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In this paper, we try to evaluate whether the presence of abnormal observations in the returns may afiect the higher-moment portfolio allocation. From the methodology developed by Charles and Darne (2005), we detect, explain and correct outliers on 50 shares among the most liquid on the French Stock Market. Then we determine optimal portfolio allocation according to the investor select assets from a unique moment preference - global minimum variance, global maximum skewness, global minimum kurtosis - or that he considers multi- moments preference. Moreover, portfolio evaluation is better for adjusted data rather than for unadjusted data when we consider the global minimum variance, the global minimum kurtosis and the multi-moments portfolios. Consequently, it seems important to take into account outliers in portfolio optimisation because they may afiect the portfolio evaluation.
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Influence of the Outliers in the Higher-order
Portfolio Allocation
Am´elie CharlesaErwan Le Saouta,b Ghislain Yanouc
April 25, 2008
aUniversit´e Paris 1 Panth´eon-Sorbonne, 17 rue de la Sorbonne, 75006 Paris.
bESCEM, 1 rue L´eo Delibes, 37200 Tours.
cUniversit´e Paris 1 Panth´eon-Sorbonne, 106-112 Bld de l’hˆopital, 75013 Paris.
Abstract
In this paper, we try to evaluate whether the presence of abnormal observations in the
returns may affect the higher-moment portfolio allocation. From the methodology developed
by Charles and Darn´e (2005), we detect, explain and correct outliers on 50 shares among the
most liquid on the French Stock Market. Then we determine optimal portfolio allocation
according to the investor select assets from a unique moment preference - global minimum
variance, global maximum skewness, global minimum kurtosis - or that he considers multi-
moments preference. Moreover, portfolio evaluation is better for adjusted data rather than
for unadjusted data when we consider the global minimum variance, the global minimum
kurtosis and the multi-moments portfolios. Consequently, it seems important to take into
account outliers in portfolio optimisation because they may affect the portfolio evaluation.
Keywords: outliers, portfolio allocation, higher moments.
Classification J.E.L. : G11, G12.
Corresponding author: E-mail: amelie.charles@univ-paris1.fr
1
1 Introduction
The mean-variance methodology originally proposed by Markowitz (1952) plays a crucial
role in the theory of portfolio selection and gains widespread acceptance as a practical tool
for portfolio optimization. Numerous studies on portfolio selection have been made based
on only the two first moments of return distributions. Despite its universality, the standard
mean-variance portfolio theory suffers from well-known limits; asset and portfolio return
distributions are characterized by strong departures from Gaussian distribution (skewness
and leptokurticity), investors tend to display preferences for non-convex option-like pay-
offs (downside risk aversion and outer risk aversion), volatility might not be a sufficient
statistic for measuring market risk under some circumstances. These facts advocate for an
extension of the mean-variance framework for taking into account individual preferences for
higher-order moments. Several alternative approaches have been developed in the financial
literature to incorporate the individual preferences for higher-order moments into optimal
asset allocation problems (Lai, 1991, Chunhachinda et al, 1997, Wang and Xia, 2002, Chang
et al, 2003, Sun and Yan, 2003, Jondeau and Rockinger, 2004 and Jurczenko and Maillet,
2006). These studies show that it seems more pertinent for an individual investor to take
into account the higher moments in their individual choice rather than only the two first
moments.
Nevertheless, these studies do not take into account the presence of outliers, which are
defined as economic, political and financial events and which have been observed in financial
data (Charles and Darn´e, 2006). More precisely, outliers may cause excess kurtosis in time
series (Fiorentini and Maravall, 1996 and Ruiz, Lorenzo and Carnero, 2001, among others.).
In general, outliers may affect the returns distributions and thus the computation of the mo-
ments of the distribution. Verhoeven and McAleer (2000) have shown that adjusting outliers
makes the distribution more normal, reducing negative skewness and excess kurtosis.
