Journal of Financial Economics - NYUpages.stern.nyu.edu/~dbackus/BCZ/bakshi_chabi-yo_jfe2012...equity portfolio or the equity market. Such an asset captures the equity risk premium,

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Journal of Financial Economics

Journal of Financial Economics ] (]]]]) ]]]–]]]

0304-40

doi:10.1

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Variance bounds on the permanent and transitory componentsof stochastic discount factors$

Gurdip Bakshi a,1, Fousseni Chabi-Yo b,n

a Smith School of Business, University of Maryland, College Park, MD 20742, USAb Fisher College of Business, Ohio State University, Columbus, OH 43210, USA

a r t i c l e i n f o

Article history:

Received 6 November 2010

Received in revised form

30 June 2011

Accepted 7 July 2011

JEL classification:

C51

C52

G12

Keywords:

Stochastic discount factors

Permanent component

Transitory component

Variance bounds

Eigenfunction problems

5X/$ - see front matter Published by Elsevier

016/j.jfineco.2012.01.003

are especially grateful for the comments and

William Schwert. The authors also acknowle

ith Andrew Ang, Geert Bekaert, Xiaohui G

ysels, Steve Heston, Bryan Kelly, Ravi Jaganna

, Lars Lochstoer, Dilip Madan, George Panay

gleton, Georgios Skoulakis, Rene Stulz, Jessic

ch, Ingrid Werner, Wei Yang, and Lu Zhang. Se

hio State University, the 2011 Jackson Hole F

1 FIRS Conference (Sydney, Australia) provi

ny remaining errors are our responsibility

vided superb research assistance. All compu

m the authors. Details on the proofs and mo

d in the SSRN version of our paper and on o

smith.umd.edu/faculty/gbakshi/ and http:/

chabi-yo/).

esponding author. Tel.: þ1 614 292 8477; fax:

ail addresses: [email protected] (G. B

[email protected] (F. Chabi-Yo).

l.: þ1 301 405 2261; fax: þ1 301 405 0359

e cite this article as: Bakshi, G., Chastic discount factors. Journal of Fin

a b s t r a c t

In this paper, we develop lower bounds on the variance of the permanent component

and the transitory component, and on the variance of the ratio of the permanent to the

transitory components of SDFs. Exactly solved eigenfunction problems are then used to

study the empirical attributes of asset pricing models that incorporate long-run risk,

external habit persistence, and rare disasters. Specific quantitative implications are

developed for the variance of the permanent and the transitory components, the return

behavior of the long-term bond, and the comovement between the transitory and the

permanent components of SDFs.

Published by Elsevier B.V.

B.V.

advice of a referee

dge helpful discus-

ao, Anisha Ghosh,

than, Ralph Koijen,

otov, Eric Renault,

a Wachter, Michael

minar participants

inance Conference,

ded useful sugges-

alone. Kyoung-hun

ter codes are avail-

del parameters are

ur website (http://

/fisher.osu.edu/fin/

þ1 614 292 2359.

akshi),

.

abi-Yo, F., Variance bancial Economics (20

1. Introduction

In an important contribution, Alvarez and Jermann(2005) lay the foundations for deriving bounds whenthe stochastic discount factor (hereby, SDF), from an assetpricing model, can be decomposed into a permanentcomponent and a transitory component [see also thecontribution of Hansen and Scheinkman (2009) andHansen, Heaton, and Li (2008)]. Our objective in thispaper is to propose a bounds framework, in the contextof permanent and transitory components, and we showthat our bounds are fundamentally different from Alvarezand Jermann (2005). A rationale for developing ourbounds is that the bounds postulated in Alvarez andJermann (2005) hinge on the return properties of thelong-term discount bond, the risk-free bond, and a genericequity portfolio. Our approach generalizes to an assetspace with a dimension greater than three, and it departs

ounds on the permanent and transitory components of12), doi:10.1016/j.jfineco.2012.01.003

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http://www.rhsmith.umd.edu/faculty/gbakshi/

http://www.rhsmith.umd.edu/faculty/gbakshi/

http://fisher.osu.edu/fin/faculty/chabi-yo/

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G. Bakshi F. Chabi-Yo / Journal of Financial Economics ] (]]]]) ]]]–]]]2

by relying on the variance-measure (as in Hansen andJagannathan, 1991).

Building on our treatment, we address the key ques-tions of (i) how useful our bounds are in assessing assetpricing models, and (ii) how our bounds compare, in thetheoretical and empirical dimension, with the counter-parts in Alvarez and Jermann (2005) (wherever applic-able). The first question is pertinent to economicmodeling, while the second question is pertinent to theincremental value-added and the tightness of our boundsin empirical applications.

In our setup, we develop a lower bound on thevariance of the permanent component of the SDF and alower bound on the variance of the transitory componentof SDFs, and then a lower bound on the variance of theratio of the permanent to the transitory components ofSDFs. The lower bound on the variance of the permanent/transitory component can be viewed as a generalization ofthe Alvarez and Jermann (2005) bounds. Our lower boundon the variance of the ratio of the permanent to thetransitory components of SDFs allows us to assesswhether asset pricing models are capable of describingthe additional dimension of joint pricing across markets,and has no analog in Alvarez and Jermann (2005).A salient feature of our bounds is that they incorporateinformation from average returns as well as the variance-covariance matrix of returns from a generic set of assets.We show that the bound implications for the permanentcomponent of SDFs in the three-asset case is considerablyweaker than those reported in our multiple-asset context.

An essential link between our variance bounds andasset pricing is the eigenfunction problem of Hansenand Scheinkman (2009), which facilitates an analyticalexpression for the permanent and transitory componentsof the SDF for an asset pricing model. The posited bounds,in conjunction with an analytical solution to the eigen-function problem, can provide a setting for discerningwhether the time-series properties underlying an assetpricing model are consistent with observed data fromfinancial markets.

To provide wider foundations for our empirical exam-ination, we focus on the eigenfunction problem for abroad class of asset pricing models, namely, (i) long-runrisk [Bansal and Yaron (2004) as also parameterized inKelly (2009)), (ii) external habit persistence (Campbelland Cochrane (1999) as also parameterized in Bekaert andEngstrom (2010)], and (iii) rare disasters (as parameter-ized in Backus, Chernov, and Martin, 2011). Then we areled to ask a question of economic interest: What can belearned about asset pricing models that consistently pricethe long-term discount bond, the risk-free bond, theequity market, and a multitude of other assets. Theimportance of studying long-run risk, external habitpersistence, and rare disasters under a common platformis also recognized by Hansen (2009).

A number of new insights can be garnered about theperformance of asset pricing models in the context ofequity and bond data. One implication of our findings isthat the variance of the permanent component of the SDFin models is of an order lower than the correspondingbound reflected in returns of bonds, equity market, and

Please cite this article as: Bakshi, G., Chabi-Yo, F., Variance bstochastic discount factors. Journal of Financial Economics (20

portfolios sorted by size and book-to-market. The modelwith rare disasters exhibits the highest variance of thepermanent component, a feature linked to occasionalconsumption crashes.

Next, we observe that the transitory component of theSDF in models fails to meet the lower variance boundrestriction. This bound is tied to the Sharpe ratio of thelong-term bond. We further characterize the expectedreturn (variance) of a long-term bond, and find that whileeach model quantitatively depicts the real return of a risk-free bond, they fall short in reproducing the long-termbond properties. This metric of inconsistency matterssince prospective models typically offer reconciliationwith the equity premium, while often ignoring the returnof long-term bonds. Our inquiry uncovers that the mis-specified transitory component is a source of the limita-tion in describing the return behavior of long-term bonds.

We make two key observations with respect to thejoint dynamics of permanent and transitory componentsof SDFs. First, we show that it is possible to recover thecomovement between the permanent and the transitorycomponents from historical data without making anydistributional assumptions. While the data tell us thatthe two components should move in the same direction,our analysis reveals that this feature is not easily imitatedby the models. Second, we show that the model-basedvariance of the ratio of the permanent to the transitorycomponents of the SDFs is insufficiently high, conveyingthe need to describe more plausibly the joint dynamics ofthe returns of bonds and other assets.

Taken all together, our non-parametric approach high-lights the dimensions of difficulty in reconciling returnsdata under some parameterizations of asset pricing mod-els. Our work belongs to a list of studies that searches forthe least misspecified asset pricing model, and explorestheir implications, as outlined, for example, in Bansal,Kiku, and Yaron (2009), Beeler and Campbell (2009), andYang (2010, 2011) Our variance bounds could be adoptedas a complementary device to examine the validity of amodel, apart from matching sample moments, slopecoefficients from predictive regressions, and correlations.The eigenfunction problem offers further guidance forasset price modeling.

2. Bounds on the permanent and transitory components

This section presents theoretical bounds related to theunconditional variance of permanent and transitory com-ponents of SDFs, and the ratio of the permanent to thetransitory components of SDFs.

Since asset pricing models often face a hurdle ofexplaining asset market data based on unconditionalbounds, we develop our results in terms of unconditionalbounds, instead of the sharper conditional bounds.

2.1. Motivation for developing bounds based on the

properties of a generic set of asset returns

We adopt notations similar to that in Alvarez andJermann (2005), and let fMtg be the process of strictlypositive pricing kernels. As in Hansen and Richards



G. Bakshi F. Chabi-Yo / Journal of Financial Economics ] (]]]]) ]]]–]]] 3

(1987), we use the absence of arbitrage opportunities tospecify the current price of an asset that pays Dtþk at timetþk as

Vt½Dtþk� ¼ EtMtþk

MtDtþk

� �, ð1Þ

where Etð:Þ represents the conditional expectation opera-tor. The SDF from t to tþ1 is represented by Mtþ1=Mt .

To differentiate returns offered by different types ofassets, we first define Rtþ1,k as the gross return fromholding, from time t to tþ1, a claim to one unit of thenumeraire to be delivered at time tþk. Then, the returnfrom holding a discount bond with maturity k from time t

to tþ1, and the long-term discount bond are, respectively,

Rtþ1,k ¼Vtþ1½1tþk�

Vt ½1tþk�and Rtþ1,1 � lim

k-1Rtþ1,k: ð2Þ

The case of k¼1 in Eq. (2) corresponds to the gross returnof a risk-free bond.

Next we denote by Rtþ1, the gross return of a broadequity portfolio or the equity market. Such an assetcaptures the equity risk premium, and plays a key rolein the formulations of Alvarez and Jermann (2005).

To build on the asset space, it is of interest to defineRtþ1 ¼ ðRtþ1;1,Rtþ1,Rtþ1,aÞ

0, which constitutes an nþ2-dimensional column vector of gross returns. The n-dimen-sional vector Rtþ1,a contains a finite number of riskyassets that excludes the equity market and the long-termdiscount bond.

Consider the set of SDFs that consistently price nþ3assets, that is, the long-term discount bond, the risk-freebond, the equity market, and additionally n other riskyassets,

S�Mtþ1

Mt: E

Mtþ1

MtRtþ1,1

� �¼ 1 and E

Mtþ1

MtRtþ1

� �¼ 1

� �,

ð3Þ

where 1 is a column vector of ones conformable with Rtþ1,and Eð:Þ represents the unconditional expectation operator.The mean of the SDF is given by mm � EððMtþ1=MtÞ1tþ1Þ.

