Title INTERNATIONAL INTEREST-RATE DIFFERENTIALS AND …

Title INTERNATIONAL INTEREST-RATE DIFFERENTIALS AND THE VOLATILITY OF EXCHANGE RATESSub TitleAuthor HAMADA, Koichi

MUTOH, TakahikoPublisher Keio Economic Society, Keio University

Publication year 1984Jtitle Keio economic studies Vol.21, No.2 (1984. ) ,p.17- 36

JaLC DOIAbstract

NotesGenre Journal ArticleURL https://koara.lib.keio.ac.jp/xoonips/modules/xoonips/detail.php?koara_id=AA00260492-1984000

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INTERNATIONAL INTEREST-RATE DIFFERENTIALS

AND THE VOLATILITY OF EXCHANGE RATES

Koichi HAMADA* and Takahiko MUTOH*

I. INTRODUCTION

One of the major concerns for the current world economy is the question of how exchange rates react to changes in interest rate differentials between countries . By observing recent movements in exchange rates , one cannot help but suspect that the high level of interest rates in the United States may be the reason for the recent weakness of the Deutsche-mark and the yen . It seems that we still need a theoretical explanation of the economic mechanism through which interest differentials influence exchange rates , and the reason why exchange-rate move-ments are so volatile.

This paper is an attempt to present a simple framework that clarifies these issues. We would like to emphasize the analysis of the reason why the response of exchange rates to changes in interest rate differentials is often quite significant .

We shall make use of the model of simultaneous determination of spot and forward exchange rates, a framework developed by Tsiang (1959) and Sohmen

(1966). The model has gained wide acceptance and it has been applied by many authors to their analyses [e.g. Black (1973), Komiya-Suda (lg8oa) , Otani (1982), K. Fukao (1982) etc.]. In this paper, we shall introduce rational expectations into the Tsiang-Sohmen model and develop our analysis in a stochastic framework . Although this has already been done by Shinkai (1969) and Driskill-McCafferty

(1982), they were mainly concerned with the stationary state properties of the model. We shall, on the other hand , analyze the way in which the steady state is eventually reached. We shall study the properties of the expected path of future exchange rates. The path towards the steady state differs depending on the nature of parameter changes. It will be shown, in particular , that the properties of the expected path of exchange rates depend crucially on whether the shock to the system is considered to be temporary or permanent , on whether the shock occurs in international interest differentials or in the import-export function , and so forth.

In Section II, we shall introduce endogenous expectations into the Tsiang-Sohmen model. A simultaneous formulation of the spot and forward exchange rates will be reduced to a stochastic difference equation . In Section III, we shall consider the impact of changes in traders' demand and in interest differentials . We

* We are much indebted to Dis. Kyoji Fukao, Mitsuhiro Fukao, Yoichi Shinkai, Hajime Hort and Kazuo Ueda for their helpful discussions.

17

18 KOICHI HAMADA and TAKAHIKO MUTOH

shall analyze the effect of temporary as well as permanent changes in traders' demand and in interest rate differentials, and the effect of permanent changes anticipated several periods in advance. In Section IV, we shall discuss the economic implications of the results obtained in Section III.

The main results of this paper will be summarized below. We use the qualifying

phrase "tend to" because we can only make statements on the projected path of exchange rates from the present. The actual path of exchange rates is determined by random disturbances that occur in the future. Therefore the direction of exchange rates can be projected only in the sense of expected value.

1) Temporary changes in interest rates as well as in net absorption tend to be well smoothed out by speculative activities.

2) An unexpected permanent shift in net import schedule tends to cause a sudden jump in exchange rates.

3) An unexpected permanent change in interest rates tends to cause a very sharp change in exchange rates. The degree of sensitivity of exchange rates is

positively related to the intensity in speculation. The more intense speculative activities are, the more volatile will be the movement of exchange rates.

4) The effect of a permanent future change in the net import schedule or in the interest rate differential that is anticipated will be less than the effect of unanticipated changes that are in effect from the present.

II. A SIMPLE FRAMEWORK

Let us construct a simple stochastic model of simultaneous determination of spot and forward exchange rates. Suppose at period t there is a single forward exchange market for delivery at period (t+ 1). Following the tradition of Tsiang

(1959) and Sohmen (1966), activities of the participants in the foreign exchange market are classified into three categories according to their functions: first, activities to hedge the value of foreign assets or liabilities arising from import and export commitments. We shall call these activities traders' activities. In our discussion, traders' activities are assumed to be carried out solely in the forward market. Second, speculation is assumed to be carried out by taking uncovered open positions in the forward market. Speculation in the spot market is of course admitted, but they are decomposed into the forward speculation and the covered interest arbitrage for analytical purposes. The intensity of speculation is assumed to be proportionally related to the difference between the forward exchange rate now and the expected spot exchange rate for the next period. Third, arbitrage funds are assumed to move by swap operations in the spot and forward market in such a way that they equate the interest differential between the home country and the foreign country to the forward premium (or discount) of the foreign exchange rate.'

