- 19 -
Thus in equilibrium z will always be small enough so that
i
fo > 0; i =H, F,

and bankruptcy cannot occur.


- 40 -

Peled, D., "Government Indexed Bonds -- Dothey Improve Matters?," Ph.D.
Dissertation, University of Minnesota, (1980).

Richard, S.F. and M. Sundaresan, "A Continuous Time Equilibrium Model of
Forward Prices and Future Prices in a Multigood Economy," Journal
of Financial Economics 9 (December, 1981) 347-376.

Siegel, J., "Risk, Interest Rates and the Forward Exchange," Quarterly
Journal of Economics 86 (May, 1972) 303-309.

Stockman, A.C., "Risk, Information and Forward Exchange Rates," in J.

Frenkel and H.J. Johnson, eds., The Economics of Exchange Rates:

Selected Studies, (Reading, Mass.: Addison-Wesley, 1978), 159-
177.

Viswanath, P.V., "On the Pricing of Forward Contracts in Foreign Exchange,"
Unpublished Manuscript, (1982).

Wallace, N., “Overlapping Generations Model of Fiat Money," in J.H. Kareken

and N. Wallace, eds., Models of Monetary Economics, (Minneapolis:

Federal Reserrve Bank of Minneapolis, 1980), 49-82.


~ 36 -

for every | 4"(+) - +) £6. gh ° uniformly; hence for every 6 > 0,

there exists Ny such that

fete) = oe)

|cs

for every n > N,. Choose N = max {Ny No}.

Thus there exist functions

n_ an
po = 6, (S)

p” = af (Ss)

where 8 (*) > of (+) pointwise, 0,95 <M, j= 1, ..., 4; hence {el} is
equicontinuous. Thus 84 > a uniformly. This completes the proof of claim
1

The argument for 85 is simiar, Thus is continuous. The

preceding argument also showed (0,°, a) =M(6") is well-defined.


:
(16) E, lets" Vi(etre MF 4 e'-f 2*)] =e ur(xFe eM’)
Pp Pp p Pp p
' H' \H! H
(17) Fle f vam + os fz )]=0

op
(18) E,[e'- f v' ett? Wie ete fo zk =0
t=

(19) z +z =0,

for every possibte realization of {S(t)}.

It is well known that models involving rational expectations give
rise to a multiplicity of equilibria.o/ This is because there may be
equilibria where extraneous variables, such as variables unrelated to the
model or lagged values of state variables, are important only because agents
believe they are important. In order to render a rational expectations
model with multiple equilibria useful it is necessary to single out a
Particular equilibrium. McCallum [19811] suggests that an appropriate
candidate is the equilibrium depending on the information set of smallest
dimesionality.

In this model the economy at time t is described by the vector of

Hq wF rH, , xt Fy,

time t state variables, {M Since the economy at time
t is completely summarized by this vector, the path to the current state
should not matter in determining prices. Thus, the equilibria focused upon
are equilibria where prices are solely a function of the current state.
Given knowledge of the joint distribution function of the random

variables consumers should be able to determine the pricing function, and


- 23 -
He l= o& 14, (S)|-
1

The set of pricing functions to which attention is restricted is
cl(#,) X cl (0). cl(#)) is the set of admissable candidates for the good
pricing function and c1(#5) is the set of admissable candidates for the
exchange rate pricing functions. A generic element of o, will be denoted
oj; 5 = 1.2.

It will first be shown that the appropriate bounds for the pricing
functions exist. Then it will be shown that cl(#,) Xx cl(#) is convex and
compact. Finally, equations (23) and (24) will be used to construct a
continuous map from cl(%)) x c1(%5) into itself. By Shauder's theorem such
a map has a fixed point which will be the equilibrium pricing function.

To establish the existence of the appropriate bounds on the
pricing functions, consider first the good price. For every Se 8 and every
continous do(*)s equation (23) gives the relationship between o,(+)> the
time t+l pricing function, and $,(S)» the time t price. The right-hand side

of equation (23) is strictly increasing in $,(S)-

H H
t 1 Hot >
AS (5) res gasy UTE - gel +O

is 465) * Ses Sui ee
*1 x? O05) #305)
Let

4
(8) = {oye cals < (+) <e}.


2. The Forward Foreign Exchange Market

A foreign exchange contract is an agreement where each party
agrees to pay the other an agreed upon amount of a specified currency on an
agreed upon date. In a spot transaction the payment or "value" date
coincides with the contract date.3/ In a forward contract the value date
occurs after the contract date.

Suppose an investor enters into a forward contract to purchase
1000 pounds with dollars at an exchange rate of 1.7 $/pound in three months.
The investor, who is said to be "long" in sterling, pays nothing at the time
of purchase. At the value date he pays $1700 and the seller, who is said to
be "short" in sterling, delivers 1000 pounds.

Forward rates are tied to known, current spot prices and the
distributions of unknown, future spot prices through two different types of
behavior in the forward exchange market: covered interest arbitrage and
speculation. Covered interest arbitrage, which entails no foreign exchange
risk, works as follows. An agent purchases riskless, interest-bearing
financial assets denominated in foreign currency and at the same time sells
the foreign currency forward for delivery at the time of the sale of the
interest-bearing assets. The relevant opportunity cost is the foregoing of
purchasing interest-bearing assets denominated in the home currency. Under
the assumption that there are no transactions costs, the rates of return on
the alternative investment strategies must be equal. Hence the interest
parity theorem, that the ratio of the forward price to the current spot
price equals the ratio of the returns on the two countries’ risk free
financial assets, is obtained. Thus a relationship between the forward

price and the current spot price is given in terms of known interest rates.


