The Economic Value of Volailiy Timing Using a Range-based Volailiy Model Ray Yeuien Chou * Insiue of Economics, Academia Sinica & Insiue of Business Managemen, Naional Chiao Tung Universiy Nahan Liu Deparmen of Finance, Feng Chia Universiy Forhcoming, Journal of Economic Dynamics and Conrol Absrac There is growing ineres in uilizing he range daa of asse prices o sudy he role of volailiy in financial markes. In his paper, a new range-based volailiy model was used o examine he economic value of volailiy iming in a mean-variance framework. We compared is performance wih a reurn-based dynamic volailiy model in boh in-sample and ou-of-sample volailiy iming sraegies. For a risk-averse invesor, i was shown ha he predicable abiliy capured by he dynamic volailiy models is economically significan, and ha a range-based volailiy model performs beer han a reurn-based one. JEL classificaion: C5; C5; G Keywords: Asse allocaion; CARR; DCC; Economic value; Range; Volailiy iming. * Corresponding auhor: Insiue of Economics, Academia Sinica, #8, Yen-Jio-Yuan Road, Sec, Nankang, Taipei, Taiwan. Telephone: 886--7879 ex 3, fax: 886--7853946, email: rchou@econ.sinica.edu.w.
. Inroducion In recen years, here has been considerable ineres in volailiy. The exensive developmen of volailiy modeling has been moivaed by relaed applicaions in risk managemen, porfolio allocaion, asses pricing and fuures hedging. In discussions of economeric mehodologies for esimaing he volailiy of individual asses, ARCH and GARCH have been emphasized mos. Various applicaions in finance and economics are provided as a review in Bollerslev e al. (99; 994), and Engle (004). Several sudies have noed ha range daa based on he difference of high and low prices in a fixed inerval can offer a sharper esimae of volailiy han he reurn daa. Range daa are available for mos financial asses and inuiively have more informaion han reurn daa for esimaing volailiy. They uilize he wo pieces of informaion (high and low) comparing wih he reurn daa ha use only he close o close price. Parkinson (980) showed ha i reduced he variance of he volailiy esimaor by 5 imes comparing wih he radiional reurn-based volailiy esimaor. Furhermore, range is an unbiased esimaor of he sandard deviaion. There are quie a few exensions of Parkinson s original resuls. More recenly, Brand and Jones (006), Chou (005, 006), and Marens and van Dijk (007). In paricular, Chou (005) proposed a condiional auoregressive range (CARR) model which can easily capure he dynamic volailiy srucure, and provides sharper volailiy forecass comparing wih he reurn-based GARCH model. The CARR model is a condiional mean model and i is easily o incorporae oher explanaory variables. However, he lieraure above jus focuses on he volailiy forecas of a See for example, Garman and Klass (980), Wiggins (99), Rogers and Sachell (99), Kuniomo (99), Yang and Zhang (000), and Alizadeh e al. (00).
univariae asse. I should be noed ha here have been some aemps o esablish a relaionship beween muliple asses, such as VECH [see Bollerslev e al. (988)], BEKK [see Engle and Kroner (995)] and a consan condiional correlaion model (CCC) [see Bollerslev (990)], among ohers. VECH and BEKK allow ime-varying covariance processes which are oo flexible o esimae, and CCC wih a consan correlaion is oo resricive o apply o general applicaions. Seminal work on solving he puzzle was carried ou by Engle (00a). A dynamic condiional correlaion (DCC) model proposed by Engle (00a) provides anoher viewpoin o his problem. The esimaion of DCC can be divided ino wo sages. The firs sep is o esimae univariae GARCH, and he second is o uilize he ransformed sandardized residuals o esimae ime-varying correlaions [see Engle and Sheppard (00), Cappiello e al. (006)]. A new mulivariae volailiy, recenly proposed by Chou e al. (009), combines he range daa of asse prices wih he framework of DCC, namely range-based DCC 3. The range-based DCC model is flexible and easy o be esimaed hrough he wo-sep esimaion. I also has he relaive efficiency of he range daa over he reurn daa in esimaing volailiy. Through he saisical measures RMSE and MAE, based on four benchmarks of implied and realized covariance, 4 hey concluded ha he range-based DCC model performs beer han oher reurn-based models (MA00, EWMA, CCC, reurn-based DCC, and diagonal BEKK). See Tsay (00) and Tse and Tsui (00) for oher relaed mehods for esimaing he ime-varying correlaions. 3 Fernandes e al. (005) propose anoher kind of mulivariae CARR model using he formula Cov( X, Y ) = [ Var( X + Y ) Var( X ) Var( Y )]/. However, his mehod can only apply o a bivariae case. 4 Daily daa are used o build four proxies for weekly covariances, i.e. implied reurn-based DCC, implied range-based DCC, implied DBEKK, and realized covariances. 3
Asse allocaion efficiency is closely linked o he predicions of asse reurns and volailiies. Wes e al. (993) was he firs o focus on his insigh and devise a way o use he uiliy funcion o derive he economic value of dynamic volailiy models. The economic inuiion is simple. A more accurae volailiy predicion will render he invesors a way o adapively adjus heir porfolio posiions o achieve a higher uiliy level. Hence invesors will be willing o pay a fee o swich from a fund manager wih poor volailiy predicion skill (or model) o anoher manager offering beer volailiy predicions. The maximum of such a swiching fee is a measure of he difference of economic values of he wo compeing volailiy models. The above described sraegy of adjusing porfolio weighs according o he predicion of volailiy changes is called volailiy iming. This is differen from he oher ype of marke iming echnique in which he porfolios are adjused following he predicion of changes in expeced reurns. Marke iming is usually no an effecive ool given ha an efficiency marke implied he reurns are unpredicable. Following Wes e al.(993), some sudies have concenraed on wheher some newly devised volailiy models have sufficienly high economic values [see Busse (999), Fleming e al. (00, 003), Marquering and Verbeek (004), Thorp and Milunovich (007), and Core, Sarno, and Tsiakas, (009)]. The quesions upon which we focused were wo: firs, wheher he range-based DCC model conains economic value comparing wih a benchmark model using a saic or buy-and-hold sraegy; and second, wheher economic value of range-based volailiy model sill exiss comparing wih a reurn-based DCC model. In comparing he economic value of reurn-based and range-based models, i is helpful o use a suiable measure o capure he rade-off beween risk and reurn. Mos lieraure evaluaes volailiy models hrough error saisics and relaed 4
