The Risk-Return Relationship and Financial Crises
Eric Ghysels
Alberto Plazzi
Rossen Valkanov
UNC∗
University of Lugano and SFI†
UCSD‡
This Version: January, 2013
This Version: April 15, 2013
ABSTRACT
The risk-return trade-off implies that a riskier investment should demand a higher expected return relative to
the risk-free return. The approach of Ghysels, Santa-Clara, and Valkanov (2005) consisted of estimating the
risk-return trade-off with a mixed frequency - or MIDAS - approach. MIDAS strikes a compromise between
on the one hand the need for longer horizons to model expected returns and on the other hand to use high
frequency data to model the conditional volatility required to estimate expected returns. Using the approach
of Ghysels, Santa-Clara, and Valkanov (2005), after correcting a coding error pointed out to us, we find that
the Merton model holds over samples that exclude financial crises, in particular the Great Depression and/or
the subprime mortgage financial crisis and the resulting Great Recession. We find that a simple flight to safety
indicator separates the traditional risk-return relationship from financial crises which amount to fundamental
changes in that relationship. For those months or quarters we characterize as flight to safety we find there is no
(i.e. neither negative nor positive) risk-return trade-off.
∗
Department of Finance, Kenan-Flagler Business School and Department of Economics, UNC, Gardner Hall
CB 3305, Chapel Hill, NC 27599-3305, phone: (919) 966-5325, e-mail: [email protected].
†
Via Buffi 13 Lugano, 6900, Switzerland, email: [email protected].
‡
Rady School of Management, Otterson Hall, 9500 Gilman Drive, La Jolla, CA 92093, phone: (858) 534-0898,
e-mail: [email protected].
1
Introduction
The writing of this paper was prompted by recent exchanges of the authors with Esben Hedegaard and Bob Hodrick who discovered a key - albeit subtle - mistake in the Matlab code that
was written for the estimation results reported in Ghysels, Santa-Clara, and Valkanov (2005)
on the risk-return trade-off.1 Our purpose is two-fold: (1) to report corrected estimation results and (2) to provide a re-examination of the risk-return relationship in the context of the
current financial crisis.
The risk-return trade-off implies that a riskier investment should demand a higher expected
return relative to the risk-free return. One version of the ICAPM of Merton (1973) states that
the expected excess equity market return is positively related to its conditional variance:
Et (Rt+1 ) = µ + γVart (Rt+1 )
However, the literature often finds conflicting results as it is not clear whether the coefficient
γ is positive and significant.2 A possible reason for such results is the lack of conditioning
variables in the ICAPM equation. In fact, the ICAPM is a broader theory, as it allows for
expected returns to be driven by conditional covariances of returns with additional state variables. Scruggs (1998) finds that it is crucial to control for shifts in investment opportunities
to obtain a positive risk-return relation. Lettau and Ludvigson (2001) point out the relevance
of the consumption-wealth ratio as a conditioning variable in the ICAPM. Similarly, Guo and
Whitelaw (2006) argue that the specification misses the hedge component of Merton’s model.
They include additional predictors to account for changing investment opportunities and uncover a positive risk-return relation. In the same vein, Ludvigson and Ng (2007) show that
including factors in the ICAPM equation leads to a positive relation between risk and return.
Finally, the literature review by Lettau and Ludvigson (2010) emphasizes the importance of
conditioning variables in the estimation of the risk-return trade-off. In particular, they find a
positive conditional correlation between risk and return that is strongly significant (conditional
on lagged mean and lagged volatility), whereas the unconditional risk-return relation is weakly
negative and statistically insignificant.
The approach of Ghysels, Santa-Clara, and Valkanov (2005) consisted of estimating the riskreturn trade-off with a mixed frequency - or MIDAS - approach. MIDAS strikes a compromise
1
Esben Hedegaard wrote independently original code that yielded results different from ours. Ultimately he
found the error in our code which related to the actual sample size being used in the estimation route. We are
particularly grateful to Esben for all his insightful feedback and help.
2
For example, French, Schwert, and Stambaugh (1987) find a strong negative relation between the unpredictable component of volatility and expected returns, whereas expected risk premia are positively related to
the predictable component of volatility. Guo and Whitelaw (2006) and Ludvigson and Ng (2007) find a positive
risk-return trade-off. However, Glosten, Jagannathan, and Runkle (1993), using different GARCH specifications, find a negative relation between risk and return. Brandt and Kang (2004) model both the expected
returns and conditional variance as latent variables in a multivariate framework and find a negative trade-off.
1
between on the one hand the need for longer horizons to model expected returns and on
the other hand to use high frequency data to model the conditional volatility required to
estimate expected returns.3 Using the approach of Ghysels, Santa-Clara, and Valkanov (2005),
after correcting the aforementioned coding error, we find that the Merton model holds over
samples that exclude financial crises, in particular the Great Depression and/or the subprime
mortgage financial crisis and the resulting Great Recession. We find that a simple flight to
safety indicator separates the traditional risk-return relationship from financial crises which
amount to fundamental changes in that relationship. We follow the spirit of Baele, Bekaert,
Inghelbrecht, and Wei (2012) who attempt to provide an empirical characterization of flight
to safety. Their measure is based on the tail behavior of the unconditional return distribution.
We extend their approach - using tail-measures of the conditional distribution rather than the
unconditional one. We do so via quantile regression methods. The latter share features with
the specification of conditional volatility, namely we use daily information in the formulation
of conditional quantiles for monthly and quarterly returns. The quantile regressions are also
of the MIDAS type.
In a first section we document revised estimates of Ghysels, Santa-Clara, and Valkanov
(2005). A second section flight to safety characterization, econometric issues and empirical
results. A concluding section follows.
