MODELING THE CURRENCY FORWARD RISK PREMIUM: THEORY AND EVIDENCE

Transcription

1 MODELING THE CURRENCY FORWARD RISK PREMIUM: THEORY AND EVIDENCE Ramapraad Bhar *, Car Chiarea and Toan M. Pham ** * Schoo of Finance and Economic Univeriy of Technoogy, Sydney PO Box 123, Broadway, NSW 2007 AUSTRALIA Fax ** Schoo of Baning and Finance The Univeriy of New Souh Wae Sydney 2052 AUSTRALIA * Correponding auhor: Emai:ramapraad.bhar@u.edu.au 1

2 MOEDELING THE CURRENCY FORWARD RISK PREMIUM: THEORY AND EVIDENCE Abrac: There i a huge ieraure on he exience of ri premia in he foreign exchange mare and i infuence in expaining he divergence beween he forward exchange rae and he ubequeny reaied po exchange rae. In hi paper, we ee o mode direcy he ri premium a a meanrevering diffuion proce. Thi i done by maing ue of he poforward price reaionhip and auming a geomeric Brownian proce for he po exchange rae. We are abe o obain a ochaic differenia equaion yem for he po exchange rae, he forward exchange rae and he ri premium which we eimae uing Kaman fiering echnique. The mode i hen appied o he French Franc/USD and Japanee Yen/USD exchange rae from 1 January 1990 o 31 December For boh currencie our main finding how (i) he perience of ubania poiive ime variaion in he forward ri premium and i aernaing regime; and (ii) he preence of a erm rucure of he forward ri premia. 2

3 MODELING THE CURRENCY FORWARD RISK PREMIUM: THEORY AND EVIDENCE 1. Inroducion Thi paper focue on a opica and imporan area of finance heory and pracice ha anaye ri premia in he foreign exchange mare. The noion ha he forward exchange rae migh be he opima predicor of he fuure po exchange rae ha been inveigaed by a number of reearcher. Thi noion deveoped a a coroary o he efficien mare hypohei. For mare paricipan i i, herefore, an imporan iue o monior wheher he forward exchange rae i an unbiaed foreca of he fuure po exchange rae. Thi unbiaedne hypohei ha been he ubjec of evera reearch paper (Enge (1996)). The ypica aring poin in hee anaye i o conider he foowing regreion of he change in he og of he po exchange rae on he forward premium: ( f ) u = a b, (1) where i he og of he po price (S) of foreign currency a ime, f, i he og of he period forward price (F) a ime, and u i he regreion error erm. The nu hypohei generay eed i ha a = 0, b = 1 and he error erm ha a condiiona mean zero. Thu, under he nu hypohei he og of he forward rae i an unbiaed predicor of he og of he fuure po exchange rae. Severa paper over he year have examined he regreion (1) wih variou improvemen in economeric echnique empoyed and he overa reu may be decribed a mixed. For exampe, Wu and Zhang (1997) empoy a nonparameric e and no ony rejec he unbiaedne hypohei bu ao concude ha he forward premium eiher conain no informaion or wrong informaion abou he fuure currency depreciaion. On he oher hand, Bahi and Naa (1997) derive an error correcion mode under he aumpion ha he po and he forward rae are coinegraed and concude uing he generaied mehod of momen ha he unbiaedne hypohei canno be rejeced. Phiip and McFarand (1997) deveop a 3

4 robu e and rejec he unbiaedne hypohei bu concude ha he forward rae ha an imporan roe a a predicor of he fuure po rae. I ha been uggeed ha he unbiaedne hypohei may be faiing empirica e due o he exience of a foreign exchange ri premium. Thi ha ed o a grea dea of reearch on he modeing of he ri premia in he forward exchange rae mare. However, mode of ri premia have been unuccefu in expaining he magniude of he faiure of unbiaedne (Enge (1996), page 124). Under raiona expecaion, ( ) = E e (2) where E i he mahemaica expecaion condiiona on informaion a and uncorreaed wih informaion a ime. We define he erm f E ( ) e i rp º, a he foreign exchange ri premium. Under rineuraiy he mare paricipan woud behave in uch a way ha f, equa ( ) forward mare pecuaion woud be zero. E and he expeced profi from Thi definiion of ri premium i baed on he raiona expecaion of he mare paricipan. Even hen, he meaure of rp may uffer from ma ampe biae. If rp coud be reaed o underying economic variabe hen i heoreica foundaion woud be firmy baed upon economic heory. Severa arice (ee for exampe a urvey in Suz (1994)) dicu he mode of foreign exchange ri premium baed on opimiing behaviour of inernaiona inveor. However, aongide uch heoreica deveopmen pure ime erie udie of rp aume a renewed imporance. In paricuar, hey are uefu in decribing he behaviour of ( ) f, E, which mode of foreign exchange ri premium ha aume raiona expecaion need o be abe o expain. Exampe of uch udie incude Bacu e a (1993) and Beaer (1994). Some reearcher, Woff (1987, 2000) and Chung (1993), have modeed hi ri premium a an unoberved componen in ae pace form and eimaed i uing he Kaman fier. The advanage of hi igna exracion approach i ha he reearcher 4

