The Risk Premium in the Market for Forward Foreign Exchange

International Finance Disucssion Papers Number 211

Necember 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to International Finance Discussion Papers (other than an acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

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 holds 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

1/

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

-?2-

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 [19811] 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

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.

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.2/ 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.

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.

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, nonstorable, 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.

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.

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

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

Cx! (t) - 1) m' (t) := monetary transfer from the government of country i to the generation t agent in country i. c; (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) Ule}(t)] + &, {vfep(t)I}

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

U:R + 7 =», » [ and

ViR > J -e, © £ are strictly increasing, twice differentiable, strictly concave functions. The Inada conditions are assumed to hold, thus |

(2) U +e as Cc, 70, V +@ as Cy > 0 t ' ,

U +0 as Cc) > V +0 as Cy > The function V (cy )c, 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 fei} , m' and z' subject to

a

fm" ifiifi

(3) p(x! = ch) > \ oF Lem’ if i =F

and

(4) pich (ere tyetic ft m' if i =H (chem? if i= F,

where an unprimed variable is a time t variable and a primed variable is a

time t+1 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"}, ("Hest assumed to be the realization of a known,

stationary, first-order Markov process with

4 i] ° i (5) Pr[xt (tary < xO, xFoteny < xP) cP(tady < ot,

Petey = xt Feey = xP, Fety = ot) cPety = oF] -

« (t41) <t H

i ‘ t ' e(x4 xF : H : oF |. xP oH, a),

- 10 - The stochastic process takes its values on the set Sse OP GPR] Ct, MD x (aha 2]

where 0 < (x4, x, Pan t) < (x4

oxy, 7) <8,

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 8,

Vv AGS, where u(+) denotes Lebesque measure. 3.4 Solutions to the Consumers’ Problems

u' is strictly concave in m' and E.(v') is a Lebesque integral of strictly concave functions in m' and z', where U’ and vy" 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

- ll -

H'47/.H 7H p" _#—p F

(9) E,[tF e'v'(ch)] = ur(ch)

p! p

' 17eF (10) e,[ © V2) ] = 0, p"

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 {r"}.

supplies grow according to:

ii

(11) Mee wets de Hy F.

- 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

(12) ch +C, +C, + ch =X +X

(13) m =Mdt3i=H, F.

(14) z +z =0. 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),

zit), (they such that f(t) > 0, {p(t)} and {e(t)} are bounded away from zero and

Hie A! WHY) yg Hy) a ap HH! (15) Elz vir te'-fz)j/=1lu'(x -™ )

p p p Dp )

- 13 -

(16) E, lets" Vi(etdt Ma et. # zy] = e U'(xF. emt) p p p p Pp (17) E,let-f vi(rt ms et # z4)] = 9 p p po (18) E.le!- f Vi(etrt Me eta 2*))] = 0

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

It is well known that models involving rational expectations give rise to a multiplicity of equilibria.©/ 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

time t state variables, {mt uF rH i, xP xP,

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

- 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"9,(S),

the exchange rate may take the form

H

(21) e= a o5(S) M

and the forward price may take the form

H! A

(22) f= M f(S),

“FT where it is recalled that S:= fot rf, xt xh, Definition 2. An equilibrium pricing function is a continuous function

g:= ($)> 9) 7S > Ruy

- 15 -

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

the exchange rate (weighted by mF mt) in state S and

(23) ef tt yep aH 4g 2 (S') - 0S) qygp eH uel Hy ie ee ie ee wer $05) A)

(24) E{ r*9,(S") VE of 9 (S') + ¢5(S') - f(S) zt ]\s} 6,57) OG) $, 05") mas

= t6,(S) u'[xF . 65(S)r" ] “$4 (8) $05)

bo(S*) = F(S) yep HY, G95) = US) OH rey | (25) E{ *2 Vitor 2 z |{[S} = 0 ; o, 65 ) @, 05 ) 6,05 ) M*

(26) ef #205") - #(S) ve bo(S')e” , 5(S*) - F(S) GF 1 s}=0

$057) 5) 0, 0S") Mer

(27) za 2F = 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.

