Decomposition of portfolio VaR and expected shortfall based on

ISSN 1750-9653, England, UK International Journal of Management Science and Engineering Management, 7(2): 153-160, 2012 http://www.ijmsem.org/

Decomposition of portfolio VaR and expected shortfall based on multivariate Copula simulation∗ Guobin Fan1 † , Yong Zeng1 , Woon K. Wong2 1

School of Management and Economics, University of Electronic Science and Technology of China, Chengdu 610054, P. R. China 2 Bristol Business School, University of the West of England, Bristol BS16 1QY, U. K. (Received 7 June 2011, Revised 15 December 2011, Accepted 12 March 2012)

Abstract. Portfolio risk-adjusted performance measurement involves the calculation of the risk contribution for each asset it contains. This paper uses multivariate Copula functions to model the dependence structure among the assets in a portfolio, then, based on a simulation, decomposes the portfolio VaR and Expected Shortfall. The research shows this simulation approach provides a way to test if the risk contributions of various assets are significantly different, and also displays results insusceptible to confidence level and risk measures. Furthermore, with this approach, the risk contribution calculated using Expected Shortfall is more robust, and its estimation error can be reduced by increasing the simulation sample size. For the equally-weighted portfolios of five Shanghai industrial stock indices, empirical evidence shows that the Real Estate Index has the largest risk contribution of the whole portfolio. Keywords: Value-at-Risk, expected shortfall, risk contribution, multivariate Copula

1 Introduction In risk management, measuring the risk of the whole portfolio is only the first step, as this can only tell us “how much the portfolio risk is”. In order to achieve truly effective risk management, it is necessary to further calculate each individual asset’s risk contributions to the whole portfolio by decomposing the portfolio risk and diagnosing “where the portfolio risk comes from”. Then, based on the calculated risk contributions, capital can be allocated to different assets for risk coverage (Glasserman, 2005 [8]). Moreover, an accurate calculation of risk contributions can also help investors choose which assets are best when constructing their investment portfolios, thereby attaining the best risk-return tradeoff (Yamai and Yoshiba, 2002 [26]). However, research on the evaluation of individual assets risk contributions are quite limited compared to the abundant research dedicated to seeking the methods of portfolio risk forecast. Tasche (2002) [22] and Yamai and Yoshiba (2002) [26] proposed theories on risk contribution calculations. Nevertheless, in the following empirical researches (Gourieroux et al., 2000 [9]; Dietsch and Petey, 2002 [6]; Huang et al., 2007 [13]; Tasche, 2009 [23]; Rosen and Saunders, 2010 [18]; Copeland, 2008 [25]; Hu and Fang, 2003 [11]; Shao and Zhang, 2003 [20]) on risk decompositions, risk contributions are mostly calculated based on observed historical data. In these studies, only one-set of estimated values of risk contributions are obtained and thus the significance of the differences in the risk contributions of various individual assets cannot be tested.

In order to overcome this defect, an improved approach for the calculation of risk contributions is proposed in this paper: First, multivariate Copulas are employed (Cherububu et al., 2004 [5]; Nelson, 2006 [17]) to model the dependence structure across the assets in the portfolio, and then the model with the best fit is chosen to generate simulation data, and finally this simulation data is used to calculate an individual asset’s risk contribution to the whole portfolio. This approach has two advantages: First, multi-sets of estimated values of risk contributions can be calculated by running many simulations, which allows us to test whether the risk contributions of various assets are significantly different. Second, the traditional Monte Carlo simulation often assumes that asset returns are normally distributed, and the cross-asset relation is described by a linear variance-covariance matrix. In contrast, Copulas can well describe non-linear dependence structures and capture tail dependence across assets, i.e., the propensity of different financial assets to crash (or boom) together. Hence, our approach of calculating risk contributions based on a Copula simulation is more accurate, and the empirical analysis indeed verifies that the improved approach is effective in calculating the risk contributions of different assets and testing the significance of their differences. Besides, in existing studies, most research (Gourieroux et al., 2000 [9]; Dietsch and Petey, 2002 [6]; Glasserman, 2005 [8]; Huang et al., 2007 [13]; Tasche, 2009 [23]; Rosen and Saunders, 2010 [18]) has focused on the decomposition of credit risk, while only few papers (Wong and Copeland,

∗

This research has been supported by International Technical Cooperation Project of Sichuan Province (No. 2008HH0014).

