Pricing Kernels with Coskewness and Volatility Risk
Fousseni CHABI-YO∗
Fisher College of Business, Ohio State University
This Version : March 16, 2009
Abstract
I investigate a pricing kernel in which coskewness and the market volatility risk factors are
endogenously determined. I show that the price of coskewness and market volatility risk are
restricted by investor risk aversion and skewness preference. The risk aversion is estimated to
be between two and five and significant. The price of volatility risk ranges from -1.5% to -0.15%
per year. Consistent with theory, I find that the pricing kernel is decreasing in the aggregate
wealth and increasing in the market volatility. When I project my estimated pricing kernel on
a polynomial function of the market return, doing so produces the puzzling behaviors observed
in pricing kernel. Using pricing kernels, I examine the sources of the idiosyncratic volatility
premium. I find that nonzero risk aversion and firms’ non-systematic coskewness determine
the premium on idiosyncratic volatility risk. When I control for the non-systematic coskewness
factor, I find no significant relation between idiosyncratic volatility and stock expected returns.
My results are robust across different sample periods, different measures of market volatility
and firm characteristics.
∗
Fousseni Chabi-Yo is from the Fisher College of Business, The Ohio State University. I am grateful to Gurdip
Bakshi, Turan Bali, Karl Diether, Rene Garcia, Eric Ghysels, Pete Kyle, Dietmar Leisen, Kewei Hou, Hong Liu, Andrew Karolyi, Mark Loewenstein, Lemma Sembet, Georgios Skoulakis, Rene Stulz, Liuren Wu, Harold Zhang, Guofu
Zhou and seminar participants at Baruch College, the University of Maryland, HEC Montreal, Rutgers University,
The Ohio State University, the University of Washington in St Louis, the University of Texas in Dallas, and the
Stockholm School of Economics for helpful comments. As usual, they are exonerated with respect to the paper’s
residual shortcomings. I thank Kenneth French for making a large amount of historical data publicly available in his
online data library. I thank Bollerslev Tim, George Tauchen and Hao Zhou for making the monthly time-series of
realized volatility available. I thank the Dice Center for Financial Economics for financial support. Correspondence
Address: Fousseni Chabi-Yo, Fisher College of Business, Ohio State University, 840 Fisher Hall, 2100 Neil Avenue,
Columbus, OH 43210-1144. email: chabi-yo 1@fisher.osu.edu
Electronic copy available at: http://ssrn.com/abstract=1361926
Pricing Kernels with Coskewness and Volatility Risk
Abstract
I investigate a pricing kernel in which coskewness and the market volatility risk factors are
endogenously determined. I show that the price of coskewness and market volatility risk are
restricted by investor risk aversion and skewness preference. The risk aversion is estimated to
be between two and five and significant. The price of volatility risk ranges from -1.5% to -0.15%
per year. Consistent with theory, I find that the pricing kernel is decreasing in the aggregate
wealth and increasing in the market volatility. When I project my estimated pricing kernel on
a polynomial function of the market return, doing so produces the puzzling behaviors observed
in pricing kernel. Using pricing kernels, I examine the sources of the idiosyncratic volatility
premium. I find that nonzero risk aversion and firms’ non-systematic coskewness determine
the premium on idiosyncratic volatility risk. When I control for the non-systematic coskewness
factor, I find no significant relation between idiosyncratic volatility and stock expected returns.
My results are robust across different sample periods, different measures of market volatility
and firm characteristics.
Electronic copy available at: http://ssrn.com/abstract=1361926
I.
Introduction
In a setting with standard preferences and static prices of risk, the pricing kernel can be interpreted
as a scaled marginal utility. As a result, under these assumptions, to be consistent with positive
marginal utility and the no arbitrage condition, the pricing kernel should be positive, and to be
consistent with decreasing absolute risk aversion it should be decreasing in the aggregate wealth
(market return). Over a reasonable range of wealth states, these pricing kernels increase as the
aggregate wealth increases. Jackwerth (2000), Rosenberg et al. (2002) and Aı̈t-Sahalia and Lo
(2000) document the puzzling behavior of the pricing kernel, which affects the absolute risk-aversion
function and renders it negative over a reasonable range of wealths. Previous studies ignore the
possibility that the market volatility risk could affect the behavior of the pricing kernels. Recent
papers such as Ang et al. (2006) find that the market volatility risk is priced. Bakshi and Madan
(2006) show that the market volatility premium is determined by a nonzero risk aversion and higher
moments such as market skewness. Their finding suggests that investor skewness preferences should
also impact the volatility risk premium as well. Ang et al. (2006) show that, in addition to the
market volatility, the idiosyncratic volatility is priced. In contrast to Merton’s (1987) prediction,
Ang et al. (2006) find that on average, stocks with high idiosyncratic volatility earn low returns.
They call their result a puzzle. This puzzle has attracted recent attention and there has been
increasing interest in explaining their results.
My contribution to these studies is three-fold. First, I build a partial equilibrium model in
which investors trade in a multi-period market. I go beyond the representative agent utility models
by allowing heterogeneity of preferences among agents. I make no assumptions about the functional form of investor utility functions and the distribution of asset returns. Instead, I provide a
general framework that maps heterogeneity of preferences into the pricing kernel. Thus, I provide a
structural interpretation to the pricing kernel involving coskewness and volatility risk. My pricing
kernel is a function of two deep structural parameters, the average value of investor risk-tolerances
(inverse of the Arrow-Pratt measure of investor risk aversions) and skewness preference. I extend
the frameworks of Samuelson (1970) and; Judd and Guu (2001) to a dynamic market model in
which each investor maximizes his or her expected utility. My intertemporal portfolio choice is a
dynamic problem in which each investor chooses asset allocations conditional on current wealth.
My model setup is a dynamic extension of Harvey and Siddique (2000), Kraus and Litzenberger
(1976), Rubinstein (1973) and is also related to Brandt et al. (2005). As a result, the aggregate
pricing kernel in equilibrium depends on both coskewness and market volatility risk. I show that
the price of coskewness and market volatility are restricted by investor preferences. Further, I provide a closed-form solution for the prices of coskewness and market volatility risk in terms of mean
average of investor risk aversions and skewness preferences. The price of coskewness and volatility
risk are highly nonlinear functions of the mean risk aversion and mean skewness preference. I use
two sets of independent data. I first use the 30 industry portfolio returns. Second, I use the 30
1
Electronic copy available at: http://ssrn.com/abstract=1361926
Dow Jones stock returns. I estimate both the risk aversion and skewness preference parameters
with different proxies for the market volatility. I use the implied market volatility measures VIX
and VXO, and the realized market volatility (RV) computed with high frequency intraday squared
returns. I also consider different sub-sample periods. When I use the 30 industry portfolio returns,
the risk aversion is estimated to be between 2.5 and 4.75 while the skewness preference ranges from
1.05 to 2.25. The parameters are mostly statistically significant. The implied price of coskewness
associated to these estimates ranges from -3.2% to -0.72% per year . When I use the VIX and VXO
as my proxy for the market volatility, the implied price of the market volatility ranges from -1.2%
to -0.19% per year. The implied price of the market volatility ranges from -1.51% to -0.77% per
year when I use the realized volatility. When I use the 30 Dow Jones Stock returns, the risk aversion is estimated to be between 3.75 and 5.75 while the skewness preference ranges from 1.003 to
1.71. The parameters are all statistically significant. The implied price of coskewness associated to
these estimates ranges from -2.7% to -2.07% per year and the implied price of the market volatility
ranges from -1.30% to -0.15% per year. These estimates are in a reasonable range and consistent
with the literature. I also investigate the impact of the risk aversion and skewness preference on
the price of the market volatility risk over time. I find that periods of high price of volatility risk is
sometimes associated with high risk aversion and low or stable skewness preference, sometimes to
high skewness preference and low and stable risk aversion, and sometimes to both high risk aversion
and high skewness preference.
Second, I examine the puzzling behaviors of the pricing kernel. I show that my estimated pricing
kernel is consistent with economic theory. It is decreasing in the aggregate wealth and increasing in
the market volatility. When I project my estimated pricing kernel on a polynomial function of the
market return, doing so produces the puzzling behaviors observed in pricing kernel. The missing
market volatility priced factor in the pricing kernel and the lack of a structural interpretation of the
price of coskewness and volatility risk in terms of investor risk aversion and skewness preference,
in previous studies, could be the cause of the puzzling behaviors of the pricing kernel.
Third, I study the negative relation between idiosyncratic volatility and expected returns, and
ask why do low idiosyncratic volatility firms earn higher future returns than ones with higher idiosyncratic volatility? To answer this question, I use pricing kernels to examine the sources of the
idiosyncratic volatility premium. I find that the premium on idiosyncratic volatility risk is determined by a nonzero risk aversion and firms’ non-systematic coskewness. I define non-systematic
coskewness as the non-systematic component of asset skewness that is related to the market portfolio’s skewness. I find that when this non-systematic component is positive, the difference in
Fama-French (1993) alpha between the valued-weighted decile portfolio with the highest and lowest non-systematic coskewness has a significant alpha of -1.30% per month (with a t-statistic of
-2.82). In contrast, when the non-systematic coskewness is negative, a long-short portfolio holding the highest non-systematic coskewness decile of stocks and shorting the lowest non-systematic
coskewness decile of stocks has a highly significant alpha of 1.30% per month (with a t-statistic
2
of 2.94). To study the negative relation between idiosyncratic volatility and expected returns, I
sort stocks on idiosyncratic volatility and form value-weighted decile portfolios. Consistent with
previous studies, I find that the Fama-French (1993) alphas of the high-idiosyncratic decile exceeds
the alpha of the low-idiosyncratic decile by -1.38% per month (with a t-statistic of -3.02). When
I control for long-short portfolios that hold the highest non-systematic coskewness decile of stocks
and short the lowest non-systematic coskewness decile of stocks, I find that the Fama-French (1993)
alphas of the high-idiosyncratic decile exceeds the alpha of the low-idiosyncratic decile by 0.04%
per month (with a t-statistic of 0.17). I also relate my findings to recent studies that use GARCH
specification of idiosyncratic volatility risk. I show that by assuming GARCH specifications, these
papers restrict the relation between idiosyncratic volatility and stock returns to be positive. Therefore, given our sample, it is not possible to use a GARCH type of specification and arrive at a
negative relation between idiosyncratic volatility and expected returns.
The paper is organized as follows. In Section II, I describe the features of my partial equilibrium
model. In Section III, I discuss the equilibrium pricing kernel and its implication for asset pricing.
In Section IV, I describe the data set and discusses the empirical results of my basic model. In
Section V, I investigate the sources of the idiosyncratic volatility premium. In Section VI, I relate
my findings to studies on idiosyncratic volatility that uses GARCH specification of idiosyncratic
volatility risk. Section VII concludes.
II.
The Model
I develop an approximated equilibrium model with heterogeneous investors. I do so to characterize
the pricing kernel and endogenously link the pricing kernel to both coskewness and volatility risk
with a structural interpretation of the market prices of the aggregate volatility and coskewness risk.
I keep the model standard, and summarize it by a finite number of investors who make the optimal
allocation decisions in a multi-period market. I exclude intermediate consumption from the model.
The heart of the model is the return decomposition. This decomposition is crucial to obtain the
closed-form solution for the optimal portfolio weight at any trading date and the aggregate pricing
kernel in equilibrium.
A.
Investor Preferences and Portfolio Optimization
I consider an economy in which there are many investors with heterogeneous preferences and endowments. In this economy, investors are indexed by i = 1, ..., I and trade in n risky assets and a
safe asset at times τ = t, t + 1, ...,T − 1. I use Rkτ +1 , k = 1, ...n to denote, the return from investing
$1 at time τ in each security. All assets are traded in competitive markets without transaction
costs and taxes. I consider the portfolio choice at time t of investor i, who maximizes the expected
utility of wealth at some terminal date T < ∞ by trading in n risky assets and a safe asset at times
3
τ = t, ..., T − 1. Formally, the investor’s problem is:
h ³
´i
(i)
(i)
Vt = n max
E
u
W
t
i
T
o
(i)
ωτ
τ =T −1
i = 1, ..., I
(1)
τ =t
subject to the sequence of budget constraints
³
´
(i)
Wτ +1 = Wτ(i) Rf + ωτ(i)| Reτ +1 ∀ τ ≥ t
(i)
where ωτ
is the vector of portfolio weights on the risky assets chosen at time τ , and Rτe +1 =
Rτ +1 − Rf 1n is the n-dimensional portfolio weight vector of excess returns on the risky assets from
(i)
time τ to τ +1. The function ui (.) measures the investor’s utility of terminal wealth WT . I assume
that the individual asset allocation shares fulfill the market clearing conditions
I
X
Wτ(i) ωτ(i) = θτ ∀ τ ≥ t
(2)
i=1
>
where θτ −1 Rτ represents the aggregate future wealth. I use the small noise expansion approach
to solve the portfolio choice (1). To solve for the optimal asset allocation, I follow Samuelson
(1970) and Judd and Guu (2001), assuming that the distribution of the returns belongs to a family
of “compact”or “small-risk” distributions. I define the small-risk distribution as some specified
parameter ε goes to zero, all the distributions converge to a sure outcome. I can decompose any
random vector Rkτ +1 that belongs to this family as follows:
Rkτ +1 = Rf + ε2 akτ (ε) + εYkτ +1 .
(3)
Here, the coefficient akτ (ε) is a function of the ε parameter, which characterizes the scale of risk.
In terms of Brownian motion, ε is the square root of time, and the drift and diffusion terms are
¡
¢
given by Rf + ε2 akτ (ε) and (εYkτ +1 ) respectively. Throughout this paper, ε refers to the scale
risk parameter in the small noise expansions framework. I note that there are no restrictions on
the distribution of Ykτ +1 . Hence, the distribution of Rkτ +1 is general and is not restricted to a
specific distribution. All asset returns are not correlated through ε, since the correlation between
two assets k and j is independent of ε. The term ε2 akτ (ε) is the risk premium on the risky asset.
The function akτ (ε) that characterizes this premium is unknown.
The return’s decomposition (3) shows that the return is a function of the scaled risk parameter
ε. As a result, it follows that the first-order conditions of (1) implicitly define the portfolio weight
as a function of ε. Like most Taylor expansion series1 , I assume that the small noise expansion of
(i)
ωt
does converge, that is
(i)
ωt ≈
j=Q
X
j=0
1
See Brandt et al. (2005).
4
1 (i) [j]
ω
,
j! t
(4)
(i) [j]
where Q = 1 and ωt
represents the jth derivative of the portfolio weight evaluated at ε = 0. I
note that an approximation with Q = 1 works well when investors have homogeneous preferences.
Approximating the portfolio weight with Q = 1 is sufficient to show that, in equilibrium, the
pricing kernel depends on three factors: the market return, the coskewness factor and the volatility
of the market return. My approach is more intuitive than the standard contingent state approach to
equilibrium. As shown in Hart (1975) and Elul (1995), the incomplete markets paradigm focuses on
the difference between the number of contingent states and the number of assets. It depends on how
many assets are missing and the number of agents in the economy. It is difficult to interpret such
indices of incompleteness. The main reason is that we can count neither the number of contingent
states nor the number of different kinds of agents in a real economy2 . To characterize investor
0
preferences, I assume that all agents have utility functions that exhibit non-satiation (ui > 0),
00
000
risk aversion (ui < 0), and a preference for positive skewness (ui > 0). At ε = 0, the following
parameters characterize the investor’s preferences:
³
´
³
´
0
000
(i)
(i)
ui WT
u
W
2
T
℘ i
³
´ ], ρi = lim [ i
´ ],
℘i = lim [− 00 ³
0
(i)
(i)
ε7→0
ε7→0 2
ui WT
ui WT
(5)
where 1/℘i is the Arrow-Pratt absolute measure of risk aversion and ρi represents the skewtolerance.
Next, I consider the average values:
I
P
℘=
1
I
I
X
℘i ,
ρ=
i=1
ρi ℘i
i=1
I
P
i=1
,
(6)
℘i
where ℘ is the cross-sectional average of investor risk tolerances and ρ represents the weighted
average of investor skewness preferences. The intertemporal portfolio choice in equation (1) is a
(i)
dynamic problem. At time τ , each investor chooses his asset allocation ωτ conditional on having
(i)
wealth Wτ . To make his or her decision at time τ , investor i takes into account the fact that at any
(i)
future date τ , the portfolio weight will be optimally revised conditional on the wealth Wτ . I solve
backward the dynamic portfolio choice in Equation (1) by expressing the multiperiod problem (1)
as single-period problems. In my model, all investors have homogeneous expectations. The source
of heterogeneity in this economy comes from investor preferences and endowments.
2
The impact of asset incompleteness on economic performance is related to the diversity of investor objectives
than to the number of states and the number of agents. Therefore, the number of different agents in an economy is
a poor measure of agent diversity because an economy with 1000 types of investors with different risk aversions and
skewness preferences close to the mean risk aversion and mean skewness preferences is less diverse than an economy
with 20 types of investors with substantially different risk aversions and skewness preferences.
5
III.
Pricing Kernels with Coskewness and Volatility Risk
I use the small noise expansion approach proposed by Judd and Guu (2001), without knowledge of
investor utility functions. Using this approach makes it possible for me to derive the closed-form
solution for the optimal portfolio weights by assuming that risky asset returns can be decomposed
according to equation (3). In the Appendix, I show how I derive the optimal shares of wealth
invested in risky securities. I then examine the asset pricing implications of the investor optimization
problem. By using the optimal portfolio weights, I can recover the functional form for the parameter
aτ (.) appearing in the return decomposition, and use it to derive the functional form of the pricing
kernel in equilibrium3 .
PROPOSITION 1 : The pricing kernel for the period [t, T ] is
mTt
=
T
Y
mν−1,ν
(7)
ν=t+1
with
mν−1,ν
−1
£ 2
¤ TX
1
2
2
2
=
+ D0 rM ν + D1 rM ν − Eν−1 rM ν +
D2τ [Eν σM
τ − Eν−1 σM τ ]
Rf
τ =ν
(8)
where:
ν−1
X
ρ [2(T −ν)−1]
(ρ − 1) [2(T −τ )−1]
1 [T −ν−1]
, D2τ =
Rf
1τ ≤T −1 , D0 = − Rf
D1 = 2 Rf
+
Dτ ν rM τ
℘
℘
℘2
τ =t+1
and
Dτ ν =
(2ρ − 1) [(T −τ )+2(T −ν)−1]
Rf
℘2
(9)
(10)
where rM ν is the demeaned market return. The indicator function 1τ ≤T −1 equals one if τ ≤ T − 1
2
and zero otherwise. σM
τ represents the volatility of the market return.
Proof See the Appendix.
The pricing kernel derived in Proposition 1 gives a structural interpretation of the market
prices of risk in terms of the cross-sectional average of investor risk tolerance, ℘, and the crosssectional average of investor skewness preferences ρ. If investors have identical preferences and
different endowments, and if the portfolio weight is approximated by (4) with Q = 1, then my
model predicts that the aggregate pricing kernel will remains unchanged. Moreover, the aggregate
pricing kernel is still valid if the representative agent assumption applies. What really matters in
equilibrium is the mean average of investor preferences4 .
3
In the previous version of the paper, I derived the pricing kernel with the restriction that the risk-free return
' Rf . In this version of the paper, see proposition 1, there is no restriction on the risk-free return. As shown in
the empirical section, this restriction has no impact on the estimated preference parameters.
4
When I include an additional high-order term into the portfolio weight (4), that is Q = 2, the aggregate pricing
kernel is different from the one obtained under the representative agent assumption. The theoretical results are
available on request. I will investigate this issue in future research.
Rf2
6
In nonlinear pricing kernels literature, the usual route is to assume the existence of a representative agent, then expand the agent’s marginal utility in Taylor series and then drop the higher
moments that are economically unimportant for explaining returns. This approach produces a
pricing kernel that is a polynomial function of the market return, and does not depend on the
volatility of the market return. If the volatility of the market return is priced, it should be relevant
for explaining the cross-section of risky returns. To my knowledge, this paper is the first to show
how the market volatility risk factor enters the pricing kernel. To compute the risk premium on
the risky asset from time t to t + 1. I assume that the investor invests at time t in the asset k for
T − t periods. Then, the pricing kernel mTt should price correctly returns, that is
Et mTt Rkt+1 ....RkT = 1.
(11)
To examine the pricing implications of my model from time t to t + 1 in a one and two-period
market, I assume that the investor invests in the risky asset from t to t + 1 and reinvests the payoff
