Portfolio risk evaluation: An approach based on dynamic conditional ...

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Portfolio risk evaluation: An approach based on dynamic conditional correlations models and wavelet multiresolution analysis R Khalfaoui and M Boutahar GREQAM d’Aix-Marseille, IML universit´e de la m´editerran´ee

24. September 2012

Online at https://mpra.ub.uni-muenchen.de/41624/ MPRA Paper No. 41624, posted 1. October 2012 13:34 UTC

Portfolio risk evaluation: An approach based on dynamic conditional correlations models and wavelet multiresolution analysis R. Khalfaoui∗,a, , M. Boutahar,b, , a GREQAM, b IML

2 rue de la charite, 13236, Marseille cedex 02, FRANCE. , Campus de Luminy, Case 907, 13288, Marseille cedex 09 FRANCE.

Abstract We analyzed the volatility dynamics of three developed markets (U.K., U.S. and Japan), during the period 2003-2011, by comparing the performance of several multivariate volatility models, namely Constant Conditional Correlation (CCC), Dynamic Conditional Correlation (DCC) and consistent DCC (cDCC) models. To evaluate the performance of models we used four statistical loss functions on the daily Value-at-Risk (VaR) estimates of a diversified portfolio in three stock indices: FTSE 100, S&P 500 and Nikkei 225. We based on one-day ahead conditional variance forecasts. To assess the performance of the abovementioned models and to measure risks over different time-scales, we proposed a wavelet-based approach which decomposes a given time series on different time horizons. Wavelet multiresolution analysis and multivariate conditional volatility models are combined for volatility forecasting to measure the comovement between stock market returns and to estimate daily VaR in the time-frequency space. Empirical results shows that the asymmetric cDCC model of Aielli (2008) is the most preferable according to statistical loss functions under raw data. The results also suggest that wavelet-based models increase predictive performance of financial forecasting in low scales according to number of violations and failure probabilities for VaR models. Key words: Dynamic conditional correlations, Value-at-Risk, wavelet decomposition, Stock prices

1. Introduction Measuring market risk is the most interest of financial managers and traders. The most widely used measure of risk managment is the Value-at-Risk (VaR), which was introduced by Jorion (1996). Forecasting VaR is based on the volatility models. The well-known volatility model is the generalized autoregressive conditional heteroskedasticity (GARCH) model of Engle (1982) and Bollerslev (1986). The success of this model has subsequently led to a family of univariate and multivariate GARCH models which can capture different behavior in financial returns. The development of this family of models has led to the development of conditional VaR forecasts. ∗ Corresponding

author Email addresses: [email protected] (R. Khalfaoui), [email protected] (M. Boutahar) Preprint submitted to The Journal of Derivatives September 24, 2012

The literature on multivariate GARCH models is large and expanding. Engle and Kroner (1995) defined a general class of multivariate GARCH (MGARCH) models. The popular ones are diagonal VECH model of Bollerslev et al. (1988), the BEKK model of Engle and Kroner (1995). While, popular, these models have limitations.1 In particular, diagonal VECH lacks correlation between the variance terms, BEKK can have poorly behaved likelihood function (making estimation difficult, especially for models with more than two variables), and VECH has a large number of free parameters (which makes it impractical for models with more than two variables). To deal with the curse of the dimensionality Engle (2002) proposed the dynamic conditional correlations (DCC) model which generalizes the specification of Bollerslev (1990) by assuming a time variation of correlation matrix. Tse and Tsui (2002) defined a multivariate GARCH model which includes time-varying correlations and yet satisfies the positive-definite condition. Ling and McAleer (2003), McAleer et al. (2009) and Caporin and McAleer (2009, 2010) proposed another family of multivariate GARCH models which assume constant conditional correlations, and do not suffer from the problem of dimensionality, by comparison with the VECH and BEKK models. The convenient of these models is that modeling conditional variances allows large shocks to one variable to affect the variances of the other variables. Another important topic in the financial econometrics is the asymmetric behavior of conditional variances. The basic idea is that negative shocks have a different impact on the conditional variance evolution than do positive shocks of similar magnitude. Cappiello et al. (2006), Aielli (2008) and Palandri (2009) extended the DCC model of Engle (2002) to an asymmetric DCC model which is a generalization of the DCC model (authors develop a model capable of allowing for conditional asymmetries not only in volatilities but also in correlations). Authors used several asymmetric versions of the DCC model. In our article, we based on version which allows asymmetry in conditional variance. This phenomenon was raised by Nelson (1991) in introducing the Exponential GARCH (EGARCH) model, and was also considered by Glosten et al. (1993) (GJR-GARCH models), Rabemananjara and Zakoian (1993) and Zakoian (1994) (Threshold GARCH models) for the univariate case. More recently, in the area of finance, several empirical studies based on symmetric and asymmetric multivariate GARCH models have been employed. For instance, Cappiello et al. (2006) used the asymmetric generalized-DCC (AG-DCC) specification to investigate asymmetries in conditional variances and correlation dynamics of three groups of countries (Europe, Australia and North America). They provide evidence that equity returns show strong evidence of asymmetries in conditional volatility, while little is found for bond returns. Chiang et al. (2007) applied a DCC model to nine Asian stocks and confirm a contagian effect during the Asian crisis. Ho et al. (2009) applied various multivariate GARCH models to investigate the evidence of asymmetry and time-varying conditional correlations between five sectors of Industrial Production of the United States. They provide also, strong evidence of asymmetric conditional volatility in all sectors and some support of 1 See

