Expected Stock Returns and Variance Risk Premia
Tim Bollerslev
Duke University
George Tauchen
Duke University
Hao Zhou
Federal Rerve Board
Motivated by the implications from a stylized lf-contained general equilibrium model incorporating the effects of time-varying economic uncertainty,we show that the difference between implied and realized variation,or the variance risk premium,is able to explain a nontrivial fraction of the time-ries variation in post-1990aggregate stock market returns, with high(low)premia predicting high(low)future returns.Our empirical results depend crucially on the u of“model-free,”as oppod to Black–Scholes,options implied volatil-ities,along with accurate realized variation measures constructed from h
igh-frequency intraday as oppod to daily data.The magnitude of the predictability is particularly strong
at the intermediate quarterly return horizon,where it dominates that afforded by other popu-lar predictor variables,such as the P/E ratio,the default spread,and the consumption–wealth ratio.(JEL C22,C51,C52,G12,G13,G14)
Is the return on the stock market predictable?This age-old question still ranks among the most studied and contentious in all of economics.To the extent that a connsus has emerged,it ems to be that the predictability is the strongest over long multi-year horizons.There is also evidence that the degree Bollerslev’s work was supported by a grant from the NSF to the NBER and CREATES funded by the Danish National Rearch Foundation.The paper combines results of an earlier paper with the same title by thefirst and the third authors,and a paper by the cond author titled“Stochastic V olatility in General Equilibrium.”Excellent rearch assistance was provided by Natalia Sizova.We would also like to thank an anonymous referee,John Ammer,Torben Andern,Federico Bandi,Ravi Bansal,Oleg Bondarenko,Craig Burnside,Robert Hodrick, Pete Kyle,David Lando,Benoit Perron,Monika Piazzesi,Raman Uppal,Tuomo Vuolteenaho,Jonathan Wright, Amir Yaron,Motohiro Yogo,Alex Ziegler,and minar participants at the Federal Rerve Boar
d,the2007 conference on“Return Predictability”at Copenhagen Business School,the2007SITE conference at Stanford, the2007NBER Summer Institute,the2007conference on“Measuring Dependence in Finance”at Cass Business School,and the2008Winter Meetings of the American Finance Association for helpful discussions.The views prented here are solely tho of the authors and do not necessarily reprent tho of the Federal Rerve Board or its staff.Send correspondence to Tim Bollerslev,Department of Economics,Duke University,Durham,NC 27708;telephone:919-660-1846;fax:919-684-8974.E-mail:boller@econ.duke.edu.
C The Author2009.Published by Oxford University Press on behalf of The Society for Financial Studies. All rights rerved.For Permissions,plea e-mail:journals.
doi:10.1093/rfs/hhp008Advance Access publication February12,2009 at Fuqua School of Business Library on August 13, 2014/ Downloaded from
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of predictability has diminished somewhat over the past two decades.1In lieu of this,we show that the difference between“model-free”implied and realized variances,which we term the variance risk premium,explains a nontrivial fraction of the variation in post-1990aggregate stock market returns wit
h high (low)values of the premium associated with subquent high(low)returns. The magnitude of the predictability is particularly strong at the quarterly return horizon,where it dominates that afforded by other popular predictor variables, such as the P/E ratio,the default spread,and the consumption–wealth ratio (CAY).
Our empirical investigations are directly motivated by the implications from a stylized lf-contained general equilibrium model.The model may be en as an extension of the long-run risk model pioneered by Bansal and Yaron (2004),who emphasized the importance of long-run risk in consumption growth for explaining the equity premium and the dynamic dependencies in returns over long multi-year horizons.In contrast,we explicitly exclude predictability in consumption growth,focusing instead on the implications of allowing for richer and empirically more realistic volatility dynamics.Our model generates a two-factor structure for the endogenously determined equity risk premium in which the factors are directly related to the underlying volatility dynamics of consumption growth.Different volatility concepts defined within the model load differently on the fundamental risk factors.In particular,the difference
between the risk-neutralized expected return variation and the realized return variation effectively isolates the factor associated with the volatility of con-sumption growth volatility.Conquently,the var
iance risk premium should rve as an especially uful predictor for the returns over horizons for which that risk factor is relatively more important.In a reasonably calibrated version of the model,this translates into population return predictability regressions that show the most explanatory power over intermediate“quarterly”return horizons.
