GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 301

Asset allocation under higher moments with the GARCH filter

Ryo Kinoshita

March 2013

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Asset allocation under higher moments with the
GARCH ﬁlter

Abstract
Except for several pairs of utility functions and distribution functions, expected utility maximization problems do not have closed form solutions so that these problems
often require complex numerical optimizations. The paper proposes an approximatedsolution to this problem using higher moments of returns. Utility functions are approximated by polynomials by the Taylor expansion, and thus expected utility functions areapproximated by linear combinations of moments. With the GARCH eﬀects, a simple
approach to estimate conditional higher moments is given. In an empirical study, thestrategy is compared to alternative strategies such as mean variance optimization andstatic optimization.

JEL Classiﬁcation Code: G11, C58, C61
Keywords : Portfolio allocation, GARCH model, non-normality1

1

Introduction

Markowitz (1952) proposed the now well known mean variance portfolio selection. Whenasset returns follow the multivariate normal distribution, this approach is a good approximation of the expected utility maximization for risk averse investors. However, when returns are
non-normally distributed higher moments may aﬀect the optimal portfolios since the meanvariance criteria cannot capture the investor’s preference about the asymmetric property or
the heavy tailedness of distributions of asset returns. Recently, many papers have reported
that asset returns are asymmetric and heavy tailed distribution. In addition, many papers
reported that asset returns have conditional heteroskedasticities. Asset allocation strategies
using the conditional heteroskedasticities are known as volatility timing. In this paper, I
consider a simple asset allocation problem with non-normalities and volatility timing.
Previous studies have used two model speciﬁcations where asset allocation problems arestudied under non-normalities. The ﬁrst is the regime switching model. Ang and Bekaert(2002) and Guidolin and Timmerman (2008) consider asset allocation problems where returns
are regime switching in international stock markets. In this speciﬁcation, the conditional
distributions that investors face are the mixtures of normal distributions. The other speciﬁcation is the Generalized conditional heteroskedasticity model (GARCH) with non-normal
innovations. Harvey et al (2010) and Jondeau and Rockinger (2012) focused on the skew
normal distribution proposed by Sahu, Dey and Branco (2003), of which the co-skewness
matrix and co-kurtosis matrix can be computed explicitly from the parameters of the distribution function to demonstrate the optimizations. Patton (2004) and Jondeau and Rockinger(2006) used the copula-GARCH model. With an assumption of a location-scale mixture of
the multivariate normal distributions, Mencia and Sentana (2009) derived a mean-varianceskewness optimization problem. Under the GARCH model with non-normal innovations,conditional distribution is derived by scaling innovations with GARCH eﬀects. Other related studies include the following, Adcock (2010) demonstrated an asset allocation problem
under multivariate skew normal and skew-t distribution which were proposed by Azzalini
and Capitanio (2003). Konno et al. (1991) derived a mean-absolute deviation-skewness op2

timization problem that can handle a very large number of assets. Theoretical propertiesof the mean-variance-skewness frontier were derived by Athayde and Flores (2004). Nonnormalities of asset returns have been reported in many studies not directly related to asset
allocation problems. Longin and Solnik (2001) reported non-normalities of international asset returns using the exceedance correlation measure. Bauwens and Laurent (2005) adapted
a multivariate skew-t distribution for the multivariate GARCH model. For univariate series,
Hansen (1994) considered skewed distributions for innovations of the GARCH model. Jondeau and Rockinger (2003) studied conditional shapes of distributions of asset returns using
distributions proposed by Hansen (1994). Harvey (1999) proposed the GARCHS model,
which models the conditional skewnesses in the manner of the GARCH formula.
Except for several pairs of distributions and utility functions, the expected utility functions cannot be computed analytically. When non-normalities are introduced, this is true formost cases. One of approaches for calculating expected utility function are numerical integrations called quadrature methods proposed by Tauchen and Hussey (1991). Ang and Chen
(2002) employed this approach under the regime switching model. The second approach isMonte Carlo integration, which is used when random generations are possible. Patton (2004)
used this approach under the copula-GARCH model. These two approaches require heavy
computation to obtain the expected utility function, and the optimization is often unstable.
The third approach is an approximation of the utility function by the Taylor expansion.
Expansion up to the second moments results in the quadratic utility function, so it only
focuses on the ﬁrst two moments (mean variance approach). In the cases of non-normal
distributions of returns, an approximation using higher moments may be valuable. Jondeau
and Rockinger (2006) derived an optimal asset allocation problem with the expansion up to
the ﬁrst four moments using sample moments. Jondeau and Rokinger (2012) extended thisto incorporate the dynamic conditional correlation GARCH (DCC-GARCH) model. They
assume speciﬁc distribution for innovations, which restricts the structures of the moments.
In this paper, I propose a simple expected utility maximization problem using an approximation by the Taylor expansion under the GARCH model without assuming any speciﬁc3

