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Working Paver 8908
INTERVENTION AND THE RISK PREMIUM
IN FOREIGN EXCHANGE RATES
by William P. Osterberg
William P. Osterberg is an economist at
the Federal Reserve Bank of Cleveland. .
Working papers of the Federal Reserve Bank
of Cleveland are preliminary materials
circulated to stimulate discussion and
critical comment. The views stated herein
are those of the author and not necessarily
those of the Federal Reserve Bank of
Cleveland or of the Board of Governors
of the Federal Reserve System.
August 1989
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ABSTRACT
The shift from a fixed-exchange-rate regime to a flexible regime, in
which central-bank exchange-market intervention has been hlghly visible, has
renewed interest in studying the effects of intervention. In separate work
started by Engle (1982), new techniques have been developed to analyze risk
premia in asset returns and particularly in exchange rates. We utilize a
framework developed by Hodrick (1989) to show how central-bank intervention
can affect both the level of exchange rates and the risk premium. We assume
specific f o m s for preferences and for the stochastic processes of the
exogenous variables and show how the risk premium is related to the
conditional variances of intervention and the other exogenous processes. This
approach differs from previous analyses of intervention by explicitly relating
intervention to the risk premium. This lays the groundwork for future tests
of the theory's implications for the intervention/ris<kpremium relati~~nship.
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I. Introduction
Central-bank intervention in exchange markets has increased
markedly since 1985, renewing interest among economists in understanding the
effects of this activity. Although the current regime is ostensibly one in
which rates are permitted to float, central banks commonly intervene to
influence the level of exchange rates as well as to reduce the rates'
volatility. Continued intervention is based on the belief that such actions
indeed have the desired effect.
A more general interest in discerning the effects of intervention
results from the potential significance of this activity as a policy
instrument. If sterilized intervention (intervention that has no impact on
monetary policy) can influence exchange rates, then policymakers have a third
instrument (in addition to monetary and fiscal policy) with which to achieve
their targets.
Determining the effectiveness of intervention also has implications
for other policies. If bonds that differ only in currency denomination are
perfect substitutes for one another, then intervention may be ineffective.
However, this may imply that fiscal policy would be ineffective in a small,
open economy with floating exchange rates (Siebert
[1989]).
Intervention may influence the risk premium in exchange rates as well
as the level of exchange rates. Although reducing exchange-rate volatility is
a somewhat different objective than influencing the level of exchange rates,
intervention for this purpose may indirectly influence the level of exchange
rates, because changes in volatility may influence the risk premium that
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investors require in their return on foreign exchange.
Most recent studies of exchange-rate determination give the risk
premium a prominent role. This can be traced partly to the failure of earlier
theories that did not explicitly consider risk. The presence of a risk
premium can explain a divergence of the rates of return between domestic and
foreign assets, measured in the same currency (that is, a violation of
uncovered interest parity).
As a result of such findings, we now have theories to explain how such
a risk premium could arise.
In addition, largely as a result of the work of
Engle (for example [ 1 9 8 2 ] ) , new techniques are now available to analyze time
variation in conditional variances. Conditional variances may be closely tied
to perceptions of future volatility and, thus, risk.
11. Channels of Influence in Central-Bank Intervention
To understand the mechanics of a typical spot-market intervention,
consider a transaction designed to offset a dollar depreciation. In this
case, the Federal Reserve would purchase dollars for marks on the spot
market from a commercial bank. This would typically give the Federal Reserve
two business days for delivery of marks. To finance the transaction, the
Federal Reserve would sell mark securities held in accounts with the
Bundesbank. The Bundesbank would act as the agent for the Federal Resenre,
establishing an account for the U.S. central bank with the proceeds of the
security transactions. The Federal Reserve would then settle the spot
transaction with the commercial bank by drawing on its account with the
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Bundesbank. The net effect is to decrease U.S. reserves and the monetary
base.
