1. The Currency Composition of Foreign Reserves

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1 Inroducion Cenral banks hold foreign reserves o faciliae inernaional rade and as a mean of financing exchange rae inervenions in he foreign exchange marke. Reserves can herefore be hough of as a buffer sock, held for precauionary purposes. 1 Cenral banks' oal demand for reserves should herefore depend on facors such as he magniude of foreign rade, he marginal propensiy o impor, he opporuniy cos of holding reserves, curren accoun volailiy and exchange rae arrangemens. 2 The relaive demand for reserves, i.e. he composiion of reserves, depends however on such facors as he real rae of reurn on currencies and he volailiy of hese reurns, he cenral banks' risk aversion, ransacion coss and, possibly, addiional noneconomical facors and insiuional arrangemens. The purpose of his paper is o sudy he relaive demand of cenral banks for currency asses, aking he Cenral Bank of Iceland as an example. 3 The paper is divided ino five secions, plus appendices. The firs secion describes he hisorical developmen of currency composiions of cenral banks and discusses he main deerminans of relaive currency demand. The second secion analyses in more deail he mean-variance and he ransacion cos models, which are he wo main models for explaining he currency composiion of reserves. In he hird secion we calculae opimal porfolios for he Cenral Bank of Iceland in he period 1987 o 1993, using he mean-variance model, and compare he resuling porfolios wih acual daa. The fourh secion conains esimaion of demand funcions for relaive currency demand, using wo alernaive modelling procedures. The fifh secion concludes. 1. The Currency Composiion of Foreign Reserves 1.1. Hisorical Developmen Looking a he currency composiion of reserves of all counries we see ha he share of he U.S. dollar has gradually been decreasing, alhough he dollar is sill by far he mos imporan reserve asse. However, hese changes in he composiion of foreign reserves have encompassed somewha conrasing behaviour on he par of indusrial 1 This is he sandard assumpion in his lieraure. I seems a highly plausible assumpion for any cenral bank, especially for small cenral banks such as he Cenral Bank of Iceland. See, for example, Ben-Bassa (1980), Rikkonen (1989) and Dellas and Yoo (1991). 2 There are number of sudies available on he oal demand for reserves. See, for example, Frenkel and Jovanovic (1981), Heller and Kahn (1978), Landell-Mills (1989), Leho (1994), Lizondo and Mahieson (1987) and Roger (1993). 3 Even hough oal and relaive demand for reserves are in some ways linked, i is commonly assumed ha hese decisions are independen. This radiion is followed here.

2 and developing counries. For indusrial counries, he dollar share has decreased, bu for developing counries here is acually a small increase in he dollar share. The drop in he dollar share has largely been mached by an increase in he share of he German mark. Comparison of currency composiions of indusrial and developing counries also shows ha he currency composiion in he developing counries is more sable han in he indusrial counries. 4 This can largely be explained by flucuaions in he currency composiions of indusrial counries, reflecing he effecs of inervenions relaed o exchange rae movemen raher han fundamenal shifs in porfolio preferences. Table 1. The currency composiion of foreign reserves of he Cenral Bank of Iceland in per cen Counries Currency Average Iceland USD DEM GBP JPY Ohers Indusrial counries USD DEM GBP JPY Ohers Developing counries USD DEM GBP JPY Ohers All counries USD DEM GBP JPY Ohers Source: Cenral Bank of Iceland and IMF: Annual Repor, various publicaions. The currency composiion of he foreign reserves of he Cenral Bank of Iceland has generally developed in a similar fashion o hose of oher indusrial counries. The share of he dollar has been decreasing, mached by an increase in he share of he mark. The relaive share of he dollar has, however, always been smaller han in oher indusrial counries and he Japanese yen share larger, alhough i has also decreased recenly. I can also be seen from he able ha he volailiy of currency shares of he Cenral Bank of Iceland is greaer han for oher counries (he raio of 4 The raio of sandard deviaion o he mean share of currencies is on average 13 per cen for he developing counries bu 25 per cen for he indusrial counries. 2

3 sandard deviaion o mean currency share is on average 38 per cen for Iceland). This can probably be explained by large foreign loans ha can change he currency composiion quie subsanially The Deerminans of Relaive Currency Demand There are mainly wo views concerning which facors are imporan in deermining relaive currency demands of cenral banks. The firs is based on he mean-variance model from porfolio heory, which saes ha cenral banks choose he composiion of reserves ha minimises he risk of he reserves for a given reurn on he reserves. 5 The laer view argues ha cenral banks have broader objecives han porfolio opimisaion. This view argues ha ransacion coss are a key facor in deermining currency shares, leading o he so-called "vehicle currency moive", which saes ha he currencies wih he larges porfolio shares are hose who are mos imporan for financing foreign rade and foreign deb obligaions. We call his moive he ransacion cos model. These alernaive models are discussed in more deail in he nex secion. 2. Theoreical Framework 2.1. The Mean-Variance Model The mean-variance model was developed o describe how privae invesors could diversify heir asse holdings so as o minimise heir porfolio risk in he seminal papers by Markowiz (1952) and Tobin (1958). Here we use his model o describe he opimal composiion of foreign reserves. In his case he cenral bank chooses he relaive currency weighs so ha he oal risk on he porfolio is minimised for a given reurn on reserves. 6 Le m i be he ex pos real rae of reurn on currency i, i = 1,...,G, from ime o ime +1. m i is unknown a ime, bu we can denoe is condiional mean as µ i, is condiional variance as σ ii and he condiional covariance beween currency reurns i and j as σ ij. Expecaions are condiional on he informaion se Θ, which is known a ime, i.e. E ( m ) E( m Θ ) = µ, where E( ) is he mahemaical expecaions operaor. i i i 5 This approach is being adoped by an increasing number of cenral banks. See, for example, Allen (1994) and Downes (1989). 6 As foreign exchange markes are assumed o be efficien, small cenral banks can be viewed as price akers who canno affec exchange raes. They can however reduce heir exposiion o currency risk by diversifying heir reserves as he following resuls indicae. 3