in the risk-free asset for the remaining period. I simplify the Euler equation (11) as:
Et mt,t+1 Rkt+1 = 1
(12)
where
mt,t+1
T
−1
X
£ 2
¤
1
2
2
2
=
+ D0 rM t+1 + D1 rM t+1 − Et rM t+1 +
D2τ [Et+1 σM
τ − Et σM τ ]
Rf
(13)
τ =t+1
where D0 , D1 and D2τ are defined in (9).
In a one-period model, T = t + 1, I use (12) to derive the risk premium on the risky asset from
time t to t + 1 is
Et Rkt+1 − Rf = (λM t /Rf )βM t + (λSKDt /Rf )βSKDt .
with
λM t =
and
βM t =
1
2
Rf σM
t,
℘
ρ 2 3
R σ ,
℘2 f M t
(15)
2
Covt (rM
t+1 , Rkt+1 )
,
3
σM
t
(16)
λSKDt = −
Covt (rM t+1 , Rkt+1 )
,
2
σM
t
βSKDt =
(14)
Equation (14) is the asset pricing model derived in Rubinstein (1973), Kraus and Litzenberger
(1976), and Harvey and Siddique (2000). The main contribution of my model is that in a twoperiod model, the asset risk premium depends on the co-movement between the market volatility
and the return on the risky asset. In a two-period model, T = t + 2, I use (12) to show that the
risk premium on the risky asset from time t to t + 1 is
Et Rkt+1 − Rf = λM t βM t + λSKDt βSKDt + λV OLt βV OLt
with
λV OLt =
(1 − ρ) 2
2
Rf V art (σM
t+1 ),
℘2
βV OLt =
7
2
Covt (σM
t+1 , Rkt+1 )
.
2
V art (σM
t+1 )
(17)
(18)
where λM t and λSKDt are defined in (15) and λV OLt is defined in 18.
For T = t + 2, the resulting asset pricing model is a generalization of Rubinstein (1973), Kraus
and Litzenberger (1976), Harvey and Siddique (2000), and Ang et al. (2006). Since the mean
risk aversions
1
℘
and mean skewness preferences ρ are positive, λSKDt is negative and assets with
negative βSKDt (coskewness) have higher expected returns than assets with positive coskewness
and identical characteristics. The asset’s risk premium decomposition shows that the prices of
coskewness and volatility risk have a structural interpretation in terms of the average value of
investor risk tolerance and skewness preferences. My results indicate that the price of coskewness
and volatility risk are restricted by investor risk aversion and skewness preference. Moreover, the
sign of the price of market volatility risk depends on the average values of investor preferences.
If the cross-sectional average of investor skewness preference ρ is larger than one, then the price
λSKDt of the market volatility risk is negative.
The main prediction of this model is that stocks with different loadings on volatility risk have
different average returns, and high volatility is associated with high future expected returns. In
contrast, when ρ is lower than one, the price of the market volatility risk is positive. Ang et al.
(2006) use the risk premium specification to estimate the price of the market volatility. However,
their approach leaves open the question of what value of investor risk aversion and skewness preferences are consistent with the prices of coskewness and volatility risk. My theoretical model makes
it possible to address this issue. I am interested in estimating investor risk aversion and skewness
preference, and then using these structural preference parameters to estimate both the market price
of coskewness and volatility risk.
To examine the sources of the variance risk premium, I consider the pricing kernel (13) generated
by my two-period model with T=t+2. I use (13) to derive the market variance premium as5
2
∗2
2
σM
t − σM t = αM t + λM t σM t SM t + λSKDt σM t (KM t − 1) + λV OLt υM t ,
with
υM t =
(19)
2
Covt (σM
, r2
)
¡t+12 M¢t+1 ,
V art σM t+1
∗2 is the variance of the market return under the risk neutral measure, α
where σM
M t is the
t
expected excess return on the market, σM t , SM t and KM t represent the standard deviation, the
skewness and the kurtosis of the market return. Carr and Wu (2008) propose a direct and robust
method for quantifying the variance risk premium on financial assets. As shown in equation (19)
my model examines the sources of the variance premium. The variance premium can be attributed
to exposure to higher moments such as skewness, kurtosis and the correlation between the market
volatility and the squared of the market return; the risk-averse behavior of the investors; and
investor preferences for skewness. At a basic level, equation (19) shows that the variance premium
is related to nonzero risk aversion ( ℘1 ). The result in equation (19) has special relevance. At
5
See the proof in the Appendix.
8
a theoretical level, the equation states that there are three sources of negative market variance
premium. First, a negative skewness (SM t ) and high positive kurtosis (KM t ) cause the variance
premium to be more negative. Second, if the average skewness preference ρ is higher than one,
then a high correlation of the market volatility with the squared market return causes the variance
risk premium to be even more negative. Third, for a given risk aversion level, raising the level of
skewness preferences could generate a more pronounced negative variance premium.
Comparing the expression in equation (19) to the expression for the equity premium in equation
(17), suggests that the variance risk premium should serve as useful predictor for the actual realized
returns. My paper is one of several recent papers that intend to explain the relation between
volatility risk and expected returns. Bollerslev et al. (2008) present a stylized general equilibrium
model designed to illuminate theoretical linkages between financial market volatility and expected
returns. Their model involves a standard endowment economy with Epstein-Zin-Weil recursive
preferences. Drechsler and Yaron (2008) propose a more elaborate model. Nyberg and Wilhelmsson
(2008) test if innovations in investor risk aversion are a priced factor in the stock market as predicted
by models incorporating habit formation in preferences. Their proxy for time-varying risk aversion
is based on the volatility risk premium series constructed by Bollerslev et al. (2008).
My paper is also related to recent theories that examine theoretical link between the idiosyncratic skewness and the asset expected excess return. To compare my model’s prediction to the
predictions in these papers, I assume that the assets j = 1, ...n for j 6= k are normally distributed but asset k is not. I denote ISkt = Et ε3kt+1 , the idiosyncratic skewness and assume
that E(Rkt+1 − Et Rkt+1 )(Rjt+1 − Et Rjt+1 )(Rit+1 − Et Rit+1 ) = ISkt for i = j = k and 0 otherwise.
Similar to Barberis and Huang (2007) and Mitton and Vorkink (2007), this restriction on the asset’s skewness allows to isolate the potential impact that idiosyncratic skewness may have on the
expected excess return. Under these restrictions, my two-period model predicts that the expected
excess return on asset k is6 .
Et Rkt+1 − Rf = λM t βM t −
ρ 2 2
R ω ISkt + λV OLt βV OLt
℘2 f k
(20)
where ω k is the weight of asset k in the market portfolio. Since ρ > 0, equation (20) suggests that
stocks with high idiosyncratic skewness have low expected excess return on average. This is consistent with Mitton and Vorkink (2007). Recent theories investigate whether idiosyncratic skewness is
related to the expected excess return. Each theory starts from a different set of assumptions. Boyer,
Mitton and Vorkink (2008) find that expected skewness helps explain the phenomenon that stocks
with high idiosyncratic volatility have low expected returns. Barberis and Huang (2007) show
that when investors have cumulative prospect theory preferences, stocks with greater idiosyncratic
skewness may have lower average returns. Brunnermeier and Parker (2005) and Brunnermeier,
P
The proof is straightforward from (17). I write the market return RM t+1 as n
j=1 ω j Rjt+1 . I replace the market
return in the asset’s coskewness that appears in equation (17), and use the underlying assumptions to simplify this
coskewness and obtain the final result.
6
9
Gollier, and Parker (2007) solve an endogenous probabilities model that produces similar asset
pricing implications for individual asset skewness.
IV.
A.
The Empirical Framework
Estimation Methods
My main goal is to estimate the cross-sectional average of risk aversions and skewness preferences,
then check whether these preference parameters are reasonable, and then use these values to recover
the price of coskewness and volatility risk. To compare my results with those in recent studies on
the pricing of volatility risk, I consider the two-period model with T = t + 2. I assume that the
investor invests at time t in the asset k for one period and then reinvests the payoff in the risk-free
asset for the remaining period. This assumption leads me to collect the vector of errors
²t+1 = mTt RtT − In ,
(21)
where RtT is the vector of risky asset returns over this two periods. This vector contains elements
of the form Rkt+1 Rf where Rkt+1 is the return from time t to t + 17 . The Euler equation will be
Et mt,t+1 mt+1,t+2 Rkt+1 Rf = 1
(22)
This allows to simplify the Euler equation as Et mt,t+1 Rkt+1 = 1 with
mt,t+1 =
£ 2
¤ (ρ − 1)
£ 2
¤
1
ρ
1
2
2
− rM t+1 + 2 Rf rM
Rf σM
t+1 − Et rM t+1 +
t+1 − Et σM t+1 ,
2
Rf
℘
℘
℘
(23)
Equation (21) implies E [²t+1 |Zt ] = 0, which forms a set of moment conditions that I can utilize
to test the asset pricing model via Hansen’s (1982) generalized method of moments (GMM); E[]
denotes the unconditional expected operator. Zt represents the set of instrumental variables. If
the pricing kernel prices correctly returns, then the sample version of the moment conditions is:
gT (Θ) =
T
1 X (T )
²τ ⊗ Zτ .
T
(24)
τ =1
³
where Θ =
´
1 1
Rf , τ , ρ
is the set of parameters to be estimated. As the sample size T increases,
gT (Θ) should be sufficiently close to zero. Hansen (1982) shows that a test of model specification
can be obtained by minimizing the quadratic form:
|
J = arg min gT (Θ) WT gT (Θ) ,
Θ
where WT is the GMM weighting matrix. However, Chapman (1997) shows that the standard
GMM estimator in a Euler equation test may result in acceptance of the pricing kernel due to
7
I also use two-period returns in the form Rkt+1 Rjt+2 . The results are qualitatively similar. The results are
available on request.
10
noise in the pricing kernel. In a different framework, Hansen and Jaganathan (1997) use the same
criterion function as in the standard GMM approach but specify the weighting matrix as the second
moment of the returns. I follow Dittmar (2002) and use the Hansen and Jagannathan weighting
matrix in the estimation process.
B.
Data
I use the 30 monthly industry portfolios obtained from Kenneth French’s website8 . The sample
period is from January 1986 to December 2006, and yields a total of 252 observations. For the
market portfolio, I use the value-weighted NYSE/AMEX/NASDAQ index. This index is also known
as the value-weighted index of the Center for Research in Security Prices (CRSP).
As a proxy the volatility of the market return, I use the options implied volatility estimators.
The Chicago Board Options Exchange (CBOE)s VXO implied volatility index provides investors
with up-to-the-minute market estimates of expected volatility by using the real-time S&P 100 index
option bid/ask quotes. The VXO is a weighted index of American implied volatilities calculated
from eight near-the-money, near-to-expiry, S&P 100 call and put options based on the BlackScholes (1973) pricing formula.
I also use the CBOE’s newer VIX index, which is obtained from the European style S&P 500
index option prices and which is based on “model-free”implied volatilities. The VIX incorporates
information from the volatility skew by using a wider range of strike prices rather than just at-themoney series. I use historical monthly data on the VIX from 1990 to 2006.
As an alternative to the VXO and VIX indexes, I use the Realized Volatility (RV). Several
recent studies have argued for the use of so-called “model-free”realized variances computed by
the summation of high-frequency intraday squared returns. These measures generally afford much
more accurate expost observations on the actual return variation than the more traditional sample
variances based on daily or coarser frequency return observations. I follow Bollerslev et al. (2008)
to construct my “model-free”realized volatility measure. Bollerslev et al. (2008) use intraday data
for the SP500 composite index to construct the realized variance measure9 .
C.
Results
I estimate the mean of the pricing kernel, the cross-sectional average of investor risk aversions
and the cross-sectional average of investor skewness preferences. I then use these parameters to
compute the price of the market, coskewness and volatility risk. I investigate three different subsample periods.
Table I presents results of GMM tests when I estimate the pricing kernel (23). I estimate the
preference parameters by using the Hansen and Jagannathan (1997) weighting matrix. Column
(1) shows the mean of the pricing kernel and columns (2) and (3) present the risk aversion and
8
9
I thank Kenneth French for making a large amount of historical data publicly available in his online data library.
I thank Bollerslev et al. (2008) for making the monthly time-series of realized volatility available.
11
skewness preferences, respectively. Column (4) gives the Hansen and Jagannathan (1997) distance
measure with p-values for the test of model specification in parentheses. Columns (5) through (7)
present the implied price of market, coskewness and volatility risk that I obtain when I use the
estimated preference parameters. P -values for tests of the coefficients appear in parentheses.
The set of returns used covers different sample periods. As my proxy for the volatility of the
market return, I use the CBOEs VXO and VIX implied volatilities and the realized volatility (RV).
Panel A presents the results when I use the VXO. I show that the estimated coefficients for the
risk aversion are reasonable and range from two to four. The p-values indicate that most of the
estimated risk aversions are statistically significant at the 5% level, except for the short sample
period January 1996 to December 2006. The distance measure and p-values suggest that the
estimated pricing kernel cannot be rejected at the 5% level when I use the sample periods January
1986 to December 2000 and January 1986 to December 2006. Panel A also reports the implied
price of the market, coskewness and market volatility risk. The price of the market is positive and
ranges from 6% to 10% per year. The price of coskewness risk is negative and ranges from -1.87%
to -1.3% per year and the price of the market volatility ranges from -0.4% to -0.2% per year. The
sign of the prices of market, coskewness, and volatility risk is consistent with my model’s prediction
and the results in previous studies. For the sample period from January 1986 to December 2000,
I find that the price of the volatility risk is about -0.38% annually. To test whether these prices
of risk are statistically significant, I use the Delta-method to compute t-statistics of the prices and
find that the price of the market risk is significant (with p-value of 0.044), but that the prices of
coskewness and volatility risk are not significant. The results are not reported10 .
However, this result does not suggests that the prices of volatility and coskewness risks are not
significant. The main reason for this finding is that the Delta method is based on a linear approximation of the price of risk in terms of risk aversion and skewness preference. Linear approximations
do not incorporate nonlinear components of the price of risk. While the price of the market is linear
in the risk aversion, the price of coskewness and volatility risk are highly nonlinear functions of
both risk aversion and skewness preference. My results indicate that the price of the volatility risk,
λV OLt , is a nonlinear function of the risk aversion and skewness preference parameters that turn
to be significant at the 5% level. As a result, the volatility risk premium defined in equation (19)
is significantly different from zero and is time-varying.11
To investigate how changes in skewness and risk aversion parameters cause changes in the
prices of the volatility and coskewness risk, I examine ten-years windows, yielding a total of 120
observations. Every year, from 1996 to 2006, I use the past ten years’ data and estimate the pricing
kernel. The risk aversion and skewness preference are mostly statistically significant12 .
Figure 1 plots the time series of the estimated risk aversion and skewness preference. As is
10
Results are not tabulated but are available on request.
They are two sources of time-variation in the volatility premium (19), (i) time-variation in the price of the
volatility risk λV OLt and/or (ii) time-variation in the conditional moments of the market return.
12
Results are not tabulated but are available on request.
11
12
evident from the figure, both the risk aversion and the price of market volatility risk are somewhat
higher in magnitude during the 1998 to 2000 part of the sample. This result suggests that high risk
aversion may imply a high price of market variance risk and consequently a high volatility premium.
This figure also shows that the risk aversion estimated is stable during the 2002 to 2006 part of the
sample, but the price of the market volatility is somewhat higher in magnitude, particularly during
the 2004 to 2006 part of the period. Over the same period, the estimated skewness preference
is higher (ranging from one to 2.4). This result suggests that changes in the price of the market
volatility and hence in the volatility premium, could also be caused by changes in investor skewness
preference while their risk aversion is stable. Figure 1 also plots the price of coskewness risk, and
shows that the price of coskewness preference and volatility risk tend to move together during the
1998 to 2001 and 2002 to 2006 periods. There is at least one explanation to this co-movement. The
price of volatility risk is a sum of two quantities: the first is the price of coskewness risk and the
second is the square of the risk aversion parameter.
Panel B in Table I reports the results when I use the VIX. With this measure, the risk aversion
parameter ranges from two to 4.75, which represent a marginal increase compared to the estimated
values with the old volatility measure VXO. Both the skewness preference and risk aversion are
statistically significant. The prices of both the coskewness and volatility risk are slightly higher in
magnitude compared to the implied prices of risks when I use the VXO, except for the sample period
January 1996 to December 2006. The price of the volatility risk ranges from -1.2% to -0.62% per
year, and the price of coskewness risk ranges from -2.86% to -1.7% per year. The distance measure
and p-values suggest that the estimated pricing kernel is rejected at 5% level.
In Figure 2, I plot the time-series of risk aversion, skewness preference, prices of coskewness
and volatility risk. The results in this figure confirms my previous findings that periods of high
volatility premium is due to high risk aversion or high skewness preference or both.
Panel C in Table I reports the results when I use the realized volatility measure (RV). With
this new measure, the risk aversion parameter ranges from 2.50 to 4.75. The prices of coskewness
(volatility) risk are lower (higher) in magnitude compared to the implied prices of risks when I use
the VXO and VIX, except for the sample period January 1990 to December 2000. The distance
measure and p-values suggest that the estimated pricing kernel is rejected at 5% level, except for
the sample period January 1990 to December 2006.
C.1.
Controlling for Size, Book-to-Market, Momentum Factors
I investigate the robustness of my results by estimating an alternative specification of the pricing
kernel that incorporates the Fama and French (1993) size and book-to-market characteristics. I augment the pricing kernel with the size and book-to-market factors. I also control for the momentum
factor of Jegadeesh and Titman (1993). The augmented pricing kernel has the form
m∗t,t+1 = mt,t+1 + D3 rSM Bt+1 + D4 rHM Lt+1 + D5 rM OM t+1 .
13
(25)
In this pricing kernel, rSM Bt+1 represents the excess return on a portfolio of small-cap stocks over
large-cap stocks, rHM Lt+1 represents the excess return on a portfolio of high market-to-book stocks
over low market-to-book stocks, and rM OM t+1 represents the return on the momentum portfolio of
Jegadeesh and Titman (1993).13
Table II reports the estimated risk aversion and skewness preference, and the implied price of
market, coskewness and volatility risk14 . Panel A presents the results when I use the VXO as my
proxy for the market volatility. The table shows that the estimated risk aversion ranges from four to
4.3, and that the results are significant at 10% level. The skewness preferences are also statistically
significant, except for the sample period January 1996 to December 2006. The implied price of
the volatility risk ranges from -1.04% to -0.19%, while the price of coskewness ranges from -3.2%
to -1.96% per year. Panel B reports the results when I use the VIX as my proxy for the market
volatility. The estimated risk aversion ranges from 3.4 to 4.6 and is significant at the 5% level
when the full sample is used in the estimation process. The implied price of coskewness risk ranges
from -2.95% to -1.7% per year, while the implied price of the market volatility risk ranges from
-0.94% to -0.89% per year. After controlling for the size, book-to-market and momentum factors,
I find that both the estimated risk aversions and skewness preferences and the implied prices of
coskewness and market volatility risk are in a reasonable range. Panel C in Table I reports the
results when I use the realized volatility measure (RV). With this new measure, the risk aversion
parameter ranges from 3.62 to 3.99. Both the skewness preference and risk aversion coefficients are
in majority statistically significant. The prices of coskewness (volatility) risk are lower (higher) in
magnitude compared to the implied prices of risks when I use the VXO and VIX, except for the
sample period January 1990 to December 2000. The distance measure and p-values suggest that
the estimated pricing kernel is rejected at 5% level, except for the sample period January 1990 to
December 2006.
C.2.
Explaining the Puzzling Behavior of Pricing Kernels
To gauge the ability of the estimated pricing kernel to explain recent puzzles documented in studies
mentioned earlier, I use a setting with standard preferences and static prices of risk. By doing so,
I can interpret the pricing kernel as a scaled marginal utility. Under these assumptions, to be
consistent with positive marginal utility and the no arbitrage condition, the pricing kernel should
be positive, and be consistent with decreasing absolute risk aversion it should be decreasing in the
aggregate wealth (market return).
I plot my estimated pricing kernel as function of the market return and volatility of the market return. The estimated mean pricing kernel, risk aversion and skewness preference, are those