applicaions bu neglecs he influence of asse expeced reurns. A more precise measuremen should consider boh of hem, bu only a few such sudies have been made a his poin. However, a uiliy funcion can easily connec hem and build a comparable sandard. Before enering ino a deailed discussion for he economic value of volailiy iming, i was necessary o clarify is definiion in his paper. In shor, he economic value of volailiy iming is he gain compared wih a saic sraegy. Our concern was o esimae he willingness of he invesor wih a mean variance uiliy o pay for a new volailiy model raher han a saic one. In ligh of he success of he range-based volailiy model, he purpose of his paper was o examine is economic value in volailiy iming by using he condiional mean-variance framework developed by Fleming e al. (00). We considered an invesor wih differen risk-averse levels using condiional volailiy analysis o allocae hree asses: socks, bonds and cash. Fleming e al. (00) exended he uiliy crierion derived from Wes e al. (993) o es he economic value of volailiy iming for shor-horizon invesors wih differen risk olerance levels 5. In addiion o he shor-horizon forecas of seleced models, we also examined he economic value of longer horizon forecass and an asymmeric range-based volailiy model in our empirical sudy. This sudy may lead o a beer undersanding of range volailiy. The reminder is laid ou as follows. Secion II inroduces he asse allocaion mehodology, economic value measuremen, and he reurn-based and he range-based DCC. Secion III describes he properies of daa used and evaluaes he performance of he differen sraegies. Finally, he conclusion is showed in secion IV. 5 They found ha volailiy-iming sraegy based on one-sep ahead esimaes of he condiional covariance marix [see Foser and Nelson (996)] significanly ouperformed he uncondiional efficien saic porfolios. 5
. Mehodologies To carry ou his sudy we used he framework of a minimum variance sraegy, which was conducive o deermining he accuracy of he ime-varying covariances. We waned o find he opimal dynamic weighs of he seleced asses and he implied economic value of a saic sraegy for a risk-adverse invesor. Before applying he volailiy iming sraegies, we needed o build a ime-varying covariance marix. The deails of he mehodology are as follows.. Opimal Porfolio Weighs in a Minimum Variance Framework Iniially, we considered a minimizaion problem for he porfolio variance subjeced o a arge reurn consrain. To derive our sraegy, we le R be he k vecor of spo reurns a ime 6. Is condiional expeced reurn μ and condiional covariance marix Σ were calculaed by E[ R Ω ] and E[( R μ)( R μ ) Ω ], respecively. Here, Ω was assumed as he informaion se a ime. To minimize porfolio volailiy subjec o a required arge reurn μ arge, i can be formulaed as: min w w Σ w, w μ + w R = μ, () s. ( ) arg f e where w is a k vecor of porfolio weighs for ime. R f is he reurn for he risk-free asse. The opimal soluion o he quadraic form () is: w = ( μarge Rf) Σ ( μ Rf) ( Rf ) ( Rf ) μ Σ μ. () 6 Through ou his paper, we have used blackened leers o denoe vecors or marices. 6
Under he cos of carry model, we regarded he excess reurns as he fuures reurns by applying regular no-arbirage argumens7. I is clear ha he covariance marix Σ of he spo reurns is he same as ha of he excess reurns. Equaion () can be simply expressed as: w μ = Σ μ arge μσ μ, (3) where he vecor μ and he marix Σ are redefined in erms of fuures. A bivariae case ( k = ) of Equaion (3) can be wrien as: w w μ ( μ σ μ σ ) arg e,,,,, = μ, σ, + μ, σ, μ, μ, σ, μ ( μ σ μ σ ) arge,,,,, = μ, σ, + μ, σ, μ, μ, σ,,, (4) μ, μ where, and, are he fuures reurns of S&P 500 index (S&P 500) and 0-year Treasury bond (T-bond) in our empirical sudy. In addiion, fuures conracs are easy o be raded and have lower ransacion cos compared o spo conracs. The above analysis poined ou ha he opimal porfolio weighs were ime-varying. Here we assumed he condiional mean μ was consan 8. Therefore, he dynamics of weighs only depend on he condiional covariance Σ. In his sudy, he opimal sraegy was obained based on a minimum variance framework subjec o a given reurn. 7 There are no coss for fuures invesmen. This means he fuures reurn equals he spo reurn minus he risk-free rae. 8 The changes in expeced reurns are no easy o deec. Meron (980) poins ou ha he volailiy process is more predicable han he reurn series. 7
The mean-variance framework above is used o derive he opimal porfolio weighs under differen arge reurns. In he following secion, we wan o build crierion9 o compare means and variances of he porfolios from he saic and dynamic sraegies. However, i is no easy o decide he bes sraegy, especially for he invesors wih differen risk aversions. In his sudy, we wan o apply he quadraic uiliy funcion o calculae economic value under some seings.. Economic Value of Volailiy Timing Fleming e al. (00) uses a generalizaion of he Wes e al. (993) crierion which builds he relaionship beween a mean-variance framework and a quadraic uiliy o capure he rade-off beween risk and reurn for ranking he performance of forecasing models. According o heir work, he invesor s uiliy can be defined as: U αw = p, p,, (5) ( W ) W R R where W is he invesor s wealh a ime, α is his absolue risk aversion, and he porfolio reurn a period is R p, = wr. For comparisons across porfolios, we assumed ha he invesor had a consan relaive risk aversion 0 (CRRA), γ = αw /( αw ) = γ. This implies α W is a consan. The CRRA seing means an invesor s loss olerance increases in proporion o he invesor s wealh. I implies ha he expeced uiliy is linearly relaed o wealh. 9 The Sharpe raion is one of candidaes for comparison. However, i may underesimae he performance of dynamic sraegies, see Marquering and Verbeek (004). 0 Wes, e al. (993), Fleming e al. (00), and Core e al. (009) also applied CRRA o heir sudies. 8
