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Skewness and the Relation Between Risk and Return
Panayiotis Theodossiou, Christos S. Savva
To cite this article:
Panayiotis Theodossiou, Christos S. Savva (2015) Skewness and the Relation Between Risk and Return. Management Science
Published online in Articles in Advance 28 Sep 2015
. http://dx.doi.org/10.1287/mnsc.2015.2201
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MANGEMENT SCIENCE
Articles in Advance, pp. 1–12
ISSN 0025-1909 (print) ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2015.2201
© 2015 INFORMS
Skewness and the Relation Between Risk and Return
Panayiotis Theodossiou, Christos S. Savva
Department of Commerce, Finance and Shipping, Cyprus University of Technology, Limassol 3603, Cyprus
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The relationship between risk and return has been one of the most important and extensively investigatedissues in the financial economics literature. The theoretical results predict a positive relation between the two.
Nevertheless, the empirical findings so far have been contradictory. Evidence presented in this paper shows that
these contradictions are the result of negative skewness in the distribution of portfolio excess return and the fact
that the estimation of intertemporal asset pricing models are based on symmetric log-likelihood specifications.
Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2015.2201.
Keywords: risk–return trade-off; SGT distribution; GARCH-M
History: Received February 4, 2014; accepted February 24, 2015, by Jerome Detemple, finance. Published
online in Articles in Advance.
1. Introduction
The financial and economic literature on the rela-
tionship between risk and return is voluminous and
the findings thus far have been inconclusive. Many
well-known scholars have found a positive relation-
ship, others have found a negative relationship, and
an equal number have found no relationship. For
example, a significant positive risk–return relation for
the United States is reported in French et al. (1987),
Lundblad (2007), and Lanne and Saikkonen (2007); a
significant negative relation in Glosten et al. (1993);
an insignificant one in Nelson (1991), Campbell and
Hentschel (1992), Glosten et al. (1993), Theodossiou
and Lee (1995), and Bansal and Lundblad (2002); and
mixed findings in Baillie and DeGennaro (1990).1
In standard intertemporal capital asset pricing
models, stochastic factors influence the investment
opportunity set and through that the equilibrium risk
premia of financial assets, e.g., Merton (1973). These
factors trigger fluctuations in the risk–return trade-off
and, as such, they are a source of skewness and kur-
tosis when returns are computed over discrete time
intervals. Because investors hedge constantly against
such fluctuations, higher moments are likely to be
priced.
This paper investigates the impact of skewness and
kurtosis on the risk–return relationship using an ana-
lytical framework based on the popular skewed gen-
eralized t (SGT) distribution, e.g., Theodossiou (1998).
1 Contradictory findings are also reported in studies using other
methodologies, such as Scruggs (1998), Harrison and Zhang (1999),
Bali and Peng (2006), Ludvigson and Ng (2007), Pástor et al. (2008),
and Chan et al. (1992).
The SGT distribution is chosen because of its flexibil-
ity in modeling fat tails, peakness and skewness, often
observed in financial data.2 Furthermore, it includes
several well-known symmetric distributions used in
the finance literature, such as the generalized t (GT),
generalized error (GED), Student’s t (T ), and normal
(N), e.g., Bali and Theodossiou (2008) and Hansen
et al. (2010).
2. Impact of Skewness on the
Pricing of Risk
2.1. SGT Framework
The intertemporal relationship between risk and
returns is investigated using the GARCH-in-mean
process, which has been the standard in the literature,
e.g., Engle et al. (1987) and Glosten et al. (1993). That
is, a portfolio’s excess returns are specified as
rt Dc t CaCbrt 1 Cut1 (1)
where 2t D var4rt It 15 is the conditional variance of
rt based on the information set It 1 available prior to
the realization of rt; rt 1 is the past value of excess
returns included in It 1; a and b are typical regres-
sion coefficients; and c, also known as the GARCH-in-
mean coefficient, links t to t. For practical purposes
and without loss of generality, a single lag value of rt