Then

2 34-

Proof of Proposition 10.

Let {o"} @ where 3" > 9° uniformly, Note that is not closed;

hence 9” may not be in 6. I show (6") > (6°) uniformly.

For every Se S$, pe [4 (S)s $,(S)]5 Tet

(wtp) = stot! - eg oy Js),
mes 005") |

, ,
where U" is evaluated at p and Vis evaluated at 9°(S').

v"(p) = sul’  epst va! 5),
p #4 (S")

' '
where Ti is evaluated at p and vi is evaluated at "(s').

Let p” be the unique p such that y"(p") = 0, p" € fo, (S), $4 (S)1:

for every n; hence by the Bolzano-Weierstrass theorem, {p"} has a convergent

subsequence. W.1.0.g., p" +p.

0


- 20 -

4. The Nature of the Risk Premium

Adding and subtracting from equation (17) or equation (18) gives

(33) # = E(et) + (& s

+ [ee Ee]: i= 4, Fe

This says that the forward rate is equal to the expected future
spot rate plus a term which can loosely be thought of as a risk premium,
because it is only for risk neutral utility functions that it equals zero,
and a convexity term arising from Siegal's paradox.

This decomposition is similar to the. one derived in
Stockman [1978]. A difficulty with equation (33) is that it suggests that
the variable of interest is the. expected difference between the future spot
and current forward rates, or the expected nominal profit of a forward
contract. Actually agents are interested in the expected difference between
the real future spot rate, e'/p', and the real forward rate, f/p'.

Ele! -f)/p], rather than E,(e! - f), is the appropropriate measure of the
risk premium. Thus, a more interesting way to formulate the equations

describing equilibrium in the market for forward contracts is


- 26 -

Proposition 6. For every Se 3, o © , there exists a unique @ € @ such

that
(36) Ef a VE wt + $2(S') ~ f(S) z(S)J} |S} = wt u' [xt a ]
Rey GT Tas ay BLS)

FY iss Fie iss ar?
(37) eft o2(S') vit $(S') _  9(S') - f(S) 2(S)]|S} =
$, 05") 4,035") 4,05")

re,(s) uaF . fe(s)),
aS On

If this proposition is true then 6 = constitutes an equilibrium.
Thus it will be sufficient to.show that WM: ® > o, such M(o) = 6, has a
fixed point. To prove proposition 6, propositions 7-9 are employed.
Proposition 7. For every Se 3, o © ¢, there exists a unique
84(S) © [o,(S)s $,(S)1, such that (36) holds.
Proof. 8,(S) = PCS, o,(-)1. 8(S) € [9,(S)> $,(S)] by propositions 1
and 2. .
Proposition 8. Let 8, (S) be given by proposition 7. For every S « 3,
oe 0, there exists a unique 84(S) € [eo(S)> $o(S)1, such that (37) holds.
Proof. 85(S) = RCS, $o(+)]. 8,(S) € [eo(S), $,(S)] by propositions 3 and 4.
Proposition 9. 6c %.
Proof See the appendix.
Proposition 10. WW: # + @ is continuous.

Proof. See the appendix.


5. Existence of Equilibrium

Defining an equilibrium to be a pricing function has been done
previously by Bental [1979] and Lucas [1972] among others. Proving such an
equilibrium exists amounts to finding a fixed point in a function space.
This is done in Lucas by applying the contraction mapping theorem and in
Bental by applying Schauder's generalization of the Brower fixed point
theorem to Banach spaces. Here I follow Bental in applying Schauder's
theorem. The proof is similar to Bental's in structure.

The first step in the proof is to restrict the collection of
functions over which the search must be conducted to a compact, convex set.
Then the equilibrium conditions are used to construct a continuous mapping
from the set into itself. Schauder's theorem ensures that such a map has a
fixed point, which will be the desired equilibrium pricing function.

sc Rt is the sample space of the random state variable over which
the pricing function is to be defined. 3 is closed and bounded; hence & is
compact in R*

For the moment, assume bounds on the pricing functions exist. Let

a, 5 fee C7lg: 3 * Rs o5(+) < (+) < O5()5

Dole) <M 5 k= 1, .., 45521, 2,

where (9; (+), $5(*)), j = 1, 2 are the appropriate bounds, D, denotes the

kth partial derivative and


- 35 -

Claim 1. |y(p)| = 0.

Proof of Claim 1.

