- Email: [email protected]
The optimal composition of foreign exchange reserves
Jo Jrnal of International
Economics
10 (1980) 285-295. 0 North-Holland
Publishing
Company
THE OPTIMAL COMPOSITION OF FOREIGN EXCHANGE RESERVES Avraham Bank
of Israel
und
BEN-BASSAT*
The Hebrew Uniaersity, Jcwsalem.
Received April 1978, revised version received October
Isr~rel 1979
This paper presents a model for selecting an optimal foreign exchange reserves portfolio for semi-indu:;trial and developing countries, using the mean-- variance approach. The model described here focuses on the relationship between the composition of reserves and that of imports, :ts well as the impact of return and risk of thk investments in each currency. The empirical imporiance of these factors is demonstrated by investigating the optimal policy for Israel in the period 1972-76. in comparing the actual and the efficient portfolio of different groups of countries, we find that proiit considerations play a greater role in semi-industrial and developin: countries than in industrial ones.
1. Iatraduction The ;iirn 01;this study is to describe an analytical approach to the optimal selection of a country’s foreign-exchange reserve portfolio. We concentrate on the molives for holding foreign reserves in semi-industrialized and developing countric:s and compare their actual behavior with that of the industrial countril:s. Previous studies on international portfolio selection have dealt mainly with the individual or the firm. A few economists, such as Kenen (1967), Officer and Willett (1969), Hagemann (1969), Steckler and Pickarz (1970), and Makin (197:U), have studied the co~lposition of national foreign currency reserve:;, bul: they deal only with Ihe optimum ratio of gold to foreign currenc ies. These studies relate: to a period in which the international monets.ry sy~jtem was based on gold ;Ind the typical pattern was one of fixed exchange rates adjusted at relatively long intervals; this helps to explain w-hy studies of fo’reign-reserves composition dealt only with the share of gold in total reserves, other currencies being represented by the dollar. After the shift *This paper is part of March 1978. I would like for their useful gui,dance from whose c~~mmcnts I fruitful 1iscussion of many
a Ph.D. thesis submitted to the Hebrew University of Jerusalem in to thank my supervis,ors. Professors Najav Halevi and David Levhari. and comments, and Dr. Zvi Sussman who suggested the study and benefited greatly. I am also indebted to Prof. Giora Hanoch for a points.
186
A. Ben-Bussut,
Foreign
e.uchun,ge
reserves
from fixed to floating exchange rates, gold and the dollar iost their special place as the chief reserve assets, and today they are only twrl out of a whole array of assets in a central bank’s portfolio. This study is the first attempt LO investigate the selection of such a portfolio. In the first part of the paper the objective function for foreign exchange reserves is formulated and considerations bearing on the choice of a foreign reserve portfolio are discussed. In the second part the application of the model is demonstrated by estimating optimum portfolios for Israel in the period 1972-76. In the final part we arialyze the reserves investment policy of the industrial countries in comparison vq4h that of the :;erm-industrial and dev,Aoping countries and compare the actual portfolio of each group with Its effiic.ient portfolio.
2. The analytical anproach
The choice of a portfolio by a central bank is intimately related to the objects served by the holding of reserves. These have been extensively discussed in numerous studies of the demand for reserves, and many aspects are mentioned, among them political and economic ones. While economists ,differ widely on the determinants of the size and composition of reserves and on their relatilve importance, they all mention the instability of balance-ofpayments flows as the main reason for holding reserves.. Foreign currency receipts (including capital imports) are not necessarily equal to foreign currency outla.ys at any given point in time; countries therefore keep a foreign exchange reserve in order to ‘pay for imports when receipts fall short of outlays. This is the precautionary motive [see, for example, Heller (1966, pp. 300-301)]. in constructing an investment mod.el for a central bank it is important to specity not on1.y the purposes for which the foreign reserves are held, but also the country’s attitude to profitability. Two groups of countries may be distinguished: (1) Industrialized countries with floating exchange rates and whose currencies may serve as reserves in the portfolios of other countries. The reserve composition of these countries is influenced more by the stability of the international monetary system, while profit considerations are of secondary im;sortance. (2) The developing al,d semi-industrialized countries, in most of which rates of exchange are subject to administrative control. Unlike the rich nations, thes? countries are free from concern over international monetary stability, 50 that their allocation of reserves among alternativt: assets will be influenced mainly by profit and liquidity considerations. This stuldy deals chiefly wit’h the second group of countriks. In view of the
uncertainty prevailing m the foreign exchange market, their decisions regarding reserve composition are determined according to the rates of return and risk of the various currencies, with the object being to maximize the reurn in terms of a basket of imports at a given risk. The approach adopted here for finding the rate of substitution between return and risk is the mr,an-variance model developed by Markowitz (1352) and Tobin (19%). Application of this model to a central bank demonstrxto< the importance of determ.ining the objective function. In two studies of international portfolio selection - Solnik (1973) and Levy and Sarnat (1975) -~ it has been proved that if different economic agents (countries) face different prices, because of non-tr;tded goods, tariffs, taxes, etc. then optima1 international porIfo1ios will depencl on consumption patterns. As mentioned, foreign reserves are, inter alia, meant to finance imports. The nation’s imported consumption, unlike the inc.ividual’s, is paid for in a variety of currencies, depending on the origin of the imports. Since the consumption currency plays an important role in determining the optima1 portfolio, the optimum composition of reserves will be obtained by minimizing variance at a given return, where risk and return are exl’ressed in terms of the basket of import currencies.’ The problem is then formulated as:
i=l
i=l
j=l
jai
subject to f-J=
i i=
Uiyi,
1
CUi=l,
where L+ is the optimum share of currency i in the portfolio, 0: is the variance of the returns on currency i, R,, is tb,e c0,rre:ation coefficient between returr;s on currencies i and j, and i.1;is the return nn currency i. Fl’he rate of return on each currency is a functior. of the interest rato (Qi), ;Lnd the rate of change of the exchange rate of currency i 1t1 relation 10 the import currency basket (Ei ):
‘For a formal proof of this argument see Ben-Bawt
(19’5).