distributions. As in Jondeau and Rockinger (2006), Jondeau and Rockinger (2012) andGuidolin and Timmerman (2008), I expand an investor’s utility function up to the ﬁrst four
moments. This paper can be considerd an extension of Jondeau and Rockinger (2006). For
simplicity, it is assumed that asset returns follow the multivariate GARCH model with thecorrelation matrix, co-skewness matrix co-kurtosis matrix of innovations constant over time;
however, the DCC GARCH is also applicable. The main idea is a two step estimation. I ﬁrst
estimate the multivariate GARCH model by variance targeting Gaussian Quasi-maximum
likelihood estimation (QMLE). Given the estimated GARCH parameters, I estimate thestandardized innovation series and the co-skewness matrix of innovations by those sample
moments. Changing the location and scale of the correlation matrix, co-skewness matrix andco-kurtosis matrix of innovations by the GARCH eﬀects, conditional covariance matrices, coskewness matrices and co-kurtosis matrices are estimated. This two step estimation is easy
to compute relative to the previous studies and is adaptive for many studies of the GARCH
model. With the estimated conditional moments, optimal asset allocations are estimated by
maximizing the approximated expected utility function in each period.
The eﬀects of the degree of approximation for the expected utility functions are measured
by the certainty equivalent returns (performance fee) as in Fleming, Kirby and Ostdiek(2001), Patton (2004) and Jondeau and Rockinger (2012). To evaluate these diﬀerences
statistically, the bootstrap method is employed. The remainder of this paper is organized asfollows. In the section 2, a model of the asset allocation problem and underlying asset returns
processes are described. Derivation of optimal portfolio weights with approximated expectedutility functions is explained. In the section 3, alternative portfolios and the methods tocompare their performances are given. In section 4, an example of an empirical applicationis shown. and the section 5, concludes.

4

2

Model

In this section, an approach to the expected utility maximization problem for a risk averse
investor is explained. In the ﬁrst subsection, assumptions for return processes are given.
Returns are assumed to be the multivariate GARCH model and innovations are identicallyand independently distributed and have ﬁnite third and fourth moments. Several previous
studies assumed speciﬁc distributions for innovations (Jondeau and Rockinger (2012), Mencia and Sentana (2009) and Harvey et al. (2010)). In contrast to these studies, this paper
does not assume any speciﬁc distributions for innovations. Under this setting, this paper
demonstrates how to estimate time varying conditional covariance matrices, co-skewness
matrices and co-kurtosis matrices which are induced by the GARCH eﬀect. In the second
subsection, an expected utility maximization problem and approximations are given. Except for several cases, the expected utility cannot be calculated analytically and numericalintegrations are required. In this paper, utility functions are approximated by the Taylor
expansion to evaluate expected utility. This makes it possible to approximate the expectedutility function by a weighted sum of the elements of the moment matrices. The approach iseasy to compute relative to numerical integration methods such as the quadrature method
and Monte Carlo integration. By maximizing the approximated utility functions under theestimated conditional moments, conditional optimal portfolio weights are estimated for eachperiod.

2.1

Return process

2.1.1

Estimations of the GARCH and central moment matrices

In this subsection, a model of asset returns is given, and the dynamics of conditional covariance matrices and co-skewness matrices and estimations are explained. There are a risk freeasset and n risky assets with return vector rt = (r1t , r2t , . . . , rnt )0 and those processes are multivariate GARCH with constant conditional correlation. That is, the individual return series
follow univariate asymmetric-GARCH (1,1) which is proposed by Glosten, Jagannathan and5

Runkle (1993) and the correlation matrix of innovations ²t = (²1t , ²2t , . . . , ²nt )0 is constantover time. In addition, it is assumed that the co-skewness matrix and co-kurtosis matrix ofthe innovations are constant over time:rit = µi + σit ²it ,(1)

2
σit2 = wi + αi σit−1+ βi ²2it−1 + γ²2it−1 I{²it−1 <0} ,(2)²t ∼ iid(0, Ω, S, K),(3)

where S and K are the co-skewess matrix (n×n2 ) and co-kurtosis matrix (n×n3 ) respectively,
the same as in Jondeau and Rockinger (2006) such that:S = E[(²t − µ) ⊗ (²t − µ)0 ⊗ (²t − µ)0 ],(4)

K = E[(²t − µ)(²t − µ)0 ⊗ (²t − µ)0 ⊗ (²t − µ)0 ].