4 The cenral bank's problem is o choose a porfolio of currencies in each period ha minimises he variance of he porfolio for a given expeced real reurn: (1) min σ wi G G 2 R = wiw jσij i = 1 j= 1, so ha: (2) µ = w µ G R i i i= 1 G ; w = i 1; wi 0; i = 1,..., G, i= 1 2 where σ R is he variance of he porfolio, µ R is he expeced real reurn on ha porfolio and w i is he relaive share of currency i in he porfolio. This problem does no have a unique soluion, as he opimal soluion depends on he risk aversion of he cenral bank. The se of porfolios ha solve his problem are described by he efficien fronier. To ge a unique soluion, he preferences of he cenral bank mus be explicily formulaed wih an expeced uiliy curve. We assume ha expeced uiliy is posiively relaed o expeced real reurn on he porfolio bu negaively relaed o he porfolio risk. A simple example of such an expeced uiliy curve is: 7 (3) U ( µ R, σ 2 R ) = µ R ρσ 2 R / 2, where ρ is he Arrow-Pra measure of risk aversion, which also measures he marke price of risk. 8 When he expeced uiliy curve has been formulaed, a single soluion o he 2 porfolio problem can be found by maximising (3) wih respec o µ R, for a given σ R ha lies on he efficien fronier. The soluion o his problem can be described as: 9 ** * (4) W = W + ρ 1 Ω 1 E ( X ), 7 This expeced uiliy funcion can be derived from a concave uiliy funcion, V(R), where R are reserves and V V = ρ is he Arrow-Pra measure of risk aversion. If we assume a densiy funcion for normal disribuions, f ( R; µ R, σ 2 R ), expeced uiliy can be found as E( V ( R)) = 2 f ( R; µ, σ ) V ( R) dr = U ( µ R R R, σ 2 R ) if V saisfies V ( R) ae br 2 for any a, b > 0. See Sargen (1987). 8 The opimal porfolio composiion can be derived wihou explicily formulaing he preferences of he cenral bank if a risk free invesmen is available o he bank, according o he CAPM. The problem wih his approach is ha risk free invesmens for cenral banks are generally no available. This is why we do no proceed in ha direcion. See Roger (1993) and Ben-Bassa (1980). 9 See appendix B for a derivaion of (4). 4

5 where W * is a vecor of currency share, describing he minimum variance porfolio, Ω is he variance-covariance marix of currency reurns and E ( X ) is a vecor of expeced reurns on currencies in excess of he reurn on he minimum variance porfolio, condiional on he informaion se Θ. From equaion (4) we see ha he opimal porfolio can be divided ino wo porfolios: he minimum variance porfolio, W * 1, and a speculaive porfolio, Ω E ( X ). 10 ρ is herefore he relaive weigh ha deermines he corresponding weighs of he minimum variance and he speculaive porfolios in he oal porfolio. 2.2 Calculaing Real Reurns on Currencies Levy and Sarna (1975) have shown ha if invesors face differen inernaional prices (because of non-raded goods, ariffs, axes, ec.), opimal porfolios will depend on consumpion paerns. Consequenly, he problem for cenral banks is more complicaed han ha of privae invesors. For privae invesors i is usually assumed ha he porfolio is valued in he currency of his or her consumpion, bu for cenral banks he choice of he consumpion currency is no so obvious. As menioned in he inroducion, i is usually assumed ha foreign reserves are, iner alia, mean for financing impors. Reserves should herefore be valued by heir real purchasing power and real reurns calculaed by deflaing nominal reurns wih an impor price index. 11 Expeced real rae of reurn is herefore calculaed as: ( 1+ ri )( E ( ei + 1) ei ) (5) µ i = 1, ( E ( P ) P ) + 1 where r i is he nominal reurn on bonds denominaed in currency i, beween ime and ime +1, e i is he exchange rae of he domesic currency per foreign currency, N ai P = ( * ei P ) i= 1 i is he impor price index, P * i is he impor price level in counry i, a i is he weigh of currency i and N is he number of currencies in he impor price index. 12 If we accep ha reserves are ulimaely used o finance impors, heoreically 10 I should be noed, ha when he uncovered ineres rae pariy (UIP) holds he efficien fronier collapses o a single poin, ha poin being he minimum variance porfolio. The inuiion for his is ha when UIP holds, all expeced reurns are equal. 11 The impor price index is sill he appropriae deflaor for real reurns, even hough reserves are never used for financing of impors, provided we are ineresed in he real value of he reserves in he purchasing power sense. See Rikkonen (1989). 12 An alernaive invesmen scheme would be o preserve he nominal value of reserves. In his case we would usually measure he reserves in erms of a vehicle currency, which is mos ofen he dollar. Ben-Bassa (1980) calculaes he opimal currency composiion for Israel using dollar denominaed reurns. His resuls show ha in he minimum variance porfolio he opimal share of he dollar is 96 per cen. A similar model for Iceland gives a minimum variance porfolio of 98 per cen 5

