DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND

Transcription

1 DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND Risk Managemen of Precious Meals Shawka Hammoudeh, Farooq Malik, and Michael McAleer WORKING PAPER No. 37/2010 Deparmen of Economics and Finance College of Business and Economics Universiy of Canerbury Privae Bag 4800, Chrischurch New Zealand

2 Risk Managemen of Precious Meals* Shawka Hammoudeh LeBow College of Business Drexel Universiy Farooq Malik College of Business Universiy of Souhern Mississippi Michael McAleer Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam and Tinbergen Insiue The Neherlands and Deparmen of Economics and Finance Universiy of Canerbury New Zealand May 2010 * The hird auhor wishes o acknowledge he Ausralian Research Council, Naional Science Council, Taiwan, Cener for Inernaional Research on he Japanese Economy (CIRJE), Faculy of Economics, Universiy of Tokyo, and a Visiing Erskine Fellowship, College of Business and Economics, Universiy of Canerbury.

3 Absrac This paper examines volailiy and correlaion dynamics in price reurns of gold, silver, plainum and palladium, and explores he corresponding risk managemen implicaions for marke risk and hedging. Value-a-Risk (VaR) is used o analyze he downside marke risk associaed wih invesmens in precious meals, and o design opimal risk managemen sraegies. We compue he VaR for major precious meals using he calibraed RiskMerics, differen GARCH models, and he semi-parameric Filered Hisorical Simulaion approach. Differen risk managemen sraegies are suggesed, and he bes approach for esimaing VaR based on condiional and uncondiional saisical ess is documened. The economic imporance of he resuls is highlighed by assessing he daily capial charges from he esimaed VaRs. The risk-minimizing porfolio weighs and dynamic hedge raios beween differen meal groups are also analyzed. Keywords: Precious meals, condiional volailiy, risk managemen, value-a-risk. JEL Classificaions: G1 2

4 1. Inroducion Financial and commodiy markes have been highly volaile in recen years. Volailiy brings risk and opporuniy o raders and invesors, and should herefore be examined. There are many reasons, oher han changes in supply and economic use, for volailiy o occur in commodiy markes. Inroducion of new financial innovaions, such as fuures, opions and ETFs (exchange-raded funds), can affec precious meals volailiy. Selling and buying of gold by he Inernaional Moneary Fund (IMF) and cenral banks can also change volailiy. Changes in demand for he produc of an indusry ha uses commodiies as an inpu may lead o flucuaions in prices of commodiies. Marke paricipans form differen expecaions of profiable opporuniies, perform cross-marke hedging across differen asse classes, process informaion a differen speeds, and build and draw invenories a differen levels. These facors conribue o volailiy of commodiies over ime and across markes. In addiion o policy makers and porfolio managers, manufacurers are also ineresed in his informaion because precious meals have imporan and diversified indusrial use in jewelry, medicine, elecronic and auo caalyic indusries. Quanificaion of he predicable variaions in precious meals price changes is fundamenal in designing sensible risk managemen sraegies. Value-a-risk (VaR) has become an imporan insrumen wihin financial markes for quanifying and assessing he porfolio marke risk associaed wih financial asse and commodiy price movemens. There is a cos of inaccurae esimaion of he VaR in financial markes which affecs efficiency and accuracy of risk assessmens. Surprisingly, despie he imporance of precious meals and 3

5 heir volaile naure, here is no sudy on he analysis of VaR for precious meals. One of he primary purposes of he paper is o fill his void in he risk managemen lieraure. Specifically, we compue VaR for gold, silver, plainum and palladium using RiskMerics, he GARCH model (using normal and -disribuion), and he recen Filered Hisorical Simulaion (FHS) approach. The ou-of-sample forecas performance indicaes ha he GARCH wih - disribuion produces a VaR wih he mos accurae and robus esimaes of he acual VaR hresholds for all four precious meals. This quanificaion is fundamenal in designing sensible risk managemen sraegies. The uncondiional coverage es of Kupiec (1995) and he condiional coverage es of Chrisoffersen (1998) are used o assess he performance of he various models in regards o VaR, and differen risk managemen sraegies based on he empirical resuls are discussed. The economic imporance of he esimaion resuls is highlighed by calculaing he capial requiremens using differen VaR models o assess marke risk exposure for all precious meals. Finally, he economic significance of he esimaes is underscored by esimaing he riskminimizing porfolio weighs and dynamic hedge raios beween differen meal groups, which may be used o formulae opimal risk managemen sraegies for he four precious meals. 2. Review of Lieraure The commodiies lieraure is expanding and gaining imporance as a resul of he increasingly significan role ha commodiies play in inernaional financial markes and economies. More ETFs are being creaed for specific commodiies. The recen mos promising ETFs have been creaed for plainum and palladium. Alhough he 4

