Proceedings of he Firs European Academic Research Conference on Global Business, Economics, Finance and Social Sciences (EAR5Ialy Conference) ISBN: 978--6345-028-6 Milan-Ialy, June 30-July -2, 205, Paper ID: I53 Measuring he Downside Risk of he Exchange-Traded Funds: Do he Volailiy Esimaors Maer? Hung-Chun Liu, Deparmen of Finance, Minghsin Universiy of Science & Technology, Taiwan. E-mail: hungchun65@gmail.com Jying-Nan Wang, Deparmen of Applied Economics, Fo Guang Universiy, Taiwan. E-mail: jyingnan@gmail.com Yuan-Teng Hsu, College of Managemen, Yuan Ze Universiy, Taiwan. E-mail: yuaneng.hsu@gmail.com Absrac This paper aims o propose he augmened GJR-GARCH (GJR-GARCH M) model which exends he GJR-GARCH model by including hree volailiy esimaors, overnigh volailiy (ONV), daily prices range (PK), and implied volailiy (VIX) as explanaory variables for he variance equaions in GJR. The proposed value-a-risk (VaR) models are used o esimae he daily VaR values and evaluae heir downside risk managemen performance for he SPDRs over he period from 2009 o 204. Empirical resuls indicae 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 conained in ONV, PK and VIX. In addiion, daily prices range and implied volailiy are far more informaive han he overnigh volailiy esimaor for improving he GJR-GARCH-based VaR forecass. Marke praciioners can adop he proposed model for esimaing and managing he poenial loss of ETFs in he face of caasrophic evens. Key words: Exchange-raded funds, SPDRs, Value-a-Risk, Volailiy esimaor, GJR JEL Classificaion: C52; C53; G32
Proceedings of he Firs European Academic Research Conference on Global Business, Economics, Finance and Social Sciences (EAR5Ialy Conference) ISBN: 978--6345-028-6 Milan-Ialy, June 30-July -2, 205, Paper ID: I53. Inroducion Exchange-raded funds (ETFs) are very popular and have become exensively adoped invesmen insrumens among global invesors over recen years. ETFs are aracive as invesmens because of heir low expense raios, ax efficiency, diversified-porfolio and sock-like feaures. In he early 990s, he American Sock Exchange (AMEX) inroduced Sandard & Poor s Deposiary Receips (SPDRs, or Spider), which are backed by a sock porfolio ha closely racks he S&P 500 index. By far, he Spider is he mos acively raded and he larges passive ETF worldwide, wih US$25.9 billion under managemen as a January, 205. Researchers have long been aware ha reurns volailiy changes over ime and ha period of high volailiy end o be found in clusers. The auoregressive condiional heeroskedasic (ARCH) model of Engle (982) and he generalized auoregressive condiional heeroskedasic (GARCH) model advocaed by Bollerslev (986) respond o address 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. (993) propose he so-called GJR-GARCH model which is a simple class of GARCH-ype models ha can capure asymmeric effecs of good news and bad news on condiional volailiy. Brooks e al. (2000) propose he overnigh volailiy (ONV) in order o capure accumulaed overnigh informaion which would be useful for capuring he persisence in he condiional heeroscedasiciy of sock reurns. Moivaed by he daily price range, on he one hand, Parkinson (980) uses he scaled high-low price ranges o develop he daily PK volailiy esimaor based on he assumpion ha inraday prices follow a Brownian moion process. On he oher hand, Garman and Klass (980) develop he GK esimaor by using opening and closing prices in addiion o price range, wih assumpions similar o hose of he PK esimaor. In addiion, Rogers and Sachell (99) propose an esimaor, RS, by including he drif in he price process. In 993, Chicago Board Opions Exchange (CBOE) proposes he implied volailiy index (VIX) which is derived from he S&P 500 index opion prices daa wih an opion pricing model. Recenly, he broad availabiliy of inraday rading daa has inspired researchers o explore heir informaion value in modeling and forecasing he volailiy of financial markes (Blair e al., 200; Koopman e al., 2005; Corrado and Truong, 2007; Vipul and Jacob, 2007; Fueres e al., 2009). However, despie an exensive 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. This sudy aims o propose he augmened GJR model which exends he radiional GJR- 2