(5)

The co-skewness matrix consists of n submatrices such that[
S = S1 S2 · · ·

]
Sn ,

where




s
s
· · · si1n
 i11 i12
 si21 si22 · · · si2n Si =  ..
..
..  .
...
 .
.
. 


sin1 sin2 · · · sinn
The co-kurtosis matrix consists of n submatrices such that[
K = K1 K2 · · ·]Kn ,where Ki consists of submatrices such that[
Ki = Ki1 Ki2 · · ·

6

]
Kin ,

wherekk· · · kij1n ij11 ij21 kij21 kij22 · · · kij2n 
Kij =  ..
..
..  ,
..
.
 .
.
. 


kijn1 kijn1 · · · kijnn
where sijk = E[²it ²jt ²kt ] and kijkl = E[²it ²jt ²kt ²kt ]. The co-skewness matrix and co-kurtosis
matrix have n(n + 1)(n + 2)/6 and n(n + 1)(n + 2)(n + 3)/24 non-redundant components
respectively. Using these matrices, Jondeau and Rockinger (2006) proposed a framework
to use higher moments for asset allocation problems with sample moments. I extend theirapproach to incorporate the GARCH eﬀects. Mencia and Sentana (2009) also applied the
GARCH model assuming that the distributions of ²t are location-scale mixture of the multivariate normal distributions. Jondeau and Rockinger (2012) and Harvey et al. (2010) also
used the GARCH model and higher moments assuming the innovations to be the skewed
t-distribution proposed by Sahu, Dey and Branco (2003). In contrast to these previous studies, this paper does not assume any speciﬁc distributions for innovations so it has ﬂexibility
for the shape of ²t . In this paper, a two step semiparametric estimation is employed. The
ﬁrst step is the estimation of GARCH parameters. The second step is the estimation of the
shape of innovations, that is, an estimation of higher moment matrices.
For the GARCH estimation, variance targeting estimation is employed. Unconditional
means and variances are estimated by the sample moments for each asset:
1∑
µ̂i =
rit ,
T t=1

(6)

1∑
σ̄i =
(rit − µ̂i )2 .
T t=1

(7)

T

T

Given these unconditional moments, the GARCH parameters are estimated by the Gaussian7

QMLE which uses the likelihood function of the multivariate normal distribution as follows:
max l(αi , βi , γi , Ω; r) = −

T ∑
n
T
∑
nT
T
1 ∑ 0 −1
log 2π − log |Ω| −log |σit | −²t Ω ²t ,22
2
t=1 i=1i=1(8)2s.t σit2 = ωi + ασit−1+ βi ²2it−1 + γi ²2it−1 I{²it−1 <0} ,ωi= σ̄i2 ,1 − αi − βi − γ2iwhere ²it = (rit − µ̂i )/σit . When the dynamics of scales are correctly speciﬁed, the Gaussian
QMLE gives consistent estimations for {wi , αi , βi , γi , Ω}. Given the estimated parameters,
innovations for the GARCH are estimated:2
σ̂it2 = ŵi + α̂i σ̂it−1+ β̂i ²̂2it−1 + γ̂i ²2it−1 I{²it−1 <0} ,²̂it =

rit − µ̂i
.
σ̂it

(9)
(10)

Using these standardized innovations, the co-skewness matrix and co-kurtosis matrix ofinnovations are estimated as follows:
1 0²̂t ²̂ ⊗ ²0t ,T t
1K̂ = ²̂t ²̂0t ⊗ (²0t ⊗ ²0t ).
T
Ŝ =

(11)
(12)

∑
These are (n×n2 ) and (n×n3 ) matrcies, and their components are simply ŝijk = T1 Tt=1 ²̂it ²̂jt ²̂kt ,∑k̂ijkl = T1 Tt=1 ²̂it ²̂jt ²̂kt ²̂lt , Using the estimated GARCH parameters, volatility series and standarized innovation series are estimated. Since S and K are the third and fourth momentmatrices of standardized innovations respectively, conditional moment matricies also vary because of the change in scale induced by the GARCH eﬀects. Taking into account the GARCH
eﬀects, the conditional covariance matrix, co-skewness matrix and co-kurtosis matrix are as
follows:
Vt = Gt ΩGt ,[]St = σ1t Gt S1 Gt σ2t Gt S2 Gt · · · σnt Gt Sn Gt ,[]Kt = σ1t σ1t Gt K11 Gt σ1t σ2t Gt K12 Gt · · · σnt σnt Gt Knn Gt ,
8

