Returns and GARCH Component Estimated
Variance
Thierry C.J. Parihala1
Faculty of Economics and Business Administration
Tilburg University, NL
Final version: 08-12-2010
Abstract
This paper tries to contribute to a better understanding of the general risk-return
relation by examining the relation between the market variance and market return
and the relation between the market variance and the return of a small portfolio.
First, the market variance is measured using the GARCH Component model of
Engle and Lee (1993). Second, two new asset pricing models are constructed by
adding variance factors to the CAPM and to the model of Fama and French
(1993), creating the models CAPM+GCOMP and FF+GCOMP respectively.
Third, these models are used in regressions on the market and portfolio return.
Regressions using CAPM+GCOMP show there is a statistically and economically
significant positive relation between the market variance and the returns.
Regressions using FF+GCOMP do not give conclusive results. Several
motivations are examined that are likely to drive these findings. This paper
concludes that the market variance and market returns appear positively related.
the market variance and portfolio returns also appear positively related, but this
relation seems weaker and does not seem robust.
1
Thierry Parihala, student number 753274, has written this research paper as a student at Tilburg University for
the Master Financial Management – enrollment period: 2008-2010. This research paper is written to serve as the
Master Thesis for this master and may be freely distributed, copied, and edited. I am grateful to Prof.dr. B.J.M.
Werker for mentoring, commenting, Thomas van Steenis for reviewing and commenting, and Esther
Vorstenbosch for coaching. The views expressed in this paper are those of the author and do not necessarily
represent those of Tilburg University or Prof.dr. B.J.M. Werker. This paper is digitally available via the Library
Services of Tilburg University. For comments, questions or a digital copy, please contact the author at
[email protected].
1
I Introduction
The financial literature on the relation between risk and return is abundant. Theoretical works
like that of Sharpe (1964) and Black and Scholes (1974) indicate a relation between stock
returns and stock variance or between stock returns and the covariance between stock returns
and market returns. The empirical literature remains ambiguous, both regarding the existence
of this relation as regarding its nature. For example, French, Schwert and Stambaugh (1987),
Adrian and Rosenberg (2008) and Bollerslev and Tauchen (2009) find a positive relation
between variance and returns. Glosten, Jagannathan, and Runkle (1993) and Ang, Hodrick,
Xing and Zhang (2006) find a negative relation, while Baillie and DeGennaro (1990) find no
significant relation between variance and returns. The financial literature shows various
approaches have been used to measure the relation between variance and returns. Different
methods of measuring variance have been employed such as GARCH, GARCH-M,
component models, and “model-free” methods. Despite this, clear consensus has yet to
emerge. To complement the various approaches in this strand in literature, and to gain more
insight in the risk-return relation, this paper uses a different approach to measure the relation
between variance and returns.
In the first stage, a version of the GARCH Component model (GCOMP) developed by
Engle and Lee (1993) is used to measure the conditional variance. The GCOMP is a twocomponent model which distinguishes between the long-term trend and the short-term trend
in variance. Engle and Rosenberg (2000) perform hedging tests on several variance models to
compare their performance in explaining realized variance. They rank the ‘GCOMP with
leverage’ higher than other specifications of ARCH models, therefore this paper uses this
particular version of the GCOMP. In the second stage, the GCOMP variance is used as a
pricing factor in several specifications of asset pricing models. Adrian and Rosenberg (2008)
find that a pricing model in which short- and long-term variances are added to the model of
Fama and French (1993) (FF) performs better in explaining the cross-section of expected
returns, than other established pricing models – including the CAPM and FF model. In this
context, variance factors are added to the CAPM and to the FF model. The first factor which
is added to these models is the conditional variance specified by the GCOMP. The second
factor is the permanent component of the GCOMP, q, which is analogous to the long-term
trend in variance. Hence, the models CAPM+GCOMP and FF+GCOMP are created by
adding these two factors to the CAPM and FF model respectively. In the third stage, these
pricing models are used to perform regressions on the market return. Scope limitations of this
paper demarcate the area of interest of the empirical part to the effect of market variance on
2
market returns. In this way, this paper tries to gain more insight in the risk-return relation on a
broader, market level. The results should answer the question, how is market risk related to
market return? The results obtained to answer this question can contribute to a better
understanding of the general risk-return relation. In addition, to explore if the contingent
effects of market risk on market return are also generalizable to a smaller selection of stocks,
the effect of market variance on the return of a small portfolio of randomly selected stocks is
also examined.
The results of the regressions using CAPM+GCOMP show that market variance is
positively related to the market return and the portfolio return. The explanatory power of
market variance is stronger for the market return than for the portfolio return. The coefficient
on market variance is statistically and economically significant for both the market as well as
the portfolio return. One standard deviation increase in market variance results in 1.6 basis
points increase in the daily market return and 1.5 basis points increase in the daily portfolio
return. The results regarding the market return appear robust using different specifications of
ARCH models to measure the variance. The regressions using FF+GCOMP do not give
conclusive results regarding the explanatory power of market variance in stock returns.
Several motivations for these findings are examined. The possibility that the FF factors do a
better job in explaining the effects of risk on return than the conditional market variance is not
excluded. Accuracy issues in estimating the variance are also considered as a driver of its low
explanatory power regarding asset returns, as financial literature, in the meanwhile, has
suggested better performing alternatives to model the conditional variance. For example,
Adrian and Rosenberg use a more advanced two-component variance model and do find a
statistically and economically significant role for short- and long-term variance when these
factors are added to the FF model. The conclusion of this paper is the market variance and
market returns appear positively related. The market variance and portfolio returns also
appear positively related but this relation seems weaker and does not seem robust, which
creates interesting possibilities for further research.
The remainder of this paper is organized as follows. Section II provides an overview
of literature related to this subject to provide the context for this research. This section
describes several theoretical and empirical works regarding the relation between risk and
return and reviews various variance models. Section III is the empirical part in which the
methodology, the model, and the results are presented. Section IV presents robustness results
and gives interpretations to the results of section III. Section V concludes.
3
II Literature Review
2.1 Theories on risk and return
Assuming investors are risk averse and rational, the intuitive consequence with respect to the
relation between risk and return is that higher risk makes investors require a higher expected
return. Obviously, this means that investors who seek higher expected returns must undertake
riskier investments. In order to be able to quantify this positive relation between risk and
return, both variables must be measurable. For returns this is obvious. For risk we use the
widely accepted measure of variance (or standard deviation), which shows the variation from
the mean. Assuming investors’ mean-variance utility function, variance in returns is seen as
synonymous for risk. Higher variance implies higher fluctuations in asset prices. The more the
value of an asset fluctuates, the less certainty there is about its value at the time the investor
wishes to sell it. This lack of certainty is, thus, perceived as risk and is, based on the
assumptions, unwanted.
A paradigm in finance which tries to describe this relation between risk and return
more precisely is the Capital Asset Pricing Model (CAPM), as developed by Treynor (1961,
1962), Sharpe (1964), Lintner (1965a,b) and Black (1972). The model assumes all investors
are risk averse and rational. Therefore, those who do take risk, expect to be rewarded. Second,
it assumes markets are efficient and perfect. And third, investors can borrow and lend
unlimitedly against the risk-free rate.2 The CAPM essentially divides risk into systematic risk
and idiosyncratic risk. It assumes the latter to be diversifiable through a properly structured
portfolio. Hence, investors will only be compensated for an asset’s systematic risk. That is,
investors will only be compensated for volatility which cannot be reduced through
diversification. The CAPM defines the volatility of the total market portfolio as the unit for
systematic risk, because the market portfolio is, by definition, the most diversified portfolio.
Inherently, the return on the market in excess of the risk-free rate is the price of one unit of
systematic risk. As common in economic theories, this price arises from an equilibrium
determined by the supply of and the demand for risk. Here, the demand is a function of
investors’ risk aversion. The logic reasoning of the CAPM that follows is that an asset’s
return in excess of the risk-free rate is therefore linked to an asset’s exposure to systematic
risk and the price of systematic risk. The exposure to systematic risk is determined by
2
To complete the analysis, CAPM also assumes that investors only care about expected returns and volatility,
that they have homogeneous beliefs about the market, that there is only one common risk factor which is the
systematic risk, and that returns are normally distributed.