Consequently, in this paper, we tried to understand the influence of outliers in the port-
folio allocation. We suppose that market is composed by four class of investors. The first
group only care about the reduction of their portfolio volatility, and will invest on the Global
Minimum Variance Portfolio (henceforth GMVP). The second group of investors have pref-
erences for assets which historically have more returns on the right hand rather than left
of the sample mean, for them the Global Maximum Skewness Portfolio (henceforth GMSP)
will be the most appropriate investment. The third group of investors have preferences only
for the temperance and consequently will invest on the Global Minimum kurtosis Portfo-
lio (henceforth GMKP). The three investor’s class above care only for an unique moment,
contrary of the last group of investors who care as good as for the risk aversion than the
prudence and temperance. Therefore, we consider a non-parametric optimization for the
portfolio selection problem from Jurczenko et al. (2006) the multi-moment approach.
The outline of this paper is as follows. Section 2 describe the method developed by Charles
and Darn´e (2005) to detect and correct the outliers which are present in the series. Section
3 describes different class of investments : the case of investors who select a portfolio from
2
an unique moment preference and the case of investors who consider multi-moment prefer-
ences. The empirical results are presented in Section 4. The last section provides concluding
remarks.
2 Detection of Outliers in GARCH Models
Consider the returns series εt, which is defined by εt= log ptlog pt1, where ptis the
observed price at time t, and consider the GARCH (1,1) model
εt=ηtpht,(1)
εt|εt1, εt2,··· ∼ N(0,pht),
ηti.i.d.N(0,1),
ht=α0+α1ε2
t1+β1ht1,(2)
where α0>0, α10, β10 and α1+β1<1, such that the model is covariance-stationary.
The GARCH (1,1) model can be rewritten as an ARMA (1,1) model for ε2
t(see Bollerslev,
1986)
ε2
t=α0+ (α1+β1)ε2
t1+νtβ1νt1,(3)
where νt=ε2
tht. As shown by Franses and Ghijsels (1999), we exploit this analogy of
the GARCH model with an ARMA model to adapt the method of Chen and Liu (1993)
to detect and correct additive outliers [AO] and innovative outliers [IO] in GARCH models.
Specifically, suppose that instead of the true series εtone observes the series etthat is defined
by
e2
t=ε2
t+ωiξi(B)It(τ) with i= 1,2,(4)
where It(τ) is the indicator function defined as It(τ) = 1 if t=τand zero otherwise, with τ
the date of outlier occurring, ωiand ξi(B) denote the magnitude and the dynamic pattern
of the outlier effect, respectively. For the additive and innovative outliers, their dynamic
pattern is defined as
AO: ξ1(B)=1
IO: ξ2(B) = 1β1B
1(α1+β1)B=ψ(B) = π(B)1
An AO is related to an exogenous change that directly affects the series and only its
level of the given observation at time t=τ. An IO is possibly generated by an endogenous
change in the series, and affects all the observations after time τthrough the memory of the
process.
We can write the equation (3) as
νt=α0
1β1B+π(B)ε2
t(5)
3
Similarly, the observed residuals ηtare given by
ηt=α0
1β1B+π(B)e2
t=νt+π(B)ωiξi(B)It(τ) (6)
The expression (6) can be interpreted as a regression model for ηt, i.e.
ηt=ωixit +νt(7)
with xit = 0 for i= 1,2 and t < τ ,xit = 1 for i= 1,2 and t=τ,x1+k=πk(for AO) and
x2+k= 0 (for IO) for t > τ and k > 0.
Detection of outliers is based on the following statistics :
AO: ˆτ1(τ) = ¡ˆω1(τ)¢/ˆσν¢³n
X
t=τ
x2
1t´1/2=³³ n
X
t=τ
x1tηt´.ˆσν´³ n
X
t=τ
x2
1t´1/2
IO: ˆτ2(τ) = ˆω2(τ)/ˆσν=ητ/ˆσν(8)
where ˆσ2
νdenotes the estimated variance of the residual process.
The outlier detection method for GARCH (1,1) models then consists of the following
steps.
1. Estimate the GARCH (1,1) model for the observed series etand obtain estimates of
the conditional variance ˆ
htand ˆηt=e2
tˆ
ht;
2. Estimate ˆτi(τ) (i= 1,2) for all possible τ= 1, . . . , n, and compute ˆτmax = max1τn|ˆτi(τ)|.
If the value of the test-statistic exceeds the pre-specified critical value C, an outlier is
detected at the observation for which ˆτi(τ) is maximized.
