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Inﬂuence of the Outliers in the Higher-order

Portfolio Allocation

Am´elie Charlesa∗Erwan 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 pt−log pt−1, where ptis the

observed price at time t, and consider the GARCH (1,1) model

εt=ηtpht,(1)

εt|εt−1, εt−2,··· ∼ N(0,pht),

ηt∼i.i.d.N(0,1),

ht=α0+α1ε2

t−1+β1ht−1,(2)

where α0>0, α1≥0, β1≥0 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

t−1+νt−β1νt−1,(3)

where νt=ε2

t−ht. 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) = 1−β1B

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

1−β1B+π(B)ε2

t(5)

3

Similarly, the observed residuals ηtare given by

ηt=−α0

1−β1B+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,τ+k=−πk(for AO) and

x2,τ+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

t−ˆ

ht;

2. Estimate ˆτi(τ) (i= 1,2) for all possible τ= 1, . . . , n, and compute ˆτmax = max1≤τ≤n|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 e∗2

τ=e2

τ−ˆω1for an AO, and e∗2

τ+j=e2

τ+j−ˆω2ψjwith j > 0 for an IO.

The outlier corrected series e∗

tis deﬁned as

AO: e∗

t=½etfor t6=τ

sign(et)pe∗2

tfor t=τ

IO: e∗

t=½etfor t < τ

sign(et)pe∗2

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

N≥4) 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;Ωbe 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 A⊗Bis called the

Kronecker product of Aand B:

A⊗B=

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

wpI≥0, 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(wp⊗wp)¤

s.t w0

p1=1

wpI≥0, i = 1, ..., N

(11)

and the GMKP is obtained by the following optimization program:

Min

(wp)£w0

pΓ(wp⊗wp⊗wp)¤

s.t w0

p1=1

wpI≥0, i = 1, ..., N

(12)

We have:

w0

pΩwp=N

i=1 N

j=1wpiwpj σij

w0

pΣ(wp⊗wp) =N

i=1 N

j=1N

k=1wpiwpj wpksijk

w0

pΓ(wp⊗wp⊗wp) =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

N≥4) 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+δg∈Dp,g∈IR+×IR−×IR+×IR−}(14)

with: (mp= ( κ4(Rp)s3(Rp)σ2(Rp)E(Rp))

0

g= (−gκ+gs−gσ+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(wp⊗wp)

κ4(Rk)−δ κ4(Rk)≥w0

pΓ(wp⊗wp⊗wp)

w0

p1= 1

wp≥0

(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 -0.30∗5.41∗

Adjusted -0.13∗4.28∗

LVMH Unadjusted 0.19∗5.02∗

Adjusted 0.08∗3.13∗

∗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¢f−1, 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

T−1

T

i=1 ¡Ri−¯

R¢2i1/2∗√f, 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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