6 i is more appropriae o use currency weighs han impor weighs. However, due o daa availabiliy mos sudies have used impor weighs as a proxy for currency weighs. 13 This is, however, no a good proxy, especially no for he dollar, as he dollar is used more exensively in inernaional rade han is capured by bilaeral rade weighs wih he Unied Saes. Dellas and Yoo (1991) were he firs o use currency weighed impor prices o calculae real raes of reurn and heir work shows a significan difference in resuls of he mean-variance model when he wo alernaive measures of weighs are used. Mos significanly, is an increase in he opimal weigh of he dollar when currency weighs are used. As currency weighs are also available in our case, we base our calculaions of real reurns for he Cenral Bank of Iceland on hose weighs. 14 To calculae mean reurns we proceed in wo ways. Firs we calculae he ex pos real reurns, m i, which is equivalen o assuming perfec foresigh on behalf of he cenral bank. 15 For comparison, we also calculae he ex ane real reurns, by assuming ha exchange rae reurns follow a random walk, and forecas impor prices wih a parsimonious ARIMA model. 16 These wo alernaives measures of real reurns lead o quie subsanial differences in he opimal composiion of foreign reserves Theoreical Assumpions of he Mean-Variance Model There are wo criical assumpions of he mean-variance model worh menioning. The firs one concernes he sabiliy of he variance-covariance marix of currency reurns. This is a common assumpion in he finance lieraure, bu has increasingly been quesioned, especially for high frequency daa, such as daily reurns. Recen empirical findings indicae ha variances of high frequency reurns follow an auoregressive process. According o hese resuls i would be incorrec o assume a consan Ω, leading o subopimal currency holdings. One approach ha akes accoun of his phenomenon is he ARCH model (and is generalisaions), which direcly uses he auoregressive properies in variances of asse reurns. 17 dollar and 2 per cen mark. We also calculaed he opimal currency shares when he vehicle currency was measured as he exchange rae index: 76 per cen ECU, 18 per cen dollar and 6 per cen yen. These resuls gave ye anoher opimal composiion: 33 per cen dollar, 38 per cen mark, 28 per cen pound serling and 1 per cen yen. 13 See, for example, Ben-Bassa (1980), Rikkonen (1989) and Horii (1986). 14 Furher descripion of he daa is in Appendix A. 15 Mos sudies are based on ex pos reurns. 16 The ARIMA model provides a beer ou of sample forecas of he log-differenced impor price level han a simple random walk model. See Appendix C. 17 See Engle (1982) and Bollerslev, Engle and Nelson (1993). 6

7 In Table 2 we esimae reurns wih ARMA models o es for homoscedasisic residuals in krona denominaed reurns. We also compare he resuls of using real reurns on currencies (deflaed wih impor prices) wih nominal exchange rae reurns. The ARCH ess for auocorrelaion up o 5h degree indicae ha here is no significan auocorrelaion in he condiional variances of krona denominaed real reurns, whereas here is evidence in favour of such effecs in krona denominaed nominal exchange rae reurns, in he case of he mark and he yen. We can hus rejec he hypohesis of ime-varying variances in krona denominaed real reurns in favour of he hypohesis of consan variances. For exchange rae reurns his resul does, however, no hold. This differs from he resuls of Papaioannou and Temel (1994). They use an ARCH model, as is done here, and find ha dollar denominaed reurns on currencies can be described by an ARCH realisaion. These effecs do no show up in he krona denominaed real reurns used here. As he resuls in Table 2 indicae, he use of he impor price level o calculae real reurn on currencies seems o filer he ARCH effec ou of he daa. Table 2. Saisical diagnosics of reurns Monhly daa ln( 1+ m i ) ln( e e ) USD DEM GBP JPY USD DEM GBP JPY Skewness Kurosis Bera-Jarque ** ** ** ** ARCH(1) ARCH(5) CHOW 83:3 CHOW 90:1 HF(12) * 3.150* * ** * 5.426** 2.818* 2.527* i+1 i * 3.415** Model ARMA ARMA ARMA ARMA ARMA ARMA ARMA ARMA for reurns (1,2) (1,3) (1,2) (3,3) (2,4) (1,3) (2,3) (3,3) Noe: * (**) indicaes saisical significance a he 5 per cen (1 per cen) level. The number of observaions is 168. The Bera-Jarque es for normaliy is disribued as χ 2 (2) under he null hypohesis of normaliy. ARCH(r) is disribued as χ 2 (r) under he null hypohesis of non-significan auhocorrelaion in he condiional variance up o order r. The Chow es for srucural break is disribued as F(n, n ) under he null of no srucural change. HF(m) is he Hendry ou of sample 1 2 forecas es for a m period horizon, asympoically disribued as χ 2 (m) under he null of equal variances in he esimaion and he forecasing periods. Alhough ime-varying variances in real reurns can be rejeced i could be ha he hypohesis of a consan variance-covariance marix is rejeced due o srucural 7

