A Portfolio Index GARCH model

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International Journal of Forecasting 24 (2008) 449 – 461 www.elsevier.com/locate/ijforecast

A Portfolio Index GARCH model Manabu Asai a,⁎, Michael McAleer b b

a Faculty of Economics, Soka University, Japan School of Economics and Commerce, University of Western Australia, Australia

Abstract This paper develops the structure of a parsimonious Portfolio Index (PI) GARCH model. Unlike the conventional approach to Portfolio Index returns, which employs the univariate ARCH class, the PI-GARCH approach incorporates the effects on individual assets, leading to a better understanding of portfolio risk management, and achieves greater accuracy in forecasting Value-at-Risk (VaR) thresholds. For various asymmetric GARCH models, a Portfolio Index Composite News Impact Surface (PI-CNIS) is developed to measure the effects of news on the conditional variances. The paper also investigates the finite sample properties of the PI-GARCH model. The empirical example shows that the asymmetric PI-GARCH-t model outperforms the GJR-t model and the filtered historical simulation with a t distribution in forecasting VaR thresholds. © 2008 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. Keywords: Risk management; Portfolio Index model; Multivariate volatility; Asymmetry; Composite news; Value-at-Risk thresholds; Monte Carlo simulations

1. Introduction Over the past two decades, numerous multivariate volatility models have been developed to accommodate and manage financial portfolios with large numbers of assets. In particular, such models have enabled more efficient estimation of parameters, led to a greater understanding of portfolio risk management, and achieved greater accuracy in forecasting Value-atRisk (VaR) thresholds. A comprehensive discussion of alternative univariate and multivariate, conditional and stochastic, financial volatility models for calculat-

⁎ Corresponding author. E-mail address: [email protected] (M. Asai).

ing VaR is given by McAleer (2005) and Kuester, Mittnik, and Paolella (2006), while Li, Ling, and McAleer (2002) and Bauwens, Laurent, and Rombouts (2006) provide recent surveys of the multivariate conditional volatility literature. One of the major concerns of this paper is modeling the conditional volatility of the Portfolio Index. A straightforward approach is to use the estimated conditional covariance matrix of all the assets in the portfolio, while the conventional way to deal with the model is to treat the Portfolio Index as a univariate process. The latter approach loses useful information, while the former method suffers from the computational burden associated with having a large number of parameters in the conditional covariance and/or conditional correlation matrices.

0169-2070/$ - see front matter © 2008 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.ijforecast.2008.06.006

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A primary purpose of this paper is to bridge the gap between the two methods by developing a parsimonious Portfolio Index GARCH (PI-GARCH) approach. Instead of modeling the multivariate volatility itself, as in standard multivariate GARCH models, the PIGARCH approach models the conditional volatility of a portfolio directly by using information from each individual asset. For the case where the number of assets is very large, we also propose a hybrid method based on the portfolio return and the returns of some assets. Compared with the conventional univariate GARCH class, the PI-GARCH models can be used to obtain more efficient conditional volatility estimates and more accurate forecasts of VaR thresholds. The plan of the paper is as follows. Section 2 presents the parsimonious Portfolio Index GARCH (PIGARCH) approach for modeling the conditional volatility of a portfolio of financial assets. While the PI-GARCH is based on the diagonal GARCH model of Ding and Engle (2001), our approach is also applied to the BEKK (Baba, Engle, Kraft and Kroner) model of Engle and Kroner (1995) in order to derive the PI-BEKK model. We also propose asymmetric models using the PI approach. For a large number of assets, we discuss two methods for decreasing the number of parameters. Section 3 presents Monte Carlo simulation results for three different processes for comparing the PI-GARCH model with the GARCH and diagonal GARCH models for estimating conditional covariances and correlations, and for forecasting VaR thresholds. In addition, we also investigate rejection frequencies of the likelihood ratio test for the GARCH model against the PI-GARCH model. As an application of the PI-GARCH approach, Section 4 extends the filtered historical simulation (FHS) method proposed by Barone-Adesi, Giannopoulos, and Vosper (1999). For alternative asymmetric GARCH models, a Portfolio Index Composite News Impact Surface (PI-CNIS) is developed in Section 5 to measure the effects of news on the conditional variances. An empirical example is given in Section 6 to illustrate the usefulness of the asymmetric PI-GARCH models with the standardized t distribution, compared with the GJR model of Glosten, Jagannathan, and Runkle (1993) and the FHS model of Barone-Adesi et al. (1999). Some concluding remarks are given in Section 7. In Appendix A, we discuss and compare the PIGARCH model with Engle's (2002) dynamic conditional correlation model.

