Revsa Conaduría y dmnsracón Edada por la Dvsón de Invesgacón de la Faculad de Conaduría y dmnsracón de la UNM hp://conadurayadmnsraconunam.mx rículo orgnal acepado (en correccón) Tíulo: Model of forecas medum erm exchange rae n an open economy. The case of he Mexcan peso uor: Mosqueda lmanza, Rubén M. y Jorge Gullén. Fecha de recepcón: 06.06.2012 Fecha de acepacón: 09.10.2012 El presene arículo ha sdo acepado para su publcacón en la revsa Conaduría y dmnsracón. cualmene se encuenra en el proceso de revsón y correccón snácca, razón por la cual su versón fnal podría dferr susancalmene de la presene. Una vez que el arículo se publca ya no aparecerá más en esa seccón de arículos de próxma publcacón, por lo que debe carse de la sguene manera: Mosqueda lmanza, Rubén M. y Jorge Gullén. (2012). Model of forecas medum erm exchange rae n an open economy. The case of he Mexcan peso, Conaduría y dmnsracón, próxma publcacón. Conaduría y dmnsracón
Model of forecas medum erm exchange rae n an open economy. The case of he Mexcan peso Fecha de recepcón: 06.06.2012 Fecha de acepacón: 09.10.2012 bsrac Rubén M. Mosqueda lmanza 1 Jorge Gullén 2 Keynes (1930) and Samuelson (1965) proposals open he possbly of machng predcably and effcency, as evdenced by he semnal sudy by Fsher (1930). Recen fndngs sugges ha he foregn exchange marke ncorporaes gradually relevan nformaon allowng he formaon of prces n a raonal manner bu no randomly. Models of exchange rae by erm based on he valuaon of asses sugges ha he ncluson of rsk n he spo rae ncreases he degree of predcably. The purpose of he paper s o show ha afer ncorporang an accurae measure of rsk, predcably of medum erm foregn exchange rae ncreases. JEL Classfcaon: F31 Keywords: Forecas Exchange Rae, Forex Marke, sse Valuaon, Rsk Premum, Modelo de pronósco del po de cambo a plazo en una economía abera. El caso del peso mexcano. Resumen. Las propuesas de Keynes (1930) y Samuelson (1965) abren la posbldad de compalblzar efcenca con predcbldad según se deduce del esudo semnal de Fsher (1930). Recenes hallazgos sugeren que el mercado de dvsas ncorpora gradualmene la nformacón relevane favorecendo la conformacón de precos de manera raconal y no aleaora. Los modelos del po de cambo a plazo basados en la valoracón de acvos sugeren que la nclusón del resgo al po de cambo spo aumena el grado de predcbldad. Los resulados muesran que ras ncorporar una medda precsa de resgo se aumena susancalmene la predcbldad de las dvsas. Clasfcacón JEL: F31 Palabras clave: Pronósco del Tpo de cambo, Mercado de dvsas, Valoracón de acvos, Prma de resgo 1 ITESM, Campus guascalenes; e-mal: ruben.mosqueda@esm.mx 2 ESN, Perú; e-mal: jgullen@esan.edu.pe Conaduría y dmnsracón
Inroducon The FOREX marke s he mos mporan fnancal marke n he world, s daly radng volume of negoaon mpacs on he behavor of oher fnancal markes or goods and servces. More recen daa show ha he daly volume of rade hs 9 rllon dollars, whch exceeded by more han one hundred mes he daly average value of he shares of Wall Sree. Therefore, s a very profable marke. Snce he FOREX marke lberalzaon and he adopon of he sysem of flexble exchange rae, n 1973, exchange raes have become ncreasngly errac and volale. Tha s why s frequen ha corporae decsons are ncreasngly adopng exchange rae forecasng echnques. Even speculave posons, proecons or arbrage of nvesors have o ake no accoun he behavor of exchange rae. The leraure shows sudes ha begn o explan, no only he composon of he curren exchange rae, bu o fnd hose elemens ha deermne her value over me. So, for more han a cenury, economss have red o esablsh and hen model he facors ha deermne he exchange rae. Precsely he economc leraure documens he exsence of fve heores ha explan he deermnans of exchange rae. One of hese models assumes effcen markes, and hen always equlbrum s acheved. lhough n pracce, s a fac dffcul o acheve due o arbrage operaons (Eun e al., 2007). The laer pon brngs an neresng dscusson: Eher exchange raes are gven a random or rraonal movemens gven ha operaors are nvolved n effcen markes or, nvesors make raonal decsons ha allow hem o ake advanage of arbrage opporunes. The cerany of he erm arbrage opporunes wll depend on he robusness of he exchange rae forecass. In hs regard, one of he frs sudes on exchange rae arbrage condons erm was proposed by Samuelson (1965) and emprcally developed by Cornell and Derch (1978) whou conclusve resuls acheved n her models. ha me he common denomnaor of hese sudes was o consder he heorecal forward rae as an unbased predcor of he exchange rae erm. However, he semnal work of Messe and Rogoff (1983) showed he poor predcve ably of models o deermne he exchange rae compared o a nave random walk model (random walk). Snce hen here has been an enormous effor boh deepen and unravel he causes of he exreme dffculy of forecasng exchange raes. They also provded alernave procedures of some predcve mprovemen over he random walk model. Conaduría y dmnsracón
Thus, recen models sugges ha he ncluson of rsk predcon models o he exchange rae wll provde more robus models, bu for hs, s essenal o capure properly. We wll develop our model by followng he lne of he laer approach. Ths sudy has been dvded no four secons. The followng secon gves parcular aenon on models based on he valuaon of asses gvng rse o he hrd secon ha analyzes he sascal robusness accordng wh he classcal rsk models ha wll perm o conclude a he end. Dfferen pproaches asc approaches abou exchange rae s predcons symmeres comes frequenly n profable arbrage operaons, herefore, hs value wll flucuae no only beween he conrbuons ha are beng made n dfferen currences (spo marke), bu also change over me. Thus, exchange rae s gven by he rao of one currency agans anoher n he followng way: S Q 1 Where, Q refers o he number of uns of currency requred o conver n erms of currency, and 1 refers o he couner-currency. (1) For more han a cenury, economss have red o esablsh and model he facors ha deermne he exchange rae 3. Recen heores ha show beer resul are hose based on he valuaon of asses, due o s ably o explan he behavor of exchange raes from moneary marke expecaons for nflaon and neres raes. There are varous models developed whn hs approach whch vary only n he degree of recognon of her mpac on exchange raes. In pracce, companes generae her own forecass, whle ohers pay o specalzed frms o do he job. Forecasng echnques would be classfed n any of he followng approaches: a) Effcen Marke Model, b) Techncal or Chars Model and c) Economc or Fundamenal Model. Table 1 collec he man feaures of he models above. [See Table 1] The model based on effcen marke, consdered by many as a purely heorecal assumpon (see Campbell e al.,1997), has had an enormous nfluence on fnancal sudes of 3 Of course, we wll refer o he exchange rae of free marke. Fxed exchange rae and managed floa regme are ruled, for example, by ssues drecly relaed o moneary polcy raher han o marke condons. Conaduría y dmnsracón
valuaon of asses, ncludng exchange rae ssue. The laer model s based on he dea of Fama (1970) where s assumed ha all he nformaon s relevan and dscouned by he agens. Thus, he curren exchange rae wll reflec all he relevan nformaon, such as nflaon, rade balance, economc growh and money supply condons; hen, he exchange rae would be affeced only f receves new nformaon, even unexpeced. s a resul, he new exchange rae s fxed ndependen of s hsorcal performance. From hs perspecve, s no surprsng ha he exchange rae follow a random walk and, herefore, he curren exchange rae would be he bes predcor of he exchange rae erm ( 1 Fw ). Hence he predcor s defned as: 1 Fw = E( S I ) 1 (2) Where I s he avalable nformaon, 1 Fw s he heorecal forward exchange rae, S s he spo exchange rae a maury. In general, s assumed ha dfferences beween 1 neres raes causes ha exchange rae 1 Fw vares wh respec o he curren exchange rae.4 In addon, anoher model called Chars, based on he approach of Caballer (1998) and effcen marke esablshes ha nformaon s auomacally refleced n prce and herefore does no jusfy he developmen of prevous sudes,.e.: If I have no dea of he value of a company, bu oher more powerful han me have and hose bg, acng n he marke, leavng a ral of s conclusons I can reach hs own conclusons (Ello, 2000: 47). Ths approach analyses fgures and sascal chars o ancpae he movemen of he currency. Followng hs lne of research, several ndcaors have been desgned (such as RSI, Fbonacc) o predc, wh accepable accuracy, he rend and srengh of he currency. I has been acceped, by he academc communy, ha he use of hese ools deermnes he mng of nvesmen or asse sale (ndrada-félx e al., 2004). However, beween hese dfferen approaches, he bes predcor can be based on economc fundamenals. Ths s because, he economc agen s raonal gven ha he relevan nformaon s beng gradually nroduced o he expecaons of nvesors. In he nex secon, we wll sudy advanced approaches o buld up our own model o fnd he bes predcor of exchange rae. 4 The man problem n hs heory s ha he acual exchange rae as he Fw or heorecal exchange rae should be good predcors of erm exchange rae. However, gmon and mnud (1981), for example, could no fnd emprcal evdence ha demonsrae ha he Fw exchange rae follows a random behavor approach. Conaduría y dmnsracón
The forward exchange rae under models based on asse valuaon The classcal heory s based on asse valuaon whch begns o calculae he pons pps o be added o he spo rae n order o fnd he forward exchange rae,.e.: Fw 1 S pps (3) We can see ha he predcve accuracy s subjec o exchange rae flucuaons wh he spo rae (S) a maury,.e., a he me of he operaon. Moreover, mprovng he mng problem deeced n he classcal model of Fsher (1930), he general equaon may be specfed by: Fw 1 S ( Spread ) n 360 Where, n s he erm forward n days; spread s he dfferenal beween he neres raes of he currences on each counres,.e. pps have o add or subrac he curren exchange rae; here he neres raes should mach o he exchange rae erm calculaed n he forward. In any case, dfferenal n neres raes represens he gap beween he neres rae n counry ( ) and he neres rae n counry ( ),.e., spread 1 (5) 1 (4) Kozkowzk (2000) shows an adjusmen o he formula (4) ha capures dfference n raes and erms 5 : Fw 1 S..( n /12) S 100 (6) nomales and rsk premums Fnancal economss have analyzed exensvely wheher he currency forward markes reflec all relevan nformaon. In hs sense, n an envronmen whou arbrage, he assumpons of rsk neuraly and raonal expecaons lead o beleve ha he neres rae should be an unbased predcor of forward exchange raes (). Ths asseron has been called unbased forward exchange rae Hypohess (UFER). 5 Emprcal Resuls are shown n Table 2. We use monhly nformaon of he foregn exchange and money markes for a perod from Sepember 2009 o January 2010. The resuls of he hree classcal proposals may sugges ha he beer equaons are specfed n equaon (4) and equaon (6). However, he DM es for non lnear specfcaon, and s mplyng here ha he possbly of consderng a rsk varable ha wll be developed furher. See ha no all he perods can be perfec forecased by he wo bes equaons. Conaduría y dmnsracón
From he begnnng, o ess he UFER we have he followng specfcaon: ln (ln Fw (7) 1 ) 1 Even hough, he null hypohess esablshed n equaon (10) defnes he condons of equaon (11),.e., 0, 1 subsequen nvesgaons abou un roo ess (un roos) have rased serous doubs abou he properes of equaon (11 ), as he convenonal wsdom abou he behavor of un roo would connue o exchange raes. Ths concern abou spurous regressons promped researchers o explore alernaves such saonary behavor n equaon (11) 6 ha led o he followng equaon: ln (ln Fw ln S ) (8) 1 1 Where ln (ln ln S ). The early esmaes were based on hs equaon (8) ha connued o rejec he UFER boh n ess of forward exchange raes a varous maures as n expermens ha used dfferen currences. In hs suaon, he sudes showed fndngs regardng he paradox of capal premum, conradcng, n hs way, he classcal heory of Fsher ha assumed ha he posons of he premum or dscoun affecng prces under he same paern. boom, seems unlkely ha he smple rule of nvesng n he hghes rae guaranee good resuls and be predcve and, hen, s an ssue ha has no been esed or fully explored. lhough an neresng fndng of hs research s o oban frequen negave esmaes 7. negave means ha nvesors would be happy o nves n currences wh hgher neres rae 8. See Table 3 ha shows he resuls of esng he dsperson of forward exchange rae as a predcor of he spo exchange rae. Researchers have come o conclude ha he nconssences found n a recen connecon o he classcal heores on exchange raes could be due o economc facors rrelevan o arac he aenon of speculaors and herefore dffcul o deec. s a resul, rases an neresng possbly o fnd hs relaonshp n whch he ncluson of rsk would be gven dependng on 6 very common nerpreaon of spurous regressons would be ha he varables are sgnfcanly assocaed bu experencng a low value of R 2, hs suggess ha addonal varables are mssng n he equaon, he absence of whch n urn explans he low value of Durbn-Wason ha experence. 7 Furher research made by Fron and Thaler (1989), for example, repored ha he values of should be a -0.88. 8 Tha mgh resul conradcory o he recommendaons of he economc heory ha says ha he nvesmen n a weak currency (for he hgh neres raes) orgn a decrease n he nvesor s capal because he currency of nvesmen s devaluaed. lhough, have evdence ha shows ha a hgher neres rae brng a guaranee o he nvesors because he currency wh hgher neres rae end o apprecae. Conaduría y dmnsracón
he behavor and level of neres raes of he currences nvolved. The possbly of a Premum Wh lle srengh on he emprcal fndngs on he UFER, Fama (1984) and Neuwland e al. (2000) are only a few aemps o reconcle he anomalous resuls found n pracce wh effcen marke heory. The exsence of a rsk premum for hem would be he varable ha would explan he exchange rae erm. The argumen s ha rsk averson nvesors have requred compensaon o movae hem o ake ha rsk, and afer relaxaon of he neuraly ha could be a rsk whle mananng he raonaly, he formaon of expecaons denoe as follows: E ( ) Fw (9) 1 Fw 1 (10) Equaon (8) emphaszes he mporance of deecng he presence of a rsk premum, capurng many economc relaons heory ofen prescrbes hose varables ha are unobservable. Ths observaon s mporan because whle economc models can be consdered robus when hey ncorporae rsk premums. Falng o deec he rsk premum means ha hese models are nadequae raher han when hey provde nformaon whou consderng he premum. For hs reason, researchers have been focusng, hrough numerous echnques for he deecon of a rsk premum n he forward marke. For example, arnhar and Szakmary (1991) ncorporaed Kalman fler exracon mehod approaches sgnals, deecng some deermnsc componens n he error. ccordng o her fndngs, he error can be characerzed n an R (1) whch can be removed wh sysemac paerns. However, he presence of a sysemac componen s no suffcen o conclude ha hs bas compensaes he rsk of speculaon agens down because hey would be leavng ou he formaon of raonal expecaons. Even he mehod proposed by arnhar and Szakmary (1991) can no deermne he possbly of rsk compensaon because does no explan he ncrease n predcably. In hs regard he semnal work of Engle (1982) on auoregressve condonal heerocedascy (RCH)-whch nally was used o verfy f he volaly of he forecas error could explan he assumpons abou he behavor of neres raes, resuled n RCH-ype models n he mean (RCH-M), and would be useful o denfy and analyze he performance of fnancal asses durng perods of urbulence bu also of peace. mong he conclusons n hs ype of works shows ha a greaer varaon n yelds he greaer creaes a clmae of uncerany Conaduría y dmnsracón
for nvesmen, so nvesors wh some degree of rsk averson wll demand compensaon for above average durng he perods of uncerany. Domowz and Hakko (1985) found ha afer a ceran level n he condonal varance by nvesors, here s an ncrease n rsk premums as compensaon rses. In hs sense, he RCH-M model deec he nfluence of condonal volaly on he condonal mean and, herefore, can measure he expecaons and nfluence of sakeholders on hgher rsk premums n more urbulen mes. However, Domowz and Hakko (1985), alle and ollerslev (1990) and ekaer and Hodrck (1993) found lle evdence o renforce he dea ha he premum depends on he condonal varance of error n he forecas. Fundamenally, he sense of volaly has been he lmng facor for explanng abnormaly. I s rue ha agens, acng on boh sdes of he marke, are subjec o he same volaly bu ha volaly alone can no explan wha makes a premum currency. Then, he RCH-M model does no explan why a currency aracs premum 9 as a naural progresson of he ssue would be o explan he relaonshp ha exs beween he sysemac rsk and he parcpan's expecaons n he currency marke. Sysemac rsk and he rsk premum I has been found ha he Fsher effec generaes a non-symmercal relaon beween nomnal neres raes and ncreased nflaon. Consequenly, he real neres rae does no correspond o he rsk-free rae, snce hey ncorporae oher ypes of rsk. The arcles ha suded he relaonshp beween sysemac rsk and he rsk premum does no make explc why a currency aracs premum as does. Cumby (1988) and Kamnsky and Peruga (1990) found some correlaon beween rsk o he deermnaon of fuure awards bu faled o ry o deec he percenage ha, a pror, be dsrbued among parcpans. Forunaely, whn he fnancal economcs here are models ha esmae wh a any degree of precson ceran dsrbuon. The crera ha follows hs knd of approach s ha s n equlbrum, hen he rsk s assocaed wh hgh performance. Thus, assumng ha nvesors have an averson o rsk and expec a premum o compensae for he hold posons n oher currences, he balance s presened as: E( R m ) E( R0 ) 0 (11) 9 alle and ollerslev (1989) concluded n her emprcal sudy ha afer usng monhly daa no found srong condonal heerocedascy ha, a pror, hey fel would be n connecon wh fner samplng nervals. Conaduría y dmnsracón
mong hs approach hghlghs he semvarance models o es he balance UFER by he equlbrum ha he nvesors ncorporae o her expecaons. Despe he resrcve assumpons under whch was orgnally desgned, even wh weak emprcal suppor, one of he mos powerful models of semvarance s he CPM. Ths model s, however, one of he ools mos used n fnancal research of asse prcng because has he good sense o propose a lnear relaonshp beween he expeced reurn on rsky asses (asses k) 10. Several researchers have red o fnd precsely modelng he rsk premum n foregn exchange markes by CPM. Whle early sudes gnored he possbly of me-varyng resuls, recen research has shown ha hey are an mporan feaure of fnancal markes. In hs sense, he CPM can be re-specfyng he expecaons for me, condonal on nformaon avalable a me -1. Harvey (1991) deeced some varaon n rsk across dfferen conexs and mes. The fnancal leraure clearly shows ha no only he rsk premum assocaed wh currency deermne he varaon n rsk over me, bu mus ake no accoun he sysemac marke rsks. Drawng on he saus of he opc, we gve he ask o derve he classcal equaon (4) n our own model. In he orgnal verson, as seen n equaon (5), he spread on neres raes (P fw ) added o he curren exchange rae s deermned by he dfference beween exchange raes subjec o exchange rae erm fw,.e., P fw 1 1 n 360 (12) However, dong a smple nspecon of he relaonshp gven n equaon (12) follows a lkely problem of non-normaly assumpon of he orgnal daa exraced from he money marke. To elmnae hs problem, he relaonshp beween he fees s reurned o s naural logarhm, hs leads us o modfy he equaon (12) by a se of represened by he followng expresson: P ndcaor we call fw, whch can be ln(1 ln(1 ) ) n 360 (13) 10 Fernández (2008) analyzes a hundred of books abou fnancal maers, and found ha 89 of hs 100 books recommend explcly CPM o calculae he requred reurn o shareholders from he rsk consdered n balance funcon. Conaduría y dmnsracón
We noe ha boh P as fw capure he so-called Inernaonal Fsher Effec, only adjus, a pror, o a less dspersed. In a prelmnary ral, as expeced, he addon of he logarhmc funcon o equaon (13) allows he daa grouped around a behavor close o normal (see Table 4). Normaly ess were done on he Fw exchange rae o a perod of one monh and for an Fw of 3 monhs. Pary analyzed n he able was abou Mexcan Peso/U.S. Dollar. The rae for he Mexcan marke was he CETE o 28 days and 91 days, whle for he U.S. marke, we consdered reasury blls wh a maury of one monh and hree monhs. The analyss of panel n Table 4, on he exchange rae a one monh, shows ha boh curves are asymmercally posve. lso, equaon (13) generaes a kuross closer o 0 n comparson o he equaon (12) whch means a more symmercal dsrbuon. Ths s rafed by he normal dsrbuon ndex ha shows greaer symmery (2.23541) n equaon (13) n conras o he based dsrbuon from equaon (12). These resuls sugges he use of equaon (13) can be more effcenly and capure he dfferenal n neres rae beween counres. In wha follows we wll denfy, precsely, he change n he behavor of unadjused spreads, consequence of even a seady behavor 11. The daa confrms he superory of λ agans P fw n ha fosers a greaer degree of normalcy. kuross (0.3761 versus -1.4916) closes o 0 and a normal dsrbuon coeffcen (0.5764 versus 1.3569) more n lne wh a symmercal curve adjusmens show ha he logarhms of he dfferences n ypes are needed o work wh values less based. Takng a look a he radonal equaon (4), we undersand ha s subjec o he spo rae bu also he behavor of neres raes n an open economy adjused sascally by a logarhmc funcon, however, we noe he absence of a permanen measure and unexpeced volaly of exchange raes 12. Snce lber (1973) nroduced he concep of porfolo nvesmen o explan he behavor of exchange raes, aenon has been focused on quanfyng 11 Engel (2000) found ha he exchange rae experenced a saonary process and, herefore, he devaons were large and perssen bu saonary, sll, even n he presence of ransacon coss. 12 In hs work, Fernandez argues ha here are 4 ypes of Rsk Premum of Marke generaed conroversy and confuson as hey obey very dfferen reales, namely: a) Rsk Premum of Hsorcal Marke: s he dfference beween he hsorcal performance of Sock Marke (of a sock ndex) and fxed ncome, b) Rsk Premum of Marke Expeced: s he expeced value of he fuure profably of he Sock Marke above he fxed ncome, c) Rsk Premum Requred Marke: s he ncremenal reurn ha an nvesor requres he sock marke (o a dversfed porfolo) over he rsk free rae (requred equy premum). Is o be used o calculae he requred reurn o acon, d) Marke Rsk Premum Implc: he marke rsk premum requred o mach he marke prce. Conaduría y dmnsracón
he errors of esmaon and marke rsk as facors ha wll ncrease he precson models o predc exchange raes. There have been several aemps o esmae correcly he level of rsk whou a srong advance n he feld. For example, realey e al. (2005) ell us he wsdom o defne low rsk premum; whle for 1996, U.S. Treasury blls were he approprae proxy for all marke rsk premums, however, some me laer, hey hemselves acknowledged ha "we have no offcal poson on he marke rsk premum, bu we beleve ha a range beween 6% and 8.5% s reasonable for he Uned Saes". In 2005, hese people changed her mnds and accep ha he nerval should be "beween 5 and 8%." I was such dsrus among he polymakers opnons ha realey e al. (2005: 154) argue: "Ths debae comes jus a frm concluson: do no rus anyone who clams o know wha reurn nvesors expec". Incluson of he rsk premum under he CPM approach s long as we focus cornersone n he developmen of our predcve equaon seems no more han a chmera, hen, our company s mpossble. However, we mus accep ha here are also mporan and sgnfcan advances n he desgn of he rsk premum. In our case we say ha he ncluson of he rsk premum n he funcon of he forward exchange rae can be denoed by. From he equaon (9) we fel he need o ncorporae a rsk premum or premum assumed by nvesors n he exchange rae so he ncluson of lke a rsk measure s reasonably necessary because perms o measure he ably of counres o comply wh s oblgaons under delays, whch, nvarably, wll be refleced n he cos of money. If he rsk premum of a counry decrease, we would expec a drop n neres raes as a sgnfcan ncrease n growh. Then, hs rsk premum would affec he relaonshp n he adjused neres rae and he spo rae as follows: Fw 1 S ln(1 ln(1 ) ) n 360 The reduced form of he forward exchange rae would be expressed as: Fw S (15) 1 (14) Conaduría y dmnsracón
To shape our rsk premum proxy ( ), a CPM approach mehodology has been used. djusmens o he CPM approach To adjus he CPM approach accordng o he parculares of he FOREX marke we wll follow some recommendaons: a) García (2009), wh respec o he adjusmens would be made o he varables ha make up he nal proposal of he CPM, b) Damodaran (2002) n relaon o an exended CPM, and c) Madura (2004) ha analyzes he addonal rsk n hs approach and ha nvesors ncorporae her expecaons. Thus, equaon (15) under he proposed Damodaran (2002) be exended o ncorporae an underlyng rsk ha s calculaed from a spread beween soveregn bonds and he relave sandard desvaon, RP, ha s: E ( R ) R ( E( R ) R ) RP (16) f m f y addng hs underlyng rsk would be solvng he problem caused by he mehods of rsk measuremen ha only added rsk o he counry rsk dscoun rae (bond spread). The assumpon ha makes Damodaran (2002) s fundamenal because s assumed, correcly, ha he whole counry rsk s no "non-dversfable. Damodaran proposal o measure he counry rsk (RP ) ncorporaes he relave sandard devaon as follows: RP spread * ONOS (17) Where: σ ONOS σ spread : The sandard devaon of he soveregn bonds for he Uned Saes. : Sandard devaon of soveregn bonds for counry : The dfference beween soveregn bonds. s proposed by Damodaran, he spread of soveregn bonds would be a useful ndcaor o approxmae quanfcaon of counry rsk, bu, no enough. The spread beween neres raes on soveregn bonds s a based ndcaor due o nvesmen polces and s no adjused for nvesmen decsons of parcpans. We herefore, propose a more specfc proxy ha wll redefne he spread accordng o he proposal made n equaon (13) snce hey correspond drecly o he expecaons se by nvesors, bu n hs case: n : The rsk-free rae of counry 's currency : The rsk-free rae of counry 's currency : Term whch corresponds boh exchange rae and depos raes fw Conaduría y dmnsracón
Then, subsung he spread for ρ n equaon (17) we have: n ln( 1 ) 360 ln(1 ) * ONOS (18) Wh respec o he adjusmens ha are made o oher varables n he CPM, we propose he followng: R f : Gven ha hs s he rsk free rae, adaps o he rsk-free rae of he base currency, rae. : The varable ha measures marke rsk. We use he proxy EMI + because corresponds o a more represenave measure of rsk for weak markes, as s he relaonshp of he Mexcan peso agans oher currences. lso, he subsuon of he classcal can be due, also o he favorable resuls obaned from he emprcal es for he EMI+ 13 (see Fgure ). E ( R m ) R f Snce our nenon s o fnd he excess marke, o capure hs varable usng he mehod of movng averages o welve monhs on he depos rae. Ths s conssen because assumes ha nvesors prefer o shape her mnmum performance expecaons from hsorcal behavor of he local rsk-free rae (counry ). Remember ha s he proxy of he rsk premum s added/subraced from he spo rae and ha hs rsk has been formed from an approach adaped from he CPM. However, n our aemp o refne he rsk varable, we should observe he followng resrcons: 1. The elemens of he rsk premum s dvded by 100 o gve hem a margnal approach, 2. The rsk premum he un. s calculaed addve bu no a resdual, so ha marke varables jon 13 I s noeworhy ha he EMI s used when dealng wh he relaonshp of currences from developed counres. In a sascal es, we checked a daa span of hree years, and we fnd ha he EMI plus EMI + were beer specfed han oher ndcaors for soveregn rsk. Research suggess ha he EMI + (along wh wo oher varables ha esed her monhly performance over he las 5 years), explaned more han 50% of he performance for dfferen emergng markes, whch dd no happen f we have aken no accoun he "ypcal ". Conaduría y dmnsracón
3. Fac_EMI + s used whch comes from dvdng he EMI + /1,000. Ths approach provdes a facor whch s nerpreed drecly as he percenage ha he nvesor n counry would have n erms of he suaon n he Uned Saes. Then, s composed as follows: 1 f Fac _ EMI ( R R m 100 R f ) ONOS nd redefnng he equaon n erms of ρ, he rsk premum can be expressed as: R Fac _ EMI ( R R ) (20) f m f 1 100 (19) Emprcal Model We wll prove f he model proposed here can wheher or no predc prces of fnancal asses. To demonsrae he degree of precson, we focus on he behavor of random errors ha explan he dfference beween he forecas 1 Fw and he exchange rae erm. Our daa ncludes Mexcan peso (MXP) agans he U.S. dollar (USD); he MXP agans he pound serlng (GP) and he MXP agans he euro (EUR) for a perod ha goes from one monh o hree monhs. The daa spans from January 2006 o May 2010, whch s colleced from loomberg. The neres raes comes from he cenral banks of he nvolved counres. The raes used are: for MXP he CETE of 28 and 91 days, for he dollar, reasury blls looked a one and hree monhs, for he pound, he one monh LIOR and hree-monh GP, and fnally for he EUR, s consdered he one-monh EURIOR rae o hree monhs. Moreover, he long-erm bonds can be approprae for he dollar Treasury onds; for he case of GP, he rsh GS o 20 years, and for he EUR onds, European governmen. Furhermore, he EMI + ndcaor s also consdered and aken from JP Morgan Chase. Esmaon errors necessary es o measure he degree of success of he funcon (14) hrough a praccal assessmen es whch measures he accuracy of forward smulaon calculaed agans he exchange rae a maury,.e.: Error Fw 1 1 (21) Conaduría y dmnsracón
Fgure s gven he behavor of he esmaon errors found for forward exchange raes o a monh. Thus, we have for he MXP agans USD a mnmum error forecas rae of -6.05%, whch means an overvaluaon wh respec o he exchange rae a maury. Moreover, as long as me goes furher he error hs 19.62% whch means ha he formula underesmaes he exchange rae. lso, MXP/GP experence greaer volaly han he prevous nerm pary. We noe ha he handcap s se a 14% and, n general, we also noe ha we never overvalued he peso agans he pound over me. However, for he case of he exchange rae MXP/Eur, we noced ha a ceran pon, he formula recognzes a full apprecaon of 7.86 and a maxmum underesmaon around 12.89%. Table 5 shows he relave values of he predcon error for he peso/currency forward whn a monh. lhough, sgnfcan levels warn volaly n he exchange rae behavor, noe ha he predcve model (14) succeeds, much hgher han oher models based only on neres raes. The ess were made on random errors and shows a predcve ably o forecas 89.81% for a monh on he MXP/USD, a 94.18% for he rae MXP/GP and a 93.18% of he MXP agans he Euro. Forecas Success n he Predcor The resuls shown by equaon (14) sugges a very good level of accuracy for he years under assessmen (close o or above 90% of accuracy). Conrary o he fndngs of ekaer and Hodrck (1993) n he sense ha, for hem, he premum does no depend on he condonal varance of forecas error, hs s ncluded wh success n equaon 14. Table 6, shows he accuracy of he classc formulas suded n hs paper o forecas he forward exchange rae of he peso agans he dollar. The resuls sugges, agan, less predcve n classcal models compared o equaon (14). Much of hs predcve falure can be explaned by he lack of a rsk premum whch does conan he equaon (18). Whle he wsdom of he classcal formula s around 52%, equaon (14) s locaed a 89; so does he prognoss of hree monhs of beng an average accuracy level of 44% n classcal equaons compared o 87% of he model (14). In he case of forecass for he pound serlng and he euro, he precson of he equaon (14) ncreases a a level above 90% vs. 50% obaned by he classc formulas. How relable are hese resuls? For hs, we apply valdaon model ha allows us o oban objecve resuls. Such s he case of U-Thel ndex whose resuls are dsplayed n Table 6 and show ha n all he predcons of equaon (14) produces conssen resuls and nvesors make raonal decsons as our predcors have a beer predcve performance han random walk snce he ndex s less han 1. Correcly predc he prognoss of adjusng he rsk premum Conaduría y dmnsracón
Despe he progress shown on he robusness of he equaon (14) s sll possble he presence of a margnal rsk. Noe he hgh "peaks" ha reach he esmaon errors: from 12.89% and 19.62%. These fndngs, however, gve us he opporuny o connue refnng our proposal, whch bascally seeks o have a greaer degree of modelng and conrol of random errors. In hs sense, Mosqueda (2008) fnds ha he earnng power heory s an approach ha would help o explan he random behavor of he varables ha make up he prospecve model. Indeed, hs heory esablshes he relaonshp beween prof and capal resources o generae and promoe profs. Here, he formaon of expecaons ress on hsorcal daa whch s esmaed from pas nformaon enough o predc fuure evens. In hs sense, equaon (14) would follow from he modelng of a rsk premum se gven s hsorcal performance: Fw 1 (15) We show he resuls on he forward exchange rae calculaed by he adjused rsk premum (see Table 7). In all cases rsk premum was oversaed. a confdence level of 95%, he regresson on he possbly of reward (15) n a forward of one monh, sugges an adjusmen o he rsk premum -0.00074, -0.0006 and -0.0019 for he USD, GP and he EUR respecvely. In hs case he forecas of he peso agans he euro demanded a greaer degree of adjusmen. lso, he regressons on he adjused premum s observed n a mulple correlaon coeffcen ha s very hgh: a R 2 of 0.9031 and an adjused R 2 of 0.8991, whch corresponds o he ypcal error levels as low. Wh respec o forward hree monhs, also recognzes he need for a greaer degree of adjusmen on he rsk premum because he behavor of he esmaon errors may become, mosly, n a volale and errac behavor over me. Thus, he same way as n he calculaon of he exchange rae one monh, he resuls sugges a correcon o he rsk premum, gven a confdence level of 95%, from -0.0347, -0.0058 and -0.0232 respecvely. The ncrease n he coeffcen of deermnaon (R 2 and R 2 dj. of 0.9370 and 0.9275 compared wh 0.9031 R 2 and adjused R 2 of 0.8991 obaned wh he equaon (14) whou adjusng for rsk) s nerpreed as an ncrease n he ably of expecaons and o dscoun he value of he spo rae. o collec he nvesor's In prevous sudes, Mark (1995), for example, found mulple correlaon coeffcens wh levels rangng from 0.179 (for he Canadan marke) up o 0771 (for he Swss franc), whou coeffcens sascally sgnfcan. y oher sde, Neely and Sarno (2002), ncorporaed he value a rsk approach, replcang he sudy of Mark (1995) wh coeffcen of Conaduría y dmnsracón
deermnaon smlar o Mark s: 0,065 for he Canadan dollar, 0.0675 o he German mark dsappeared, 0,173 for he yen and a 0,137 n he case of he Swss marke. ased on he above, we can nfer ha he adjusmen o he rsk premum for he possbly of Premum and gven a non-seasonal hsorcal behavor, s possble o acheve a hgher degree of predcve ably. Fgure C shows, by way of example, an almos perfec correlaon beween he margnal behavor of he forward rae, 1 Fw, esmaed by equaon (14), wh he percenage changes ha occur n he spo rae a maury,. In hs sense, s clear he relevance of ncorporang he rsk premum adjused,, because capures, as suggesed by Harvey (1991), he performance varan of he premum demanded by nvesors across dfferen conexs and mes. Conclusons In lgh of hese fndngs we can conclude ha he resuls are encouragng because no only models he behavor of he rsk premum bu he funcon gven n equaon (14) has a dynamc characersc ha makes more powerful. Noe, for example, ha he hgh degree of accuracy and specfcaon perm o renforce he dea ha neres raes can be an unbased proxy for he exchange rae subjec o adjusmen under he CPM approach. We noced a close and srong correlaon beween changes n spo raes and forward esmaes for he equaon (14) ha allows us o nfer ha s possble o keep prces accurae wh conssen forecas regardng he erms. The spo exchange rae self s an unbased proxy for forward exchange raes when s adjused o he expecaons ha are formed by nvesors. The laer goes along he lne of Jensen (1978) who already esablshes he need for a prme o offse rsk. Despe he fndngs of oher sudes, we found no randomly relaonshp n he approprae level of rsk ha nvesors expec, bu on he conrary, we see some raonaly n shapng expecaons of rsk ha would occur from he dfferen adjused economc posons of counres. The resuls provded by economc models, manly hose based on neres raes, are beer specfed han hose obaned n a sudy of random walk. Conaduría y dmnsracón
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Fgures and Tables Table 1. Mos relevan sudes abou he behavor of he exchange rae Source: y he auhors Conaduría y dmnsracón
RTÍCULO ORIGINL CEPTDO Table 2. Goodness conras of he classcal predcors MXP/USD usng DM Smulaed exchange Rae Fw Ec (3) Ec (4) Ec (6) Sep-2009 5.5752 ** 1.932 6.8292 ** Oc-2009 1.0209-0.9432 2.30922 * Nov-2009 5.2002 ** 1.726-0.928 Dec-2009 1.0912-0.5287 0.8427 Jan-2010 5.6039 ** 3.72628 * 3.6373 * ** Sgnfcan o 1% * Sgnfcan o 5% Tabla 3. Esmaon of spread n he Forward Exchange Rae Source: y he auhors Conaduría y dmnsracón
RTÍCULO ORIGINL CEPTDO Tabla 4. Normaly Tes on he spread n neres rae Panel. Forward one monh Panel. Forward hree monhs Source: y he auhors Fgure. Correlaon beween Mexcan Prce Index (IPyC) and counry rsk (EMI) Conaduría y dmnsracón
RTÍCULO ORIGINL CEPTDO Source: Ponce (2007) Fgure. Error Predcon over me Fw1 MXP/USD Fw1 MXP/GP Conaduría y dmnsracón
RTÍCULO ORIGINL CEPTDO Fw1 MXP/EUR Source: y he auhor Table 5. Esmaon of forecasng error Source: y he auhor Conaduría y dmnsracón
RTÍCULO ORIGINL CEPTDO Table 6. Valdaon of predcve models Conaduría y dmnsracón
RTÍCULO ORIGINL CEPTDO Tabla 7. Predcve ably of adjused rsk premum MXP/USD MXP/GP MXP/EUR R2 djused R2 No rsk premum adjusmen 0.89810 0.94180 0.93190 Typcal error 0.39544 0.61315 0.61944 Fw1 Rsk premum adjusmen 0.89884 0.9424 0.9338 0.9031 0.8991 Typcal error 0.33844 0.44359 0.47263 djusmen on he rsk premum No rsk premum adjusmen -0.00074-0.0006 0.47263 0.8714 0.9449 0.9321 Fw3 Rsk premum adjusmen 0.9061 0.9507 0.9553 Typcal error 0.38635 0.40098 0.40837 0.9370 0.9276 djusmen on he rsk premum -0.0347-0.0058-0.0232 FW1: Forward rae of one monh FW3 : Forward rae of hree monhs Conaduría y dmnsracón
RTÍCULO ORIGINL CEPTDO Fgure C. Monhly Varaon n he Exchange rae MXP/USD for a monh. Varaon n Exchange rae Fw 1 Varaon n he spo Exchange rae a maury Source: loomberg and Cenral anks Conaduría y dmnsracón