks are more exposed to external financing conditions than value stocks and

are hence more exposed to discount rate shocks. This point will remain relevant in our later analysis.

4.5. News Estimation

Table 4 on the next page documents the VAR estimation results as well as summary statistics on

the news terms. The results add further weight to the previous observation that our state variables

are highly persistent, with most variables exhibiting a lagged coefficient in excess of 90% . Expected

Volatility also has a large lagged coefficient of approximately 61%.

In addition, the VAR results indicate that many of the state variables have reasonable predictive

power for future market excess returns. The PE ratio negatively predicts future returns at a 1%

significance level, a result that is consistent with a long literature documenting the link between

the PE ratio and discount rates (Campbell and Shiller 1988b, 1998; Campbell 1987; Fama and

French 1989). Volatility also negatively predicts future returns, in line with recent evidence that

investors engage in volatility hedging behaviour (Ang et al 2006; Adrian and Rosenberg 2008;

Lettau, Ludvigson and Wachter 2008)

13

Table 4. VAR Parameter Estimates and News Terms

Variables Constant ReM,t+1 σt TSt RFt PEt VSt R2 F

ReM,t+1 0.082 -0.102 - 0.870 0.004 -0.54 -0.018 -0.011 2.70% 5.538

(-0.352) (3.232) (-2.023) (1.398) (-0.606) (-3.580) (-1.88)σt+1 0.001 -0.003 0.615 0.001 0.292 0.00 0.001 49.80% 1.578 x 103

(0.857) (-1.489) (24.504) (3.629) (5.535) (-1.490) (4.080)TSt+1 -0.0369 -0.001 1.604 0.934 -1.440 0.001 0.047 88.67% 1.280 x 104

(-0.352) (-0.003) (0.748) (73.837) (-0.319) (-0.032) (1.579)RFt+1 0.001 -0.001 0.002 -0.008 0.954 0.001 -0.002 95.22% 3.234 x 104

(1.39) (-1.357) (0.499) (-3.208) (102.08) (0.383) (-1.776)PEt+1 0.024 0.519 -0.243 0.002 0.611 0.991 -0.001 99.03% 1.666 x 104

(1.67) (24.389) (-0.848) (1.396) (1.010) (292.297) (-0.292)VSt+1 0.019 -0.001 1.267 -0.003 -1.479 0.001 0.985 98.24% 9.087 x 104

(1.019) (-0.326) (3.340) (-1.507) (-1.856) (0.120) (183.919)

Summary Statistics µ σ 25% Mean 75% Skewness Kurtosis

NCt+1 0.00% 0.03% - 0.01% 0.00% 0.01% 0.004 19.827NDt+1 0.00% 0.57% -0.02% 0.00% 0.02% -0.01 33.889NVt+1 0.00% 0.19% 0.00% 0.00% 0.00% 0.43 20.903ECt -2.60% 1.29% -2.96% -2.19% -1.81% -2.745 14.795EDt -2.60% 1.28% -2.92% -2.20% -1.80% -2.703 14.369EVt+1 2.60% 0.20% 0.14% 0.20% -1.80% 2.900 14.749

News Correlation NCt+1 NDt+1 NVt+1 ECt EDt EVtNCt+1 1 0.046 - 0.609 0.001 -0.003 -0.016

(1.450) (-24.040) (0.030) (-0.100) (-0.490)NDt+1 0.046 1 -0.057 -0.02 -0.01 0.03

(1.450) (-1.780) (-0.590) (-0.420) (0.940)NVt+1 -0.609 -0.057 1 0.050 0.056 -0.043

(-24.040) (-1.780) (1.560) (1.750) (-1.330)ECt 0.001 -0.02 0.050 1 0.998 -0.897

(0.030) (-0.590) (1.560) (535.640) (-63.430)EDt -0.003 -0.01 0.056 0.998 1 -0.895

(-0.100) (-0.420) (1.750) (535.640) (-62.590)EVt+1 -0.016 0.03 -0.043 -0.897 -0.895 1

(-0.490) (0.940) (-1.330) (-63.430) (-62.590)