3. Replace e2
twith e2
τ=e2
τˆω1for an AO, and e2
τ+j=e2
τ+jˆω2ψjwith j > 0 for an IO.
The outlier corrected series e
tis defined as
AO: e
t=½etfor t6=τ
sign(et)pe2
tfor t=τ
IO: e
t=½etfor t < τ
sign(et)pe2
tfor t=τ+j,j > 0
4. Return to step (1) to estimate a GARCH (1,1) model for the series e
t, and repeat all
steps until no ˆτmax test-statistic exceed the critical value C.
A critical value C= 10 is used. This choice for Cis based on simulation experiments
proposed by Franses and van Dijk (2000). The authors simulate some percentiles of the
distribution of the ˆτmax-statistic under the null hypothesis that no outliers are present for
several values of ARCH and GARCH parameters and for two sample sizes (250 and 500). It
is seen that the value of C= 10 is reasonably close to the 90th percentile of this distribution
for most parameter combinations.
4
3 Portfolio Selection
We propose different methodologies to measure the effect of outliers on investors allocation
who care only for a unique moment and for investors who care for the whole moments1.
3.1 Case of Investors with an Unique Moment Preference
We consider the problem of many investors selecting a portfolio from Nrisky assets (with
N4) in the variance-skewness-kurtosis framework. The first group of investors care
about the minimization of the portfolio volatility, while the second group is interested by
the maximization of the portfolio asymmetry and the third, care about the temperance and
want to minimize the portfolio kurtosis. We assume that investors do not have access to a
riskless asset, implying that the portfolio weights must sum to one. In addition we impose
a no short-sale portfolio constraint: asset positions must be non-negative. Let wpand E
denote respectively the (N×1) vector of weights and of expected returns for the Nrisky
assets in the portfolio p;be the non-singular (N×N) variance-covariance matrix of the
risky assets; and Σand Γrepresent respectively the (N×N2) skewness-coskewness matrix
and the (N×N3)kurtosis-cokurtosis matrix of the Nrisky asset returns, defined such as
(Athayde and Flˆores, 2004 and 2005 ):
Σ
(N×N2)= (Σ1Σ2···ΣN)
Γ
(N×N3)= (Γ11Γ12 ···Γ1N|Γ21Γ22 ···Γ2N|...|ΓN1Γ12 ···ΓN N )(9)
where Σkand Γkl are the (N×N) associated sub-matrixes of Σand Γ, with elements (sijk)
and (κijkl ), with (i, k, l, k)(I N )4, and the sign stands for the symbol of the Kronecker
product2. It should be noticed that, because of the symmetries, not all the elements of
these matrixes need to be computed. The dimension of the variance-covariance matrix is
(N×N) but only N(N+ 1) /2 of its elements must be computed.Similarly the skewness-
coskewness and kurtosis-cokurtosis matrices have dimensions (N×N2) and (N×N3), but
only N(N+ 1) (N+ 2) /6 and N(N+ 1) (N+ 2) (N+ 3) /24 elements are independent3.
The allocation program of the first pool of investors derives directly from the most known
Markowitz (1952) portfolio optimization program which consider a mean-variance frame-
1Understand by ”whole moments”, the mean, the variance, the skewness and the kurtosis.
2Let Abe an (n×p) matrix and Ban (m×q) matrix. The (mn ×pq) matrix ABis called the
Kronecker product of Aand B:
AB=
a11Ba12 B··· a1NB
a21Ba22 B··· a2NB
.
.
..
.
.....
.
.
aN1BaN2B··· aN N B
where the sign stands for the Kronecker symbol product.
3For N= 4, whereas these matrices have respectively 16, 64 and 256 terms, 10 different elements for
the variance-covariance matrix, 20 elements for the skewness-coskewness matrix and 35 elements for the
kurtosis-cokurtosis matrix are to be computed.
5
work, by no considering the mean. The GMVP is obtained by the following optimization
program:
Min
(wp)¡w0
pΩwp¢
s.t w0
p1=1
wpI0, i = 1, ..., N
(10)
where 1denotes the (N×1) of ones.
