OUTLIERS AND PORTFOLIO OPTIMISATION

Amélie CHARLES ∗

University of Montpellier I

LAMETA-CNRS

charles@lameta.univ-montp1.fr

Abstract

This paper studies the impact of the outliers on portfolio optimisation. Firstly,

we propose to detect and model four types of outliers from the CAC40 index and

three French stocks included in it (Bnp, Carrefour and Total) using TRAMO program

developed by Gomez and Maravall (1997, 2000). This method is based on the approach

proposed by Tsay (1988) and Chen and Liu (1993). We apply it on the daily closing

prices of the index and the shares from 02/21/2000 to 07/04/2002 (620 observations).

All series present outliers, some of them may be explained by economic events.

Secondly, from an estimated GARCH(1,1) model with 440 observations, we calculate

three months one-step ahead (60 business days), six months one-step ahead (120

business days) and nine months one-step ahead (179 business days) volatility forecasts.

To evaluate the volatility forecasts between unadjusted and adjusted series, we consider

the mean squared prediction error and the mean absolute error. Finally, we compute a

global minimum variance portfolio assuming a diagonal variance matrix from the three

(unadjusted and adjusted) stocks for each forecasting period. To compare the portfolio

performances with the CAC40 index for the out-of-sample period we use several

criteria as the mean return per year, the risk per year, the cumulative returns and the

return per unit of risk. Results show that it seems important to take into account outliers

in portfolio optimisation because they affect portfolio variance, portfolio structure and

portfolio performance.

∗University of Montpellier I, LAMETA, Faculté des Sciences Economiques, Espace Richter, Avenue

de la Mer, BP 9606, 34054 Montpellier cedex 1, France.

1

OUTLIERS AND PORTFOLIO OPTIMISATION

Amélie CHARLES 1

University of Montpellier I

LAMETA-CNRS

Abstract

The purpose of this paper is to study the impact of the outliers on porfolio

optimisation. We show that outliers disturb the volatility estimates and thus, the

portfolio structure. We use the method developed by Gomez and Maravall (1997,

2000) based on the approach proposed by Tsay (1988) and Chen and Liu (1993) to

correct and detect outliers in financial time series. We apply it on the daily closing

prices of the index and 3 French stocks. Then, to evaluate the volatility forecasts from

GARCH(1,1) model between unadjusted and adjusted series, we consider the mean

squared prediction error and the mean absolute error. Finally, we compute a global

minimum variance and we compare the portfolio performances using the mean return

per year, the risk per year, the cumulative returns and the return per unit of risk.

1. Université de Montpellier I, LAMETA, Faculté des Sciences Economiques, Espace Richter, Avenue

de la Mer, BP 9606, 34054 Montpellier cedex 1, France. Mail: charles@lameta.univ-montp1.fr

2

1 Introduction

According to Kendall (1953), the stock returns in the London Stock Exchange

follow a random walk. This empirical observation is the starting point of the modern

financial theory called efficient market theory. Several studies both in the american

markets (Cootner (1964), Fama (1965) and Moore (1963)) and in the european

markets (Solnik (1973), Echard (1972) and Hawawini (1985)) confirm this assumption.

Nevertheless, there are some anomalies which are unexplained, one of the most studied

in the literature is the fat-tailed observed in the statistical distributions. It is well

known that financial series put on display volatility clustering that is to say periods

where volatility is larger than in other periods. To capture some excess kurtosis, Engle

(1982) proposed the Auto-Regressive Conditional Heteroscedasticity (ARCH) model

followed by the popular Generalized Auto-regressive Conditional Heteroscedasticity

(GARCH) model developed by Bollerslev (1986)2. But, it is possible that after fitting

an ARCH model to some financial time series, the standardized residuals are not

normal and still present excess kurtosis. Two explanations may be formulated (Tolvi

(1998)): one is to assume a heavier-tailed distribution (such as Student distribution),

another is to detect and correct outliers which are present in the series. According

to Van Dijk, Franses and Lucas (1999, 2002) and Van Dijk and Franses (1999)

the observed excess kurtosis may be caused by additive outliers. More recently,

Franses and Ghijsels (1999) examine the forecasting performance of GARCH models

for unadjusted and adjusted of outliers financial series. Their results suggest that

neglecting additive outliers in series leads to bias parameter estimates and thus the

volatility forecasts.

