CADERNOS DO IME Sére Esaísca Unversdade do Esado do Ro de Janero - UERJ Ro de Janero RJ - Brasl ISSN 1413-90 / v. 4, p. 15-8, 008 ANALYSIS OF CRUDE OIL AND GASOLINE PRICES THROUGH COPULAS Rcardo de Melo e Slva Accoly Unversdade do Esado do Ro de Janero Peróleo Braslero S.A. rcardo.accoly@gmal.com Fernando Anono Lucena Aube Ponfíca Unversdade Caólca do Ro de Janero - Peróleo Braslero S.A. aube@puc-ro.br Absrac In hs paper we nvesgae he dependence of crude ol and gasolne prces. The undersandng of he behavor of hs dependence s useful for modelng he porfolo of nvesmens n an negraed ol company. An accurae smulaon of he behavor of hese prces reveals precsely he rsk and reurn of he porfolo. Morover he movemens of hese prces s crucal for govermen planng snce hey affec he overall economy of developed and developng counres. The classcal approach whch uses ellpcal dsrbuons o model he rsk facors can be msleadng snce hey are acually no ellpcal. We used copula o esablcsh such dependence snce hs mehodolgy precludes he use of ellpcal dsrbuons. We found a change n he behavor of prces n he recen perod compared o hose n begnnng of he decade and hs fac s also repored n he leraure. Ths change s observed hrough dfferen copula models ha were adjused. These resuls were confrmed wh a boosrap analyss. Key-words: Crude ol prces, Gasolne prces, Copulas.
Cadernos do IME Sére Esaísca Analyss of Ol and Gasolne Through Copula 1. Inroducon Crude ol s by far he mos mporan commody n he world. Is relevance s relaed o s mporance as he man source of energy n developed and n developng economes. Is prce can affec he economy of many counres for long perods. Recenly crude ol and gasolne prces have shown a pecular behavor. They had been an upward movemen snce 003 unl he end of he frs semeser of hs year. In he second semeser of 008 hey had a sudden drop manly due o he fear of a global recesson. The comprehenson of he dynamcs of crude ol and gasolne prces and her relaonshp s relevan for: () governmenal economc and energec plannng; () ol ndusry as a suppler; () consumer ndusry; and (v) consumer n general. Tradonally when modelng he dynamcs of prces one assumes hey have a Gaussan or ellpcal dsrbuon. Ths s a common hypohess n fnance leraure boh from sochasc processes perspecve as well as from economercs pon of vew (see Mendes e Souza (004)). The key pon n hs framework s ha he sascal dependence or he co-movemen of he varables can be modeled hrough a lnear correlaon as he Pearson coeffcen or a covarance marx a n a mulvarae case. Unforunaely, mos of he random varables are no ellpcally dsrbued. For example, prces (or reurns) whch represen he fundamenal rsk facor n economcs and fnance are non ellpcal, srcly speakng. As a consequence he srucure of dependence of hese varables could be compromsed f hs smplfed approach s adoped. The mehodology whch s sued n capurng he whole dependence of varables s provded by copulas. Roughly speakng he copula s a funcon ha jons he margnal dsrbuons of random varables n a mulvarae dsrbuon descrbng he jon behavor. Or sascally speakng, he copula allows he descrpon of a mulvarae dsrbuon n erms of a specfc dependence srucure of he margnal dsrbuons of hese varables. Copulas were nroduced n 1959 n a conex of probably, (see Frees and Valdez (1998)). Is use n nsurance ndusry sared n 1995, (see Embrechs (008)) and spread o dfferen felds of fnance such as rsk, decson analyss, porfolo managemen, and prcng dervaves. The use of copulas n hese felds s well dealed n McNel e al. (005) and Cherubn e al. (004), among ohers. Despe he growng neres of copula n fnance research, has been used n dfferen 16
