A Skewed Studentt ValueatRisk Approach for Long Memory Volatility Processes in Japanese Financial Markets

This paper investigates the relevance of skewed Student‐t distributions in capturing long memory volatility properties in the daily return series of Japanese financial data (Nikkei 225 Index and JPY‐USD exchange rate). For this purpose, we assess the performance of two long memory Value‐at‐Risk (VaR) models (FIGARCH and FIAPARCH VaR model) with three different distribution innovations: the normal, Student‐t, and skewed Student‐t distributions. From our results, we find that the skewed Student‐t distribution model produces more accurate VaR estimations than normal and Student‐t distribution models. Thus, accounting for skewness and excess kurtosis in the asset return distribution can provide suitable criteria for VaR model selection in the context of long memory volatility and enhance the performance of risk management in Japanese financial markets.

This paper investigates the relevance of skewed Student‐t distributions in capturing long memory volatility properties in the daily return series of Japanese financial data (Nikkei 225 Index and JPY‐USD exchange rate) For this purpose, we assess the performance of two long memory Value‐at‐Risk (VaR) models (FIGARCH and FIAPARCH VaR model) with three different distribution innovations: the normal, Student‐t, and skewed Student‐t distributions From our results, we find that the skewed Student‐t distribution model produces more accurate VaR estimations than normal and Student‐t distribution models Thus, accounting for skewness and excess kurtosis in the asset return distribution can provide suitable criteria for VaR model selection in the context of long memory volatility and enhance the performance of risk management in Japanese financial markets

A Skewed Student-t Value-at-Risk

Processes in Japanese Financial Markets

Seong‐Min YoonProfessor, Division of Economics, Pukyong National University

smyoon@pknu.ac.krSang‐Hoon KangPhD Candidate, School of Commerce, University of South Australia

sang.kang@unisa.edu.au

Keywords: Value‐at‐Risk, Japanese financial markets, volatility, asymmetry, long memory,

skewed Student‐t distribution

본 연구의 목적은 왜도와 두터운 꼬리를 반영한 비대칭(skewed) Student‐t 분포가 장 기기억 변동성과정에서 어떠한 역할을 하는지를 일본 금융시장의 시계열자료(Nikkei

225 주가지수 및 엔‐달러 환율)를 이용하여 분석하는 것이다 VaR 분석기법을 이용하

여 오차항에 대한 세 가지 다른 분포(정규분포, 대칭적 t 분포 그리고 비대칭 t 분포)를 가정하는 두 가지 대표적 장기기억 모형(FIGARCH 및 FIAPARCH)의 적합성을 비교 분 석하였다 실증분석 결과 비대칭 Student‐t 분포가 정규분포나 대칭적 t 분포에 비해서 장기기억 변동성 모형의 적합도를 높일 수 있음을 발견하였다 이는 일본 금융시장의 경우에는 왜도와 두터운 꼬리 분포를 반영한 VaR 모형이 금융자산 위험관리 측면에서

姜 商 勳

School of Commerce, University of South Australia 박사과정

sang.kang@unisa.edu.au

핵심용어: VaR, 일본금융시장, 변동성, 비대칭성, 장기기억, 비대칭 Student‐t 분포

I Introduction

Due to ubiquitous risks in financial markets, Value‐at‐Risk (VaR) has become

a crucial issue in measuring market risks and assessing the accuracy of asset portfolio volatility risks in financial economics VaR models simply calculate the maximum possible loss of an investment with a specified significance level over

a given time period.1) That is, VaR is defined as a quantile of a probability distribution, which is used to quantify market risks and set capital reserves for market risks (Duffe and Pan 1997)

The RiskMetrics model of the J P Morgan Group is one of the most popular tools for measuring the market volatility risk of asset portfolios under the assumption of normality.2) Financial asset returns undeniably suffer from excess skewness and kurtosis, implying that the assumption of normal (or Gaussian) distribution is inappropriate for explaining the skewed and fat‐tailed characteristics

of return distribution (Fang and Lai 1997; Harvey and Siddique 2000; Smith 2006; Theodossiou 1998) By identifying the appropriate shape of return distributions, investors or risk managers are able to measure investment risk exposure in financial markets (Hogg and Klugman 1983; Premaratne and Bera 2005)

