Modeling the fat tails in Asian stock markets

International Review of Economics and Finance 20 (2011) 430–440

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International Review of Economics and Finance j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i r e f

Modeling the fat tails in Asian stock markets☆ Jullavut Kittiakarasakun, Yiuman Tse ⁎ Department of Finance, University of Texas at San Antonio, USA

a r t i c l e

i n f o

Article history: Received 12 November 2010 Available online 25 November 2010 JEL classification: F30 G15

Keywords: Asian stock markets Fat tails Value-at-Risk

a b s t r a c t We test whether stock returns in the Asian markets are characterized by infinite variance or just large variance, which has an important implication for the applicability of many financial models in Asian market data. Employing the extreme value framework, we find that the Asian index return distributions are fat-tailed but have finite variance. However, the tails of the distributions behave similarly to those in the U.S. and the MSCI World index returns, suggesting that any financial model or risk management tool that incorporates the second moment would work equally well for the Asian market data as it does for developed market data. We apply the Value-at-Risk method using Asian and U.S. data and find no significant difference in performance. © 2010 Elsevier Inc. All rights reserved.

1. Introduction There is a growing interest in emerging market research due to rapid growth in size and unique characteristics of the markets. In the past two decades, Asian emerging markets have experienced astonishing growth rates that are much higher than the growth rate of the U.S. markets. From 1990 to 2007, the growth rates in market capitalization for Hong Kong, Indonesia, and Singapore are approximately 6.4, 5.2, and 3.4 times that for the New York Stock Exchange (see the World Federation of Exchanges www.worldexchanges.org). These impressive growth rates are mainly driven by liberalization in the late 1980s and the early 1990s, which attracted large inflows of foreign capital into the markets (Bekaert & Harvey, 1998; Devereux, 2009). Emerging markets are so different from developed markets that some financial models may not be applicable. Empirical studies based on data from the U.S. and other G7 countries do not necessarily apply to emerging market data. For example, many emerging countries impose restrictions on foreign investments and, to some extent, exert controls on cross-border capital flows. Harvey (1995) and Bekaert (1995) find that emerging-market stocks with high CAPM betas are usually associated with low expected returns, which contradicts the risk-return framework. However, this finding is consistent with the notion that emerging markets are segmented from developed markets. The emerging equity market remains a challenge to existing financial theories and models (Bekaert & Harvey, 2002). The behavior of asset returns in emerging markets differs substantially from that in developed markets. There are at least four distinguishing features of asset returns in emerging markets: high average returns, low correlations with asset returns in developed markets, high volatility, and high predictability in asset returns (Bekaert & Harvey, 1997). It is also well documented that asset returns in emerging markets are far from normally distributed. This characteristic can hinder the effectiveness of financial models since most of those models rely on the assumption of normality. For this reason, Bekaert et al. (1996, 1998) suggest that portfolio managers should incorporate higher (statistical) moments into their portfolio decisions. In this paper, we examine the statistical behavior of stock index returns in Asian markets for the period October 1989 through August 2009. Despite their close geographic proximity to each other, Asian countries are exposed to different levels of ☆ Tse acknowledges the financial support from a summer research grant of U.S. Global Investors, Inc. ⁎ Corresponding author. One UTSA Circle, College of Business, Department of Finance, University of Texas at San Antonio, TX 78249, USA. Tel.: + 1 210 458 5314l. E-mail addresses: [email protected] (J. Kittiakarasakun), [email protected] (Y. Tse). 1059-0560/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.iref.2010.11.013