Cadernos do IME Sére Esaísca Accoly & Aube areas of scence such as hydrology, see Genes and Frave (007), ol feld developmen, see Accoly and Chyosh (004), for example. The man goal of hs paper s o nvesgae he dependence srucure of crude ol prces and gasolne prces. From he perspecve of an ol frm hs dependence s crucal for he porfolo managemen of real projecs. The undersandng of he acual rsks nvolved n such porfolo and he reurn predced are dependen on he correc modelng of prces (or rsk facors) embedded n hs analyss. Ths way we wll nvesgae he behavor of hese varables based on pas nformaon. We wll adjus auo regressve GARCH (AR-GARCH) models o fler he lnear and he non lnear me dependence n he seres of reurns. The resdual of hese models wll be suded hrough copulas. Hence we are able o capure he rue nerdependence of he varables. The paper s organzed as follows: secon presens an overvew of copula funcons, secon 3 presens he daa and he economerc modelng of AR-GARCH models, secon 4 analyzes he dependence of he varables and secon 5 presens he concluson.. Copula bascs Our sarng pon s as smple as mos real problems. We know, for example, wo varables or wo rsk facors. Then we wan o know how hese wo varables are relaed or how o descrbe he sensvy of one o he oher. To acheve hs goal we are gong o consruc he jon dsrbuon usng he concep of copula. A hs pon we refer o he exbooks of Nelsen (1999) and Joe (1997), among ohers. Consder X 1 and X wo varables ha represen wo rsk facors. The jon dsrbuon funcon of hese wo rsk facors s gven by. F X ( x) = P( X x R 1 x1, X x ) Consder ha we know he nformaon abou he dsrbuon funcon of hese wo varables F X1 and F X, n oher words, we know he margnal dsrbuons of hese wo rsk facors X 1 and X, such as F X ( x) = P( X x) x R. Assume ha hese margnal dsrbuons are connuous and srcly ncreasng. The copula s a funcon C 17
Cadernos do IME Sére Esaísca Analyss of Ol and Gasolne Through Copula defned n [,1] 0 wh unform margnals n ( 0,1) whch provdes he dependence lnk beween F and he margnals F X1 and F X so ha F X ( x ) = C( F1 ( x1 ),F ( x )) (1) Where F = F X. We can also wre equaon (1) n he converse form 1 1 C( u1,u ) = F( F1 ( u1 ),F ( u )) () Sklar (1959) showed wha s essenal n hs formulaon: f he margnals are connuous and srcly ncreasng, he funcon C s unque. Sklar's Theorem s absoluely general: any jon dsrbuon can be wren n copula form. 3. Tme seres modelng There are dfferen ypes of crude ol negoaed n he world. Among hem he WTI (Wes Texas Inermedae) s he mos lqud raded crude ol. We are gong o use he hsorcal WTI and gasolne frs fuure conracs o proceed wh hs sudy. We ook he daly close prces for he frs fuure conrac for gasolne and crude ol (WTI) from 05/01/1990 unl 09/6/008. Table 1 presens he man sascs of hese varables. The sample conans 4614 values and hey are repored n US$/bbl. Table 1: Man Sascs of he Seres Sasc Gasolne Crude ol Mean 40,68 33.93 Medan 8.95 4.13 Maxmum 149.98 145.9 Mnmum 13.68 10.7 Sd. Dev. 6.3 3.99 Skewness 1.77.0 Kuross 5.71 7.08 18
Cadernos do IME Sére Esaísca Accoly & Aube There s emprcal evdence of he exsence of a srucural break a end of 003, see Mura and Toka (008). Our sudy has some smlares wh Grégore e al. (008), bu hey looked only o he dependence beween ol prces and naural gas prces. They used a sample ha encompasses he perod of July, 003 unl July, 006. In hs arcle we dvded he enre sample no wo samples followng he evdences of a srucural break n 003. The frs perod begns n 05/01/1990 and fnshes n 1/31/003. The second perod covers he res of he sample from 01/05/004 o 09/6/008. We proceeded adjusng an AR-GARCH model o he log reurns of hese