2
Risk-return trade-off: MIDAS estimates revised and updated
Ghysels, Santa-Clara, and Valkanov (2005) estimate via QMLE the parameters θi jointly with
µ and γ in:
Rt+1 = µ + γVtMIDAS + εt+1
VtMIDAS = Vart [Rt+1 ]
MIDAS
Vt
= A
D−1
X
2
w(d, θ1 , θ2 )rt−d
(2.1)
d=0
where
exp{θ1 d + θ2 d2 }
w(d, θ1 , θ2 ) = P∞
2
i=1 exp{θ1 i + θ2 i }
and Rt refers to low frequency excess returns - say monthly or quarterly - whereas rt−d are
daily returns for day t − d. The scaling constant A is either 22 or 66, respectively for monthly
and quarterly returns - linking daily returns r to either monthly or quarterly R. Moreover, we
3
Hedegaard and Hodrick (2013) carry the MIDAS argument further. Namely, they develop an overlapping
data inference methodology that uses all of the data while maintaining the monthly or quarterly forecasting
period, and apply it to the conditional CAPM.
2
use the past D equal to 260 days as the maximum lag length as in the original paper.
p
Originally, we only considered Gaussian conditional distributions, i.e. εt+1 / VtMIDAS ∼
N (0, 1) which we treat as QMLE estimates - similar to the estimation of ARCH-type models.
p
We also add to this a MLE estimation by assuming that εt+1 / VtMIDAS ∼ t(DF ). In the former
case, we compute QMLE robust standard errors (see e.g. Bollerslev and Wooldridge (1992) for
the discussion in the context of ARCH-type models), whereas in the latter we use standard
ML standard errors. Note also that in the former case we estimate four parameters, i.e. µ, γ,
θ1 and θ2 , whereas in the latter we add ’DF’ as an extra parameter.
The empirical findings are reported in Tables 1, 2 and 3. We consider samples covering
monthly and quarterly returns from 1964-2000 (as in the original Ghysels, Santa-Clara, and
Valkanov (2005) paper), then we extend the original sample to include the subprime mortgage
crisis, i.e. 1964-2011. Finally, we also consider a sample 1928-2011 which covers both the
Great Recession and the subprime mortgage crisis.4 We start with the monthly results. For
the 1964-2000 in Table 1 an estimate of γ of 3.4656 that is significant at 10 % with a t-stat
of 1.867 (compared to a point estimate of 3.748 with a t-stat of 8.612 reported originally in
Table 2 of Ghysels, Santa-Clara, and Valkanov (2005)) for the QMLE procedure and 3.4275
with a t-statistic of 1.7816 for the MLE procedure with highly significant degrees of freedom of
roughly 8. With quarterly returns, for the same sample (reported in Table 2) we find a QMLE
estimate for γ of 4.3269 with t-stat 2.0364 (compared to an estimate of 3.476 with t-stat 5.028
in Table 3 of the original paper) whereas the MLE estimate is 5.4831 with t-stat of 2.7296.5
The point estimates all imply a positive risk-return trade-off - as originally found - albeit with
larger standard errors and therefore smaller t-statistics.
Considering monthly returns again, and including the subprime crisis, we obtain estimates
for γ of 1.3464 (QMLE) and 1.4040 (MLE) that are both insignificant. Hence, the parameters
remain positive, but much lower and insignificant. With quarterly returns we obtain estimates
that remain positive and low with QMLE - namely equal to 1.7816 and insignificant - but
remain similar to the previous sample estimates - equal to 2.2673 with t-stat 1.7566 and thus
significant at 10 % with MLE. Finally, when we consider the sample starting in 1928 and
ending in 2011 - covering at least two major financial crises - we obtain estimates of γ that are
negative and insignificant.
To conclude we also report some updates for Tables 2 and 3 of Ghysels, Santa-Clara, and
Valkanov (2005) which appear in Table 3. The results reveal a pattern similar to findings
4
Other sample configurations have also been considered - we cover these as they relate to the original paper
as well as updated samples.
5
Under suitable regularity conditions QMLE and MLE are consistent - and therefore should yield similar
point estimates. The differences should appear in standard errors. The estimated degrees of freedom in Tables
1 and 2 are 7.5 or larger. Although the t-distribution formally converges to the normal as the DF go to infinity,
the two distributions are actually quite similar when the DF get to be around 8. Hence, it is not too surprising
that the standard errors for QMLE and MLE are quite close, although for the quarterly returns in Table 2 we
note considerable reductions in standard errors with MLE.
3
reported in this section. The samples that include financial crises typically yield zero or
negative risk-return trade offs. In addition, we also consider the sample studied by Hedegaard
and Hodrick (2013). While our estimates are numerically slightly different from theirs the
main conclusions remain as well.6
3
Flight to Safety
It is often said that crises are characterized by flights to safety (henceforth FTS). Several attempts have been made to characterize the notion of FTS, see e.g. Vayanos (2004), Caballero
and Krishnamurthy (2008), Brunnermeier and Pedersen (2009), among others. A plethora
of econometric techniques to transform these features are used to identify such episodes, involving either threshold or regime-switching models.7 We follow the spirit of Baele, Bekaert,
Inghelbrecht, and Wei (2012) who attempt to provide an empirical characterization of FTS. In
particular, a large and positive bond return, a large and negative equity return, and negative
high-frequency correlations between bond and stock returns (which are typically otherwise
positively correlated) are used to characterize FTS. More specifically we use an augmented
ICAPM equation:
FTS
MIDAS
Rt+1 = µ + γVtMIDAS + 1FTS
+ εt+1
t+1 × µFTS + 1t+1 × γFTS Vt
(3.1)
where VtMIDAS is specified as in equation (3.1) and 1FTS
t+1 equals one during flight to safety
episodes and zero otherwise. The key is how to define 1FTS
t+1 . We follow two approaches - the
first is close to Baele, Bekaert, Inghelbrecht, and Wei (2012) and relies on the unconditional
distribution of returns, whereas the second approach is novel as it entails an attempt to use
characteristics from the conditional distribution of returns. The latter is an improvement of
Baele, Bekaert, Inghelbrecht, and Wei (2012) which is of independent interest. Two subsections
describe the details, preceded by a subsection covering econometric issues.