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development. Japan is a developed market and Hong Kong, Singapore, Taiwan, and Korea are more advanced than other Asian emerging markets. By focusing on Asian markets, we can compare characteristics of the return distributions across different levels of market development with less influence from other unrelated factors. Studying the tails of the distributions of stock returns provides information regarding how likely it is that extreme price changes will occur, which is of central interest to portfolio managers. As we often observe large price movements in Asian markets, it is natural to suspect that the shape of their return distributions may differ from those in other developed markets, which has an important economic implication with respect to the types of market in which securities are traded (Longin, 2005; also see Fama, 1963; Mandelbrot, 1963; McCulloch, 1978). In a continuous market, an asset return over a fixed interval is the sum of many small returns on non-overlapping subintervals, but in a discontinuous market, an asset return results from only a few price jumps in some subintervals.1 Most of the existing financial models assume a continuous market (i.e., that asset prices follow a Brownian process), which may not be true if the assets often exhibit extreme price changes. Hence, to evaluate the applicability of the models, it is important to understand the underlying characteristics of the data. Many papers, e.g., Cha and Oh (2000), Climent and Meneu (2003), and Gutierrez et al. (2009), have examined the relationship between the Asian emerging stock markets and the U.S. market. We focus on the extreme returns in the Asian and the U.S. stock markets. In the first part of the paper, we employ the extreme value theory (EVT) to examine the distribution tails. Estimates of the shape parameter, which is the inverse of the tail index, can be used to distinguish types of distributions. We find that the shape parameters of Asian stock index returns are mostly positive and less than 0.5, suggesting that the return distributions are fat-tailed and have finite variances. Despite differences in levels of development, we find similar characteristics of stock index returns in Asian markets. Their shape parameters are also close to those of the U.S. S&P 500 and the MSCI World index returns. Thus, the difference between the Asian market index returns and developed market index returns could be merely a matter of scale. Although Asian markets are more volatile than developed markets, occurrences of extreme returns (relative to returns observed in the same market) are as likely to be observed in Asian markets as they are in developed markets. This has the immediate implication that financial models and risk management tools incorporating the second moment would work as well in developed markets as in the Asian markets. We examine this assumption in the second part of our paper. We employ a commonly used risk management tool, Value-at-Risk (VaR), and examine its effectiveness when applied to Asian market data. We use two types of VaR calculations, EVT-based VaRs and ARCH-based VaRs. EVT-based VaRs rely on estimates of the distribution tails and are considered a static approach, while ARCH-based VaRs allow for the time-varying second moment of asset returns and are considered a dynamic approach. Ho et al. (2000) also examine the tails in Asian stock returns for an earlier period, 1984–1996, focusing on the EVT–based VaR model. We find some support for our conjecture that VaR performance does not deteriorate when applying VaR estimations to Asian market data. We evaluate the VaR performance on the basis of the proportion of times that actual losses exceed VaR estimates. In our out-of-sample tests, we find that the proportion of times that this occurs is quite close to the pre-specified probability p used in VaR estimation, and that the difference is usually less than 2%. Based on comparisons across methods, ARCH-type models provide better VaR performance than do EVT models. Comparing across datasets, VaR models applied to U.S. market data do not exhibit better performance than that of VaR models applied to Asian market data. 2. The extreme value theory (EVT) EVT studies the statistical behavior of extreme values and their limiting distributions.2 Consider a collection of n independent and identically-distributed random variables, {x1,x2,…,xn}, and let u and l be the upper and the lower bounds, respectively, of the random variable. Let Fx(x) be the cumulative distribution function (CDF) of xi and y be the maximum value, y = max(x1,x2,…,xn). By the ordered statistics principle, the CDF of y is n

Fy ðxÞ = ½Fx ðxÞ :

ð1Þ

In general, Fx(x) and Fy(x) are unknown. However, from Eq. (1), one can see that as n approaches infinity, Fy(x) approaches zero if x is less than its upper bound, and unity if x is greater than its upper bound. The maximum random variable y can be transformed such that for some (αn N 0, βn), the distribution of the transformed variable y⁎ ≡ (y-βn)/αn converges to a non-degenerate distribution. The coefficients αn and βn are, respectively, the scale parameter and the location parameter. The subscription n indicates that the two parameters are estimated from the sample size n. Under the independence assumption, the limiting distribution of the transformed variable, Gy*(y), is given by:
 −1 = k : Gy ð yÞ = exp −ð1 + kyÞ

1

ð2Þ

Longin (1996, 2001) shows that the U.S. stock market is characterized by something in between the two markets. The results for the minimum value can be deduced directly with the signs being reversed: max(x1, …, xn) = min(-x1, …, -xn). See Tsay (2005), Longin (1996, 2005), and Kofman and De Vries (1990). 2

