Hedging Foreign Currency Portfolios under. Switching Regimes

Transcription

1 Hedging Foreign Curreny Porfolios under Swihing Regimes Hsiang-Tai Lee Assoiae Professor Deparmen of Banking and Finane,, Universiy Rd., Puli, Nanou Hsien, Naional Chi Nan Universiy, Taiwan 5456 (886) Absra This paper invesigaes he effeiveness of hedging urreny porfolios wih muliple urreny fuures when he marke ondiion is subje o regime shifs. A Regime Swihing Dynami Condiional Correlaion GARCH model is applied o apure he sae-dependen dynamis of boh volailiy and orrelaion proesses. Ou-of-sample hedging exerises reveal ha regime swihing porfolio hedging sraegy is superior o boh sai hedging and sae-independen dynami hedging mehods. Resuls of Diebold, Mariano and Wes (DMW) ess wih adjused MCraken s riial values also show he saisial superioriy of sae-dependen hedging mehods, illusraing he imporane of inorporaing regime swihing effes in onsruing a muli-urreny porfolio hedging. Keywords: Porfolio Fuures Hedging, GARCH models, Markov Regime Swihing, Minimum Variane Hedge Raio

2 I. Inroduion Single ommodiy hedging problem has been widely invesigaed in he fuures hedging lieraure. A ommon approah o single ommodiy ondiional minimumvariane hedging is o model he seond momens of spo and fuures reurn series wih a variey of bivariae GARCH models (Baillie and Myers, 99; Kroner and Sulan, 993; Park and Swizer, 995; Gagnon and Lypny, 995; Brooks e al., 00; Bysröm, 003; Lien and Yang, 008). However, when an agen holds a porfolio of ommodiies and aemps o hedge he enire porfolio, separae hedging eah ommodiy wih i orresponding fuures onra ends o overesimae he number of fuures holding and he derived minimum variane hedge raios for separae hedging will no be he opimal hedge raios for he enire ommodiies porfolio (Gagnon e al., 998). Several reen ariles have sudied he hedging effeiveness of muli-asse porfolio wih muliple fuures onras (Gagnon e al., 998; Mun and Morgan, 003; Fernandez, 008). Gagnon e al. (998) sudy foreign urreny porfolio hedging wih a rivariae BEKK- GARCH model and found ha aouning for porfolio effes in onsruing a muliurreny hedging improves he hedging effeiveness. Mun and Morgan (003) apply a four-dimensioned BEKK GARCH model o invesigae he hedging effeiveness for large US banks wih exposure o boh ineres rae and foreign exhange risks. I is found ha a simulaneous hedging for boh ineres rae and exhange risks is superior o he separae hedging sraegy. Fernandez (008) proposes a wavele based mehod o sudy he hedging performane of porfolio of meals raded in he London Meal Exhange (LME). Empirial resuls also sugges he diversifiaion gains of porfolio hedging relaive o separae hedging. Compared o single ommodiy hedging, hese sudies

3 invesigae he enire ovariane sruure aross ash porfolio and muliple fuures onras in deermining he minimum variane hedge raios for hedging porfolio. Reen sudies reognize ha he relaionship beween spo and fuures reurns may be haraerized by regime shifs (Sarno and Valene, 000, 005a, 005b) and researhers sar o inorporaing he regime shifing properies in deermining he minim variane hedge raios. Alizadeh and Nomikos (004) apply a regime swihing leas square model o apure he sae-dependen propery beween spo and fuures reurn series. This mehod, however, limis he hedge raio o be a onsan wihin eah regime. To apure he ime-varying dynamis of hedge raio wihin regime, various regime swihing mulivariae GARCH models have been proposed for opimal fuures hedging. Lee and Yoder (007a) propose a regime swihing Varying Correlaion GARCH (VC- GARCH) model, Lee and Yoder (007b) apply a regime swihing BEKK-GARCH model and Alizadeh e al. (008) sugges a regime swihing veor error orreion BEKK-GARCH model for fuures hedging and find ha he hedging effeiveness is improved ompared o sae-independen hedging sraegies. Furher exensions in his line of researh inlude inorporaing he effes of unaniipaed jumps in deermining of opimal hedge raios (Lee, 009a), releasing he assumpion of join normaliy and hedging wih regime swihing opula GARCH models (Lee, 009b) and invesigaing he muli-regime hedging problem wih independen swihing dynami ondiional orrelaion GARCH (IS-DCC-GARCH) model (Lee, 00). All hese sae-dependen hedging sudies, however, fous only on he single ommodiy hedging problem. This sudy aemps o fill his lieraure gap by invesigaing he porfolio hedging problem when he marke ondiion is subje o regime swihing. A full swihing IS- 3