2. Portfolio Index approach Let the returns on m (≥ 2) financial assets be given by yit = μit + εit, i = 1,…, m, t = 1,…, T, or yt = μt + εt, where yt, μt and εt are m-dimensional column vectors, lt ¼ Eðyt jIt1 Þ, and It is the past information available at time t. The return of the portfolio consisting of m assets is denoted as yP;t ¼ w Vyt ¼ w Vlt þ w Vet ;

ð1Þ

where w = (w P1m,…, wm)′ denotes the portfolio weights, such that i¼1 wi ¼ 1. For the portfolio returns, the conditional mean vector and of the  disturbance  portfolio are defined by lP;t ¼ E yP;t jIt1 ¼ w Vlt and εP,t = yP,t − μP,t, respectively. The condition for the portfolio weights can be relaxed in order to model a variety of portfolios, such as equally weighted, balanced weighted and hedge portfolios. The conditional covariance matrix of yt is given as: Qt ¼ V ðyt jIt1 Þ ¼ E ð et eVt jIt1 Þ;

ð2Þ

and the conditional volatility of the portfolio is given by:   hP;t ¼ V yP;t jIt1 ¼ w VQt w: ð3Þ In the framework of multivariate GARCH models, the constant conditional correlation (CCC) model of Bollerslev (1990) is based on multiple univariate GARCH processes for the components of yt, and hence does not incorporate significant information on potential spillover effects of the remaining m − 1 assets. More general multivariate specifications, such as the VARMA-GARCH model of Ling and McAleer (2003), the asymmetric VARMA-GARCH (or VARMAAGARCH) model of McAleer, Hoti and Chan (2008), the GARCC model of McAleer, Chan, Hoti, and Lieberman (2008), the diagonal GARCH model of Ding and Engle (2001), and the BEKK model of Engle and Kroner (1995), can suffer from the fact that the number of parameters increases considerably with the number of financial assets. This rapid increase in the number of parameters can cause serious problems for convergence of the appropriate estimation algorithms, especially for a portfolio with many assets, and is also known as the “curse of dimensionality”. Once the estimates and forecasts of Qt are obtained, it is straightforward to obtain those for the portfolio by

M. Asai, M. McAleer / International Journal of Forecasting 24 (2008) 449–461

using Eq. (3). However, this is not the method used in the literature. One of the reasons for this is that, due to the number of parameters, only the portfolio return is modeled, instead of all the assets. In this sense, modeling yp,t as a simple univariate process may lose valuable information. In order to capture such spillover effects, this paper proposes the Portfolio Index GARCH (PI-GARCH) model, which is given as follows: hp;t ¼ x þ w Vð ABet1 eVt1 Þw þ bhp;t1 ;

ð4Þ

where ω and β are scalar parameters, A is an mdimensional symmetric positive definite matrix of parameters, and ‘∘’ represents the Hadamard elementby-element product. The number of parameters to be estimated is 0.5m (m + 1) + 2. It should be noted that, for the portfolio returns, εP,t =w′εt and hP,t = w′Qtw. In the PI-GARCH model, the conditional volatility for the Portfolio Index is modeled by using the return of each asset. It is straightforward to establish which is the higher-order PI-GARCH(p,q) model. It is worth examining the structural relationships among the diagonal GARCH, GARCH and PI-GARCH models. For the simplest case, the diagonal GARCH model can be written as Qt ¼ X þ ABet1 eVt1 þ BBQt1 ;

ð5Þ

where Ω, A and B are m-dimensional parameter matrices. Under the restriction that B = βιι′, using Eq. (3) and defining ω = w′Ωw, where ι is an m-vector of unit elements, leads to the PI-GARCH model. The conventional GARCH model is obtained from the PI-

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GARCH model by setting A = αιι′. For convenience, Table 1 summarizes the model specifications and the numbers of parameters in various constant and dynamic correlation multivariate GARCH models that are available in the literature. Appendix A discusses and compares the PI-GARCH and DCC models. As the PI-GARCH model can be considered as a special case of the diagonal GARCH model, we can derive the conditions for ht N 0 and for stationarity. In order to guarantee ht N 0, given h0 N 0, it is sufficient to assume that ω ≥ 0, β ≥ 0, and A is a positive semidefinite matrix, and that one of the three must be strictly positive (or positive definite). The stationarity condition is given by aij + β b 1 for all i and j. For a large number of assets, we suggest two methods for decreasing the number of parameters. One way is to use the vector diagonal specification for A, such that A = aa′, where a is the m × 1 parameter vector. This approach was suggested by a referee, and is motivated by the vector diagonal GARCH model of Ding and Engle (2001). An alternative approach is a hybrid of the GARCH and PI-GARCH models. Let ε⁎t be an n × 1 vector consisting of the elements of εt (n b m). Define w⁎ as the corresponding weights. Note that the sum of the elements of w⁎ is less than one. We then propose the hybrid model as   hp;t ¼ x þ ae2p;t þ w⁎ V CBe⁎t1 e⁎t1 V w⁎ þ bhp;t1 ; ð6Þ where C is a n × n positive semi-definite matrix. As before, we can set C = cc′ when n is large.

