Heavy-Tailed Distribution and Risk Management of Gold Returns Hao Shen, Xuanjin Meng, Rongjie Guo, Yuyan Zhao, Siyi Ding, and Xiaojin Meng

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1 Heavy-Tailed Distributio ad Risk Maagemet of Gold Returs Hao She, Xuaji Meg, Rogjie Guo, Yuya Zhao, Siyi Dig, ad Xiaoji Meg DOI: /IJAREMS/v6-i3/3147 URL: Abstract: Gold has bee recogized as the most importat precious metals i the huma society. Other tha as a medium of exchage, gold has bee a cosumptio ad ivestmet product for a log history. It has bee recogized a well-positive role i portfolio performace by may fiacial market practitioers. Durig the recet fiacial crisis, gold spot prices have exhibited sigificat volatility. Thus, effective risk maagemet of gold spot prices play a crucial role for the idustry. I this paper, we cosider several types of heavy-tailed distributios ad compare their performace i risk maagemet of gold spot prices. Our results show the Skewed t distributio has the best goodess-of-fit i modellig the distributio of daily gold spot returs ad geerates suitable Value at Risk measures. Keywords: Skewed t Distributio, Goodess of Fit, Risk Maagemet 1. Itroductio Perhaps, gold has bee i a loger history as moey tha ay other types of currecy, icludig commodities. From 3600 BC to the preset day, from aciet past to the preset day, gold has played a major role i the world's developmet ad ecoomy. Nowadays, very few people still use gold as a medium of exchage, but gold is still oe of the most popular ivestmet products. Accordig to the World Gold Coucil (WGC), as of the ed of 014 eve after discoutig jewellery, total market capitalizatio of the gold market stads aroud $3 trillio. Gold raks higher tha all Europea sovereig debt markets, ad trails oly US Treasuries ad Japaese govermet bods. If oe sets both gold price ad Dow Joes Idustrial Average Idex to be oe i 1970, oe could see actually returs of ivestmet i the gold market is eve higher tha returs of ivestmet i the equity market as show i Figure 1 (yearly data). Ivestors also buy gold as a way of diversifyig risk, especially through the use of futures cotracts ad derivatives. The world gold market is subject to speculatio ad volatility as other fiacial markets. Although gold is cosidered risk immuized i face of political risk ad Hao She, School of Ecoomics ad Maagemet, Northwest Uiversity, Xia, Shaxi, Chia; Xuaji Meg, School of Busiess, Chogqig Uiversity, Chia; Rogjie Guo, School of Ecoomics, Xiame Uiversity, Xiame, Fujia, Chia; Yuya Zhao, Departmet of Ecoomics, Uiversity of Adelaide, Adelaide, Australia; Siyi Dig, Deparmet of Ecoomics, Uiversity of Newcastle, Newcastle, Australia; ad Xiaoji Meg, School of Busiess, Taiyua Uiversity, Taiyua, Shaxi, Chia. The authors aloe are resposible for this article. 15

2 sovereig risk, risk maagemet of the gold market is still very importat for ivestors ad market practitioers. Accordig to Hammoudeh, Malik ad McAleer (011), most of idustry participats apply some risk maagemet tools based o the cocept of Value at Risk (VaR) for risk maagemet of precious metals. I this paper, we follow the ewly developed method by Guo (017a). We compared several widely-used heavy-tailed distributios ad discuss how these statistical distributios perform i risk maagemet, especially i VaR calculatio. Figure 1: Gold price vs. DJIA Idex (1970=1) Literature Review The fiace literature has bee focusig o studies of heavy-tailed distributios for a log time. I 1994, Hase itroduced the Skewed t distributio, as a geeralized versio of the Studet s t distributio ito the fiace literature, ad the Skewed t distributio gaied its popularity quickly. Guo (017a) cosidered the Skewed t distributio, ad compared it with four other types of statistical distributios i fittig the SP 500 idex returs: ormal, Studet s t, Skewed t, ormal iverse Gaussia (NIG), ad geeralized hyperbolic (GH) distributios. Guo showed the Skewed t distributio has the best goodess of fit ad geerates suitable hypothetical rare scearios. Although the four heavy-tailed distributios have existed i the literature for a while, to the best of our kowledge Guo (017a) is the first oe who empirically compare them for regulatory risk maagemet practice. I 1977, Bardorff-Nielse developed the geeralized hyperbolic (GH) distributio ito the fiace literature, ad it soo becomes popular. Sice the differet types of subclass of the GH distributio have bee ivestigated, icludig the very popular ormal iverse Gaussia (NIG) distributio (see Figueroa-Lopez, et al., 011, for a survey). I this paper, we follow Guo (017a) ad recosider these five distributios but focus o risk maagemet of the world gold market. 16

