The Economic and Social Review, Vol. 22, No. 1, October, 1990, pp. 57-63
Speculative Rational Expectations' Exchange Rate
Dynamics with Risk Averse Wealth Owners and a
Monetary Authority Averse to Exchange Rate
Variability
P E T E R G. D U N N E
The Queen's University of Belfast
I I N T R O D U C T I O N
I
n the analysis t o f o l l o w an i m p o r t a n t aspect o f speculative exchange rate movements is examined i n isolation f r o m macroeconomic feedback effects. The over-riding objective here is t o investigate the effect o f adherence to an interest rate rule b y the monetary a u t h o r i t y w h e n international specu lators behave i n a way such that there is full rational expectations w i t h respect to b o t h the mean and variance o f the future exchange rate.The m o d e l is most relevant w h e n there is free capital m o b i l i t y across inter national capital markets; a small domestic capital market relative to competing foreign capital markets and w h e n the domestic monetary a u t h o r i t y uses the interest rate as a means o f adhering t o a d i r t y fixed exchange rate mechanism. A cautious monetary a u t h o r i t y raises the domestic interest rate above the foreign rate so as to allow a risk p r e m i u m for the expected variance around the expected future exchange rate. The effect o f adhering t o an interest rate rule o f this sort (when speculators are w e l l i n f o r m e d about the rule), results i n a less variable b u t weaker exchange rate the more cautious the monetary a u t h o r i t y tries to be.
The weakness i n the exchange rate occurs because although the monetary a u t h o r i t y is more cautious, this i n itself reduces the expected exchange rate variability to such an extent that a lower interest rate p r e m i u m is expected.
This i n t u r n reduces the demand for domestic treasury bills at any given exchange rate. Since the supply o f treasury bills is assumed constant there is a j u m p depreciation i n the expected e q u i l i b r i u m exchange rate.
The origin o f the present analysis is an elegant paper by Begg ( 1 9 8 4 ) , where there is presented a rational expectations m o d e l o f e q u i l i b r i u m b o n d pricing i n w h i c h risk averse lenders make p o r t f o l i o choices between short and l o n g assets so that expectations are self fulfilling i n b o t h the mean and variance o f future b o n d prices. The i n t r o d u c t i o n o f the expected variance i n t o the analysis results i n a non-linear difference equation for the e v o l u t i o n o f b o n d prices ( w h i c h w h e n appropriately transformed is the logistic equation). Begg showed that for appropriate levels o f uncertainty and o f risk aversion the e q u i l i b r i u m b o n d price w o u l d be locally unstable so that a unique rational expectation e q u i l i b r i u m b o n d price w o u l d exist. Begg associated financial panic w i t h parameter values for w h i c h the e q u i l i b r i u m b o n d price was stable since this w o u l d i m p l y that there existed an infinite amount o f possible paths t o e q u i l i b r i u m . V a n der Ploeg ( 1 9 8 6 ) , showed that the same m o d e l could give rise t o unique non-convergent f o r w a r d l o o k i n g r a t i o n a l expectations paths i n the f o r m o f l i m i t cycles or erratic trajectories.
Specifically, the b o n d pricing m o d e l considers a p o r t f o l i o choice over t w o assets one o f w h i c h has a r e t u r n w i t h certainty b y way of a capital gain on a one-period treasury b i l l . The other asset is a conventional b o n d the capital gain o n w h i c h is uncertain. This f o r m u l a t i o n allows all of the uncertainty o f the expected r e t u r n o n the p o r t f o l i o to be due t o the uncertain capital gain o n the b o n d . I n the exchange rate m o d e l presented here i t is assumed that foreign w e a l t h holders make a p o r t f o l i o choice between the one-period treasury bills o f t w o countries w h i c h have certain capital gains i n own-currency terms. This makes i t possible to a t t r i b u t e all o f the uncertainty o f the expected r e t u r n o n the i n t e r n a t i o n a l p o r t f o l i o t o the uncertain future exchange rate.
I I A N A L Y S I S
I t is assumed that the small c o u n t r y monetary a u t h o r i t y sets its one-period interest rate o n treasury bills above the foreign rate by a fixed p r o p o r t i o n o f the expected variance o f the end of period exchange rate.
Thus: rt = r * + T ^ . t + l •
(A c i r c u m f l e x hereafter represents the expectations operator.)
