Download - Portfolio substitution and exchange rate volatility

ELSEVIER Journal of Monetary Economics 39 (1997) 517 534

JOURNALOF Monetary ECONOMICS

Portfolio substitution and exchange rate volatility

Anne Sibert a'*, J i m ing H a b

aDepartment of Economics, Birkbeck College, London W1P IPA, UK; and University of London and CEPR

blnternational Monetary Fund, Washington, DC 20431, USA

Abstract

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.

Keywords: Exchange rates; Financial markets JEL classification: F31; G15

1. Introduction

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 markets have all raised the ability and willingness of investors to respond promptly

*Corresponding author.

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.

0304-3932/97/$17.00 @, 1997 Elsevier Science B.V. All rights reserved Pll S 0 3 0 4 - 3 9 3 2 ( 9 7 ) 0 0 0 2 8 - 7

518 A. Sibert, J. Ha / Journal o f Monetary Economics 39 (1997) 517 534

to a change in perceptions. 1 Both academics and policy makers have expressed fears that as these innovations continue, exchange rates will respond increasingly 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.

This paper introduces a model with imperfect portfolio substitution. We employ a two-country, two-good overlapping-generations model where consumers must buy goods in the sellers' currencies. Young consumers are uncertain 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.

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.

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.

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.

2. The model

Exchange rates react sharply to news and are more volatile than goods prices. This suggests they are determined in financial asset markets. Karaken and

ISee Goldstein et al. (1993) for a discussion of this.

A. Sibert, J. Ha / Journal of Monetary Economics 39 (1997) 517-534 519

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.

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.

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.

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.

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.

2.1. The c o n s u m e r s

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,

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.

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.

4The model is related to Sibert and Liu (forthcoming).

520 A. Sibert, J. Ha /Journal of Monetary Economics 39 (1997) 517-534

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.

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.

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.

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.

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.

The preferences of the generation-t consumer with location c~ are

W = U(cY, cY;0.5) + flE[U(c~,c~:;a)], 0 < fl < 1, (1)

O' where c y and cj are his consumption o f g o o d j when young (at time t) and old (at time t + 1), respectively, E is the expectations operator, conditioned on time-t

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.

A. Sibert, J. Ha /Journal qf Moneta~ Economics 39 (1997) 517-.534 521

consumption of the home good

A home real balances saved

lower transactions costs the budget flatten

........,. "-. / constraint

. . . . . . . . . .

\\\ \\\\~

\\\\\ \\\\\

p.

foreign real C consumption balances of the foreign saved good

Fig. 1. The budget constraint of the old.

information, and

~(C~/CI-'x) 1 -~'/(1 -- p) if 1 # /~ > O, U(CH, CF~

1 2 1 n O n + (1 - - :~) lnc e i f f l = 1. (2)

We follow the notational convention of denoting all variables evaluated at

r ime t + 1 w i t h a pr ime. T h o s e e v a l u a t e d at t are w r i t t e n w i t h o u t a pr ime.

When young, the agent's budget constraint is

pHCYH "4- pFc y + mH + emv = p n x n + pFXF, (3)

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.

When old, his budget constraint is

ot , ! hp'nc~ + pI.'CF = hmn + e'mv if pncn > ran, (4) t ,or pncn + hp'vc'F = rnn + he'mv otherwise,

where h - 1 > 0 is the proportional cost of spot market trading.

522 A. Sibert, J. Ha /Journal of Monetary Economics 39 (1997) 517-534

At time t + 1, he learns his ~ and the time-t + 1 variables and maximizes U(c~, c~'; ~) subject to (4). The solution is

/ - o ' _ ~ m n i f c t > ~ ' , [a~' ( 1 - c 0 d c n - ~ ' p ' n = ( - 1 ~ m ~ if ~>0~,

' cn _~,p~ , [ (1 - --~)e_~,)pkm F if ~ < c(, c ~ = / ° ' - ~ m u i f ~ < ~ ' , c ~ = c~' ( 1 - -

/

[ mn/p'u otherwise, e'mv/p'v otherwise,

(5) where

t . 0 ~ t ~' = hmn/ (hmn + e mr), _ = mn/ (mn + he'mv). (6)

His consumption when young satisfies

c y = 0 .5 (pnxn + pFXF -- mu -- eme)/pi. (7)

Thus, by (1) and (5), he chooses mn and mF to maximize

(fl ) W = U(c y , c~;0.5) + fiEf _U'd~ + U'd~ + d_~ (8)

subject to (5)-(7), where U ' = U(_c~, o.. U . . . . . _ c_F, o0, - U(m~i/pn, e mr /pe , oO and - o ' ' • ' =- U (c ~, ~ ; ~).

