This article was downloaded by: [128.227.226.126] On: 09 May 2016, At: 08:17
Publisher: Institute for Operations Research and the Management Sciences (INFORMS)
INFORMS is located in Maryland, USA
Management Science
Publication details, including instructions for authors and subscription information:
http://pubsonline.informs.org
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
Full terms and conditions of use: http://pubsonline.informs.org/page/terms-and-conditions
This article may be used only for the purposes of research, teaching, and/or private study. Commercial use
or systematic downloading (by robots or other automatic processes) is prohibited without explicit Publisher
approval, unless otherwise noted. For more information, contact .
The Publisher does not warrant or guarantee the article’s accuracy, completeness, merchantability, fitness
for a particular purpose, or non-infringement. Descriptions of, or references to, products or publications, or
inclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, or
support of claims made of that product, publication, or service.
Copyright © 2015, INFORMS
Please scroll down for article—it is on subsequent pages
INFORMS is the largest professional society in the world for professionals in the fields of operations research, management
science, and analytics.
For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org
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
{, }
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
is used.
2 The SGT distribution has been used widely in finance for comput-
ing value-at-risk (VaR) measures, pricing options, and estimating
asset pricing models. It is also incorporated in econometric pack-
ages such as GAUSS.
1
Downloaded from informs.org by [128.227.226.126] on 09 May 2016, at 08:17 . For personal use only, all rights reserved.
Theodossiou and Savva: Skewness and the Relation Between Risk and Return
2 Management Science, Articles in Advance, pp. 1–12, © 2015 INFORMS
Under the SGT framework, rt is modeled as
f4rt It 15
D k2
nC 1
k
1=k
B
1
k1
n
k
1
1t
1 C ut
k
44nC 15=k541 Csign4ut5 5k kt
4nC15=k
1 (2)
where
ut rt mt Drt 4c t CaCbrt 15 (3)
are deviations of returns rt from their conditional
mode mt (in the case of the symmetric GT the mean
and the mode are equal). The scaling parameter t is
a time-varying dispersion measure related to t when
it exists, k and n are positive kurtosis parameters con-
trolling respectively the peakness around the mode
and the tails of the distribution, is a skewness param-
eter with domain the open interval 4 1115, sign4ut5 is
the sign function (i.e., sign4ut5 D 1 for ut 0 and 1
for ut > 0), and B4w1z5 D 4w5 4z5= 4w Cz5 is the
beta function. Values of k 2, the conditional mean and variance of
rt (see Equations (27) and (29) in the appendix) are
t DE4rt It 15Dmt CE4ut It 15Dmt Cp t (4)
and
2t D var4ut It 15D4A2 A215 2t1 (5)
where
p DA1=
p
A2 A211 (6)
A1 D 2
nC 1
k
1=k
B
2
k1
n 1
k
B
1
k1
n
k
1
1 (7)
and
A2 D41 C 3 25
nC 1
k
2=k
B
3
k1
n 2
k
B
1
k1
n
k
1
0 (8)
It follows easily from Equation (4) that the parame-
ter p D4 t mt5= t. This measure, known as Pearson’s
skewness, is a symmetric function of the skewness
parameter and a highly nonlinear function of the
Figure 1 (Color online) Skewness–Kurtosis Price of Risk
–1.0
–0.5
0
0.5
1.0
1.0
1.5
2.0
–1.5
–1.0
–0.5
0
0.5
1.0
1.5
P n = 2.1
P n = 4
P n = 30
Skewness parameter, SLlambdaKurtosis parameter, k
SK price of risk
p
kurtosis parameters k and n. This is a key measure
for the issues investigated in this paper.
Figure 1 provides a graphical illustration of the
parameter p for various values of , k and n. It is clear
from the figure that as increases in magnitude, p
also increases in magnitude. Negative values of are
associated with negative values of p and vice versa.