seres for boh perods. Ths way we flered he lnear and he quadrac dependences (he dependence on varance) n each seres. The resduals of each seres were esed. The Box-Perce es showed ha hey are uncorrelaed. We used he ARCH-LM and we could no rejec he null hypohess ha here s no ARCH effec. For perod 1 we modeled he log reurns for each one of he seres as: Gasolne: r = 0.0078 + 0.00874d h ( 1.96 ) = 8.97 10 ( 5.80 ) 6 ( 3.46 ) + 0.07691ε (18.14 ) w 0.0578r 1 (.97 ) 6 + 0.9134h ( 166.86 ) 1 + h 1 ε Crude ol: r = 0.05347r h ( 3.01) = 3.99 10 ( 4.70 ) 6 0.04473r (.57 ) + 0.07531ε (15.0 ) 6 1 + h 1 ε + 0.93h ( 161.15 ) 1 where d w accouns for wner seasonaly and ε ~ N( 0,1 ) and he fgures n brackes are -Sascs. For perod we proceeded n he same way and go he followng resuls: 19
Cadernos do IME Sére Esaísca Analyss of Ol and Gasolne Through Copula Gasolne: r = 0.1354d + h ε h ( 8.84 ) = 6.99 10 (.10 ) Crude ol: h ( 3.9 ) =.55 10 (.33 ) 5 5 1 r = 0.0131d + h ε + 0.0577ε 1 ( 3.00 ) ( 5.81) 1 + 0.0764ε 1 + 0.8359h ( 13.57 ) ( 7.75 ) 1 + 0.8740h 1 where d accouns for oulers pons. Once we had exraced he lnear and quadrac dependences, he resduals wll rean he rue dependence of he varables. So he copula wll be adjused usng he resduals of he models above, for perod 1 and perod. 4. Copula modelng In hs secon he deals of copula modelng are presened. We used he sofware R verson.7. for Wndows wh he QRMlb by Alexander McNel (see McNel e al. (005)). We also use he Resample Lbrary of Splus. Our frs sep o selec a copula model begns wh he analyss of he dependence beween he resduals from economerc models of gasolne and WTI. As poned by Frees and Valdez (1998), Pearson correlaon coeffcen s a good choce when he dependence beween he varables s lnear. So s always a good pracce o use also a nonparamerc dependence measure lke Kendall's au. In Table one can observe ha here s a consderable posve dependence beween gasolne and WTI resduals. I seems ha he dependence has ncreased from he frs perod o he second. In boh perods, as could be expeced, Pearson's coeffcen ndcaes a greaer dependence. 0
Cadernos do IME Sére Esaísca Accoly & Aube Table : Correlaon coeffcens Perod Pearson coefcen Kendall s τ Frs 0.708 0.549 Second 0.743 0.588 Snce we have wo perods of daa, wll be neresng o check he uncerany of hese dependence measures n boh perods. For hs purpose we wll use he boosrap mehod (see Davdson and Hnckley (1997)), a resamplng procedure ha s very useful o consruc confdence nervals when here s no a drec procedure. Because of he compuaonal cos of he Kendall's τ measure, only one housand boosrap samples were used o oban he percenle confdence nervals (95%). Table 3 presens he resuls. From hese confdence nervals one can observe ha here s an nersecon beween boh perods. A permuaon es for he dfference of Pearson correlaon coeffcen on boh samples wll ndcae f here s a sascally sgnfcan dfference (see Pesarn (001)). The resuls from wo housand permuaons are presened n Table 4. The resuls showed no evdence of a sascally sgnfcan dfference beween boh perods. Table 3: Confdence nervals Perod Pearson coefcen Kendall s τ Percenle Percenle Frs (0.681 0.738) (0.53 0.567) 1
Cadernos do IME Sére Esaísca Analyss of Ol and Gasolne Through Copula Second (0.703 0.78) (0.56 0.616) Table 4: Sascs for he dfference of Pearson coeffcen Sasc Observed Mean Sd. Dev. Alernave p-value Pearson -0.0349 0.0007153 0.03176 Two sded 0.679 The nex sep s o ransform each par of observaon (X 1, Y 1 ),, (X n, Y n ) no s rank based represenaon. The pars (R x, R y ), represen pseudo observaons from he underlyng copula ha characerzes he dependence srucure, were calculaed by R x = rank( X n + 1 ) and R y = rank( Y n + 1 ) These pars of resduals are shown n Fgure 1 for boh perods (perod 1 - P1 and perod - P) and hey confrm a posve dependence ha was prevously verfed hrough he compuaon of Pearson's and Kendall's measures. The paern s smlar and he percepble dfference s due o he number of pons: 3419 n perod 1 and 1187 n perod. Fgure 1: Normalzed resduals for perod 1 and perod
