Value-At-Risk for Long and Short Positions Of

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International Research Journal of Finance and Economics

ISSN 1450-2887 Issue 22 (2008)© EuroJournals Publishing, Inc. 2008

http://www.eurojournals.com/finance.htm

Value-at-Risk for Long and Short Positions of

Asian Stock Markets

Anthony H. Tu

Department of Finance, National Chengchi Universit

Woon K. Wong

Cardiff Business School, Cardiff University, Aberconway Building

Colum Drive, Cardiff CF10 3EU, U.K

Matthew C. Chang

Department of Finance and Banking, Hsuan Chuang University, No. 48

Hsuan Chuang Rd., Hsinchu City, 300, Taiwan E-mail: [email protected]

Tel: +886-3-530-2255, Ext. 6024; Fax: +886-3-539-1292

Abstract

Empirical research shows that stock market returns distributions can be asymmetricand the auto-correlations of its absolute returns are stronger than that of its squared returns.

An asymmetric returns distribution means that the left and right tails would have different

thickness, whereas higher auto-correlation of absolute returns implies that better volatilityforecasts are possible. This has significant economic implication for risk management that

requires accurate estimation of Value-at-Risk (VaR). In this paper, we therefore investigatethe performance of VaR models that take into consideration of the skewness of the

innovation process and utilize the Box-Cox transformation of conditional variance in aflexible ARCH-type model. Specificaly, we use the Asymmetric Power Autoregressive

Conditional Heteroscedasticity (APARCH) model based on the skewed Student density to

model the VaRs of daily returns of those Asian markets found to exhibit both skewness andkurtosis in the innovation process. We apply the likelihood ratio tests of proportional

failure rates to such VaR model and compare the results with other VaR models, in

particular, APARCH with symmetric distributions and APARCH with skewed Studentdistributions. It is found that the APARCH model with skewed Student distribution

performs the best for the Asian markets considered.

Keywords: Value-at-Risk, APARCH, GJR, RiskMetrics, Skewed Student distribution,

Box-Cox transformation, proportional failure rates, likelihood ratio tests

JEL Classification Codes: G32

1. IntroductionMost Value-at-Risk (VaR) models focus on the computation of the VaR for negative returns (Jorion

(2000)). It is assumed that traders or portfolio managers have long trading position or they bought theasset and are concerned when the price of the asset falls. Since it is common to have short trading



International Research Journal of Finance and Economics - Issue 22 (2008) 136

position, Giot and Laurent (2003) is amongst the first to present the VaR models that are used to model

both the long and short sides of daily trading positions.

Empirical research shows that stock return distributions can be asymmetric. It means that theleft and right tails would have different tickness, implying the “usual” VaR models of the ARCH class

have a tough job in modelling correctly the left and right tails of the distributions of stock returns. To

alleviate these problems, we follow Giot and Laurent (2003) to introduce the skewed Student

Asymmetric Power ARCH (APARCH) model (Ding et. al. (1993)) to model the one-day-ahead VaR

for portfolios of long and short trading positions on Asian stock markets1. In this paper, we consider only the APARCH model as it is a very general ARCH-type model, in that it encompasses

RiskMetrics, GARCH and GJR as special cases. We compare the performance of the skewed StudentAPARCH model by computing Kupiec’s (1995) LR tests on the empirical failure rates with the ones of

the Student APARCH and the normal models. The result shows that the skewed Student models in

general brings about significant improvement in correctly forecasting one-day ahead VaR for both longand short trading positions on ten daily Asian stock index returns (Hong Kong, Singapore, Australia,

Korea, Malaysia, Philippines, Thailand, Indonesia, China and Japan), but its performance is not

satisfactory in all cases.