The complementary slackness conditions are

(;: f: ) - - ! t UY/(flpv) >_ Et U_'~/(hp'e)da + U'~/p'ndct + h U 2 / p F d ~ (9)

with equality ifmn > O,

eUY2/(flpF) > E, e 'U'z/p 'vd~ + e'U'2/p'rdct + e ' lYz /p 'vd~' (10)

with equality if mF > 0,

where U y - U(c y , cY;0.5). 6

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.

A. Sibert, J. Ha /Journal of Monetary Economics 39 (1997) 517-534 523

2.2. The government

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.

2.3. The stochastic structure

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, x r , 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).

2.4. Equil ibrium

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:

f : ' f : ' ; f cYu + c ~ d ~ + , mn/p'udo~ + , ?~dc~ (11)

(h - - 1 ) f ) ((~j - - mn/p'u)do~ + (M'u - + Mn) /p 'o X ttl .

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 =- e m r / M u = 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:

if c~ > ~,

if ct <g ,

otherwise,

C° F --

o c v = c°v - c~ = c~ = ~zu /grn

ZH/FH

(1 - ~)zvq if c~>&

(1 - ~)rv

( l - c 0 z e q i f~<_~, (1 - ~_)rv

otherwise, z~'q/rF

(12)

524 A. Sibert, ~ Ha / Journal of Monetary Economics 39 (1997) 517-534

= hzn / (h zu + zeq), ~_ = ZH/(ZH + hzvq) , (13)

c y = 0 .5(rnxn + rFxv -- 1 -- q)/rj, (14)

money marke t clearing:

[(fo ; ; )] UYz/rv = fiE z'n/r'v U_'z/hd~ + r'vV'~/r'ud~ + h l J z d a , (15)

I (;o ; )] qUYz/re = fiE z'vq'/r'v U'2da + U'zda + O'2da . (16)

and marke t clearing for good H,

fo ;: c~ + c~dc~ + (1 - ~ - h + h~)ZH/rL, + h g~dc~ + (1 - zn) / rn = XH,

(17)

where s = s l , . . . , sN.

3. A simple economy

In this section we look at a simple case with symmetric countries and log-linear preferences. Shocks may be serially, but not contemporaneous ly , 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.

Mult iplying both sides o f ( 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

q = E(1 -- 0')/E(1 + 0'), (18)

where

0 = [ZH/(ZH + hzvq)] 2 -- [zvq/(hzn + zvq)] 2 .

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*.

Let ~ be the probabil i ty a country 's mone ta ry policy is unchanged from one period to another. If there is no serial correlat ion (zr = ½), q* = 1 solves (18). Suppose, g 4:½ and let ¢ = ~/_z. Then (18) implies

llq,( )2(q,)2 R(q*) :-- 2 ~ z ~ 1 + q~ -- (b q--hq* - hc~ + q* =- B(q*;h) , (19)

A. Sibert, J. Ha /Journal of Moneta~ Economics 39 (1997) 517-534 525

Result 1. There exists q* = q*(h, ~9, n) which solves (19}. I f n > ( < , = ) ½, then q* < ( > , = )1. A sufficient condition for q* to be unique is n < 1/2.

Proof. See the Appendix.

The intuit ion is as follows. If the expected growth of money i is greater than that of money j ~ i, we refer to money i as the 'weaker ' currency. Consumers want to allocate less than half their portfolios to the weaker currency. Suppose the good home and bad foreign money shocks have occurred (state (5, _z)). The foreign currency is weaker and q* < 1, if and only if foreign m o n e y growth is expected to exceed home money growth next period. This is true if and only if 7r~½.

Result 1 does not guarantee global uniqueness for n > ½, but the equil ibrium is locally unique. In our numerical exper iments we were unable to find mult iple equilibria.

Result 2. ~q*/8h > ( < ) 0 !f n > ( < ) 1//2. As h .qoes to zero, the value o f q* remains (locally) unique.

Proof See the Appendix.

With uniformly dis tr ibuted preferences, a rise in h causes agents to want a more diversified portfolio. Tha t is, a rise in h increases the demand for the weaker money; we call this increasing the 'p recau t ionary ' demand for that currency. If n > ( < ) ½, the foreign (home) currency is weaker and a rise in h causes q* to rise (fall).

When h = I, the model produces the Kareken and Wallace (1981) result. The exchange rate is not determined. However , in the limit as h approaches one, the indeterminacy result is not obtained.

