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

• Date: 01 November 2016
• Published date: 01 November 2016
• Author: Thomas C. Chiang,Cathy Yi‐Hsuan Chen
• DOI: http://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 five advanced markets. Using Value-at-Risk (VaR) as a

measure of downside risk, we find a positive and significant relationship between

VaR and the expected return before the world financial crisis (September 2008).

However, when we estimate the model using a sample after this date, the results

show a negative risk–return 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 conflicting signs in the risk–return

relationship.

Keywords: downside risk, Value-at-Risk, transition probability model, risk–return

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, 749–796

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 investors’portfolio 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¼gEt½s2

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

Merton’s model, researchers have carried out numerous studies that investigate this

risk–return 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 find evidence of a positive and statistically significant 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 find 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) find

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 risk–return relationship is positive at long horizons,

but insignificant at short horizons. Brandt and Kang (2004) find that the conditional

correlation between the mean and volatility is negative; however, the unconditional

correlation is positive.

In recent years, influenced by significant changes in stock return volatility triggered by

financial market crises, from the US subprime crisis (2007–2009) to the European

sovereign risk (2009–2011) 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¼gEt½VaRtþ1ð2Þ

1

Merton’s (1973) original article includes a hedging component that captures the investor’s

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 market’s 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 Et½VaRtþ1is 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