Proceedings of he Firs European Academic Research Conference on Global Business, Economics, Finance and Social Sciences (EAR5Ialy Conference) ISBN: 978--6345-028-6 Milan-Ialy, June 30-July -2, 205, Paper ID: I53 GARCH model by including hree volailiy esimaors, overnigh volailiy (ONV), daily prices range (PK), and implied volailiy (VIX) as explanaory variables for he variance equaions in GJR model. The proposed value-a-risk (VaR) models are used o esimae heir daily VaR values and evaluae heir downside risk managemen performance for he SPDRs reurns spanning from 2009 o 204. The remainder of his sudy is organized as follows. The daa and mehodology are provided in Secion 2, followed in Secion 3 by he empirical resuls of daily VaR forecass performance for SPDRs across alernaive confidence levels. Conclusions drawn from his sudy are summarized in he final Secion. 2. Mehodology 2. Daa and Preliminary Analysis The daa examined in his sudy 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 2 January 2009 o 3 December 204 for a oal of,50 rading days. The firs four years (,006 observaions) are used as he in-sample period for esimaion purpose, while he remaining wo years (504 observaions) are aken as he ouof-sample for forecas evaluaion. Table provides he descripive saisics of he daily reurns for he Sandard & Poor s Deposiary Receips. As showed in Table, he average daily reurn is posiive, and approaches close o zero. The reurns series exhibis significan evidence of skewness and kurosis, which means ha he series is skewed o he lef, and he disribuion of he daily reurns is more fa-ailed and high-peaked han normal disribuion. The J-B es saisic furher confirms ha he daily reurns are non-normal disribued. Finally, he Ljung-Box es saisic displays linear dependence for he squared reurns and srong ARCH effecs. Table : Descripive saisics of daily reurns for he SPDRs 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 (987) s normal disribuion es. Q s (2) refers o he Ljung- Box Q es saisic of he squared reurn series for up o he 2h order serial correlaion. * indicaes significance a he % level. Mean (%) Sd. Min Max Skew Kur J-B Q s(2) 0.053.42-6.734 6.960-0.270 * 4.484 * 283.09 * 796.50 * 2.2 Augmened GJR Model We propose he augmened GJR model which exends he GJR-GARCH model of Glosen e al. (993) by including various volailiy esimaors (ONV, PK and VIX), respecively, for is variance equaion as follows: 3
Proceedings of he Firs European Academic Research Conference on Global Business, Economics, Finance and Social Sciences (EAR5Ialy Conference) ISBN: 978--6345-028-6 Milan-Ialy, June 30-July -2, 205, Paper ID: I53 R z, z ~ NID(0,) (), 2 2 2 2 d (2) Where R is daily SPDRs reurn; denoes he condiional mean of reurns; is he 2 innovaion process; z is he sandardized residual wih zero mean and uni variance; is he condiional variance. d denoes he indicaor funcion ha akes he value of uniy if, and 0 oherwise. The indicaor variable differeniaes beween posiive (good news) 0 and negaive (bad news) shocks, so ha asymmeric effecs in he daa 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 effec on volailiy if 0 ( 0 ). Finally, is a volailiy esimaor made a day, including ONV (overnigh volailiy), PK (daily PK price