Following the approach of Athayde and Flˆores (2004, 2005), the GMSP is obtained by
the following optimization program:
Max
(wp)£w0
pΣ(wpwp)¤
s.t w0
p1=1
wpI0, i = 1, ..., N
(11)
and the GMKP is obtained by the following optimization program:
Min
(wp)£w0
pΓ(wpwpwp)¤
s.t w0
p1=1
wpI0, i = 1, ..., N
(12)
We have:
w0
pΩwp=N
i=1 N
j=1wpiwpj σij
w0
pΣ(wpwp) =N
i=1 N
j=1N
k=1wpiwpj wpksijk
w0
pΓ(wpwpwp) =N
i=1 N
j=1N
k=1N
l=1wpiwpj wpkwplκij kl
(13)
with (i, j, k, l)[1, ..., N ]4: where (wpi), (σij ), (sij k) and (κijkl) represent, respectively, the
weight of the asset iin the portfolio p, the covariance between the returns of asset iand
j, the coskewness between the returns of asset i,jand kand the cokurtosis between the
returns of asset i,j,kand l, with (i×j×k×l) = (IN)4.
For the optimization programs above, there is no exist a parametric solution, the optimal
portfolio need to be approached. In the case of the GMVP we can use a quadratic optimiza-
tion procedure and following Lai (1991) and Chunhachinda et al. (1997), we can approach
optimal portfolio for the higher moment framework by a PGP procedure.
3.2 Case of Investors with a Multi-Moment Preference
We consider the problem of many investors selecting a portfolio from Nrisky assets (with
N4) in the variance-skewness-kurtosis framework. These investors haven’t a specifical
moment preference, they would be interested by the first, second, third and fourth moments.
It’s more difficult in this case to measure influence of outliers on the investment preferences.
Jurczenko, Maillet, and Merlin (2006)4proposes a multi-moment efficient frontier based on
4Jurczenko, Maillet, and Yanou (2007) proposes an extension of this approach to the L-Moments which
is a more robust measure of moments, see Hosking (1989).
6
a shortage function coming from the production theory, where investors have simultaneous
a preference for greediness, risk aversion, prudence and temperance.
In production theory, the shortage function measures the distance between some point of
the production possibility set and the efficient production frontier (Luenberger, 1995). The
properties of the set of portfolio return moments on which the shortage function is defined
have been already discussed in the mean-variance plane by Briec et al. (2004) and in the
mean-variance-skewness space by Briec et al. (2007). It is now possible to extend their
definitions to get a portfolio efficiency indicator in the four-moment case.
The shortage function associated to a feasible portfolio pwith reference to the direc-
tion vector gin the mean-variance-skewness-kurtosis space is the real-valued function Sg(.)
defined as:
Sg(wp) = sup {δ:mp+δgDp,gIR+×IR×IR+×IR}(14)
with: (mp= ( κ4(Rp)s3(Rp)σ2(Rp)E(Rp))
0
g= (gκ+gsgσ+gE)0
where gis the directional vector in the four-moment space, κ4(Rp), s3(Rp), σ2(Rp), E(Rp)
denote respectively the fourth, third, second and first moment of a feasible portfolio repre-
sented by Dp. They propose an optimal allocation in an mean-variance-skewness-kurtosis
framework by the following program:
w
p=Arg
wp{Max δ}
s.t.
E(Rk) + δ E (Rk)w0
pE
σ2(Rk)δ σ2(Rk)w0
pΩ wp
s3(Rk) + δ s3(Rk)w0
pΣ(wpwp)
κ4(Rk)δ κ4(Rk)w0
pΓ(wpwpwp)
w0
p1= 1
wp0
(15)
4 Data and Empirical Results
In this section, we apply the previous procedure to detect and correct AOs and IOs in the
returns. We extract from the database EcoWin Reuters daily prices of 50 liquid shares
included in SBF120 - on the period from January 1st, 1999 to December 31st, 2006 (that it
represents 102 500 quotes). No completion have been made, and we only take into account
the data returns adjusted from dividend available from the starting date to the end date,
this is why we just have in our database 50 assets.