reported in Table I. Figures 3 and 4 depict the estimated pricing kernel when I use the VXO and
13
I also use the Pastor and Stambaugh (2003) liquidity factor. The results are qualitatively similar. Therefore I do
not report the results with the liquidity factor. These results are available on request.
14
I do not report the coefficients of the size, book-to-market and momentum priced factors. Results are available
on request.
14
VIX as my proxies for market volatility. The support for the graphs is the range of the returns on
the value-weighted index and the implied volatility difference. These figures show that the pricing
kernel is decreasing as the market return increases and is increasing when the market volatility
increases. This result makes my pricing kernel consistent with preference theory.
To further examine this suggestion, I project the estimated pricing kernel on the polynomial function of the market return alone. The projected pricing kernel has the form mt,t+1 =
Pn
j
j=0 bj rM t+1 . I use different values for n, as in n equals three, four, and five, and find that the
results remain unchanged. Therefore, I present only the result for n equals five. Figures 5 and 6
depict the projected pricing kernel when I use both the VXO and VIX. The support for the graphs
is the range of the returns. These Figures show that for various sub-samples, the projected pricing
kernel, increases when the aggregate wealth (market return) increases.
As an alternative measure to the VXO and VIX indexes, I also use the realized volatility measure
(RV). The results are qualitatively similar but are not reported15 . My finding suggests that the
missing market volatility factor in the pricing kernel and a lack of a structural interpretation of
the pricing kernel in terms of investor preferences are plausible explanations to the pricing kernel
puzzle. My estimated risk aversion and skewness preference are reasonable, and more importantly,
the implied prices of coskewness and market volatility have the expected sign and are within a
reasonable range.16
D.
Robustness to the Test Assets
For robustness check, I use the latest stock composition of the 30 Dow Jones Industrial Average.
The company names and summary statistics are presented in Table III. I use monthly returns
on Dow Jones 30 stocks from January 1990 to December 2006. Table IV reports the estimated
risk aversion and skewness preference, and the implied price of market, coskewness and volatility
risk. Panel A presents the results when I use the VXO, VIX, and RV as my proxy for the market
volatility. The table shows that the estimated risk aversion ranges from 3.78 to 5.75, and that
the results are significant at 5% level. The skewness preferences are also statistically significant.
The implied price of the volatility risk ranges from -0.91% to -0.14%, while the price of coskewness
ranges from -2.58% to -2.09% per year. Panel B reports the results when I control for the Fama
and French and momentum factors. The estimated risk aversion ranges from 4.61 to 5.4 and is
significant at the 5% level. The implied price of the volatility risk ranges from -1.30% to -0.87%,
while the price of coskewness ranges from -2.69% to -2.34% per year.
15
The results of the projected pricing kernels are available on request.
Chabi-Yo et al. (2008) argue that state dependence in preferences and fundamentals could be the cause of the
pricing kernel puzzle. Brown and Jackwerth (2004) provide a model of generating the pricing kernel puzzle, albeit
only for parameter constellations which are not typically observed in the real word.
16
15
V.
Sources of the Idiosyncratic Volatility Premium
In the previous sections, I examine the sources of the market volatility risk premium. In this
section, I first examine the sources of the idiosyncratic volatility premium. In an economy where
the pricing kernel is a linear function of priced factors, I find that nonzero risk aversion and firms’
non-systematic coskewness determine the premium on idiosyncratic volatility risk. I interpret the
non-systematic coskewness as the non-systematic component of asset’s skewness that is related to
the market’s portfolio skewness. I empirically show the relevance of this non-systematic coskewness
in explaining the idiosyncratic volatility puzzle put forward in Ang et al. (2006, 2008).
I define idiosyncratic risk as the risk that is unique to a specific firm, so I also refer to it as
firm-specific risk. By definition, idiosyncratic risk is independent of the common movement of the
market. To understand the idiosyncratic volatility puzzle, it is first important to understand how
it is priced. Given an asset pricing model with ft+1 as the set of risky factors, the pricing kernel is
mt,t+1 =
1
dt
−
ft+1
Rf
Rf
(26)
where the coefficient dt depends on investor preferences17 . In a single factor model, dt represents the
risk aversion coefficient. I define the idiosyncratic variance risk premium as the difference between
the idiosyncratic volatility under the objective and risk neutral measure18 . Given the pricing kernel
(26), the idiosyncratic variance premium is
2
∗2
ivpkt = σεkt
− σεkt
= −Rf Covt (mt,t+1 , ε2kt+1 )
(27)
where εkt+1 represents the asset’s k idiosyncratic shock in the linear regression
rkt+1 = αkt + βkt ft+1 + εkt+1 .
(28)
2 represents the idiosyncratic volatility of asset k under the objective measure, σ ∗2 is the idiosynσεkt
εkt
cratic volatility of asset k under the risk neutral measure. If the idiosyncratic volatility is priced,
the component (27) which is the premium on the idiosyncratic volatility risk should be significantly different from zero. Proposition 2 below gives the conditions under which this idiosyncratic
volatility premium component is zero.
PROPOSITION 2 : The idiosyncratic volatility premium, ivpkt , is
ivpkt = λt .γkt
(29)
with
γkt = [V art (ft+1 )]−1 Covt (ft+1 , ε2kt+1 )
and
λt = dt [V art (ft+1 )].
(30)
I refer γkt to as the non-systematic coskewness. If the idiosyncratic shock εkt+1 and the risk factor
ft+1 are jointly and normally distributed, then the non-systematic coskewness is zero. As a result,
the non-systematic variance premium is zero and the idiosyncratic volatility risk is not priced.
17
18
Notice that the pricing kernel derived in Section 1 is a linear function of the risky factors.
Carr and Wu (2008) adopt a similar definition for individual stock variance risk premium.
16
Proof See the Appendix for the proof.
In regression (28), the idiosyncratic shock is uncorrelated with the risk factor ft+1 . However, the
regression does not tell me whether the higher-order components of the idiosyncratic shock are
uncorrelated to the risk factor ft+1 . If the idiosyncratic shock and the stock return are jointly and
normally distributed, then by using Stein Lemma19 , I can show that the idiosyncratic volatility
risk is not priced and that γkt = 0. Idiosyncratic volatility is priced due to the presence of higher
moments in the stock returns and a non-zero risk aversion via dt . To access how the asset’s
expected excess return vary across stocks with different levels of non-systematic coskewness, I
consider a single factor model in which the market excess return is a the only priced factor20 . I do
not make any assumption about the distribution of the market return. Under the restriction that
cov(εkt+1 εjt+1 , rmt+1 ) = 0 for j 6= k, I show that
∂γkt
∂αkt
= −
2
2 $k γk ,
dt σM
t
where $k represents the weight of asset k in the market portfolio and dt is the risk aversion
coefficient21 . Equation (31) states that the non-systematic coskewness γkt is negatively (positively)
related to the asset’s expected excess return if it is positive (negative). In the subsequent sections,
I show that the non-systematic coskewness is priced, and is helpful in explaining the idiosyncratic
volatility anomaly.
A.
Estimating Idiosyncratic Volatility
To investigate this prediction, I use all the NASDAQ, AMEX, and NYSE stocks and consider
industrial firms. The sample period is from January 1971 to December 2006. To reduce the impact
of infrequent trading on idiosyncratic volatility estimates, I require that firms have a minimum of
120 trading days (non-zero observations) in a year. I also exclude equity prices lower than one dollar.
I first compute the idiosyncratic volatility at the end of each month using the past 12 months daily
observations. I use different models to compute the idiosyncratic volatility: the CAPM model, the
Fama and French (1993) three-factor model, the Fama and French three-factor model augmented
with the Jegadeesh and Titman (1993) momentum factor and the Harvey and Siddique (2000)
market coskewness model. I then rank stocks based on their idiosyncratic volatility to form valueweighted decile portfolios and then hold the portfolios over the next month. I rank the stocks based
on their past idiosyncratic volatility risk into ten groups and form ten value-weighted portfolios in
10% increments from 10% to 100%. Figure 7 depicts the mean average return across deciles. The
figure shows that on average, regardless of the model used to compute the idiosyncratic volatility
risk, stocks with high idiosyncratic risk earn lower returns than do stocks with low idiosyncratic
volatility risk.
19
See the Stein Lemma in the Appendix.
With two additional priced factors, the quantitative results are more complicated, but the conclusions are qualitatively similar. They are available on request.
21
See the proof in the Appendix.
20
17
The mean average returns reported in Panel A of Table V are strongly and almost monotonically
declining in idiosyncratic volatility risk regardless of which model I use. The mean average returns
for the value-weighted portfolio return with the lowest-idiosyncratic volatility risk (10% Low) are
positive and ranges from 1.04% to 1.07% per month. The mean average returns for the valueweighted portfolio return with the highest-idiosyncratic volatility risk (10% High) are significantly
negative and range from -0.22% to -0.16% per month. A long-short portfolio holding the volatile
decile of stocks and shorting the safest decile has a mean average return ranging from -1.29% to
-1.19% with robust Newey-West (1987) t-statistics ranging from -2.64 to -2.48. Panel B reports
each model alpha, showing robust Newey-West t-statistics in square brackets. A long-short portfolio
holding the volatile decile of stocks and shorting the safest decile has an alpha ranging from -1.47%
to -1.37% and robust Newey-West t-statistics ranging from -3.15 to -3.10. When I correct for risk
using either the CAPM model, the Fama and French (1993) model, the Fama and French (1993)
model, the Fama and French (1993) model augmented with the momentum factor of Jegadeesh
and Titman (1993) and the Harvey and Siddique (2000) models, I worsen the anomalous poor
performance of volatile stocks rather than correcting it.
I also correct for the market volatility. I use different measures of the market volatility as
described in Section IV. I use the VXO from January 1986 to December 2006, the VIX from
January 1990 to December 2006 and the realized volatility RV from January 1990 to December
2006. The results are qualitatively similar to those reported in Table V. Consistent with Ang et al.
(2006), the market volatility cannot explain the anomalous poor performance of volatile stocks. I
do not report the results.22
Since the idiosyncratic anomaly is robust to the models used including the market volatility,
I use the CAPM model as a benchmark to compute the idiosyncratic volatility in the rest of the
paper.
B.
Non-Systematic Coskewness and Expected Returns
I find a strong positive cross-sectional relation between the average returns and non-systematic
coskewness when the non-systematic coskewness is negative and there is a strong negative crosssectional relation between the average returns and non-systematic coskewness when the nonsystematic coskewness is positive. This finding is consistent with my theoretical motivation. In
this section, I assess how average returns vary across stocks with different levels of non-systematic
coskewness.
B.1.
Decile Portfolios in 10% increments from 10% to 100%
My model predicts that if stocks’ non-systematic coskewness is negative (positive), then on average,
stocks with high non-systematic coskewness earn higher (lower) returns than do stocks with low nonsystematic coskewness. To verify my prediction, I use the past 12 months daily returns to compute
22
The results are available on request.
18
the non-systematic coskewness at the end of each month. I form two groups of stocks, those with
negative and those with positive non-systematic. Within each group, I rank the stocks based on
their past non-systematic coskewness into ten groups and form ten value-weighted portfolios in 10%
increments from 10% to 100%.
When the non-systematic coskewness is positive, the mean average returns reported in the
Panel A of Table VI are strongly and almost monotonically declining in non-systematic coskewness
risk. The mean average returns for the value-weighted portfolio with the lowest non-systematic
coskewness risk (10% Low) is positive at 1.07% per month, and the mean average returns for the
value-weighted portfolio with the highest non-systematic coskewness (10% high) is significantly
lower at 0.62% per month. However, a long-short portfolio holding the highest non-systematic
coskewness decile of stocks and shorting the lowest non-systematic coskewness decile has an average
return of -0.45% per month which is not statistically significant (the t-statistic is -1.04).
When the non-systematic coskewness is negative, the mean average returns reported in the Panel
A of Table VI are strongly and almost monotonically increasing in non-systematic coskewness risk.
The average returns for the value-weighted portfolio with the lowest non-systematic coskewness risk
(10% Low) is positive at 0.32% per month, and the average returns for the value-weighted portfolio
with the highest non systematic coskewness (10% High) is significantly higher at 1.13% per month.
A long-short portfolio holding the highest non-systematic coskewness decile of stocks and shorting
the lowest non-systematic coskewness decile has an average return of 0.81% per month which is
statistically significant (the t-statistic is 2.3). These results confirm my model’s prediction that on
average, stocks with high non-systematic coskewness earn on average higher returns than do stocks
with low non-systematic coskewness.
To correct for the CAPM or Fama and French (1993) three-factors, for each of the ten valueweighted portfolio returns and the “10-1 ”portfolio returns formed by ranking stock based on
non-systematic coskewness, I run the regression:
rp = α + βf + η.
(31)
where f represents the market excess returns when I use the CAPM and the Fama and French
(1993) three-factor when I use the Fama and French (1993) model. I define 10-1 as the difference in
returns between portfolio 10 and portfolio 1. When I correct for risk using either the CAPM or the
Fama-French (1993) three-factor model, there is a striking variation in alpha across the portfolios in
Table IV. First, when the non-systematic coskewness is positive, the value-weighted portfolios with
low non-systematic coskewness have highly significant alphas, but the value-weighted portfolios with
high non-systematic coskewness portfolio have non significant alphas. In contrast, when the nonsystematic coskewness is negative, the value-weighted portfolios with low non-systematic coskewness
have non significant alphas and the value weighted portfolios with high non-systematic coskewness
have highly significant alphas. Moreover, when the non-systematic coskewness is negative, a longshort portfolio holding the highest non-systematic coskewness decile of stocks and shorting the
19
lowest non-systematic coskewness decile has a highly significant alpha of 1% per month (the tstatistic is 3.101) when I control for the CAPM, and a highly significant alpha of 0.96% per month
(the t-statistic is 2.86) when I control for the Fama and French three-factor. These results suggest
that when the non-systematic coskewness is negative, the alphas of the value-weighted portfolio
with the highest non-systematic coskewness exceeds the Fama-French alphas of the value-weighted
portfolio with the lowest non-systematic coskewness by about 1% per month.
B.2.
Decile Portfolios and the Tails of the Distribution
Because the non-systematic coskewness is related to the skewness of the return’s distribution, I
construct value-weighted portfolios that pays greater attention to the tails of stocks’ distribution. I
form two groups of stocks, those with negative and those with positive non-systematic coskewness.
Within each group, I rank the stocks in percentiles 0-5, 5-20, 20-30, 30-40, 40-50, 50-60, 60-70,
70-80, 80-95, and 95-100 based on their past non-systematic coskewness. Within each percentile, I
form the value-weighted return.
When the non-systematic coskewness is positive, the mean average returns reported in the Panel
A of Table VII are strongly and almost monotonically declining in non-systematic coskewness
risk. The mean average returns for the value-weighted portfolio with the lowest non-systematic
coskewness risk (5% Low) is positive at 1.17% per month, and the mean average returns for the
value-weighted portfolio with the highest non-systematic coskewness (5% high) is significantly lower
at -0.05% per month. A long-short portfolio holding the highest non-systematic coskewness decile
of stocks and shorting the lowest non-systematic coskewness decile has an average return of -1.22%
per month which is statistically significant (the t-statistic is -2.65).
When the non-systematic coskewness is negative, the mean average returns reported in the Panel
A of Table VII are strongly and almost monotonically increasing in non-systematic coskewness risk.
The average returns for the value-weighted portfolio with the lowest non-systematic coskewness risk
(5% Low) is negative at -0.05% per month, and the average returns for the value-weighted portfolio
with the highest non systematic coskewness (5% High) is significantly higher at 1.24% per month.
A long-short portfolio holding the highest non-systematic coskewness decile of stocks and shorting
the lowest non-systematic coskewness decile has an average return of 1.29% per month which is
statistically significant (the t-statistic is 2.88).
When I correct for risk using either the CAPM or the Fama-French (1993) three-factor model,
there is a striking variation in alpha across the portfolios in Table VII. First, when the nonsystematic coskewness is positive, the value-weighted portfolios with low non-systematic coskewness
have highly significant alphas, but the value-weighted portfolios with high non-systematic coskewness portfolio have non significant alphas. In contrast, when the non-systematic coskewness is
negative, the value-weighted portfolios with low non-systematic coskewness have non significant alphas and the value weighted portfolios with high non-systematic coskewness have highly significant
alphas.
20
When the non-systematic coskewness is positive, a long-short portfolio holding the highest nonsystematic coskewness decile of stocks and shorting the lowest non-systematic coskewness decile
has a highly significant alpha of -1.37% per month (the t-statistic is -3.17) when I control for the
CAPM, and a highly significant alpha of -1.30% per month (the t-statistic is -2.82) when I control
for the Fama and French three-factor. Further, when the non-systematic coskewness is negative,
a long-short portfolio holding the highest non-systematic coskewness decile of stocks and shorting
the lowest non-systematic coskewness decile has a highly significant alpha of 1.49% per month (the
t-statistic is 3.52) when I control for the CAPM, and a highly significant alpha of 1.30% per month
(the t-statistic is 2.94) when I control for the Fama and French three-factor.
C.
Explaining the Low Returns of High Idiosyncratic Volatility Stocks
In this section, I assess how the non-systematic coskewness explains the low returns of high idiosyncratic volatility stocks. I first consider only stocks with positive non-systematic coskewness.
Second, I consider stocks with negative non-coskewness. Third, I consider all stocks.
C.1.
Positive Non-Systematic Coskewness
To explain the anomalous underperformance of high idiosyncratic volatility stocks, I consider the
following explanation. I consider stocks with positive non-systematic coskewness. I construct a
long-short portfolio relative to non-systematic coskewness when it is positive. I construct this
portfolio by holding the highest non-systematic coskewness decile of stocks and shorting the lowest
non-systematic coskewness decile. For each of the ten value-weighted portfolio returns, and the
10-1 portfolio returns formed by ranking stocks based on the idiosyncratic volatility, I run the
regression:
rp = α + βf + βν (ν + − ν − ) + η,
(32)
where f represents the market excess returns when the CAPM is used and the Fama and French
(1993) three-factor when I use the Fama and French (1993) three-factor model. I refer ν − to as
the value-weighted portfolio return formed with the 5% lowest non-systematic coskewness stocks
and ν + the value-weighted portfolio return formed with the 5% highest non-systematic coskewness
stocks. I then formulate the following hypothesis based on my theoretical model’s prediction.
H1: Because the expected excess return on the long-short portfolio (ν + − ν − ) is negative, my
model predicts that the non-systematic coskewness factor will explain the idiosyncratic volatility
anomaly if the value-weighted portfolio with high idiosyncratic volatility risk has significant positive loadings on the long-short portfolio (ν + − ν − ); and that the difference in returns between
the value-weighted portfolio with the highest idiosyncratic volatility risk and the value-weighted
portfolio with the lowest idiosyncratic volatility risk has a non significant alpha and significant
positive loading on the long-short portfolio (ν + − ν − ).
21
Panel A of Table VIII reports the average returns on value-weighted idiosyncratic portfolios. I
measure the statistics in the columns labeled Mean and Std Dev (standard deviation) in monthly
percentage terms. Standard errors appear in parentheses. Robust Newey-West (1987) t-statistics
appear in square brackets. A long short portfolio holding the highest idiosyncratic decile of stocks
and shorting the lowest idiosyncratic decile of stock has an average of -1.29 (wit a t-statistic of
-2.64).
Panel B of Table VIII reports the regression alphas and betas when I use the market excess
return and the long-short portfolio (ν + − ν − ) as explanatory variables. The table reports robust
t-statistics in brackets. The alphas reported in the first row of Panel A are all positive. More
importantly, the alpha for the 10-1 portfolio return is -0.16% per month (the t-statistic is -0.39),
