Empirical Analysis of the Intertemporal Relationship between Downside Risk and Expected Returns: Evidence from Time‐varying Transition Probability Models

Date01 November 2016
Published date01 November 2016
DOIhttp://doi.org/10.1111/eufm.12079
Empirical Analysis of the
Intertemporal Relationship between
Downside Risk and Expected Returns:
Evidence from Time-varying
Transition Probability Models
Cathy Yi-Hsuan Chen
Department of Finance, Chung Hua University, Hsinchu, Taiwan
Ladislaus von Bortkiewicz Chair of Statistics, Center for Applied Statistics and Economics Humboldt-
Universit
at zu Berlin, Germany
E-mail: chencath@hu-berlin.de
Thomas C. Chiang
Department of Finance, Drexel University, Philadelphia, PA, USA
E-mail: chiangtc@drexel.edu
Abstract
This paper examines the intertemporal relationship between downside risks and
expected stock returns for ve advanced markets. Using Value-at-Risk (VaR) as a
measure of downside risk, we nd a positive and signicant relationship between
VaR and the expected return before the world nancial crisis (September 2008).
However, when we estimate the model using a sample after this date, the results
show a negative riskreturn relationship. Evidence from a two-state Markov
regime-switching model indicates that as uncertainty rises, the sign of the risk
return relationship turns negative. Evidence suggests that the Markov regime-
switching model helps to resolve the conicting signs in the riskreturn
relationship.
Keywords: downside risk, Value-at-Risk, transition probability model, riskreturn
relationship
JEL classification: G11, G15
Thomas C. Chiang would like to thank the Marshall M. Austin Chair for financial support.
The authors have benefitted from discussions with Wayne Ferson and Malcolm Baker. The
authors also gratefully acknowledge the constructive comments of the Editor John Doukas
and two anonymous referees on an earlier version of the paper. Any remaining errors are
those of the authors.
European Financial Management, Vol. 22, No. 5, 2016, 749796
doi: 10.1111/eufm.12079
© 2015 John Wiley & Sons, Ltd.
1. Introduction
The intertemporal relationship between risk and expected stock returns has long played a
central role in explaining investorsportfolio behaviour. Merton (1973, 1980) formally
developed an intertemporal capital asset pricing model (ICAPM) that postulates a
positive relationship between expected excess returns E[R
tþ1
]and risk.
1
This notion can
be simply expressed as:
Et½Rtþ1¼gEt½s2
tþ1;ð1Þ
where R
tþ1
is the excess return of the market portfolio at time tþ1; Et½ is an
expectation operator at time t;gis the relative risk aversion parameter of the
representative agent; and s2
tþ1is the market return variance at time tþ1, which is
usually used as a proxy for risk (Ghysels et al., 2005; Bali and Peng, 2006). To test
Mertons model, researchers have carried out numerous studies that investigate this
riskreturn relationship. French et al. (1987), Baillie and DeGennaro (1990), Campbell
and Hentschel (1992), Scruggs (1998), Ghysels et al. (2005), Bali and Peng (2006),
Lundblad (2007), Ludvigson and Ng (2007) and Bali and Cakici (2010) test the null
hypothesis by relating the conditional mean of stock returns to the conditional variance.
They nd evidence of a positive and statistically signicant relationship. However,
Breen et al. (1989), Nelson (1991), Glosten et al. (1993) and Ang et al. (2006) test the
same hypothesis and document a negative relationship. Some research papers nd that
the sign of the test relationship is often conditioned on the methods (models and
exogenous variables) being used. Along this line, Koopman and Uspensky (2002) nd
evidence of a weak negative relationship with a stochastic variance-in-mean model but
a weak positive relationship with an ARCH-based volatility-in-mean model. Harrison
and Zhang (1999) report that the riskreturn relationship is positive at long horizons,
but insignicant at short horizons. Brandt and Kang (2004) nd that the conditional
correlation between the mean and volatility is negative; however, the unconditional
correlation is positive.
In recent years, inuenced by signicant changes in stock return volatility triggered by
nancial market crises, from the US subprime crisis (20072009) to the European
sovereign risk (20092011) crisis, investors are more sensitive to market risk involving
extreme losses. As a result, much attention has been paid to downside risk, and tests of
the trade-off hypothesis have shifted from the mean-variance relationship to the mean-
downside risk relationship. Simply put, it can be written as:
Et½Rtþ1¼gEt½VaRtþ1ð2Þ
1
Mertons (1973) original article includes a hedging component that captures the investors
motive to hedge future investment opportunities. However, a later article by Merton (1980)
indicates that the hedging component can be negligible under certain conditions. Thus, it is
convenient to write that the conditional expected excess return can be approximated by a
linear relationship with the markets conditional variance. Some researchers, such as De
Santis and Gerard (1997) and Bali et al. (2009), prefer to include an additional covariance
term between expected excess return and state variables to capture other risks besides market
risk in their analyses of the conditional capital asset pricing model (CAPM).
© 2015 John Wiley & Sons, Ltd.
750 Cathy Yi-Hsuan Chen and Thomas C. Chiang
where Et½VaRtþ1is the expected VaR of the market portfolio obtained from the
conditional VaR daily (monthly) index returns. In the empirical analysis, the conditional
VaR is either approximately measured by the lagged realised VaR ðEt½VaRtþ1¼VaRtÞ
or generated by an ARIMA process or a GARCH-type model with higher moments (Bali
et al., 2008; and Bali et al., 2009). As shown in Figures 1 and 2, the time-series plots of
the realised variance and the VaR exhibit a very similar pattern. It appears that both the
conditional variance and the conditional VaR may be characterised by similar time-
series properties or driven by some common economic forces.
2
Fig. 1. Time variations of realised variance.
Fig. 2. Time variations of downside risk.
The value on the vertical line is the square of the monthly Value-at-Risk.
2
If VaR is estimated using a GARCH method, the time-series behaviour between VaR and
the conditional variance should be similar, since both are based on similar historical
information.
© 2015 John Wiley & Sons, Ltd.
Empirical Analysis of the Intertemporal Relationship 751

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