4
measuring the sensitivity of an asset’s returns to the sensitivity of market returns, which is
defined by:
βi = cov (ri, rm) / var(rm)
(1)
Where, βi is asset i's sensitivity to the market returns, Cov (ri, rm) is the covariance of the
returns of asset i and the returns of the market, and var(rm) is the variance of the returns of the
market. The expected return that the CAPM derives is:
E(ri) = rf + βi(E(rm) – rf)
(2)
Where, E(ri) is the expected return of asset i, E(rm) is the return on the market portfolio, and
Rf is the risk-free rate.3 According to the CAPM formula, asset returns are related to the
volatility of the market to a degree expressed by the beta. A higher beta implies more
sensitivity to market returns, a larger part of the volatility that cannot be reduced by
diversification, more exposure to systematic risk, and thus a higher expected return. In its
attempt to describe the interaction between risk and return, the CAPM proves a positive linear
relation with a constant rf and a reward for risk which is a function of the exposure to
systematic risk and the price of systematic risk. The CAPM provides a comprehensible
analysis of the risk-return relation in a straightforward way. However, the simplicity of the
analysis is only allowed for by defining certain assumptions which also form the weakness of
the model. Because the derivation of the model is flawless, criticism covers the assumptions.
Such as the fact that perfect markets do not exist, one can’t borrow and lend unlimitedly at the
risk-free rate, and there is no true risk-free asset. Critique has also reached a conceptual level.
Ross (1976b) argues it is difficult to justify the assumption of normality of returns as well as
investors’ mean-variance preferences. Furthermore, Roll (1977) explains that the CAPM
cannot be tested unless the complete market portfolio, including all assets, is known, which
cannot be identified.
An alternative to the CAPM is the Arbitrage Pricing Theory (APT), as described by
Ross (1976a, 1976b). Like the CAPM, the APT is a one-period model in which stochastic
properties of returns are consistent with a factor structure. Both models share the assumptions
of perfect capital markets and homogenous expectations. However, the APT does not require
assumptions of normally distributed returns and merely one common risk factor. Instead, the
model assumes that the returns of asset i can be predicted using the linear relationship
3
The original CAPM is based on arithmetically calculated returns, whereas this study will use log normal
returns.
5
between the returns of asset i on the one hand, and k common risk factors and idiosyncratic
(firm-specific) risk on the other. Furthermore, the APT assumes the non-existence of arbitrage
opportunities. Under these assumptions, Ross argues that the return of a risky asset can be
defined as:
i=
Here,
i
i
+ bi1
1
+ bi2
2
+ … + bik
is the expected return on asset i,
k
+
1,
(3)
i
…,
k
are the risk factors, the extent to which the
return depends on the these factors is given by the factor loadings bi1, …, bik, and i is the
idiosyncratic risk. This idiosyncratic risk is assumed to be zero on average, uncorrelated
between assets, and uncorrelated with the factors. As such, Ross proves that, in large
numbers, the average effect of the idiosyncratic risk can be neglected. The conclusion of the
APT is therefore, that the expected risk premium of a risky asset merely depends on the factor
loadings:
i
Where (
– rf = bi1(
fk
f1
– rf) + bi2(
f1
– rf)… + bik(
fk
– rf)
(4)
– rf) is the premium on factor k. The intuition behind this is that of Arrow Debreu
security pricing (Arrow and Debreu (1954)). If the unexpected part of asset i’s return is
linearly related to a set of k factors, then the expected return of that same asset must also be
linearly related to that same set of k factors. The model does not specify which factors might
have explanatory powers. But after finding these factors and determining the factor loadings
through regression, the APT will indicate what an asset’s price should be in accordance to the
most recent factor values - assuming the model is well specified. If the indicated price were to
deviate from the actual price, an arbitrage opportunity would present itself – which is assumed
not to be possible. In addition to having less stringent assumptions, one other advantage of the
APT over the CAPM is the absence of the direct need to identify the market portfolio.
Numerous studies have tried to identify appropriate common risk factors in an effort to
specify a factor model with a good fit. Fama and French (FF) show that the relations between
average return and size, and average return and book to market equity are strong and that the
explanatory power of the CAPM beta is actually weak (Fama and French (1992)). Fama and
French (1993) argue that the expected excess return is largely explained by the sensitivity to
three factors: 1) the risk premium on the market; 2) the return on a long position in a portfolio
of small stocks, financed with a short position in a portfolio with large stocks (SMB); and 3)
the return on a long position in a portfolio of high book-to-market stocks, financed with a
6
short position in a portfolio with low book-to-market stocks (HML). Therefore, they suggest a
factor model which accounts for these factors:
E(rm) – rf = α + bi(E(rm) – rf) + siE(SMB) + hiE(HML)
(5)
Here, (E(rm) – rf), E(SMB), and E(HML) are the expected premiums, and bi,, si, and hi are the
factor loadings determined by regression. In theory, the constant α is equal to zero as this
would otherwise point to the existence of possible other pricing factors. However, empirical
evidence does not support this hypothesis. The factor loading of the market risk premium bi is
analogous to the CAPM beta, but it is not equal to it due to the addition of the other two
factors. The extensions to the CAPM, as the SMB and HML factor can be regarded, are a
reaction to observed anomalies with respect to the CAPM. The CAPM can be viewed as a
particular one-factor model of the APT. This three-factor model can be seen as Fama and
French’s interpretation of the APT and is their way to explain anomalies found in the crosssection of average returns. The fact that the model originated from empirical findings makes it
quite applicable for practitioners who require estimates of expected stock returns. However,
the theoretical justification for the common risk factors is subject to debate. 4 It can thus be
said that the Fama and French model is more descriptive than explanatory. All the same, the
discussion is manifested at a conceptual level whereas the value of the SMB and HML risk
factors is empirically well proven.5 Likewise, empirical observations, that complement the
theories of the CAPM and the APT, have contributed to consensus in literature about the
existence of an important relation between asset returns and market risk.6 However, the exact
shape of this relation remains subject to the methodology used.
2.2 Modeling Variance
There is a conceptual discussion in science about what is the true variance in stock returns.
Variance can be estimated using data with different kinds of frequency. Most of the literature
stemming from the previous century is based on monthly or daily data. However, advances in
technology have made it possible for research to use intra-day data or even transaction data to
4
One economic interpretation of the explanatory power of SMB and HML that is in line with the Effcient
Market Hypothesis (EMH), as described by Fama (1969), is that the higher returns on these portfolios are due to
the higher risk which is inherent. Fama and French (1996) argue that a high book-to-market ratio signals distress
and doubts about future earnings, as do Chan and Chen (1991). A small firm could be less diversified making it
more exposed to business risk. Both result in a higher cost of capital. Adversaries of the EMH argue that the
higher returns on these portfolios are due to corrections following undervaluation.
5
See Banz (1981), Basu (1983), Rosenber, Reid, and Lanstein (1985), and Lakonishok, Shleifer and Vishny
(1994).
6
For empirical evidence see Douglas (1969), Miller and Scholes (1969), Black, Jensen, and Scholes (1972),
Fama and French (1993, 1996), Carhart (1997).
7
estimate variance. The discussion can subsequently be shifted to whether to use bid, ask or
mid prices. Researchers are given a lot of options in choosing the ‘best’ data frequency and
type and there is no convention or consensus about what is appropriate. Data used in financial
literature thus lacks certain uniformity. This paper will treat the results of different types of
data to be generalizable to results generated by other types of data as to avoid the discussion.
In modeling the conditional variance, the only real constant seems to be the continuous
evolution of stochastic variance models over time. To motivate the variance model used in
section III, the following paragraph will provide an overview of variance models working up
to the GCOMP model with leverage of Engle and Lee (1993). Based on the ARCH, as
proposed by Engle (1982), Bollerslev (1986) introduced the GARCH(p,q) model which may
be written as:
rit - rft = µt - 0.5σ
ෝ2t + εt
εt ~ N(0, σ
ෝ2t)
(6)
σ
ෝ2t = α + βσ
ෝ2t-1 + Φε2t-1
where rit is ln(pit /pit-1), where pit is the price of stock i at time t, rit - rft is the return in
excess of the risk free rate, µ is a constant risk premium, σ
ෝ2t is the conditional variance, and
Rft is the risk-free rate. This model allows for both autoregressive and moving-average
components in the heteroskedastic variance, making it an ARMA process. The parameter β
represents the rate of mean reversion, and the parameter Φ determines the relative importance
of the lagged squared error. Through the years, the widely used GARCH has uncovered one
stylized fact, which is the fact that variance in asset returns is highly persistent, i.e. the sum of
the parameters Φ and β is close to one. Despite this however, the mean reversion as modeled
in the GARCH also implies an exponential decay of a shock’s persistence. This is contrast
with findings in many empirical studies, which have pointed out that shocks to the conditional
variance endure in an extreme degree7. In other words, the GARCH assumes markets have a
short term memory, whereas empirical findings suggest markets have a long term memory.
An extension to the GARCH handling this issue is the IGARCH, as introduced by
Bollerslev (1986). When the GARCH model is integrated in variance, the innovations sum up
to one:
7
See for example, Bollerslev, Chou, and Kroner (1992).