14

NC

Tim

e

NC

19401960

19802000

−0.003 −0.001 0.001 0.003

ND

Tim

e

ND

19401960

19802000

−0.006 −0.002 0.002 0.004

NV

Tim

e

NV

19401960

19802000

−0.0015 −0.0005 0.0005 0.0015

EC

Tim

e

EC

19401960

19802000

−0.12 −0.10 −0.08 −0.06 −0.04 −0.02

ED

Tim

e

ED

19401960

19802000

−0.12 −0.10 −0.08 −0.06 −0.04 −0.02

EV

Tim

e

EV

19401960

19802000

0.000 0.005 0.010 0.015

Evolution of N

ews Term

s

The time series graphs of the news terms indicates that two very interesting patterns. Firstly, expecta-

tion terms ECt and EDt exhibit right skewed distributions that experience large negative asymmetric

shocks centred around 1987, 2001 and 2007-2008. EVt on the other hand has a left skewed distribu-

tion with large positive asymmetric shocks occuring around the same dates. These trends also have an

intuitive explanation: investor expectations about the short term future are on average negative and

subject to large negative shocks because risk aversion implies that investors apply greater weight to bad

news than to good news of the same magnitude. Similarly expected volatility is on average positive with

large positive shocks given that negative shocks to cash flows and discount rates have a larger asym-

metric effect on volatility than positive shocks (Black 1976). In addition major systemic events such as

Black Monday (1987), the Technology Bubble (2000-2001) and the Global Financial Crisis (2007-2008)

dramatically shift these short term expectations, resulting in large volatility in the short term which

accounts for the opposite directions of the asymmetric shocks for ECt and EDt and EVt. Thus systemic

risk events dramatically shift short term expectations about cash flows, discount rates and volatility.

Secondly the shock terms NCt+1, NDt+1 and NVt+1 are largely centred around zero, though the dis-

tributions exhibit significant kurtosis as documented by table 4. This is captured by large spikes in

the time series for each of these terms at the beginning (1930s) and end (2007-2008) of the samples.

Since these time intervals correspond with the Great Depression and the recent Global Financial Crisis,

a natural interpretation is that systemic risk events dramatically shift investor expectations about long

term cash flows, discount rates and volatility. This indicates that asset prices are not only affected by

short term expectations but also long run expectations about the future which is entirely consistent with

the Epstein and Zin utility framework that my model is seeking to test.

4.6. Multivariate Regression Estimation

Adding further credence to the relevance of the long run risk framework are the results of the multivari-

ate beta regressions which are documented in table 5. Table 5 documents the multivariate regressions

for all 10 momentum sorted portfolios available on the Kenneth French Data library. Portfolio I is the

’loser’ portfolio and contains the stocks whose previous 12 month market performance ranked in the

bottom 10% of stocks across the NASDAQ and NYSE at any point in time. Portfolio X is the ’winner’

portfolio and represents the top 10% of stock performers across the NASDAQ and NYSE at any point in

time. The reported R2 is the implied measure defined by equation 10. Estimation results are reported

for both the full sample and a later sub-sample from January 1970-December 2010.

The regression results indicate four main findings. Firstly, The no-arbitrage model of momentum outlined

in this paper does a reasonably good job at explaining the cross-section of momentum sorted portfolios.

For all portfolios, the R2 ranges between 50-75% with all the alphas statistically insignificant at the 10%

level. The plot of the predicted momentum return using the regression coefficients against the realised

momentum returns also paint a similar story with the best fitted line for each portfolio approximating

a 45 degree line.

16

Table 5: Panel A: Multivariate Regressions

Model α ∆Nc ∆Nd ∆Nv ∆Ec ∆Ed ∆Ev R2 N

Panel A: 1929-2010

I -0.004 0.926∗∗∗ −1.166∗∗∗ 0.077 0.102∗∗∗ −0.060∗ 0.328∗∗∗ 60.60% 898(-1.59) (15.33) (-32.35) (0.52) (3.18) (-1.94) (4.72)

II −0.002 0.902∗∗∗ −0.988∗∗∗ 0.061 0.030 −0.0134 0.109∗∗ 66.55% 898(-1.17) (19.14) (-35.11) (0.53) (1.20) (-0.55) (2.00)

III -0.002 0.762∗∗∗ −0.898∗∗∗ −0.090% 0.024 0.001 0.179∗∗∗ 70.39% 898(-0.87) (19.46) (-38.46) (-0.95) (1.16) (0.06) (3.97)