6 commodiies lieraure is focusing more and more on imporan issues, he pace and coverage remains narrow, paricularly in relaion o precious meals, including plainum and palladium, and commodiy risk managemen. In his secion, we presen a review of exising sudies and highligh he economic significance of he relaively sparse lieraure relaed o precious meals. Jensen a al. (2002) find ha commodiy fuures subsanially enhance porfolio performance for invesors, and show ha he benefis of adding commodiy fuures accrue almos exclusively when he Federal Reserve is following a resricive moneary policy. Overall, heir findings indicae ha invesors should gauge moneary condiions o deermine he opimal allocaion of commodiy fuures wihin a porfolio. Draper e al. (2006) examine he invesmen role of precious meals in financial markes using daily daa for gold, plainum, and silver. They show ha all hree precious meals have low correlaions wih sock index reurns, which suggess ha hese meals may provide diversificaion wihin broad invesmen porfolios. They also show ha all hree precious meals have hedging capabiliy for playing he role of safe havens, paricularly during periods of abnormal sock marke volailiy. Hammoudeh and Yuan (2008) apply univariae GARCH models o invesigae he volailiy properies of wo precious meals, gold and silver, and one base meal, copper. They found in he sandard univariae GARCH model ha gold and silver had almos he same volailiy persisence, while he persisence was higher for he pro-cyclical copper. Conover e al. (2009) presen new evidence on he benefis of adding precious meals (gold, silver and plainum) o U.S. equiy porfolios. They find ha adding a 25% meals allocaion o he equiies of precious meals firms improves porfolio performance 5

7 subsanially, and ha gold relaive o plainum and silver has a beer sand-alone performance and appears o provide a beer hedge agains he negaive effecs of inflaionary pressures. They also show ha while he benefis of adding precious meals o an invesmen porfolio varied somewha over ime, hey prevailed hroughou much of he 34-year period. Chng (2009) examines cross-marke rading dynamics in fuures conracs wrien on seemingly unrelaed commodiies ha are consumed by a common indusry. He finds such evidence in naural rubber, palladium and gasoline fuures markes. The paper offers new insighs ino how commodiy and equiy markes relae a an indusry level and documens implicaions for muli-commodiy hedging. Khalifa e al. (2010) sugges ha he characerizaion of reurn disribuions and forecass of asse-price variabiliy plays a criical role in he analysis of financial markes. They esimae differen measures of volailiy for gold, silver and copper. They find ha he reurn disribuions of he hree markes are no normal and he applicaion of financial ime sampling echniques is helpful in obaining a normal disribuion. Using he auoregressive disribued lag approach, Sari e al. (2010) examine he co-movemens and informaion ransmission among he spo prices of four precious meals (gold, silver, plainum, and palladium), oil price, and he US dollar/euro exchange rae. They find evidence of a weak long-run equilibrium relaionship, bu srong feedbacks in he shorrun. They conclude ha invesors may diversify a porion of he risk by invesing in precious meals, oil, and he euro. Hammoudeh e al. (2010) examined he condiional volailiy and correlaion dependence and inerdependence of four major precious meals (gold, silver, plainum and palladium), while accouning for geopoliics wihin a mulivariae sysem. The resuls indicae significan shor-run and long-run dependencies 6

8 and inerdependencies o news and pas volailiy. The empirical resuls become more pervasive when exchange rae and federal funds rae are included. Baur and Lucey (2010) examine relaions beween inernaional socks, bonds and gold reurns o evaluae gold as a hedge and a safe haven. They find ha gold is a hedge agains socks, on average, and a safe haven in exreme sock marke condiions. Prices of precious meals have been highly volaile in he pas, and even more so recenly. The volaile precious meal price environmen requires risk quanificaion. VaR has become an essenial ool wihin financial markes for quanifying and assessing porfolio marke risk, ha is, he risk associaed wih price movemens [see Chrisoffersen (2009) for a deailed overview of VaR]. VaR deermines he maximum loss a porfolio can generae over a cerain holding period, wih a pre-deermined probabiliy value. Therefore, VaR can be used, for insance, o evaluae he performance of porfolio managers by providing risk quanificaion, ogeher wih porfolio reurns. Moreover, VaR can help porfolio managers o deermine he mos suiable risk managemen sraegy for a given siuaion. VaR has become a sandard measure of downside marke risk and is widely used by financial inermediaries and banks [see Basel Commiee on Banking Supervision, (1988, 1995, 1996)], sock markes [McAleer and da Veiga (2008a, b), McAleer (2009), McAleer e al. (2009, 2010)], oil markes [see Cabedo and Moya (2003)], among ohers. As menioned above, despie he imporance of precious meals and heir volaile naure, here is no sudy of VaR using precious meals. One of he primary purposes of our paper is o fill his void in he lieraure. 7

