Analyzing the Downside Risk of Exchange-Traded Funds: Do the Volatility Estimators Matter?

Transcription

1 Inernaional Journal of Economics and Finance; Vol. 8, No. 1; 016 ISSN X E-ISSN Published by Canadian Cener of Science and Educaion Analyzing he Downside Risk of Exchange-Traded Funds: Do he Volailiy Esimaors Maer? Jying-Nan Wang 1, Lu-Jui Chen, Hung-Chun Liu 3 & Yuan-Teng Hsu 4 1 Deparmen of Applied Economics, Fo Guang Universiy, Taiwan Deparmen of Inernaional Business, Ming Chuan Universiy, Taiwan 3 Deparmen of Finance, Minghsin Universiy of Science & Technology, Taiwan 4 College of Managemen, Yuan Ze Universiy, Taiwan Correspondence: Hung-Chun Liu, Deparmen of Finance, Minghsin Universiy of Science & Technology, No. 1, Xinxing Rd., Xinfeng Hsinchu 30401, Taiwan (R.O.C.). Tel: hungchun65@gmail.com Received: Sepember 0, 015 Acceped: Ocober 8, 015 Online Published: December 5, 015 doi: /ijef.v8n1p1 URL: hp://dx.doi.org/ /ijef.v8n1p1 Absrac This paper aims o propose he augmened GJR-GARCH (GJR-GARCH M ) model ha exends he GJR-GARCH model by comprising overnigh reurns volailiy (ONV), daily high-low prices range (PK), and fear index (VIX) as explanaory variables for he GJR s variance equaion, respecively. The proposed models are used o esimae he daily value-a-risk values and evaluae heir downside risk managemen performance for he SPDRs covering he period from 009 o 014. Empirical resuls show ha he GJR-GARCH M model ouperforms he GJR-GARCH model for mos cases, suggesing ha he GJR-GARCH-based VaR forecass can be moderaely improved wih he addiional informaion embodied in he ONV, PK and VIX volailiy esimaors. In addiion, daily high-low prices range and VIX are far more informaive han he overnigh volailiy esimaor for improving he GJR-GARCH-based VaR forecass. Risk managers can employ he proposed models for esimaing and conroling he poenial loss of ETFs in he face of financial caasrophes. Keywords: exchange-raded funds, SPDRs, value-a-risk, volailiy esimaor, GJR 1. Inroducion Exchange-raded funds (ETFs) are very prevalen and have become popularly adoped invesmen ools among common invesors over recen years. ETFs are aracive as invesmens because hey have many advanages, such as low fee raios, ax efficiency, diversified-porfolio and sock-like characerisics. The American Sock Exchange inroduces he Sandard and Poor s Deposiary Receips (SPDRs, or called Spider) in he early 1990s, which are backed by an equiy porfolio ha closely races he S&P 500 index. By far, he Spider is he mos acively raded and he larges passive ETF worldwide, wih US$15.91 billion marke value as a January, 015. Researchers have long been observed ha volailiy of financial asses reurns are ofen described by several sylized facs, such as ime-varying, clusering, and persisence feaures.the ARCH (auoregressive condiional heeroskedasic, ARCH) model proposed by Engle (198) and he GARCH (generalized auoregressive condiional heeroskedasic, GARCH) model advocaed by Bollerslev (1986) respond o deal wih hese sylized phenomena. Since hen, volailiy forecasing echnique has been dominaed by a variey of he GARCH genre of models, especially for he asymmeric GARCH model. Glosen e al. (1993) propose he so-called GJR-GARCH model which is a simple class of GARCH-ype models which can capure leverage effecs of posive news and negaive news on condiional volailiy. To he bes of our knowledge, a large volume of recen sudies have been invesigaed and wrien abou he value-a-risk (VaR) issue for various financial markes by using GARCH echniques, such as Angelidis e al. (004), Huang and Lin (004), Liu and Hung (010), Orhan and Köksal (01) and So and Yu (006) for sock markes, Al Janabi (006), Bams e al. (005) and So and Yu (006) for foreign exchange rae markes, Chan and Gray (006), Sadeghi and Shavvalpour (006) and Sadorsky (006) for energy markes. However, despie he imporance of VaR on financial risk managemen and he populariy of ETFs for common invesors, here seems o have been relaively lile work endeavored on ETFs. 1