8
௣
௤
௜ୀଵ
௜ୀଵ
෍ߚ + ෍ߔ = 1
(7)
This way, the IGARCH incorporates infinite persistence of shocks. A side effect of the unit
root in this model is that it is non-stationary. That means that there are no mean reverting
characteristics. As mean reversion is no longer a property of the variance, predictions using
IGARCH will tend to resemble recent variation more than the average historic variation. For
example, a large shock in 1929 will be expected to have less effect than an equally sized
shock in 1982. Another implication of the infinite persistence is the fact that the variance will
increase linearly with the forecast horizon. As Baillie, Bollerslev, and Mikkelsen (1996)
argue, such a relation between variance and horizon would cause the pricing of assets to be
extremely dependent on the initial conditions, or the current state of the economy, which both
contradict with observed pricing behavior. Besides Baillie et al, Bollerslev and Engle (1993)
also argue the infinite persistence contradicts to stylized facts. They argue financial markets’
variance depends on long run dependencies, and seem to have long term memory, but no
infinite memory. Hence, both the GARCH’s exponential decay and the IGARCH’s infinite
persistence seem to contradict with empirical findings. Or to quote Baillie et al., “the knifeedge distinction between I(0) and I(1) processes can be far too restrictive”.
In reaction, Baillie et al propose another extension to the GARCH, the FIGARCH, which is a
GARCH model, fractionally integrated I(d), with 0 < d < 1. As a first step in separating shortrun and long-run variance, the FIGARCH captures the short run through the GARCH
components, and accounts for the long-run through the fractional differencing parameter (d).
This model implies a slow hyperbolic rate of decay of the shocks. Hence, it is more in line
with empirical findings.
To cope with the leverage effect, first mentioned by Black (1976), Glosten et al.
(1993) generalize the GARCH(p,q), to allow for an asymmetric effect of negative return
shocks to variance. The theoretical explanation for this effect is that the debt-to-equity ratio
increases, when the market capitalization of the firm decreases, following a negative return
shock. The Glosten et al. model is defined by:
rit - rft = µt - 0.5σ
ෝ 2t + ε t
εt ~ N(0, σ
ෝ2t)
(8)
σ
ෝ2t = α + βσ
ෝ2t-1 + Φε2t-1 + γMax(0,-εt-1)2
9
Here, γ measures the effect of a negative return shock to the variance. The other coefficients
serve the same purpose as in the GARCH without leverage.
Engle and Lee (1999) further develop the distinction between variance in the short-run
and the long-run, by modeling the long memory behavior of the variance process as the sum
of two components. The transitory component, aimed at capturing the short-run, is
characterized by an almost complete integration. The permanent component is aimed at
capturing the long-run and has much faster time decay. Their model, known as the GCOMP,
short for GARCH Component model, is defined by:
rit - rft = µt 0.5σ
ෝ2t + εt
ε t ~ N(0, σ
ෝ2t)
σ
ෝ2t = qt + θ + α(εt-12 – q2t-1)+ β(σ2-1 – q2t-1) + γ[Max(0,-εt-1)2 –qt-1 ]
(9)
qτ = ω + ρq2t-1 + Ф(εt-12 - σ2t-1)
Here, q is the permanent long-run component, α is the shock on the short-run component of
variance, β reflects the influence of the lagged variance factor, γ reflects the short-run
leverage effect, ρ is the persistence of the long-run component, and Ф reflects the effect of a
variance shock on the long-run component. Engle and Rosenberg (2000) rank the
performance of stochastic variance models in explaining ‘true’ variance use hedging tests.
They derive and implement volatility hedges using different variance models and rank these
models based on their performance in hedging shifts in the volatility term structure.
According to their findings, this latter model of Engle and Lee outperforms all other
stochastic variance models.
2.3 Empirics on the relation between variance and return
As the CAPM argues market risk plays an important role in explaining asset returns, a more
direct way of examining this relationship is to assess the relation between market variance and
returns. Assuming variance varies trough time; changes in variance will change expectations
of future market returns and can affect the risk-return trade-off. In this respect, variance can
be seen as a systematic risk factor, which, according to the APT, should be priced in the
cross-section of stock returns. Merton (1980) was the first to argue that the market risk
premium depends on market variance. Based on the exploratory work of Merton (1980),
French, Schwert, and Stambaugh (1987) use daily values to predict monthly variance of stock
returns. To correct for both autocorrelation caused by non-synchronous trading of securities
10
and heteroskedasticity, French et al. first model the variance using an adjusted GARCH model
which allows for both these effects8. Second they estimate the following relation:
E( rmt - rft |
ଶ
୫୲ ,
σumt) = µ + b1
ଶ
ଶ୳
୫୲ + b2 ୫୲
(10)
where rm,t is the return on the stock market portfolio, rft is the risk-free rate,
ଶ
௠௧
is the
predicted variance of the market, the constant µ is can be viewed as an average risk premium,
and where
ଶ୳
௠௧
= ߪଶ௠௧ -
ଶ
௠௧
is the unpredicted variance, in which ߪଶ௠௧ is the observed
variance. These regressions do support the hypothesis that the unpredicted variance is
negatively related to the realized excess holding period return (b2 < 0). This, indirectly,
provides evidence for a positive relation between predictable variance and risk premiums9. As
French et al. argue, higher than expected observed variance, ߪଶ௠௧ >
ଶ
௠௧
, implies predictions
of variance will be revised upward for all future time periods. A positive relation between
standard deviation and risk premiums will cause the discount rate for future cash flows to
increase. Ceteris paribus, both the discounted present values as well as the current stock prices
will decrease. Therefore, French et al. argue: “a positive relation between the stock market
variance and the risk premium induces a negative relation between the unpredicted
component of variance and excess holding period returns”.10 However, they argue that the
effect of the negative relation between the unpredicted component in observed variance and
risk premiums is likely to be larger than the positive relation between the prediction of
volatility and risk premiums.
Ballie and DeGennaro (1990) use GARCH-M models, like explained in part 2.2,
which are adjusted to allow for the implications of delayed delivery. At the time of their
8
To measure the variance French et al use a GARCH type model specified as:
ଶ
୫୲
N
N
౪
౪షభ ଶ
= ቄ෌୧ୀଵ
r୧୲ଶ + 2 ෌୧ୀଵ
r୧୲ r୲ାଵ,୲ ቅ
(11)
Here, there are Nt daily returns, rit, in month t. The product of the daily return with an adjacent return serves to
capture the autocorrelation effect. French et al. do not subtract the average daily return in the variance
calculation because this adjustment so small that it does not impact the results. To prove this, French et al. have
experimented with model adjustments that feature the subtractions of within-month mean return from each
observation and experienced little effect on the results. Second, omitting this subtraction of the sample mean
when values are small is consistent with Merton (1980).
9
Initial regressions on the variance without the unpredicted variance component, provides weak evidence that
the standard deviation is positively related to the expected risk premium.
10
To strengthen their argument, French et al. use the following GARCH-in-mean model,
E( rmt - rft) = α + b3
m,t +
εt – b4εt-1,
(12)
as proposed by Engle, Lilien and Robins (1987), with which they show a significant positive relation (b3 > 0 )
between predicted volatility and risk premium in a more direct way.
11
research, delivery of stocks usually took place six days after the purchase. Baillie and
Degenarro argue that the opportunity costs of six days of interest could play a role in the riskreturn relation and that unexpected changes in this rate could influence variance. Therefore,
they use the change in Federal Funds Rate as a regressor in both the mean and variance
equation. Using daily CRSP data from 1970 to 1984 they investigate eight specifications of
GARCH-M models, in which the mean equation and variance equation are estimated at the
same time, but find only one specification to have a significant coefficient on variance at the
five percent level. The lack of evidence of a relationship between the mean returns and the
variance leads them to conclude that: “simple mean-variance models are inappropriate”. They
do not reject the hypothesis that a relation exists between risk and return, but they suspect
investors to use other types of risk measures which are more important than the variance of
portfolio returns.
Glosten, Jagannathan, and Runkle (1993) also investigate the tradeoff between risk
and return. They recognize “the general agreement that investors, within a given time period,
require a larger expected return from a security that is riskier”. But they stress that such
agreement does not exist concerning the intertemporal relation between risk and return. They
argue that, in theory, riskier periods could coincide with periods in which investors are better
able to bear risk. In such a case, a higher risk premium would not be a necessity. They also
suppose that in a world without true risk free assets, the risk premium on assets would be
reduced as the price of risky assets would bid up. They note that most of the empirical works
that find a positive risk-return relation use GARCH and GARCH-M models, while empirical
works that find a negative relation often use other techniques. Hence, they suggest that the
standard GARCH and GARCH-M models may not be rich enough to capture the time series
properties of monthly excess returns. Glosten et al. use an adapted GARCH-M model in
which they allow for seasonal effects, by adding a monthly dummy to the GARCH-M, and by
allowing for asymmetries to control for the leverage effect. Indeed their model suggests a
significant negative relation. They also find that the unexpected part of the returns and the
following period’s variance are negatively related.