9 3. Esimaing and Forecasing Value-a-Risk In his secion, we explicily define VaR followed by descripion of differen mehods we use o esimae VaR for precious meals Defining Value-a-Risk Le he asse reurn process be denoed by R (1) where I -1 (0, h ), I -1 is he informaion se a ime -1 and h is he variance a ime. The VaR measure wih coverage probabiliy, p, is defined as he condiional quanile, VaR (p), where 1 Pr ( R VaR (p) I- 1) 1 = p (2) This means he proporion of excepions, or days when he acual loss exceeds he 99% VaR, is a mos 1%. The condiionaliy of he VaR measure is imporan. Throughou he paper, we will assume ha 0, so ha R. This is a reasonable assumpion for daily daa and is consisen wih he lieraure [see Chrisoffersen (2009)]. However, volailiy is presumed o be ime-varying. The probabiliy, p, is aken wih respec o he disribuion funcion of he porfolio reurns, condiional on he informaion se a -1. Throughou he paper, we focus on he porfolio VaR wih he coverage probabiliy p = 1%, which is consisenly used in he lieraure for compuing risk exposure [see Basel Commiee on Banking Supervision (1988, 1995, 1996)]. One can esimae VaR using informaion obained from univariae or mulivariae models. Mos sudies [see, for example, Gio and Lauren (2004) and Kueser e al. (2006)] analyze VaR forecasing performance for univariae models, while ohers [see, 8

10 for example, McAleer and da Veiga (2008a)] have used mulivariae models o check for he impac of volailiy spillovers on esimaing VaRs. Berkowiz and O'Brien (2002) conclude ha a simple univariae model is able o improve he accuracy of porfolio VaR esimaes delivered by large US commercial banks. Brooks and Persand (2003) also concluded ha here are no gains from using mulivariae models while, more recenly, McAleer and da Veiga (2008b) found mixed evidence regarding volailiy spillovers across financial asses. Chrisoffersen (2009) argues ha univariae models are more appropriae if he purpose is risk measuremen as in compuing VaR forecass, while mulivariae models are more suiable for risk managemen as in porfolio selecion. Based on his evidence, we use only univariae models in he empirical analysis. 1 There are many ways of specifying univariae volailiy o capure VaR. This paper uses he following four specificaions of volailiy RiskMerics The benchmark measure advocaed in J.P. Morgan s (1996) RiskMerics (RM) ses he condiional mean o be consan and specifies he variance as an exponenial filer. Under he RiskMerics approach, he variance is calibraed using an Exponenially Weighed Moving Average, which corresponds o he following Inegraed GARCH model: 1 We also esimaed mulivariae GARCH models incorporaing all precious meals using differen parameerizaions. The empirical resuls are no repored for he sake of breviy bu are available on reques. 2 All VaR calculaions repored in he paper are calculaed wih he help of files which were graciously provided by Peer Chrisoffersen. We also calculaed VaR wih he hisorical simulaion approach, which is a naive mehod bu is sill popular among banks and financial insiuions [see Perignon and Smih (2010)]. The empirical resuls are no repored bu are available on reques. 9

11 h 2 ( 1 ) 1 h 1 (3) where he conribuion o he long-erm persisence of uniy, namely λ, is se o 0.94 for daily daa, and hence is no esimaed. RiskMerics assumes ha he sandardized residuals are normally disribued, so ha he VaR measure is given by VaR RM (p) Z p 1 h (4) where Z p denoes he p-h percenile of a sandard normal variable. For p = 0.01, i follows ha Z p = GARCH evolves as: In he Gaussian GARCH(1,1) model of Bollerslev (1986) he condiional variance h h (5) where > 0, > 0, > 0, and + < 1 are sufficien condiions o guaranee he posiiviy of he condiional variances and he saionariy of reurns. The one-sep ahead condiional quanile wih coverage probabiliy p is given as VaR GARCH (p) Z p 1 h (6) where he forecas of h is obained from Eq. (5) GARCH wih disribuion Mos empirical applicaions of VaR assume ha asse reurns are normally disribued as his assumpion considerably simplifies he compuaion of VaR. However, normaliy is inconsisen wih he empirical evidence of asse reurns which finds he 10

12 disribuion o be skewed, fa-ailed, and peaked around he mean. This implies ha exreme evens are more likely o occur in pracice han would be prediced by he symmeric and hinner-ailed normal disribuion. This fac is painfully obvious in ligh of he recen financial crisis. Thus normaliy assumpion can produce VaR esimaes ha are inappropriae measures of he rue risk faced by financial insiuions or porfolio managers. Thus, in our paper we also esimae VaR hresholds assuming a -disribuion given as: vˆ 2 GARCH VaR T 1(p) Tp v ).. vˆ (ˆ h (7) where T p ( vˆ ) denoes he p-h percenile of a suden random variable wih vˆ degrees of freedom, and h is he forecas obained from he GARCH model GARCH - Filered Hisorical Simulaion We know ha he assumpion of a normal disribuion is no appropriae for mos speculaive asses a he daily frequency. Choosing an alernaive disribuion is difficul. Raher han imposing such a choice, we can also rely on a simple resampling scheme which in financial risk managemen is referred as Filered Hisorical Simulaion (FHS). The erm filered refers o he fac ha we are no simulaing from he se of raw reurns bu from a se of shocks, z, which are reurns filered by he GARCH model. In GARCH-FHS mehod, a parameric GARCH model is iniially filered which generaes a sequence of sandardized reurns, zˆ R / hˆ, where ĥ denoes he insample fied condiional volailiy esimae from he GARCH model. VaR is hen esimaed as: 11