A.
288
Ben-Bnssat,
Foreign
exchange
resertres
The problem does not have a unique solution for the vector of weights since the choice of optimal composition depends on the desired degree of rise. Therefore there is an infinite set of solutions, each representing an alternative optimum combination of return and variance. 3. The findings The efficiency curve representing the relationship between the maximum return for a given risk was estimated by a quadratic programming model2 The model was specified in the preceding section; here we examine the empirical zesuits obtained from ii. The application of the mode1 is illustrated by estimating the efficiency curve for Israel for 1972-76. The estimate is based on monthly observations c,f returns on the major currencies and gold; :he distribution of returns resuling from these observations is assumed to approximate the distribution of the ex ante returns.3 In order to understand the optimum composition obtained for each level of returns, data are presented on the mean and the standard deviation of the returns on various poGble assets (see table 1). The variance of the total return is explained mainly by fluctuations in the exchange rate: the monthly fluctuation in the exchange rate varies between 15 and 30 percent, cornpa~~d Table Retu::n
and
standard
deviation
1
of investments (percent).
in selected
currencies.”
1972 -76 -_
In import terrns
~Dollar Sterling Canadian dollar Deutschemark Gi_ulder Swiss f?snc French franc Yen tiold aBased on monthly
Mean 88 4.8 9.2 13.5 13.6 15.6 12.4 7.4 23.3 ~-___
-
In dollar terms
Standard deviation
Mean
15.6 20.0 20.6 28.1 27. I 30.8 26.5 25.1 94.0
7.8 4.1 82 12.9 13.0 15.0 11.8 6.7 26.7
Standard deviation 2.3 30.8 13.6 40.2 39.3 41.9 38.8 33.2 101.5 .~
data.
‘The computations were performed wi1.h a computer program constructed at the Department uf B?lsiness Administration of the Hebrew University. I am indebted to Prof. Haim Levy for putting this program at my disposd!. 3Calculation of the returns, variances, and covariances is based on the beginr.ing-of-monrh changes in exchange rates and on the beginning-of-mo+h interest rate on monthly EurcIdzposits.
A. Ben-Bassat,
Foreign
cxchmge
r. serve:
289
with no more than 2-3 percent for the interest rate. The interest rates and the rates of exchange are not of collrse entirely indepe.ldent of one another: currencies with a high probability of devaluation usuaily have higher interest rates; that is to say, the correlation between the interest rate and the rate of change of the exchange rate is negative, and we obtain R = -0.85. This inverse relatio*lship narrows the differences i:i r.otal returns between the various currenciC:s Some remarks on gold are in order, since it is still a major reserve asset. In the period reviewed, the average return on gold was 6-16 percentage points above that on other currencies; +ut the high return was accompanied by a standard deviation of 6@-75 percentage points above that of other currencies.4 3.1. The efficiency curce in terms of imports Estimating the efficiency frontier of the Bank of Israel in 1972-76 in terms of imports, it is found to be a concave= curve; this means that the return increases at a diminishing rate with the increases in the variance. The alternative return-risk combinations on the curve apply to different currency baskets. To illustrate this point, selected combinations are presented in table 2. The first two rows show optimum combinations of rzldrn and standard Table 2 Composition -.-
-.___-
of optimum portfolios for selected returns,” 1972 76 (percent ). ~-.-~
In import terms --
-~
In dti!lar terms ____ __~
__ _
Selected optimum combinations Mean return
9.0
12.0
14.0
x.0
12.0
13.0
Standard deviation
3.3
10.5
19.5
1.3
20.2
30.0
62.0 1).Y
l-l.6 0.4
Currency composition of selected combinatiorts Dollat C.madiar. dollar Sterling Deutschemark Guilder Swiss franc French franc Yen Gold Tota:
46.0
CO.3
30.3
96.4 1.9
21.5 10.6 4.8 1.8 7.1 7.0 0.3
19.4 19.3
16.3 33.4
0.2
24.2
36.0
6.0 __ 5.0
10.0
0.3 0.7
12.9
19.0
100.0
100.0
100.0
100.0
100.0
100.0
0.5
“Based on monthly data. ‘The rate of return on gold was estimated as the change III the p-ice of gold in the Jolla~denominated market, converted into terms of the Israeli import currency basket.
290
f‘oreigrlcwhangr
A. Ben-Bow&
re.serws
deviation, .while the corresponding porrfolios are shown in the lower part of :he tat-le. The optimum portfolios depend on the desired risk, lower-risk strategies involving greater diversification. For example, for the lowest retu.-n-risk combination [column ( J-)-J1the optimum portfolio contains seven of tre eight possible currencizs as well as gold. But although all but one of the possibilities appear, three of them account for 78 percent of tie total. The optimum low return-risk portfolio turns out to be quite similar to the currency composition of the Israeli irlport basket.’ This was to be expected Table Distribution cuuntry
3
of Israel’s imports of or:& 1972 (percent).