4 DCC GARCH (FIS-DCC) model is applied in his arile o sudy he problem of urreny porfolio hedging. In FIS-DCC model, he swihing behavior of boh volailiy and orrelaion dynamis ould be invesigaed. A general finding is ha imevarying porfolio hedging sraegy migh no superior o sai porfolio hedging. However, afer aking aoun of he regime swihing effes, sae-dependen imevarying porfolio hedging is superior o boh sai and sae-independen ime-varying hedging sraegies ou-of-sample. The remainder of he arile is organized as follows. Seion II gives he heoreial framework of opimal porfolio hedging. The full swihing IS-DCC model is presened in seion III. In seion IV, he porfolio minimum variane hedge raios under regime swihing and hedging performane measures are disussed. This is followed by daa desripion and empirial resuls. A onlusion ends he arile. II. Theoreial Framework of Opimal Porfolio Hedging Consider a wo-period world in whih an agen holds a porfolio of n foreign urrenies wih Q i, unis of urreny i in he porfolio a ime, i,,, n. Le P i, be he prie of urreny i a ime, he one-period reurn on ash posiion a ime is given by n Q P i n i, i, i, i, i i p,, () n r or equivalenly, Q i, P Q i, P Lee s IS-DCC (00) model resris he swihing dynamis only for orrelaion proess. The advanage of his resried model is ha i avoids he problem of over parameerizaion and is less ompuaional inensive. A full swihing IS-DCC (FIS-DCC), however, is more flexible and allows swihing behaviors no only in he orrelaion bu also in he volailiy proesses. Boh hedging models are onsidered in his paper. 4

5 where r w p, wi, ri, ω r,, () i n Q i, i, i, is he weigh on urreny i held a ime n i Q P i, P i, ω is he porfolio weighing veor, r i, is he one- ime, w, w, wn, period reurn on urreny i a ime and r r r and arry o, r,, n, is he spo reurn veor. Wihou loss of generaliy, i is assumed ha he agen holds one uni of eah urreny, namely, Q, i,,, n. The weigh on urreny i an be expressed as w i n i, Pi, / Pi, whih is ime-varying. Suppose ha he agen would like o hedge i he ash porfolio wih muliple fuures onras, he reurn for he enire hedging porfolio r p is hen given by r, ω r βr, (3) p where f,, f, β is n veor of hedge raios and r f, rf, rfn, n r is n veor of reurns on he fuures posiions. The variane of he agen s hedging porfolio is given by (Fernandez, 008) where are r ωω ω βω β ωω β Var, (4) p,,,n,,,n Ω and, n, n, nn f f f, f, f,n f, f, f,n Ω f, f, n f, n f, nn n n variane-ovariane maries of he reurns on he ash and fuures posiions, respeively, and Ω f f f f,,, n f, f, f, n f,n f,n f, nn 5

6 is an n n marix of ross ovarianes of spo and fuures reurns. To derive he opimal hedge raios for he hedging porfolio, derive he porfolio variane wih respe o β and solve he equaion dvar rp f f dβ he opimal hedge raio is given by βω ωω 0 Ω Ω ω, (5) β*. (6) f f Alernaively, he opimal hedge raio an be expressed using he relaionship of ash porfolio and individual fuures onras as in Gagnon e al. (998). Tha is Ω f Σf β*, (7) where Σ Ω ω f is an n veor of ross ovarianes of he enire ash porfolio f and fuures onras. III. Full Swihing IS-DCC (Independen Swihing Dynami Condiional Correlaion) GARCH Model To esimae he opimal hedge raios derived in equaion (7), a full swihing version of Lee s independen swihing DCC-GARCH (IS-DCC) is applied. IS-DCC (Lee, 00) is a modifiaion of he Markov regime swihing dynami ondiional orrelaion GARCH (MS-DCC; Caporin and Billio, 005) suh ha he problem of pah-dependeny is avoided and is free from he umbersome reombining proedure. Lee s IS-DCC resris he swihing dynami only o he orrelaion proess o make he invesigaion of muli-regime hedging problem feasible. To invesigae he swihing behaviors of boh volailiy and orrelaion dynamis, a full-swihing IS-DCC (FIS-DCC) is also applied in his paper. In his seion, he speifiaion of sae-independen DCC-GARCH is given firs and he desripion of FIS-DCC is followed. Suppose ha he observed -dimensioned eonomi proess R is given by 6

7 R μ, (8) e μ D ε, (9) where T μ is a n veor of ondiional means, T sands for ranspose, mean of he h f normally disribued fn is he ondiional mean of spo porfolio and i urreny fuures, e T e e e D ε H f fn fi is he ondiional is assumed o be e ~ N 0,, (0) where ε D R μ is he normalized residual veor, N sands for mulivariae normal disribuion, is he informaion se up o ime and H is he imevarying variane-ovariane marix whih an be deomposed ino volailiy and orrelaion omponen given by H D Γ D, () where diag, D, i f, f,, h i, f n is a diagonal marix wih he volailiies of spo and fuures reurns on he i h elemen. Eah ondiional variane dynami are assumed o follow a GARCH(,) proess h e h, i f, f,,. () i, i i i, i i,, f n speified as The sae-independen orrelaion sruure Γ proposed by Engle (00) is Γ Q / Q diagq / diag, (3) where given by Q is he ime-varying ondiional sandardized residual ovariane marix and is 7