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The parameter k is called the shape parameter, and τ = 1/k is the tail index that characterizes the distribution. The negative shape parameter k suggests that the distribution has no tail and belongs to the Weibull family. When k = 0, the distribution has an exponentially-declining tail and belongs to the Gumbel family, i.e. the normal distribution and the log-normal distribution. When k N 0, the distribution has a power-declining tail and belongs to the Fréchet family. In addition, the tail index τ can be used to infer the number of existing moments of the random variable. When k approaches zero, τ approaches ∞, implying that all moments exist, which is the case for a normal distribution. If k is between 0 and 0.5, τ is greater than 2, suggesting that the second moment exists and that the distribution is fat-tailed with finite variance, e.g. a Student'st distribution. De Haan et al. (1989) prove that the ARCH process also has a limiting Fréchet distribution with k between 0 and 0.5. When k is greater than 0.5, τ is less than 2, suggesting that the second moment does not exist, which corresponds to a stable Paretian distribution. Longin (2005) proposes that two hypotheses can be tested from the shape parameter: H1 : k = 0 and H2 : k ≤ 0:5: The first null hypothesis tests whether the distribution belongs to the Gumbel family, including the normal distribution. Rejecting H1 indicates that the return distribution is not normal. The second null hypothesis tests for finite variance. Rejecting H2 indicates that the return has infinite variance and that the second moment of the return does not exist. In addition, the rejection of H2 suggests that the return follows a stable Paretian distribution and that the asset price is characterized by a discontinuous process, or jumps. 3. Estimation methods 3.1. Estimation of the shape parameter and EVT-based VaRs We next implement the parametric maximum likelihood (ML) estimation and the non-parametric Hill (1975) method. We discuss the estimations only for the right tail (corresponding to positive extreme returns), as the estimation of the left tail index can be implied easily by reversing the signs of the observations. Although it is well-known that daily stock returns are not independent, Berman (1963) shows that the limiting distribution of extreme values is still obtained if the squared correlation coefficients are finite.3 Let rt be the daily (log) return on a stock portfolio at time t. For a sample of T observed returns, we divide the sample into g nonoverlapping subsamples, each of which contains m observations. If the last subsample contains fewer than m observations, it is discarded. We collect the maximum return from each subsample. Assuming that the subperiod maxima are drawn from the extreme value distribution, the probability density function of the maxima can be derived from Eq. (2). Under the independence assumption, the likelihood function is a product of the density function. We then apply an optimization algorithm numerically to choose parameters that maximize the objective log likelihood function. Under the assumption that the limiting distribution of the extreme values holds, the ML estimate is unbiased and asymptotically normal (Tsay, 2005; Longin, 2005). According to Jorion (2006), VaR describes the quantile of the projected distribution of the portfolio returns over a target horizon. With a (1-p) percent confidence level, the probability of losses greater than the VaR should be less than p percent: Pr½loss N VaR ≤ p

ð3Þ

Investors with long positions in the portfolio face losses when the portfolio value declines and, hence, are concerned with the left tail of the return distribution. On the other hand, investors with short positions in the portfolio face losses when the portfolio value increases, so they are concerned with the right tail of the return distribution. In this paper, we measure losses in terms of percentage change in the portfolio value and evaluate VaR performance at the one-period horizon. For a given probability p, the VaR estimate based on the ML estimate is:

VaRML

8  αn  τ  > 1− −m·lnð1−pÞ < βn + τ = > : β − αn 1−−m·lnð1−pÞτ  n τ

for along position ð4Þ for a short position

where αn, βn, and τ are parameters estimated from the ML estimator and m is the length of the subperiod. The non-parametric Hill estimator is based on the order statistics and requires a pre-specified parameter q, supposed to be the number of extreme values in the sample. There are at least two approaches that can be used in selecting the value of q. The first approach we use is based on a Monte Carlo simulation (see Longin, 2005; Jensen & De Vries, 1991). Our second approach is a

3

Previous studies directly applying EVT estimations to price returns include Longin (2001, 2005) and Kofman and De Vries (1990).

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heuristic approach that selects a value of q at which the estimate of k becomes stable and does not change in a small neighborhood of q (Tsay, 2005). We explore both approaches, and the details of the Monte Carlo experiment are discussed in Appendix A. For a sample of T observations, we sort the sample in ascending order such that r(1) ≤ r(2) ≤ … ≤ r(T). The subscription in parentheses signifies the rank of each observation in the sample; for example, r(1) and r(T) are, respectively, the smallest and the largest values in the sample. For a given integer q, the Hill estimator for the shape parameter k is kHill =

1 q−1 ∑ ln rðT−iÞ −ln rðT−qÞ : q i=0

ð5Þ

The statistic is asymptotically normal with mean k and variance k2. The Hill estimator is applicable only when returns follow a Fréchet distribution (k N 0) and is an efficient method to test the null hypothesis of infinite variance. The VaR estimate based on the Hill estimator is given by: VaR =

 ðv=pT Þk −1
rðT−vÞ −rðT−2vÞ +rðT−vÞ 1−2−τ

ð6Þ

where v is the largest integer but less than q/2. The estimate can be viewed as a linear interpolation of two extreme values (Dekkers & De Haan, 1989; Kofman & De Vries, 1990). Table 1 Descriptive statistics. The sample comprises returns on country stock indices during the period from Oct 1, 1989 to Aug 31, 2009 (sample size is 5196). The noncrisis sample excludes observations from 06/01/97 to 12/31/98 and from 04/01/08 to the end of the sample. The residual return sample contains residuals from regressing excess country index return on excess MSCI World index returns and exchange rate returns. The regression also includes one lead and one lag of each independent variable. The equally–weighted portfolio sample contains average return of the US S&P 500 returns and the country index returns. Mean

Median

Panel A: raw returns Hong Kong (HK) Singapore (SG) Taiwan (TW) Thailand (TH) Indonesia (IN) Malaysia (MA) Korea (KO) Japan (JP) US S&P 500 MSCI World

St. Dev.