In the discussions to follow, we refer to Var½u� ¼

Eðu2Þ�ðEðuÞÞ2 as the variance-measure for some randomvariable u, and use it to quantify the importance oftransitory and permanent components of SDFs.

The L-measure-based bounds framework in Alvarezand Jermann (2005) is intended specifically for a long-term discount bond, a risk-free bond, and a single equityportfolio, whereas Eq. (3) allows one to expand the assetspace to a dimension beyond three. Our motivation forstudying variance-measure-based bounds, as opposed toL-measure-based bounds, will be articulated in Section 3.

There are reasons to expand the set of assets in ouranalysis. Note that the crux of the bounds approach is thatthe bounds are derived under the assumption that (i) thetransitory component prices the long-term bond, (ii) thepermanent component prices other assets, and (iii) theSDF correctly prices the entire set of assets, whileaccounting for the relation between the transitory andthe permanent components. In this context, we show thatthe bounds implication for the three-asset case of Alvarezand Jermann (2005) is considerably weaker than those


reported in our empirical illustrations involving manyrisky assets.

Equally important, our variance bounds treatment isintended for all sorts of assets, which is in the vein of,among others, Shanken (1987); Hansen and Jagannathan(1991); Snow (1991); Cecchetti, Lam, and Mark (1994);Kan and Zhou (2006); Luttmer (1996); Balduzzi and Kallal(1997), and Bekaert and Liu (2004). In this sense, theviability of asset pricing models can now be judged bytheir ability to satisfactorily accommodate the risk pre-mium on a spectrum of traded assets, and not just theequity premium.

2.2. Bounds on the permanent component of SDFs

Alvarez and Jermann (2005, Proposition 1), andHansen and Scheinkman (2009, Corollary 6.1) show thatany SDF can be decomposed into a transitory componentand a permanent component. Inspired by their analyses,we presume that there exists a decomposition of thepricing kernel Mt into a transitory and a permanentcomponent of the type:

Mt ¼MTt MP

t , with EtðMPtþ1Þ ¼MP

t : ð4Þ

The permanent component MtP

is a martingale, while thetransitory component Mt

Tis a scaled long-term interest

rate. In particular,

Rtþ1,1 ¼MT

tþ1

MTt

!�1

, ð5Þ

which follows from Alvarez and Jermann (2005,Assumptions 1 and 2, and their proof of Proposition 2).The transitory component prices the long-term bond withEððMT

tþ1=MTt ÞRtþ1,1Þ ¼ 1. Completing the description of

the decomposition (4), the transitory and permanentcomponents of the SDF can be correlated.

We assume that the variance-covariance matrix of Rtþ1,Rtþ1=Rtþ1,1, Rtþ1=R2

tþ1,1 are each nonsingular. Our prooffollows.

Proposition 1. Suppose the relations in Eqs. (4) and (5) hold.

Then the lower bound on the unconditional variance of the

permanent component of SDFs Mtþ1=Mt 2 S is

VarMP

tþ1

MPt

" #Zs2

pc � 1�ERtþ1

Rtþ1,1

� �� 0O�1 1�E

Rtþ1

Rtþ1,1

� �� ,

ð6Þ

where O� Var½Rtþ1=Rtþ1,1�.

Proof. See Appendix A.

We note that our variance bound is general and notspecific to the Alvarez and Jermann (2005) or Hansen andScheinkman (2009) decomposition. The bound applies toany SDF that can be decomposed into a permanent and atransitory component.

The s2pc in inequality (6) is computable, given the

return time-series of long-term bond, risk-free bond,equity market, and other assets. Our development facil-itates a variance bound on the permanent component of




the SDF that can accommodate the return properties ofthe desired number of assets contained in Rtþ1.

Inequality (6) bounds the variance of the permanentcomponent of the SDF, which can be a useful object forunderstanding what time-series assumptions are neces-sary to achieve consistent risk pricing across a multitudeof asset markets. In addition to using the informationcontent of returns across different asset classes, the boundis essentially model-free and can be employed to evaluatethe empirical relevance of the permanent component ofany SDF, regardless of its distribution. The permanentcomponent of the SDF from any asset pricing modelshould respect the bound in (6).

The quadratic form for the lower bound in (6) departsfundamentally from the corresponding L-measure-basedbound for the three-asset case in Alvarez and Jermann(2005, Eq. (4)):

Lower bound on the permanent component in

Alvarez and Jermann is EðlogðRtþ1=Rtþ1,1ÞÞ: ð7Þ

Under their approach, it is the expected return spread thatunpins the bound. In contrast, the stipulated bound in (6)combines information from the average returns and thevariance-covariance matrix of returns.

Although the Hansen and Jagannathan (1991) boundwas not developed to differentiate between the perma-nent and the transitory components of the SDF, thevariance bound on the permanent component s2

pc isreceptive to an interpretation, as in the Hansen andJagannathan (1991) bound (see also Cochrane andHansen, 1992). To appreciate this feature, note thatRtþ1=Rtþ1,1C1þ logðRtþ1Þ�logðRtþ1,1Þ1, and supposethat Rtþ1 consists of a risk-free bond, equity market,and equity portfolios. Then following Hansen andJagannathan (1991) and Cochrane (2005), the variancebound on the permanent component of SDFs can beinterpreted as the maximum Sharpe ratio when theinvestment opportunity set is composed of the risk-freebond with excess return relative to the long-term bond,the equity market with excess return relative to the long-term bond, and the equity portfolios with excess returnrelative to the long-term bond.

In a manner akin to the lower bound on the volatilityof the SDF in Hansen and Jagannathan (1991), we estab-lish, via Eq. (18), that s2

pc corresponds to the volatility ofthe permanent component of SDFs exhibiting the lowestvariance. Thus, there is a key difference between thelower bound on the variance of SDFs and the lower boundon the variance of the permanent component of SDFs.

Koijen, Lustig, and Van Nieuwerburgh (2010) highlightthe economic role of Var½MP

tþ1=MPt �=Var½Mtþ1=Mt� in affine

models. Our formulation leads to a bound on the ratio(proof is in Bakshi and Chabi-Yo, 2011):

VarMP

tþ1

MPt

" #

VarMtþ1

Mt

� �Z s2pc

s2pcþ1�m2

m

: ð8Þ

The upshot is that the lower bound on the size of thepermanent component of the SDFs is also analytically


distinct from its L-measure and three-asset based coun-terpart in Alvarez and Jermann (2005, Eq. (5)).

The relative usefulness of our bounds in empiricalapplications is the focal point of the exercises inSections 3.3, 5.1, and 5.6. Our contention in Section 3.3is also that the bounds appear quantitatively stable tohow the return of a long-term bond is proxied.

2.3. Bounds on the transitory component of SDFs

While a central constituent of any SDF is the perma-nent component, a second constituent is the transitorycomponent, which equals the inverse of the gross returnof an infinite-maturity discount bond and governs thebehavior of interest rates. Absent a transitory component,the excess returns of bonds are zero, which contradictsempirical evidence (Cochrane and Piazzesi, 2005).To gauge the ability of SDFs to explain aspects of thebond market data, while consistently pricing the remain-ing set of assets Rtþ1, as described in Eq. (3), we provide alower bound on the variance of the transitory componentof SDFs.


Then the lower bound on the unconditional variance of the

transitory component of SDFs Mtþ1=Mt 2 S is

VarMT

tþ1

MTt

" #Zs2

tc �

1�EMT

tþ1

MTt

!EðRtþ1,1Þ

!2

Var½Rtþ1,1�: ð9Þ


The variance of the transitory component of any SDFthat consistently prices the long-term bond should behigher than the bound depicted in Eq. (9). There is noanalog in Alvarez and Jermann (2005) to our analyticalbound on Var½MT

tþ1=MTt �.

We can also characterize the upper bound on the size ofthe transitory component, namely, the counterpart toAlvarez and Jermann (2005, Proposition 3), as:

VarMT

tþ1

MTt

" #

VarMtþ1

Mt

� � rVar

1

Rtþ1,1

� �ð1�mmEðRtþ1ÞÞ

0ðVar½Rtþ1�Þ

�1ð1�mmEðRtþ1ÞÞ

,

ð10Þ

where the denominator of (10) is the lower bound on thevariance of the SDFs, as in Hansen and Jagannathan (1991,Eq. (12)).

The quantity on the right-hand side of Eq. (9) istractable and computable from the returns data. A parti-cular observation is that the bound in (9) is a parabola inðEðMT

tþ1=MTt Þ,s2

tcÞ space, and s2tc is positively associated

with the square of the Sharpe ratio of the long-term bond.We potentially contribute by using our bound (9) toassess the bond market implications of asset pricingmodels.




2.4. Bounds on the ratio of the permanent to the transitory

components of SDFs

A third feature of SDFs is their ability to link thebehavior of the bond market to other markets. In thisregard, a construct suitable for understanding cross-mar-ket relationships is the variance of ðMP

tþ1=MPt Þ=ðM

Ttþ1=MT

t Þ,which captures the importance of the permanent compo-nent relative to the transitory component.


Then the lower bound on the unconditional variance of the ratio

of the permanent to the transitory components of SDFs is

Var

MPtþ1

MPt

MTtþ1

MTt

266664

377775Zs2

pt � 1�mptERtþ1

R2tþ1,1

! !0S�1 1�mptE

Rtþ1

R2tþ1,1

! !,

ð11Þ

where S� Var½Rtþ1=R2tþ1,1� and mpt � EððMP

tþ1=MPt Þ=ðM

Ttþ1=

MTt ÞÞ.


For a given EððMPtþ1=MP

t Þ=ðMTtþ1=MT

t ÞÞ, Proposition 3provides the lower bound on Var½ðMP

tþ1=MPt Þ=ðM

Ttþ1=MT

t Þ�,the purpose of which is to assess whether the SDFs canexplain joint pricing restrictions across markets. Theimportance of linking markets has been emphasized byCampbell (1986); Campbell and Ammer (1993), Baele,Bekaert, and Inghelbrecht (2010), Colacito, Engle, andGhysels (2011), and David and Veronesi (2008). Ourbound in Proposition 3 is new, with no counterpart inAlvarez and Jermann (2005).

Further note that Var½ðMPtþ1=MP

t Þ=ðMTtþ1=MT

t Þ� can becast in terms of the mixed moments of the permanent andthe transitory components of the SDF. Later we derive anexplicit implication regarding a form of dependencebetween MP

tþ1=MPt and MT

tþ1=MTt that can be inputed from

the data. In one extreme, a pricing framework devoid of acomovement between MP

tþ1=MPt and MT

tþ1=MTt amounts to

a flat term structure, a feature falsified by the data.

2.5. Further discussion and relation to Hansen and

Jagannathan (1991)

Recapping our results so far, the first type of boundswe propose is on the variance of the permanent compo-nent of SDFs. Such a bound is beneficial for characterizingthe restrictions imposed by time-series assumptions inasset pricing models. Next, we derive a lower bound onthe variance of the transitory component of SDFs, which isa potentially useful tool for disentangling which time-series assumptions on the consumption growth processcan more aptly capture observed features of the bondmarket. Finally, we provide a lower bound on the varianceof the relative contribution of the permanent to thetransitory component of SDFs, which can establish thelink between bond pricing and the pricing of other assets.Our bounds, thus, provide a set of dimensions alongwhich one could appraise asset pricing models.