8 Q Q ε ε Q, (4) where Q is he unondiional ovariane marix of he sandardized residuals and an be replaed by he sample ovariane marix T Q ε ε o simplify he esimaion T i i i (Bauwens, Lauren, and Rombous, 006). When he marke ondiion is subje o regime shifing, o implemen a regime swihing hedging sraegy, he opimal minimum variane hedge raios are esimaed wih FIS-DCC. The speifiaion of FIS-DCC is given below: R s e s μ, (5) where, s s s μ D ε, (6) s is he sae variable assumed o follow a firs-order, wo-sae Markov proess, μ s s s s is f fn T n sae-dependen ondiional means veor and he sae-dependen residual veor s D s ε s be normally disribued s ~ N0 H s e is assumed o e,, (7) where he sae-dependen ime-varying variane-ovariane marix deomposed as s s Γ s D s H D diag D, i f, f,, s h i,., f n H an be furher (8) is he sae-dependen diagonal volailiy marix. To avoid he pah-dependeny problem inheren in he regime swihing GARCH volailiy proess (Cai, 994; Hamilon and Susmel, 994; Gray, 996; and Lee e al., 007a and 007b), eah ondiional variane dynami is assumed o follow an 8

9 independen swihing GARCH(,) proess suggesed by Hass e al. (004) and Lee (009b): h s i s i s ei, i s hi,, i, f, f,, f n i,. (9) To inorporae regime shifs in he orrelaion proess of Engel s DCC model, Caporin and Billio (005) inrodue a Markov regime swihing dynami ondiional orrelaion (MS-DCC) model. In MS-DCC, he ondiional sandardized residual ovariane marix is speified as Q s s Q s ε ε s Q, (0) Given he join presene of regime swihing and ime-varying orrelaion in eah regime in equaion (0), reombining proedure is required o solve he well-known pahdependeny problem. Analog o Kim s filer (994), Caporin and Billio (005) propose a modified Hamilon filer (Hamilon, 989 and 994) for esimaing MS-DCC. MS-DCC, however, requires umbersome reombining proedure (Lee, 00). Following Hass e al. s idea (004), Lee (00) proposes an independen swihing DCC-GARCH (IS-DCC) o solve his problem. The independen swihing ovariane proess in IS-DCC is given by s s Q s ε ε s Q s Q s, () and he orresponded orrelaion dynami is s diag Q s / Q s diagq s / Γ. () Wih his speifiaion, he ovariane proess in eah regime evolves independenly whih avoids he pah-dependeny problem. Furhermore, beause eah ovariane proess evolves in parallel aording o differen se of parameers, he 9

10 eonomi signifian of he ovariane dynamis are preserved (Hass e al., 004; Lee, 00). The esimaion proedure of FIS-DCC is given in he appendix A. IV. Opimal Fuures Holdings under Regime Swihing and Measuring Hedging Performane For single ommodiy hedging, he opimal ime-varying minimum variane hedge raio is esimaed wih where Cov r, r, r f,, (3) Var f, r, and r, are single ommodiy spo and fuure reurns. Under he independen f swihing ondiion, Lee (009b) derives a formula for single holding, sae-dependen hedge raio given by r,,, rf,, p Covr,,, rf,, Varr p Varr p Cov, (4) p f,, f,, where p is he regime probabiliy of being in sae one and, and r f, i,, i, are r i, sae-dependen reurns on ash and fuures posiions, respeively. The hedge raios for an independen swihing muli-ommodiy porfolio an be derived as χ p Ω p Ω p Σ p ff, ff, f, Σ f,, (5) r p This an be proved as follows. Le r p be he hedging porfolio reurn whih is given by porfolio reurn in sae p porfolio reurn in r s r s p r s χr s p sae p p f p f χ, (6) Deriving he variane of he sae-dependen hedging porfolio reurn using he assumpion of independen swihing gives equaion (5). r p, wih respe o χ and 0

11 where Ωff, and Ω ff, are he ovariane maries of fuures holdings when he marke is in he sae one and wo, respeively, and Σ f, and Σ f, are he ovariane maries of ash porfolio and fuures holdings when he marke is in he sae one and wo, respeively. Hedging performane is evaluaed from boh a risk-minimizaion and a uiliy sandpoin. To minimize he hedging porfolio risk, agen hooses a hedging sraegy o minimize he variane of he enire hedged porfolio reurn. Under he porfolio hedging framework, he hedging effeiveness is denoed as r ωω ω χ p Var rp Ωf χ Varr ωω ω Var ωω f χ HE, (7) p where χ is he opimal fuures holdings esimaed from FIS-DCC, IS-DCC and saeindependen DCC-GARCH models. Besides regime swihing effes, he dynami hedging effes are also invesigaed by omparing he hedging performanes of FIS- DCC, IS-DCC and DCC wih sai mulivariae ordinary leas square (OLS) mehod. As is usomary in he fuures hedging lieraure, he eonomi benefis of hedging wih dynami models over sai model are invesigaed wih mean-variane expeed uiliy funion (Kroner and Sulan, 993; Gagnon e al., 998; Lafuene and Novales, 003; Alizadeh and Nomikos, 004; and Lee e al., 006): E U r Er Varr p, p, p,, (8) where is he oeffiien of absolue risk aversion and E sands for expeaion operaor. A dynami hedging sraegy is onsidered o be superior o a sai ordinary leas square (OLS) mehod if i has higher expeed uiliy ne of ransaion oss.

