Expected Stock Returns and Variance Risk
Premia
Tim Bollerslev
Duke University
George Tauchen
Duke University
Hao Zhou
Federal Reserve Board
Motivated by the implications from a stylized self-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-series variation in post-1990 aggregate stock market returns,
with high (low) premia predicting high (low) future returns. Our empirical results depend
crucially on the use of “model-free,” as opposed to Black–Scholes, options implied volatil-
ities, along with accurate realized variation measures constructed from high-frequency
intraday as opposed 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 consensus has emerged, it seems 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 Research Foundation. The paper combines results of an earlier paper with the same title by the first and
the third authors, and a paper by the second author titled “Stochastic Volatility in General Equilibrium.” Excellent
research assistance was provided by Natalia Sizova. We would also like to thank an anonymous referee, John
Ammer, Torben Andersen, 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 seminar participants at the Federal Reserve Board, the 2007
conference on “Return Predictability” at Copenhagen Business School, the 2007 SITE conference at Stanford,
the 2007 NBER Summer Institute, the 2007 conference on “Measuring Dependence in Finance” at Cass Business
School, and the 2008 Winter Meetings of the American Finance Association for helpful discussions. The views
presented here are solely those of the authors and do not necessarily represent those of the Federal Reserve 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
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The Review of Financial Studies / v 22 n 11 2009
of predictability has diminished somewhat over the past two decades.
1
In 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-1990 aggregate stock market returns with high
(low) values of the premium associated with subsequent 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 self-contained general equilibrium model. The model may be seen
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 these 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. Consequently, the variance risk premium should
serve as an especially useful 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 these
theoretical relations are both fairly new. On the one hand, several recent studies
have argued for the use of so-called model-free realized variances computed
by the summation of high-frequency intraday squared returns. These types of
measures generally afford much more accurate ex post observations on the
actual return variation than the more traditional sample variances based on
daily or coarser frequency return observations (see, for example, Andersen
et al. 2001a; Barndorff-Nielsen and Shephard 2002; Meddahi 2002).
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 based on the Black–Scholes
pricing formula, or some variant thereof, the “model-free” implied variances
1
For recent discussions in support of return predictability, see, for example, Lewellen (2004) and Cochrane (2008).
2
Earlier empirical studies exploring similar ideas include 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 use of a specific
pricing model (see, for example, Carr and Madan 1998; Britten-Jones and
Neuberger 2000; Jiang and Tian 2005).
Our main empirical finding 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 the 1990–2007 sample period
is new and easily dominates that afforded by other more commonly employed
predictor variables.
3
Moreover, combining the variance risk premium with some
of these 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. Section 1 outlines the basic
theoretical model and corresponding predictability regressions that motivate
our empirical investigations. Section 2 discusses the “model-free” implied and
realized variances that we use in empirically quantifying the variance risk
premium along with practical data considerations. Section 3 presents our main
empirical findings and robustness checks. Section 4 concludes.
1. Volatility in Equilibrium
The classical intertemporal CAPM model of Merton (1973) is often used 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 search for a significant time-invariant expected return–
volatility tradeoff type relationship has largely proven elusive.
4
In this section,
we present a stylized general equilibrium model designed to illuminate new
and more complex theoretical linkages between financial market volatility and
expected returns. The model involves a standard endowment economy with
Epstein–Zin–Weil recursive preferences.
5
The basic setup builds on and extends the discrete-time long-run risk
model pioneered by Bansal and Yaron (2004) by allowing for richer volatility
3
Related empirical links between stock market returns and various notions of variance risk have been informally
explored by finance professionals. For example, Beckers and Bouten (2005) report that a market timing strategy
based on the ratio of implied to historical volatilities doubles the Sharpe ratio relative to that of a constant
S&P 500 exposure. Many equity-oriented hedge funds also actively trade variance risk in the highly liquid OTC
variance swap market (see, for example, Bondarenko 2004).
4
A 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-sectionally idiosyncratic volatility
is negatively related with average stock returns.
5
The Epstein and Zin (1991) and Weil (1989) preferences are rooted in the dynamic choice theory of Kreps and
Porteus (1978).
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The Review of Financial Studies / v 22 n 11 2009
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 themselves 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.