3.1
Econometric Issues
Unfortunately, the estimation of equation (3.1) is not as straightforward, as it features what
is known as a truncated regression problem. To proceed, we will assume for the moment that
FTS
1FTS
t+1 is based on a fixed threshold, such as for example 1t+1 equals one when Rt+1 < −1.5 ×
std(Rt ), which roughly corresponds to the 5 % left tail of the unconditional return distribution.
6
The numerical differences are due to the fact that Hedegaard and Hodrick (2013) use a Beta polynomial
for the estimation of the MIDAS weights (see Ghysels, Sinko, and Valkanov (2007)). In the process of writing
our paper, we exchanged codes with Esben Hedegaard and Bob Hodrick. Using exponential Almon weights, as
in Table 3, yielded the same empirical results with both our and their programs.
7
Some recent papers have adopted a regime switching model approach to the risk-return trade-off, see e.g.
Guérin (2011), Nyberg (2012), among others.
4
Note that such a formulation implies a look ahead bias - a topic we will discuss later. We can
therefore write (3.1) as:
E[Rt+1 |VtMIDAS , 1FTS
t+1 ] =

MIDAS
+ E[εt+1 |VtMIDAS , 1FTS

t+1 = 0]
 µ + γVt


MIDAS
(µ + µF T S ) + (γ + γF T S )Vt
(3.2)
MIDAS
+ E[εt+1 |Vt
, 1FTS
t+1
= 1]
Not surprisingly, the above equation tells us that distributional assumptions matter. Unlike the
estimation of (2.1), we are now faced with the need to make explicit distributional assumptions
beyond the typical convenience of QML estimation of risk-return relationships. Assuming
normality for the purpose of tractability (but not an empirically tenable assumption), we
have:
MIDAS
]
E[εt+1 |VtMIDAS , 1FTS
t+1 = 0] = E[εt+1 |εt+1 > −1.5 × std(Rt ) − µ − γVt
MIDAS
)/σε )
φ((−1.5 × std(Rt ) − µ − γVt
= σε
MIDAS
Φ((−1.5 × std(Rt ) − µ − γVt
)/σε )
≡ λ1 (µ, γ, σε )
(3.3)
where λ1 (µ, γ, σε ) is some times referred as the inverse Mills ratio, and
MIDAS
E[εt+1 |VtMIDAS , 1FTS
]
(3.4)
t+1 = 1] = E[εt+1 |εt+1 ≤ −1.5 × std(Rt ) − µ − γVt
MIDAS
)/σε )
φ((−1.5 × std(Rt ) − (µ + µF T S ) − (γ + γF T S )Vt
= σε
MIDAS
1 − Φ((−1.5 × std(Rt ) − (µ + µF T S ) − (γ + γF T S )Vt
)/σε )
≡ λ2 (µ, µF T S , γ, γF T S , σε )
where φ(.) and Φ(.) are the normal pdf and cdf, respectively. Moreover, let:
FTS
λ(µ, µF T S , γ, γF T S , σε ) ≡ (1FTS
t+1 = 1)λ2 (µ, µFTS , γ, γFTS , σε ) + (1t+1 = 0)λ1 (µ, γ, σε )
Then, the above implies a bias-correction to regression (3.1), namely:
MIDAS
Rt+1 = µ + γVtMIDAS + 1FTS
+ λ(µ, µFTS , γ, γFTS , σε ) + ut+1 (3.5)
t+1 × (µFTS + γFTS )Vt
Hence, the bias correction is a nonlinear function of µ, µF T S , γ, and γF T S , and the distributional assumption (i.e. εt+1 ∼ N (0, σε2 )) plays a key role in determining this function. Put
differently, the bias correction involves the distributional characteristics of the residuals, and as
a consequence, the bias correction is potentially biased as a result of the wrong distributional
assumption.8
8
Some semi-parametric procedures, estimating the inverse Mills ratio via kernel methods have been proposed,
see e.g. Heckman (1990), Andrews and Schafgans (1998) among others. Such procedures are more suitable
for large cross-sectional i.i.d. settings and therefore do not apply to our relatively small sample time series
application.
5
Given that we cannot proceed with the assumption that the errors are Gaussian, we rely
on a bootstrapping approach to correct for the truncation bias in equation (3.1). First, it is
worth noting some appealing features of MIDAS volatility estimation in this particular case.
Ghysels, Santa-Clara, and Valkanov (2005) noted that, as MIDAS involves higher frequency
data (daily versus monthly or quarterly in the conditional mean), it is known from Merton’s
work that higher frequency data is more desirable, compared to low frequency returns when
it comes to estimating conditional variances. There is a second appealing reason hitherto not
well spelled out in the literature and it relates to the aforementioned arguments. The fact that
daily returns are used in MIDAS implies that the conditional mean specification affects to a
lesser degree than the estimation of the conditional variance (say compared to GARCH). We
therefore consider a bootstrap setting based on the normalized returns:
zt+1 ≡
(Rt+1 − R̄T )
p MIDAS
Vt
(3.6)
with R̄T the sample mean of Rt+1 , from which we draw B bootstrap samples {ztb }Tt=2 for b =
p MIDAS
b
b
Vt
. It might
1, . . . , B, where we set B = 1000. This yield bootstrap returns Rt+1
≡ zt+1
b
FTS
MIDAS
be tempting then to run regression (3.1), using Rt+1
and VtMIDAS , 1FTS
as
t+1 and 1t+1 × Vt
regressors plus a constant. The drawback, however, is that such a regression implies that the
true volatility process is known to the econometrician. To add volatility uncertainty, we also
bootstrap errors ζtb ≡ VtMIDAS /RVt+1 where the denominator is the realized volatility, i.e. the
sum of daily squared returns over the next month/quarter.9 We therefore estimate (3.1), using
b
FTS
b
MIDAS
Rt+1
and ζtb × VtMIDAS , 1FTS
as regressors, with bootstrap samples
t+1 and 1t+1 × ζt × Vt
drawn from (3.6).