- 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, 7H 2}, 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 $(+) is an element of an arbitrary set P(S) of continuous functions on S$ such that 9$($) > 0 for every Se S, o(+) © B(S). Then the left-hand sides of equations (25) and (26) are strictly decreasing in 24 and

7 respectively. Thus, there exist functions

(28) ch: 3x BS) x [-», @] + [- @, @] ch: 3x BS) x[- ©, @] + [-«, @], such that (29) zt = cHs, Fy zt = cfs, #)

solve equations (25) and (26).

By the assumption that Vile, )e, is an increasing function, both

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

- 17 -

contracts increases, the demand for forward contracts by both countries

decreases,

A inf, H F As f+ Se S$ ¢5(S), z + e@and -z + =,

~ SUP. H F as f+SeS ¢,(S), Zz += and -z +o,

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

rR inf_ sup) fe ]SeS8 ¢5(S), Se § ¢5(S)[.

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

contracts clears Hi.” Fre fy | (30) C (S, f) +C (S, f) = 0.

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

Aa f with

- 18 -

HF a inf, C +C +easfr+SeS§ ¢5(S),

HL OF a fUP x C +C +-masfr+SeS§ ¢,(S).

Thus, there exists a continuous function

A inf sup f:3+[Se38 $,(S). Se S ¢5(S)[ such that (31) f = f(S) > 0

solves equation (30). Substituting #(S) into equation (29) gives a

continuous function

z: SX We R, where W is the set of all possible values of M!, such

that

(32) ze - 2) = 2(5).

M M

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

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

74 and z to be small enough in absolute value so that with probability one

- 19 . Thus in equilibrium z will always be small enough so that cl > 0; i = H, F 2 9 9 EY

and bankruptcy cannot occur.

290

4, The Nature of the Risk Premium

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

(33) f= E(e') + [‘t p™ - “pr

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'.

E,E(e' -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

- 21 -

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) Covs(e f ,V )< 0; i =H, F.

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.

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.

4 is the sample space of the random state variable over which

SCR the pricing function is to be defined. 3 is closed and bounded; hence § is compact in Rt

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

0; = [oe Clo: S > Rs aslo) < OC) <8 (45

where (o5(+), $5(*))s j = 1, 2 are the appropriate bounds, D, denotes the

kth partial derivative and

- 23- He ll= gcR 10, (S)1. 1

The set of pricing functions to which attention is restricted is cl(#,) X (#5) 2! cl1(®,) is the set of admissable candidates for the good pricing function and cl(,) is the set of admissable candidates for the exchange rate pricing functions. A generic element of °, will be denoted 5» j = 1,2.

It will first be shown that the appropriate bounds for the pricing functions exist. Then it will be shown that cl(s,) X cl(a,) is convex and compact. Finally, equations (23) and (24) will be used to construct a continuous map from cl(#,) X c1(%,) 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 § and every continous bo(*)s equation (23) gives the relationship between o,(+)> the time t+l pricing function, and o,(S). the time t price. The right-hand side

of equation (23) is strictly increasing in o,(S)- As ,(S) + a U'fxt _ a ] +0 ; 1 ; Ae) 6,05)

H H t T if JH wo As o(S) > are artsy be = aesy I >

Let

~ |x" a(S) = {o4€ CTI < (+) < },

- 24 -

H

Then there exists a function P: $ x #(S) > 14 , 2] where P[S, x

o,(+)] = 0,(5), which solves equation (23).

H

Proposition 1. There exists 0) € {¢,| 1s 61 (+) < @} such that for X

every ¢,(+) © {,€ c7| = < o(+) < O(+)}, PLS, 6,(+)] < $,(S), for every Se 8,

If proposition 1 is true, then $,(°) constitutes an upperbound for o,(°). Proof. See the appendix. The proof of tnis proposition is based on a proof in Peled [1980]. |

H

Proposition 2. There exists 9, (+) € (6 < o,() < o,(-)I. such that for every 6,(+) © (6) © C*Las(+) < o:(+) < dy(e)bs PLS. 64(4)7 > Oy(e)s For every Se 3. Proof, See the appendix. o,(°) constitutes a lower bound for oi (+).