†

Correspondence to: Tel.: +86-15928613697. E-mail address: [email protected].

1

Recently, there also appears the research which is dedicated to estimate the systemic risk contributions of financial institutions and analyze their applications in the design of a systemic capital surcharge or Pigouvian tax; see Acharya et al. (2009) [1] and Adrian and Brunnermeier (2010) [2]. The authors appreciate the referees’ pointing out this branch of literature. ©International Society of Management Science And Engineering Management®

Published by World Academic Press, World Academic Union

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G. Fan & Y. Zeng & W. Wong: Decomposition of portfolio VaR and expected

2008 [25]; Hu and Fang, 2003 [11] and Shao and Zhang, 2003 [20]) studied the securities portfolio risk decomposition1 . Therefore, in this paper the above methods of calculating risk contributions are applied to a portfolio constructed using the five industrial indices in the Shanghai stock market. This study can help to investigate the differences in the risk characteristics amongst the various industries on China’s stock market. Some domestic studies in China have analyzed the industrial characteristics of the stock market from other perspectives. For example, Lao and Shao, 2005 [15] found that the volatilities of some industries are relatively larger. However, volatility is only a measure of overall risk. In contrast, this study focuses on the risk brought by extremely negative returns. We analyze from the perspective of risk management, which industry contributes most when investors are facing the risk of extreme losses. This is rather important for risk managers, and can provide helpful guidance for investors’ portfolio choice. Finally, in this paper it is recommended that risk contributions be calculated using both Value-at-Risk (VaR) and Expected Shortfall (Artzner et al., 1997 [3]; Artzner et al., 1999 [4]) (ES). Most previous studies employed only VaR to calculate risk contributions and lack of comparison with the results based on ES. Although Fermanian and Scaillet, 2005 [7] used both VaR and ES, their focus was on the credit risk decomposition. Yamai and Yoshiba, 2005 [27] compared these two risk measures, showing that ES is a coherent measure whereas VaR does not satisfy sub-additivity. Also, as a quantile measure, VaR may face “tail risk” (i.e., VaR tends to underestimate the risks when the returns distribution is fat-tailed (Yamai and Yoshiba, 2005 [27]; Leippold, 2004 [16])) as it disregards any extreme losses beyond the VaR boundary, but ES does not face such tail risk as it considers the amounts of extreme losses. Tab. 1 provides an intuitive example based on Normal and Student t distributions: at a confidence level of 95%, the Student t distribution shares a lower VaR, but it is well-known that its tail is fatter than Normal. Such a contradiction does not happen when using ES. Hence, for underlying heavy-tail distributions, ES is a more accurate risk measure. Previous studies like Wong (2010) [24] have already showed that it is hard to capture all risk information using only VaR, and the analysis in this paper can further suggest that it is also necessary to employ both VaR and ES to calculate risk contributions and compare results based on these two measures. Table 1 The comparison of two risk measures Confidence

Standard Normal

Student t (d.v. = 5)

level 95%

VaR

ES

VaR

ES

1.645

2.063

1.561

2.239

99%

2.326

2.667

2.608

3.451

Note: VaR is calculated as the maximum possible loss at a given confidence level (95% or 99%), while ES is the average losses beyond the VaR boundary.

The rest of this paper is organized as follows. Section 2 demonstrates the method used to calculate risk contributions based on a multivariate Copula simulation; Section 3 explains the main empirical results and Section 4 provides several sensitivity analyses; Section 5 concludes. 2