IV 0.000 0.717∗∗∗ −0.781∗∗∗ −0.021 0.016 0.011 0.213∗∗∗ 66.45% 898(-0.49) (18.84) (-34.40) (-0.22) (0.81) (0.55) (4.86)

V 0.00 0.650∗∗∗ −0.790∗∗∗ −0.124∗ −0.002 0.020 0.131∗∗∗ 74.25% 898(-0.05) (20.70) (-42.13) (-1.62) (-0.11) (1.22) (4.86)

VI 0.00 0.677∗∗∗ −0.751∗∗∗ −0.068 −0.002 0.024 0.131∗∗∗ 73.46% 898(0.31) (21.74) (-40.41) (-0.90) (-0.09) (1.48) (4.66)

VII 0.00 0.638∗∗∗ −0.701∗∗∗ −0.101 −0.010 0.046∗∗∗ 0.265∗∗∗ 71.57% 898(0.41) (20.39) (-37.58) (-1.32) (-0.63) (2.85) (7.36)

VIII 0.0012 0.593∗∗∗ −0.653∗∗∗ −0.083 0.002 0.021 0.168∗∗∗ 65.13% 898(0.82) (17.91) (-33.01) (-1.02) (0.11) (1.21) (4.41)

IX 0.001 0.582∗∗∗ −0.704∗∗∗ −0.077 −0.022 0.040∗∗ 0.141∗∗∗ 65.09% 898(0.79) (16.53) (-33.54) (-0.90) (-1.19) (2.23) (4.41)

X 0.002 0.569∗∗∗ −0.693∗∗∗ −0.089 −0.038 0.052∗∗ 0.141∗ 49.83% 898(0.93) (11.93) (-24.34) (-0.77) (-1.49) (2.13) (1.93)

Panel B: 1970-2010

I -0.005 1.023∗∗∗ −1.296∗∗∗ 0.354∗ 0.271∗∗∗ −0.247∗∗∗ 0.193∗ 49.90% 451(-1.35) (9.96) (-19.13) (1.78) (4.02) (-3.80) (1.73)

II -0.003 0.894∗∗∗ −1.071∗∗∗ 0.141∗∗∗ −0.136∗∗∗ −0.013 0.027 58.15% 451(-1.11) (11.92) (-21.65) (1.21) (2.86) (-2.87) (0.33)

III -0.002 0.708∗∗∗ −0.919∗∗∗ -0.037 0.080∗ −0.072∗ 0.039% 59.83% 451(-0.88) (11.19) (-22.01) (-0.30) (1.93) (-1.79) (0.56)

IV 0.00 0.728∗∗∗ −0.847∗∗∗ 0.056 0.088∗∗∗ −0.074∗∗ 0.105∗ 64.18% 451(-0.52) (13.53) (-23.86) (0.54) (2.50) (-2.18) (1.80)

V 0.00 0.646∗∗∗ −0.816∗∗∗ −0.050∗ 0.078∗∗∗ −0.072∗∗ 0.034 67.98% 451(-0.16) (13.68) (-26.21) (-0.54) (2.51) (-2.39) (0.66)

VI 0.00 0.729∗∗∗ −0.834∗∗∗ 0.039 0.059∗∗ −0.050∗ 0.070 71.19% 451(-0.05) (15.85) (-27.51) (0.43) (1.96) (-1.72) (1.41)

VII 0.00 0.724∗∗∗ −0.828∗∗∗ −0.025 0.074∗∗ −0.055∗ 0.135 68.75% 451(-0.29) (14.98) (-25.98) (0.26) (2.33) (-1.81) (2.57)

VIII 0.00 0.690∗∗∗ −0.828∗∗∗ 0.007 0.063∗∗% −0.058∗∗ 0.026 69.63% 451(0.13) (14.76) (-26.85) (0.07) (2.06) (-1.97) (4.41)

IX 0.00 0.714∗∗∗ −0.891∗∗∗ −0.010 −0.060∗ −0.056∗ 0.034 67.84% 451(0.21) (13.66) (-25.85) (-0.10) (1.76) (-1.69) (0.61)

X 0.00 0.816∗∗∗ −1.053∗∗∗ 0.057 0.086∗ −0.081∗∗ 0.048∗ 58.33% 451(0.19) (10.95) (-21.44) (0.39) (1.76) (-1.72) (0.19)