2 Inernaional Journal of Economics and Finance Vol. 8, No. 1; 016 Brooks e al. (000) advocae he overnigh reurn volailiy (ONV) in order o capure accumulaed overnigh informaion ha would be beneficial for capuring he persisence in he condiional heeroscedasiciy of sock reurns. Moivaed by he daily price range, on he one hand, Parkinson (1980) explois he scaled high-low price ranges o consruc he daily PK esimaor based on he assumpion which inraday rading prices follow a random walk process. On he oher hand, Garman and Klass (1980) propose he so-called GK esimaor by including opening and closing rading prices in addiion o price range, wih similar assumpions o he PK. In addiion, Rogers and Sachell (1991) develop he RS volailiy esimaor by considering he drif in he price process. In 1993, Chicago Board Opions Exchange develops he fear index (VIX, or called implied volailiy) ha is acquired from he S&P 500 index opion prices daa via an opion pricing model. In recen years, he wide availabiliy of inraday daa has encouraged researchers o explore heir informaion value in modeling and predicing he volailiy of financial asses, such as Blair e al. (001), Corrado and Truong (007), Fueres e al. (009), Koopman e al. (005), Vipul and Jacob (007) and he reference herein. However, despie a large volume of exising lieraure on volailiy forecasing, none of hem invesigaes he prices informaion which is embodied in he ONV, PK and VIX volailiy esimaors for improving predicive accuracy of daily value-a-risk forecass in ETF. Thus, his paper aims o propose he augmened GJR model ha exends he radiional GJR-GARCH model by comprising hree volailiy esimaors, overnigh volailiy (ONV), daily prices range (PK), and fear index (VIX) as explanaory variables for he variance equaions in GJR model. The proposed models are used o esimae heir daily VaR values and evaluae heir downside risk managemen performance for he SPDRs reurns spanning from 009 o 014. The remainder of his paper is organized as follows. The daa and economeric mehodology are provided in Secion, followed in Secion 3 by he empirical resuls of daily VaR forecass performance for SPDRs across alernaive confidence levels. The final Secion summarizes he conclusions drawn from his paper.. Mehodology.1 Daa and Descripive Analysis The daa analyzed in his paper comprises of he daily open, high, low, and closing prices daa on SPDRs as well as he VIX daa obained from he Yahoo Finance websie. The sample period for hese daily daa covers from January 009 o 31 December 014 for a oal of 1,510 rading days. The firs four years (1,006 observaions) are used as he in-sample period for esimaion purpose, while he remaining wo years (504 observaions) are lef for ou-of-sample forecas evaluaion. Table 1. Descripive saisics of daily reurns for he SPDRs Mean (%) Sd. Min Max Skew Kur J-B Q s(1) * * * * Noe. This able presens he descripive saisics of daily reurns for he Sandard & Poor s Deposiary Receips. J-B represens he saisics of Jarque and Bera (1987) s normal disribuion es. Q s(1) refers o he Ljung-Box Q es saisic of he squared reurn series for up o he 1h order serial correlaion. * indicaes significance a he 1% level. Table 1 provides he descripive saisics of he daily reurns for he Sandard & Poor s Deposiary Receips. As showed in Table 1, he daily average reurn of SPDR is 0.053%, and very closes o zero. The skewness and kurosis of reurns series exhibi significan evidence, meaning ha he reurns have a lef skewness, and he disribuion of he reurns is much fa-ailed and high-peaked han normal disribuion. The Jarque and Bera (1987) es saisic also shows ha he daily SPDR reurns are no normal-disribued. Moreover, he Q s (1) es saisic exhibis linear dependence for he squared reurns and exiss siginifican ARCH effecs.. Augmened GJR Model We propose he augmened GJR model which exends he GJR-GARCH model of Glosen e al. (1993) by including various volailiy esimaors (ONV, PK and VIX), respecively, for is variance equaion as follows: R, z, z ~ NID(0,1) (1) 1 1 d ()