' We assume that the "long-term" capital movements carried out on an uncovered basis are

negligible, and that there are no inflationary expectations.

INTERNATIONAL INTEREST-RATE DIFFERENTIALS 19

Let st be the (natural) logarithm of the spot exchange rate at period t i.e., the price of foreign currently expressed in terms of domestic currency, (say, yen per dollar). Let f; be the logarithm of the forward exchange rate at period t for delivery at period (t+ 1) similarly expressed. Also let Etst + 1 be the expected value of the logarithm of the spot exchange rate at period (t+ 1), projected from the information set available at period t. We follow the Hicksian tradition of assuming that markets open at the beginning of the period . We shall denote the domestic interest rate as rt and the foreign interest rate as r;' , where t indicates that the rate prevails during the period t.

We shall express the excess demand for the foreign currency arising from traders' demand as D(t) and D c (t), where subscript c refers to traders' demand; superscripts s and f refer to spot and forward respectively. Let us assume that traders' demand appears mainly in the forward market, and that it is a decreasing function of the forward exchange rate. That is

(1)Dl(t)=al—blf+wt, al>0, bl>0.

Here, wt indicates the random disturbance appearing in traders' demand; wt is assumed to be distributed with mean zero and constant variance a! , and is serially independent. That is

E [wt] = 0 ,

E [wt • wt,] = Q u if t= t'

=oil tOt' .

For simplicity we shall assume that traders' demand in the spot market is negligible so that

(2)D(t)=0.

Let us turn to the excess demand arising from speculation . Speculators are assumed to take deliberately long or short positions , and to Qperate solely in the forward market. They buy foreign currency in the forward market if they expect the future spot rate to be numerically higher than the current forward rate . Specifically, we shall assume

(3)D"(t)=p(Etst+1—f)

where Etst+1 is the value of the spot rate at period (t+ 1) expected at period t , and the subscript p refers to speculation. Since the exchange market is under random disturbances we shall consider the case where p is not plus infinity . Speculators are taking risks so that they cannot be in an infinitely long (or short) position even if the expected future spot rate exceeds (falls short of) the forward rate . Speculators who are committed through decision in the last period to deliver foreign currency in the present period would meet their obligation by buying in the spot market; those who committed themselves to buy foreign currency would do the reverse .


Therefore

(4) Dp(t)= —Dp(t-l) . Finally for the covered interest arbitrage, we assume that arbitrage funds Da(t)

move to exploit the difference between interest rate differentials and the forward premium. Thus

(5)Da(t)= Art —rt+f —st) . Since arbitrage is a swap activity involving simultaneous transactions in the spot as well as in the forward market, and since in the forward market we have to take care of interest payments for arbitrage funds, we must have

(6)Da(t)= —(1 +r*)Da(t) .

Arbitrage is risk less with respect to the exchange risk because one can always hedge the exchange risk by engaging in swap operations. Therefore we may assume that the following interest parity condition is always satisfied2

(7)rt—r*=ft—st .

In the following discussion, we shall suppress the explicit form of the arbitrage function (5) and use equation (7) instead.

The equilibria in the spot and forward market are respectively given by

(8)Dsc(t)+Dp(t)+Da(t)=0 ,

and

(9)D .(t)+Dp(t)+Da(t)=0 .

Substituting (6) into (8) and (9) we get

(10)(1 +r*){D'(t)+D p(t)}+ID- (t)+Dp(t)}=0 . If we substitute (1), (2), (3), and (4) into (10), we obtain the following equation:

(11) (1 +r*)p(Et-lst—f -1)+{al—bl f +wt+p(Etst+1—ft)}=0 .

Let us analyze the properties of solutions of equation (11), when the perfect arbitrage relationship (7) holds and the two interest rates are given as constants. Let us assume rt = F, and r * = F * for all t, and use the notation F— r* - 0. Then st =

f —0, Etst +1= Erl + 1 — 0 and so forth by (7). Thus equation (11) is reduced to

(12) p[Etf +1—ft-o]—bl f —(1 +F*)p[Et-lf -1—f -1 —0]+al= —wt .

2 If r , and r;'` are instantaneous interest rates, then the interest parity implies

er= er' F, S,

where F, and S, are forward and spot rates such that f = In F, and s,= In S,. Equation (7) follows directly from this.


This equation describes the dynamic performance of the forward exchange rate

provided that expected values of forward rates are given. We first solve the equation (12) for statioary values when wt = 0 for all t. Putting ft= Et _ i f = f for all t, we obtain

The stationary spot

forward rate is

al

f=b +

1 rate corresponding to

P -----0. bl

this stationary state solution of the

s=bb l+rPo-o. ll

Next we shall introduce the rational expectation formation into the model . Although our basic model essentially equivalent to Shinkai (1969) and Driskill-Cafferty (1980; 1982), these authors are mainly concerned with the stationary state

property of the underlying stochastic system. We are rather interested in the way how the stationary state is achieved. We concentrate our analysis on the dynamic behavior of the system towards the steady state. Thus, the spirit of our analysis resembles that of Black (1973),3 although we shall solve the model in detail in order to fully exploit the implications.