range), and VIX (implied volailiy). Table 2 provides a synopsis of hese volailiy esimaors: Table 2: The synopsis of various volailiy esimaors This able presens he various volailiy esimaors employed in his sudy. O, H, L and C denoe he opening, high, low, and closing prices a day, respecively. Abbreviaion of Sudies Formula or explanaion volailiy esimaors 2 2 ONV Brooks e al. (2000) ˆ (ln( O / C )) (3) PK Parkinson (980) VIX - ONV, ˆ (4ln 2) (ln(h / L )) (4) 2 2 PK, 2.3 Downside Risk Measuremen and Performance Evaluaion VIX is a popular measure of he implied volailiy 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 oher daily volailiy esimaors, all VIX indexes are squared and divided by 252. The GJR-based VaR forecass for a one-day holding period can be calculaed as follows: VaR Z ˆ (5) Where Z denoes he corresponding quanile of he sandard normal disribuion a, while ˆ is he volailiy forecas generaed from eiher GJR, GJR-ONV, GJR-PK or GJR- VIX model. To backes he VaR resuls, his sudy firs employs a likelihood-raio es by Kupiec (995) o es wheher he rue failure rae is saisically consisen wih he VaR model s heoreical failure rae. The null hypohesis of he failure rae P is esed agains he alernaive hypohesis ha he failure rae is differen from P, in which saisics is given by: 4
Proceedings of he Firs European Academic Research Conference on Global Business, Economics, Finance and Social Sciences (EAR5Ialy Conference) ISBN: 978--6345-028-6 Milan-Ialy, June 30-July -2, 205, Paper ID: I53 n n 0 ˆ ( ˆ ) 2 2ln ~ () P( P) LR uc n0 Where ˆ n /( n0 n ) is he maximum likelihood esimae of P, and n denoes a Bernoulli random variable represening he oal number of VaR violaions. Chrisoffersen (998) developed a condiional coverage es (LR cc) ha joinly invesigaes wheher he oal number of failures 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 defined as follows:,if R VaR I (7) 0,if R VaR Since accurae VaR esimaes display he propery of correc condiional coverage, he I series mus exhibi boh correc uncondiional coverage and serial independence. The LR cc es is a join es of hese wo properies, and he corresponding es saisics is LR cc = LR uc + LR ind as we condiion on he firs observaion. Consequenly, under he null hypohesis ha he failure process is independen and he expeced proporion of violaions equals P, he appropriae likelihood raio is represened as follows: ( P) P n0 n 2 LR cc 2ln ~ (2) n00 n0 n0 n ( ˆ ˆ ˆ ˆ 0) 0 ( ) Where n i, j he number of observaions wih value i followed by value j (i, j 0, ), P{ I j I i}( i, j 0,), ˆ 0 n0 /( n00 n0), ˆ n /( n0 n). ij 3. Empirical resuls and analysis 3. Empirical 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 overpredic 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, (6) (8) If he prediced VaR canno cover he realized dollar loss, his is ermed as a violaion. 5
Proceedings of he Firs European Academic Research Conference on Global Business, Economics, Finance and Social Sciences (EAR5Ialy Conference) ISBN: 978--6345-028-6 Milan-Ialy, June 30-July -2, 205, Paper ID: I53 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 203~204. 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. Table 3: Daily VaR forecass resuls 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 0% significance level are 2.7 and 4.6, respecively. Figures in bold ex indicae rejecion of he null hypohesis of correc VaR esimaes a he 0% significance level. Model Mean VaR Violaion Failure prob. LR uc LR cc Panel A: 90% Confidence Level GJR -.092 36 7.4% 5.02 5.5 GJR-ONV -0.9864 37 7.34% 4.32 4.5 GJR-PK -0.942 42 8.33%.63 2.52 GJR-VIX -0.8268 53 0.5% 0.4 0.63 Panel B: 95% Confidence Level GJR -.366 25 4.96% 0.00 0.05 GJR-ONV -.2744 26 5.5% 0.02 0.0 GJR-PK -.838 26 5.5% 0.02 0.9 GJR-VIX -.070 34 6.74% 2.92 2.95 Panel C: 99% Confidence Level GJR -.8745 5 0.99% 0.00 3.73 GJR-ONV -.846 5 0.99% 0.00 3.73 GJR-PK -.6897 7.38% 0.68 3.27 GJR-VIX -.5265 2.8% 5.32 6.48 6
