Conditional Heavy Tails, Volatility Clustering and Asset Prices of the Precious Metal

Transcription

1 Condiional Heavy Tails, Volailiy Clusering and Asse Prices of he Precious Meal Wei Ma, Keqi Ding, Yumin Dong, and Li Wang DOI: /IJARBSS/v7-i7/3131 URL: hp://dx.doi.org/ /ijarbss/v7-i7/3131 Absrac: The world palladium demand has increased seadily and dramaically in he years of 1h cenury. However, is spo price sill has no reached he peak level observed in January 17, 001. In 008, a single-day increase of he palladium spo price has exceeded 37%, which winesses significan risk for invesmens in he world palladium marke. In his paper, we apply he GARCH model wih heavy-ailed disribuions ino he palladium spo reurns series for risk managemen purpose. We compare empirical performance of he Suden s disribuion and he normal reciprocal inverse Gaussian (NRIG) disribuion. Our resuls show he newlydeveloped disribuion, he NRIG, canno ouperform he older fashion one, he Suden s disribuion. Neverheless, our resuls do demonsrae ha i is imporan o incorporae condiional heavy ails for precious meals spo reurn modelling. Key words: Suden s disribuion, GARCH model, palladium 1. Inroducion Palladium is a rare elemen widely-used in auomoive, chemical, elecrical, jewelry and denal indusries. Of he four precious meals, palladium is he leas known. As one of he invesmen vehicles, he world palladium price has drawn a lo of aenions over he world. On he supply side, he wo counries, Russia and Souh Africa, accoun for more han 75% of annual global mine supplies. On he demand side, recen increases mainly come from rapidly growing gasoline-powered vehicle producion in he U.S., China, India and elsewhere, which requires palladium in he caalyic converers o conrol exhaus emissions. Also, a recen inroducion of a Unied Saes palladium coin is likely o draw an endorsemen of palladium as a bullion alernaive and increases he demand. World demand for palladium increased from 100 ons in 1990 o nearly 350 ons in 010. The global producion of palladium from mines was slighly more han 50 ons in 010 according o he Unied Saes Geological Survey. Recenly, alhough he palladium spo price is growing in rend, he price becomes very volaile. Many marke paricipans are hesian o invesmen in he palladium, alhough many analyss claim here are increasing demands. Thus, people are demanding effecive quaniaive risk Wei Ma, Deparmen of Economics, Universiy of Leiceser, Leiceser, Unied Kingdom; Keqi Ding, School of Business, Shanghai Universiy, Shanghai, China; Yumin Dong, School of Business, Yunnan Universiy, Kunming, China; and Li Wang, School of Economics, Shenzhen Universiy, Guangdong, China. 686

2 managemen ools. In his paper, we follow his demand rend and ry o invesigae an exising widely-used model, he generalized auoregressive condiional heeroscedasiciy (GARCH) model, in risk managemen of palladium spo reurns. We follow he work in Guo (017a) and compare wo ypes of heavy-ailed disribuion, he Suden s disribuion and he normal reciprocal inverse Gaussian (NRIG) disribuion, under he GARCH framework. We are ineresed in if he wo disribuions have differences in empirical performance of quanifying palladium spo volailiies. Lieraure Review The GARCH models have been adoped o invesigae he palladium marke in he recen decade. Adrangi and Charah (00) provided evidence of nonlinear dependencies in palladium and plainum fuures markes and found ha ARCH-ype processes, wih conrols for seasonaliy and conrac-mauriy effecs, generally explain he nonlineariies in he daa. Diaz (015) invesigaed he spos prices of he wo scarce precious meals, plainum and palladium. Diaz found inermediae memory in he reurn srucures of boh precious meals, which implies he insabiliy of plainum and palladium reurns persisency in he long run. Moreover, Diaz showed boh he ARFIMA-FIGARCH and he ARFIMA-FIAPARCH models confirm longmemory properies in he volailiy of he wo spo prices and he leverage effecs phenomenon is no also presen based on he ARFIMA-APARCH and ARFIMA-FIAPARCH models. Auer (015) used dummy-augmened GARCH models o invesigae he impac of he specific calendar day on he condiional means of palladium reurns. Auer illusraed ha during he period from July 1996 o Augus 013 here is no significan impac of he specific calendar day observed. Lucey and Li (014) analyzed wha and when precious meals could ac as safe havens using he US daa. Lucey and Li showed ha for he period examined silver, plainum and palladium could ac as a safe haven bu gold could no. Lucey and Li provided evidence ha a imes palladium could serve as he sronges and safes haven among he four precious meals. All he above lieraure did no consider condiional heavy ails and hus canno be direcly used for risk managemen purpose. Here, we wan o develop a quaniaive risk managemen ool based on he GARCH framework. We consider wo differen heavy-ailed disribuions, he Suden s disribuion as in Bollerslev (1987) and he NRIG disribuion as in Guo (017b, 017c). Mos of he exising sudies on he GARCH models wih heavy-ailed disribuions have been focusing on he US daa. For insance, Tavares, e al. (007) invesigaed he heavy ails and asymmeric effec on socks reurns volailiy in he GARCH framework, and found he Suden s and he sable Pareian disribuion clearly ouperform he Gaussian disribuion in fiing S&P 500 reurns. Su and Hung (011) sudied a range of socks reurns in he NYSE marke during he period of he U.S. Subprime morgage crisis, and show ha he GARCH model wih normal, generalized error disribuion (GED) and skewed normal disribuions provide accurae VaR esimaes. In his paper, we follow he model framework in Guo (017a) and compare he empirical performance of he Suden s disribuion and he NRIG disribuion. The remaining secions 687

3 of he paper are organized as follows. In Secion, we discuss he models. Secion 3 summarizes he daa. The esimaion resuls are in Secion 4. Secion 5 concludes.. The Models Here, we lis a simple GARCH(1,1) process as: e where he hree posiive numbers , 1 and. The assumpion of a consan mean reurn (.1) (.) 1 are he parameers of he process and is purely for simplificaion and reflecs ha he focus of he paper is on dynamics of reurn volailiy insead of dynamics of reurns. The variable e is idenically and independenly disribued (i.i.d.). Two ypes of heavyailed disribuions are considered: he Suden s and he normal reciprocal inverse Gaussian (NRIG) disribuions. The densiy funcion of he sandard Suden s disribuion wih degrees of freedom is given by: 1 1 ( ) e ( 1) 1 1/ ( ) f e where 1 1., 4. (.3) ( )[( ) ] denoes he -field generaed by all he available informaion up hrough ime The NRIG is a special class of he widely-used generalized hyperbolic disribuion. The generalized hyperbolic disribuion is specified as in Prause (1999): ( / ) K 1/( ( e ) ) f ( e,,, ) exp( ( e )), (.4) 1/ ( ( e ) / ) K ( ) where K () is he modified Bessel funcion of he hird kind and index and: 0, 1 0. When, we have he normalized NRIG disribuion as: f K0( ( 1) ) ( 1) exp( 1). (.5) 688

4 3. Daa and Summary Saisics Figure 1: Daily palladium spo prices We colleced he daa from he London Plainum and Palladium Marke (LPPM). The LPPM is he mos imporan over-he-couner rading marke for plainum and palladium and one of he world's major commodiy rading associaions. The rade in LPPM was esablished in he early 0h cenury, ypically by exising dealers of gold and silver. The daa includes he period from November 17, 1994 o June 30, 017 and in oal 6459 observaions. Figure 1 illusraes he daily palladium spo prices in he LPPM. We can he palladium spo prices have never researched he peak level of $110.5 per ounce in January 7, 001 in he las decade. Figure illusraes he dynamics of he palladium spo reurns. There are significan volailiy clusering phenomenon and wo huge posiive and negaive spikes are observed in he recen financial crisis. Figure : Daily palladium spo reurns The summary saisics of he daa is presened in Table 1. The daa presen he sandard se of well-known sylized facs of asse prices series: non-normaliy, limied evidence of shor-erm 689