3.2
Unconditional FTS Analysis
We adopt an empirical definition of FTS inspired by Baele, Bekaert, Inghelbrecht, and Wei
(2012), albeit a simplified version of their suggested measure. Namely, since Rt is an excess
return, we define an indicator function 1FTS
t+1 as equal to one when Rt+1 < −1.5 × std(Rt ). As
noted before, this roughly amounts to looking at the left 5 % tail of excess returns, i.e. episodes
where excess returns are extremely low (and typically bond returns are high and equity returns
are low).10 It is also important to note that we do not use Rt+1 to formulate FTS and not
Rt . The latter would in fact imply that extreme outcomes are predictable using only the past
return series - which is not the case. Hence, the FTS dummy allows us to assess the risk return
9
To guarantee that the volatility remains unbaised for each bootstrap sample we rescale ζtb so it has mean
equal to one.
10
This is close to the univariate regime-switching model used by Baele, Bekaert, Inghelbrecht, and Wei (2012)
to characterize FTS episodes. Of course, left tail observations of Rt may be due to high interest rates and normal
equity returns or normal interest rates and extremely negative equity returns. It would take a bivariate analysis
to determine the full extend of the co-movements. Baele, Bekaert, Inghelbrecht, and Wei (2012) consider both
univariate and bivariate schemes. We follow the former in spirit.
6
trade-off ex post, not ex ante in a predictive sense. The estimates appear in the top panels of
Tables 4 and 5. The samples covered are the two crisis-contaminated samples, i.e. 1964-2011
and 1928-2011.
The top panels in Tables 4 and 5 report the estimates with respectively Monthly/Daily and
Quarterly/Daily return combinations for the regression model appearing in (3.5). Since the
point estimates and the significance of the QMLE and MLE were similar, we report only the
former. Taken at face value the conclusions are quite simple and remarkable. The estimates of
γ are positive and significant - with a simple exception perhaps for the 1928-2011 sample with
quarterly returns. Moreover, the estimates of γF T S are large, negative and highly significant.
Likewise, the intercept µF T S is, when compared to µ, also large in magnitude, negative and
significant. Recall that 1FTS
t+1 picks out 5 % of the data. For those months or quarters we have
extremely negative estimates if we combine γ and γF T S , typically cancel each other out - and
therefore no risk-return trade-off. The latter is also a finding reported in Table 3. For the
other 95 % of the data, we obtain γ estimates that coincide with levels of risk aversion one
would expect in a Merton-type model.
Of course, we know from the discussion in the previous subsection that the aforementioned
results are biased. The last column in Tables 4 and 5 reports the p-value of the bootstrap
tests. They tell is how likely it is to observe values for γ above γ̂ in regression models (3.5)
with data generated according to (3.6). Specifically, call γ̂ b the bootstrap sample estimates for
regression model (3.5) using the bootstrap sample b using returns void of a risk-return trade-off
(the null of interest) and consider the p-value of γ̂ reported in Tables 4 and 5 obtained from
the actual historical data. The results in the last column tells us that the estimated values γ̂
are far out in the right tail of the γ̂ b bootstrap distribution, i.e. unlikely, under the bootstrap
scheme of no risk-return trade-off. The only exception is again the long sample with quarterly
returns, which as noted earlier, yields a lower and insignificant value of γ. While not reported
in the tables, it is also worth noting that the bootstrap p-values for γ̂F T S are also in the tails
as well. Note that the bootstrap test only tells us to reject the null, implying there is a positive
risk-return trade-off, even after taking into account the potential bias in its estimation. Yet,
we cannot take the point estimates of γ at face value. Indeed, the reported values most likely
feature an upward bias, i.e. the population value of γ is positive and significantly different from
zero but smaller than the reported estimates.
In Figure 1 we report the monthly and quarterly volatility MIDAS weights with and without
FTS, i.e. weights w(d, θ1 , θ2 ) for VtMIDAS obtained from estimating respectively equations (2.1)
(no FTS) and (3.1) over the sample 1964-2011 using monthly (top panel) and quarterly returns.
The FTS weights appear as the circled line. The plots reveal that taking into account FTS
yields shorter memory volatility weighting schemes. Some of the weighting schemes we obtain,
however, are more hump-shaped - a feature of the exponential Almon that can be remedied by
using for instance beta weighting schemes (see Ghysels, Sinko, and Valkanov (2007) for more
7
details).
3.3
Conditional FTS Analysis
Formulating the notion of FTS as discussed in the previous subsection involves the full sample
unconditional distribution of returns and therefore is subject to look ahead bias. In this subsection we extend the analysis of Baele, Bekaert, Inghelbrecht, and Wei (2012) to a conditional
setting. To do so, we need to model the conditional quantile θ which we denote by qθ,t (Rt ) .