Consider now the exchange rate. For every Se 3,

oyio Je [oi(+)> o(*)INN\ C2, equation (24) gives the relationship between

o5(*), 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

F F

As ¢,(S) > x*9,(5) ; t 9,(S) urfxt _ (5) e,

a 9165) 9, (5)

- 25 -

Thus, there exists a function R: SX {4 € c?

0 < ¢5(*) <

x os (2)/ oF} > (0, xHy (tL, where R[S, o5(+)] = ¢5(S) which solves equation (24).

Proposition 3. For every $, € 1, € c? | o,(+) < o,(+) < o,(-)]; there exists 9,(+) where 0 < $5(*) < Foe) such that for every | bo © {oy © CAlbol+) < dole) <x oy(+)/t1}s RES, o9(+)] > g9(5) for every Proof. Similar to the proof of proposition 2.

Proposition 4. For every $, € 1, € C71 44(+) < 6, (+) < o,(-)]5 there exists o4(*) where 9.(+) < $o(*) < Oia such that for every ¢o(*) €

{45 © C*ldo(+) < dp(*) < Fole)} » RES dp(+)] < 9, (5) for every Se S, Moreover sup ¢5(S) < xFo,(S)/t'

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(#,) x cl(#,) is convex and compact. Proof, See the appendix.

S is compact and %, is bounded; hence by Arzela's theorem, o, is relatively compact .©/ By Tychonoff's theorem cl(#,) x c1(%) is compact. Theorem 1. There exists $ € cl(#,) X c1(@,) such that » is an equilibrium pricing function.

The strategy will be to show that for every admissable time t+l 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.

- 26 -

Proposition 6. For every Se 8, ¢ € 0, there exists a unique 6 € ® such

that

(36) ea vee a 205) = FS) a¢syyigp = et pte ety

6 ») rE ry 6ST) 6, (5) 6, (5) oF '6.(S') yap te 65(S') _ do(S') = FlS) (37) &{ 2 vif 2 - £2 z(S)]|S} = eS 4 5") O05) F

80(5) yiiyF F80(S)), ay tS

If this proposition is true then 6 = $ constitutes an equilibrium. Thus it will be sufficient to.show that Wt: 6 + 6, 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 6,(S) © £9, (S). 6, (S)], such that (36) holds. Proof. 6,(S) = PCS, o,(-)]. 6, (S) € Co, (S)> $,(S)] by propositions 1 and 2. Proposition 8. Let 6, (S) be given by proposition 7. For every Se §, ¢ ¢ ®, there exists a unique 64(S) € [¢5(S), $5(S)], such that (37) holds. Proof. 6,(S) = R(S, ¢o(+)1. 6,(S) € [¢,(S). $,(S)] by propositions 3 and 4. Proposition 9. 6c 4. Proof See the appendix. Proposition 10. WM: ® + ® is continuous.

Proof. See the appendix.

- 27 -

Proof of Theorem 1. is continous; hence

M(c1(44)> UC) = (My (C1044) AG (C1 (49)))

in

Cc (cl(o,)> c1(,) } by the argument in the proof of proposition 10.

Hence, M(c1(44)s C1(4,))S (c1(44)> cM(O,))- CoC) c1($5)) is convex and compact. Hence there exists a fixed point.

Thus, it is established that equilibrium pricing functions

6,(S) = p/M

= eM*/M

=a ine) ~~ ” ~— 1

- 28 -

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. Timevarying 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.

- 29 -

7. Appendix

Proof of Proposition 1

Claim 1. For every Se S$ there exists y(S) e€ qo 74 eo [such that

PLS, o,(+)1 < $4(S) for every 4 © {o, € C 21y(S) < o,(+) < ef.