Hive 0) Hyg
Iw(p°)| <|c u'(p )- xu (0")| +
p P

atye (pt) - ELH ve(g") |s}| +
n Tet
P 4, (S')

E(t’ vice") |s] - ext ve(s |s)|
on (5') #y (S')
= | Mu) = cue f+ | ef atv Joi - ef x
P p”

7s") ¢

Hye is continuous in p on (4, (S). $,(5)15 hence for every ¢/2 > 0, there
Pp ‘ ‘ ,
exists 6 > 0, such that

| “ee ou n) | cs.

for every \p° - p” | <6. p” > ps hence for every 6 > 0, there exists Ny
such that |p? - p"| < 6 for every n > Ni.
"ony H + : : 2 -
ELV (e )e" [8] is continuous in $, on {, © C]oy(+) < (+) < Oy (+)}5
$, (S')

hence for every ¢/2 > 0, there exists 6 > 0, such that

| ef out! (oy |s} - ef stat co) M9] « eve
ey (S') #1 (5)



-~ 17 -

contracts increases, the demand for forward contracts by both countries

decreases.

Aa inf, H F
As f+ Se S $5(S), z +eand -z +- 2,

A sup
as f+ Se $4¢,(s), 24+ - «and -2° +=,

There is no opportunity cost to entering a forward contract, thus
as profit on a forward contract becomes positive with certainty, demand by
the home country becomes infinitely positive and supply by the home country
becomes infinitely negative. As profit on a forward contract becomes
negative with certainty, demand by the home country becomes infinitely
negative and supply by the foreign country becomes infinitely positive.

Thus in equilibrium

n inf_ sup
fe J]Se3 to(S), Se 8 o(S)[.

Equations (27) and (29) imply that if the market for forward

.

contracts clears
Hie % Fie
(30) c'(s, f) + C'(S, f) = 0.

The left-hand side of equation (30) is monotonically decreasing in

A
f with


ia (m4 ip a ata
(3) pQxt-cly> |
<

(en if i=

and

, , HOH eg]
(4) picy = (e' - f)z' < tm ifi=uH

tem’ if i =F,
where an unprimed variable is a time t variable and a primed variable is a
time t+] variable.

Equation (3) says that the nominal value (in terms of country H's
currency) of savings must be less than or equal to the nominal value of
endownment minus first period consumption. Equation (4) says that the
nominal value (in terms of country H's currency) of second period
consumption of the agent minus his profit on his forward contract must be

less than or equal to the nominal value of his savings.

3.3. The Underlying Stochastic Process

{s(t)} = {{x'}, {c"Hesy assumed to be the realization of a known,

stationary, first-order Markov process with

' ' '
(5) pr[x"(t+1) < x4 > xf (T41) < xf > tN (ts1) < ot ;

ety act cFety oF) Fcty = ot Pty et

a (t+1) ot Fy

' ' ' '
a(x4 xF JF |x4, xf, af, af),


- 32 -

= 4 (ur + e,u" af
eo a.

1 %
their minima on 3. Prices are strictly positive and finite; hence

) (are continuous functions) on 3; hence they reach

consumption is strictly positive and finite and the minima must be greater

than zero. Denote the smallest of the minima ¥.

wae
HLH 6,U -TtU H" H
a U 1
(At) | Se] = ( sz 0, - YF, 0)
. 1 oy 1
9 tT 8yU" our - Fou uF F
(A.2) | Se¢ )| = (0, gE
1

All of the components of the vectors (A.1) and (A.2) are

continuous functions of S on the compact set 3; hence they reach their

maxima on 8. Choose M, to be the largest such bound

a ef ctv [5] = e(TM [eo(S') - F0S)] yt |}
BRST —aeT

[az - 2(s)] af(s) +1 § to vt" ag(s,s') dS"
aF 8, Ty HS) 85

- [ ag(s) / o(s)] e[ stv4  [sy.
3 AST


- 29 -

7. Appendix

Proof _of Proposition 1

Claim 1. For every Se 3 there exists y(S) € rp, eo [such that

PLS, #4(+)] < 44(8) for every 4, © {41 © C7ly(S) < ay(+) <a}.

Proof of Claim 1. Suppose that the claim is not true for some Se 3. Then

for every sequence {y"} on qt, @ [ such tht y" + ©, there exists a

sequence of functions {oy(-)} such that 1 € {91 € chy" < e (+) < =} and

Ps, 4 (-)]> 1 (Ss) > y" for every n. Then there exists a neighborhood,
MS), of S such that PLS, + (*)] > 1 (S) for every S' eN(s). By

assumption, the probability measure of N(S) is strictly greater than zero.

As n+ @, y"> @; hence + (+) + © and P[S, ef (-)] +

utod - w/pcs, CD) + utoxty, a strictly positive, finite constant.

vi(ct7osM(St) + Le(S') = £(S)1/4] (SD). V'(0) = 2. Thus,

H '
e,[ 2PE oTOM se

———_——————_

6" )u!

at S. This is a contradiction; hence Claim 1 is proved.

Choose such a {y(S)}c,%-

Claim 2. For every Se 3, PLS, $,(-)] is bounded from above by some e(S) ¢€
cot, » [ for every $, € { ed Pa < o(+) < y(S)}.

Proof of Claim 2. Suppose not for some Se 3. Then there exists a sequence
foN(e)s PESs a8 (e)I} where of © {oy © C717 x7 < ay(+) < ¥(5)} such that
Pcs, a] +O, V'/o,0°) evaluated at a strictly positive, bounded price
is strictly positive; hence this is a contradiction and claim 2 is proved.

