Leisure, Growth and Long Run Risk.∗

Harald UhligUniversity of Chicago

PRELMINARYCOMMENTS WELCOME

First draft: March 19th, 2006This revision: November 11, 2007

∗A previous version has circulated under the title “Asset Pricing with Epstein-ZinPreferences.”. I am grateful to Wouter den Haan, Fernando Alvarez, Lars Hansen and toparticipants at seminars at MIT, the Federal Reserve Bank of Chicago, GSB Chicago, andKen Judd for useful comments. This research was supported by the Deutsche Forschungs-gemeinschaft through the SFB 649 “Economic Risk”. Address: Prof. Harald Uhlig,Department of Economics 1126 East 59th Street Chicago, IL 60637, USA. e-mail: huh-lig@uchicago.edu, fax: (773) 834-3452.

Abstract

Bansal and Yaron (2004) and Hansen, Heaton and Li (2005) haveshown how news about future consumption and dividend growth im-pacts on current asset prices, if agents have Epstein-Zin preferences.This paper shows that leisure must be taken into account when pric-ing assets with Epstein-Zin preferences, if leisure affects per-periodfelicity, even if it enters separably. The compatibility of preferenceswith long-run growth in consumption and stationarity in leisure isexamined. A log-linear approximation for asset pricing is provided,showing how news about future consumption and leisure matter forasset prices. The asset pricing formulas are evaluated empirically. Itis shown, that the long-run risk price due to the evolution of leisurecan be large.

Keywords: consumption-based asset pricing, business cycle, calibra-tion, equity premium, Sharpe ratio, nonseparability between consumptionand leisure, long-run growth, Epstein-Zin preferences, long-run risk

JEL codes: E32, G12, E22, E24

2

1 Introduction

This paper examines asset pricing with Epstein-Zin preferences, allowing fornonseparabilities between consumption and leisure as well as trend growth inconsumption. A log-linear approximation for the asset pricing formula is pro-vided, showing how news about future consumption growth as well as leisurematters for asset prices. The asset pricing formulas are evaluated empirically.It is shown that the term due to incorporating leisure is quantitatively large.

Since Mehra and Prescott (1985) have pointed out the puzzling equity pre-mium, many approaches have been undertaken to provide preference-based(or production-based) theories consistent with observed asset-pricing facts.Surveys of the literature are e.g. in Cochrane (2001) or Campbell (2003), anda list of “exotic preferences” generated by the ensuing research can be foundin Backus, Routledge and Zin (2004). Recently, the approach of separatingintertemporal elasticity of substition from risk aversion, i.e. Kreps-Porteus(1978), Weil (1990) or Epstein-Zin (1989) preferences1, has received renewedinterest. In a path breaking contribution, Bansal and Yaron (2004) haveshown how these preferences give rise to considerations of long run risk forpricing assets. More precisely, the correlation between current returns andnews about future growth rates of consumption matter. Bansal and Yaron(2004) argue that the data is consistent with a small, but very persistentcomponent in long run consumption growth. Moderate levels of risk aversionthen suffice to explain the observed equity premium.

Their research has triggered a literature, investigating these issue in fur-ther depth. Building on the operator framework of Hansen and Scheinkman(2005), Hansen, Heaton and Li (2005) extract the long-run risk component byanalyzing persistent eigenvalues in the evolution of the state space. Piazzesiand Schneider (2007a) use the approach to link inflation and consumption tothe term structure of interest rates. Bansal, Kiku and Yaron (2006) provideempirical tools. Ang and Bekaert (2007) investigate the statistical fragilityof inferences regarding the low-frequency component of consumption growth.Bansal (2007) provides an application to several international asset markets.More broadly, the long run risk perspective is shown to be important for un-derstanding asset prices, see Alvarez and Jermann (2005), for modelling the

1These preferences can be reinterpreted within a robust-control perspective, see Hansen,Sargent and Tallarini, 1999.

value premium, see Lettau and Wachter (2007) or the impact of housing onasset prices, see Piazzesi and Schneider (2007b) and Jaccard (2006). Com-bining a special role of housing with Epstein-Zin preferences is taken up inFillat (2007), who shows that the housing share in total spending generatesa third asset pricing factor.

With the exception of Fillat (2007), the Epstein-Zin-based literature prac-tically exclusively uses consumption to price assets. From the perspective oflinking it up with macroeconomics, it is natural to consider preferences thatalso depend on leisure, however. While one may ignore parts stemmingfrom the leisure term in standard asset pricing exercises, provided leisureand consumption enter the felicity function separably, this is no longer truewith Epstein-Zin preferences. If felicity depends at all on leisure, so will theEpstein-Zin aggregator, and therefore the asset pricing equations.

This is a key insight of this paper. It and the resulting asset pricing equa-tion will be discussed in section 2, where I focus first on the special case of aseparable logarithmic felicity function. A number of researchers have stressedthat nonseparabilities between consumption and leisure are important to ex-plain key facts regarding labor markets, see e.g. Hall (2006). Likewise, manyaggregate models typically feature trending variables, e.g. due to a unit rootor trend growth in the log of total factor productivity. I therefore investigatea more general approach, allowing for trends and externalities in consump-tion growth as well as nonseparabilities between consumption and leisure insection 3.

Most approaches to pricing assets, using Epstein-Zin preferences, typi-cally substitute out some unobserved value function by an expression, involv-ing the value of the market or the value of the future consumption streamin order to derive testable implications, see in particular Epstein and Zin(1989) and Campbell (1996). This makes it both hard to apply this frame-work to the case of non-representative agents as well as applying it in thesolution of business cycle models, as in e.g. Tallarini (2000), Gomes andMichaelides (2006) or Kaltenbrunner and Lochstoer (2006). This paper in-stead follows the lead of Hansen, Heaton and Li (2005) to derive asset pricingequations from a log-linear expansion around some steady state or some well-understood benchmark (such as the case of unitary intertemporal elasticityof substitition). I derive an approximate infinite-horizon formula for pric-ing assets, involving future news about leisure and consumption. Combinedwith assumptions regarding the evolution of the state of the economy, an

2

operator-based pricing equation can then be re-derived and compared to theresults in Hansen-Heaton-Li (2005) as well as Campbell (1996). An empiricalapplication is provided.

Rather than evaluating the resulting asset pricing equations with theavailable apparatus of numerical techniques, see e.g. Judd (1998), I provideasset pricing formulas, using a log-linear framework. The log-linear frame-work has become a useful point of departure for understanding key featuresof asset pricing more deeply, and is therefore a useful and perhaps evennecessary complement to an entirely numerical evaluation. For a numberof reasons, in particular the reasons emphasized in Weitzman (2005), theapproximation here should be regarded as a “small-shock” approximation,rather than an asset pricing framework that properly handles tail-events.

2 Motivation and Preview of Results

I use capital letters to denote the original variables, and small letters to de-note log-deviations from a steady-state growth path (unless explicitly statedotherwise).

Consider recursive preferences given by

Vt = (1− β) (log(Ct) + κ log(Lt)) +β

1− νlog (Et [exp((1− ν)Vt+1)]) (1)

where Ct is consumption and Lt is leisure. Alternatively - as is more commonin the literature - define the recursive utility in terms of the transformation

Vt = exp(Vt)

so that

Vt = exp ((1− β) (log(Ct) + θ log(Lt)) + β log (R(Vt+1))) (2)

where the risk-adjusted expectation operator R(Vt+1) is given by

R(Vt+1) =(Et

[V1−ν

t+1

]) 11−ν

Either by direct calculation or per section 4.3 in Hansen, Heaton and Li(2005), one can show that the stochastic discount factor for valuing a claim

3

to consumption next period is given by

Mt+1 = βCt+1

Ct

exp((1− ν)Vt+1)

Et[exp((1− ν)Vt+1)]= β

Ct+1

Ct

V1−νt+1

(R(Vt+1))1−ν (3)

If κ 6= 0, then Mt+1 will be a function of leisure at date t + 1 and beyond,since this is true for Vt+1, despite the fact that leisure enters separably in thefelicity function

U(Ct, Lt) = log(Ct) + κ log(Lt)