2. Three competing ARCH-type Models

For all price series t P, daily returns (in %) are defined as

)]ln()[ln(100 1−−= t t t PP y. To characterize

the APARCH models, we consider a collection of daily returns, t y, with t = 1,2,…T, which is known

to be conditional heteroscedastic. It is typically modelled as follows:

t t y ε μ +=(1)

and

t t t zσ ε = (2)

We now consider several specifications for the conditional variance of t ε .

2.1 Normal APARCH

The normal APARCH (Ding et. al. (1993)) is an extension of the GARCH model of Bollerslev (1986).

It is very flexible ARCH-type model as it nests at least seven GARCH specifications. The normalAPARCH (1, 1) is defined as:

)1,0(...~ N d ii zt (3)

andδ δ δ σ β ε α ε α ω σ 11111 )( −−− +−+= t t nt t (4)

whereω 1α , nα 1 β andδ are parameters to be estimated. δ (δ >0) plays the role of a Box-Cox

transformation of t σ , while nα (-1< nα

<1), reflects the so-called leverage effect. A positive (negative)

value of nα means that past negative (positive) shocks have a deeper impact on current conditional

volatility than past positive shocks (Black (1976), French et. al. (1987)).

2.2 Student APARCH

Previous empirical studies on VaR forecasts have shown that models based on the normal distributionusually cannot fully take into account the “fat tails” of the return distribution. To alleviate the problem,

the Student APARCH is formulated by using the standardized student distribution (with zero mean,unit variance and υ degree of freedom)

1 Recently, Giot and Laurent (2003) have shown that the skewed Student APARCH model (unlike the RiskMetrics model or more simple ARCH-type

models) does provide more accurate VaR forecasts, both for the right and left tails of the distribution of returns.



137 International Research Journal of Finance and Economics - Issue 22 (2008)

),1,0(...~ υ t d ii zt (5)

and t σ is defined as in equation (4).

2.3 Skewed Student APARCH

To account for the excess skewness and kurtosis, Fernández and Steel (1998) propose to extend theStudent distribution by adding a skewness parameter

2. The innovation process z is said to be

(standardized) skewed Student distributed if:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−≥++

−<++

=

s

m zif mszsg

s

m zif mszsg

z f

]/)[(1

2

])([1

2

),(

υ ξ

ξ ξ

υ ξ

ξ ξ

υ ξ (6)

where)( υ ⋅gis the symmetric (unit variance) Student density andξ is the asymmetry coefficient

3. In

addition, m and2s are respectively the mean and the variance of the non-standardized skewed Student:

)1

(

)2

(

2)2

1(

ξ ξ

υ π

υ υ

−Γ

−−Γ=m

(7)

2

2

22 )11

( ms −−+=ξ

ξ

(8)

Notice also that with respect to the (zero) mean),(),1( υ ξ υ ξ t t z f z f −=

,Thus, the sign of indicates the direction of the skewness: the third moment is positive (negative), and the density is

skewed to the right (left), if 0)ln( >ξ )0(<

For the three competing models, the VaR for long and short positions have been described inGiot and Laurent (2003). They are summarized in the Table 1.

2 The main weakness of Fernández and Steel (1998) is that its density is expressed in terms of the mode and the dispersion. In order to keep it in theARCH “tradition”, Lambert and Laurent (2001) re-expressed the skewed Student density, in terms of the mean and the variance, in such a way that the

innovation process has zero mean and unit variance.3 The asymmetry coefficient ( >0ξ ) is defined such that the ratio of probability masses above and below the mean

is2)0Pr()0Pr( ξ ξ ξ =<≥ z z




Table 1: VaR for long and short under three competing ARCH-type models)( ,1, ν α ν α −St St denotes the left (right) quantile at α% for the (standardized) Student distribution with