Min

Max

Skewness

Kurtosis

0.04 0.01 −0.01 0.00 0.03 0.02 0.01 −0.02 0.02 0.01

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.05

1.68 1.29 1.85 1.72 1.54 1.45 1.86 1.53 1.15 0.94

−14.74 −9.73 −10.29 −16.06 −12.73 −24.20 −12.81 −12.11 −9.47 −7.33

17.25 8.77 12.84 11.35 13.13 20.69 11.28 13.24 10.96 9.10

0.01 −0.16 −0.10 −0.02 0.02 0.25 −0.11 −0.03 −0.22 −0.33

12.82 8.54 6.51 9.59 12.64 45.35 7.43 8.85 12.72 12.33

Panel B: non-crisis sample Hong Kong (HK) Singapore (SG) Taiwan (TW) Thailand (TH) Indonesia (IN) Malaysia (MA) Korea (KO) Japan (JP) US S&P 500 MSCI World

0.06 0.02 0.00 0.01 0.05 0.04 0.02 −0.01 0.02 0.02

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.04

1.39 1.11 1.86 1.57 1.28 1.13 1.66 1.39 0.96 0.78

−9.29 −9.73 −10.29 −16.06 −10.93 −12.61 −12.81 −7.23 −6.32 −5.21

10.18 5.76 12.84 10.58 10.87 10.34 8.19 12.43 5.57 4.77

−0.35 −0.41 −0.10 −0.24 0.15 −0.54 −0.22 0.10 −0.07 −0.07

7.82 8.24 6.82 10.39 11.60 14.53 6.70 6.60 6.39 6.28

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

−0.01 −0.01 −0.01 −0.02 −0.02 −0.01 −0.03 0.00

1.48 1.16 1.77 1.63 1.47 1.36 1.71 1.28

−13.06 −9.07 −10.00 −15.75 −12.04 −25.27 −12.26 −7.63

15.55 8.11 12.15 10.82 12.51 19.99 9.84 9.31

0.21 0.01 −0.07 0.03 0.17 −0.24 −0.06 0.10

11.81 7.51 6.63 9.38 12.06 53.28 6.78 6.02

Panel D: equally-weighted portfolio Hong Kong (HK) 0.03 Singapore (SG) 0.02 Taiwan (TW) 0.01 Thailand (TH) 0.01 Indonesia (IN) 0.03 Malaysia (MA) 0.02 Korea (KO) 0.02 Japan (JP) 0.00

0.05 0.02 0.02 0.02 0.05 0.04 0.02 0.01

1.09 0.93 1.12 1.09 0.98 0.94 1.15 1.01

−8.41 −6.62 −6.24 −7.92 −7.25 −9.62 −7.34 −7.04

11.83 8.90 6.85 8.10 7.62 10.58 7.83 8.23

0.12 −0.09 −0.15 −0.16 −0.26 0.20 −0.11 −0.15

13.57 10.34 6.26 8.43 9.56 16.10 7.72 8.47

Panel C: residual returns Hong Kong (HK) Singapore (SG) Taiwan (TW) Thailand (TH) Indonesia (IN) Malaysia (MA) Korea (KO) Japan (JP)

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3.2. ARCH-based VaRs We employ two Engle-ARCH-type models to calculate VaRs. The first is the RiskMetrics model, a simple model that is used widely by practitioners. Second is the APARCH model with skewed-t innovations. Following Laurent and Peters (2002) and Giot and Laurent (2003), we characterize the models as follows. First, we fit the daily return to an ARMA(n,s) structure. ΦðLÞðrt −μ Þ = ΘðLÞεt

ð7Þ

where Φ(L) = 1 − /1L −... − /nLn is a AR polynomial of order n and Θ(L) = 1 − θ1L −... − θsLs is an MA polynomial of order s. The RiskMetrics model is equivalent to the normal integrated GARCH model with zt as i.i.d N(0,1) and 2

2

2

σ t = ð1−λÞεt−1 + λσ t−1

ð8Þ

where λ is fixed to 0.94. The VaR is the one-step-ahead forecast and is given by μt + z(1-p) σt for the short position and μt + zp σt for the long position, where zp is the pth quantile of the standard normal distribution. The APARCH model was proposed by Ding, Granger, and Engle (1993) and is a flexible model that encompasses many ARCH extensions, including the “long memory” persistence and the “leverage” effect. The specification for APARCH(1,1) model is

δ

δ

δ

σ t = ω + γ 1 ðjεt−1 j−γ n εt−1 Þ + β1 σ t−1

ð9Þ

where ω, γ1, γn, β1, and δ are the parameters to be estimated. Lambert and Laurent (2001) extended the model by allowing the innovation zt to follow the skewed-t distribution. The VaR for the long-position investment and the short-position investment,

Table 2 Estimates of shape parameter k. For Hill estimator, q is selected based on the Monte Carlo experiment and reported in square bracket. In parenthesis are the 95% confidence interval obtained from a bootstrap procedure (with 5000 bootstrap samples). For ML estimator, we arbitrarily choose m equal to 10, 21, and 63 which respectively correspond to two-week, one-month, and one quarter intervals. The standard error is reported in parenthesis. * indicates statistically significant at 95% confidence level for the null hypothesis of normality (k = 0). Indices

HK

SG

TW

TH

IN

MA

KO

JP

SP500

MSCI

Left tail

Right tail

Hill [q]

ML [m = 10]

ML [m = 21]

ML [m = 63]

0.322 (0.24,0.39) [60] 0.288 (0.23,0.37) [60] 0.114 (0.07,0.15) [35] 0.257 (0.19,0.37) [35] 0.290 (0.22,0.37) [60] 0.384 (0.29,0.48) [60] 0.232 (0.16,0.30) [35] 0.245 (0.17,0.35) [35] 0.354 (0.27,0.43) [60] 0.382 (0.29,0.45) [60]

0.226 (0.04)*

0.271 (0.06)*

0.440 (0.12)*

0.210 (0.04)*

0.248 (0.06)*

0.167 (0.09)

0.164 (0.04)*

0.082 (0.06)

−0.042 (0.09)

0.222 (0.04)*

0.224 (0.06)*

0.220 (0.10)*

0.269 (0.04)*

0.249 (0.06)*

0.175 (0.09)

0.306 (0.03)*

0.473 (0.07)*

0.457 (0.12)*

0.164 (0.04)*

0.222 (0.06)*

0.275 (0.10)*

0.075 (0.03)*

0.065 (0.05)

0.074 (0.08)

0.203 (0.03)*

0.260 (0.06)*

0.272 (0.09)*

0.204 (0.04)*

0.226 (0.05)*

0.251 (0.10)*

Hill [q]

ML [m = 10]

ML [m = 21]

ML [m = 63]

0.325 (0.24,0.41) [60] 0.192 (0.14,0.30) [35] 0.183 (0.11,0.24) [35] 0.262 (0.20,0.34) [35] 0.324 (0.24,0.39) [60] 0.419 (0.32,0.54) [60] 0.204 (0.14,0.26) [35] 0.305 (0.22,0.37) [60] 0.246 (0.17,0.35) [35] 0.320 (0.24,0.40) [60]

0.174 (0.03)*

0.292 (0.05)*

0.333 (0.10)*

0.206 (0.04)*

0.262 (0.06)*

0.162 (0.10)

0.142 (0.04)*

0.105 (0.05)*

0.062 (0.08)

0.165 (0.03)*

0.178 (0.05)*

0.219 (0.10)*

0.265 (0.04)*

0.252 (0.05)*

0.268 (0.10)*

0.190 (0.02)*

0.355 (0.06)*

0.419 (0.11)*

0.149 (0.04)*

0.225 (0.06)*

0.236 (0.12)

0.168 (0.03)*

0.172 (0.05)*

0.226 (0.10)*

0.246 (0.04)*

0.270 (0.06)*

0.306 (0.11)*

0.146 (0.03)*

0.261 (0.05)*

0.345 (0.12)*

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respectively, are given by μt + tp,ν,ξ σt and μt + t(1-p),ν,ξ σt, where tp,ν,ξ is the pth quantile of the skewed-t distribution with ν degrees of freedom and asymmetry coefficient ξ. In this study, we implement the [email protected] software to estimate the parameters (see Laurent & Peters, 2002). 4. Data and empirical results We downloaded daily country indices between October 1, 1989 and August 31, 2009 from the DataStream database. Our sample indices include Hong Kong (HK), Singapore (SG), Taiwan (TW), Thailand (TH), Indonesia (IN), Malaysia (MA), South Korea (KO), and Japan (JP). Because the level of development differs across countries, we categorize these countries into developed Asian markets (Japan), advanced markets (Hong Kong, Singapore, Taiwan, and Korean), and developing markets (Thailand, Indonesia, and Malaysia). We also include the U.S. S&P500 index and the MSCI World index for comparison purposes. 4.1. Descriptive statistics Table 1 reports the descriptive statistics of daily returns on country stock indices. Panel A reports the statistics based on raw returns. The average returns are approximately zero, with the standard deviation ranging from 0.94 to 1.86%. The standard deviation of the S&P 500 index return and the MSCI World index return are lower than that of Asian market returns. The standard

Fig. 1. Hill estimates of the shape parameter k for the left tail. The vertical line is drawn at the optimal q chosen from the Monte Carlo experiment. The horizontal line is at k = 0.5.