Let some asset have the return process Rt+1 and assume that (rt+1, ct+1, lt+1) =(log Rt+1, log Ct+1, log Lt+1) is jointly normal and covariance stationary. LetRf

t be the risk-free return from t to t+1. The calculations below imply, thatthe equity premium is approximately given by

log Et [Rt+1]− log Rft = Covt (rt+1, ∆ct+1)

+(ν − 1)Covt

rt+1,

∞∑

j=0

βj∆ct+1+j

+(ν − 1)(1− β)κCovt

rt+1,

∞∑

j=0

βjlt+1+j

The first two terms are familiar from the literature. While the first termmeasures the usual risk premium due to one-period intertemporal substitu-tion, the second term arises from the Epstein-Zin specification, highlightingthe role of long-term risk. In particular, current news about future consump-tion growth matters for asset pricing, if these news are correlated with theone-period ahead equity returns: this is the key insight of Bansal and Yaron(2004) and analyzed in greater depth in Hansen, Heaton and Li (2005). Iflog consumption is exactly a random walk, then this long-term risk piece willnot matter, however. Since log consumption appears to be at least close toa random walk, much of the debate in the literature has been whether theretruly is a low-frequency component to consumption growth or whether justthe possibility of such a low-frequency component should make investors becautious regarding that possibility, see Hansen (2007) or Ang-Bekaert (2007).Figures 1 and 2 show what is at stake: both the autocorrelation of consump-tion growth at higher lags as well as the correlation of returns with futureconsumption growth may be zero and it is hard to reject zero for both2.

4

0 2 4 6 8 10 12−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Autocorrelations for ∆ c

j (Quarters)

Figure 1: Autocorrelation of consumption changes, ∆c

0 2 4 6 8 10 12−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Correlations corr(rt,∆ c

t+j)

j (Quarters)

Figure 2: Correlations: stock returns, future first-diff. log cons.

5

0 2 4 6 8 10 12−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2Autocorrelations for ∆ l

j (Quarters)

Figure 3: Autocorrelation of Leisure changes, ∆l

0 2 4 6 8 10 12−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Correlations corr(rt,∆ l

t+j)

j (Quarters)

Figure 4: Correlations: stock returns, future first-diff. log leisure

6

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4Autocorrelations for l

j (Quarters)

Figure 5: Autocorrelation of leisure levels

0 2 4 6 8 10 12−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Correlations corr(rt,l

t+j)

j (Quarters)

Figure 6: Correlations: stock returns, future log leisure levels

7

If κ 6= 0 however, i.e. if leisure does not enter the felicity function above,then the third term in the expression above enters the risk premium. Sincethere is considerably more predictability in leisure, taken as the differencebetween a fixed time endowment and observable hours worked, the effectof pricing assets based on news about leisure in the future is likely to benontrivial.

Figures 3 and 4 show the autocorrelation of leisure growth as well as thecorrelation of current returns and future changes in leisure. Note, however,that it is not the first difference of log leisure that shows up in the assetpricing equation, but the level of log leisure. Therefore, figures 5 and 6 showthe autocorrelation of the levels of leisure as well as the correlation of currentreturns and future levels of leisure.

Since currently high returns tend to predict upswings in economic activity,i.e., the correlation between returns now and leisure changes in the futureis negative. As a result, the leisure term in the equity premium calculationabove depresses rather than raises the equity premium, compared to the caseof not taking leisure into account.

The development so far leaves several issues open. First, if consumption isa random walk, it is no longer covariance stationary. More generally, in manymacroeconomic applications, one may wish to rescale consumption with someexogenously given trend Φt in order to render it stationary. It may be that Φt

represents a slow-moving and exogenous drift to consumption, a time trendor a random walk with drift in the log of total factor productivity, or it maybe that Φt represents a consumption externality, capturing “catching up withthe Joneses”, see Abel (1990).

Second, one may wish to specify preferences more generally than above.In particular, one may not wish to fix the intertemporal elasticity of substitu-tion at 1, but should allow for that value as a special case. Furthermore, thereis no a priori reason to assume that the cross elasticity between consumptionand leisure is zero, as assumed here.

Third, the asset pricing formulas below will be derived from a first-orderapproximation to the first-order conditions (or second-order approximationto utility functions). To that end, it is not necessary to completely specify

2Shown here are simply the raw correlations. In terms of the theory, the correlation ofnews and not the raw correlations are relevant. This is taken up in section 8 with resultsin section 9, where also the data is described in more detail.

8

the functional form. Nonetheless, in order to allow for consumption growthwithout changing either the intertemporal elasticity of substitution or riskaversion, one needs that per-period felicity with respect to consumption aswell as the recursive preference specification exhibits some form of constantelasticity. When preferences only depend on consumption, this is also centralto the theory developed in Dolmas (1996) and Farmer-Lahiri (2005). Here,preferences also depend on leisure, assumed stationary, where we thus canourselves more liberty.

Let log(Γ) be the mean of the difference of log consumption. The bench-mark example for the theory to follow shall be

Vt = (1−β)U(Ct, Lt)+β

(1

1− η

(E

[((1− η)(Vt+1 + χ))

1−ν1−η

]) 1−η1−ν − χ

)(4)

where

U(Ct, Lt) =(Ctv(Lt))

1−η

1− η− χ

χ =1− β

1− βΓ1−η

η

1− η

(Cv(L)

)1−η

Subtracting a suitable constant in the felicity function as well as addingand subtracting it in suitable place in the definition of recursive preferencesallows me to incorporate the case of logarithmic preferences for η → 1, whichI discussed above. The particular constant chosen here is a normalizationand makes some of the algebra more convenient. In particular, one extramarginal unit of utility corresponds to a permanent increase of consumptionby a marginal unit.

Generally, I shall allow the felicity function U(Ct, Lt; Φt) to depend on theexogenous and possibly stochastic trend Φt in such a way, that everythingcan be rendered stationary after a suitable transformation.

3 Preferences

The preferences I wish to examine are given by

Vt = (1− β)U(Ct, ΦtLt; Φt) (5)

+βH−1 (Et [H(Vt+1)])

9

where Ct denotes consumption and Lt denotes leisure, and where Φt is an ex-ogenous and possibly trending variable, and where I assume that (Ct/Φt, Lt) ∈D for some open convex subset D of IR2

++. I make several assumptions.

Assumption A. 1 (Φt/Φt−1, Ct/Φt, Lt) is a stationary and strictly positivestochastic process. Furthermore, the logarithmic means Γ, C and L satisfying

log(Γ

)= E [log Φt − log Φt−1]

log(C

)= E [log Ct − log Φt]

log(L

)= E [log Lt]

are well defined with (C, L) ∈ D.

This assumption is satisfied in many stochastic models, where the drivingprocess as well as the resulting economic variables, including consumptionand leisure, are stationary and their logs have finite means. It is also satis-fied in many models with a stochastically trending total factor productivity,provided log Φt is cointegrated with the log of total factor productivity. Thisincludes models with preference shocks and stochastic trend growth, wherethe productivity of leisure grows with the productivity of market labor in thelong run. Interestingly, it also includes models with catching-up-with-the-Joneses preferences, where Φt is some average of present and past aggregateconsumption so that the ratio of private consumption Ct to the aggregationvariable Φt is stationary.

Assumption A. 2 U(·, ·; ·) is twice continously differentiable. It is concaveand strictly increasing in its first two arguments.

The role of the third argument in U(·, ·; ·) will become clearer below. Forthe other two arguments, this is a standard assumption.

Assumption A. 3 The functions obey the following functional form restric-tions.