The dual variance concepts underlying our empirical investigations of the theoretical relations are both fairly new.On the one hand,veral recent studies have argued for the u of so-called model-free realized variances computed by the summation of high-frequency intraday squared returns.The types of measures generally afford much more accurate ex post obrvations on the actual return variation than the more traditional sample variances bad on daily or coarr frequency return obrvations(e,for example,Andern et al.2001a;Barndorff-Nieln and Shephard2002;Meddahi2002).2
On the other hand,the recently developed so-called model-free implied variances provide ex ante risk-neutral expectations of the future return variation. In contrast to the standard option-implied variances bad on the Black–Scholes pricing formula,or some variant thereof,the“model-free”implied variances 1For recent discussions in support of return predictability,e,for example,Lewellen(2004)and Cochrane(2008). 2Earlier empirical studies exploring similar ideas inclu
de Schwert(1990)and Hsieh(1991).
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Expected Stock Returns and Variance Risk Premia
are computed from a collection of option prices without the u of a specific pricing model(e,for example,Carr and Madan1998;Britten-Jones and Neuberger2000;Jiang and Tian2005).
Our main empiricalfinding that the difference between the“model-free”implied and realized variances is able to explain a nontrivial fraction of the variation in quarterly stock market returns over the1990–2007sample period is new and easily dominates that afforded by other more commonly employed predictor variables.3Moreover,combining the variance risk premium with some of the other predictor variables,most notably the P/E ratio,results in even greater return predictability and joint significance of the predictor variables. This in turn suggests that volatility and consumption risk both play important roles in determining the returns,with their relative contributions varying across return horizons.
The plan for the rest of the paper is as follows.Section1outlines the basic theoretical model and corresponding predictability regressions that motivate our empirical investigations.Section2discuss the“model-free”implied and realized variances that we u in empirically quantifying the variance risk premium along with practical data considerations.Section3prents our main empiricalfindings and robustness checks.Section4concludes.
1.Volatility in Equilibrium
The classical intertemporal CAPM model of Merton(1973)is often ud to motivate the existence of a traditional risk–return tradeoff in aggregate market returns.Despite an extensive empirical literature devoted to the estimation of such a premium,the arch for a significant time-invariant expected return–volatility tradeoff type relationship has largely proven elusive.4In this ction, we prent a stylized general equilibrium model designed to illuminate new and more complex theoretical linkages betweenfinancial market volatility and expected returns.The model involves a standard endowment economy with Epstein–Zin–Weil recursive preferences.5
The basic tup builds on and extends the discrete-time long-run risk model pioneered by Bansal and Yaron(2004)by allowing for richer volatility 3Related empirical links between stock market return
s and various notions of variance risk have been informally explored byfinance professionals.For example,Beckers and Bouten(2005)report that a market timing strategy bad on the ratio of implied to historical volatilities doubles the Sharpe ratio relative to that of a constant S&P500exposure.Many equity-oriented hedge funds also actively trade variance risk in the highly liquid OTC variance swap market(e,for example,Bondarenko2004).
4A significant equilibrium relationship,explicitly allowing for temporal variation in the price of risk,has recently been estimated by Bekaert,Engstrom,and Xing(2008).Also,Ang et al.(2006)find that innovations in aggregate volatility carry a statistically significant(negative)risk premium and that cross-ctionally idiosyncratic volatility is negatively related with average stock returns.
5The Epstein and Zin(1991)and Weil(1989)preferences are rooted in the dynamic choice theory of Kreps and Porteus(1978).
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dynamics in the form of stochastically time-varying volatility-of-volatility.6 This in turn results in an empirically more realistic two-factor structure for the aggregate stock market volatility,and importantly suggests new and interesting channels through which the endogenously generated time-varying risk premia on consumption and volatility risk might manifest themlves empirically.To simplify the analysis and focus on the role of time-varying volatility,we ex-plicitly exclude the long-run risk factor in consumption growth highlighted in the original Bansal and Yaron(2004)model.