(13)
(14)
(15)

where Gt = diag{σ1t · · · σnt }. The elements of the conditional co-skewness matrix areŝijkt = σ̂it σ̂jt σ̂kt ŝijk and the elements of the conditional co-kurtosis matrix are k̂ijklt =σ̂it σ̂jt σ̂kt σ̂lt kijkl . The conditional covariance matrix is the same as the constant conditionalcorrelation model (CCC). Since the focus is not on the dynamics of the second moments, theconstant correlation GARCH is assumed for simplicity. However the dynamic conditionalcorrelation model (DCC) proposed by Engle (2002) can be applied. The conditional coskewness matrix and co-kurtosis matrix are also scaled by the GARCH eﬀects as well as thecovariance matrix. In this way, conditional moments with the GARCH eﬀects are estimated.
2.1.2

Transformation into noncentral moments from central moments

As showen later, when the utility function is approximated by a polynomial, the expectedutility function is the weighted sum of the noncentral moments of a portfolio return. Since
GARCH models are those of central moments, moment matrices estimated above are also central moments. Preliminary to an asset allocation problem, I transform central moments into
noncentral moments using binomial expansion. Let M2,t , M3,t ,and M4,t be the ﬁrst,second,third, and fourth conditional noncentral moment matrices, respectively. Elements of thenoncentral moments are computed from central moments as follows:
m2,ijt = Et [rit+1 rjt+1 ] = m2,ijt + µi µj ,(16)m3,ijkt = Et [rit+1 rjt+1 rkt+1 ] = s3ijkt + m2,ijt µk + m2,ikt µj + m2,jkt µi − 2µi µj µk ,(17)4m4,ijklt = Et [rit+1 rjt+1 rkt+1 rlt+1 ] = kijklt+ m3,ijkt µl + m3,ijlt µk + m3,iklt µj + m3,jklt µi (18)− m2,ijt µk µl − m2,ikt µj µl − m2,ilt µj µk − m2,jkt µi µl − m2,jlt µi µk − m2,klt µi µj+ 3µi µj µk .

2.2

Asset allocation

In this subsection, a conditional expected utility maximization problem for a risk averse
investor is explained under the assumptions of the return process described in the previous
subsection. There are n risky assets with return vector rt = (r1t , . . . , rnt )0 and a corresponding
9portfolio vector is denoted by wt = (w1t , . . . , wnt )0 and the weight for the risk free asset is∑denoted by wf t = 1 − wt0 1 =1 − ni=1 wit ≥ 0. Given the initial wealth is Wt at t, the myopic
optimization problem is as follows:max
wt

Et [U (Wt+1 )] = Et [U (Wt (1 + wt 0 rt+1 + wf t rf t ))].

(19)

Except for some pairs of a utility function and a distribution function, this problem does nothave a closed form solution. For instance, a pair of the exponential utility function and themultivariate normal distribution has a closed form solution so that a numerical optimizationis required to evaluate the integration. When speciﬁc distributions are assumed, the integration can be calculated with the quadrature methods proposed by Tauchen and Hussey(1991). Ang and Bekaert (2002) employed this approach to calculate the expected utility
function where returns are regime switching. Another approach is Monte Carlo integration
which was employed by Patton (2004) when random generations were possible. However,both approaches involve heavy computation. In this paper, another approach is employed toevaluate the expected utility function. The utility function is approximated by the Taylor
expansion up to a ﬁnite number of moments as follows:
Et [U (Wt+1 )] ≈

K
∑
θk
k=1

k!

Et [(Wt+1 − Wt )k ].

(20)

When the expansion is up to the second moment, the problem results in mean varianceoptimization. When returns follow a multivariate normal distribution, the ﬁrst two moments
suﬃce for the expected utility maximization problem. However, when non-normalities are
introduced, higher moments aﬀect the investor’s optimization problem. A great deal of
empirical research suggests that asset returns are sometimes asymmetric and heavy tailed(Longin and Solnik (2001), Bauwen and Laurent (2003)). In this paper, the utility function
is expanded up to the ﬁrst four moments. Jondeau and Rockinger (2006) and Harvey et al(2010) also employed this approach. A case of Taylor expansion up to the fourth moments

10can be written as follows:
1
Et [U (Wt+1 )] ≈ e(wt ) =U (1) (Wt )m1,P t+1 (wt ) + U (2) (Wt ) · m2,P t+1 (wt )
2
1 (3)
1
+ U (Wt )m3,P t+1 (wt ) + U (4) (Wt )m4,P t+1 (wt )
6
24