13 VaR GARCH FHS 1(p) Zˆ p hˆ (8) where Ẑ p is he empirical p-h percenile of he fied sandardized reurns, Ẑ, over he previous 250 rading days [see Chrisoffersen (2009), Barone-Adesi e al. (1999, 2002) for furher deails]. 4. Daa We used daily reurns based on closing spo prices for he four precious meals (gold, silver, plainum, and palladium) for he period January 4, 1995 o November 12, Our sample period is paricularly ineresing o sudy since i includes he financial crisis of All precious meals are raded a COMEX in New York, and heir prices are measured in US dollars per roy ounce. The descripive saisics are given in Table 1, which shows ha palladium has he highes sandard deviaion, while gold has he lowes. The Jarque-Bera Lagrange muliplier saisic indicaes ha all series are no normally disribued. All series also have high kurosis, which implies ha a GARCHype model is appropriae. These saisics show ha he seemingly close precious meals can be quie differen. The low volailiy of he gold price is consisen wih he fac ha he annual demand and producion of gold are less han 10% of is above-ground supply, and is sock is a supply buffer agains fundamenal shocks. The low volailiy of he gold price is also consisen wih he fac ha gold has an imporan moneary componen, and is no used frequenly in exchange marke inervenions. Silver is more commodiy-driven han gold as is moneary elemen has been gradually phased ou. However, he wo precious meals are closely relaed. Silver ouperforms gold when he marke is up and does worse 12

14 when he marke is down. In erms of conemporaneous correlaions (no repored bu available on reques), he correlaion beween plainum and palladium reurns is posiive and he highes among all he pairs of precious meals, followed by he correlaion beween gold and silver reurns. 5. Empirical Resuls In his secion, we provide empirical resuls for he ou-of-sample VaR forecass followed by he resuls of he uncondiional and condiional coverage ess Ou-of-Sample VaR Forecasing In order o assess he ou-of-sample performance of he VaR measures, we proceed as follows: A 10-year rolling sample, saring from January 4, 1995, is used o esimae he VaR measures and a 1-year holdou sample (year subsequen o he esimaion) is used o evaluae he performance. Specifically, he firs rolling (esimaion) sample includes he reurns for he years 1995 o 2004 and he firs holdou sample includes he reurns for he year Nex, he esimaed sample is rolled forward by removing he reurns for he year 1995 and adding he reurns for he year Consequenly, he new holdou sample includes he reurns for he year The procedure coninues hrough o he end of he sample. As he precious meals price reurns span he period January 4, 1995 o November 12, 2009, he 10-year rolling esimaion procedure yields a oal holdou sample of 5 individual years. As menioned before, his sample period includes he global financial crises and a mehod which can predic accuraely during his financial urmoil will be indispensable. 13

15 The resuls for he ou-of-sample VaR for he one-day ahead forecas a he 1% level for he four esimaion mehods for he four previous meals are provided in Figure 1. The esimaed VaR for he hold-ou period was volaile for all four precious meals, wih palladium having he highes VaR volailiy. One hing which clearly sands ou is he high variance and corresponding VaR for all precious meals during he lae 2008 financial crisis period. 3 As he financial markes were in urmoil and risk was rising, financial marke paricipans were invesing heavily in safe reasuries and precious meals (gold in paricular) which conribued o high volailiy. Figure 1 also shows he relaively posiive reurns of gold during ha ime period. We also noe he high VaR in 2006 for all precious meals, paricularly for silver. 4 Silver reurn and is corresponding VaR experienced a spike in April 2006 as he firs Exchange-Traded Fund (ETF) for silver was launched on he American Sock Exchange. 5 Meanwhile, palladium had a huge negaive reurn in June 2006 largely aribued o he correcion in he marke fueled by earlier speculaion ha palladium may also have is own ETF, which maerialized a he end of The VaR resuls of he differen approaches for all precious meals show ha he VaR based on GARCH- gives a fairly conservaive VaR and he VaR based on RiskMerics gives he mos aggressive. We conclude ha when he volailiy of reurn is 3 Precious meals prices in 2008 were also volaile because of power shorages and labor srikes in Souh Africa, he world s second larges gold producer and he firs larges plainum producer. (see hp:// ) 4 There was also exreme volailiy in Demand for gold in dollars Gold hi a record in ha period. Demand in 2007 was much like in 2006, wih seady in he firs eigh monhs before seeing a sharp urn and experiencing some exreme bous of volailiy in he final quarer. (see hp:// 5 See hp:// 14

16 low like he early par of our forecasing sample, one can use any mehod since all give similar resuls. However, when markes experience high volailiy, like during he 2008 financial crisis, hen VaR esimaes among differen models diverge considerably, underscoring he imporance of a conservaive mehod like GARCH Uncondiional Coverage Tes This es checks he percenage of violaions (i.e. acual loss exceeds prediced loss) agains wha is expeced under he null, namely 1%. The null hypohesis is ha he proporion of excepions, or days when he acual loss is greaer han he 99% VaR, equals 1%. In a sample of T daily VaRs a he 99% confidence level, we check wheher we observe 0.01 T excepions. A rejecion of he null hypohesis means ha he model is no adequae. We employ he Likelihood Raio es of Kupiec (1995), known as he uncondiional coverage es, as follows: LR ˆ) ˆ) T X x T X X UC 2ln (1 p) p 2ln (1 p ( p (9) where p = 0.01 is he arge excepion rae, pˆ he sample proporion of excepions, X is he oal number of excepions, T is he oal number of observaions, and LR is asympoically disribued as chi-square wih one degree of freedom. This uncondiional es couns violaions over he enire period. The resuls are presened in Table 4, which shows ha he RiskMerics and GARCH models perform poorly while GARCH-FHS does well, and GARCH- performs he bes as i does no fail he uncondiional es for any of he four meals Condiional Coverage Tes 15