(estimated) number of degree of freedom ν.)( ,,1,, ξ ν α ξ ν α −skst skst denotes the left (right) quantile at α%

for the skewed Student distribution with ν degree of freedom and asymmetry coefficient ξ. Lambert

and Laurent (2000) show that the quantile function ξ ν α ,,*

skst of a non-standardized skewed Student

density is

⎪⎪⎩

⎪⎪⎨

⎧

+≥⎥⎦

⎤⎢⎣

⎡ +−

−

+<⎥⎦

⎤⎢⎣

⎡+

=−

2

2

,

2

2

,

,,*

1

1)1(

2

1

1

1

)1(2

1

ξ α ξ

α ξ

ξ α ξ

α

ξ

ν α

ν α

ξ ν α

if st

if st skst

where ν α ,St is the quantile function of the (unit variance) Student-t density. It is straightforward to

obtain the quantile fuction of the standardized skewed Student:

s

mskst skst

−=

*

,,

,,

ξ ν α

ξ ν α a

ARCH Model VaR

long Short

Normal APARCHt z σ μ α + t z σ μ α −

Student APARCH1 t st σ μ ν α ,+ t st σ μ ν α ,1−+

Skewed APARCH2 t skst σ μ ξ ν α ,,+ t skst σ μ ξ ν α ,,1−+

3. Data and Empirical ResultsThe data consist of daily closing prices for the ten major Asia-Pacific equity market indices: Hong

Kong Hang-Seng, Singapore Straits Times, Sydney All Ordinaries, Seoul Composite Index, Kuala

Lumpur Composite Index, Manila Composite Index, Bangkok Composite Index, Jakarta CompositeIndex, Tokyo Nikkei 225 and Shanghai B-shares Index.

In order to perform the VaR analysis, the normal APARCH, Student APARCH and skewed

Student APARCH are estimated first. Table 2 presents the results for the (approximate) maximum

likelihood estimation of the skewed Student APARCH model on the ten equity indices4.

4 We do not report full estimation results of the normal and Student APARCH models as they are quite similar. Furthermore, these specifications are

encompassed by the skewed Student APARCH model which we fully detail in Table 2.




Table 2: Estimated Results of the Skewed Student APARCH

The t-ratios (t*) are reported in parentheses. Thus,2t* >denotes significance at 95% confidence

level. ξ is the asymmetry coefficient. If 1)(or 0)ln( << ξ ξ the density is skewed to the left. Thus, the

VaR for long positions will be larger (for the same conditional variance) than the VaR for short

positions. If 1)(or 0)ln( >> ξ ξ , we have the opposite result.

ω α1 β1 αn δ ν ln(ξ)

Hong Kong0.0182 0.0687 0.9353 0.4052 1.1808 6.6304 0.0081

(2.997) (7.586) (100.216) (5.376) (7.043) (8.415) (0.386)

Singapore0.0293 0.1251 0.8776 0.2309 1.2873 6.7202 0.011

(3.513) (7.286) (46.563) (4.366) (6.358) (8.759) (0.449)

Australia0.0121 0.0616 0.9342 0.5472 1.1146 10.274 -0.0860

(3.290) (6.068) (80.832) (5.677) (7.196) (6.101) (-3.358)

Korea0.0207 0.0729 0.9292 0.2388 1.5704 8.507 -0.0074

(2.561) (4.642) (66.116) (4.154) (7.407) (6.524) (-0.333)

Malaysia0.0206 0.1312 0.8852 0.2230 1.1506 5.288 0.0605

(2.616) (5.513) (38.260) (4.501) (6.074) (11.477) (2.291)

Philippines0.0913 0.1604 0.8157 0.1523 1.4651 5.705 0.0169

(3.211) (6.018) (22.320) (3.332) (5.632) (9.325) (0.674)

Thailand0.0486 0.1149 0.8832 0.1738 1.3760 7.360 0.0881

(2.300) (5.375) (32.912) (3.548) (5.486) (8.443) (3.721)

Indonesia0.0294 0.1721 0.8562 0.1354 1.3430 4.355 0.0227

(2.227) (4.252) (23.651) (3.334) (7.615) (12.502) (0.937)