(21)

where m1,P t+1 (wt ), m2,P t+1 (wt ), m3,P t+1 (wt ) and m4,P t+1 (wt ) are the ﬁrst, second, third andfourth noncentral moments of the portfolio, respectively. The noncentral moments of theportfolio are computed from the weighted sum of noncentral moment matrices as follows:
m1,P t+1 (wt ) = wt0 µt+1 =n∑wit µi + wf t rf t+1 ,i=1m2,P t+1 (wt ) = wt0 M2,t+1 wt =

n ∑
n
∑

wit wjt m2,ijt+1 ,i=1 j=1
0

m3,P t+1 (wt ) = wt M3,t+1 (wt ⊗ wt ) =

n ∑
n ∑
n
∑

wit wjt wkt m3,ijkt+1 ,i=1 j=1 k=1
n
n ∑
n ∑
n ∑
∑

m4,P t+1 (wt ) = wt 0 M4,t+1 (wt ⊗ wt ⊗ wt ) =wit wjt wkt wlt m4,ijklt+1 .

i=1 j=1 k=1 l=1

To maximize this approximated utility function by standard techniques, the derivative
vector and the Hessian matrix are required. The elements of the derivative vector andthe Hessian matrix can be written using submatrices of the higher moment matrices. The
elements of the derivative vector are as follows:∂m1,P t+1
= µi − rf t+1 ,∂witn∑∂m2,P t+1
=2
wit m2,ijt+1 ,
∂wit
j=1
n ∑
n
∑
∂m3,P t+1
0
wjt wkt m3,ijkt+1 ,= 3wt M3,t+1 wt = 3∂wit
j=1 k=1n ∑n ∑n∑∂m4,P t+1
0
= 4wt M4,t+1 (wt ⊗ wt ) = 4
wjt wkt wlt m4,ijklt+1 .
∂wit
j=1 k=1 l=1

11

The elements of the Hessian matrix are as follows:∂ 2 m2,P t+1= 2m2,ijt+1∂wit ∂wjtn∑∂ 2 m3,P t+1=6m3,ijkt+1 wkt∂wit ∂wjt
k=1
n ∑
n
∑
∂ 2 m4,P t+1= 12wt 0 M4,t+1 wt = 12m4,ijklt+1 wkt wlt .
∂wit ∂wjt
k=1 l=1

In this paper, I consider an investor whose preference is represented by a constant relative
1−λrisk aversion (CRRA) utility function (i.e. U (W ) = W1−λ ) and normalize the initial wealth
W = 1. The approximated utility function is as follows:λEt [U (1 + wt0 rt+1 + wf t rf t+1 )] ≈ m1,P t+1 − m2,P t+1 (wt )(22)
2
λ(λ + 1)λ(λ + 1)(λ + 2)+
m3,P t+1 (wt ) −m4,P t+1 (wt ).
6
24
For risk averse investors, the eﬀects of the third and fourth moments are positive and negative
respectively. This is consistent with the intuition that an investor tries to avoid a large
negative return and a fat-tailed distribution. It is logical to assume that the CRRA utility
function is useful since optimizations are not aﬀected by the level of initial wealth. If theinitial wealth is W , then the optimal weight is W ωt∗ where ωt∗ is the optimal portfolio when
the initial wealth is unity. To sum up, replacing the moments matrices with estimated series,
the conditional optimal portfolio weights given information up to t−1 are selected by solving
following problem:wt

λ(λ + 1)λ(λ + 1)(λ + 2)
λm̂3,P t+1 (wt ) −m̂4,P t+1 (wt )ê(wt ) = m̂1,P t+1 − m̂2,P t+1 (wt ) +
2
6
24

s.t

m̂1,P t+1 (wt ) = wt0 µ̂i + wf t rf t+1 ,max

m̂2,P t+1 (wt ) = wt 0 M̂2,t+1 wt 0 ,m̂3,P t+1 (wt ) = wt 0 M̂3,t+1 (wt ⊗ wt ),m̂4,P t+1 (wt ) = wt 0 M̂4,t+1 (wt ⊗ wt ⊗ wt ),12

The optimization problem is solved by the Newton-Rapthon method for each t. Since theproblem is an approximation of the expected utility maximization, the performance of an
approximation is of interest. By casting estimated portfolio weights in the argument ofutility function, realized utilities can be calculated. In the following section, I measure thediﬀerence between alternative strategies.

3

Comparison of performance

In section 2, an approximated expected utility maximization problem is introduced. When
returns are non-normally distributed, including higher moments will better approximate theexpected utility function. In this section I determine whether including the higher moments
and volatility timing is valuable or not.

3.1

Alternative portfolios

In this subsection, alternative asset allocation strategies are deﬁned. To study the eﬀect ofhigher moments and volatility timings, I compare static and dynamic portfolios changing thedegree of approximations. The ﬁrst strategy is the static portfolio with mean and variancewhich maximizes the approximated expected utility function up to ﬁrst two moments. The
approximated expected utility function is as follows:
Et [U (1 + wt 0 rt+1 + wf t rf t+1 )] ≈ mP t+1 −λ2 2m(wt ).
2 P t+1(23)

The second strategy is the static portfolio with ﬁrst four moments. This portfolio captures
the non-normalities of returns but the GARCH eﬀects are not considered. The third strategy
is the dynamic mean-variance portfolio which changes portfolio weights depending on the
GARCH eﬀects. This portfolio captures the GARCH eﬀects but non-normalities are notconsidered.