1.1Model tup and assumptions
To begin,suppo that the geometric growth rate of consumption in the econ-omy,g t+1=log(C t+1/C t),is unpredictable,
g t+1=μg+σg,t z g,t+1,(1) whereμg denotes the constant mean growth rate,σg,t refers to the conditional variance of the growth rate,and{z g,t}is an i.i.d.N(0,1)innovation process.7
Furthermore,assume that the volatility dynamics are governed by the following discrete-time versions of continuous-time square root-type process,
σ2g,t+1=aσ+ρσσ2g,t+√
q t zσ,t+1,(2)
q t+1=a q+ρq q t+ϕq √
q t z q,t+1,(3)
where the parameters satisfy aσ>0,a q>0,|ρσ|<1,|ρq|<1,ϕq>0,and {zσ,t}and{z q,t}are independent i.i.d.N(0,1)process jointly independent of {z g,t}.The stochastic volatility processσ2g,t+1reprents time-varying economic uncertainty in consumption growth with the volatility-of-volatility process q t in effect inducing an additional source of temporal variation in that same process. Both process play a crucial role in generating the time-varying volatility risk premia discusd below.The assumption of independent innovations across all three equations explicitly rules out any return–volatility correlations that might otherwi ari via purely statistical channels.8
6Empirical evidence in support of time-varying consumption growth volatility has recently been prented by Bekaert and Liu(2004);Bansal,Khatchatrian,and Yaron(2005);Bekaert,Engstrom,and Xing(2008);and Lettau,Ludvigson,and Wachter(2008),among others.
7The growth rate of consumption is identically equal to the dividend growth rate in this Lucas-tree economy.
8Direct estimation of the stylized model defined by Equations(1)–(3)would require the u of latent variable techniques.Instead,as a way to gauge the specification,we calculated a robust estimate forσ2g,t by exponentially smoothing the squared(de-meaned)growth rate in al expenditures on nondurable goods and rvices (g t−ˆμg)2over the1947:Q2to2007:Q4sample period using a smoothing parameter of0.06.Consistent with the basic model structure in Equation(2),the rial dependencies in the resultingˆσ2g,t ries appear to be well described by an AR(1)model withρσclo to unity.Consistent with the Great Moderation,the variances are generally also much lower over the latter part of the sample.Moreover,on estimating an AR(1)-GARCH(1,1) model forˆσ2g,t,the estimates for the two GARCH parameters equal0.238and0.655,respectively,and the Wald test for their joint significance and the abnce of any ARCH effects(129.9)has a p-value of virtually zero,thus strongly supporting the notion of time-varying volatility-of-volatility in consumption growth or Var(q t)>0.
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We assume that the reprentative agent in the economy is equipped with Epstein–Zin–Weil recursive preferences.Conquently,the logarithm of the intertemporal marginal rate of substitution,m t+1≡log(M t+1),may be ex-presd as
m t+1=θlogδ−θψ−1g t+1+(θ−1)r t+1,(4) where
θ≡(1−γ)(1−ψ−1)−1,(5)δdenotes the subjective discount factor,ψequals the intertemporal elasticity of substitution,γrefers to the coefficient of risk aversion,and r t+1is the time t to t+1return on the consumption ast.We will maintain the assumptions thatγ>1andψ>1,which in turn implies thatθ<0.9The restrictions ensure,among other things,that volatility carries a positive risk premium,and that ast prices fall on news of positive volatility shocks consistent with the so-called leverage effect.Importantly,the effects are not the result of any direct statistical linkages between return and volatility,but instead ari endogenously within the model.
1.2Model solution and equity premium
Let w t denote the logarithm of the price–dividend ratio,or equivalently the price–consumption or wealth–consumption ratio,of the ast that pays the consumption endowment,{C t+i}∞i=1.The standard solution method forfind-ing the equilibrium in a model like the one defined above then consi
sts in conjecturing a solution for w t as an affine function of the state variables,σ2g,t and q t,
w t=A0+Aσσ2g,t+A q q t,(6) solving for the coefficients A0,Aσ,and A q,using the standard Campbell and Shiller(1988)approximation r t+1=κ0+κ1w t+1−w t+g t+1.The resulting equilibrium solutions for the three coefficients may be expresd as
A0=logδ+(1−ψ−1)μg+κ0+κ1[Aσaσ+A q a q]
(1−κ1)
,(7)
Aσ=
(1−γ)2
2θ(1−κ1ρσ)
,(8)
A q=1−κ1ρq−
(1−κ1ρq)2−θ2κ41ϕ2q A2σ
θκ21ϕ2q
.(9)
9The assumption thatγ>1is generally agreed upon,but the assumption thatψ>1is a matter of some debate (e,for example,the discussion in Bansal and Yaron2004).
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