The volatility modelling and bond fund price time series forecasting of VUB bank: Statistical approach

Size: px

Start display at page:

Download "The volatility modelling and bond fund price time series forecasting of VUB bank: Statistical approach"

Conrad Morris Boone
5 years ago
Views:

1 8 h Inernaonal scenfc conference Fnancal managemen of frms and fnancal nsuons Osrava VŠB-TU Osrava, faculy of economcs, fnance deparmen 6 h 7 h Sepember The volaly modellng and bond fund prce me seres forecasng of VUB bank: Sascal approach Dusan Marcek Absrac The arcle descrbes he developmen and he use of he ARCH-GARCH (AuoRegressve Condonally Heeroskedasc) ype models. We provde nformaon abou he examnaon of he ARCH-GARCH models for he forecasng of he bond prce me seres provded by he VUB bank and compare of he forecas accuracy wh he models based on he resduals analyss of he developed models.. We show some new aspecs of me seres analyss whch may mprove he forecasng ably of sascal models so ha her can beer capure he characerscs and dynamcs of a parcular me seres and promse beer goodness of f. Key words Tme seres, classes ARCH-GARCH models, volaly, forecasng, forecas accuracy. JEL Classfcaon: C3, G3.. Inroducon Ths paper dscusses and compares he forecass from volaly models whch are derved from sascal heory. The am of he paper s o provde nformaon abou he examnaon of he ARCH-GARCH (AuoRegressve Condonally Heeroskedasc) models for he forecasng of he volaly and bond prce me seres provded by he VUB bank and comparsons of he forecas accuracy wh he models based on he resduals analyss of he developed models. Volaly s an mporan facor n asses radng. By volaly we mean he condonal sandard devaon of he underlyng asse prce. Volaly has many fnancal applcaons. For example volaly modellng provdes a smple approach for calculang value a rsk of a fnancal poson n rsk managemen. I also plays an mporan role n asse allocaon under he mean-varance framework. To model he volaly for he commercal VUB bank of he Slovak Republc by means of hgh sascal heory, we developed several ARCH/GARCH models. In he basc ARCH/GARCH model economc shocks have no effecs on condonal volaly. However, a sylzed fac of fnancal volaly s ha bad news (negave shocks) end o have a larger mpac on volaly han good news (posve shocks). For hs reason we sudy also EGARCH/PGARCH (Exponenal/Power) models whch allow for leverage effecs and exhausve sudy of he underlyng dynamcs. The sudy was performed for he daa avalable a hp:// The daa s also lsed a hp://fra.fr.unza.sk/fles/daa_vub. The paper s organzed as follows. In Secon we brefly descrbe he basc ARCH- GARCH model and s varans: EGARCH, PGARCH models. In Secon 3 we presen he Dusan Marcek- Unversy of Zlna, Deparmen of Macro and Mcro Economcs, Unverzna, 6, Zlna, Slovak Republc, e-mal: dusan.marcek@fr.unza.sk; Slesan Unversy Opava, Faculy of Phlosophy and Scence, Czech Republc, e-mal: dusan.marcek@fpf.slu.