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Fig. 2. In-sample VaR performance for long position investment (left tail). The VaRs are calculated from Hill estimate with q equal 300, ML estimate with m equal to 10, RiskMetrics-ARMA(2,1) model, and APARCH-ARMA(2,1) model with skewed-t innovation. VaRs are estimated with (1-p) percent confidence level where p is equal to 0.05, 0.025, 0.01, and 0.005. VaR performance is measured by the proportion of times that the actual portfolio losses exceed the VaR estimate. On the y-axis is the difference between the proportion of the exceedances and the pre-specified probability p used in estimating VaR.

deviation of Japanese index returns is not lower than that of other Asian markets despite being a developed market. Similarly, the standard deviation of advanced emerging markets is not lower than that of other emerging markets. Stock returns in Taiwan and Korean have the highest standard deviation. All return distributions are highly negatively skewed and leptokurtic suggesting nonnormality of the distributions. These results may be driven by observations in crisis periods. Panel B reports the statistics based on a sample that excludes observations during the two crisis periods of 6/01/1997 to 12/31/1998 and 04/01/2008 to the end of the sample. The standard deviation and the range (maximum minus minimum) of the returns become smaller. Except for the S&P 500 index and the MSCI World index, we find that the returns become more negatively skewed for five of the Asian indices and the kurtosis does not change for four of the Asian indices. The market index returns in Asian emerging markets (Thailand, Indonesia, and Malaysia) have the highest kurtosis when the crash periods are excluded. The standard deviation and kurtosis of the Japanese market are similar to those of other Asian markets. We test whether the return characteristics are systematic or idiosyncratic to the countries in our study. In keeping with the international CAPM (ICAPM) relationship, we regress the excess country index returns on the excess MSCI World index return and the changes in exchange rates. We take the perspective of U.S. investors so that the excess returns can be approximated by the index returns less the interest rate of the U.S. 1-month Treasury bill.4 To account for non-synchronous trading, we include one lead and one lag of each independent variable.

4 The interest rate and the exchange rate data are obtained from Datastream database, except for Taiwanese exchange rates, which we downloaded from the U.S. Federal Reserve Web site.

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Fig. 3. Out-of-sample VaR performance for long position investment (left tail). The VaRs are calculated from Hill estimate with q equal 300, ML estimate with m equal to 10, RiskMetrics-ARMA(2,1) model, and APARCH-ARMA(2,1) model with skewed-t innovation. VaRs are estimated with (1-p) percent confidence level where p is equal to 0.05, 0.025, 0.01, and 0.005. VaR performance is measured by the proportion of times that the actual portfolio losses exceed the VaR estimate. On the y-axis is the difference between the proportion of the exceedances and the pre-specified probability p used in estimating VaR.

Panel C in Table 1 reports statistics based on the regression residuals. Compared to the raw returns in Panel A, we find that the standard deviation is slightly lower but that the range statistics do not change significantly. We also do not observe any significant changes in skewness and kurtosis statistics, which shows that return characteristics are more likely to be country-specific. The coefficients of the contemporaneous MSCI returns and the lagged MSCI returns are highly statistically significant for all Asian indices.5 The R-square ranges from 0.08 to 0.30, depending on the level of development of the markets. This is reasonable given that the market development level is typically positively correlated with the market capitalization and the portfolio weight in the MSCI World index. However, the low R-square statistics indicate that the majority of the variations in the Asian index returns are not systematic. In unreported results, we find that Asian index returns are more correlated to one another than to the S&P 500 index returns and the MSCI World index returns. Furthermore, the correlation coefficients are only slightly smaller when excluding the crash periods from the sample. A small correlation suggests potential diversification benefits. We examine whether the distributions of the Asian index returns will change when forming a portfolio including the U.S. index. For each Asian market index, we create an equally-weighted portfolio of the market index and the S&P 500 index. Panel D in Table 1 reports the statistics for the portfolio returns. We find that the dispersion of the returns is significantly lower; however, there is no significant change in the skewness and kurtosis statistics. In summary, we find that the returns on Asian market indices are far from normality and this characteristic is country-specific and cannot be alleviated with a portfolio strategy. Despite differences in levels of market development, there are some similarities in the degrees of dispersion and the negative skewness. The kurtosis statistic is lowered when the crash periods are excluded from

5

The results are not reported but available upon request.