When modeling such quantiles we want to be more explicit in our notation and denote them
by qθ,t (Rt ; δθ ) where the unknown model parameters are collected in vector δθ . The notation
reflects the fact that the function q can be estimated for each quantile θ and the parameters
δθ may differ across quantiles and horizons. For instance, we can model qθ,t (rt,n ; δθ,n ) as an
affine function of state variables, collected in a vector Zθ,t−1 :
qθ,t (Rt ; δθ ) = αθ + βθ Zθ,t−1
(3.7)
where δθ = (αθ , βθ ) are unknown parameters to be estimated. In the above specification,
we allow the conditioning variables Zθ,t−1 to also differ across quantiles. The choice of the
functional form and conditioning variables in the estimation of conditional quantile regressions
is similar to that of any regression, whether we are estimating a conditional mean, conditional
variance, or a conditional quantile. Therefore, the parametrization of qθ,t (Rt ; δθ ) and the type
of conditioning information that is used are of primary importance.
To capture fluctuations in the quantiles of Rt , we use daily conditioning variables similar
to the MIDAS specification of conditional volatility. Hence, we formulate conditional quantiles
qθ,t (Rt ; δθ ) for next months/quarter excess returns, given past high frequency data. The
notion of MIDAS quantile regressions was introduced in Ghysels, Plazzi, and Valkanov (2011).
In particular, we characterize a MIDAS quantile regression - where the conditional quantile
pertains to Rt returns and the regressors are daily conditioning information - as follows:
qθ,t (Rt ; δθ ) = αθ + βθ Zt−1 (κθ )
Zt−1 (κθ ) =
D
X
wd (κθ ) xt−1−d
(3.8)
(3.9)
d=0
where δθ = (αθ , βθ , κθ ) are unknown parameters to estimate. The quantiles are an affine
function of Zt−1 (κi,θ ) which consists of linearly filtered xt−1−d representing daily conditioning
information with lag of d days. The weights wd (κθ ) are parameterized as a lag polynomial
function whose shape is captured by a low-dimensional parameter vector κθ .
The parsimonious specification of the MIDAS weights wd (κθ ) again greatly reduces the
number of lag coefficients to estimate to only a few. The parameters κθ governing the filtering
8
of the daily observations appearing in equation (3.9) and the parameters αθ and βθ in the
quantile regression equation (3.8) are estimated jointly as further discussed below. The MIDAS
regression framework allows us to use high-frequency data in the estimation of quantile forecasts
at various horizons.
To estimate the quantile function (3.8), we need to specify the functional form of wd (κθ )
and the conditioning variables xt−1−d in the definition of Zt−1 (κθ ). We address these model
specifications briefly, as they are fairly standard in the literature. We follow Ghysels, SantaClara, and Valkanov (2006) and specify wd (κθ,n ) as a so called “Beta” polynomial.11 Ghysels,
Plazzi, and Valkanov (2011) provide the technical details regarding estimation of MIDAS
quantile regressions.
We also need to clarify the choice of conditioning variables xt−1−d . We follow Engle and
Manganelli (2004), who find that absolute returns successfully capture time variation in the
conditional distribution of returns, and use |rt−1−d | as the conditioning variable in (3.9). While
we could have used any conditioning information, the |rt−1−d | specification provides the most
robust results. Alternative specifications based on transformations of daily returns yield similar, but slightly noisier estimates.12
Having specified the conditional quantiles, we now characterize 1FTS
t+1 as equal to one when
Rt < q0.05,t (Rt ; δ0.05 ) , i.e. when returns (monthly or quarterly) are in the left 5 % tail of their
conditional distribution.
We turn our attention again to Tables 4 and 5, and consider the panels other than the
top one discussed earlier. Before turning out attention to the results it may be worth taking
first a look at Figure 2 where we plot the monthly and quarterly excess returns on the S&P
500 over the 1928-2011 sample together with (1) the 5 % unconditional (red line) and (2) the
5% conditional quantiles (green line) of monthly/quarterly Rt . For the conditional quantiles,
we use daily conditioning variables obtained via the MIDAS quantile regression appearing in
(3.9) using absolute high frequency returns. The top figure displays the monthly case and the
lower figure displays the quarterly returns and quantiles. The parameter estimates for the 5
% conditional quantile models appear in Table 6. In the latter table we only report the 5 %
quantile since the 95 % which also appears in the plot is not used for the purpose of estimating
the FTS models. We note from the figure that the excess returns cross the lower quantile on
a few occasions - during turbulent times. It is these left tail events - which again amount to 5
% of the sample that drive the estimation results reported in Tables 4 and 5. It is also clear
from the figure that there is clearly variation in the conditional quantiles, hence the appeal to
look into such a specification for FTS.
11
We could also use the exponential Almon - as we do in the volatility specification. Both polynomial
specifications yield similar results. Both involve only two parameters and have similar features - see Ghysels,
Sinko, and Valkanov (2007) for more details.
12
Results from regressions based on simple, squared, and cubed returns are available upon request.
9
The empirical results reported in Tables 4 and 5 involving conditional FTS indicators
indicate that overall the results for samples starting in 1964 we continue to find evidence of
positive and significant estimates of γ, using both squared and absolute daily returns in the
conditional quantile models. For this we rely on the estimates as such - although we know
that there are bias concerns indicated earlier. Unfortunately, there is no bootstrap version for
the conditional results reported in Tables 4 and 5. Indeed, bootstrapping quantiles estimates
is a bit more complicated.13 The results for the sample starting in 1928 are less convincing,
however. While all the estimates of γ remain positive, they are not significant - again using
the standard inference criteria which do not take into account the presence of biases. The
bootstrap results with unconditional FTS may give us some guidance, since the ’significant’
estimates according to standard asymptotics also appear to be significant according to the
bootstrap test. As far as this guidance goes, it appears that the conditional FTS still yields
significant risk-return trade-offs for the 1964-2011 sample.