13

3.2

Method of the comparison

To measure the diﬀerence of alternative strategies, I use the concept of certainty equivalent
return. Let wt ∗ and wt ∗∗ be allocations under diﬀerent strategies. Then an unconditional
certainty equivalent return is m such that00E[U (1 + wt ∗ rt+1 + m)] = E[U (1 + wt ∗∗ rt+1 )].

(24)

Certainty equivalennt return m is a required extra return where investors become indiﬀerent
to both strategies. This can be interpreted as the investor being willing to pay m to switch
his strategy from wt ∗ to wt ∗∗ . This enables an investor to compare the cost of calculating
the higher moments and its beneﬁt. The certainty equivalent return is estimated by solving
sample analogs of (24):
T
T
1∑
1∑
0
U (1 + ŵt rt+1 + wf t rf t+1 + m̂) =U (1 + w̃t0 rt+1 + wf t rf t+1 )
T t=1
T t=1

(25)

by the grid search method.

3.3

Bootstrap

To compare the expected utilities of two alternative strategies statistically, I employ a bootstrap method. To derive a bootstrap distributions of certainty equivalent returns, the following procedures are iterated.
(k)

1. Estimate the GARCH model (1), (2) and (3) using the return series {rt }.
2. Calculate the certainty equivalent return m(k) using the original return series rt andthe estimated parameters.
3. Using the original return series rt and the estimated parameters, the mean, variance

14

(k)

2(k)

(k)

and innovation series {µi , σit , ²t } are estimated as in section 2
(k)
µ̂i =

1 ∑ (k)r ,T t=1 it2(k)σ̂i0 =1 ∑ (k)(k)(rit − µ̂i )2 ,
T t=1

T

T

2(k)

σ̂it

(k)

(k) 2(k)(k) 2(k)(k) 2(k)= ŵi + α̂i σ̂it−1 + β̂i ²̂it−1 + γ̂i ²̂it−1 I{²it−1 <0} ,
(k)

(k)

²̂it =

rit − µ̂i
(k)

.

σ̂it

(k)

(k∗)

4. Resampling ²t , the bootstrap series of innovations {²t} is computed.

5. A bootstrap series of returns {rt (k+1) } are computed as follows:
(k+1)

rit

(k)

(k) (k∗)= µ̂i + σ̂it ²it .

The null hypothesis is m = 0 and the alternative is m 6= 0. The null hypothesis implies
that the two strategies are indiﬀerent, at least unconditionally. Demeaning the bootstrap
sample of m, the distribution of the certainty equivalent return under the null is simulated.
The p-value for m̂ is computed as
}
{
B
B
∑
1
1 ∑
I(m(k) − m̄ ≤ m̂),I(m(k) − m̄ > m̂)p(m̂) = 2 min
B k=1
B k=1
where m̄ = B1

∑B

k=1 m

(k)

(26)

. In the following section, an illustrative empirical application isdemonstrated.

4

Empirical Application

In this section, an empirical application for the Tokyo Stock Exchange (TSE) is demonstrated. The most popular index of the TSE is TOPIX which consists of all the stocks in theﬁrst section of the TSE. In the TSE, TOPIX includes 33 categorizes industries. I constructportfolios using several indexes and a risk-free asset. The selected categories are Mining, Iron
15

& Steel, Transportation Equipments and Securities & Commodity Futures and the risk freerate is proxied by the commercial paper rate. The data are weekly returns from 1999/12/28
to December 2011/4/29.

