REVISA INVESIGACIÓN OPERACIONAL Vol., 30, No. 1, 11-19, 009 MULIVARIAE RISK-REURN DECISION MAKING WIHIN DYNAMIC ESIMAION Josip Arnerić 1, Elza Jurun, and Snježana Pivac, 3 Universiy of Spl Faculy of Economics, Croaia ABSRAC Risk managemen in his paper is focused on mulivariae risk-reurn decision making assuming ime-varying esimaion. Empirical research in risk managemen showed ha he saic "mean-variance" mehodology in porfolio opimizaion is very resricive wih unrealisic assumpions. he objecive of his paper is esimaion of ime-varying porfolio socks weighs by consrains on risk measure. Hence, risk measure dynamic esimaion is used in risk conrolling. By risk conrol manager makes free supplemenary capial for new invesmens. Univariae modeling approach is no appropriae, even when porfolio reurns are reaed as one variable. Porfolio weighs are ime-varying, and herefore i is necessary o re-esimae whole model over ime. Using assumpion of bivariae Suden's - disribuion, in mulivariae GARCH(p,q) models, i becomes possible o forecas ime-varying porfolio risk much more precisely. he complee procedure of analysis is esablished from Zagreb Sock Exchange using daily observaions of Pliva and Podravka socks. MSC: 6P0, 91B6, 91B8, 91B84 KEY WORDS: MGARCH model, dynamic porfolio weighs esimaion, bivariae Suden's -disribuion, posiive-definie covariance marix RESUMEN El manejo del riesgo es enfocado en ese rabajo como la oma de decisión sobre la esimación de un riesgo del reorno mulivariado asumiendo variabilidad en el iempo.. Invesigaciones empíricas del manejo del riesgo han mosrado que la meodología esáica de "media-varianza" en la opimización de porafolio es muy resriciva bajo asunciones poco realisas. El objeivo de ese rabajo es la esimación de pesos para los porafolios de acciones con variación en el iempo con resricciones sobre la medida de riesgo. Enonces, la esimación dinámica de la medida de riesgo es usada en el conrol del riesgo. Conrolando el riego el manager libera capial suplemenario para nuevas inversiones. El enfoque univariado no es apropiado, aun cuando los reornos de porafolio sean raados como una variable. Los pesos para los porafolios varían en el iempo, por ano es necesario re-esimar odo el modelo en el iempo. Asumiendo una disribución - Suden bivariada, en el modelo de GARCH(p, q) muivariado, es posible predecir el riesgo del porafolio que varia en el iempo mucho mas precisamene. El procedimieno compleo de análisis es desarrollado para el Zagreb Sock Exchange usando las observaciones diarias de los socks de Pliva y Podravka. 1. INRODUCION he aim of his paper is esimaion of ime-varying socks weighs in porfolio opimizaion using mulivariae approach. he purpose of his approach, realized using mulivariae GARCH(p,q) model, is forecasing of imevarying condiional expecaion and condiional variance. hose forecased values of firs and second order momens are inpus for porfolio reurn maximizaion by consrains on sandard deviaion as risk measure. he assumpion of bivariae Suden's -disribuion is used in maximizaion of he likelihood funcion o esimae parameers of DVEC-GARCH(1,1) model wih Cholesky facorizaion. Sample properies, especially lepokurosis of disribuion wih fa ails, shows ha assumpion of Suden disribuion is he mos appropriae (Arnerić, Jurun, Pivac 007). Modern financial analysis requires forecasing he dependences in he second order momens of porfolio reurns. Forecasing is necessary, apar from oher reasons, because financial volailiies moves ogeher over ime across asses and markes. According o hese requiremens mulivariae modeling framework leads o more relevan empirical models in comparison o esimaions based on separae univariae models. his appears from he pracical need for opions pricing, porfolio selecion, hedging and Value-a-Risk esimaion. 1 jarneric@efs.hr elza@efs.hr 3 spivac@efs.hr 11
