Modeling Volatility of Exchange Rate of Chinese Yuan against US Dollar Based on GARCH Models

Transcription

1 013 Sixh Inernaional Conference on Business Inelligence and Financial Engineering Modeling Volailiy of Exchange Rae of Chinese Yuan agains US Dollar Based on GARCH Models Marggie Ma DBA Program Ciy Universiy of Jiangze Du Deparmen of Managemen Sciences Ciy Universiy of Kin Keung Lai Inernaional Business School Shaanxi Normal Universiy Xi an, China Deparmen of Managemen Sciences Ciy Universiy of Absrac This sudy aims o model he characerisics of volailiy of he exchange rae of he Chinese Yuan, based on he daily daa of and over he period from Augus 3, 010 o Sepember 10, 013, in he backdrop of RMB inernaionalizaion. By inroducing boh symmeric and asymmeric models of he generalized auoregressive condiional heeroscedasic (GARCH) family, we use he daily daa o esimae he parameers of each model. Finally, he paper assesses hese wo models by concluding ha hese wo models can capure mos characerisics of he exchange rae volailiy and boh are adequae o model he exchange rae volailiy series. Keywords- Volailiy modeling; GARCH; Foreign exchange rae I. INTRODUCTION Foreign exchange rae movemen and analysis of facors ha influence i have always been an imporan issue in he area of financial economics during he pas few decades, since he failure of he Breon Wood sysem of fixed exchange rae among currencies of major indusrial counries. The move from a fixed exchange rae sysem o a floaing exchange rae sysem has resuled in significan and exensive research and discussions among academics, policy makers and financial economiss abou exchange rae volailiy and is relaionship wih oher economic facors and indicaors such as inflaion, rade balance and exchange rae policies. As a resul, several models have been formulaed o sudy his kind of financial ime series wih daa from differen counries and periods. For example, Harman [1], Engel and Kennehen and Mark [3] have indicaed he connecion beween he exchange rae and economeric variables such as a naion s GDP, money supply and foreign reserves. The subjec of modeling of foreign exchange rae volailiy has araced wide ineres in academic research afer many counries shifed from fixed exchange rae regime o floaing exchange rae in Mos of hese sudies were conduced o analyze he characerisics of he financial reurns of exchange rae volailiy such as volailiy clusering and leverage effec. Engle firs modeled asse volailiy by he so-called Auoregressive Condiional Heeroscedasiciy (ARCH). Then Bollerslev [4] generalized his model o GARCH. From hen on, a variey of models abou he GARCH family have been developed. A review of he exended models can be found in Francq and Zakoian [5]. Alhough modeling of exchange rae volailiy has been exensively sudied in developed counries, sudies in developing counries have been few and far beween. From July 1, 005 onwards China adoped he floaing exchange rae sysem which means he marke deermines he exchange rae. This sep is considered as one of he mos imporan developmens in he Chinese foreign exchange marke. The remainder of his paper is organized as follows. Secion II reviews some sylized facs abou exchange rae volailiy and he background of RMB inernaionalizaion. In Secion III, hree GARCH family models are proposed and he empirical experimen is conduced in Secion IV. Finally, Secion V concludes he paper. II. EXCHANGE RATE VOLATILITY AND RMB INTERNATIONALIZATION A. Exchange rae volailiy Volailiy is known as a measure of risk or uncerainy of an asse s price, marke index, commodiy price and so on. Exchange rae volailiy refers o he amoun of flucuaion and is reaed as a measure of risk of he exchange rae. There are mainly wo measures of volailiy, i.e. hisorical volailiy and implied volailiy. Hisorical volailiy assesses he fuure change from he pas period. This kind of volailiy is mosly obained by he sandard deviaion of pas exchange rae movemens. However, implied volailiy is based on he expecaion of he paricipans for he fuure marke. More generally, implied volailiy can be calculaed from he price of he specified opion based on he Black Scholes model since he opion price reveals he rader s /14 $ IEEE DOI /BIFE