17

−0.4

0.00.2

0.40.6

0.8

−0.5 0.0 0.5

M1

PM1

−0.4

0.00.2

0.40.6

0.8

−0.6 −0.2 0.2 0.6

M2

PM2

−0.2

0.00.2

0.40.6

−0.4 0.0 0.4

M3

PM3

−0.2

0.00.2

0.40.6

−0.4 0.0 0.2 0.4

M4

PM4

−0.2

0.00.2

0.40.6

−0.4 0.0 0.2 0.4

M5

PM5

−0.2

0.00.2

0.4

−0.4 0.0 0.2 0.4

M6

PM6

−0.3

−0.1

0.10.2

0.3

−0.4 0.0 0.2 0.4

M7

PM7

−0.2

−0.1

0.00.1

0.20.3

−0.4 −0.2 0.0 0.2 0.4

M8

PM8

−0.2

0.00.1

0.20.3

−0.4 0.0 0.2 0.4

M9

PM9

−0.3

−0.2

−0.1

0.00.1

0.20.3

−0.4 −0.2 0.0 0.2 0.4

M10

PM10P

redicted vs Realised P

lot

Secondly there is a clear declining relationship between the R2 and the decile of our momentum portfolio,

with the R2 maximised for the losers and minimized for the winners. A similar phenomenon was also

evident in the R2 of our multivariate regressions. This indicates that our momentum model has greater

predictive power for the losers than the winners.

Thirdly, the signs of the betas largely conform to the predictions of the model. βNC is positive whereas

βND is negative, confirming that cash flow (discount rate) risk attracts a positive (negative) risk pre-

mium. The expectation terms for cash flow and discount rates βEC and βED are not statistically

significant in the full sample and their signs fluctuate across momentum portfolios. However this is

corrected in the later sample, as the coefficients for βEC and βED turn positive (negative) and are all

statistically significant at the 5-10% level. This result may make sense given well documented evidence

that many of the prevalent asset pricing anomalies today such as size, value and betting against the beta

are not observable in the data before the 1970s (Campbell, Vuolteenaho 2004). Thus state variables

that predict expected cash flows and discount rates are likely to have far less predictive power for the

early sample (1929-1970), explaining the fluctuating sign of the coefficients on Ec,t, Ed,t.

Fourthly, the signs of the volatility betas fluctuate across the momentum portfolios for both the full

sample and the later sample. For both samples, the sign of βNV fluctuates and is largely insignificant at

all significance levels. The sign of βEV is largely significant in the full sample but loses its significance in

the later sample. This indicates that βEV was capturing the noise in the earlier sample resulting from

the reduced predictability of asset returns discussed earlier. Improved predictability of asset returns in

the later sample has therefore largely removed the significance of βEV , indicating that both our volatility

terms Nv,t+1 and Ev,t do not predict momentum returns.

Finally and most significantly, the central prediction of our model, that the winner portfolio should

have higher cash flow betas, is inconsistent with our results. We find that for both samples, the loser

portfolio attracts higher betas than the winner portfolio. Table 8 on the next page documents this

formally with the beta differential between the loser portfolio (I) and the winner portfolio (X) ranging

between 0.3 - 0.5 for all the respective betas. This is robust even when we apply the lag adjusted beta

measurement defined by equations 11 and 12. There is a clear descending relationship between cash

flow, discount rate and volatility betas and the rank of the momentum portfolio, whereas our model

implies the exact opposite: winner stocks attract higher betas as compensation for cash flow, discount

rate and volatility risk. Whilst the magnitude of the R2 still suggests that our model has a lot to say

about the cross-section of momentum returns, the failure of the model prediction indicates that there

are some important ingredients missing from our model.