Cadernos do IME Sére Esaísca Accoly & Aube Table 5 conans he famles of copulas ha wll be used o model he dependence beween he resduals n hs sudy. Famles N.1 and N.14 receved her names from Nelsen (1999). The doman of he dependence parameer shows ha half of hem are only sued for posve dependences. The parameer esmaon wll be carred ou hrough he maxmzaon of log-lkelhood funcon usng he canoncal maxmum lkelhood (CML) process. As shown n Accoly (005) hs mehod avods problems n model selecon due o mproper specfcaon of margnal dsrbuons. Genes e al. (1995) proposed such esmaon procedure ha s approprae when one does no wan o specfy any paramerc model o descrbe he margnal dsrbuons. In hese cases one could use a nonparamerc esmae for he margnal, so he nference abou he dependence parameer α should be margn-free. Table 5: Copulas Famles used Famly C ( u, v) α Clayon Gumbel Frank 1 α α u + v 1) α [-1, ) \ {0} ( α α 1 α exp( [( lnu) + ( ln v) ] [1, ) 1 ( e ln 1 + α αu 1)( e e α αv 1 N.1 1 α 1 α 1 α ( ) 1 1) (-, ) \ {0} 1+ [( u 1) + ( v 1) ] [1, ) N.14 1 α α 1 α α 1 α ( 1+ [( 1) + ( v 1) ] ) α Gaussan 1 ( ( ) 1 N Φ u, Φ ( v ) ) 1 ( ( ) 1 T T u T ( v ) ) Placke [1 + ( α 1)( u + v) u [1, ) α [-1,1], γ γ, γ α [-1,1] [{ 1+ ( α 1)( u + v) } 4uvα ( 1 α )] 1/ ]/ ( α 1) { } (0, ) \ {1} To consruc he lkelhood funcon our man concern was he paramerc represenaon of he copulas, specfcally he copula densy. Gven a random sample {X 1k,X k } :k=1, n} from a dsrbuon Fα ( x1,x ) = Cα ( F 1( x1 ),F ( x )), he usual procedure s he selecon of parameer α ha maxmzes he pseudo log-lkelhood funcon: 3
Cadernos do IME Sére Esaísca Analyss of Ol and Gasolne Through Copula L( α ) = n k= 1 ln [ c ( F ( x ),F ( x ] α 1 1k k )) Once we esmaed he dependence parameerα hrough he procedure above for each copula model, we need o selec he bes model. Ths s addressed usng he AIC crera gven by AIC = L( α ) + P where P s he number of esmaed parameers and works as a penaly. Ths mehod s one of he mos used n model selecon, see for example Frees and Valdez (1998). Burnham and Anderson (00) recommended he compuaon of AIC dfferences, = AIC AIC mn, over all canddaes. For a specfc model, he larger s, less lkely s o be he bes model. They also suggesed ha a beer nerpreaon can be obaned wh Akake weghs, gven by w = R r = 1 exp( 0.5 ) exp( 0.5 ) r Gven R canddaes, w s he wegh of evdence ha model s he bes model. To ensure ha he model s properly seleced we wll use he boosrap procedure descrbed n Burnham and Anderson (00). Usng hs mehodology we can verfy how dfferen samples can affec he model selecon. Table 6 presens he MLE (Maxmum Lkelhood Esmaon) resuls for he frs perod. I also shows he compuaon of and w. From he resuls one can observe ha copula s he bes possble model. The dfference beween copula and he ohers s so huge ha he Akake weghs of he ohers can be consdered as zero. Due o numercal problems he Gaussan copula was no esmaed. Table 6: MLE resuls for he frs perod Famly α ) L(α) 0.75 1456 0 1,0 Placke 15.34 146 58 0 w 4