For D 0, p D 0. Interestingly, larger values of k and
n are also associated with larger values of p. Clearly,
p is a monotonic function of , k, and n.
2.2. Intertemporal Pricing Model
The substitution of mt c t CaCbrt 1 into the condi-
tional mean excess return Equation (4) gives
t Dc t Cp t CaCbrt 11 (9)
where c t is the “pure” risk premium, which is
expected to be positive, and p t is the skewness–
kurtosis premium. The latter, depending on the direc-
tion of skewness in the distribution of excess returns,
can be negative, zero, or positive.
The regression in (1) can be written in the following
equivalent form.:
rt D t C t D4c Cp5 t CaCbrt 1 C t0 (10)
Unlike ut, the error term t D ut p t has a zero
expected value. Note that the term 4c C p5 t D
t measures the combined impact of “pure” and
skewness–kurtosis risk on the mean of a portfolio’s
excess returns. This equation provides the foundation
for exploring and explaining the contradictory find-
ings in the literature regarding the risk–return rela-
tionship. In case of negative skewness, depending on
the size of p, the value of can be positive, zero, or
negative.
2.3. Conditional Variance
The conditional variance of excess returns is speci-
fied as a function of the past regression errors, includ-
ing their squared, absolute, and standardized values
Downloaded from informs.org by [128.227.226.126] on 09 May 2016, at 08:17 . For personal use only, all rights reserved.
Theodossiou and Savva: Skewness and the Relation Between Risk and Return
Management Science, Articles in Advance, pp. 1–12, © 2015 INFORMS 3
and past conditional variances. The following four
popular generalized autoregressive conditional het-
eroscedasticity (GARCH) models are considered.
GARCH of Bollerslev (1986):
2t DvC 2t 1 C 2t 13 (11a)
GJR-GARCH of Glosten et al. (1993):
2t DvC4 Nt 1 C 5 2t 1 C 2t 13 (11b)
QGARCH of Sentana (1995):
2t DvC t 1 C 2t 1 C 2t 13 (11c)
EGARCH of Nelson (1991):
ln 2t DvC zt 1 C g4zt 15C ln 2t 11 (11d)
where Nt D 0 for t 0 and Nt D 1 for t 0 and 1 1,
M1 DEuDA1 1 (27)
where
A1 D 1264 1541 52 C41 C 527G1 D 2 G10
Variance
For j D 2 and n> 2,
M2 DEu2 DA2 21 (28)
where
A2 D 1264 1541 53 C41 C 537G2 D41 C 3 25G20
The variance of u is
2 D var4u5DM2 M21 D4A2 A215 21 (29)
where A2 A21 > 0. Substitution of D =
q
A2 A21 into (27)
gives
M1 DEuD4A1=
q
A2 A215 Dp 0 (30)
Third Centered Moment and Skewness
For j D 3 and n> 3,
M3 DEu3 DA3 31 (31)
where
A3 D 1264 1541 54 C41 C 547G3 D 4 41 C 25G30
The third centered moment of u is
E4u Eu53 D Eu3 3Eu2EuC 24Eu53
D 4A3 3A2A1 C 2A315 30 (32)
The standardized skewness is
SK D E4u Eu5
3
3 D
4A3 3A2A1 C 2A315
4A2 A2153=2 0 (33)
Fourth Centered Moment and Kurtosis
For j D 4 and n> 4,
M4 DEu4 DA4 41 (34)
where
A4 D 1264 1541 55 C41 C 557G4 D41 C 10 2 C 5 45G40
The fourth centered moment of u is
E4u Eu54 D Eu4 4Eu3EuC 6Eu24Eu52 34Eu54
D 4A4 4A3A1 C 6A2A21 3A415 40 (35)
The standardized kurtosis is
KU D E4u Eu5
4
4 D
4A4 4A3A1 C 6A2A21 3A415
4A2 A2152 0 (36)
References
Baillie RT, De Gennaro RP (1990) Stock returns and volatility.