Then, in order to sterilize the intervention(that is, offset its impact
on reserves), the Federal Reserve may sell the equivalent amount of U.S.
government securities, leaving as the only net effect of the two transactions
a change in the Federal Reserve's and the private sector's portfolios of
domestic and foreign assets. If the initial transaction is not sterilized,
then it is equivalent to an open market operation. Since the impact of open
market operations is presumably better understood than the impact of
intervention, most studies of intervention focus on sterilized interventions.
Sterilized intervention could matter if the currency composition of
debt influenced the exchange rate. In the portfolio-balance approach,
exchange rates are determined by expected nominal rates of return on debt of
different currency denominations. If investors care about portfolio risk and
expected rates of return, and if bonds of different denominations are
imperfect substitutes, then shifts in asset supplies will alter portfolio risk
and induce changes in rates of return and in the exchange rate. This was the
predominant approach to analyzing the effects of intervention in the 1970s.
Even if foreign and domestic assets are imperfect substitutes,
intervention may not matter under Ricardian equivalence (see Obstfeld [1982]).
In that case, agents do not regard the government bond holdings as part of net
wealth, and fully capitalize future tax effects, neutralizing the impact of
intervention. Backus and Kehoe (1988) emphasize the key role played by the
government budget constraint in analyses of intervention. If other government
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policies are changed, then the impact of the overall operation depends on the
structure of the economy and on the exact nature of the policy change.
However, under Ricardian equivalence, exchange rates are unaffected by
intervention if lump-sum taxes are levied on the representative consumer.
Another channel through which intervention may matter is its effect on
expectations of economic conditions or policies. In particular, intervention
may provide a credible signal of changes in future monetary and/or fiscal
policies.
Exactly why intervention would be chosen as the signal is unclear.
However, once the central bank has intervened, it may stand to lose money by
not following through on the expected policy. For example, if the U.S. central
bank purchases dollar-denominated bonds and sells foreign currency bonds to
signal its intention to allow the price of dollars to rise, it has an
incentive to increase the price of dollars and thus the value of its holdings.
Recent research analyzing other possible incentive effects of central-bank
intervention is sumnarized by Obstfeld (1989a).
111. Does Intervention Matter?
Most empirical studies conclude that intervention does not influence
exchange rates. Many of these studies indirectly examine the influence of
intervention by testing the hypothesis of perfect substitutability of bonds
that differ in currency denomination. The usual technique is to regress
either exchange rates or the difference between the rates of return on foreign
and domestic bonds (the covered-interest parity condition) on measures such as
relative supplies of debt denominated in different currencies. Numerous
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studies, summarized by Weber (1986) and Henderson (1984), include asset
supplies as explanatory variables and find evidence against imperfect
substitutability. On the other hand, Danker et al. (1985), Loopesko (1984),
and Johnson (1988) find evidence for imperfect substitutability. However,
little of the variation in the dependent variable can be explained by relative
debt supplies. This, in turn, implies that intervention is not likely to have
much impact, since it is small relative to the debt aggregates.
The previous discussion of the role of the government budget
constraint and the tenuous link between perfect substitutability and the
effects of intervention should make us cautious in interpreting these results.
Without having specified and controlled for possible effects operating through
the budget constraint, these empirical studies may be misspecified.
Recent investigations have implied a role for intervention as a
signal. Domingues (1988) finds that U.S. intervention has played a role in
signaling changes in monetary policy, but that the effectiveness of
intervention depends on the credibility of the monetary policy. When actual
and announced monetary policies are inconsistent, intervention may be used to
send a false signal to the market. Thus, intervention should be considered
part of overall monetary policy. Humpage (1988) finds that intervention has
an initial, one-time impact if it is supported by consistent statements of
changes in monetary and fiscal policy and by coordinated action of central
banks.
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There is some evidence that Canadian central-bank intervention has
systematically reduced short-run exchange-rate fluctuations(Pippenger and
Phillips [1973]). However, this conclusion is disputed by Sweeney (1981).