Wih his assumpion, he average realized uiliy U ( ) can be used in esimaing he expeced uiliy wih a given iniial wealh W 0. U ( ) = W T 0 = γ R p, R ( + γ ) p, where W 0 is he iniial wealh., (6) Therefore, he value of volailiy iming calculaed by equaing he average uiliies for wo alernaive porfolios is expressed as: T = ( R ) b, T γ γ = γ ( ) ( ) R b, = R a, + γ ( + ) R a,, (7) where is he maximum expense ha an invesor would be willing o pay o swich from he sraegy a o he sraegy b. R a, and R b, are he reurns of he porfolios from he sraegy a and b. If he expense is a posiive value, i means he sraegy b is more valuable han he sraegy a. In our empirical sudy, we repored as an annualized expense wih hree risk aversion levels of γ =, 5, and 0..3 Reurn-based and Range-based DCC We used he DCC model of Engle (00a) o esimae he covariance marix of muliple asse reurns. I is a direc exension of he CCC model of Bollerslev (990). The covariance marix Η for a vecor of k asse reurns in DCC can be wrien as: H = D Γ D, (8) Γ, (9) / / = diag{ Q } Q diag{ Q } In our seing, we le he sraegy pair (a,b) be (OLS, reurn-based DCC), (OLS, range-based DCC) and (reurn-based DCC, range-based DCC), respecively. Because he rolling sample mehod was adoped in he ou-of-sample comparison, his ype of OLS was named by rollover OLS. 9
where, D is he k k diagonal marix of ime-varying sandard deviaions from univariae GARCH models wih h i, for he i h reurn series on he i h diagonal. Γ is a ime-varying correlaion marix. The covariance marix Q = ] of he [ q ij, sandardized residual vecor Z = z, z )' is denoed as: (,, Q, (0) = ( a b) Q + az Z + bq where Q = q } denoes he uncondiional covariance marix of Z. The coefficiens, { ij a and b, are he esimaed parameers depicing he condiional correlaion process. The dynamic correlaion can be expressed as: ρ, = [( a b) q ( a b) q + az, + bq + az, z,, + bq ][( a b) q, + az, + bq,. () ] We esimaed he DCC model wih a wo-sage esimaion hrough quasi-maximum likelihood esimaion (QMLE) o ge consisen parameer esimaes. The log-likelihood funcion can expressed as = L Vol LCorr, where L Vol L +, he volailiy componen, is ( k log(π ) + log D + r ' D r ), and L Corr, he correlaion componen, is ( log R + Z ' R Z Z ' Z ). The explanaion is more fully developed in Engle and Sheppard (00) and Engle (00a). In addiion o using GARCH o consruc sandardized residuals, we can also build hem by oher univariae volailiy models. In his paper, CARR was used as an alernaive o verify wheher he specificaion seleced adequaely suis DCC or no. The CARR model is a special case of he muliplicaive error model (MEM) of Engle (00b). I can be expressed as: R =, u I ~ exp(, ), i =,, i, λ i, u i, i, 0
i, = ωi + α iri, + β iλi, λ, () z * = r λ,where λ i, = adj i λi,, adj i c * i, i, / i, σ i =, ˆ λ i where he range R i, is calculaed by he difference beween logarihm high and low prices of he i h asse during a fixed ime inerval, and i is also a proxy of sandard deviaion. λ i, and λˆ i are he condiional and uncondiional means of he range, respecively. disribuion. u i, is he residual which is assumed o follow he exponenial σ i is he uncondiional sandard deviaion for he reurn series. In considering differen scales in quaniy, he raio adj i was used o adjus he range o produce he sandardized residuals. 3. Empirical Resuls The empirical daa employed in his paper consiss of he sock index fuures, bond fuures and he risk-free rae. As o he above-menioned mehod, we applied he fuures daa o examine he economic value of volailiy iming for reurn-based and range-based DCC. Under he cos of he carry model, he resuls in his case can be exended o underlying spo asses [see Fleming e al. (00)]. In addiion o avoiding he shor sale consrains, his procedure reduces he complexiy of model seing. To address his issue, we used he S&P 500 fuures (raded a CME) and he T-bond fuures (raded a CBOT) as he empirical samples. According o Chou e al. (009), he fuures daa were aken from Daasream, sampling from January 6, 99 o December 9, 006 (5 years, 78 weekly observaions). Daasream provided he Parkinson (980) derived he adjusmen raio as a consan, 0.36, bu an asse price was required o follow a geomeric Brownian moion wih zero drif, which is no ruly empirical.
neares conrac and rolls over o he second nearby conrac when he nearby conrac approaches mauriy. We also used he 3-monh Treasury bill rae o subsiue for he risk-free rae. The Treasury bill rae is available from he Federal Reserve Board. < Figure is insered abou here > Figure shows he graphs for close prices (Panel A) reurns (Panel B) and ranges (Panel C) of he S&P 500 and T-bond fuures over he sample period. Table shows summary saisics for he reurn and range daa on he S&P 500 and T-bond fuures. The reurn was compued as he difference of logarihm close prices on wo coninuous weeks. The range was defined by he difference of he high and low prices in a logarihm ype. The annualized mean and sandard deviaion in percenage (8.0, 5.3) of he sock fuures reurns were boh larger han hose (0.853, 6.68) of he bond fuures reurns. This fac indicaed ha he more volaile marke may have a higher risk premium. Boh fuures reurns have negaive skewness and excess kurosis, indicaing a violaion of he normal disribuion. The range mean (3.34) of he sock fuures prices was larger han ha (.306) of he bond fuures prices. This is reasonable because he range is a proxy of volailiy. The Jarque-Bera saisic was used o es he null of wheher he reurn and range daa were normally disribued. Boh reurn and range daa rejeced he null hypohesis. The simple correlaion beween sock and bond reurns was small 3 (-0.03), bu his does no imply ha heir relaionship was very weak. In our laer analysis, we showed ha he dynamic relaionship of socks and bonds will be more realisically revealed by he condiional correlaions analysis. < Table is insered abou here > 3 The resuls are differen from he posiive correlaion value (sample period 983-997) in Fleming, e al. (00). Afer 997, he relaionship beween S&P 500 and T-bond presened a reverse condiion.