8 changes in he esimaion period. 18 To es for his we consruced several Chow ess. The residuals from he above ARMA models indicae ha such as break could have occured in early A formal es for a srucural break in 1983:3 suppors his conjecure. This suggess ha he residuals from he ARMA model for real and nominal reurns are heeroscedasisic, alhough his is rejeced a he 1 per cen level for real reurns. Bu as we are only calculaing he opimal currency porfolios for he period 1987 o 1993, his is no of major imporance. More imporan is o es for a srucural break in he period 1987 o To do his we simply consruced a Chow es for a srucural break in he middle of he period, 1990:1. The resuls from his es indicae ha he real reurns of he dollar and he yen do no have consan variances. Finally, we calculaed he Hendry forecas es for he hypohesis of consan variances in he esimaion period up o 1993 and he 12 monh forecas period, 1993:1 o 1993:12. On he basis of his es we canno rejec he hypohesis of equal variances in he esimaion and forecas periods. Concluding from his analysis, here is no evidence of ime-varying variances in real reurns on currencies, alhough such effecs are significan in nominal reurns. This indicaes ha he currency shares from he mean-variance model in secion 3 are opimal, in he sense of efficien use of available informaion. There is, however, some evidence of heeroscedasiciy in real reurns due o srucural breaks in he daa, suggesing shifs in he variance-covariance marix in he period 1987 o This indicaes ha he currency shares from he mean-variance model are no necessarily sable when differen ime periods are used, possibly resuling in poor ou of sample performance of he porfolio. Anoher implici assumpion of he mean-variance model is ha he rae of reurns and risk of currencies can be adequaely described by he firs wo momens of he probabiliy disribuion, i.e. he mean and he variance. For normally disribued reurns his would be sufficien, bu non-normally disribued reurns call for he use of oher momens of he probabiliy disribuion, such as he hird and fourh momen, i.e. he skewness and he kurosis. Mos economiss accep ha exchange rae reurns, as mos oher asse reurns, are no normally disribued bu are lepokuric, ha is, he disribuion has faer ails and higher peaks han he normal disribuion. 19 In Table 2 we es wheher he assumpion of normaliy is rejeced by he daa. The Bera-Jarque normaliy es indicaes ha real reurns on currencies are normally disribued, whereas he normaliy of exchange rae reurns is srongly rejeced. 18 I should be menioned ha he ARCH es has power agains a variey of heeroscedasiciy alernaives. 19 See, for example, de Vries (1994). 8

9 2.4. The Transacion Cos Model Dooley (1987) and Dooley, Lizondo and Mahieson (1989) argue ha a clear disincion mus be made beween he currency composiion of gross and ne reserves (ne of foreign liabiliies). Dooley, Lizondo and Mahieson show ha he currency composiion of ne reserves is a funcion of reurns and risk, whereas he composiion of gross reserves is relaed o he counry's exchange rae arrangemen and he currency composiion of foreign rade and liabiliies. They posulae an expeced uiliy funcion ha is negaively relaed o expeced ransacion coss: 2 2 (6) U ( µ, σ, C ) = µ ρσ 2 C, R R R R where C is he expeced ransacion cos in period, represening he cos of using he foreign exchange marke for currency subsiuion and possible emergency borrowing coss. If H denoes he currency shares of ne reserves, Dooley, Lizondo and Mahieson (1989) show ha: * (7) H = H( µ R, σ R ); Hµ > 0 Hσ < 0. For, currency shares in gross reserves, however, we can posulae he following funcional relaionship: * (8) W = W( T, D ), W, W > 0, T D where T is he share of currencies in foreign rade and D is he share of currencies in foreign deb in period. From equaion (7) i can be seen ha he currency composiion of ne reserves is deermined by expeced real reurns on currencies and currency risk, whereas, from equaion (8), currency shares in gross reserves are independen of hese variables. Currency shares in gross reserves are deermined by he ransacion cos model, represened here by currency shares in foreign rade and deb servicing paymens. The problem wih his approach, as Dooley (1987) poins ou, is ha he insiuional arrangemens in mos counries may well preven porfolio opimisaion of ne reserves, he reason being ha, in mos cases, differen insiuions are in charge of gross reserves (usually he cenral bank) and foreign loans (ofen he minisry of finance). Furher, he abiliy o manage currency posiions on he liabiliy side may be 9

10 much more consrained as liabiliies ypically have longer mauriy. Anoher reason for quesioning his approach is ha ransacion coss in foreign exchange markes are neglecable, especially for he mos imporan currencies. 20 Therefore, we argue ha he mean-variance model is sill he appropriae model for currency composiions of gross reurns. Table 3 shows he currency composiion of foreign deb and foreign rade for Iceland in he period 1987 o I is clear from he able ha he currency composiion of foreign deb is very similar on average o he composiion of reserves in his period (see Table 1). The same applies o foreign rade, alhough he share of he mark and yen are relaively smaller in foreign rade and he share of he Briish pound larger. Table 3. Currency composiion of foreign rade and deb for Iceland in per cen Table 3.a. Currency composiion of foreign deb Currency Average USD DEM GBP JPY Ohers Currency Table 3.b. Currency composiion of foreign rade Average USD DEM GBP JPY Ohers Source: Cenral Bank of Iceland. 3. The Opimal Composiion of Foreign Reserves In his secion we presen resuls on he opimal composiion of reserves for he Cenral Bank of Iceland, using he mean-variance model, for he period 1987 o We analyse four currencies, he dollar, he mark, he pound, and he yen Formally, ransacion coss should lie beween he spread beween bid and ask prices, on he one hand, and deviaions from cross exchange raes, on he oher. Mos sudies indicae ha his cos is neglecable. See discussions in Péursson and Helgason (1994). 21 The opimal porfolios for he period 1980 o 1986 was also calculaed and are available from he auhor upon reques. 22 Originally boh he Duch guilder and he Swiss frank were also analysed. I was however decided o drop he guilder from he porfolio, due o a very high correlaion wih he mark (abou 0.99). The Swiss frank had a zero share in he opimal composiion in all cases and is herefore no repored here. 10