To accommodate the characteristics of skewed and fat‐tailed distributions of financial returns, various empirical studies have developed a generalized autoregressive conditional heterosedaticity (GARCH)‐type framework with different distribution innovations For instance, Bollerslev (1987) proposes a Student‐t distribution to capture excess kurtosis of stock returns In addition, the skewed Student‐t distribution of Hansen (1994) allows for asymmetry and tail‐fatness distributions

of asset returns; this innovation was extended by Lambert and Laurent in 2001

1) The Bank for International Settlement (BIS) imposes confidence level 99% and the time horizon at 10 days for the purpose of measuring the adequacy of bank capital 2) The RiskMetrics model is equivalent to a normal integrated GARCH (IGARCH) model where the autoregressive parameter is set at a pre‐specified value λ (0.94), whereas the coefficient of ε is equal to λt2− 1 − (0.06) for optimal back-testing VaR forecasts (RiskMetrics Group 1996).

Subsequent studies dealing with VaR have widely adopted the skewed Student‐t distribution innovation in order to model the appropriate shape of return distributions for financial time series data (Giot and Laurent 2003; Tang and Shieh 2006; Bali and Theodossiou 2007)

Another growing issue in financial economics is that of financial asset return volatility, which often exhibits long memory properties where the autocorrelations

of the absolute and squared returns are characterized by a very slow decay (Baillie 1996) In order to circumvent this problem, the fractionally integrated GARCH (FIGARCH) model of Baillie, Bollerslev and Mikkelsen (1996) takes into consideration the fractional integration (long memory) of the conditional variance, which dates back to Granger (1980), Granger and Joyeux (1980) and Hosking (1981) Wu and Shieh (2007) compare the VaR performance of the GARCH and FIGARCH models with normal, Student‐t, and skewed Student‐t innovation distributions Their evidence suggests that the FIGARCH model with

a skewed Student‐t innovation distribution outperforms the GARCH model with different distribution innovations for US Treasury bond returns

However, although the FIGARCH model can capture persistence in conditional variance, it is unrealistic to assume that positive and negative shocks have the same effects on volatility in the FIGARCH specification (Tse 1998; Hwang 2001) It is well known that volatility tends to increase more following a large price fall (i.e bad news) than following a price rise (i.e good news) of the same magnitude (Black 1976; Nelson 1991; Engle and Ng 1993; Hentschel 1995) To incorporate both long memory and asymmetry in volatility, Tse (1998) develops a fractionally integrated asymmetric power ARCH (FIAPARCH) model Degiannakis (2004) finds that the FIAPARCH model with a skewed Student‐t distribution provides more accurate VaR predictions than other variants

of GARCH‐class models for the three European stock markets

This paper considers the relevance of skewed Student‐t distributions in estimating volatility persistence for daily returns data in Japanese financial markets (stock market and foreign exchange market) using two long memory volatility models, namely the FIGARCH and FIAPARCH To further enhance

the robustness of the estimation results, we compare the performance of various VaR models with normal, Student‐t, and skewed Student‐t distribution innovations The contribution of this paper is twofold First, we examine the process of long memory volatility with different distribution innovations in Japanese financial markets The empirical results reveal that the FIAPARCH (1 d, , 1) model

is suitable for the Nikkei 225 Index to capture asymmetric long memory properties, while the FIGARCH (1 d, , 1) model outperforms in interpreting symmetric long memory properties for the Japanese Yen–US Dollar (JPY‐USD) exchange rate The second contribution is that VaR analyses offer the relevance of asymmetries and tail‐fatness in the return distribution of Japanese financial data For instance, the models with skewed Student‐t distributions provide more accurate volatility forecasting results than normal and Student‐t distribution VaR models for both long and short positions Consequently, our VaR analyses imply that the assumption of normal distribution is inappropriate for evaluating the accuracy of VaR estimates in Japanese financial markets

The rest of this paper is organized as follows Section Ⅱ describes the theoretical properties of symmetric and asymmetric long memory VaR models Section Ⅲ provides the statistical characteristics of sample data and the empirical results The final section contains some concluding remarks