IV. Risk in Exchange Rates
Evidence
A wide variety of evidence suggests that there is a risk premium
component to exchange rates (see Hodrick [1987]).
Violation of the uncovered-
interest parity condition (expected profits to forward speculation should be
zero) and the poor out-of-sample predictive performance of log-linear
exchange-rate models relying on first moments suggest a risk premium.
However, evidence of a risk premium has been synonymous with the failure of
previous theories of exchange-rate determination. Not all investigators are
convinced that a risk premium exists (for example, Froot and Frankel [1989]).
Expectational errors may explain the above anomalies. Tests of the parity
condition involve the joint hypothesis of market efficiency, perfect
substitution, and capital mobility. Such considerations further complicate
interpretation of the results.
Many empirical investigations into the risk premium in
foreign-exchange rates model risk with time variation in conditional variance
using Autoregressive Conditional Heteroscedasticity (ARCH).
Useful
discussions of this literature are found in Hodrick (1987) and Frankel(1989).
Pagan and Hong (1988) and Nelson (1987) question the appropriateness of the
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ARCH formulation. Other investigators(for example, Lyons [1988]) extract
variances implied by options-pricing formulas and find time variation in
"risk." However, the significance of the magnitude and time variation in the
risk premium is unclear.
Theory
Exchange rates have been at various times viewed as the relative
prices of currencies, the relative prices of domestic versus foreign goods,
and the relative price of assets denominated in different currencies.
However, as Dornbusch (1985) states, "...it becomes readily apparent that in
most instances real, monetary, and financial considerations interact in the
determination of exchange rates."
In models of the risk premium that incorporate optimization and
equilibrium behavior under uncertainty, the risk premium will depend on the
risk preferences of the consumers, on other parameters of the model, and on
the stochastic properties of exogenous variables such as money. Lucas (1982)
and Siebert (1989) present contrasting theoretical approaches to the
determination of exchange rates in general equilibrium under uncertainty.
Tests of theoretical models of the risk premium are growing in number.
In international capital asset pricing models of mean-variance optimizing
consumers, time variation in risk should be related to time variation in the
covariance matrix of asset returns. Examples of this approach are Engel and
Rodrigues (1987), Giovannini and Jorion (1989), and Mark (1988).
Hodrick
(1989), Cumby (1988), and Obstfeld (1989b) test consumption-based asset
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pricing models in which the risk premium is related to time variation in the
stochastic processes of the exogenous variables, including money. Both
approaches have had limited success in explaining risk premia.
The role of intervention in explaining foreign exchange risk is
largely unexplored. One reason may be that early investigations focused on
the ability of debt variables to explain the deviation from interest-rate
parity, with that deviation being a measure of risk. However, there is
evidence that the volatility of exchange rates has varied across monetary
policy regimes (Lastrapes [1989]) and that the impact of intervention is
related to monetary policy (Domingues
[I9881 and Humpage [1988]).
V. The Model
The theoretical model we present provides testable hypotheses about
the influence of intervention on the risk premium in foreign exchange rates.
The consumption-based asset pricing model of Hodrick (1989) is modified for
this task. In his model, the risk premium in the exchange rate is a function
of the conditional variances of money, government's share in production, and
production itself. Simplifying assumptions about preferences and about the
stochastic properties of exogenous variables are necessary in order to derive
closed-form solutions indicating the relations among the exchange rate, the
risk premium, and the first and second moments of the exogenous pr0cesses.l
Without such assumptions, it is difficult to say much about the likely impacts
of intervention on the risk p r e m i ~ m . ~
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Our model differs from Hodrick's mainly by including intervention. In
Hodrick's model, consumers and governments each face cash-in-advance (CIA)
constraints, and the total stock of each currency is split between private and
governmental holdings. We model intervention in terms of governments' holding
of foreign currencies. Intervention is actually variation in the stock held,
influencing the amount of currency available for private or government
consumption. In Hodrick's model, the variability, as well as the level, of
private money influences exchange rates and the risk premium. Thus, in our
model, the level, as well as the variability, of intervention influences the
rate and its risk premium. In effect, knowledge of the stochastic process
describing intervention improves the ability of monetary aggregates to predict
exchange rates.