U.S.A. U.K. Canad; West Germany Nether land:, Swi:zerland Belgium Franc: Italy Japan Swedl:n Denmark Othe 7s’ Total
by
26.2 17.9 0.9 12.5 3.2 3.8 47 3.8 63 I .9 1.8 0.5 16.5 1000
“IIllports from the group of ‘other’ tour tries are assumed to have been paid in dl)llars
since a reserve portfolic, based on the composition of imports reflects a strategy of hedging against fluctuations, in the exchange rates of import cl:rrencies and.. as mentioned earlier, exchange rate fluctuations were the drlminant clement in the total return vat-lance in the period reviewed. Thus, th: lowest risk is obtained bith a portfolio wtm;t: composition is similar to rhSit of the currency basket in the objective function, in our case that of Israel’s imports. As the level of return and risk i> increased, the number of currencies in the portfolio declines steadily, until at a return of 12.5 percent we are left with only three currencies, apart from gold: the dolliai., the Swiss franc. and the guiider. From this point on, any increase in return and risk involves a rapid
‘lmport~ to ihr,lel were c,la;;sified by twelve countries of origin, which account for about 84 ;xx
A. Ben-Buswt,
FoIxigr~
r.xchange
rewrws
291
drop in the share of the dollar and a corresponding rise in the share of the Swiss franc and goId. Note that because of its riskiness the share of gold is very small at the low risk level, and it rises with an increase in the desired return and risk.6 Besides the return and risk of each asset, the optimal portfolio is determined by the coefficients of correlation between the returns on the various assets. Thus, in choosing an opZma1 portfolio, preference must be given to sets of assets with low correlations, and particularly to assets wh:ch correlate negatively since such combinations reduce’ the total risk. The lowest correlations are obtained between the U.S. and Canadian dollars on the cne hand and ail other currencies on the other. In particular, the correlation between the dollar and the European currencies is highly negative (around -‘0.5). This is one of the main reasons why low-risk portfolios consist of roughly half dollars and half other currencies (table 2). The large effec-t of a negative correlation of returns on the standard deviation of the portfolio can be gauged from the fact that the lowest standard deviation for any single currency is 16 percent for the dollar at a return of 9 percent, where#ls the standard deviation of the portfolio yielding this return is only 3 perce;::. The most important (as reserve currency) of the European currencies is t4e SWISS franc. The reason for the smaller weight of the other European currencies in spite of their very similar distribution of returns is that they arc highly and positively correlated among themselves. This implies that they are substitutes rather than complements within the portfolio. The Swiss franc is predominant among the European currencies because of its better performance (ratio of return to standard deviation).
The major argument of this paper is that the efficiency curve must be estimated in terms of the country’s basket of imports. Is this conclusion empirically relevant or is it merely of theoretical interest’! To answer this question we estimate the efficiency curve in terms of dollars, since this currency is used by most countries as a unit of account for foreign reserves. and the corresponding portfoi’os. The optimum return risk combinations where the objective function is specified in dollars, are presented in thl: second part of table 2. It can bra seen that the results differ considerably from those in import terms. The mait! currencies are still the dollar and the Swiss franc but their weights are different along the entire curve. For example, the lowest risk level for both the dollar and the import-basket curves is obtained at a standard deviation of around 2.5 percent. Gn the imports efficiency “In the highest and return
risk
portfolio.
oold will c
of the assets considered.
remain
as the sole asset. since it has the maximum
r Sk
292
A. Ben-Bassat, Foreign
excharlge resw!es
curve th,is risk is obtained with 45 percent dollars, the remainder of the portfolio consisting of the various European currencies. When the target is dollars, the same ris;lt is obtained with 96 percent dollars, the remainder of the portfolio being in Swiss francs and Canadian dollars. Several of the currencies in the low return-risk portfohos of the imports efficiency curves are not repre:;ented at all (or only negligibly) in the portfolio of the investor who wishes to maximize his return in terms of the dollar. These .substantial differences between the two efficiency curves indicate that specificatioln ok‘ the central banks target currency is not merely a theoretical issue.. Tn the modlIe we did not take into aoccunt the volume of trade in each of the currencies. Since the portfolio of most central banks is fairly large compared wiih the market for some currencies, central banks will probably limit their holdings of some currencies because of convertibility constraints, even if these currencies are attractive from the return-risk point of view. For example, the volume of assets denominated in guilders, Canadian dollars and yens in the international money market is fairly small compared with the size of central b,ank reserves. To demonstrate the effects of market size, we found that if the shares of the Swiss, franc and the guilder were restricted, the deutschemark would take their place. 4. Efficient and actual reserves investment policy Orle of the interesting iTSpeCt.S of reserves investment policy is a comparison cf the optimal currency composition derived from the model with the actu.11 compositioia In each country. Such a comparison would enable us to test the at\-itude of central banks towa.rd profit and risk and the performance of portfolio management. But the fact that reserve portfolio data are secret precluded such a rest. In an article written recently by two TMF staff-members, Heller and Knight (1978), aggregate data were for the first time presented on the internat.ional reserve portfolios of 69 countries categorized according to exchange rate regime. These daty. enabled us io test the model at an aggregate level, as well as t;‘le assumption tha’i the industrial countries’ attitude toward profit and risk differs from that of the semi-industrialized and developing countries.’ To calculate the optrmum reserve portfolio If all the central banks or of each group of countries we first h.ave 1.0 cstirnate the eOiciency frontier for -The countrie:; in the IMF study are classified by exchange rate regime: 11 with a floating exchang: rate, 6 in the European snake, and 52 wir h a fixed rate. This classification corresponds with that in th,: IMIF reports 211 intr.rr.ational reserves by level of development (the first two groups :ire categorized as industrial co lrltries)
A. Ben-Bussat,
Foreign
exchange
reserrzs
293
each of them separately, since each country has a different import mix. The estimation of the efficiency curve is based on the monthly return disl.ribution in the period 1972-76, with the returns in each case being expressed in terms of the objective function of each country in 1976. The options consisted only of investments in th=: principle reserve currencies dollars, sterling, deutschemarks, French francs, and Swiss francs. Estimating the eniciency curve does not in itself leave us with a single solution, and information is needed on the utility functions in order to select the most d.:sirable of all the optimal portfolios. Since the utility function is not given, we soived this problem by using Sharpe and Lintner’s market price theory (CAPM). According to this theory the optimum pc\ytfolio of risky assets t&ould be determined at the point of tangency betwe::n the efficiency curve. and the straight line, which rises from the riskiess asse;: return. When the objective function is defined in terms of a single currency, the rate of interest on treasury bills in that currency is generally taker1 as the riskless asset rate. Since in our case the objective function is a basket of currencies, we used the average of interest rates on treasury bills in these currencies, expressed in terms of the import basket.* In each country the portfolio selected according to CAPM also represents a relatively low return-variance strategy. Thus, even if we would prefer to se&t an efficient portfolio using a specific utility function that assumes a relatively high degree of risk aversion on the part of the central banks, we woufd obtain a portfolio very similar in composition to that presented in this section. The optimum portfolio of the industrial countries and that of the developing and semiindustrial countries (as a group) were calculated by weighting the effllzient set of each country in the group by its international reserves. Table 4 sets forth the efficient and the actual portfolio for the two groups at the end of 1976, with the degree of detail dictated by that of the data on the actual portfolio. The table also lists the pates of return and variance for each portfolio. Several con;lusians can be drawn from the table. Fi? it, it wi!l be seen that there is a fairly close correspondence between the actual and the efficient portfolio of the s,emi-industrial and developing countries. They do not completely tit, but it should be borne in mind that our argument is that profit and risk are the main but not the sole considerations. It may ‘Jet-ywell be that political and other factors explain why the weight of the doHar is 14 percent greater than the optimum. Part of this difference may also be due to the performance of portfolio management. On the other hand, in th: case of the industrial countries the correspondence between the actual and the
‘The rate of interest on the import mix of each country ranged around 7 percen;. In our C:IW deviation of 1 percent on riskless asset interest lacks sip-lificawe in selecting :fic efficient portfolio.