2 Methodology This section outlines the methods to decompose portfolio risk and calculate risk contributions of individual assets, and then introduces the Copula properties and outlines the methods for choosing the best Copula function and generating a simulation sample. 2.1 Risk decomposition The portfolio loss X (opposite number of returns) is the linear combination of individual assets’ losses Xi , calculated as X = ω1 X1 + · · · + ωn Xn , where ωi is the weight of asset i. Since the portfolio VaR is a linear homogeneous function of the individual weights, the following equation holds: n ∂VaRα ( X ) · ωi , (1) VaRα ( X ) = ∑ ∂ωi i =1 where α denotes the confidence level (95% or 99%). In this way, the portfolio VaR is decomposed into ∂VaR/∂ωi multiplied by individual weights ωi . ∂VaR/∂ωi is defined as the “Marginal VaR” of individual assets. Tasche (2000) [21] proved that under the conditions of continuity of distributions and integrability of expectations, Marginal VaR can be represented by a conditional expectation E[ Xi | X = VaRα ( X )]. Thus, the risk contributions of individual assets to the portfolio VaR are represented by a “Component VaR” and calculated as: ∂VaR C − VaRi = · wi = E[ Xi | X = VaRα ( X )] · ωi . (2) ∂ωi It is difficult to estimate the conditional expectation on the right-hand side of Eq. (2) directly, so the Hallerbach (2003) [10] procedure is used to gain an approximation. Suppose there is a sample of T observations (either from observed data or simulation data). A data window within which the portfolio losses are close to the portfolio VaR level is chosen, i.e., X ∈ [VaRα ( X ) − εd , VaRα ( X ) + εu ], where εd and εu are small positive numbers2 . Assume that observations are chosen in the data window. Let X j , 1 < j < 1 denote the portfolio loss of j-th sample in this data window, and the corresponding losses of the i-th individual asset of j the j-th sample is denoted by Xi (1 ≤ i ≤ n, 1 ≤ j ≤ T ∗ ), then the Marginal VaR of the individual asset i can be estimated using 1 ∂VaRα ( X ) j = E[ Xi | X = VaRα ( X )] ∗ ∑ Xi . M − VaRi = ∂ωi T j (3) This was shown by Hallerbach (2003) [10] to be the best approach to approximately calculate conditional expectations when the distribution is discrete. Similar to the risk decomposition bases on the VaR above, ESα ( X ) is also a linear homogeneous function of the individual weights, so the following equation holds: n ∂ESα ( X ) ESα ( X ) = ∑ · ωi . (4) ∂ωi i =1 The portfolio is thus decomposed into ∂ES/∂ωi multiplied by the individual weights ωi . Tasche (2000) [21] also proved that ∂ES/∂ωi , called the “Marginal ES” of individual assets, could be represented by a conditional expectation E[ Xi | Xi ≥ VaRα ( X )] under the conditions of the continuity of distributions and the integrability of expectations.

To approximate the condition X = VaRα ( X ), there should not be too many observations in the chosen data window. However, the sample mean of Xi cannot be estimated accurately without sufficient observations. As a result of the trade-off between the accuracy of conditioning on X = VaRα ( X ) and the accuracy of the sample mean of Xi , equal εd and εu are chosen so that the data window includes 31 observations in this paper.

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International Journal of Management Science and Engineering Management, 7(2): 153-160, 2012

So the contribution of the individual assets to the portfolio ES is represented by the “Component ES” and calculated as: ∂ESα ( X ) · ωi = E[ Xi | Xi ≥ VaRα ( X )] · ωi . (5) C − ESi = ∂ωi Compared to the risk decomposition based on VaR, estimating the conditional expectation in Eq. (5) is somewhat easier: From a total sample of T observations, a data window within which the portfolio losses are larger than or equal to the portfolio VaR boundary is chosen. Again let T ∗ be the number of chosen observations in this data window, let X j (1 ≤ j ≤ T ∗ ) denote the portfolio losses of the j j-th sample, and Xi (1 ≤ i ≤ n, 1 ≤ j ≤ T ∗ ) denote the corresponding losses of the i-th individual asset, then the Marginal ES of the individual asset i can be estimated using the following equation: 1 j (6) M − ESi = E[ Xi | Xi ≥ VaRα ( X )] ∗ ∑ Xi . T j From the discussion above: Marginal VaR or ES can tell us how low the individual assets returns will be when whole portfolio returns are extremely low, so it has considered one asset’s linkage with other assets in portfolio, while the stand-alone VaR or ES of individual assets neglects this linkage. Besides this, Eqs. (1) and (4) suggest that the sum of the individual assets’ Component VaR (or ES) (defined in Eqs. (2) and (5)) is exactly equal to the portfolio VaR (or ES), hence the risk contributions represented by Component VaR or ES can intuitively show us how much of the total portfolio risk is brought by the various individual assets. However, the sum of the individual assets’ stand-alone VaR (or ES) is not necessarily equal to the portfolio VaR (or ES). 2.2 Multivariate Copula Unlike risk decomposition based on observed historical data in previous research, in this paper the risk contributions are calculated based on simulation data generated using multivariate Copulas. A Copula is a function which can incorporate marginal distributions into a joint distribution. The pivotal Sklar’s theorem (Nelson, 2006 [17]) in the Copula theory demonstrates that: Suppose X = ( X1 , · · · , Xn ) are a vector of n variables with marginal distributions denoted by F1 , · · · , Fn , a Copula C would exist which can link the joint ndimensional distribution F to their marginals using the equation F ( x1 , · · · , xn ) = C ( F1 ( x1 ), · · · , Fn ( xn )). The information of marginal distributions is contained in Fi ( xi ), while all the information of dependence structure is captured by Copula. This key property allows the marginal distribution to be fitted separately in the first step, and then correct Copulas can be chosen to model the dependence structure in a second step. The Copula has another very important advantage for risk management as they can describe the non-linear dependence structure. Unlike a linear variance-covariance matrix, Copulas allow the dependence degree in the tail to be different from that in the middle of distribution. More importantly, Copula functions can be easily related to the tail dependence coefficient, which measures the propensity of different financial assets to crash (or boom) together (Cherububu et al., 2004 [5]). Therefore, Copula functions have been widely used in the integration of risk: Rosenberg