U(ΦC, L; Φ) = Φ1−η(U(C,L) + χ

)− χ (6)

andH(V ) = ((1− η)(V + χ))

1−ν1−η (7)

10

for some function U(·, ·) and parameters η > 0, ν > 0, χ, χ satisfying

ν > 0

(1− η)(U(C, L) + χ

)≥ 0, all (C, L) ∈ D

β = βΓ1−η < 1

χ =1− β

1− βχ

Note that U(·, ·) must be concave.This may appear at first to be a strange assumption. Intuitively, this

assumption assures that the curvatures do not change as the economy growsricher. Instead, risk aversion with respect to relative gambles in consumptionremains stationary. Likewise, the tradeoff between leisure and consumptionremains stationary (i.e. income and substitution effects balance), when wagesgrow with consumption. The same considerations have led many researchersto assume CRRA specifications in growing economies with separable pref-erences: the assumption above constitutes a generalization to the case ofnonseparabilities across time. These assertions follow from the following fun-damental property of the preference formulation above. Define

Ct =Ct

Φt

Vt = Φη−1t (Vt + χ)− χ

Proposition 1 Equation (5) can be rewritten as

Vt = (1− β)U(Ct, Lt) + (8)

βH−1

(Et

[(Φt+1

ΓΦt

)1−ν

H(Vt+1)

])

Proof: Note that for any ϕ and x, one has ϕ1−η (H−1(x) + χ) =H−1(ϕ1−νx) + χ, as long as everything is well-defined. Note that χ =(1− β)χ + βχ. Finally, note that

H(Vt+1) =((1− η)Φ1−η

t+1 (Vt+1 + χ)) 1−ν

1−η = Φ1−νt+1 H(Vt+1)

11

Thus,

Vt + χ = Φη−1t (Vt + χ)

= (1− β)Φη−1t (U + χ)

+β

(ΓΦt)1−η

(H−1

(Et

[Φ1−ν

t+1 H(Vt+1)])

+ χ)

= (1− β)(U(Ct, Lt) + χ

)

+β

(H−1

(Et

[(Φt+1

ΓΦt

)1−ν

H(Vt+1)

])+ χ

)

Subtract χ to obtain (8).•

DefineV = U = U(C, L)

and note that V is that value for Vt and Vt+1, which satisfies (8) for Ct = C,Lt = L, Φt+1/Φt = Γ.

I shall also often impose the following, additional assumption.

Assumption A. 4

1 =UC

(C, L

)C

U(9)

This can always be assured by shifting the intercept of the felicity functionU(·, ·) without affecting economic choices, i.e., this is a normalization ofthe preference function. The assumption is convenient, since I can theneasily calculate random shifts in the value function V in terms of equivalentpermanent increases in consumption.

It is time to examine some examples.

Example 1 Suppose that H(·) is linear, i.e. suppose that ν = η 6= 0. Inthat case, (5) becomes

V0 = E

[ ∞∑

t=0

βtU(Ct, Lt; Φt)

]

i.e., the standard formula for time-separable preferences, except that it moregenerally also allows for Φt to enter the felicity function directly.

12

Example 2 A standard specification is

U(C,L; Φ) =C1−η

1− η= U(C, L)

0 = χ = χ

Note that U(C,L; Φ) does not depend3 on L. In that case, equation (5) reads

Vt =1

1− η

(C1−η + β

(E

[((1− η)Vt+1)

1−ν1−η

]) 1−η1−ν

)

which is a standard specification for Epstein-Zin preferences. It is easy toverify assumption 3. Note that U(·, ·; ·) does not depend on its third argument.Note, though, that assumption 4 is violated.

Example 3 Let

U(C,L) =(Cv(L))1−η

1− η− χ∗ (10)

for some strictly positive and strictly increasing function v(L) so that U(·, ·)is concave. In order to achieve (9), I need

χ∗ =η

1− η

(Cv(L)

)1−η(11)

Consequently,

χ∗ =η

1− ηU (12)

To assure (1−η)(U(C, L)+χ) > 0, as required in (3), one needs χ to satisfy

(1− η)(χ∗ − χ) ≥ 0

The simplest assumption is to set

χ = χ∗ (13)

3Strictly speaking, this violates assumption 2 and my theory below does not apply tothis case. However, it is easy to see that a simplified version of that theory, dropping allterms involving leisure, applies here as well.

13

This assumption has an additional role, see the remarks following proposi-tion 2. With this and assumption 3,

U(Ct, Lt; Φt) =(Ctv(Lt))

1−η − 1−β

1−βη

(Cv(L)

)1−η

1− η

and U(·, ·; ·) again does not depend upon its third argument. For v(L) con-stant, one obtains a version of example 2, additionally satisfying assumption4.

Example 4 With the definition of the previous example 3, let v(L) = Lθ,θ > 0. Then,

U(C,L; η) → log(C) + θ log(L)−(log(C) + θ log(L)

)+ 1 (14)

as η → 1. This thus delivers the preference specification in Tallarini (2000)except for the constant intercept. The intercept is due to my normalization(9), which one can also check directly. Furthermore

H−1 (E[H(Vt+1]) → 1

1− νlog (E [exp ((1− ν)Vt+1)])

One way to check this is to define f(η) = (1−η)χ(η) = η(C1−αLα

)1−η. Note

that f(1) = 1 and write (1 − η)(Vt+1 + χ(η)) ≈ 1 + (1 − η)(Vt+1 − f ′(1)) ≈exp((1−η)(Vt+1−f ′(1))) to see that H(Vt+1; η) → exp((1−ν)(Vt+1−f ′(1))).Likewise, H−1(x; η) → log(x)/(1 − ν) + f ′(1). Combining, the terms f ′(1)drop out.

Example 5 For a case where Φt will enter the utility function U(·, ·; ·), con-sider the GHH-preferences, see Greenwood, Hercowitz and Huffman (1988),

U(C, L) =

(C − κ(A− L)1+φ

)1−η

1− η− χ (15)

where A is the total time endowment and thus, A − L is working time, andκ, φ, η are parameters and where

χ = −(C − κ(A− L)1+φ

)−η

1− ηκ(A− L)1+φ

14

in order to fulfill assumption (4). Now,

U(Ct, Lt; Φt) =

(Ct − Φtκ(A− Lt)

1+φ)1−η

1− η− χ

which amounts to letting the “productivity” of leisure grow as the economygrows, and which is familiar from the literature employing GHH preferences.

Example 6 For a simple catching-up-with-the-Joneses example, considerΦt = C

aggrt to be aggregate consumption at date t, and U(·, ·) any of the

specifications above. Note though, that for Φt to enter the original problem,one needs a specification such that Φt does not drop from U(Ct, Lt; Φt). E.g.with (15) of example 5, one obtains

Vt = (1− β)

(Ct − C

aggrt κ(A− Lt)

1+φ)1−η

1− η− χ + βH−1 (Et [H(Vt+1)])

4 Local Characterizations

Introduce

ηcc = − UCC(C, L)C

UC(C, L)(16)

ηll = − ULL(C, L)L

UL(C, L)(17)

ηcl,c =UCL(C, L)C

UL(C, L)(18)

ηcl,l =UCL(C, L)L

UC(C, L)(19)

which characterize the curvature properties of the felicity function U . Notethat ηcc ≥ 0 is the usual risk aversion with respect to consumption and ηll ≥ 0is risk aversion with respect to leisure. Due to the Epstein-Zin formulation,e.g. the role for ηcc will be the characterization of intertemporal substitution,rather than risk aversion.

15

Define

κ =ULL

UCC

If preferences are nonseparable in consumption and leisure, then ηcl,c 6= 0and consequently

κ =ηcl,l

ηcl,c

and hence, κ can be calculated from ηcl,l and vice versa, given a value forηcl,c. To provide some further intuition on κ, consider a stochastic neoclas-sical growth model with a Cobb-Douglas production function, where wagetimes labor equals the labor share (1 − θ) times output Yt. The usual first-order condition with respect to leisure then shows κ to be the ratio of theexpenditure shares for consumption to leisure, and is equal to

κ =L

(1− L)

(1− θ)Y

C

The following proposition may be useful, if one wishes to avoid depen-dency of U(·, ·; ·) on Φ.