Endowments and Timing
Two countries, indicated with subscripts 1 and 2, each produce one
good, which is also the endowment of each country. The realizations of the
two exogenous, nonstorable goods are denoted Y1, and Yzt. We assume that
the goods markets are open at the start of the period and that asset markets
are open at the end. It is convenient to think of each household as comprised
of two agents, one that takes the accumulated cash out for shopping, and
another that subsequently enters the asset market to purchase cash, bonds, and
equities.
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Information about the state of the world (detailed below) becomes
available at the start of the period. The government and the private shopper
enter the goods market with available cash balances. The government's cash
balance can be augmented through new currency issue and is also influenced by
intervention. Any remaining cash balances, in addition to the gross returns
on bonds and stocks, become available to the consumer for the subsequent asset
markets. Lump-sum taxes or transfers are also levied in the second half of
the period.
Government
Each government purchases some of the endowment of its own country,
collects lump-sum taxes, supplies state-contingent claims to its own currency,
prints its own currency, and intervenes in the foreign exchange market by
purchasing some of the foreign currency. The real q,uantityof government i p s
purchases of good i at time t is Gi,.
Because consumers do not value
government spending, variation in Git affects the amount of the endowment
available for consumption. rib is the lump-sum tax levied by government i
in the asset market.
Bit+l(xt+l)is the amount of money i that government i
promises to pay if state x,+~ occurs. Its currency i value at time t in state
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xt is ni(xt+,,xt).
The gross growth rate of money i over period t, Mit+JMit,
is denoted nit. The outstanding amount of money i and the amount held by
:,
the foreign government at the end of period t-1 are denoted Mit and M
respectively. Nominal government purchases of endowment i in the time t goods
market are constrained by the government's holding of currency i cash balances
at the start of period t,
q,,,
plus any additional currency i to be supplied.
In Hodrick (1989), the additional amount represents the amount printed
by government i and supplied in the asset market. Here, however, governments
purchase foreign currency and do not spend it. So, the additional amount of
currency i to be made available is the amount printed net of the increase in
foreign holdings of the currency. This CIA constraint can be expressed as
The holdings of the foreign currency have no effect other than to reduce the
amount of currency available to purchase foreign goods. For simplicity, we
ignore any effect of govemment earnings on foreign reserves.
Expression (2) is the government budget constraint.
(2)
'
i
t
= 'it
+
J
ni(xt+l,xt)Bit+l(xt+l)&t+1
Pit
-
Bit(xt)
+
,+ti'(
Pit
Mit)
, i=1,2,
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12
where Pit is price in currency i of the good/endowment of country i.
Agents' Preferences and Constraints
Following Hodrick (1989), we assume that all agents' preferences are
homothetic and, thus, that there is a representative consumer in each country.
Preferences and initial wealth levels of the two consumers are assumed to be
identical and each consumer is taxed equally by the two countries. Each
representative consumer maximizes expected lifetime utility as in
by choosing C1, and C2, and by making her savings decisions.
The consumer in each country faces two constraints: a CIA constraint
and a budget constraint. The CIA constraint, expressed in real terms, shows
that purchases of good i are constrained to be no greater than the amount of
currency i held by the consumer when she enters the goods market:
(4)
cl, 5 ~~,,n,,,
(5) @,C,,
5
M ~ ~ ~ $ ~ .
Here IIlt = l/P1, is the good one purchasing power of currency one, and I12t = St/Pl,
is the good one purchasing power of currency two. St is the exchange rate
of currency one per unit of currency two, and 8, = StP2,/P1, is a "real terms of
trade," although goods cannot be exchanged directly in the model. Note also
that monies cannot be exchanged directly in the goods markets.