3 Inernaional Journal of Economics and Finance Vol. 8, No. 1; 016 where R is daily SPDRs reurn; is he condiional mean of reurns; represens he innovaion process; z denoes he sandardized residual wih zero mean and uni variance; is he condiional variance. d 1 denoes he indicaor funcion ha akes he value of uniy if 1 0, and 0 oherwise. The indicaor variable differeniaes beween good news and bad news impacs, so ha leverage effecs are capured by. Thus, in he augmened GJR model, good news has an impac of, and bad news has an impac of ( ), wih bad (good) news having a greaer shock on volailiy if 0 ( 0 ). Finally, denoes a volailiy esimaor made a 1 day 1, including ONV (overnigh volailiy), PK (daily high-low price range), and VIX (fear index). Table provides a synopsis of hese volailiy esimaors: Table. The synopsis of various volailiy esimaors Abbreviaion of volailiy esimaors Sudies Formula or explanaion ONV Brooks e al. (000) ˆ (ln( O / C )) (3) PK Parkinson (1980) VIX - Noe. This able presens he various volailiy esimaors employed in his paper. closing prices a day, respecively. ONV, 1 ˆ (4ln ) (ln(h / L )) (4) 1 PK, VIX is a popular measure of he fear index of S&P 500 index opions, which represens one measure of he marke's expecaion of sock marke volailiy over he nex 30 day period. For consisen scaling wih ONV and PK, he VIXs are squared and divided by 5. O, H, L and C denoe he opening, high, low, and.3 Downside Risk Measuremen and Performance Evaluaion The GJR-based VaR forecass for a one-day holding period can be calculaed as follows: VaR Z ˆ (5) Where Z a1 denoes he corresponding quanile of he sandard normal disribuion a a 1, while ˆ is he volailiy forecas generaed from eiher GJR, GJR-ONV, GJR-PK or GJR-VIX model. In order o backesing, his paper uses a likelihood-raio es of Kupiec (1995) o examine wheher he rue failure rae is saisically consisen wih heoreical failure rae of he VaR model. The null hypohesis of he failure rae P is esed agains he alernaive hypohesis ha he failure rae is differen from P. The LR uc saisic can be formulaed as follows: n1 n 0 ˆ (1 ˆ ) LR uc ln ~ (1) n (6) 0 P(1 P) where ˆ n1 /( n0 n1 ) denoes he maximum likelihood esimae of P, and n 1 is a Bernoulli random variable ha represens he oal number of VaR violaions. (Noe 1) Chrisoffersen (1998) consrucs he sophisicaed condiional coverage es (LR cc ) which joinly examines wheher he oal number of violaions is equal o he expeced one, and he VaR violaions are independenly disribued. Given he realizaions of he SPDRs reurns series R and he se of VaR esimaes, he indicaor variable I can be formulaed as follows: 1,if R VaR I (7) 0,if R VaR Since accurae VaR esimaes exhibi he propery of correc condiional coverage, he I series mus display boh correc uncondiional coverage and serial independence. The LR cc es is a join es of hese wo properies, and he relaed es saisic is LR cc = LR uc + LR ind as we condiion on he firs observaion. Thus, under he null hypohesis ha he failure process is independen and he expeced proporion of violaions equals P, he corresponding likelihood raio can be formulaed as follows: 1 3

4 Inernaional Journal of Economics and Finance Vol. 8, No. 1; 016 (1 P) P n0 n1 LR cc ln ~ () n00 n01 n10 n11 (1 ˆ ˆ ˆ ˆ 01) 01 (1 11 ) 11 where n i, j he number of observaions wih value i followed by value j (i, j 0, 1), P{ I j I i}( i, j 0,1), ˆ 01 n01 /( n00 n01), ˆ 11 n11 /( n10 n11 ). ij 1 3. Empirical Resuls and Analysis Table 3 presens ou-of-sample daily VaR forecass performance across he various models by reporing mean VaR, violaion, failure prob., LR uc and LR cc saisics, under 90%, 95% and 99% confidence levels. As shown in Table 3, he GJR model generaes he highes average absolue VaR esimaes a every confidence level, and followed by he GJR-ONV, GJR-PK and GJR-VIX models. Thus, he GJR and he GJR-VIX models generae he lowes and highes numbers of VaR violaions, respecively. Panel A of Table 3 provides daily VaR forecass resuls for SPDRs