3. The In-sample Comparison To obain an opimal porfolio, we used he dynamic volailiy models o esimae he covariance marices. The parameers fied for reurn-based and range-based DCC, were boh esimaed and arranged in Table. We divided he able ino wo pars corresponding o he wo seps in he DCC esimaion. In Panel A of Table, one can use GARCH (fied by reurn) or CARR (fied by range) wih individual asses o obain he sandardized residuals. Figure provides he volailiy esimae of he S&P 500 fuures and he T-bond fuures based on GARCH and CARR. Then, hese sandardized residuals series were brough ino he second sage for dynamic condiional correlaion esimaing. Panel B of Table shows he esimaed parameers of DCC under he quasi-maximum likelihood esimaion (QMLE). < Table is insered abou here > < Figure is insered abou here > The correlaion and covariance esimaes for reurn-based and range-based DCC are shown in Figure 3. I seems ha he correlaion became more negaive a he end of 997. This means ha i is more desirable o diversify in his period because he bond holding will help offse he volailiy caused by he equiy componen in he porfoilo. This conjecure is confirmed in our laer analysis of he esimaed porfolio weighs. A deeper invesigaion is also given in Connolly e al. (005). < Figure 3 is insered abou here > Following he model esimaion, we consruced he saic porfolio (buil by OLS) using he uncondiional mean and covariance marices o ge he economic values of dynamic models. Under he minimum variance framework, he weighs of he porfolio were compued by he given expeced reurn and he condiional 3
covariance marices esimaed by reurn-based and range-based DCC. Then, we compared he performance of he volailiy models on differen arge annualized reurns (5% - 5%, % in an inerval). < Table 3 is insered abou here > Table 3 shows how he performance comparisons varied wih he arge reurns and he risk aversions. Panel A of Table 3 shows he annualized means ( μ ) and volailiies (σ ) of he porfolios esimaed from hree mehods, reurn-based DCC, range-based DCC and OLS. A a quick look, he annualized Sharpe raios 4 calculaed from reurn-based DCC (0.680) and range-based DCC (0.699) were higher han he saic model (0.560). Panel B of Table 3 shows he average swiching fees ( ) from one sraegy o anoher. The value seings of CRRA γ were, 5, and 0. As for he performance fees wih differen relaive risk aversions, in general, an invesor wih a higher risk aversion should be willing o pay more o swich from he saic porfolio o he dynamic ones. Wih higher arge reurns, he performance fees increased seadily. In addiion, Panel B of Table 3 also repors he performance fees for swiching from reurn-based DCC o range-based DCC. Posiive values for all cases show ha he range-based volailiy model can give more significan economic value in forecasing covariance marices han reurn-based ones. In he real pracice, he ransacion coss should be considered when he dynamic sraegies are compared o he saic one. For S&P 500 fuures, he bid/ask spread and round-rip commission oally cos abou $0.0 index uni. The annualized cos of a one-way ransacion in our sudy can be calculaed by 0.05/94.55 5=0.8%, where 94.55 is an average index level from 99 o 006. I means he advanage of he r 4 The Sharpe raio is consan wih differen arge mulipliers. For he furher deails, see Engle and Colacio (006). 4
dynamic sraegies will no be offse by he ransacion coss. For example, wih a fixed arge reurn 0%, he economic advanage is abou 6% for an invesor wih relaive risk aversion of 5. Figure 4 plos he weighs of an in-sample minimum volailiy porfolio derived from wo dynamic models. OLS has consan weighs for cash, socks, and bonds, i.e. -0.934, 0.7079, and 0.4855. < Figure 4 is insered abou here > I is ineresing o observe he dynamic paerns of he porfolio weighs implied by he wo dynamic models. In conrary o he OLS (buy-and-hold sraegy), hey have subsanial flucuaions across he sample periods. The wo sraegies (panel A for reurn-based DCC and panel B for range-based DCC) have roughly similar paerns in movemens bu wih noiceably quaniaive differences. The sock porfolio weigh is mos sably flucuaing around less han 0.8 before 997 afer which i drops o a lower level of abou 0.65 wih larger variaions. I is ineresing o observe ha he bond weighs have been negaive or zero before 997 and become posiive afer lae 997. The zero or negaive weighs is a resul of he booming equiy marke in he mid nineies hence i s desirable o inves mosly in he equiy marke. The mid-crash in he lae 997 has caused an increase of volailiy which would cause a drop in invesor s uiliy and hence should be hedged away. As is seen in Figure 3, his is a period when he correlaion beween sock and bond reurns became negaive. The negaive correlaion in bond/equiy reurn suggess an increase in he bond posiion would help o reduce he oal porfolio volailiies. The lower level and higher variaions of sock weighs since hen is also a reflecion of he fac ha he sock/bond correlaions in he laer periods are mosly negaive bu wih wide swings. Finally, he cash posiion serves as a residual in he porfolio since he hree asse 5
