Asymmetric correlations of equity portfolios

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Journal of Financial Economics 63 (2002) 443–494
Asymmetric correlations of equity portfolios
$
Andrew Ang
a,
*, Joseph Chen
b
a
Columbia Business School, Columbia University, New York, NY 10027, USA
b
Marshall School of Business, University of Southern California, Los Angeles, CA 90089, USA
Received 11 May 2000; received in revised form 4 April 2001
Abstract
Correlations between U.S. stocks and the aggregate U.S. market are much greater fordownside moves, especially for extreme downside moves, than for upside moves. We develop anew statistic for measuring, comparing, and testing asymmetries in conditional correlations.Conditional on the downside, correlations in the data differ from the conditional correlationsimplied by a normal distribution by 11.6%. We ﬁnd that conditional asymmetric correlationsare fundamentally different from other measures of asymmetries, such as skewness and co-skewness. We ﬁnd that small stocks, value stocks, and past loser stocks have more asymmetricmovements. Controlling for size, we ﬁnd that stocks with lower betas exhibit greatercorrelation asymmetries, and we ﬁnd no relationship between leverage and correlationasymmetries. Correlation asymmetries in the data reject the null hypothesis of multivariatenormal distributions at daily, weekly, and monthly frequencies. However, several empiricalmodels with greater ﬂexibility, particularly regime-switching models, perform better atcapturing correlation asymmetries.
r
2002 Elsevier Science B.V. All rights reserved.
JEL classiﬁcation:
C12; C15; C32; G12
Keywords:
Stock return asymmetries; Correlation; Dispersion; Model bias; GARCH; Jump model;Regime-switching
$
Any errors or omissions are the responsibility of the authors. The authors wish to thank LarryGlosten, Charlie Himmelberg, Harrison Hong, and seminar participants at Columbia University, theFederal Reserve Board, Ohio State University, University of California at Riverside, University of Colorado at Boulder, University of Southern California, Vanderbilt University, and WashingtonUniversity. We are especially grateful for suggestions from Geert Bekaert, Bob Hodrick, and KenSingleton. We also thank an anonymous referee whose comments and suggestions greatly improved thepaper.*Corresponding author. Tel.: +1-212-854-9154; fax: +1-212-662-8474.
E-mail address:
aa610@columbia.edu (A. Ang).0304-405X/02/$-see front matter
r
2002 Elsevier Science B.V. All rights reserved.PII: S030 4- 405X(0 2)00068 -5
1. Introduction
Correlations conditional on ‘‘downside’’ movements, which occur when both aU.S. equity portfolio and the U.S. market fall, are, on average, 11.6% higher thancorrelations implied by a normal distribution. In contrast, correlations conditionalon ‘‘upside’’ movements, which occur when both an equity portfolio and the marketrise, cannot be statistically distinguished from those implied by a normaldistribution. Asymmetric correlations are important for several applications. Forexample, in optimal portfolio allocation, if all stocks tend to fall together as themarket falls, the value of diversiﬁcation may be overstated by those not taking theincrease in downside correlations into account. Asymmetric correlations have similarimplications in risk management. In this paper, we examine this correlationasymmetry in several ways.We begin by formally deﬁning downside correlations as correlations for whichboth the equity portfolio and the market return are below a pre-speciﬁed level.Similarly, upside correlations occur when both the equity portfolio and the marketreturn are above a pre-speciﬁed level. Downside correlations in U.S. markets aremuch larger than upside correlations as shown by the plots of downside and upsidecorrelations presented in Longin and Solnik (2001). These graphs demonstrate that,on the downside, portfolios are much more likely to move together with the market.Second, we measure this asymmetry by developing a summary statistic,
H
:
The
H
statistic quantiﬁes the degree of asymmetry in correlations across downside andupside markets relative to a particular model or distribution. This measurement of asymmetry is different from other measurements established in the literature.Covariance asymmetry has usually been interpreted within a particular generalizedautoregressive conditional heteroskedasticity (GARCH) model, where covarianceasymmetry is deﬁned to be an increase in conditional covariance resulting from pastnegative shocks in returns.
1
In contrast, our statistic measures correlation asymmetryby looking at behavior in the tails of the distribution. Our statistic is not speciﬁc toany model. Hence, we can apply the statistic to evaluate several different models. Weshow that conditional correlations differ from other measures of higher moments,such as skewness and co-skewness, and from risk measured by beta.The
H
statistic corrects for conditioning biases. Boyer et al. (1999), Forbes andRigobon (1999), and Stambaugh (1995) note that calculating correlations condi-tional on high or low returns, or high or low volatility, induces a conditioningbias in the correlation estimates. For example, for a bivariate normal distributionwith a given unconditional correlation, the conditional correlations calculated on joint upside or downside moves are different from the unconditional correlation.Ignoring these conditioning biases may lead to spurious ﬁndings of correlationasymmetry.