Proposition 2 Do not necessarily impose assumption 4. U(C, L; Φ) doesnot depend on Φ for all (C, L, Φ), iff

(1− η)(U(C, L) + χ) ≡ UC(C,L)C (20)

U(C, L; Φ) does not depend on Φ for (C,L) locally around (ΦC, L) up to asecond-order approximation, iff

(1− η)(U + χ) = UC(C, L)C (21)

η = ηcc (22)

1− η = ηcl,c (23)

Proof: For equation (20), differentiate the right-hand side of the rela-tionship between U(·, ·; ·) and U(·, ·) in assumption 3 with respect to Φ. Forthe local approximation, differentiate again with respect to C, with respectto L and with respect to Φ. Evaluate these as well as (20) at the steadystate. Noting that one equation is implied by the three others, one obtains(21) to (23). •

16

When assumption 4 is imposed, equation (21) can be rewritten as

χ =η

1− ηU (24)

Note that this coincides with (12), provided (13) holds. Thus, (13) is neces-sary in example 3 to assure that U(C, L; Φ) does not depend on Φ. Equation(22) links the relative risk aversion with respect to consumption of the auxil-iary felicity function U(·, ·) to the risk aversion parameter η of the functionalform assumption 3. It may be natural to impose this condition anyhow.Equation (22) shows, that using the same η in (10) as in (6) was necessaryfor that example in order for U(·, ·; ·) not to depend on Φ. Equation (23)is an equation effectively familiar from imposing the equality of income andsubstitution effects in balanced growth models, see King and Plosser(1989).It is easy to verify directly that (22) and (23) are satisfied in example 3.

Introduce

ζ = −H ′′(V )V

H ′(V )=

ν − η

1− η

V

V + χ(25)

as the elasticity of the function H(·), measuring the degree of curvature indeparting from the benchmark expected discounted utility framework. Notethat ζ = 0 iff H(·) is linear, i.e., if the benchmark expected discountedutility framework applies. If the normalization assumption 4 together withlocal independence of U(C, L; Φ) from Φ around the steady state path isimposed, then

ζ = ν − η (26)

as can be seen from equation (24) together with V = U .The next proposition shows, that these values together with some steady

state values completely determine our preference specification up to a secondorder approximation.

Proposition 3 1. Assume values for V > 0, C > 0, L > 0, η > 0,ηcc ≥ 0, ηll ≥ 0, κ > 0, ηcl,c, ζ and χ. Iff these values satisfy

0 < (1− η)(1 +

χ

V

)ζ + η (27)

0 < (1− η)(V + χ) (28)

0 ≤ ηccηll − η2cl,cκ (29)

17

then there is a concave utility function U(C, L), U(C,L, Φ) defined onsome open domain D ⊂ IR2

++ containing (C, L) as well as ν > 0, H(·)satisfying (25), (16) to (19) as well as assumptions 2, 3 and 4. Thefunction U(C, L), U(C,L, Φ) is unique up to a second order approxi-mation.

2. Assume values for V > 0, C > 0, L > 0, η > 0, ηll ≥ 0, κ > 0 as wellas ζ. Iff these values satisfy

0 < η + ζ (30)

0 ≤ ηηll − (1− η)2κ (31)

then there is a concave utility function U(C, L), U(C,L; Φ) defined onsome open domain D ⊂ IR2

++ containing (C, L) as well as ν > 0, H(·)satisfying (25), (16) to (19) as well as assumptions 2, 3 and 4 such thatU(C,L; Φ) does not depend on Φ locally around (ΦC, L). The functionU(C,L), U(C, L, Φ) is unique up to a second order approximation.

Proof:

1. Suppose such functions exist. Then ν > 0 implies (27), concavity ofU(·, ·) implies (29) and the positivity of (1− η)(U(C,L)+χ) evaluatedat (C, L) implies (28).

Conversely, suppose these conditions hold. Let

ν = (1− η)(1 +

χ

V

)ζ + η

and define H(V ) per the functional form in assumption 3. Define

UC(C, L) =V

C> 0

exploiting assumption 4. Note that

UL(C, L) = κC

LUC(C, L) > 0

Define

C =C − C

C, L =

L− L

L

18

The second-order approximation of U around (C, L) must be the fol-lowing quadratic function, which I shall conversely use for providing aconstructive example,

U(C, L)

U= 1 +

UC

U(C − C) +

UL

U(L− L)

+1

U

(1

2UCC(C − C)2 + UCL(C − C)(L− L) +

1

2ULL(L− L)2

)

= 1 + C + κ L− 1

2ηccC

2 + ηcl,lCL− 1

2κ ηllL

2 (32)

exploiting the normalization of assumption 4. The assumptions are nowsatisfied. Tracing through this construction, one can see that there isno choice, demonstrating uniqueness.

2. Suppose, the conditions hold. Define

ν = ζ + η

χ =η

1− ηV

ηcc = η

ηcl,c = 1− η

Complete the construction as in the previous step. Note that

(1− η)(V + χ) = V > 0

Per proposition 2 and equation 26, the result now follows. The converseis now easily established as well.

•

Since ηcl,l = κηcl,c, the concavity condition (29) can be rewritten as

0 ≤ ηccηll − ηcl,cηcl,l (33)

which may be slightly more reminiscent of the formula for a determinant.Define c = log(C)−log(C), l = log(L)−log(L) and u = log(U(C, L))−log(U)to be the log-deviations, and notice that c ≈ C up to first order, etc.. It may

19

be tempting to directly replace these terms in (32). Since that is a second-order approximation, however, one needs to be more careful. A second-orderTaylor expansion of f(c, l) = log U(C exp(c), L exp(l)) yields instead

u ≈ c + κl − 1

2ηccc

2 + (ηcl,l − κ)cl − 1

2κ (ηll + 1− κ)l2

Example 7 To be specific, example 3 gives

ηcc = η

ηcl,c = 1− η

κ =v′(L)L

v(L)

ηcl,l = (1− η)κ

ηll = ηκ− v′′(L)L

v′(L)

For the logarithmic case, i.e. for example 4, one obtains

ηcc = 1

ηcl,c = 0

κ = θ

ηcl,l = 0

ηll = 1

which one could have also obtained from the previous set of equations, notinge.g. that ηll = ηθ − (θ − 1) = 1.

5 The Investment Problem

To proceed towards asset pricing, consider the investment problem of anagent maximizing V0 subject to the evolution of preferences, (5) as well as arecursively defined budget constraint of the form

Ct + St + . . . = RtSt−1 + . . .

where St is the wealth invested in some asset with a gross return (measuredin consumption units) of Rt from period t− 1 to t. Let Λt be the Lagrange

20

multiplier on the budget constraint, and let Ωt be the Lagrange multiplieron (5). We obtain the standard Lucas (1978) asset pricing equation,

Λt = βEt[Λt+1Rt+1] (34)

as well as two further first-order condition from differentiation with respectto Vt and with respect to Ct.

A “period” here shall be interpreted to be the relevant investment hori-zon. For example, while trading costs (and, in some countries, Tobin taxes)probably are a major friction for short investment horizons such as a fewmonths, they presumably matter less, if the horizon is several years. Thus,I shall abstract from trading costs, despite the considerable attention theyhave attracted, see e.g. Luttmer (1999), and instead investigate a variety ofinvestment horizons. A further reason for considering different investmenthorizons is the return predictability, which has been observed at longer ratherthan shorter horizons.

Since these variables are trending, it is more convenient to restate theinvestment problem in terms of the detrended variables4. Define

Γt =Φt

Φt−1

St =St

Φt

The maximization problem now reads

max V0 s.t.

Vt = (1− β)U(Ct, Lt) + (35)

βH−1

(Et

[(Γt+1

Γ

)1−ν

H(Vt+1)

])

Ct + St + . . . =Rt

Γt

St−1 + . . . (36)

Let Ωt be the Lagrange multiplier for the first constraint (35) and Λt be theLagrange multiplier for the second constraint (36). The first-order conditions

4Equivalently, take the first-order condition from the original problem, and detrendthem.

21

are

∂

∂Vt

: Ωt = Ωt−1

(H−1

)′(Et−1

[(Γt

Γ

)1−ν

H(Vt)

]) (Γt

Γ

)1−ν

H ′(Vt)(37)

∂

∂Ct

: Λt = (1− β)ΩtU1(Ct, Lt) (38)

∂

∂St

: Λt = βEt

[Λt+1

Γt+1

Rt+1

](39)

All variables in these equations are now stationary, allowing the possibilityfor approximation around some fixed value. While this can in principle bedone with high-powered numerical tools such as Judd (1998), we seek tounderstand the implications of these equations, using loglinearization here,in order to obtain some “first-order, small-noise” insights.