Furhermore, a fundamenal characerisic of modern capial markes leads o applicaion of mulivariae volailiy models. his can be summarized in following facs: he volailiy of one marke leads o volailiy of oher markes; he volailiy of an sock ransmis o anoher sock; he correlaion beween sock reurns change over ime; he correlaion beween sock reurns increases in he long run because of he globalizaion of financial markes. In order o respec all hese facs and modern financial requiremens as he mos useful mehodological ool financial economerics offers Mulivariae Generalized Auoregressive Condiional Heeroscedasiciy models - MGARCH (Bauwnes, Lauren, Rombous 006). he complee procedure of analysis is esablished using daily observaions of Pliva and Podravka socks, as he mos frequenly raded socks from CROBEX index a Zagreb Sock Exchange. Daa on observed socks was provided by Croaia capial marke. In his paper daily daa of compound reurns of named socks from 1s January 005 o 16h Ocober 007 are used (www.zse.hr). his paper is organized as follows. Afer inroducion, assumpion of he bivariae Suden's -dusribuion is presened. Nex secion presens he mehodology employed in MGARCH model selecion. Following par gives esimaion of condiional variances and correlaion. he opic of he nex secion is opimal dynamic porfolio weighs forecasing. he final secion is dedicaed o conclusion remarks.. ASSUMPION OF HE BIVARIAE SUDEN'S -DISRIBUION Reurns from financial insrumens such as exchange raes, equiy prices and ineres raes measured over shor ime inervals, i.e. daily or weekly, are characerized by high kurosis. In pracice, he kurosis is ofen larger han hree, leading o esimaion of non-ineger degrees of freedom. hus, degrees of freedom can easily be esimaed using he Broyden-Flecher-Goldfarb-Shanno (BFGS) algorihm. BFGS is a mehod o solve an unconsrained nonlinear opimizaion problem (Bazarra, Sheral Shey 1993) he BFGS mehod is derived from he Newon's opimizaion mehods, as a class of hill-climbing opimizaion echniques ha seeks he saionary poin of a funcion, where he gradien is zero. Newon's mehod assumes ha he funcion can be locally approximaed as a quadraic in he region around he opimum, and use he firs and second derivaives o find he saionary poin. In Quasi-Newon mehods he Hessian marix of second derivaives of he funcion o be opimized does no need o be compued a any sage. he Hessian is updaed by analyzing successive gradien vecors insead. he BFGS mehod is one of he mos successful algorihms of his class. In his paper i will be used in bivariae Suden's -disribuion degrees of freedom esimaion by maximizaion of he likelihood funcion. he sandardized mulivariae Suden densiy from he k-dimensional vecor of reurns X is given as: f ( X ) df + k Γ = df Γ ( π ( df ) ) k 1+ k+ df ( X X ) df, (1) where Γ ( ) is gamma funcion and df is a shape parameer - degrees of freedom. I is imposed ha df is larger han o ensure he exisence of he variance marix. here are no mahemaical reasons why he degrees of freedom should be an ineger, however when df he ails of he densiy become heavier (Heikkinen, Kano 00). Hence, he assumpion ha reurns are independenly and idenically normally disribued is unrealisic. According o BFGS algorihm degrees of freedom of he bivariae Suden -disribuion for Pliva and Podravka sock reurns are esimaed a level 3.. Sandard error of esimaed degrees of freedom is 0.00705, which means ha parameer disribuion is saisically significan. Figure 1 shows ellipic conour plo of he bivariae 1
Suden's -disribuion densiy wih esimaed degrees of freedom. Namely, as conour of bivariae Suden's - disribuion densiy is more ellipic, he less are esimaed degrees of freedom (Bauwnes, Lauren 005). he Suden densiy has become widely used due o i's simpliciy and i's inheren abiliy o fi excess kurosis. Figure 1: Conour plo of he Bivariae Suden's -disribuion Densiy ( df = 3. ) Podravka sock reurns -0.10-0.05 0.00 0.05 0.10-0.10-0.05 0.00 0.05 0.10 Pliva sock reurns Source: According o daa on ZSE 3. APPROPRIAE MULIVARIAE GARCH(p,q) MODEL SELECION ypical example when i is necessary o simulaneously observe reurn movemens of several socks is marke porfolio reurn. If marke porfolio consiss of k socks, expeced reurn ( r, ) and