Table 1 Constant and dynamic correlation multivariate GARCH models Model Diagonal GARCH: Ding and Engle (2001) BEKK: Engle and Kroner (1995) CCC: Bollerslev (1990) DCC: Engle (2002)

PI-GARCH: this paper PI-BEKK: this paper

Specification

Number of parameters

m=5

m = 20

Qt = Ω + A ∘ εt − 1εt −′ 1 + B ∘ Qt − 1 Qt =C′C + Aεt − 1εt −′ 1A′ + BQt − 1B′ Qt = DtPDt, Dt = [diag{ht}]1/2, ht = ω + α ∘ εt − 1 ∘ εt − 1 + β ∘ ht − 1 Qt = DtPDt, Dt = [diag{ht}]1/2, ht = ω + α ∘ εt − 1 ∘ εt − 1 + β ∘ ht − 1, Pt = diag{Rt}− 1/2 Rt = diag{Rt}− 1/2, Rt = (1 − γ −δ)S + γ(ηt − 1ηt − 1′) + δRt − 1 ηt = D−t 1εt hp,t = ω + w′(A ∘ εt − 1εt −′)1 w + βhp,t − 1 hp,t = ω + w′(Aεt − 1εt −′ 1A′)w + βhp,t − 1 A is a lower triangular matrix.

3m2 þ3m 2 5m2 þm 2 2 m þ5m 2

45 65 25

630 1010 250

m2 þ5m 2

27

252

17 17

212 212

m2 þm 2 m2 þm 2

þ2

þ2 þ2

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The Portfolio Index approach can also be applied to other multivariate GARCH models. For example, a standard BEKK model is given by Qt ¼ C VC þ Aet1 eVt1 A Vþ BQt1 B V; where C, A and B are m-dimensional parameter matrices. Under the restrictions pffiffiffiffiffiffiffi that A is a lower triangular matrix and that B ¼ bIm , using Eq. (3) and defining ω = w′C′Cw, leads to the PI-BEKK model, as follows: hp;t ¼ x þ w Vð Aet1 e Vt1 A VÞw þ bhp;t1 :

ð7Þ

normal and t distributions. For more accurate VaR analysis, there are several alternatives (see, for example, McAleer, 2005; Kuester et al., 2006), but a further analysis is beyond the scope of the current paper. Assuming normality of the conditional distribution of the standardized residuals, we can estimate the parameters using the maximum likelihood (ML) method for any Portfolio Index model, as follows: b h ¼ argmaxLp ðhÞ; where

As a result, the Portfolio Index approach drastically reduces the number of parameters in the standard BEKK model, from 0.5 (5m2 + m) to 0.5 (m + 1) + 2 (see Table 1), and thereby significantly reduces the number of parameters typically associated with the standard BEKK model. It is worth noting that the (i,j)th element of Aεt − 1εt′− 1 A′ is a function of all the elements of εt − 1ε′t −1, unlike the diagonal GARCH model, which is one reason why A is specified as a triangular matrix. The PI-GARCH model can be modified to capture asymmetric effects, as follows:  hp;t ¼ x þ w Vð ABet1 e Vt1 Þw þ gdp;t ep;t þ bhp;t1 ; ð8Þ − where dp,t takes the value one if εp,t is negative, and zero otherwise. The GJR model of Glosten et al. (1993) is obtained by setting A = αιι′ in Eq. (8). An alternative extension is given by    hp;t ¼ x þ w V A þ CBdt dt V Bet1 eVt1 w þ bhp;t1 ;

ð9Þ − − where the vector dt− = (d1t ,…, dmt )′ denotes a set of − indicator variables, and dit takes the value one if εit is negative, and zero otherwise. In the following, we will use the latter model to capture asymmetric effects. Similar extensions can be developed for the PIGARCH variants and the PI-BEKK model (see also Kroner & Ng, 1998).

3. Estimation, testing, and model comparison 3.1. Maximum likelihood estimation We consider two kinds of distributions for the conditional distribution of returns, namely the standard

L p ð hÞ ¼

T X

lp;t ;