The four equations (35) and (37) to (39) are the key equation for assetpricing. Equation (39) is the standard asset pricing equation (34), modifiedwith a term due to detrending. To make practical use of this equation,I need to rewrite Λt in terms of observables. Equation (38) relates Λt tothe slope of the felicity function U(·, ·), evaluated at observed consumptionand leisure, which suffices, if preferences are time-separable. Here, however,there is an additional term indicated from Ωt, if H(·) is nonlinear. Thesecan be obtained from (37) in principle, except that now unobservables interms of the value function V seem to arise, which is given by equation(35). This has led researchers in the past to seek ways to substitute out Vusing observables such as wealth or some proxy thereof. Here, I pursue adifferent route. As shall be shown below for loglinearization, (37) can berelated back to observables on consumption and leisure with an additionalparameter characterizing preferences.

6 Loglinearization

As before, let the log of variables with a bar denote the expectation of thelog of the corresponding stochastic variables. I.e., introduce also

log(Ω) = E[log Ωt]

log(Λ) = E[log Λt]

log(R) = E[log Rt]

22

where I assume from now on, that Rt is strictly positive. Note that Ωt ≡ 0,if H(·) is linear, i.e, strictly speaking, the calculations below only apply tothe case of non-separability over time. It is easy to infer the appropriateequations, if there is separability, and I shall include comments to that end,when appropriate.

Use small letters to denote the loglinear deviation of some variable fromits steady state. It is not hard to see5, that (35) loglinearizes to

vt = (1− β)(ct + κlt) + βEt

[ν − η

ζγt+1 + vt+1

](40)

exploiting (9). Note that the coefficient on γt+1 is equal to 1 if (26) holds.I.e., risk-aversion or intertemporal substitution does not enter this equation.Rather, it converts temporary changes in consumption and leisure and ex-pected preference shifts into shifts of the value function. In particular, notethat a permanent increase in consumption compared to the steady state,cs ≡ c, s ≥ t, and with all other variables equal to zero, results in vt = c.I.e., vt measures the shift in welfare in terms of an equivalent permanentpercentage increase in consumption.

Equation (40) shows, that vt can be related back to observables, i.e.,to ct, lt, as well as γt, as long as the parameters η,ν and ζ characterizingpreferences are known. The parameter γt can be inferred either from thefirst-order condition with respect to leisure or - if e.g. denoting total factorproductivity or a smoothed version thereof - observed directly (to the extentthat total factor productivity is observable).

Equation (37) loglinearizes to

ωt − ωt−1 = −ζ (vt − Et−1[vt]) (41)

+(1− ν) (γt − Et−1[γt]) + (1− η)Et−1[γt]

where I have sorted the terms conveniently. Note that the change in the La-grange multiplier on the value function equation is driven by news about thevalue function and the preference shift parameter. Additionally, predictablemovements in the preference shift parameter lead to changes in the multiplier

5This can be shown by noting that, generally for variables Yt = f(Xt), one has yt =(f ′(X)X)/(f(X)xt. Calculate (H ′(V )V )/(H(V )) = ((1− ν)/(1− η))(V /(V +χ)) = ((1−ν)/(ν−η))ζ, and likewise for (H ′)−1(·). Also, compare the result to directly loglinearizing(eq:logprefs), noting that U = V = 1− α there.

23

ωt, if the intertemporal elasticity of substitution η is different from unity. Ifζ = 0, which is the benchmark case of welfare as the discounted sum of ex-pected utilities, and if there are no preference shocks, γt ≡ 0, then ωt ≡ 0,starting at the steady state ω−1 = 0.

Finally, equations (38) and (39) loglinearize to

λt − ωt = −ηccct + ηcl,llt (42)

0 = Et [λt+1 − λt + rt+1 − γt+1] (43)

In a model without stochastic long-run growth, i.e. where γt ≡ 0, thefour equations (40) to (43) simplify to

vt = (1− β)(ct + κlt) + βEt [vt+1]

ωt − ωt−1 = −ζ (vt − Et−1[vt])

λt − ωt = −ηccct + ηcl,llt

0 = Et [λt+1 − λt + rt+1]

7 Asset price implications

7.1 Preliminaries

Introduce the abbreviation

mt+1 = λt+1 − λt − γt+1 (44)

for the log-deviation of the stochastic discount factor

Mt+1 = βΛt+1

ΛtΓt+1

from its nonstochastic counterpart, M = β/Γ.Rewrite the Lucas asset pricing equation (39) as

0 = log(MR

)+ log (Et [exp (mt+1 + rt+1)]) (45)

Note that there is no approximation involved so far.

24

Assume that, conditionally on information at date t, mt+1 and rt+1 arejointly lognormally distributed, conditional on information up to and includ-ing t. Let

Covt(Xt+1, Yt+1) = Et[(Xt+1 − Et[Xt+1])(Yt+1 − Et[Yt+1])]

denote covariances, conditional on information up to and including t. Intro-duce the abbreviated notation

covm,r,t = Covt(mt+1, rt+1)

σ2m,t = Covt(mt+1,mt+1)

σ2r,t = Covt(rt+1, rt+1)

ρm,r,t =covm,r,t

σm,t σr,t

These variances, covariances and correlations may generally depend on time,as indicated above.

Using the standard formula for the expectation of lognormally distributedvariables, equation (45) can be rewritten as

0 = log(MR

)+ Et[mt+1] + Et[rt+1] +

1

2

(σ2

m,t + σ2r,t + 2ρm,r,tσm,tσr,t

)(46)

For the risk-free rate

rft = log Rf

t+1 = log Rf + rt+1 = log Rf + Et [rt+1]

i.e. for an asset with σ2r = 0, I have

rft = − log

(M

)− Et[mt+1]− 1

2σ2

m,t (47)

As usual, the risk-free rate varies over time either due to variations in the ex-pected growth rate of the shadow value of wealth, Et[mt+1], or its conditionalvariance, σ2

m,t.For any risky asset, note that

log Et[Rt+1] = Et[rt+1] +1

2σ2

r,t

25

Let SRt denote the Sharpe ratio of that asset, calculated as the ratio of therisk premium or equity premium and the standard deviation of the log return,expressed in terms of log-returns rather than percent returns,

SRt =log Et[Rt+1]− rf

t

σr,t

The Sharpe ratio is the “price for risk”, and generally a more useful numberthan the equity premium itself, see Lettau and Uhlig (2002) for a detaileddiscussion. Since it is the difference of the log returns that matters, it usuallydoes not much matter whether both returns are calculated in real terms or innominal terms. The calculations in nominal terms are usually easier due tothe availability of suitable data. Obviously, if rf

t is a safe nominal return, itwill not be a safe real return. This matters if unpredictable inflation volatilityis substantial: the Sharpe ratio would then not fully reflect the excess returnof a risky over a safe asset. I find that

SRt = −ρm,r,tσm,t (48)

Moreover, the maximally possible Sharpe ratio SRmaxt for any asset is

SRmaxt = σm,t (49)

This expression only depends on elements of the preference specification, i.e.preference parameters as well as data on consumption, leisure and growth,but not on the underlying asset structure.

7.2 Consumption, leisure and growth

Up to this point, the asset pricing calculations above were exact, given theassumption of lognormal distributions. I shall now proceed to use the loglin-ear approximations to the first order conditions, i.e. equations (40) to (43).I assume that all logdeviations have a joint normal distribution, conditionalon information available at date t. I also assume that all variables dated tare in the information set at date t.

I now apply this standard logic to the preference specification above.Since the model was formulated such that there is a steady state, the resultsabove stay valid, if I replace the logarithms of the Lagrange multiplier with

26

the log-deviations, etc.., except that for comparison to the data, one ought tokeep in mind (and possibly correct the formulas with) the average expectedconsumption growth rate.