When a diagnostic test, for instance, the Hansen andJagannathan (1991) variance bound, rejects an asset pri-cing model, it often fails to ascribe model failure specifi-cally to the inadequacy of the permanent component ofSDFs, the transitory component of SDFs, or to a combina-tion of both. Thus, our framework offers the pertinentmeasures to investigate which dimension of the SDF canbe modified, when the goal is to capture return variationeither in a single market or across markets. We elaborateon this issue by analytically solving an eigenfunctionproblem for asset pricing models in the class of long-runrisk, external habit persistence, and rare disasters, andthen invoking our Propositions 1, 2, and 3.

Each of the bounds derived in Propositions 1, 2, and 3are unconditional bounds. In Appendix B, we scale thereturns by conditioning variables and propose boundsthat incorporate conditioning information.

3. Comparison with Alvarez and Jermann (2005) bounds

Germane to our bounds are two central questions: Inwhat way are these bounds distinct from the correspond-ing bounds in Propositions 2 and 3 in Alvarez andJermann (2005)? How useful are our proposed bounds,and how do they compare, say, along the empiricaldimension, with Alvarez and Jermann (2005)?

To address these questions, we first note that Alvarez andJermann define the L-measure (entropy) of a random variableu as

L½u� � f ½EðuÞ��Eðf ½u�Þ, with f ½u� ¼ logðuÞ: ð12Þ

Using L½u� as a measure of volatility, Alvarez and Jermanndevelop their bounds in terms of the L-measure. Under theircharacterizations, a one-to-one correspondence existsbetween the L-measure and the variance-measure of logðuÞ,when u is distributed lognormally, as in L½u� ¼ 1

2 Var½logðuÞ�.Still, discrepancies between the two measures can getmagnified under departures from lognormality, for example,Kelly (2009); Bekaert and Engstrom (2010), and Backus,Chernov, and Martin (2011), as we show, where neither theSDF nor the permanent component are lognormally distrib-uted, as captured by 9L½u��1

2Var½logðuÞ�940.The distinction between our treatment and in that of

Alvarez and Jermann (2005) is hereby studied from threeperspectives. First, we illustrate some differences inMP

tþ1=MPt and MT

tþ1=MTt across the L-measure and the

variance-measure in an example economy. Second, wehighlight some conceptual differences in the character-ization of bounds, focusing on the three-asset setting ofAlvarez and Jermann, i.e., we rely on the return propertiesof the long-term bond, the risk-free bond, and the equitymarket. Third, we provide a comparison of bounds in thedata dimension while maintaining the three-asset setting.

3.1. L-measure versus the variance-measure in an example

economy

Suppose the log of the pricing kernel evolves accordingto an AR(1) process,

logðMtþ1Þ ¼ logðbÞþB logðMtÞþetþ1, where etþ1 �N ð0,s2e Þ, ð13Þ




with 9B9o1 as in Alvarez and Jermann (2005, page 1997).In this setting, the log excess bond return at maturity k is

logRtþ1,k

Rtþ1;1

� �¼s2e

2ð1�B2ðk�1ÞÞ:

It can be shown that the permanent component of the SDF is

MPtþ1

MPt

¼ exp �B2 s2e

2þBetþ1

� �, and hence,

MPt is a martingale E

MPtþ1

MPt

!¼ 1: ð14Þ

The transitory component is

MTtþ1

MTt

¼ exp �s2e

2ð2�B2ðk�1ÞÞ�logðbÞ�ðB�1ÞlogðMtþ1Þ

� �,

for a large k: ð15Þ

Now,

LMP

tþ1

MPt

" #¼ B2 s2

e2

, and LMT

tþ1

MTt

" #¼ ð1�BÞ2 s

2e

2

1�B2ðtþ1Þ

1�B2

� �:

ð16Þ

Further, under our treatment, the variance of the permanentcomponent is Var½MP

tþ1=MPt � ¼ expðB2s2

e Þ�1, and

VarMT

tþ1

MTt

" #¼ exp ðB�1Þ2

s2e

2

1�B2ðtþ1Þ

1�B2

� �� 1

� �

�exp �s2e ð2�B

2ðk�1ÞÞ�2Bðtþ1ÞlogðbÞ�

þð1�BÞ2 s2e

2

1�B2ðtþ1Þ

1�B2

� ��, ð17Þ

and they do not coincide with the L-theoretic counterparts in(16). The basic message, namely, that there are intrinsicdifferences between the two dispersion measures, alsoobtains under conditioning information.

3.2. Under what situations, the framework based on the

variance-measure may be preferable?

As can be inferred from the properties of theL-measure (see Appendix A in Alvarez and Jermann,2005), the bounds they derive are designed for thesituation in which the SDFs correctly price at most threeassets: the long-term discount bond, the risk-free bond,and the equity portfolio, with no obvious way to general-ize to the dimension of asset space beyond three.

We offer bounds that are derived under the conditionthat the permanent and the transitory components of theSDFs correctly price a finite number of assets. Our evi-dence in Section 5.1 puts on a firmer footing the notionthat the variance bound on the permanent component isalso considerably sharper. The novelty of our bounds isthat they exploit the information in both the averagereturns and the variance-covariance matrix of assetreturns.

Even in the setting of the long-term discount bond, therisk-free bond, and the equity portfolio, some conceptualdifferences between the two treatments can be high-lighted. Among a set of MP

tþ1=MPt that correctly price

asset returns, we denote by MnPtþ1=MnP

t the permanent


component of SDFs with the lowest variance. It is

MnPtþ1

MnPt

¼ 1þ

1�ERtþ1;1

Rtþ1,1

� �

1�ERtþ1

Rtþ1,1

� �0BBB@

1CCCA0

Var

Rtþ1;1

Rtþ1,1

Rtþ1

Rtþ1,1

0BBB@

1CCCA

26664

37775

0BBB@

1CCCA�1

�

Rtþ1;1

Rtþ1,1�E

Rtþ1;1

Rtþ1,1

� �Rtþ1

Rtþ1,1�E

Rtþ1

Rtþ1,1

� �0BBB@

1CCCA: ð18Þ

Hence, in the three-asset setting, Eq. (18) can be viewedas the solution to the problem:

minMP

tþ 1=MPt

VarMP

tþ1

MPt

" #subject to E

MPtþ1

MPt

Rtþ1;1

Rtþ1,1

!¼ 1,

EMP

tþ1

MPt

Rtþ1

Rtþ1,1

!¼ 1,and E

MPtþ1

MPt

!¼ 1: ð19Þ

In general, the variance of MnPtþ1=MnP

t equals the lowerbound on the variance of the permanent component ofSDFs. Therefore, our analysis makes it explicit that,regardless of the probability distribution of MP

tþ1=MPt ,

our results pertain to bounds on variance. The messageworth conveying is that the variance of the permanent(and transitory) component from an asset pricing model isdenominated in the same units of riskiness as the boundsrecovered from the data, a trait that our framework sharesalso with the studies of Hansen and Jagannathan (1991);Bernardo and Ledoit (2000), and Cochrane andSaa-Requejo (2000).

Exhibiting a specific distributional property, theAlvarez and Jermann (2005) lower bound is the L-measureof the permanent component: EðlogðRtþ1=Rtþ1,1ÞÞ. Yet itis not possible to find an analytical expression for thepermanent component of SDFs, namely, ~M

P

tþ1=~M

P

t such asL½ ~M

P

tþ1=~M

P

t � ¼ EðlogðRtþ1=Rtþ1,1ÞÞ. To establish this argu-ment, we use the definition of the L-measure, whereby

L~M

P

tþ1

~MP

t

" #¼ logð1Þ�E log

~MP

tþ1

~MP

t

! !,

since E~M

P

tþ1

~MP

t

!¼ 1

!: ð20Þ

Therefore, we can at most deduce that Eðlogð ~MP

tþ1=~M

P

t ÞÞ

¼ EðlogðRtþ1,1=Rtþ1ÞÞ, and it may not be possible to

recover ~MP

tþ1=~M

P

t analytically in terms of asset returns.

It seems that the lower bound on the L-measure of thepermanent component of SDFs does not satisfy thedefinition of the L-measure.

3.3. Lessons from a comparison with Alvarez and Jermann

(2005) bounds in the data dimension

Still, some empirical questions remain with respect toobserved data in the financial markets: What is gainedby generalizing the L-measure-based setup in Alvarezand Jermann (2005)? In what sense do our proposedbounds quantitatively differ from Alvarez and Jermann(2005)? How sensitive are our variance bounds when one




surrogates Rtþ1,1 by the return of a bond with a reason-ably long maturity?

To facilitate these objectives, here we follow Alvarez andJermann (2005) in the choice of three assets, the monthlysample period of 1946:12 to 1999:12 (637 observations), aswell as expressing the point estimates from the L-measureand, hence, from the variance-measure in annualized terms inTable 1. Specifically, we rely on data (http://www.econometricsociety.org/suppmat.asp?id=61&vid=73&iid=6&aid=643)on the return of a long-term bond, the return of a risk-freebond, and the return of a single equity portfolio (optimalgrowth portfolio based on ten CRSP size-decile portfolios andthe equity market).

The maturity of a long-term bond is guided by dataconsiderations and allowed to take a value of 20, 25, or 29years. Also reported in Table 1 are the 90% confidenceintervals (in square brackets), which are based on 50,000random samples of size 637 from the data and a blockbootstrap.

There are three lessons that can be drawn. First, ourexercise suggests that our bounds are broadly differentfrom Alvarez and Jermannn. Second, when the maturity ofthe bond is altered from 20 to 29 years, the bound onL½MP

tþ1=MPt � varies little, whereas the bound s2

pc onVar½MP

tþ1=MPt � varies somewhat from 0.799 to 0.933.

Table 1Comparison of our variance bounds with the L-measure-based bounds in Alvare

with maturity of 20, 25, or 29 years.

Here we follow Alvarez and Jermann (2005), both in the choice of three asse

long-term discount bond, risk-free bond, and single equity portfolio (optimal g

market). We proxy Rtþ1,1 by bond returns with maturity ranging from 20 years

from 1946:12 to 1999:12, and taken from http://www.econometricsociety.org/

(annualized, in bold) of the bounds from the data, along with the 90% confiden

create 50,000 random samples of size 637 (number of observations) from the da

on Var½MPtþ1=MP

t � and Var½MPtþ1=MP

t �=Var½Mtþ1=Mt � are, respectively, based on E

based on Eq. (10). The expressions for the bounds on the L-measure are presen

Rtþ

20

years

Lower bound on VarMP

tþ 1

MPt

h i0.799

[0.646, 0.942]

Lower bound on LMP

tþ 1

MPt

h i0.201

[0.178, 0.226]

LMP

tþ 1

MPt

h i� 1

2 VarMP

tþ 1

MPt

h i�0.198

(Difference in bounds) [�0.274, �0.118]

Lower bound on VarMP

tþ 1

MPt

h i=Var Mtþ 1

Mt

h i0.884

[0.866, 0.904]

Lower bound on LMP

tþ 1

MPt

h i=L Mtþ 1

Mt

h i1.018

[0.964, 1.075]

Upper bound on VarMT

tþ 1

MTt

h i=Var Mtþ 1

Mt

h i0.039

[0.029, 0.047]

Upper bound on LMT

tþ 1

MTt

h i=L Mtþ 1

Mt

h i0.079

[0.066, 0.090]


This differential sensitivity can be ascribed to the fact thats2

pc (as in Eq. (6)) is determined by both average returnsand the variance-covariance matrix of asset returns. Finally,the discrepancy between the bounds on L½MP

tþ1=MPt � and

12Var½MP

tþ1=MPt � is large in the data, implying that the

permanent component is far from being distributed nor-mally in logs. This finding suggests that higher-moments ofMP

tþ1=MPt may be relevant to asset pricing.