weighs add up o one. The movemens will be relaed o he erm spread or he erm srucure of ineres raes and he bond volailiy. I s also useful o conras he ime-varying paern of he bond posiion o he fixed weigh suggesed by OLS. The laer sugges ha roughly 48% should be invesed in he bond marke regardless of he movemens in he volailiy and correlaion srucures. This is obviously oo naïve given our discussion above ha volailiies and correlaions of sock and bond reurns do vary over ime. A buy-and hold sraegy will herefore yield a poor performance. 3. The Ou-of-sample Comparisons For robus inference, a similar approach was uilized o esimae he value of volailiy iming in he ou-of-sample analysis. Here he rolling sample approach was adoped for all ou-of-sample esimaions. This mean ha he rollover OLS mehod replaced he convenional OLS mehod used in he in-sample analysis. Each forecasing value was esimaed by 5 observaions over abou 0 years. Then, he rolling sample mehod provided 6 forecasing values for he one period ahead comparison. The firs forecased value occurred he week of January 4, 00. < Table 4 is insered abou here > Table 4 repors how he performance comparisons varied wih he arge reurns and he risk aversions for one period ahead ou-of-sample forecas. We obained a consisen conclusion wih Table 3. The esimaed Sharpe raios calculaed from reurn-based DCC, range-based DCC and rollover OLS were 0.540, 0.586 and 0.36, respecively. The performance fees swiching from rollover OLS o DCC were all posiive. In oal, he ou-of-sample comparison suppored he former inference. Figure 5 plos he weighs ha minimize condiional volailiy while seing he expeced annualized reurn equal a 0%. 6
< Figure 5 is insered abou here > In addiion o examining he performance of shor-horizon invesors, we furher repored he resuls of he long-horizon asse allocaions. Table 5 repors one o hireen periods ahead of ou-of-sample performance for hree mehods. Here he rolling sample approach provided 49 forecasing values for each ou-of-sample comparison. The porfolio weighs for all sraegies were obained from he weekly esimaes of he ou-of-sample condiional covariance marices wih a fixed arge reurn (0%). In general, he Sharpe raios aken from range-based DCC were he larges, and reurn-based DCC were he nex. For each sraegy, however, we could no find an obvious rend in he Sharpe raios forecasing periods ahead. As for he resul of he performance fees, i seems reasonable o conclude ha an invesor would sill be willing o pay o swich from rollover OLS o DCC. Moreover, he economic value seems o indicae a decreasing rend for forecasing periods ahead. For a longer forecasing horizon (-3 weeks), however, he resuls of esimaed swiching fees were mixed. Swiching from reurn-based DCC o range-based DCC always remains posiive. < Table 5 is insered abou here > Thorp and Milunovich (007) show ha a risk-averse invesor holding seleced inernaional equiy indices, wih γ =, 5, and 0, would pay lile for symmeric o asymmeric forecass. In some cases, he swiching fees would even be negaive. In order o furher undersand his argumen, we examined i based on he range-based volailiy model. Chou (005) provides an asymmeric range model namely CARRX: λ = ω+ αr + βλ + φre. The lagged reurn in he condiional range equaion was used o capure he leverage effec. For building an asymmeric range-based volailiy model, CARR in he firs sep of range-based DCC can be replaced by 7
CARRX. Cappiello e al. (006) inroduced asymmeric DCC: Q = ( a b) Q cn+ az Z + bq + cn n. n is he k vecor calculaed by I( Z < 0) o Z o allow correlaion o increase more in boh falling reurns han in boh rising reurns, and N= E ( nn ), where o denoes he Hadamard marix produc operaor, i.e. elemen-wise muliplicaion. Table 6 shows he one period ahead performance of he volailiy iming values for asymmeric range-based DCC compared wih rollover OLS. The swiching fees from rollover OLS o asymmeric range DCC seem o be smaller han he fees from rollover OLS o symmeric range DCC in Table 4. One of he reasons for his may be he poor performance of he bond daa. In his case, i is no valuable o swich he symmeric sraegy o he asymmeric one. < Table 6 is insered abou here > 4. Conclusion In his paper, we examined he economic value of volailiy iming for he range-based volailiy model in uilizing range daa which combines CARR wih a DCC srucure. Our analysis is carried ou by uilizing S&P 500 and T-bond fuures in a mean-variance framework wih a no-arbirage seing. By means of he uiliy of a porfolio, he economic value of dynamic models can be obained by comparing i o OLS (a buy-and-hold sraegy). Boh he in-sample and ou-of-sample resuls show ha a risk-averse invesor should be willing o swich from OLS o DCC wih subsanial high swiching fees. Moreover, he swiching fees from reurn-based DCC o range-based DCC were always posiive. We concluded ha he range-based volailiy model has more significan economic value han he reurn-based one. The resuls gave robus inferences for supporing he range-based volailiy model in forecasing volailiy. Fuure sudies can consider more general ype of uiliy 8
funcions and also include oher asse classes such as commodiy fuures, REIT s and VIX s. 9