Suppose we are now in period t. Let us write (12) into the future for t = t, t + 1, t+ 2 etc. Then we have

(12)

(12)'

Rational they engage these agents known solving the mathematical expectations conditional on the information set available in period t. Thus, ft +j and f+- i will be replaced by Et f + j and Et.f + j _ 1 and wt +j will be replaced by Et(wt + j) = O. Further, Et + if + j+1 and Et + j _ll;+jwill be replaced by Et.f + j+ 1 and Et f + j, because the iterative law of conditional expectations implies that

P[EtJt+''1 -ft—0]-blf—(1+ 'Y*)P[Et-iJt—It-l —0]+al = —wt

P[Et+if+j+1 -f+j-o]-biff+j

—(1 +r*)PLEt+j-lf +j—f +j-i —0]+al

_ —wt+j for j .�1.

1 agents will take account of these market equilibrium conditions when

;age in the market in period t. From the standpoint of period t, in which ents actually behave, equation (12)' contains some variables that are not

the equation (12)', the rational agents will replace these variables by their

Et[Et+jft+j+1]=EtJt+j+1 •

With these replacements, we now have the following nonstochastic difference

3 See Black (1973), esp. pp. 22-27.


equation for E, f +;, and that equation is going to be the basic equation of our exercises to follow.

(12)"P[Erl +;+1 —Etf +;-0]—blEtf +; —(1 +r*)P[Elf +;—Et f -1— 01+ a,

= 0 for j� 1 .4

Equations (12) and (12)" describe the dynamic path of the forward rate as projected from period t by using the currently available informations. In order to solve the system we first choose the stable root of the characteristic equation of (12)":

(13)P).2 —[(2+r*)P+bl]A+(1 +r*)P=0 .

If we denote the smaller solution by A,, it is easy to see that 0 < A, < 1 and A2 > 1.5 Therefore, we employ Al as the stable root.

Now that the stable root is found, the solution of the system is rather straightforward. We first note that the solution of (12)" in this particular case must be written as:

(14)E,f +;=f+c1A (j~)

where f has been defined above, and cl is a constant to be determined as follows. The solution (14) applies for period t +j (j� 1), and accordingly for E, f; + (j� 0) by our convention that Et f = ft. We may now determine the value off; (= Et ft) by equation (12), given wt, f;_, and E _ 1 f . In order to do that, one substitutes j = 0 and 1 into (14) to get E, f; =1+ cl and E, f + 1 = f + cl Al. Substituting these back into (12), one obtains, after rearranging the terms:

wtA l wt (15)alp(1 — 1)+bl P(1+r*)(1—~,1)'

where

wt=wt—(1 +r*)P[Et-l ff —f -1]

Here w, is the displacement at period t due to the disturbance at that period and the forecast error at the previous period.

We have now derived a solution to our basic system. What are the properties of this solution which, incidentally, is essentially equivalent to the ones derived by Shinkai (1969) and Driskill-McCaffery (1982)? It must be noted that our solution has been derived as the projected path when the random term wt (or, w,) happens to take on a nonzero value. In other words, such a path is the one which is obtained

If j =1, (12)" contains the term E,f . We assume for convention .that E,f =f .

(z s In fact,=C2+r*+bt~±161)+2(2+r*)b, +r*z . P P P


when the system has encountered an unexpected-temporary' shock. Thus, a temporary shift in the traders' demand curve from al— bl lo to al + d al— bl lo

WI, > 0 for period t and 4a1= 0 for periods t +1, t + 2, • • •) is analytically equivalent to the case where the random term tit (for t= 0) happens to be positive. In both cases the forward rate is affected by a factor of )Ll/(p(1 + r*)(1 — ).1)), multiplied by the magnitude of the temporary shock; further, the effect will decay exponentially at the rate A.

These are the effects of unexpected-temporary shocks on the foreign exchange rates. When one turns to analyze the effect of permanent shocks on the foreign exchange rates in the next section, the analysis becomes more complex.

III. THE IMPACT OF PERMANENT SHOCKS

Let us study the response of exchange rates in the system to permanent changes in demand functions al — bl f and interest differentials.'

Since some of the cases to be considered in this section are rather complicated, we shall proceed with the following simplifying assumptions. Suppose the

participants are now in period 0. Before the change in parameters, the rate of interest at home as well as abroad is assumed to be zero. That is r = r * = 0 = 0 before period 0, for t� —1. Thus the characteristic equation becomes

(13)'/12—(2+/3)1.+1=0, where /3=bl/p.

(i) Unanticipated permanent changes in structural parameters Let us first consider the effect of an unexpected permanent change in traders'

demand function from al— b, f, to al + dat— bf f from t� 0. At period zero this change may be due to a demand shift towards import goods or to an expansionary macroeconomic policy. Let us assume that all the participants recognize the structural change from al to al + dat, and that they all believe that this change will be permanent.