Choose such a {e(S)}c.%


3. The Model
3.1 The Enviorment

There are two countries, the home country, H, and the foreign
country, F. In every period a single representative agent is born in each
country. The agents live for two periods; thus there is a constant
population of two agents in each country. At the start of the first period
of life an agent is endowed with his country's output of the single, non-
storable, costlessly-transportable consumption good. This output is
exogenous and stochastic.

There also exist two country-specific fiat monies. The original
stocks of the monies are held by the agents of generation zero at the start
of period one. Additional money is: injected into the economy by means of a
stochastic, proportional transfer to the agents of each generation. Each
agent receives an amount proportional to his savings at the beginning of his
second period of life.

During their first period of life the agents of generation t trade
with agents of generation t-1., The young agents allocate their endownment
between consumption and savings. They trade part of their output to the old

agents for fiat money. They may also enter into forward contracts /

It is
assumed that their are no margin requirements. By convention, forward
contracts have zero value at the time of signing; hence there are no
opportunity costs associated with entering into a forward contract.

In their second period of life the agents of generation t receive
their proportional monetary transfers from the governments and settle their

forward contracts. They then trade with the agents of generation t+1 to

obtain the consumption good in. exchange for fiat money.


-5-

Speculative purchasing does entail foreign exchange risk. By
convention the value of the contract at its signing date is zero; hence in
the absence of margin requirements, the amount invested may be unlimited and
there is no opportunity cost. The expected nominal return on this
alternative depends on the forward rate and the spot rate expected to
Prevail in the delivery period. Thus the forward rate depends on, among
other things, the distribution of the future spot rate.

The foreign exchange contract entails the trade of one currency
for another. If one currency is, in some relevant sense, “riskier” than the
other, and the risk is not completely diversifiable, a premium must be paid
for the purchase of risk. Even in the absence of a risk premium, the
forward price cannot be an unbiased estimator of the future price for both
countries because the exchange rate for one country is the reciprocal of the
exchange rate for the other country. By Jensen's inequality it will not
generally be true that it is possible for the expected value of the future
spot rate to equal the forward rate and for the expected value of the
reciprocal of the future spot rate to equal the reciprocal of the forward
rate. This phenomenon is known as Siegal's paradox./ Thus, standard
intuition has it that the forward rate is a combination of the expected
future spot rate, a risk premium and a convexity term arising from Siegal's

paradox.


-18-

inf
has was fs se 8 4,(5),

H F A sup_
c +c + seas f+SeS 4,(S).

Thus, there exists a continuous function

A inf sup
f:3+[Se3 $9(S). Se 3 6,(S)[

such that
A A
(31) f = f(S)>0

solves equation (30), Substituting (s) into equation (29) gives a

continuous function

2: 3 xis R, where WF is the set of all possible values of M Fe such

that

H F
(32) z= -z = 2(S).
wow

The shapes of #(+) and z(+*) depend upon the shape of 9(+).

By the Inada conditions, it must be the case that agents choose

2H and Pa to be small enough in absolute value so that with Probability one

i

“2

>0; i =H, F.


tx! (t) - 1] m'(t) := monetary transfer from the government
of country i to the generation t agent
in country i.
<j (t) := time t-1+j consumption by the generation t agent in
of country i; i =H, F, j=1, 2.
All individuals of generations t > 1 have the identical utility

function

(1) Ufe}(t)] + € {veeg(ty I} ,

where Be is the expectations operator conditioned on current and past values
of all variables and

U:R + 7 - ©, » [ and

V:iR + J-e, o£
are strictly increasing, twice differentiable, strictly concave functions.

The Inada conditions are assumed to hold, thus

' , :
(2) U +eascy +o, V + ascy+0
'

U > Oasc, +s, v > Oascy+TM,

The function V (cy)ey is assumed to be strictly increasing. This
implies that the substitution effect of a price change will dominate the
income effect.

In period t > 1 the generation t agent of country i, i =H, F,

maximizes (1) with respect to {e5} »m' and z! subject to


- 24 -

H
Then there exists a function P: 3 x 7(3) > Tay , @] where P| S,
x
O00) = 04{5)s which solves equation (23).

H
Proposition 1. There exists $,(+) € {o,| 4 < ¢,(+) <=} such that for
x

H
every $,(+) € {#¢ cy a < oy (+) < d(+)}s PLS, 0) (+)] < 3,(S), for every
x
Se.
If proposition 1 is true, then $,(+) constitutes an upperbound for
(+).
Proof. See the appendix. The proof of this proposition is based on a proof

in Peled [1980].

H
Proposition 2. There exists oy(+) € {6 15r < 4) < $,(+)}> such that for
x

every o4(+) © {61 © C7las(+) < oy(e) < Syl}, PLS. 64(+)7 > ay(+), For
every Se S,
Proof, See the appendix.
$,(°) constitutes a lower bound for o4(+)-

Consider now the exchange rate. For every Se 3,
oy{eJe [4,(+)> a()IN 02, equation (24) gives the relationship between
bo(+)> the time t+1 exchange rate function, and $,(S), the time t exchange

rate. The right-hand side of equation (24) is strictly increasing in

$o(S).