ð10Þ

t¼1

lp;t ¼ 0:5lnð2pÞ  0:5loghP;t  0:5e2P;t =hP;t ; and θ denotes the vector of parameters to be estimated in the conditional log-likelihood function. If the assumption of normality does not hold for the standardized residuals, the procedure is defined as the quasimaximum likelihood estimator (QMLE). For the case of the standardized t distribution, we have lp;t ¼ 0:5ln½pðm  2Þ þ lnCð0:5ðm þ 1ÞÞ  lnCð0:5mÞ  h0:5loghP;t  i  0:5ðm þ 1Þln 1 þ e2p;t = hP;t ðm  2Þ ;

mN2; ð11Þ

where v represents the degrees of freedom and Γ(·) denotes the gamma function. The t distribution approaches the standard normal distribution when v → ∞. We calculated ML estimates using the OX language. In order to guarantee the positive definiteness of A, we use A =LL′, where L is the lower triangular matrix. In order to check the usefulness of the PI approach, the following likelihood ratio tests can be used: (i) Testing H0 :A = αιι′ (GARCH) against H 1: A ≠ αιι′ (PI-GARCH); (ii) Testing H0:C = O (GARCH) H1:C ≠ O against (Hybrid); (iii) Testing the Hybrid model against the PIGARCH model. Asymptotically, the LR test statistics will have the χ2 distribution under the null hypothesis. The degrees

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of freedom for the LR tests are: (i) 0.5m (m + 1) − 1, (ii) 0.5n (n + 1), and (iii) 0.5m (m + 1) − 0.5n (n + 1) − 1. When the vector diagonal specifications are used, the degrees of freedom will be different. 3.2. Monte Carlo experiments In this sub-section, we conduct Monte Carlo experiments in order to investigate the finite sample properties of the ML estimator of the PI-GARCH model and the likelihood ratio test, and to forecast VaR thresholds. A bivariate GARCH model with a normal conditional distribution is simulated 200 times for 1000 observations, in a setting where the true correlation structure is known. We use an equally weighted portfolio, with w= (0.5, 0.5)′. The data generating process (DGP) is based on the diagonal GARCH model, which is given as follows: yt fN ð0; Qt Þ; Qt ¼ ðii V A  BÞBS þ ABð yt1 y Vt1 Þ þ BBQt1 ; where the unconditional covariance matrix is specified as S = {sij}, with s11 = s22 = 1 and s12 = s21 = − 0.4. The diagonal GARCH models are assumed to follow three different DGPs, namely: Case 1: A = 0.15ιι′ and B = 0.83ιι′;

0:14 0:08 Case 2: A ¼ and B = 0.83ιι′; 0:08 0:11 Case 3: A ¼ B¼

0:15 0:12

0:12 0:20

and

0:80 0:72 . 0:72 0:75

Case 1 indicates that the portfolio returns follow the GARCH process, while Case 2 corresponds to the PI-GARCH specification. Case 3 is the general diagonal GARCH model. The following results from the three simulation experiments are presented: (i) the finite sample properties of the MLE for the PI-GARCH model; (ii) the rejection rates of the likelihood ratio tests for the GARCH against the PI-GARCH; and

453

(iii) the mean absolute errors of the forecasts of the VaR thresholds for the GARCH and PI-GARCH models. Some explanation may be necessary for experiment (ii). Under normality, the 1% VaR threshold is defined pffiffiffiffiffiffiffiffiffiffiffiffi as VaRt ¼ 2:33 w VQt w, with the estimates based on the PI-GARCH q and by ffiffiffiffiffiffiffiGARCH models being given qffiffiffiffiffiffiffi PI GARCH b b ¼ 2:33 h p;t , VaRt ¼ 2:33 h p;t and VaRt respectively. For each simulation, the mean absolute errors, MAE = (1/T) Σ|VaRtPI − VaRt|, are calculated. Table 2 shows the means, standard deviations and root mean squared errors (RMSE) of the MLE for the PI-GARCH model. Cases 1 and 2 can be captured by the PI-GARCH specification, while Case 3 is beyond the model. For Cases 1 and 2, the means are close to the true values. As the standard deviations and RMSEs are close to each other, the bias is negligible. Regarding the RMSE for Case 3, the average of the elements of B, 0.7475, is used as a proxy for the true value of β. However, the mean of β is close to the average of the diagonal element s of B, 0.775, with the result that the difference between the standard deviation and the RMSE for β is relatively large. In spite of the bias in β, the biases for A are relatively small.

Table 2 Means, standard deviations and root mean squared errors of MLE for the PI-GARCH model Parameters

Case 1

Case 2

Case 3

ω

0.0072 (0.0032) [0.0034] 0.1491 (0.0298) [0.0298] 0.1468 (0.0288) [0.0289] 0.1505 (0.0305) [0.0305] 0.8207 (0.0315) [0.0328]

0.0055 (0.0039) [0.0040] 0.1361 (0.0343) [0.0345] 0.0756 (0.0282) [0.0285] 0.1081 (0.0282) [0.0282] 0.8271 (0.0342) [0.0343]

0.0016 (0.0024) [0.0090] 0.1555 (0.0319) [0.0324] 0.1009 (0.0306) [0.0360] 0.1810 (0.0401) [0.0444] 0.7791 (0.0347) [0.0469]

a11

a12

a22 β

Note: Standard deviations are in parentheses, while the root mean squared errors (RMSE) are in square brackets. For calculating the RMSE of β in Case 3, we used the average of the elements of B as a proxy for the true value.