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

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

sequence of functions {on(°)} such that $1 € {4 € c7]y" < $y (+) < @} and

PLS, 1 (*)] > $5 (S) > y" for every n. Then there exists a neighborhood, AS), of S such that P[S, $4 (-)] > $4 (S) for every S' eN(S). By assumption, the probability measure of N(S) is strictly greater than zero.

As n+, y"+ ©; hence by (+) + » and P[S, oy (*)] +.

ur(xt - xyes, (+) > ur(xty, a strictly positive, finite constant.

ve(rA' 7g s2(8") + [e,(S') - f(S)V/oq(S')) > v'(0) =o, Thus,

H ; e,[ 1 PES: of C)N yy.

Leen eee

6,"(-)u!

- at §. This is a contradiction; hence Claim 1 is proved. Choose such a {y(S)}o.z Claim 2. For every Se $, PLS, ,(+)] is bounded from above by some e(S) ¢ coy o [ for every $, € { 1 € C224 7,4 < 6, (+) < y(S)}. Proof of Claim 2. Suppose not for some Se 3. Then there exists a sequence {oh (+), PLS, #7 (+) I} where o) e {4 © c2iaty x4 < gs(+) < v(S)} such that P[S, a(-)] 7%, V'/o44*) 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)} oz

-~ 30 -

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

Claim 3. For every S e« S, for every o,€ {o4€ 2c iy yH < o,(+) < o,(+)}; $,(S) bounds P[S, o,(-)] from above. Proof of Claim 3.

PLS, $,(+)] < €(S) < 34(5)

H for every $,(+) € {, € c2| “a o,(+) < ¥(S)} x

PES, 44(*)1 < 4,(S) < 3,(S) for every $1 (+) € {$4 € C7} ¥(S) < $1 (>) < »}

This proves claim 3.

Proof of Proposition 2.

Suppose not for some Se S. Then there exists a sequence [e}(+)s PLS, 6] (+)]} where of © (6, © c°|e%ax" cro (+) < 3y(+)} and gf > 6 € {$y € C24 xt y,H < $1 (+) < o,(-)} such that Pfs, 4° (°)] > ot yh V fo, (+) evaluated at a strictly positive finite function is finite; hence this is a contradiction. |

Suppose inf $4 (S) = oly then there exists a sequence s" 5° such that for some ¢,(+) (0, © C7las(+) < o)(+) < 3:(+)}, PLS", oN(s)] > prs?, e7(°)] > yd % Gs closed; hence $° « & and this is a contradiction.

Proof of Proposition 5. Convexity is obvious.

I now show o,; i = 1, 2, is equicontinuous. Applying the mean value theorem to each component of S, and using the bounds on the partial derivatives of O53 gives 4 {5

Jo,(S°) - 6,(8')| <M 2!

- st j=l j j

- 31 -

_ sup _ .

<M j | ss - s; I for every 5°, sl e S, ¢; € oi I 50. s) ih < 6 implies sup, 0 gl

j | Sj - S3 | < 8; hence choose 6 = €/(4M). Then for every e > 0,

1

1s? - si < 6 implies lo, (8°) - 6,(S")| < e, for every 5°, S' e 8, o, € O.-

i Thus , is equicontinuous and the proof is complete.

Proof of Proposition 9.

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

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

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

1 4! ' ‘ ea fe vt ysy - tol yy a (otut ) aS; aS; $05") a, 30, 8) ' tyyF tel F' F. F! 3. OS. (ST) . 00, a5, a5, $4 S 85 98. 85 984 F, yr! (Liem. 85 TF where v4 and uf are V and UV evaluated home levels of consumption and yF and

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

H enue ) |

1

"9 } and | a ( i Lo 0

ja (tut) [= wt ey 6° 2

1

- 32 -

= a (ut + eu TT ) (are continuous functions) on 8; hence they reach 8 8 1 1

their minima on $. Prices are strictly positive and finite: hence

—n

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

than zero. Denote the smallest of the minima M:.