4
Conclusion
We revisited the risk-return trade-off using the approach of Ghysels, Santa-Clara, and Valkanov (2005). After correcting a coding error pointed out to us by Esben Hedegaard and Bob
Hodrick, we find that the ICAPM model with the market volatility as risk factor holds over
samples that exclude financial crises, in particular the Great Depression and/or the subprime
mortgage financial crisis and the resulting Great Recession. We find that a simple flight to
safety indicator separates the traditional risk-return relationship from financial crises which
amount to fundamental changes in that relationship. While our findings still involve some
unresolved challenging econometric issues, our results suggest an appealing future research
program on the pricing of volatility risk.
13
Hahn (1995) considers bootstrapping for a generic quantile regression setting. However, the MIDAS setting
brings about additional complications that we leave for future research. Notably, the trick we used to bootstrap
volatility estimation error through ζtb cannot be used since we do not have realized conditional quantiles. This
means we need to have a mixed frequency bootstrapping scheme - a topic we leave for future research.
10
References
Andrews, D., and M. Schafgans (1998): “Semiparametric estimation of the intercept of a
sample selection model,” Review of Economic Studies, 65, 497–517.
Baele, L., G. Bekaert, K. Inghelbrecht, and M. Wei (2012): “Flight to Safety,”
Discussion paper, Columbia, Ghent and Tilburg Universities.
Bollerslev, T., and J. Wooldridge (1992): “Quasi-maximum likelihood estimation and
inference in dynamic models with time-varying covariances,” Econometric reviews, 11(2),
143–172.
Brandt, M. W., and Q. Kang (2004): “On the relationship between the conditional mean
and volatility of stock returns: A latent VAR approach,” Journal of Financial Economics,
72(2), 217–257.
Brunnermeier, M., and L. Pedersen (2009): “Market liquidity and funding liquidity,”
Review of Financial studies, 22, 2201–2238.
Caballero, R., and A. Krishnamurthy (2008): “Collective risk management in a flight
to quality episode,” The Journal of Finance, 63, 2195–2230.
Engle, R., and S. Manganelli (2004): “CAViaR,” Journal of Business and Economic
Statistics, 22(4), 367–381.
French, K. R., G. W. Schwert, and R. F. Stambaugh (1987): “Expected stock returns
and volatility,” Journal of Financial Economics, 19(1), 3–29.
Ghysels, E., A. Plazzi, and R. Valkanov (2011): “Conditional Skewness of Stock Market Returns in Developed and Emerging Markets and its Economic Fundamentals,” Swiss
Finance Institute Research Paper No. 11-06.
Ghysels, E., P. Santa-Clara, and R. Valkanov (2005): “There is a risk-return trade-off
after all,” Journal of Financial Economics, 76(3), 509–548.
(2006): “Predicting volatility: getting the most out of return data sampled at different
frequencies,” Journal of Econometrics, 131(1-2), 59–95.
Ghysels, E., A. Sinko, and R. Valkanov (2007): “MIDAS regressions: Further results
and new directions,” Econometric Reviews, 26(1), 53–90.
Glosten, L. R., R. Jagannathan, and D. E. Runkle (1993): “On the Relation between
the Expected Value and the Volatility of the Nominal Excess Return on Stocks,” Journal of
Finance, 48(5), 1779–1801.
11
Guérin, P. (2011): “Essays in Applied Time Series Econometrics,” Ph.D. thesis, European
University Institute.
Guo, H., and R. F. Whitelaw (2006): “Uncovering the Risk-Return Relation in the Stock
Market,” Journal of Finance, 61(3), 1433–1463.
Hahn, J. (1995): “Bootstrapping quantile regression estimators,” Econometric Theory, 11,
105–121.
Heckman, J. (1990): “Varieties of selection bias,” American Economic Review, 102, 313–318.
Hedegaard, E., and R. Hodrick (2013): “Estimating the Conditional CAPM with Overlapping Data Inference,” Discussion paper Arizona State University and Columbia University.
Lettau, M., and S. Ludvigson (2001): “Resurrecting the (C)CAPM: A Cross-Sectional Test
When Risk Premia Are Time-Varying,” Journal of Political Economy, 109(6), 1238–1287.
(2010): “Measuring and Modeling Variation in the Risk-Return Trade-off,” Handbook
of Financial Econometrics, pp. 617–690.
Ludvigson, S. C., and S. Ng (2007): “The empirical risk-return relation: A factor analysis
approach,” Journal of Financial Economics, 83(1), 171–222.
Merton, R. C. (1973): “An Intertemporal Capital Asset Pricing Model,” Econometrica,
41(5), 867–87.
Nyberg, H. (2012): “Risk-Return Tradeoff in US Stock Returns Over the Business Cycle,”
Journal of Financial and Quantitative Analysis, 1(1), 1–50.
Scruggs, J. T. (1998): “Resolving the Puzzling Intertemporal Relation between the Market
Risk Premium and Conditional Market Variance: A Two-Factor Approach,” Journal of
Finance, 53(2), 575–603.
Vayanos, D. (2004): “Flight to quality, flight to liquidity, and the pricing of risk,” Discussion
paper, National Bureau of Economic Research.
12
Table 1: Risk-Return with MIDAS Volatility Model - Estimates with Monthly/Daily Returns
The entries to the table are parameter estimates for equations (2.1) using monthly low frequency returns. The
scaling constant Ap
is 22 and D is equal to 260 days as the maximum lag length. Two distributional assumptions
are used for εt+1 / VtMIDAS : (1) N (0, 1) which we treat as QMLE estimates and (2) t(DF ) viewed as MLE. In
the former case we estimate in equation (2.1) four parameters, i.e. µ, γ, θ1 and θ2 , whereas in the latter we add
’DF’ as an extra parameter. The numbers appearing below the estimates are the t-stat valid for respectively
QMLE and MLE.
Sample
Obs.