Thus: et +. - et + j + Ue > t + j.
Where U£t+j is i i d ( 0 , a2) .
(The expected variance o f the exchange rate w i l l be endogenously determined.) We consider the p o r t f o l i o choice o f a foreign wealth holder between a r e t u r n w i t h certainty on their o w n country's treasury bill and an uncertain r e t u r n o n the treasury bill issued by the small-country monetary a u t h o r i t y . V a l u e d i n foreign currency terms the r e t u r n o n such a p o r t f o l i o w o u l d actually be:
it = at (et+l " et ) + at (r* + Tae , t+l )et+l + (W~ atet )r*
-Here the exchange rate is the foreign currency cost o f domestic currency. at
is the q u a n t i t y demanded o f small-country treasury bills w h i c h are valued at a u n i t a r y price. The r e d e m p t i o n value o f the treasury bills is set at the begin ning and this determines the interest rate for the p e r i o d . W is wealth assumed constant over t i m e .
The first term i n the equation gives the capital gain/loss o n the p o r t f o l i o . The second term is the end o f period value o f the interest r e t u r n o n at. The
t h i r d term is the r e t u r n o n remaining wealth invested i n foreign treasury bills. The expected r e t u r n o n the p o r t f o l i o is:
{ = M V i - et ) + at (r* + ^2 e ) t + 1) et + 1 + (w- atet )r*
-We can n o w proceed to calculate the variance o f the r e t u r n as follows:
K ' { = M 1 + r* + 7& e , t+l ) (et+l ' 6t+l )•
Squaring and t a k i n g expectations we o b t a i n :
ol = at 2( l+r * + T 52 e ) t + 1)2^t + 1.
N o w suppose foreign wealth holders have the f o l l o w i n g expected u t i l i t y func t i o n w i t h respect to the r e t u r n and variance associated w i t h a given p o r t f o l i o .
t = \ - ( l / 2 ) b a2 t.
(The parameter b is a constant coefficient o f absolute risk aversion.)
The p o r t f o l i o choice problem o f foreign wealth holders is therefore t o choose the a m o u n t o f small-country treasury bills consistent w i t h maximising expected u t i l i t y . F o r m a l l y this is:
M a x ( w r t at) at( 8t + 1 - et) + at( r * + T&2 ) t + 1 ) et + 1 + (W - atet) r *
- ( l / 2 ) b a2( l+r * + T52 e ) t + 1)252 e > t + 1.
The first order c o n d i t i o n gives:
(*t+i - et ) + ('* + T < t+i )8t+i - etr * - b at( l +r * + 7a2 t + 1)2a2 e t + 1 = 0.
Which on rearranging gives:
_ ( l + r * + r o 2 )
et 7 T - r ^ v — (et+i - bat ( ! + r* + y°it+l) < t + i )•
This equation describes the demand, a£, for the small-country treasury bills.
A l t e r n a t i v e l y , i f we assume that the monetary a u t h o r i t y keeps the supply o f treasury bills constant, this equation then describes the equilibrium exchange rate as a f u n c t i o n o f b o t h the expected future rate and the expected future variance.
We can n o w proceed t o derive the expected future variance. Recall that i n any given period the present period's interest rate on treasury bills is k n o w n since i t is simply the difference between the issue and r e d e m p t i o n prices. Assuming that the monetary a u t h o r i t y follows its interest rate rule w i t h some random f l u c t u a t i o n and that foreign interest rate's remain constant over t i m e , we can conclude that:
r
t + 1= r* + 7 6
2 ) t + 2+ Z
t + 1.
Where exogenously determined Zt+. is i i d ( 0 , o2) .
Agents i n time t w i l l , therefore, h o l d the f o l l o w i n g expectation for e j :
( l + r * + T62 )
*t+l = ( 1+ r * } (*t+2 " b M 1 + r * + < t + 2 ) < t + 2 ) .