a
A. Ben-Bassar,
291
Foreig,l
exchange
Table
E05cient and actc:il currency
composition
--
4
of fo1:;;.1 exchange 1976 (p zrcent). ------
Semi-industrial and developing cot!nlries _ _ -.---_
Dollar Sterling Deutschemark 01 hers Total Mean return’ Standard deviation”
wsvws
te:.crve~ by IeI-el of development,
Industrial
countries -____
Efficient portfolio
Actual portfolio
EflYcient portfolio
Actllal portfolio
58.2 2.0 12.5 27.4 lWJ.O
72.2 2.5 10.0 15.:: 1OO.O
56.1 0.1 13.6 29.6 100.0
85 6 1.3 4.8 8.3 100.0
10.1 8.0
9,; 6.7
9.6 6.1
8.0 9.1
---
‘The calculations of returns and standard devalion by the reserves oi the clountries in each group.
--
are in terms of an import
basket weighted
efficient portfolio is very low, which supports our argument that in these countries the profit-risk factor is of sec:ond.tiry importance. Additional proof of the disparate behavior o!Pthe industrial countries can be found by comparing the return arlc! variance of the actual portfolio of these two groups, when the two moments are calculated using the portfolio composition of each group in 1970 and the distribution of the returns in 1972-76. It was found that the averaye return in the industrial countrres is lower thas that in the others, while the ~~~arianceof the return is much higher (see table 4).9 5. Conclusion
In this paper it is shown that the composition of foreign exchange reserves in the semi-industrialized and ilevelopi:lg countries depends on the return and risk of their investment in each currezlcy and on the currency composition of their imports. Calculating the optimum portfolio for the Israeli econo1-,13/,we found that the composition of the currency basket in terms of which the variables are expressed has not t)nly theoretmcal but also practical impor:ance. Estimating the e?%cient portfqiio for g,roups of lcolmtries classified by level of development shows a closer correspcmdence bletween ths actual and the ‘In the actual currency- composition there is a rr:sidu.il, which t!cller and KnigLt claim breaks down mainly between French francs, Swiss francs, and Dsutch guiiders. In calculating the return and variance for each portfolio we allocated rhe rf sidual among these currencies in an arbitrary but proportionally equal manner in both groups.
portTolL? for developing and semi-industrial countries than for industrial countries. Evaluation of the actual portfolio by the mean-variance criterion also shows a better performance for the former group. These findings prove our argument that the profit consideration plays a smaller role in the industrial than in the other countries. Another conclusion relates to the impact of international capital movements on the demand for currencies. Since the currency composition of reserves Mers among countries, it is clear that changes in the country distribution of the worlds reserves may alter the central banks’ demand for the various currencies. effxient
References Ben-Bassat. A., 1978, The optimal composition cf Israel’s foreign exchange reserves, Unpublished Ph.D. dissertation, The Hebrew L’nivenity. Hagemann, H.A., 1969, Reserve policies of central banks and implications for U.S. balance of payments policy, American Economic Review 59, 62 77. Heller, H.R., 1966. Optimal international reserves, The E:onomic Journal 76, 296311. Heller, H-R. and M. Knight, 1978, Reserve currency :>referenl.es of central banks, Essays in international finance, no. 131 (Princeton). Kenen. P.B., 1967. Reserve-asset preferences of central banks and stabtiity of the &d-exchange standard, Princeton Studies in International Finance, no. 10. Levy, H. and M. Sarnat, 1,975, Devaluation risk and the portfolio of intern.-itional investment. capital markers (North-Holland, in: E.J. Elton and M.J. Gruber, eds.. International Amsterdam) 177-206. Makin, J.H., 1971. The composition of international reserve holdings: A problem of choice involving risk, American Economic Review 61, 819-832. Markowitz, H.M.. 1952. Portfolio selection, Journal of Finance 7. 77-91. Officer, L.H. and I.D. Willett, 1969. Reserve asset preferences and :he conhdcncc problem in the crisis zone, Quarterly Journal of Economics 83, 688-695. Solnik, B., 1973, European capital market; Toward a general theory of an international Zapitai market (Heath. Lexington). Stekler, I.. and R. Pickarz. 1970. Reserve asset composition for major central banks, Oxford Economic Papel-s 22, 260-274. Tobin, J., 1958, Liquidity preference as behavior towards risk, Review of Economic Stud.es 26. 65 --86.