2 expecaion of he fuure movemen. In his paper, we only consider hisorical volailiy. [6] B. Some sylized fac abou exchange rae volailiy 1) Volailiy clusering The phenomenon of volailiy clusering was firs observed by Mandelbro [7]; large volailiy is more likely o be followed by large volailiy and small volailiy ends o be followed by small volailiy. In fac, volailiy clusering is accumulaion and clusering of informaion. ) Leverage effec The leverage effec means volailiy of a downward movemen of a financial ime series is higher han he upward movemen when he magniude of he movemen is he same. Pas sudies have revealed ha price movemen has a negaive correlaion wih volailiy. This fac was firs defined by Black [8] and he aribued he asymmery o he leverage effec. Oher researchers opined ha he reason of he leverage effec is he risk of holding a declining currency is larger han holding a rising currency. 3) Fa ails Fa ail also refers o excess kurosis. This means he disribuion of exchange rae reurn exhibis a fa ail when compared wih normal disribuion. One way o disinguish fa ail is ha he sandardized fourh momen for he disribuion is above 3. [8-10] C. RMB inernaionalizaion Yuan, he Chinese currency, has experienced 3 sages since China s economic reforms began in In he firs period, RMB was devalued wih an exchange rae of USD/RMB from in 1979 o in 1994 when he wo-ier foreign exchange rae sysem was adoped. Then he official exchange rae of RMB remained consan in he second period when some counries and many economiss hough he RMB was undervalued by abou 15% o 35%. From July 1, 005 onwards, RMB sared rising since China sared he marke-oriened floaing exchange rae sysem which is based on a baske of currencies. This policy allowed he flucuaion of RMB agains USD by 0.3% per day in eiher direcion. Obviously, an inernaional currency means a currency ha is acceped all over he world. RMB inernaionalizaion refers o he process of circulae he currency across borders, in oher counries, o reach a sage where he RMB becomes accepable for pricing and selemen and as a reserve currency worldwide. RMB is ypically considered as an inernaional currency when i can be reaed as an exchange medium for cross border rading and a currency o deposi. China has me he prerequisies for RMB s inernaionalizaion. China has mainained he growh of is economy and has played an imporan role in he inernaional economy. China has been perceived as a world economic gian especially because is GDP growh rae remained above 8% for more han en years from 000 o 01 and is GDP has surpassed ha of Japan, making i he second larges economy in he world wih a GDP of 5.8 rillion US dollar. Many analyss even predic ha China will become he world s larges economy in 00s. In 010, China s GDP growh was 10.3%, which accouned for one quarer of global GDP growh. On he oher hand, he RMB had a marke share of only 0.9% in he global foreign exchange business in 010. I is obvious ha he RMB s role in he global economy is currenly undersaed compared wih he large size of China s economy. Inernaionalizaion of RMB is expeced o resul in many benefis for China. Domesic corporaes can reduce exchange fees and risk. Inernaional companies will have more business in China if hey underake business ransacions denominaed in RMB. In his way, China s economy can also prosper. So inernaionalizaion of RMB is an ineviable resul of economic developmen. The Chinese governmen sared o ake a number of acions o promoe inernaional use of RMB. The main offshore RMB clearing cener was se up in and he concep of was inroduced o indicae foreign exchange rades in offshore markes which have differen exchange raes. III. METHODOLOGY Volailiy models can be classified