China

0.0910 0.2553 0.7980 0.0900 1.1920 3.403 0.0473

(3.643) (7.816) (31.046) (2.197) (5.993) (16.960) (2.117)

Japan0.0231 0.0712 0.9257 0.4844 1.3296 8.337 -0.0169

(3.663) (7.626) (96.261) (4.554) (6.629) (7.079) (-0.682)

The ten equity indices feature some relatively similar volatility specifications. First, the

autoregressive effects are strong ( 1 β are around 0.9) for eight indices, suggesting strong memory

effects. Second, nα is positive and significant for all ten equity indices, indicating a leverage effect for

negative returns in the conditional variance specification. Third, δ are between 1.1146 and 1.5704 and

all significantly different from 2.

On the other hand )ln(ξ is significant for only four indices: one (Australia) is negative (thedensity is skewed to the left) and three (Malaysia, Thailand and China) are positive (the density isskewed to the right)

5. They imply that the asymmetry in the Student distribution is not necessary for

all equity indices to fully model the distribution of returns.

The estimated parameters based on 1000 returns observations t yare used to forecast the one-

day-ahead VaR for observation 1001. For observation 1002, the VaR is forecasted using the same

estimated model parameters and return observation 1001 y. In this manner, five one-day-ahead VaRs are

forecasted before the model parameters are re-estimated. This procedure is repeated until the VaRs of

5 One possible explanation for the density skewed to the “right” is due to the presence of short-sales restrictions.




all observations are forecasted. Re-estimating VaR models at every five-day frequency instead of every

day does not affect the results but have the benefit of less computation time.6

All models are tested with a VaR levelα (which ranged from 5% to 0.5%) and their

performance is then assessed by computing the failure rate for the returns, t yIf the VaR model is

correctly specified, the failure rate should be equal to pre-specified VaR level. Thus, the failure rate isdefined as the number of times returns exceed (in absolute value) the forecasted VaR. In our empirical

application, we define a failure rate )( s f f l for the long (short) trading positions, which is equal to the percentage of negative (positive) returns smaller (larger) than one-step-ahead VaR for long (short) positions.

We assess the models’ performance by first computing their empirical failure rates, both for the

left and right tails of the distribution of returns. Because the computation of the empirical failure ratedefines a sequence of yes (VaR violation)/ no (no VaR violation) observations, it is possible to test

α = f H :0 against α ≠ f H :1 , where f is the failure rate (estimated by f ̂ , the empirical failurerate), by so-called Kupiec LR test (see Kupiec (1995)). Under the null hypothesis, the LR test statistic

is asymptotically distributed as2

)1( χ .

The p-values for the Kupiec’s (1995) LR test on the empirical failure rates for skewed Student

(SKST) APARCH, Student (ST) APARCH and normal (N) APARCH are reported in Table 3. Theempirical failure rates are also presented in parentheses. The skewed Student APARCH model ingeneral improves on all other specifications for both long and short positions, but its performance isstill not satisfactory in all cases. For Australia, Malaysia, Thailand and China, the improvement issubstantial as the switch to a skewed Student distribution alleviates all violations (except China). For Hong Kong, Singapore, Korea and Philippines, the Student based (symmetric or skewed) modelsoutperform the normal based model. The symmetric and skewed Student models perform equally well.In Indonesia, even though the Student based models improve the performance of normal based model,the skewed Student model performs even worse than that of symmetric one. For Japan, all three models(normal, Student or skewed Student) correctly model on all VaR levels for long and short positions.

6 In Giot and Laurent (2003), the VaR models is also estimated at every 5-day frequency.




Table 3: Out-of-sample VaR Results for 10 Equity Indices

P-values of Kupiec’s (1995) LR test for the null hypotheses l f α = (α is the failure rate for the long

positions) and(α is the failure rate for the short positions). Empirical failure rates are reported in

parentheses. They are defined as the number of times returns exceed (in proportion) the forecasted

VaR.