4.1

Estimation of the GARCH and moment matrices

In Table 1, the estimated parameters of the GARCH model are reported. The co-skewness
matrix is 4 × 42 and the co-kurtosis matrix is 4 × 43 and only non-redundant elements are
displayed. As the correlation matrix shows, the indexes are positively correlated so that there
is little gain from diversiﬁcation that reducing the variance of a portfolio. However, there isan another type of gain from diversiﬁcation that increases a skewness or reduces a kurtosis.
Under the multivariate normal distribution, E[²3it ] = 0, E[²it ²2jt ] = 0, E[²4it ] = 3, E[²2it ²2jt ] =1, E[²it ²3jt ] = 0. Co-skewnesses E[²1t , ²22t ] can be interpreted as correlation of ²1t and ²22t . The
negative co-skewness E[²1t ²22t ] = −0.11 implies that when the Mining index experience large
return in absolute value, the Iron & Steel index tends to experience negative return. Since
rare event of Mining and negative return of Iron & Steel are correlated, this combination
raises the possibility of large negative return. This has negative eﬀect on skewness of aportfolio return, and thus this combination can be interpreted as risky. Co-kurtosis E[²21 ²22 ]can be interpreted as the covariance of ²21t and ²22t . The positive co-kurtosis E[²21t ²22t ] = 1.69
implies that when the Mining index experiences a rare event, Iron & Steel index also tends
to experience rare event. This combination raises the possibility of very rare event, and so
can be interpreted as risky. These bootstrap resamplings are repeated 100 times with theprocedure shown in section 3. Bootstrap standard errors are reported in parentheses.
Table 1: GARCH parameters and moment matrices of innovations
With the estimated GARCH parameters and higher moment matrices of innovations,conditional moment matrices are estimated every week. Given the conditional moments,
optimal asset allocations are estimated for each strategy. As in Jondeau and Rockinger(2012), I use the parameters of risk aversion λ = 5, 10, 15. When this value is large the16investor is more risk averse. In the following subsections the performance of each strategy isreported.

4.2

Descriptive statistics of the realized return

Table 2 represents the descriptive statistics of the realized return for each strategy. Eachrow reports the summary statistics of the portfolio return of the dynamic higher moment
portfolio, dynamic mean-variance portfolio, static higher moment portfolio and static meanvariance portfolio. Two static portfolios shows very similar performance. This implies thathigher moment does not has value unconditionally. The dynamic higher moment portfolio
have large mean, large variance, large skewness and large kurtosis than that of dynamic meanvariance portfolio. In terms of the mean variance criterion, the portfolio may be interpreted
as a high risk, high return portfolio; however, in terms of the expected utility, the risk is not
so large since relatively large skewness makes the probability of negative return small.
Table 2: Summary statistics of the realized returns

4.3

Expected utility and certainty equivalent returns

For several values of risk aversion, certainty equivalent returns are estimated. Certainty
equivalent returns are measured between the dynamic higher moment portfolio and otherstrategies respectively. Table 3 presents the computation results. The ﬁrst column shows
the certainty equivalent return for each strategy and the dynamic higher moment portfolio.
The second column shows the bootstrap p-values explained in section 3. Figure 1 presents
the bootstrap distribution of the certainty equivalent returns with λ = 15. The result showsthat the dynamic higher moment portfolio signiﬁcantlly improve the performance when it iscompared to dynamic mean-variance portfolio. This impies that considering higher moments
is valuable in term of the expected utility when it is combined to the volatility timing. The
value 0.0000056 of the certainty equivalent return between the dynamic higher moment
portfolio and the dynamic mean variance portfolio implies that investors are willing to pay
170.0000056 to switch the strategy from dynamic mean variance portfolio to dynamic higher
moment portfolio. Since average risk free rate during this period is 0.0000385, it can beinterpreted that considering higher moments has about seventh part of value of the risk freerate when λ = 15. Also ﬁgure 1 shows, the most of simulated certainty equivalent returns are
positive. Comparing to static mean variance, or static higher moment portfolio, certainty
equivalent returns are 0.000020 but is not statistically signiﬁcant. As ﬁgure 2 and 3 show,
the value of the certainty equivalent returns are volatile. This imply that the value of volality
timing is ambiguous.
Table 3: Certainty equivalent returns and bootstrap p-values
Figure 1: Bootstrap distribution of the certainty equivalent returns (a) Dynamic meanvariance portfolio
Figure 2: Bootstrap distribution of the certainty equivalent returns (b) Static higher
moment portfolio
Figure 3: Bootstrap distribution of the certainty equivalent returns (c) Static meanvariance portfolio

5Conclusion and future work

In this paper a simple asset allocation problem with higher moments and GARCH eﬀects
is introduced. The approach is easy to estimate since the time series estimations are basedon the standard estimation of the GARCH and portfolio optimizations are solved by the
Newton-Rapthon method with explicitly computed derivative vectors and Hessian matrices.
The performance of the strategies is measured by the bootstrap method. The empiricalresults demonstrated that conditional higher moments with GARCH eﬀects improve an
investor’s utility. For most risk aversion parameters, the certainty equivalent return thatinvestors are willing to pay to switch their strategy to the portfolio with higher moments
and GARCH eﬀects is positive. This implies that information of the higher moments of the18

return has value at some situations where non-normalities are observed. The paper considers
only a simple model with constant moment matrices of the GARCH innovations. However
the dynamic conditional correlation model is applicable. Additionally, conditional skewness
and kurtosis should be topics for future research.