Economics
10 (1980) 285-295. 0 North-Holland
Publishing
Company
THE OPTIMAL COMPOSITION OF FOREIGN EXCHANGE RESERVES Avraham Bank
of Israel
und
BEN-BASSAT*
The Hebrew Uniaersity, Jcwsalem.
Received April 1978, revised version received October
Isr~rel 1979
This paper presents a model for selecting an optimal foreign exchange reserves portfolio for semi-indu:;trial and developing countries, using the mean-- variance approach. The model described here focuses on the relationship between the composition of reserves and that of imports, :ts well as the impact of return and risk of thk investments in each currency. The empirical imporiance of these factors is demonstrated by investigating the optimal policy for Israel in the period 1972-76. in comparing the actual and the efficient portfolio of different groups of countries, we find that proiit considerations play a greater role in semi-industrial and developin: countries than in industrial ones.
1. Iatraduction The ;iirn 01;this study is to describe an analytical approach to the optimal selection of a country’s foreign-exchange reserve portfolio. We concentrate on the molives for holding foreign reserves in semi-industrialized and developing countric:s and compare their actual behavior with that of the industrial countril:s. Previous studies on international portfolio selection have dealt mainly with the individual or the firm. A few economists, such as Kenen (1967), Officer and Willett (1969), Hagemann (1969), Steckler and Pickarz (1970), and Makin (197:U), have studied the co~lposition of national foreign currency reserve:;, bul: they deal only with Ihe optimum ratio of gold to foreign currenc ies. These studies relate: to a period in which the international monets.ry sy~jtem was based on gold ;Ind the typical pattern was one of fixed exchange rates adjusted at relatively long intervals; this helps to explain w-hy studies of fo’reign-reserves composition dealt only with the share of gold in total reserves, other currencies being represented by the dollar. After the shift *This paper is part of March 1978. I would like for their useful gui,dance from whose c~~mmcnts I fruitful 1iscussion of many
a Ph.D. thesis submitted to the Hebrew University of Jerusalem in to thank my supervis,ors. Professors Najav Halevi and David Levhari. and comments, and Dr. Zvi Sussman who suggested the study and benefited greatly. I am also indebted to Prof. Giora Hanoch for a points.
186
A. Ben-Bussut,
Foreign
e.uchun,ge
reserves
from fixed to floating exchange rates, gold and the dollar iost their special place as the chief reserve assets, and today they are only twrl out of a whole array of assets in a central bank’s portfolio. This study is the first attempt LO investigate the selection of such a portfolio. In the first part of the paper the objective function for foreign exchange reserves is formulated and considerations bearing on the choice of a foreign reserve portfolio are discussed. In the second part the application of the model is demonstrated by estimating optimum portfolios for Israel in the period 1972-76. In the final part we arialyze the reserves investment policy of the industrial countries in comparison vq4h that of the :;erm-industrial and dev,Aoping countries and compare the actual portfolio of each group with Its effiic.ient portfolio.
2. The analytical anproach
The choice of a portfolio by a central bank is intimately related to the objects served by the holding of reserves. These have been extensively discussed in numerous studies of the demand for reserves, and many aspects are mentioned, among them political and economic ones. While economists ,differ widely on the determinants of the size and composition of reserves and on their relatilve importance, they all mention the instability of balance-ofpayments flows as the main reason for holding reserves.. Foreign currency receipts (including capital imports) are not necessarily equal to foreign currency outla.ys at any given point in time; countries therefore keep a foreign exchange reserve in order to ‘pay for imports when receipts fall short of outlays. This is the precautionary motive [see, for example, Heller (1966, pp. 300-301)]. in constructing an investment mod.el for a central bank it is important to specity not on1.y the purposes for which the foreign reserves are held, but also the country’s attitude to profitability. Two groups of countries may be distinguished: (1) Industrialized countries with floating exchange rates and whose currencies may serve as reserves in the portfolios of other countries. The reserve composition of these countries is influenced more by the stability of the international monetary system, while profit considerations are of secondary im;sortance. (2) The developing al,d semi-industrialized countries, in most of which rates of exchange are subject to administrative control. Unlike the rich nations, thes? countries are free from concern over international monetary stability, 50 that their allocation of reserves among alternativt: assets will be influenced mainly by profit and liquidity considerations. This stuldy deals chiefly wit’h the second group of countriks. In view of the
uncertainty prevailing m the foreign exchange market, their decisions regarding reserve composition are determined according to the rates of return and risk of the various currencies, with the object being to maximize the reurn in terms of a basket of imports at a given risk. The approach adopted here for finding the rate of substitution between return and risk is the mr,an-variance model developed by Markowitz (1352) and Tobin (19%). Application of this model to a central bank demonstrxto< the importance of determ.ining the objective function. In two studies of international portfolio selection - Solnik (1973) and Levy and Sarnat (1975) -~ it has been proved that if different economic agents (countries) face different prices, because of non-tr;tded goods, tariffs, taxes, etc. then optima1 international porIfo1ios will depencl on consumption patterns. As mentioned, foreign reserves are, inter alia, meant to finance imports. The nation’s imported consumption, unlike the inc.ividual’s, is paid for in a variety of currencies, depending on the origin of the imports. Since the consumption currency plays an important role in determining the optima1 portfolio, the optimum composition of reserves will be obtained by minimizing variance at a given return, where risk and return are exl’ressed in terms of the basket of import currencies.’ The problem is then formulated as:
i=l
i=l
j=l
jai
subject to f-J=
i i=
Uiyi,
1
CUi=l,
where L+ is the optimum share of currency i in the portfolio, 0: is the variance of the returns on currency i, R,, is tb,e c0,rre:ation coefficient between returr;s on currencies i and j, and i.1;is the return nn currency i. Fl’he rate of return on each currency is a functior. of the interest rato (Qi), ;Lnd the rate of change of the exchange rate of currency i 1t1 relation 10 the import currency basket (Ei ):
‘For a formal proof of this argument see Ben-Bawt
(19’5).