11 These four currencies represen more han 80 per cen of he reserves on average in his period. 23 Two alernaive composiions are calculaed, using differen assumpions for he Cenral Bank's informaion se, Θ. Firs, we simply assume perfec foresigh and calculae he ex pos opimal shares. Secondly, we calculae he ex ane opimal shares, according o a simple forecasing model for real reurns. This model assumes ha currencies are described as random walk processes, which indicaes ha he bes forecass for fuure exchange raes are curren exchange raes. Recen research clearly suggess ha his simple forecasing scheme has beer success han more complicaed srucural models in forecasing ou of sample exchange raes in low frequency daa. 24 To forecas he impor price level, a simple ARIMA model for he period 1980 o 1986 is esimaed and used o forecas real reurns for he period 1987 o This is described in more deail in Appendix C. The opimal currency shares are shown in Table 4. Four differen composiions are shown, using differen values for he Cenral Bank's risk aversion, ρ. The firs composiion is he minimum-variance composiion. The second composiion is calculaed assuming ha he Cenral Bank assigns equal weighs o he minimum variance and he speculaive porfolios. The hird composiion assumes a relaively risky invesmen scheme and he las composiion is he speculaive porfolio. Table 4. Opimal currency composiions for he Cenral Bank of Iceland Ex pos real reurn Ex ane real reurn Risk aversion Real reurn Sandard deviaion USD 48% 47% 40% 0% 62% 0% 0% 0% DEM 28% 28% 20% 0% 32% 0% 0% 0% GBP 24% 24% 30% 0% 0% 100% 100% 100% JPY 0% 1% 10% 100% 6% 0% 0% 0% Toal 100% 100% 100% 100% 100% 100% 100% 100% Using ex pos real reurns indicaes ha he opimal composiion is no sensiive o he assumpion of risk aversion, excep when we assume very risky porfolios such as he speculaive porfolio. The opimal composiion using ex ane real reurns is, on he oher hand, very sensiive o he Cenral Bank's level of risk aversion. 23 As here are only four currencies used here, we proceed by runcaing he acual shares so ha hey sum o 1 o ease all comparison. 24 See, for example, de Vries (1994). 11

12 I can also be seen ha, using ex pos reurns, higher reurn can be aained by increasing he porfolio risk only slighly, which indicaes ha he efficien fronier is quie seep near he minimum variance porfolio. This can be seen from Figure 1 which shows he efficien fronier using ex pos real reurns. Figure 1. The Efficien fronier for he Cenral Bank of Iceland Ex pos real reurn 5,50 5,40 EE 5,30 µ 5,20 5,10 5,00 4,90 A 4,80 2,80 3,00 3,20 3,40 3,60 3,80 4,00 σ The difference in he opimal ex pos minimum variance composiion and he ex ane minimum variance composiion lies mainly in he share of he pound serling. The opimal ex pos minimum variance share of he pound is quie high whereas he ex ane share of he pound is zero. The more risky ex ane porfolios, however, only include he pound. The explanaions for hese differen resuls can be seen in Table 5, which shows he reurn and sandard deviaion of each currency and he correlaions of reurns under he wo alernaive assumpions. Table 5. Reurns and correlaions Table 5.a. Ex pos real reurn Reurn Sandard USD DEM GBP JPY deviaion USD DEM GBP JPY Table 5.b. Ex ane real reurn Reurn Sandard USD DEM GBP JPY deviaion USD DEM GBP JPY In Table 5.a i can be seen ha he dollar and mark are he leas risky asses and ha he correlaion beween he dollar, on one hand, and mark and pound, on he 12

13 oher hand, is high and negaive. A he same ime here is lile correlaion beween he mark and he pound. This clearly indicaes ha i is desirable o use hese hree currencies in he reserves o reduce he porfolio risk. For more risky invesmens he share of he yen should clearly rise as he yen has he highes reurn of all he currencies. In Table 5.b compleely differen paerns are seen. Now, he yen has he lowes reurn and he correlaions are posiive beween reurns on all currencies. There is however a small correlaion beween he mark and he dollar, suggesing he use of boh hese currencies in he opimal minimum variance porfolio. As he porfolio risk increases, he share of he pound increases as he pound has he highes reurn. A comparison of ex pos opimal currency shares wih acual, average currency shares for Iceland in Table 1 shows very similar currency shares of he dollar and he mark, bu a significan difference in he pound and he yen shares. As he yen is a relaively risky asse in his period, his indicaes ha he acual, average porfolio was riskier han he ex pos opimal porfolio. The real reurn on he average porfolio was 4.89 per cen, wih a sandard deviaion of This suggess ha he mean-variance model is no sufficien o explain acual relaive currency demand of he Cenral Bank of Iceland. 4. Esimaing Relaive Currency Demands Comparing acual currency shares wih hose prediced by he mean-variance model indicaes ha oher facors are insrumenal in explaining he currency composiion of he Cenral Bank of Iceland's reserves. I is however ineresing o es he meanvariance model more formally and, simulaneously, es he ransacion cos model. In his secion we do his using wo approaches. The firs one is simply o esimae a model where he real rae of reurn on currencies, currency risk, currency share in foreign rade and currency share in foreign deb explain he composiion of foreign reserves of he Cenral Bank of Iceland. The second mehod uses an approach suggesed by Frankel (1982), which direcly ess heoreical resricions implied by he mean-variance model. In boh cases daa for he period 1980 o 1993 is used The Mean-Variance Model Versus he Transacion Cos Model In he empirical model ha is esimaed in his secion, we ry o ake ino accoun boh he mean-variance model and he ransacion cos model. We hope o be able o say which model has beer explanaory power in explaining he relaive currency demand of he Cenral Bank of Iceland. Formally he model can be described as: 13