The theory of asset pricing is based upon the reward for bearing risk. To improve

the volatility forecasts is also important for the portfolio theory. Indeed, Markowitz

(1952) proposed the expected return and the variance of the return as criteria for

optimal portfolio selection. The performance of the portfolio depends on the quality of

the forecasts of its two first moments. Nevertheless, the mean-variance analysis is very

sensitive to errors in the estimates of the inputs. Chopra and Ziemba (1993) show that

it is important to distinguish between errors in variances and covariances. According

to them, errors in covariances are the least important in terms of their influence on

portfolio optimisation. More recently, study of Pojarliev and Polasek (2001) indicates

that the weights of a global minimum variance (GMV) portfolio are very sensitive with

2. Over the past few years, several variants of the basic GARCH model have been proposed (Nelson

(1991), Glosten et al. (1993) and Zakoian (1990), among others).

3

respect to the predicted variance matrix. More precisely, the quality of the volatility

forecasts more than that of the covariance forecasts, determine the portfolio structure.

This sensitivity to the inputs may be called the butterfly effect3. In fact, small changes

in inputs may cause large changes in the optimal asset weightings.

In this paper, we propose to study the impact of the outliers on porfolio

optimisation using the method developed by Gomez and Maravall (1997, 2000) based

on the approach proposed by Tsay (1988) and Chen and Liu (1993). We show on the

daily closing prices of the index and 3 French stocks from 02/21/2000 to 07/04/2002,

that outliers bias the volatility estimates and thus, the portfolio structure. Then, to

evaluate the volatility forecasts from GARCH(1,1) model between unadjusted and

adjusted series, we consider the Mean Squared Predictions Error (MSPE) and the

Mean Absolute Error (MAE). Finally, we compute a global minimum variance and

we compare the portfolio performances using the mean return per year, the risk per

year, the cumulative returns and the return per unit of risk.

2 Identification of outliers

In this section, we present the method used to detect and correct outliers in time

series and the results obtained with financial series.

2.1 Method

Outliers may take several forms in time series. Fox (1972) is the first to propose

formal definitions and a classification of these points. The first and the most usually

studied is the additive outlier (AO) which only affects a single observation. In constrast,

an innovative outlier (IO) affects several observations. Tsay (1988) defines three other

types of outliers: a level shift (LS), a transient change (TC) and a variance change

(VC), which are more considered as structural changes.

Among the different methods used to detect and correct outliers4, we choose those

developed by Gomez and Maravall (1997, 2000) based on the approach proposed by

Tsay (1988) and Chen and Liu (1993). This procedure is incorporated in the TRAMO5

program.

Consider a univariate time serie y∗t which may be described by the ARIMA (p,d,q)

3. The butterfly effect has been an interesting concept in the development of chaos theory.4. See Tolvi (1998) for a survey of these methods.5. TRAMO: Time Series Regression with ARIMA Noise, Missing Observations, and Outliers.

4

model:

α(B)φ(B)y∗t = θ(B)at (1)

where B is the lag operator, at is a white noise process, α(B), φ(B), θ(B) are the

lagged polynomial with order d, p, q, respectively. The outliers may be modelled by

regression polynomials as follows:

yt = y∗t +∑I

ωiνi(B)It(t) (2)

where y∗t is an ARIMA process, νi(B) is the polynomial characterizing the outlier

occuring at time t = τ, ωi represents its impact on the series and It(t) is an indicator

function with the value of 1 at time t = τ and 0 otherwise. An AO may be modelled by

setting νi(B) = 1, an IO with νi(B) = θ(B)/(α(B)φ(B)), a LS using νi(B) = 1/(1−B),

and a TC by νi(B) = 1/(1−δB) where δ represents the speed of decay (0 < δ < 1).

An ARIMA model is fitted to y∗t in (1) and the residuals are obtained :

at = π(B)yt (3)

where π(B) = α(B)φ(B)/θ(B) = 1−π1B−π2B2 − . . . .