1.1 Model setup and assumptions
To begin, suppose 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
= μ
g
+ σ
g,t
z
g,t+1
, (1)
where μ
g
denotes the constant mean growth rate, σ
g,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 processes,
σ
2
g,t+1
= a
σ
+ ρ
σ
σ
2
g,t
+
√
q
t
z
σ,t+1
, (2)
q
t+1
= a
q
+ ρ
q
q
t
+ ϕ
q
√
q
t
z
q,t +1
, (3)
where the parameters satisfy a
σ
> 0, a
q
> 0, |ρ
σ
| < 1, |ρ
q
| < 1, ϕ
q
> 0, and
{z
σ,t
} and {z
q,t
} are independent i.i.d. N(0, 1) processes jointly independent of
{z
g,t
}. The stochastic volatility process σ
2
g,t+1
represents 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 processes play a crucial role in generating the time-varying volatility risk
premia discussed below. The assumption of independent innovations across all
three equations explicitly rules out any return–volatility correlations that might
otherwise arise via purely statistical channels.
8
6
Empirical evidence in support of time-varying consumption growth volatility has recently been presented by
Bekaert and Liu (2004); Bansal, Khatchatrian, and Yaron (2005); Bekaert, Engstrom, and Xing (2008); and
Lettau, Ludvigson, and Wachter (2008), among others.
7
The growth rate of consumption is identically equal to the dividend growth rate in this Lucas-tree economy.
8
Direct estimation of the stylized model defined by Equations (1)–(3) would require the use of latent variable
techniques. Instead, as a way to gauge the specification, we calculated a robust estimate for σ
2
g,t
by exponentially
smoothing the squared (de-meaned) growth rate in U.S. real expenditures on nondurable goods and services
(g
t
−
ˆ
μ
g
)
2
over the 1947:Q2 to 2007:Q4 sample period using a smoothing parameter of 0.06. Consistent with
the basic model structure in Equation (2), the serial dependencies in the resulting
ˆ
σ
2
g,t
series appear to be well
described by an AR(1) model with ρ
σ
close 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 for
ˆ
σ
2
g,t
, the estimates for the two GARCH parameters equal 0.238 and 0.655, respectively, and the Wald
test for their joint significance and the absence 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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Expected Stock Returns and Variance Risk Premia
We assume that the representative agent in the economy is equipped with
Epstein–Zin–Weil recursive preferences. Consequently, the logarithm of the
intertemporal marginal rate of substitution, m
t+1
≡ log(M
t+1
),maybeex-
pressed as
m
t+1
= θ log δ − θψ
−1
g
t+1
+ (θ − 1)r
t+1
, (4)
where
θ ≡ (1 − γ)(1 − ψ
−1
)
−1
, (5)
δ denotes the subjective discount factor, ψ equals the intertemporal elasticity
of substitution, γ refers to the coefficient of risk aversion, and r
t+1
is the time
t to t + 1 return on the consumption asset. We will maintain the assumptions
that γ > 1 and ψ > 1, which in turn implies that θ < 0.
9
These restrictions
ensure, among other things, that volatility carries a positive risk premium, and
that asset prices fall on news of positive volatility shocks consistent with the so-
called leverage effect. Importantly, these effects are not the result of any direct
statistical linkages between return and volatility, but instead arise endogenously
within the model.
1.2 Model 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 asset that pays the
consumption endowment, {C
t+i
}
∞
i=1
. The standard solution method for find-
ing the equilibrium in a model like the one defined above then consists in
conjecturing a solution for w
t
as an affine function of the state variables, σ
2
g,t
and q
t
,
w
t
= A
0
+ A
σ
σ
2
g,t
+ A
q
q
t
, (6)
solving for the coefficients A
0
, A
σ
, and A
q
, using the standard Campbell and
Shiller (1988) approximation r
t+1
= κ
0
+ κ
1
w
t+1
− w
t
+ g
t+1
. The resulting
equilibrium solutions for the three coefficients may be expressed as
A
0
=
log δ + (1 − ψ
−1
)μ
g
+ κ
0
+ κ
1
[A
σ
a
σ
+ A
q
a
q
]
(1 − κ
1
)
, (7)
A
σ
=
(1 − γ)
2
2θ(1 − κ
1
ρ
σ
)
, (8)
A
q
=
1 − κ
1
ρ
q
−
(1 − κ
1
ρ
q
)
2
− θ
2
κ
4
1
ϕ
2
q
A
2
σ
θκ
2
1
ϕ
2
q
. (9)
9
The assumption that γ > 1 is generally agreed upon, but the assumption that ψ > 1 is a matter of some debate
(see, for example, the discussion in Bansal and Yaron 2004).
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