7
Figure 1: The SBF 120 index
1 500,00
2 000,00
2 500,00
3 000,00
3 500,00
4 000,00
4 500,00
5 000,00
31/12/97
30/06/98
31/12/98
30/06/99
31/12/99
30/06/00
31/12/00
30/06/01
31/12/01
30/06/02
31/12/02
30/06/03
31/12/03
30/06/04
31/12/04
30/06/05
31/12/05
30/06/06
31/12/06
30/06/07
31/12/07
The evolution of the French stock market can be decomposed into three movements. The
first phase, a bullish trend, was initiated in October 1998 following the Russian and Asian
financial crisis which took place in summer 1998. Index SBF 120 strongly progressed until
September 2000 and reached an historical level. This period corresponds to the rise of the
shares belonging to the sector of new technologies. The second phase, a bearish tendency,
corresponds to the bursting of Internet bubble and the consequences of attacks against the
World Trade Centre. This phase ended in March 2003. Finally the last period correspond
to an increase of the stock exchange which will be completed during the year 2007 and the
first sudden starts of the subprimes’ crisis. The existence of various tendencies during study
period allows us to consider that our results could be generalized to all periods.
As expected with financial data, the statistics for normality indicate that none of the
time series is normally distributed (Table 1)5. After the outlier correction these measures
of non-normality are decreased, sometimes quite dramatically. Adjusting outliers makes
the distribution of the standardized residuals more normal, reducing skewness and excess
kurtosis. This is an important result for portfolio optimization.
5All results are not given to save space but they are available from the authors upon request.
8
Table 1: Descriptive statistics
Series Type Sk Ku
Accor Unadjusted -0.305.41
Adjusted -0.134.28
LVMH Unadjusted 0.195.02
Adjusted 0.083.13
and ∗∗ mean significant at the 5% and 10% level, respectively.
4.1 Detection of abnormal returns
The procedure employed has detected 175 outliers over our period of study. This corresponds
to a rate of appearance of 0,171%. Table 1 indicates, in the one hand, that there are as many
negative outliers than positive outliers. On the other hand, we can notice that there are more
AO than IO (Table 2).
Table 2: Descriptive statistics of the outliers
Sign of return AO IO Total
Positive 63 27 90
Negative 55 30 85
Total 118 57 175
We gathered the origin of the outliers in three categories: the outliers which are specific to
the company, the outliers occurring in a noisy market and the outliers which have a sectoral
origin. Table 3 proposes an extract of our work of classification.
The explanations of the outliers which are specific to the company can be divided into five
subcategories: the first subcategory gathers the outliers occurring at the time of the publi-
cation of PL or the sales turnovers (realized or prospect). Other subcategories correspond
to the outliers appearing with an advertisement of a modification of the business portfolio of
the company (transfer or acquisition), announcement of corporate actions (capital increase,
rights issue, debt offering, stock-repurchase program...), advertisement or rumours relative
to modification of the shareholding (takeover bid, arrival of new shareholders). Finally, a
last subcategory gathers other events such as index additions and deletions, the change of
president o the company, strikes, lawsuits lost or wined, or recommendations of financial
analysts.
We decomposed the outliers which occur in a noisy market in three subcategories. The first
subcategory corresponds to the outliers known as event-driven: the second war in Iraq, the
Asian crisis, attacks against World Trade Centre and the war in Afghanistan. The second
9
Table 3: Outliers
Date Return Company t-stat Type Origin Explanation
04/18/2001 13,22% Dassault 10,0913 AO Market FED lowers interest rates
09/11/2001 -10,14% Suez -11,2650 AO Market World Trade Center
25/07/2002 10,28% Eurazeo 13,9279 IO Market Pull back
30/09/1998 -8,48% Michelin 10,2505 IO Sector Portfolio rebalancement
10/08/1998 -10,03% L’Or´eal 11,2724 AO Sector Lower dollar
03/14/2003 9,49% Carrefour 11,4730 AO Sector P&L concurrent Casino
06/26/2002 -18,00% Alcatel 13,8330 IO Share Profit Warning
08/22/2002 16,49% PPR 26,0510 AO Share PPR sells Guilbert
09/04/2002 24,46% Soitec 14,4621 AO Share Earned trial
01/28/2003 -9,25% Fimalac 12,2982 IO Share Capital increase
07/25/2005 -7,82% Danone 11,9400 AO Share Contradicted Takeover Bid
subcategory corresponds to the variations of market which occur following publication of
statistics or speeches leading to anticipations on the evolution of the interest rates (Index of
Michigan, ISM, Beige Book...) and to strong variations of the exchange rates (Monica gate
for example). The last subcategory gathers other phenomena such as strong profit-takings
or strong pull back following a long period of rise or fall of the markets.