As market return is defined as a weighted average return on all assets, the relation
between market variance and market returns could be induced by a relation between market
variance and individual asset returns. Ang, Hodrick, Xing and Zhang (2006) study this
conjecture by examining the effect of market risk on the cross-section of stock returns,
investigating relations of the form:
12
E(rit – rft | (rmt – γmt),(σmt – γvt)) = α + ܾଵ௜ (rmt – γmt) + ܾଶ௜ (σmt – γvt)
(13)
where (rmt – γmt) is the excess return on the market, where γmt is the conditional mean of the
market, and (σmt – γvt) is the innovation in market volatility, where γvt is the conditional mean
of the market volatility.11 Ang et al. find lower average returns for stocks that have higher
sensitivities, ܾଶ௜ , to innovations of market volatility.12 Ang et al. explain this effect using
several economic theories. Investment opportunities deteriorate with a rise in volatility,
leaving investors with a demand for a hedge (Chen, 2002). French et al. (1987) show that
large volatility shocks usually coincide with a downward movement of the market. Hence,
stocks with higher sensitivities to innovations in market volatility risk form a hedge (Bakshi
and Kapadia (2003)) which results in a higher demand and thus a lower return. Ang et al.
thus find a negative relation between risk and return. Though their model outperforms the
CAPM, it is outperformed by the Fama and French model.
Adrian and Rosenberg (2008) observe that recent component models for variance, in
which the variance is divided into a short-run part and a long-run part, demonstrate unique
performance in option pricing. They use the following model to estimate volatility:
Market return:
rit - rft = µt-1 + σ
ෝtηt
Market volatility:
σ
ෝt = exp(St + Lt)
ηt ~ N(0, 1)
(14)
Short-run volatility: St = Ф1St-1 + Ф2Max(0, -εt) + Ф3(|ηt |-√(2/π)
Long-run volatility: Lt = Ф4 + Ф5Lt-1 + Ф6Max(0, -εt) + Ф7(|ηt |-√(2/π)
where, (|ηt |-√(2/π) are the shocks to the volatility components, with expected value zero, Ф4
is the mean of the long-run component, and Ф2 and Ф6 are the sensitivities to the leverage
effect. The main difference between the short and long run component is that each component
has a different rate of mean reversion, thus capturing shocks to systematic risk at different
horizons. The asset pricing model they construct uses both the variance’s short-run and longrun component:
To measure the innovations in aggregate volatility, (σm,t – γv,t), Ang et al. use changes in the VIX index,
∆VIX, from the Chicago Board Options Exchange as a proxy. The VIX index is a derivative of traded options,
whose prices directly reflect volatility risk. Therefore, ∆VIX makes a good proxy for innovations in volatility
risk. However, use of options also implies implied volatility from the Black-Scholes (1973) model which is a
derived volatility. As a robustness test, Ang et al. have also used sample volatility like French, Schwert and
Stambaugh (1987), as well as other proxies for innovations in volatility risk but found little difference in their
results.
12
These results are adjusted for several known cross-sectional effects, such as size, value, momentum and
liquidity.
11
13
E( rit - rft |(E(rm) – rf), St, Lt) = b1(E(rm) – rf) + b2St + b3Lt
(15)
where (E(rm) – rf) is the market risk premium, St is the short term volatility and Lt is the long
term volatility, and b1 to b3 are the factor loadings. St and Lt are both derived using (14). To
motivate the use of St and Lt in their model, they relate them to tightness of financial
constraints and to the business cycle respectively, both for which they present empirical
evidence. Adrian and Rosenberg find evidence for a positive relation between risk and return.
More remarkably, they construct an asset pricing model which performs better than the FF
model. Their model reports lower pricing errors than the Fama and French model suggesting a
better specified model. A second model, in which the St and Lt factors are added to the FF
model, outperforms the first model of Adrian and Rosenberg. This does not contradict the
conjecture that the completion of the FF model with variance factors can lead to a better
specified model to explain stock returns.
III Empirical Research
3.1 Context
Section 2.1 shows the basis of asset pricing models that try to identify the factors which drive
common stock returns. This subject still is within the limelight of financial literature as
thorough understanding of this relation remains unattained. Section 2.3 shows some of the
work that approaches the risk-return relationship empirically. French et al. find a positive
relation between variance and returns using a GARCH model. Glosten finds this relation to be
negative, using a GARCH-M model. Meanwhile Baillie finds no convincing evidence
suggesting that this relation should be examined using other more precise measures of risk.
More recently, Ang et al. find a negative relation using another type of risk measure, changes
in the VIX, and Adrian and Rosenberg find a positive relation using a complicated short- and
long run variance model. The latter find that a model with the Fama and French factors
supplemented with the short- and long-run variance as pricing factors outperforms all other
model specifications. The discussion continues. And though one might think of the prevailing
thought to relate to the positive relation between risk and return, empirics finding a negative
relation are abundant13.
13
For empirical work finding a negative relation between risk and return Glosten et al. refer to Fama and
Schwert (1977), Campbell (1987), Pagan and Hong (1991), Breen, Glosten, and Jagannathan (1989), Turner
Startz, and Nelson (1989), and Nelson (1991).
14
To complement this strand of financial literature and improve the understanding of the
general risk-return relation, this paper will examine the relation between market risk and
market return using an approach different to those in the aforementioned studies. In this way,
this paper tries to answer the question; how is market risk related to market return? The
results obtained to answer this question can contribute to a better understanding of the general
risk-return relation. To explore the generalizability of the contingent effects of market
variance to smaller selections of stock returns, this paper also investigates the effects of
market variance on a small portfolio.
3.2 Methodology
Adrian and Rosenberg have found that asset pricing models in which variance factors are
priced factors exhibit good performance in explaining realized stock returns. To investigate
the relation between market risk and market return, an asset pricing model is constructed in
which market variance is a complementary pricing factor to both the CAPM and the Fama and
French Model. Engle and Rosenberg have evaluated the different variance models using
volatility term structure (VTS) hedging tests. These test asses the performance of different
variance models in predicting variance (or its square root) over the complete VTS.
Furthermore, Engle and Rosenberg provide several arguments to emphasize why this method
is preferred over the comparison of forecasted VTS to realized variance VTS.14 They rank the
GCOMP model with leverage (9) higher than other models, which is the reason that a
particular version of this model is chosen to measure variance in this research. Particularly,
both the conditional variance (σ2t) as well as the permanent component (qt) will be used in the
regressions on returns, because Adrian and Rosenberg also distinguish between trends in
variance at different horizons. Hence, the two asset pricing models used in the regressions are:
CAPM+GCOMP:
E( ri,t - rf,t |(E(rm) – rf), σ2, q) = bi1(E(rm) – rf) + bi2σ2t + bi3qt
FF+GCOMP:
E( ri,t - rf,t |(E(rm) – rf), HML, SMB, σ2, q)
= bi1(E(rm) – rf) + bi2HMLt + bi3SMBt + bi4σ2t + bi5qt
Here, E(rm) – rf is the market excess return, HML and SMB are the Fama and French factors, σ2
is the conditional variance of the market, qt is the permanent component of the conditional
14
First, hedging test may be able to better distinguish between models with different term structure shapes but
similar levels of unconditional variance. Second, “hedging tests may be superior at identifying omitted variables
or interrelationships in the volatility model, because hedging performance depends on eliminating sensitivity to
all sources of volatility” – Engle and Rosenberg (2000).
15
variance of the market, and bi…bj are the factor loadings on the dependent variables.
Hypotheses with respect to both models are:
H 0:
b1 = b2 = … = bj = 0
H 1:
bj ≠0 for at least one j
Though theory to justify the role of the factors in both my models is existent, the choice to
include them into these models is mainly driven by empirics dealt with in section II. This
pragmatic approach brings the use of these models into the realm of the APT. Hence, the
framework and the assumptions of the APT are applicable to this research. The methodology
uses the following sequence. First, the GCOMP model will be estimated using Maximum
Likelihood Estimation (MLE), so that the conditional variance series and permanent
component series for the market can be calculated. Second, using Ordinary Least Square
regression (OLS), both models of interest will be estimated and the significance of their factor
loadings will be determined. As depicted in the specification of both models, the variance of
stock returns, not the standard deviation, will be used in the regressions. French et al. have
shown that using variance or standard deviation in regressions does not influence the results
and that conclusions on effects of variance as dependant variable in regressions can be
generalized to effects of volatility – and vice versa. These regressions can be performed on
lots of types of stocks, portfolios, and the market. As said, scope limitations demarcate the
area of interest on the market return, for which the S&P 500 Composite is a proxy. In
addition, a portfolio of six randomly selected stocks will be used to explore if the results
obtained using market returns are somewhat generalizable to a smaller selection of stock
returns.