4. Eigenfunction problem and the transitory andpermanent components of SDFs in asset pricing models

This section contributes by deriving an analytical solu-tion to the eigenfunction problem of Hansen andScheinkman (2009) to determine the transitory and perma-nent components of the SDF. Featured is a strand of assetpricing models that have drawn considerable support intheir ability to depict stylized properties of aggregate equitymarket returns and risk-free bond returns.

Specifically, we focus on models in (a) the long-runrisk class (Bansal and Yaron, 2004), (b) the external habitpersistence class (Campbell and Cochrane, 1999), and(c) the rare consumption disasters class (Rietz, 1988;and Barro, 2006). Within each class we adopt ageneralization of the SDF that invokes departure from

z and Jermann (2005), when Rtþ1,1 is surrogated by the return of a bond

ts and the sample period. The bounds hinge on the return properties of

rowth portfolio based on ten CRSP size-decile portfolios and the equity

to 29 years. The monthly data used in the construction of the bounds is

suppmat.asp?id=61&vid=73&iid=6&aid=643. Reported are the estimates

ce intervals (in square brackets). To obtain the confidence intervals, we

ta, where the sampling is based on 12 blocks. The displayed lower bound

qs. (6) and (8), while the upper bound on Var½MTtþ1=MT

t �=Var½Mtþ1=Mt � is

ted in Alvarez and Jermann (2005, Propositions 2 and 3).

1,1 is proxied by the return of a bond with maturity:

25 29

years years

0.909 0.933

[0.710, 1.089] [0.675, 1.151]

0.205 0.217

[0.179, 0.232] [0.184, 0.251]

�0.249 �0.249

[�0.343, �0.145] [�0.364, �0.116]

0.896 0.898

[0.878, 0.917] [0.876, 0.923]

1.039 1.098

[0.964, 1.117] [0.977, 1.222]

0.055 0.102

[0.041, 0.067] [0.075, 0.125]

0.111 0.199

[0.092, 0.129] [0.165, 0.232]


http://www.econometricsociety.org/suppmat.asp?id=61&vid=73&iid=6&aid=643











log-normality, the purpose of which is to enrich thesetting for researching the broader relevance of boundsin Propositions 1, 2, and 3. The asset pricing modelsdelineated next, along with those of Bansal and Yaron(2004) and Campbell and Cochrane (1999), are at the coreof the empirical investigation.

4.1. Solution to the eigenfunction problem for a model

incorporating long-run risk

To determine the transitory and permanent compo-nents of the SDF through the eigenfunction problem,consider the modification of the long-run risk modelproposed in Kelly (2009). The distinguishing attribute isthat the model incorporates heavy-tailed shocks to theevolution of (log) nondurable consumption growth gtþ1,which are governed by a tail risk state variable Lt .

gtþ1 ¼ mþxtþsgstzg,tþ1þffiffiffiffiffiffiLt

pWg,tþ1,

xtþ1 ¼ rxxtþsxstzx,tþ1, ð21Þ

s2tþ1 ¼ s

2þrsðs

2t�s

2Þþsszs,tþ1,

Ltþ1 ¼LþrLðLt�LÞþsLzL,tþ1, ð22Þ

zg,tþ1,zx,tþ1,zs,tþ1,zL,tþ1 � i:i:d N ð0;1Þ,Wg,tþ1 � Laplaceð0;1Þ: ð23Þ

Following Bansal and Yaron (2004), xt is a persistentlyvarying component of the expected consumption growthrate, and s2

t is the conditional variance of consumptiongrowth with unconditional mean s2.

The z shocks are standard normal and independent. Inaddition to gaussian shocks, the consumption growthdepends on non-gaussian shocks Wg, where the Wg shocksare Laplace-distributed variables with mean zero andvariance 2, and independent. Wg shocks are independentof z shocks. The model maintains the tradition of Epsteinand Zin (1991) recursive utility.

Proposition 4. The transitory and permanent components of

the SDF in the model of Kelly (2009) are

MTtþ1

MTt

¼ u expð�c1ðxtþ1�xtÞ�c2ðs2tþ1�s

2t Þ�c3ðLtþ1�LtÞÞ,

andMP

tþ1

MPt

¼Mtþ1

Mt

MT

tþ1

MTt

, ð24Þ

where u is defined in (C.13), and the coefficients c1, c2, and c3

are defined in (C.14). The expression for Mtþ1=Mt is

presented in (C.1) of Appendix C.

Proof. See Appendix C.

Eq. (24) is obtained by solving the eigenfunctionproblem of Hansen and Scheinkman (2009, Corollary 6.1):

EtMtþ1

Mt

1

Metþ1

!¼

uMe

t

, ð25Þ

where the parameter u is the dominant eigenvalue. The

conjectured Metþ1 that satisfies (25) determines MT

t ¼ utMet

and MPt ¼Mt=MT

t .

While the transitory component of the SDF is lognor-mally distributed, the permanent component of the SDF,


and the SDF itself, are not lognormally distributed.The non-gaussian shocks Wg are meant to amplify thetails of the permanent component of the SDF and the SDF.Eq. (24) makes it explicit how the sources of the varia-bility in the transitory and permanent components of theSDF can be traced back to (i) the specification of thepreferences, and (ii) the dynamics of the fundamentals.


incorporating external habit persistence

Bekaert and Engstrom (2010) propose a variant of theCampbell and Cochrane (1999) model where (i) thedynamics of consumption growth gt consist of two fat-tailed skewed distributions, and (ii) the SDF is

Mtþ1

Mt¼ b expð�ggtþ1þgðqtþ1�qtÞÞ, ð26Þ

where b is the time preference parameter, g is thecurvature parameter, and qt � logðCt=ðCt�HtÞÞ ¼ logð1=StÞ.For external habit Ht and consumption Ct, the variable St isthe surplus consumption ratio and, hence, qt representsthe log of the inverse surplus consumption ratio (see alsoSantos and Veronesi, 2010; and Borovicka, Hansen,Hendricks, and Scheinkman, 2011). Uncertainty in thiseconomy is described by

gtþ1 ¼ gþxtþsgpop,tþ1�sgnon,tþ1,

xt ¼ rxxt�1þsxpop,tþsxnon,t , ð27Þ

qtþ1 ¼ mqþrqqtþsqpop,tþ1þsqnon,tþ1,

op,tþ1 ¼ getþ1�pt and on,tþ1 ¼ betþ1�nt , ð28Þ

pt ¼ pþrpðpt�1�pÞþsppop,t ,

nt ¼ nþrnðnt�1�nÞþsnnon,t , ð29Þ

getþ1 �Gammaðpt ,1Þ, betþ1 �Gammaðnt ,1Þ, ð30Þ

where pt and nt are the conditional mean of the goodenvironment and bad environment shocks denoted bygetþ1 and betþ1, respectively. The distinguishing attributeof the model is that it incorporates elements of externalhabit persistence together with long-run consumptionrisk.


the SDF in Bekaert and Engstrom (2010) are

MTtþ1

MTt

¼ u expðgðqtþ1�qtÞ�c2ðxtþ1�xtÞ�c3ðptþ1�ptÞ

�c4ðntþ1�ntÞÞ, andMP

tþ1

MPt

¼Mtþ1

Mt

MT

tþ1

MTt

,

ð31Þ

where u is defined in (D.2), the coefficients c2 through c4 are

defined in (D.3), and Mtþ1=Mt is as in (26).

Proof. See Appendix D.

Besides generating the statistically observed risk-freereturn, the equity premium, and moments of consump-tion growth, among the noteworthy model features are itsability to generate time-variation in risk premiums andconsistency with risk-neutralized equity return moments.




One implication of the solution (31) is that the modelembeds a transitory component of the SDF whichcomoves with changes in qt. In contrast, the permanentcomponent is detached from variations in qt.


incorporating rare disasters

We consider a version of the asset pricing model ofRietz (1988); Barro (2006), and Gabaix (2009). Uncer-tainty about consumption growth is modeled followingBackus, Chernov, and Martin (2011) as

logðgtþ1Þ ¼wtþ1þztþ1, with wtþ1 � i:i:d: N ðm,s2Þ,

ztþ19Jtþ1 � i:i:d: N ðyJtþ1,d2Jtþ1Þ, and ð32Þ

Jtþ1 is i:i:d: Poisson random variable with densityoj

j!e�o

for j 2 f0;1,2, . . .g, and mean o: ð33Þ

The distinguishing attribute of this model is that ztþ1

produces sporadic crashes in consumption growth andserves as a device to produce a fat-tailed distribution ofconsumption growth. In the model, ðwtþ1,ztþ1Þ aremutually independent over time, and the SDF is of theform (under time-separable power utility):

Mtþ1

Mt¼ bg�gtþ1 ¼ expðlogðbÞ�gwtþ1�gztþ1Þ, ð34Þ

where b is time preference parameter, and g is thecoefficient of relative risk aversion.


the SDF in the rare disasters model are

MTtþ1

MTt

¼1

Rtþ1,1, and

MPtþ1

MPt

¼ Rtþ1,1expðlogðbÞ�gwtþ1�gztþ1Þ, ð35Þ

where

R�1tþ1,1 ¼ exp logðbÞ�gmþ 1

2g2s2

� �

�X1j ¼ 0

e�ooj

j!exp �gyjþ

1

2g2d2j

� �

is a constant.

Proof. See Appendix E in Bakshi and Chabi-Yo (2011).

Within the setting of (32) and (33), the model inheritsthe property that Mtþ1=Mt and MP

tþ1=MPt are not lognor-

mally distributed. It can be shown that EðMPtþ1=MP

t Þ‘

¼ ðbRtþ1,1Þ‘expðoðe0:5g2d2‘2�gy‘�1Þþ0:5g2s2‘2�gm‘Þ, for

‘¼ 2;3, . . ., which furnishes the moments of the perma-nent component, while maintaining EðMP

tþ1Þ ¼MPt .

5. Empirical application to asset pricing models

To lay the groundwork for the empirical examination,the analysis of this section starts by highlighting thetightness of our lower bound on the variance of thepermanent component in the context of a finite number


of assets. Then we elaborate on the performance of assetpricing models under our metrics of evaluation, includingthe lower bound restrictions on the permanent andtransitory components of SDFs.