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Although they are not reported in this paper, we also conducted Monte Carlo experiments for trivariate GARCH models. The results are similar to the bivariate case, except for the elements of A. The estimates of A are unbiased, while the standard deviations are larger than those of the bivariate case. Table 3 reports the rejection frequencies of the LR test for the GARCH model against the PI-GARCH specification. For the null of the GARCH specification, the rejection frequency of Case 1 is less than 0.05, but is reasonable in practice. The results for Cases 2 and 3 show that the LR tests have sufficient power to detect the difference between the GARCH and more general models. Table 4 presents the average MAE values for the VaR forecasts based on 200 simulations for the GARCH and PI-GARCH models. For Case 1, which is essentially the GARCH DGP, the GARCH has a smaller MAE than the PI-GARCH model. For Cases 2 and 3, the MAEs of the GARCH model are more than double those of the PI-GARCH model. 4. Application to filtered historical simulation Recent studies, including Kuester et al. (2006), show that extreme value theory (EVT) and filtering historical simulation (FHS) are effective approaches for predicting VaR thresholds. As an application to PIGARCH models, this section applies the PI-GARCH model to FHS. The PI-GARCH is also applicable to EVT. Before introducing FHS, we introduce the (naïve) historical simulation (HS) method. The HS method gives an estimate of VaR thresholds by using the sample quantile estimate based on historical returns data. More precisely, the HS method calculates the VaR for t + 1 as the empirical λ-quantile (0 b λ b 1) of a moving window of N observations to date t. For example, with a moving window of length, say, N = 1000 observations, the 5% VaR estimate is the Table 3 Rejection frequencies of the LR test for GARCH against PI-GARCH Test

Case 1

LR

0.025

Case 2

Case 3

0.965

1.000

Note: The LR test statistic is χ (2) asymptotically. The nominal significance level is 5% and the critical value is 5.99. 2

Table 4 Mean absolute errors of VaR forecasts Model

Case 1

Case 2

Case 3

GARCH PI-GARCH

0.0354 0.0430

0.0982 0.0483

0.1247 0.0521

50th sample from the lowest value. The HS method captures the non-normality of the returns distribution, while it is inadequate for capturing recent movements. The FHS method overcomes such drawbacks. Barone-Adesi et al. (1999) suggested filtering the returns data, before calculating the sample quantile estimates by using the GARCH model. The FHS method generates VaR forecasts from paths simulated using draws from the filtered residuals. Barone-Adesi et al. (1999) showed that this method performs rather well. Recently, Kuester et al. (2006) applied the normal, Student t and generalized asymmetric t distributions to the conditional density of the GARCH model. For the case of Portfolio Index returns, we suggest using the PI-GARCH model for pre-filtering rather than the GARCH model. A special merit of the PIGARCH approach is that it recovers information from each asset. Below we compare the FHS-t, asymmetric PI-GARCH-t and FHS models with the asymmetric PI-GARCH-t model. 5. Portfolio Index composite news impact surface Engle and Ng (1993) introduced the news impact curve (NIC), which is a useful tool for measuring the effects of news on the conditional variances. They graphically showed the asymmetric reactions of the conditional variances to positive and negative shocks of equal magnitude for the GJR model of Glosten, et al. (1993) and the EGARCH model of Nelson (1991). Recently, Caporin and McAleer (2006) extended the NIC by developing news impact surfaces (NIS) for multivariate conditional volatility models, specifically the dynamic asymmetric multivariate GARCH (DAMGARCH) model. For a bivariate GARCH model, the NIS presents a 3-dimensional graph for each volatility. For the m-variate case, the NIS gives 0.5m (m − 1) 3-dimensional graphs for each volatility, a total of 0.5m 2 (m − 1) graphs. Thus, for m = 5 (m = 20), there would be 75 (210) 3-dimensional graphs, which is

M. Asai, M. McAleer / International Journal of Forecasting 24 (2008) 449–461

not an efficient way of presenting the impact of news on volatility. In this section we present an alternative method for capturing the impact of news, which is based on the Portfolio Index approach for multivariate GARCH models. To illustrate this approach, consider an asymmetric diagonal GARCH model:   Qt ¼ X þ A þ CBdt dt V Bet1 e Vt1 þ BBQt1 ; − − ,…, dmt )′ denotes a set of where the vector dt− = (d1t − indicator variables, and dit takes the value one if εit is negative, and zero otherwise. Regarding this model, we define the Portfolio Index Composite News Impact Surface (PI-CNIS) as    hp;t ¼ w þ w V A þ CBdt dt V Bet1 eVt1 w;

where Ψ = w′[Ω + B ∘ Ω% (ιi′ − A − D⁎ ∘ C − B)]w with, ‘%’ representing the Hadamard element-by-element division, and D⁎ is an m-dimensional matrix with the diagonal elements given by 1/2 and each of the off-