For period 0, our basic equation can be written (noting /7* = 0 and 0 = 0 for any t)

(16)P[Eoft —lo] + [al + d al — bl lo] = — we .8

For t>_ 1, our basic equation looks like

(16)' P[Eof+1—Ea.f}+[al +dat—blEof]—P[Ea.f —Ea.f -1]=0 .

It is easy to see that the solution for (16) and (16)' is

(17) Eof =al + Jai

bl+cltt,

6 "Temporary" here means that the shock itself is expected "not to continue ." Readers who are only interested in policy implications may skip to Section IV.

8 we is equal to — p[E_ 1 J —f_ 1]+no, and it is the given initial condition for the system.


where cl and /11 have been defined previously. This implies that the structural shift of the traders' demand will immediately

raise the stationary solution by dat/bl. The forward rate (as well as the spot rate) at time zero jumps by dat/bl, keeping the effect of the random and historical term we unchanged. As shown by the solid line in Fig. 1, the stationary value jumps from al/bl to (al + dat)lb1 at period 0. The actual path of exchange rate shifts from, for example, a path like the dotted line to the broken line, because the actual

path is influenced by realized values of disturbances.

Ed

a,{da,

~ actual pat~---

/--------------------- stationa

b,/1---------------------- stauona

a'actual pail

1),//

stationary value of Eao f

Fig. 1.

This analysis implies that the structural shift in traders' demand schedule does not tend to give trade deficit or surplus. On the average (i.e., if we set aside the effect of random disturbances), the structural shift is absorbed by the movement in exchange rates. In this sense the traders demand curve plays the role of "anchor" in the foreign exchange market.

Next consider the impact of a permanent change in the domestic interest rate. Assume as before, r= f* =0=0 for time t < —1. Suppose, however, that from time 0, the domestic interest rate becomes positive (4 r > 0 andAc0 = 0 >0).

Then for t= 0

(18)pEao.ft—fo-ol1+[a,—bitf,]= weo .9

For t >1 our basic equation becomes

(18)' p[EoJ+t —Eof —0]+[at biEo0 f]—pEao.f —E0f_1-01=0 .

Because r* is assumed to be zero, the stationary solution (at/b,) remains unchanged. Therefore we have the stable branch of the solution

Ig9)Eof =a, +c/ i . (t 0)

t Substituting (19) into (18), one obtains 9weo is again equal to — p[E_ 1loo —f- 1 ] +no0. Notice that there is no effect of 0 inweo, because the

change in the interest rate differential was not expected in the previous period.


—p l-oA, Al 10 c=

p(1—Al)+bio+p(l-2l)+b1w o_1—Al +p(1—Al)w° •

Therefore, (19) can be written

alA!+1 At+1 (20)Eo~t=—1 —~0+11w°t>0 blip(1)

t+It+1 (21)East=bl—11 + 10+---------11w°t>-0 . 11 P(i)

Since w° itself is a random variable for t < 0, the deterministic component of E° ft (i.e., the projected path from the previous period) is regarded as (all bl)— (A i+ 1 /(1—.,1))0. This is drawn in Figure 2. The deterministic parts of E° ft and E°st are drawn by solid curves. The actual paths of E° f and E°st could be drawn around these solid curves depending on the values of w°. Moreover, actual

paths off; and sr in the future are random movements around E° f and E°st curves, even when expectations on future values of et happen to be true.

Projected paths of Ea lo and Eos,

- stationary

Eof,

stationary path of s, Eos,

01

Fig. 2.

In summary, even though the stationary value of Ea f remains unchanged, and even though the displacement of the stationary value of Eos, is equal to the change in the interest rate differential, the divergence in the actual path of exchange rates from what it would have been if there were no change in interest rate differentials may be quite large when speculation is active. This is because if p is large )1/(1—A.1) can be very large. Notice that the stationary state value of Eof, does not change because of the assumption that r * = 0.

(il) Anticipated permanent changes in structural parameters Let us next consider the case where a permanent shift in the demand curve

10 Notice that 1/(p(1—~1)+bl) is equal to )1/p(1—A.1) if r*=0.


al— bl f or the permanent change in the interest differential 0 is anticipated in advance. We shall consider the problem in the following two steps.

(ita) In period 0, parameter shift in period 1 is anticipated Consider first a shift in the traders' demand schedule. It is anticipated in period

0 that from period 1 al will shift to al + dat. In period 0, we have as before

(22)P(Eofi —.lo) + (al — blfo) _ —00 •11

For t >_ 1, we have

(22)' P(Eof+1—Ea.f)+(al +4a1—blEoff)—P(Eof—Eof-l)=0

The general (stable) solution that satisfies the initial condition can be written:

al+dat 2i+1 (23)Eof (=East)=----------b

l +p(1—~)(—dat+we).