and therefore is not significant. As the results indicate, it is apparent that compared to the same
alpha in Panel A, in Panel B, the difference in alphas between the portfolio with the highest
idiosyncratic volatility risk and the value-weighted portfolio with the lowest idiosyncratic volatility
risk reduces from -1.29 (the t-statistic is -2.64) to -0.16 (the t-statistics is -0.39). It is apparent
from Panel B that alphas exhibit a reverse symmetric U-shaped across deciles.
In contrast almost all betas on the long-short portfolio (ν + − ν − ) are positive and highly significant. These betas increase as I move from the value-weighted portfolios with lower idiosyncratic
volatility to the value-weighted portfolios with higher idiosyncratic volatility. The betas ranges
from 0.03 to 1.05. With the exception of the value-weighted portfolio with the lowest idiosyncratic volatility, all t-statistics range from 2.60 to 16.34. More importantly, the 10-1 portfolio loads
positively on the long-short portfolio (ν + − ν − ) with a value 1.01 and a t-statistic of 13.89.
These results indicate that stocks with high idiosyncratic volatility risk have high positive betas
on the long-short return. Since the expected return on the long-short portfolio is negative, it follows
that on average, stocks with high idiosyncratic volatility earn lower returns. Another interpretation
is that stocks with high idiosyncratic volatility have high positive betas on the long-short return and
hence highly positive non-systematic coskewness. As noted in section B, when the non-systematic
coskewness is positive, stocks with highly positive non-systematic coskewness earn, on average lower
returns. Panel C of Table VIII presents similar results when I control for the Fama and French
(1993) three factors.
C.2.
Negative Non-Systematic Coskewness
In this section, I consider stocks with negative non-systematic coskewness. I construct a longshort portfolio relative to non-systematic coskewness when it is negative. I construct this portfolio
by holding the highest non-systematic coskewness decile of stocks and shorting the lowest nonsystematic coskewness decile. For each of the ten value-weighted portfolio returns, and the 10-1
portfolio returns formed by ranking stocks based on the idiosyncratic volatility, I run the regression:
rp = α + βf + βν (υ + − υ − ) + η,
22
(33)
I refer υ − to as the value-weighted portfolio return formed with the 5% lowest non-systematic
coskewness stocks and υ + the value-weighted portfolio return formed with the 5% highest nonsystematic coskewness stocks. I then formulate the following hypothesis based on my theoretical
model’s prediction.
H2: Because the expected excess return on the long-short portfolio (υ + − υ − ) is positive, my
model predicts that the non-systematic coskewness factor will explain the idiosyncratic volatility
anomaly if the value-weighted portfolio with high idiosyncratic volatility risk has significant negative loadings on the long-short portfolio (υ + − υ − ); and that the difference in returns between
the value-weighted portfolio with the highest idiosyncratic volatility risk and the value-weighted
portfolio with the lowest idiosyncratic volatility risk has a non significant alpha and significant
negative loading on the long-short portfolio (υ + − υ − ).
Panel A of Table IX reports the average returns on value-weighted idiosyncratic portfolios. I
measure the statistics in the columns labeled Mean and Std Dev (standard deviation) in monthly
percentage terms. Standard errors appear in parentheses. Robust Newey-West (1987) t-statistics
appear in square brackets. A long short portfolio holding the highest idiosyncratic decile of stocks
and shorting the lowest idiosyncratic decile of stock has an average of -1.35 (wit a t-statistic of
-2.44).
Panel B of Table IX reports the regression alphas and betas when I use the market excess
return and the long-short portfolio (υ + − υ − ) as explanatory variables. The table reports robust
t-statistics in brackets. The alpha for the 10-1 portfolio return is -0.54% per month (the t-statistic
is -1.24), and therefore is not significant. As the results indicate, it is apparent that compared to
the same alpha in Panel A, in Panel B, the difference in alphas between the portfolio with the
highest idiosyncratic volatility risk and the value-weighted portfolio with the lowest idiosyncratic
volatility risk reduces from -1.35 (the t-statistic is -2.44) to -0.54 (the t-statistics is -1.24).
In contrast almost all betas on the long-short portfolio (υ + − υ − ) are negative and highly significant. These betas decrease as I move from the value-weighted portfolios with lower idiosyncratic
volatility to the value-weighted portfolios with higher idiosyncratic volatility. The betas ranges
from -0.84 to -0.09. With the exception of the value-weighted portfolio with the lowest idiosyncratic volatility, all t-statistics range from -12.99 to -2.09. More importantly, the 10-1 portfolio
loads negatively on the long-short portfolio (υ + − υ − ) with a value -0.84 and a t-statistic of -12.63.
These results indicate that stocks with high idiosyncratic volatility risk have negative betas on
the long-short return. Since the expected return on the long-short portfolio is positive, it follows
that on average, stocks with high idiosyncratic volatility earn lower returns. Panel C of Table IX
presents similar results when I control for the Fama and French (1993) three factors.
23
C.3.
Negative and Positive Non-Systematic Coskewness
I consider all stocks with positive and negative non-systematic coskewness. I use the value-weighted
decile portfolios that I construct in Section A. For each of the ten value-weighted portfolio returns,
and the 10-1 portfolio returns formed by ranking stocks based on the idiosyncratic volatility, I run
the regression:
rp = α + βf + βν (ν + − ν − ) + βυ (υ + − υ − ) + η,
(34)
where f represents the market excess returns when the CAPM is used and the Fama and French
(1993) three-factor when I use the Fama and French (1993) three-factor model. I then formulate
the following hypothesis based on my theoretical model’s prediction.
H3: My model predicts that the non-systematic coskewness factor will explain the idiosyncratic volatility anomaly if the value-weighted portfolio with high idiosyncratic volatility risk has
significant positive loadings on the long-short portfolio (ν + − ν − ), significant negative loading on
the long-short portfolio (υ + − υ − ); and that the difference in returns between the value-weighted
portfolio with the highest idiosyncratic volatility risk and the value-weighted portfolio with the
lowest idiosyncratic volatility risk has a non significant alpha and significant positive loading on
the long-short portfolio (ν + − ν − ) and negative loading on the long-short portfolio (υ + − υ − ).
Table X reports the regression alphas and betas and robust t-statistics in brackets. Panel
A reports the result when I use the full sample of industry firms to construct the ten valueweighted portfolio returns and the 10-1 portfolio returns based on idiosyncratic volatility. The
alphas reported in the first row of Panel B are all positive. It is apparent from Panel B that alphas
exhibit a reverse symmetric U-shaped across deciles. More importantly, the alpha for the 10-1
portfolio return is 0.04% per month (the t-statistic is 0.17), and therefore is not significant. In
contrast almost all betas on the long-short portfolios (ν + − ν − ) and (υ + − υ − ) have the expected
sign and are highly significant. When I use the 10-1 value-weighted idiosyncratic portfolio return
as my dependent variable in the regression, the adjusted R-square is about 74%. The correlation
between the two long-short portfolio returns (ν + − ν − ) and (υ + − υ − ) is 0.55.
As my results indicate, it is apparent that compared to the same alpha in Panel A, in Panel
B, the difference in alphas between the portfolio with the highest idiosyncratic volatility risk and
the value-weighted portfolio with the lowest idiosyncratic volatility risk reduces from -1.47% (the
t-statistic is -3.15) to 0.04% (the t-statistics is 0.17). Panel C of Table X presents similar results
when I control for the Fama and French (1993) three factors.
VI.
Relation to other research on idiosyncratic volatility
Recent papers such as Fu (2008), Brockman and Schutte (2007) assume that risky assets’ return
follow an asymmetric GARCH model. These papers use EGARCH method to estimate conditional
24
idiosyncratic volatility and confirm that the relation between stock returns and conditional idiosyncratic volatility is positive in both U.S. and international data. Similarly Spiegel and Wang (2006)
and Eiling (2006) adopt the EGARCH models to estimate conditional idiosyncratic volatility and
also find the positive relation in the U.S. data.
To explain why these authors find a positive relation between the idiosyncratic volatility and
expected excess return when GARCH models and its extensions are used to compute the idiosyncratic volatility risk, I begin by assuming that the asset’s return is described by Equation (28) with
the market return as a single risky factor. In addition, I assume that the idiosyncratic risk εkt+1
is normally distributed with conditional variance described by a model that belongs to a family of
GARCH models. Further, I assume that the idiosyncratic volatility is described by an asymmetric
GARCH model based on Glosten, Jagannathan and Runkle (1993):
hkt+1 = β0 +
j=p
X
βj hkt+1−j +
αj ε2kt+1−j +
j=1
j=1
where εkt+1−j =
j=q
X
j=q
X
δj It+1−j ε2kt+1−j
(35)
j=1
p
2
hkt+1−j zt+1−j with zt+1−j ∼ N (0, 1) and hkt+1−j = σεkt−j
. The indicator
function It+1−j equals 1 if εkt+1−j < 0, and zero otherwise. When δ1 > 0, the model (35) accounts
for the leverage effect, that is, that bad news (εkt+1−j < 0) raises the future volatility more than
does good news (εkt+1−j ≥ 0) of the same absolute magnitude. Under assumption (35), the nonsystematic coskewness γkt is23
γkt
1
=
(α1 +
2
Covt (rM t+1 , σεkt+1
)
2
δ1
σM
t
2)
(36)
where the coefficients α1 and δ1 are both positive. Thus a negative correlation of the idiosyncratic
volatility with the market return causes the non-systematic coskewness to be negative. In my
sample, the average correlation of idiosyncratic volatility with the market return is negative (2%). According to my model’s prediction, if the non-systematic coskewness is negative, then on
average, stocks with high non-systematic coskewness earn high returns. In my sample, the average
correlation of idiosyncratic volatility with the non-systematic coskewness is positive which implies
that stocks with high idiosyncratic volatility risk have high non-systematic coskewness. Thus, if
I assume a GARCH specification for individual stock returns, the non-systematic coskewness in
Equation (36) will be negative and stocks with high idiosyncratic volatility will have high nonsystematic coskewness and therefore would earn in average higher returns. This finding could
explain why under GARCH specifications, recent studies find that stocks with high idiosyncratic
volatility earn high expected returns. This reasoning is valid even under a simple GARCH model
in which δ1 = 0.
23
The proof of this expression appears in the Appendix.
25
VII.
Conclusion
Recent papers such as Ang et al. (2006) estimate the price of the market volatility risk and find
that the volatility of the market is priced. Its price is about -1% per annum. Bollerslev, Tauchen
and Zhou (2008) use a model-free approach and show that the difference between the volatility
of the market under the risk-neutral measure and the volatility of the market under the physical
measure is significant, and that the magnitude of return predictability of the variance premium
easily dominates that afford by standard predictor variables. These authors suggest that temporal
variation in the risk and risk aversion play an important role in determining the variance premium
and also argue that period of high volatility premium is intimately associated with high risk aversion.
However, these papers leave unanswered the question of what value of risk aversion is consistent
with the estimated price of market volatility or the observed volatility premium.
I build a partial equilibrium model in which investors trade in a multi-period market. As a result,
the aggregate pricing kernel in equilibrium depends on both coskewness and market volatility risk
factor. I show that the price of coskewness and market volatility are restricted by investor risk
aversion and skewness preference, and I provide a closed-form solution for the prices of coskewness
and market volatility risk in terms of investor risk aversion and skewness preference. I use two
sets of independent data. I first use the 30 industry portfolio returns. Second, I use the 30 Dow
Jones stock returns. When I use the 30 industry portfolio returns, the risk aversion is estimated to
be between 2.5 and 4.75 while the skewness preference ranges from 1.05 to 2.25. The parameters
are mostly statistically significant. The implied price of coskewness associated to these estimates
ranges from -3.2% to -0.72% per year . When I use the VIX and VXO as my proxy for the market
volatility, the implied price of the market volatility ranges from -1.2% to -0.19% per year. The
implied price of the market volatility ranges from -1.51% to -0.77% per year when I use the realized
volatility. When I use the 30 Dow Jones Stock returns, the risk aversion is estimated to be between
3.75 and 5.75 while the skewness preference ranges from 1.003 to 1.71. The parameters are all
statistically significant. The implied price of coskewness associated to these estimates ranges from
-2.7% to -2.07% per year . When I use the VIX, VXO, and RV as my proxy for the market volatility,
the implied price of the market volatility ranges from -1.30% to -0.15% per year. These estimates
are in a reasonable range and consistent with the literature. I also investigate the impact of the
risk aversion and skewness preference on the price of the market volatility risk over time. I find
that periods of high price of volatility risk is sometimes associated to high risk aversion and low or
stable skewness preference, sometimes to high skewness preference and low and stable risk aversion,
or sometimes to both high risk aversion and high skewness preference.
I also examine the puzzling behaviors of the pricing kernel. I show that my estimated pricing
kernel is consistent with economic theory, in that it is decreasing in the aggregate wealth (market
return) and increasing in the market volatility. When I project my estimated pricing kernel on the
polynomial function of the market return alone, doing so produces the puzzling behaviors observed
26
in pricing kernel. I argue that the missing market volatility priced factor in the pricing kernel and
the lack of a structural interpretation of the price of coskewness and volatility risk in terms of
investor risk aversion and skewness preference noted in previous studies could be the cause of the
puzzling behaviors of the pricing kernel.
Finally, I examine the negative relation between idiosyncratic volatility and expected returns,
and ask why do low idiosyncratic volatility firms earn higher future returns than ones with higher
idiosyncratic volatility? To answer this question, I study the source of idiosyncratic volatility premium by using pricing kernels. I find that the premium on idiosyncratic volatility risk is determined
by a nonzero risk aversion and firms non-systematic coskewness. I define non-systematic coskewness as the non-systematic component of asset skewness that is related to the market portfolio’s
skewness. I find two results. First, when this non-systematic component is positive, the difference in Fama-French (1993) alphas between the valued-weighted decile portfolio with the highest
non-systematic coskewness and the value-weighted decile portfolio with the lowest non-systematic
coskewness has a significant alpha of -1.30% per month. In contrast, when the non-systematic
coskewness is negative, a long-short portfolio holding the highest non-systematic coskewness decile
of stocks and shorting the lowest non-systematic coskewness decile of stocks has a highly significant
alpha of 1.30% per month.
I also study the negative relation between idiosyncratic volatility and expected returns. My
results show that the non-systematic coskewness is helpful in solving the idiosyncratic volatility
anomaly. I relate my findings to recent studies that use GARCH specification of the idiosyncratic
volatility risk. I show that by assuming GARCH specifications, these studies restrict the relation
between idiosyncratic volatility and stock returns to be positive. Therefore, given our sample
period, it is not possible to use a GARCH type of specification and arrive at a negative relation
between idiosyncratic volatility and expected returns.
27
Appendix
Appendix A:
The Optimal Asset Allocation
To give a formal proof of all propositions, I first use the bifurcation theorem (see Therorem 4, Page 8 in Judd and
Guu (2001)) to solve the optimization problem (1). Following the bifurcation theorem, the optimal portfolio weight
is a function of the small noise expansion parameter ε and is given by:
(i)
(i)0
(i)
ων−1 = ων−1 (0) + ων−1 (0)ε,
(i)
(i)
(A1)
0
where ων−1 (0) and ων−1 (0) represents the level and slope of the portfolio weights. To determine these quantities, I
solve backward the optimization problem (1). Appendix A contains the proof of the optimal portfolio weight (A1)
and the risk premium function aτ (.) appearing in the return decomposition (3). Appendix B contains the proof of
all propositions.
Proof I consider the First-Order Conditions (hereafter FOCs)
(i)
ων−1 : Eν−1 u(i)
0
³
(i)
WT
T
´ Y
³
´
(i)|
Rf + ωτ −1 Reτ (εaν−1 (ε) + Yν ) = 0.
(A2)
τ =ν+1
at time ν − 1 ∈ {t, ..., T − 1}. I discuss below the steps needed to solve (A2) for the optimal portfolio weight.
1. First Step: I proceed in an intuitive fashion to arrive at a solution validated by the bifurcation theorem. I
(i)
(i)
want to solve ων−1 as functions of ε near 0. I first compute what ων−1 is the correct solution to the ε = 0
case. I compute:
(i)
lim ων−1 .
ε7−→0
In the rest of this proof, I denote:
(i)
(i)
ων−1 (0) = lim ων−1 .
ε7−→0
(i)
To solve the FOCs for ων−1 (0), I consider the FOCs as shown in Equation (A2) and denote:
T
³
´
³
´ Y
³
´
0
(i)
(i)
(i)|
H ων−1 , ε = Eν−1 u(i) WT
Rf + ωτ −1 Reτ (εaν−1 (ε) + Yν ) .
(A3)
τ =ν+1
³
´
(i)
(i)
The choice of ων−1 is a function of ε implicitly defined by H ων−1 (ε) , ε = 0. Implicit differentiation of
Equation (A3) with respect to ε implies:
³
´
³
´
(i)
(i)0
(i)
Hω ων−1 (ε) , ε ων−1 (ε) + Hε ων−1 (ε) , ε = 0.
(A4)
³
´
(i)
(i)
Differentiating H ων−1 , ε with respect to ε and ων−1 respectively, I find:
³
´
(i)
Hε ων−1 , ε
=
Eν−1 u(i)
00
³
(i)
WT
+Eν−1 u(i)
0
³
+Eν−1 u(i)
0
³
T
´ ∂W (i) Y
³
´
(i)|
T
Rf + ωτ −1 Reτ (εaν−1 (ε) + Yν )
∂ε τ =ν+1
(i)
´
WT
RfT −ν−1
T
³
X
(A5)
´
(i)|
(i)0 |
ωτ −1 Yτ + ωτ −1 Reτ (εaν−1 (ε) + Yν )
τ =ν+1
T
´ Y
³
(i)
WT
(i)|
Rf + ωτ −1 Reτ
´³
´
0
aν−1 (ε) + εaν−1 (ε) ,
τ =ν+1
and
T
´ ∂W (i) Y
³
´
³
³
´
00
(i)
(i)
(i)|
T
Hω ων−1 , ε = Eν−1 u(i) WT
Rf + ωτ −1 Reτ (εaν−1 (ε) + Yν ) .
(i)
∂ων−1 τ =ν+1
³
´
(i)
(i)
Now, I look for a bifurcation point ων−1 (0) defined by Hε ων−1 (0) , ε = 0.To do this, notice that:
(i)
lim
ε7→0
∂WT
∂ε
= Wt RfT −t−1
T
³
X
τ =t+1
28
(i)|
ωτ −1 (0) Yτ
´
(A6)
(i)
and lim
ε7→0
∂WT
(i)
∂ων−1
=0
(A7)
(i)
I replace limε7→0
∂WT
(i)
∂ων−1
in (A6), and take the limit to get
00
Eν−1 u(i)
³
(i)
lim WT
ε7→0
´ ∂W (i)
³
´
0
(i)
T
Yν + Eν−1 u(i) lim WT aν−1 (0) = 0
ε7→0
∂ε
(A8)
(i)
(i)
for all ων−1 . In addition, I replace limε7→0
∂WT
∂ε
Eν−1 Wt RfT −t−1
which simplifies to:
³
in (A5) and substitute the result in Equation (A4) to derive:
´
(i)|
ων−1 (0) Yν Yν = τi aν−1 (0) .
³
´
(i)
(i)|
RfT −ν Covν−1 lim Wν−1 ων−1 (0) Yν , Yν = τi aν−1 (0) .
(A9)
ε7→0
Recall that, the market clearing conditions take the form as expressed in Equation (2). Thus, near 0, I write
the market clearing conditions as
I
X
(i)
(i)
(A10)
lim Wν−1 ων−1 (0) = θν−1 ,
i=1
ε7→0
I take the sum of Equation (A9) for i = 1, ..., I and use these market clearing conditions (see Equation A10))
to obtain:
¡
¢
1 T −ν
Rf Covν−1 ω |ν−1 Yν , Yν = aν−1 (0)
(A11)
℘
I
P
with ω ν−1 = I1 θν−1 and ℘ = I1
℘i .
i=1
Now, I plug Equation (A11) in Equation (A9) and get:
(i)
ων−1 (0) =
1
(i)
limε7→0 Wν−1
¢
¡
1
℘i −1
℘i
ω
.
Σ
Covν−1 ω |ν−1 Yν , Yν =
(i) ℘ ν−1
℘ ν−1
limε7→0 Wν−1
(A12)
where Σν−1 is the variance-covariance matrix defined by:
Σν−1 = Eν−1 Yν Yν| .
(i)
2. Second Step: I want to solve for the slope of the weights ων−1 near ε = 0. Specifically, I want to solve for
(i)0
ων−1
near ε = 0. To do this, I consider again the FOCs at date ν − 1.
³
´
³
´
(i)
(i)
(i)
For all ων−1 , it is straightforward to show that limε→0 H ων−1 , ε = H ων−1 (0) , 0 = 0.
³
´
(i)
Furthermore, for ων−1 (0) , 0 , I have:
³
´
(i)
H ων−1 (0) , 0 = 0,
³
´
(i)
Hε ων−1 (0) , 0 = 0,
(A13)
(A14)
where Hε represents the first derivative
of H
³
´ (., .) with respect to ε.
(i)
Now, I check whether det Hωε ων−1 (0) , 0 6= 0 where Hωε (., .) represents the second derivative of H with
respect to ω and ε respectively. It can show that
³
´
³
´
00
(i)
(i)
2(T −ν)
(i)
Hωε ων−1 (0) , 0 = u(i) lim WT Rf
lim Wν−1 Σν−1
(A15)
ε7→0
and
ε7→0
³
´
³
´
00
(i)
(i)
2(T −ν)
(i)
det Hωε ων−1 (0) , 0 = u(i) lim WT Rf
lim Wν−1 det Σν−1 6= 0
ε7→0
(A16)
ε7→0
Since Equations (A13), (A14) and (A16) are satisfied, I use the bifurcation theorem (see Theorem 4 in Judd
(i)0
and Guu (2001)) to solve the FOCs for ων−1 .
³
´
(i)
(i)
Following this , there exists an open neighborhood N of ων−1 (0) , 0 and a function hν−1 (ε) : R → Rn ,
(i)
hν−1 (ε) 6= 0 for ε 6= 0, such that
³
´
³
´
(i)
(i)
Hεε hν−1 (ε) , ε = 0 for hν−1 (ε) , ε ∈ N
(A17)
(i)
where Hεε (., .) represents the second derivative of H (., .) with respect to ε. Furthermore hν−1 is analytical
and can be approximated by a Taylor series. In particular the first order derivative equals
´
³
´
(i)0
1 −1 ³ (i)
(i)
(i)0
ων−1 (0) , 0 Hεε ων−1 (0) , 0 = lim ων−1 .
(A18)
hν−1 (0) = − Hωε
ε7→0
2
29
In the rest of this proof, I denote
(i)0
(i)0
ων−1 (0) = lim ων−1 .
ε7→0
Now, I compute the second derivative of H with respect to ε:
³
´
(i)
Hεε ωT −1 , 0
T
³
´´00 Y
³
´
³
0
(i)
(i)|
Rf + ωτ −1 Reτ (εaν−1 (ε) + Yν )
Eν−1 u(i) WT
=
τ =ν+1
+Eν−1 u
(i)0
³
(i)
WT
´
Ã
T
³
Y
Rf +
(i)|
ωτ −1 Reτ
´
!00
(εaν−1 (ε) + Yν )
τ =ν+1
+Eν−1 u(i)
0
³
(i)
WT
T
´ Y
³
´
00
(i)|
Rf + ωτ −1 Reτ (εaν−1 (ε) + Yν )
τ =ν+1
³
+2Eν−1 u
(i)0
³
(i)
WT
Ã
´´0
T
³
Y
Rf +
´
(i)|
ωτ −1 Reτ
!0
(εaν−1 (ε) + Yν )
τ =ν+1
T
´
³
³
´´0 Y
³
0
0
(i)
(i)|
+2Eν−1 u(i) WT
Rf + ωτ −1 Reτ (εaν−1 (ε) + Yν )
τ =ν+1
+2Eν−1 u
(i)0
³
(i)
WT
´
Ã
T
³
Y
Rf +
(i)|
ωτ −1 Reτ
´
!0
0
(εaν−1 (ε) + Yν ) .
τ =ν+1
I expand expression above and take its limit as ε approaches zeros to get:
³
´
(i)
Hεε ωT −1 , 0
µ
¶
³
´³
´2
³
´
000
00
(i)
(i)0
(i)
(i)00
= Eν−1 u(i)
lim WT
lim WT
+ u(i) lim WT
lim WT
RfT −ν Yν
ε7→0
ε7→0
ε7→0
ε7→0