ino wo caegories: symmeric models and asymmeric models. The fuure volailiy of symmeric models is only based on he magniude of financial reurn series. However, in an asymmeric model, he fuure volailiy depends on boh he magniude and he sign of financial reurn series. In his paper, we explore boh symmeric (GARCH) and asymmeric (EGARCH) models and compare he wo. A. The Generalized Auoregressive Condiional Heeroscedasic (GARCH) Model The GARCH model is he exension of Engle s ARCH model [11]. In his paper, we use he GARCH model o sudy he characerisics of exchange rae volailiy. The condiional variance in he GARCH model is based on is own pas lags. Generally, he GARCH (p,q) express he curren condiional variance as dependen on p pas condiional variances and q pas squared innovaions. The form of GARCH (p,q) model is as follows: q p j j i i j1 i1 (1) where p is he number of pas lags and q is he number of pas lags which deermine he curren. In his sudy, we choose he widely used GARCH (1,1) model as follows: () r (3) where 0, 1 0 and 1 0, which ensure is sricly posiive. r is he reurn of an asse a curren ime. is he average reurn. is he residual series, defined as, 96

3 where z (4) z is an iid random variable wih mean 0 and variance 1. The GARCH model specifies ha he condiional variance consiss of a consan erm, las period squared residuals 1 and las period forecas variance 1. Thus, one can predic curren variance by forming a long erm average, he las forecas variance and he informaion of volailiy from he las period. B. The Exponenial GARCH (EGARCH) model Alhough GARCH model can capure some of he volailiy characerisics like clusering and fa ail, i is poor a modeling he leverage effec because he condiional variance is jus a funcion of he pas magniude and does no consider he role of sign. To capure he asymmery of he leverage effec, we inroduce he widely used EGARCH model. This model was firs developed by Nelson [13]. p ilog i i1 log (5) q q j j j j E j j1 j j j1 j In his model, coefficien is expeced o be negaive. Then he posiive reurn brings less influence o volailiy han he negaive wih he same magniude of variaion. In his paper, we assume he innovaion follows Gaussian disribuion. Then, j E E z j (6) j To capure he asymmery of he exchange rae volailiy, his sudy specifies he EGARCH (1,1) model as follows: log log 1 1 j 1 (7) r (8) IV. EMPIRICAL EXPERIMENT A. Basic summary of he saisics In his paper, we use daily reurns of he exchange rae of and agains USD from Augus 3, 010 o Sepember 10, 013, making a oal of 787 observed daa series, all from Bloomberg. In he area of financial economics, mos researchers choose o model he log reurn of he financial ime series. In his paper, we also use he log reurn by aking he naural logarihm of he firs difference of he curren exchange rae and he previous exchange rae. The following formulaion shows he way o calculae log reurn. where E and 1 E r 100 E 1 (9) E are he exchange rae a ime and -1, respecively, r is he log reurn of he exchange rae. Table I shows he summary of he daily exchange rae of and. TABLE I. STATISTICS OF EXCHANGE RATE OF AND AGAINST USD Mean Max Min Sd.D Table II describes he properies of daily reurns of exchange rae. I is obvious ha he wo reurns do no follow normal disribuion since skewness is no equal o 0 and kurosis is greaer han 3. Also, he Jarque-Bera (J-B) es also confirms none of he wo series are normal. In order o coninue our experimen, we need o apply he Augmened Dickey-Fuller (ADF) es o sudy wheher he reurn series are saionary. Table also shows ha he ADF es rejecs he null hypohesis ha he uni roo exiss which means he wo series are all saionary. Figures 1 and presen he exchange rae series and he reurn series of and. Figure 3 plos he reurns of and. TABLE II. STATISTICS OF DAILY RETURN OF AND Mean Max Min Sd.D Skewness Kurosis J-B ADF B. Resuls of Heeroscedasiciy es Table 3 summarizes resuls of he ARCH-LM es which invesigaes he residuals o examine wheher he series exhibi condiional heeroscedasiciy. The es resuls rejec he null hypohesis and confirm he exisence of ARCH effecs. 97