Bauwens et al. (2006) for more details.

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time-varying correlations in various sectoral pairs. Büttner and Hayo (2011) used a bivariate DCC model to extract dynamic conditional correlations between European stock markets. Chang et al. (2011) employed several multivariate GARCH models (CCC, DCC, VARMA-GARCH and VARMA-AGARCH) to model conditional volatility in the returns of rubber spot and futures in major rubber futures and rubber spot Asian markets, in order to investigate volatility transmissions across these markets. Their results provide presence of volatility spillovers and asymmetric effects of positive and negative return shocks on conditional volatility. Kenourgios et al. (2011) applied the AG-DCC model and a multivariate regime-switching Gaussian copula model to capture non-linear correlation dynamics in four emerging equity markets (Brazil, Russia, India and China) and two developed markets (U.S. and U.K.). Arouri et al. (2011) employed a VAR-GARCH approach to investigate the return linkage and volatility transmission between oil and stock markets in Gulf Cooperation Council countries. Lahrech and Sylwester (2011) used DCC multivariate GARCH models to examine the dynamic linkage between U.S. and Latin American stock markets. Their results show an increase in the degree of co-movement between Latin American equity markets and U.S. equity ones. In this study, we employ three multivariate GARCH models, such as the CCC model of Bollerslev (1990), the DCC model of Engle (2002) and the cDCC model of Aielli (2008). These models impose a useful structure on the many possible model parameters. However, parameters of the model can easily be estimated and the model can be evaluated and used in straightforward way. Our empirical methodology follows a two-step approach. The first step applies these dynamic conditional correlation models to model conditional volatility in the returns of three developed stock markets (U.K., U.S. and Japan stock markets), in order to examine the evidence of time-varying conditional variances and correlations between stock markets. Moreover, in order to show the asymmetric effects of positive and negative return shocks on conditional volatility the EGARCH and GJR-GARCH models of Nelson (1991) and Glosten et al. (1993), respectively, were employed for modeling univariate conditional volatility. In the second step, we re-examine the dynamic conditional correlation analysis among the three major developed stock markets through a novel approach, wavelet analysis. This technique is a very promising tool as it is possible to capture the time and frequency varying features of co-movement within an unified framework which represents a refinement to previous approaches. This wavelet-based analysis takes account the distinction between the short and long-term investor. From a portfolio diversification view, there exist a kind of investors whose are more interested in the co-movement of stock returns at higher frequencies (lower scales), that is, short-term fluctuations, and also, there exist a kind of investors whose focuse on the relationship at lower frequencies (higher scales), that is, long-term fluctuations. The study of the co-movement of stock market returns, i.e. dynamics of variances and correlations, across scales is crucial for risk assessment of a portfolio. In terms of risk management, a higher co-movement (higher covariances) among assets of a given portfolio implies lower gains. According to investors or traders, evaluating the co-movement of assets is a 3