References Alizadeh, S., Brand, M., Diebold, F., 00. Range-based esimaion of sochasic volailiy models. Journal of Finance 57, 047-9. Bollerslev, T., 986. Generalized auoregressive condiional heeroskedasiciy. Journal of Economerics 3, 307-38. Bollerslev, T., 990. Modeling he coherence in shor-run nominal exchange raes: A mulivariae generalized ARCH model. Review of Economics and Saisics 7, 498-505. Bollerslev, T., Chou, R.Y., Kroner, K., 99. ARCH modeling in finance: A review of he heory and empirical evidence. Journal of Economerics 5, 5-59. Bollerslev, T., Engle, R.F., Nelson, D., 994. ARCH Models, in Handbook of Economerics, IV, 959-3038, ed. Engle, R.F., and McFadden, D.C., Amserdam: Norh-Holland. Bollerslev, T., Engle, R.F., Wooldridge, J.M., 988. A capial asse pricing model wih ime varying covariances. Journal of Poliical Economy 96, 6-3. Brand, M., Jones, C., 006. Volailiy forecasing wih range-based EGARCH models. Journal of Business and Economic Saisics 4, 470-86. Busse, J.A., 999. Volailiy iming in muual funds: Evidence from daily reurns. Review of Financial Sudies, 009-04. Cappiello, L., Engle, R.F., Sheppard, K., 006. Asymmeric dynamics in he correlaions of global equiy and bond reurns. Journal of Financial Economerics 4, 537-7. Chou, R.Y., 005. Forecasing financial volailiies wih exreme values: The condiional auoregressive range (CARR) model. Journal of Money Credi and Banking 37, 56-8. 0
Chou, R.Y., 006. Modeling he asymmery of sock movemens using price ranges. Advances in Economerics 0, 3-58. Chou, R.Y., Wu, C.C., Liu, N., 009. Forecasing ime-varying Covariance wih a range-based dynamic condiional correlaion model. Review of Quaniaive Finance and Accouning 33, 37-345. Connolly, R., Sivers, C., Sun, L., 005. Sock marke uncerainy and he sock-bond reurn relaion. Journal of Financial and Quaniaive Analysis 40, 6-94. Core, P.D., Sarno, L., Tsiakas, I., 009. An economic value evaluaion of empirical exchange rae models, Review of Financial Sudies, 349-3530. Engle, R.F., 98. Auoregressive condiional heeroskedasiciy wih esimaes of he variance of Unied Kingdom inflaion. Economerica 50, 987-008. Engle, R.F., 00a. Dynamic condiional correlaion: A simple class of mulivariae GARCH models. Journal of Business and Economic Saisics 0, 339-50. Engle, R.F., 00b. New froniers for ARCH models. Journal of Applied Economerics 7, 45-446. Engle, R.F., 004. Risk and volailiy: Economeric models and financial pracice. American Economic Review 94, 405-0. Engle, R.F., Colacio, R., 006. Tesing and valuing dynamic correlaions for asse allocaion. Journal of Business and Economic Saisics 4, 38-53. Engle, R.F., Kroner, K., 995. Mulivariae simulaneous GARCH. Economeric Theory, -50. Engle, R.F., Sheppard, K., 00. Theoreical and empirical properies of dynamic condiional correlaion mulivariae GARCH. Working Paper (Universiy of California, San Diego). Fernades, M., Moa, B., Rocha, G., 005. A mulivariae condiional auoregressive
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Table : Summary Saisics for Weekly S&P 500 and T-bond Fuures Reurn and Range Daa, 99-006 The able provides summary saisics for he weekly reurn and range daa on S&P 500 sock index fuures and T-bond Fuures. The reurns and ranges were compued close close high low by 00 log( p / p ) and 00 log( p / p ), respecively. The Jarque-Bera saisic is used o es he null of wheher he reurn and range daa are normally disribued. The values presened in parenheses are p-values. The annualized values of means (sandard deviaion) for S&P 500 and T-bond fuures were 8.0 (5.3) and 0.853 (6.68), respecively. The simple correlaion beween sock and bond reurns was -0.03. The sample period ranges form January 6, 99 o December 9, 006 (5 years, 78 observaions) and all fuures daa were colleced from Daasream. S&P 500 Fuures T-Bond Fuures Reurn Range Reurn Range Mean 0.58 3.34 0.06.306 Median 0.4.607 0.033.94 Maximum 8.4 3.556.46 4.55 Minimum -.395 0.690-4.050 0.30 Sd. Dev...809 0.855 0.560 Skewness -0.503.756-0.498.390 Kurosis 6.455 7.3 4.7 6.46 Jarque-Bera 4.37 985.454 80.44 64.367 (0.000) (0.000) (0.000) (0.000) 4
Table : Esimaion Resuls of Reurn-based and Range-based DCC Model Using Weekly S&P500 and T-bond Fuures, 99-006 = c +, hk, = ω k + α kε k, i + β k hk,, ε ~ N(0, h ), r i, ε i, R i, = u i,, k, = ω k + α kr k, + β k λk, k, I k, λ, R k, I ~ exp(, ), k =,. Q Q Z Q = ( a b) + a + b Z, and hen ρ, = [( a b) q ( a b) q + az, + bq + az, z,, + bq ][( a b) q, + az, + bq,, ] where R is he range variable, Z is he sandard residual vecor which is sandardized by GARCH or CARR volailiies. Q = { q ij, } and Q = { q ij } are he condiional and uncondiional covariance marix of Z. The hree formulas above are GARCH, CARR and he condiional correlaion equaions respecively of he sandard DCC model wih mean reversion. The able shows esimaions of he hree models using he MLE mehod. Panel A is he firs sep of he DCC model esimaion. The esimaion resuls of GARCH and CARR models for wo fuures were presened here. Q() is he Ljung Box saisic for he auocorrelaion es wih lags. Panel B is he second sep of he DCC model esimaion. The values presened in parenheses are -raios for he model coefficiens and p-values for Q(). Panel A: Volailiies Esimaion of GARCH and CARR models S&P500 Fuures T-bond Fuures GARCH CARR GARCH CARR c 0.88 0.008 (3.56) (0.4) ωˆ 0.09 0.03 0.08 0.075 (.49) (.93) (.533) (.80) αˆ 0.05 0.48 0.060 0.57 (3.698) (9.090) (.03) (5.08) βˆ 0.946 0.79 0.90 0.785 (7.36) (3.67) (8.645) (8.04) Q() 6.3 5.647 5.87 3. (0.00) (0.933) (0.97) (0.07) Panel B: Correlaion Esimaion of Reurn- and Range-based DCC Models S&P500 and T-bond Reurn-based DCC Range-based DCC â 0.037 0.043 (4.444) (4.679) bˆ 0.955 0.95 (85.6) (80.4) 5