We define condi t ional volatility, Vt, as the condi t ional s tandard deviat ion of the exchange rate divided by its condi t ional mean, stdt (e~+ 1)/E,(et ~ 1). 7

Result 3. ©V,/©h < ( >_ , ~- )O if n > ( < , = )½.

Proof See the Appendix.

~Volatility could also be measured as the unconditional mean of the squared percentage change in exchange rates. It is easy to show Result 3 holds in this case as well.

526 A. Sibert, J. Ha / Journal of Monetary Economics 39 (1997) 517-534

Suppose the time-t state is (L z). If n = ½, this conveys no information about the future; consumers allocate half their portfolios to each currency. Because time-t foreign money growth exceeds home money growth, the time-t foreign money supply is larger, relative to the home supply, than was expected at t - 1. Thus, the exchange rate falls below its expected level. If n > 1/2, consumers expect low home and high foreign money growth to persist. They allocate less than half their portfolios to foreign money. The exchange rate falls more than if n = ½. If n < ½, consumers expect high home money growth and low foreign money growth in t + 1. They allocate more than half their portfolios to foreign currency and the fall in the exchange rate is less than

1 8 with n - 2. Suppose h rises. This increases the precautionary demand for the weaker

currency. If n = ½, no currency is weaker and there is no effect. If n > ½, the demand for home currency is lower and the fall in the exchange rate is less pronounced. If n < ½, the demand for foreign currency is lower and the fall in the exchange rate is greater. Thus, in state (~, _z) the exchange rate is lower than expected at t - 1 and it is rising (falling) in h if n > ( < ) ½. The reverse is true for state (~, f). An increase in h lowers the tendency of the exchange rate to rise above its expected level i fn > ½, magnifies it i fn < ½, and has no impact ifn - I- In states (L z-) and (_z, _z), q = 1 and h has no effect.

This contrasts with Canzoneri and Diba's (1992) result that substitutability stabilizes. It is similar to Woodford's (1991), but for a different reason. We view greater substitutability as a lessening of frictions creating a precautionary demand for a weaker currency. Canzoneri and Diba, and Woodford view it as a property of preferences.

They assume consumers get direct or indirect utility from holding money. Increased substitutability is seen as a flattening of indifference curves. Flatter indifference curves require smaller exchange rate changes to accommodate disturbances; this explains Canzoneri and Diba's result.

Actual exchange rates may be the result of a dynamic adjustment toward equilibrium. Greater substitutability may make this process more volatile because it causes shocks to lead to greater disequilibrium. This is in the spirit of Woodford's suggestion that, if agents lack perfect foresight, more substitutability may make the dynamic path subject to greater fluctuations.

The results here require an intertemporal elasticity of substitution of one. Do the results generalize to other elasticities? It seems to us that if /z is sufficiently different from one, the results may not hold. Log-linear preferences are special in that exchange rates do not depend upon output. With other preferences, output shocks matter; these may interact differently with h than do money shocks.

8q. < q5 ensures it still falls.

A. Sibert, J. Ha / Journal o f Monetary Economics 39 (1997) 517-534 527

Our intuition is as follows. Suppose output shocks are positively autocorrelated and uncorrelated with money shocks. By money demand equations (9) and (10), portfolio shares do not depend directly on current output. But, they depend on future prices, which depend on future output. Current output helps predict future output, influencing demand indirectly. If home output is high today, the home good price is expected to be low next period. With transactions costs, this tends to make the future purchasing power of home money high relative to that of foreign. This has both a substitution and an income effect. If current and future consumption are gross substitutes (/~ < 1), the former dominates and the share of portfolios allocated to home currency rises. If they are gross compliments (# > 1), the latter dominates and the share of portfolios allocated to home currency falls. Thus, the exchange rate tends to fall if # < 1 and to rise if p > l .

An increase in the h has two effects. First, it makes output shocks more important in exchange rate determination. The higher cost of converting currencies makes a currency's rate of return more dependent on goods prices. This effect magnifies exchange rate changes caused by output shocks and tends to make a rise in h increase exchange rate volatility.

The second effect of a rise in h is to increase agents' desire to diversify. Suppose home output is high. All other things being equal, home money is expected to make up a greater share of future wealth. Thus, a rise in h increases the demand for money F. If # < 1 this dampens the fall in the exchange rate. If

> 1, this magnifies its rise. The situation is analogous when foreign output is high. Thus, this effect tends to make exchange rate volatility rise with h if/~ > 1 and fall if ~ < 1.

Suppose there are no money shocks. Our intuition is that if tt > 1, exchange rates become less volatile with a rise in substitutability. If/~ < 1, we are unsure what will happen.