The stationary solution jumps by dat/bl, but lo does not jump as much as dat/bl if we neglect the effect of we.

Similarly, suppose that the change in interest differential in period 2 from 0 to 0

(> 0) is anticipated in period zero. We have to assume that the change in interest differential takes place in period 2 rather than in period one: otherwise, we would have exactly the same equations of motion as (18) and (18)', due to the fact that we assumed D(t)=O. Then in period 0, we have as before

(24)P(Eofi —.lo) + (al— blfo) _ —we •

For time t = 1,

(24)'P(Eof2 — Eofl— 0) + (a1- bl Eofi) — P(Eofi —.lo) = 0

and for t >_ 2

(24)" p(Eof+1—Eof —0)—(al—blEof)—P(Eof—Ea .f -1-0)=0 .

The stable solution satisfying (24), (24)' and (24)" is written

al 212 ~1 (25) .lo =bl—-----0+1—~tp(1— Al)we

a~2+1At+t (25)'Ea.f=bl—11 —2PIO+---------two (t~1) 11 P(11)

where

we=no—P(E-l .lo—.f-l) .12

If we compare (25) with (20), the initial jump in lo is mitigated if the shock is an

11 we= —no+PIE 110-1 11 12 Notice also + 1= (1 +/3)) .


anticipated change in interest rate differential in advance. This analysis makes it possible for us to study the effect of a temporary change

in the interest rate differentials. In Section (i) we analyzed the effect of an unanticipated permanent change in the interest differential on the exchange rate. In the present section we have analyzed the effect of an anticipated permanent change in the interest differential on the exchange rate. Combining these two analyses, we are able to analyze the effect of a temporary change in the interest differential on the exchange rate; to be sure, a temporary increase in the interest differential is nothing but an unanticipated permanent increase in the interest differential that is anticiptated to be followed by a permanent reduction in the interest differential later. Then the total effect is given by combining the coefficient for Jo in equation

(20) for t = 0, and the negative of the coefficient of Jo in equation (25). Accordingly, the total effect due to a temporary interest change at home, obtained by subtracting (20) (for t= 0) from (25), can be written as

2 (26)dfo=— +------~149=—~dots 1 — 1—~1

and

(26)'4s0= —(Al+ 1)4 0 .

Therefore a temporary increase in the domestic interest rate gives a temporary decline in the exchange rates. The spot rate declines more than the interest rate differential. From (20) and (25)' the time path of the expected exchange rate in response to this temporary increase can be written as for t_� 1

(27)EofP(=bl+(1—~,1)~id0+ 1~~ we. 1)

In Fig. 3, the response of the forward and spot exchange rates due to the temporary increase in the domestic interest rate is drawn.

(lib) Finally we come to the most interesting case where in period 0, the change in parameters from period k (k� 2) is anticipated. Because we are dealing with the second order difference equation, we need to deal with the case (ita) separately from this general case.

Let us start with the case where an increase in al from period k is anticipated at

period 0. For t= 0 we have

(28)P(Eofi —.lo) + (al — bl lo) = — we

where we is defined as above. For 1 < t < k —1

(28)' P(Eof+t —Ea.fr)+(al—bl ft) —P(Ea.fr—Ea. -1)=0

and for t > k, we have 13 We are neglecting the effect of we that exists anyway and does not need to be superimposed.


Ea;

0

Fig. 3.

(28)" p(Eof+1—Eoft)+(al+da—blEof,)—p(Eof—Eof-l)=0.

The general solution for (28)" is written for t� k —1

a+da Eof =1+c,t+c"A2 bl

where Al and A.2 are characteristic roots of equation (13)'. However, by the same reasoning we adopted earlier, we shall assume that c"= 0 and

al+da, (29)Eof = b+ ctr , for t >_ k —1 .

1

With respect to the intermediate (1 _< t �_k— 1) periods, it is not possible to apply the same reasoning to drop the unstable characteristic root. Because k is finite , there is no reason to assume that the unstable branch of the solution actually explodes. We therefore have to assume that the general solution

(30) Eof=bl+cAi+cit2 applies for 0 �t �k. We have to find the values of c, c and c' that satisfy the initial condition (28) and the condition that the two solutions expressed by (29) and (30) must coincide for t = k —1 and t = k. These values are easily calculated as in Appendix A. We then have the following solutions (writting A. A and A.2 = A-l for the economy of notation):

At+IA.k+1 /11c da (31) Eof` b

l+p(1—.)we+ 1+A.At+l+AA-t b 1 for 0�t—k.

al+daAt+1Ak+1—Ak+1 da (31) E°fbll+p(1 —~)w°+ -----------1+~bit>k

1

INTERNATIONAL INTEREST-RATE DIFFERENTIALS2g

In particular,

aAda (31)"Jo=bl+------1 —Aw°+~.bl• tp()1

If k is large, the initial jump will be not so great. Next let us turn to the impact of interest rate changes k (k� 2) periods ahead. At

period 0, market participants are assumed to predict correctly the change in interest differentials from 0 = 0 to 0 = 0 > 0 k period ahead. al is assumed to remain constant.