A.
288
Ben-Bnssat,
Foreign
exchange
resertres
The problem does not have a unique solution for the vector of weights since the choice of optimal composition depends on the desired degree of rise. Therefore there is an infinite set of solutions, each representing an alternative optimum combination of return and variance. 3. The findings The efficiency curve representing the relationship between the maximum return for a given risk was estimated by a quadratic programming model2 The model was specified in the preceding section; here we examine the empirical zesuits obtained from ii. The application of the mode1 is illustrated by estimating the efficiency curve for Israel for 1972-76. The estimate is based on monthly observations c,f returns on the major currencies and gold; :he distribution of returns resuling from these observations is assumed to approximate the distribution of the ex ante returns.3 In order to understand the optimum composition obtained for each level of returns, data are presented on the mean and the standard deviation of the returns on various poGble assets (see table 1). The variance of the total return is explained mainly by fluctuations in the exchange rate: the monthly fluctuation in the exchange rate varies between 15 and 30 percent, cornpa~~d Table Retu::n
and
standard
deviation
1
of investments (percent).
in selected
currencies.”
1972 -76 -_
In import terrns
~Dollar Sterling Canadian dollar Deutschemark Gi_ulder Swiss f?snc French franc Yen tiold aBased on monthly
Mean 88 4.8 9.2 13.5 13.6 15.6 12.4 7.4 23.3 ~-___
-
In dollar terms
Standard deviation
Mean
15.6 20.0 20.6 28.1 27. I 30.8 26.5 25.1 94.0
7.8 4.1 82 12.9 13.0 15.0 11.8 6.7 26.7
Standard deviation 2.3 30.8 13.6 40.2 39.3 41.9 38.8 33.2 101.5 .~
data.
‘The computations were performed wi1.h a computer program constructed at the Department uf B?lsiness Administration of the Hebrew University. I am indebted to Prof. Haim Levy for putting this program at my disposd!. 3Calculation of the returns, variances, and covariances is based on the beginr.ing-of-monrh changes in exchange rates and on the beginning-of-mo+h interest rate on monthly EurcIdzposits.
A. Ben-Bassat,
Foreign
cxchmge
r. serve:
289
with no more than 2-3 percent for the interest rate. The interest rates and the rates of exchange are not of collrse entirely indepe.ldent of one another: currencies with a high probability of devaluation usuaily have higher interest rates; that is to say, the correlation between the interest rate and the rate of change of the exchange rate is negative, and we obtain R = -0.85. This inverse relatio*lship narrows the differences i:i r.otal returns between the various currenciC:s Some remarks on gold are in order, since it is still a major reserve asset. In the period reviewed, the average return on gold was 6-16 percentage points above that on other currencies; +ut the high return was accompanied by a standard deviation of 6@-75 percentage points above that of other currencies.4 3.1. The efficiency curce in terms of imports Estimating the efficiency frontier of the Bank of Israel in 1972-76 in terms of imports, it is found to be a concave= curve; this means that the return increases at a diminishing rate with the increases in the variance. The alternative return-risk combinations on the curve apply to different currency baskets. To illustrate this point, selected combinations are presented in table 2. The first two rows show optimum combinations of rzldrn and standard Table 2 Composition -.-
-.___-
of optimum portfolios for selected returns,” 1972 76 (percent ). ~-.-~
In import terms --
-~
In dti!lar terms ____ __~
__ _
Selected optimum combinations Mean return
9.0
12.0
14.0
x.0
12.0
13.0
Standard deviation
3.3
10.5
19.5
1.3
20.2
30.0
62.0 1).Y
l-l.6 0.4
Currency composition of selected combinatiorts Dollat C.madiar. dollar Sterling Deutschemark Guilder Swiss franc French franc Yen Gold Tota:
46.0
CO.3
30.3
96.4 1.9
21.5 10.6 4.8 1.8 7.1 7.0 0.3
19.4 19.3
16.3 33.4
0.2
24.2
36.0
6.0 __ 5.0
10.0
0.3 0.7
12.9
19.0
100.0
100.0
100.0
100.0
100.0
100.0
0.5
“Based on monthly data. ‘The rate of return on gold was estimated as the change III the p-ice of gold in the Jolla~denominated market, converted into terms of the Israeli import currency basket.
290
f‘oreigrlcwhangr
A. Ben-Bow&
re.serws
deviation, .while the corresponding porrfolios are shown in the lower part of :he tat-le. The optimum portfolios depend on the desired risk, lower-risk strategies involving greater diversification. For example, for the lowest retu.-n-risk combination [column ( J-)-J1the optimum portfolio contains seven of tre eight possible currencizs as well as gold. But although all but one of the possibilities appear, three of them account for 78 percent of tie total. The optimum low return-risk portfolio turns out to be quite similar to the currency composition of the Israeli irlport basket.’ This was to be expected Table Distribution cuuntry
3
of Israel’s imports of or:& 1972 (percent).