F F
As 4p(s) +0, 7 9205) urpsF - 9205)" | 5 0
0,05) $105)

Fore F F
As 4,(8) + * 94(S) |e e0(S) yeryF _ (SIT) yo
F $1 (5 6, (5)
1 1

T


International Finance Disucssion Papers
Number 211
December 1982

THE RISK PREMIUM IN THE MARKET FOR
FORWARD FOREIGN EXCHANGE

by

Anne C. Sibert

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publications to International Finance Discussion Papers (other than an
acknowledgment by a writer that he has had access to unpublished
material) should be cleared with the author or authors.


-h-

(7) ef Hye (cy 4) le uv (ct)

a

(a) ef (e: FVN(Ch) Sg
p

F!

(9) Eg[xF erve(e5)] = Ur(eh)

p p

' 1eF
(10) e,[ 2'7 FV (2) ] = 0
r

Equations (8) and (10) are similar to the equation derived in the
Richard and Sunderasen [1981] model of commodity forward markets, and have
the same interpretation. The expected marginal utility of consumption of
the home good at time t+1, from entering into a forward contract at time t,

is zero.

3.5 The Government

Under floating exchange rates the governments control the money
supply by means of the exogenous, stochastic policy variables {c}. Money
supplies grow according to
ito ili

(11) we M't'; d= Hy, Fe


-10-
The stochastic process takes its values on the set
See (a, Gh, Py (ot THY x [ch 2]

Mee AF) c ts A)

can

=
>
®
5
o
o
a
I>,
Ix
I,
kal

> ’

G(+|*) is a cumulative distribution function with the continuous, strictly
positive density function g(+|+), which has a continuous derivative. In
addition
(6) u(A) > 0 implies Pr[S(t+1)eA S(t) = S] > 0,

Y S(t), S(t+l)e 3,

vy AGS,

where u(+) denotes Lebesque measure.

3.4 Solutions to the Consumers' Problems

ui is strictly concave in mn and E,(V') is a Lebesque integral of
strictly concave functions in mi and 2, where ul and vi are U and V
evaluated at country i consumption levels. Hence, the objective function is
strictly concave, and the young agents' problems have unique solutions which

are given by


-31-

_ SUP
<TM j | s§ - sil for every 5°, si e S$, 6, € %,. r 59

F -sbacs implies

sup
j " sj - 55 | <6; hence choose 6 = €/(4M). Then for every € > 0,

ql sh < 6 implies 16, (s°) - 4; (S 1) ¢ ©, for every S°, ste 3, 6, %-

Thus ¢, is equicontinuous and the proof is complete.
1

Proof of Proposition 9.

The continuity of both sides of equations (23) and (24) in S
guarantees that 6 is a continuous function in Ss.

It is now shown that the partial derivatives of @ are uniformly

bounded. Let sj be the jth component of S, j = 1,..., 4.

M2 a fet ve sj - tut yy (atyt)

as a8; 6,05") oy 38, om
Ft Flogy' Fo Ft F. Ft
> 9 fe[t oo(s')V |S] - 1 85U bra ( $ 8 5U )+ 2
33; a8; $405) 85 385 on 38;
(rant) 6,
. ’
a es

F

where vi and uf! are V and U evaluated home levels of consumption and V and

uP are V and U evaluated at foreign levels of consumption.

” F Ft
ia (att | =f cut + Hey) and fo (E82!) |
1 0,3 8 8



- 30 -

Let $,(S) = max{y(S), (S)} for every Se 8.

Claim 3. For every S « 3S for every $, © {oye 02 eA x4 < o(+) < $,(+)},
$,(S) bounds P[S, $,(°)] from above.

Proof of Claim 3.

PLS, (+) < €(S) < 3,(5)

H
for every 4,(+) © {6 © C7] Ty < ay(+) < ¥(S)}
x

PES. #)(+)] < 04(S) < 3,(5)

for every #(+) © {1 © Cl¥(S) < o4(+) <o}
This proves claim 3.
Proof of Proposition 2.

Suppose not for some Se 3. Then there exists a sequence
Loq(+)> PES» of (+)]} where #7 © {6 © C7|xMix¥ cig (+) < 3,(+)} and o” >
ay © fo, 6 C8 cM pH « #1(+) < ,(+)} such that PCs, ¢,"(-)7 > text,

V' 144 (+) evaluated at a strictly positive finite function is finite; hence
this is a contradiction.

Suppose inf $,(S) = run then there exists a sequence sts 3°
such that for some (+) « {o) € c7les(+) < o(+) < o,(+)}5 pcs”, ote)] >
pcs?, eT (e)] » tx, 8 is closed; hence $9 © % and this is a
contradiction.

Proof of Proposition 5.
Convexity is obvious.

I now show o3 i= 1, 2, is equicontinuous. Applying the mean
value theorem to each component of S, and using the bounds on the partial
derivatives of $53 gives

4 S02 s 4

0 leg
lo(S)) - (8°) «M25, IS; - 8;


There are no riskless interest-bearing assets. Interest arbitrage
in this case would require that current spot and forward rates be identical,
and thus agents would be indifferent between saving their own country's
currency or entering into a spot-forward swap for the other country's
currency.

In this case the home and foreign currency would be perfect
substitutes, as in Karaken and Wallace [1977], and the exchange rate would
not be determined. As the focus is on speculative behavior, the restriction
that agents save only their own money will be imposed. It is this

restriction which allows exchange rates to be determined.

