# Portfolio substitution and exchange rate volatility

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<ul><li><p>ELSEVIER Journal of Monetary Economics 39 (1997) 517 534 </p><p>JOURNALOF Monetary ECONOMICS </p><p>Portfolio substitution and exchange rate volatility </p><p>Anne Sibert a'*, J iming Ha b </p><p>aDepartment of Economics, Birkbeck College, London W1P IPA, UK; and University of London and CEPR </p><p>blnternational Monetary Fund, Washington, DC 20431, USA </p><p>Abstract </p><p>Legal and institutional changes are making it easier to adjust foreign exchange portfolios. This has raised fears that exchange rates will become increasingly volatile. This paper presents an optimizing, equilibrium model where varying degrees of portfolio substitutability are possible. Our results suggest that if preferences are nearly log linear, or transactions costs are small, exchange rate volatility rises as portfolios become more substitutable. With empirically reasonable parameter values, however, volatility is little affected by substitutability. An implication is that a transactions tax on foreign exchange trading would have little impact. </p><p>Keywords: Exchange rates; Financial markets JEL classification: F31; G15 </p><p>1. Introduction </p><p>An array of operational changes has recently occurred in the foreign exchange market. Improved information systems have enabled market participants to receive information more quickly and have made their beliefs more homogene- ous. The liberalization of cross-border financial flows, technological advances lowering transactions costs, and the growth of liquid domestic securities mar- kets have all raised the ability and willingness of investors to respond promptly </p><p>*Corresponding author. </p><p>This paper was begun when the first author was a consultant in the Research Department at the International Monetary Fund. We are grateful to Allen Drazen and Jianbo Zhang for helpful comments. </p><p>0304-3932/97/$17.00 @, 1997 Elsevier Science B.V. All rights reserved Pll S0304-3932(97)00028-7 </p></li><li><p>518 A. Sibert, J. Ha / Journal of Monetary Economics 39 (1997) 517 534 </p><p>to a change in perceptions. 1 Both academics and policy makers have expressed fears that as these innovations continue, exchange rates will respond increasing- ly sharply to news about economic fundamentals. Foreign exchange markets are growing rapidly and the relationship between these changes and speculative volatility is an important issue. The intent of this paper is to examine whether advances that make portfolio substitution easier also make exchange rates more variable. </p><p>This paper introduces a model with imperfect portfolio substitution. We employ a two-country, two-good overlapping-generations model where con- sumers must buy goods in the sellers' currencies. Young consumers are uncer- tain what their demand for the goods will be when they are old. This matters because old agents must pay a transactions fee to convert currencies. This friction is intended as a proxy for anything that makes portfolio adjustment costly. </p><p>We obtain some intuition-and analytical results by considering a special case where consumers have log-linear preferences and there is a simple stochastic structure. We show that a decrease in transactions costs increases exchange rate volatility if and only if money growth exhibits positive autocorrelation. This is in contrast to Canzoneri and Diba's (1992) result that greater substitutability is stabilizing. It is similar to Woodford's (1991) result, but does not require his assumption of myopic investor behavior. </p><p>We calibrate a more general model with US data and solve it numerically. If the intertemporal elasticity of substitution is close to one, or if transactions costs are small, the results of our analytical model are obtained. For other parameter values, the opposite result occurs. We explain how this can result from the interaction between output shocks and transactions costs. Whatever the size of the intertemporal elasticity of substitution, however, our numerical results suggest that in the empirically relevant range of transactions costs, increased portfolio substitutability has little impact on exchange rate variability. A policy implication of this is that a Tobin-type transactions tax would have little effect. </p><p>We describe the model in Section 2. Section 3 contains the special case of log-linear preferences and a simple stochastic structure. Numerical results for the more general case are presented in Section 4. Section 5 is the conclusion. </p><p>2. The model </p><p>Exchange rates react sharply to news and are more volatile than goods prices. This suggests they are determined in financial asset markets. Karaken and </p><p>ISee Goldstein et al. (1993) for a discussion of this. </p></li><li><p>A. Sibert, J. Ha / Journal of Monetary Economics 39 (1997) 517-534 519 </p><p>Wallace (1981) provide a frictionless model where currencies are viewed solely as financial assets. Startlingly, the exchange rate is not determined; any constant is an equilibrium. </p><p>Lucas (1982) avoids this problem by insisting consumers buy goods with the sellers' currencies. This yields a model where the exchange rate