µ
γ
θ1
θ2
DF
1964.01-2000.12
432
-0.0007
-0.2519
3.4656
1.8670
-0.0303
-1.9661
0.0001
1.3616
-
0.0014
0.5266
3.4275
1.7816
-0.0288
-2.0983
0.0001
1.3926
8.4851
4.7763
0.0014
0.5859
1.3464
1.1844
-0.0373
-2.4501
0.0001
1.5223
-
0.0031
1.5170
1.4040
1.1653
-0.0319
-2.3176
0.0001
1.2280
8.5740
5.0045
0.0057
3.0639
-0.0212
-0.0224
-0.0271
-2.1723
0.0001
2.0571
-
0.0076
5.1173
0.0839
0.1008
-0.0344
-3.5598
0.0001
3.0406
8.9116
6.7694
1964.01-2011.12
1928.01-2011.12
564
997
13
Table 2: Risk-Return with MIDAS Volatility Model - Estimates with Quarterly/Daily Returns
The entries to the table are parameter estimates for equation (2.1) using quarterly low frequency returns. The
scaling constant Ap
is 66 and D is equal to 260 days as the maximum lag length. Two distributional assumptions
are used for εt+1 / VtMIDAS : (1) N (0, 1) which we treat as QMLE estimates and (2) t(DF ) viewed as MLE. In
the former case we estimate in equation (2.1) four parameters, i.e. µ, γ, θ1 and θ2 , whereas in the latter we add
’DF’ as an extra parameter. The numbers appearing below the estimates are the t-stat valid for respectively
QMLE and MLE.
Sample
Obs.
µ
γ
θ1
θ2
DF
1964.Q1-2000.Q4
144
-0.0051
-0.6028
4.3269
2.0364
-0.0208
-1.4842
0.0000
0.2556
-
-0.0042
-0.5559
5.4831
2.7296
-0.0192
-0.9604
-0.0000
-0.1051
9.0914
3.0404
0.0019
0.2709
1.7816
1.4306
-0.0224
-1.6583
0.0000
0.3628
-
0.0042
0.6849
2.2673
1.7566
-0.0202
-1.1157
-0.0000
-0.0022
9.9609
2.9399
0.0197
3.3010
-0.4021
-0.3683
0.0175
0.6720
-0.0001
-0.6589
-
0.0233
5.1489
-0.1435
-0.1681
0.0078
0.3952
-0.0000
-0.5673
7.5318
4.8331
1964.Q1-2011.Q4
1928.Q1-2011.Q4
188
332
14
Table 3: Updated Risk-Return Estimates with MIDAS Volatility Model
The table updates Tables 2 and 3 of Ghysels, Santa-Clara, and Valkanov (2005). Daily returns are used in the
construction of the conditional variance estimator. Monthly returns are used in the estimation of the risk-return
trade-off parameter. The t-statistics are computed using Bollerslev and Wooldridge standard errors.
Sample
µ
γ
θ1
θ2
Monthly/Daily
1928-2000
0.0057
2.8168
0.1472
0.1249
-0.0237
-1.7259
0.0001
1.6743
1928-1963
0.0103
3.5334
-1.0615
-0.7297
-0.0202
-1.1857
0.0001
1.3140
1955-2000
0.0009
0.3457
3.0255
1.5303
-0.0354
-2.5298
0.0001
1.9907
1964-2000
-0.0007
-0.2519
3.4656
1.8670
-0.0303
-1.9661
0.0001
1.3616
1955-2011
0.0026
1.2328
1.1525
1.0309
-0.0401
-3.0275
0.0001
2.2413
Bi-Monthly/Daily
1928-2000
0.0117
2.7029
0.0633
0.0502
-0.0043
-0.2316
0.0000
0.1425
1928-1963
0.0212
3.6353
-1.2221
-0.8249
0.0061
0.2332
-0.0000
-0.0267
1955-2000
0.0012
0.2033
3.1863
1.5304
-0.0308
-1.2556
0.0001
0.9698
1964-2000
-0.0012
-0.1684
3.3907
1.5263
-0.0057
-0.2961
-0.0000
-0.0462
1955-2011
0.0040
0.8606
1.4027
1.1566
-0.0291
-1.0872
0.0001
0.7893
Quarterly/Daily
1928-2000
0.0202
3.0931
-0.3230
-0.2388
0.0104
0.4260
-0.0000
-0.3765
1928-1963
0.0332
3.7885
-1.3691
-0.8554
0.2123
0.8410
-0.0009
-0.8999
1955-2000
-0.0045
-0.4759
4.6932
1.9821
-0.0123
-0.8966
0.0000
0.5373
1964-2000
-0.0051
-0.6028
4.3269
2.0364
-0.0208
-1.4842
0.0000
0.2556
1955-2011
0.0024
0.3383
2.0007
1.5490
-0.0111
-0.8508
0.0000
0.4737
15
Table 4: Risk-Return with MIDAS Volatility Model and Flight to Safety - Estimates with
Monthly/Daily Returns
The entries to the table are parameter estimates for equation (3.1) using monthly low frequency returns. The
scaling constant
A is 22 and D is equal to 260 days as the maximum lag length. QMLE estimates are reported
p
with εt+1 / VtMIDAS ∼ N (0, 1). involving five parameters, i.e. µ, γ, θ1 and θ2 , and γF T S . The numbers appearing
below the estimates are the t-stat valid for QMLE.
Sample
Obs.