12 To es he saisial superioriy of FIS-DCC and IS-DCC over DCC and OLS, Diebold-Mariano (995) and Wes (996) (DMW) ess are performed. The DMW es saisi is given by DMW d, (9) N V Where d f f A, B, T d R, d N ˆ T, V N dˆ d R denoes he lengh of esimaion period, N is he lengh of he prediion period, T is he, R sample size, f is he square error loss funion and v r χ ˆ and A, p, A, rf, v χ ˆ B, rp, B, rf, wih χˆ A, and χˆ B, he hedge raios esimaed from model A and model B, respeively. The es is one-sided wih he null hypohesis ha he prediive abiliy of model B is no superior o model A H f f 0 0 E A, B,. (30) Beause hedging sraegies onsidered in his paper are nesed models, MCraken s riial values (MCraken, 007) whih depend on he N / R raio and he number of addiional esimaed parameers in he more flexible model are applied. V. Daa Desripion and Empirial Resuls Three foreign urrenies, Briish pound (PB), Canadian dollar (CD) and Ausralian dollar (AD) denominaed in US dollar are invesigaed in his arile o illusrae he porfolio hedging effeiveness under regime swihing irumsane. Three urreny porfolios are onsidered inluding he porfolio of BP and AD, he porfolio of BP and CD and he porfolio of BP, AD and CD. Spo and fuures pries are Wednesday

13 pries of nearby urreny fuures onras for he period of January 99 o Deember 009 obained from Daasream. Tuesday s losing prie is used when a holiday ours on Wednesday. Esimaion of all models was ondued using daa from January 99 o Deember 008; he remaining daa are used for ou-of-sample analysis. The reurns are ompued as he hanges in he naural logarihms of pries muliplied by 00 and esimaion of alernaive models are ondued wih GAUSS program. Summary saisis of he prie and reurn series for BP, AD and CD are given in able I. All unondiional mean reurns are small. The larges unondiional mean reurns are 0.09% and 0.087% in absolue values for spo and fuures daa, respeively. Ausralian dollar has he larges unondiional volailiies whih are equal o.5498% and.59% for spo and fuures daa, respeively. Based on he skewness, lepokurosis and Jarque-Bera saisis, mos of he unondiional disribuions of urreny spo and fuures reurns are asymmeri, fa-ailed, and non-gaussian. Parameer esimaes of DCC, IS-DCC and FIS-DCC are presened in able II. The parameer esimaion seps for FIS-DCC are illusraed in Appendix A. As shown in able II, all esimaed ondiional mean ' s of DCC and IS-DCC are small. This is onsisen wih he small average reurns repored in able I. As for he FIS-DCC, he ' s refle he ondiional mean reurns in eah regime. The implied ondiional reurns are equal o he weighed average of sae-dependen ' s, weighed by he regime probabiliies. As a onsequene, he ' s in FIS-DCC have wider ranges. The value of ondiional reurn ould be as large as 0.988% ourred in he porfolio of BP, CD and AD. In he volailiy equaion, refles he degree of volailiy persisene. Mos of he values of 3

14 are lose o indiaing ha volailiy persisene for all daa is high and modeling volailiy dynamis wih GARCH proesses is adequae. In he orrelaion equaion, refles he memory in orrelaion. Take he porfolio of BP and CD for insane, he orrelaion memory of sae-independen DCC is As for he sae-dependen FIS-DCC, he orrelaion memory is in he sae and in he sae. Beause for he porfolio of BP and CD, sae is he sae of higher reurn volailiy, i is found ha when he marke is urmoil, he orrelaion memory among spo porfolio and differen fuures onras are also higher. In general, mos of he parameers esimaed in he IS-DCC and FIS-DCC models are signifian. This jusifies he using of mixure disribued regime swihing models for fiing daa. Table III repors he in and ou-of-sample hedging effeiveness of alernaive hedging sraegies. A general finding is ha sai mulivariae OLS has he bes hedging effeiveness for he in-sample exerises. For he porfolio of BP and CD, he OLS esimae of he variane of hedged porfolio reurn is equal o This provides a 9.05% variane reduion ompared o unhedged spo posiion. Among he dynami hedging models, FIS-DCC has he bes hedging effeiveness wih 9.6% variane reduions for he porfolio of BP and CD. Generally speaking, boh IS-DCC and FIS- DCC ouperform sae-independen DCC for all hree porfolios invesigaed. This illusraes he imporane of inorporaing regime swihing properies in he urreny porfolio hedging. Ou-of-sample performane is also invesigaed beause for he hedger wha maers mos is he hedging performane in he fuure no in he pas. I is found ha ouof-sample, dynami DCC model does no neessarily provide beer hedging 4

15 effeiveness ompared o sai OLS model. However, by aking aoun of he regime swihing effes, IS-DCC and FIS-DCC are superior o boh sae-independen DCC and sai OLS hedging mehods. In he porfolio of BP and AD, DCC has a variane of or a variane reduion of 83.38%. This is superioriy o OLS wih a variane reduion of 8.99%. In he porfolio of BP and CD, however, DCC has a variane reduion of 93.99% whih is inferior o OLS wih a 94.37% variane reduion. In boh wo-urreny porfolios, allowing he orrelaion dynami o be subje o regime swihing provides beer hedging performanes. IS-DCC has 94.70% and 83.78% variane reduions in he porfolio of BP and CD and porfolio of BP and AD, respeively. Besides he orrelaion dynami, if we also allow he volailiy dynamis o be sae-dependen, FIS-DCC provides even higher variane reduions. FIS-DCC is he bes performer for boh he porfolio of BP and CD and he porfolio of BP and AD wih respeively 96.% and 86.7% variane reduions. The ou-of-sample improvemens of FIS-DCC over OLS are.85% and 3.74% for he porfolio BP and CD and he porfolio of BP and AD, respeively. In he hree-urreny porfolio, IS-DCC is he bes performer. I has a variane reduion of 89.75%. This is followed by FIS-DCC, OLS and DCC wih variane reduions of 89.30%, 88.74% and 89.75%, respeively. In general, regime swihing models, eiher IS-DCC or FIS-DCC exhibi superior ou-of-sample hedging effeiveness ompared o sae-independen and sai hedging mehods. The eonomi signifiane is also invesigaed as is usomary in he fuures hedging lieraure o ake aoun of he rebalaning os inurred in he dynami hedging models. Based on he expeed uiliy funion given in equaion (8), i is found ha hedger an always benefi from regime swihing dynami hedging ompared o sai 5