455

diagonal elements given by 1/4. In a similar manner, it is possible to evaluate the PI-CNIS for any multivariate GARCH model, and it is straightforward to determine the PI-CNIS for the alternative Portfolio Index models. Fig. 1(a)–(c) present an example of a simulated PICNIS for the trivariate asymmetric diagonal GARCH model, under the equally weighted portfolio. The parameters are given by 0 1 0:20 0:10 0:15 A ¼ @ 0:10 0:20 0:20 A; 0:15 0:20 0:20 1 0 0:75 0:75 0:70 B ¼ @ 0:75 0:75 0:65 A; 0:70 0:65 0:75 0 1 0:095 0:080 0:065 C ¼ @ 0:080 0:050 0:035 A; 0:065 0:035 0:020 and Ω = (ιι′−A−D⁎∘C−B)∘S, where S = {sij} is the unconditional covariance matrix, with sii = 1 and

Fig. 1. Simulated PI-CNIS and NIC for equally weighted portfolio.

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sij = 0.3(i ≠ j). It is clear that the graphs in Fig. 1(a)–(c) have asymmetrically quadratic surfaces. It is also clear that there is asymmetric behaviour in the three NICs associated with each of the three PI-CNIS, as is shown in Fig. 1(d). In the NIC, each value of εit has the same positive news impact, while the negative shocks are different from each other. Fig. 2 presents the PI-CNIS and NIC for the trivariate hedge portfolio. As compared with Fig. 1(b), Fig. 2(b) reflects the negative relationship between ε2t and ε3t, which arises because of hedging. Fig. 2(c) shows a similar co-movement through hedging in comparison with Fig. 1(c). The differences between an equally weighted portfolio and a hedge portfolio are also captured in the three NICs, as is shown in Fig. 2(d). 6. Empirical example In this section we examine the MLE and out-ofsample forecasts of the GARCH-t, GJR-t, PI-GARCHt, and asymmetric PI-GARCH-t models for three

sets of empirical data, namely the Nikkei 225 Index (Nikkei), Hang Seng Index (Hang Seng) and Straits Times Index (Straits Times) returns. The sample period for the three series is 1/April/2003 to 5/Feb/2007, giving T = 1000 observations. Returns, yit, are defined as 100 × {log Pit  log Pi,t1}, where Pit is the closing price on day t for stock i. We use the threshold AR(1) model for μit. For convenience, we apply the two step estimation for the mean and variance equations. Table 5 shows the ML estimates for each of the four models, where equal weights are used. For all models, the estimates of v are less than 8.5, indicating heavytails and the rejection of normality for the conditional distribution. For the GARCH model, the estimate of α + β is 0.964, which is typical of the values obtained in the literature. With respect to the GJR model, the estimate of γ is positive and significant, implying the existence of asymmetric effects. The LR test for the GARCH-t against the GJR-t also rejects the null hypothesis of no asymmetry at the 5% significant level. The estimate of v is larger than that of the

Fig. 2. Simulated PI-CNIS and NIC for hedge portfolio.

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Table 5 ML estimates Parameters ω α

GARCH-t 0.062973 (0.028558) 0.057461 (0.018372)

GJR-t 0.056244 (0.027253) 0.023268 (0.015988)

Parameters ω

0.057677 (0.033412) 0.31135 (0.14991) − 0.078856 (0.10983) − 0.021971 (0.095074) 0.096998 (0.11815) − 0.071938 (0.12214) 0.26981 (0.19181)

a11 a21 a31 a22 a32 a33

γ

0.068224 (0.028759)

PI-GARCH-t

c11 c21 c31 c22 c32 c33

β v LogLike AIC BIC

0.90708 (0.027393) 6.5901 (1.3030) − 1629.63 3267.26 3286.89

0.91089 (0.026998) 6.7173 (1.3241) − 1626.00 3262.00 3286.54⁎

β v LogLike AIC BIC

0.88246 (0.038867) 7.0895 (1.4569) −1621.88 3261.77 3305.94

Asymmetric PI-GARCH-t 0.041991 (0.036462) 0.19018 (0.15876) −0.063006 (0.14748) −0.10386 (0.17190) 0.041142 (0.15005) −0.015942 (0.18258) 0.18181 (0.26058) 0.46030 (0.43259) −0.013121 (0.40192) −0.13294 (0.39587) 0.86748 (0.44428) −0.79205 (0.50872) 0.89681 (0.73956) 0.85591 (0.052510) 8.4667 (2.0751) − 1613.92 3257.84⁎ 3331.46

Notes: Standard errors are given in parentheses. ‘⁎’ denotes the minimum AIC and BIC among the four models.