4 TABLE III. SUMMARY OF ARCH-LM TEST ARCH-LM es Figure 1. PLOTS OF SERIES AND RETURNS C. Esimaion of GARCH (1,1) Model Through he Maximum Likelihood mehod, he hree parameers, he consan, ARCH erm and GARCH erm are esimaed (Table IV). The significance of he ARCH erm reveals volailiy clusering for boh variance series. As boh ARCH erm and GARCH erm are significan, he curren volailiy can be influenced by he pas condiional variance and pas squared residuals. This means ha he fuure volailiy can be forecas by pas and curren volailiy. TABLE IV. ESTIMATION RESULTS OF GARCH (1,1) MODEL Figure. PLOTS OF SERIES AND RETURNS Figure 4. CONDITIONAL VARIANCES OF & BASED ON GARCH(1,1) Also he sum of ARCH erm and GARCH erm is smaller han bu very close o 1 which indicaes ha Figure 3. PLOTS OF AND RETURNS 98

5 volailiy shocks in Chinese exchange marke are quie persisen. Since he variance is no saionary in GARCH (1,1) model, shocks of exchange rae volailiy are high and endless. The inferred volailiy of GARCH (1,1) Model is ploed in Figure 4. D. Esimaion of EGARCH (1,1) Model Table 5 presens he esimaed EGARCH (1,1) parameers, including he consan, ARCH erm, GARCH erm and he EGARCH erm. The EGARCH erm is negaive for boh cases, which means exchange rae volailiy is asymmeric. The negaive EGARCH erm suggess ha he negaive shocks influence he volailiy more han posiive shocks, which means he exisence of leverage effec of he exchange rae volailiy series. The inferred volailiy of EGARCH (1,1) Model is ploed in Figure 5. TABLE V. ESTIMATION RESULTS OF EGARCH (1,1) MODEL e V. CONCLUSION This paper aemps o sudy he behavior of Chinese exchange rae sysem and characerisics of is volailiy by applying boh he symmeric and asymmeric GARCH family models based on 787 daily observaions from Augus 3, 010 o Sepember 10, 013. Boh he GARCH and EGARCH model can capure mos of he sylized facs abou exchange rae series such as fa ail and volailiy clusering. From he empirical experimen, we can see ha he volailiy of Chinese exchange rae is highly persisen in boh cases. Furhermore, he asymmeric EGARCH (1,1) Model shows he exising leverage effec in boh Chinese exchange rae series. Finally, boh he GARCH (1,1) and EGARCH (1,1) can model he exchange rae volailiy sufficienly. REFERENCES [1] Harman, Richard. "The Effecs of Price and Cos Uncerainy on Invesmen." Journal of Economic Theory 5, no. (197): [] Engel, Charles and Kenneh D. Wes. "Exchange Raes and Fundamenals." Journal of Poliical Economy 113, no. 3 (005): [3] Mark, Nelson C. "Changing Moneary Policy Rules, Learning, and Real Exchange Rae Dynamics." Journal of Money, Credi and Banking, Blackwell Publishing 41, no. 6 (009): [4] Bollerslev, Tim. "Generalized Auoregressive Condiional Heeroskedasiciy." Journal of Economerics 31, no. 3 (1986): [5] Francq, Chrisian, Jean-Michel Zakoian and ebrary Inc. Garch Models Srucure, Saisical Inference, and Financial Applicaions. Hoboken, NJ: Wiley, 010. [6] Abdalla, Suliman Zakaria Suliman. "Modelling Exchange Rae Volailiy Using Garch Models: Empirical Evidence from Arab Counries." Inernaional Journal of Economics and Finance 4, no. 3 (01): [7] Mandelbro, Benoi. "The Variaion of Cerain Speculaive Prices." The Journal of Business 36, no. 4 (1963): [8] Black, F. "Sudies of Sock Marke Volailiy Changes." In Proceedings of he American Saisical Associaion, , [9] Fama, Eugene F. "Mandelbro and he Sable Pareian Hypohesis." The Journal of Business 36, no. 4 (1963): [10] Fama, Eugene F. "The Behavior of Sock-Marke Prices." The Journal of Business 38, no. 1 (1965): [11] Engle, Rober F. "Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion." Economerica 50, no. 4 (198): [1] Nelson, Daniel B. "Condiional Heeroskedasiciy in Asse Reurns: A New Approach." Economerica 59, no. (1991): Figure 5. CONDITIONAL VARIANCES OF & BASED ON EGARCH(1,1) 99