1 5 ty au" - ttyl ed (Al) | ssa) = (———_——__ » 0», - ~4—— , 9) S' 8) 0. 1 wFe uh e,UY = tPe_yr” FY F (A.2) (4+) | = (0 oe ne 2 0 U_t )

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 $. Choose M, to be the largest such bound

A t H! ' aA ; iy a ef ct vA sy = egt_ LoS") - F(S)] |s} a3, o,(S") o,(S")

-{ ag(s) / g(S)] Ef thyh Is], [ ag(s) / 9(S)] € | sist |S]

- 33 -

All terms are continuous functtons of S. 9(S) reaches its minima on $ and is strictly positive at the minima, and 29(S) and ag(S.S') are as as

continuous, and hence, bounded.

af (a2y" ) x, E(aut ) + E(azur ) 35, E(at' ) 385 E(au )E[ YU - zu ] + €(acu elu +z U Ne

7 o9(S') = £05) where A = aan To? which is also uniformly bounded,

hence

C)

| Fe Serer [51 | is bounded from above. Similarly

FY pes yyF! 1

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

F, F 6U t 6 F" F Ft a (* “2 = 2 (U1 e,+U 6,), 1

which is bounded from above by some finite Me

2 34 -

‘Then | a6) | Mot M3 _ ql por as < a J J 1 | 2 : (1 + My) M, + M3) 2

Proof of Proposition 10. ca tas.e) Let {o"} ® where 6” > 6” uniformly. Note that is not closed; hence 9° may not be in 6. I show ($") > (6°) uniformly.

For every Se S, pe [¢,(S)s 6, (S)], let

HH! H' HH! ‘ (v(p) = tu -€E[ tr Vv S]), p 6,0 ) |

where U” is evaluated at p and V"is evaluated at 9°(S").

: a , a5 ' my ‘ c — v"(p) = yt E[ 4 vi |s}, n ' P oy (S') where Ty is evaluated at p and vi is evaluated at o"(S').

Let p" be the unique p such that. y"(p") = 0, p” £4, (S}s $,(S)I

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

subsequence. W.1.0.g., p” > p?,

- 35.

Claim 1. |¥(p”)| = Proof of Claim 1.

Bs) et n p

p

| zv'(p") - Efe vce") |s]] +

p" 6," (S')

Fees “Ste

65 (S')

- | a - Mem) | + | et shyt") [sy - ef xMivece%) |s]].

ni a (S*)

thu is continuous in p on (4, (S), 3, (Ss hence for every ¢/2 > 0, there D or a es pas exists 6 > 0, such that

| ur”) _ tut ip) | ‘§ Lo 7 :

p Pp

for every \p° - p" | <6. p" > p”; hence for every 6 > 0, there exists N

such that ip” - p"| < 6 for every n > Nye E[V ach )t H js] is continuous in $1 on {o) e C 16) (*) < 1 (*) < O° )}; $5 (S')

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

Jef su Go) |} - ef ey” (oly) < e/2

of (S') 61 (5)

- 36 -

for every | o'(s) - 6+) «6, on > ¢” uniformly; hence for every 6 > 0,

there exists Ny such that [lerc-) - oy] [<5

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

Thus there exist functions

3 Mt

ey (S)

oy (8)

mo] tt

where 8 (+) > ey (+) pointwise. D0 <M, j= 1, ..., 45 hence {ey} is

equicontinuous. Thus oh > ay uniformly. This completes the proof of claim 1. | The argument for 8, is simiar. Thus is continuous. The

preceding argument also showed (0,", 05”) =M(¢") is well-defined,

- 37 - FOOTNOTES

1/ 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).

7/ Actually elements of a are pricing functions of the good weighted by

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

2 by Mut The term “pricing function" will be used for converience,

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

- 38 - KEFEREWCES

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Bilson, J.F.6., "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, "Internationa! Interest-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 61 (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).

- 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.