Long position Short position No. of VaR forecasts

0.5% 1. 0% 2.5% 5.0% 99.5% 99.0% 97.5% 95.0%

Hong Kong

SKST APARCH0.130 0.539 0.328 0.928 0.273 0.275 0.304 0.302

2301

(0.007) (0.011) (0.028) (0.05) (0.003) (0.008) (0.022) (0.055)

ST APARCH0.210 0.681 0.471 0.770 0.884 0.386 0.734 0.161

(0.007) (0.011) (0.027) (0.049) (0.005) (0.008) (0.024) (0.056)

N APARCH0.001* 0.020* 0.328 0.241 0.130 0.681 0.555 0.510

(0.01) (0.015) (0.028) (0.045) (0.007) (0.011) (0.027) (0.053)

Singapore

SKST APARCH0.594 0.745 0.514 0.651 0.819 0.230 0.138 0.459

2355

(0.004) (0.009) (0.023) (0.048) (0.005) (0.008) (0.02) (0.047)

ST APARCH0.398 0.450 0.514 0.402 0.819 0.330 0.104 0.868

(0.004) (0.008) (0.023) (0.046) (0.005) (0.008) (0.02) (0.049)

NAPARCH0.152 0.200 0.514 0.011* 0.242 0.371 0.229 0.185

(0.007) (0.013) (0.023) (0.039) (0.007) (0.012) (0.021) (0.044)

Australia

SKST APARCH0.097 0.793 0.479 0.072 0.578 0.313 0.209 0.895

2372

(0.008) (0.011) (0.023) (0.042) (0.004) (0.008) (0.021) (0.051)

ST APARCH0.016* 0.793 0.135 0.412 0.059 0.016* 0.007* 0.311

(0.009) (0.011) (0.03) (0.046) (0.003) (0.005) (0.017) (0.046)

NAPARCH0.004* 0.071 0.387 0.133 0.545 0.567 0. 035* 0.227

(0.01) (0.014) (0.028) (0.043) (0.006) (0.009) (0.019) (0.045)

Korea

SKST APARCH0.200 0.294 0.603 0.136 0.863 0.407 0.333 0.689

2283

(0.007) (0.012) (0.027) (0.057) (0.005) (0.008) (0.022) (0.048)

ST APARCH0.200 0.294 0.797 0.114 0.645 0. 407 0.211 0. 689

(0. 007) (0.012) (0.026) (0.057) (0.006) (0.008) (0.021) (0.048)

N APARCH0.005* 0.103 0.992 0.935 0.200 0.807 0.267 0.028*

(0.01) (0.014) (0.025) (0.05) (0.007) (0.011) (0.021) (0.04)

Malaysia

SKST APARCH0.886 0.829 0.851 0.080 0.074 0.677 0.196 0.251

2296

(0.005) (0.01) (0.024) (0.042) (0.003) (0.009) (0.021) (0.045)

ST APARCH0.654 0.392 0.250 0.024* 0.153 0.993 0.851 0.576

(0.004) (0.008) (0.021) (0.04) (0.003) (0.01) (0.024) (0.047)

N APARCH0.006* 0.224 0.851 0.003* 0.003* 0.007* 0.634 0.122

(0.01) (0.013) (0.024) (0.037) (0.01) (0.016) (0.027) (0.043)




Table 3: continued

Long position Short position No. of VaR forecasts

0.5% 1. 0% 2.5% 5.0% 99.5% 99.0% 97.5% 95.0%

Philippines

SKST APAECH0.682 0.704 0.070 0.337 0.863 0.704 0.904 0.645

2316

(0.006) (0.011) (0.031) (0.054) (0.048) (0.025) (0.011) (0.005)