2 8 h Inernaonal scenfc conference Fnancal managemen of frms and fnancal nsuons Osrava VŠB-TU Osrava, faculy of economcs, fnance deparmen 6 h 7 h Sepember daa, conduc some prelmnary analyss of he me seres and demonsrae he forecasng ables of ARCH-GARCH modes of an applcaon. In Secon 4 oher sascal ools for mprovng of he forecasng ably of ARCH-GARCH models are presened and we pu an emprcal comparson. Secon 6 brefly concludes.. Some ARCH-GARCH Models for Fnancal Daa ARCH-GARCH models are desgned o capure ceran characerscs ha are commonly assocaed wh fnancal me seres. They are among ohers: fa als, volaly cluserng, perssence, mean-reverson and leverage effec. As far as fa als, s well know ha he dsrbuon of many hgh frequency fnancal me seres usually have faer als han a normal dsrbuon. The phenomena of faer als s also called excess kuross. In addon, fnancal me seres usually exhb a characerscs known as volaly cluserng n whch large changes end o follow large changes, and small changes end o follow small changes. Volaly s ofen perssen, or has a long memory f he curren level of volaly affecs he fuure level for more me perods ahead. Alhough fnancal me seres can exhb excessve volaly from me o me, volaly wll evenually sele down o a long run level. The leverage effec expresses he asymmerc mpac of posve and negave changes n fnancal me seres. I means ha he negave shocks n prce nfluence he volaly dfferenly han he posve shocks a he same sze. Ths effec appears as a form of negave correlaon beween he changes n prces and he changes n volaly. The frs model ha provdes a sysemac framework for volaly modellng s he ARCH model of Engle (98). Bollerslev (986) proposes a useful exenson of Engle s ARCH model known as he generalzed ARCH (GARCH) model for me sequence { y } n he followng form y = ν h, m h = α + α y + β h j = s j= j where { v } s a sequence of d random varables wh zero mean and un varance. α a β are he ARCH and GARCH parameers, h represen he condonal varance of me seres condonal on all he nformaon o me -, I -. In he leraure several varans of basc GARCH model () has been derved. In he basc GARCH model () f only squared resduals ener he equaon, he sgns of he resduals or shocks have no effecs on condonal volaly. However, a sylzed fac of fnancal volaly s ha bad news (negave shocks) ends o have a larger mpac on volaly han good news (posve shocks). Nelson (99) proposed he followng exponenal GARCH model abbrevaed as EGARCH o allow for leverage effecs n he form p + γ q log h = α + α + β h () j j = σ j= Noe f s posve or here s good news, he oal effec of () s ( + γ ). However conrary o he good news,.e. f s negave or here s bad news, he oal effec of s ( ) γ. Bad news can have a larger mpac on he volaly. Then he value of γ would be expeced o be negave (see Zvo and Wang (5, p. 4)).

3 8 h Inernaonal scenfc conference Fnancal managemen of frms and fnancal nsuons Osrava VŠB-TU Osrava, faculy of economcs, fnance deparmen 6 h 7 h Sepember The basc GARCH model can be exended o allow for leverage effecs. Ths s performed by reang he basc GARCH model as a specal case of he power GARCH (PGARCH) model proposed by Dng, Granger and Engle (993): p d + α + γ ) + = q d σ = α ( β σ (3) j= j d j where d s a posve exponen, and γ denoes he coeffcen of leverage effecs (see Zvo and Wang (5, p. 43)). Anoher ARCH-GARCH models as he ARCH-GARCH regresson and ARCH-GARCH mean model can be found n Marcek, (, p. 55 and 56). 3. An Applcaon of ARCH-GARCH Models We llusrae he ARCH-GARCH mehodology on he developng a forecas model. The daa s aken from he commercal VUB bank of he Slovak Republc and are avalable a hp:// The daa s also lsed a hp://fra.fr.unza.sk/fles/daa_vub. The daa conss of daly observaons of he prce me seres for he bond fund of VUB (BPSVUB). The daa was colleced for he perod May 7, 4 o February 8, 8 whch provded of 954 observaons (see Fgure and 4). To buld a forecas model he sample perod (ranng daa se denoed Α ) for analyss r,..., r 9 was defned,.e. he perod over whch he forecasng model was developed and he ex pos forecas perod (valdaon daa se denoed as Ε ) r 9,..., r 954 as he me perod from he frs observaon afer he end of he sample perod o he mos recen observaon. By usng only he acual and forecas values whn he ex pos forecasng perod only, he accuracy of he model can be calculaed. Inpu selecon s crucal mporance o he successful developmen of an ARCH-GARCH model. Poenal npus were chosen based on radonal sascal analyss: hese ncluded he raw BPSVUB and lags hereof. The relevan lag srucure of poenal npus was analysed usng radonal sascal ools,.e. usng he auocorrelaon funcon (ACF), paral auocorrelaon funcon (PCF) and he Akake/Bayesan nformaon creron (AIC/BIC): we looked o deermne he maxmum lag for whch he PACF coeffcen was sascally sgnfcan and he lag gven he mnmum AIC. Accordng o hese crerons he ARMA(5) model was specfed as r = ξ + φ r + φ r + φ r + φ r + φ r (4) where ξ, φ, φ,..., φ are unknown parameers of he model, s ndependen random 5 varable drawn from sable probably dsrbuon wh mean zero and varance σ. As we menoned early, hgh frequency fnancal daa, lke our BPSVUB, reflec a sylzed fac of changng varance over me. An approprae model ha would accoun for condonal heeroscedascy should be able o remove possble nonlnear paern n he daa. Varous procedures are avalable o es an exsence of ARCH or GARCH. A commonly used es s he LM (Lagrange mulpler) es. The LM es assumes he null hypohess H : α = α =... = α = ha here s no ARCH. The LM sascs has an asympoc χ p dsrbuon wh p degrees of freedom under he null hypohess. For calculang he LM sascs see for example, Bollerslev (986 Eqs. (7) and (8)). The LM es performed on he BPSVUB ndcaes presence of auoregressve condonal heeroscedascy. For esmaon he parameers of an ARCH or GARCH model he maxmum lkelhood procedure was used. The quanfcaon of he model was performed by means sofware R.6. a hp://cran.rprojec.org and resuled no he followng mean equaon:

4 8 h Inernaonal scenfc conference Fnancal managemen of frms and fnancal nsuons Osrava VŠB-TU Osrava, faculy of economcs, fnance deparmen 6 h 7 h Sepember r + e = r r +.75r +.58r r and varance equaon 8 = e h (5) h where e are esmaed resduals of from Eq. (4). Fnally o es for nonlnear paerns n prce bond me seres he fed sandardzed resduals ˆ = e / h were subjeced o he BDS es. The BDS es (a dmensons Ν =, 3, and olerance dsances =.5,.,.5,.) fnds no evdence of nonlneary n sandardzed resduals of he BPSVUB. The fed vs. acual values are graphcally dsplayed by means sofware Evews (hp:// n Fgure The volaly was esmaed by means sofware R.6. and s dsplayed n Fgure. Fgure : Acual and fed values of he VUB fund: ARMA(5,)+GARCH(,) ARMA(5,)+GARCH(,) Resdual Acual Fed ARCH-GARCH model (5). Resduals are a he boom. Acual me seres represens he sold lne, he fed vales represens he doed lne Fgure : The esmaed volaly for ARMA(5,)+GARCH(,) process, model (5) Condonal SD x Index 4. Oher sascal ools for mprovng of he forecasng ably of ARCH-GARCH models In many cases, he basc GARCH model wh normal Gaussan error dsrbuon () provdes a reasonably good model for analyzng fnancal me seres and esmang condonal volaly. However, here are some aspecs of he model whch can be mproved

5 8 h Inernaonal scenfc conference Fnancal managemen of frms and fnancal nsuons Osrava VŠB-TU Osrava, faculy of economcs, fnance deparmen 6 h 7 h Sepember so ha can beer capure he characerscs and dynamcs of a parcular me seres. For hs purpose he Quanle-Quanle (QQ) plos are used. For example, he R sysem (hp://cran.r-projec.org/) asss n performng resdual analyss (compues he Gaussan, sudensed and generalzed resduals wh generalzed error dsrbuon GED). In Fgure 3 he QQ-plo reveals ha he normaly assumpon of he resduals may no be approprae. A comparson of QQ-plos n fgure 3 shows ha GED dsrbuon promse beer goodness of f. Ths s confrmed by AIC and BIC crerons and Lkelhood funcon dsplayed n Table. The GED error dsrbuon provdes he bes f because AIC and BIC crerons are he smalles. Model model.n (Gaussan) model. (sudensed) model.g ED AIC creron BIC creron Lkelhood func Table : AIC, BIC and lkelhood funcon for varous ypes error dsrbuon (model (4)) Fnally, for cachng he leverage effec, he model ARMA(5,)+EGARCH(,) was esmaed. The coeffcen for leverage effec γ from equaon () s sascal sgnfcan and equals , and s negave whch means ha bad news have larger mpac o volaly. If we compare he esmaed volaly n Fgure wh he VUB fund n Fgure, we can see ha n perod of depresson he leverage effecs and he bad news cause he asymmerc jump n he volaly. Fgure 3: QQ-plo of Gaussan sandard resduals (lef), sudensed (mddle) and generalsed(ged) (rgh) QQ-Plo of Sandardzed Resduals QQ-Plo of Sandardzed Resduals QQ-Plo of Sandardzed Resduals 6 4 QQ-Plo 39 5 QQ-Plo QQ-Plo 39 Sandardzed Resduals - Sandardzed Resduals Sandardzed Resduals Quanles of gaussan dsrbuon Quanles of dsrbuon Quanles of normal dsrbuon As we menoned above, he esmaon of EGARCH and PGARCH models has showed he presence of leverage effecs. The assumpon of normal error dsrbuon s also volaed because he alernave error dsrbuons provde beer goodness of her f. These fndngs ndcae he chance of ganng beer resuls n forecasng wh usng some of hese models. Our suspcon was confrmed by compung he sascal summary measure of he model s forecas RMSE. As we can see n Table he smalles errors have jus he GARCH wh GED dsrbuon. Model AR(5)+ GARCH(,) AR(5)+ EGARCH(,) AR(5)+ PGARCH(,)