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the sample, particularly for the developed markets and the advanced Asian markets. For all return series, the Ljung–Box test can easily reject the null hypothesis of no series correlation, and the Lagrange multiplier test suggests a strong ARCH effect. In a subsequent section, we use the raw return sample in our analysis because it is the extreme event that we aim to characterize. 4.2. Tests for types of distribution Table 2 reports the estimates of the shape parameter k which is the inverse of the tail index. For the Hill estimate, we use the optimal q suggested by the simulation (reported in square brackets). For hypothesis testing, we obtain the 95% confidence interval (reported in parenthesis) by bootstrapping 5000 samples with the same sample size. For the ML estimates, we arbitrarily choose the block size m equal to 10, 21, and 63, equivalent to an assumption that an extreme value can be observed once every two weeks, one month, and one quarter, respectively. Although the estimates of the shape parameter k vary across the methods, most estimates lie between 0 and 0.5. None of the estimates is greater than 0.5 so we cannot reject the null hypothesis of infinite variance. The ML estimates also can be used to test the null hypothesis of normality. In most cases, we are able to reject the null hypothesis. Particularly, when m is equal to 10, the null hypothesis of normality is rejected for all indices. When m is set to 63, the number of observations used in estimation is reduced, resulting in large standard error and, hence, the null hypothesis cannot be rejected for some markets. The estimates of the shape parameter are for comparably sized markets. We do not find the estimates for the U.S. market index and the MSCI World index that differ from those of Asian markets. The Hill estimate can be sensitive to the pre-specified parameter q. To assure the above results, we plot the Hill estimates over a range of q. For brevity, we report only results for the left tail of the distribution. The results for the right tail are similar and available upon request. Fig. 1 plots the Hill estimates of the shape parameter k along with its asymptotic 95% confidence interval over a range of q. A horizontal line is drawn at k = 0.5 to mark the null hypothesis of infinite variance (k ≤ 0.5) and a vertical line is drawn at the optimal q as if the simulation method is used. For some markets, the Hill estimates at the optimal q and its neighborhood are not stable. However, most estimates are below 0.5 and, hence, we cannot reject the null hypothesis of finite variance. Over the range of q, the Hill estimates for the U.S. market index and the MSCI World index are between 0.3 and 0.4, comparable to those for Asian markets. 4.3. VaR performance We estimate VaR using four methods. The first two methods are based on EVT: the Hill estimator with q equal to 300 and the ML estimator with m equal to 10. The values for q and m are ex-post selected to maximize the VaR performance. Although this expost selection can bias toward the EVT-based models, it cannot guarantee superior performance in out-of-sample tests. The last two methods are based on ARCH-type models: the RiskMetrics and APARCH(1,1) models. We impose an ARMA(2,1) structure to accommodate serial correlation. Fig. 2 illustrates in-sample VaR performance for the long position investment (left tail). The results for the short position investment (right tail) are similar and available upon request. By definition, with a (1-p) percent confidence level, losses exceeding VaR should occur less than p percent of the time, so we measure VaR performance by the proportion of times that actual losses exceed the VaR estimate. A high performance VaR should have the proportion of exceedances (or exceptions) close to the prespecified probability p used in estimating VaR. The y-axis in Fig. 2 represents the difference between the proportion of the exceedances and the pre-specified probability p (abbreviated as DPP). A small DPP (in absolute value) suggests good VaR performance. When DPP is positive, the proportion of the exceedances is greater than the pre-specified probability p, suggesting that the VaR model underestimates portfolio risk. When DPP is negative, the proportion of the exceedances is less than the prespecified probability p, suggesting the VaR model overestimates portfolio risk. We choose four levels of the pre-specified probability, p = {0.05, 0.025, 0.01, and 0.005}, reported in rows 1through 4. Overall in-sample performance of the VaR models is reasonably good. Except for the ML method (in column 2), the DPPs are often less than 1% (in absolute value). The Hill model and the APARCH model outperform the other two models. The ML model (in column 2) and the RiskMetrics model (in column 3) tend to underestimate the risk, as the proportion of exceedances is mostly greater than the pre-specified probability p. We do not find any significant deterioration in VaR performance when applying the models to Asian market data. Each cell in a given row and a given column contains a bar graph presenting performance of a VaR model applied to a different index return series. The last two bars in each figure present the VaR performance when the model is applied to the U.S. market index and the MSCI World index, respectively. Compared to other bars in the same figure, the VaR performance does not significantly deteriorate when the VaR models are applied to Asian market data. The largest differences in VaR performances are observed in the RiskMetrics model (see column 3 for Indonesia and Malaysia). However, the differences are less than 1%. We then evaluate the VaR performance for out-of-sample tests. For a sample of size T, we use the first c observations (where c b T) to estimate VaR and compare it to the realized return at time c + 1. If the actual loss of the portfolio at time c + 1 exceeds the VaR estimate, we mark it as one exceedance. Then we extend the estimation sample by one period, use the first c + 1 observations to estimate VaRs, and compare it to the realized return at time c + 2. We roll our estimation over in this fashion until the end of the sample period and record the number of exceedances. The proportion of the exceedances is the number of the recorded exceedances divided by (T-c). We arbitrarily set c equal to 3716, giving us 1480 observations for the out-of-sample tests.