14 (9) W = W( µ, σ, T, D); Wµ, WT, WD > 0, Wσ < 0. The resuls shown here are calculaed using ex pos real reurns. The main resuls did no change when ex ane real reurns were used. The model uses annual daa from 1980 o 1993 for he dollar, he mark, he pound and he yen. 25 We have herefore only 14 observaions for each currency, which are oo few o esimae a significan empirical model. We herefore assume ha reurn, risk, he currency share in rade and deb all affec each currency in he same way, which seems no so economically unreasonable. This allows us o pool ogeher ime series and cross-secion daa o increase he number of observaions o 56 (he resuls for each currency are shown in able 9 for furher informaion). To allow for currencydependen effecs, a currency-specific consan for each currency is developed. The esimaed model is a linear approximaion of (9): (10) W = β + β + β + β + β µ + β σ + β T + β D USD DEM GBP JPY We firs esimae he model using he OLS rule. This mehod is simple o implemen bu ress on raher sric assumpions. These assumpions are ha of no conemporary correlaions beween currency shares and consan volailiy across currency shares. The second assumpion can possibly be acceped bu he firs one is obviously inappropriae in he model o be esimaed here. This indicaes ha he sandard errors of he OLS parameer esimaes will be incorrec which could lead o inappropriae inference rules. Therefore, we esimae he model again using he GLS rule, which allows for conemporary correlaion across currency shares and differen volailiy in hose shares. Dooley, Lizondo and Mahieson (1989) ry o explain currency shares in reserves wih he ransacion cos model. They poin ou ha currency shares can only ake values in he inerval (0,1) and he use of leas squares rules (LS) would herefore be inappropriae, as LS mehods assume ha currency shares can ake on all values on he real line. This would herefore lead o biased and inefficien esimaors. They use he Tobi esimaion echnique insead o esimae currency shares. For comparison hey also esimae he model using OLS. The comparison indicaes ha he OLS echnique underesimaes he explanaory power of he ransacion cos model. This is ineresing since all empirical resuls of he ransacion cos model have been based on OLS esimaions. I is possible o verify wheher he use of LS echniques leads o a serious bias in he esimaed coefficiens and sandard errors by esing wheher he LS residuals are normally disribued. Failure o rejec normaliy indicaes ha he problem of using LS 25 Daa on higher frequency is no available for he currency composiion of foreign rade and deb. 14

15 mehods is no so serious in our case. For our model he Bera-Jarque saisic is 0.85, ha is well below all commonly used saisical levels. We can herefore assume ha he residuals from he LS esimaions are normally disribued and ha he problems of using his esimaion mehod are no so serious. Table 6 repors he esimaion resuls. The able shows he esimaed parameer values for four consans (one for each currency) and for ex pos real reurn, risk, rade shares and deb shares. I should be noed ha R 2 can be used here, as he variances of currency shares are relaively similar as indicaed by he upper line in Table 7 ha shows he diagonal elemens of Σ, he residual variance-covariance marix. From Table 6 we see ha boh esimaion echniques give pracically he same parameer values, as should be expeced since OLS is sill unbiased. The explanaory power and he sandard errors of he models are also he same. Table 6. Esimaion of currency demand Ex pos real reurns β USD β DEM β GBP β JPY β 1 β 2 β 3 β 4 R 2 SE OLS -values ** ** 0.059* ** 0.679** GLS ** ** 0.055* ** 0.600** values Noe: * (**) indicaes saisical significance a he 5 per cen (1 per cen) level. R 2 is he coefficien of deerminaion, correced for degrees of freedom. SE is he sandard error of he equaion. The consans for he dollar, pound and yen are all significan from zero. They also seem quie differen from each oher, indicaing some specific currency affecs. The coefficien on real reurn has a negaive sign, bu is no significan from zero a he 1 per cen level. The coefficien on risk is also negaive, as should be expeced, bu i is also insignifican. The coefficiens on rade and deb shares boh have, however, correc signs and are well significan. Table 7 shows he esimaed variance-covariance marix. As menioned above volailiy in currency shares seem o be quie similar. Correlaions are however differen, bu all negaive. These resuls indicae ha i is more appropriae o use GLS han OLS. Table 7. Esimaed variances in Σ and correlaions USD DEM GBP JPY $Σ ii USD DEM GBP JPY

16 These resuls srongly sugges ha he ransacion cos model has more explanaory power han he mean-variance model in explaining relaive currency demand of he Cenral Bank of Iceland in he period 1980 o The Cenral Bank seems o have based is currency composiion on he relaive shares of currencies in he counry's foreign rade and is foreign liabiliies raher han on porfolio opimisaion. These resuls are similar o hose of Dooley (1987), Heller and Knigh (1978) and Dooley, Lizondo and Mahieson (1989), who find ha foreign rade, foreign deb (for developing counries) and exchange rae arrangemens have significan explanaory power in explaining cenral banks' currency composiions. 26 Finally, in Table 8 we show he esimaed model for each currency. In hese esimaions here are only 14 observaions, or 9 degrees of freedom, so one should be careful in inerpreing he resuls. Table 8. Esimaion of currency demand for each currency Ex pos real reurn w i Consan β i1 β i2 β i3 β i4 R 2 SE DW F USD ** values (0.001) DEM values (0.053) GBP * values (0.223) JPY ** values (0.009) Noe: * (**) indicaes saisical significance a he 5 per cen (1 per cen) level. The F-es is a es for he significance of each equaion (p-values in parenhesis). DW is he Durbin-Wason firs order serial correlaion es saisic. The equaions for he dollar and he yen have he larges explanaory power, while he equaions for he pound and he mark have very lile. The hypohesis ha all parameers (excep he consan) are non-significan canno be rejeced for he wo laer currencies. In he dollar equaion, he dollar share in foreign rade is significan a he 1 per cen level. The pound share in foreign rade is also significan a he 5 per cen level in he pound equaion, bu only marginally. In he yen equaion, he yen share in foreign deb is significan a he 1 per cen level. In he mark equaion here are, however, no significan parameers. This indicaes ha he dollar share in he reserves can be explained largely by he dollar's share in foreign rade. The same applies o he pound, bu o lesser exen. 26 In all hese sudies he counry composiion of foreign rade is used as a proxy for he currency composiion of foreign rade. 16