T
P
(i)|
2RfT −ν−1
ωτ −1 (0) aτ −1 (0)

τ
=ν+1

T
P
³
´
(i)0 |

(i) 
(i)0
+2RfT −ν−1
ωτ −1 (0) Yτ
+Eν−1 u
lim WT 
τ
=ν+1
ε7→0

τ 6=ν

³
´³
´
T
P

(i)|
(i)|
T −ν−2
+2Rf
ωτ −1 (0) Yτ
ωτ ∗ −1 (0) Yτ ∗
0
+2Eν−1 u(i)
³
(i)
(A19)





 Yν




ν+1≤τ <τ ∗ ≤T
´
0
RfT −ν aν−1 (0)
Ã
!
T
³
³
´
´
X
00
(i)
(i)0
(i)|
+2Eν−1 u(i) lim WT
lim WT
RfT −ν−1
ωτ −1 (0) Yτ Yν
lim WT
ε7→0
ε7→0
³
+2Eν−1 u
+2Eν−1 u
(i)
(i)0
00
³
³
ε7→0
(i)
lim WT
ε7→0
(i)
lim WT
ε7→0
´
´
Ã
(i)0
lim WT
ε7→0
RfT −ν−1
´
τ =ν+1
RfT −ν aν−1
T
X
(0)
!
(i)|
ωτ −1
(0) Yτ
aν−1 (0) .
τ =ν+1
Notice that:
T
³
X
(i)
lim
ε7→0
∂WT
∂ε
=
Wt RfT −t−1
τ =t+1

(i)
∂ 2 WT
lim
ε7→0
∂2ε
=




´
(i)|
ωτ −1 (0) Yτ
2
T
P
τ =t+1
(i)
(i)|
RfT −t−1 Wt ωτ −1 (0) aτ −1 (0) + 2
+2
T
P
t+1≤τ <τ ∗ ≤T
(i)
RfT −t−2 Wt
30
³
(i)|
T
P
τ =t+1
τ 6=ν
ωτ −1 (0) Yτ
(i)
(i)0 |
RfT −t−1 Wt ωτ −1 (0) Yτ
´³
(i)|
´
ωτ ∗ −1 (0) Yτ ∗ .





(i)
(i)
I then replace analytical expressions of ωτ −1 (0), aτ −1 (0) and ωτ ∗ −1 (0) in expression above and deduce:
T
¢
℘i X T −τ ¡ |
ω τ −1 Yτ
R
℘ τ =t+1 f

T
T
¢
¡
P
P
2(T −τ )
(i) (i)0 |
RfT −t−1 Wt ωτ −1 (0) Yτ
2 ℘i
R
V arτ −1 ω |τ −1 Yτ + 2
 ℘2 τ =t+1 f
τ =t+1

τ 6=ν

T
(i)

2
¡ |
¢¡
¢
P
T −τ ∗ +t−τ ℘i /Wt
+2
Rf
ω τ −1 Yτ ω |τ ∗ −1 Yτ ∗
℘2
(i)
lim
ε7→0
∂WT
∂ε
=
(i)
lim
ε7→0
∂ 2 WT
∂2ε
=



.

t+1≤τ <τ ∗ ≤T
I replace the two equations above in Equation (A19), use the definition of preference
in Equations (5) and (6) to get
h
³
´i−1
³
´
00
(i)
(i)
u(i) lim WT
Hεε ωT −1 , 0
ε7→0
µ
¶
´2
ρi ³
(i)0
(i)00
= Eν−1 −2
lim WT
+ lim WT
RfT −ν Yν
ε7→0
℘i ε7→0

T
P
(i)|
2RfT −ν−1
ωτ −1 (0) aτ −1 (0)

τ =ν+1


T
P
(i)0 |

+2RfT −ν−1
ωτ −1 (0) Yτ
−Eν−1 ℘i 

τ =ν+1

´³
´
³
T
P

(i)|
(i)|
T −ν−2
ωτ ∗ −1 (0) Yτ ∗
+2Rf
ωτ −1 (0) Yτ
ν+1≤τ <τ ∗ ≤T
0
−2Eν−1 ℘i RfT −ν aν−1
(0) +
Ã
(i)0
2Eν−1 lim WT
ε7→0
RfT −ν−1
(0) − 2Eν−1 τi
(A20)





 Yν



!
T
X
(i)|
ωτ −1
(0) Yτ
τ =ν+1
Ã
(i)0
+2Eν−1 lim WT RfT −ν aν−1
ε7→0
parameters as specified
RfT −ν−1
T
X
(i)|
ωτ −1
Yν
!
(0) Yτ
aν−1 (0)
τ =ν+1
which simplifies to
³
´
(i)
Hεε ωT −1 , 0
ε7→0
´
³¡
¢2
ρi ℘i 3(T −ν)
−2 2 Rf
Covν−1 ω |τ −1 Yτ , Yν
℘
¸
·
T
X
¡
¡
¢
¢
2℘i (1 − ρi ) 2(T −τ )+(T −ν)
+
R
Covν−1 V arτ −1 ω |τ −1 Yτ , Yν
f
2
℘
τ =ν+1