5.1. Description of the set of asset returns and the tightness

of the variance bound

Stepping outside of the three-asset setting in Table 1,recall that Rtþ1 is the return vector that is correctly pricedby the SDF along with Rtþ1,1 (see Eq. (3)). Whereas ourvariance bound on the permanent component s2

pc (see Eq.(6)) exhibits dependence on EðRtþ1=Rtþ1,1Þ andVar½Rtþ1=Rtþ1,1�, tractable expressions are not yet avail-able for L-measure-based bounds when there are morethan three assets.

Two questions are pertinent to our development andto our empirical comparison of asset pricing models: (1)Does the variance bound s2

pc get incrementally sharperwhen the SDF is required to correctly price more assets?(2) How does the tightness of the new bound fare relativeto Alvarez and Jermann (2005)?

To answer these questions, we consider a set of monthlyequity and bond returns over the period 1932:01 to 2010:12(our choice of the start date circumvents missing observa-tions). The source of risk-free bond and equity returns is thedata library of Kenneth French, while the source of inter-mediate and long-term government bond returns is Mor-ningstar (Ibbotson). Real returns are computed by deflatingnominal returns by the Consumer Price Index inflation. Forour illustration, we consider four different Rtþ1:

(i)

ound12),

SET A: Risk-free bond, equity market, intermediategovernment bond, and 25 Fama-French equity port-folios sorted by size and book-to-market;

(ii)
SET B: Risk-free bond, equity market, intermediategovernment bond, ten size-sorted, and six size andbook-to-market, sorted equity portfolios;
(iii)
SET C: Risk-free bond, equity market, intermediategovernment bond, and six size and book-to-market,sorted equity portfolios; and,
(iv)
SET D: Risk-free bond, equity market, and intermedi-ate government bond.
The lower bound on the permanent component, s2pc ,

reported below, imparts two conclusions:

s on the permanedoi:10.1016/j.jfine

nt andco.201

transit2.01.00

ory co3

mpone

Set A
Set B Set C Set D AJ
Lower bound s2pc (Eq. (6))
0.1254 0.0835 0.0674 0.0296 0.0041
First, s2pc declines from SET A to SET D, suggesting that

expanding the number of assets in Rtþ1 leads to a boundthat is intrinsically tighter. Second, the estimates of s2

pc

generated from SET A through D are sharper relative toAlvarez and Jermann (2005, Eq. (4)), which is represented bythe entry marked AJ. The gist of this exercise is that s2

pc=2based on the return properties of SET A (SET D) is about 15(four) times sharper than the L-measure-based lower boundon the permanent component, thereby substantiating itsincremental value in asset pricing applications.

nts of



5.2. A setting for model evaluation, granted that each model

calibrates to some data attributes

Turning to the themes of our study, we exploit theexactly solved eigenfunction problems in Propositions4–6, to provide the building blocks for our empiricalstudy in a few ways:

�

Ps

Our interest is in assessing the ability of a model toproduce realistic permanent and transitory compo-nents of the SDF and whether their variance respectsthe proposed lower bounds. We rely on a statisticgenerated from a simulation procedure and the asso-ciated p-value;
� Analytical solutions to the eigenfunction problem
facilitate a quantitative implication regarding thereturn of a long-term bond, whereby

Eðrtþ1,1Þ¼ EðRtþ1,1�1Þ ¼ EMT

t

MTtþ1

�1

!,

Var½rtþ1,1� ¼ EMT

t

MTtþ1

�EMT

t

MTtþ1

! !2

: ð36Þ

The return moments of a long-term bond implicit in anasset pricing model are, thus, computable, and couldbe benchmarked to those from a suitable proxy toascertain their plausibility. Campbell and Viceira(2001), among others, provide an impetus to developasset pricing models that also generate reasonablereturn behavior for the long-term bond.

The bounds yardstick, when combined with (36), couldserve as a differentiating diagnostic when competingmodels (i) each calibrate closely to the mean and standarddeviation of consumption growth, and (ii) offer confor-mity with the historical real return of risk-free bond andthe real return of equity market.

Additionally, and equally relevant, we extract theimplication of each model for the comovement betweenthe permanent and transitory components of the SDF, in amanner to be described shortly via Eqs. (37) and (38), andexamine its merit relative to the one imputed fromthe data.

Primitive parameters are chosen consistently accord-ing to the models of Kelly (2009), Bekaert and Engstrom(2010), and Backus, Chernov, and Martin (2011), as dis-played in Tables Appendix-I through Appendix-III ofBakshi and Chabi-Yo (2011), respectively, together withthose of Bansal and Yaron (2004) and Campbell andCochrane (1999). Even though we refrain from presentingthe full-blown solution to the eigenfunction problem forthe last two models to save space, our objective inimplementing all five models is to offer a unifying pictureof how each asset pricing model performs under ouryardsticks of evaluation.

While the conditional variances are amenable toclosed-form characterization, the unconditional variancesare tractable only via simulations, except for the raredisaster model given i.i.d. uncertainties. Accounting forthis feature, each model is simulated using the dynamicsof consumption growth and other state variables, for

lease cite this article as: Bakshi, G., Chabi-Yo, F., Variance btochastic discount factors. Journal of Financial Economics (20

instance, as in (21)–(23), for the long-run risk model ofKelly (2009), over a single simulation run of 360,000months (30,000 years). Then we build the time-series ofmodel-specific fMP

tþ1=MPt g and fMT

tþ1=MTt g, according to

the solution of the eigenfunction problem, and we calcu-late the unconditional moments. Using a single simula-tion run to infer the population values for the entities ofinterest is consistent with, among others, the approach ofCampbell and Cochrane (1999) and Beeler and Campbell(2009).

5.3. Models could appeal to utility specifications and

dynamics of fundamentals that magnify the permanent

component of SDFs

Our thrust is to examine whether models producesensible dynamics for the permanent component of theSDF. Such an analysis can enable insights into howeconomic fundamentals are linked to SDFs, and how theperformance of an asset pricing model could be improvedby altering the properties of the permanent componentof SDFs.

At the outset, we report, for each model, the varianceof the permanent component of the SDF, Var½MP

tþ1=MPt �,

in Panel A of Table 2. Reported in the final column is thelower bound s2

pc calculated based on SET A, along with the90% confidence intervals, shown in square brackets, froma block bootstrap, when sampling is done with 15 blocks.It is helpful to think in terms of s2

pc from SET A, since thisset corresponds to a universe of equity portfolios whosereturn properties are the subject of much scrutiny in theempirical asset pricing research (e.g., Malloy, Moskowitz,and Vissing-Jorgensen, 2009).

Our implementations reveal that the monthlyVar½MP

tþ1=MPt � implied by the models of Kelly (2009) and

Bansal and Yaron (2004) is 0.0374 and 0.0342, respec-tively, while that of the models of Bekaert and Engstrom(2010) and Campbell and Cochrane (1999) is 0.0280 and0.0234, respectively. The most pronounced value of0.0580 is obtained under the model of Backus, Chernov,and Martin (2011).

Going further, we formulate the restriction: s2pc�

Var½MPtþ1=MP

t �r0 for a candidate asset pricing model,which allows one to elaborate on whether a modelrespects the lower bound [beyond eye-balling estimates;see also Cecchetti, Lam, and Mark, 1994]. Then inferenceregarding this restriction can be drawn via repeatedsimulations. For this purpose, we rely on a finite-samplesimulation of 948 months (1932:01 to 2010:12), and wechoose 200,000 replications. The proportion of the repli-cations satisfying s2

pc�Var½MPtþ1=MP

t �r0 can be inter-preted as a p-value for a one-sided test. This p-value isshown in curly brackets in Table 2, and a low p-valueindicates rejection. Pertinent to this exercise, our evidencereveals that the variance of the permanent component ofthe SDF from each model fails to meet the lower boundrestriction of 0.1254 per month. Importantly, the reportedp-values provide some support for the contention thatthe model-based variances of the permanent componentare reliably lower than s2

pc . A likewise conclusionemerges when s2

pc is computed from SET B and SET C



Table 2Assessing the restriction on the variance of the permanent component of SDFs from asset pricing models.

We compute the variance of the permanent component of the SDF, Var½MPtþ1=MP

t �, via simulations, respectively, for the models that incorporate long-

run risk, external habit persistence, and rare disasters. All calculations are based on model parameters tabulated in Bakshi and Chabi-Yo (2011, Tables

Appendix-I through Appendix-III), and the reported values are the averages from a single simulation run of 360,000 months. The reported lower bound

s2pc on the permanent component (see Eq. (6) of Proposition 1) is based on the return properties of Rtþ1, corresponding to SET A, and the long-term bond.

We keep the maturity of the long-term bond to be 20 years, as also in Alvarez and Jermann (2005, p. 1993). The monthly data used in the construction of

s2pc is from 1932:01 to 2010:12 (948 observations), with the 90% confidence intervals in square brackets. To compute the confidence intervals, we create

50,000 random samples of size 948 from the data, where the sampling in the block bootstrap is based on 15 blocks. Reported below the estimates of

Var½MPtþ1=MP

t � are the p-values, shown in curly brackets, which represent the proportion of replications for which model-based Var½MPtþ1=MP

t � exceeds s2pc

in 200,000 replications of a finite sample simulation over 948 months. Real returns are computed by deflating the nominal returns by the Consumer Price

Index inflation. Reported annualized mean, standard deviation, and first-order autocorrelation of consumption growth in Panel B are obtained by

following the convention of aggregating monthly consumption series to annual. Shown in Panel C are the average real return of the risk-free bond and the

average return of equity market, all based on a single simulation run. The 90% bootstrap confidence intervals on the return of risk-free bond and equity

are ½0:0013,0:0048� and ½0:0527,0:1189�, respectively.

Long-run risk External habit Rare Lower

Kelly Bansal- Bekaert- Campbell- disasters bound, s2pc

Yaron Engstrom Cochrane (based on SET A)

Panel A: Permanent component of the SDF, monthly

Var½MPtþ1=MP

t �0.0374 0.0342 0.0280 0.0234 0.0580 0.1254

f0:000g f0:000g f0:001g f0:000g f0:099g [0.045, 0.131]

Data

Panel B: Consumption growth, annualized

Mean 0.0245 0.0189 0.0380 0.0189 0.0205 0.0341

Standard deviation 0.0494 0.0298 0.0175 0.0150 0.0298 0.0285

First-order autocorrelation 0.248 0.517 0.243 0.000 0.000 0.227

Panel C: Real return of a risk-free bond and equity, average (annualized)

Real return of risk-free bond, Rtþ1;1�1 0.0240 0.0252 0.0040 0.0094 0.0200 0.0017

Real return of equity, Rtþ1�1 0.0603 0.0802 0.0583 0.0724 0.0593 0.0864

2 Recent studies include, among others, Bansal, Dittmar, and Kiku

(2009), Constantinides and Ghosh (2008), Croce, Lettau, and Ludvigson

(2008), Ferson, Nallareddy, and Xie (2010), Kaltenbrunner and Lochstoer

(2010), Koijen, Lustig, Van Nieuwerburgh, and Verdelhan (2010), Lettau

and Ludvigson (2004), Piazzesi and Schneider (2006), and Zhou and Zhu

(2009).