Three subcategories are also proposed for the outliers which have a sectoral origin. A first cor-
responds to the outliers appearing following announcements made by a competitor (turnover,
PL, mergers and acquisitions). A second corresponds to portfolio rebalancements which in-
tervened during Internet bubble and its bursting. Finally the last subcategory corresponds
to other cases such as variations observed on values involved in debt or subjected to exchange
rates.
10
Table 4 reports the distribution of the outliers. This enables to note that the outliers
appear mainly for reasons specific to the company (61%) in particular at the time of the
publication of turnovers or PL (34%). A quarter of the outliers appear when the markets
know a strong level of volatility. It is within this category that one finds the second principal
origin of outliers: international conflicts in particular the attack of World Trade Centre which
led to high volatility during one month. Events represent nearly 19%. The sectoral origin
represents a relatively weak percentage (11%). This is explained by the fact that a variation
in one or two sector often involves a variation of the market as a whole. In this category,
outliers caused by Internet bubble is the most representing with more than 7%.
Table 4: Distribution of outliers following origins
Category Subcategory Numbers Percentage
Share Business Portfolio 11 6.29%
Share Corporate actions 2 1.14%
Share Others 26 14.86%
Share Publication 59 33.71%
Share Shareholding 8 4.57%
Share Subtotal 106 60.57%
Market Event 33 18.86%
Market Macro 7 4.00%
Market Others 9 5.14%
Market Subtotal 49 28.00%
Sector Concurrence 5 2.86%
Sector Internet 13 7.43%
Sector Others 2 1.14%
Sector Subtotal 20 11,43%
Total 175 100%
4.2 Portfolio Allocation
An illustration of the GMVP, GMSP, GMKP and the multi-moments portfolios (henceforth
MMP) computed from unadjusted data and adjusted data is presented. An estimation
window of two years is considered6. We also compare a set of portfolio indicators computed
from unadjusted and adjusted data. The notation GMVP(0), GMSP(0), GMKP(0), MMP(0)
are used when we refer to the global portfolio computed from adjusted data and GMVP(1),
GMSP(1), GMKP(1) , MMP(1) when we consider unadjusted data. The MMP are obtained
by using the shortage function, with a goal sets by the GMVP(0), GMSP(0), GMKP(0) for
6The choice of this window prevents that the results are skewed by the existence of a bullish or a bearish
trend.
11
unadjusted and adjusted sample data.7The empirical protocol is the following:
consider data returns from 01/04/1999 to 01/03/2000, we compute the optimal allo-
cation and buy the corresponding portfolio one month (twenty days)8after,
slide the estimation window for one week (five days), that is we have a new estimation
window from 01/11/1990 to 01/10/2000,
compute a new optimal allocation from the new estimation window,
perform the algorithm until the 12/31/20069.
We compute three statistics indicators which characterize the portfolio performance, diver-
sification and stability. For the portfolio performance we consider the Sharpe Ratio denotes
by SR and defines like:
SR =AM R
ASD (16)
with AMR10 and ASD11 represent the annualized mean return and the standard deviation
of the optimal portfolio, respectively
Concerning the portfolio diversification, we use the effective size denotes by ES, which
measures the effective number of assets taken into account in the optimal allocation:
ES =1
T
T
X
i=1
1
N
P
j=1 ¡w
i,j¢2
(17)
where w
ij denotes for the date of estimation i, the optimal allocation for asset j, and N
denotes the number of assets in the investment universe. If the optimal allocation is the
naive allocation, the effective size set to N, and at the contrary, in the case where only one
asset constitutes the optimal portfolio, th effective size set to one.
The stability of portfolio is measure by the turnover denotes by T R:
7Intuitively, the multi-moment portfolio obtained by unadjusted data should perform less than the one
obtained by the adjusted data by construction, and the illustration of this portfolio does not give us any
information.