great importance to best assess the risk of a portfolio. Several applications of wavelet studying the co-movement of stock indices have been recently applied by Sharkasi et al. (2005), Rua and Nunes (2009), Rua (2010) and Masih et al. (2010). In this paper, we investigate also the selection of the multivariate GARCH models used in the two approaches (with and without wavelet analysis) to identify which model has the best out-of-sample forecasting performance. The assessment of the forecast performance of these models is based on out-of-sample one-day ahead conditional matrix forecasts. However, to measure model performance we used four statistical loss functions.2 The empirical evidence showed that the conditional variances and correlations of U.K., U.S. and Japan stock market returns were dynamic and the three markets were highly correlated. We showed also, that the cDCC model of Aielli (2008) is preferable than the CCC and DCC models based on one-day ahead out-of-sample forecasts. With regard to wavelet-based multivariate conditional volatility approach, our findings suggested a multi-scale behavior of the three markets under study, which decomposes the total spillover into three sub-spillovers and decomposes the market risk measured by VaR into wavelet VaR (WVaR) estimates. In addition, this new approach help traders and investors to reduce risk management on their investing time horizons. The purpose of the paper is four-fold. First, we estimate multivariate conditional volatility for stock market returns using several recent models of multivariate conditional volatility. Second, we investigate the importance of volatility spillovers on the conditional variance across the three developed markets. Third, we focuse on the forecasting performance of the multivariate conditional volatility models under study for the last 250 days of the data set. Forecast comparison is based on four different loss functions including the mean squared error, the mean absolute error, the mean absolute percentage error and the logarithm loss error. Fourth, we propose a wavelet-based multi-resolution analysis in order to combine between traditional multivariate conditional volatility models and wavelet decomposition. The combination of wavelet decomposition and dynamic conditional correlation models was introduced to analyze the comovements and volatility spillovers between three developed markets on multi-scale framework, based on behavior of investors. Finally, we compare the performance of wavelet-based multivariate conditional volatility model against the traditional one for one-day ahead forecast. The structure of the paper is organized as follows: Section 2 presents the data used for the empirical analysis and the multivariate conditional volatility methodology. Section 3 reports the empirical results under raw data. Wavelet analysis is discussed in section 4. Finally, concluding remarks are stated in section 5. 2 Hansen et al. (2003), Hansen and Lunde (2005), Becker and Clements (2008) and others, showed that evaluation of univariate volatility forecast is well understood, while for an applied point of view ther are no clear guidelines available on model evaluation and selection in multivariate setting (see Laurent et al. (2011)).

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2. Methodology and empirical specifications 2.1. Data Our data on stock market prices consist of the S&P 500, FTSE 100 and NIKKEI 225 composite indices for U.S., U.K. and Japan. We collect daily data over the period from January 01, 2003 to February 04, 2011. Indices are obtained from DataStream. We use daily data in order to retain a high number of observations to adequately capture the rapidity and intensity of the dynamic interactions between markets. Returns of market i (index i) at time t are computed as ri,t = log (Pi,t /Pi,t−1 ) × 100, where Pi,t and Pi,t−1 are the closing prices for day t and t − 1, respectively. 2.2. Descriptive statistics The summary statistics of the data are given in Table 1. In panel A (Table 1), for each return series, the mean value is close to zero. For each return series the standard deviation is larger than the mean value. Each return series displays a small amount of skewness and large amount of kurtosis and the returns are not normally distributed. In panel B (Table 1), unconditional correlation coefficients in stock market index returns indicate strong pairwise correlations. The correlation between S&P 500 and FTSE 100 is positive and larger than the correlation between NIKKEI 225 and FTSE 100. This could be due to the high trade share between the two markets. Table 1 Descriptive statistics of stock market returns. Panel A: Descriptive statistics.

FTSE 100 S&P 500 NIKKEI 225

Mean

Max

Min

Std.dev

Skewness

Kurtosis

Jarque-Bera

0.0199 0.0187 0.0097

9.384 10.957 13.234

-9.265 -9.469 -12.111

1.252 1.314 1.544

-0.0914 -0.256 -0.466

9.019 11.410 8.889

7180.19∗ 11508.39∗ 7048.35∗

Panel B: Unconditional (market return) correlation matrix

FTSE 100 S&P 500 NIKKEI 225

FTSE 100

S&P 500

NIKKEI 225

1.000

0.55 1.000

0.34 0.11 1.000

Notes: ∗ denotes the rejection of the null hypothesis of normality at the 1% level of the Jarque-Bera test. The data frequency is daily and covers the period from 01 January 2003 to 04 February 2011.