3 There are quite extesive researches o statistical distributios ad risk maagemet of the gold commodity. Hammoudeh, et al. (011) examied volatility ad correlatio dyamics i price returs of gold, silver, platium ad palladium, ad explores the correspodig risk maagemet implicatios for market risk ad hedgig. Hammoudeh, et al. used the cocept of Value-at-Risk (VaR) ad showed the best approach for estimatig VaR based o coditioal ad ucoditioal statistical tests. Hammoudeh, et al. focused o volatility modelig istead of statistical distributios fittig, which differ from our paper. Similarly, Hammoudeh, Satos ad Al-Hassa (013) cosidered the market dowside risk associated with ivestmets i six key idividual assets icludig four precious metals, oil ad the S&P 500 idex, usig VaR, but focused o volatility modelig istead of statistical distributios fittig. Batte, Cier ad Lucey (010) modeled the mothly price volatilities of four precious metals, icludig gold, silver, platium ad palladium prices, ad ivestigated the macroecoomic determiats of these volatilities. Batte, et al. showed that gold volatility ca be explaied by moetary variables, but ot true for silver. Agai, Batte, et al. focused o volatility modelig istead of statistical distributios fittig. Tully ad Lucey (007) ivestigated macroecoomic iflueces o gold usig the asymmetric power GARCH model. Reboredo (013) studied whether gold could be used as hedgig tool for the US dollar for risk maagemet purpose. Almost all of the above cited papers focus o volatility modelig, while our mai iterest is statistical fittig ad risk measures calculatio, which differ from the existig literature. The paper is structured as follows. I Sectio, we itroduce the heavy-tailed distributios. Sectio 3 summarizes the data. The estimatio results are i Sectio 4. Fially, we coclude i Sectio 5.. Heavy-tailed Distributios We cosider four types of widely-used heavy-tailed distributio i additio to the ormal distributio: (i) the Studet s t distributio; (ii) the Skewed t distributio; (iii) the ormal iverse Gaussia distributio (NIG); ad (iv) the geeralized hyperbolic distributio (GH). All the distributios have bee stadardized to esure that their mea ad stadard deviatio equal to zero ad oe respectively. Their probability desity fuctios are give as follows. (i) Studet s t distributio: 1 1 ( ) ( ) e t f et 1, (1) 1/ ( )[( ) ] ( ) idicates degrees of freedom ad is daily equity market idex retur. where (ii) Skewed t distributio: e t 17

4 e t ( 1)/ 1 bet a bc 1 et a / b 1 f( et, ), () ( 1)/ 1 bet a bc 1 et a / b 1 where is the stadardized log retur, ad the costats a, ad are give 1 by a 4c ( ) 1, b 1 3 a, ad c. The desity fuctio has a mode ( ) ( ) of, a mea of zero, ad a uit variace. The desity fuctio is skewed to the right whe, ad vice-versa whe. The Skewed t distributio specializes to the stadard Studet s t distributio by settig the parameter. a/ b b c (iii) Normal iverse Gaussia distributio (NIG): K1( ( et ) ) f ( et,,, ) exp( ( et )), (3) ( e ) t where K () is the modified Bessel fuctio of the third kid ad idex 0 ad 0. The NIG distributio is specified as i Prause (1997). The NIG distributio is ormalized by settig 3 ad, which implies Ee ( ) 0 t ad Var( et ) 1. (iv) Geeralized hyperbolic distributio: p,, g et m p, b, g,, ;,, f et p b g q p b g, (4) d p b g,, where R K p p g K g g 1 1 p b g d p b g K p g 1, d p b g R1 b R R1,, 0 m p, b, g b d p, b, g R., ad 1 p, b ad g are parameters. The geeralized hyperbolic distributio is a stadardized versio of Prause (1997). 3. Data We collected the daily gold spot prices from Yahoo Fiace for the period from Jue 7, 1991 to Jue 30, 017, coverig all the available data i Yahoo Fiace. There are i total 7503 observatios. Figure idicates gold price icreased quite sigificatly sice 1991 except the Great Recessio period. 18

5 Figure : Gold spot prices dyamics We fit the heavy tailed distributios with the ormalized gold spot returs. The world gold market is the most importat precious metal markets. I this paper, Yahoo Fiace origially collected the gold prices data from the Chicago Mercatile Exchage (CME). Curretly, CME is the largest commodity exchage i the world. It merged with the Chicago Board of Trade i July 007 to become the largest commodity derivative exchage. Figure 3 illustrates the dyamics of the gold spot returs. There are sigificat volatility clusterig pheomeo ad high volatilities are observed i the Great Recessio period. Figure 3: Gold spot returs 19