a he 90% confidence level. We observe ha eiher he GJR or he GJR-ONV model fails o pass he uncondiional coverage es (LR uc ), indicaing ha boh radiional GJR and GJR-ONV models end o over-predic VaR values for SPDRs reurns. Moreover, he GJR model has been rejeced by he condiional coverage es (LR cc ), indicaing ha clusered violaions were generaed. Tha is, he GJR model is very slow a updaing he VaR value when marke volailiy changes rapidly. By conras, boh he GJR-PK and he GJR-VIX models pass he coverage ess, suggesing ha he empirical failure probabiliy is saisically consisen wih he prescribed one for each of hem, especially for he laer model. Meanwhile, wih any sudden change in marke volailiy, he GJR-PK and he GJR-VIX models are beneficial for rapidly updaing he VaR value. Thus, he rading prices informaion which is implied in PK and VIX volailiy measures is crucial for producing adequae daily VaR forecass for SPDRs reurns a he 90% confidence level. For he case of 95% confidence level, we observe ha he LR uc es saisic is insignifican for he GJR, GJR-ONV and GJR-PK models, indicaing ha he sample poin esimae is saisically consisen wih he prescribed confidence level of hese hree VaR models. The LR cc saisic furher shows ha he aforesaid hree models also can pass he condiional coverage es, indicaing ha hese models performance is quie sable over ime during he ou-of-sample forecasing period 013~014. However, he GJR-VIX model fails o offer adequae VaR forecass according o he LR uc es saisic. The VaR forecass resuls a he 99% confidence level are very similar o hose obained a he 95% confidence level. Tha is, he LR uc and LR cc saisics repored in Panel C of Table 3 are all insignifican, excep for he GJR-VIX model, indicaing ha he GJR, GJR-ONV and GJR-PK models are able o produce adequae Daily VaR forecass for SPDRs reurns. (8) Table 3. Daily VaR forecass resuls Model Mean VaR Violaion Failure prob. LR uc LR cc Panel A: 90% Confidence Level GJR % GJR-ONV % GJR-PK % GJR-VIX % Panel B: 95% Confidence Level GJR % GJR-ONV % GJR-PK % GJR-VIX %.9.95 Panel C: 99% Confidence Level GJR % GJR-ONV % GJR-PK % GJR-VIX % Noe. This able presens daily VaR forecass resuls for SPDRs a hree confidence levels. The criical values of he LR uc and LR cc saisics a he 10% significance level are.71 and 4.61, respecively. Figures in bold ex indicae rejecion of he null hypohesis of correc VaR esimaes a he 10% significance level. 4

5 Inernaional Journal of Economics and Finance Vol. 8, No. 1; Conclusions This paper proposes he augmened GJR-GARCH model ha exends he GJR-GARCH model of Glosen e al. (1993) by respecively comprising overnigh volailiy, daily high-low prices range, and fear index as explanaory variables for he GJR-GARCH s variance equaion. These VaR models are used o esimae heir daily VaR values and evaluae heir downside risk managemen performance for he SPDRs reurns under 90%, 95% and 99% confidence levels. Empirical resuls show ha he augmened GJR model ouperforms he GJR model for mos cases, suggesing ha he GJR-GARCH-based VaR forecass can be moderaely improved wih he addiional informaion embodied in he ONV, PK and VIX volailiy esimaors. In addiion, daily high-low prices range and fear index are far more informaive han he overnigh volailiy for improving he GJR-GARCH-based VaR forecass. Risk managers can employ he proposed VaR models for esimaing and conroling he poenial loss of ETFs in he face of financial caasrophes. References Al Janabi, M. A. M. (006). Foreign-exchange rading risk managemen wih value a risk: Case analysis of he Moroccan marke. Journal of Risk Finance, 7(3), hp://dx.doi.org/ / Ana-Maria, F., Izzeldin, M., & Kaloychou, E. (009). On forecasing daily sock volailiy: The role of inraday informaion and marke condiions. Inernaional Journal of Forecasing, 5(), hp://dx.doi.org/ /j.ijforecas Angelidis, T., Benosa, A., & Degiannakis, S. (004). The use of GARCH models in VaR esimaion. Saisical Mehodology, 1, hp://dx.doi.org/ /j.same