5 predicabiliy and srong evidence of predicabiliy in volailiy. All series are presened in daily percenage growh raes/reurns. The Bera Jarque es conclusively rejecs normaliy of raw reurns in all series, which confirms our assumpion ha he model seleced should accoun for he heavy-ail phenomenon. The smalles es saisic is much higher han he 5% criical value of The marke index is negaively skewed and has fa ails. The asympoic SE of he skewness saisic under he null of normaliy is 4 /T, where T heavy ails. 6/T, and he SE of he kurosis saisic is is he number of observaions. The daa exhibis saisically significan Series Obs. Mea n Palladium spo reurns 9 % Sd..01 % Skewnes Kurosi s s 0.66* 8.84* * BJ Q(5) Q ARCH (5 ) Q (5) 91.7** * 41.8* * * Table 1: Summary saisics. BJ is he Bera-Jarque saisic and is disribued as chi-squared wih degrees of freedom, Q(5) is he Ljung-Box Pormaneau saisic, Q ARCH (5) is he Ljung-Box Pormaneau saisic adjused for ARCH effecs following Diebold (1986) and Q (5) is he Ljung- Box es for serial correlaion in he squared residuals. The hree Q saisics are calculaed wih 5 lags and are disribued as chi-squared wih 5 degrees of freedom. * and ** denoe a skewness, kurosis, BJ or Q saisically significan a he 5% and 1% level respecively. We use he Ljung-Box pormaneau, or Q, saisic wih five lags o es for serial correlaion in he daa, and adjus he Q saisic for ARCH models following Diebold (1986). The resuls ha no serial correlaion is found confirm our assumpion of a consan mean reurn in Equaion (.1). The evidence of linear dependence in he squared demeaned reurns, which is an indicaion of ARCH effecs, is significan. 4. Esimaion Resuls We esimae he GARCH(1,1) model wih he heavy-ailed disribuions by maximizing he loglikelihood funcion of equaion: T ˆ argmax log( f (,, )). (4.1) Table repors esimaion resuls of he GARCH(1,1) model wih he wo ypes of heavy-ailed disribuion for all he daily palladium spo reurn series. We also include he normal disribuion as he benchmark saisical disribuion. All he parameers are significanly differen from zero. There resuls show ha i is crucial o inroduce heavy-ailed disribuions ino he GARCH framework and he Suden s disribuion has he bes in-sample performance. Since he wo disribuions has he same number of parameers, he Akaike informaion crierion (AIC) and he Bayesian informaion crierion (BIC) also indicae he Suden s disribuion has bes empirical performance. 690

6 alpha1 bea1 1/nu (1/alpha) log-likelihood AIC BIC Normal 0.05** 0.897** Suden's 0.041** 0.901** 0.159** NRIG 0.047** 0.905** 0.69** Table : Esimaion of he GARCH model wih heavy-ailed innovaions * and ** denoe saisical significance a he 5% and 1% level respecively. 5. Conclusion In he recen decades, he world demand of palladium has increased dramaically. However, is spo price sill has no reached he peak level observed in 001. In 008, a single-day increase of he palladium spo price has exceeded 37%, which indicaes significan risk for invesmens in he world palladium marke. In his paper, we apply he GARCH model wih heavy-ailed disribuions ino he palladium spo reurns series for risk managemen purpose. Our resuls show he newly-developed disribuion, he NRIG, canno ouperform he older fashion one, he Suden s disribuion. Neverheless, our resuls do demonsrae ha i is imporan o incorporae condiional heavy ails for precious meals spo reurn modelling. References 1. Adrangi, B. and A. Charah (00), The dynamics of palladium and plainum prices. Compuaional Economics, vol. 19, no., pp Auer, B. (015), Supersiious seasonaliy in precious meals markes? Evidence from GARCH models wih ime-varying skewness and kurosis. Applied Economics, vol. 47, pp Bollerslev, T. (1987), "A condiional heeroskedasic ime series model for securiy prices and raes of reurn daa." Review of Economics and Saisics, vol. 69, pp Diaz, J. (016), Do scarce precious meals equae o safe harbor invesmens? he case of plainum and palladium. Economic Research Inernaional, vol. 016, pp Diebold, F. (1986), "Tesing for serial correlaion in he presence of ARCH." Proceedings of he Business and Economic Saisics Secion of he American Saisical Associaion, vol. 3, pp Guo, Z. (017a), A Sochasic Facor Model for Risk Managemen of Commodiy Derivaives, Proceedings of he 7h Economic and Finance Conference, pp. 6-4; 7. Guo, Z. (017b), Models wih Shor-Term Variaions and Long-Term Dynamics in Risk Managemen of Commodiy Derivaives, mimeo. 8. Guo, Z. (017c), How Informaion Is Transmied across he Naions? An Empirical Invesigaion of he US and Chinese Commodiy Markes, Global Journal of Managemen and Business Research, vol. 17, no. 1, pp Lucey, B. and S. Li (015), Wha precious meals ac as safe havens, and when? Some US evidence. Applied Economics Leers, vol., no. 1, pp Prause, K. (1999) "The generalized hyperbolic model: esimaion, financial derivaives, and risk measures." Ph.D. Disseraion. 691

7 11. Su, J. and J. Hung (011), Empirical analysis of jump dynamics, heavy-ails and skewness on value-a-risk esimaion. Economic Modelling, vol. 8, no. 3, pp Tavares, A., J. Curo and G. Tavares (008), Modelling heavy ails and asymmery using ARCH-ype models wih sable Pareian disribuions. Nonlinear Dynamics, vol. 51, no. 1, pp