Use (42) to replace λt+1 in the expression (44) for mt+1,

mt+1 = −ηccct+1 + ηcl,llt+1 − γt+1 + ωt+1 − λt (50)

and rederive the expression for the risk free rate (47) and the Sharpe ratio(48) in terms of the individual components of mt+1. I obtain

Proposition 4 To a first-order approximation

rft = − log

(M

)+ ηccEt[ct+1− ct]− ηcl,lEt[lt+1− lt] + ηEt[γt+1]− 1

2σ2

m,t (51)

andSRt = ηccρc,r,tσc,t − ηcl,lρl,r,tσl,t + ργ,rσγ,t − ρω,r,tσω,t (52)

One needs to be careful, how the term “first-order approximation” is to beunderstood in this proposition. The equation is exact, when all variablesare jointly and conditional log-normal and the equation (42) holds exactly,i.e. not just as an approximation. However, if (42) is a first-order approx-imation to (38), then joint log-normality of the log-deviations with the logreturn would imply, that the first-order approximation is applied globally onthe entire real line for log Ct, log Lt and log Γt+1, not just locally around C,L and Γ, as pointed out by Samuelson (1970). Judd and Guu have shownthat the first-order approximation above holds, as the standard deviationsof the shocks converge to zero, provided the underlying asset pricing equa-tion is analytic, which essentially means that it can be represented by aninfinite-order Taylor expansion. Whether analyticity can be established, ifthe underlying random variables have unbounded support (as is the case for anormal distribution) or whether weaker but provable conditions exist, whichvalidate the mean-variance analysis here for lognormally distributed shocks,as the volatility of shocks converge to zero, is an as-of-yet unsolved problem.Analyticity can be established, if the random variables have bounded sup-port, however. Thus, the “first-order approximation” means that it holds, asthe variance of the underlying shocks and the diameter of their support setconverges to zero, while at the same time approach a normal distribution,

27

when comparing the demeaned shocks, divided by their standard deviation.A precise statement will be included in a future version of this paper.

For a practical procedure to calculate σ2m,t in equation (51), see equation

(63) below. If mt is homoskedastic, then the risk-free rate is only relatedto the predictable growth rates in consumption and leisure, as usual. As aconsequence and if e.g. ηcc = η, then its inverse is the intertemporal elasticityof substitution.

Equation (52) is a key equation for asset pricing, which I shall examinemore closely. First, it is useful to consider some special cases.

Example 8 Consider first the benchmark case of time-separable preferenceswithout trend growth Φt ≡ Φ and a constant-relative-risk-aversion utilityfunction in consumption only,

U(Ct, Lt; Φ) =C1−η

t

1− η

In that case, ηcc = η, ηcl,l = 0 and (52) reads

SRt = ηρc,r,tσc,t (53)

This equation has been emphasized by Lettau and Uhlig (2002) and provides asummary statement of the equity premium observation of Mehra and Prescott(1985). Assuming t to denote years, and given observations of the Sharperatio srt ≈ 0.3 (which is lower than the typical number given in the litera-ture, since I am using log-returns here), a conditional standard deviation foraggregate annual consumption growth of 0.015 and a conditional correlationbetween stock returns and consumption of near 0.4 implies η = 50. For amore detailed discussion, see section 9 or e.g. Cochrane (2001).

Example 9 Consider the same specification, but now allow for stochasticconsumption growth as well. Consider rederiving equation (52) directly fromthe original Lucas asset pricing equation (34), noting that

Ct = ΦtC exp(ct)

Ct+1 = ΦtΓC exp(ct+1 + γt+1)

One obtainsSRt = ηρc,r,tσc,t + ηργ,r,tσγ,t (54)

28

In particular, both terms contain the relative risk aversion η as factor, be-cause ct+1 and γt+1 both enter the asset pricing equation in the same way. Ittherefore may appear puzzling, that the term involving γt+1 only enters witha unitary coefficient in (52). There is no contradiction here, however. Withtime-separability, ζ = 0 and ν = η, so that (41) becomes

ωt+1 − ωt = (1− η)(γt+1 − Et[γt+1]) + (1− η)Et[γt+1]

and hence,

σω,t = (1− η)σγ,t

ρω,r,t = ργ,r,t

Replacing these terms in (52) delivers equation (54).

Example 10 Consider time-separable preferences without trend growth Φt ≡Φ, but where the utility function is not separable in consumption and leisure,i.e., where ηcl,l 6= 0. Suppose further, that ρl,r,tσl,t 6= 0. Then, given observa-tions on correlations, standard deviations and the Sharpe ratio, for any valuefor ηcc > 0, there is a value ηcl,l solving (52),

ηcl,l =ηccρc,r,tσc,t − SRt

ρl,r,tσl,t

(55)

Thus, there is a large class of preferences which deliver the observed Sharperatio for any given value of ηcc > 0, given the observations on consumptionand leisure and returns, see proposition 3: there is no need to impose ηcc = 50.While one can assume low values for the relative risk aversion with respectto gambles in consumption, risk aversion is not really gone per se: it is justshifted to leisure instead. Note in particular, that the relative risk aversionfor leisure needs to satisfy

ηll ≥η2

cl,cκ

ηcc

(56)

due to concavity of preferences, see equation (29). This is discussed in greaterdetail in section 9 below.

These examples show, that the Sharpe ratio is related to news about theeconomic variables during the holding period for the asset, but not beyond

29

that. This changes with nonseparable preferences. To see this and to derivea Sharpe ratio formula in terms of observables, I need a bit more algebra.For any variable x, define the date-t news about xt+j per

εx,j,t = Et[xt+j]− Et−1[xt+j]

With (40), note that

εv,j,t = Et[vt+j]− Et−1[vt+j]

= (1− β) (εc,j,t + κεl,j,t) + βν − η

ζεγ,j,t+1 + βεv,j+1,t

With this and equation (41), note now that

ωt+1 − Et[ωt+1] = (1− ν)εγ,0,t+1 − ζεv,0,t+1

= (1− η)εγ,0,t+1

−∞∑

j=0

βj(ζ(1− β)εc,j,t+1 + κζ(1− β)εl,j,t+1 + (ν − η)εγ,j,t+1

)

where I have telescoped out the previous equation and split the εγ,0,t+1-term.Note that the first term (1 − η)εγ,0,t+1 is a term which already arises withseparable preferences, and which was crucial in example 9. The other termsonly arise due to nonseparabilities across time, however, i.e. only if ν 6=η. One can see, that current news about future consumption fluctuationsand, in particular, current news about future changes in the growth rate ofconsumption impact on the surprise in ωt+1 and therefore impact on assetpricing. It is this component which is the center of attention in Hansen,Heaton and Li (2005): (future) consumption strikes back.

For a given asset with returns rt+1, define the covariances between futureconsumption-, leisure- and growth-changes with the next period-return,

τc,r,t =∞∑

j=0

βjEt [εc,j,t+1εr,0,t+1]

τl,r,t =∞∑

j=0

βjEt [εl,j,t+1εr,0,t+1]

τγ,r,t =∞∑

j=0

βjEt [εγ,j,t+1εr,0,t+1]

The Sharpe ratio equation (52) can then be restated as follows.

30

Proposition 5 To a first-order approximation

SRt = ηccρc,r,tσc,t − ηcl,lρl,r,tσl,t + ηργ,rσγ,t (57)

+1

σr,t

(ζ(1− β)τc,r,t + ζκ(1− β)τl,r,t + (ν − η)τγ,r,t

)

For the interpretation of “first-order approximation”, see the discussionfollowing proposition 4. There are obviously alternative restatements of (57).Let e.g.

σ2ε,c,j,t = Covt(εc,j,t+1, εc,j,t+1) = Et[ε

2c,j,t+1]

ρε,c,j,r,t =Covt(εc,j,t+1, εr,j,t+1)

σε,c,j,tσr,t

=Et[εc,j,t+1εr,t+1]

σε,c,j,tσr,t

If σε,c,j,t 6= 0 for all j, then one can rewrite equation (57) by replacing theleft-hand-side with the right-hand-side of the equation

τc,r,t

σr,t

=∞∑

j=0

βjρε,c,j,r,tσε,c,j,t

and thereby express this term in the same manner as the first three terms in(57).

The derivations above were always concerning a particular asset. Onecan equally well find out the maximally possible Sharpe ratio per (49). Notethat

εm,0,t+1 = −ηccεc,0,t+1 + ηcl,llt+1 − ηγt+1

−∞∑

j=0

βj(ζ(1− β)εc,j,t+1 + κζ(1− β)εl,j,t+1 + (ν − η)εγ,j,t+1

)

To calculate the maximal Sharpe ratio, one needs to calculate the varianceof εm,0,t+1, which involves not only the variances of the terms in (58), butalso their covariances. I provide a tractable expression in (63) below.