19

Table 6: Lag Adjusted Betas

Model βNC βND βNV βEC βED βEVPanel A: 1929-2010 (Full Sample)

I 0.557 −1.549 −0.164 3.044 −2.589 −0.078II 0.481 −1.297 −0.142 2.600 −2.268 −0.111III 0.414 −1.134 −0.140 2.358 −2.090 −0.158IV 0.378 −1.030 −0.122 2.283 −2.034 −0.145V 0.361 −0.981 −0.118 1.965 −1.733 −0.151VI 0.341 −0.954 −0.120 2.00 −1.785 −0.139VII 0.300 −0.881 −0.109 1.950 −1.783 −0.165VIII 0.292 −0.836 −0.105 1.953 −1.768 −0.200IX 0.263 −0.843 −0.097 1.757 −1.613 −0.155X 0.284 −0.856 −0.091 2.145 −1.974 −0.279

Panel B: 1929-1969 (Early Sample)

I 0.800 −1.799 −0.218 2.202 −1.637 0.153II 0.603 −1.524 −0.181 3.088 −2.664 −0.081III 0.532 −1.352 −0.177 2.949 −2.593 −0.177IV 0.466 −1.237 −0.146 2.605 −2.263 −0.133V 0.445 −1.176 −0.143 2.480 −2.172 −0.179VI 0.420 −1.119 −0.147 2.789 −2.500 −0.217VII 0.356 −1.029 −0.133 2.703 −2.470 −0.244VIII 0.347 −0.946 −0.130 2.524 −2.274 −0.264IX 0.308 −0.926 −0.126 2.300 −2.111 −0.228X 0.319 −0.843 −0.120 2.837 −2.643 −0.365

Panel B: 1970-2010 (Later Sample)

I 0.311 −1.154 −0.976 4.066 −3.787 −0.443II 0.285 −0.932 −0.080 1.889 −1.705 −0.161III 0.225 −0.779 −0.081 1.809 −1.685 −0.136IV 0.237 −0.692 −0.083 2.013 −1.912 −0.171V 0.227 −0.667 −0.079 1.123 −1.014 −0.107VI 0.213 −0.689 −0.076 0.771 −0.688 −0.016VII 0.210 −0.643 −0.071 0.755 −0.693 −0.038VIII 0.203 −0.659 −0.065 1.092 −1.012 −0.100IX 0.191 −0.710 −0.050 0.825 −0.747 −0.036X 0.228 −0.876 −0.045 1.044 −0.910 −0.141

20

Table 7 Panel A: GMM Betas (Full Sample)

Model (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Portfolio I II III IV V VI VII VIII IX X

βNC 0.919∗∗∗ 0.918∗∗∗ 0.670∗∗∗ 0.700∗∗∗ 0.778∗∗∗ 0.800∗∗∗ 0.743∗∗∗ 0.641∗∗∗ 0.761∗∗∗ 0.869∗∗∗

(4.80) (7.95) (4.64) (7.74) (7.51) (9.18) (9.40) (4.94) (7.25) (3.96)βND −1.14∗∗∗ −0.924∗∗∗ −0.772∗∗∗ −0.650∗∗∗ −0.690∗∗∗ −0.801∗∗∗ −0.723∗∗∗ −0.575∗∗∗ −0.718∗∗∗ −0.828∗∗∗

(-9.67) (-17.20) (-10.59) (-7.93) (-8.24) (-14.94) (-12.43) (-7.79) (-8.04) (-4.19)βNV 0.038 0.45 −0.19 0.250∗ −0.087 0.124 0.200 0.02 0.101 0.010

(1.12) (0.41) (-1.20) (1.87) (-0.57) (1.01) (1.25) 0.08 (0.52) (0.05)

N 467 503 516 538 544 548 564 564 566 550R2 59.40% 69.25% 62.79% 63.94% 65.28% 66.56% 52.88% 48.77% 47.47% 24.97%

Panel B: GMM Betas (Later Sample)

Model (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Portfolio I II III IV V VI VII VIII IX X

βNC 0.880∗∗∗ 0.707∗∗∗ 0.454∗∗∗ 0.841∗∗∗ 0.646∗∗∗ 0.647∗∗∗ 0.697∗∗∗ 0.613∗∗∗ 0.791∗∗∗ 1.272∗∗∗

(2.60) (4.34) (4.08) (7.63) (6.52) (5.97) (6.18) (4.03) (6.40) (5.57)βND −1.07∗∗∗ −0.9793∗∗∗ −0.500∗∗∗ −0.759∗∗∗ −0.498∗∗∗ −0.794∗∗∗ −0.682∗∗∗ −0.431∗∗∗ −0.621∗∗∗ −1.125∗∗∗