variance of complee porfolio ( σ P, ) can be defined in he following way: where w is he vecor of porfolio weighs, Σ is he variancecovariance marix of he reurns. r P, σ P = w µ, () = w Σ w P, µ is he vecor of expeced reurns and According o equaion () i's obvious ha he mean vecor µ and he variance-covariance marix Σ are no consan over ime. Univariae GARCH(p,q) model for porfolio variance can be esimaed for a given weighs vecor. Because he weighs vecor changes over ime, i is necessary o esimae he model again and again. ha's way i is wisely o inroduce he possibiliies of MGARCH model. If a MGARCH model is fied, he mulivariae disribuion of he reurns can be direcly used o compue ime varying mean vecor and variancecovariance marix of he reurns. Hence here is no need o re-esimae he model for a differen weighs vecor. r of dimension 1. Dynamic model wih ime varying means, variances and covariances for componen can be denoed by: r = µ + ε Consider a bivariae vecor sochasic process { } In above relaions µ = E( r I ε = Σ Σ 1/ = V ( r I u 1 1 ) ) µ is mean vecor condiioned on pas informaion, and Σ is condiional variance marix. Furhermore, i is assumed ha expeced value of u is null-vecor wih variance equal o ideniy marix, as well (3) 13
as ha variance Σ is posiive definie marix. his assumpion is no generally saisfied auomaically. Posiive definie variance-covariance marix can be obained by he Cholesky facorizaion of Σ (say, 005). Before he concree modeling procedure i is useful o esimae auocorrelaion and parial auocorrelaion coefficiens of squared reurns. herefore correlograms and cross-correlograms are presened in Figure for hiry ime lags. I is obvious from Figure ha esimaed auocorrelaion coefficiens are saisically significan in many ime lags when exceed criical value presened by doed lines. his confirms hypohesis ha observed reurn ime series conain ARCH effecs (heeroscedasiciy). Figure : Correlograms and cross-correlograms beween Pliva and Podravka square reurns rpliva rpliva and rpodravka ACF 0.0 0. 0.4 0.6 0.8 1.0-0.05 0.00 0.05 0.10 0 5 10 15 0 5 0 5 10 15 0 5 rpodravka and rpliva rpodravka ACF -0.05 0.00 0.05 0.10 0.0 0. 0.4 0.6 0.8 1.0-5 -0-15 -10-5 0 Lag 0 5 10 15 0 5 Lag Source: According o daa on www.zse.hr According o given assumpions a diagonal VEC model wih Cholesky facorizaion is esimaed. Univariae GARCH(1,1) model can be generalized ino bivariae, as follows: vech( Σ ) = C + Avech( ε 1ε 1) + Bvech( Σ 1) (4) In equaion (4) vech ( ) operaor denoes he column-sacking operaor applied o he lower porion of he symmeric marix. he model (4) requires he esimaion of 1 parameers (C marix has 3 elemens, A and B marices each 9 elemens). o overcome dimensionaliy problem in VEC mehodology A and B marices can be resriced o diagonal elemens. herefore, bivariae DVEC-GARCH(1,1) model is resriced o 3 ( k + 1) = 9 parameers. Furhermore, DVEC model can be simplified by resricing A and B o be a vecors or a posiive scalars (Zivo, Wang 006). A disadvanage of he DVEC model is ha here is no guaranee of a posiive definie variance-covariance marix. Hence, differen DVEC(1,1) models wih differen resricions on coefficiens marices will be analyzed, underaking Cholesky facorizaion. able 1. VAR(1)-DVEC(1,1) model wih Cholesky facorizaion 14
Esimaed Coefficiens: Value Sd.Error value Pr(> ) C(1) 0.000519 0.0003415 1.5019 0.066800 C() 0.0004966 0.0004663 1.0649 0.143660 AR(1; 1, 1) 0.1047563 0.039044.6831 0.003740 AR(1;, ) 0.0508917 0.0448776 1.1340 0.18605 A(1, 1) 0.0036373 0.00011 17.39 0.000000 A(, 1) 0.000860 0.0004458 0.6416 0.60690 A(, ) 0.004537 0.0004084 11.0764 0.000000 ARCH(1; 1, 1) 0.7506377 0.034564 3.0013 0.000000 ARCH(1;, 1) 0.174671 0.0569743 3.0658 0.001131 ARCH(1;, ) 0.4859494 0.033898 15.0031 0.000000 GARCH(1; 1) 0.775338 0.0133463 58.0935 0.000000 GARCH(1; ) 0.87569 0.0157466 5.5551 0.000000 Informaion crieria: AIC(1) = -7406.841 BIC(1) = -7353.080 Source: According o daa on www.zse.hr For complee esimaion procedure i is necessary o esimae mean vecor of reurns using VAR(1) sysem. Also