ST APARCH0.490 0.862 0.092 0.386 0.429 0.973 0.781 0.716

(0.006) (0.01) (0.031) (0.054) (0.004) (0.01) (0.026) (0.048) N APARCH

0.006* 0.008* 0.119 0.577 0.003* 0.008* 0.884 0.019*

(0.009) (0.016) (0.03) (0.047) (0.01) (0.016) (0.025) (0.04)

Thailand

SKST APAECH0.636 0.956 0.700 0.179 0.636 0.205 0.132 0.296

2274

(0.006) (0.01) (0.024) (0.044) (0.006) (0.007) (0.02) (0.045)

ST APARCH0.465 0.047* 0.017* 0.010* 0.304 0.875 0.909 0. 613

(0.004) (0.006) (0.018) (0.039) (0.007) (0.01) (0.025) (0.052)

N APARCH0.636 0.956 0.099 0.018* 0.002* 0.027* 0.675 0.397

(0.006) (0.01) (0.02) (0.04) (0.01) (0.015) (0.026) (0.046)

Indonesia

SKST APAECH0.304 0.502 0.038* 0.046* 0.636 0.502 0.774 0.752

2274

(0.007) (0.011) (0.032) (0.059) (0.006) (0.011) (0.026) (0.051)

ST APARCH0.119 0.639 0.088 0.086 0.912 0.383 0.984 0.548

(0.007) (0.011) (0.031) (0.058) (0.005) (0.012) (0.025) (0.053)

N APARCH0.000* 0.002* 0.038* 0.649 0.000* 0.017* 0.878 0.397

(0.012) (0.017) (0.032) (0.048) (0.012) (0.015) (0.026) (0.046)

China

SKST APARCH0.496 0.605 0.890 0.843 0.034* 0.179 0.285 0.011*

2239

(0.004) (0.009) (0.025) (0.051) (0.008) (0.013) (0.029) (0.062)

ST APARCH0.088 0.153 0.997 0.774 0.034* 0.124 0.115 0.001*

(0.003) (0.007) (0.025) (0.049) (0.008) (0.013) (0.03) (0.066)

N APARCH0.000* 0.001* 0.590 0.138 0.000* 0.000* 0.014 0.498

(0.011) (0.018) (0.027) (0.043) (0.016) (0.022) (0.033) (0.053)

Japan

SKST APARCH0.892 0.193 0.759 0.661 0.892 0.193 0.156 0.604

2291

(0.005) (0.007) (0.024) (0.048) (0.005) (0.007) (0.021) (0.052)

ST APARCH0.892 0.532 0.913 0.966 0.892 0.125 0.118 0.806

(0.005) (0.009) (0.025) (0.05) (0.005) (0.007) (0.02) (0.049)

N APARCH0.126 0.155 0.717 0.464 0.316 0.404 0.156 0.185

(0.007) (0.013) (0.026) (0.047) (0.007) (0.012) (0.021) (0.044)

4. SummaryThe performance of VaR forecasts by three competing models—normal APARCH, Student APARCHand skewed Student APARCH is investigated by using daily returns of ten Asian equity indices. Weapply the Kupiec’s LR tests of proportional failure rates to such VaR models. The results show that theskewed Student APARCH model in general outperforms all other specifications for both long and short positions.




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returns and a new model. Journal of Empirical Finance 1, pp.83-106.

[4] Fernández, C. and Steel, M.F.J. 1998. On Bayesian modelling of fat tails and skewness. Journalof the American Statistical Association 93, pp.359-371.

[5] French, K.R. and Schwert, G..W. and Stambaugh, R.F. 1987. Expected stock returns andvolatility. Journal of Financial Economics 19, pp.3-29.

[6] Giot, P. and S. Laurent, 2003. Value-at-risk for long and short trading positions. Journal of

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[9] Lambert, P. and Laurent, S. 2000. Modelling skewness dynamics in series of financial data.Discussion Paper, Institut de Statistique, Louvain-la-Neuve.

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Value-At-Risk for Long and Short Positions Of