(-4.95) (-6.23) (-5.06) (-8.97) (-5.64) (-10.86) (-9.01) (-4.35) (-6.09) (-6.89)βNV 0.176 0.09 −0.27∗ 0.040 −0.1597 0.100 0.020 −0.349 −0.432∗∗ −0.370

(0.39) (1.13) (-1.64) (0.39) (-1.15) (0.72) (0.10) -1.52 (-2.04) (-1.01)

N 226 245 250 264 270 267 280 282 285 274R2 40.00% 46.38% 35.71% 58.61% 44.91% 63.66% 54.66% 39.80% 39.21% 19.15%

Panel C: GMM Betas (Earlier Sample)

Model (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Portfolio I II III IV V VI VII VIII IX X

βNC 1.05∗∗∗ 0.992∗∗∗ 0.888∗∗∗ 0.759∗∗∗ 0.821∗∗∗ 0.945∗∗∗ 0.741∗∗∗ 0.683∗∗∗ 0.707∗∗∗ 0.614∗∗∗

(6.09) (6.470) (6.53) (6.13) (5.18) (7.92) (7.17) (7.54) (4.98) (5.22)βND −1.145∗∗∗ −0.952∗∗∗ −0.864∗∗∗ −0.700∗∗∗ −0.822∗∗∗ −0.858∗∗∗ −0.773∗∗∗ −0.6135∗∗∗ −0.833∗∗∗ −0.600∗∗∗

(-8.20) (-18.35) (-15.46) (-7.00) (-18.03) (-10.35) (-8.52) (-10.17) (-4.64) (-5.48)βNV 0.616 −0.018 −0.120 0.6620∗∗∗ −0.104 0.278 0.550∗∗∗ 0.286∗ 0.481∗∗ 0.106

(1.31) (-0.06) (-0.63) (2.77) (-0.54) (1.37) (2.50) 1.92 (2.23) (0.36)

N 241 258 266 274 274 281 284 282 281 276R2 72.76% 77.98% 77.79% 71.10% 77.51% 70.20% 52.81% 59.82% 55.48% 40.44%

21

Table 8 Panel A: Beta Differentials (Multivariate)

Full Sample βNC βND βNV βEC βED βEVX 0.569 −0.693 0.089 −0.038 −0.052 0.140I 0.926 −1.166 0.072 0.103 −0.060 0.329|D| -0.357 -0.473 0.017 -0.075 -0.008 -0.189

Early Sample βNC βND βNV βEC βED βEVX 0.515 −0.535 −0.339 −0.003 0.007 0.182I 0.904 −1.111 −0.247 0.0599 −0.001 0.366|D| -0.389 -0.576 0.092 -0.063 0.008 -0.184

Late Sample βNC βND βNV βEC βED βEVX 0.815 −1.053 0.057 −0.086 −0.081 0.048I 1.024 −1.296 0.354 0.271 −0.247 0.193|D| -0.204 -0.243 -0.297 -0.357 -0.166 -0.145

Panel B: Beta Differentials (Lag Adjusted Betas)

Portfolio βNC βND βNV βEC βED βEVX 0.284 −0.856 −0.091 2.145 −1.974 −0.279I 0.557 −1.550 −0.164 3.044 −2.589 0.078|D| -0.273 -0.694 -0.074 -0.899 -0.615 -0.357

Early Sample βNC βND βNV βEC βED βEVX 0.319 −0.843 −0.120 2.837 −2.643 0.365I 0.800 −1.799 −0.218 2.202 −1.637 −0.153|D| -0.481 -0.956 -0.098 0.635 1.006 0.518

Late Sample βNC βND βNV βEC βED βEVX 0.228 −0.876 −0.045 1.044 −0.910 −0.141I 0.311 −1.154 −0.976 4.066 −3.787 −0.443|D| -0.083 -0.278 -0.931 -3.022 -2.877 -0.298

Panel C: Beta Differentials (GMM)

Full Sample βNC βND βNVX 0.869 −1.140 0.010I 0.919 −0.828 0.038|D| -0.050 -0.312 -0.028

Early Sample βNC βND βNVX 0.614 −0.600 0.101I 1.050 −1.145 0.616|D| -0.436 -0.545 -0.515

Late Sample βNC βND βNVX 1.272 −1.125 −0.370I 1.050 −1.070 0.176|D| 0.222 0.055 -0.526

22

4.7. Lagged Adjusted Betas and GMM estimation

Tables 6 and 7 document the estimation results of our lag adjusted and GMM beta estimation. Table

6 reports lag adjusted betas computed in accordance with equations 9 and 10. Table 7 documents

my GMM beta estimation using the moment condition specified in equation 17. Table 8 documents the

beta differentials between the winner and loser portfolios using our beta estimates from our 3 approaches.