8We consider monthly portfolio rebalancement. More frequent rebalancement involves high transaction
costs.
9Because the decision of investment is made one month before, the computation ends before the end date
of the sample considered.
10The annualized mean return can be obtained by the following expression: AM R =¡1 + ¯
R¢f1, where
¯
Rdenotes the sample mean return and fthe frequency period per year.
11The annualized standard deviation can be obtained by the following expression: ASD =
h1
T1
T
i=1 ¡Ri¯
R¢2i1/2f, where Ridenotes the portfolio returns along the estimation period, and T
the size of the estimation period.
12
T R =1
2T
T
X
i=1
N
X
j=1 ¯¯w
i+1,j w
i,j¯¯(18)
where w
i+1,j denotes the optimal allocation at the date i+1 and w
i,j the optimal portfolio at
the date ifor asset j. We also compute the skewness and the kurtosis of the whole portfolio
4.3 Comments
Table 5 displays the results of the portfolio analysis. Results indicate that the MMP0,
computed from adjusted data, performs well than the others portfolios in term of Sharpe
ratio (the value is equal to 0.67), the diversification (the corresponding average effective size
is equal to 50 per cents) and the portfolio stability (the corresponding average effective size
is equal to 49 per cents). For the same level of diversification and portfolio stability, the
GMVP(0), GMKP(0) portfolios display a more interesting Sharpe ratio than the GMVP(1),
GMSP(1) obtained from unadjusted data12.
Concerning the effet size and turnover ratio, we made a test of equality. Results which are
not reported here, indicate that these ratio are not different when they are computed from
adjusted or unadjusted data.
Table 5: Portfolio analysis
Annualized Annualized Sharpe Skewness Kurtosis Effet Turnover
mean st. dev. ratio size
GMVP13.76% 6.83% 0.55 -0.55 4.07 0.311 1.412
GMVP04.06% 6.82% 0.60 -0.26 2.26 0.305 1.423
GMSP110.60% 21.92% 0.48 -1.14 10.34 0.052 1.902
GMSP012.36% 28.74% 0.43 -0.02 2.33 0.053 1.894
GMKP13.36% 7.10% 0.47 -0.54 3.25 0.280 1.470
GMKP04.08% 7.05% 0.58 -0.26 1.96 0.275 1.483
MMP15.26% 8.80% 0.60 -0.13 7.15 0.526 0.495
MMP06.05% 9.02% 0.67 -0.08 5.46 0.523 0.496
The best information criteria are given in bold face.
An interesting observation comes from the comparison of the moment which character-
izes the global portfolio computed from adjusted and unadjusted data. It seems that the
GMVP(0) displays the most lower annualized standard deviation. The GMSP(0) displays the
most higher skewness. This result is not surprising because we have detected more positive
outliers than negative outliers. In fact, we may explain this result by the fact that return (in
absolute value) of the negative outliers are higher than return of the positive outliers. More-
over, we note that the lowest kurtosis is done for the GMKP(0). This result was expected
12This observation is not true concerning the GMSP(0), which performs less than the GMSP(1) .
13
because the tails of distributions of the values composing GMKP(0) are less thick (Table 1).
Consequently the GMKP (1) seems more influenced by the abnormal observations.
5 Conclusion
In this paper, we analyse the effects of outliers in the higher-order portfolio allocation. In
a fist step, following the methodology developed by Charles and Darn´e (2005), we detect
and correct outliers on shares listed in Paris Stock Exchange. We note that the outliers are
mainly explained by reasons specific to the company appearing in our sample. The outliers
belong more the individual risk of than to the market risk. In a second step, we consider
several cases of investor’s preferences in order to identify the influence of outliers on different
class of investment: the case of investors with a unique moment preference - global minimum
variance, global maximum skewness, and global minimum kurtosis - and the case of investors
with multi-moments preference. We observe that the performance of the monthly wallets,
measured by a Sharpe Ratio, is higher for the portfolios built from the corrected data in
three cases out of the four cases studied. Consequently, it seems important to take into
account outliers in portfolio optimisation because they may affect the portfolio evaluation.
14
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