17 The LR UC given in he previous equaion is an uncondiional es saisic as i simply couns violaions over he enire period. However, in he presence of volailiy clusering, he VaR models ha ignore mean-volailiy dynamics may have he correc uncondiional coverage, bu a any given ime, hey may have incorrec condiional coverage. In such cases, he LR UC es will be of limied use as i will classify inaccurae VaR esimaes as accepably accurae. The condiional coverage es developed by Chrisoffersen (1998) inspecs serial independence of VaR esimaes. For a given VaR esimae, he indicaor variable, I, is consruced such I is 1 if a violaion occurs, and I is 0 if no violaion occurs. Chrisoffersen (1998) proposes he following likelihood raio es saisic for he null hypohesis of serial independence agains he alernaive of firs-order Markov dependence: LR IND n ln( /(1 )) n ln((1 )/ ) n ln( /(1 )) n ln((1 / ) (10) where n ij is he number of observaions wih value i followed by j, П 00 = n 00 /(n 00 +n 01 ), П 10 = n 10 /(n 10 +n 11 ), and П = (n 01 +n 11 )/N, respecively. The LR IND saisic has an asympoic chi-square disribuion wih one degree of freedom. In essence, Chrisoffersen (1998) argues ha violaions should be independen and idenically disribued over ime. However, wha we really care abou is simulaneously esing if he VaR violaions are independen and he average number of violaions is also correc. We can es joinly for independence and correc coverage using he condiional coverage es. The join es (LR CC ) of condiional coverage can be calculaed by simply summing he wo individual ess for uncondiional coverage and independence [see Chrisoffersen (2003) for deails]. 16

18 The resuls for boh LR IND and LR CC are presened in Table 4. The LR IND es shows ha RiskMerics and GARCH produce an inadequae VaR in he case of plainum as he evidence indicaes ha he violaions are no independen. Focusing on LR CC, we see ha RiskMerics fails his imporan es for all meals, GARCH fails for all meals excep palladium, GARCH-FHS fails only for gold, and GARCH- does no fail for any meal. Overall, our resuls indicae ha he GARCH- model no only performs he bes on average bu also is violaions are independen. This is quie remarkable, given he fac ha he sample period includes he global financial crisis, where one migh expec repeaed violaions. 6. Calculaing Daily Capial Charges Based on VaR Forecass The aim in his subsecion is o compare he saisical resuls obained above wih he requiremens esablished by he curren regulaory framework se by he Basel II Accord. Under he framework of Basel II, he VaR esimaes of he banks mus be repored o he domesic regulaory auhoriy. These esimaes are used o compue he amoun of regulaory capial requiremens in order o monior and conrol a financial insiuion s marke risk exposure, and o ac as a cushion agains adverse marke condiions. The marke risk capial requiremens are obviously a funcion of he forecas VaR hresholds. The Basel Accord sipulaes ha he daily capial charge mus be se a he higher of he previous day s VaR or he average VaR over he las 60 business days, muliplied by a facor k (see Table 2). The muliplicaive facor k is se by he local regulaors bu mus no be lower han 3. Thus he Basel Accord imposes penalies in he form of a higher muliplicaive facor k on banks which use models ha lead o a greaer 17

19 number of violaions han would be expeced given he specified confidence level of 1%. I is ineresing o noe ha he Basel II penaly srucure is concerned only wih he frequency of violaions and no he magniude of any violaion. The empirical evidence presened by Berkowiz and O'Brien (2002) and Perignon e al. (2008) show ha banks sysemaically overesimae heir VaR which leads o excessive amoun of regulaory capial which affecs banks profiabiliy. Therefore, using models ha deliver accurae esimaes of his capial can lead o an increase in efficiency and accuracy of risk assessmens made by invesors and porfolio managers. McAleer e al. (2010) propose a decision rule for calculaing daily capial charges in ligh of hese compeing forces [for furher deails see, for example, McAleer and da Veiga (2008a, b)]. We calculae he daily capial charges by using our VaR forecass and he resuls are repored in Table 3. The able shows ha he mean daily capial charge, which is a funcion of boh he penaly and he forecas VaR, implied by GARCH- is he larges for all meal cases, and also yields he lowes violaions. This is consisen boh wih inuiion and he empirical resuls repored in McAleer e al. (2010). A high capial charge is undesirable as i reduces profiabiliy while large violaions may lead o bank failures, as he capial requiremens implied by he VaR hreshold forecass may be insufficien o cover he realized losses. This exercise shows ha porfolio managers engaged in precious meals who wan o follow a conservaive sraegy should calculae VaR using GARCH- as his will yield fewer violaions, hough wih lower profiabiliy. 7. Esimaing Porfolio Designs and Hedging Sraegies 18