3.3 Data
The S&P 500 Composite is the proxy for the market. Hence, it will be used for calculating the
market variance and it will be used as a dependent variable in the regressions. The stocks used
to form a portfolio are randomly selected. For the return data, DataStream ‘Total Return’
indexes are used for both the S&P 500 Composite and the US stock returns as to incorporate
the effects of dividends. To reduce estimation errors resulting from a wrongly specified
volatility model, Nelson (1992) suggest using higher frequency data. Therefore this study uses
daily values. Higher frequency data, such as intraday data, is not available over the sample
16
period. The Fama and French Factors are obtained from Kenneth French’s online database15.
To save time, this study uses the market return and risk free rate from French’s database for
the regressions as these figures are ready available16.
Table 1 shows yearly expected returns of the selection of US stocks vary between 6%
and 17 % and standard deviations vary between 25% and 37%. The stock returns are
negatively skewed and have a high Kurtosis17. One implication is that extreme negative
values are more likely than in a normal distribution, but this is a normal empirical finding in
stock return data. The size factor (SMB) shows an average return of -0,04%. If this is the true
value of the SMB factor then this would go against Fama and French’s arguments for its
relevance. However, the 95% confidence interval ranges from approximately -17% to +17%,
which does not reject Fama and French’s observation that small stocks outperform large
stocks. Meanwhile, empirics also suggest a changing strength of the SMB factor through the
years, underpinning the fact that the SMB factor is not found significantly positive in this
sample18. The skewness of the portfolio which consists of six randomly selected stocks, each
with an equal weight, is positive. A large part of literature argues the shape of stock return
distribution to be negatively skewed, these studies are mainly focused on the aggregate of
stock market returns.
19
Singleton and Wingender (1986) find empirical evidence for
portfolios of stock returns to be positively skewed for certain periods in time, but not
persistently. Without persistence of skewness and in efficient markets this skewness should
not explain much about the pricing of stocks. This does not explain the positives skewness of
15
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
Though it is not likely that the market return of French’s database, which is based on the total CRSP database,
is exactly equal to that of the S&P, I assume that the difference is of such a magnitude that it will not influence
the results.
17
Kurtosis is high relative to that of a normal distribution which is 3. Also, another measure of Tabachnik and
Fidell (1996), who argue that data with a Kurtosis of more than two times the standard error of Kurtosis can be
considered not normally distributed, considers these Kurtosis values high.
18
As Fama and French (1993) argue, their observation that size is related to profitability stems mostly from the
1980s part of their data. Fama and French (1993):
16
“Until 1981, controlling for BE/ME, small firms are only slightly less profitable than big firms. But for small
firms, the 1980-1982 recession turns into a prolonged earnings depression. For some reason, small firms do not
participate in the economic boom of the middle and late 1980s”.
The difference in earnings between small and big firms creates a common risk factor, size, which is negatively
related to returns. However, this strong negative relation also seems not as strong after the 1980s. Zhang (2008)
observes that, during the 1992-2007 period, the Fama and French data shows that the SMB factor is much less
strong than during the 1963-1991 period. For lower book-to-market portfolios, the size effect even changes its
sign. As the majority of my sample period covers the period after 1990, Zhangs’ observations might underpin
why the SMB factor is not found significantly positive over the 1981-2005 sample period.
19
I refer to Fama (1965), Black (1976), Christie (1982), Blanchard and Watson (1982), Pindyck (1984), French
et al. (1987) and Hong and Stein (2003).
17
Table I
Descriptive Statistics
S&P 500
Portfolio
SMB
HML
MKT -/RF
RF
0,1307
0,1309
-0,0004
0,0605
0,0702
0,0554
Std. Dev.
0,1665
0,1883
0,0887
0,0823
0,1538
0,0019
Mean
0,0005
0,0003
0,0000
0,0002
0,0003
0,0002
Std. Dev.
Median
0,0105
0,0006
0,0119
0,0004
0,0056
0,0002
0,0052
0,0001
0,0097
0,0005
0,0001
0,0002
Maximum
Minimum
0,0831
-0,2121
0,1138
-0,2419
0,0643
-0,1124
0,0388
-0,0489
0,0863
-0,1716
0,0006
0,0000
Skewness
Kurtosis
-1,45
33,99
1,35
34,29
-1,43
34,46
0,04
9,44
-1,02
23,83
0,88
4,12
6311,00
6311,00
6311,00
6311,00
6311,00
6311,00
Coca Cola
General
Dynamics
General
Electric
Goodyear
Merck
Walt
Disney
0,1606
0,1353
0,1603
0,0631
0,1323
0,1340
Std. Dev.
0,2617
0,2822
0,2582
0,3617
0,2684
0,3218
Mean
0,0006
0,0005
0,0006
0,0003
0,0005
0,0005
Std. Dev.
Median
0,0165
0,0000
0,0178
0,0000
0,0162
0,0000
0,0228
0,0000
0,0169
0,0000
0,0203
0,0000
Maximum
Minimum
0,1796
-0,2836
0,1823
-0,1542
0,1174
-0,1922
0,1654
-0,3365
0,1225
-0,3117
0,1748
-0,3439
Skewness
Kurtosis
-0,61
22,21
0,10
10,38
-0,17
9,69
-0,47
15,58
-1,13
24,85
-1,10
25,05
6311,00
6311,00
6311,00
6311,00
6311,00
6311,00
Yearly Mean
Daily
Observations
Yearly Mean
Daily
Observations
This table shows the descriptive statistics for the sample used in the empirical part. I use daily return
data over the sample period 02-01-1981 to 30-12-2005 which gives 6311 observations. The Fama and
French factors, small minus big (SMB), high minus low (HML), and market return minus risk free rate
(mkt-/-rf), are obtained from Kenneth French's online database. The remaining data is obtained from the
DataStream database. I have used 'Total Return' indexes for both the S&P 500 Composite as the
individual stocks, to incorporate the effects of dividends. The portfolio consists of the six individual
stocks, in which all are equally weighted.
General Dynamics (GD). GD is the only stock for which the maximum return is higher than
its minimum return. Though it would be easy to impute the positive skewness to this positive
outlier, the histogram (see Figure 2 in the appendix) shows a combination of more positive
outliers as well as more positive observations in the proximity of the mean are accountable for
this positive skewness20.
20
The positive outliers are market responses to the sales of GD units or the announcements of restructuring.
This was the part of the strategy of downsizing, restructuring, and liquidation put in place by the new
management which was installed early in 1991. The highest outlier of 18 percent is the market response to the
sale of the Fort Worth division to Lockheed in March 1991. Subsequently, this management and its strategy
18
Table II
GARCH Component Model Estimates
return equasion
Variable
µ
Coefficient
0,0006
Std. Error
0,0001
z-Statistic
6,3216
Prob.
0,0000
variance equasion
ω
0,0001
0,0000
11,1112
0,0000
ρ
φ
0,9943
0,0281
0,0011
0,0035
879,3999
7,9501
0,0000
0,0000
θ
α
-0,0223
0,1204
0,0073
0,0072
-3,0486
16,7727
0,0023
0,0000
β
0,8484
0,0157
54,1002
0,0000
R-squared
-0,0001
Adjusted R-squared
S.E. of regression
-0,0001
0,0105
Sum squared resid
Log likelihood
0,6934
20713,55
Durbin-Watson stat
1950657
To estimate the variance of the market which will be used in the regressions, the GARCH
Component Model with leverage is used as shown in (16):
rit - rft = µ t + εt εt ~ N(0, σ2)
σ2 = qt + θ + α[ Max(0,-εt-1)2 – qt-1]+ β(σ2t-1 – q2t-1)
qt = ω + ρ(q2t-1 – ω) + Ф(ε2t-1 - σ2t-1)
Using Maximum Likelihood Estimation, I estimate the GCOMP model in Eviews on daily
return data of the S&P 500 between 02-01-1981 and 30-12-2005. The estimates of the
coefficients are shown in Table II. The alpha and beta sum up to 0.97 and are thus close to one
like is often seen in a GARCH type variance model of stock return data. All variables are
statistically significant at the 1 percent level.
3.4 GARCH Component Model
The daily variance series for the S&P 500 which will be used in the regressions are estimated
using Maximum Likelihood Estimation in Eviews. The variance of the S&P is estimated in
using the following version of the GARCH Component model of Engle and Lee (1993):
rit - rft = µt + εt
ε t ~ N(0, σ2)
(16)
σ2 = qt + θ + α[ Max(0,-εt-1)2 –qt-1 ]+ β(σ2t-1 – q2t-1)
qτ = ω + ρ(q2t-1 – ω) + Ф(ε2t-1 - σ2t-1)
The coefficients are estimated on daily log normal returns on DataStreams’ Total Return
index of the S&P 500 for the period 1981-2005. Estimates of the coefficients of the GCOMP
realized a dividend-reinvested return of 533% by 1993 – though the influence of US participation in the Gulf
War on GD’s stock returns should not be undervalued.