(2℘i −4ρi ℘i ) (T −τ )+2(T −ν)
ν−1
X
R
¡
¢
¡¡
¢
¢
2
f
℘

 ω |τ −1 Yτ Covν−1 ω |ν−1 Yν , Yν
(i)
2(T −ν)+t−τ ℘2
i /Wt
+2Rf
τ =t+1
℘2
h
=
00
u(i)
³
(i)
lim WT
´i−1
0
−2Eν−1 ℘i RfT −ν aν−1 (0)
I replace expression above and Equation (A15) in Equation (A18) to deduce
(i)
(i)0
lim Wν−1 ων−1 (0)
ε7→0
=
³¡
´
¢2
ρi ℘i (T −ν) −1
|
Σ
R
Cov
ω
Y
,
Y
ν−1
ν
ν
ν−1
ν−1
℘2 f
T
X
¢
¢
¡
¡
℘i (ρi − 1) 2(T −τ )−(T −ν) −1
Σν−1 Covν−1 V arτ −1 ω |τ −1 Yτ , Yν
Rf
2
℘
τ =ν+1
Ã
!
ν−1
(i)
X
¢
¡¡ |
¢
¢
¡ |
℘i (2ρi − 1) T −τ
℘2i /Wt
t−τ
+
Rf
−
Rf
Σ−1
ω ν−1 Yν , Yν
ν−1 ω τ −1 Yτ Covν−1
2
2
℘
℘
τ =t+1
+
−(T −ν)
+℘i Rf
0
Σ−1
ν−1 aν−1 (0)
31
which simplifies to:
(i)
(i)0
lim Wν−1 ων−1 (0)
ε7→0
(i)
π2 Σ−1
ν−1 Covν−1
=
T
X
+
³¡
ω |ν−1 Yν
¢2
´
, Yν
(A21)
¡
¡ |
¢
¢
(i)
π1τ Σ−1
ν−1 Covν−1 V arτ −1 ω τ −1 Yτ , Yν
τ =ν+1
ν−1
X
+
¢
¡¡ |
¢
¢
(i) ¡
ω ν−1 Yν , Yν
π0τ ω |τ −1 Yτ Σ−1
ν−1 Covν−1
τ =t+1
−(T −ν)
+℘i Rf
0
Σ−1
ν−1 aν−1 (0)
with:
(i)
(i)
π2 =
℘i (ρi − 1) T −ν 2(ν−τ ) (i)
℘i (2ρi − 1) T −τ
℘2 /W
ρi τi T −ν (i)
, π1τ =
Rf Rf
, π0τ =
Rf
− i 2t
2 Rf
2
2
℘
℘
℘
℘
1
.
Rfτ −t
I assume that the market clearing conditions hold, in the neighborhood
N
³
´ , and I differentiate the market
(i)
clearing conditions with respect to ε and evaluate the result at ων−1 (0) ,0 , then get:
lim
ε7→0
I
X
(i)0
(i)
Wν−1 ων−1 (ε) +
i=1
I
X
i=1
Notice that:
(i)
(i)0
lim Wν−1 ων−1 (ε) = 0
(A22)
ε7→0
ν−1
¢
℘i X ν−1−τ ¡ |
ω τ −1 Yτ
R
℘ τ =t+1 f
(i)0
lim Wν−1 =
ε7→0
I take the sum of Equation (A21) for i = 1, ..., I and use the market clearing conditions (A22) to get
³¡
´
¢2
0
aν−1 (0) = δ 2 Covν−1 ω |ν−1 Yν , Yν
+
T
X
(A23)
¡
¡
¢
¢
δ 1τ Covν−1 V arτ −1 ω |τ −1 Yτ , Yν
τ =ν+1
+
ν−1
X
¡
¢
¡¡
¢
¢
δ 0τ ω |τ −1 Yτ Covν−1 ω |ν−1 Yν , Yν
τ =t+1
with
δ2 = −
(1 − ρ) 2(T −ν) 2(ν−τ )
(1 − 2ρ) T −τ 2(T −ν)
ρ 2(T −ν)
R
Rf
Rf Rf
, δ 1τ =
Rf
.
, δ 0τ =
℘2 f
℘2
℘2
Now, I replace Equation (A23) in Equation (A21) and get
´
³¡
¢2
(i)
(i)0
(i)
lim Wν−1 ων−1 (0) = ψ2 Σ−1
ω |ν−1 Yν , Yν
ν−1 Covν−1
(A24)
ε7→0
+
T
X
¢
¢
¡
¡ |
(i)
ψ1τ Σ−1
ν−1 Covν−1 V arτ −1 ω τ −1 Yτ , Yν
τ =ν+1
+
ν−1
X
¢
¢
¢
¡¡ |
(i) ¡
ψ0τ ω |τ −1 Yτ Σ−1
ω ν−1 Yν , Yν
ν−1 Covν−1
τ =t+1
with:
(i)
ψ2 =
(i)
(ρi − ρ) ℘i T −ν (i)
2℘i (ρi − ρ) T −τ
℘i (ρi − ρ) T −ν 2(ν−τ ) (i)
℘2 /W
, ψ0τ =
Rf , ψ1τ =
R f Rf
Rf
− i 2t
2
2
2
℘
℘
℘
℘
0
1
Rfτ −t
In the following proofs, I will use the analytical expressions aν−1 (0) , aν−1 (0), derived in Equations (A11) and
(A23).
32
Appendix B:
Proof of Propositions
0
Proof of Proposition 1 I first use analytical expressions of aν−1 (0) and aν−1 (0) derived in Equations (A11) and
(A23) respectively to derive the risk premium on the risky assets from time ν − 1 to ν (see the return decomposition
(3)). This premium is
Eν−1 Rν − Rf l
=
ε2 aν−1 (ε)
´
³
0
ε2 aν−1 (0) + εaν−1 (0)
=
T
X
¡
¢
B 2 Covν−1 (RM ν − Eν−1 RM )2 , Rν +
B 1τ Covν−1 (V arτ −1 (RM τ ) , Rν )
=
τ =ν+1
+B 0 Covν−1 (RM ν , Rν )
with:
B2 = −
ν−1
X
(1 − ρ) [2(T −τ )]
ρ 2[T −ν]
1 T −ν
,
B
=
,
B
=
R
+
δ 0τ (RM τ − Eτ −1 RM τ )
R
R
1℘
0τ
f
f
2 f
2
℘
℘
℘
τ =t+1
[(T −τ )+2(T −ν)]
with δ 0τ = (1−2ρ)
Rf
and RM τ = ω |τ −1 Rτ . Now, I use the risk premium result to derive the pricing
℘2
kernel. The pricing kernel for the time period [t, T ] is
mTt =
T
Y
mν−1,ν .
ν=t+1
where mν−1,ν represents the pricing kernel for the period [ν − 1, ν]. By definition, the risk premium on the risky
asset for the time period [ν − 1, ν] is given by
Eν−1 Rν − Rf l = −Rf Covν−1 (mν−1,ν , Rν ) .
(B1)
I identify the risk premium in Equation (B1) with the analytical expression of asset risk premium and deduce the
analytical expression of the pricing kernel which is of the form.
mν−1,ν =
where:
D1 =
T
−1
X
£ 2
1
2 ¤
2
2
+ D0 rM ν + D1 rM
D2τ [Eν σM
ν − Eν−1 rM ν +
τ − Eν−1 σM τ ]
Rf
τ =ν
ν−1
X
(ρ − 1) [2(T −τ )−1]
ρ [2(T −ν)−1]
1 [T −ν−1]
, D2τ =
Rf
1τ ≤T −1 , D0 = − Rf
+
D τ ν rM τ
2 Rf
2
℘
℘
℘
τ =t+1
and
Dτ ν =
(−1 + 2ρ) [(T −τ )+2(T −ν)−1]
Rf
℘2
(B2)
(B3)
(B4)
Since Rf ≈ 1, the coefficients, D0 , D1 , and D2 can be written as:
D1 =
ν−1
(ρ − 1)
(2ρ − 1) X
1
ρ
D
=
D
=
−
+
rM τ
,
1
,
2τ
0
τ
≤T
−1
℘
℘2
℘2
℘2
τ =t+1
(B5)
Proof of the Analytical Expression of the Volatility Premium I denote rM t+1 = RM t+1 − Rf and note
33
that
∗2
σM
t
=
=
=
=
=
=
=
=
=
Thus
Et∗ (RM t+1 − Rf )2
mt,t+1 2
Et
rM t+1
Et mt,t+1
¶
µ
mt,t+1
2
2
Covt
, rM
+ Et rM
t+1
t+1
Et mt,t+1
¡
¢
2
2
Rf Covt mt,t+1 , rM
t+1 + Et rM t+1
¢
¡
2
2
Rf Covt mt,t+1 , rM
t+1 + Et (RM t+1 − Rf )
¡
¢
2
2
Rf Covt mt,t+1 , rM
t+1 + Et (RM t+1 − Et RM t+1 + Et RM t+1 − Rf )
¡
¢
2
Rf Covt mt,t+1 , rM t+1 + Et (RM t+1 − Et RM t+1 + Et RM t+1 − Rf )2
¡
¢
2
2
2
Rf Covt mt,t+1 , rM
t+1 + Et (RM t+1 − Et RM t+1 ) + (Et RM t+1 − Rf )
¡
¢
2
2
2
Rf Covt mt,t+1 , rM
t+1 + σM t + (Et RM t+1 − Rf )
¡
2
2¢
2
∗2
σM
t − σM t = − (Et RM t+1 − Rf ) − Rf Covt mt,t+1 , (RM t+1 − Rf )
I then replace the excess market squared return by its expression
(RM t+1 − Rf )2 = (RM t+1 − Et RM t+1 )2 + (Et RM t+1 − Rf )2 + 2(RM t+1 − Et RM t+1 )(Et RM t+1 − Rf ),
and the pricing kernel mt,t+1 by its expression to get the final result.
Proof of Proposition 2 I Note that the covariance of the pricing kernel and the square of the idiosyncratic
shock is
mt,t+1 2
εkt+1 − Et mt,t+1 Et ε2kt+1 .
(B6)
Covt (mt,t+1 , ε2kt+1 ) = [Et mt,t+1 ]Et
Et mt,t+1
m
Since Et mt,t+1 = Rf−1 and Et Et mt,t+1
[X] = Et∗ [X] where the operator Et∗ [.] denotes the expectation taken under
t,t+1
the risk neutral measure, I simplify (B6) and get:
∗2
2
Rf Covt (mt,t+1 , ε2kt+1 ) = Et∗ ε2kt+1 − Et ε2kt+1 = σεkt
− σεkt
(B7)
I then replace mt,t+1 in (B7) and get the final result.
Lemma B1: (Stein’s Lemma) If X an Y are jointly normally distributed, then Cov(g(X), Y ) = E(g(X)0 )Cov(X, Y ).
If the idiosyncratic shock and the factor ft+1 are jointly normally distributed, Using Stein Lemma, it can be shown
that
Covt (ft+1 , ε2kt+1 ) = 2Et εkt+1 Covt (ft+1 , εkt+1 ).
(B8)
Since Et εkt+1 = 0, it follows that Covt (ft+1 , ε2kt+1 ) = 0.
Proof of the relationship between the nonsystematic coskewness and the expected excess return
I notice that in a single factor model, the risk premium on the asset’s k is
αkt
=
Et Rkt+1 − Rf
=
−Rf covt (mt,t+1 , Rkt+1 )
¶
µ
dt
1
−
(RM t+1 − Et RM t+1 ) , Rkt+1
−Rf covt
Rf
Rf
=
=
2
dt σM
t βkt
where
βkt =
Let
covt (Rkt+1 , RM t+1 )
2
σM
t
ISkt = Et ε3kt+1
denote the idiosyncratic skewness with
εkt+1 = Rkt+1 − Rf − αkt − βkt (RM t+1 − Et RM t+1 )
34
Therefore,
∂γkt
∂αkt
=
=
=
=
=
¡
¢
covt ε2kt+1 , rM t+1
σ2
³ Mt
´
∂εkt+1
covt εkt+1 ∂α
, rM t+1
kt
2
2
σM
t
³
´
∂εkt+1
covt εkt+1 ∂α
, rM t+1
kt
2
2
σM
¡t
¢
2
∂βkt+1 covt εkt+1 , rM t+1
−2
2
∂αkt
σM
t
¡
¢
2
2 covt εkt+1 , rM t+1
−
2
2
dt σM
σM
t
t
Since
rM t+1 = RM t+1 − Et RM t+1 =
n
X
$jt εjt+1
j=1
where $jt represents the weight of asset j in the market portfolio. Therefore, under the restrictions covt (εjt+1 εkt+1 , rM t+1 ) =
0 for j 6= k , we have
∂γkt
∂αkt
=
=
∂γkt
∂αkt
=
n
covt (εkt+1 , rM t+1 εjt+1 )
2 X
$jt
2
2
dt σM
σM
t j=1
t
"
#
n
X $j covt (εjt+1 εkt+1 , rM t+1 )
2
−
$k γ k +
2
2
dt σM
σM
t
t
j=1
−
−
2
$k γ k
2
dt σM
t
Proof of the Analytical Expression of Non-Systematic Coskewness with GARCH Specification I
take the expectation of (35) under the physical measure, and the expectation of (35) under the risk neutral measure
and show that the spread of these expected values equals
δ1 2
)σεkt (1 − V art∗ [zt+1 ]).(B9)
2
2
2
Et [σεkt+1
] − Et∗ [σεkt+1
] = (α1 +
I use the idiosyncratic risk εkt+1 = σεkt zkt+1 and express the non-systematic volatility premium as
2
∗2
2
σεkt
− σεkt
= (1 − V art∗ [zt+1 ])σεkt
.
(B10)
I recover 1 − V art∗ [zt+1 ] from (B9) and replace the result in (B10) to get
2
∗2
σεkt
− σεkt
=
2
2
Et [σεkt+1
] − Et∗ [σεkt+1
]
δ1
(α1 + 2 )
I note that
2
2
2
Et [σεkt+1
] − Et∗ [σεkt+1
] = −Rf Covt (mt,t+1 , σεkt+1
)=
1
2
Covt (rM t+1 , σεkt+1
)
℘
(B11)
(B12)
and I replace this last expression in (B11) to get
2
∗2
σεkt
− σεkt
=
2
Covt (rM t+1 , σkt+1
)
℘(α1 +
δ1
)
2
(B13)
From equation (??), I can also write the non-systematic volatility risk premium as.
2
∗2
σεkt
− σεkt
=
I equate equations (B13) and (B14) to get the final result.
35
1 2
σM t γkt
℘
(B14)
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37
Table I: Preference Parameters and Implied Prices of Risk Using Industry Portfolio Returns
Table I presents results of GMM tests of the Euler equation condition, EmT
t Rt+1 Rf = 1 using the pricing kernel derived in
Proposition 1 when the investment horizon h = T − t = 2. I estimate the preference parameters by using the Hansen and
0
Jagannathan (1997) weighting matrix ERt+1 Rt+1 . Column (1) presents the mean of the pricing kernel, Columns (2) and (3)
present the risk aversion and skewness preference respectively. Column (4) presents the HansenJagannathan distance measure
with p-values for the test of model specification in parentheses. Columns (5) - (7) present the annualized price of market,
coskewness and market volatility risk, using the estimated preference parameters. The P -values for tests of the coefficients
appear in parentheses. The set of returns I use in my estimations are those of 30 industry-sorted portfolios augmented by the
return on a one-month Treasury bill, covering the sample periods 01/1986-12/2000, 01/1996-12/2006 and 01/1986-12/2006.
For the market portfolio, I use the value-weighted NYSE/AMEX/NASDAQ index, also known as the value-weighted index of
the Center for Research in Security Prices (CRSP). As my proxy for the volatility of the market return, I use the Chicago
Board Options Exchange (CBOE)s VXO, the VIX implied volatilities and the realized volatility RV, respectively. In Panel A,
I present the results when I use the VXO. Panel B presents the results when I use the VIX. Panel C presents the results when
I use the realized volatility RV.
Panel A: Market Volatility: VXO
Subperiod: 01/1986-12/2000
Coefficient
P-value
Subperiod: 01/1996-12/2006
Coefficient
P-value
Subperiod: 01/1986-12/2006
Coefficient
P-value
1
Rf
1
τ
ρ
HJ Dist
λM KT (%)
λSKD (%)
λV OL (%)
0.996
0.000
3.989
0.044
1.096
0.001
0.150
0.045
9.552
-1.861
-0.384
0.997
0.000
2.756
0.197
2.254
0.345
0.194
0.000
6.669
-1.857
-0.741
0.996
0.000
3.439
0.033
1.080
0.017
0.094
0.081
7.934
-1.290
-0.192
1
Rf
1
τ
ρ
HJ Dist
λM KT (%)
λSKD (%)
λV OL (%)
0.996
0.000
4.748
0.067
1.583
0.023
0.248
0.012
9.671
-2.857
-1.207
0.997
0.000
2.656
0.219
2.207
0.403
0.196
0.000
6.428
-1.689
-0.621
0.997
0.000
3.782
0.053
1.705
0.082
0.119
0.003
7.761
-2.066
-0.906
1
Rf
1
τ
ρ
HJ Dist
λM KT (%)
λSKD (%)
λV OL (%)
0.996
0.000
4.753
0.049
1.105
0.000
0.272
0.036
9.178
-1.930
-0.978
0.997
0.000
2.504
0.239
1.183
0.003
0.202
0.000
5.629
-0.721
-1.320
0.997
0.000
3.236
0.098
1.175
0.000
0.124
0.060
6.277
-0.958
-1.380
Panel B: Market Volatility: VIX
Subperiod: 01/1990-12/2000
Coefficient
P-value
Subperiod: 01/1996-12/2006
Coefficient
P-value
Subperiod: 01/1990-12/2006
Coefficient
P-value
Panel C: Market Volatility: RV
Subperiod: 01/1990-12/2000
Coefficient
P-value
Subperiod: 01/1996-12/2006
Coefficient
P-value
Subperiod: 01/1990-12/2006
Coefficient
P-value
38
Table II: Preference Parameters and Implied Prices of Risk Using Industry Portfolio Returns (Robustness to
Size, Book-to-Value and Momentum Factors):
Table II presents results of GMM tests of the Euler equation condition, EmT
t Rt+1 Rf = 1 using the pricing kernel derived in
Proposition 1 when the investment horizon h = T − t = 2 augmented with Fama and French (1993) size and book-to-market
0
factors. I estimate the preference parameters by using the Hansen and Jagannathan (1997) weighting matrix ERt+1 Rt+1 .
Column (1) presents the mean of the pricing kernel, Columns (2) and (3) present the risk aversion and skewness preference
respectively. Column (4) presents the HansenJagannathan distance measure with p-values for the test of model specification
in parentheses. Columns (5) - (7) present the annualized price of market, coskewness and market volatility risk, using the
estimated preference parameters. The P -values for tests of the coefficients appear in parentheses. The set of returns I use in
my estimations are those of 30 industry-sorted portfolios augmented by the return on a one-month Treasury bill, covering the
sample periods 01/1986-12/2000, 01/1996-12/2006 and 01/1986-12/2006. For the market portfolio, I use the value-weighted
NYSE/AMEX/NASDAQ index, also known as the value-weighted index of the Center for Research in Security Prices (CRSP).
As my proxy for the volatility of the market return, I use the Chicago Board Options Exchange (CBOE)s VXO, the VIX implied
volatilities and the Realized Volatility RV, respectively. In Panel A, I present the results when I use the VXO. Panel B presents
the results when I use the VIX. Panel C presents the results when I use the realized volatility RV.
Panel A: Market Volatility: VXO
Subperiod: 01/1986-12/2000
Coefficient
P-value
Subperiod: 01/1996-12/2006
Coefficient
P-value
Subperiod: 01/1986-12/2006
Coefficient
P-value
1
Rf
1
τ
ρ
HJ Dist
λM KT (%)
λSKD (%)
λV OL (%)
0.996
0.000
4.211
0.032
1.095
0.002
0.116
0.162
10.082
-2.072
-0.424
0.997
0.000
4.009
0.169
1.827
0.182
0.175
0.002
9.701
-3.185
-1.033
0.996
0.000
4.318
0.023
1.050
0.001
0.085
0.081
9.961
-1.976
-0.187
1
Rf
1
τ
ρ
HJ Dist
λM KT (%)
λSKD (%)
λV OL (%)
0.996
0.000
3.403
0.268
1.741
0.270
0.199
0.037
6.931
-1.689
-0.896
0.997
0.000
3.755
0.206
1.913
0.264
0.177
0.003
9.086
-2.926
-0.939
0.997
0.000
4.608
0.061
1.482
0.027
0.113
0.006
9.457
-2.667
-0.920
1
Rf
1
τ
ρ
HJ Dist
λM KT (%)
λSKD (%)
λV OL (%)
0.996
0.000
3.990
0.201
1.118
0.000
0.208
0.016
7.705
-1.376
-0.774
0.997
0.000
3.636
0.286
1.138
0.000
0.172
0.003
8.174
-1.461
-1.487
0.997
0.000
3.623
0.195
1.215
0.001
0.097
0.076
7.027
-1.241
-1.519
Panel B: Market Volatility: VIX
Subperiod: 01/1990-12/2000
Coefficient
P-value
Subperiod: 01/1996-12/2006
Coefficient
P-value
Subperiod: 01/1990-12/2006
Coefficient
P-value
Panel C: Market Volatility: RV
Subperiod: 01/1990-12/2000
Coefficient
P-value
Subperiod: 01/1996-12/2006
Coefficient
P-value
Subperiod: 01/1990-12/2006
Coefficient
P-value
39
Table III: Descriptive Statistics on Dow 30 Stocks:
This table presents summary statistics for the monthly returns on Dow 30 Stocks. Maximum, minimum, mean, standard
deviation (Std), skewness, and kurtosis are reported for each stock. The descriptive statistics are computed for the sample
period from Januray 1990 to December 2006.
Dow Jones Stocks
Microsoft
Honeywell
AT&T Inc
Coca Cola
E.I. DuPont de Nemours
Exxon Mobil
General Electric
General Motors
International Business Machines
Altria (was Philip Morris)
United Technologies
Procter and Gamble
Caterpillar
Boeing
Pfizer
Johnson & Johnson
3M Corporation
Merck
Alcoa
Walt Disney Co.
Hewlett-Packard
McDonalds
JP Morgan Chase
Wal-Mart Stores
Intel Corp
Verizon Communications
Home Depot
American Int’l Group
CitiGroup
American Express IBM
International Business Machines
Minimum
Maximum
Mean
Std
Skewness
Kurtosis
-0.3435
-0.3840
-0.1876
-0.1910
-0.1699
-0.1165
-0.1765
-0.2403
-0.2619
-0.2656
-0.3202
-0.3570
-0.2146
-0.3457
-0.1707
-0.1601
-0.1578
-0.2577
-0.2387
-0.2678
-0.3199
-0.2567
-0.3468
-0.2080
-0.4449
-0.2099
-0.2059
-0.2310
-0.3401
-0.2933
0.4078
0.5105
0.2900
0.2228
0.2174
0.2322
0.1924
0.2766
0.3538
0.3427
0.2461
0.2509
0.4079
0.1949
0.2655
0.1881
0.2580
0.2276
0.5114
0.2415
0.3539
0.1826
0.3257
0.2643
0.3382
0.3901
0.3023
0.2387
0.2608
0.2031
0.0251
0.0141
0.0097
0.0115
0.0091
0.0127
0.0134
0.0072
0.0123
0.0164
0.0154
0.0135
0.0157
0.0120
0.0151
0.0142
0.0108
0.0114
0.0115
0.0100
0.0176
0.0114
0.0161
0.0136
0.0223
0.0076
0.0193
0.0129
0.0214
0.0139
0.1024
0.0901
0.0717
0.0660
0.0673
0.0465
0.0626
0.0951
0.0890
0.0830
0.0711
0.0631
0.0836
0.0788
0.0735
0.0629
0.0581
0.0773
0.0907
0.0775
0.1099
0.0697
0.1004
0.0730
0.1213
0.0721
0.0849
0.0665
0.0889
0.0754
0.3978
-0.0367
0.2064
-0.2103
0.1367
0.6183
0.1297
0.1984
0.3456
-0.2708
-0.6543
-0.7790
0.4065
-0.5865
0.1404
0.0878
0.4040
-0.0737
0.7795
-0.0686
0.0874
-0.2540
-0.1720
0.1671
-0.3005
0.8210
0.2564
0.0223
-0.1032
-0.8057
4.6155
9.3331
4.1836
4.0537
2.8528
5.4267
3.5668
3.1706
4.3209
5.0895
6.1925
8.7933
4.7004
4.4729
2.9849
3.1813
4.8393
3.3652
6.9529
3.9232
3.6069
3.5148
4.8161
3.4579
3.6254
6.7551
3.5188
4.2054
4.3008
4.6244