in vecor auoregression model of order 1, only diagonal coefficiens will be esimaed. Esimaion resuls of VAR(1)-DVEC(1,1) model wih Cholesky facorizaion are presened in able 1. According o able 1, in which marix A is resriced o be diagonal and marix B a vecor, he resuls are presened as follows: r r PL PD 0.000513 0.104756 = + 0.000497 0.05089 r r PL 1 PD 1 0.000013 0.563457 Σ 0.000001 0.13111 = + vech( ε 0.00001 0.66656 1ε 1 0.601141 ) + 0.641637 vech 0.684860 ( Σ ) 1 (5) In able he resuls of normaliy es, Ljung-Box es and Lagrange muliplier es are presened, indicaing ha in esimaed model (5) remains no auocorrelaion and no ARCH effecs, i.e. here is no heeroscedasiciy lef. 4. ESIMAION OF CONDIIONAL VARIANCES AND CORRELAION On Figure 3 here is eviden similariy beween Pliva and Podravka sock reurn movemens. I can be seen increase in volailiy movemen in las quarer of 006. he main precondiion for his rise was an inensive increase of invesmen in invesmen funds. Especially he increase of invesmen in he sock funds, caused by increased public awareness of he advanages of such invesmen (above average yield, higher ineres raes offered by banks) as well as he broadening of marke supply allowing beer risk diversificaion, has significanly affeced he rading volume on he Croaian capial marke and hus he volailiy of Pliva and Podravka socks. hese marke movemens are characerisic of he emerging capial marke. herefore, posiive correlaion beween hem can be expeced. Considering ha correlaion is condiioned on pas informaion, i is ime-varying. Namely, correlaion coefficiens are calculaed using daa of covariances and variances of sock reurns in marix Σ. Daa are ime-varying as i is shown on Figure 4. able : Diagnosic ess of sandardized residuals 15
Normaliy es: Jarque-Bera P-value Shapiro-Wilk P-value rpliva 1779.3 0 0.9114 0.000e+000 rpodravka 704.7 0 0.9619 4.441e-016 Ljung-Box es for sandardized residuals: Saisic P-value Chi^-d.f. rpliva 10.55 0.5677 1 rpodravka 10.17 0.6010 1 Ljung-Box es for squared sandardized residuals: Saisic P-value Chi^-d.f. rpliva 9.737 0.6390 1 rpodravka 7.8 0.8385 1 Lagrange muliplier es: R^ P-value F-sa P-value rpliva 8.810 0.7191 0.811 0.7380 rpodravka 6.468 0.8907 0.5940 0.969 Source: According o daa on www.zse.hr I is obvious ha condiional correlaion values of sock reurns are posiive in mos rading days. his may be an indicaor of emerging bull marke. Figure 3: Reurns of Pliva and Podravka socks wih heir condiional volailiies Pliva sock reurns Podravka sock reurns -0.10-0.0 0.06-0.1-0.04 0.04 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 005 006 007 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 005 006 007 Condiional volailiy of Pliva socks Condiional volailiy of Podravka socks 1 3 4 5 6 7 8 9 1 3 4 5 6 7 8 9 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 005 006 007 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 005 006 007 Source: According o daa on www.zse.hr Figure 4: Condiional correlaion of Pliva and Podravka sock reurns 16
Condiional correlaion of socks reurns -0.3-0. -0.1 0.0 0.1 0. 0.3 0 100 00 300 400 500 600 Daily observaions Source: According o daa on www.zse.hr 5. FORECASING OPIMAL DYNAMIC PORFOLIO According o esimaed model (5) prediced values of mean vecor and variance-covariance marix are compued and presened in able 3 for 10 days-ahead. able 3: en-days predicion of reurns, sandard deviaions, and correlaion beween Pliva and Podravka socks Prediced Pliva reurns Prediced Podravka Prediced Pliva Prediced Podravka Prediced correlaion reurns -0.001574771 0.0004609494 0.0003811753 0.00050030 0.000558344 0.000530389 0.0005708168 0.000531919 0.000577006 0.000531997 0.000578979 0.00053001 0.000579186 0.00053001 0.00057907 0.00053001 0.00057910 0.00053001 0.00057910 0.00053001 Source: According o daa on www.zse.hr S.dev. S.dev. 0.01877445 0.0098619 0.0058464 0.0101907 0.051005 0.01087058 0.045687 0.01153195 0.0675579 0.01178 0.091013 0.0166881 0.03161594 0.0131694 0.0343116 0.01361647 0.037067 0.0140344 0.0403166 0.0144085 