16 OLS hedging in erm of uiliy improvemen. Assuming a oeffiien of absolue risk aversion equal o 4 (Lafuene and Novales, 003; Alizadeh and Nomikos, 004; and Lee e al., 006), shown in able III, FIS-DCC provides larges uiliy gains ompared o sai OLS for wo-urreny porfolio and IS-DCC provides larges uiliy gain for he hree-urreny porfolio. Taking porfolio of BP and CD for insane, he average weekly varianes of hedged porfolio reurns from OLS and FIS-DCC hedging are 0.9 and 0.44, respeively. The hedged porfolio reurns from OLS and FIS-DCC hedging are 0.006% and -0.00%, respeively. Based on equaion (8), if an invesor adops OLS hedging, he average weekly uiliy is.006% 40.9% 0.84% U OLS 0. Wih FIS-DCC, he average weekly uiliy is 40.44% 0.57% U -0.00%. FIS DCC The hedger s ne benefi from using FIS-DCC hedging over OLS hedging is equal o U FISDCC U OLS C 0.70% C, where C is he rebalaning os from dynami rebalaning. Sine he ypial round rip ransaion os is around 0.0% o 0.04% whih is lower han 0.70%, hedger will benefi from dynami FIS-DCC hedging even afer aking aoun of ransaion oss. To hek he saisial superioriy of regime swihing hedging sraegies, he Diebold-Mariano (995) and Wes (996) (DMW) es is performed wih adjused riial values repored by MCraken (007) for nesed models. As repored in able IV, i is found ha FIS-DCC is saisially superior o IS-DCC, DCC and OLS a 95% in he porfolio of BP and CD and is saisially superior o IS-DCC, DCC and OLS a 90% in he porfolio of BP and AD. As for he hree-urreny porfolio of BP, CD and AD, IS- DCC is saisially superior o DCC and OLS a 95% and 90%, respeively. IS-DCC and FIS-DCC, however, are saisially indifferen. 6

17 Figures o 3 show he OLS and FIS-DCC esimaes of opimal fuures holding for BP, AD and CD, respeively. 3 The OLS hedge raios are onsan and he ondiional hedge raios of FIS-DCC are ime-varying and very volaile revealing ha adjusmen of he hedging porfolio using dynami hedging sraegies is highly required. Figure 4 7 show he sae-dependen volailiies of ash porfolio, BP fuures, AD fuures and CD fuures, respeively. In general, sae is a higher volailiy sae and sae has relaively low and less volaile volailiies. For example, he ash porfolio has a mean variane of.409 and a variane on variane of.335 in he high volailiy sae and has a mean variane of 0.69 and a variane on variane of 0.79 in he low volailiy sae. The regime probabiliies of being in he lower volailiy regime are ploed in figures 8. Figure 9-3 illusrae he sae-dependen orrelaions among ash porfolio, BP fuures, AD fuures and CD fuures. In general, in he high volailiy sae, namely sae, he orrelaion among differen asses is more volaile. Taking he orrelaions of CP and CD fuures for insane, he orrelaion volailiy are equal o 0.4 for sae whih is muh higher han ha in he sae wih a orrelaion volailiy of VI. CONCLUSIONS The fous of his arile has been invesigaing he effes of regime swihing on urreny porfolio hedging. Alhough single urreny hedging problem has been widely invesigaed in he hedging lieraure, urreny porfolio hedging problem under swihing regimes, however, are rarely sudied. This paper invesigaes he effeiveness of hedging urreny porfolios wih muliple urreny fuures using a sae-dependen IS- 3 here. To save spae, only hose figures for hree-urreny porfolio of BP, CD and AD are illusraed 7

18 DCC model. In-sample hedging exerises show ha among dynami hedging sraegies, sae-dependen IS-DCC model is superior o sae-independen DCC GARCH model. Ou-of-sample, dynami DCC does no neessarily provide hedging benefis ompared o sai OLS mehod. However, by aking aoun of he regime swihing effes, IS-DCC and FIS-DCC are superior o boh sae-independen DCC and sai OLS hedging sraegies. Diebold, Mariano and Wes (DMW) ess and evidenes on uiliy gains also suppor he superioriy of IS-DCC and FIS-DCC over DCC and OLS ou-of-sample. This illusraes he imporane of inorporaing regime swihing properies in onsruing he urreny porfolio hedging sraegies. 8