6 8 h Inernaonal scenfc conference Fnancal managemen of frms and fnancal nsuons Osrava VŠB-TU Osrava, faculy of economcs, fnance deparmen 6 h 7 h Sepember Dsrbuon Gaussan dsrbuon GEDdsrbuon Table Ex pos forecas RMSEs for varous exensons of GARCH models and granular RBF NN. See ex for deals. Afer hese fndngs we can make predcons for nex 54 radng days usng he model wh he smalles RMSE,. e. by he ARMA(5,) + GARCH(,) wh GED error dsrbuon. These predcons are calculaed by means sofware Evews (hp:// and shoved n Fgure 4. Fgure 4: Acual (sold) and forecas (doed) values of he VUB fund.33 ARMA(5,)+GARCH(,) wh GED dsrbuon SERIES SERIES_F Acknowledgemen Arcle has been made n connecon wh projec IT4Innovaons Cenre of Excellence, reg. no. CZ..5/../.7 suppored by Research and Developmen for Innovaons Operaonal Programme fnanced by Srucural Founds of Europe Unon and from he means of sae budge of he Czech Republc. The work of hs conrbuon was also suppored by he gran GACR P4//7.

7 8 h Inernaonal scenfc conference Fnancal managemen of frms and fnancal nsuons Osrava VŠB-TU Osrava, faculy of economcs, fnance deparmen 6 h 7 h Sepember References [] BOLLERSLEV, D Generalzed Auoregressve Condonal Heeroscedascy, Journal of Economercs 3: [] ENGLE, R.F. 98. Auoregressve Condonal Heeroscedascy wh Esmaes of he Varance of Uned Kngdom Inflaon, Economerca 5 (4): [3] NELSON, D.B. 99. Condonal Heeroskedascy n Asse Reurns: a New Approach, Economerca 59 (): [4] ZIVOT, E., WANG, J. 5. Modelng Fnancal Tme Seres wh S-PLUS, NY: Sprnger Verlag. Summary Článek opsuje vývoj a aplkac ARCH-GARCH modelů. Poskyuje návod na vývoj ARCH-GARCH modelů pro predkc dluhopsů VUB banky a porovnává predkční přesnos s modely rozšířené o analýzu rezíduí z vyvnuých modelů. Modely založené na analýze rezíduí jsou schopné lépe zachy vývojovou dynamku časových řad a ím vylepšují predkční schopnos.

RELATIONSHIP BETWEEN VOLATILITY AND TRADING VOLUME: THE CASE OF HSI STOCK RETURNS DATA

RELATIONSHIP BETWEEN VOLATILITY AND TRADING VOLUME: THE CASE OF HSI STOCK RETURNS DATA Mchaela Chocholaá Unversy of Economcs Braslava, Slovaka Inroducon (1) one of he characersc feaures of sock reurns