8 A VAR-Approach

These quantities can be calculated from the data by calculating the newsabout future variables from the impulse responses of a vector autoregression.

31

This also provides for a link to the operator-approach in Hansen, Heaton andLi (2005).

Let Yt be a vector of variables and their lags up to some maximal laglength, containing in particular the log of the ratio of consumption to somechosen trend variable log(Ct/Φt), log leisure logLt, the log of the growth rateof the chosen trend variable log Γt = log Φt−log Φt−1 and log returns logRt asthe first, second, third and forth variable. Suppose that one can summarizethe correlation structure in form of a VAR,

Yt = BYt−1 + ut, Et−1[utu′t] = Σt−1 (58)

where the ut are uncorrelated across time and where one lag in the VARsuffices due to “stacking” lags of the variables into the vector Yt. I assumethat βB has all its eigenvalues inside the unit circle. Note that I used thesomewhat unconventional date t−1 for the variance-covariance matrix of theinnovation dated t: that way, the VAR notation is consistent with the nota-tion for conditional covariances above. Note that I allow for heteroskedas-ticity in the innovations but not in the VAR coefficients. The news aboutconsumption (detrended with Φt), leisure and growth is now given by

εc,j,t+1 =(Bjut+1

)1

εl,j,t+1 =(Bjut+1

)2

εγ,j,t+1 =(Bjut+1

)3

The covariance with the news about returns is now

Et[εc,j,t+1εr,0,t+1] =(BjΣt

)14

Et[εl,j,t+1εr,0,t+1] =(BjΣt

)24

Et[εγ,j,t+1εr,0,t+1] =(BjΣt

)34

Summing up, one now obtains

τc,r,t =

∞∑

j=0

βjBj

Σt

14

= Q(β, t)14

τl,r,t = Q(β, t)24

τγ,r,t = Q(β, t)34

32

whereQ(β, t) =

(I − βB

)−1Σt

and where the dependence on the preference parameter β and time t hasbeen indicated via the argument. Alternatively, it is useful to rewrite (50) as

εm,0,t+1 = ~a′ut+1 − ~b′(I − βB

)−1ut+1 (59)

where the vectors ~a, ~b and, additionally, the vector ~|e4, are defined as

~a =

−ηcc

ηcl,l

−η0...0

, ~b =

ζ(1− β)ζ(1− β)κ

ν − η0...0

, ~|e4 =

00010...0

(60)

Together with proposition (5), the following proposition follows.

Proposition 6 Given the VAR representation (58) of the data, to a first-order approximation

SRtσr,t = ηcc (Σt)14 − ηcl,l (Σt)24 + η (Σt)34 (61)

+ζ(1− β)Q(β, t)14 + ζκ(1− β)Q(β, t)24 + (ν − η)Q(β, t)34

whereσr,t =

√(Σt)44

Alternatively,

SRtσr,t = −~a′Σt~|e4 + ~b′Q(β, t)~|e4 (62)

To calculate the maximal Sharpe ratio per equation (49), use the followingproposition.

Proposition 7 To a first-order approximation,

(SRmaxt )2 = σ2

m,t = ~a′Σt~a− 2~b′Q(β, t)~a + ~b′Q(β, t)(I − βB′)−1 ~b (63)

33

8.1 k-Period Asset Holdings

Since there may be frictions and preventive trading costs in adjusting port-folios on a quarterly basis, it may be more sensible to apply the asset pricingformulas to a holding period of k periods rather than one period, i.e. for alonger period than the time distance between subsequent observations of thedata.

The derivation is quite similar, and a sketch suffices. The Lucas assetpricing formula is

1 = Et

[Λt+k

Λt

Rt+1

Γt+1

Rt+2

Γt+2

. . .Rt+k

Γt+k

]

Define the compounded log-deviation of the stochastic discount factor as

mt,t+k =k∑

i=1

mt+i = λt+k − λt −k∑

i=1

γt+i

Define the averaged6 compounded log-deviation of the return as

rt,t+k =1

k

k∑

i=1

rt+i (64)

Similarly, let rft,t+k be the log-deviation for the k−period risk free at date

t. Since there generally is unexpected future variation in the risk free rate,it is not the case that (64) (or a version, where expectations are taken onthe right hand side) holds for the k-period and 1-period risk-free rates . Putdifferently, the compounding of the one-period risk-free rate is a risky return,to which the asset pricing formulas below apply. For the k-period risk-freerate itself, the following generalization of (51) is obtained immediately.

Proposition 8 To a first-order approximation

rft,t+k = − log

(M

)(65)

+1

k

(ηccEt[ct+k − ct]− ηcl,lEt[lt+k − lt] + ηEt[

k∑

i=1

γt+i]− 1

2σ2

mt,t+k,t

)

6I average the return rather than add them up, since that corresponds more closely tothe way returns are usually reported.

34

For the calculation of σ2mt,t+k,t, see proposition 10 below.

Define the normalized Sharpe ratio as

SRk,t =Et[rt,t+k]− rf

t,t+k√k σrt,t+k,t

where and σrt,t+k,t is the conditional standard deviation of the difference

between rt,t+k and rft,t+k.

Note that if the expected k-period log excess return can be written as asum of per-period excess returns re

t+i,(

k∑

i=1

rt+i

)− rf

t,k =k∑

i=1

ret+i

in such a way that the per-period excess returns are iid, then SRk,t = SR1,t =SRt, i.e. in that case, the normalized Sharpe ratio is independent of theholding horizon. Since excess returns are known to be somewhat predictable,and since there are unpredictable variations in the safe rate, one would notgenerally expect this independence from the horizon k to hold exactly, butit provides a benchmark for comparison.

For the general case and under joint log-normality, equation (48) nowbecomes √

k SRt,k = −ρmt,t+k,rt,t+k,tσmt,t+k,t (66)

or, alternatively,√

k SRt,kσrt,t+k,t = −covt(mt,t+k, rt,t+k) (67)

Note that

mt,t+k − Et[mt,t+k] = λt+k − Et[λt+k]−k∑

i=1

(γt+i − Et[γt+i])

= −ηcc(ct+k − Et[ct+k]) + ηcl,l(lt+k − Et[lt+k])

−k∑

i=1

(γt+i − Et[γt+i]) + ωt+k − Et[ωt,t+k] (68)

The decomposition of the compounded log returns and the k−period changein log consumption and leisure into news is

ct+k − Et[ct+k] =k∑

i=1

εc,k−i,t+i

35

lt+k − Et[lt+k] =k∑

i=1

εl,k−i,t+i

k∑

i=1

(γt+i − Et[γt+i) =k∑

i=1

k−i∑

j=0

εγ,j,t+i

rt,t+k − Et[rt,t+k] =k∑

i=1

(rt+i − Et[rt+i])

=k∑

i=1

k−i∑

j=0

εr,j,t+i

Furthermore

ωt+k − Et[ωt+k] =k∑

i=1

(ωt+i − ωt+i−1)−k∑

i=1

Et [ωt+i − ωt+i−1]

= −ζk∑

i=1

εv,0,t+i

+(1− ν)k∑

i=1

εγ,0,t+i

+(1− η)k∑

i=1

(Et+i−1[γt+i]− Et[γt+i])

= −k∑

i=1

∞∑

j=0

βj(ζ(1− β)εc,j,t+i + κζ(1− β)εl,j,t+i + (ν − η)εγ,j,t+i

)

+(1− η)k∑

i=1

(γt+i − Et[γt+i])

Note that the terms εγ,0,t+i stemming from εv,0,t+i have been split across thelast two last lines in this formula.