Bams, D., Lehner, T., & Wolff, C. (005). An evaluaion framework for alernaive VaR-models. Journal of Inernaional Money and Finance, 4, hp://dx.doi.org/ /j.jimonfin Blair, B. J., Poon, S. H., & Taylor, S. J. (001). Forecasing S&P 100 volailiy: The incremenal informaion conen of implied volailiies and high frequency reurns. Journal of Economerics, 105, 5-6. hp://dx.doi.org/ /s (01) Bollerslev, T. (1986). Generalized auoregressive condiional heeroskedasiciy. Journal of Economerics, 31(3), hp://dx.doi.org/ / (86) Brooks, C., Clare, A. D., & Persand, G. (000). A word of cauion on calculaing marke-based minimum capial risk requiremens. Journal of Banking and Finance, 4, hp://dx.doi.org/ /s (99) Chan, K. F., & Gray, P. (006). Using exreme value heory o measure value-a-risk for daily elecriciy spo prices. Inernaional Journal of Forecasing,, hp://doi: /j.ijforecas Chrisoffersen, P. F. (1998). Evaluaing inerval forecass. Inernaional Economic Review, 39, hp://dx.doi.org/10.307/57341 Corrado, C., & Truong, C. (007). Forecasing sock index volailiy: Comparing implied volailiy and he inraday high-low price range. Journal of Financial Research, 30(), hp://dx.doi.org/ /j x Engle, R. F. (198). Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kindom inflaion. Economerica, 50(4), hp://dx.doi.org/10.307/ Garman, M., & Klass, M. (1980). On he esimaion of securiy price volailiies from hisorical daa. Journal of Business, 53(1), hp://dx.doi.org/ /9607 Glosen, L., Jagannahan, R., & Runkle, D. (1993). On he relaion beween he expeced value and he volailiy nominal excess reurn on socks. Journal of Finance, 46, hp://dx.doi.org/ /j b0518.x Huang, Y. C., & Lin, B. J. (004). Value-a-Risk analysis for Taiwan sock index fuures: Fa ails and condiional asymmeries in reurn innovaions. Review of Quaniaive Finance and Accouning,, hp://dx.doi.org/10.103/b:requ a9 Jarque, C. M., & Bera, A. K. (1987). A es for normaliy of observaions and regression residuals. Inernaional Saisics Review, 55, hp://dx.doi.org/10.307/ Koopman, S., Jungbacker, B., & Hol, E. (005). Forecasing daily variabiliy of he S&P 100 sock index using hisorical, realised and implied volailiy measuremens. Journal of Empirical Finance, 1(3),

6 Inernaional Journal of Economics and Finance Vol. 8, No. 1; 016 hp://dx.doi.org/ /j.jempfin Kupiec, P. (1995). Techniques for verifying he accuracy of risk measuremen models. Journal of Derivaives, 3, hp://dx.doi.org/ /jod Liu, H. C., & Hung, J. C. (010). Forecasing S&P-100 sock index volailiy: The role of volailiy asymmery and disribuional assumpion in GARCH models. Exper Sysems wih Applicaions, 37(7), hp://dx.doi.org/ /j.eswa Orhan, M., & Köksal, B. (01). A comparison of GARCH models for VaR esimaion. Exper Sysems wih Applicaions, 39(3), hp://dx.doi.org/ /j.eswa Parkinson, M. (1980). The exreme value mehod for esimaing he variance of he rae of reurn. Journal of Business, 53(1), hp://dx.doi.org/ /96071 Rogers, L. C. G., & Sachell, S. E. (1991). Esimaing variance from high, low and closing prices. Annals of Applied Probabiliy, 1(4), hp://dx.doi.org/10.114/aoap/ Sadeghi, M., & Shavvalpour, S. (006). Energy risk managemen and Value a Risk modeling. Energy Policy, 34, hp://dx.doi.org/ /j.enpol Sadorsky, P. (006). Modeling and forecasing peroleum fuures volailiy. Energy Economics, 8, hp://dx.doi.org/ /j.eneco So, M. K. P., & Yu, P. L. H. (006). Empirical analysis of GARCH models in Value a Risk esimaion. Journal of Inernaional Financial Markes, Insiuions & Money, 16, hp://dx.doi.org/ /j.infin Vipul, & Jacob, J. (007). Forecasing performance of exreme-value volailiy esimaors. Journal of Fuures Markes, 7(11), hp://dx.doi.org/10.100/fu.083 Noe Noe 1. If he forecased VaR can no cover he realized dollar loss, his is defined as a violaion. Copyrighs Copyrigh for his aricle is reained by he auhor(s), wih firs publicaion righs graned o he journal. This is an open-access aricle disribued under he erms and condiions of he Creaive Commons Aribuion license (hp://creaivecommons.org/licenses/by/3.0/). 6