1
Authors such as Cho and Engle (2000), Bekaert and Wu (2000), Kroner and Ng (1998), and Conradet al. (1991) document the covariance asymmetry of domestic stock portfolios using multivariateasymmetric GARCH models.
A. Ang, J. Chen / Journal of Financial Economics 63 (2002) 443–494
444
Third, we establish several empirical facts about asymmetric correlations in theU.S. equity market. We ﬁnd the level of asymmetry, measured at the daily, weekly,and monthly frequencies, produces sufﬁcient evidence to reject the null hypothesis of a normal distribution. To investigate the nature of these asymmetric movements, weexamine the magnitudes of correlation asymmetries using portfolios sorted onvarious characteristics. Returns on portfolios of either small ﬁrms, value ﬁrms, orlow past return ﬁrms exhibit greater correlation asymmetry. We ﬁnd signiﬁcantcorrelation asymmetry in traditional defensive sectors, such as petroleum andutilities. We also ﬁnd that riskier stocks, as reﬂected in higher beta, have lowercorrelation asymmetry than lower beta stocks. After controlling for size, themagnitude of correlation asymmetry is unrelated to the leverage of a ﬁrm. Previouswork focuses on asymmetric movements of leverage-sorted portfolios of Japanesestocks (Bekaert and Wu, 2000), and size-sorted portfolios of U.S. stocks (Kroner andNg, 1998; Conrad et al., 1991) using asymmetric GARCH models.Finally, we analyze asymmetric correlations by asking if several reduced-formempirical models of stock returns can reproduce the asymmetric correlations foundin the data. These candidate models are used by various authors to capture theincrease in covariances on downside movements. We discuss four models that allowasymmetric movements between upside and downside movements in returns. Thesemodels are an asymmetric GARCH-in-Mean (GARCH-M) model, a Poisson Jumpmodel, for which jumps are layered on a bivariate normal distribution, a regime-switching normal distribution model, and a regime-switching GARCH model. Weﬁnd the most successful models in replicating the empirical correlation asymmetryare regime-switching models. However, none of these models completely explain theextent of asymmetries in correlations.Our study of asymmetric correlations is related to several areas of ﬁnance. There isa long literature documenting the negative correlation between a stock’s return andits volatility of returns.
2
Other studies analyze patterns of asymmetries in thecovariances of stock returns in domestic equity portfolios.
3
This literatureconcentrates on documenting covariance asymmetry within a GARCH framework.Our approach uses a different methodology to document asymmetric correlations,interpreting asymmetries more broadly than simply within the class of GARCHmodels. We examine a much wider range of portfolio groups than previously used inthe literature, and investigate if other classes of empirical models can replicate thecorrelation asymmetry found in data.Our approach of creating portfolios sorted by ﬁrm characteristics creates a verydifferent view of the determinants of conditional correlations than previouslyobtained in the literature. The
H
statistic uses the full sample of observations
2
For example see, among others, French et al. (1987), Schwert (1989), Cheung and Ng (1992), Campbelland Hentschel (1992), Glosten et al. (1993), Engle and Ng (1993), Hentschel (1995), and Duffee (1995).Bekaert and Wu (2000) provide a summary of recent GARCH model applications with asymmetricvolatility.
3
Some papers documenting asymmetric betas are Ball and Kothari (1989), Braun et al. (1995), and Choand Engle (2000). Conrad et al. (1991), Kroner and Ng (1998), and Bekaert and Wu (2000) documentasymmetric covariances in multivariate GARCH models.
A. Ang, J. Chen / Journal of Financial Economics 63 (2002) 443–494
445
measured over time to calculate the correlation at the extreme tails of the jointdistribution. By employing time-series data, we use as many observations as possibleto calculate correlations for events for which there are relatively few observations.We also focus on the cross-sectional determinants of correlation asymmetry in stockreturns, whereas Erb et al. (1994) and Dumas et al. (2000) use conditioning oninstrumental variables such as business cycle indicators, rather than on theobservations, to determine the characteristics of time-varying correlations.Work in international markets has found that the correlations of internationalstock markets tend to increase conditional on large negative, or ‘‘bear market’’,returns.