As in section 8, suppose that the dynamics of the data can be summarizedby the VAR in equation (58) with the conventions adopted there. Introducethe following matrices

Σt,i = Et[Σt+i]

Q(β, t, k) = (1− βB)−1k∑

i=1

k−i∑

j=0

Σt,i−1(B′)j

36

S(t, k) =k∑

i=1

k−i∑

j=0

Bk−iΣt,i−1(B′)j

P (t, k) =k∑

i=1

k−i∑

j1=0

k−i∑

j2=0

Bj1Σt,i−1(B′)j2

Recall the definition of ~b and ~|e4 in equation (60). Define vectors ~a∗, ~|e3 asfollows:

~a∗ =

−ηcc

ηcl,l

00...0

, ~|e3 =

0010...0

(69)

Calculation of the conditional covariances now yields the following gen-eralization of proposition 6:

Proposition 9 Given the VAR representation (58) of the data, to a first-order approximation

k3/2SRt,kσrt,t+k,t = ηcc (S(t, k))14 − ηcl,l (S(t, k))24 + η (P (t, k))34 (70)

+ζ(1− β)Q(β, t, k)14 + ζκ(1− β)Q(β, t, k)24 + (ν − η)Q(β, t, k)34

or alternatively

k3/2SRt,kσrt,t+k,t =(− ~a∗

′S(t, k) + η~|e′3P (t, k) + ~b′Q(β, t, k)

)~|e4 (71)

where

σrt,t+k,t =

√(P (t, k))44

k

To similarly calculate the maximal Sharpe ratio, rewrite (68) as

mt,t+k − Et[mt,t+k] =k∑

i=1

~c′iut+i (72)

where

~c′i = ~a∗′Bk−i − η~|e′3

k−i∑

j=0

Bj

− ~b′(I − βB)−1 (73)

The following proposition follows.

37

Proposition 10 To a first-order approximation,

SRmaxt,k =

1√kσmt,t+k,t

where

σ2mt,t+k,t =

k∑

i=1

~c′iΣt,i−1~ci (74)

9 An empirical implementation

TO BE REDONE IN A FUTURE VERSION OF THE PAPER

We use data 1970 - 2003 from the Federal Reserve Bank of St Louis database. We use non-durable consumption for Ct. Leisure shall be Lt = 1−Nt,where Nt is measured as fraction of total time and where we use the seriesAWHI (total hours worked) for labor. For assets, we use the excess returnsof SP 500 over 1-yr T-Bill.

In order to investigate the role of the long run trend, I proceed somewhatartificially as follows. Construct γt per:

γt = (1− ργ)(log Ct − log Ct−1) + ργγt

with γ1 = (log C1 − log C0). Find ργ such that

∆ct = log Ct − log Ct−1 − γt

has zero autocorrelation: ργ = 0.71. I also try alternatives for consump-tion growth trend, per ργ = 0.9 and ργ = 0.5. One needs to be careful ininterpreting γt as a low-frequency or trend in consumption growth andI donot encourage to read it as a low frequency component extracted from anappropriate filtering exercise. Rather, it should be seen as just applying amedium frequency filter to consumption. Figure 7 and 8 show the result. Itry various β = βΓ1−η.

For a Bayesian Vectorautoregression, I include a constant, and 2 or 4 or 6lags, but do not incorporate time variation in volatilities. I use a fairly “unin-formative” Normal-Wishart prior. Thus, the posterior is Normal-Wishart aswell. I take 1000 draws and compute 16%, 50% and 84% quantiles. Figures

38

1970 1975 1980 1985 1990 1995 2000 2005680

690

700

710

720

730

740

750

760

770level cons. vs. phi

consphi

Figure 7: Trends in Consumption Growth

1970 1975 1980 1985 1990 1995 2000 2005−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5cons. vs. growth

cons growthgamma

Figure 8: Trends in Consumption Growth

39

0 2 4 6 8 10−1

0

1

2

3

4

5

6

7R news −> Excess stock returns

quarter

perc

ent

Figure 9: Impulse Response of Excess Returns

0 2 4 6 8 10−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1R news −> gamma (cons.growth)

quarter

perc

ent

Figure 10: Impulse Response of Consumption Trend Growth

40

0 2 4 6 8 10−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0R news −> leisure

quarter

perc

ent

Figure 11: Impulse Response of Leisure

9, 10 and 11 show the impulse responses to excess returns ordered first in aCholesky decomposition.

I next plot the long-Run risk components. The long-run risk for con-sumption is

Q(β, c, r) = Q(β, t)14 +

(ν − η

ζ

1

1− β

)Q(β, t)34

whereas the long-run risk for leisure is

Q(β, l, r) = κQ(β, t)24

The results are shown in figures 12, 13 and 14, when varying the numberof lags for the benchmark choice of ργ. The figures 15, 16 and 17 containthe results for ργ = 0.90. The figures 18, 19 and 20 contain the results forργ = 0.50.

10 Conclusions

This paper has provided a loglinear framework for pricing assets, using ageneralization of Epstein-Zin preferences, allowing for nonseparabilities be-tween consumption and leisure. As in Bansal and Yaron (2004) or Hansen,

41

consumption + leisure = long run risk

0.94 0.95 0.96 0.97 0.98 0.99 10

0.2

0.4

0.6

0.8

1Q(beta,c+g,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

0.94 0.95 0.96 0.97 0.98 0.99 1−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0Q(beta,l,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

0.94 0.95 0.96 0.97 0.98 0.99 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Q(beta,c+g+l,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

Figure 12: The long run risk components: 4 lags.

consumption + leisure = long run risk

0.94 0.95 0.96 0.97 0.98 0.99 10

0.5

1

1.5Q(beta,c+g,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

0.94 0.95 0.96 0.97 0.98 0.99 1−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0Q(beta,l,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

0.94 0.95 0.96 0.97 0.98 0.99 10

0.2

0.4

0.6

0.8

1Q(beta,c+g+l,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

Figure 13: The long run risk components: 2 lags.

consumption + leisure = long run risk

0.94 0.95 0.96 0.97 0.98 0.99 10

0.2

0.4

0.6

0.8

1Q(beta,c+g,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

0.94 0.95 0.96 0.97 0.98 0.99 1−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0Q(beta,l,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

0.94 0.95 0.96 0.97 0.98 0.99 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Q(beta,c+g+l,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

Figure 14: The long run risk components: 6 lags.

42

1970 1975 1980 1985 1990 1995 2000 2005680

690

700

710

720

730

740

750

760

770level cons. vs. phi

consphi

Figure 15: Trends in Consumption Growth: ργ = 0.9

1970 1975 1980 1985 1990 1995 2000 2005−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5cons. vs. growth

cons growthgamma

Figure 16: Trends in Consumption Growth: ργ = 0.9

43

consumption + leisure = long run risk

0.94 0.95 0.96 0.97 0.98 0.99 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Q(beta,c+g,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

0.94 0.95 0.96 0.97 0.98 0.99 1−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0Q(beta,l,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

0.94 0.95 0.96 0.97 0.98 0.99 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Q(beta,c+g+l,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

Figure 17: The long run risk components: 4 lags, ργ = 0.9.

1970 1975 1980 1985 1990 1995 2000 2005680

690

700

710

720

730

740

750

760

770level cons. vs. phi

consphi

Figure 18: Trends in Consumption Growth: ργ = 0.5

44

1970 1975 1980 1985 1990 1995 2000 2005−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5cons. vs. growth

cons growthgamma

Figure 19: Trends in Consumption Growth: ργ = 0.5

consumption + leisure = long run risk

0.94 0.95 0.96 0.97 0.98 0.99 10

0.2

0.4

0.6

0.8

1

1.2

1.4Q(beta,c+g,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

0.94 0.95 0.96 0.97 0.98 0.99 1−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0Q(beta,l,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

0.94 0.95 0.96 0.97 0.98 0.99 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Q(beta,c+g+l,r)/std(r)

β

Sha

rpe

ratio

con

trib

.

Figure 20: The long run risk components: 4 lags, ργ = 0.5.

45

Heaton and Li (2005), I find that news about future consumption matterfor current asset prices, if preferences are non-separable across time. I showthat if preferences also depend on leisure, then also news about future leisureand its correlation with current news about returns matter for asset pric-ing. The relationship between future news and the current Sharpe ratio hasbeen calculated, using a log-normal framework. A VAR formulation has beenprovided, allowing the calculation of the news component based on currentinnovations. An empirical analysis shows the resulting term to be of similarorder of magnitude as the term arising from consumption.

46

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