Table 8 documents a broadly similar result to our multivariate betas: the loser portfolio has higher cash

flow and discount rate betas than the winner portfolio, again inconsistent with our model. This result is

relatively robust across both the full sample and the two sub-samples. The only exception to this result

is the late sample of our GMM estimation which yields higher cash flow and discount rate betas for

the winner portfolio, consistent with our model. Whilst such a development is promising for our model,

in light of all other contrary evidence we must treat this evidence as an anomaly rather any concrete

evidence that the model predictions are correct.

Another interesting result is that the lag adjusted betas reported in table 6 are considerably lower than

the multivariate and GMM betas for both the full sample and the two sub-samples. This indicates that

the illiquidity effect of stale prices is not only biasing upwards our beta estimates in the earlier sample

but also in the later sample. This result is consistent with recent literature documenting the existence of

a liquidity risk premium (Amihud and Mendelson 1986; Brennan and Subrahmanyan (1996) and Easley

et al 2002). Also the signs flip for βEV indicating that after illiquidity is controlled for momentum

returns are decreasing in expected volatility, a finding that is consistent with the observed volatility

hedging motivation amongst investors (Bakshi and Kapadia 2003).

23

5. Concluding Remarks

The primary contribution of this paper has been to test whether or not long run risk models of the

Stochastic Discount Factor can account for the momentum anomaly. Using a model framework that

largely borrows from Campbell and Vuolteenaho (2004) and Campbell et al (2014), I derive a no-arbitrage

model of the momentum anomaly based on a representative investor with Epstein and Zin recursive util-

ity preferences. Whilst I find that this model can explain a large portion of the cross-sectional variation

in momentum returns, I do not find that the main predictions of the model are consistent with the data.

Loser stocks have higher cash flow, discount rate and volatility betas than winner stocks, whereas my

model indicates that the converse should be true.

Whilst the main predictions of the model are not supported by the data, the strong in-sample perfor-

mance of the model indicates that the model contains some informative predictive power for momentum

returns. I interpret the high R2 for our multivariate regressions and the GMM beta estimations for the

loser portfolio as evidence that our model does a good job of explaining return variation in the loser

portfolio. However the lower R2 for the winner portfolio indicates that the model is still missing a key

ingredient in the pricing of the momentum anomaly.

One possible area of future research is to look at higher moments of the news terms Nc,t+1, Nd,t+1 and

Nv,t+1. As pointed out by Kalev, Saxena and Zolotoy (2017), the standard derivation of ICAPM risk

prices in the long run risk framework of Campbell and Vuolteenaho (2004) defined by equation 2 relies

on a first order taylor approximation to the Epstein and Zin pricing kernel. This first order approx-

imation by construction removes higher moments of the news terms from the model set-up. Kalev,

Saxena and Zolotoy (2017) demonstrate that exclusion of these higher order news terms may be costly

for pricing momentum given empirical evidence that the squared news terms N2c,t+1, N

2d,t+1 and the

covariance news term Ncd,t+1 are priced for the 25 size and momentum sorted portfolios available on the

Kenneth French data library. Thus it may very well be possible that if our model were to include these

higher order news moments, the results would become consistent with the main predictions of the model.

Another possible area of future research would be to examine some of the methodological approaches

to estimating the news terms employed by this paper as well as by Campbell and Vuolteenaho (2004)

and Campbell et al (2016). In particular it may be necessary to rethink alternative approaches to the

VAR system which has not been proven to be robust to changes in the state variables used in the VAR

system (Chen and Zhao 2008). In this regard Kozak (2017) provides a novel alternative to the VAR

system which involves using future realized returns as a proxy for future expected returns which yields

consistent estimates of the news terms without requiring an estimate for future returns. Re-examining

the work in this paper using this alternative construction to estimate our news and expectation terms

may very well also yield results more consistent with the predictions of the model.

24

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