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Purchases of assets are constrained by the agent's wealth at the time
she enters the asset market, after having made her consumption choices, net of
taxes. Wealth includes unspent monies, realizations on previous purchases of
state-contingent bonds, and realizations on equity shares. Agents in each
country can buy and trade titles to the endowments of each country. The
number of titles to the endowment of country i purchased in the asset market
The associated currency one price is denoted Qit.
at time t is denoted Zit+1.
For convenience, we assume that there is just one share of the endowment for
each country. The period t budget constraint, identical to Hodrick(1989) , is
reproduced here:
(6)
n l t ~ ~ l t ++l %t%t+l
+
~ 1 S n(xt+1,
1
xt)Btlt+, (xt+,)dxt+,
+ ~~2J%~(
'~t)%t+l
~t+l(~t+l) %+l
+
$ltZ1t+l
+ +ztZzt+l "
+
($lt+Y1t)Z1t
+ (+2t*tY2t)Z2t - (1/2) (rltMtrzt) , where
(nlt~tlt-cl,) + (%t%t
-
@tCzt)
+
~~lt~:t(xt) +nzt~~2t(xt)
$it
' Qit/Plt.
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Agent's Solutions
The agent chooses consumption of both goods, holdings of both
currencies, state-contingent claims to both currencies, and titles to both
endowments. The future states of the world are uncertain to the consumers,
but there is a known, first-order Markov density, F(X,+~~X,), between the
states of the world at times t and t+l. Utility maximization is subject to
the wealth constraint and the two CIA constraints. The optimality conditions,
listed in appendix A, are identical to those in Hodrick(1989).
The marginal utility of consumption is not necessarily equated to the
t
marginal value of wealth unless the CIA constraint is assumed binding. The
choice of money holding will equate the current real value of wealth to the
expected marginal utility of money in the next period, which will depend on
the marginal values of wealth and money then. The Euler equations for the
nonmoney assets differ from those for money, since bonds and stocks provide no
return until consumption in the next period has occurred.
Equilibrium
The definition of the equilibrium is identical to Hodrick but for the
inclusion of intervention as an additional exogenous process. The equilibrium
i=1,2), the
is defined as the initial stocks of monies and bonds (Mio,Bio,
stochasic processes for the exogenous variables (Yit
,Git
, T~~
'it+l
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Mi,+, , i=1,2,t=O to
t=O to
a),
a),
P
choice variables (C,, ,Mit+,,
B:,+,,
the prices (llit,8,,$,,,i=l,2,t=O to
a),
Zit+,
, i=1,2,
and the pricing functions
i=1,2 such that 1) budget constraints are satisfied, 2) the
ni(x,+l,x,),
household's decisions solve the maximization problem, and 3) the following
market-clearing conditions are satisfied:
(7b) Bit+,(xt+,)
=
(7c) 2Ci, + G,,
=
(7d) Mi,+,
=
M:,+
~B~,+,(x~+,), i = 1'2,
Y,,, i = 1,2, and
+
+
MY,,,,
i=1,2.
Closed-Form Solutions
In order to show explicitly how intervention can influence the
exchange rate and the risk premium in the exchange rate, we assume particular
stochastic processes for the exogenous variables. We follow Hodrick regarding
the assumed processes, noting the key role played by assumptions about the
stochastic independence of exogenous variables. Hodrick examines variation in
government's share of output as an independent exogenous variable. Government
expenditures influence the amount of output available for consumption.
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Below we define the relevant variable as consumption's share of
output, which, given the assumptions of the theory, just equals one minus the
government's share.5 Lower-case letters denote logarithms, and wit+, denotes
We assume
the logarithm of the gross growth rate of currency i , nit+l=Mit+2/Mit+l.
conditional log-normality for outputs and gross money-growth rates.
define the proportion of currency i held by the foreign government by
rit
=
F
Mit/Mit
and assume that the ritsand the consumption shares
xit
(defined as [Yit-Git]
pit)
are conditionally uniform in distribution.