19
Table III
Comparison of S&P 500 sample variance estimated by different models
Var (daily)
GCOMP
GARCH(1,1)
GARCH(1,1) with
leverage
Descriptives
0,0001
0,0001
0,0001
0,0001
Var (yearly)
0,0270
0,0280
0,0286
0,0277
Std.Dev. (yearly)
16,44%
16,74%
16,90%
16,65%
Table III compares the sample variance of the S&P 500 estimated by different models. The first model is
the model of interest and is the GCOMP model as shown in (16), which will be used in the regressions.
The second model is a GARCH(1,1) model as shown in (6):
rit - rft = µ t + εt
εt ~ N(0, σ2t)
σ2t = α + βσ2t-1 + Φε2t-1
The third model is a GARCH(1,1) model adjusted to cope with the asymmetric effects of leverage as
shown in (8):
rit - rft = µt - 0.5σ2t + εt
εt ~ N(0, σ2t)
2
2
2
σ t = α + βσ t-1 + Φε t-1 + γMax(0,-εt-1)2
The descriptives are derived from the sample descriptives as generated by Eviews. All measures of the
sample variance are reasonably in line with each other showing yearly standard deviations of
approximately sixteen percent.
Table IV
Descriptive Statistics of the S&P 500 daily variance estimated by the GCOMP
Model
σ2
q
Mean
0,0270
0,0253
Std. Dev.
0,29%
0,14%
Mean
Median
0,0001
0,0001
0,0001
0,0001
Maximum
Minimum
0,0062
0,0000
0,0015
0,0000
Std. Dev.
Skewness
0,0002
18,55
0,0001
5,80
Kurtosis
Sum
469,41
0,6743
63,78
0,6313
Sum Sq. Dev.
Observations
0,0002
6311
0,0000
6311
Horizon
Descriptive
yearly
daily
Table 4 depicts the descriptive statistics of the daily variance series of the S&P 500 from 1981 to 2005 as
measured with the GCOMP model. These series are created using the model estimates from Table 2. The σ2
column depicts the descriptives for the conditional variance series and the q column depicts the descriptives
of the series of the permanent component of the GCOMP model. The yearly mean of the conditional
variance of 0,0270 is analogous to the yearly var of the S&P 500 of Table 3.
20
model are shown in Table 2. All coefficients are significant at a one percent level. The alpha
and beta sum up to 0.97 and are thus close to one like we would expect from a GARCH type
variance model.
Table 3 shows the mean variance and standard deviation of the S&P 500 estimated
with the GCOMP model. The values are in line with the GARCH(1,1) variance and the
GARCH(1,1) with leverage variance which I also estimated in Eviews using MLE. The
values also compare to the descriptives of table 1.
Table 4 shows the descriptive statistics of the daily variance series estimated with the
GCOMP. Obviously, a large part of the conditional variance is explained by the long run
component. Inherently to the way the GCOMP is constructed, q’s standard deviation is lower
than that of the conditional variance.
Though the GCOMP model contains a long run
component, there is no assignable short run component. In this respect the model is
sometimes referred to as a GARCH(2,2) model. Extreme values are controlled for by the
conditional variance, hence its higher skewness and Kurtosis.
3.5 Regression Results
Using the GCOMP variance time series, CAPM+GCOMP and FF+GCOMP are estimated on
the returns of the S&P 500 and the returns of the portfolio of randomly selected stocks using
OLS regression technique. Table 5 shows the results with t-statistics indented. All coefficients
of the benchmark models CAPM and Fama and French are found to be statistically
significant. Though the Fama and French factors increase the R-square of the model for both
the S&P and the portfolio, the increase is not as large as reported in Fama and French (1996).
For CAPM+GCOMP the coefficients on the GCOMP factors are significant for both the S&P
500 and the portfolio. The coefficient on the conditional market variance is positive,
suggesting a positive relation between risk and return. The permanent component of the
GCOMP model, q, shows a significant negative coefficient. At first sight, this might seem
contradicting to a positive relation between risk and return. However, the variable q is also a
constant in the variance model (see (9)). In this respect, the variable q is thus priced twice in
CAPM+GCOMP; once as part of the conditional variance, and once individually. Another
implication of the fact that q is part of the conditional variance is a high correlation between
both variables of .85 percent (see table 6). Therefore, the negative coefficient on q is likely to
offset part of the correlation. The 95% confidence ellipses of the coefficients of
CAPM+GCOMP for both the S&P 500 as the portfolio confirm this, see Figure 1. It shows
that the coefficient on q (C3 in Figure1) is likely to increase with a decrease in the market
21
Table V
OLS Results
Rm - Rf
HML
SMB
σ2
q
S&P 500
CAPM+
CAPM+σ2
GCOMP
CAPM
σ2
1,07*
…
1,07*
1,06*
1,01*
1,01*
1,01*
…
(455,05)
…
(439,21)
…
(451,67)
-0,02*
(449,27)
-0,02*
(451,64)
-0,02*
…
(-4,1)
-0,27*
(-3,99)
-0,27*
(-4,1)
-0,27*
(-89,22)
…
(-86,76)
0,15
(-88,51)
0,01
…
(1,02)
-0,25
(0,14)
…
(454,3)
…
…
…
…
…
2,71*
2,92*
0,92*
…
(4,38)
…
(13,77)
-3,78*
(8,33)
…
FF
(-11,01)
FF+GCOMP
FF+σ2
(-1,11)
R-squared
0,9704
0,0022
0,9690
0,9684
0,9867
0,9867
0,9867
Sum squared
residuals
0,0198
0,6919
0,0215
0,0219
0,0092
0,0092
0,0092
FF
FF+GCOMP
FF+σ2
Rm - Rf
HML
SMB
σ2
q
Portfolio
CAPM+
CAPM+σ2
GCOMP
CAPM
σ2
0,98*
…
0,98*
0,98*
0,98*
0,98*
0,98*
…
(106,99)
…
(106,85)
…
(79,79)
0,20*
(79,39)
0,20*
(79,81)
0,20*
…
(9,00)
-0,37*
(9,00)
-0,37*
(9,01)
-0,37*
(-21,65)
-0,51
(-22,08)
-0,67
(-0,66)
-0,30
(1,7)
…
(106,97)
…
…
…
…
…
2,46*
3,36*
0,81*
(-22,04)
…
…
(3,52)
…
(4,14)
-4,81*
(1,94)
…
…
(-3,67)
R-squared
Sum squared
residuals
(-0,24)
0,6443
0,0013
0,6453
0,6446
0,6849
0,6850
0,6850
0,3151
0,8849
0,3143
0,0071
0,2792
0,2791
0,2791
Table 5 shows the results of various Ordinary Least Squares regressions on the daily excess returns of the S&P 500
(upper half) and the daily excess returns of the small portfolio (lower half) between 1981 and 2005. The columns
show the composition of the model used in the regressions and the coefficients estimated for the independent
variables with t-statistics in parentheses. Rm-Rf is the market return minus the risk free rate, HML and SMB are
the high-minus-low and small-minus-big factors of Fama and French (1993), σ2 is the conditional market variance
measured by the GCOMP model as depicted in (16) of which the model estimates are shown in Table 2, and q is
the permanent component of the GCOMP model also explained in (16). Coefficients tagged with a (*) are
significant at the one percent level.
22
Figure 1
95% Confidence elipses for coefficients of the CAPM+GCOMP model
S&P 500
Portfolio
3.4
5
3.2
C(2)
C(2)
4
3.0
2.8
2.6
3
2
-2
-3.2
-3.6
C(3)
C(3)
-4
-4.0
-6
-4.4
0
2
4
3.
3.
8
C(1)
3.
6
1.070
2.
1.065
2.
-8
1.060
0.97
0.98
0.99
1.00
C(1)
C(2)
2
3
4
5
C(2)
Figure 1 shows the 95% confidence ellipses for the coefficients of the CAPM+GCOMP model, which are
estimated using OLS on the daily excess returns of the S&P 500 and the portfolio. The regressions cover
the sample period 1981 to 2005. C(1) is the coefficient on mkt-/-rf, C(2) is the coefficient on the
conditional market variance measured by the GCOMP model, and c(3) is the coefficient on q which is the
long term component of the market variance as measured by GCOMP model. The ellipse in the lowerright quadrant shows the interaction between the conditional market variance and the long term
component of the market variance q. It shows that the coefficient on q is likely to be less negative when
the conditional market variance is less positive.