4. Numerical experiments

In this section we solve the model of Section 2 numerically. To do this, we must specify the vector of state variables, {si}, the matrix of transition probabilities, {nik}, the transactions cost, h - 1, and the preference parameters fi and I~.

To choose the technology parameters, we follow the common procedure of supposing variables are independent and imposing a two-state specification for each random variable. 9 Let xi e { x + a, x - a } and zi e {z + b, z - b} , i = H , F.

9See, for example, Mehra and Prescott (1985).

528 A. Sibert, J. Ha /Journal of Monetary Economics 39 (1997) 517 534

Let ~u be the probabil i ty u, u = x, z, takes on the same value it did the previous period. Let

U ~ X , Z . Au : = 1 - - ~u Otu

Then the transit ion matrix is A x ® A x ® Az ® A z . ~° As is typical in numerical studies of the foreign exchange market, we calibrate

our baseline model using US data. We interpret the parameters of the two-state M a r k o v specification as the parameters of an AR(1) process, la They are chosen to replicate quarterly US M2 growth and real G N P for the period i 973Ql -1996Q2 .

We estimate the mean, s tandard deviation, and autocorre la t ion coefficient of zi = 1/(1 + M2 growth) to be 0.98, 1.06, and 0.77, respectively. We normalize the mean of xi to be 100. We used a H o d r i c k - P r e s c o t t (1980) filter to detrend the G N P data and find a s tandard deviation of 1.64. M a n y authors find real G N P has a unit root, but our model does not allow a first-order autocorre la t ion coefficient of one. To introduce persistence to output , we let first-order correla- tion be 0.90.

The choice of/3 had little qualitative impact on our experiments. As is usual when using quarterly data, it is chosen to be 0.95. Mehra and Prescott (1985) cite m a n y studies suggest ing/ t is between zero and two; we let it range from 0.5 to 5.0. We let h - 1 range from 0.0005 to 100.

The equat ions we solve are (18)-(20), where consumpt ion levels, _~, and ~ are given by (15)-(17). We solve for q(s), rn(s), and rv(s). There are 16 possible values of s; hence, this amounts to solving 3 x 16 = 48 nonlinear equations for 48 unknowns. The integrals in (18)-(20) are evaluated with a 10-point G a u s s - L e g e n d r e quadra ture rule.

Fig. 2 depicts exchange rate volatility as a function of the percentage transactions cost for different values of/~. Volatility is defined as the uncondi t ional

1 ~-16 {E2]__61Kik(q(Si)ZFi/ZHi)2] mean of std,(e,+l)/E, (e,+l) and is equal to 16/~=1 _ (~,=v~ ~ 1 =,k q (s~) z~,/z , , )2,2/Y ] ~ =,~ q (s,) z~,/z,,i }.1

If/z : 1.0 or 0.75, the intuition from the analytical model is confirmed. Falling transactions costs make exchange rates more volatile. When/~ = 0.5 and 2.0, the result holds except for a small region of intermediate values of h, in which

1°We suppose the countries are symmetric.

XlThis is an approximation which does not take account of the dichotomous nature of the variables. It is a common procedure in the asset-pricing literature; see, for example, Mehra and Prescott (1985), Backus et al. (1993), and Hakkio and Sibert (1995).

12We did the experiments with volatility defined as the unconditional percentage change. The results were similar.


.07

.06

.05

.04

.03

.02

.01

.00

p.=5.0

~ g=4.0

~ - - ~ ~ g=3.0

g=2.0 p.=0.5 p~=.75 Ix=l.O

I I I

0 20 40 60 80 100

Percentage transactions cost

Fig. 2. Exchange rate volatiliy.

smaller transactions costs decrease volatility. For larger values of /~, lower transactions costs decrease volatility unless h is very small ~3, or possibly very large.

To get an idea of where the results come from, we decompose our experiment. Fig. 3 depicts the results when the standard deviation of output is zero and Fig. 4 depicts the results when the standard deviation of money growth is zero. With no output shocks, volatility rises as transactions costs fall, with ~ having little impact. In Fig. 3, exchange rates are constant with /~ = 1. Otherwise. volatility falls with the transactions cost. This effect becomes more important as /~ rises above, or falls below, one. When transactions costs approach zero, the exchange rate becomes delinked from real variables and becomes constant.

Thus, a fall in transactions costs appears to raise the part of volatility resulting from monetary shocks and decrease the part resulting from real shocks. When /~ is near one or very small, real shocks play relatively little role and exchange rate volatility rises as transactions costs fall.