Formally, these assumptions are
(8a) Ylt+l = Ply,, + (~-P,)Y,+
El,+,,
(8b) yzt+,= p2yzt + (~-P,)Y~+
Jzt+,,
( 8 ~ ) wit+,
=
P3Wlt+ (1-P3)w1 +
J3t+l,
(8d) w,,+,
=
P4WZt+ (1-p4)w2 +
+4t+l'
(8e) XI,+, - P5Xlt+ ('-P,)X,
(8f)
Xzt+,
+ <5t+l'
- PBXzt+ (1-pc,)x2+
<6t+l,
we
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(8g)
C1t+l
=
p7Clt + ('-p7)C1
+
(8h) C2t+l
=
p8Czt + ('-p8)C2
+
where 0 s lpil s
f7t+l
1
I, i-1 to 8, and each
<it+l,
i-1 to 4 is normally distributed
with conditional mean equal to zero and conditional variance denoted hit.
However, each fit+l,
i=5 to 8 is distributed uniformly on the interval [-hit
,hit]
with conditional mean of zero but conditional variance given by (hit12/3. We
also assume that the fit+lsare independent of each other. The conditional
variances are described by the following autoregressive processes:
(9) Et(hit+l)= dihit+ (1-di)hi,i=1,2,3,4.
Here the term on the left-hand side is just E~[E~+~(~~,:)],
and the his are
the unconditional variances. The conditional and unconditional variances of
both the foreign money shares and the consumption shares are denoted (hit)'/3
and (hi)'/3,
respectively. The state of the economy, xt, is defined as
(yit
,mit+l
,wit,xit,
Cit+l,rit,hjt,i=l,2, j=l,8, t=O to -1,
and the rit and
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x, vectors are
Markov processes.
As in Hodrick (1989), we assume the following utility function:
(10) U(Clt ,Czt)
=
[l/(l-7)
1~i-T+
[l/(l-6) ]~f-t.
Here we have assumed constant relative risk aversion. The magnitude of the
parameter of risk aversion (which is also equal to the parameter expressing
intertemporal substitution) will influence the response of prices such as the
exchange rate to shocks from processes such as intervention.
In addition, we assume that the CIA constraints hold with equality,
implying constant unitary velocity of money.'
However, Hodrick,
Kocherlakota, and Lucas (1989) indicate that relaxing the constraint is not
likely to alter velocity greatly. When combined with market clearing, the
binding constraints imply the following key relations:
(11)
n,,
=
Yit/[M1t+l(l-r,t+,)I
(12)
n,,
=
~,Y2,/[~,+,(~-r,,+,~1.
9
Here, since endowments must be consumed, changes in end-of-period-t
foreign holdings of currency one impact the price of good one in that period
by reducing money available for purchases, given the total available, MI,+,.
Although set in the goods market before the money is injected, the goods price
is influenced by intervention, since the government's purchases indicate the
amount of money (net of the amount absorbed by the foreign government) that
the government must inject into the asset market.
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St, the spot market currency one price of currency two, can be
expressed as
Use of the optimality conditions yields the general form of St:
Assuming that money (net of intervention) is independent of the growth
rate of money (net of intervention) and the other variables in (14) yields
expression (15) for the natural logarithm of the exchange rate
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Expression(15) shows clearly that increases in
in
rlt+, depreciate currency one
or decreases
(St is the currency one price of currency two).
Either way, the purchasing power of currency one falls.
The effect of a
higher endowment of good one depends on the parameter 7 , which indicates
intertemporal substitutability. An increase in the endowment of good one will
increase the value of currency one, since cash must be accumulated in advance
of purchases. An increase in the expected foreign holdings of currency one in
the next period will reduce the amount expected to be available for purchases,
increase its future expected value, and thus induce increased demand now,
leading to appreciation of currency one.
To arrive at an expression for the logarithm of the exchange rate in
terms of observable variables and conditional variances, we utilize the
distributions of the exogenous processes and assume that the Mit+lsand the
s are independent and known at time t.& In addition, we replace
rit+l
ln(1-c.l t +.~) by its first-order approximation,
-cit+j, to yield expression (16)
The theoretical values of the coefficients in (16) are given in appendix B.