19 Table I Summary Saisis for Spo and Fuures Pries and Reurns Briish Pound Canadian Dollar Ausralian Dollar Spo Fuures Spo Fuures Spo Fuures Level % Reurn Level % Reurn Level % Reurn Level % Reurn Level % Reurn Level % Reurn Mean Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera 68.6*** 837.7*** 68.7*** 44.6*** 67.35*** 06.4*** 68.9*** 04.85*** *** 4.989* *** Noe:. * and *** indiae signifiane a he 0% and % levels, respeively.. Reurns are alulaed as he differenes in he logarihm of prie muliplied by 00. 9

20 Table II Esimaes of Unknown Parameers of Alernaive Models. Daa period is from January 99 o Deember 008 Porfolio of BP and CD Porfolio of BP and AD DCC IS-DCC FIS-DCC DCC IS-DCC FIS-DCC Mean Equaion () (0.0) (0.0) (0.089)*** (0.06) (0.04) 0.37 (0.049)*** ( ) (0.089) (0.049)*** f () (0.03) (0.030) (0.6)*** 0.00 (0.06) (0.04) 0.3 (0.05)*** f () 0.05 (0.036)* (0.0)* f () -0.0 (0.04) (0.04)* (0.093) 0.07 (0.046) (0.054) 0.0 (0.089)*** f () (0.06) -0.6 (0.6)** Volailiy Equaion () (0.0)*** (0.009)***.497 (0.790)** (0.08)*** (0.06)*** (0.6)*** ( ) 0.05 (0.007)*** 0.08 (0.09)*** f () (0.04)*** 0.04 (0.06)***.39 (4.84) 0.07 (0.09)*** (0.0)** (0.30)*** f ( ) (0.0)*** 0.09 (0.06)*** f () 0.04 (0.0)*** (0.00)*** 0.44 (0.9)* 0.0 (0.033)*** 0.7 (0.03)*** 0.07 (0.035)** f ) 0.00 (0.008)*** 0.84 (0.80) () 0.6 (0.04)*** (0.0)*** 0.00 (0.0)** 0. (0.08)*** 0.00 (0.06)*** (0.09)*** ( ) 0.06 (0.007)*** 0.5 (0.03)*** f () 0.0 (0.0)*** (0.06)*** (0.04) 0.08 (0.03)*** (0.09)*** (0.003) f ( ) 0.07 (0.009)*** 0.04 (0.04)*** f () 0.0 (0.0)*** 0.7 (0.07)*** 0.04 (0.094)** 0.35 (0.033)*** 0.30 (0.09)*** 0.09 (0.07)** f ( ) 0.04 (0.0)*** 0.90 (0.060)*** ( ) 0.80 (0.0)*** (0.08)*** (0.6)*** 0.8 (0.08)*** (0.04)*** 0.40 (0.87)*** ( ) 0.93 (0.05)*** (0.06)*** f ( ) 0.84 (0.030)*** (0.03)*** (0.49)* (0.036)*** (0.09)*** (0.94) f ( ) (0.07)*** (0.0)*** f ( ) 0.86 (0.04)*** (0.0)*** (0.53)*** 0.85 (0.07)*** 0.8 (0.06)*** (0.03)*** f ( ) 0.99 (0.0)*** 0.80 (0.8)*** Correlaion Equaion (0.03)*** (0.054)*** (0.008) (0.03)*** (0.0)*** (0.067) (0.00)*** 0.07 (0.00)*** 0.79 (0.044)*** (0.034)** (0.033)*** 0.09 (0.047) (0.00)*** (0.03)*** 0.97 (0.07)*** (0.405) (0.04)*** (0.9)*** (0.04)*** 0.93 (0.047)*** LL Noe:. DCC, IS-DCC, FIS-DCC sand for Dynami Condiional Correlaion GARCH, Independen Swihing DCC GARCH and Full Swihing IS-DCC GARCH models, respeively. Figures in parenheses are sandard errors, LL sands for he log-likelihood values and *, ** and *** indiae signifiane a he 0% level, 5% level and % level, respeively. 0

21 f () ( ) () f () f () f () f 3 () f 3 () () ) ( f () f ( ) f () f () f 3 () f 3 ) Table II Coninue Esimaes of Unknown Parameers of Alernaive Models Daa period is from January 99 o Deember 008 Porfolio of BP, CD and AD DCC IS-DCC FIS-DCC Mean Equaion (0.00) (0.04) 0.03 (0.03) (0.74) (0.08) (0.040) (0.035) 0.64 (0.574) (0.034) 0.04 (0.040) 0.06 (0.034) (0.404)*** (0.033) (0.03) (0.0) Volailiy Equaion (0.40) 0.05 (0.00)*** (0.00)*** 0.95 (0.46)*** 0.03 (0.009)*** 0.04 (0.06)*** (0.00)***.476 (0.966)* 0.04 (0.05)*** 0.7 (0.035)*** 0.36 (0.037)***.350 (0.559)*** 0.05 (0.036)* (0.0)*** (0.0)*** 0. (0.097)** 0.07 (0.008)** () (0.0)*** 0.07 (0.009)*** 0.80 (0.073)*** ( ) 0.04 (0.006)*** f () (0.04)*** 0.04 (0.009)*** (0.05) () f 0.03 (0.006)** f () 0.7 (0.08)*** 0. (0.09)*** (0.93)*** f () (0.0)*** f 3 () 0.3 (0.08)*** 0. (0.08)*** (0.089)*** f 3 () (0.008)*** ( ) 0.85 (0.09)*** 0.86 (0.08)*** (0.073)*** ( ) 0.97 (0.09)*** f ( ) 0.96 (0.0)*** (0.04)*** (0.3)*** f ( ) (0.07)*** f ( ) 0.8 (0.07)*** 0.83 (0.030)*** 0.7 (0.6)*** f ( ) 0.97 (0.06)*** f 3 ( ) 0.80 (0.05)*** 0.80 (0.06)*** 0.57 (0.090)*** f 3 ( ) 0.96 (0.03)*** Correlaion Equaion 0.03 (0.006)*** 0.03 (0.007)*** 0.08 (0.0)** 0.6 (0.063)*** 0.05 (0.006)*** 0.93 (0.07)*** (0.03)*** (0.08)*** (0.06) (0.87)** LL Noe:. DCC, IS-DCC, FIS-DCC sand for Dynami Condiional Correlaion GARCH, Independen Swihing DCC GARCH and Full Swihing IS-DCC GARCH models, respeively.. Figures in parenheses are sandard errors, LL sands for he log-likelihood values and *, ** and *** indiae signifiane a he 0% level, 5% level and % level, respeively.