Current transactions costs in the foreign exchange market are small. As Fig. 4 shows, volatility flattens for small values of h. This suggests that further increases in substitutability of currencies will have an unimportant impact on exchange rate volatility. One type of friction would be a transactions cost on transactions. Fig. 4 suggests that such a tax on foreign exchange would have little impact.

~3This is difficult to see in the figure, but can be confirmed numerically.

530 A. Sibert, d. Ha /Journal of Monetary Economics 39 (1997) 517-534

.07

.06

.05

.04

.03

.02

.01

.00 . . . . 0 20 40 60 80 100

~ ~" 0 .5

21o 3.0 4.0 5.0


Fig. 3. Exchange rate volatility: no output shocks.

.07

.06 p.=5.0

ta=4.0 .05

.04 p.=3.0

.03 ~t=2.0

.02 ~t=0.5

.01 gt=.75

.00 ~t=1.0 0 20 40 60 80 100


Fig. 4. Exchange rate volatility: no monetary shocks.


5. Conclusion

Frictions make portfol io adjustment costly. This paper considers the impact of frictions on exchange rate volatility. We study a model with log-linear preferences and a simple stochastic structure analytically. We calibrate and solve a more general model numerically.

In the simple model, if money shocks are positively autocorrelated, less friction implies more volatility. The intuit ion is as follows. States where a country's money growth is low relative to foreign money growth are states where the value of that money is high relative to foreign money. With persistence of shocks, consumers want to allocate more than half their portfolio to this 'strong' currency. Frictions temper this desire; if it is costly to adjust portfolios, consumers like diversification. Thus, in states where a currency is strong, decreased frictions increase demand for it; this increases its value further. Similarly, in states where it is weak, increased frictions decrease demand for it, lowering its value further.

Log-linear preferences are special; ou tput shocks do not affect exchange rates. If ou tpu t shocks matter , a decrease in frictions appears to decrease the influence of goods markets on exchange rates. Thus, real shocks become less important , tending to lower volatility.

The interact ion between moneta ry shocks and frictions implies higher substitutability raises volatility; the interaction between real shocks and frictions implies the reverse. Which effect dominates? Our results suggest that if frictions are sufficiently small or preferences are nearly log linear, it is the former. However , in the empirically relevant range, frictions have little impact on volatility. Increased substitutability leads to little change in exchange rate variability.

One interpretat ion of the friction is that it is a tax on foreign exchange transactions. Thus, an impor tan t policy implication is that such a tax will have little, and possibly a perverse, affect on volatility.

Appendix

Proof of Results I and 2. R and B have the following properties: (i) dR/dq* < ( > )0 if n > ( < ~2; (ii) R(0) = 1/(2n - 1); (iii) R(1) = 0; (iv) R ~ - 1/(2n - 1) as q* ~ ~ ; (v) dB/dq* < 0; (vi) B(0; h) = 1; (vi) B(~b; h) = 0; (vii) B -~ - 1 q* ~ ~ ; and (viii) a rise in h increases B if and only if q* > ~b.

Fig. 5 depicts the case of n > ½. F r o m the figure, there exists a q* < 1 such that R = B. R(~b) < - 1 ensures there are no q* > l such that this is true. Fig. 6 depicts the case of rc < ½. F r o m the figure, there is a unique q* such that R = B and q* > 1. By Figs. 5 and 6, an increase in h causes q* to rise (fall) if there is n > ( < )½. As h ~ 1, the qualitative features of the figures do not change.

532 A. Si bert, J. Ha /Journal o f Monetary Economics 39 (1997) 517-534

1/(2=- 1 )

\ ,,,. ° c r e ° s e

q*o q*~ 1 " ~ ~

Fig. 5. Positively autocorrelated money shocks.

1/[2~-1 ]

, an increase ~ in h R '. Bo

q*

Fig. 6. Negatively autocorrelated money shocks.

Proof of Result 3. dV, > 0 if and only ifEt(et+OE,(et+lde,+l) > Et(det+l)E,(eZ+l). This is true if and only if

i j m n ZHi \ Znm ZHi /I \ ZHm /I

A. Sibert, J. Ha /Journal of Monetary Economics 39 t1997) 517-534 533

where Znl = zFt = ~, Zn2 = zv2 = z and rtkk = 7Z, k = 1, 2. This is true if and only if

i j Zni L \ ZHI ZHi /I \ ,2HI J

~ d q * ~ rcik~jlqiJz~'J ~lkrC2t i j zni -~ zm /

q*2 ZH i / ]

This is true if and only if (q* - qS)dq* > 0. By Figs. 5 and 6, q* < 4~; hence dV, > 0 if and only if dq* < 0. O q * / ~ h > ( < )0 ifrt > ( < ~ ; hence Result 3 is true.

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