U.S.A. U.K. Canad; West Germany Nether land:, Swi:zerland Belgium Franc: Italy Japan Swedl:n Denmark Othe 7s’ Total
by
26.2 17.9 0.9 12.5 3.2 3.8 47 3.8 63 I .9 1.8 0.5 16.5 1000
“IIllports from the group of ‘other’ tour tries are assumed to have been paid in dl)llars
since a reserve portfolic, based on the composition of imports reflects a strategy of hedging against fluctuations, in the exchange rates of import cl:rrencies and.. as mentioned earlier, exchange rate fluctuations were the drlminant clement in the total return vat-lance in the period reviewed. Thus, th: lowest risk is obtained bith a portfolio wtm;t: composition is similar to rhSit of the currency basket in the objective function, in our case that of Israel’s imports. As the level of return and risk i> increased, the number of currencies in the portfolio declines steadily, until at a return of 12.5 percent we are left with only three currencies, apart from gold: the dolliai., the Swiss franc. and the guiider. From this point on, any increase in return and risk involves a rapid
‘lmport~ to ihr,lel were c,la;;sified by twelve countries of origin, which account for about 84 ;xx
A. Ben-Buswt,
FoIxigr~
r.xchange
rewrws
291
drop in the share of the dollar and a corresponding rise in the share of the Swiss franc and goId. Note that because of its riskiness the share of gold is very small at the low risk level, and it rises with an increase in the desired return and risk.6 Besides the return and risk of each asset, the optimal portfolio is determined by the coefficients of correlation between the returns on the various assets. Thus, in choosing an opZma1 portfolio, preference must be given to sets of assets with low correlations, and particularly to assets wh:ch correlate negatively since such combinations reduce’ the total risk. The lowest correlations are obtained between the U.S. and Canadian dollars on the cne hand and ail other currencies on the other. In particular, the correlation between the dollar and the European currencies is highly negative (around -‘0.5). This is one of the main reasons why low-risk portfolios consist of roughly half dollars and half other currencies (table 2). The large effec-t of a negative correlation of returns on the standard deviation of the portfolio can be gauged from the fact that the lowest standard deviation for any single currency is 16 percent for the dollar at a return of 9 percent, where#ls the standard deviation of the portfolio yielding this return is only 3 perce;::. The most important (as reserve currency) of the European currencies is t4e SWISS franc. The reason for the smaller weight of the other European currencies in spite of their very similar distribution of returns is that they arc highly and positively correlated among themselves. This implies that they are substitutes rather than complements within the portfolio. The Swiss franc is predominant among the European currencies because of its better performance (ratio of return to standard deviation).
The major argument of this paper is that the efficiency curve must be estimated in terms of the country’s basket of imports. Is this conclusion empirically relevant or is it merely of theoretical interest’! To answer this question we estimate the efficiency curve in terms of dollars, since this currency is used by most countries as a unit of account for foreign reserves. and the corresponding portfoi’os. The optimum return risk combinations where the objective function is specified in dollars, are presented in thl: second part of table 2. It can bra seen that the results differ considerably from those in import terms. The mait! currencies are still the dollar and the Swiss franc but their weights are different along the entire curve. For example, the lowest risk level for both the dollar and the import-basket curves is obtained at a standard deviation of around 2.5 percent. Gn the imports efficiency “In the highest and return
risk
portfolio.
oold will c
of the assets considered.
remain
as the sole asset. since it has the maximum
r Sk
292
A. Ben-Bassat, Foreign
excharlge resw!es
curve th,is risk is obtained with 45 percent dollars, the remainder of the portfolio consisting of the various European currencies. When the target is dollars, the same ris;lt is obtained with 96 percent dollars, the remainder of the portfolio being in Swiss francs and Canadian dollars. Several of the currencies in the low return-risk portfohos of the imports efficiency curves are not repre:;ented at all (or only negligibly) in the portfolio of the investor who wishes to maximize his return in terms of the dollar. These .substantial differences between the two efficiency curves indicate that specificatioln ok‘ the central banks target currency is not merely a theoretical issue.. Tn the modlIe we did not take into aoccunt the volume of trade in each of the currencies. Since the portfolio of most central banks is fairly large compared wiih the market for some currencies, central banks will probably limit their holdings of some currencies because of convertibility constraints, even if these currencies are attractive from the return-risk point of view. For example, the volume of assets denominated in guilders, Canadian dollars and yens in the international money market is fairly small compared with the size of central b,ank reserves. To demonstrate the effects of market size, we found that if the shares of the Swiss, franc and the guilder were restricted, the deutschemark would take their place. 4. Efficient and actual reserves investment policy Orle of the interesting iTSpeCt.S of reserves investment policy is a comparison cf the optimal currency composition derived from the model with the actu.11 compositioia In each country. Such a comparison would enable us to test the at\-itude of central banks towa.rd profit and risk and the performance of portfolio management. But the fact that reserve portfolio data are secret precluded such a rest. In an article written recently by two TMF staff-members, Heller and Knight (1978), aggregate data were for the first time presented on the internat.ional reserve portfolios of 69 countries categorized according to exchange rate regime. These daty. enabled us io test the model at an aggregate level, as well as t;‘le assumption tha’i the industrial countries’ attitude toward profit and risk differs from that of the semi-industrialized and developing countries.’ To calculate the optrmum reserve portfolio If all the central banks or of each group of countries we first h.ave 1.0 cstirnate the eOiciency frontier for -The countrie:; in the IMF study are classified by exchange rate regime: 11 with a floating exchang: rate, 6 in the European snake, and 52 wir h a fixed rate. This classification corresponds with that in th,: IMIF reports 211 intr.rr.ational reserves by level of development (the first two groups :ire categorized as industrial co lrltries)
A. Ben-Bussat,
Foreign
exchange
reserrzs
293
each of them separately, since each country has a different import mix. The estimation of the efficiency curve is based on the monthly return disl.ribution in the period 1972-76, with the returns in each case being expressed in terms of the objective function of each country in 1976. The options consisted only of investments in th=: principle reserve currencies dollars, sterling, deutschemarks, French francs, and Swiss francs. Estimating the eniciency curve does not in itself leave us with a single solution, and information is needed on the utility functions in order to select the most d.:sirable of all the optimal portfolios. Since the utility function is not given, we soived this problem by using Sharpe and Lintner’s market price theory (CAPM). According to this theory the optimum pc\ytfolio of risky assets t&ould be determined at the point of tangency betwe::n the efficiency curve. and the straight line, which rises from the riskiess asse;: return. When the objective function is defined in terms of a single currency, the rate of interest on treasury bills in that currency is generally taker1 as the riskless asset rate. Since in our case the objective function is a basket of currencies, we used the average of interest rates on treasury bills in these currencies, expressed in terms of the import basket.* In each country the portfolio selected according to CAPM also represents a relatively low return-variance strategy. Thus, even if we would prefer to se&t an efficient portfolio using a specific utility function that assumes a relatively high degree of risk aversion on the part of the central banks, we woufd obtain a portfolio very similar in composition to that presented in this section. The optimum portfolio of the industrial countries and that of the developing and semiindustrial countries (as a group) were calculated by weighting the effllzient set of each country in the group by its international reserves. Table 4 sets forth the efficient and the actual portfolio for the two groups at the end of 1976, with the degree of detail dictated by that of the data on the actual portfolio. The table also lists the pates of return and variance for each portfolio. Several con;lusians can be drawn from the table. Fi? it, it wi!l be seen that there is a fairly close correspondence between the actual and the efficient portfolio of the s,emi-industrial and developing countries. They do not completely tit, but it should be borne in mind that our argument is that profit and risk are the main but not the sole considerations. It may ‘Jet-ywell be that political and other factors explain why the weight of the doHar is 14 percent greater than the optimum. Part of this difference may also be due to the performance of portfolio management. On the other hand, in th: case of the industrial countries the correspondence between the actual and the
‘The rate of interest on the import mix of each country ranged around 7 percen;. In our C:IW deviation of 1 percent on riskless asset interest lacks sip-lificawe in selecting :fic efficient portfolio.