20 We now provide wo examples using mulivariae GARCH models for precious meals o analyze porfolio design and hedging sraegies as par of our examinaion of risk managemen. Ewing and Malik (2005) underake a similar exercise for small cap and large cap sock reurns Porfolio Weighs The firs example follows Kroner and Ng (1998) by considering a porfolio ha minimizes risk wihou lowering expeced reurns. If we assume he expeced reurns o be zero, he opimal porfolio weigh of one commodiy (or asse) relaive o he oher in a wo commodiy (asse) porfolio is given by: w 12, h22, h12, h 2h h 11, 12, 22, (11) 0, if w12, 0 w12, w12,, if 0 w12, 1 1, if w 1 12, (12) where w 12, is he porfolio weigh for asse 1 relaive o asse 2 in a one-dollar porfolio of he wo asses a ime, h 12, is he condiional covariance beween asse reurns, and h 22, is he condiional variance of he asse 2. The porfolio weigh of he second asse in he one dollar porfolio is 1-w 12,. The average values of w 12, based on our precious meals are repored in Table 5. For insance, he average value of w 12, of a porfolio comprising gold and silver is This suggess ha he opimal holding of gold in one dollar of gold/silver porfolio is 80 cens and 20 cens for silver. These opimal porfolio weighs sugges ha invesors should own more gold han oher commodiies in heir porfolios. 19

21 7.2. Hedge Raios As a second illusraion, we follow Kroner and Sulan (1993) regarding riskminimizing hedge raios and apply i o our precious meals. Kroner and Sulan (1993) show ha in order o minimize he risk of he porfolio ha is $1 long in asse 1, an invesor should shor $β of he asse 2, where β is given as: h 12, (13) h22, here β is he risk-minimizing hedge raio for wo asses, h 12, is he condiional covariance beween asse 1 and 2, and h 22, is he condiional variance of he second asse. Table 5 repors he average values of β for he differen commodiy markes. For example, when holding a long posiion for $100 in he silver porfolio, invesors should shor $6.90 of gold. 8. Conclusion This paper examines he volailiy dynamics in precious meals and explores he corresponding risk managemen implicaions. The condiional volailiy and correlaion dynamics in he price reurns of gold, silver, plainum and palladium are modeled using daily daa from January 1995 o November Value-a-Risk (VaR) is used o analyze he risk associaed wih precious meals, and o design opimal risk managemen sraegies. We compue he VaR for all precious meals using he calibraed RiskMerics, alernaive empirical GARCH models, and he semi-parameric Filered Hisorical Simulaion approach. 20

22 Differen risk managemen sraegies are suggesed based on condiional and uncondiional saisical ess. The economic imporance of our resuls is highlighed by calculaing he daily capial charges from he esimaed VaRs using differen mehods for all precious meals. This exercise shows ha porfolio managers engaged in precious meals who wan o follow a conservaive sraegy should calculae VaR using GARCH- as his will yield fewer violaions, hough wih lower profiabiliy. The risk-minimizing porfolio weighs and dynamic hedge raios beween differen meal groups are documened. The porfolio weighs sugges ha he precious meal porfolios should have more gold han any of silver, plainum and palladium in order o minimize risk wihou affecing expeced reurns. The hedging raios indicae ha using a shor posiion o hedge a long posiion for precious meals is no expensive excep for plainum/palladium. Our resuls are very imely and useful for financial marke paricipans as he global financial markes coninue o experience unprecedened volailiy and he need for invesmen in precious meals remains high. 6 6 On May 6, 2010 he Dow Jones Indusrial Average plunged by nearly 1,000 poins (mosly due o Greek deb concerns) in weny minues making i he larges inra-day poin decline in he marke s hisory. No surprisingly, gold prices among oher precious meals increased dramaically as volailiy in he marke remained high. (See Economis aricle iled America's sock marke plunge: A few minues of mayhem on May 13, 2010). Such evens remind us ha financial markes remain unnerved. 21