In period t we have the initial condition

(32)P(Ea.fi +.lo) + (al — bl.lo) _ — we .

For period 1 < t < k —1

(32)' P(Eof+1—Ea ft) +(al —b,Ea ft) —P(Ea.fr—Ea.f -1)=0 .

For t= k, a mixed (or bridge) equation

(32)" P(Ea.fk+1—Ea.fk-o)+(al—blEo.fk)—P(Eofk—Eofk-l)=0

and for t >— k + 1

(32) — p(Eof +1 — Ea ft — 0) + (al — bl Eoft) — P(Ea.f — Eof -1 — 0) = 0 .

Therefore the solution for the expected path of forward and spot exchange rates is

given by'

For 0<t<k

a,It +1Ak+t+2Ak-t+1 (33)Ea.f(=East)=b+p(1------—A)w° 1—,120, and for t>—k

t

alAt+1Ak+2+A—k+1 (33)Eof(=East+0)=bl+P(1 —A)"'°1—A2----------- 0 .

In particular

al AAk .lo=b

l+p(1-------—A)we 1—Ac.

Thus it can be seen that a larger value of k implies a smaller adjustment in the initial period.

Figure 4 illustrates the projected response of exchange rates to the shift of demand schedule for foreign currencies by trader from al = dat after period 3, i.e. the case of k = 3. In other words we are neglecting the effect of the initial condition we. Similarly, Fig. 5 illustrates the response of forward and spot exchange rates to

14 See Appendix B for details.


Projected Ed; and Eos,

a,- da

b,

a, --

h, -

x---g" EoJr=EoJs

0 1 2 3

Fig. 4.

Projected Ea f and Eos,

4 t

0 1 2 3 4 t k=3

Fig. 5.

the expected increase in interest differentials from 0 = 0 to 0 = 0 > 0 from period 3.

IV. ECONOMIC IMPLICATIONS OF THE RESULTS

Let us summarize the results obtained through the quite intricate calculations

presented in the previous section. 1) The simultaneous determination of spot and forward exchange rates can be

formulated as a solution to a stochastic difference equation. The nature of the rational expectation equilibrium path can be studied as a type of stochastic equilibrium. This formulation can be interpreted as a simplified version of the asset approach to exchange rate determination with speculators with constant absolute risk aversion.

2) A temporary shift in traders' demand for foreign currencies will affect the

path of exchange rates. However, with active speculation the effect of this temporary shock will be smoothed out, so that the current values of spot as well as forward exchange rates will not be much influenced by a temporary shock in traders' demand.

A temporary change in the interest rate differential, starting from the situation of zero interest rate differential will affect the spot exchange rate to a moderate


extent. The effect on the spot exchange rate of 1% rise in the domestic interest rate will be more than 1% (to be exact (A + 1) %) decline of the price of the foreign currency. The effect on the forward exchange rate will be less than one percent

()%). 3) An unexpected permanent shift in the demand function for foreign

currencies arising from traders' activities will cause a sudden jump in the expected

path of forward as well spot exchange rates to the level where excess demand is cleared. In this sense, the demand for foreign currencies from traders' activities

plays the role of an anchor in the exchange rate determination. An unexpected permanent change in the interest rate differential, on the other

hand, causes rather significant movements in the forward as well as in the spot exchange rates. As far as the stationary state level of exchange rates is concerned, the change in the interest rate differential does not alter them very much. However, when speculation is active, a sudden unexpected change in the interest differential may give a dramatic movement in exchange rates on the outset of new infor-mation. 1 percent change in the domestic interest rate will cause exchange rates to change by Al/(1 — A,) percent which can be very large when the intensity of speculation p is large relative to the slope bl of traders' demand curve. In this sense the market "overshoots" the long-run equilibrium value [cf. Dornbusch (1976)].

The reason of this instability seems to stem partly from the fact that the market should handle the large amount of commitment already made by speculators in the last period. Also from the standpoint of the asset approach, the change in interest rate means that one asset suddenly becomes more attractive relative to others as store of value. Therefore this suddenly gives incentives for asset holders to accumulate or de cumulate the preferred asset as rapidly as possible through incurring trade surplus or deficit. However, since the amount of the outstanding foreign asset cannot be changed in the short-run, one observes a volatile movement in exchange rates.

4) If the change in parameters such as the demand schedule or the interest differential is anticipated some periods in advance, then the adjustment of exchange rates will take a gradual form. Naturally when the shock is expected to occur in the longer future, the adjustment of exchange rates will be more gradual. Nevertheless, there is a jump in the exchange rate path when new information on the future changes comes into the minds of market participants.

5) The greater the intensity of speculative activities, the more stable will be the movement of exchange rates, provided that other structural parameters remain the same. However, when these parameters do change, the more intense speculative activity implies more erratic movements in exchange rates. Speculators' activities make the system very efficient and delicate. They may turn the system even more vulnerable to unexpected change in the structure.