Cadernos do IME Sére Esaísca Accoly & Aube N.14 1.66 1394 11 0 N.1 1.43 1347 15 0 Frank 6.81 1339 31 0 Gumbel.07 197 315 0 Clayon 1.61 1110 689 0 As menoned before we confrmed he concluson above usng boosrap. Afer each boosrap sample we evaluaed he AIC o verfy he bes model. Because of compuaonal cos we lmed hs analyss o one housand samples. Table 7 presens he resuls. Alhough he copula model s somemes worse han Placke copula, here s no doub ha he former s he bes possble model n hs group. Table 7: Boosrap resuls for he frs perod Famly % Seleced hrough mn AIC 95,1% Placke 4,9% For he second perod he resuls are shown n Table 8. One can observe ha he Placke copula s he bes possble model. The dfference beween he Placke copula and he ohers s no as bg as he resuls of he frs perod, bu we sll can consder he Akake weghs from all oher models approach zero. Agan we used he same analyss as before. Table 9 shows he resuls from boosrap analyss. One can observe ha he resuls are no as srong as n he frs perod. The Placke copula had been overcome by copula and Frank copula 30% of he mes. Comparng he dependence beween perod 1 and perod, we can say ha he ncrease n dependence combned wh oher (undeeced) facor has led o he change n he order of model selecon. Table 8: MLE resuls for he second perod Famly ) α L(α) w Placke 17.4 50 0 0,99978 0.78 495 17 0,00017 Frank 7.66 49 0 0 N.14 1.79 474 56 0 Gumbel.0 459 87 0 Gaussan 0.76 458 89 0 5
Cadernos do IME Sére Esaísca Analyss of Ol and Gasolne Through Copula N1 1.49 448 109 0 Clayon 1.6 359 87 0 Table 9: Boosrap resuls for he second perod Famly % of Seleced hrough mn AIC Placke 70,7 % 1,9 % Frank 7,4 % Fgure presens smulaed pars and he orgnal pars for perod 1. I s clear ha he dependence behavor s well represened by copula. We also presen he bdensy plo of copula wh dependence parameer 0.754 and 4 degrees of freedom, correspondng o he dependence beween gasolne and WTI resduals n he frs perod. Fgure 3 presens he smulaed and he orgnal resduals for perod. I s clear ha he behavor of he dependence beween gasolne and WTI has a good represenaon by Placke copula. The bdensy of Placke copula wh parameer 17.4 s also presened. Analyzng he resuls for gasolne n boh perods, one can observe he srong dependence on he als. The exreme evens observed n he varable reurn s responsble for he fa als of hese dsrbuons, hs s a sylzed fac of fnancal me seres. Our resul s n accordance wh he fac ha dsrbuon and fa al dsrbuons are sued for modelng fnancal seres. Fgure 3: Orgnal and smulaed resduals and bvarae copula 6
Cadernos do IME Sére Esaísca Accoly & Aube Fgure 4: Orgnal and smulaed resduals and he bdensy Placke copula 5. Conclusons Ths paper analyzed he dependence of ol and gasolne prces. The man goal was o esablsh he real dependence nsead of he classcal approach of lnear dependence. Prce dependences have mporan mplcaons on ol ndusry manly n he decson makng process for nvesmens. In each seres we adjused an AR-GARCH model, flerng he lnear and quadrac dependences. Then we analyzed he dependence hrough he resduals of hese models n each perod. We found for he frs perod ha he dependence s well represened by he bvarae copula. The boosrap analyss showed ha n 95% of he cases he copula s he bes choce. In he second perod he behavor of prces changed and he Placke copula bes descrbed he behavor. The boosrap analyss confrmed ha s he bes choce bu no as good as he copula n he frs perod. I can be observed n he second perod ha here s an ncrease n he dependence and hs fac s n accordance wh he Pearson coeffcen. Ths ncreased dependence n he second perod can be one of he reasons ha he compeng models ( copula and Placke copula) have changed he selecon order. One naural drecon o exend hs analyss s he ncluson of oher refned producs such as 7
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