.9
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-
2
aS13clt+l + aS14c2t+l + aslS(h7t+l)
-
ag16(h8t+l)2-
Here we define El, as ln( Et[ x ~ ~ +) ~
and
- ~Zzt
] as ln(Et [x,~+,-~]
).
In expression (16) there are multiple channels through which current
monetary conditions influence the exchange rate. An increase in either money
supply (Mit+l)directly affects st and provides information about future
money, since the logs of the gross growth rates of money are autocorrelated.
An increase in the conditional variance of the endowment for good one,
(hit),
will increase the value of currency one to the extent that consumers are
risk-averse. An increase in the conditional variance of the growth rate of
currency one causes it to appreciate, since the conditional variance
influences expectations of future purchasing power. The intervention
variables,
rit+l, do not have the one-for-one influence of the money stock,
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because they also impact the expected growth rates of money available for
purchases. Conditional variance in intervention helps predict variability in
the purchasing power of money, since the endowment must be consumed in
equilibrium.
Intervention and the Risk Premium in the Exchange Rate
A n expression for the risk premium can be developed from the
interest-rate parity condition, expressed in equation (17).
Arbitrage implies
equality between the rates of return on investing currency one in bonds of
country one, then converting to currency two and investing in country one
bonds, and then selling the proceeds forward.
Ft is the forward price at time t of delivery and payment in time t+l. A
commonly studied expression for the risk premium is Et(st+l)-ft, which (17)
implies is equal to E,(S,+~-S,)
- (ilt-izt)
. lo
Expression (18), derived
from the optimality conditions, yields the interest rate in country one:
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Assuming independence between total money supplies, intervention variables,
and endowment processes and taking logarithms of both sides, we can derive
expression (19).
Utilizing the assumed stochastic processes of the exogenous variables,
we arrive at expression (20) for the interest rate in country one. The
theoretical values of the coefficients are presented in appendix B.
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(20) iit =
ail^
+
An
+
aillElt
ailshlt
+
ai12(1nEt.[E1t+l-7~ +
413(~lt-~1)
+ ai14(~1t-w1)
2
+ ai16h3t
increase in y
+
ai17(Clt+l-Cl)
+
~ile(h7~)-
It increases the interest rate in country one if p1
and y are between 0 and 1. The increased demand for money will increase
the current purchasing power of money. However, the endowment will return
toward its unconditional mean, and the purchasing power of money will fall in
the next period. This increase in expected inflation increases ilt. However,
the increase in current consumption decreases current marginal utility and
leads to intertemporal substitution, which may amplify or reduce this effect.
An above-average money growth rate will be followed by another increase in the
money supply (although a smaller increase in the growth rate) and thus an
increase in expected inflation. An increase in intervention in currency one
(increased foreign holding of that currency) increases the purchasing power of
the remaining currency one, but will be followed by a decrease in purchasing
power as, in the next period,
Clt+, declines towards its average. Unless
swamped by intertemporal substitution, an increase in the conditional variance
of good one increases ilt. Risk-averse consumers would desire to hold
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less of currency one, since good one is more risky. Thus, the purchasing
power of currency one rises but is anticipated to fall in the next period.
Increased variance of the intervention in currency one increases the interest
rate because it implies that the purchasing power of that currency is likely
to fall.
Utilizing the analogous expression for iZt and an updated version of
st+l,we derive the expression (21) for the risk premium. The theoretical values
of the coefficients are found in appendix B.
21)
tt+l-ft =
-
arlhl,- ar2h2t + ar3h3t
a
r
+
+ ar7 (h7t+1)2
~
l
-
t
+
ar4h4t
7 + ar6(~t'2t+l-1n[Et(~2t+226)
2
ar8 (h8t+1)2
-
1)
'
If the conditional variances of both endowments increase by the same
amount, the risk premium is unaffected if pl
=
p2. Analogous statements
can be made for the conditional variances of money-growth rates. The extent
to which equal changes in conditional variances offset one another depends on
the extent to which such changes are expected to be propagated into the
future. Increasing the conditional variance of foreign holdings of currency
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one increases the risk premium, here defined in terms of currency one/currency
two. Increasing the other conditional variance has the opposite effect.