23 References Basel Commiee on Banking Supervision, (1988), Inernaional Convergence of Capial Measuremen and Capial Sandards, BIS, Basel, Swizerland. Basel Commiee on Banking Supervision, (1995), An Inernal Model-Based Approach o Marke Risk Capial Requiremens, BIS, Basel, Swizerland. Basel Commiee on Banking Supervision, (1996), Supervisory Framework for he Use of Backesing in Conjuncion wih he Inernal Model-Based Approach o Marke Risk Capial Requiremens, BIS, Basel, Swizerland. Baen, J.M. and B.M. Lucey (2010). Volailiy in he gold fuures marke. Applied Economics Leers, 17(2), Barone-Adesi, G., K. Giannopoulos and L. Vosper (1999). VaR wihou correlaions for porfolios of derivaive securiies. Journal of Fuures Markes, 19, Barone-Adesi, G., K. Giannopoulos and L. Vosper (2002). Backesing derivaive porfolios wih filered hisorical simulaion. European Financial Managemen, 8, Baur, D.G. and B.M. Lucey (2010). Is gold a hedge or a safe haven? An analysis of socks, bonds and gold. The Financial Review, 45(2), Berkowiz, J. and J. O'Brien (2002). How accurae are value-a-risk models a commercial banks? Journal of Finance, 57(3), Bollerslev, T. (1986). Generalized Auoregressive Condiional Heeroskedasiciy. Journal of Economerics, 31: Brooks, C. and G. Persand (2003). Volailiy forecasing for risk managemen. Journal of Forecasing, 22 (1), Cabedo, J.D. and I. Moya (2003). Esimaing oil price value a risk using he hisorical simulaion approach, Energy Economics, 25(3), Chng, M.T. (2009). Economic linkages across commodiy fuures: Hedging and rading implicaions. Journal of Banking and Finance, 33(5), Chrisoffersen, P. (1998). Evaluaing inerval forecass. Inernaional Economic Review, 39(4), Chrisoffersen, P. (2003). Elemens of Financial Risk Managemen, San Diego: Academic Press. 22

24 Chrisoffersen, P. (2009). Value-a-risk models. In T. Andersen, R. Davis, J.-P. Kreiss, and T. Mikosch (Eds.), Handbook of Financial Time Series. Springer Verlag. Conover, C.M., G.R Jensen, R.R Johnson and J.M Mercer (2009). Can Precious Meals Make your Porfolio Shine? Journal of Invesing, 18(1), Draper, P., R.W. Faff and D. Hillier (2006). Do precious meals shine? An invesmen perspecive. Financial Analyss Journal, 62(2), Ewing, B.T. and F. Malik (2005). Re-examining he asymmeric predicabiliy of condiional variances: The role of sudden changes in variance. Journal of Banking and Finance, 29(10), Gio, P. and S. Lauren (2004). Modelling daily value-a-risk using realized volailiy and ARCH ype models. Journal of Empirical Finance, 11(3), Hammoudeh, S. and Y. Yuan (2008). Meal volailiy in presence of oil and ineres rae shocks. Energy Economics, 30(2), Hammoudeh, S., Y. Yuan, M. McAleer and M. Thompson (2010). Precious mealsexchange rae volailiy ransmissions and hedging Sraegies. Inernaional Review of Economics and Finance (forhcoming). Jensen, G.R., R.R. Johnson and J.M. Mercer (2002). Tacical asse allocaion and commodiy fuures. Journal of Porfolio Managemen, 28(4), JP Morgan (1996). RiskMerics, Technical Documen, 4 h Ediion, New York. Khalifa, A.A., H. Miao and S. Ramchander (2010). Reurn disribuions and volailiy forecasing in meal fuures markes: Evidence from gold, silver, and copper. Journal of Fuures Markes, forhcoming. Kroner, K.F. and V.K. Ng (1998). Modeling asymmeric movemens of asse prices. Review of Financial Sudies, 11, Kroner, K.F. and J. Sulan (1993). Time dynamic varying disribuions and dynamic hedging wih foreign currency fuures. Journal of Financial and Quaniaive Analysis, 28(4), Kueser, K., S. Minik, and M. Paolella (2006). Value-a-risk predicion: A comparison of alernaive sraegies. Journal of Financial Economerics, 4(1), Kupiec, P. (1995). Techniques for verifying he accuracy of risk measuremen models. The Journal of Derivaives 3(2),

25 McAleer, M. (2009). The en commandmens for opimizing value-a-risk and daily capial charges, Journal of Economic Surveys, 23(5), McAleer, M. and B. da Veiga (2008a). Forecasing value-a-risk wih a parsimonious porfolio spillover GARCH (PS-GARCH) model. Journal of Forecasing, 27(1), McAleer, M. and B. da Veiga (2008b). Single index and porfolio models for forecasing value-a-risk hresholds. Journal of Forecasing, 27(3), McAleer, M., J.-A. Jimenez-Marin and T. Perez-Amaral (2009). The Ten Commandmens for managing value-a-risk under he Basel II Accord. Journal of Economic Surveys, 23(5), McAleer, M., J.-A. Jimenez-Marin and T. Perez-Amaral (2010). A decision rule o minimize daily capial charges in forecasing value-a-risk. Journal of Forecasing, forhcoming. Perignon, C., Z. Deng, and Z. Wang (2008). Do banks oversae heir value-a-risk? Journal of Banking and Finance, 32(5), Perignon, C. and D. R. Smih. (2010). The level and qualiy of Value-a-Risk disclosure by commercial banks. Journal of Banking and Finance, 34(2) 2010, Sari, R., S. Hammoudeh and U. Soyas (2010). Dynamics of oil price, precious meal prices, and exchange rae. Energy Economics, 32(2),

26 Table 1: Descripive Saisics on Precious Meal Reurns Gold Silver Plainum Palladium Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Noes: All saisics are for daily reurns from January 4, 1995 o November 12, 2009, yielding 3665 observaions. 25