II Methodology

1 FIGARCH model

Baillie, Bollerslev and Mikkelsen (1996) extend the general GARCH model through the introduction of a fractionally integrated process, I (d), otherwise known as the FIGARCH model Unlike the knife‐edge distinction between ( 0 )

and ( 1 ) processes, the fractionally integrated process, I (d), distinguishes between short memory and long memory in the time series The FIGARCH (p ,,d q)

model is defined by:

y t=µ+εt,

εt=z tσt, z t ~ N( )0 , 1, ( )( ) t [ ( ) ] t

t

Lελ

λ and 0≤ d≤ 1 For the FIGARCH (p, d, q) process

to be well‐defined and the conditional variance to be positive for all t, all coefficients in the infinite ARCH representation must be non‐negative, i.e

1 2

1

βφβφ

(3)

For the FIGARCH model in Equation (3), the persistence of shocks to the conditional variance or the degree of long memory is measured by the fractional differencing parameter d Thus, the attractiveness of the FIGARCH model is that for 0< d< 1, it is sufficiently flexible to allow for an intermediate persistence range

2 FIAPARCH model

The FIAPARCH model extends the FIGARCH model with the APARCH

model by Ding, Granger and Engle (1993) to capture asymmetry in the conditional variance The FIAPARCH (p ,,d q) model is specified as follows:

δ [ ( ( ) ) ( ( ) )( ) ] ( )δ

γεεφ

βω

t = + 1 − 1 − L − 1 1 − L 1 −L −

, (4)

where δ> 0, − 1 <γ < 1, and 0< d< 1 When γ > 0, negative shocks give rise to

higher volatility than positive shocks, and vice versa The FIAPARCH model also nests the FIGARCH model, when δ= 2 and γ = 0 Thus, we can argue the

FIAPARCH model is superior to the FIGARCH model since the former model can capture asymmetric long memory features in the conditional variance (Tse 1998)

3 Model densities

The parameters of the volatility models can be estimated by using non‐linear optimization procedures to maximize the logarithm of the Gaussian likelihood function Under the assumption that the random variable is z t ~ N( )0 , 1, the log‐likelihood of Gaussian or normal distribution (L Norm) can be expressed as

∑ [ ( ) ( ) ]

=

+ +

−

t

t t

L

1

2 2

ln 2 ln 2

, (5)

where T is the number of observations However, it is widely recognized that residuals suffer from excess kurtosis In order to capture fat‐tails, the Student‐t distribution is incorporated into this study If the random variable is

2

1 2

ln 2

1

T

L Stud

− T

t t

z

2 2

2 1

ln 1 ln 2

1

υσυ

σ

, (6)

where 2 <υ≤ ∞ and Γ ⋅ ) is the gamma function In contrast to the normal distribution, the Student‐t distribution is estimated with an additional parameter

υ, which stands for the number of degrees of freedom measuring the degree of

fat‐tails in the density

Despite accounting for tail‐thickness, a Student‐t distribution alone cannot capture the asymmetric feature of density To incorporate excess skewness and kurtosis, we consider a skewed Student‐t distribution proposed by Lambert and Laurent (2001) If z t~SKST(0 , 1 ,k,υ), the log‐likelihood of the skewed Student‐t distribution (L SkSt) is as follows:

k k T

2

1 2 ln 2

− T

t

I t

1

2 2 2

2 1

ln 1 ln 2

1

υυ

σ

, (7)

where I t = 1 if z t ≥ −m s or I t = − 1, if z t < −m s, k is an asymmetry parameter

The constants m=m( )k,υ and s= s2( )k,υ are the mean and standard deviations

of the skewed Student‐t distribution:

=

k k k

2

2 2

1 ,

υπ

υυυ

2 2

k k k

When k= 1, the skewed Student‐t distribution equals the general Student‐t distribution, i.e z t ~ ST(0 , 1 ,υ) in Equation (6)

4 VaR models and tests

1) VaR models

Today, traders or portfolio managers are increasingly finding that their portfolios change dramatically from one day to the next, and must concern themselves with not only long trading positions, but also short trading positions

So the performance of each VaR model should be compared on the basis of both long trading positions (the left tail of distribution) and short trading positions (the right tail of distribution)