These results under rational expectations lead us to a few interesting obser-vations about the nature of stochastic equilibrium of forward and spot exchange rates.


First, in these exercises, market participants are assumed to possess perfect knowledge of the value of structural parameters and of the future changes in

parameters. However, in reality, market participants do not necessarily know enough about parameter values, and much less about their changes. Particularly, it is hard for the participants to tell whether observed movements in demand curves or interest rates are of temporary or permanent nature [Barro (1978), Ueda

(1980)]. Therefore, the effect of any movements in traders' demand schedule or interest

rates will create such impacts as caused by a mixed combination of temporary and

permanent shocks. In any case, our result suggests that a permanent changes in interest rates may induce rather drastic exchange rate movements.

The second observation concerns the role of government interventions in the exchange market. We know that if there is no change in parameters a larger value of p implies a more stable exchange market. The value of p is limited by the capacity of speculators to assume or to "absorb" risks. When there is a shortage of risk taking activities, then the monetary authorities may as well take the risk themselves. Of course, in such a case the criterion for their intervention is whether or not intervention creates profits in the long-run. Presumably the government can expose itself to a greater degree of risk and can make profit calculations based on longer time perspectives.

Thus one can make a case for government intervention in the foreign exchange market from this ground. The supply of risk-taking activities will increase stability of exchange market with given parameters. However, our analysis indicates at the same time that more risk-taking activities also increases vulnerability of the system to changes in structural parameters, particularly in interest differentials . Therefore, in practice, whether or not the monetary authorities can do anything about this fragility of the exchange market due to speculation is a question that remains to be pursued.' 5"6

The third application of this model concerns the effect of fiscal and monetary

policy under flexible exchange rates. Mundell (1968) pointed out a drastic contrast of the effect of fiscal policy and monetary policy under alternative exchange rate regimes. In the case of flexible rates, fiscal policy does not influence national income just because the effect of fiscal expansion is exactly matched by a decline in the current account of the balance of payments. Monetary policy works through the channel of an export expansion rather than through a domestic investment boom.

15 We are not yet convinced of the necessity of any forms of interest equalization tax when supply of

speculation is too large. But one might make a case for them if structural discurbances are so frequent. 16 One can pursue the analysis to determine the value of p if one assumes the constancy of the degree

of absolute risk aversion by speculators. If speculators have a constant absolute risk aversion, and if

random disturbances are normally distributed, then the amount of foreign assets with exchange risks

that the speculators will hold is a linear function of the expected rate of return with a coefficient that is

reciprocally related to the forecast error. Therefore the value of p could be determined by the variance

of random disturbances.


His clear-cut classification of the exchange regimes depends, however, on the specific assumption that the forward exchange rates is equal to the spot exchange rates all the time. Mundell assumed a static expectation on the spot exchange rate so that the forward rate is always equal to the spot rate. Thus the generality of his conclusion is regarded to be doubtful in many cases.

With our partial equilibrium analysis of the spot and flexible exchange rates, where expectations are formed rationally by market participants, one cannot say much about Mundell's conclusions on national income. However, one can say on the effect of fiscal and monetary policies on the balance of payments. Fiscal policy, as long as it increases domestic net absorption, could be interpreted in our framework as an increase in absorption in the domestic economy, and accordingly as an increase in al to al + da. For, the increase in fiscal policy will increase domestic demand so that the total absorption given by al— bl f for a given f will increase. Our analysis shows that an unexpected permanent increase in absorption due to fiscal expansion will create a sudden jump in the forward exchange rate. So long as the domestic interest rate is kept constant, fiscal expansion will keep the realized net absorption unchanged, because exchange rates adjust to the changes in al. Thus the trade balance will not be affected through time on the average (we used the term "on the average" because we are only dealing with the expected value of trade balances in our stochastic framework).

Next consider the case of decline in the domestic interest rate due to monetary policy when total absorption schedule remains unchanged. As pointed out in the last section, a temporary as well as permanent decrease in the interest rate differential will impart a substantial shock to the path of forward as well as spot exchange rates. The amount of capital movements is, by the assumption of the equilibrium in overall balance of payments under flexible rates, equal to the reciprocal of the trade balance. Suppose that the delivery lag for traders' demand is equal to unity. In other words, suppose that the export or import contract in period t that is covered by a forward exchange contract also in period t, is executed in period (t + 1). Then the path of capital movements B(t) can be expressed as the excess supply of traders' demand in the forward market in period (t — 1) and the interest payment accruing to the arbitrage capital made by swap agreement at period (t — 1).17

B(t)=De(t-l)—r*Da(t-l) .