µ
γ
θ1
θ2
µF T S
γF T S
Bootstrap
p-value
Unconditional FTS
1964.01-2011.12
564
0.0031
1.2291
3.9693
3.2480
-0.2625
-2.3390
0.0010
2.3207
-0.1020
-10.4016
-4.0999
-2.3928
0.01
1928.01-2011.12
997
0.0047
2.6177
3.0833
3.8027
-0.1648
-7.5301
0.0006
7.6184
-0.1251
-10.8571
-4.5896
-2.6843
0.02
Conditional Quantiles using Squared Returns
1964.01-2011.12
564
0.0038
1.6407
2.6116
2.2444
-0.0372
-1.5788
0.0001
0.7040
-0.0965
-9.7730
-6.1431
-1.7260
-
1928.01-2011.12
997
0.0095
5.6203
0.7813
0.9138
-0.0329
-2.3345
0.0001
2.1175
-0.0880
-13.1525
-9.9627
-2.9720
-
Conditional Quantiles using Absolute Returns
1964.01-2011.12
564
0.0052
2.1544
2.0080
1.6785
-0.0465
-1.5572
0.0001
1.0676
-0.0856
-8.3898
-9.7310
-1.8473
-
1928.01-2011.12
997
0.0101
5.5445
0.4643
0.5065
-0.0087
-0.3902
0.0000
0.3688
-0.0867
-12.8024
-9.9928
-2.3507
-
16
Table 5: Risk-Return with MIDAS Volatility Model with Flight to Safety - Estimates with
Quarterly/Daily Returns
The entries to the table are parameter estimates for equation (3.1) using quarterly low frequency returns. The
scaling constant
A is 66 and D is equal to 260 days as the maximum lag length. QMLE estimates are reported
p
with εt+1 / VtMIDAS ∼ N (0, 1). involving five parameters, i.e. µ, γ, θ1 and θ2 , and γF T S . The numbers appearing
below the estimates are the t-stat valid for QMLE.
Sample
Obs.
µ
γ
θ1
θ2
µF T S
γF T S
Bootstrap
p-value
Unconditional FTS
1964.Q1-2011.Q4
188
0.0042
0.6317
4.1899
3.2853
-0.0144
-0.5042
-0.0002
-0.6810
-0.1799
-10.1065
-4.2297
-2.2836
0.06
1928.Q1-2011.Q4
332
0.0197
3.3666
1.1978
1.5174
-0.2137
-2.0323
0.0008
2.0420
-0.2396
-8.6261
-4.0713
-1.8465
0.28
Conditional Quantiles using Squared Returns
1964.Q1-2011.Q4
188
0.0048
0.7384
3.1697
2.5495
-0.0052
-0.1318
-0.0002
-0.4468
-0.1903
-9.9282
-3.9873
-2.0474
-
1928.Q1-2011.Q4
332
0.0195
3.7348
1.2379
1.3628
-0.0252
-1.0286
0.0001
0.7151
-0.2084
-7.8136
-9.0733
-3.9283
-
Conditional Quantiles using Absolute Returns
1964.Q1-2011.Q4
188
0.0067
1.0167
2.6263
2.0967
-0.0108
-0.3778
-0.0002
-0.6502
-0.1724
-4.5761
-7.7621
-0.8822
-
1928.Q1-2011.Q4
332
0.0267
5.0849
0.2371
0.2575
0.0478
1.6258
-0.0002
-1.7885
-0.1549
-8.8975
-9.7866
-4.2081
-
17
Table 6: Conditional MIDAS Quantile Regression Estimates with Quarterly-Monthly/Daily
Returns
Conditional quantiles of Rt , - representing either monthly or quarterly returns - are estimated with daily conditioning variables reported in equation (3.9) where δθ = (αθ , βθ , κθ ) are unknown parameters to estimate. The
quantiles are an affine function of Zt−1 (κi,θ ) which consists of linearly filtered xt−1−d representing daily conditioning information with lag of d days. The MIDAS weights wd (κθ ) are parameterized by κθ . We follow Ghysels,
Santa-Clara, and Valkanov (2006) and specify wd (κθ,n ) as a so called “Beta” polynomial. The parameters αθ
and βθ in the quantile regression equation (3.8) are estimated jointly with κθ . We report the quantile estimates
for θ = 0.05, which corresponds to the lower 5 % tail. Quantile estimates for quarterly and monthly returns are
provided over two samples: (1) 1928-2011 and (2) 1964-2011, using either absolute or squared daily returns as
conditioning variables.
Sample
αθ
βθ
κ1θ
κ2θ
Monthly - Squared Returns
1964.01-2011.12
-0.0627
-4.0837
0.8976
10.5619
1928.01-2011.12
-0.0524
-10.5946
0.8971
1.1346
Monthly - Absolute Returns
1964.01-2011.12
-0.0487
-0.7824
0.9624
36.9225
1928.01-2011.12
-0.0279
-1.5070
21.0346
17.1768
Quarterly - Squared Returns
1964.01-2011.12
-0.1103
-5.6459
75.0000
59.4547
1928.01-2011.12
-0.1335
-3.6882
0.8663
1.3388
Quarterly - Absolute Returns
1964.01-2011.12
-0.0708
-1.2331
2.0839
94.4618
1928.01-2011.12
-0.02625
-2.1649
13.1921
10.1580
18
Figure 1: Monthly and Quarterly Volatility MIDAS weights with and without FTS
The figures display the weights w(d, θ1 , θ2 ) for VtMIDAS obtained from estimating respectively equations (2.1) (no
FTS) and (3.1) over the sample 1964-2011 using monthly (top panel) and quarterly returns. The FTS weights
appear as the circled line.
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
50
100
150
200
250
300
0
50
100
150
200
250
300
0.06
0.05
0.04
0.03
0.02
0.01
0
19
Figure 2: Monthly and Quarterly Returns S&P 500 1928-2011 with Conditional Quantiles
The figures display (1) returns of the S&P 500 (blue line), (2) the 5% unconditional (red line), and (3) conditional
(green line) quantiles of monthly/quarterly Rt . For the conditional quantiles we use daily conditioning variables
obtained via the MIDAS quantile regression appearing in (3.9). The top figure displays the monthly case and
the lower figure displays the quarterly returns and quantiles.
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
0
100
200
300
400
500
600
700
800
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0
50
100
150
20
200
250
300
350
900
1000