However, actual e j w i l l be:
( 1 + r * + 7 a2 t + 2 + Zt + 1)
V l =
T J T ^ j
(g t + 2 - ba t (1 +r * + 7&e , t+2 )5e , t+ 2)S P E C U L A T I V E E X C H A N G E R A T E D Y N A M I C S 6 1
t+i " gt + i = Zt+i (at+ 2 - ba( l + r * + yoltt+2)o2e>t+2)
or
f l + r * I
L ( l + r *
+ 7a 2i t + 2)J
et + l ^t+l Zt + 1 , 5 \ I
Squaring and t a k i n g expectations gives:
52 = a 2 [ ( l + r * )2 i( e ) 2 ae , t + l CTz n , * , .2 \2 h t+1 ^ "
U 1 + r * + 7 < t + 2e,t+2> ) J
I n this equation the t i m e label o n the i n f o r m a t i o n set is understood t o be p e r i o d t . The equations for future periods may be based u p o n the i n f o r m a t i o n set o f any previous t i m e period. This is possible because the driving variable Z{ +j is i i d and agents learn n o t h i n g f r o m the realised disturbance w h i c h can
help t h e m revise beliefs about the future. We may therefore w r i t e d o w n the t w o interacting equations o f the system as f o l l o w s :
et + j
1 + r * + 7 a2
= e,t+j+l I - b a n + r * + <va2 )a2
( l + r * ) ( t +J+ 1 b M 1 + r +TC Te , t + j + l ^ e>t + j + l .
and
I n e q u i l i b r i u m :
b a ( l + r * + 7&I)2
- a2
z
/ r 2 a2( l + r * )3b2a2l \
( l + r * )2b2a2( a2)2 + I 1 - z V ' - J Jo]
a2( l + r * )4b2a2
z l "; = 0.
72
By the i m p l i c i t f u n c t i o n theorem we o b t a i n :
6
8y
^ ^ - 2 a2( l+r * ) 3 b2a2
j"^
+( l
+r * ) j
j
1 - 2 a2( l + r * )2b2a2
j ^ 2 + i l t l ! L j
I t w i l l be shown below that-the sign o f the denominator w i l l be positive (and the derivative therefore negative), for most relevant solutions t o the qudratic above. T o see this notice that to achieve non-complex roots i n the variance quadratic, the f o l l o w i n g c o n d i t i o n must h o l d :
4 a2( l + r * )3b2a2
1 > — •
7
R e w r i t i n g the denominator o f the derivative as
4 az 2( l + r * )3b2a2
1
-L
2(1 + r * ) 2 |t h e n , i f
ya2< ( 1 + r * )
and i f complex roots are excluded then the entire derivative is unambiguously negative. The left hand side o f the inequality is simply the domestic interest rate p r e m i u m and this is very u n l i k e l y to equal or exceed (1 + r * ) .
F u r t h e r m o r e , the partial derivative o f the expected e q u i l i b r i u m exchange rate w i t h respect to the parameter o f caution is:
~ = b a f - - I L i p l + 2 ( l + r * + Ta2)
Here, the m i d d l e term w i t h i n the parentheses is negative i f the partial deriva tive o f the e q u i l i b r i u m expected variance w i t h respect to the parameter o f caution is negative. This implies that i f the absolute value o f the last t e r m is less than or equal t o that o f the first t e r m then the entire derivative is negative. This i n fact reduces to the same c o n d i t i o n sufficient t o ensure that:
5 a2 „ Se* .
-— - < 0 namely: ya2 < (1 + r * ) = > " T — < °
-6 7 e ° T
F i n a l l y , t o o b t a i n a rational expectations s o l u t i o n for this system we need t o have:
5 e _ . 5 a2
- 1 < L t L . ^ L . < 0 S ef + ; +. ' 5 a 2
t+J+1 e,t+j+l
w i t h the derivatives evaluated when the expected exchange rate and its expected variance are at their e q u i l i b r i u m values. As i n the b o n d pricing m o d e l i t can be shown that for large enough values o f the risk aversion parameter, the supply o f treasury bills and variance o f the driving variable r, the system either gives rise to w h a t Begg termed "financial panic", or t o unique b u t non-convergent solutions o f a chaotic nature similar t o the V a n der Ploeg case.
T o conclude, i t should be clear that i f speculators are interested i n the stance o f a m o n e t a r y a u t h o r i t y viz. exchange rate stability and i f the monetary a u t h o r i t y uses the interest rate rule suggested above, then a p o l i c y trade-off w i l l exist between e q u i l i b r i u m exchange rate variability and the e q u i l i b r i u m exchange rate level.
REFERENCES
B E G G , D A V I D H . , 1984. "Rational Expectations and Bond Pricing: Modelling the Term Structure With and Without Certainty Equivalence", Economic Journal, Conference Papers Supp., pp. 45-58.