a
A. Ben-Bassar,
291
Foreig,l
exchange
Table
E05cient and actc:il currency
composition
--
4
of fo1:;;.1 exchange 1976 (p zrcent). ------
Semi-industrial and developing cot!nlries _ _ -.---_
Dollar Sterling Deutschemark 01 hers Total Mean return’ Standard deviation”
wsvws
te:.crve~ by IeI-el of development,
Industrial
countries -____
Efficient portfolio
Actual portfolio
EflYcient portfolio
Actllal portfolio
58.2 2.0 12.5 27.4 lWJ.O
72.2 2.5 10.0 15.:: 1OO.O
56.1 0.1 13.6 29.6 100.0
85 6 1.3 4.8 8.3 100.0
10.1 8.0
9,; 6.7
9.6 6.1
8.0 9.1
---
‘The calculations of returns and standard devalion by the reserves oi the clountries in each group.
--
are in terms of an import
basket weighted
efficient portfolio is very low, which supports our argument that in these countries the profit-risk factor is of sec:ond.tiry importance. Additional proof of the disparate behavior o!Pthe industrial countries can be found by comparing the return arlc! variance of the actual portfolio of these two groups, when the two moments are calculated using the portfolio composition of each group in 1970 and the distribution of the returns in 1972-76. It was found that the averaye return in the industrial countrres is lower thas that in the others, while the ~~~arianceof the return is much higher (see table 4).9 5. Conclusion
In this paper it is shown that the composition of foreign exchange reserves in the semi-industrialized and ilevelopi:lg countries depends on the return and risk of their investment in each currezlcy and on the currency composition of their imports. Calculating the optimum portfolio for the Israeli econo1-,13/,we found that the composition of the currency basket in terms of which the variables are expressed has not t)nly theoretmcal but also practical impor:ance. Estimating the e?%cient portfqiio for g,roups of lcolmtries classified by level of development shows a closer correspcmdence bletween ths actual and the ‘In the actual currency- composition there is a rr:sidu.il, which t!cller and KnigLt claim breaks down mainly between French francs, Swiss francs, and Dsutch guiiders. In calculating the return and variance for each portfolio we allocated rhe rf sidual among these currencies in an arbitrary but proportionally equal manner in both groups.
portTolL? for developing and semi-industrial countries than for industrial countries. Evaluation of the actual portfolio by the mean-variance criterion also shows a better performance for the former group. These findings prove our argument that the profit consideration plays a smaller role in the industrial than in the other countries. Another conclusion relates to the impact of international capital movements on the demand for currencies. Since the currency composition of reserves Mers among countries, it is clear that changes in the country distribution of the worlds reserves may alter the central banks’ demand for the various currencies. effxient
References Ben-Bassat. A., 1978, The optimal composition cf Israel’s foreign exchange reserves, Unpublished Ph.D. dissertation, The Hebrew L’nivenity. Hagemann, H.A., 1969, Reserve policies of central banks and implications for U.S. balance of payments policy, American Economic Review 59, 62 77. Heller, H.R., 1966. Optimal international reserves, The E:onomic Journal 76, 296311. Heller, H-R. and M. Knight, 1978, Reserve currency :>referenl.es of central banks, Essays in international finance, no. 131 (Princeton). Kenen. P.B., 1967. Reserve-asset preferences of central banks and stabtiity of the &d-exchange standard, Princeton Studies in International Finance, no. 10. Levy, H. and M. Sarnat, 1,975, Devaluation risk and the portfolio of intern.-itional investment. capital markers (North-Holland, in: E.J. Elton and M.J. Gruber, eds.. International Amsterdam) 177-206. Makin, J.H., 1971. The composition of international reserve holdings: A problem of choice involving risk, American Economic Review 61, 819-832. Markowitz, H.M.. 1952. Portfolio selection, Journal of Finance 7. 77-91. Officer, L.H. and I.D. Willett, 1969. Reserve asset preferences and :he conhdcncc problem in the crisis zone, Quarterly Journal of Economics 83, 688-695. Solnik, B., 1973, European capital market; Toward a general theory of an international Zapitai market (Heath. Lexington). Stekler, I.. and R. Pickarz. 1970. Reserve asset composition for major central banks, Oxford Economic Papel-s 22, 260-274. Tobin, J., 1958, Liquidity preference as behavior towards risk, Review of Economic Stud.es 26. 65 --86.