4
Longin and Solnik (2001) use extreme value theory to show that thecorrelation of large negative returns is much larger than the correlation of positivereturns. However, in their work, Longin and Solnik do not provide distribution-speciﬁc characterizations of downside and upside correlations. Our paper uncoversstrong correlation asymmetries that exist in domestic markets and emphasizes thatsuch asymmetries are more than an international phenomenon in aggregate markets.In our domestic focus we examine which individual ﬁrm characteristics are mostrelated to the magnitude of correlation asymmetry.Other related studies by Campbell et al. (2001), Bekaert and Harvey (2000), andDuffee (1995) examine cross-sectional dispersion of individual stocks, which hasincreased in recent periods. Duffee (2000) and Stivers (2000) document anasymmetric component in the cross-sectional dispersion. Chen et al. (2001) andHarvey and Siddique (2000) analyze cross-sectional differences in conditionalskewness of stock returns. However, these authors have not examined therelationships between ﬁrm characteristics and asymmetric correlations. We ﬁndthat stocks which are smaller, have higher book-to-market ratios, or have low pastreturns exhibit greater asymmetric correlations. Stocks with higher beta risk showfewer correlation asymmetries. We also show that correlation asymmetry is differentfrom skewness and co-skewness measures of higher moments.The remainder of this paper is organized as follows. Section 2 demonstrates theeconomic signiﬁcance of asymmetries in correlations within a portfolio allocationframework. Section 3 shows that correlation asymmetries exist in domestic U.S.equity data. We deﬁne and characterize conditional upside and downsidecorrelations and betas of a bivariate normal distribution in closed-form, anddiscuss how to correct for conditioning bias. Section 4 measures the correla-tion asymmetries, and analyzes their cross-sectional determinants. In Section 4, wedevelop the
H
statistic measure of correlation asymmetry, and demonstrateasymmetric correlations in equity portfolios using the normal distribution asthe benchmark. In Section 5, we ask if several models incorporating asymmetry intothe conditional covariance structure can replicate the asymmetry found empiricallyin the data. Section 6 contains our conclusions. Proofs are reserved for theappendices.
4
See Erb et al. (1994), Lin et al. (1994), Longin and Solnik (1995, 2001), Karolyi and Stulz (1996), DeSantis et al. (1999), Forbes and Rigobon (1999), Boyer et al. (1999), St
&
aric
&
a (1999), Ang and Bekaert(2000), Bae et al. (2000), and Das and Uppal (2001).
A. Ang, J. Chen / Journal of Financial Economics 63 (2002) 443–494
446
2. Economic signiﬁcance of asymmetric correlations
In this section, we demonstrate the economic signiﬁcance of asymmetriccorrelations using a simple asset allocation problem. Appendix A details thesolution and the calibration method used in this example. Suppose an investor canhold amounts
a
1
and
a
2
of two assets with continuously compounded returns
x
and
y
;
respectively. The remainder of her wealth is held in a riskless asset. Let
*
x
and
*
y
denote the standardized transformations of
x
and
y
;
respectively.
5
The agentmaximizes her expected end-of-period constant relative risk aversion (CRRA) utilityas follows:max
a
1
;
a
2
E
W
1
g
1
g
:
ð
1
Þ
In Eq. (1), the end-of-period wealth is given by
W
¼
e
r
f
þ
a
1
ð
e
x
e
r
f
Þ þ
a
2
ð
e
y
e
r
f
Þ
;
r
f
¼
0
:
05 is a constant continuously compounded risk-free rate, and
g
is the agent’scoefﬁcient of risk aversion. We set
g
equal to 4.To abstract from the effects of means and variances on portfolio weights, supposeboth assets have the same mean and volatility. We denote the expected continuouslycompounded excess return of both
x
and
y
as
m
¼
0
:
07
;
and the volatility of thecontinuously compounded excess return as
s
¼
0
:
15
:
For illustration, we set theunconditional correlation of
x
and
y
to be
r
¼
0
:
50
:
Suppose that the agent believes
x
and
y
are normally distributed. Since each assethas the same mean and volatility, the investor holds equal amounts of either asset.Let
a
w
denote this portfolio position. With normal distributions, lower unconditionalcorrelations imply greater beneﬁts from diversiﬁcation.We examine the joint behavior of the two assets conditional on downside moves,which can also be called bear-market moves. We deﬁne this bear-market move to bea draw that is below each asset’s mean by more than one standard deviation. If
x
and
y
are normally distributed with unconditional correlation
r
¼
0
:
5
;
the correlationconditional on
x
o
m
s
and
y
o
m
s
is
%
r
¼
corr
ð
x
;
y
j
x
o
m
s
;
y
o
m
s
Þ¼
corr
ð
*
x
;
*
y
j
*
x
o
1
;
*
y
o
1
Þ ¼
0
:
1789
:
ð
2
Þ
Note that the downside correlation for a normal distribution is less than theunconditional correlation. This difference arises from the conditioning bias of viewing returns based on contemporaneous events of both
x
and
y
being below aﬁxed level. Appendix B demonstrates how to calculate this conditional correlation inclosed-form.Suppose the actual distribution of
x
and
y
is a regime-switching (RS) model,although the agent erroneously believes that
x
and
y
are normally distributed. Under
5
To standardize a variable
x
;
we perform the transformation
*
x
¼ ð
x
m
Þ
=
s
;
where
m
is theunconditional mean of
x
and
s
is the unconditional standard deviation of
x
:
Throughout the paper, weuse tildes to denote standardized returns. Variables without tildes are not standardized.
A. Ang, J. Chen / Journal of Financial Economics 63 (2002) 443–494
447

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