If we neglect the effect of foreign interest rate r*, B(t) = D c (t — 1). From our solutions for the effect of a temporary or a permanent interest rate differential, we can conclude that when there is a decline in the domestic interest rate , the expected path of the forward exchange rates will increase (domestic currency will depre-

17 Also capital movements can be expressed as the difference betwe en the returning capital that is the Da(t-l) and repatriating capital Da(t) due to the new arbitrage operation at period t, that is,

B(t)=Da(t-l)—Da(t) .

s4KOICHI HAMADA and TAKAHIKO MUTOH

crate), so that the trade account will improve substantially, starting from the next

period if we consider the delivery lag. Therefore the decrease in the domestic interest rate will cause a capital outflow and a depreciation of exchange rates that accompany the trade balance surplus.

So far as monetary policy is concerned, Mundell's conclusion still holds. As for fiscal policy, Mundell's conclusion does not exactly hold. However, if fiscal expansion without monetary expansion tightens the domestic financial market so that an increase in domestic interest rates takes place, then it can affect capital inflow that will lead to appreciation of the exchange rates from the level that equilibrates traders' demand. With this consideration one could say that the policy conclusion by Mundell applies to the effect of fiscal policy in a limited sense. Thus capital moves to the country that raises the domestic interest rate from the

previous level. The extent to which these capital movements are active under flexible exchange rates is a matter which should be verified through empirical analysis [see e.g., for the case of the yen, Mutoh and Hamada (1982)].

APPENDIX A

For the economy of notation let us write A1- 2, and 22 - 2 -1. Then by substituting (30) into (28), and using the relationships that (1 — 1) + #=2-1— 1 and

(1-1-1)+/3=A-l, one obtains

(A-l) — 1)c+(A-l)c=we/P

By the coincidence of solutions at t = k —1 and t = k, one also obtains

(A-2)lk-lc+2-k+lE-2k-lcr= bl

1 (A-3)2kc+l-kc—Akc'_~al 1 From these equations, one obtains the following solution

2k+1 dat c=

p(1-2)w°+1+2 bl

2k jai c=

1 +;, bl

2 1k+l-2-k+1 dat c'=

p(1— A) +------------ 1+1 bl

APPENDIX B

By the reasoning similar to that used before, we have a general (stable) solution of (32)—


(A-4) Eof =al +c'At for t>_k,

1 and a general solution of (32)'

(A-s)Eof =bl+c,,`+c2—'for 0—t<k 1 (A-s) must satisfy the initial condition (32), which gives

(A-6)(.1-1-1)c+(•1.-1)c= we/p

Also (32)' and (32)"' must satisfy the bridge equation (32) ",

(A-7)~k-IC+~ -k+IC—/~k-lC~= e

by utilizing the fact

C'Ak-l(1.2—(2+01)=

For t=k, solutions (32)' and (32)" must coincide. Therefore

(A-8) A —kc — I,kC' =0 .

Again (A-6), (A-7) and (A-8) give the values of coefficients as

Ailk+2 c=

p(1--------—A)we 1-------—A2 0

)lc+1

c--l—,120

r _----------- A A.k+2+A—k+1

c,=

p(1—A)wot—A2----------------o.

which reduced to

Tokyo UniversityTokyo Keizai University

REFERENCES

Barro, R. J. (1978), "A Stochastic Equilibrium Model of an Open Economy under Flexible Exchange Rates," Quarterly Journal of Economics, Vol. 92, No. 366, 149-64.

Black, S. W. (1973), International Money Markets and Flixible Exchange Rates . Princeton Studies in International Finance, No. 32.

Dornbusch, R. (lgi6a), "Expectations and Exchange Rate Dynamics," Journal of Political Economy, Vol. 84, 1161-76.

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--------- (1980), Open Economy Macroeconomics, Basic Books, New York. Driskill, R. and S. McCafferty (1980), "Exchange-Rate Variability, Real and Monetary Shocks, and

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Fukao, K. (1982), "An Asset Approach to the Spot and Forward Exchange Markets," (in Japanese), in H. Uzawa and Y. Onizuka (eds.).

Fukao, M. (1980), The Risk Premium in the Foreign Exchange Market, Ph. D. dissertation, University of Michigan.

Komiya, R. and M. Suda (lg8oa), "The Short Term Capital Movements under the Managed Float,"

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46, No. 3, 25-56. McKinnon, R. (1979), Money in International Exchange, Oxford University Press. Mundell, R. A. (1963), "Capital Mobility and Stabilization Policy under Fixed and Flexible Exchange

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Rates," Journal of International Economics, Vol. 5, 275-81. Otani, K. (1982), "A Macroeconomic Model with the Forward Exchange Market," Economic Studies

Quarterly, Aug., pp. 97-116. Shinkai, Y. (1969), "Speculation on Foreign Exchange and Expectation Formations," (in Japanese),

Economics Studies Quarterly, Vol. 20, No. 1, 1-10. Sohmen, E. (1966), The Theory of Forward Exchange, Princeton Univ. Press. Tsiang, S. C. (1959), "The Theory of Forward Exchange and Effects of Government Interventions on

the Forward Exchange Market: IMF Staff Papers, Vol. 7. Ueda, K. (1980), "Dynamic Interactions between Trade Flows and Exchange Rates: Theory and

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