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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
3000
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7/00
18/0
8/00
18/0
9/00
18/1
0/00
18/1
1/00
18/1
2/00
18/0
1/01
18/0
2/01
18/0
3/01
18/0
4/01
18/0
5/01
18/0
6/01
18/0
7/01
18/0
8/01
18/0
9/01
18/1
0/01
18/1
1/01
18/1
2/01
18/0
1/02
18/0
2/02
18/0
3/02
18/0
4/02
18/0
5/02
18/0
6/02
cac cac modified
30
35
40
45
50
55
60
65
70
18/0
2/00
18/0
3/00
18/0
4/00
18/0
5/00
18/0
6/00
18/0
7/00
18/0
8/00
18/0
9/00
18/1
0/00
18/1
1/00
18/1
2/00
18/0
1/01
18/0
2/01
18/0
3/01
18/0
4/01
18/0
5/01
18/0
6/01
18/0
7/01
18/0
8/01
18/0
9/01
18/1
0/01
18/1
1/01
18/1
2/01
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
18/0
4/00
18/0
5/00
18/0
6/00
18/0
7/00
18/0
8/00
18/0
9/00
18/1
0/00
18/1
1/00
18/1
2/00
18/0
1/01
18/0
2/01
18/0
3/01
18/0
4/01
18/0
5/01
18/0
6/01
18/0
7/01
18/0
8/01
18/0
9/01
18/1
0/01
18/1
1/01
18/1
2/01
18/0
1/02
18/0
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
6/00
18/0
7/00
18/0
8/00
18/0
9/00
18/1
0/00
18/1
1/00
18/1
2/00
18/0
1/01
18/0
2/01
18/0
3/01
18/0
4/01
18/0
5/01
18/0
6/01
18/0
7/01
18/0
8/01
18/0
9/01
18/1
0/01
18/1
1/01
18/1
2/01
18/0
1/02
18/0
2/02
18/0
3/02
18/0
4/02
18/0
5/02
18/0
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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