is determined solely in goods markets and does not depend on expectations of future events. By assuming a different timing of transactions, Stockman (1980) and Svensson (1985) introduce a precautionary demand for currencies, allowing financial asset markets to play a role. </p><p>Our goal is to produce a model with three properties. First, the exchange rate should be (locally) unique. Second, it should be primarily determined in financial asset markets. Third, we would like our model to have a parameter measuring the degree of portfolio substitutability. To do this, we make the following assumptions. Consumers must buy goods with the sellers' currencies; they are uncertain what their individual circumstances will be when old; and there is a cost to trading in spot markets before buying goods. </p><p>We suppose the following sequence of events. First, agents sell their en- dowments for money and trade in the spot market for foreign exchange trading. 2 Second, they learn the state of the world. Third, they use their accumulated money to buy goods and may trade in the spot market at a cost. </p><p>This cost is viewed as a measure of substitutability. It could also be inter- preted as measuring the extent to which exchange rates are determined in financial asset markets. The influence of goods markets on the exchange rate can be made arbitrarily small by making the cost to spot market trading after the state of nature is revealed sufficiently low? If the cost is zero, our model reproduces the Karaken and Wallace result; if it is infinite, goods market considerations become important and our model is similar to Stockman's and Svensson's. </p><p>2.1. The consumers </p><p>Overlapping generations of two-period-lived consumers inhabit a home (H) and foreign (F) country. 4 The two goods and monies are country-specific, but, </p><p>2As in Stockman (1980) and Svensson (1985), spot market trading of money from endowment sales is costless. Alternatively, we could assume the young face transactions costs too. With asymmetric country-specific consumers, at most one currency would be held in both countries. </p><p>3The exchange rate is the relative price of two nominal assets. In a frictionless model nothing ties it to anything real. Thus, it is reasonable that the goods market must have some influence for the exchange rate to be determined. </p><p>4The model is related to Sibert and Liu (forthcoming). </p></li><li><p>520 A. Sibert, J. Ha /Journal of Monetary Economics 39 (1997) 517-534 </p><p>following Woodford (1991), we assume consumers are not. This is equivalent to assuming that, although consumers in the two countries may differ, their differences do not affect aggregate demand curves. This allows us to avoid distributional issues that are unimportant here and would complicate the analysis. </p><p>The life of a typical agent is as follows. When young, he is endowed with both goods. He consumes some and trades the rest to the old for money. When he is old, he can trade in the spot market at a cost and buys the country-j good with money j, j = H, F. Both the supplies of the goods and the money growth rates are stochastic; hence, the young regard future prices as random as well. </p><p>We suppose the young are also uncertain about their future preferences. They must decide how much of each money to hold before knowing how much of each good they wish to consume: Alternatively, we could suppose old agents' preferences over consumption bundles depend on noneconomic factors, such as where they live, their health, and the weather. These variables are unknown to the young. Thus, young agents' uncertainty can be in- terpreted as uncertainty about random variables affecting their future preferences. </p><p>The kinked budget set in Fig. 1 represents the opportunities of the old generation-t consumer. If he does not trade in the spot market for foreign exchange, his consumption is shown by point B. If he does trade in the spot market, he incurs a constant proportional real cost. If he buys home money with foreign money, he can consume along line segment AB; if he trades home for foreign money he can consume along BC. The existence of the cost ensures his budget constraint is kinked. A decrease in the cost is reflected in a flattening of the constraint, as indicated in the figure. </p><p>The consumers' uncertainty about their preferences is represented formally as follows. In each period t, a unit interval of agents is born. Agents are indexed by their location, :~, in the interval. They do not know their location when young, but correctly believe it is distributed uniformly on (0, 1). All young agents have the same known preferences. </p><p>The preferences of the generation-t consumer with location c~ are </p><p>W = U(cY, cY;0.5) + flE[U(c~,c~:;a)], 0 < fl < 1, (1) </p><p>O' where c y and cj are his consumption ofgoodj when young (at time t) and old (at time t + 1), respectively, E is the expectations operator, conditioned on time-t </p><p>5The model is related to Goldman (1974). There, agents hold money in addition to bonds because they are unsure of their discount rate and portfolio adjustment is costly. </p></li><li><p>A. Sibert, J. Ha /Journal qf Moneta~ Economics 39 (1997) 517-.534 521 </p><p>consumption of the home good </p><p>A home real balances saved </p><p>lower transactions costs the budget flatten </p><p>........,. "-. / constraint </p><p>. . . . . . . . . . </p><p>\\\ \\\\~ </p><p>\\\\\ \\\\\ </p><p>p. </p><p>foreign real C consumption balances of the foreign saved good </p><p>Fig. 1. The budget constraint of the old. </p><p>information, and </p><p>~(C~/CI-'x) 1 -~'/(1 -- p) if 1 # /~ > O, U(CH, CF~ </p><p>121nOn + (1 -- :~)lnc e i f f l = 1. (2) </p><p>We follow the notational convention of denoting all variables evaluated at r ime t + 1 w i th a pr ime. Those eva luated at t are wr i t ten w i thout a pr ime. </p><p>When young, the agent's budget constraint is </p><p>pHCYH "4- pFc y + mH + emv = pnxn + pFXF, (3) </p><p>where x; is his endowment of good j, m; is his savings of money j, e is the time-t price of money F in terms of money H, and p; is the time-t price of good j in terms of money H. </p><p>When old, his budget constraint is </p><p>ot , ! hp'nc~ + pI.'CF = hmn + e'mv if pncn > ran, (4) t ,or pncn + hp'vc'F = rnn + he'mv otherwise, </p><p>where h - 1 > 0 is the proportional cost of spot market trading. </p></li><li><p>522 A. Sibert, J. Ha /Journal of Monetary Economics 39 (1997) 517-534 </p><p>At time t + 1, he learns his ~ and the time-t + 1 variables and maximizes U(c~, c~'; ~) subject to (4). The solution is </p><p>/ -o ' _~mn i fc t>~' , [a~' (1 -c0d cn-~ 'p ' n =(-1~ m~ if ~>0~, </p><p>' cn _~,p~ , [ (1 - --~)e_~,)pkm F if ~ < c(, c~=/ ' -~mu i f~ 0, </p><p>where U y - U(c y, cY;0.5). 6 </p><p>6Sibert and Liu (forthcoming) show that with no aggregate uncertainty, consumers demand both monies if and only if the depreciation rate of the weaker currency is less than the transactions cost. In the examples in this paper demand for both currencies is always positive. </p></li><li><p>A. Sibert, J. Ha /Journal of Monetary Economics 39 (1997) 517-534 523 </p><p>2.2. The government </p><p>The government produces a public good using inputs of its domestic good purchased with increases in the money supply. Let z) = My/M~ be the reciprocal of country-j money growth where My is the stock of money j. </p><p>2.3. The stochastic structure </p><p>The stochastic structure is chosen to be similar to that of other equilibrium exchange rate models (e.g. Lucas, 1982; Svensson, 1985). The random variables are the outputs and money growth rates. The time-t realization of the random variables, s - {Xn, xr , zn, zv } has a finite, discrete support {sl, . . . , sN } of strict- ly positive, finite elements. The process {st} is a known, first-order Markov process with the time-invariant transition probability rcik= Prob(s' = sils = Sk). </p><p>2.4. Equilibrium </p><p>Money market clearing requires mj = M j, j = H, F. Market clearing for the home good requires the sum of private and public spending on good H and real resources lost in costly spot market trading equal the amount of good H avail- able: </p><p>f: ' f : ' ; f cYu + c~d~ + , mn/p'udo~ + , ?~dc~ (11) </p><p>(h -- 1 ) f ) ((~j -- mn/p'u)do~ + (M'u - + Mn)/p'o X ttl . </p><p>Monetary rational expectations models can have a plethora of equilibria; we focus on the one we find most believable. The current realization of the random variables summarizes the current state of the world and is the only useful information for predicting the future; hence it is natural to consider equilibria where time-t real variables depend solely on st. Thus, let qt =- emr /Mu = q(s) = q and rjt =- p~/MH = rj(s) = r , j = H, F, if st = s. Substituting money market clearing into (5)-(7) and (9)-(11) and rewriting gives the demand for goods: </p><p>if c~ > ~, </p><p>if ct & </p><p>(1 - ~)rv </p><p>( l - c0zeq i f~</p></li><li><p>524 A. Sibert, ~ Ha / Journal of Monetary Economics 39 (1997) 517-534 </p><p>= hzn/(hzu + zeq), ~_ = ZH/(ZH + hzvq), (13) </p><p>c y = 0.5(rnxn + rFxv -- 1 -- q)/rj, (14) </p><p>money market clearing: </p><p>[(fo ; ; )] UYz/rv = fiE z'n/r'v U_'z/hd~ + r'vV'~/r'ud~ + h l J zda , (15) I (;o ; )] qUYz/re = fiE z'vq'/r'v U'2da + U'zda + O'2da . (16) </p><p>and market clearing for good H, </p><p>fo ;: c~ + c~dc~ + (1 - ~ - h + h~)ZH/rL, + h g~dc~ + (1 - zn)/rn = XH, (17) </p><p>where s = sl, . . . , sN. </p><p>3. A simple economy </p><p>In this section we look at a simple case with symmetric countries and log-linear preferences. Shocks may be serially, but not contemporaneously, correlated. Money growth takes on two values. Denote the higher value by Z and the lower value by _z. Since z is the reciprocal of money growth, we call these states 'good' and 'bad', respectively. </p><p>Multiplying both sides of( l 5) by q, subtracting the corresponding sides of(16) from both sides of the result, and substituting in p = 1 and (12)-(14) yields </p><p>q = E(1 -- 0')/E(1 + 0'), (18) </p><p>where </p><p>0 = [ZH/(ZH + hzvq)] 2 -- [zvq/(hzn + zvq)] 2 . </p><p>By (18), q depends only on the money shocks and q = q(zn, ZF). Let q(i, _z) = q*. By (18), q(5, z3 = q(_z, _z) = 1 and q(_z, 5) = 1/q*. </p><p>Let ~ be the probabil ity a country's monetary policy is unchanged from one period to another. If there is no serial correlation (zr = ), q* = 1 solves (18). Suppo...</p></li></ul>

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