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and Schuermann (2006) [19] specified different distributions for market risk, credit risk and operational risk in banks, and then used Copula functions to integrate different risks to estimate the VaR of the total risk. Kole et al. (2007) [14] employed Copula functions to describe the dependence structure across three assets (stock, bond and real estate) and used VaR to measure the total risk. Different Copula functions imply different patterns of dependence structure, and hence have different features for tail dependence. This paper employs three multivariate Copula functions: Gaussian, Student t and Clayton3 . The Gaussian Copula is usually used as a benchmark, and has no tail dependence; the Student t Copula captures symmetric upper and lower tail dependence; and the Clayton Copula only has lower tail dependence. Their formulas and properties can be found in Cherububu et al. (2004) [5] and Nelson (2006) [17]. The IFM (inference for the marginal) approach (Nelson, 2006 [17]) was applied to estimate the model’s parameters in two stages: First, the parameters of marginal distributions were estimated: θˆ1 = ArgMaxθ1

T

n

∑ ∑ ln fi (xit ; θ1 ),

(7)

t=1 i =1

where f i ( xit ; θ1 ) denotes the density functions of the marginal distributions. Given the estimation of θˆ1 , the parameters of the Copula were estimated: θˆ2 = ArgMaxθ2

T

∑ ln c( F1 (xit ), F1 (xit ), · · · , F1 (xit ); θ2 , θˆ1 ).

t=1

(8) After the estimation of the different Copula functions, the one which describes the cross-asset dependence structures best was chosen, based on the AIC value, the KS (Kolmogorov Smirnov) and AD (Anderson Darling) distance proposed by Kole et al. (2007) [14]. The following four equations are used: D − KSm = max | FE ( xt ) − FH ( xt )|, D − KS a =

Z x

| FE ( xt ) − FH ( xt )|dFH ( xt ),

F ( xt ) − FH ( xt ) , D − AD m = max pE FH (1 − FH ( xt )) F ( x ) − FH ( xt ) pE t dFH ( xt ). (9) FH (1 − FH ( xt )) where FE and FH denote the empirical and hypothesized distributions respectively. Eq. (9) shows that the KS distance is sensitive to deviations in the middle of the distribution, while AD distance gives more weight to the deviations in the distribution tail. This feature of AD distance is very useful to this study, as the VaR and ES contributions here are exactly related to extreme losses in the tails. Based on the best Copula function, sample data can be generated repeatedly through simulation. Using the simulated data a multi-set of estimated risk contributions can be calculated, and thus the risk contributions of individual assets that are significantly different can be tested. D − AD a =

Z

x

3 Main empirical results This section provides the main results of the empirical analysis on different industries’ risk contributions to the whole stock market in China. Our study was based on

3

Unlike bivariate Copula, the theory of multivariate Copula has not well developed. Due to the difficulty in estimation and simulation, we cannot choose more multivariate Copulas.