Table VI
Correlation Matrix of Independent Variables
Rm - Rf
hml
smb
σ2
q
Rm - Rf
1
-0,6226
-0,3007
0,0262
0,0235
hml
smb
σ2
-0,6226
-0,3007
1
-0,0804
-0,0804
1
-0,0019
-0,1619
-0,0154
-0,0638
0,0262
0,0235
-0,0019
-0,0154
-0,1619
-0,0638
1
0,8477
0,8477
1
q
Table VI shows the correlation between the various independent variables used to construct the asset
pricing models. The correlation is determined based on daily data between 1981 and 2005. Rm - Rf,
hml, and smb are the Fama and French factors for this period of which the daily data is obtained from
Kenneth French's database. The GCMP variance factors, σ2 and q, are the conditional variance of the
S&P 500 and the permanent component of that variance, respectively. These daily series are
calculated using (16) and its estimates in Table 2.
23
variance. Estimations of CAPM+GCOMP without the permanent component q also do not
reject this theory. These regressions show the values of the coefficients on market variance
remain significantly positive but become much smaller, 0.92 and 0.81 for the S&P 500 and
the portfolio respectively, with almost equal R-square values for the model. Hence, a positive
relation between risk and return is not rejected by the negative coefficient on q. The R-square
value of CAPM+GCOMP is almost equal to that of the CAPM for both the S&P 500 and the
portfolio, suggesting that adding variance to the CAPM as a regressor does not drastically
increase the performance of the asset pricing model. Statistical significance in finance is of
course of less relevance without economic significance. To isolate any effects of
multicollinearity between the variance and q, we consider the coefficient on variance of the
model CAPM+σ2 on the S&P 500, which is 0,91. Table 4 shows the standard deviation of the
variance is 0,000185. One standard deviation increase in the variance results in 1,6 basis
points increase in the return of the S&P 500 on a daily basis21. That is equivalent to a yearly
increase in risk premium of 420 basis points. This is a considerable economic value.
Assuming transaction costs of 25 basis points, this leaves 395 basis points per year as a
trading opportunity, emphasizing the role of market variance in returns.
In contrast to CAPM+GCOMP, the coefficients on the GCOMP variables are not
found significant in FF+GCOMP. When observed singly, this result could be an argument for
the insignificance of variance in explaining asset returns. But when observed in conjunction
with the result on CAPM+GCOMP, one interpretation could be that most of the variation
explained by the GCOMP factors in CAPM+GCOMP is accounted for by the Fama and
French factors in FF+GCOMP, thus making the GCOMP factors insignificant. The strongly
declined coefficient on variance and q in FF+GCOMP, in comparison to CAPM+GCOMP,
does not falsify this conjecture. Furthermore, the coefficients on HML and SMB show
virtually no decrease in FF+GCOMP, when compared with the Fama and French Model. This
suggests that their explanatory power is not affected by the addition of variance to the model.
However, Table 6 shows the GCOMP factors and the Fama and French factors are almost
uncorrelated. Besides, the very small correlation coefficients show negative values which are
not supportive to this conjecture. One other explanation for the loss of significance of the
GCOMP factors in FF+GCOMP could be the lack of economic significance. However,
CAPM+GCOMP shows that that the variance does play an significant role economically.
21
In the sample data, a one percent return is denoted as 0,01. Hence, the impact of one standard deviation
increase in variance is 0,91*0,000185 = 0,000168 on the daily return, which, multiplied with 252 trading days, is
0,042 percent or 420 basis points annually.
24
With respect to the hypotheses, I reject H0 for both models. Consistent with a large part of
literature, these regression results suggest a positive relation between risk and return.
IV Analysis
4.1 CAPM+GCOMP
The results for CAPM+GCOMP are consistent with French et al. and Adrian and Rosenberg,
as variance is found positively related to returns. In addition to the 95% confidence ellipse of
the coefficients of CAPM+GCOMP, a Wald test also rejects the hypothesis that the
coefficient on variance is zero both in CAPM+GCOMP as in the CAPM+σ2 model at the one
percent level. The coefficient on variance is economically significant, but adding market
variance to the CAPM does not increase the model’s performance drastically. The effect of
adding market variance to the CAPM on the R-square of the model is small, which is in line
with the regression on variance only. In contrary, the R-square of the traditional CAPM factor
confirms its reputation for being a model which can explain a large part of variation in stock
returns. For CAPM+GCOMP, one standard deviation increase in the CAPM factor results in
103 basis points increase in the daily return for the S&P 500 and 116 basis points increase in
the daily return for the portfolio. Correspondingly, the R-square of the CAPM model is
already at 0.97 and 0.64 for the S&P 500 and the portfolio respectively.
To verify the robustness of these results, these regressions are tested using other, less
complicated variance models. The models used in the tests for robustness are a standard
GARCH(1,1) model (see (6) for the specification) and a GARCH(1,1) ‘with leverage’ model
(see (8) for the specification). The means of the time series estimated with these models are
reported in Table 3. Table 7 shows the regression results. For the regressions on the S&P 500,
CAPM+σ2 shows significant positive coefficients on the variance for both variance
specifications. Both coefficients are smaller than those reported for CAPM+GCOMP and the
CAPM+σ2 model in Table 5. However, Engle and Rosenberg have already emphasized the
weaker performance of these variance models in comparison to the GCOMP. The R-squares
are also smaller, though with a minimal amount. The results using the portfolio do not pass
the robustness test. The coefficients on variance are also smaller than those reported for
CAPM+GCOMP and the CAPM+σ2 model in Table 5, and remain positive, however they are
not significant. One possible explanation for this result could be the weaker variance models
used. One other explanation could be the weaker explanatory power of market variance in
explaining small portfolio returns compared to their explanatory power with respect to market
25
Table VII
Testing for robustness using different variance specifications in the
regressions
CAPM+σ2
Rm - Rf
HML
SMB
σ2garch(1,1)
σ2garch(1,1) with leverage
S&P 500
CAPM+σ2
FF+σ2
FF+σ2
1,07*
1,07*
1,01*
1,01*
(437,63)
…
(438,62)
…
(451,66)
-0,02*
(451,66)
-0,02*
…
…
(-4,08)
-0,73*
(-4,10)
-0,27*
0,49*
…
(-89,08)
-0,10
(-88,72)
…
(4,28)
…
0,65*
(-1,36)
…
-0,06
(6,91)
(-0,90)
R-squared
0,9681
0,9683
0,9867
0,9867
Sum squared residuals
0,0229
0,0019
0,0092
0,0092
FF+σ2
FF+σ2
Portfolio
CAPM+σ
CAPM+σ2
2
Rm - Rf
HML
SMB
2
σ
garch(1,1)
σ2garch(1,1) with leverage
R-squared
Sum squared residuals
0,98*
(106,88)
0,98*
(106,88)
0,98*
(79,82)
0,98*
(79,80)
…
…
0,20*
(9,03)
0,20*
(9,00)
…
…
-0,37*
(-22,11)
-0,37*
(-22,09)
0,29
(0,67)
…
-0,72
(-1,76)
…
…
0,55
…
-0,59
(1,53)
(-1,73)
0,9681
0,9683
0,9867
0,9867
0,0229
0,0019
0,0092
0,0092
Table 7 shows results of various Ordinary Least Squares regressions on the daily excess returns of
the S&P 500 (upper half) and the daily excess returns of a portfolio of six randomly selected stocks
(lower half) over the period 1981 to 2005. These regressions serve as tests for robustness for the
results obtained in Table 5. The columns show the composition of the model used in the
regressions and the coefficients estimated for the independent variables with t-statistics in
parentheses. Rm-Rf is the market return minus the risk free rate, HML and SMB are the highminus-low and small-minus-big factors of Fama and French (1993), σ2GARCH(1,1) is the conditional
variance of the S&P 500 as measured by (6), and σ2GARCH(1,1) with leverage is the conditional variance of
the S&P 500 as measured by (8). Coefficients tagged with a (*) are significant at the one percent
level.
returns. The results in Table 5 do not reject this conjecture, showing variance values for the
portfolio are smaller than for the S&P 500 return. It is possible that a combination of these
effects explain the results on the robustness test of the portfolio. The robustness tests reject
26
the hypothesis that the coefficient on market variance is significantly different from zero for
the FF+σ2 model, which is in line with the results from Section 3. In addition, Granger
causality tests show that the hypothesis that the conditional variance does not drive the returns
can be rejected for both the S&P 500 and the portfolio. Breusch-Godfrey serial correlation
LM tests on both CAPM+GCOMP and FF+GCOMP show no signs of serial correlation or
autocorrelation, strengthening the reliability of the results. More tests for robustness is also
considered. For example, the variance measured by both models shows extreme values during
the crash of 1987. Another test for robustness could therefore exclude these extreme values
out of the sample to examine the degree in which the results are driven by these extreme
values. However, these tests are not performed due to limitations in time. Based on all tests
on CAPM+GCOMP, variance appears to be positively related to returns. Specifically, a
statistically significant positive relation between the market variance and market return is
found as well as between market variance and portfolio return. The latter of these appears not
as strong as the first and does not pass tests for robustness.