Proceedings of he Firs European Academic Research Conference on Global Business, Economics, Finance and Social Sciences (EAR5Ialy Conference) ISBN: 978--6345-028-6 Milan-Ialy, June 30-July -2, 205, Paper ID: I53 4. Conclusions This sudy proposes he augmened GJR model which exends he GJR-GARCH model of Glosen e al. (993) by including overnigh volailiy, daily prices range, and implied volailiy as explanaory variables for he variance equaions in he GJR model. These VaR models are used o esimae heir daily VaR values and evaluae heir downside risk managemen performance for he SPDRs reurns covering from 2009 o 204. Empirical resuls indicae ha he augmened GJR models ouperform he GJR model for mos cases, suggesing ha he GJR-GARCH-based VaR forecass can be moderaely improved wih he addiional informaion conained in ONV, PK and VIX volailiy esimaors. In addiion, daily prices range and implied volailiy are far more informaive han he overnigh volailiy for improving he GJR-GARCH-based VaR forecass. Marke praciioners can adop he proposed model for esimaing and managing he poenial loss of ETFs in he face of caasrophic evens. References Blair, B.J., Poon, S.H. and Taylor, S.J., 200, Forecasing S&P 00 volailiy: he incremenal informaion conen of implied volailiies and high frequency reurns. Journal of Economerics 05, 5-26. Bollerslev, T., 986, Generalized auoregressive heeroskedasiciy. Journal of Economerics 3(3), 307 327. Brooks, C., Clare, A.D. and Persand, G., 2000, A word of cauion on calculaing markebased minimum capial risk requiremens. Journal of Banking and Finance 24, 557-574. Chrisoffersen, P.F., 998, Evaluaing inerval forecass. Inernaional Economic Review 39, 84-862. Corrado, C. and Truong, C., 2007, Forecasing sock index volailiy: comparing implied volailiy and he inraday high-low price range. Journal of Financial Research 30(2), 20-25. Engle, R. F., 982, Auoregressive condiional heeroscedasiciy wih esimaes of he variance of U.K. inflaion. Economerica 50(4), 987 008. Fueres, Ana-Maria, Izzeldin, M. and Kaloychou, E., 2009, On forecasing daily sock volailiy: The role of inraday informaion and marke condiions. Inernaional Journal of Forecasing 25(2), 259-28. Garman, M. and Klass, M. 980, On he esimaion of securiy price volailiies from hisorical daa. Journal of Business 53(), 67-78. Glosen, L., Jagannahan, R., Runkle, D., 993, On he relaion beween he expeced value and he volailiy nominal excess reurn on socks. Journal of Finance 46, 779-80. Jarque, C.M. and Bera, A.K., 987, A es for normaliy of observaions and regression 7
Proceedings of he Firs European Academic Research Conference on Global Business, Economics, Finance and Social Sciences (EAR5Ialy Conference) ISBN: 978--6345-028-6 Milan-Ialy, June 30-July -2, 205, Paper ID: I53 residuals. Inernaional Saisics Review 55, 63-72. Koopman, S., Jungbacker, B. and Hol, E., 2005, Forecasing daily variabiliy of he S&P 00 sock index using hisorical, realised and implied volailiy measuremens. Journal of Empirical Finance 2(3), 445-475. Kupiec, P., 995, Techniques for verifying he accuracy of risk measuremen models. Journal of Derivaives 3, 73-84. Parkinson, M., 980, The exreme value mehod for esimaing he variance of he rae of reurn. Journal of Business 53(), 6-65. Rogers, L.C.G. and Sachell, S.E., 99. Esimaing variance from high, low and closing prices. Annals of Applied Probabiliy (4), 504-52. Vipul and Jacob, J., 2007, forecasing performance of exreme-value volailiy esimaors. Journal of Fuures Markes 27(), 085-05. 8