0.040395556 0.031090784 0.0473091 0.00176599 0.01681666 0.01458517 0.01611 0.010666151 0.00936695 0.0089039 Using resuls from able 3 in equaion () vecor µ and marix Σ are obained in following opimizaion problem: max w µ i = 1, w w w i= 1 ( ) Σ w i = 1 SD In accordance wih opimizaion problem in (6) dynamic porfolio selecion is defined on daily basis, where SD is desired porfolio sandard deviaion in momen. In his paper opimizaion is done under assumpion * p, of consan risk preference a level of % and 3%. Maximizaion problem (6) is nonlinear programming problem solved using Solver by generalized reduced gradien mehod (Rombous, Rengifo 004). en days forecas resuls are given in able 4. * p, (6) 17
able 4: en days forecas opimal porfolio weighs a he sandard deviaion level of % and 3% Sock weighs of opimal porfolio selecion k-days forecas 1s nd 3rd 4h 5h 6h 7h 8h 9h 10h 0,948 0,87 0,80 w plv, 0 0 1 1 1 1 4 3 0,7374 0,051 0,17 0,197 SD p, 0,03 w pod, 1 1 0 0 0 0 6 7 8 0,66 0,885 0,807 0,735 0,670 0,609 0,553 0,501 w plv, 0 0 4 4 9 0 4 4 9 0,4545 0,114 0,19 0,64 0,330 0,390 0,446 0,498 SD p, 0,0 w pod, 1 1 6 7 1 0 6 6 1 0,5455 Source: According o daa on www.zse.hr From able 4 i can be seen ha porfolio weighs change daily wih consan risk preference. I is obvious ha porfolio weighs are changing a he hird day, and more inensively in given ime frame a lower risk level (%). In oher words, if invesor is willing o accep higher risk his porfolio weighs would changed afer sixh day (risk level 3%). According o presened daa in ables 3 and 4 i can be concluded ha sock wih higher reurn has higher weigh in opimal porfolio. If we compare 9h and 10h prognosic days porfolio is chosen according o defined sandard deviaion level. Namely, regardless o equal socks reurns higher weigh is associaed o he sock wih lower risk measure. 6. CONCLUSION his paper is focused on dynamic esimaion of socks weighs in porfolio opimizaion by mulivariae approach. For he purpose of he risk conrolling i is imporan o forecas he rae of reurn and is variance over he holding period and o esimae he risk associaed wih holding a paricular asse. By risk conrol manager makes free supplemenary capial for new invesmens. In comparison o he convenional approach, empirical research in risk managemen showed ha he saic "mean-variance" mehodology is very resricive wih unrealisic assumpions. So, mulivariae GARCH(p,q) volailiy modeling is more adequae for he purpose of forecasing of ime-varying condiional firs and second order momens. From MGARCH family models, DVEC(1,1) model wih resricions o posiive-definie covariance marix was used. he complee procedure of analysis is esablished using daily observaions of Pliva and Podravka socks, as he mos frequenly raded socks from CROBEX index a Zagreb Sock Exchange. he analysis shows ha socks wih negaive prediced reurns have no weigh in porfolio regardless invesor risk preference. Socks wih posiive prediced reurns will have relevan weighs in porfolio aking ino accoun profi maximizing wih proposed risk measure, i.e. sandard deviaion. Dynamic porfolio opimizaion for en days forecas is solved as nonlinear programming problem by generalized reduced gradien mehod. Finally, respecing differen invesor risk preference a chosen sandard deviaion level up o % and 3% forecass of opimal porfolio weighs for en days ou of sample are made. Received March 008 Revised June 008 REFERENCES [1] ARNERIĆ, J., JURUN, E. and PIVAC, S.(007.): heoreical Disribuions in Risk Measuring on Sock Marke, Proceedings of he 8h WSEAS Inernaional Conference on Mahemaics & Compuers in Business & Economics, Canada, Vancouver, 194-199. [] BAUWNES, L., LAUREN, S. and ROMBOUS, J. V. K. (006): Mulivariae GARCH Models: A Survey, Journal of Applied Economerics, 1, 79-109. [3] BAUWNES, L. and LAUREN, S. (005.): A New Class of Mulivariae Skew Densiies, wih Applicaion o Generalized Auoregresive Condiional Heeroscedasiciy Models, Journal of Business & Economic Saisics, 3, 346-353. 18
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