Table 3: In-sample Comparison of he Volailiy Timing Values in he Minimum Volailiy Sraegy Using Differen Targe Reurns, 99-006 The able repors he in-sample performance of he volailiy iming sraegies wih differen arge reurns. The arge reurns were from 5% o 5% (annualized). The weighs for he volailiy iming sraegies were obained from he weekly esimaes of he condiional covariance marix and he differen arge reurn seing. Panel A shows he annualized means ( μ ) and volailiies (σ ) for each sraegy. The esimaed Sharpe raios for he reurn-based DCC model, he range-based DCC model, and he OLS sraegy were 0.680, 0.699, and 0.560, respecively. Panel B shows he average swiching annualized fees ( r ) from one sraegy o anoher. The values of he consan relaive risk aversionγ were, 5, and 0. Targe reurn(%) Panel A: Means and Volailiies of Opimal Porfolios Reurn-based DCC Range-based DCC OLS μ σ μ σ μ σ 5 5.0.00 5.4.00 5.000.90 6 6.366 3.84 6.438 3.83 6.000 3.977 7 7.530 5.57 7.635 5.56 7.000 5.764 8 8.694 7.4 8.83 7.39 8.000 7.55 9 9.859 8.954 0.08 8.95 9.000 9.338 0.03 0.668.5 0.665 0.000.5.87.38.4.378.000.9 3.35 4.095 3.69 4.09.000 4.699 3 4.56 5.808 4.85 5.804 3.000 6.486 4 5.680 7.5 6.0 7.57 4.000 8.73 5 6.845 9.35 7.09 9.30 5.000 0.060 Panel B: Swiching Fees wih Differen Relaive Risk Aversions Targe OLS o Reurn DCC OLS o Range DCC Reurn o Range DCC reurn(%) 5 0 5 0 5 0 5 0.303 0.376 0.393 0.343 0.47 0.434 0.040 0.04 0.04 6 0.703 0.950.008 0.777.05.084 0.074 0.076 0.076 7.44.77.897.353.883.009 0.09 0. 0. 8.99.845 3.063.073.994 3.3 0.44 0.49 0.5 9.76 4.73 4.507.940 4.360 4.696 0.80 0.89 0.9 0 3.739 5.753 6.4 3.956 5.979 6.453 0.7 0.30 0.33 4.866 7.578 8.06 5. 7.846 8.477 0.55 0.73 0.77 6.4 9.64 0.44 6.434 9.95 0.754 0.94 0.38 0.34 3 7.565.93.94 7.897.83 3.70 0.334 0.365 0.373 4 9.35 4.436 5.609 9.507 4.83 6.009 0.375 0.44 0.44 5 0.85 7.4 8.509.6 7.580 8.95 0.48 0.466 0.479 6
Table 4: Ou-of-sample Comparison for he One Period Ahead Volailiy Timing Values in he Minimum Volailiy Sraegy wih Differen Targe Reurns, 99-006 The able repors he one period ahead ou-of-sample performance of he volailiy iming sraegies wih differen arge reurns. There were 5 observaions in each of he esimaed models and he rolling sample approach provided 6 forecasing values for each ou-of-sample comparison. The firs forecased value occurred he week of January 4, 00. The arge reurns were from 5% o 5% (annualized). The weighs for he volailiy iming sraegies were obained from he weekly esimaes of he one period ahead condiional covariance marix and he differen arge reurn seing. Panel A shows he annualized means ( μ ) and volailiies (σ ) for each sraegy. The esimaed Sharpe raios for he reurn-based DCC model, he range-based DCC model, and he rollover OLS sraegy were 0.540, 0.586, and 0.36, respecively. Panel B shows he average swiching annualized fees ( r ) from one sraegy o anoher. The values of he consan relaive risk aversion were, 5, and 0. Panel A: Means and Volailiies of Opimal Porfolios Targe Reurn-based DCC Range-based DCC Rollover OLS reurn(%) μ σ μ σ μ σ 5 4.69.698 4.747.66 4.344.749 6 5.438 3.083 5.540 3.06 4.808 3.76 7 6.86 4.468 6.333 4.370 5.73 4.603 8 6.933 5.853 7.7 5.75 5.737 6.030 9 7.68 7.39 7.90 7.080 6.0 7.456 0 8.48 8.64 8.74 8.435 6.667 8.883 9.76 0.009 9.507 9.790 7.3 0.30 9.93.394 0.300.45 7.596.737 3 0.67.779.094.500 8.060 3.64 4.48 4.65.887 3.854 8.55 4.59 5.66 5.550.680 5.09 8.990 6.08 Panel B: Swiching Fees wih Differen Relaive Risk Aversions Targe OLS o Reurn DCC OLS o Range DCC Reurn o Range DCC reurn(%) 5 0 5 0 5 0 5 0.393 0.45 0.433 0.48 0.537 0.550 0.089 0. 0.8 6 0.78 0.890 0.96 0.99.76.0 0.0 0.89 0.308 7.3.463.58.606.998.090 0.377 0.545 0.585 8.746.44.39.38 3.00 3.59 0.589 0.88 0.953 9.33.935 3.079 3.56 4.85 4.45 0.848.303.43 0.963 3.834 4.039 4.09 5.545 5.88.54.80.967 3.667 4.84 5.6 5.33 7.077 7.5.509.40.67 4.435 5.956 6.309 6.80 8.774 9.338.93 3.083 3.363 3 5.67 7.74 7.64 7.53 0.69.3.366 3.85 4.06 4 6.6 8.495 9.09 8.885.634 3.460.869 4.707 5.46 5 7. 9.94 0.548 0.340 4.78 5.746 3.4 5.65 6.8 7
Table 5: Ou-of-sample Comparison for One o Thireen Periods Ahead Volailiy Timing Values in he Minimum Volailiy Sraegy, 99-006 The able repors he one o hireen periods ahead ou-of-sample performance of he volailiy iming sraegies wih he fixed 0% (annualized) arge reurn. The weighs for he volailiy iming sraegies were obained from he weekly esimaes of he one o hireen periods ahead condiional covariance marix. There were 5 observaions in each of he esimaed models and he rolling sample approach provided 49 forecasing values for each ou-of-sample comparison. The firs forecased mean value occurred he week of January 4, 00. Panel A shows he annualized means ( μ ), volailiies (σ ), and Sharpe raios (SR) for each sraegy. Panel B shows he average swiching annualized fees ( r ) from one sraegy o anoher. The values of he consan relaive risk aversion were, 5, and 0. Panel A: Means and Volailiies of Opimal Porfolios Periods Reurn-based DCC Range-based DCC Rollover OLS Ahead μ σ SR μ σ SR μ σ SR 7.77 8.74 0.45 8.060 8.540 0.50 6.0 8.956 0.5 7.868 8.830 0.464 8.56 8.556 0.560 6.068 8.933 0.57 3 7.37 8.807 0.408 8.3 8.57 0.59 6.660 8.93 0.33 4 8.7 8.838 0.49 8.750 8.604 0.578 7.03 8.98 0.373 5 8.464 8.860 0.59 9.00 8.653 0.67 6.869 8.989 0.344 6 9.088 8.903 0.597 9.600 8.637 0.674 7.3 8.973 0.385 7 9.36 8.840 0.63 0.033 8.69 0.75 7.87 8.945 0.458 8 8.853 8.897 0.57 9.49 8.683 0.65 7.644 8.975 0.43 9 9.806 8.878 0.679 0.093 8.664 0.79 8.476 9.03 0.5 0 9.746 8.887 0.67 9.576 8.695 0.667 8.89 8.983 0.49 9.436 8.908 0.636 8.986 8.7 0.598 8.03 8.90 0.478 8.737 9.003 0.55 8.076 8.79 0.489 7.44 8.853 0.4 3 8.73 9. 