3.2 The Consumers' Problems
Time t variables:

p(t) := the good price in country H currency.

e(t) := the exchange rate, or the price of country F's
currency in terms of country H's currency.
f(t) := the forward price of country F's currency at

time t for delivery at time t+1, in terms of
country H's currency.

z (t) := the number of units of country F's country
purchased forward by the generation t agent
in country i; i =H, F.

ni (t) := the number of units of country i's currency
saved by the generation t agent in country i;
i; i =H, Fe

x! (t) := output in country i; i =H, F.


- 21 -

a
Cov,(e'- f , V' )

(34) & (et f) = - Lai
> —r

BR (W')

This is similar to the equilibrium condition in Richard and
Sundarsasen [1981].
By equation (34), a positive premium or a positive expected profit

on a forwrd exchange contracts exists when

(35) Cov,(e'- fv) <0; 4 =H, F
r)

This occurs when higher than average are profits on long positions
and lower than average profits on short positions are associated with lower
than average marginal utilities, and hence, higher than average consumption
levels. In this case purchasing currency forward does not provide
protection from risk, but selling currency forward does. The forward buyer
of foreign currency is insuring the forward seller of foreign currency.
Hence, the buyer must make a positive expected profit and the seller will

make a positive expected loss.


- 38 -

Bental, B. "Capital Controls in a Two Sector Model of Growth and Trade,"
Ph.D Dissertation, University of Minnesota (1979).

Bilson, J.F.0., "The ‘Speculative Efficiency’ Hypothesis," Journal of
Business 54 (July, 1981), 435-52,

Brock, W.A., "Asset Pricing in an Economy with Production: ‘Selective!
Survey of Recent Work on Asset Pricing Models," in P. Lin, ed,

Dynamic Optimization and Mathematical Economics, (New York:

Alenum Press, 1980).

Cumby, R.E. and M. Obstfeld, "International Tyterest-Rate and Price -Level
Linkages under Fiexible Exchange Rates: A Review of Recent
Evidence," in J.F.0. Bilson and R.C. Marston, eds., Exchange

Rates: Theory and Practice, (Chicago: University of Chicago

Press, forthcoming, 1983).

Dooley, M.P. and J.R. Shafer, “Analysis of Short-Run Exchange Rate Behavior:
March, 1973 to November, 1981," International Monetary Fund
Research Paper, (April, 1982).

Geweke, J. and E. Feige, "Some Joint Tests of Efficiency of Markets for

Forward Foreign Exchange" Review of of Economics and Statistics 6]

(August, 1979), 334-41.
Hakkio, C., “Expectations and the Forward Exchange Rate," International

Economic Review 22 (October, 1981), 663-678.


- 39 -

Hansen, L.P. and R.J. Hodrick, "Forward Exchange Rates as Optimal Predictors

of Future Spot Rates," Journal of Political Economy 88 (October,

1980) 829-53.
» "Risk Averse Speculation in the Foreign Exchange Market: An
Econometric Analysis of Linear Models," in J.A. Frenkel, ed.,

Exchange Rates and International Macroeconomics, (Chicago:

University of Chicago Press, forthcoming).

Hsieh, D.A., "Tests of Rational Expectations and No Risk Premium in Forward
Exchange Markets," Working Paper, No. 843, National Bureau of
Economic Research (1982).

Kareken, J.H. and N. Wallace, "Samuelson's Consumption-Loan Model with
Country Specific Fiat Monies," Research Department Staff Report,
No. 24, Federal Reserve Bank of Minneapolis (1977).

Kolmogorov, A.N. and S.V. Fomin, Introductory Real Analysis, R.A. Silverman,

trans., (new York: Dover Press, 1975).
Lucas, R.E., Jr., “Asset Prices in an Exchange Economy," Econometrica 46
(November, 1978), 1429-45.
> “Expectations and the Neutrality of Money," Journal of
Economic Theory 4 (April, 1972), 102-124.
» "Interest Rates and Currency Prices in a Two Country
World," Unpublished Manuscript, (July, 1981).
McCallum, B.T., "On Non-Uniqueness in Rational Expectations Models: An
Attempt at Perspective," Working Paper, No. 684, National Bureau
of Economic Research, (June, 1981).
McCulloch, J.H., "Operational Aspects of the Siegal Paradox" Quarterly
Journal of Economics 89 (February, 1975), 170-175.
Nickelsburg, G., "On the Multiplicity of Stochastic Dynamic Exchange Rate

Equilibria," Working Paper, No. 8031, (1980).


- 27 -

Proof of Theorem 1. is continous; hence

JU(CM(Gy)> 1%) = (My (€1(44)) AQ (EM49)))

S (VM (04))> AG (45))

c (c1(4), c1($5)) by the argument in the proof of proposition 10.

Hence, Mel (44) CH 4_))& (CMg)> V(p))- (eM 4) X 6142)
js convex and compact. Hence there exists a fixed point.

Thus, it is established that equilibrium pricing functions

"

#,(8) = P/M

and

"

#,(S) = eM*/M

exist.


forward markets as a means of risk-sharing, they prefer to save their own
currency. In fact, the extreme case, that they will save only their own
currency, will be assumed.