The results of the unit root tests for all sample of level prices (in logs) and returns in each market are summarized in Table 2. The Augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) tests are used to explore the existence of unit roots in individual series. The results show that all returns are stationary. 5

Table 2 Unit root tests. Panel A: Stock market level prices (in log) Prices

FTSE 100 S&P 500 NIKKEI 225

ADF test

Phillips-Perron test

None

Constant

Constant and trend

Constant

Trend

0.724 0.644 0.247

-1.870 -1.908 -1.744

-1.920 -1.879 -1.807

-1.847 -1.951 -1.670

-1.866 -1.912 -1.733

None

Constant

Constant and trend

Constant

Trend

-35.171 -37.754 -34.817

-35.177 -37.758 -34.811

-35.171 -37.754 -34.833

-49.933 -52.593 -46.359

-49.924 -52.587 -46.382

Panel B: Stock market returns Returns

FTSE 100 S&P 500 NIKKEI 225

ADF test (t-statistic)

Phillips-Perron test

Notes: Entries in bold indicate that the null hypothesis is rejected at 1% level.

2.3. Model specifications The econometric specification used in our study has two components. To model the stock market return we used a vector autoregression (VAR). To model the conditional variance we used a multivariate GARCH model. A VAR of order p, where the order p represents the number of lags, that includes N variables can be written as the following form: p

Yt = Φ0 + ∑ ΦiYt−i + εt ,

t = 1, . . . , T

(1)

i=1

′

where Yt = (Y1t , . . . ,YNt ) is a column of observations on current values of all variables in the model,Φi is N × N ′

matrix of unknown coefficients, Φ0 is a column vector of deterministic constant terms, εt = (ε1t , . . . , εNt ) is a column vector of errors.3 Our basic VAR will have the three stationary variables, first log differences of FTSE 100, S&P 500 and NIKKEI 225 stock market prices (will be defined in empirical section). We focused on the modelling of multivariate time-varying volatilities. The most widely used model is DCC one of Engle (2002) which captures the dynamic of time-varying conditional correlations, contrary to the benchmark CCC model (Bollerslev (1990)) which retains the conditional correlation constant.4 3 Following Brooks (2002), the main advantage of the VAR is that there is no need to specify which variables are the endogenous variables and which are the explanatory variables because in the VAR, all selected variables are treated as endogenous variables. That is, each variable depends on the lagged values of all selected variables and helps in capturing the complex dynamic properties of the data. Note that selection of appropriate lag length is crucial. If the chosen lag length is too large relative to the sample size, the degrees of freedom will be reduced and the standard errors of estimated coefficients will be large. If the chosen lag length is too small, then the selected lags in the VAR analysis may not be able to capture the dynamic properties of the data. The chosen lag length should be free of the problem of serial correlation in the residuals. 4 The CCC specification can be presented as: p Ht = Dt RDt , where, Dt = diag hi,t is a diagonal matrix with square root of the estimated univariate GARCH variances on the diagonal. R is

6

6500

0.075

6000

0.050

5500

0.025

5000

0.000

4500

-0.025

4000

-0.050 -0.075

3500 2003

2004

2005

2006

2007

2008

2009

2010

2011

2003

2004

2005

(a) FTSE 100 price

2006

2007

2008

2009

2010

2011

2009

2010

2011

2009

2010

2011

(b) FTSE 100 returns 0.100

1500

0.075

1400 1300

0.050

1200

0.025

1100

0.000

1000

-0.025

900 -0.050

800 -0.075

700 2003

2004

2005

2006

2007

2008

2009

2010

2011

2003

2004

2005

(c) S&P 500 price

2006

2007

2008

(d) S&P 500 returns

18000 0.10 16000 0.05 14000 0.00

12000

-0.05

10000

-0.10

8000

2003

2004

2005

2006

2007

2008

2009

2010

2011

2003

2004

(e) NIKKEI 225 price

2005

2006

2007

2008

(f) NIKKEI 225 returns

Figure 1 Time series plots of FTSE 100, S&P 500 and NIKKEI 225 stock market indices. We plot the daily level stock market indices (left panel) and corresponding returns (right panel) in the period January 01, 2003 to February 04, 2011. The specification of the DCC model is as follows: p

rt = µ + ∑ Φs rt−s + εt ,

t = 1, . . . , T,

s=1

′

1/2

εt = (εUK,t , εUS,t , εJP,t ) = Ht

zt ,

εt |Ωt−1 ∼ N (0, Ht ),

(2)

zt ∼ N (0, I3 ),

(3)

′

Ht = E(εt εt |Ωt−1 ),

(4)

where, rt is a 3 × 1 vector of the stock market index return, εt is the error term from the mean equations of stock market indices (Equation 2), zt is a 3 × 1 vector of i.i.d errors and Ht is the conditional covariance matrix. Equation 2 the time-invariant symmetric matrix of the correlation returns with ρii = 1. In CCC model, the conditional correlation coefficients are constant, but conditional variances are allowed to vary in time.