4.2 FF+GCOMP
Section 3 shows that H0 can be rejected for FF+GCOMP due to the statistically significant
Fama and French factors. The GCOMP factors do not seem to play a significant role in
explaining returns - judging from their t-statistic and judging from the R-square of
FF+GCOMP in comparison to that of the Fama and French model. Based on the results of
CAPM+GCOMP, it is unlikely that the economic significance of the market variance is a
determinant of that fact. The reason why the GCOMP factors are insignificant in
FF+GCOMP could originate in the interaction between the GCOMP factors and the Fama
and French factors. A theoretical approach on how this interaction manifests might emphasize
Fama and French’s motivation for their relation with returns:
“Stocks with low long-term past returns (losers) tend to have positive SMB and HML slopes (they are
smaller and relatively distressed) and higher future average returns. Conversely, long-term winners
tend to be strong stocks that have negative slopes on HML and low future returns.”
- Fama and French (1993) –
According to Fama and French the HML and SMB factors do a good job in pricing risk which
mainly originates from higher probability of distress due to lower earnings. Combining the
good performance of these factors with the possibility of a relative weak model specification
of the GCOMP creates a basis for the insignificance of the GCOMP factors in FF+GCOMP.
27
Though Engle and Lee (1993) have found the GCOMP better performing in modeling
variance than other GARCH types, more recent option literature favors ‘real’ two component
models, like the one used by Adrian and Rosenberg (2008), as well as other methods of risk
measuring22. One such other measure, implied volatility from options, is compared to the
performance in forecasting variance of ARCH models by Blair, Poon, and Taylor (2001).
They find that implied variance forecasts are a better predictor of realized variance than
ARCH forecasts.
There are more possible explanations for the results in section 3 that demand
consideration. The varying results between CAPM+GCOMP and FF+GCOMP could be
generated by insufficient accuracy of the estimations due to the frequency of the data. Nelson
(1992) argues higher frequency of data results in less estimation errors. Andersen and
Bollerslev (1998) prove that the use of intraday returns significantly improves the forecasting
performance of ARCH models. French et al., Glosten et al., and Ang et al also use daily data,
but their methods use a monthly return horizon for which the monthly variance is calculated
as an aggregate of the daily variances. And the selected return horizon can contribute to the
varying results between CAPM+GCOMP and FF+GCOMP. This research uses daily data and
also asses the explanatory contribution of variance in the daily return. Bollerslev and Tachen
(2009) observe that the explanatory power of their measure of variance is strongest at the
quarterly horizon, and that the explanatory value of variance across other horizons varies
strongly. Adrian and Rosenberg (2008) find “wide dispersion in sensitivity to the volatility
components, which generates cross sectional variation in the risk premia attributed to these
facts”. So the explanatory power of variance has shown to vary in the cross section of returns,
suggesting the sample period characteristics might play a role. The fact that the SMB’s mean
of the sample used for this research deviates from other empirical work can be recalled in this
context. Hence, issues concerning the, return horizon, accuracy stemming from data
frequency, as well as the characteristics of the sample period could influence the results.
Moreover it is likely that the difference in results between CAPM+GCOMP and FF+GCOMP
originates from the SMB and HML factors capturing risk better than the variance specified by
an model for which better alternatives have been found in the mean time23. In summary, based
22
For option literature favoring two-component variance models, Adrian and Rosenberg refer to Xu and Taylor
(2004), Bates (2000), and Christofferssen, Jacobs, and Wang (2006).
23
A performance test of the GCOMP model as a complement to this analysis is not implanted due to limitations
in resources and scope of this paper.
28
on the results obtained with FF+GCOMP, conclusions cannot be drawn with certainty
regarding the relationship between market variance and stock returns.
4.3 Model adjustments
In hindsight, the initial specifications of CAPM+GCOMP and FF+GCOMP could have been
constructed different from their current structure. Like Adrian and Rosenberg (2008), this
research tried adding two volatility components to the return regressions to better differentiate
among trends in variance at different horizons. However, unlike Adrian and Rosenberg’s
model which truly distinguishes between short- and long-run variance, the GCOMP model
generates a long term trend which is added to the conditional variance. The correlation and
multicollinearity effects that are likely to drive some of the results could therefore have been
anticipated by omitting the variable q from both models. This would leave only the
conditional variance as an addition to the CAPM (CAPM+σ2) and the FF model (FF+ σ2).
The risk-return relation could also have been examined in several other ways. French
et al. use a prediction of market variance, predicted at t-1, as a regressor on stock returns at
time t. The model used to predict the variance inherently exerts a lot of influence on the
results. This could be one reason why French et al. do not find a direct positive relation
between their predicted variance and returns. With his modified GARCH-M model, Baillie
also uses a direct way to measure this relation and finds no significant relation. However, the
indirect measure used by French et al., the difference between their prediction of variance and
realized variance, does give some indications about the relation between risk and return. The
difference between what variance derived from ex ante data and variance derived from ex
post data also plays a role in Bollerslev and Tauchen. They use the difference between the
“model-free” implied variance, derived from a large collection of option prices, and the
“model-free” realized variance, computed by summating high-frequency intraday squared
returns, which they call the variance risk premium. This variance risk premium is significantly
positively related to the returns. In this respect, another way of examining the risk-return
relationship would be to examine the difference between variance derived from ex ante and ex
post data.
The focus of this paper is on the influence of market variance at time t on the market
return at time t, in the context of the CAPM and FF model. The regressions on the portfolio
support the results obtained with regressions on the S&P 500, but show signs of a weaker
relationship between market variance and portfolio returns than between market variance and
market returns. Besides the other possible modifications described in this part, it would be
29
interesting to see if portfolio returns perhaps exhibit a stronger relationship to portfolio
variance. It would also be interesting to examine how the individual stock returns are related
to their idiosyncratic risk.
V Conclusions
To gain more insight into the general risk-return relation, this paper examines the relation
between market risk and market return and market risk and portfolio return. In the first stage,
the conditional variance of the market using the GCOMP with leverage model is estimated. In
the second stage, two factors are added to well known asset pricing models. The first factor is
the conditional market variance as modeled by the GCOMP. The second factor is the long
term trend of the market variance, q, which is the ‘permanent’ variable within this GCOMP
model. These factors are added to the CAPM and the FF model, thus creating model
CAPM+GCOMP and model FF+GCOMP respectively. In the third stage, regressions using
these models are performed on the market return and the return of a portfolio of six randomly
selected US stocks. Consistent with a large part of the literature, the regressions using
CAPM+GCOMP show that the market variance is positively related to market returns and the
portfolio return. The explanatory power of the market variance is larger for market returns
than for the portfolio return, but both regressions indicate that market variance plays an
economic significant role in explaining returns. To be precise, one standard deviation increase
in the variance of the market results in 1.6 basis points increase in the daily market return and
1.5 basis points increase in the daily portfolio return. In contrary to the results obtained with
CAPM+GCOMP, the regressions using FF+GCOMP do not exhibit statistically significant
coefficients for the GCOMP factors. Several motivations for these findings are examined. I
theorize these results are possible when the FF factors do a better job in explaining the effects
of risk on return than the conditional variance. However, Adrian and Rosenberg do find the
coefficients on short- and long-term market variance to be significantly positive when added
to the FF model. The results could therefore also be influenced by the use of a variance model
for which literature, in the meanwhile, has suggested better performing alternatives. This
paper concludes that variance appears to be positively related to returns. Specifically market
returns and the market variance have a significant positive relation, which also passes several
tests for robustness. The relation between the market variance and portfolio return is also
significantly positive but seems weaker than the first. The tests for robustness on this relation
cannot provide conclusive results about its true nature. Perhaps smaller selections of stocks
and/or individual stocks are more sensitive to idiosyncratic risk. Further research might
30
therefore be focused on the relation between portfolio variance and portfolio returns or
idiosyncratic risk and individual stock returns. In doing so, it is recommended to use more
recent and advanced ways of modeling the conditional variance to avoid ambiguities found
using FF+GCOMP.
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Appendix
Figure 2
Histogram of General Dynamic's daily stock return 1981-2005
This histogram shows the distribution of the General Dynamics daily stock return between 1981 and
2005. Over this period, the skewness of the stock returns for GD is positive which contrasts to what is
normally observed in the empirics of individual stock returns. This figure shows that the positive
skewness is not only due to more positive outliers, but also due to more positive observations in the
proximity of the mean. The highest outlier is the market response to the sale of the Fort Worth division to
Lockheed in March 1991. This was the part of the strategy of downsizing, restructuring, and liquidation
put in place by the new management which was installed early in 1991. Subsequently, this management
and its strategy realized a dividend-reinvested return of 533% by 1993 – though the influence of US
participation in the Gulf War on GD’s stock returns should not be undervalued.
34