Section two contains a general discussion of forward foreign
exchange markets. In section three the model is set up, the forward market
is described and the equilibrium is defined. In section four the roles of
risk and hedging in the determination of a time-varying risk premium are

discussed. In section five the existence of an equilibrium is proved.


-14-

hence their optimal decision rules. Rational expectations will ensure that
the prices that agents believe will prevail if a given state is realized,
are in fact the prices which will prevail.

At this point it is conjectured that goods prices must vary
proportionately with the money stocks and the spot and forward exchange
rates must vary proportionately with the ratio of the money stocks. Then

the price of the good may take the form
H
(20) p = M"4,(s),

the exchange rate may take the form

H

(21) e= MY 9(s)

M
M
and the forward price may take the form

H!

22) f= MM Fs),
(22) He £(S)

where it is recalled that S:= ta i, xf x,

Definition 2. An equilibrium pricing function is a continuous function

:= (45 oo) 2 S> Rs


- 16 -

In the next section the market for forward contracts is discussed.

It will be shown that bankruptcy cannot occur and that equations (25) - (26)
a

can be solved for a unique triplet {f, 2H zh, which is a function of S and

o(+). In section 5 the existence of an equilibrium is proved.

3.8. The Market for Forward Contracts

Given the demand for savings, equations (25) - (26) give the home
and foreign country demand for forward purchases of the currency of country
F. Suppose that (+) is an element of an arbitrary set B(3) of continuous
functions on S such that $(3) > 0 for every Se 3, (+) © B(S). Then the
left-hand sides of equations (25) and (26) are strictly decreasing in 24 and

2F respectively. Thus, there exist functions

wn

(28) cl: 3x B(3) x [-*, @] + [- 2, 2]

C: 3x BS) x [- ©, ©] + [-», «],

such that
A
(29) zt = cMs, fy
zfs ccs, f)

solve equations (25) and (26).

By the assumption that Vi(cy)ey is an increasing function, both

cH and cf are strictly decreasing in f. Thus, as the price of forward


a testable equation, Hansen and Hodrick [forthcoming] and Viswanath [1982]
rely on the real intertemporal asset pricing models of Brock [1980] and
Lucas [19781], which give the result that the risk in a forward contract is
caused by the covariance of the real profit of the contract and the
intertemporal marginal rate of substitution.

A challenge for monetary theorists is to produce models of nominal
assets which are analgous to the intertemporal pricing models for real
assets. Such models would integrate monetary theory with modern financial
theory.

In order to model forward markets in this paper a Samuelsonian
overlapping generations model is employed. Overlapping generations models
have been previously used to model foreign exchange markets by Karaken and
Wallace [1977] and Nickelsburg [1980].

In both Karaken and Wallace's deterministic model and
Nickelsburg's stochastic model, country-specific fiat monies are regarded as
perfect substitutes. Hence, under laisez-faire the exchange rate is not
determined. The assumption of perfect substitutability does not, however,
seem to agree with reality. Americans prefer to hold dollars and the French
prefer to hold francs. Lucas [1981] claims that the reason for this ". .
must have something to do with the local nature of information people have,
but it is difficult to think of models that even make a beginning on
understanding this issue."2/

Part of the allure of overlapping generations models is that
valued fiat money arises endogenously in them. It would be desirable if the
lack of perfect substitutability between currencies arose endogenously in a
model of exchange rate determination. Such a phenomenon, however, will

not occur here. It will instead he postulated that, while agerts enter


- 37 -
FOOTNOTES

i/ See Bilson[1981], Cumby and Obstfeld [1983], Dooley and Shafer [1981],
Geweke and Feige [1979], Hakkio [1981], Hansen and Hodrick [1980, forth.],
Hsieh [1981] and Viswanath [1982].

2/ Lucas [1981], p. 41.

3/ Actually the value date is usually one or two days after the contract
date. This technicality will be ignored.

4/ See Siegal [1972]. McCulloch [1975] claims this is emprically trivial.
The magnitude, however, obviously depends on the shape of the distribution
function.

5/ In reality there are both forward and futures markets for foreign
exchange. The fine differences between the two are discussed in Richard and
Sundarasen [1981]. All contracts here are one period contracts and thus
forward and futures contracts are equivalent.

6/ See, for example, McCallum [1981] or Wallace [1980].

J/ Actually elements of % are pricing functions of the good weighted by

1/M* and elements of ¢, are pricing functions for the exchange rate weighted

2
by wt The term “pricing function" will be used for convenience,

8/ See Kolmogorov and Fomin [1975], p. 102.


- 2 -

Thus, there exists a function R: 3 X {o © c2|o < oo(+) <
Ma) Fy > (0, He C/E where RS, ¢9(+)] = $5(S) which solves
equation (24).

Proposition 3. For every $, © {9 2 C2 | (+) < o,(+) < $,(-)}, there
exists $o(*) where 0 < bol) < Fest such that for every

$y © {op © c*loo(+) < b9(+) < OEE RES, $9(+)] > go(S) for every
se.

Proof. Similar to the proof of proposition 2.

Proposition 4. For every $, € {9 © c719,(+) < o(*) < $,(+)}, there exists
$o(>) where $o(*) < $5(+) < Fey such that for every $0(*) €

{oy © C*laal+) < dol+) < Bol+)} » RESs b9(+)] < ¥2(S) for every Se 8.
Moreover sup (5) < xFgi(s)/ct

Proof. Similar to the proof of proposition 1.