0.54 8.7 8.94 0.505 7.794 8.867 0.453 Panel B: Swiching Fees wih Differen Relaive Risk Aversions Periods OLS o Reurn DCC OLS o Range DCC Reurn o Range DCC Ahead 5 0 5 0 5 0.77 3.546 3.77 3.944 5.89 5.599.96.83.983.8.633.76 4.3 5.448 5.73.970.94 3.37 3.93.7.83 3.308 4.495 4.77.09.830 3.09 4.440.758.834 3.5 4.44 4.499.78.544.738 5.0.665.773 3.900 5.03 5.97.7.446.6 6.9.44.503 3.938 5.078 5.345.775.730.958 7.993.373.464 3.647 4.740 4.997.674.440.65 8.58.86.98 3.6 4.7 4.40.597.369.555 9.08.556.683 3.39 4.578 4.875.33.03.95 0.09.370.455.753 3.767 4.007 0.753.465.638.46.44.46.89.59.758 0.489.09.383 0.593 0.037-0.00 0.945.64.7 0.358.8.33 3-0.69 -.0 -.436 0.5 0.078 0.035 0.58.43.47 8
Table 6: The One Period Ahead Performance of he Volailiy Timing Values for he Asymmeric Range-based Volailiy Model, 99-006 The able repors he one period ahead ou-of-sample performance of he volailiy iming sraegies for he asymmeric range-based volailiy model wih differen arge reurns. There were 5 observaions in each of he esimaed models and he rolling sample approach provided 6 forecasing values for each ou-of-sample comparison. The firs forecased value occurred he week of January 4, 00. The arge reurns were from 5% o 5% (annualized). The weighs for he volailiy iming sraegies were obained from he weekly esimaes of he one period ahead condiional covariance marix and he differen arge reurn seing. The annualized means ( μ ) and volailiies (σ ) of he opimal porfolio are shown here. r is he average swiching annualized fee from he rollover OLS model o he asymmeric range-based volailiy model. The esimaed Sharpe raio for he asymmeric range-based DCC model was 0.5. The values of he consan relaive risk aversion were se as, 5, and 0. Targe reurn(%) Means and Volailiies of Opimal Porfolios for Asymmeric Range-based DCC Swiching Fees from Rollover OLS o Asymmeric Range-based DCC μ σ 5 0 5 4.643.666 0.373 0.45 0.438 6 5.35 3.05 0.787 0.96.003 7 6.060 4.384.30.670.757 8 6.769 5.744.95.550.699 9 7.478 7.03.630 3.60 3.87 0 8.87 8.46 3.445 4.88 5.36 8.895 9.8 4.36 6.99 6.6 9.604.80 5.377 7.738 8.74 3 0.33.540 6.49 9.48 0.087 4.0 3.899 7.703.6.050 5.730 5.58 9.0 3.3 4.55 9
Panel A: Close Prices S&P 500 Tbond,600 30,400,00 0,000 0 800 00 600 400 90 00 9 94 96 98 00 0 04 06 80 9 94 96 98 00 0 04 06 Panel B: Reurns S&P 500 Tbond 0 4 5 0 0-5 - -0-4 -5 9 94 96 98 00 0 04 06-6 9 94 96 98 00 0 04 06 Panel C: Ranges S&P 500 Tbond 6 5 4 3 8 4 0 9 94 96 98 00 0 04 06 0 9 94 96 98 00 0 04 06 Figure : S&P 500 Index Fuures and T-bond Fuures Weekly Closing Prices, Reurns and Ranges, 99-006. This figure shows he weekly close prices, reurns, and ranges of S&P 500 index fuures and 0-year Treasury bond (T-bond) fuures over he sample period. 30
Panel A: Volailiy Esimaes for he GARCH Model 0 6 8 4 0 99 994 996 998 000 00 004 006 S&P 500 Tbond Panel B: Volailiy Esimaes for he CARR Model 8 7 6 5 4 3 0 99 994 996 998 000 00 004 006 S&P 500 Tbond Figure : In-sample Volailiy Esimaes for he GARCH and CARR Model 3
Panel A: Correlaion Esimaes.8.6.4..0 -. -.4 -.6 -.8 99 994 996 998 000 00 004 006 reurn DCC range DCC Panel B: Covariance Esimaes 0 - - -3 99 994 996 998 000 00 004 006 reurn DCC range DCC Figure 3: In-sample Correlaion and Covariance Esimaes for he Reurn-based and Range-based DCC Model 3
Panel A: In-sample Porfolio Weighs Derived by he Reurn-based DCC Model.0.6. 0.8 0.4 0.0-0.4-0.8 -. -.6 99 994 996 998 000 00 004 006 Cash S&P 500 Tbond Panel B: In-sample Porfolio Weighs Derived by he Range-based DCC Model 3 0 - - 99 994 996 998 000 00 004 006 Cash S&P 500 Tbond Figure 4: In-sample Minimum Volailiy Porfolio Weigh Derived by he Dynamic Volailiy Model. Panels A and B show he weighs ha minimize condiional volailiy while seing he expeced annualized reurn equal a 0%. The OLS model had consan weighs for cash, sock, and bond, i.e. -0.934, 0.7079, and 0.4855. 33
Panel A: Ou-of-sample Porfolio Weigh Derived by he Reurn-based DCC Model.4.0.6. 0.8 0.4 0.0-0.4-0.8 -. -.6 00 003 004 005 006 Cash S&P 500 Tbond Panel B: Ou-of-sample Porfolio Weigh Derived by he Range-based DCC Model 3 0 - - 00 003 004 005 006 Cash S&P 500 Tbond 34
Panel C: Ou-of-sample Porfolio Weigh Derived by he Rollover OLS Model.5.0 0.5 0.0-0.5 -.0 -.5 00 003 004 005 006 Cash S&P 500 Tbond Figure 5: Ou-of-sample Minimum Volailiy Porfolio Weigh Derived by he Dynamic Volailiy Model for One Period Ahead Esimaes. Panels A, B, and C show he one period ahead weighs ha minimize condiional volailiy while he expeced annualized reurn is se a 0%. Differen from he in-sample case, he rolling sample mehod was used in esimaing he porfolio weighs. The porfolio weighs in he rollover OLS model (Panel C) also vary wih ime. The firs forecased weighs occurred he week of January 4, 00. 35