Cointegration and

Consumption Risk

in Asset Returns

RaviBansal, Robert Dittmar, and Dana Kiku

Introduction

�How does cointegration between consumption and dividends alter the risk-return

tradeoff across investment horizons?

�The cointegrating relation between dividends and consumption provides a measure

of long-run risks in dividends. Is this source of risk important in explaining the

cross-section of equity returns?

�Find that dividends’ exposure to long-run consumption risks is an important

explanatory variable in accounting for differences in mean returns across portfolios.

�Motivated by Bansal and Yaron (2004); Bansal, Dittmar, and Lundblad(2001,

2005); Hansen, Heaton, and Li (2006); and Bansal, Gallant, and Tauchen (2007)

who show that long-run risks can be important in explaining risk premia.

Risk Premia

11

1

1

11

1

SDF:

log()

Euler equation: [exp(

)]1, where

Assume that and are jointly conditionally normal and homoskedstic. Let

(), the innovatio

tt

s

tt

ts

tts

tts

tj

j

tt

tt

mc

Em

rm

m

mr

cE

c

ργ

η

+→+

+→+

+→+

+=

++

+

=−∆

+=

=

=∆

−∆

∑

11

1

,1

11

n in consumption growth;

(), the innovation in the cumulative consumption growth;

(), the innovation in the -horizon return.

The conditional e

tts

tts

tt

ts

rt

ts

tts

tt

ts

cE

c

rEr

s

η

η

+→+

+→+

+→+

+→+

+→+

+→+

=∆

−∆

=−

2

1,

,1

,1

1,

12 ,

2 ,

xpected return at any horizon is

11

(0.5

)cov(

,)

s

cov(

,)

1

f

tt

ts

rs

ts

tts

rt

ts

tts

rt

ts

cs

cs

ss

s

Er

srs s

σγ

ηη

ηη

γσ

σ

λβ

+→+

+→+

+→+

+→+

+→+

+−

= = =

Risk Premia

()

2 ,s

2 ,2

1,

,

1 is the price of risk and is the conditional consumption risk measure.

Taking unconditional expectations:

11

E(

0.5

)(

).

s2

Expected returns are de

scs

rs

f

tt

ts

rs

rts

ss

s

Er

Er

s

λγσ

β

σσ

µβλ

+→+

=

+=

+=

+

termined by the market price of horizon risk ,

and the horizon-dependent beta, .

s

s

λ

β

Campbell-ShillerApproximation

11

11

11

1

11

1

1

1

1

1

Multiplying both sides by

, and rearranging

1.

Taking logs, with lowercase letters denoting logs of uppercase letters,

tt

tt

t

t

tt

tt

tt

tt

t

tt

t

PD

RR

RP

PD

PD

PR

DD

D

pd

r−−

++

++

+

−+

++

+

+

+=

=

=+

−=−

+1

1

1

11

11

10

1

log(1

).

Taking a Taylor expansion of the last term about a point

,

log1

[(

)].

1

Let

,,

log1

. 1

tt

pd

t

pd

tt

rt

tt

t

tt

t

de

PD

e

PPD

pd

rd

dp

dpd

DPD

PD

Pz

pdk

kkz

PD

D++

−

+

−

++

++

∆+

+

=

−=−

+−+

++

−−−

+

=−

==

+−

+

10

11

10

11

11

Then

(1)

tt

tt

tt

tt

rk

dd

kz

zk

dz

kz

++

++

++

=+

−+

−=+∆

+∆

+−

Betas by Horizon

10

11

11

11

1

,1

,1

,1

(1)

Innovation in s-horizon return is determined by innovations to

,,

, denoted by

,, and

respectively.

The covariance o

tt

tt

tts

tts

tts

dt

ts

zt

ts

zt

ts

rk

dz

kz

dz

z

ηη

η

++

++

+→+

+→+

+→+

+→+

∆+→+

+→+

=+∆

+∆

+−

∆∆

1,

11

,1

12

2

,,

f these components with consumption growth will determine

the overall risk compensation. That is,

is the sum of three components:

cov(

,)

cov(

,)

(

s

tts

dt

ts

tts

zt

ts

s

cs

cs

k

β

ηη

ηη

βσ

σ

+→+

+→+

+→+

∆+→+

=+

+−

1,

1

2 ,

,,

1,

cov(

,)

1)

(1)

The joint dynamics of consumption growth, dividend growth, and price-dividend

ratios contribute to the determination of assets' risk.

tts

zt

ts

cs

sds

zs

zs

k

ηη

σ

ββ

ββ

+→+

+→+

∆⇒

=+

+−

Deriving Mean Returns &

Betas by Horizon

�We want to estimate assets’ consumption betas for different investment

horizons.

�First estimate cointegrating relation between dividends and consumption by

regressing de-trended cash flows onto the stochastic trend in consumption.

�Model the dynamics of the resulting cointegrating residual jointly with the

portfolios, price-dividend ratios, and consumption growth via a EC-VAR.

�The error correction state variable alters the information set used to predict

returns relative to other specifications of return dynamics. Hence it alters

the return innovation and conditional betas.