27 Table 2: Backesing VaR for Precious Meals Gold RiskMerics GARCH GARCH- GARCH-FHS LR uc * * * LR ind LR cc * * * Silver RiskMerics GARCH GARCH- GARCH-FHS LR uc * * LR ind LR cc * * Plainum RiskMerics GARCH GARCH- GARCH-FHS LR uc * * LR ind * * LR cc * * Palladium RiskMerics GARCH GARCH- GARCH-FHS LR uc 4.209* LR ind LR cc 4.875* Noes: * denoes ha we rejec he null hypohesis a he 10% level implying ha he model is inadequae. LR uc is he uncondiional coverage es given by Kuipic (1995) while LR ind and LR cc are condiional coverage ess given in Chrisoffersen (1998, 2003). Criical values for rejecing he null hypohesis for LR uc, LR ind and LR cc a he 10% level are 2.70, 2.70, and 4.60, respecively. The degree(s) of freedom are 1 for he firs wo ess and 2 for he hird es. If he calculaed es saisic is greaer han he criical value, we rejec he VaR model. A 10% level is ypically used as he consequences of acceping a poor VaR model are very large. 26

28 Table 3: Basel Accord Penaly Zones Zone Number of Violaions k Green 0 o Yellow Red Noe: The number of violaions is given for 250 business days. 27

29 Table 4: Daily Capial Charges for Precious Meals Panel A: Gold Model Number of Daily Capial Charges Violaions Mean Maximum Minimum RiskMerics GARCH GARCH GARCH-FHS Panel B: Silver Model Number Daily Capial Charges Of Violaions Mean Maximum Minimum RiskMerics GARCH GARCH GARCH-FHS Panel C: Plainum Model Number Of Violaions Daily Capial Charges Mean Maximum Minimum RiskMerics GARCH GARCH GARCH-FHS Panel D: Palladium Model Number Daily Capial Charges Of Violaions Mean Maximum Minimum RiskMerics GARCH GARCH GARCH-FHS Noes: The daily capial charge is he higher of he negaive of he previous day s VaR or he average VaR over he las 60 business days imes (3+k), where k is he penaly given in Table 3. 28

30 Table 5: Opimal Porfolio Weighs and Hedge Raios for Precious Meals Porfolio Average w 12, Average β Gold/Silver Gold/Plainum Gold/Palladium Silver/Plainum Silver/Palladium Plainum/Palladium Noes: w 12, is he porfolio weigh of wo asses holdings a ime and average β is he risk-minimizing hedge raio for wo precious meals. We used he BEKK parameerizaion for he mulivariae GARCH model for compuaions. 29

31 1/4/05 3/4/05 5/4/05 7/4/05 9/4/05 11/4/05 1/4/06 3/4/06 5/4/06 7/4/06 9/4/06 11/4/06 1/4/07 3/4/07 5/4/07 7/4/07 9/4/07 11/4/07 1/4/08 3/4/08 5/4/08 7/4/08 9/4/08 11/4/08 1/4/09 3/4/09 5/4/09 7/4/09 9/4/09 11/4/09 1/4/05 3/4/05 5/4/05 7/4/05 9/4/05 11/4/05 1/4/06 3/4/06 5/4/06 7/4/06 9/4/06 11/4/06 1/4/07 3/4/07 5/4/07 7/4/07 9/4/07 11/4/07 1/4/08 3/4/08 5/4/08 7/4/08 9/4/08 11/4/08 1/4/09 3/4/09 5/4/09 7/4/09 9/4/09 11/4/09 Figure 1: VaR esimaes 8.00% 6.00% 4.00% 2.00% 0.00% -2.00% -4.00% Gold Reurn Risk Merics GARCH GARCH- FHS -6.00% -8.00% % % Panel A: Gold 15.00% 10.00% 5.00% 0.00% -5.00% % Silver Reurn Risk Merics GARCH GARCH- FHS % % % Panel B: Silver 30

32 1/4/05 3/4/05 5/4/05 7/4/05 9/4/05 11/4/05 1/4/06 3/4/06 5/4/06 7/4/06 9/4/06 11/4/06 1/4/07 3/4/07 5/4/07 7/4/07 9/4/07 11/4/07 1/4/08 3/4/08 5/4/08 7/4/08 9/4/08 11/4/08 1/4/09 3/4/09 5/4/09 7/4/09 9/4/09 11/4/09 1/4/05 3/4/05 5/4/05 7/4/05 9/4/05 11/4/05 1/4/06 3/4/06 5/4/06 7/4/06 9/4/06 11/4/06 1/4/07 3/4/07 5/4/07 7/4/07 9/4/07 11/4/07 1/4/08 3/4/08 5/4/08 7/4/08 9/4/08 11/4/08 1/4/09 3/4/09 5/4/09 7/4/09 9/4/09 11/4/09 Figure 1: VaR esimaes (coninued) 15.00% 10.00% 5.00% 0.00% Plainum Reurn Risk Merics GARCH GARCH- FHS -5.00% % % Panel C: Plainum 15.00% 10.00% 5.00% 0.00% -5.00% % Palladium Reurn Risk Merics GARCH GARCH- FHS % % % Panel D: Palladium 31