For the four types of outliers in (2), the equation in (3) becomes :

AO : at = at + ω1π(B)It(τ)

IO : at = at + ω2It(τ)

LS : at = at + ω3[π(B)/(1−B)]It(τ)

TC : at = at + ω4[π(B)/(1−δB)]It(τ)

These expressions may be viewed as a regression model for at , i.e.,

at = ωixi,t +at

with

xi,t = 0 ∀i and t < τ

xi,t = 1 ∀i and t = τ

5

and for t > τ and k ≥ 1

AO : x1,t+k = −πk

IO : x2,t+k = 0

LS : x3,t+k = 1−k

∑j=1

π j

TC : x4,t+k = δk −k−1

∑j=1

δk− jπ j −πk

The impact ωi at time t = τ of the outliers may be estimated :

AO : ω1(τ) =n

∑t=τ

at x1,t/n

∑t=τ

x21,t (4)

IO : ω2(τ) = aτ (5)

LS : ω3(τ) =n

∑t=τ

at x3,t/n

∑t=τ

x23,t (6)

TC : ω4(τ) =n

∑t=τ

at x4,t/n

∑t=τ

x24,t (7)

Chang et al. (1988) suggest to standardize ωi such that one may test for the

significance of an outlier. For this standardization, we need an estimate of the variance

of the residual process6. Now, we may calculated the standardized statistic:

AO : τ1(τ) = [ω1(τ)/σa]/( n

∑t=τ

x21,t

)1/2

IO : τ2(τ) = ω2(τ)/σa

LS : τ3(τ) = [ω3(τ)/σa]/( n

∑t=τ

x23,t

)1/2

TC : τ4(τ) = [ω4(τ)/σa]/( n

∑t=τ

x24,t

)1/2

An outlier is identified at time t = τ when the t-value of ωi exceeds a critical

value7. The different t-values of the estimators at time t = τ are compared in order

to identify the type of outlier. The one chosen has the greatest significance8such as

τmax = max|τi(τ)|. When an outlier is detected, we may adjust the observation yt at

time t = τ to obtain the corrected y∗t via (2) using the ωi obtained from (4-7), i.e.

y∗t = yt − ωiνiIt(τ). Finally, the procedure is repeated until no outlier is detected.

6. Chang et al. (1988) consider three methods for obtaining a better estimate of σ2a: (i) the median

absolute deviation method, (ii) the α% trimmed method and (iii) the omit-one method.7. The critical value is determined by the number of observations in the series.8. A multiple regression on y∗t is performed on the various outliers detected to identify spurious

outliers.

6

2.2 Application and results

We investigate the daily returns of the CAC40 French index and of 3 French

stocks included in it (Bnp, Carrefour and Total) from 02/21/2000 to 07/04/2002 (620

observations). We apply the previous procedure to detect and correct 4 types of outliers

(AO, IO, LS, TC). The results are given in Table 1. All detected outliers are given

by series, with their timing, type, value and t-statistic. The last column presents one

possible economic and financial explanation of outliers occuring. Effectively, we try

to link the date of each outlier to an economic event that occured at or near that date.

TAB. 1 – Outliers detected by series

Series Date Type Value t-Stat Events

Cac 09/11/2001 LS -324.91 -4.24 World Trade Center Attack

Bnp 04/04/2000 AO 2.31 3.94 e-banking Invesment

11/10/2000 TC -3.14 -4.11 Anxiety about loans given banking groups

09/11/2001 LS -4.46 -5.39 World Trade Center Attack

07/03/2002 IO -5.52 -6.67 Vivendi Crisis

Carrefour 03/31/2000 LS -0.11 -5.84 Internet Investment

02/25/2000 LS 0.92 4.67 Announcement of results group

Total 04/05/2000 AO -11.23 -4.26

11/15/2001 TC 8.75 4.21

The World Trade Center attack is one of the most important, it appears in 2 series

(Cac and Bnp). The French crisis supported by Vivendi Universal may justify the

IO identified 07/03/2002 in the Bnp serie. Indeed, Bnp is one of the most creditor

banks of Vivendi Universal. The AO observed 04/04/2000 may be explained by the

e-banking investment started by Bnp. Finally, the anxiety about loans given to banking

groups may explain the TC detected in Bnp. For Carrefour, it seems that internet

investments and the announcement of results in 1999 justify outliers observed in the

data. Nevertheless, no cause may be found to explain outliers detected in Total asset.

Figures 1-2 represent the closing daily prices before and after controlling for outliers.

7

Real closing daily prices are denoted by the dotted line, the full line characterize the

adjusted data.

The presence of outliers may affect the distributional properties of the series

(Franses and Ghijsels (1999)). Table 2 gives us the values of skewness, kurtosis and

Jarque-Bera statistics of the original and corrected series.

TAB. 2 – Distributional properties

Series Type Ska Kb JBc LB2(10)d

Cac unadjusted -0.10 4.16∗ 35.56∗ 119.29∗

adjusted 0.05 3.33 2.98 67.90∗

Bnp unadjusted -0.27∗ 6.33∗ 292.59∗ 36.60∗

adjusted 0.08 4.00∗ 26.59∗ 39.49∗

Carrefour unadjusted 0.03 5.06∗ 110.05∗ 48.90∗

adjusted 0.12 3.61∗ 11.03∗ 34.35∗

Total unadjusted -0.08 3.92∗ 22.53∗ 69.57∗

adjusted -0.10 3.49∗ 7.22∗ 46.51∗

∗ Significant at 5%.a The Skewness statistic is computed as follow : ν1 = SK/(

√6/n). If | ν1 |> 1.96, the null hypothesis of zero skewness

may not be accepted. b The Kurtosis statistic is computed as follow : ν2 = K/(√

24/n). If | ν2 |> 1.96, the null

hypothesis of a kurtosis of 3 may not be accepted. c The Jarque-Bera statistic is compared with χ2(2). If, JB > χ2(2),

the null hypothesis of normality may not be accepted. d The LB2(10) represent the Ljung-Box statistic for 10 lags for

the squared returns. We compare LB2(10) with χ2(10) = 18.31 at 5%.

The statistics are better if we consider adjusted series rather than unadjusted series.

We observe a drop of the kurtosis and the Jarque-Bera when the data are corrected.

Nevertheless, returns are not normal, except for the French index. It seems that outliers

cause a part of the non-normality observed in this serie9. Note also that the Ljung-Box

statistics of squared returns of adjusted and unadjusted data clearly show the existence

of conditional heteroscedasticity10 .

9. Same results have been found by Balke and Fomby (1994) among others.10. The time plots of original and corrected asset returns which are not reported to save space and

8

3 Study of Volatility forecasts

Now, we examine the impact of outliers in the volatility forecasts from a GARCH

(1,1) model. We evaluate the forecasts performance using 2 criteria : the Mean Squared

Predictions Error (MSPE) and the Mean Absolute Error (MAE).

3.1 Volatility model

The volatility forecast is one of the most important factor for modern portfolio

theory (Chopra and Ziemba (1993), Pojarliev and Polasek (2001)). Many statistical

models have been proposed to describe the behaviour of stock markets volatility

(Engle (1982), Bollerslev (1986), Nelson (1991), Glosten et al. (1993) and Zakoian

(1994), among others). In this study, we consider the GARCH model to investigate

the volatility of the financial series. The conditional variance σ2t is a linear function of

lagged conditional variances (p) and lagged squared innovations (q) defined as follows:

rt = c+at

at ∼ N(0,σt)

σ2t = α0 +

q

∑i=1

αia2t−i +

p

∑i=1

βiσ2t−i

Where rt represents the asset returns, c a constant and at is the error process given

information at time t − 1 which is normally distributed with mean zero and variance

σ2t . The parameters should satisfy α0 > 0, αi > 0 and βi ≥ 0 to guarantee that σ2

t ≥ 0.

We model the first 440 observations (from 02/21/2000 to 10/26/2001) with a

GARCH (1,1) model for each series (unadjusted and adjusted). The parameters are

estimated from the maximum likehood procedure using the BHHH algorithm.

Results of estimates are presented in the Table 3. The columns noted α1 and β1

represent the GARCH coefficients of the one-lag squared innovations and the one-lag

conditional variance respectively. All estimates are significant at 5% or at 10%. The

coefficients on the conditional variance are greater than on the squared innovations in

all cases. This implies that large market surprises induce relatively small revisions in

future volatility. If we compare the estimates of GARCH coefficients, we may observe

that they differ. For instance, the value of α1 of carrefour returns drops from 0.12 to

0.06 whereas the value of β1 increase from 0.82 to 0.91 when the serie is cleaned of

outliers. These results are similar to those obtained by Franses and Ghijsels (1999).

display volatility clustering, suggest that data have some ARCH effects.

9

TAB. 3 – GARCH (1,1) estimates

Parameters Standardized

residuals

Series α0 α1 β1 Sk K JB LB2(10)

Cac 1.09E−5

(1.26)0.07(2.26∗)

0.88(13.72∗)

-0.34∗ 3.83∗ 20.82∗ 10.95

Cacm† 6.92E−6

(0.70)0.05(2.05∗)

0.92(14.95∗)

-0.13 2.93 1.32 6.25

Bnp 1.56E−4(3.03∗)

0.17(2.43∗)

0.37(1.89∗∗)

-0.25∗ 6.25∗ 198.51∗ 4.07

Bnpm 7.44E−5(1.99∗)

0.13(2.54∗)

0.61(3.62∗)

-0.08 4.03∗ 19.88∗ 5.97

Carrefour 2.76E−5

(2.25∗)0.12(4.01∗)

0.82(17.98∗)

-0.02 3.90∗ 14.83∗ 6.18

Carrefourm 1.07E−5

(1.71∗∗)0.06(2.76∗)

0.91(30.69∗)

0.01 3.58∗ 6.28∗ 8.05

Total 4.04E−5

(1.46)0.07(2.12∗)

0.82(8.95∗)

-0.05 3.35 2.42 4.13

Totalm 4.43E−5

(1.27)0.06

(1.77∗∗)0.82(7.34∗)

-0.16 3.14 2.21 4.83

In brackets are given the Student statistic.∗ Significant at 5%, ∗∗ Significant at 10%.† It means that we consider the data corrected.

Whatever the series, the values of kurtosis and Jarque-Bera statistics are better than

those in Table 2. The Ljung-Box statistics for the squared standardized innovations are

not significant at 5%, thus the volatility equation seems adequate.

The other observations are used for the out-of-sample evaluation. We compute

three months (60 business days), six months (120 business days) and nine months

(180 business days) one-step-ahead forecasts of the conditional variance.

3.2 Forecast evaluation

To evaluate and compare the forecasts for the uncorrected and corrected data, two

forecast evaluation criteria are computed, with true volatility measured by the squared

realized returns. We summarize the forecast performance by considering the mean

10

squered prediction error (MSPE) and the mean absolute error (MAE) which are defined

as follows :

MSPE =1T

T−1

∑t=0

(σ2t − σ2

t )2

MAE =1T

T−1

∑t=0

| (σ2t − σ2

t ) |

Table 4 presents results of out-of-sample forecasting of volatility of Cac index and

the 3 stocks based on GARCH (1,1) model for the unadjusted and adjusted series and

for the 3 forecasting periods. The best criterium is given in bold face.

TAB. 4 – Forecast evaluation

Series Criteria∗ 60 120 180

business days business days business days

Cac MSPE 1.20E−3 7.17E−4 1.10E−3

MAE 2.48 1.87 2.16

Cacm† MSPE 9.03E−4 5.43E−4 8.18E−4

MAE 2.16 1.65 1.89

Bnp MSPE 9.24E−4 8.90E−4 6.38E−3

MAE 2.44 2.44 3.50

Bnpm MSPE 6.28E−4 6.16E−4 1.27E−3

MAE 1.95 2.00 2.41

Carrefour MSPE 2.62E−3 2.62E−3 2.38E−3

MAE 3.19 3.18 3.12

Carrefourm MSPE 2.63E−3 2.63E−3 2.39E−3

MAE 3.20 3.21 3.14

11

Continued

Series Criteria∗ 60 120 180

business days business days business days

Total MSPE 5.96E−3 3.29E−3 2.87E−3

MAE 4.24 3.23 3.18

Totalm MSPE 2.82E−3 1.73E−3 1.82E−3

MAE 3.51 2.89 2.96

† It means that we consider the corrected data.∗ The criteria are given for 10−4.

The best criterium is given in bold face.

An examination of results reveals that it is more usefull to use adjusted series rather

than unadjusted series for all series except for Carrefour. Indeed, the volatility forecasts

are better if this serie is not adjusted of outliers. Nevertheless, the values of the MSPE

and MAE are very close if we considered unadjusted and adjusted data. Therefore, the

LSs detected in this serie do not really affect the data. The values of these outliers

(−0.11, 0.92) are smaller than that detected in the French index (−342.91). It is

interesting to note also that the corrected data outperform the unadjusted data in 9

of the 12 cases (75%), whatever the forecasting horizons.

4 Portfolio construction

We treat in this section the problem of portfolio selection proposed by Markowitz

(1952) and we examine the portfolio evaluation.

4.1 Portfolio selection

The portfolio selection problem suggests a trade-off between returns and risk.

Generally, investors prefer large returns and lower risk and they will accept more

risk only if they get higher returns as compensation. The portfolios chosen by them

will be on the mean-variance frontier (Markowitz (1952)). We consider the global

minimum variance portfolio [henceforth GMV portfolio] which gives the smallest

possible variance and does not depend on the expected returns. In this case, it is

12

important to have good estimates of variances and covariances11. Chopra and Ziemba

(1993) show that errors in covariance are the least important in terms of their influence

on portfolio optimisation. Pojarliev and Polasek (2001) indicate that the weights

of a GMV portfolio depend more on volatility forecasts than covariance forecasts.

Consequently, we assume a diagonal variance matrix (by setting covariances equal to

zero) for the 3 stocks to compute the weights in portfolios. This yields a quadratic

programming problem12:

Min α′ Σ α

under constraints α′I = 1

α ≥ 0

where α represents the stock weights, Σ the covariance matrix and I a unit vector.

We avoid short positions applying α ≥ 0.

We construct GMV portfolios, named GMV, using the volatility forecasts for each

forecasting period and for both unadjusted and adjusted series.

Figures 3-5 (given in Appendix) show the weights of GMV portfolios for the

out-of-sample periods. The dotted lines represent the weights of unadjusted data and

the full line the weights of adjusted data. We note that weights are not stable over

time, they fluctuate. They seem sensitive to the forecasted volatility. Bnp share has a

weight more important in the GMV portfolio if the data are beforehand corrected of

outliers. Outliers which are present in the Carrefour and Total shares tend to increase

the weights of these two shares in the GMV portfolio. If the data are cleaned of outliers,

the weight of each share in the porfolio decreases. Notice that for a long forecasting

period, these fluctuations are more important than for a short forecasting period. Table

5 gives, in average, the weights for each stock in the GMV portfolio for each out-of-

sample.

The structure of portfolio is not the same if we use unadjusted or adjusted data.

The weights of Bnp in the GMV portfolio increase if the data are corrected. Outliers

detected in Carrefour and Total tend to decrease weights of these series in the GMV

portfolio. The weights are really sensitive to the presence of outliers in the data.

Therefore, the forecasting of the conditional variance is very important to determine

the portfolio structure and to evaluate its performance (Pojarliev and Polasek (2001)).

11. Modeling volatility of returns is fundamental for risk management, ARCH models have been

developed to measure the price of risk.12. Additional constraints may be added to further restrict the investment choices.

13

TAB. 5 – Structure of portfolios

Portfolio Bnp (%) Carrefour (%) Total (%)

GMV60 38.44 32.02 29.54

GMV60m∗ 42.90 28.63 28.47

GMV120 35.21 31.56 33.23

GMV120m 39.85 28.81 31.34

GMV180 33.63 32.69 33.68

GMV180m 38.41 29.96 31.63

∗ 60m means that we use the corrected data to built the portfolio from 60 volatility forecasts.

4.2 Portfolio evaluation

We compare the performance of the portfolios for the forecasting periods. Table 6

summarizes the following criteria used by Pojarliev and Polasek (2000):

1. mean return per year (%),

2. risk per year (%),

3. cumulative return (%),

4. return per unit of risk.

From Table 6, we may make several remarks. Firstly, all GMV portfolio beat the

Cac40 index whatever the number of forecasts. Secondly, the criteria used to evaluate

the portfolio performance are not equal if we consider unadjusted and adjusted series.

They are better if data are adjusted to outliers rather than if they are not unadjusted.

Consequently, the outliers detected in financial series have an impact on criteria.

Indeed, these points tend to decrease mean return per year, to increase mean risk per

year and thus, to decrease the return per unit of risk. Note that we do not reported

historical portfolio based on the same out-of-sample to save space13. Nevertheless,

GMV portfolios give us the best results.

13. Results of historical portfolio are available from the author upon request.

14

TAB. 6 – Portfolio evaluation for each out-of-sample

Series Mean return Risk Cumulative Return

per year (%) per year (%) returns (%) per unit of risk

Cac60 0.16 29.36 0.03 0.005

GMV60 10.89 19.79 1.79 0.550

GMV60m† 11.57 18.65 1.90 0.620

Cac120 2.45 25.24 0.80 0.097

GMV120 11.98 19.47 3.94 0.615

GMV120m 12.70 18.67 4.17 0.680

Cac180 -35.18 28.01 -17.25 -1.256

GMV180 11.69 19.53 5.73 0.598

GMV180m 12.50 18.79 6.13 0.665

† It means that we use the corrected data to built the portfolio from 60 volatility forecasts.

The best criterium is given in bold face.

5 Conclusion

In this paper we study the impact of outliers on GMV portfolio. From the method

developed by Gomez and Maravall (1997, 2000), we detect and correct outliers

in Cac40 French index and in 3 French stocks included in it. It appears that all

financial data present outliers whose some of them may be explained by economic

and financial events. We calculate the conditional volatility forecast for 60 business

days, 120 business days and 180 business days, using GARCH(1,1) model with 440

observations. As suggested by Franses and Ghijsels (1999), we show that outliers

disturb the volatility estimates. Indeed, it seems better to correct outliers before

forecasting volatility than used unadjusted series. To evaluate the error, we compute the

MSPE and the MAE. Finally, we examine the impact of ouliers on the global variance

minimum portfolio structure. The weights of each stocks are significantly different if

the series are beforehand ajusted or not. Moreover, portfolio evaluation is better for

adjusted data rather than for unadjusted data. Consequently, it seems important to take

into account outliers in portfolio optimisation because they affect portfolio variance,

weights of portfolio and portfolio evaluation.

15

Appendix

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2/00

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5/02

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6/02

cac cac modified

30

35

40

45

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55

60

65

70

18/0

2/00

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18/0

1/02

18/0

2/02

18/0

3/02

18/0

4/02

18/0

5/02

18/0

6/02

bnp bnp modified

FIG. 1 – Closing daily prices for the CAC40 index and bnp share

16

40

45

50

55

60

65

70

75

80

85

9018

/02/

00

18/0

3/00

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1/02

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2/02

18/0

3/02

18/0

4/02

18/0

5/02

18/0

6/02

carrefour carrefour modified

120

130

140

150

160

170

180

190

18/0

2/00

18/0

3/00

18/0

4/00

18/0

5/00

18/0

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2/02

18/0

3/02

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4/02

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5/02

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6/02

total total modified

FIG. 2 – Closing daily prices for carrefour and total shares

17

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

wei

gh

ts

Bnp Bnpm

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

wei

gh

ts

Carrefour Carrefourm

0.15

0.2

0.25

0.3

0.35

0.4

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

wei

gh

ts

Total Totalm

FIG. 3 – Weights of GMV for 60 forecasting

18

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101

105

109

113

117

wei

gh

ts

Bnp Bnpm

0.15

0.2

0.25

0.3

0.35

0.4

0.45

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101

105

109

113

117

wei

gh

ts

Carrefour Carrefourm

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101

105

109

113

117

wei

gh

ts

Total Totalm

FIG. 4 – Weights of GMV for 120 forecasting

19

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101

106

111

116

121

126

131

136

141

146

151

156

161

166

171

176

wei

gh

ts

Bnp Bnpm

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101

106

111

116

121

126

131

136

141

146

151

156

161

166

171

176

wei

gh

ts

Carrefour Carrefourm

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101

106

111

116

121

126

131

136

141

146

151

156

161

166

171

176

wei

gh

ts

Total Totalm

FIG. 5 – Weights of GMV for 180 forecasting

20

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23