GARCH-t model, showing that some extreme values are removed by introducing the asymmetric effect. Regarding the PI-GARCH-t model, the LR test for the GARCH-t model against the PIGARCH-t model rejects the null hypothesis of the GARCH-t model. The PI-GARCH-t model has a smaller AIC than the GJR-t model, while BIC favors GJR-t. The estimate of β is smaller than those of the GARCH-t and GJR-t models, while the estimate of v is larger. For the asymmetric model, the LR test favors the asymmetric PIGARCH-t model over the PI-GARCH-t model. The asymmetric PI-GARCH-t model has the smallest AIC and largest BIC values. The BIC criterion selected GJR-t among the four models. The estimate of v for

the asymmetric PI-GARCH-t is the largest among the four, implying that the model has successfully removed the unexplained components. Next, we compare the out-of-sample forecasts for the VaR thresholds for the four models. Fixing the sample size of estimation at 800, we re-estimate the model, and forecast the one-step-ahead VaR thresholds for the last 200 observations, where the 1% VaR qffiffiffiffiffiffiffiffiffiffiffiffiffi s threshold is given by lˆ tjt1 þ t0:01 ðvÞ  hˆ p;tjt1, where μ̂t|t − 1 and ĥt|t − 1 are the one-step-ahead predics tions of the mean and variance, respectively, and t0.01 (v) is the 1st percentile of the standardized t distribution, with degrees of freedom given by v. Note that μ̂t|t − 1 is the same for all models, which makes the effects of each volatility forecast clearer.

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Table 6 Failure percentages for % VaR forecasts

Table 7 Failure percentages for % VaR forecasts based on the FHS method

Model

1%

2%

3%

4%

5%

Model

1%

2%

3%

4%

5%

GARCH-t GJR-t PI-GARCH-t Asymmetric PI-GARCH-t

0.000 0.000 0.000 0.010

0.010 0.005 0.010 0.015

0.015 0.025 0.020 0.025

0.035 0.035 0.035 0.045

0.045 0.045 0.035 0.055

FHS-t FHS with asymmetric PI-GARCH-t

0.005 0.015

0.010 0.015

0.015 0.020

0.025 0.045

0.040 0.050

Table 6 gives the failure percentages for the VaR forecasts based on the four models with respect to the true percentage from 1%–5%. While the asymmetric PI-GARCH-t model captured the tail behavior well, the other three models failed, especially for the 1% and 2% VaR thresholds. The nominal value for the 5% VaR for the PI-GARCH-t is relatively low, due to the combination of the lack of an asymmetric effect and the sensitivity of the model. Fig. 3 provides the 1% VaR threshold forecasts for the GJR-t and asymmetric PI-GARCH-t models, compared with the portfolio returns, ypt. The estimates for the GJR-t model are less volatile than those for the asymmetric PI-GARCH-t model, showing that the GJR-t model is less sensitive. As was stated in Section 5, we compare the FHS-t, asymmetric PI-GARCH-t and FHS with the asymmetric PI-GARCH-t models. Using the same frame-

work as in Table 6, we calculated the failure percentages for the VaR forecasts based on the FHS method, which are shown in Table 7. We use the results of the asymmetric PI-GARCH-t in Table 6 as the benchmark model. The results show that the asymmetric PIGARCH-t model outperforms the FHS-t in all cases. With respect to FHS with the asymmetric PI-GARCHt model, it provides poorer estimates than the asymmetric PI-GARCH-t model for the 1% and 3% thresholds, while it provides better estimates for the 5% threshold. The remaining two cases provide similar results. It should be noted that the results for the larger window size for FHS are open to question. It is found that the asymmetric PI-GARCH-t model is preferred for the 1% VaR threshold, while FHS with the asymmetric PI-GARCH-t model is preferred for the 5% VaR threshold.

Fig. 3. 1% VaR Thresholds.

M. Asai, M. McAleer / International Journal of Forecasting 24 (2008) 449–461

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Fig. 4. Empirical PI-CNIS and NIC for the asymmetric PI-GARCH Model.

Fig. 4 presents the PI-CNIS and NIC for the trivariate asymmetric PI-GARCH model. Fig. 4(a)– (c) display asymmetric quadratic surfaces, reflecting the relationships between the combinations of positive and negative shocks in the returns. The NIC in Fig. 4(d) shows the asymmetric news impacts of the three shocks on the volatility in the portfolio. 7. Conclusion Financial portfolios with large numbers of assets present a challenge to the development of optimal risk management strategies. Numerous multivariate volatility models have been developed in recent years to accommodate financial portfolios with large numbers of assets. These developments have enabled more efficient estimation of parameters, led to a greater understanding of portfolio risk management, and achieved greater accuracy in forecasting Value-atRisk (VaR) thresholds.