22 Table III In- and Ou-of-Sample Hedging Effeiveness of Alernaive Models for Porfolio of Briish Pound and Canadian Dollar, Porfolio of Briish Pound and Ausralian Dollar and Porfolio of Briish Pound, Canadian Dollar and Ausralian Dollar Porfolio of BP and CD In-sample Ou-of-sample Variane of Hedged Porfolio Reurns Perenage Variane Reduion Improvemen of FIS-DCC over Oher model 4 Variane of Hedged Porfolio Reurn Perenage Variane Reduion Improvemen of FIS-DCC over Oher model Hedged Porfolio Reurn Expeed Weekly Uiliy 5 Uiliy Gain of Dynami Hedging Models over OLS 6 Unhedged OLS % -0.43% %.85% DCC %.84% %.% IS-DCC %.4% %.5% FIS-DCC % % Porfolio of BP and AD In-sample Ou-of-sample Variane of Hedged Porfolio Reurns Perenage Variane Reduion Improvemen of FIS-DCC over Oher model Variane of Hedged Porfolio Reurn Perenage Variane Reduion Improvemen of FIS-DCC over Oher model Hedged Porfolio Reurn Expeed Weekly Uiliy Uiliy Gain of Dynami Hedging Models over OLS Unhedged OLS % -.35% % 3.74% DCC % 0.00% % 3.35% IS-DCC % -0.54% %.95% FIS-DCC % % Porfolio of BP, CD and AD In-sample Ou-of-sample Variane of Hedged Porfolio Reurn Perenage Variane Reduion Improvemen of IS-DCC over Oher model 7 Variane of Hedged Porfolio Reurn Perenage Variane Reduion Improvemen of IS-DCC over Oher model Hedging Porfolio Reurn Expeed Weekly Uiliy Uiliy Gain of Dynami Hedging Models over OLS Unhedged OLS % -0.46% %.0% DCC % 3.59% % 5.00% IS-DCC % % FIS-DCC %.87% % 0.45% Noe:. Varianes of hedged porfolio reurns are in perenage.. Perenage variane reduions are alulaed as he differenes of variane of unhedged posiion and esimaed variane of aleraive models over variane of unhedged posiion muliplied by DCC, IS-DCC, FIS-DCC sand for Dynami Condiional Correlaion GARCH, Independen Swihing DCC GARCH and Full Swihing IS-DCC GARCH models, respeively. 4. Improvemen of FIS DCC over oher hedging sraegies is defined as he differene of he perenage variane reduion of FIS DCC and he perenage variane reduion of alernaive hedging sraegies 5. Expeed weekly uiliy is alulaed based on equaion (8) 6. Uiliy gains of dynami hedging models over OLS are defined as he differenes of he expeed uiliies of alernaive dynami models and he expeed uiliy of OLS. 7. Beause IS-DCC has he bes ou-of-sample hedging effeiveness for he porfolio of Briish pound, Canadian dollar and Ausralian dollar, he improvemens of IS-DCC over oher hedging sraegies are repored.

23 Table IV Diebold-Mariano-Wes Tes Saisis of No Superioriy of FIS-DCC and IS-DCC over DCC and OLS Sample Period is from January 009 o Deember 009 Porfolio of BP and CD Porfolio of BP and AD Porfolio of BP, CD and AD Addiional Addiional Addiional DMW Saisis Esimaed Parameers DMW Saisis Esimaed Parameers DMW Saisis Esimaed Parameers FIS-DCC vs. IS-DCC.377** 3 (k =).76* (k =) (k =6) FIS-DCC vs. DCC.738** (k =6).59* (k =6).7* (k =0) FIS-DCC vs. OLS.94** (k =8) 4.94* (k =8) (k =35) 4 IS-DCC vs. DCC (k =4) 0.54 (k =4).66** (k =4) IS-DCC vs. OLS (k =6) 0.68 (k =6).* (k =9) DCC vs. OLS (k =) 0.43 (k =) -.68* (k =5) MCraken s Adjused Criial Values %-ile k 4 99% % % k 99% % % k 5 99% % % k 6 99% % % k 9 99% % % k 0 99% % % Noe:. The DMW saisi is shown in equaion (9) wih he adjused riial values for nesed models abulaed in MCraken (007).. The N / R raio is and he number of addiional esimaed parameers for more sophisiae models is denoed as k. The riial values are abulaed for N / R 0 and 0., and he values for N / R are onsrued by inerpolaion. 3. *, ** and *** indiae signifiane a he 0% level, 5% level and % level, respeively. 4. MCraken abulaes he riial values wih a maximum k equal o 0. If DMW is signifian a 0, i will be also signifian for k 0. If DMW is insignifian a k 0, however, we anno onlude ha i will be also insignifian for k 0. 3