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Fig. 3 compares the out-of-sample VaR performance for the long position (left tail). The y-axis is the difference between the proportion of the exceedances and the pre-specified probability p (again, abbreviated as DPP). VaR performance is worsened for out-of-sample tests, as DPPs are roughly twice those for the in-sample tests. In comparing across the models (across columns), VaRs estimated from the ARCH-type models tend to outperform VaRs estimated from the EVT-based models as DPPs are generally smaller. Surprisingly, despite a high degree of complication in parameterization and computation, the APARCH model does not outperform the RiskMetrics model in many cases. The differences between the DPP from the RiskMetrics model and the DPP from the APARCH model are only marginal. However, the APARCH model provides quite conservative VaR estimates, as the proportion of the exceedances is greater than the pre-specified probability p in only one case (for the Indonesian market data, in the first row and last column). In comparing across datasets (across bars in each figure), there is no clear evidence of deterioration in VaR performance when applying the models to Asian market data. For the APARCH model (in column 4), the VaR performance for the U.S. market data is comparable to that for the Asian market data. The VaR performance for the MSCI World index data is slightly better than others. For the RiskMetrics model (in column 3), although we observe some differences in VaR performance when comparing the U.S. market to Indonesian market, the differences in VaR performance still are less than 1%. Furthermore, for EVT-based models (in columns 1 and 2), in many instances the VaR performance for the U.S. market data is worse than that for the Asian market data. In general, the differences in VaR performances are usually less than 2%. In unreported results, we find the RiskMetrics-based VaR performance worsens when predicting losses for the short position investment (right tail). But the general conclusion remains the same; we do not find that the VaR performance significantly deteriorated when the VaR models are applied to Asian market data. 5. Conclusions It is well-known that asset prices in emerging markets are highly volatile. A natural question is whether the asset returns are characterized by extreme price jumps or merely large volatility. This question has an important implication because many existing risk management techniques rely on the continuity assumptions of asset returns. We apply the extreme value theory (EVT) to characterize the return distributions of Asian stock indices for the period of 1989 through 2009. Our results show that the return distributions of the Asian market indices are fat-tailed with finite variance. However, the shape parameter of the stock return distributions in emerging markets is not significantly different from those of developed markets (represented by the U.S. S&P 500 index and the MSCI World index). The Asian stock markets are more volatile than those in developed markets, but the occurrences of extreme price changes (relative to other price changes in the same market) are just as likely in Asian markets as they are in developed markets. Therefore, any risk management tools that incorporate the second moment in the consideration would work equally well in emerging markets as in developed markets. In particular, the performance of the VaR model does not deteriorate when applying the model to Asian market data compared to applying it to developed market data. Appendix A. Monte Carlo simulation for selecting the optimal q for the Hill estimator Following Jensen and De Vries (1991) and Longin (2005), we conduct a Monte Carlo simulation based on a sample size T equal to 5196. We experiment on four distributions: one Cauchy distribution and three Student's-t distributions with degrees of freedom equal to 2, 3, and 4. Respectively, the shape parameter k for the four distributions are 1, 0.5, 0.33, and 0.25 and the tail index τ are 1, 2, 3, and 4. For each distribution, we simulate a sample of T observations and calculate the Hill estimate using various values of q ranging from 5 to 1035 (about 20% of the sample) with a step size of 5. We retain the estimates for each distribution and for each q and repeat these steps 10,000 times. The optimal qi for distribution i (for i = 1, 2, 3, and 4) is the q that minimizes the mean square error (MSE). Longin (2005) gives the MSE of the S simulated estimates { k˜ s } for the true parameter ki as MSE


n

o 
– 2 2 1 S
k˜ s ; ki = k−ki + ∑ k˜ s −ki S s=1

– where k is the average of { k˜ s }. The first part measures bias and the second part measures inefficiency in the estimates. Similar to Jensen and De Vries (1991) and Longin (2005), we find a U-relationship between MSE and q. Results are available upon request. To select the optimal q for our real data, we follow Longin (2005) by applying the optimal qis from the simulation to the real data and estimate the shape parameter k. The optimal q for our real data is the qi that gives an estimate closest to the true parameter ki by computing (ki(qi) − ki)/σi, where ki(qi) is the estimate for ki using qi, and σi is the asymptotic standard error of the estimate. References Bekaert, G. (1995). Market integration and investment barriers in emerging equity markets. World Bank Economic Review, 9, 75−107. Bekaert, G., Erb, C. B., Harvey, C. R., & Viskanta, T. E. (1996). The behavior of emerging market returns. Working paper. : Duke Univ. Bekaert, G., Erb, C. B., Harvey, C. R., & Viskanta, T. E. (1998). Distributional characteristics of emerging market returns and asset allocation. Journal of Portfolio Management, 102−116. Bekaert, G., & Harvey, C. R. (1997). Emerging equity market volatility. Journal of Financial Economics, 43, 29−77.

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