As one of the major problems in the analysis of multivariate volatility models for large numbers of assets has been the computational burden in estimating the conditional covariance matrix, the univariate ARCH class is conventionally used for modeling the Portfolio Index. In order to incorporate the effects of individual assets in the univariate framework, this paper presents a parsimonious Portfolio Index GARCH (PI-GARCH) approach to modeling conditional volatility. Instead of modeling multivariate volatility as in standard GARCH models, the PIGARCH approach models the volatility of a portfolio directly by using integrated information from each individual asset. We also suggest a hybrid model based on the portfolio return and the returns of the some assets. For various asymmetric GARCH models, a Portfolio Index Composite News Impact Surface (PI-CNIS) was developed to measure the effects of news on the conditional variances. We conducted Monte Carlo experiments for investigating the finite

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sample properties of the MLE for the PI-GARCH model. An empirical example showed the benefits of the asymmetric PI-GARCH-t model for the purposes of forecasting VaR thresholds. Acknowledgements The authors are most grateful to the Editor and three referees for very helpful comments and suggestions. The first author acknowledges the financial support of the Japan Society for the Promotion of Science and the Australian Academy of Science. The second author is grateful for the financial support of the Australian Research Council. Appendix A In this appendix we discuss the features of the PIGARCH approach by comparing it with the dynamic conditional correlation (DCC) model of Engle (2002). As an intermediate approach for deriving dynamic conditional correlations, namely one that parsimoniously incorporates conditional correlation spillover effects, Engle (2002) proposed the DCC model, which is, in its simplest form, based on multiple individual GARCH(1,1) models, as follows: Qt ¼ Dt Pt Dt ; Dt ¼ ½diag fht g1=2 ; b Bht1 ; xþ P a Bet1 Bet1 þ P ht ¼ P

ð12Þ

with the dynamic conditional correlation matrices given by: Pt ¼ diag fRt g1=2 Rt diag fRt g1=2 ; Rt ¼ ð1  g  dÞS þ gðgt1 g Vt1 Þ þ dRt1 ; gt ¼ D1 t et ; ð13Þ where, ω, α and β are m-dimensional parameter vectors, S is the (unconditional and conditional) correlation matrix of ηt, γ and δ are scalar parameters, ht is an m-dimensional vector of conditional variances, and Pt is a dynamic conditional correlation matrix. The DCC model extends its CCC counterpart by incorporating time-varying conditional correlations, and reduces to the CCC model when γ = δ = 0. In other words, with the addition of only two parameters, the DCC model can parsimoniously capture elementary

dynamic effects of the remaining m − 1 assets. A drawback of the DCC model is that Rt is not, strictly speaking, a process of conditional correlation matrices, so that Rt must be standardized to obtain Pt in Eq. (12). Moreover, it is highly restrictive to require each element of the matrix of updating news effects in Eq. (13) to be weighted by the same constant parameter. For the purposes of more efficient portfolio risk management and more accurate forecasting of VaR thresholds, the PI-GARCH model concentrates on integrating the available information. For example, the parameter vector, ω, in the DCC model has m parameters, while the PI-GARCH model needs only one additional parameter, as it is not affected by the constant term. As in the DCC model, the PI-GARCH model takes the dynamic correlation structure into account through the addition of news, namely ABet1 eVt1 : Furthermore, as this additional term is the same as in the diagonal GARCH model of Ding and Engle (2001), PI-GARCH is more parsimonious than DCC, which is more parsimonious than the diagonal GARCH model. The number of parameters in the DCC model is 0.5(m2 + 5m) + 2, whereas the PIGARCH model has 0.5m(m + 1) + 2 parameters. Thus, PI-GARCH is more parsimonious than DCC by 2m parameters, which can be substantial for large portfolios. As is explained in Section 2, the imposition of restrictions on the diagonal GARCH model is one reason for expecting the PI-GARCH model to outperform its DCC counterpart. References Barone-Adesi, G., Giannopoulos, K., & Vosper, L. (1999). VaR without correlations for portfolios of derivative securities. Journal of Futures Markets, 19, 583−602. Bauwens, L., Laurent, S., & Rombouts, J. K. V. (2006). Multivariate GARCH models: A survey. Journal of Applied Econometrics, 21, 79−109. Bollerslev, T. (1990). Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH approach. Review of Economics and Statistics, 72, 498−505. Caporin, M., & McAleer, M. (2006). Threshold, news impact surfaces and dynamic asymmetric multivariate GARCH. Working Paper N.06.01, Gruppi di Ricerca Economics Teorica e Applicata. Ding, Z., & Engle, R. F. (2001). Large scale conditional covariance matrix modeling, estimation and testing. Academia Economic Papers, 1, 83−106.

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