EC-VAR

0,

,

Given estimates of the parameters and residuals in the cointegrating relation

model the dynamics of the resulting cointegrating residual

jointy with the

portfolio's price dividend

tt

dt

dt

dt

cττδ

ε

ε

=++

+

1

,,

1 1

1

ratio, , and consumption growth,

, by the EC-VAR

00

00 0 0

10

10

tt

tt

c

dt

dt

cz

d

zcz

zzd

tt

cc

zd

tt

zcz

zzd

tt

zc

cc

aa

a

aa

az

z

aa

ad

d

aa

az

z

εε

εε

ε

εε

εε

ε

ρ

εε

ρ

ρ

δρρ

ρ

− − −

− −

∆

∆

∆

=

+

−∆

∆

−

∆∆

, ,

,

,1

t t

zt

tt

zt

ε

ε

η η η

ηδη

η

+

+

EC-VAR

1

1, ,

We can rewrite the EC-VAR compactly as

,

where

is defined above, is a 31 vector of shocks:

10

0

01

0

.0

01

10

00

1

The scaled covarianc

tt

t

t

tt

tt

zt

XAX

Gu

Au

x

XAX

Gu

εη η ηδ

−

−

=+

−=

=

'

s1

1'

1e

e matrix of the innovation in the sum of

consecutive

' is given by

11

1,

where

and

.

se

ss

s

ss

u

sXs

BB

ss

BB

AG

G

−

−−

Σ=

Σ+−Σ

=+

Σ=Σ

Consumption Betas

s

s1

,

The -period covariance matrix

allows us to calculate the

-period beta of an asset:

(1,4)

(1,5)

(1,3)

(1)

.(1,1)

(1,1)

(1,1)

The first component corresponds to cash flow risk,

, th

ss

s

ss

s

ds

s

s

kβ

β

Σ

ΣΣ

Σ=

++

−Σ

ΣΣ

,,

e second

component,

, corresponds to price change risk, and

corresponds to level risk.

zs

zs

ββ

∆

Data

�Annual returns on decile portfolios sorted by size and book-to-market.

�Construct dividends per share monthly, sum to get annual dividend series,

and convert to real by the personal consumption deflator.

�Log growth rates are constructed by taking log first difference of cash flow

series.

�Profile of expected returns can be read from EC-VAR representation by

adding half the variance of the return innovation for a given horizon to the

mean log return.

Summary Statistics

Cointegration Evidence

Dividend Predictability

Return Predictability

Betas & Expected Returns by Horizon

Betas & Expected Returns by Horizon

Betas & Expected Returns by Horizon

�Risk measures implied by EC-VAR exhibit substantial cross-sectional

variation –declining pattern in size dimension & increasing pattern in

book-to-market.

�As horizon increases, the pattern of betas remains roughly the same.

�Betas in VAR specification do not reflect the cross-sectional differences in

mean returns on size and book-to-market sorted portfolios, implying that

there is crucial information contained in the cointegrating residual for

computing assets’ exposures to consumption risk.

Cross-Sectional Risk & Return

2

,

,0,

1,

,

1Cross-sectional regression:

2

Market prices of risk are estimated jointly with the time-series

parameters via GMM.irs

is

ssis

Er

s

σλ

λβ

+=

+

Long-Run Risks Compensation

Long-Run Risks Compensation

�BN decomposes difference stationary consumption series into a deterministic trend,

stochastic trend, and transitory component.

�The covariance of returns with consumption (i.e., beta) can be broken into two

parts: covariance with trend shocks and covariance with transitory shocks

�Market price of risk reflects the premium for both very long runrisks and short run

fluctuations in consumption.

�Compensation for long run risks is about 60% of the overall compensation at the

one-year horizon, and increases with the horizon

�Long run fluctuations in consumption are the dominant source of the premium for

consumption risks in asset markets

Conclusions

�Cointegration between consumption and dividends is important for

understanding the dynamics of asset returns and their risk compensations

across investment horizons.

�At the one-year horizon, the cross-sectional explanatory power rises from

an adjusted R-squared of 22% for a C-CAPM with a VAR forecasting

model to 75% for a C-CAPM based on a cointegration specification.

�Long-run consumption risks are pivotal in understanding asset markets.

Discussion

�Evidence that the authors cite of cointegration between consumption and

dividends is not very compelling.

�Rely on sample ACF and weak unit root tests (in which the null is not rejected

across the board).

�Do not consider the possibility of long memory in the cointegrating residual,

nor the possibility of structural breaks.

�Ignore y–the claim that consumption, dividends, and labor income are

cointegrated is much more suggestive –i.e., they should include cayin their

EC-VAR.

�Testing by implication seems to be a theme in the long-run risk literature!

Discussion

�Test (informally) the EC-VAR asset pricing model against very weak

alternative models, known to fair abysmally at explaining the cross-section

of returns.

�Also would have liked them to use a wider set of test assets (e.g., 25

portfolios sorted on size and book-to-market, momentum portfolios, etc).

�Model relies on consumption data.

Discussion

�Thinking ahead: why might the cash flows of some assets have greater

exposure to long-run consumption risks?

�Are HML or SMB mimicking portfolios for long-run risk factors?