Propositions 1-4 have established the existence of the bounds on
the pricing functions.

Proposition 5. cl(#)) Xx c1(#) is convex and compact.
Proof, See the appendix.

3 is compact and LF is bounded; hence by Arzela's theorem, % is
relatively compact .2/ By Tychonoff's theorem cl(#)) x cl(#) is compact.
Theorem 1. There exists 9 € cl(#1) x cl(#>) such that » is an equilibrium
pricing function.

The strategy will be to show that for every admissable time t+1
price vector, there exists a time t price vector and the mapping from time

t+1 prices to time t prices has a fixed point.


The Risk Premium in the Market
for Forward Foreign Exchange
by Anne Sibert

1. Introduction

This paper attempts to prowide a theoretical link between future
spot prices and current forward prices in the market for foreign exchange.
A general equilibrium model is developed where economies experience country
specific real and nominal shocks. Risk averse agents use forward foreign
exchange markets as insurance against these shocks. The implications of
this behavior for the existence and nature of a risk premium are discussed,

The question of the existence and nature of a risk premium in the
market for foreign exchange is of enormous practical “importance. One reason
is that the efficacy of open market operations may depend upon ‘the answer.
If relative purchasing power parity nolds ex ante, and bonds, which differ
only in the currency in which they are denominated, are regarded as perfect
substitutes then expected real interest rates will be equated across
countries. In a small country open market operations will have no effect on
expected real interest rates. If, however, bonds are not regarded as
perfect substitutes, and a risk premium exists, then government policy which
influences variables affecting risk premia will influence expected real
interest rates. Thus, if risk premia depend upon stocks of debt, open
market operations may be effective.

In recent years a plethora of empirical studies have indicated
that the current forward rate is not an unbiased estimator of the future

/

spot rate. If markets are efficient this is evidence of a time-varying

risk premium, To explain the nature of a risk premium and to derive


- 3-

All terms are continuous functions of S. 9(S) reaches its minima

on 3 and is strictly positive at the minima, and ag(5) and ag(S.S') are
a a

continuous, and hence, bounded.

Fry + e(aZur ) a8 E(a

ayH" ) 5a
af E (acu ) 38; E(au
ws," ) Aad

SE (aU pe U aT AC
asy ST AST | (ST

$,(S') - f(S)
where A = as JT: which is also uniformly bounded.

is bounded from above. Similarly

SFY ouyyF!
lee | tates |
1

is bounded from above. Let a, be the largest of the two bounds.

lg. Ge }]e nC Foe uF ‘o),
54

which is bounded from above by some finite 4.


- B-

6. Conclusion

This work has attempted to model risk premia in foreign exchange
markets in a choice-theoretic, general equilibrium framework. In this
framework forward contracts are used as a method of sharing risk. Time-
varying risk premia arise and are seen to be related to the efficacy of such
contracts as hedges.

Allowing trade in forward contracts, but requiring that agents
save only their own country's currency, allows exchange rate determination
in an overlapping generations model. It is readily observable that
currencies are not actually regarded as perfect substitutes. Agents in a
given country tend to hold portfolios consisting predominately of that
country's currency. The interesting question is why this non-

substitutability should exist. This question is still to be answered.


-12-

In a more elaborate model, governments might also intervene in the
market for foreign exchange by buying and selling foreign currency. Here it

is assumed that there is no intervention.

3.6 Equilibrium
3.6.1 Market Clearing
Market clearing requires
H' F'

' 1
(12) chat tcp th =x +X

1

(13) m = wil, i=, F.

(4) zta 2F eo,
Equations (12) - (14) -are-a system of four equations, but by

Walras' Law only three of them are independent.

3.6.2 Definitions :
Substituting equations (12)- (14). into equations (7) - (10) gives

the following definition.

Definition 1. An equilibrium is a bounded sequence, -{p(t), e(t), f(t),

zt), 2 (t)}

t>1 Such that f(t) > 0, {p(t)} and {e(t)} are bounded away from

zero and

Ht Hw H HH!
(18) Eft vi( rl M+ et-f2°)} = Lux - mM")
pr pr P P P


- 15 -

where $,(S) is the price of the good (divided by M ) in state S, o9(S) is

the exchange rate (weighted by oF ty in state S and

(23) ef aM’ veg att 4 $0(S') = FS) Disp = cH ouch A]

oS) 9,157) (5°) Wr $05) $05)
a(S) yer tH an(S*) 4 dy(S') - FCS) GF
(24) ef 2 vf 2 + 92 z’ js}
$087) $,(5") ots)
- 1 4,(8) utpx® . #9(8)o* ]
9, (5) Oi 03)

(25) ef #205") - #(5) vip at 4 #2(S') - f(s) JIS} = 0
CD) qey —aSy wr

(26) ef #205") = FS) yop 9(SITY 4 4905") = FS) QF jis}e0
405) rea) ) ir

(27) 24g oF. 0,

for every possible realization of {S(t)}.
Notice that expectations in a. stationary equilibrium are
conditioned only upon S. Prices are a function of S, and hence add no

additional information.

Conjecture. There exists an equilibrium pricing function.