J. Financial Quant. Anal. 25(2):203–214.
Bali TG, Peng L (2006) Is there a risk–return trade-off? Evidence
from high-frequency data. J. Appl. Econometrics 21(8):1169–1198.
Bali TG, Theodossiou P (2008) Risk measurement performance
of alternative distribution functions. J. Risk Insurance 75(2):
411–437.
Bansal R, Lundblad C (2002) Market efficiency, asset returns, and
the size of the risk premium in global equity markets. J. Econo-
metrics 109(2):195–237.
Bollerslev T (1986) Generalized autoregressive conditional het-
eroskedasticity. J. Econometrics 31(3):307–327.
Campbell JY, Hentschel L (1992) No news is good news: An asym-
metric model of changing volatility in stock returns. J. Financial
Econom. 31(3):281–318.
Chan KC, Karolyi GA, Stulz RM (1992) Global financial mar-
kets and the risk premium on US equity. J. Financial Econom.
32(2):137–167.
Engle RF, Lilien DM, Robins RP (1987) Estimating time varying risk
premia in the term structure: The ARCH-M model. Economet-
rica 55(2):391–407.
French KR, Schwert GW, Stambaugh RF (1987) Expected stock
returns and volatility. J. Financial Econom. 19(1):3–29.
Glosten LR, Jagannathan R, Runkle DE (1993) On the relation
between the expected value of the volatility of the nominal
excess return on stocks. J. Finance 48(5):1779–1801.
Hansen BE (1994) Autoregressive conditional density estimation.
Internat. Econom. Rev. 35(3):705–730.
Hansen JV, McDonald JB, Theodossiou P, Larsen BJ (2010) Partially
adaptive econometric methods for regression and classifica-
tion. Computational Econom. 36(2):153–169.
Harrison P, Zhang HH (1999) An investigation of the risk and
return relation at long horizons. Rev. Econom. Statist. 81(3):
399–408.
Lanne M, Saikkonen P (2007) Modeling conditional skewness in
stock returns. Eur. J. Finance 13(8):691–704.
Ludvigson SC, Ng S (2007) The empirical risk-return relation: A
factor analysis approach. J. Financial Econom. 83(1):171–222.
Lundblad C (2007) The risk return trade-off in the long run: 1836–
2003. J. Financial Econom. 85(1):123–150.
McDonald JB, Newey WK (1988) Partially adaptive estimation of
regression models via the generalized t distribution. Economet-
ric Theory 4(3):428–457.
Merton RC (1973) An intertemporal capital asset pricing model.
Econometrica 41(5):867–887.
Nelson D (1991) Conditional heteroskedasticity in asset returns: A
new approach. Econometrica 59(2):347–370.
Pástor ˇL, Sinha M, Swaminathan B (2008) Estimating the intertem-
poral risk-return tradeoff using the implied cost of capital.
J. Finance 63(6):2859–2897.
Rapach D, Wohar ME (2009) Multi-period portfolio choice and the
intertemporal hedging demands for stocks and bonds: Interna-
tional evidence. J. Internat. Money Finance 28(3):427–453.
Sentana E (1995) Quadratic ARCH models. Rev. Econom. Stud.
62(4):639–661.
Scruggs TJ (1998) Resolving the puzzling intertemporal relation
between the market risk premium and conditional market vari-
ance: A two-factor approach. J. Finance 53(2):575–603.
Theodossiou P (1998) Financial data and the skewed generalized T
distribution. Management Sci. 44(12):1650–1661.
Theodossiou P (2015) Skewed generalized error distribution of
financial assets and option pricing. Multinational Finance J.
Forthcoming.
Theodossiou P, Lee U (1995) Relation between volatility and
expected returns across international stock markets. J. Bus.
Finance Accounting 22(2):289–300.
Downloaded from informs.org by [128.227.226.126] on 09 May 2016, at 08:17 . For personal use only, all rights reserved.