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21

Table 1: GARCH parameters and moment matrices of innovations
MiningIron & SteelTransportation
Equipments
Securities &Commodity Futures

µ

σ̄ 2

α

β

γ

0.00098(0.0000)
0.0011(0.0000)
0.0010(0.0000)
0.00038(0.0002)0.0021(0.0001)
0.0019(0.0002)0.0013(0.0002)0.0030(0.0001)

0.88(0.14)
0.91(0.16)
0.81(0.15)
0.91(0.14)

0.052(0.0005)
0.0738(0.0008)
0.02(0.0010)0.021(0.0005)

0.12(0.0015)0.18(0.0010)0.22(0.0017)0.017(0.0023)

1.00
0.47
0.34
0.31

(0.00)(0.00)(0.00)

1.00
0.55
0.59

(0.00)(0.02)

1.00
0.59(0.01)

1.0000

0.00
-0.11
-0.20
-0.06

(0.07)(0.02)(0.01)(0.01)

-0.04
-0.17
-0.03

(0.01)(0.01)(0.00)

-0.20
-0.06

(0.02)(0.00)

0.04(0.00)

0.21
-0.14
0.07

(0.03)(0.02)(0.01)

-0.24
-0.07

(0.01)(0.01)

0.12(0.01)

-0.18
-0.14

(0.07)(0.02)

0.06(0.01)

0.51(0.03)

Correlation matrix

Co-skewness matrix
S1

S2

S3

S4

22

Co-kurtosis matrix
K11

3.78
1.69
1.22
1.12

(3.66)(0.46)(0.28)(0.11)

1.80
1.16
1.14

(0.16)(0.19)(0.03)

1.46
0.93(0.24)(0.03)

1.38(0.01)

1.66
1.08
1.02

(0.18)(0.21)(0.03)

1.10
0.71(0.27)(0.04)

0.97(0.01)

1.17
0.74(0.48)(0.06)

0.70(0.01)

1.02(0.03)

K12

K13

K14

K21
3.70
1.93
1.96

(0.22)(0.33)(0.07)

1.93
1.36(0.45)(0.08)

1.86(0.02)

2.00
1.38(0.76)(0.12)

1.29(0.03)

1.89(0.07)

1.72(0.05)

1.85(0.03)

3.69(0.30)

K22

K23

K33
3.57
1.87

(1.51)(0.24)

K34

K44

23

Table 2: Summary statistics of the realized returns
Mean

Sd

Skew

Kurt

Max

Min

Dynamic higher moment

0.000470.0010

-0.14

1.06

0.034

-0.038

Dynamic mean-variance0.000360.00080

-0.19

0.91

0.028

-0.033

Static higher moment

0.00028

0.068

-0.41

5.55

0.043

-0.43

Static mean-variance0.00027

0.068

-0.42

5.56

0.043

-0.43

Dynamic higher moment

0.00026

0.0050

-0.13

1.06

0.017

-0.019Dynamic mean-variance0.000200.0040

-0.19

0.91

0.014

-0.016

Static higher moment

0.000160.0034

-0.41

5.55

0.043

-0.022

Static mean-variance0.000160.0034

-0.42

5.55

0.043

-0.022

Dynamic higher moment

0.000180.0033

-0.13

1.06

0.012

-0.013

Dynamic mean-variance0.000140.0026

-0.19

0.91

0.0092-0.010Static higher moment

0.000120.0023

-0.42

5.55

0.14

-0.014

Static mean-variance0.000120.0023

-0.42

5.55

0.14

-0.014λ=5

λ = 10λ = 1524

Table 3: Certainty equivalent returns and bootstrap p-values
C.E

P-value

Dynamic mean-variance0.000017

0.02

Static higher moment

0.000061

0.14

Static mean-variance0.000061

0.15

Dynamic mean-variance0.0000082

0.01

Static higher moment

0.000031

0.14

Static mean-variance0.000031

0.14

Dynamic mean-variance0.0000056

0.01

Static higher moment

0.000020

0.14

Static mean-variance0.000020

0.14

λ=5

λ = 10λ = 15

C.E means the certainty equivalent returns between
Dynamic higher moment portfolio and other portfolios.

25

20
15
Frequency
1050−5.00e−06

05.00e−06
Certainty equivalent return.00001.000015

Figure 1: Bootstrap distribution of the certainty equivalent returns

05Frequency
1015(a) Dynamic mean variance portfolio−.0001−.00005
0
Certainty equivalent return.00005.0001

Figure 2: Bootstrap distribution of the certainty equivalent returns(b) Static higher moment portfolio

26

15
10
Frequency
50−.0001−.00005

0
Certainty equivalent return.00005.0001

Figure 3: Bootstrap distribution of the certainty equivalent returns(c) Static mean variance portfolio

27