5 can empiricay characerie he empora behaviour of he premium uing ony daa on po and forward exchange rae. Thi avoid he probem aociaed wih pecifying a funciona form of he underying economic deerminan of ri premium and oher rong aumpion of he regreion baed approach. A he ame ime igna exracion mehod do no offer much inigh ino he reaionhip beween he ri premium and oher economic variabe. Woff (1987) ugge a ae pace formuaion where he ri premium and he unexpeced rae of exchange rae depreciaion are aumed uncorreaed. Cheung (1993) foow a framewor imiar o Woff and rea he unoberved ri premium a a ow order ARMA proce. In addiion, he innovaion in correaed wih E ( ) rp are aowed o be, he error from previou period foreca. Uing monhy daa Cheung (1993) find ha he fiered eimae of rp exhibi a grea dea of perience, high variabiiy and negaive correaion wih E ( ) (1991), Canova, and Io (1991) ao find high voaiiy in E ( ). Canova f,. Canova and Marrinan (1993) agree wih hee finding and furher documen high eria correaion and voaiiy cuering in he ime erie of rp. One oher common feaure of hee udie i ha he eimae of rp wiche ign during he ampe period inveigaed. For a given exchange rae, eg. USD/DEM, hi woud impy ha here are period when U.S. doar ae are conidered much afer han DEM ae and here are ime when he revere i he cae. An approach o e he hypohei ha he ri premium i a inear funcion of he condiiona variance and covariance a uggeed by andard ae pricing heory i baed on a muivariae GARCH framewor. Baiie and Boerev (1990) conider a GARCH in he mean mode uing weey daa under he aumpion of ri neuraiy and raiona expecaion. Te of heir mode fai o find uppor for hi heory. They concude a poibe vioaion of forward mare efficiency and hi coud be due o inefficien informaion proceing by mare paricipan or he fac ha oher heoreica mode are required o dea wih he ime varying ri premium. 5

6 The ri premia mode dicued above may be ermed a paria equiibrium in naure ince he ochaic proce of ae reurn i given. Duma (1993) poin ou ha a fu genera equiibrium mode wi reae hi proce o underying exogenou economic variabe. A good aring poin in hi repec i he Luca (1982) mode. Beaer (1994) dicue ome of he reaon why genera equiibrium mode canno adequaey expain he behaviour of he ri premium. The preceding anayi ugge ha he empirica evidence on he roe of he forward rae a a predicor of he fuure po rae i mixed, furhermore here eem o be an imporan infuence exered by he ri premium or even a erm rucure of ri premium. If he ize of he ri premium i unnown and i i ime varying hen he forward rae wi be a poor forecaer of he fuure po rae. I i in hi conex ha we aemp an aernaive characeriaion of he ri premium. We do hi by eeing o expoi he informaion abou he ri premium impied in he no arbirage reaionhip beween po and forward exchange rae. We ue Kaman fiering echnique o exrac hi informaion. The heoreica bacground of our approach i reviewed in Secion 2, whie a decripion of he mode i given in Secion 3. Thi i foowed by he preenaion of he Kaman fier eimaion procedure in Secion 4 and he anayi of he empirica reu in Secion 5. Finay Secion 6 concude he paper. 2. A New Framewor for he Dynamic of Ri Premia Ahough here i no unanimiy of opinion, he preceding dicuion poin ou he exience of ri premia and i infuence in expaining he divergence beween he forward exchange rae and ubequeny reaied po exchange rae. In mo cae he e rejecing he impe efficiency hypohei argumen are baed on aympoic inference. Even if he reearcher ue arge daa e, o avoid daa correaion probem wih overapped ampe, he effecive ampe ize become much maer. For exampe, when po exchange rae and onemonh forward rae are ued in he e he effecive ampe frequency become monhy. I hu eem o u ha a arge amoun of informaion in he inervening period are eiher mied or no uiied effecivey. 6

7 We propoe o adop a omewha differen approach in hi modeing exercie. We ar wih he uua aumpion in he BacSchoe opionpricing framewor and e he po exchange rae foow a geomeric diffuion proce. The andard arbirage argumen i hen appied o reae he forward exchange rae o he po exchange rae hrough he conrac period, and he reaed inere rae in he wo counrie. By appicaion of Io emma, we hen expre he dynamic of he forward price a anoher ochaic differenia equaion. I i cear ha in hi iuaion he ae underying he forward conrac i a raded ecuriy. Therefore, a dicued in Hu (1997, chaper 13) in order o price he forward conrac he inveor may be conidered rineura under he equivaen (ri neura) meaure. In operaiona erm hi impie ha under he hiorica meaure he expeced reurn par of he underying ecuriy may be repaced by anoher erm invoving he rifree rae, he mare price of ri and he voaiiy of he ecuriy. The mare price of ri, however, i no oberved in he mare and ha o be inferred from oher obervabe quaniie. Hu (1997, page 296) expain why an eimae of mare price of ri i uuay no needed o price derivaive ecuriie when he underying ae i a raded ecuriy. In our cae, however, ince we are no pricing he forward conrac a uch we incorporae he mare price of ri and rea hi an unoberved ae variabe in he yem dynamic under he hiorica meaure. Once we expre he dynamic of he mare price of ri hrough a uiabe ochaic differenia equaion, we hen have a pariay oberved yem invoving hree variabe, he po exchange rae, he forward exchange rae and he mare price of ri. Thi yem can be ca ino a ae pace form and eimaed wih he hep of he Kaman fier afer appropriae dicreiaion. The advanage of hi approach i ha we ge he fiered eimae of he mare price of ri, which can be ued o form eimae of he ri premium. I houd be noed ha we are modeing he dynamic of he mare price of ri hrough he dicreiaion period (e.g. rading day). Thu, here i no need o mach he dae for he po exchange rae wih hoe of he forward exchange rae. Thi approach herefore ha he benefi ha we are abe o uiie a of he informaion generaed hrough he rading dae, which i normay no poibe in regreionbaed approache. 7

8 For a uiabe repreenaion of he dynamic of he mare price of ri, we noe he finding in Woff (1987) and Cheung (1993). Boh hee auhor repor empirica uppor for a ow order ARMA proce for he ri premium in a number of exchange rae again he U.S. doar. Whie Woff (1987, p. 396) recognie ha he economic conen of he ri premium may be baed upon equiibrium mode of inernaiona ae pricing, i i no expiciy modeed. In hi regard our approach compimen Woff (1987) and Cheung (1993) by providing an expici modeing of he ri premium wih repec o he mare price of ri under he no arbirage condiion. We eec a mean revering form of he ochaic differenia equaion repreening he mare price of ri. By uiabe change of variabe and dicreiaion, hi mean revering form can ao be repreened a an AR (1) proce. The parameer of he ochaic differenia equaion repreening he mare price of ri are o be eimaed from he daa a we. In he nex ecion, we dicu he deai of hee modeing iue. 3. The Propoed Mode The propoed mode i a reu of hree main aumpion. Firy, under he hiorica probabiiy meaure Q (a oppoed o he ri neura probabiiy meaure ued in derivaive ecuriy pricing) he po exchange rae foow he geomeric Brownian proce () ds = m Sd SdW (3a) S where ds i he incremen of he po exchange rae, m i he expeced reurn from he po rae, S i he voaiiy of hi reurn and dw() i he incremen of a Wiener proce under he probabiiy meaure Q. The econd main aumpion i ha in efficien mare derivaive inrumen are priced in accordance wih he principe of no rie arbirage. An expreion of hi principe i ha a derivaive on foreign exchange uch a forward and opion are priced uch ha heir expeced ri adjued exce reurn i conan acro a inrumen I and foreign exchange ief and equa o he facor, he mare price of ri 1 : 1 See Ro e a (1998, pp ) for an expoiion of he fundamena reu ha in an acive, compeiive mare he mare price of ri mu be he ame for a he ae in he mare. 8

9 m (r ) = I r f I where he ubcrip I refer o he derivaive inrumen whie r and r f (3b) are he rifree inere rae in he domeic counry and he foreign counry repecivey. Conidering he foreign exchange ief we can wrie (3b) a m = r (3c) r f S which highigh he inerpreaion of a he addiiona reurn required by inveor hoding foreign currency for a uni increae in voaiiy S. Subiuing (3c) ino (3a) aow u o reexpre he dynamic of he exchange rae S a ds = ( r rf ) Sd SdW ( ) (3d) We re ha he dynamic are i under he hiorica meaure Q. The principe of no rie arbirage ao aow u o obain beween he forward exchange rae F(, T) and he po exchange rae S() he reaionhip ( rrf )( T ) F(, T) = S( ) e (3e) Uing hi, equaion (3d) for he dynamic of S and Io' emma we are abe o obain he dynamic of F under he hiorica meaure Q. The hird aumpion i ha he mare price of ri of foreign exchange ri foow a meanrevering ochaic differenia equaion. In Appendix A we how how hee hree aumpion aow u o expre he po exchange rae, he forward exchange rae and he mare price of ri a he ochaic dynamica yem ds = ( r r ) Sd SdW ( ), (4a) f S S ( ) d dw () d = (4b) df(, x ) = ( r r ) F(, x ) d F(, x ) dw( ) (4c) i f i F i p x é ù 2 æ1 e ö (, x) = 0.5 ( ( ) ) ç xú û S ë ç è ø (4d) 9

10 where: S W() = po exchange rae proce = andard Wiener proce under he hiorica probabiiy meaure = mare price of ri F, x ) = forward price F, x ) º F(, x ) wih mauriy x i ahead ( i ( i i p (, x) r r f = he ri premium for he x period ahead po rae = domeic ri free inere rae = rifree inere rae in he counry of he foreign currency = ongrun equiibrium of mare price of ri = he peed of adjumen of he proce for o i ong run equiibrium Equaion (4a), (4b) and (4c) are he ochaic procee decribing he behavior of he po exchange rae, he mare price of ri and he forward price repecivey. Equaion (4d) i he ri premium expreed in erm of he variabe embedded in Equaion (4a), (4b) and (4c). A dicued in he inroducion, auming raiona expecaion previou udie aribued he difference beween he forward rae and he ubequeny reaied po exchange rae o a ri premium and ome noie erm (eg Woff (1987), Cheung (1993)). We are, however, abe o characerie how he mare price of ri ener he expecaion formaion and hu deermine he ri premium. In fac, he noie erm idenified in Woff (1987) and Cheung (1993) can now be expained in erm of an inegra wih repec o he Wiener incremen [ee equaion (A13) and (B4) in Appendice A and B repecivey]. We woud now ie o compare he ime variaion of ri premia for differen mauriie of forward conrac obained from equaion (4d) for differen exchange rae. A a reu we wi be abe o examine he erm rucure of forward ri premia preen in he quoed forward exchange rae. To carry ou hee procedure we wi require eimae of he parameer decribing he ochaic proce for given by equaion (4b). In he nex ecion, we briefy decribe he ae pace formuaion of 10

11 he yem and eimaion of hee parameer whie a deaied echnica expoiion i conained in Appendix C. 4. Sae Space Framewor Broady peaing our empirica procedure invove he dicreiaion of he coninou yem dynamic given by he equaion (3a) hrough (3c). A number of dicreiaion cheme for ochaic differenia equaion are dicued in Koeden and Paen (1992) and we chooe o wor wih he EuerMaruyama cheme. The nex ep i o expre he dicreied yem in ae pace form. A i and, he equaion (4a) and (4c) ugge ha he diffuion erm are dependen on he ae variabe hemeve and are hu ochaic in naure. By a impe ranformaion of variabe uing he naura ogarihm and appicaion of Io emma we can ranform hee o equaion wih conan diffuion erm. Uing hee ranformaion and afer dicreiaion of he equaion (4a) hrough (4c) we obain for he ime inerva beween 1 and : ~ x (5) 2 1 = ( r rf 0.5 S ) D SD S D f ~ x (6) 2 = f 1 ( r rf 0.5 S ) D 1 SD S D ~ x (7) 1 = (1 D) D D where = ns, f = nf and x ~ ~ N(0,1) The equaion (5) (7) decribe he dynamic of he pariay oberved yem and in he ae pace framewor i i generay referred o a he ae raniion equaion. Once he yem i pecified in ae pace form a recurive agorihm uch a he Kaman fier (ee e.g. Harvey (1990)) can be appied o obain he opima eimae of he ae vecor a ime baed upon a he informaion avaiabe a ha ime. In hi ene he Kaman fier i forward ooing. However, more efficien eimae of he ae vecor and i error covariance marix can be obained if afer he iniia eimaion a he informaion up o he fina obervaion i uiied in a or of econd 11

12 pa proce. The moohing agorihm provide uch a procedure (ee Harvey (1990) for deai). In mo appicaion in finance and economic, uch a our, a he obervaion are aready avaiabe. Therefore, he moohing agorihm can be eaiy appied and ha been incorporaed in he udy of hi paper. Among oher hing, he Kaman fier provide exac finie ampe foreca. Thee foreca are ued o form predicion error a each ime ep which in urn are ued o form he og ieihood funcion from which maximum ieihood eimae of he parameer of he yem are obained. A furher by produc of hi eimaion procedure i he e of fiered eimae of he unoberved mare price of ri a each ime ep, which i hen ued o form he fiered eimae of he ri premium in equaion (4c). 5. Daa and Empirica Reu: We appy he mehodoogy ouined in Secion 4 o wo exchange rae and hree differen mauriie for he forward rae. Specificay, we ue JPY/USD and FF/USD exchangerae and forward exchange rae of 1monh, 2monh and 3monh mauriie o inveigae he variaion of he ri premia over a nine year period from January 1, 1990 o December 31, The exchange rae daa refec he daiy 4PM London quoaion obained from Daareamä and he inere rae daa are he daiy coing 3monh Treaury bi yied for he period from 1 January 1990 o 31 December Thu, he inpu o he Kaman fiering eimaion coni of he reurn on he po rae, forward rae for 1, 3, 6 monh mauriie, he bi rae in he home counry and he foreign counry (r, r f ) and he voaiiy of he reurn on he po rae,, S, being fixed a ampe vaue 2. The oupu are he parameer eimae, and he fiered eimae of { 0, 1,, T } (where T i he number of obervaion) which are hen ued o eimae he ri premium in equaion (4d). The reu are hown in Tabe 1, 2, 3 and graphed in Fig. 1, Fig. 2 and Fig. 3 from which evera obervaion and inerpreaion can be made. 2 In order o improve he abiiy of he eimaion proce he voaiiy of he reurn on he po exchange rae (ee equaion (4a)) i fixed a he vaue cacuaed a andard deviaion from he 12

13 Wih repec o he aiica ignificance of he parameer eimae (ee Tabe 1), a eimae are aiicay ignifican apar from he equiibrium mare price of ri of he French Franc,. The French Franc appear o have adjued more owy han he Japanee Yen, which i ao conien wih i ower eve of voaiiy (ee and in Tabe 1). Thi may be aribued o he fac ha ince 1979 France ha been par of he European Moneary Syem whoe purpoe i o foer currency abiiy in Europe whie he Japanee Yen ha been operaing in a reaivey free foaing environmen. From a differen poin of view he reaivey arge fucuaion of he French Franc ri premium (ee Fig. 1 and Fig. 3) in he period refec he currency urmoi in Europe which cuminaed in he currency crii of The caay for hi voaie period wa he deiberae aemp of he Bundeban o ighen moneary poicy by raiing inere rae o comba infaion (caued by he expanionary poicie o hore up he economy of Ea Germany) and o arac foreign capia o finance he reuing budge defici. We ao noice high vaue of for a hree forward exchange rae (ee Tabe 1) and hi i conien wih he finding of Canova and Io (1991) who repored high voaiiy in E ( ) f,. Furhermore, he diagnoic aiic o deermine he adequacy of he eimae of he mode mare price of ri, (ee Tabe 2) indicae ha he reidua are whie noie. Overa here i evidence upporing correc mode idenificaion wih 5 ou of he 6 parameer eimae being aiicay ignifican (ee Tabe 1) whie he behavior of he ri premium eimae of he French Franc (ee Fig. 1) refec he currency crii in Europe in he eary 1990'. Boh currencie exhibi ubania mauriy variaion in heir repecive ri premium (ee Tabe 3) and ao ign wiching beween poiive and negaive (ee Fig. 1, Fig. 2 and Fig. 3). Thu, hi finding convincingy rejec conancy of he mean of he ri premium of boh currencie paricuary for he French Franc. Whie our modeing approach i differen, hi reu of poiive eria correaion and aernaing regime ampe. Thee annuaied vaue are and for he French Franc and Japanee yen repecivey. 13

14 i conien wih previou evidence [ Enge (1996), Woff (1987, 2000), Nijman e a (1993)]. Furhermore, a 'whiene' e (ee Tabe 3) i ao performed o inveigae he behavior of he demeaned ri premia and he aiic (Tabe 3) indicae he fucuaion of he ri premia of boh currencie around heir repecive mean are nonwhie, hu furher reinforcing he perience of he poiive correaion of he ri premia. Thi feaure of he behavior of he forward ri premium i now caaogued a one of he new fac in finance (ee Cochrane (1999)). Lay he negaive ri premia (ee Tabe 3) for boh currencie are feaibe in he exane ene and conien wih recen reearch [(ee Boudouh, Richardon and Smih (1993), Odie (1998)] whie heir changing vaue acro mauriie ceary indicae a erm rucure of ri premia. On baance our empirica reu reaffirm he preence of he ime varying propery of forward ri premia whie our mode provide an inegraed framewor where he ri premium i ied o he mare price of ri in he conex of raiona expecaion and no rie arbirage. 6. Concuion: In hi paper we have preened a new approach o anaye he ri premium in forward exchange rae. Thi invove expoiing he no arbirage reaionhip ha in he po exchange rae and he forward exchange rae hrough he mare price of ri under he hiorica probabiiy meaure. By direcy modeing he mare price of ri a a mean revering proce we are abe o how how he mare price of ri ener ino expecaion formaion for a fuure po exchange rae. Thi mehodoogy aow u o quanify he ri premium aociaed wih a paricuar forward exchange rae in erm of he parameer of he proce decribing he mare price of ri. We ao demonrae how hee parameer can be eimaed in a ae pace framewor by appicaion of he Kaman fier. Thi procedure, in urn, generae he fiered and he moohed eimae he unoberved mare price of ri. We appy he procedure deveoped in he paper o French Franc/USD and JPY/USD po exchange rae and 1monh, 2monh and 3monh forward exchange rae. 14

15 For boh currencie he anayi of he reu how (i) he perience of ubania ime variaion in he forward ri premium on he poiive ide and i aernaing regime; and (ii) he preence of a erm rucure of he forward ri premia. 15

16 Tabe 1 Eimaed Parameer of he Mare Price of Ri French Franc (5.81) (1.13) (8.48) Japanee yen (12.61) (12.28) (12.88) Number in parenhee are aiic compued from andard error obained uing he heerocedaiciy conien covariance marix a he poin of convergence. The annuaied voaiiy of he po exchange rae proce i e o he ampe vaue and hee are and for French Franc and Japanee Yen repecivey. Tabe 2 Diagnoic Te of he Eimaed Mode Mare Price of Ri () 1Monh Forward 2Monh Forward 3Monh Forward French Franc Q(10) Q 2 (10) Japanee Yen Q(10) Q 2 (10) The enrie in he abe are pvaue. Q(10) meaure he LjungBox aiic (order 10) for eria correaion in he repecive reidua erie. Q 2 (10) i imiar o Q(10) bu compued wih he quared reidua. The aympoic diribuion of boh hee aiic are Chiquared wih degree of freedom 10. Tabe 3 Decripive Saiic of he Eimaed Ri Premia (p ) 1Monh Forward 2Monh Forward 3Monh Forward French Franc Mean Sd. Dev Q(10) Japanee Yen Mean Sd. Dev Q(10) Decripive aiic of he ri premia for he hree differen forward exchange rae compued from he parameer eimae in Tabe 1 and he moohed eimae of he mare price of ri. 16

17 Figure 1 Eimaed Ri Premia in French Franc Forward Exchange Rae French Franc 1M onh Forward Jan90 Jan91 Jan92 Jan93 Jan94 Jan95 Jan96 Jan97 Jan98 French Franc 2Monh Forward Jan90 Jan91 Jan92 Jan93 Jan94 Jan95 Jan96 Jan97 Jan98 French Franc 3Monh Forward Jan90 Jan91 Jan92 Jan93 Jan94 Jan95 Jan96 Jan97 Jan98 17

18 Figure 2 Eimaed Ri Premia in Japanee Yen Forward Exchange Rae Japanee Yen 1M onh Forward Jan90 Jan91 Jan92 Jan93 Jan94 Jan95 Jan96 Jan97 Jan98 Japanee Yen 2M onh Forward Jan90 Jan91 Jan92 Jan93 Jan94 Jan95 Jan96 Jan97 Jan98 Japanee Yen 3M onh Forward Jan90 Jan91 Jan92 Jan93 Jan94 Jan95 Jan96 Jan97 Jan98 18

19 Figure 3 Eimaed Ri Premia in 3Monh Forward Exchange Rae French Franc and Japanee Yen (Seeced Period) French Franc 3Monh Forward (1992) Jan92 Feb92 Mar92 Apr92 May92 Jun92 Ju92 Aug92 Sep92 Oc92 Nov92 Dec92 Jap anee Yen 3Monh Forward (1995) Jan95 Feb95 Mar95 Apr95 May95 Jun95 Ju95 Aug95 Sep95 Oc95 Nov95 Dec95 Jap anee Yen 3Monh Forward (1997) Jan97 Feb97 Mar97 Apr97 May97 Jun97 Ju97 Aug97 Sep97 Oc97 Nov97 Dec97 19

20 Appendix A Derivaion of he Sochaic Dynamica Syem Whie hi paper eeniay adop he mehodoogy of Bhar and Chiarea (2000), in order o mae i efconained he baic eemen of hi mehodoogy are ummaried in hi appendix. Le he po exchange rae foow he onedimeniona geomeric diffuion proce, ds = m Sd SdW () (A1) where m i he expeced reurn from he po ae, i he voaiiy of hi reurn, boh meaured per uni of ime and dw i he incremen of a Wiener proce under he hiorica probabiiy meaure Q. Le u define r a he domeic rifree inere rae and r f a he counerpar in he foreign currency. Since r f can be inerpreed a a coninuou dividend yied, he inananeou expeced reurn o an inveor hoding foreign exchange i ( m rf ). Thu he reaionhip beween he exce reurn demanded and he mare price of ri () houd become ( m r f ) r =, or m = ( r rf ). (A2) Thu, equaion (A1) can be rewrien a ds = ( r rf ) Sd SdW ( ), under Q. (A3) Aernaivey we may wrie ~ ds = ( r rf ) Sd SdW ( ), under Q ~ (A4) 20

21 ~ where, W ( ) = W ( ) ò 0 ( u) du and Q ~ i he ri neura probabiiy meaure. ~ We reca ha under he hiorica meaure Q, he proce W ( ) i no a andard ~ Wiener proce ince E[ dw ( )] = d ¹ 0 in genera. However, Giranov heorem aow u o obain he equivaen ri neura meaure Q ~ ~ under which W ( ) doe become a andard Wiener proce. The meaure Q and Q ~ are reaed via he RadonNiodym derivaive deai of which may be found in Koeden and Paen (1992). Uing andard arbirage argumen for pricing derivaive ecuriie (ee for exampe, Hu (1997), chaper 13), he forward price a ime for a conrac mauring a T (>), i ~ F (, T ) = E ( ST ). (A5) Bu from equaion (A4), by Io emma, d[ S( ) e ( r rf ) ~ ] = S( ) e dw ( ), ( rrf ) o ha under Q ~, he quaniy S( ) e ( rr ) f i a maringae from which i foow immediaey ha ~ ( r rf )( T ) E ( ) ST Se =, ie. ( r rf )( T ) F(, T ) = S e. (A6) If he mauriy dae of he conrac i a conan period, x, ahead hen (A6) may be wrien a, ( r r f ) x F(, x) = S e. (A7) 21

22 Le F (, x) º F(, x) and f (, x) º n F(, x), hen from (A3), (A4) and (A7) and by a rivia appicaion of Io emma we obain he ochaic differenia equaion for F under Q and Q ~. Thu, under Q ~ ~ df(, x) = ( r rf ) F(, x) d F(, x) dw ( ) (A8) whi under Q, df(, x) = ( r rf ) F(, x) d F(, x) dw ( ) (A9) wih, ( r 0 rf ) x F (0, x) = S e. We now propoe, under Q he hiorica meaure, for he mare price of ri,, he mean revering ochaic proce ( ) d dw d = (A10) where i he ongerm equiibrium mare price ri, and define he peed of mean reverion. I houd be poined ou here ha when dicreied he ochaic differenia equaion (A10) woud become a ow order ARMA ype proce of he ind repored in Woff (1987) and Cheung (1993) 3. The parameer in equaion (A10) may be eimaed from he daa uing he Kaman fier. Suppoe we have n forward price, F, x ), F(, x ),... F(, x ), hen we have a yem ( 1 2 n of (n2) ochaic differenia equaion. Thee are (under he hiorica meaure Q ), ds = ( r rf ) Sd SdW ( ), (A11a) 22

23 ( ) d dw () d = (A11b) df(, x ) = ( r r ) F(, x ) d F(, x ) dw ( ) (A11c) i f i i r f ) x i where, S ( 0) = S0, ( 0) = 0, F (0, xi ) = S0e, i = 1,2,... n. ( r I houd be noed ha he informaion conained in equaion (A11c) i ao conained in he pricing reaionhip, i ( r e r f ) x i F (, x ) = S. (A12) To eimae he parameer in he fiering framewor, however, we chooe o wor wih he equaion (A11c). From equaion (A3), we can wrie he po price a ime x a, uing ( ) = n S( ), a x x ò 2 ( x) = ( ) ( r r 0.5 ) x ( ) d dw ( ). (A13) f ò From equaion (A13) we can wrie he expeced vaue ( x) a, é E f ò ) ë x 2 ù [ ( x) ] = ( r r 0.5 ) x E ( dú û. (A14) The cacuaion ouined in Appendix B (ee in paricuar equaion (B5)) aow u o hen wrie, E 2 [ ( x) ] = ( ) ( r r 0.5 ) x ( ( ) ) f é ë æ ç1 e ç è x ö ù xú. (A15) ø úû 3 A we have poined ou in Secion 2, Woff and Cheung repor an ARMA ype proce for he ri premium ief. However, we ee from equaion (4d) ha p(, x) and () mu foow he ame ype of 23

24 The above equaion may ao be expreed (via ue of equaion (A7)) a, é æ x 1 ö ù 2 [ ( )] (, ) 0.5 ( ( ) ) ç e E x = f x xú. (A16) ç ë è ø úû Le (, x) º ( E[ ( x) f(, x)) p repreen he ri premium (under he hiorica meaure Q ) for he x period ahead po rae, hen from equaion (A16), p 2 (, x) = 0.5 ( ( ) ) é ë æ ç1 e ç è x ö ù xú. (A17) ø úû ochaic proce. 24

25 Appendix B Evauaion of Forward Expecaion The ochaic differenia equaion for (equaion 3b, Secion 3) can be expreed a, d u u u ( e ( u) ) = e e dw ( u) (B1) Inegraing (B1) from o (> ), ( e e ) ò u e ( ) e ( ) = e dw ( u), (B2) from which ( ) ( u) ( 1 e ) ò e dw ( u) ( ) ( ) = e ( ). (B3) Now inegraing (B3) from o x, x ò ) d = ( ) ò e x ( ( ) d x x ò e ( ) òò d x ( u) e dw ( u) d. The fir wo inegra in he foregoing equaion are readiy evauaed. However, in order o proceed he hird inegra need o be expreed a a andard ochaic inegra, having he dw (u) erm in he ouer inegraion. Thi i achieved by an 25

26 26 appicaion of Fubini heorem, (ee Koeden and Paen (1992)) which eeniay aow u o inerchange he order of inegraion in he obviou way. Thu, ò x d ) ( ( ) ò ò ø ö ç ç è æ ø ö ç ç è æ = x x u u x u dw d e x e ) ( 1 ) ( ) ( ( ) [ ] ò ø ö ç ç è æ = x u x x u dw e x e ) ( 1 1 ) ( ) (. (B4) Thu, ú û ù ë é ò x d E ) ( =( ) x e x ø ö ç ç è æ 1 ) (. (B5)

27 Appendix C Sae Space Framewor and he Kaman Fier Updaing Equaion For a paricuar mauriy, he dynamic of he po exchange rae, forward exchange rae and he mare price of ri are decribed by he equaion (4a) hrough (4c) in Secion 3 of hi paper. The ey concep in underanding he ae pace formuaion i he eparaion of he noie driving he yem dynamic and he obervaiona noie. Wha we oberve in pracice may no be he yem variabe direcy and hee may be maed by meauremen noie. Beide, we are deaing wih a pariay oberved yem ince he mare price of ri i no obervabe. The yem dynamic given by he equaion (4a) hrough (4c) in he paper (Secion 3) are in coninuou ime and we uuay meaure in dicree inerva, o we need o dicreie he equaion for he purpoe of impemenaion and eimaion. A number of dicreiaion cheme for ochaic differenia equaion are dicued in Koeden and Paen (1992) and we chooe o wor wih he EuerMaruyama cheme. A i and, he equaion (4a) and (4c) ugge ha he diffuion erm are dependen on he ae variabe hemeve and are hu ochaic in naure. By a impe ranformaion of variabe uing he naura ogarihm and appicaion of Io emma we can ranform hee o equaion wih conan diffuion erm. Uing hee ranformaion and afer dicreiaion of he equaion (4a) hrough (4c) (ee Secion 3) we obain for he ime inerva beween 1 and : ~ x (C1) 2 1 = ( r rf 0.5 S ) D SD S D f ~ x (C2) 2 = f 1 ( r rf 0.5 S ) D 1 SD S D ~ x (C3) 1 = (1 D) D D where x ~ ~ N(0,1) 27

28 28 The equaion (C1) (C3) decribe he dynamic of he pariay oberved yem and in he ae pace framewor i i generay referred o a he ae raniion equaion. In a muivariae iuaion i i convenien o expre hee in marix noaion and foowing Harvey (1990) hi urn ou a foow: R c a T a h = 1, (C4) where, ú ú ú û ù ë é = ú ú ú û ù ë é D D D = ú ú ú û ù ë é D D D = ú ú ú û ù ë é f = R, ) 0.5 ( ) 0.5 ( c, T, r r r r a f f and h i a ) 1 2 ( vecor of noie ource ha are eriay uncorreaed, wih expeced vaue zero and he covariance marix, ú û ù ë é D D = = h Q Cov 0 0 ).(. The obervaion in our yem are reaed o he ae variabe in an obviou way a, a Z y e = (C5) where, ú û ù ë é f = y, ú û ù ë é = = e ú û ù ë é = h h H Z 0 0 ) Cov.(, A decribed before he variance in meauremen of he obervabe are repreened by h. Anoher aumpion in hi e up i ha he noie ource in he ae and he meauremen equaion are independen of each oher. The ae pace yem require

29 pecificaion of he iniia ae vecor. A uggeed in Harvey, Ruiz and Shepherd (1994) he fir obervaion can be ued o iniiaie i if nonaionariy i upeced. Once he yem i pecified in ae pace form a recurive agorihm uch a he Kaman fier can be appied o obain he opima eimae of he ae vecor a ime baed upon a he informaion avaiabe a ha ime. A he yem given by equaion (C4) i condiionay Gauian, by recurivey cacuaing he fir wo momen of he condiiona diribuion, he Kaman fier give he minimum mean quare eimae of he ae vecor. Anoher advanage of he condiionay Gauian cae i ha he ieihood funcion can be preciey cacuaed from he predicion error and i covariance. When hi ieihood funcion i maximied wih repec o he unnown parameer of he mode heir eimae and he correponding andard error are obained. We can now wrie down he main updaing equaion of he Kaman fier for hi yem. An inuiive expanaion of he operaion of he fier can ao be found in Bhar and Chiarea (1997). ˆ If a 1i he opima eimaor of he ae vecor baed upon he obervaion up o and incuding y 1, and P 1i he covariance marix of he eimaion error hen he opima eimaor of he ae vecor a i given by, a ˆ c (C6a) 1 = T aˆ 1 and he covariance marix of he eimaion error i, P 1 = T P 1T RQ R. (C6b) The equaion (C6a) and (C6b) are he predicion equaion. Once he obervaion a become avaiabe hee eimae can be updaed a foow: 1 a aˆ P Z F ( y Z aˆ ), (C6c) ˆ = = P 1 P 1Z F Z P _ 1 P, (C6d) 29

30 where, F = Z P 1Z H. (C6e) Given he aring vaue, a 0 and P 0, Kaman fier give he opima eimaor of he ae vecor, a each new obervaion become avaiabe. The predicion error a each ep and i covariance marix can be ued o conruc he ieihood funcion, which (wihou he conan erm) for T obervaion i given by, T T = 1 = 1 1 og L = 0.5 åog F 0.5 ån F n (C7) ˆ where, n = y Z a 1. A we have een, he fier agorihm provide he opima eimae of he ae vecor, â, baed upon a he informaion up o ime. However, in our appicaion we can ao ae ino accoun of a he informaion up o T, once he maximum ieihood eimae of he parameer are obained. Thi i nown a fixed inerva moohing and we wi be uing hee moohed eimae of he mare price of ri o compue he ri premium. The moohing agorihm coni of a e of recurion aring a he fina poin and woring bacward o he aring poin. We ummarie hee equaion beow and he deai can be found in Harvey (1990, pp ) a we a Jazwini (1970, pp ): * a T ( 1 T 1 = a P a T a ), (C8a) * * P P P P P P, (C8b) T = ( 1 T 1 ) * 1 = P T 1P 1 P, for = T1, T2,, 1. (C8c) 30

31 Thee recurion require fina vaue of a and P for a and iniiaiaion a a = T T = at, PT T PT. 31

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34 34