17 The yen share in he reserves can, however, be explained by he yen's share in foreign deb. This again indicaes ha he significance of foreign rade in he pooling model can be explained by he dollar and, o some exen, he pound, and ha he significance of foreign deb can be explained by he yen Esimaing he Inverse Demand Funcions Even hough he mean-variance model seems srongly rejeced by he daa i is ineresing o es formally wheher resricions imposed on he daa by he meanvariance model can be rejeced. Such a es can be derived from he model in previous secion. 27 Equaion (4) can be wrien as an "inverse-demand funcion": (11) E ( X ) = C + ρω W, where C is a vecor of consans. 28 According o (11), for a given asse, i's expeced excess rae of reurn in equilibrium would be higher if he share of ha asse in he porfolio increased. Frankel (1982) poins ou ha if expecaions are raional, he variancecovariance marix of he forecas errors, ε X E ( X ), is Ω. Tha is: (12) E( ε Θ ) = 0 ; E( ε ε Θ ) = Ω. This provides us wih a simple es of he mean-variance model, by simply regressing ex pos excessive real reurns, X, on W. Tha should give an esimaion of he variance-covariance marix of residuals which should be proporional o he coefficien marix, wih he consan of proporionaliy equal o ρ. Therefore we can esimae he following model: (13) X = C + BW + ε, and es he resricions: 27 I was Frankel (1982) who firs suggesed his approach. See also Engel and Rodrigues (1989, 1993) and Dumas (1994). 28 In general C is no a consan bu relaed o Ω (see equaion (4)) bu we assume ha Ω is relaively consan hrough ime and is herefore reaed as a consan. Anoher reason for his assumpion is ha no consrains beween C and he vecor of errors can be imposed. See, for example, Engel and Rodrigues (1989 and 1993). 17

18 (14) B = ρω. Usually i is assumed ha he coefficien marix, B, is consan over ime, i.e. ha Ω is consan. 29 Recenly he model has been exended o allow B o vary over ime, for example according o an ARCH process. 30 Here i is assumed, however, ha Ω is consan and B is herefore reaed as a marix of fixed coefficiens. 31 The null hypohesis is b ij = ρσ ij, where b ij and σ ij are he {ij} elemens of B and Ω. A rejecion of he null indicaes ha relaive currency demand by he Cenral Bank of Iceland canno be described by he mean-variance model. Here, excess reurns are calculaed using he dollar as a numeraire. 32 Table 9 shows he resuls based on ex pos real reurns for he unresriced asse demand funcions. None of he parameer esimaes are significan from zero a he 1 per cen level. A Wald saisic for he null hypohesis in (14) is χ 2 (8) = 4.02 [p = 0.26], which indicaes ha he null canno be rejeced. A firs sigh his migh indicae a posiive resul for he mean-variance model bu his is no he case. The reason is ha all he esimaed parameers are non-significanly differen from zero and herefore we canno rejec he hypohesis ha all he ρ 's are equal. Dependen variable x DEM -values Table 9. Inverse demand funcions using ex pos real reurns Quarerly daa 1981:1-1993:4 Table 9.a. Unresriced asse demand funcions C w DEM w GBP w JPY R x GBP -values x JPY -values Noe: * (**) indicaes saisical significance a he 5 per cen (1 per cen) level Table 9.b. Variance-covariance marix of residuals DEM GBP JPY DEM GBP JPY See, for example, Frankel (1982). 30 See Engel and Rodrigues (1989 and 1993). 31 See, however, he discussion in secion See, for example, Dellas and Yoo (1991) and Engel and Rodrigues (1989, 1993). We also esimaed he model using he minimum variance porfolio shares o calculae an alernaive numeraire. The resuls from ha model were almos idenical o hose repored here. 18

19 To ge an esimae of ρ he resricions B = ρω were imposed on he model. The resuling parameer esimae is ρ $ = 4. 0 (-value = 0.616). This could indicae ha he Cenral Bank of Iceland is relaively risk averse in is porfolio managemen. However, he sandard error is very large, so a large array of parameer values canno be ruled ou. In paricular, he value of zero is no rejeced which is in accordance wih he resul ha all he parameer esimaes are non-significan. The resuls herefore srongly sugges ha he mean-variance model canno explain he relaive demand for currencies by he Cenral Bank of Iceland. This is a similar resul o ha found in oher sudies using he same approach. 33 Engel and Rodrigues (1989) poin ou ha a more appropriae approach is o allow Ω o vary over ime. Their resuls show ha his does no change he main conclusion, ha he mean-variance model is rejeced by he daa. 34 They also ry o explain his failure by allowing for measuremen errors bu o no success. Anoher possible explanaion for hese negaive resuls migh be he assumpion concerning he informaion se available o he Cenral Bank, Θ. We have assumed ha Θ conains all informaion relevan available o he Bank a ime when he Bank makes is porfolio decisions. This assumpion is someimes called he srong form of he raional expecaions hypohesis or Muh-raionaliy. As we are esing he meanvariance model and he raional expecaions hypohesis simulaneously, his srong form of he raional expecaions hypohesis migh be rejeced by he daa, leading o a failure of he model. A weaker form of he raional expecaions would be o assume ha he Cenral Bank has a differen informaion se, Φ, for which Φ Θ and E( ε Φ ) = 0, so ha Φ is a subse of he complee informaion se and is orhogonal on he forecas errors. As was described in he previous secion, we esimaed expeced reurns on currencies assuming ha exchange raes follow a random walk process, which means ha only fuure impor prices need o be forecased. We calculaed he expeced reurns by esimaing an ARIMA model for impor prices, assuming ha Φ ={ln P j 0. Using esimaions of expeced excess reurns, Table 10 shows he } j= esimaes of he inverse demand funcions. 35 The parameer esimaion changes somewha and now some of hem are saisically significan. The relaion beween currency demands and reurns is, however, mixed. The currency share of he yen is always significan, bu has a wrong sign in is own demand equaion. The Wald saisic for he null hypohesis is now 33 See, for example, Dellas and Yoo (1991). 34 They allow Ω o vary over ime according o an ARCH model and as a funcion of marke uncerainy, measured as unforeseen changes in money aggregaes and oil prices. 35 Dellas and Yoo (1991), using slighly differen argumens, also es he mean-variance model assuming ha exchange raes follow a random walk process. 19