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can be re-written as follows:      s φ11 rUK,t  µUK  p        r  s    US,t  =  µUS  + ∑ φ21  s=1     s µJP φ31 rJP,t

s φ12 s φ22 s φ32

  εUK,t           s φ23   rUS,t−s  +  εUS,t      s φ33 εJP,t rJP,t−s 



s φ13  rUK,t−s 

′

To represent the Engle’s (2002) DCC-GARCH model for the purpose of this study, let rt = (rUK,t , rUS,t , rJP,t ) be a 3 × 1 vector of stock market returns, such that, rUK , rUS and rJP are the returns of FTSE 100, S&P 500 and NIKKEI 225 indices, respectively: rt |Ωt−1 ∼ N (0, Ht ). The DCC-GARCH specification of the covariance matrix, Ht , can be written as:

Ht = Dt Rt Dt

(5)


p p hUS,t , hJP,t is 3 × 3 diagonal matrix of time-varying standard deviation from uni 2 + βi hi,t−1 , i=UK, US, JP, and Rt = ρi j is the time-varying condivariate GARCH models; i.e. hi,t = ωi + αi εi,t−1 where Dt = diag

p

hUK,t ,

tional correlation matrix.

The estimation procedure of DCC-GARCH model is based on two stages. In the first stage, a univariate GARCH p model is estimated. In the second step, the vector of standardized residuals ηi,t = ri,t / hi,t is employed to develop the DCC correlation specification as follows:

    −1/2 −1/2 −1/2 −1/2 Rt = diag q11,t , . . . , q33,t Qt diag q11,t , . . . , q33,t

(6)

where Qt = (qi j t ) is a symmetric positive define matrix. Qt is assumed to vary according to a GARCH-type process:

′ Qt = (1 − θ1 − θ2 )Q¯ + θ1 ηt−1 ηt−1 + θ2 Qt−1

(7)

The parameters θ1 and θ2 are scalar parameters to capture the effects of previous shocks and previous dynamic conditional correlation on current dynamic conditional correlation. The parameters θ1 and θ2 are positive and θ1 +

θ2 < 1. Q¯ is 3 × 3 unconditional variance matrix of standardized residuals ηi,t . The correlation estimators of equation 7 are of the form: qi j,t . ρi j,t = √ qii,t q j j,t 8

In the DCC model the choice of Q¯ is not obvious as Qt is neither a conditional variance nor correlation. Although ′

E(ηt−1 ηt−1 ) is inconsistent for the target since the recursion in Qt does not have a martingale representation.5 Aielli (2008) proposed the corrected Dynamic Conditional Correlation (cDCC) to evaluate the impact of both the lack of consistency and the existence of bias in the estimated parameters of the DCC model of Engle (2002). He showed that the bias depends on the persistence of the DCC dynamic parameters.6 In order to resolve this issue, Aielli (2008) introduces the cDCC model, which have the same specification as the DCC model of Engle (2002), except of the correlation process Qt is reformulated as follows:

∗ ∗′ ηt−1 + θ2 Qt−1 Qt = (1 − θ1 − θ2 ) Q¯ + θ1 ηt−1

(8)

where ηt∗ = diag {Qt }1/2 ηt . To investigate the asymmetric properties of stock market returns we introduce the conditional asymmetries in variance. Cappiello et al. (2006) estimate several asymmetric versions of the dynamic conditional correlation models. The version which we use is based on the following specification:

p rt = µ + ∑s=1 Φs rt−s + εt ,

Ht = Dt Rt Dt

where

t = 1, . . . , T, εt |Ωt−1 ∼ N (0, Ht ) p p
p h1,t , h2,t , h3,t Dt = diag

2 2 hit = ωi + αi εi,t−1 + γi 1{εi,t−1