We also compare the performance of the FIGARCH and FIAPARCH models estimated on the assumption of three innovation distributions including the normal, Student‐t, and skewed Student‐t distributions, as discussed above In addition, a one‐step‐ahead VaR is computed along with the results of the estimated volatility models and the given distribution The VaR of α quantiles for long and short trading positions are computed as follows

Under the assumption of the normal distribution,

VaR long =µt−zασt, and VaR short=µt+zασt, (9)

where zα is the left or right quantile at α% of the normal distribution in Equation (5)

Under the assumption of the Student‐t distribution,

VaR long=µt−stα ,υσt, and VaR short=µt +stα ,υσt, (10)

where stα , υ is left or right quantile at α% of the Student‐t distribution in

Equation (6).

Under the assumption of the skewed Student‐t distribution,

VaR long =µt−skstα ,,υkσt, and VaR short =µt+skstα ,,υkσt (11)

where st υ,k is the left or right quantile at α% of the skewed Student‐t distribution in Equation (7) If ln( )k < 0, the VaR for long trading positions will

be larger for the same conditional variance than the VaR for short trading positions When ln( )k > 0, the opposite holds true

2) Tests of accuracy for VaR estimates

We calculate the VaR at the pre‐specified significance level of α, ranging from 5% to 0.25%, and then evaluate their performance by calculating the failure rate for both the left and right tails of the distribution in the sample return series The failure rate is widely applied in evaluating the effectiveness

of VaR models (Giot and Laurent 2003; Tang and Shieh 2006) The failure rate

is defined as the ratio of the number of times (x) in which returns exceed the forecasted VaR to the sample size (T) Following Giot and Laurent (2003), testing the accuracy of the model is equivalent to testing the hypothesis

−

−

= T−x x ∧f T−x ∧f x LR

(12)

where ∧f is the estimated failure rate Under the null hypothesis, the Kupeic

LR statistic has a chi‐square distribution with 1 degree of freedom

3) The adjusted Pearson goodness‐of‐fit test

The adjusted Pearson goodness‐of‐fit test can assess the relevance of various estimated distributions, such as the normal, Student‐t and skewed Student‐t distributions, in this study The Pearson goodness‐of‐fit test compares the empirical distribution, z t, with the theoretical innovations Palm and Vlaar (1997) classify the residuals in cells corresponding to their magnitude in implementing this test For a given number of cells, denoted g, the Pearson goodness‐of‐fit statistic is defined as

En En n g P

III Empirical analysis

1 Preliminary analysis of data

This article considers two time series data sets of Japanese financial markets: the Nikkei 225 Index and JPY‐USD exchange rate.3) The daily price series are converted into the daily percentage logarithmic return series: the returns at time 3) The Nikkei 225 Index data are obtained from DataStream International, whereas the JPY‐ USD exchange rate data are sourced from PACIFIC Exchange Rate Service

Nikkei 225 JPY‐USDSample Period 1984 1 4 ‐ 2006 12 29 1990 1 2 ‐ 2006 12 29

t are calculated by y t= ln(P t P t−1)× 100 for t= 1 , 2 , Κ ,T, where P t is the current price and P t− 1 is the previous day’s price

Table 1 Descriptive statistics for the sample

Descriptive statistics for the two sample returns are summarized in Table 1 The mean of both returns are indistinguishable from zero, while the standard deviation of the Nikkei 225 Index returns is almost twice that of JPY‐USD exchange rate returns Moreover, both return series do not correspond to the normal distribution assumption For example, both skewness and kurtosis statistics indicate that returns are not normally distributed Likewise, the Jarque‐Bera (J‐B) test statistics also reject the null hypothesis of normality in both return series

We also examine the null hypothesis of an independently and identically distributed ( i.d.) process for sample returns using the Box‐Pierces test statistics

of squared return residuals, Q s( )20 Under the null hypothesis of independence, the test statistic is distributed asymptotically as a x2(chi‐square) distribution with

Note: Descriptive graphs for the daily Nikkei 225 Index: (a) level of price, (b) returns, (c) probability density of the daily returns vs the normal (Gaussian) distribution, (d) log‐log tail distributions against the normal distribution, and (e) QQ plots against the normal distribution.