20 χ 2 (8) = [p = 0.01], so he null can now be rejeced a he 1 per cen level. Table 10.b shows how he variances and covariances of he residuals have drasically reduced. This should be expeced as he volailiy of exchange rae reurns conribues by far he larges o he volailiy of real reurns. Dependen variable x DEM -values Table 10. Inverse demand funcions using ex ane real reurns Quarerly daa 1981:1-1993:4 Table 10.a. Unresriced asse demand funcions C w DEM w GBP w JPY R ** * x GBP -values 0.046** ** x JPY ** values Noe: * (**) indicaes saisical significance a he 5 per cen (1 per cen) level. Table 10.b. Variance-covariance marix of residuals DEM GBP JPY DEM GBP JPY When he model is esimaed again wih he imposed resricions B = ρω, he esimaed risk aversion is ρ $ = (-value = -4.23) and he parameer value is significan from zero. The esimaed risk aversion is very large and negaive, which would be of serious concern had he resricion no been rejeced by he daa. I seems, herefore, ha he mean-variance model can be rejeced as an explanaion for relaive currency demand of he Cenral Bank of Iceland, which is in accordance wih he resuls of he previous model and a comparison of acual currency shares and hose prediced by he mean-variance model. 5. Conclusions During he las few years cenral banks have increasingly adoped he mean-variance approach when deciding he composiion of heir foreign reserves. Here we use he mean-variance model o compose he opimal currency shares for he Cenral Bank of Iceland for he period 1987 o 1993, assuming ha he Bank is preserving he real value of reserves measured by currency weighed impors. Currency shares are calculaed using boh ex pos and ex ane real reurns, calculaed from a forecasing model. The resuling currency demands are somewha differen from he acual 20

21 demands in ha period, indicaing ha he mean-variance model is no sufficien o explain he invesmen behaviour of he Cenral Bank of Iceland. An alernaive model for explaining relaive currency demand, is he ransacion cos model. Alhough we feel ha he mean-variance model is a closer approximaion o opimal invesmen behaviour of cenral banks, formal ess srongly rejec he meanvariance model bu show srong explanaory power of he ransacion cos model. The poor resuls of he mean-variance model are furher confirmed by a direc esimaion of relaive currency demands, which clearly rejecs he model. 21

22 Appendix A. The Daa Reserves Toal reserves wih gold which, in Iceland's case, has a small and sable share in oal reserves. Source: Cenral Bank of Iceland and IMF: Annual Repor, various publicaions. Impor prices Weighed consumer prices of he 15 mos imporan rading parners of Iceland, 15 ai according o P = ( * ei P ) i= 1 i, where e i is he nominal exchange rae beween he Icelandic krona and currency i (i.e. krona per currency i), P * i is he consumer price index in counry i and a i is he weigh of currency i in rade ransacions (calculaed as he average of currency i purchases and sales by domesic commercial banks). Source: Cenral Bank of Iceland and IMF: Inernaional Financial Saisics. Reurn on currencies Real reurn on various currencies in he Eurodeposi marke, using hree monh averages. Libor raes for hree monhs deposis in each currency (for GBP he Paris inerbank rae is used). Real reurn is calculaed as: ( 1+ ri )( ei + 1 ei ) ( 1+ mi ) =, ( P P ) + 1 where m i is he ex pos real reurn on currency i, r i is he nominal reurn on currency i and P is he impor price index as defined above. All reurns are annual percenages. Source: IMF: Inernaional Financial Saisics and he auhors calculaions. Volailiy of reurns Sandard deviaions of real reurns on he Eurodeposi marke wihin each period, calculaed from monhly reurns of each currency. Source: Auhors calculaions. Appendix B. The Opimal Porfolio Composiion The mos simple way o solve he model in he main ex is o represen he problem in marix form. The variance of he reserves can hen be wrien as: 2 (B.1) σ R = W Ω W, where W = ( w1,..., wg ) is a vecor of currency shares and Ω is a (G G) variancecovariance marix of currency reurns. 22

23 Equaion (B.1) is minimised wih respec o: (B.2) µ R = W M ; W c = 1; W O, where M = ( µ 1,..., µ G ) is a vecor of expeced currency reurns, c is a (G 1) uni vecor and O is a (G 1) vecor of zeros. The soluion o his problem gives a composiion of reserves ha lies on he efficien fronier for a given reurn: 36 (B.3) W = Ω 1 ( M, c) A 1 ( µ, 1), R where A is defined as: A M Ω M Ω M Ω c Ω 1 M = 1 1 c 1 c c. 2 The soluion for σ R which lies on he fronier, σ 2* R, is: 2* 1 (B.4) σ = ( µ, 1) A ( µ, 1). R R R By minimising (B.4) wih respec o µ R, one ges he minimum variance porfolio: (B.5) W * = Ω c c Ω c 1 1. If he cenral bank is willing o increase he reurn on he reserves by aking more risk, a new porfolio can be obained by maximising an expeced uiliy funcion, such as equaion (3) in he main ex, wih respec o µ R and given equaion (B.4): ** (B.6) W = Ω c c Ω c + ρ Ω ( M c( M Ω c c Ω c)) or: ** * 1 1 * * 1 1 (B.7) W = W + ρ Ω ( M cµ ) = W + ρ Ω E ( X ). R 36 See for example. Horii (1986). 23