6 Table 1 exhibits basic statistics of the gold spot returs. The results show the gold daily returs are leptokurtotic ad margially egatively skewed. The extreme dowside move is slightly less tha the extreme upside move, which is at odds with most of major fiacial assets over the world. Table 1: Descriptive statistics mi max mea std skewess kurtosis -7.38% 10.99% 0.0% 0.9% Figure 4 is the histogram of the raw data. We fit the returs by the Gaussia distributio ad the figure clearly exhibits heavy tails. Figure 4: Histogram of Gold returs Empirical Results 4.1 Parameters Estimatio The raw retur series is ormalized to allow zero mea ad uit stadard deviatio. We use the maximum likelihood estimatio (MLE) method to fit the series ad the estimatio results of the key parameters are give i Table. All the parameters are sigificatly differet from zero at 10% sigificace level except the parameter beta i the Skewed t ad NIG distributios ad b i the geeralized hyperbolic distributio. 0

7 Table : Estimated values of key parameters Normal Studet's t Skewed t NIG Geeralized Hyperbolic Symmetric Y Y N N N Fat-tailed N Y Y Y Y Estimated Parameters Nu=.96 Nu=3.01; beta= alpha=1.13; beta= p=-1.1; b=-.013; g= Goodess of Fit As discussed i Huber-Carol, et al. (00) ad Taeger ad Kuht (014), we compare the four heavy-tailed distributios ad the bechmark ormal distributio i fittig the gold daily spot returs through four differet criteria: (i) Kolmogorov-Smirov statistic; (ii) Cramer-vo Mises criterio; (iii) Aderso-Darlig test; ad (iv) Akaike iformatio criterio (AIC) i Akaike (1973). (i) Kolmogorov-Smirov statistic is defied as the maximum deviatio betwee empirical CDF (cumulative distributio fuctio) F ( x ) ad tested CDF Fx ( ): D sup F ( x) F( x), (5) x 1 where F ( x) I[, x] ( Xi). i1 (b) Cramer-vo Mises criterio is defied as the average squared deviatio betwee empirical CDF ad tested CDF: 1 i 1 T [ F ( x) F( x)] df( x) F ( xi ) 1 i1, (6) (c) Aderso-Darlig test is defied as the weighted-average squared deviatio betwee empirical CDF ad tested CDF: ( F ( x) F( x)) A df( x), F( x)(1 F( x)) ad the formula for the test statistic A to assess if data comes from a tested distributio is give by: i 1 A l( F ( xi )) l(1 F ( xi )). (7) i1 (d) Akaike iformatio criterio (AIC) is defied as: AIC k l( L), (8) where L is the maximum value of the likelihood fuctio for the model, ad k is the umber of estimated parameters i the model. 1

8 The compariso results are showed i Table 3, idicatig the Skewed t distributio has the best goodess of fit compared with other selected types of distributio, followed by the geeralized hyperbolic distributio, ad the Studet s t distributio. Table 3: Compariso of selected types of distributio Normal Studet's t Skewed t NIG Geeralized Hyperbolic K-S Test Cv-M Test A-D Test AIC Hypothetical Rare Scearios To properly maage the market risk of gold commodity, we are iterested i how the market performs uder extreme coditios. Similarly as i Hammoudeh, et al., we adopt the cocept of Value at Risk (VaR), which has bee widely used i the idustry. I quatitative risk maagemet, VaR is defied as: for a give positio, time horizo, ad probability p, the p VaR is defied as a threshold loss value, such that the probability that the loss o the positio over the give time horizo exceeds this value is p.with the estimated parameters i Sectio 4.1, we calculate VaRs for differet cofidece levels: VaR ( et) if{ e : P( et e) 1 }, (9) where (0,1) is the cofidece level. We select the followig levels for dowside moves: {99.99%, 99.95%, 99.9%, 99.5%, 99.0%}, ad for upside moves: {0.01%, 0.05%, 0.1%, 0.5%, 1.0%}. From Equatio (9), the hypothetical rare scearios based o the VaR levels are give as i Table 4. Table 4 idicates that the Skewed t distributio has the closest VaRs to the oparametric historical VaRs compared with other types of distributios. Table 4: Scearios for Gold retur