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156

G. Fan & Y. Zeng & W. Wong: Decomposition of portfolio VaR and expected 0 0.012

RCVaR (99%) Indutrial Commercial Real Estatee Utilities Conglomerates

0 0.010 Estimation Error

Estimation Error

0.014 0.013 0.012 0.011 0.010 0.009 0.008 0.007 0.006

0.014

Indutrial

0.012

Commercial

0.011

Real Estatee

Estimation Error

0.013

0.010 0.009

In ndutrial

0 0.006

C Commercial

0 0.004

R Real Estate

0 0.002

U Utilities C Conglomerates

0 0.006

RCVaR R (95%)

Utilities Conglomerates

0.008

0 0.005 0 0.004 Estimation Error

0.015

0 0.008

0 0.000

0.016

RC CES (9 99%)

0 0.003 0 0.002

RC CES (9 95%) Ind dutrial Commercial Reaal Estate

0 0.001 0 0.000 ‐0 0.001

Utiilities Conglomerates

Fig. 1 Simulation sample size and the estimation errors of risk contributions

the five industrial indices on the SSE (Shanghai Stock Exchange): the SSE Conglomerates Index, the SSE Industrial Index, the SSE Commercial Index, the SSE Utilities Index and the SSE Real Estate Index. A sample period of 19972009 was chosen with 3138 daily observations, as a price limit of 10% has been in effect in China’s stock market since Dec 13, 1996. The indices returns were calculated as rt = 100 ∗ LN ( Pt / Pt−1 ), where Pt was the closing price of the stock indices. An equal-weighted portfolio of the five industrial indices was constructed, returns of which were the simple average of each individual index’s returns, then the VaR and ES risk contributions of the individual industrial indices to the whole portfolio are estimated4 . As shown by Eqs. (2) and (5), the risk contributions are represented by “Component VaRs or ESs”, defined as the product of portfolio weights and “Marginal VaRs or ESs”. But for equal-weighted portfolios here, the portfolio weights are equal across different industrial indices, so the risk contributions rank is mainly determined by the relative size of the estimated Marginal VaRs or ESs5 . Statistics for the five index returns series are provided in Panel A of Tab. 2. The observed kurtosis shows that the return distributions of the five indices are all fat-tailed. The non-normality of distributions and the existence of excess tail risk is important in risk management, thus it is worth conducting further analysis on these five indices. Through comparison, the Real Estate Index share returns were found to have the largest standard deviation and the lowest kurtosis. Before choosing the proper Copula functions to describe the dependence structure across the various indices, their marginal distributions were specified first. Index returns are assumed to follow the AR(1)-GARCH(1,1) model, and the distribution of standardized residuals is Student t. This specification is as follows: 4 5

rt = µ + ρrt−1 + σt zt , σt = ω + αε2t−1 + βσt−1 , zt ∼ t(υ ), (10) where rt represents the daily indices returns. The estimated marginal distributions parameters are given in Panel B of Tab. 2, from where it can be seen that the Real Estate Index’s degree of freedom is the largest, which is consistent with its lowest kurtosis in the descriptive statistics. After estimating the marginal distributions parameters, the three Copula functions were used to model the dependence structure. As shown in Panel C of Tab. 2, no matter based on the AIC value or the KS and AD distances, Student t Copula was the best specification for describing the dependence structure across the five industrial indices. This implies that stocks from different industries may both boom and crash together, which cannot be described by either of the other two Copula types. Hence, the Student t Copula was used to generate simulation data (simulate 10000 times), and estimate the risk contributions based on the simulation data. By simulating repeatedly, 10000 sets of estimated risk contributions were obtained. Tabs. 3 and 4 outline the average and standard deviation of the estimated VaR and ES risk contributions at a 99% and 95% confidence level respectively. As mentioned in Section 2.1, the sum of Component VaR (ES) is exactly equal to portfolio VaR (ES), so we divide Component VaR (ES) by the portfolio VaR (ES) to calculate their relative values, denoted by “RCVaR (RCES)”. The sum of these calculated “RCVaR (RCES)” is then equal to one. The results show that among the five industrial indices, the Real Estate Index shared the largest risk contribution to the whole market, and the estimated risk contributions rank was not affected by either the confidence level or the risk measure. The statistical test further confirmed that the risk contributions of the various indices were indeed significantly different. The finding that the Real Estate Index’s risk contribution was the largest implies that, compared to

The results for value-weighted portfolio can also be provided upon request. Intuitionally, Marginal VaRs or ESs tells us the propensity of various industrial indices crashing together with the whole market.



157

Table 2 Descriptive statistics and model estimation Industrial

Commercial

Real Estate

Utilities

Conglomerates

Panel A: Descriptive statistics Mean

0.043

0.052

0.029

0.049

0.040

St.dev

1.778

1.869

2.334

1.851

1.868 −0.055 6.990 10.057 −94.504

Skewness Kurtosis Minimum Maximum

−0.250 7.027 9.386 −9.724

µ

0.054∗

ρ

0.021

ω

0.055∗

α

0.113∗

β

0.878∗

υ

5.027∗ D-KSm

Gaussian

0.096

Student t

0.089

Clayton

0.844

−0.379 −0.108 −0.368 6.459 5.347 7.308 9.242 9.550 9.521 −9.802 −10.081 −10.190 Panel B: Estimation of marginal distributions 0.058∗ 0.000 0.029 0.065∗ 0.029 0.002 0.076∗ 0.066∗ 0.050∗ 0.125∗ 0.096∗ 0.105∗ ∗ ∗ 0.862 0.899 0.886∗ ∗ ∗ 5.277 5.708 5.177∗ Panel C: Estimation of Copula functions D-KSa D-ADm D-ADa 0.053 7.188 0.320 0.050 0.329 0.164 0.109 12.055 0.283

0.025 0.000 0.053∗ 0.103∗ 0.890∗ 4.917∗ AIC 13.254 12.710 14.294

Table 3 The results of risk decomposition based on Copula simulation (Confidence level: 99%; marginal distribution: GARCH-t) N = 10000, T = 3000 RCVaR 0.175

RCES 0.174

N = 10000, T = 10000 RCVaR 0.172

RCES 0.174

4∗ 4∗ 4∗ 4∗ (0.0090) (0.0085) (0.0085) (0.0045) 0.218 0.215 0.215 0.216 2∗ 2∗ 2∗ 2∗ Commercial (0.0097) (0.0087) (0.0088) (0.0047) 0.257 0.247 0.253 0.247 1∗ 1∗ 1∗ 1∗ Real Estate (0.0129) (0.0116) (0.0119) (0.0062) 0.170 0.168 0.167 0.167 Utilities 5 5 5 5 (0.0099) (0.0092) (0.0094) (0.0048) 0.198 0.196 0.194 0.195 Conglomerates 3∗ 3∗ 3∗ 3∗ (0.0084) (0.0074) (0.0074) (0.0039) Note: N is the simulation times, and T is the each simulation’s sample size. The values in the parentheses are the standard deviations of the estimated risk contributions. “∗” denotes that the risk contribution of a corresponding index is significantly larger than the index whose rank is just below it (at the level of 5% and based on a T test). Industrial

other industries, the Real Estate industry was more sensitive to adverse factors which may make the stock market crash, and thus the price of the stocks in this industry were more likely to fall with the market. Yamai and Yoshiba (2002) [26] found that the estimated VaR risk contributions were very sensitive to portfolio weights, and Wong and Copeland’s (2008) [25] results based on a portfolio of stocks and options suggested that the VaR risk contributions would become unreliable if a leverage effect exists. This analysis based on Copula simulations also indicated that the estimation errors (denoted by standard

deviations) of the ES risk contributions were consistently smaller than the VaR risk contributions’, no matter what the confidence levels or risk measures were. Based on such a finding, it is important to remember that the risk contribution should not be calculated only using VaR, because of the larger estimation errors in VaR risk contributions6 . So in order to obtain more reliable results, it is better to estimate the risk contribution using both risk measures.

6

The larger estimation error of VaR risk contributions is possible because that VaR disregards the information contained in the size of extreme losses and thus suffers tail risk. Moreover, in the calculation of VaR risk contribution, the conditional expectation conditioned by X = VaRα ( X ) is not estimated directly, but is approximately estimated by choosing a data window within which the portfolio loss is close to the portfolio VaR. This approximation may be another reason why the estimation errors of VaR risk contributions are larger.


158


Table 4 The results of risk decomposition based on Copula simulation (Confidence level: 95%; marginal distribution: GARCH-t) N = 10000, T = 3000 RCVaR 0.167

N = 10000, T = 10000

RCES 0.171

RCVaR 0.167

RCES 0.171

4∗ 4 4 (0.0022) (0.0106) (0.0041) (0.0106) 0.214 0.211 0.213 0.211 Commercial 2∗ 2∗ 2∗ 2∗ (0.0023) (0.0110) (0.0042) (0.0109) 0.255 0.263 0.255 0.263 1∗ Real Estate 1∗ 1∗ 1∗ (0.0032) (0.0152) (0.0058) (0.0152) 0.167 0.168 0.167 0.167 Utilities 5 5 5 5 (0.0115) (0.0045) (0.0115) (0.0024) 0.194 0.192 0.194 0.193 3∗ 3∗ 3∗ 3∗ Conglomerates (0.0019) (0.0093) (0.0035) (0.0093) Note: N is the simulation times, and T is the each simulation’s sample size. The values in the parentheses are the standard deviations of the estimated risk contributions. “∗” denotes that the risk contribution of a corresponding index is significantly larger than the index whose rank is just below it (at the level of 5% and based on a T test). Industrial

4 Sensitivity analysis 4.1 The impact of simulation sample size Yamai and Yoshiba (2002) [26] compared VaR and ES’s performance on measuring each individual asset’s univariate risk and found that when the distributions were seriously fat-tailed, the estimation error of ES could be decreased by increasing the sample size. Therefore, whether the sample size in each simulation affects the VaR and ES risk contributions estimation errors was examined. Fig. 1 depicts the estimation errors of risk contributions with simulation sample sizes ranging from 3000 to 100000, from which it was found that the estimation errors of ES risk contributions were consistently smaller than the VaR risk contributions’, and their differences became larger as the simulation sample size was increased. The estimation error of VaR risk contribution was hardly affected by the simulation sample size, but the estimation error of ES risk contribution decreased as the simulation sample size increased7 . Hence similar to stand-alone ES, the estimation errors of ES risk contribution could be decreased by increasing the simulation sample size. 4.2 The impact of the hypothesized model In our analysis above, a model with the marginal distributions of AR(1)-GARCH(1,1) and a dependence structure described by Student t Copula are employed. If the calculation is based on simulation data generated from this model, consistent results for the estimated risk contributions are obtained. To examine the consistency of the results from the hypothesized model, the model was modified slightly and the risk contributions recalculated. The Student t Copula did not change due to its flexibility to capture tail dependence but the marginal distributions were substituted by Student t. Tab. 5 shows the new results based on the changed model (simulated 10000 times with a sample size in each simulation of 10000). After the change in the model, a new rank for various industrial indices’ risk contributions was obtained and the 7

4∗

Commercial Index and Utilities Index rank were reversed. Thus, when calculating risk contributions based on simulation data, the results may be affected by the hypothesized model. From these results a model as close to the true distribution as possible is needed. However, this does not imply that the outlined approach is unreliable, as both estimations based on historical data and estimations based on simulation data may have their defects. In calculations based on historical data, the results can be easily influenced by the chosen data window, whether the data window has an extreme event (like 1987’ crisis) or not will bring quite different results. While in the estimation based on simulation data, if the hypothesized model used is incorrect, the estimated risk contributions may be biased. Our purpose of using a multivariate Copula to model the dependence structure across the various industrial indices in this paper is just to obtain an accurate estimation of risk contributions, as Copula function is well-documented as a flexible tool for modelling a nonlinear relationship across different financial assets. However, the modified model results do not alter the main conclusion in this paper: First, the Real Estate Index’s risk contribution to the stock market was still the largest. Second, the estimation of risk contributions based on ES is also more robust. Tab. 5 indicates that, the VaR risk contributions rank is affected by confidence level, but the ES risk contributions rank is not. Further, the estimation errors of ES risk contributions were still consistently smaller than the VaR risk contributions. 4.3 The impact of data window size on estimated VaR risk contribution From the method to calculate VaR risk contribution provided in Section 2.1, it is known that the conditional expectation in Eq. (2) is approximately estimated using observations from a data window, so the estimated VaR risk contribution may be affected by the size of the chosen data window. The results given in Tab. 6 show that if the VaR risk contributions are calculated based on Copula simula-

This is probably because that ES is calculated by averaging the extreme losses beyond VaR and hence is more sensitive to the appearance of extreme losses and the frequency of such appearances (Tasche, 2000 [21]). When the sample size is larger, we can use sufficient extreme observations to calculate the ES risk contribution and thus obtain more reliable estimation. 8 When using historical data, only one set of estimated risk contributions can be obtained and thus the estimation error cannot be accurately calculated. Hence, in this table only the risk contributions rank rather than the estimation errors can be compared.



159

Table 5 The results of risk decomposition based on Copula simulation (N = 10000, T = 10000; marginal distribution: Student t) Confidence level: 99% RCVaR

Confidence level: 95%

RCES

RCVaR

RCES 0.176

0.189

0.164

0.178

5 4* 5 5 (0.0041) (0.0137) (0.0081) (0.0117) 0.196 0.201 0.191 0.197 Commercial 2∗ 2 2∗ 2∗ (0.0035) (0.0123) (0.0071) (0.0107) 0.254 0.222 0.287 0.249 1∗ Real Estate 1∗ 1∗ 1∗ (0.0093) (0.0196) (0.0190) (0.0191) 0.183 0.174 0.188 0.181 Utilities 4∗ 4∗ 5 4∗ (0.0131) (0.0084) (0.0154) (0.0042) 0.193 0.201 0.185 0.195 3∗ 3∗ 3∗ 3∗ Conglomerates (0.0039) (0.0121) (0.0081) (0.0110) Note: N is the simulation times, and T is the each simulation’s sample size. The values in the parentheses are the standard deviations of the estimated risk contributions. “∗” denotes that the risk contribution of a corresponding index is significantly larger than the index whose rank is just below it (at the level of 5% and based on a T test). Industrial

Table 6 The results of risk decomposition based on Copula simulation A. based on historical data 99% Data window size

B. based on Copula simulation (N = 10000, T = 10000)

95%

99%

95%

30

50

30

50

30

50

80

100

30

50

80

100

Industrial

4

4

4

5

4

4

4

4

4

4

4

4

Commercial

2

2

2

3

2

2

2

2

2

2

2

2

Real Estate

1

1

1

1

1

1

1

1

1

1

1

1

Utilities

3

3

5

4

5

5

5

5

5

5

5

5

Conglomerates

5

5

3

2

3

3

3

3

3

3

3

3

Note: N is the simulation times, and T is the sample size of each simulation.

tions, the estimated VaR risk contributions rank is not affected by the size of the chosen data window8 . But if the VaR risk contributions are calculated based on historical data, the rank is sensitive to the size of the chosen data window at a 95% confidence level. This again highlights that the improved approach presented here for calculating the risk contributions using Copula simulations is better than calculations based on historical data. Moreover, the sensitivity of the estimated VaR risk contributions to the size of chosen data window means that when calculating the risk contribution of individual assets (especially based on historical data), VaR should not be used as the only risk measure.

5 Conclusion The accurate calculation of individual asset’s risk contributions to the whole portfolio can help risk managers achieve effective risk management (not just risk measurement), and is also useful for investors’ portfolio choice. In order to test the significance of the differences across the risk contributions of different assets, this paper employed multivariate Copula functions to model the crossasset dependence structure, and then decomposed the portfolio VaR or ES based on the simulation data generated from such models, and finally obtained the estimated risk contributions of each individual asset to the whole portfolio. The empirical analysis suggests that: First, besides

flexibility in testing the differences in the estimated risk contributions, the Copula simulation approach has consistent results which are not affected by confidence levels or risk measures, provided that the hypothesized models are as close to the true distributions as possible. Second, compared to VaR risk contributions, the estimated ES risk contributions tends to be more robust, and the estimation error of ES risk contribution can be decreased by increasing the simulation sample size. This implies that the risk contributions calculation based only on VaR in previous studies may be incorrect, and that ES risk contributions should be introduced in future studies. Finally, results based on the an equally-weighted portfolio of the five industrial indices on the SSE indicate that the Real Estate Index’s risk contribution to the stock market is the largest.

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When using the Copula simulation method, the estimation errors can be calculated. For the effect of the chosen data window size on estimation errors, we expect that there would be a trade-off as discussed in footnote 2.


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