40
Table IV: Preference Parameters and Implied Prices of Risk Using the 30 Dow Jones Returns:
Table IV presents results of GMM tests of the Euler equation condition, EmT
t Rt+1 Rf = 1 using the pricing kernel derived in
Proposition 1 when the investment horizon h = T − t = 2 augmented with Fama and French (1993) size and book-to-market
0
factors. I estimate the preference parameters by using the Hansen and Jagannathan (1997) weighting matrix ERt+1 Rt+1 .
Column (1) presents the mean of the pricing kernel, Columns (2) and (3) present the risk aversion and skewness preference
respectively. Column (4) presents the HansenJagannathan distance measure with p-values for the test of model specification
in parentheses. Columns (5) - (7) present the annualized price of market, coskewness and market volatility risk, using the
estimated preference parameters. The P -values for tests of the coefficients appear in parentheses. The set of returns I use in
my estimations are those of 30 Dow Jones returns augmented by the return on a one-month Treasury bill, covering the sample
period 01/1990-12/2006. For the market portfolio, I use the value-weighted NYSE/AMEX/NASDAQ index, also known as
the value-weighted index of the Center for Research in Security Prices (CRSP). As my proxy for the volatility of the market
return, I use the Chicago Board Options Exchange (CBOE)s VXO, the VIX implied volatilities and the Realized Volatility RV,
respectively. In Panel A, I present the results when I use the VXO, the VIX and the realized volatility RV. Panel B presents
the results when I control for the Fama and French and the momentum factors.
1
Rf
1
τ
ρ
HJ Dist
λM KT (%)
λSKD (%)
λV OL (%)
Coefficient
P-value
VIX
0.997
0.000
3.921
0.039
1.603
0.045
0.121
0.003
8.047
-2.088
-0.897
Coefficient
P-value
RV
0.997
0.000
3.782
0.053
1.705
0.082
0.120
0.003
7.761
-2.066
-0.906
Coefficient
P-value
0.997
0.000
5.750
0.008
1.003
0.000
0.101
0.319
11.154
-2.582
-0.148
Coefficient
P-value
VIX
0.997
0.000
4.771
0.046
1.394
0.008
0.115
0.004
9.791
-2.689
-0.869
Coefficient
P-value
RV
0.997
0.000
4.608
0.061
1.482
0.028
0.113
0.006
9.457
-2.667
-0.920
Coefficient
P-value
0.9967
0.000
5.4074
0.0650
1.0277
0.0000
0.0913
0.4098
10.4890
-2.3396
-1.2963
Panel A
VXO
Panel B
VXO
41
42
1.069
(3.888)
1.049
(3.897)
1.036
(3.920)
1.062
(3.884)
0.586
[3.041]
0.566
[2.917]
0.554
[2.851]
0.579
[3.013]
Mean
Std Dev
Mean
Std Dev
Mean
Std Dev
Mean
Std Dev
Panel B
Alpha
t-stat
Alpha
t-stat
Alpha
t-stat
Alpha
t-stat
Panel A
1Low
0.489
[2.162]
0.594
[2.626]
0.550
[2.439]
0.472
[2.099]
1.001
(4.671)
1.100
(4.691)
1.056
(4.703)
0.983
(4.665)
2
0.641
[2.418]
0.590
[2.233]
0.597
[2.249]
0.655
[2.471]
1.163
(5.493)
1.120
(5.531)
1.125
(5.492)
1.180
(5.476)
3
0.663
[2.278]
0.580
[1.996]
0.626
[2.145]
0.639
[2.218]
1.187
(6.115)
1.098
(6.085)
1.141
(6.133)
1.165
(6.091)
4
6
0.553
[1.643]
CAPM
0.373
[0.946]
FF
0.556
0.355
[1.630]
[0.894]
FF-M
0.517
0.341
[1.497]
[0.875]
HS
0.496
0.381
[1.495]
[0.965]
CAPM
0.938
(7.933)
FF
1.100
0.922
(7.067)
(7.870)
FF-M
1.068
0.910
(7.107)
( 7.788)
HS
1.037
0.945
(6.951)
(7.928)
1.087
(6.992)
5
0.233
[0.579]
0.338
[0.831]
0.272
[0.678]
0.210
[0.526]
0.797
(8.365)
0.901
(8.407)
0.835
(8.335)
0.775
(8.366)
7
0.066
[0.160]
0.026
[0.063]
0.041
[0.100]
0.066
[0.159]
0.645
(8.758)
0.597
(8.707)
0.612
(8.792)
0.653
(8.743)
8
-0.293
[-0.644]
-0.291
[-0.647]
-0.275
[-0.616]
-0.265
[-0.583]
0.293
(9.668)
0.321
(9.522)
0.333
(9.559)
0.031
(9.697)
9
-0.878
[-1.730]
-0.822
[-1.642]
-0.842
[ -1.686]
-0.880
[-1.728]
-0.217
(10.628)
-0.157
(10.441)
-0.178
(10.457)
-0.222
(10.621)
10 High
-1.457
[-3.146]
-1.376
[-3.016]
-1.409
[-3.081]
-1.466
[-3.153]
-1.278
[-2.624]
-1.193
[-2.480]
-1.227
[-2.552]
-1.290
[-2.642]
10-1
Table V:Portfolios Sorted on Idiosyncratic Volatility.
I sort stocks into decile portfolios based on their idiosyncratic volatility using only NYSE/AMEX/NASDAQ industrial firms. I form value-weighted decile portfolios every
month by sorting stocks based on idiosyncratic volatility relative to different models. To compute the idiosyncratic volatility, I use the CAPM model, the Fama and French
(1993) model, the Fama and French (1993) model augmented with the Jegadeesh and Titman (1993) momentum factor, and the Harvey and Siddique (2000) coskewness model.
Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) idiosyncratic volatility risk. The column titled “10-1”refers to the difference in expected returns between
portfolio 10 and portfolio 1. In Panel A, I measure the statistics in the columns labeled Mean and Std Dev (Standard Deviation) in monthly percentage terms. I use total, not
excess, returns. Std Devs are in parentheses. Robust Newey-West (1987) t-statistics (t-stat) are in brackets. Panel B reports each of the model alphas. Robust Newey-West
(1987) t-statistics appear in square brackets. The column titled 10-1 refers to the difference in alphas between portfolio 10 and portfolio 1. The sample period is from January
1971 to December 2006.
Table VI: Decile Portfolios Sorted on Non-Systematic Coskewness.
I use NYSE/AMEX/NASDAQ industrial firms and form value-weighted decile portfolios every month by sorting stocks based
on non-systematic coskewness relative to the CAPM model. My portfolios contain stocks in percentiles 0-10, 10-20, 20-30, 30-40,
40-50, 50-60, 60-70, 70-80, 80-90, and 90-100. At the end of each month, I split the sample into two groups, stocks with positive
non-systematic coskewness and stocks with negative non-systematic coskewness. Within each group, I sort stocks into decile
portfolios and then form value-weighted decile portfolios every month by sorting these stocks based on their non-systematic
coskewness. Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) non-systematic coskewness. In Panel A, I
measure the statistics in the columns labeled Mean and Std Dev (standard deviation) in monthly percentage terms. I use total,
not excess, returns. Standard errors appear in parentheses. Robust Newey-West (1987) t-statistics appear in square brackets.
The column titled 10-1 refers to the difference in expected return between portfolio 10 and portfolio 1. Panel B reports the
coefficients of the CAPM regression:
rp = α + βM [RM − Rf ] + η,
(B.15)
and Fama and French (1993) regressions:
rp = α + βM [RM − Rf ] + βSM B rSM B + βHM L rHM L + η,
(B.16)
when γk > 0. Panel C reports the regression coefficients when γk < 0. The sample period is from January 1971 to December
2006. I also report the R-Square of the regressions.
1Low
2
3
4
5
Mean
Std Dev
1.07
(4.16)
1.02
(4.47)
1.01
(4.70)
1.14
(5.09)
1.09
(5.72)
Mean
Std Dev
0.32
(8.61)
0.95
(7.38)
0.78
(6.68)
0.79
(6.46)
6
7
8
9
10 High
10-1
1.07
(6.67)
1.11
(7.74)
1.04
(8.84)
0.62
(9.93)
-0.45
[-1.04]
1.25
(5.35)
1.02
(4.96)
1.01
(4.58)
1.13
(4.40)
0.81
[2.30]
0.54
[1.75]
0.09
[1.31]
0.10
0.56
[1.45]
0.12
[1.63]
0.23
0.49
[1.10]
0.14
[1.60]
0.25
-0.01
[-0.02]
0.28
[2.70]
1.32
-0.59
[-1.49]
0.28
[3.35]
2.18
0.56
[1.73]
0.06
[0.90]
0.05
[0.56]
-0.04
[-0.34]
-0.28
0.68
[1.70]
0.05
[0.56]
0.06
[0.57]
-0.20
[-1.48]
0.39
0.61
[1.36]
0.04
[0.34]
0.15
[1.13]
-0.23
[-1.27]
0.72
0.12
[0.23]
0.18
[1.59]
0.13
[0.81]
-0.23
[-0.95]
1.52
-0.50
[-1.05]
0.23
[2.31]
0.02
[0.16]
-0.16
[-0.66]
2.02
0.77
[3.05]
-0.01
[-0.24]
-0.22
0.50
[2.11]
0.08
[1.39]
0.31
0.49
[2.16]
0.07
[ 1.25]
0.27
0.66
[3.11]
-0.03
[-0.51]
-0.15
1.00
[3.10]
-0.36
[-4.56]
4.91
0.83
[3.14]
-0.07
[-1.09]
0.13
[2.04]
-0.10
[-1.44]
0.38
0.48
[2.02]
0.06
[1.01]
0.10
[1.39]
0.00
[0.05]
0.24
0.50
[2.09]
0.03
[0.48]
0.15
[2.25]
-0.04
[-0.58]
0.97
0.66
[2.98]
-0.05
[-0.88]
0.10
[1.83]
-0.01
[-0.22]
-0.05
0.96
[2.86]
-0.32
[-3.67]
-0.07
[-0.63]
0.08
[0.50]
4.69
Panel A
γk>0
1.11
(6.11)
γk<0
0.96
0.65
(6.10)
(5.85)
Panel B
α
t-stat
βM
t-stat
R2 (%)
0.58
[2.76]
-0.01
[-0.16]
-0.22
0.53
[2.44]
0.01
[0.15]
-0.23
0.51
[2.24]
0.03
[0.59]
-0.14
0.63
[2.55]
0.05
[0.96]
0.00
α
t-stat
βM
t-stat
βSM B
t-stat
βHM L
t-stat
R2 (%)
0.62
[2.85]
-0.05
[-0.87]
0.10
[1.96]
-0.07
[-1.01]
0.30
0.57
[2.56]
-0.04
[-0.60]
0.09
[1.65]
-0.09
[-1.37]
0.16
0.54
[2.31]
-0.02
[-0.29]
0.14
[2.07]
-0.07
[-0.99]
0.57
0.68
[2.66]
0.02
[0.38]
0.02
[0.32]
-0.09
[-1.22]
-0.19
γk>0
CAPM
0.55
0.59
[1.94]
[2.01]
0.10
0.07
[1.49]
[1.07]
0.34
0.02
FF
0.58
0.65
[1.97]
[2.09]
0.05
0.03
[0.71]
[0.39]
0.14
0.06
[1.97]
[0.60]
-0.06
-0.09
[-0.70]
[-0.91]
0.64
-0.15
Panel C
α
t-stat
βM
t-stat
R2 (%)
-0.34
[-0.87]
0.33
[4.01]
2.80
0.41
[1.17]
0.10
[1.14]
0.13
0.19
[0.64]
0.21
[3.19]
1.70
0.23
[0.72]
0.15
[2.24]
0.84
α
t-stat
βM
t-stat
βSM B
t-stat
βHM L
t-stat
R2 (%)
-0.30
[-0.76]
0.27
[2.94]
0.17
[1.38]
-0.09
[-0.51]
2.91
0.27
[0.72]
0.17
[2.00]
-0.01
[-0.13]
0.24
[1.54]
0.46
0.33
[1.07]
0.11
[1.67]
0.11
[1.20]
-0.23
[-2.05]
2.64
0.31
[1.01]
0.08
[1.16]
0.10
[1.35]
-0.15
[-1.44]
1.18
γk<0
CAPM
0.46
0.11
[1.61]
[0.37]
0.03
0.11
[0.51]
[1.70]
-0.18
0.45
FF
0.51
0.13
[1.73]
[0.43]
-0.03
0.08
[-0.51]
[1.21]
0.18
0.08
[2.41]
[0.94]
-0.10
-0.04
[-0.91]
[-0.48]
0.59
0.24
43
Table VII: Portfolios Sorted on Non-Systematic Coskewness in 10 Different Groups.
I use NYSE/AMEX/NASDAQ industrial firms and form value-weighted decile portfolios every month by sorting stocks based
on non-systematic coskewness relative to the CAPM model. My portfolios contain stocks in percentiles 0-5, 5-20, 20-30, 30-40,
40-50, 50-60, 60-70, 70-80, 80-95, and 95-100. My portfolio construction procedure pays greater attention to the tails of the
stock distribution. At the end of each month, I split the sample into two groups, stocks with positive non-systematic coskewness
and stocks with negative non-systematic coskewness. Within each group, I sort stocks into decile portfolios and then form valueweighted decile portfolios every month by sorting these stocks based on their non-systematic coskewness. Portfolio 1 (10) is
the portfolio of stocks with the lowest (highest) non-systematic coskewness. In Panel A, I measure the statistics in the columns
labeled Mean and Std Dev (standard deviation) in monthly percentage terms. I use total, not excess, returns. Standard errors
appear in parentheses. Robust Newey-West (1987) t-statistics appear in square brackets. The column titled 10-1 refers to the
difference in expected return between portfolio 10 and portfolio 1. Panel B reports the coefficients of the CAPM regression:
rp = α + βM [RM − Rf ] + η,
(B.17)
rp = α + βM [RM − Rf ] + βSM B rSM B + βHM L rHM L + η,
(B.18)
and Fama and French (1993) regressions:
when γk > 0. Panel C reports the regression coefficients when γk < 0. The sample period is from January 1971 to December
2006. I also report the R-Square of the regressions.
1Low
2
3
4
Mean
t-stat
1.17
(4.43)
1.00
(4.28)
1.01
(4.70)
1.14
(5.09)
Mean
t-stat
Panel B
-0.05
(10.00)
0.87
(7.28)
0.78
(6.68)
0.79
(6.46)
5
6
7
8
9
10 High
10-1
1.07
(6.67)
1.11
(7.74)
1.07
(9.13)
-0.05
(10.18)
-1.22
[-2.65]
1.25
(5.35)
1.02
(4.96)
1.02
(4.36)
1.24
(4.66)
1.29
[2.88]
0.54
[1.75]
0.09
[1.31]
0.10
0.56
[1.45]
0.12
[1.63]
0.23
0.49
[1.08]
0.18
[1.94]
0.54
-0.69
[-1.38]
0.31
[3.12]
1.60
-1.37
[-3.17]
0.30
[3.22]
2.19
0.56
[1.73]
0.06
[0.90]
0.05
[ 0.56]
-0.04
[-0.34]
-0.28
0.68
[1.70]
0.05
[0.56]
0.06
[0.57]
-0.20
[-1.48]
0.39
0.64
[1.31]
0.08
[0.67]
0.12
[0.92]
-0.26
[-1.23]
1.01
-0.58
[-1.11]
0.21
[1.71]
0.18
[1.06]
-0.21
[-1.09]
1.91
-1.30
[-2.82]
0.25
[2.12]
0.07
[0.47]
-0.13
[-0.68]
2.00
0.77
[3.05]
-0.01
[-0.24]
-0.22
0.50
[2.11]
0.08
[1.39]
0.31
0.52
[2.38]
0.03
[0.57]
-0.12
0.76
[ 3.33]
-0.02
[-0.29]
-0.21
1.49
[ 3.52]
-0.38
[-3.15]
3.61
0.83
[3.14]
-0.07
[-1.09]
0.13
[2.04]
-0.10
[-1.44]
0.38
0.48
[2.02]
0.06
[1.01]
0.10
[1.39]
0.00
[ 0.05]
0.24
0.53
[ 2.34]
0.00
[-0.02]
0.11
[1.81]
-0.04
[-0.71]
0.17
0.75
[3.18]
-0.04
[-0.68]
0.13
[1.84]
0.00
[0.02]
0.10
1.30
[2.94]
-0.32
[-2.51]
0.13
[0.77]
0.29
[ 1.58]
4.02
Panel A
α
t-stat
βM
t-stat
R2 (%)
0.68
[3.15]
0.00
[0.08]
-0.23
0.51
[2.43]
0.00
[-0.02]
-0.23
0.51
[2.24]
0.03
[0.59]
-0.14
0.63
[2.55]
0.05
[0.96]
0.00
α
t-stat
βM
t-stat
βSM B
t-stat
βHM L
t-stat
R2 (%)
Panel C
0.72
[3.22]
-0.04
[-0.68]
0.11
[1.78]
-0.09
[-1.04]
0.37
0.56
[2.56]
-0.05
[-0.81]
0.10
[1.92]
-0.09
[-1.42]
0.32
0.54
[2.31]
-0.02
[-0.29]
0.14
[2.07]
-0.07
[-0.99]
0.57
0.68
[2.66]
0.02
[0.38]
0.02
[0.32]
-0.09
[-1.22]
-0.19
α
t-stat
βM
t-stat
R2 (%)
-0.73
[-1.54]
0.37
[3.24]
2.52
0.31
[0.91]
0.14
[1.71]
0.51
0.19
[0.64]
0.21
[3.19]
1.70
0.23
[ 0.72]
0.15
[2.24]
0.84
α
t-stat
βM
t-stat
βSM B
t-stat
βHM L
t-stat
R2 (%)
-0.55
[-1.14]
0.28
[2.34]
0.00
[0.02]
-0.29
[-1.42]
2.69
0.18
[ 0.50]
0.19
[2.34]
0.05
[0.48]
0.21
[ 1.37]
0.66
0.33
[ 1.07]
0.11
[1.67]
0.11
[1.20]
-0.23
[-2.05]
2.64
0.31
[1.01]
0.08
[1.16]
0.10
[1.35]
-0.15
[-1.44]
1.18
γk>0
1.11
(6.11)
γk<0
0.96
0.65
(6.10)
(5.85)
1.09
(5.72)
γk>0
CAPM
0.55
0.59
[1.94]
[2.01]
0.10
0.07
[1.49]
[1.07]
0.34
0.02
FF
0.58
0.65
[1.97]
[2.09]
0.05
0.03
[0.71]
[0.39]
0.14
0.06
[1.97]
[0.60]
-0.06
-0.09
[-0.70]
[-0.91]
0.64
-0.15
γk<0
CAPM
0.46
0.11
[ 1.61]
[ 0.37]
0.03
0.11
[0.51]
[1.70]
-0.18
0.45
FF
0.51
0.13
[1.73]
[0.43]
-0.03
0.08
[-0.51]
[1.21]
0.18
0.08
[2.41]
[0.94]
-0.10
-0.04
[ -0.91]
[-0.48]
0.59
0.24
44
Table VIII:Explaining the Low Returns of High Idiosyncratic Volatility Stocks when the Non-Systematic
Coskewness is Positive
I form value-weighted decile portfolios every month by sorting stocks based on non-systematic coskewness relative to the CAPM
model. I use only NYSE/AMEX/NASDAQ industrial firms. I use only stocks with positive non-systematic coskewness. My
portfolios contain stocks in percentiles 0-5, 5-20, 20-30, 30-40, 40-50, 50-60, 60-70, 70-80, 80-95, and 95-100. In Panel A, at the
end of each month, I sort stocks into decile portfolios and then form value-weighted decile portfolios every month by sorting
these stocks based on their idiosyncratic volatility. I measure the statistics in the columns labeled Mean and Std Dev (standard
deviation) in monthly percentage terms. I use total, not excess, returns. Standard errors appear in parentheses. Robust
Newey-West (1987) t-statistics appear in square brackets. The column titled 10-1 refers to the difference in expected return
between portfolio 10 and portfolio 1. In Panel B, at the end of each month, I also sort stocks into decile portfolios and then
form value-weighted decile portfolios every month by sorting these stocks based on their non-systematic coskewness. When γkt
is positive, I refer to the value-weighted portfolio formed with the lowest 5% non-systematic coskewness as ν − and the highest
5% non-systematic coskewness as ν + . I refer 10-1 to as the difference in expected return between portfolio 10 and portfolio 1.
I use the 10-1 return ν + − ν − as a control variable in the CAPM linear regression model. Panel B reports the coefficients of
the regression:
rp = α + βM [RM − Rf ] + βν (ν + − ν − ) + η.
(B.19)
To run this regression, I consider stocks with positive non-systematic coskewness. I form value-weighted decile portfolios every
month by sorting stocks based on their idiosyncratic volatility relative to the CAPM model. I then run a regression of each
decile idiosyncratic volatility portfolio return on a constant, the excess market return and the 10-1 return (ν + − ν − ). Robust
Newey-West (1987) t-statistics appear in square brackets. I also report the R-Square of the regression. In Panel C, I use the
Fama and French factors as control variables. I run the regression:
rp = α + βM [RM − Rf ] + βSM B [RSM B − Rf ] + βHM L [RHM L − Rf ] + βν (ν + − ν − ) + η.
(B.20)
The sample period is from January 1971 to December 2006.
1Low
2
3
4
5
6
7
8
9
10 High
10-1
1.07
(3.89)
0.98
(4.66)
1.18
(5.48)
1.16
(6.09)
1.09
(6.99)
0.94
(7.93)
0.77
(8.37)
0.65
(8.74)
0.30
(9.70)
-0.22
(10.62)
-1.29
[-2.64]
0.63
[3.04]
-0.04
[-0.80]
0.03
[0.84]
0.24
0.73
[3.31]
0.04
[0.67]
0.11
[2.60]
4.57
0.94
[3.79]
0.02
[0.35]
0.26
[6.54]
16.07
1.08
[4.10]
0.10
[1.54]
0.33
[7.85]
20.57
1.30
[4.25]
0.08
[1.06]
0.44
[7.54]
26.55
1.33
[4.21]
0.14
[2.04]
0.59
[12.67]
36.97
1.07
[3.07]
0.10
[1.17]
0.60
[9.74]
34.38
1.01
[2.97]
0.16
[1.89]
0.69
[14.29]
42.89
0.83
[2.42]
0.20
[2.35]
0.85
[14.79]
53.62
0.47
[1.08]
0.30
[2.39]
1.05
[16.34]
53.59
-0.16
[-0.39]
0.35
[2.87]
1.01
[13.89]
57.91