Figure 1 Descriptive graphs for the Nikkei 225 Index

20 degrees of freedom As shown in the table, due to the highly significant values of Q s( )20 , the squared residuals for both returns fail to be a white‐noise process

Note: See Figure 1.

Figure 2 Descriptive graphs for the JPY‐USD exchange rate

Descriptive graphs for the daily Nikkei 225 Index and JPY‐USD exchange rate returns are given in Figure 1 and Figure 2, respectively: (a) level of price, (b) returns, (c) probability density of the daily returns vs the normal distribution, (d) tail distributions against the normal distribution, and (e) Q‐Q plots against the normal distribution) Volatility clustering is clearly observable in the graphs

Nikkei 225 JPY‐USD

Lo’s R/S test for R t2 2.769*** 2.930***

Notes: (1) Mackinnon’s 1% critical value is ‐3.435 for ADF and PP tests (2) The KPSS critical value is 0.739 at the 1% significance level See Table 1.

Table 2 Unit root tests and Lo’s R/S analysis

The probability density graphs versus the normal distribution confirm that both return distributions exhibit fat‐tails and asymmetry In particular, the densities for both returns appear to be negatively skewed (Table 1) In addition, the tail distributions of both cases are fatter than those of the Gaussian distribution The

QQ plots also show that the fat‐tailed distributions are asymmetric against the normal distribution because negative tails are unequal to their counterparts The fat‐tailed and asymmetric properties of return distribution motivate the use of non‐normality distribution innovations in this study

Before testing for long memory properties in volatility, the two return series are subjected to three unit root tests to determine whether stationarity, integration or fractional integration should be considered for each daily data: ADF (augmented‐Dickey‐Fuller), PP (Phillips‐Peron) and KPSS (Kwiatkowski, Phillips, Schmidt and Shin) tests These tests differ from each other in terms of the null hypothesis The null hypothesis of the ADF and PP tests is that a time series contains a unit root, I( )1 process, while the KPSS test has a null hypothesis

of stationarity, I( )0 process

The empirical results of stationarity tests for both sample returns are presented

in Table 2 Large negative values for the ADF and PP tests in both returns support the rejection of the null hypothesis of a unit root at the 1% significance level Thus, both sample returns are stationary and suitable for subsequent tests

in this study Additionally, KPSS statistics indicate that the return series are insignificant and so the null hypothesis of stationarity can be rejected Thus, the implication is that both return series are stationary processes and are suitable for subsequent tests in this study.

Furthermore, the Lo’s R/S test statistics for daily returns, squared and absolute returns, are given at the bottom of Table 2.4) In this paper, squared returns and absolute returns are used as volatility proxies The modified R/S statistic supports the null hypothesis of short memory in the level of returns, while the two proxies of volatility (squared returns and absolute returns) display strong evidence of long memory Thus, it appears that the autocorrelation function for different volatility measurements (the squared returns and absolute returns) decays very slowly at the long time lags, a property defined as a process of long memory.5)

2 Long memory and asymmetry in volatility

In this subsection, we estimate two symmetric and asymmetric long memory models (FIGARCH and FIAPARCH) with normal, Student‐t, and skewed Student‐t distribution innovations for the Nikkei 225 Index and JPY‐USD exchange rate returns Table 3 and Table 4 compare the estimation results of the normal, Student‐t, and skewed Student‐t FIGARCH(1 ,d, 1) and FIAPARCH

(1 ,d, 1) models for both return series, respectively.6)

In order to verify the relevance of residuals distribution, these tables also provide a set of diagnostic tests: the lowest value of Akaike’s information criteria (AIC) indicates the best model from among the normal, Student‐t, and

4) To save space, we do not present Lo’s modified R/S analysis specifications which are done by the S‐Plus See Lo (1991) and Jacobsen (1996) for more details

5) Such volatility processes have motivated the development of long memory GARCH‐type models For more detail, see Maheu (2005).

6) The parameters of the FIGARCH and FIAPARCH models have been estimated by a quasi-maximum likelihood estimation method using G@RCH 4.0 (Laurent and Peters, 2004).