(not reported). For example, the highest p-value of 0.099(0.164) corresponds to the model with rare disasters forSET A (SET C).

While the models differ markedly in their capacity togenerate a volatile permanent component, it is note-worthy that each model parametrization reasonablymimics the equity premium and the real risk-free return,while simultaneously calibrating closely to the first-twomoments of consumption growth (see Panels B through Cof Table 2). Thus, there appears to be a tension, within amodel, between matching the sample average of equityreturns and risk-free returns, versus generating a mini-mum volatility of the permanent component stipulatedby theory and as inferred from the data. We furtherexpand on this point when discussing the implication ofthe models for long-term bond returns.

While the backbone of the Kelly (2009) model is toincorporate tails in nondurable consumption growth, ittends to elevate the variance of the permanent compo-nent, modestly relative to Bansal and Yaron (2004).Specifically, in the formulation of Kelly (2009), the dis-tributions of log SDF and the log of the permanentcomponent of the SDF are symmetric with fat tails. Fromthe documented results, we infer that long-run riskmodels could accentuate the variability in the permanentcomponent of SDFs by incorporating more flexible tailproperties in the SDF and in the permanent component ofthe SDF. We recognize nonetheless that there is insuffi-cient evidence favoring the presence of skewness in


nondurable consumption growth. Thus, an avenue toenrich long-run risk models is to incorporate durableconsumption in the dynamics of the real economy. Inthe spirit of our results, Yang (2011) provides evidencethat durable consumption growth is left-skewed andexhibits time-varying volatility.2

Judging by our results, the approach in Bekaert andEngstrom (2010) does not appear to substantially improveupon the variance of the permanent component relativeto Campbell and Cochrane (1999). This finding is some-what unexpected, given a flexible modeling of consump-tion growth as well as the term structure of interest rates,in conjunction with a modified process for marginalutility.

What could be a rationale for the highest estimate ofthe variance of the permanent component of the SDF in anasset pricing model with rare disasters? This outcomedeserves two comments. First, the high variability of thepermanent component arises from occasional crashes inthe consumption growth process, and this success comesat the expense of somewhat unrealistic consumptiongrowth higher-moments. For example, the skewness of




the annualized log consumption growth in the modelwith rare disasters is �11:02, with a kurtosis of 145.06, asdiscussed also in Backus, Chernov, and Martin (2011,Table III). Second, the model with rare disasters admitsthe most right-skewed and fat-tailed distribution of thepermanent component among all of our models.

In summary, although the list of models under con-sideration is far from exhaustive, they still embed differ-ent utility specifications and specify the long-run andshort-run risk in distinct ways (see also Hansen, 2009).Yet, a common thread among the models is their inabilityto meet the lower bound restriction on the variance of thepermanent component of SDFs. The larger lesson beingthat the SDF of the asset pricing models could be refinedto accommodate a permanent component featuring alarger variance. Our metrics of assessment can provide aperspective on how the dynamics of consumption growthand fundamentals could be modified, or preferences couldbe generalized, to improve the working of asset pricingmodels.

5.4. Success of models is confounded by their lack of

consistency with aspects of the bond market

The methods of this paper allow us to contemplate twoadditional questions: (i) Which asset pricing model con-forms with the lower bound on the transitory components2

tc (Eq. (9) of Proposition 2)?, and (ii) What are thequantitative implications of each model for the behaviorof the long-term bond returns?

It bears emphasizing that while the lower bound s2pc is

independent of the mean of the permanent component byconstruction, the lower bound on the transitory compo-nent exhibits dependence on the estimate of the meanEðMT

tþ1=MTt Þ across models. With this feature in mind, we

report s2tc , in relation to the estimate of EðMT

tþ1=MTt Þ,

which is then compared to Var½MTtþ1=MT

t � produced by a

Table 3Assessing the restriction on the variance of the transitory component of SDFs

We compute Var½MTtþ1=MT

t � via simulations, respectively, for the models that

All calculations are based on model parameters tabulated in Bakshi and Chab

values are the averages from a single simulation run of 360,000 months. The rep

Proposition 2. We keep the maturity of the long-term bond to be 20 years, and

Consumer Price Index inflation. The monthly data used in the construction

confidence intervals in square brackets. To compute the confidence intervals,

sampling in the block bootstrap is based on 15 blocks. Reported below the esti

represent the proportion of replications for which model-based Var½MTtþ1=MT

t � e

months. Because the transitory component of the SDF in the model with rare

Long-run risk

Kelly Bansal-

Yaron

Var½MTtþ1=MT

t � 1:7� 10�8 1:9� 10�3

f0:000g f0:000g

Lower bound, s2tc 9:8� 10�7 4:3� 10�1

[0.0000, 0.0000] [0.3404, 0.5026]

EðMTtþ1=MT

t Þ0.9980 1.0147


model in Table 3. The crux of our finding is that modelsgenerate insufficient Var½MT

tþ1=MTt � relative to the lower

bound s2tc . This conclusion is confirmed through the

p-values that examine the restriction s2tc�Var½MT

tþ1=

MTt �r0. We again obtain this p-value for each model by

appealing to a finite sample simulation with 200,000replications. It is seldom that Var½MT

tþ1=MTt � is greater

than the minimum volatility restriction on the transitorycomponent, and the reported p-values are all below 0.01.Our approach essentially identifies a source of the mis-alignment of asset pricing models with aspects of thebond market data.

Moving to the second question of interest, Table 4summarizes the model implications of the transitorycomponent for the moments of the long-term bond. Thekey point to note is that calibrations geared towardreplicating the equity return and the risk-free return canmiss basic aspects of the long-term bond market. Forexample, the Bansal and Yaron, 2004, and the Bekaert andEngstrom, 2010 models imply an expected annualizedlong-term bond return of �14:26% and �0:21%, respec-tively. Given the real return of a long-term bond averages2.43%, there appears to be a gap between the prediction ofthe models and the data.

That the misspecified transitory component is a sourceof the incongruity of models with bond market data isalso revealed through the standard deviation of long-termbond returns. Specifically, when the models fail to gen-erate plausible dynamics of the transitory component,they can introduce a wedge between the volatility oflong-term bond returns implied by a model versus thedata counterpart. Here, it can be seen that the Bansal andYaron (Bekaert and Engstrom) model implies an annual-ized standard deviation of 12.41% (2.36%), which deviatesfrom the data value of 8.87%. While the extant literaturehas largely focused efforts on rationalizing equity returnvolatility (see, for instance, Schwert, 1989; Wachter,

from asset pricing models.

incorporate long-run risk, external habit persistence, and rare disasters.

i-Yo (2011, Tables Appendix-I through Appendix-III), and the reported

orted lower bound s2tc on the transitory component is based on Eq. (9) of

the real returns are computed by deflating the nominal returns by the

of s2tc are from 1932:01 to 2010:12 (948 observations), with the 90%

we create 50,000 random samples of size 948 from the data, where the

mates of Var½MTtþ1=MT

t � are the p-values, shown in curly brackets, which

xceeds s2tc in 200,000 replications of a finite sample simulation over 948

disasters is a constant, its p-value entry is shown as ‘‘na.’’

External habit Rare

Bekaert- Campbell- disasters

Engstrom Cochrane

3:5� 10�5 2:2� 10�6 0.0000

f0:000g f0:000g na

7:7� 10�3 7:8� 10�4 2:0� 10�4

[0.0041, 0.0141] [0.0005, 0.0016] [0.0000, 0.0004]

1.0002 0.9987 0.9983



Table 4Implications of the transitory component of the SDF for the real return of long-term bond.

Parameters tabulated in Bakshi and Chabi-Yo (2011, Tables Appendix-I through Appendix-III), in conjunction with the solution of the eigenfunction

problems, are used to generate fMTtþ1=MT

t g via simulations, respectively, for the models that incorporate long-run risk, external habit persistence, and rare

disasters. Through a single simulation run of 360,000 months, we compute the population values corresponding to:

Eðrtþ1,1Þ ¼ EðRtþ1,1�1Þ ¼ EMT

t

MTtþ1

�1

!, Var½rtþ1,1� ¼ E

MTt

MTtþ1

�EMT

t

MTtþ1

! !2

:

For comparison, also reported are the annualized mean and standard deviation of the real return of long-term bond for the monthly data over 1932:01 to

2010:12. The 90% confidence intervals are shown in square brackets, created from 50,000 random samples of size 948 from the data where the sampling

in the block bootstrap is based on 15 blocks.

Eðrtþ1,1ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVar½rtþ1,1�

pReal return of long-term bond from models, annualized

Long-run risk, Kelly 0.0243 0.0002

Long-run risk, Bansal-Yaron �0.1426 0.1241

External habit, Bekaert-Engstrom �0.0021 0.0236

External habit, Campbell-Cochrane 0.0159 0.0069

Rare disasters 0.0201 0.0000

Real return of long-term bond in the data, annualized

1932:01–2010:12 sample 0.0243 0.0887

[0.0081, 0.0404] [0.0829, 0.0946]


2011), far less effort has been devoted to rationalizingbond return volatility.

The goal to study the behavior of the return of a long-term bond has the flavor of Beeler and Campbell(2009, Table IX), whereby we explore the possible mis-alignment of long-term bond returns across modelsusing the transitory component and by solving the eigen-function problem. Our push to consider the long-termbond return as one criterion in model assessment is alsoguided by Alvarez and Jermann (2005). Specifically, theyargue that, absent the permanent component of theSDF, the maximum risk premium in the economy isreflected in the long-term bond return. The versatility ofan asset pricing model also lies in its ability to generatecredible properties of the long-term bond returns, as alsoelaborated in a different context by Campbell and Viceira(2001).

5.5. Models face challenges capturing the relation between

MPtþ1=MP

t and MTtþ1=MT

t implicit in the data

Motivating the mixed performance of the models sofar, we further ask: What can be discerned about therelation between the transitory and the permanent com-ponents, given that they jointly price a given set of assets?In the analysis to follow, we address this question fromtwo angles.

We first examine the comovement between the tran-sitory and the permanent components embedded in anasset pricing model. More concretely, the solution to theeigenfunction problem allows us to deduce the left-hand


side below (using EðMPtþ1=MP

t Þ ¼ 1Þ:

CovMP

tþ1

MPt

,MT

tþ1

MTt

" #¼ E

Mtþ1

Mt

� ��E

1

Rtþ1,1

� �: ð37Þ

Equivalently, it imparts the following quantitative impli-cation:

CovMP

tþ1

MPt

,MT

tþ1

MTt

" #

VarMT

tþ1

MTt

" # ¼

E1

Rtþ1;1

� ��E

1

Rtþ1,1

� �Var

1

Rtþ1,1

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

can be inputed from the data; and is model�free

: ð38Þ

Note that the left-hand side of (38) can be recovered asthe slope coefficient from the OLS regression: MP

tþ1=MPt ¼

a0þb0ðMTtþ1=MT

t ÞþEtþ1 in the model-specific simulations.The variance of MT

tþ1=MTt is a convenient normalization

that enables the quantity on the right-hand side to beinputed from the data. Asset pricing theory is silent on thejoint distribution of the permanent and the transitorycomponents of SDFs.