9 Left Tail Cofidece 99.99% 99.95% 99.90% 99.50% 99.00% Empirical -7.3% -6.76% -5.55% -4.48% -3.84% Normal -4.61% -4.19% -3.98% -3.63% -3.41% T -7.05% -6.40% -5.9% -5.00% -4.35% Skewed T -7.5% -6.80% -5.37% -4.55% -3.95% NIG -6.35% -5.94% -5.8% -4.91% -4.31% GH -7.07% -6.50% -5.69% -4.74% -4.07% Right Tail Cofidece 0.01% 0.05% 0.10% 0.50% 1.00% Empirical 9.53% 8.45% 8.06% 7.5% 6.77% Normal 4.61% 4.19% 3.98% 3.63% 3.41% T 7.05% 6.40% 5.9% 5.00% 4.35% Skewed T 9.73% 8.6% 8.1% 7.3% 6.84% NIG 10.04% 9.41% 8.60% 8.04% 7.51% GH 10.46% 9.5% 8.75% 7.99% 7.1% 5. Coclusios The world gold market has grow very rapidly i the past several decades. I 1971, the gold mie productio was just 1,518 toes, but i 016 it had reached 3,169 toes. With so may idividuals, fiacial istitutios ad govermet regulators participate i the world gold market, risk maagemet plays a crucial role i facilitate the gold market developmet. I this paper, we ivestigate several widely-used heavy-tail distributios ad their performace i fittig daily gold spot returs. Our results show the Skewed t distributio has the best empirical performace ad provides suitable risk measures of VaR. As observed i Figure 3, the daily gold spot returs exhibit quite strikig volatility clusterig effects, ad thus if oe could combie the fat-tailed distributios with the geeralized autoregressive coditioal heteroskedasticity (GARCH) framework as i Guo (017b, 017c), it might be aother iterestig cotributio to the literature. Refereces 1. Akaike, H. (1973). Iformatio theory ad a extesio of the maximum likelihood priciple. i Petrov, B.N.; Csáki, F., d Iteratioal Symposium o Iformatio Theory, Tsahkadsor, Armeia, USSR, pp Bardorff-Nielse, O. (1977). Expoetially decreasig distributios for the logarithm of particle size. Proceedigs of the Royal Society, vol. 353, pp Batte, J., C. Cier ad B. Lucey (010). The macroecoomic determiats of volatility i precious metals markets. Resources Policy, vol. 35, o., pp Figueroa-Lopez, J., S. Lacette, K. Lee ad Y. Mi (011). Estimatio of NIG ad VG models for high frequecy fiacial data. i Hadbook of Modelig High-Frequecy Data i Fiace, edited by F. Vies, M.C. Mariai ad I. Florescu, Joh Wiley & Sos, Ic., USA. 3

10 5. Guo, Z. (017a). Heavy-tailed distributio ad risk maagemet of equity market tail evets. Joural of Risk & Cotrol, vol. 4, pp Guo, Z. (017b). Empirical performace of GARCH models with heavy-tailed iovatios. Workig paper. 7. Guo, Z. (017c). GARCH models with fat-tailed distributios ad the Hog Kog stock market returs. Iteratioal Joural of Busiess ad Maagemet, forthcomig. 8. Hase, B. (1994). Autoregressive coditioal desity estimatio. Iteratioal Ecoomic Review, vol. 35, pp Huber-Carol, C., N. Balakrisha, M. Nikuli ad M. Mesbah (00). Goodess-of-Fit Tests ad Model Validity, Spriger. 10. Hammoudeh, S., F. Malik ad M. McAleer (011). Risk maagemet of precious metals. Quarterly Risk of Ecoomics ad Fiace, vol. 51, o. 4, pp Hammoudeh, S., P. Satos ad A. Al-Hassa (013). Dowside risk maagemet ad VaRbased optimal portfolios for precious metals, oil ad stocks. North America Joural of Ecoomics ad Fiace, vol. 5, o. 1, pp Prause, K. (1999). The geeralized hyperbolic model: estimatio, fiacial derivatives, ad risk measures. Ph.D. Dissertatio. 13. Reboredo, J. (013). Is gold a safe have or a hedge for the US dollar? Implicatios for risk maagemet. Joural of Bakig & Fiace, vol. 37, o. 8, pp Taeger, D. ad S. Kuht (014). Goodess-of-fit tests. Statistical Hypothesis Testig with SAS ad R, Wiley Olie Library. 15. Tully, E. ad B. Lucey (007). A power GARCH examiatio of the gold market. Research i Iteratioal Busiess ad Fiace, vol. 1, o., pp

Subject CT1 Financial Mathematics Core Technical Syllabus

Subject CT1 Financial Mathematics Core Technical Syllabus Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig