Esmang nrnsc currency values Forex marke praconers consanly alk abou he srenghenng or weakenng of ndvdual currences. In hs arcle, Jan Chen and Paul Dous presen a new mehodology o quanfy hese saemens n a manner ha s conssen wh forex marke prces People alk abou he srenghenng and weakenng of ndvdual currences all he me, even hough he forex marke rades currency pars. The nuon behnd such saemens s he dea of nrnsc currency values, whch means value can be assocaed wh ndvdual currences n her own rgh. One sandard way o measure changes n an ndvdual currency s value s by referencng changes n he value of he correspondng rade-weghed baske. However, when appled smulaneously o several dfferen currences, he rade-weghed baske approach produces resuls ha are nconssen wh he forex marke. Purchasng power pary (PPP) s anoher mehodology, bu ha s also nconssen wh he forex marke. Boh hose mehods are based on qualave argumens, so he purpose of hs arcle s o presen a more quanave and praccal approach ha s conssen wh he valuaons observed n he forex marke, because marke conssency s mporan for fnance professonals. Wha currences are n play s a fundamenal queson n he forex marke. One way o ry o answer ha queson s o analyse forex raes usng mnmum spannng rees (McDonald e al, 005). However, he resul of he work presened here s a mehod ha quanfes he percenage rse or fall n he value of a sngle currency n s own rgh, whou usng anoher currency as numérare. Ths provdes a more drec answer o ha queson. In he pas, some work has been done n connecon wh nrnsc currency volaly. Maheu & Schoman (994) used he nrnsc currency value concep n he form of laen varables for modellng forex volaly, and ndependenly Dous (007) showed how o calculae he nrnsc currency covarance marx from mpled forex opon marke volales. Ths arcle follows Dous (007) and models nrnsc currency values usng geomerc Brownan moon processes, and hen uses a maxmum lkelhood esmaon (MLE) echnque o esmae nrnsc currency values gven marke forex raes. One aspec of he mehodology presened here s he expeced drf n he nrnsc currency values over me. Ths s deduced by lnkng nrnsc currency values o he prce of real goods n he money quany equaon from macroeconomcs. I shows ha nrnsc currency values should declne a he correspondng rae of nflaon, correspondng o he fac ha nflaon erodes he value of money over me. A new echnque for esmang he covarance marx of nrnsc currency values s also presened, whch uses spo forex daa nsead of he mpled forex opon marke volales n Dous (007). The new echnque works by mnmsng he correlaons beween he me seres of he dfferen nrnsc currency values. The resulng mnmum correlaon marx s a reasonable mach for he correlaon srucure found n Dous (007). Furhermore, s shown ha he correlaons calculaed from he me seres of nrnsc currency values ha he MLE echnque produces are conssen wh he correlaons obaned from boh of he oher echnques. As an example, nrnsc currency analyss s used o denfy acvy n he carry rade from June 007 o Sepember 007. As menoned above, one sandard way of measurng changes n he value of a currency s by referencng he correspondng radeweghed baske, so argumens are presened o show why he MLE mehod s beer. Basc model for nrnsc currency values and spo forex raes As dscussed n Dous (007), he goal of nrnsc currency analyss s o fnd varables X o represen he nrnsc values of currences, where he raos X /X j recover he observed spo forex raes beween currences and j, and where varaons n he X have a lmed nfluence on each oher where possble. Wh varables X ha do no nfluence each oher, he behavour of each X should represen wha s gong on n each ndvdual currency. As n Dous (007), nrnsc currency values wll be modelled usng correlaed lognormal sochasc processes so ha: dx d dw, dw dw j j d () X where a superscrp s used o denoe a quany a a parcular pon n me. Then he observed spo forex raes X are gven by: j X j = X () X j rsk.ne 89
A. Inrnsc currency correlaons j, calculaed from mpled forex opon volales, averaged over he sxmonh perod Oc 006 Mar 007 j USD.00 0.08 0. 0. 0.04 0.0 0.39 0.00 0.0 0.0 EUR 0.08.00 0.07 0.38 0.8 0.07 0.07 0.0 0.65 0.54 JPY 0. 0.07.00 0.0 0.07 0.04 0.0 0.0 0.04 0.04 GBP 0. 0.38 0.0.00 0.44 0.00 0.0 0. 0.9 0.9 CHF 0.04 0.8 0.07 0.44.00 0.0 0.0 0.05 0.56 0.47 AUD 0.0 0.07 0.04 0.00 0.0.00 0.04 0.48 0.06 0.07 CAD 0.39 0.07 0.0 0.0 0.0 0.04.00 0.05 0.04 0.08 NZD 0.00 0.0 0.0 0. 0.05 0.48 0.05.00 0.07 0.04 SEK 0.0 0.65 0.04 0.9 0.56 0.06 0.04 0.07.00 0.5 NOK 0.0 0.54 0.04 0.9 0.47 0.07 0.08 0.04 0.5.00 so ha X s he number of uns of currency j correspondng o j one un of currency. Ths scheme s conssen wh he sandard assumpon of lognormally dsrbued forex raes X j. Noe ha alhough he overall scale of X canno be deermned because we can mulply all X by any arbrary consan scale facor and sll recover he same forex raes X va (), he percenage changes j of X do no depend on he scale. The core problem s o esmae me seres of nrnsc currency values X, consraned so ha a each pon n me he spo marke forex raes X are recovered va (). Wh N currences here j are N ndependen exchange raes, so gven N such exchange raes, he N nrnsc currency values can be deermned f some way can be found o fx he one remanng ndependen degree of freedom. I s convenen o work n log space where he N consrans can be wren as lnear equaons. Hence defne: Z = ln X so ha he consrans are: ln( X j )= Z Z j ( ) (3) Then he column vecor Z = Z + Z corresponds o he percenage changes of he nrnsc currency values. Usng Iô s lemma, can be shown ha he jon dsrbuon of Z s a mulvarae normal dsrbuon wh mean and covarance marx, where: (4) j j j (5) Then he ask s o deermne Z a each me sep, gven he dsrbuon of Z and he consrans: Z Z j ln X j (6) for all currency pars and j. As shown below, an MLE echnque wll be used o esmae Z. However, before dscussng MLE he parameers n () need o be deermned, whch s he ask of he followng wo secons. The drf raes wll be deermned by makng a lnk o he money quany equaon n macroeconomcs. Parameers and j can eher be obaned by dong a bes f o he forex opon marke mpled volales (Dous, 007), or by usng he new correlaon mnmsaon echnque descrbed below. The macroeconomcs behnd nrnsc currency values In macroeconomc heory (Blankchard, 005, and Mankw, 006), f an economy reaches equlbrum, ha s, f he money supply equals he money demand, hen: MV = PY (7) where M s he quany of money n he economy (so he uns of M are money, for example, dollars, euros, yen, serlng); V s he ransacon velocy of money ha measures he rae a whch money crculaes n he economy (he uns of V are /me, correspondng o he me perod one s lookng a); P s he prce of a un of real goods (he uns of P are money/goods); and Y s he real GDP, ha s, he real ncome n erms of goods (he uns of Y are goods/me). Equaon (7) s called he money quany equaon. I descrbes how even when money sn backed by physcal commodes, s valuable because can be used o buy goods. So usng goods o measure he value of money suggess ha he nrnsc value of he money n an economy s gven by /P. Pung hs n a mulcurrency conex and usng he same goods baske o measure P n each economy, hen he resulng exchange raes beween he dfferen currences wll, by defnon, be he PPP exchange raes. Hence PPP nrnsc currency values can be defned by: X PPP = = Y (8) P M V PPP PPP so ha he rao X /X j wll be he PPP exchange rae beween currency and currency j. Ths shows ha he uns of nrnsc currency values are goods/money. Takng hs furher and wrng: ( M V )X PPP = Y (9) shows ha f changes n GDP Y are absorbed by changes n PPP nrnsc currency values X, hen correlaons beween economes wll correspond o correlaons beween nrnsc currency values. Ths s wha was observed n Dous (007), whch showed ha he nrnsc currency correlaon marx reflecs he economc lnks beween he assocaed economes. One popular example of usng goods o measure currency values s he Bg Mac ndex, nroduced by The Economs magazne n he 980s. Choosng he Bg Mac as he un of goods, he nrnsc value of each currency s he recprocal of he prce of a Bg Mac n ha currency, so ha he nrnsc currency value corresponds o he number of Bg Macs ha can be bough wh one currency un. The relaonshp of PPP o he forex marke has been analysed n deph elsewhere (Blankchard, 005, and Mankw, 006). Economc argumens show ha P grows a he nflaon rae n PPP currency, so X = /P wll fall a he rae of nflaon. Alhough hese argumens apply o he PPP nrnsc currency values, s reasonable o assume ha he forex marke nrnsc currency values X wll also drf downwards a he rae of nflaon. Inuvely hs corresponds o he fac ha money becomes less valuable over me. Furhermore, hs behavour s suppored by he exreme case of hypernflaon n a currency and he correspondng depre- Alhough wll no be used here, one of he resuls of hs work s an expresson for he real-world sochasc process for he forex rae X, whch can be wren: j dx j j j j j d dw j dw j X j Ths s dsnc from he rsk-neural sochasc process for X, where he drf s he dfference beween he j shor-erm neres raes n he wo currences Rsk July 008
caon ha s occasonally observed n he forex marke. To mplemen hs, he las assumpon s ha alhough dfferen counres use dfferen goods baskes o measure nflaon, he n () can all be se equal o he negave of he nflaon raes publshed by each counry. A correlaon mnmsaon echnque for calculang from spo forex daa A ypcal example of he correlaon marx j ha can be obaned from mpled forex volaly daa (Dous, 007) s shown n able A. In hs secon, we show ha a smlar correlaon srucure can be obaned from hsorcal daly spo forex daa. I was menoned above ha mulplyng all he X by a consan scale leaves all forex raes unchanged n (). In fac hs degree of freedom exss a each pon n me so he ransformaon: X X (0) recovers all forex raes for any. Assumng ha s ndependen of X, (0) mples ha: dx dx d () X X and s easy o see ha affecs he correlaon marx of he dx. For example, f = exp () where s a huge posve number hen he correlaons of he ransformed dx would all be very close o one. Ths shows ha he covarance marx of he dx can be used o ge a handle on he changes of hs ndependen degree of freedom, hus allowng he percenage changes of he X hemselves o be esmaed. As dscussed above, varaons n he dfferen X should have a lmed nfluence on each oher, where possble. To mplemen hs n connecon wh me seres of X we look for j X ha generae low correlaons, ha s, we look for: mn ˆ j j, j () where he weghs j are consan and where s defned as: ^j ˆ j n n 0 Z Z j Z n Z 0 n n n n Z n Z j 0 0 0 n Z j 0 n Z j 0 (3) and where he consrans (6) are mposed for all currency pars. Gven an nal guess for X ha sasfes he consrans, he mnmsaon () operaes over all possble n (0). The weghs j n () play an dencal role o he j ha were used n he prevous arcle (Dous, 007) when j was esmaed from mpled forex volales. Tha arcle dscussed wo schemes for j, namely: fully damped, where he same weghs j = > 0 are used for all currency pars; and parally damped, where j = 0 s used when s evden ha zero correlaons are no approprae beween a parcular currency par, and where he same j = > 0 s used for all oher currency pars. In pracce, hs means ha currency pars such as USD/CAD, AUD/NZD and European currences wh each oher have j = 0, and movng no he emergng markes so do currency pars such as USD/CNY and USD/RUB. Dous (007) wen on o show ha alhough many resuls are smlar beween he wo schemes, here were reasons for preferrng he parally damped model. In fac, a furher jusfcaon for he parally damped model became evden n summer 007 when, as a resul of marke urmol, here were days when he parally damped model worked fne bu where he fully damped model produced some nrnsc currency volaly esmaes = 0, whch makes no physcal sense. Hence he focus here wll ^ be on he parally damped model. In connecon wh he carry rade example below, boh he parally damped and fully damped models wll be used and wll be seen ha boh approaches produce smlar resuls. The mnmsaon () was done numercally usng he quas- Newon lne search algorhm n Malab. Usng many dfferen random sarng pons, was found ha he covarance marx and hence he correspondng correlaon marx ha sasfes () s unque. However, was found ha nfnely many produce ha covarance marx, so he nrnsc currency values hemselves canno be deermned by hs echnque. Usng hs procedure wh daly daa from 999 o 007, he mnmum correlaon marx for 0 major marke currences s shown n able B for he fully damped scheme and able C for he parally damped scheme. The effec of pullng he sgnfcanly posve correlaons owards zero n he fully damped scheme has he effec of forcng he low correlaons negave. Ths s exacly he same effec ha was observed n Dous (007). As dscussed n connecon wh (9), nrnsc currency correlaons should reflec he economc lnks beween currences, so wh, for example, a zero EUR/GBP correlaon n able B versus 0.35 n able C, he parally damped model s preferable. The correlaon marx n able A s an average of daly snapshos esmaed from mpled forex volales usng he parally damped model, and s forward lookng because mpled volales descrbe he expeced fuure behavour of he marke. The correlaons n able C, however, are long-erm hsorcal averages. Noneheless, he wo correlaon marces are n reasonable agreemen wh each oher. Ths shows ha pahs for nrnsc currency values can be found ha are conssen wh he correlaons calculaed from mpled forex volales, even hough hose pahs are no unque. However, wll be shown n he nex secon ha a unque pah can be consruced wh a MLE echnque, and ha hs pah also resuls n a correlaon marx ha s close o he marces shown n ables A and C. The MLE echnque below requres he covarance marx, and one way of calculang hs s usng Dous s mehodology To see why hs s he case, consder he sample covarance marx of a me seres of vecors Z for 0 n, gven by: where: Z n n 0 Z n Z Z Z Z Z 0 and where denoes ranspose. Then for any me seres of scalars ha has he same sample covarance c wh all enres of he vecor Z so ha: he me seres: where: sasfes (Z ~ )=(Z) n n 0 n Z Z c, n Z % Z c v v n n 0 n 0 rsk.ne 9
B. The mnmum correlaon marx calculaed from daly spo forex daa from Jan 4, 999 o Mar 5, 007 usng he fully damped scheme USD.00 0.6 0.43 0.39 0.7 0. 0.64 0.7 0.4 0.4 EUR 0.6.00 0.5 0.00 0.75 0.0 0.9 0.08 0.33 0.3 JPY 0.43 0.5.00 0.0 0.08 0.7 0.3 0. 0.8 0.3 GBP 0.39 0.00 0.0.00 0.00 0.0 0.4 0.09 0.5 0.5 CHF 0.7 0.75 0.08 0.00.00 0.9 0.9 0.6 0.5 0. AUD 0. 0.0 0.7 0.0 0.9.00 0.36 0.73 0.04 0.0 CAD 0.64 0.9 0.3 0.4 0.9 0.36.00 0.7 0. 0. NZD 0.7 0.08 0. 0.09 0.6 0.73 0.7.00 0.09 0. SEK 0.4 0.33 0.8 0.5 0.5 0.04 0. 0.09.00 0. NOK 0.4 0.3 0.3 0.5 0. 0.0 0. 0. 0..00 C. The mnmum correlaon marx calculaed from daly spo forex daa from Jan 4, 999 o Mar 5, 007 usng he parally damped scheme USD.00 0.0 0.8 0.9 0.03 0.0 0.5 0.03 0.0 0.0 EUR 0.0.00 0.00 0.35 0.89 0.04 0.07 0.06 0.70 0.68 JPY 0.8 0.00.00 0.0 0.09 0.04 0. 0.07 0.04 0.00 GBP 0.9 0.35 0.0.00 0.38 0.5 0.05 0.09 0.3 0.3 CHF 0.03 0.89 0.09 0.38.00 0.00 0.04 0.03 0.6 0.63 AUD 0.0 0.04 0.04 0.5 0.00.00 0.3 0.65 0.08 0.04 CAD 0.5 0.07 0. 0.05 0.04 0.3.00 0.06 0.0 0.0 NZD 0.03 0.06 0.07 0.09 0.03 0.65 0.06.00 0.05 0.03 SEK 0.0 0.70 0.04 0.3 0.6 0.08 0.0 0.05.00 0.63 NOK 0.0 0.68 0.00 0.3 0.63 0.04 0.0 0.03 0.63.00 wh mpled forex opon volaly daa. However, o ge he whole marx all currency par volales are requred, whch s a problem when ncludng emergng marke currences where forex opons are frequenly llqud, and f avalable, perhaps only agans USD. Ths mnmum correlaon mehod provdes an alernave way o esmae and only requres hsorcal spo forex raes agans USD, whch are always avalable. Maxmum lkelhood esmaon A convenen way o specfy he consrans and parameerse he one remanng degree of freedom s a each pon n me s o wre: Z s R (4) where s he column vecor wh every elemen se o one, and where R s defned by: R ln X X (5) In hs defnon s corresponds o Z and R corresponds o he consrans n (6). Snce he quanes X are he observed forex raes of he currences agans currency one, all he R can be mmedaely calculaed. Alhough currency one has been used as he reference currency n (5) so ha R = 0 for all, wll be shown below ha he same fnal resuls are obaned whchever currency s used as he reference. To oban unque esmaes X^ of nrnsc currency values, he probably dsrbuon of s wll be calculaed and he pon ha gves he maxmum lkelhood wll be chosen. Gven (5), he probably dsrbuon p(z ) s: pz exp Z Z (6) where denoes marx ranspose and where corresponds o he me perod from o +. Subsung (4) no (6) and rearrangng shows ha s s normally dsrbued: s ~ N R, Hence he maxmum lkelhood pon for s s: ŝ Es R and he sandard devaon s of s s gven by: s (7) (8) (9) From (4), s shfs he Z for all currences equally, so he error bar s s he same for all Z^. If and are consan over me, hen usng (4) and (9), he fnal resul can be wren: ˆX exp Ẑ ˆX 0 u exp Ẑ ˆX 0 exp u0 u0 R u R u u0 (0) where he error bar on Z^ s. Usng real daa, urns ou ha s s decreases as he number of currences beng consdered ncreases, so usng more currences mproves he esmaon accuracy. Ths resul (0) has wo mporan properes. Frs noe ha sums of he R over me can be wren as: u R u=0 0 = ln( X / X )= Q + Q () where Q = ln(x /X 0). Hence =0 Ru does no depend on any of he forex raes X u a mes u, so he maxmum lkelhood esmaor (0) s pah-ndependen. In oher words, as long j as and are consan over me, X^ only depends on he forex raes observed a me zero and me and no a any nermedae 0 me pons. Ths means ha evolvng X^ forward n me wh ck daa and daly daa gves he same resuls. Furhermore, a bad forex marke daa pon wll only nfluence one esmaon. Lasly, noe ha usng () n he form u =0 Ru = Q + Q, hen (0) can be wren as: ˆX ˆX 0 exp Q Q () Therefore, he resul no longer depends on he fac ha currency one was used as he reference currency n he defnon of R, because he Q dependence n (0) cancels ou. So even hough one currency mus be chosen o do he calculaons, all esmaes of he nrnsc currency values X^, ncludng he reference cur 9 Rsk July 008
Maxmum lkelhood esmaon example D. The hsorcal annualsed nrnsc currency volales Z Z Z Z = Z USD 7.6% EUR 6.9% JPY 8.85% GBP 5.00% ( Z, Z ) CHF 7.83% AUD 7.53% CAD 7.30% Z NZD 8.94% Z SEK 7.46% NOK 7.5% In hs llusrave example, we only have wo currences. They have he same varance, wh = 0 and = 0. The crcles correspond o he conours of dfferen probably denses. The doed lne s he consran Z Z = Z. The pon on he lne wh he maxmum lkelhood s ( / Z, / Z ), whch s he closes pon o he mean rency, are changng hrough me and would be he same for dfferen choces of he reference currency. Hence he X^ are reflec- ng srengh or weakness n each currency. The mechansm behnd MLE. The correlaons n ables A and C are ypcally eher posve or close o zero. However, n some suaons he MLE scheme can produce negave correlaons ha are no close o zero. For example, n he exreme case where here are only wo currences ha have he same volaly and = 0, hen he correlaon of he movemens gven by MLE s always mnus one, whaever he npu correlaon s n. In hs case, MLE always gves a 50% 50% movemen o each currency, as shown n fgure, whch makes sense because MLE does no know whch currency drves he movemen n he exchange rae. In pracce, usng more currences solves hs problem, because wh more currences here s more nformaon, so MLE knows more abou whch currency drves each movemen and he 50% 50% assgnmen dsspaes beween all he currences n he mos approprae way. Ths argumen s conssen wh he fac ha s s lower when more currences are used. Resuls Daly forex daa were obaned from Reuers beween 39 currences 3 from January 4, 999 o March 5, 007, a oal of,39 busness days. Mos of he resuls presened are for he frs 0 major marke currences, alhough he calculaons are based on all 39 currences. To mplemen (), he parally damped mnmum correlaon echnque was used o deermne. Inflaon raes are also requred for calculang, and hese are readly avalable. The correlaons of he frs 0 currences are shown n able C, and her sandard devaons are shown n able D. Fgure shows he hsorcal nrnsc values of USD, EUR, JPY and GBP esmaed usng MLE. The sar pons are all normalsed o so ha he movemens can be seen as percenage Normalsed nrnsc values of USD, EUR, JPY and GBP from Jan 999 o Mar 007 5 0 5 0 USD EUR JPY GBP 05 95 85 80 75 70 65 Jan 99 Jan 00 Jan 0 Jan 0 Jan 03 Jan 04 Jan 05 Jan 06 Jan 07 changes. The resulng graphcal represenaon of forex marke movemens s very nuve compared wh he alernave of showng he sx forex raes beween he currences, whch mxes he effecs of each ndvdual currency. The fac ha he nrnsc values of all currences are lower han n 007 corresponds o nflaon erodng he value of all he currences. The carry rade durng he subprme crss. In recen years, an mporan drvng force n he forex marke has been he carry rade, whch nvolves a rader borrowng money n a low neres rae currency lendng n a hgh neres rae currency. Ths acvy creaes demand and hence srengh n hgh neres rae currences, wh correspondng weakenng n low neres rae currences. Even hough he carry rade s drven by neres rae dfferences ha have nohng o do wh he nrnsc currency calculaons presened above, nrnsc currency analyss can deec acvy n he carry rade because shows he srengh and weakness of ndvdual currences. Srenghenng n hgh neres rae currences wh correspondng weakenng n low neres rae currences ndcaes radng acvy gong no he carry rade, whle radng ou of he carry rade s characersed by he oppose behavour. An example from summer 007 usng boh he fully damped and parally damped correlaon marces s shown n fgures 3 and 4. Durng hs perod, he lowes neres rae currency was 3 They are USD, EUR, JPY, GBP, CHF, AUD, CAD, NZD, SEK, NOK, CZK, SKK, PLN, HUF, TRY, ILS, ZAR, MXN, BRL, CLP, COP, ARS, THB, KRW, TWD, PHP, IDR, INR, SGD, CNY, HKD, RUB, XAU, XAG, HRK, KWD, PEN, PYG and ROL rsk.ne 93
3 Inrnsc currency values of USD, EUR, JPY, GBP and NZD from Jun 007 o Sep 007 calculaed usng he fully damped correlaon marx 0 08 06 04 0 98 96 USD 94 EUR JPY 9 GBP NZD Jun Jun 5 Jun 9 Jul 3 Jul 7 Aug 0 Aug 4 Sep 7 Noe: he way he graphs of JPY and NZD mrror each srongly suggess acvy no and hen ou of he carry rade 4 Inrnsc currency values of USD, EUR, JPY, GBP and NZD from Jun 007 o Sep 007 calculaed usng he parally damped correlaon marx 0 08 06 04 0 98 96 USD 94 EUR JPY 9 GBP NZD Jun Jun 5 Jun 9 Jul 3 Jul 7 Aug 0 Aug 4 Sep 7 Noe: hs graph s exremely smlar o fgure 3, where he fully damped correlaon marx was used JPY and he hghes neres rae currency was NZD, and s clear n boh fgures ha he movemens of JPY and NZD are mrrorng each oher. Togeher wh he relavely calm performance of USD, EUR and GBP, hs srongly suggess acvy no he carry rade up o around July 0, 007, and ou of he carry rade hereafer. Ths s conssen wh he commonly held belef ha n lae July 007 many marke parcpans were reducng her rsk appee. Furhermore, hese resuls show ha boh he fully damped and he parally damped correlaon esmaon schemes produce smlar resuls. Ths example llusraes how nrnsc currency analyss can be used o answer he queson Wha currences are n play?. Oher uses for nrnsc currency analyss nclude asse allocaon, for example helpng nvesors deermne he mos sable currency mx, ha s, he porfolos wh he lowes nrnsc volaly. The correlaons of he changes n nrnsc currency values. Havng calculaed me seres of nrnsc currency values, he E. The sample correlaons of he log movemens of nrnsc currency values from MLE usng 39 currences USD.00 0.0 0.7 0.6 0.0 0.04 0.49 0.06 0.0 0.0 EUR 0.0.00 0.03 0.3 0.89 0.0 0.0 0.03 0.69 0.68 JPY 0.7 0.03.00 0.05 0.07 0.06 0.09 0.09 0.07 0.0 GBP 0.6 0.3 0.05.00 0.36 0.9 0.09 0. 0.0 0.0 CHF 0.0 0.89 0.07 0.36.00 0.0 0.06 0.0 0.60 0.6 AUD 0.04 0.0 0.06 0.9 0.0.00 0. 0.65 0.05 0.0 CAD 0.49 0.0 0.09 0.09 0.06 0..00 0.04 0.05 0.04 NZD 0.06 0.03 0.09 0. 0.0 0.65 0.04.00 0.03 0.0 SEK 0.0 0.69 0.07 0.0 0.60 0.05 0.05 0.03.00 0.6 NOK 0.0 0.68 0.0 0.0 0.6 0.0 0.04 0.0 0.6.00 correlaons beween he daly changes n he nrnsc values of he dfferen currences can be calculaed. Wh he 0-cur- rency daa se, many negave correlaons are found, whereas he correlaons obaned from he 39-currency daa se shown n able E are a reasonable mach for he correlaons shown n ables A and C. These resuls show ha usng more currences does provde more nformaon o MLE, hus producng a more conssen correlaon marx. The error bars of he MLE echnque. Usng (9), he esmaon errors can be quanfed. Table F shows he confdence nervals for usng 0 and 39 currences n he MLE. Agan, he 39- currency case gves a much beer esmaon. Table F shows ha errors accumulae as me passes. However, he same error apples o all nrnsc currency values, so he error does no affec he relave valuaon of he dfferen currences. Also, over shor me perods he errors are relavely small, and n hese cases s possble o quanfy he percenage rses or falls n a currency s value wh reasonable accuracy. Comparson o rade weghed currency ndexes The dea behnd rade-weghed currency ndexes s o gauge he srengh or weakness of a counry s currency by lookng a s value relave o he currences of s radng parners, weghed accordng o rade volume. Trade-relaed forex ransacons are mporan because, unlke speculave ransacons, hey wll no be subsequenly unwound, and s also rade-relaed ransacons ha ac over long perods of me o drag exchange raes owards PPP exchange raes. Formally, he rade weghed ndexes I for currency a me are ofen defned (Whe, 997) usng: I I j where he rade weghs w j sasfy: = w j j X j X j w j (3) and where he choce of he nal values I 0 s arbrary. Fgure 5 compares he rade weghed ndex for he eurozone wh he nrnsc value of EUR calculaed usng MLE, and s clear ha here are mporan dfferences beween he wo graphs. The only consderaon n consrucng he I s o gve due wegh o each of he economy s radng parners, so ha hese ndexes are lkely o be good ndcaors of changes n an econo- 94 Rsk July 008
F. The confdence nervals of he MLE procedure for dfferen me perods s (0 ccys) s (0 ccys) s (39 ccys) s (39 ccys) D ( 0.9%, 0.9%) ( 0.38%, 0.38%) ( 0.08%, 0.08%) ( 0.5%, 0.5%) M ( 0.%, 0.%) (.79%,.8%) ( 0.35%, 0.35%) ( 0.70%, 0.7%) Y ( 3.05%, 3.5%) ( 6.0%, 6.39%) (.0%,.%) (.39%,.45%) 5Y ( 6.69%, 7.7%) (.93%, 4.85%) (.67%,.75%) ( 5.7%, 5.57%) Noe: he one- and wo-sandard-devaon nervals correspond o 68% and 95% confdence nervals respecvely 5 Inrnsc value of EUR, calculaed usng MLE, compared wh he correspondng rade weghed ndex 05 95 85 80 Inrnsc values Trade weghed ndex 75 Jan 99 Jan 00 Jan 0 Jan 0 Jan 03 Jan 04 Jan 05 Jan 06 Jan 07 Noe: normalsed o a he sar of January 999 my s compeveness ha resuls from changes n s exchange raes wh s radng parners. However, he problems wh usng he I o gauge nrnsc currency values are ha: The raos I /I j have no real meanng, so aken ogeher he I are no conssen wh he forex marke. Ths s an mporan flaw for marke praconers, for whom value s deermned by he marke. On he oher hand, () s used as a consran o ensure ha he resuls of he MLE mehodology are always marke-conssen. As dscussed n connecon wh he money quany equaon, he correlaons beween changes n nrnsc currency values reflec correlaons beween he underlyng economes, as seen n ables A, C and E. However, hs s no refleced n he correlaons beween he dfferen I, whch are shown n able G. The correlaon beween USD and CAD s a good example ( 0.4 for I n able G and +0.49 n able E). Inflaon erodes he value of money over me so nrnsc currency values should rend downwards n he long erm. The MLE mehodology ncorporaes hs hrough he drfs n (), whch are se o he negave of he nflaon raes. However, hs s no caered for n he rade-weghed ndex mehodology (3). The effec of nflaon s why he nrnsc value of EUR shown n fgure 5 ends up sgnfcanly below he eurozone rade-weghed ndex. For all hese reasons, he MLE echnque presened above s superor o usng rade-weghed ndexes o gauge changes n nrnsc currency values. Concluson Forex marke praconers are consanly alkng abou he srenghenng or weakenng of ndvdual currences, and G. The sample correlaons of he log movemens of rade weghed ndexes I USD.00 0.55 0.4 0.0 0.38 0.09 0.4 0.06 0.8 0.08 EUR 0.55.00 0.3 0.4 0.38 0. 0.07 0. 0.4 0.07 JPY 0.4 0.3.00 0.0 0.05 0. 0.0 0.4 0.07 0.05 GBP 0.0 0.4 0.0.00 0.09 0.0 0 0.05 0. 0.07 CHF 0.38 0.38 0.05 0.09.00 0.06 0.06 0.03 0.0 0.08 AUD 0.09 0. 0. 0.0 0.06.00 0.9 0.54 0.7 0.07 CAD 0.4 0.07 0.0 0 0.06 0.9.00 0. 0.3 0.07 NZD 0.06 0. 0.4 0.05 0.03 0.54 0..00 0.08 0.08 SEK 0.8 0.4 0.07 0. 0.0 0.7 0.3 0.08.00 0.08 NOK 0.08 0.07 0.05 0.07 0.08 0.07 0.07 0.08 0.08.00 when hey wan o quanfy such saemens he usual recourse s o rely on rade-weghed currency baskes. However, he nrnsc currency mehodology based on MLE presened above s much beer a quanfyng hese saemens because s fully conssen wh he forex marke, and because reflecs he nuve correlaon srucure ha exss beween he underlyng economes. One neresng aspec of hs work s he lnk beween nrnsc currency values and he money quany equaon MV = PY n macroeconomcs. I was shown ha nrnsc currency values corresponds o /P n hs equaon, and hence ha hey drf downwards over me as nflaon erodes he value of money. Hypernflaon s one exreme example of hs, and n hs case s clear ha forex marke behavour suppors hs connecon. A consequence of hs work s a dervaon for he real-world sochasc process for forex raes. As an example, was shown how nrnsc currency analyss was able o denfy acvy no and ou of he carry rade from June 007 o Sepember 007. Hence he mehod provdes nsgh no he forex marke, and consequenly helps marke parcpans rade he marke beer. Jan Chen s a quanave analys and Paul Dous s global head of quanave analyss a Royal Bank of Scoland. They would lke o hank Kevn Gaynor, head of economcs and raes research a RBS, for valuable help n developng hese deas, as well as an anonymous referee for some valuable suggesons n connecon wh he economc aspecs of hs work, and anoher anonymous referee for valuable suggesons n connecon wh he res. Emal: jan.chen@rbs.com, paul.dous@rbs.com References Blankchard O, 005 Macroeconomcs Fourh edon, Prence Hall Dous P, 007 The nrnsc currency valuaon framework Rsk March, pages 76 8 Maheu R and P Schoman, 994 Negleced common facors n exchange rae volaly Journal of Emprcal Fnance, pages 79 3 Mankw N, 006 Macroeconomcs Sxh edon, Worh Publshers McDonald M, O Suleman, S Wllams, S Howson and N Johnson, 005 Deecng a currency s domnance or dependence usng foregn exchange nework rees Physcal Revew E 7, 04606 Whe B, 997 The rade weghed ndex (TWI) measure of he effecve exchange rae Reserve Bank Bullen 60() rsk.ne 95