24 Figures OLS and FIS-DCC Hedge Raios for BP Fuures 0.8 OLS FIS-DCC Time Figure OLS and FIS-DCC Hedge Raios for BP Fuures OLS and FIS-DCC Hedge Raios for AD Fuures OLS FIS-DCC Time Figure OLS and FIS-DCC Hedge Raios for AD Fuures OLS and FIS-DCC Hedge Raios for CD Fuures OLS FIS-DCC Time Figure 3 OLS and FIS-DCC Hedge Raios for CD Fuures 4

25 6 Sae Dependen Volailiies of Cash Porfolio 4 0 Sae (High Volailiy Sae) Sae (Low Volailiy Sae) Time Figure 4 Sae Dependen Volailiies of Cash Porfolio Sae Dependen Volailiies of BP Fuures 0 Sae (High Volailiy Sae) Sae (Low Volailiy Sae) Time Figure 5 Sae Dependen Volailiies of BP Fuures 60 Sae Dependen Volailiies of AD Fuures Sae (High Volailiy Sae) Sae (Low Volailiy Sae) Time Figure 6 Sae Dependen Volailiies of AD Fuures 5

26 Sae Dependen Volailiies of CD Fuures Sae (High Volailiy Sae) Sae (Low Volailiy Sae) Time Figure 7 Sae Dependen Volailiies of CD Fuures Regime Probabiliies of Lower Volailiy Sae Time Figure 8 Regime Probabiliies of Lower Volailiy Sae 0.95 Sae Dependen Correlaions of Cash Porfolio and BP Fuures Sae Sae Time Figure 9 Sae Dependen Correlaions of Cash Porfolio and CD Fuures 6

27 0.9 Sae Dependen Correlaions of Cash Porfolio and AD Fuures Sae Sae Time Figure 0 Sae Dependen Correlaions of Cash Porfolio and AD Fuures Sae Dependen Correlaions of Cash Porfolio and CD Fuures Sae Sae Time Figure Sae Dependen Correlaions of Cash Porfolio and CD Fuures Sae Dependen Correlaions of BP Fuures and AD Fuures Sae Sae Time Figure Sae Dependen Correlaions of BP Fuures and AD Fuures 7

28 0.7 Sae Dependen Correlaions of BP Fuures and CD Fuures Sae Sae Time Figure 3 Sae Dependen Correlaions of BP Fuures and CD Fuures 0.8 Sae Dependen Correlaions of AD Fuures and CD Fuures Sae Sae Time Figure 4 Sae Dependen Correlaions of AD Fuures and CD Fuures 8

29 Appendix A. Esimaion and Hamilon Filer for he Full Swihing IS-DCC model The sysem parameers are esimaed by maximizing he following log-likelihood funion wih respe o he unknown parameer veor L T log f R, (A) where T is he oal number of observaions and f R is he mixure disribuion weighed by regime probabiliy. Hamilon filer (Hamilon, 989, 994) is applied o evaluae he regime probabiliy and he esimaion proedure for he full swihing IS- DCC is depied below: (i) Given he filered probabiliies ξ ˆ projes he sae probabiliies ξ ˆ, (A) Pξ ˆ where ˆ p s ξ, ˆ p s p s p s ξ, (A3) and P is a ransiion probabiliy marix wih he j i, elemen p s j s i defined as p p s s exp p 0 s, (A4) exp p 0 q 0 q exp s, (A5) exp 0 Where p 0 and q 0 are unresried parameers o be esimaed. (ii) Evaluae he regime dependen likelihood i i i Q i ε ε iq i Q, (A6) 9

30 i Q i / Q i diag Q i / Γ diag, i,, (A7) diag i h j, D, j f, f,, h, (A8) i i i e ih i, j f, f,, j, j j j, j j, i D i Γ i D i f n. (A9), H, (A0) f R s i, m / H i / exp ' R μ i H i R μ i f n, (A) where m is he number of spo ommodiies onained in he ash porfolio. Define f R s η, (A) f R s he densiy of i, a ime. R ondiional on pas observaions and being in regime (iii) Evaluae he mixure likelihood R ξ ˆ η f, (A3) where is an m veor of ones and denoes elemens-by-elemens mulipliaion. (iv) Updae he join probabiliies The sae-probabiliy is updaed wih he following equaion ξˆ ξˆ ξˆ η η (A4) 30

31 (v) Ierae (i) o (iv) unil he end of he sample and he likelihood is obained as a by-produ of his filer L T log ξˆ η (A5) To iniialize he filer, he regime probabiliies are se equal o he unondiional probabiliies. 3

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