Table 5 summarizes (i) the slope coefficient imputedfrom the data, (ii) the estimate of b0 from the regression ina single simulation run, and (iii) 95th and 5th percentilesof the b0 estimates from a finite sample simulation with200,000 replications.

Observe, however, that the numerator on the right-hand side of Eq. (38), is to first-order, the expected returnspread of the long-term bond over the risk-free bond(since 1=ð1þxÞ � 1�x). Thus, a fundamental trait of thedata is that it supports a positive covariance between the



Table 5Implication of the models for the comovement between the permanent and the transitory components of SDFs.

Parameters tabulated in Bakshi and Chabi-Yo (2011, Tables Appendix-I through Appendix-III), in conjunction with the solution of the eigenfunction

problems, are used to generate fMPtþ1=MP

t g and fMTtþ1=MT

t g via simulations, respectively, for the models that incorporate long-run risk and external habit

persistence. Through a single simulation run of 360,000 months, we perform the OLS regression MPtþ1=MP

t ¼ a0þb0ðMTtþ1=MT

t ÞþEtþ1, thereby inferring

the slope coefficient b0 ¼ Cov½MPtþ1=MP

t ,MTtþ1=MT

t �=Var½MTtþ1=MT

t �. In addition, we generate the distribution of b0 in 200,000 replications of a finite sample

simulation over 948 months, and report the values of the mean, the 95th percentile, and the 5th percentile. For comparison, reported also is the inputed

value of Eð1=Rtþ1;1Þ�Eð1=Rtþ1,1Þ=Var½1=Rtþ1,1� for the monthly data over 1932:01 to 2010:12. The 90% confidence intervals are shown in square

brackets, created from 50,000 random samples of size 948 from the data where the sampling in the block bootstrap is based on 15 blocks. Since the

transitory component in the model with rare disasters is a constant, the slope coefficient is identically zero, which renders the finite sample simulation

redundant. For this reason, some entries are shown as ‘‘na.’’

Estimate of b0 in the regression:MP

tþ 1

MPt

¼ a0þb0MT

t þ 1

MTt

� �þEtþ1

Single Finite sample simulation

simulation run (distribution of b0)

b0, population Mean 95th 5th

Long-run risk, Kelly �5.88 �5.88 57.68 �69.56

Long-run risk, Bansal-Yaron �2.43 �2.44 �2.22 �2.65

External habit, Bekaert-Engstrom �16.25 �15.55 �9.28 �22.73

External habit, Campbell-Cochrane 0.23 0.00 0.19 �0.28

Rare disasters 0.00 na na na

Imputed from the data, b0

1932:01–2010:12 sample 1.96

[1.07, 3.96]


permanent and the transitory components of SDFs. Thevalue of b0 implied from the data is 1.96.

A comparison of the slope coefficients obtainedthrough our simulations elicits the observation that, onaverage, three (one) out of five models producenegative (positive) slope coefficient whose magnitudescontrast the data counterpart. The 95th and 5th percen-tiles for the b0 distribution in the models of Bansal andYaron and Bekaert and Engstrom are negative, suggestingthat b0o0 appears to be an intrinsic feature of theirmodels. Overall, this evidence illustrates the ambivalenceof the models in replicating the association between thepermanent and the transitory components implicit inthe data.

In addition, recognize in our context that a statistic togauge the performance of models in capturing the jointpricing of bonds and other assets is the lower bound onVar½ðMP

tþ1=MPt Þ=ðM

Ttþ1=MT

t Þ�. Complementing the picturefrom Tables 2 and 3, the results in Table 6 show that assetpricing models are unable to describe the behavior of jointpricing implicit in the bound. The p-values that examinethe restriction s2

pt�Var½ðMPtþ1=MP

t Þ=ðMTtþ1=MT

t Þ�r0 aretypically small, refuting another restriction suggested bythe theory.

The relation between the permanent and the transi-tory components of the SDF has information content formodeling asset prices. A particular lesson to be gleaned isthat asset pricing models could be enriched to betterdescribe the joint dynamics of the transitory and thepermanent components of SDFs as also reflected in bond


risk premium, a positive equity risk premium, and the riskpremium on a broad spectrum of assets.

5.6. Models may reproduce some asset market phenomena,

but find it onerous to satisfy bounds

Synthesizing elements of Tables 2, 3, and 6, we posethree additional questions in turn.

First, why is it that formulations from an asset pricingmodel can come close to duplicating the observed equitypremium, and at the same time not satisfy the lowerbound on the permanent component of the SDF for abroad set of asset returns? We expand on this seeminglycontradictory observation by computing the lower boundon the permanent component when the SDF is required tosatisfactorily price the long-term bond, the risk-free bond,and the market equity [returning to the set of assets inAlvarez and Jermann (2005)]. For this set of assets,s2

pc ¼ 0:0185, and the minimum p-value examining therestriction s2

pc�Var½MPtþ1=MP

t �r0 is 0.314 across models.The nuance is that asset pricing models can meet therestrictions for a narrower set of assets, but not thebroader set of assets that includes the 25 (6) Fama-Frenchequity portfolios (as in SET A (SET C)).

As one enlarges the set of assets in Rtþ1, the perma-nent component of the SDF must be generalized to copewith the risk premiums on a wider array of assets. Thisprompts the next question: What is our incremental valuebeyond the L-measure in comparing asset pricing models?The analysis below (with p-values computed as before)



Table 6Assessing the restriction on the variance of the ratio of the permanent to the transitory components of SDFs from asset pricing models.

We compute ðVar½MPtþ1Þ=MP

t =ðMTtþ1=MT

t Þ� via simulations, respectively, for the models that incorporate long-run risk, external habit persistence, and rare

disasters. All calculations are based on model parameters tabulated in Bakshi and Chabi-Yo (2011, Tables Appendix-I through Appendix-III), and the reported

values are the averages from a single simulation run of 360,000 months. The reported lower bound s2pt on the ratio of the permanent to the transitory

components is based on Eq. (11) of Proposition 3. The bound is based on the return properties of Rtþ1, corresponding to SET A, and the long-term bond.

Monthly data used in the construction of s2pt are from 1932:01 to 2010:12 (948 observations), with the 90% confidence intervals in square brackets. To

compute the confidence intervals, we create 50,000 random samples of size 948 from the data, where the sampling in the block bootstrap is based on 15

blocks. Reported below the estimates of Var½ðMPtþ1=MP

t Þ=ðMTtþ1=MT

t Þ� are the p-values, shown in curly brackets, which represent the proportion of replications

for which model-based Var½ðMPtþ1=MP

t Þ=ðMTtþ1=MT

t Þ� exceeds s2pt in 200,000 replications of a finite sample simulation over 948 months.

Long-run risk External habit Rare

Kelly Bansal- Bekaert- Campbell- disasters

Yaron Engstrom Cochrane

VarMP

tþ 1

MPt

MT

t þ 1

MTt

� �0.0375 0.0449 0.0298 0.0234 0.0582

f0:000g f0:000g f0:001g f0:000g f0:094g

Lower bound, s2pt

0.1309 0.1842 0.1237 0.1264 0.1324

[0.0481, 0.1381] [0.0868, 0.1982] [0.0429, 0.1296] [0.0448, 0.1327] [0.0491, 0.1399]

EMP

tþ 1

MPt

MT

tþ 1

MTt

� �1.0018 0.9916 1.0008 1.0012 1.0019


speaks to the relevance of the tightness of our bounds inempirical applications:

Kelly Bansal-Yaron Bekaert-Engstrom Campbell-Cochrane Rare Disasters Lower bound

L½MP

t þ 1

MPt

�0.0176 0.0171 0.0100 0.0115 0.0032 0.0041

p-value f0:986g f0:990g f0:999g f0:994g f0:254g

In particular, our results point to a scenario in whichmodels generate high (low) p-values with the L-measure(variance-measure), as seen by comparing the entries forp-values in Table 2. It is the stringency of the lowervariance bound, together with a set of empirically rele-vant assets in Rtþ1, that enhances the flexibility of thebounds approach to differentiate between competingasset pricing models.

Building on our observations, we finally ask whetheran asset pricing model can satisfy the Hansen andJagannathan (1991) bound for a broad set of assets, forinstance, SET A, and yet fail the lower bound on thepermanent component in Proposition 1. Bakshi andChabi-Yo (2011, Lemma 1) establish the relevance of abound on Cov½ðMP

tþ1=MPt Þ

2,ðMTtþ1=MT

t Þ2�, with the implica-

tion that further theoretical work is needed to get a firmergrasp of the joint distribution of Mp

tþ1=Mpt and MT

tþ1=MTt .

More can be learned about the functioning of asset pricingmodels through the lens of permanent and transitorycomponents of SDFs.

6. Conclusions and extensions

This paper presents a variance bounds framework inthe context of permanent and transitory components of


stochastic discount factors. At the center of our approachare three theoretical results, one related to a lower bound

on the variance of the permanent component, another onthe lower bound on the variance of the transitory com-ponent, and also a lower bound on the variance of theratio of the permanent to the transitory components ofstochastic discount factors. Instrumental to the tasks athand, we establish the tightness of our variance boundsrelative to Alvarez and Jermann (2005), and we show thatour bounds can be useful in asset pricing applications. Aspecific conclusion is that bound implications for thepermanent component of the stochastic discount factorsin the setting of Alvarez and Jermann (2005) are con-siderably weaker than those reported in our context of ageneric set of assets. Our analysis furnishes bounds thatincorporate information from average returns as well asthe variance-covariance matrix of returns.

Combining the variance bounds with the eigenfunctionproblem offers guidance for asset price modeling inseveral ways. First, we present the solution to the eigen-function problem for five asset pricing models in the classof long-run risk, external habit persistence, and raredisasters. This solution justifies the calculation of themoments of the transitory and the permanent compo-nents, as well as all its mixed moments. Second, wecorroborate that models face a particular impedimentsatisfying the lower bound restrictions imposed by our




bounds, even when the models are successful in matchingthe equity premium and the return of the risk-free bond.Third, we find that the models are not compatible withthe return properties of the long-term bond. Finally, whilethe data support a positive comovement between thetransitory and the permanent components, our analysisreveals that this feature is not easily reconciled within ourparametrization of asset pricing models.

Our work could be extended. While our focus isdirected toward primary assets, one could modify theanalysis to include stochastic discount factors that alsosatisfactorily price out-of-the-money put options on themarket as well as claims on market variance. Incorporat-ing such claims can impose further hurdles on an assetpricing model.

In addition, our framework is amenable to investigatingthe suitability of stochastic discount factors to addressseveral asset pricing puzzles together, for instance, bycombining elements of the value premium, the equitypremium, the risk-free return, and the bond risk premium.In this regard, our work is related to, among others, Ai andKiku (2010), Aı̈t-Sahalia, Parker, and Yogo (2004), Cochraneand Piazzesi (2005), Koijen, Lustig, and Van Nieuwerburgh(2010), Lettau and Wachter (2007), Routledge and Zin(2010), Santos and Veronesi (2010), and Zhang (2005).Overall, our variance bounds approach, when combinedwith the eigenfunction problem, could refine the quest fora better understanding of asset returns.

Appendix A. Proofs of unconditional bounds

In the results that follow, we provide proofs ofPropositions 1–3.

Proof of Proposition 1. The proof is by construction.Recognize that

EMP

tþ1

MPt

�EMP

tþ1

MPt

! !Rtþ1

Rtþ1,1�E

Rtþ1

Rtþ1,1

� �� !

¼ 1�ERtþ1

Rtþ1,1

� �, ðA:1Þ

where the right-hand side of (A.1) is obtained by notingthat EðMP

tþ1=MPt Þ ¼ 1 and Rtþ1,1 ¼ ðM

Ttþ1=MT

t Þ�1. Denote

O� ERtþ1

Rtþ1,1�E

Rtþ1

Rtþ1,1

� �� Rtþ1

Rtþ1,1�E

Rtþ1

Rtþ1,1

� �� 0� �

and B� 1�ERtþ1

Rtþ1,1

� �: ðA:2Þ

Multiply the right-hand side of (A.1) by B0O�1 to obtain:

B0O�1B¼ EMP

tþ1

MPt

�EMP

tþ1

MPt

! !B0O�1 Rtþ1

Rtþ1,1

� �E B0O�1 Rtþ1

Rtþ1,1

� ��,

ðA:3Þ

r VarMP

tþ1

MPt

" # !1=2

� Var B0O�1 Rtþ1

Rtþ1,1

� �� 1=2

: ðA:4Þ

Given that Var½B0O�1ðRtþ1=Rtþ1,1Þ� equals B0O�1B, our

application of the Cauchy-Schwarz inequality implies thatðB0O�1BÞ1=2r ðVar½MP

tþ1=MPt �Þ

1=2. Thus, we have estab-lished the bound in Eq. (6). &


Proof of Proposition 2. For brevity, denote mtc � EðMTtþ1=

MTt Þ, which is the mean of the transitory component of the

SDF. Again the proof is by construction, whereby werecognize that

EMT

tþ1

MTt

�EMT

tþ1

MTt

! !ðRtþ1,1�EðRtþ1,1ÞÞ

!¼ 1�mtcEðRtþ1,1Þ,

ðA:5Þ

since EððMTtþ1=MT

t ÞRtþ1,1Þ ¼ 1. Multiplying both sidesof (A.5) by ð1�mtcEðRtþ1,1ÞÞ=Var½Rtþ1,1�, and using theCauchy-Schwarz inequality validates the bound in Eq. (9)of Proposition 2. &

Proof of Proposition 3. Observe that

E

MPtþ1

MPt

MTtþ1

MTt

�E

MPtþ1

MPt

MTtþ1

MTt

0BBBB@

1CCCCA

0BBBB@

1CCCCA Rtþ1

R2tþ1,1

�ERtþ1

R2tþ1,1

! !0BBBB@

1CCCCA

¼ 1�mptERtþ1

R2tþ1,1

!, ðA:6Þ

where recalling the notation mpt ¼ EððMPtþ1=MP

t Þ= ðMTtþ1=

MTt ÞÞ. We denote

S� ERtþ1

R2tþ1,1

�ERtþ1

R2tþ1,1

! !Rtþ1

R2tþ1,1

�ERtþ1

R2tþ1,1

! !0 !

and D� 1�mptERtþ1

R2tþ1,1

!: ðA:7Þ

Multiply (A.6) by D0S�1 and apply the Cauchy-Schwarzinequality to the left-hand side of (A.6). &

Appendix B. Proofs of unconditional bounds thatincorporate conditioning information

Proof of Propositions 1–3 with conditioning variables. Fortractability of exposition, we denote by zt the set ofconditioning variables. The variable zt predicts Rtþ1.We note that

EMP

tþ1

MPt

�EMP

tþ1

MPt

! !z0tRtþ1

Rtþ1,1�E

zt0Rtþ1

Rtþ1,1

� �� !

¼ Eðz0t1Þ�Ez0tRtþ1

Rtþ1,1

� �: ðB:1Þ

Eq. (B.1) follows, since the permanent component of thepricing kernel is a martingale. Now, denote

Y� Ez0tRtþ1

Rtþ1,1�E

z0tRtþ1

Rtþ1,1

� �� 2

and H� Eðz0t1Þ�Ez0tRtþ1

Rtþ1,1

� �: ðB:2Þ

Multiplying (B.1) by H0Y�1 and applying the Cauchy-Schwarz inequality, we obtain H0Y�1HrVar½MP

tþ1=MPt �.

The same approach can be used to derive Propositions2 and 3 with conditioning variables. &

Appendix C. Solution to the eigenfunction problem inKelly (2009)

Our end-goal is to present the permanent and thetransitory components of the SDF by solving the




eigenfunction problem. To save space, the intermediatesteps are provided in Bakshi and Chabi-Yo (2011).

Under the dynamics of the real economy posited in(21)–(23) and recursive utility, the SDF is,

logMtþ1

Mt

� �¼ xtþD1ðgtþ1�Etðgtþ1ÞÞþD2ðxtþ1�Etðxtþ1ÞÞ

þD3ðs2M,tþ1�Etðs2

M,tþ1ÞÞþD4ðLtþ1�EtðLtþ1ÞÞ,

ðC:1Þ

where the innovation in market volatility s2M,tþ1�

Etðs2M,tþ1Þ is displayed in (C.11). We determine

D1 ¼�lg , D2 ¼�lx, D3 ¼�ls

ðs2gþk2

1A2xs2

x Þ,

D4 ¼2ls

ðs2gþk2

1A2xs2

x Þ�lL, ðC:2Þ

with xt ¼ y logðbÞ� yc ðmþxtÞþðy�1ÞEtðrM,tþ1Þ. Furthermore,

lg ¼ 1�yþyc

, lx ¼ ð1�yÞk1Ax, ls ¼ ð1�yÞk1As,

lL ¼ ð1�yÞk1AL, and ðC:3Þ

Ax ¼

1�1

c1�k1rx

, AL ¼

y 1�1

c

� �1�

1

c

� �1�k1rx

,

As ¼y2

1�1

c

� �2

s2gþk2

1A2xs2

x

1�k1rs

0BBB@

1CCCA, ðC:4Þ

A0 ¼

logðbÞþ 1�1

c

� �mþk0þk1ðAss2

ð1�rsÞþALLð1�rLÞÞþ12 yk

21ðA

2ss2

sþA2Ls2

LÞ

1�k1: ðC:5Þ

The innovation in market volatility is derived using theexpression for excess return:

rM,tþ1�EtðrM,tþ1Þ ¼ k1ALsLzL,tþ1þsgstzg,tþ1

þffiffiffiffiffiffiLt

pWg,tþ1þk1Axsxstzx,tþ1

þk1Assszs,tþ1, ðC:6Þ

EtðrM,tþ1Þ ¼ k0þk1ðA0þAxEtðxtþ1ÞþAsEtðs2tþ1Þ

þALEtðLtþ1ÞÞ�wgtþEtðgtþ1Þ, ðC:7Þ

where

Etðxtþ1Þ ¼ rxxt , Etðs2tþ1Þ ¼ s

2ð1�rsÞþrss

2t , ðC:8Þ

EtðLtþ1Þ ¼Lð1�rLÞþrLLt , Etðgtþ1Þ ¼ mþxt : ðC:9Þ

Using the moment generating function of the Laplacevariable

Vart½rM,tþ1� ¼ s2M,tþ1 ¼ k

21ðA

2Ls

2LþA2

ss2sÞ

þðs2gþk

21A2

xs2x Þs

2tþ1þ2Ltþ1: ðC:10Þ

Then the innovation in market variance is

s2M,tþ1�Etðs2

M,tþ1Þ ¼ ðs2gþk

21A2

xs2x Þsszs,tþ1þ2sL zL,tþ1:

ðC:11Þ


Moving to the transitory component of the SDF, defineXt ¼ ðxt ,s2

t�s2,Lt�LÞ0 for notational simplicity. The aim

is to solve the eigenfunction problem

EtMtþ1

Mt

1

Metþ1

!¼

uMe

t

,

or equivalently, EtððMtþ1=MtÞe½Xtþ1�Þ ¼ ue½Xt�. Under ourassumptions, we conjecture that the solution is of the form

e½Xtþ1� ¼ expðc0Xtþ1Þ, c¼ ðc1,c2,c3Þ0: ðC:12Þ

Skipping tedious algebra, we deduce that

u¼ exp

y logðbÞþðy�1Þk0þðy�1Þk1A0�ycm

þðy�1Þk1Ass2ð1�rsÞ

þðy�1Þmþðy�1Þk1ALLð1�rLÞ

�ðy�1ÞðA0þAss2þALLÞ

þðy�1ÞALk1rLLþðy�1ÞAsk1rss2

þ 12 ðl

2ss2

sþl2Ls2

LÞ�ðlss2sc2þlLs2

Lc3Þ

þ 12 ðc

22s2

sþc23s2

LÞþl2gL

þð�lxs2x c1þ

12 c2

1s2xþ

12l

2gs2

gþ12 l

2xs2

x Þs2

0BBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCA

, ðC:13Þ

and

c1 ¼�lgþðy�1ÞAxðk1rx�1Þ

1�rx

,

c2 ¼�lxs2

x c1þ12 c2

1s2xþ

12 l

2gs2

gþ12 l

2xs2

xþðy�1ÞAsðk1rs�1Þ

1�rs,

c3 ¼l2

gþðy�1ÞALðk1rL�1Þ

1�rL: ðC:14Þ

The transitory component of the SDF is MTtþ1=MT

t ¼

u expð�c0ðXtþ1�XtÞÞ, and the permanent component isdetermined accordingly. &

Appendix D. Solution to the eigenfunction problem inBekaert and Engstrom (2010)

Write qtþ1�dq ¼ rqðqt�dqÞþsqpop,tþ1þsqnon,tþ1,with mq=ð1�rqÞ � dq. Denote the state vector as Zt ¼

ðqt�dq,xt ,pt�p,nt�nÞ0. In the eigenfunction problemEtððMtþ1=MtÞe½Ztþ1Þ ¼ ue½Zt�, conjecture that e½Zt� ¼ expðc0ZtÞ. Because getþ1 and betþ1 are independent andfollow a gamma distribution,

Etðexpðl getþ1ÞÞ ¼ expð�pt logð1�lÞÞ, and

Etðexpðl betþ1ÞÞ ¼ expð�nt logð1�lÞÞ, ðD:1Þ

for some real number l. Then we manipulate a set ofequations to determine the eigenvalue as

u¼ expðlogðbÞ�gg�a0p�b0n�p logð1�a0Þ�n logð1�b0ÞÞ,

ðD:2Þ




and

c1 ¼�g, c2 ¼�g

1�rx

, c3ð1�rpÞþa0þ logð1�a0Þ ¼ 0,

c4ð1�rnÞþb0þ logð1�b0Þ ¼ 0, ðD:3Þ

where setting a0 ��gsgp�ðgsxp=ð1�rxÞÞþc3spp and b0 �

gsgn�ðgsxn=ð1�rxÞÞþc4snn. This concludes our proof. &

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Journal of Financial Economics - NYUpages.stern.nyu.edu/~dbackus/BCZ/bakshi_chabi-yo_jfe2012...equity portfolio or the equity market. Such an asset captures the equity risk premium,