0.65
[3.03]
-0.07
[-1.10]
0.05
[0.84]
-0.05
[-0.79]
0.03
[0.80]
0.08
0.77
[3.33]
-0.01
[-0.20]
0.12
[2.20]
-0.08
[-1.12]
0.11
[2.47]
5.26
0.92
[3.56]
0.00
[0.06]
0.10
[1.59]
0.01
[0.17]
0.26
[6.37]
16.00
1.12
[3.88]
0.06
[0.85]
0.07
[ 0.84]
-0.08
[-0.78]
0.32
[7.66]
20.48
1.28
[3.97]
0.07
[0.88]
0.07
[0.86]
0.02
[0.16]
0.44
[7.47]
26.29
1.46
[4.46]
0.05
[0.65]
0.10
[1.00]
-0.23
[-1.67]
0.58
[12.76]
37.52
1.07
[2.89]
0.07
[0.77]
0.09
[0.95]
-0.01
[-0.10]
0.60
[9.65]
34.20
1.11
[3.09]
0.13
[1.37 ]
-0.08
[-0.67]
-0.14
[-1.04]
0.69
[14.11]
42.83
0.88
[2.46]
0.17
[1.75]
0.01
[0.14]
-0.09
[-0.65]
0.85
[14.58]
53.47
0.55
[1.14]
0.26
[1.59]
0.02
[0.15]
-0.13
[-0.60]
1.04
[15.90]
53.46
-0.11
[-0.24]
0.33
[2.14]
-0.03
[-0.20]
-0.08
[-0.38]
1.01
[13.65]
57.74
Panel A
Mean
Std Dev
Panel B
α
t-stat
βM
t-stat
βν
t-stat
R2 (%)
Panel C
α
t-stat
βM
t-stat
βSM B
t-stat
βHM L
t-stat
βν
t-stat
R2 (%)
45
Table IX:Explaining the Low Returns of High Idiosyncratic Volatility Stocks when the Non-Systematic Coskewness is Negative
I form value-weighted decile portfolios every month by sorting stocks based on non-systematic coskewness relative to the CAPM
model. I use only NYSE/AMEX/NASDAQ industrial firms. I use only stocks with negative non-systematic coskewness. My
portfolios contain stocks in percentiles 0-5, 5-20, 20-30, 30-40, 40-50, 50-60, 60-70, 70-80, 80-95, and 95-100. In Panel A, at the
end of each month, I sort stocks into decile portfolios and then form value-weighted decile portfolios every month by sorting
these stocks based on their idiosyncratic volatility. I measure the statistics in the columns labeled Mean and Std Dev (standard
deviation) in monthly percentage terms. I use total, not excess, returns. Standard errors appear in parentheses. Robust
Newey-West (1987) t-statistics appear in square brackets. The column titled 10-1 refers to the difference in expected return
between portfolio 10 and portfolio 1. In Panel B, at the end of each month, I also sort stocks into decile portfolios and then
form value-weighted decile portfolios every month by sorting these stocks based on their non-systematic coskewness. When γkt
is negative, I refer to the value-weighted portfolio formed with the lowest 5% non-systematic coskewness as υ − and the highest
5% non-systematic coskewness as υ + . I refer 10-1 to as the difference in expected return between portfolio 10 and portfolio 1.
I use the 10-1 return υ + − υ − as a control variable in the CAPM linear regression model. Panel B reports the coefficients of
the regression:
rp = α + βM [RM − Rf ] + βυ (υ + − υ − ) + η.
(B.21)
To run this regression, I consider stocks with positive non-systematic coskewness. I form value-weighted decile portfolios every
month by sorting stocks based on their idiosyncratic volatility relative to the CAPM model. I then run a regression of each
decile idiosyncratic volatility portfolio return on a constant, the excess market return and the 10-1 return (υ + − υ − ). Robust
Newey-West (1987) t-statistics appear in square brackets. I also report the R-Square of the regression. In Panel C, I use the
Fama and French factors as control variables. I run the regression:
rp = α + βM [RM − Rf ] + βSM B [RSM B − Rf ] + βHM L [RHM L − Rf ] + βυ (υ + − υ − ) + η.
(B.22)
The sample period is from January 1971 to December 2006.
1Low
2
3
4
5
6
7
8
9
10 High
10-1
1.05
(3.94)
1.08
(4.50)
1.08
(5.58)
1.17
(6.26)
1.23
(7.34)
1.10
(8.38)
0.79
(8.88)
0.64
(9.21)
0.27
(10.05)
-0.30
(12.46)
-1.35
[-2.44]
0.58
[2.90]
-0.01
[-0.10]
0.01
[0.20]
-0.45
0.71
[2.93]
0.01
[0.09]
-0.09
[-2.09]
2.41
0.85
[3.21]
0.11
[1.63]
-0.18
[-4.77]
8.41
1.10
[3.70]
0.07
[0.97]
-0.30
[-5.95]
16.33
0.87
[2.89]
0.08
[1.07]
-0.37
[-9.81]
20.93
0.71
[2.24]
0.13
[1.48]
-0.39
[-8.27]
20.70
0.97
[2.96]
0.23
[2.85]
-0.51
[-11.58]
30.68
0.87
[2.46]
0.24
[2.48]
-0.59
[-10.31]
33.13
0.59
[1.72]
0.15
[1.88]
-0.67
[-15.22]
40.02
0.04
[0.08]
0.55
[4.96]
-0.84
[-12.99]
44.43
-0.54
[-1.24]
0.55
[5.77]
-0.84
[-12.63]
47.12
0.56
[2.72]
-0.01
[-0.14]
0.05
[0.86]
0.02
[0.38]
0.00
[0.16]
-0.78
0.73
[2.90]
-0.04
[-0.58]
0.15
[2.41]
-0.05
[-0.70]
-0.09
[-2.14]
3.14
0.87
[3.20]
0.06
[0.83]
0.18
[2.57]
-0.05
[-0.68]
-0.18
[-4.95]
9.17
1.07
[3.52]
0.05
[0.72]
0.13
[1.58]
0.04
[0.38]
-0.30
[-6.06]
16.33
0.86
[2.73]
0.04
[0.48]
0.20
[2.06]
-0.01
[-0.08]
-0.37
[-10.14]
21.36
0.73
[2.20]
0.07
[0.69]
0.20
[1.42]
-0.07
[-0.48]
-0.39
[-8.65]
21.20
0.93
[2.80]
0.20
[2.32]
0.19
[1.83]
0.04
[0.33]
-0.52
[-11.85]
30.87
0.83
[2.27]
0.22
[2.17]
0.15
[1.26]
0.05
[0.36]
-0.59
[-10.34]
33.10
0.53
[1.49]
0.12
[1.27]
0.26
[1.80]
0.07
[0.48]
-0.68
[-14.98]
40.49
0.01
[0.01]
0.50
[3.81]
0.28
[1.54]
0.01
[0.07]
-0.84
[-12.85]
44.75
-0.55
[-1.24]
0.50
[4.13]
0.23
[1.22]
-0.01
[-0.06]
-0.85
[-12.36]
47.31
Panel A
Mean
Std Dev
Panel B
α
t-stat
βM
t-stat
βυ
t-stat
R2 (%)
Panel C
α
t-stat
βM
t-stat
βSM B
t-stat
βHM L
t-stat
βυ
t-stat
R2 (%)
46
Table X:Explaining the Low Returns of High Idiosyncratic Volatility Stocks
I form value-weighted decile portfolios every month by sorting stocks based on non-systematic coskewness relative to the CAPM
model. I use only NYSE/AMEX/NASDAQ industrial firms. My portfolios contain stocks in percentiles 0-5, 5-20, 20-30, 30-40,
40-50, 50-60, 60-70, 70-80, 80-95, and 95-100. At the end of each month, I split the sample into two groups, stocks with positive
non-systematic coskewness and stocks with negative non-systematic coskewness. Within each group, I sort stocks into decile
portfolios and then form value-weighted decile portfolios every month by sorting these stocks based on their non-systematic
coskewness. When γkt is positive, I refer to the value-weighted portfolio formed with the lowest 5% non-systematic coskewness
as ν − and the highest 5% non-systematic coskewness as ν + . When γkt is negative, I refer to the value-weighted portfolio formed
with the lowest 5% non-systematic coskewness as υ − and the highest 5% non-systematic coskewness as υ + . I refer 10-1 to as
the difference in expected return between portfolio 10 and portfolio 1. I use the 10-1 returns ν + − ν − and υ + − υ − as control
variables in the CAPM linear regression model. For comparison purpose, I report in Panel A the Fama and French (1993)
alphas of idiosyncratic decile portfolios when I use all stocks (see Panel B of Table III). Robust Newey-West (1987) t-statistics
appear in square brackets. In Panel B, I reports the coefficients of the regression:
rp = α + βM [RM − Rf ] + βν (ν + − ν − ) + βυ (υ + − υ − ) + η.
(B.23)
To run this regression, I consider all stocks and form value-weighted decile portfolios every month by sorting stocks based on
their idiosyncratic volatility relative to the CAPM model (see Table III). I then run a regression of each decile idiosyncratic
volatility portfolio return on a constant, the excess market return, the Fama and French (1993) three-factor and the 10-1 returns
(ν + − ν − ) and (υ + − υ − ). Robust Newey-West (1987) t-statistics appear in square brackets. I also report the R-Square of the
regression. In Panel C, I use the Fama and French factors as control variables. I run the regression:
rp = α + βM [RM − Rf ] + βSM B [RSM B − Rf ] + βHM L [RHM L − Rf ] + βν (ν + − ν − ) + βυ (υ + − υ − ) + η,
The sample period is from January 1971 to December 2006.
1Low
2
3
4
5
6
7
8
9
10 High
10-1
0.59
[3.04]
0.47
[2.10]
0.65
[2.47]
0.64
[2.22]
0.55
[1.64]
0.37
[0.95]
0.21
[0.53]
0.07
[0.16]
-0.26
[-0.58]
-0.88
[-1.73]
-1.47
[-3.15]
0.65
[3.14]
-0.01
[-0.13]
-0.03
[-0.88]
0.01
[0.18]
0.05
0.72
[3.09]
0.04
[0.76]
-0.07
[-1.76]
0.10
[2.01]
7.49
1.09
[4.33]
0.06
[0.97]
-0.13
[-2.70]
0.17
[3.58]
17.79
1.20
[4.46]
0.07
[1.02]
-0.20
[-3.88]
0.20
[3.75]
23.71
1.29
[4.49]
0.08
[1.06]
-0.21
[-4.34]
0.31
[5.81]
31.82
1.30
[4.62]
0.13
[1.82]
-0.26
[-4.74]
0.39
[8.48]
40.03
1.22
[3.79]
0.13
[1.65]
-0.28
[-4.82]
0.43
[7.29]
42.73
1.17
[3.50]
0.17
[1.91]
-0.32
[-5.22]
0.46
[7.52]
46.83
1.11
[3.58]
0.14
[1.85]
-0.37
[-6.87]
0.60
[10.67]
59.24
0.69
[2.12]
0.29
[2.94]
-0.42
[-6.47]
0.68
[11.13]
65.72
0.04
[0.17]
0.30
[4.09]
-0.39
[-7.01]
0.68
[11.44]
74.19
0.65
[3.07]
-0.03
[-0.46]
0.07
[1.37]
-0.02
[-0.39]
-0.03
[-0.92]
0.01
[0.14]
-0.06
0.75
[3.10]
-0.01
[-0.23]
0.15
[2.69]
-0.09
[-1.32]
-0.08
[-1.89]
0.09
[1.84]
8.59
1.08
[4.13]
0.03
[0.42]
0.15
[2.27]
-0.01
[-0.18]
-0.13
[-2.88]
0.17
[3.38]
18.22
1.18
[4.15]
0.05
[0.69]
0.10
[1.30]
0.01
[0.12]
-0.20
[-4.04]
0.19
[3.62]
23.59
1.29
[4.23]
0.06
[0.70]
0.08
[0.89]
-0.02
[-0.16]
-0.21
[-4.48]
0.30
[5.61]
31.64
1.34
[4.53]
0.06
[0.78]
0.18
[1.82]
-0.10
[-0.85]
-0.27
[-4.99]
0.38
[8.13]
40.51
1.20
[3.57]
0.10
[1.16]
0.15
[1.60]
0.01
[0.08]
-0.29
[-5.03]
0.42
[7.07]
42.77
1.18
[3.35]
0.15
[1.58]
0.03
[0.33]
-0.03
[-0.26]
-0.32
[-5.34]
0.46
[7.39]
46.61
1.11
[3.41]
0.12
[1.41]
0.10
[0.90]
-0.01
[-0.05]
-0.38
[-6.99]
0.59
[10.65]
59.15
0.69
[2.04]
0.27
[2.36]
0.10
[1.05]
-0.02
[-0.22]
-0.43
[-6.62]
0.68
[11.07]
65.66
0.04
[0.15]
0.29
[3.44]
0.04
[0.46]
0.00
[-0.02]
-0.39
[-7.03]
0.68
[11.44]
74.08
Panel A
Alpha
t-stat
Panel B
α
t-stat
βM
t-stat
βυ
t-stat
βν
t-stat
R2 (%)
Panel C
α
t-stat
βM
t-stat
βSM B
t-stat
βHM L
t-stat
βυ
t-stat
βν
t-stat
R2 (%)
47
(B.24)
Risk Aversion
Skewness Preference
Skewness Preference
Risk Aversion
6
2.4
RA
SP
2.2
5.5
2
Skewness Preference
Risk Aversion
5
4.5
4
3.5
1.8
1.6
1.4
1.2
1
0.8
3
0.6
2.5
1996
1998
2000
2002
2004
0.4
1996
2006
1998
2000
years
2002
2004
2006
years
Price of Volatility Risk
Price of Coskewness Risk
Price of Variance Risk
Price of Coskewness Risk
1.5
0
−0.5
1
−1
Price (%)
Price (%)
0.5
0
−1.5
−2
−0.5
−2.5
−1
−1.5
1996
−3
1998
2000
2002
2004
−3.5
1996
2006
years
1998
2000
2002
2004
2006
years
Figure 1: Preference Parameters and Prices of Risk
Figure 1 depicts the risk aversion, skewness preferences, price of the volatility risk and the price of coskewness risk when I
estimate the pricing kernel with constant preference parameters for different sample periods [1986 + j, 1986 + 10 + j] when
j = 0, ..., 10. I report the preference parameters for different years j. I estimate the preference parameters of the pricing kernel
via GMM utilizing the Euler equation condition Et mt,t+1 Rkt+1 = 1 where mt,t+1 represents the pricing kernel. I estimate the
parameters by using the Hansen and Jagannathan (1997) weighting matrix. The sets of returns I use in my estimations are
those of 30 industry-sorted portfolios covering the period January, 1986, through December 31, 2006, augmented by the return
on a 30-day Treasury bill. I use the Chicago Board Options Exchange (CBOE)’s VOX as my proxy for the market volatility.
48
Risk Aversion
Skewness Preference
Risk Aversion
Skewness Preference
5
2.6
RA
SP
2.4
2.2
Skewness Preference
Risk Aversion
4.5
4
3.5
2
1.8
1.6
1.4
1.2
3
1
2.5
2000
2001
2002
2003
years
2004
2005
0.8
2000
2006
Price of Volatility Risk
2001
2002
2003
years
2004
2005
2006
Price of Coskewness Risk
Price of Variance Risk
Price of Coskewness Risk
0.2
−1
−1.2
0
−1.4
−0.2
−1.6
Price (%)
Price (%)
−0.4
−0.6
−1.8
−2
−2.2
−0.8
−2.4
−1
−2.6
−1.2
−1.4
2000
−2.8
2001
2002
2003
years
2004
2005
−3
2000
2006
2001
2002
2003
years
2004
2005
2006
Figure 2: Preference Parameters and Prices of Risk
Figure 2 depicts the risk aversion, skewness preferences, price of the volatility risk and the price of coskewness risk when I
estimate the pricing kernel with constant preference parameters for different sample periods [1990 + j, 1990 + 10 + j] when
j = 0, ..., 6. I report the preference parameters for different years j. I estimate the preference parameters of the pricing kernel
via GMM utilizing the Euler equation condition Et mt,t+1 Rkt+1 = 1 where mt,t+1 represents the pricing kernel. I estimate the
parameters by using the Hansen and Jagannathan (1997) weighting matrix. The sets of returns I use in my estimations are
those of 30 industry-sorted portfolios covering the period January, 1990, through December 31, 2006, augmented by the return
on a 30-day Treasury bill. I use the Chicago Board Options Exchange (CBOE)’s VIX as my proxy for the market volatility.
49
01/1996-12/2006
01/1986-12/2000
Pricing Kernel (PK)
Pricing Kernel (PK)
2.5
3
2.5
2
PK
PK
2
1.5
1.5
1
1
0.5
0.2
0.5
0.2
0.1
0.02
0.02
−0.02
0
−0.1
−0.01
−0.2
0.01
0
0
−0.1
rm
0.1
0.01
0
−0.2
rm
∆σ2m
−0.01
−0.02
∆σ2m
01/1986-12/2006
Pricing Kernel (PK)
2.5
PK
2
1.5
1
0.5
0.2
0.1
0.02
0.01
0
0
−0.1
rm
−0.01
−0.2
−0.02
∆σ2m
Figure 3: Estimated Pricing Kernels
Figure 3 depicts point estimates of the pricing kernels estimated with constant preference parameters. The support for the
graphs is the range of the return on the value-weighted index and the implied volatility difference. The preference parameters
of the pricing kernel via GMM utilizing the Euler equation condition Et mt,t+1 Rkt+1 = 1 where mt,t+1 represents the pricing
kernel. I estimate the parameters by using the Hansen and Jagannathan (1997) weighting matrix. The sets of returns I use
in my estimations are those of 30 industry-sorted portfolios covering the period January, 1986, through December 31, 2006,
augmented by the return on a 30-day Treasury bill. I use the Chicago Board Options Exchange (CBOE)’s VOX as my proxy
for the market volatility.
50
01/1996-12/2006
01/1990-12/2000
Pricing Kernel (PK)
2.5
4
2
3
1.5
2
PK
PK
Pricing Kernel (PK)
1
1
0.5
0.2
0
0.2
0.1
0.02
0.02
−0.02
0
−0.1
−0.01
−0.2
0.01
0
0
−0.1
rm
0.1
0.01
0
−0.2
rm
∆σ2m
−0.01
−0.02
∆σ2m
01/1990-12/2006
Pricing Kernel (PK)
3
2.5
PK
2
1.5
1
0.5
0.2
0.1
0.02
0.01
0
0
−0.1
rm
−0.01
−0.2
−0.02
∆σ2m
Figure 4: Estimated Pricing Kernels
Figure 4 depicts point estimates of the pricing kernels estimated with constant preference parameters. The support for the
graphs is the range of the return on the value-weighted index and the implied volatility difference. The preference parameters
of the pricing kernel via GMM utilizing the Euler equation condition Et mt,t+1 Rkt+1 = 1 where mt,t+1 represents the pricing
kernel. I estimate the parameters by using the Hansen and Jagannathan (1997) weighting matrix. The sets of returns I use
in my estimations are those of 30 industry-sorted portfolios covering the period January, 1990, through December 31, 2006,
augmented by the return on a 30-day Treasury bill. I use the Chicago Board Options Exchange (CBOE)’s VIX as my proxy
for the market volatility.
51
01/1996-12/2006
01/1986-12/2000
Projected Pricing Kernel
3.5
2.5
3
2
2.5
PK
PK
Projected Pricing Kernel
3
1.5
2
1
1.5
0.5
1
0
−0.1
−0.08 −0.06 −0.04 −0.02
0
0.02
0.04
0.06
0.5
−0.1
0.08
−0.05
0
rm
rm
0.05
0.1
01/1986-12/2006
Projected Pricing Kernel
3.5
3
PK
2.5
2
1.5
1
0.5
−0.1
−0.05
0
rm
0.05
0.1
Figure 5: Projected Pricing Kernels
Figure 5 depicts the projection of the estimated pricing kernel estimated with constant preference parameters (see Figure 3) on
P
j
a polynomial function of the market return, mt,t+1 = 5j=0 bj rM
t+1 . The support for the graphs is the observed range of the
return on the value-weighted index.
52
01/1996-12/2006
01/1990-12/2000
Projected Pricing Kernel
3
2.5
2.5
2
2
PK
PK
Projected Pricing Kernel
3
1.5
1.5
1
1
0.5
0.5
0
−0.1
−0.08 −0.06 −0.04 −0.02
0
0.02
0.04
0.06
0
−0.1
0.08
−0.05
0
rm
rm
0.05
0.1
01/1990-12/2006
Projected Pricing Kernel
3
2.5
2
PK
1.5
1
0.5
0
−0.5
−0.1
−0.05
0
rm
0.05
0.1
Figure 6: Projected Pricing Kernels
Figure 6 depicts the projection of the estimated pricing kernel estimated with constant preference parameters (see Figure 4) on
P
j
a polynomial function of the market return, mt,t+1 = 5j=0 bj rM
t+1 . The support for the graphs is the observed range of the
return on the value-weighted index.
53
1.2
CAPM
FF
FF−M
HS
1
Expected Return (%)
0.8
0.6
0.4
0.2
0
−0.2
−0.4
1
2
3
4
5
6
7
8
9
10
Group
Figure 7: Idiosyncratic Volatility and Expected Returns
Figure 8 plots the expected return across deciles when I use different measures of idiosyncratic volatility. CAPM indicates
that the idiosyncratic volatility is computed using the CAPM model. FF indicates the Fama and French (1993) model, FF-M
indicates the Fama and French model augmented with the momentum factor of Jegadeesh and Titman (1993), and HS indicates
the Harvey and Siddique (2000) market coskewness model.
54