However, if the conditional variances of both intervention variables increase
and are propagated equally into the future, there is no effect on the risk
premium. Expression (21) makes clear the need to distinguish between the
variation in total money supplies and the components.
VI. Conclusion
In this paper we have modified a model developed by Hodrick (1989) to
show how intervention can influence the foreign-exchange risk premium. Unlike
previous studies of intervention, we specify the mechanism through which
intervention should impact the risk premium in exchange rates. While previous
studies of intervention have analyzed sterilized intervention, here we model
intervention as changes in foreign governments' holdings of domestic currency.
The proportion of currency held by the foreign government as well as the
conditional variance of that proportion can influence the level of the
exchange rate. The risk premium is shown to be a function of the conditional
variance of the intervention variable as well as the conditional variances of
the other exogenous variables, including the total money supplies. Future
work will test the theory's implications for the intervention/risk premium
relationship.
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Footnotes
1. See Siebert (1989) for an example of an analysis of the determinants of
the risk premium that avoids parameterization of preferences and distributions
of the exogenous variables.
2. Of course, the assumptions may be inappropriate for the application at
hand. Pagan and Hong (1988) discuss problems with the ARCH formulation as
Cumby (1988) cites the assumption of
employed by Hodrick(1989).
time-separability as a possible explanation of the failure of one particular
version of the consumption-based asset pricing model to explain risk premia in
forward speculation.
3. Here we do not assume sterilization. Leahy (1989) discusses the
significance of earnings on foreign reserves, indicating that such earnings
are not large enough to have much of an impact. In any case, the effects of
the disposition of such earnings involve issues similar to those raised
regarding the impact of portfolio balance effects.
4. See Stockman and Svensson (1987), p. 183 for the solution of a similar
model when currencies can be exchanged directly in the "goods" market.
5 . Of course, one can argue that these shares are not independent of
overall output. However, it may be of interest to follow other empirical work
and to examine the relation between variation in consumption and exchange
rates (for example, Cumby [I9881) .
6. Pagan and Hong (1988) claim that assuming linearity in the conditional
mean exaggerates the true volatility in such series. They claim that
nonparametric estimation of the conditional mean and conditional variance
implies different results. Diebold and Nason (1989) argue that it is unlikely
that out-of-sample predictive performance for exchange rates will be improved
by taking advantage of nonlinearities in conditional means.
7. See Stockman and Svensson (1987), p. 175 for a discussion of how
assumptions about the timing of information alters this result in related
models.
are known at the
8. The assumption that both the Mit+lsand the
start of period t is unnecessary to yield a closed-form solution. A binding
CIA constraint implies only that Mi,+l(l-(it+l) is known at the start of the period.
However, agents would presumably make use of their knowledge of this net
amount in forming their expectations of money variables dated t+2.
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9. Although the approximation error involved here may be "small" it may
have a large effect on the estimates of conditional variances. Together with
footnotes 6 and 8, this highlights the crucial role that must be played by
parameterization of the expectational terms in expression(15).
10. Derivation of a similar expression for the risk premium, Et(St+l)-Ft,
is discussed in Hodrick (1987), pp. 13-15.
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Appendix A
The first-order conditions for the agents' problem flow from the value
function
(Al) V(W, ,II1,M;,, II,,M;,
+
,x,)
=
max
U(C1,, Czt,
(
BS(wt+l ?nit+l~;t+~
9nzt+i%t+i
,~t+i)F(xt+lI xt)%+1
9
where wealth, W,, is defined as
(A21 W,
=
~,,M'I,
+
n,,~;,
+ nl,~;,(x,)
+ n,,~;,(x,)
+
(Ill,+Y1,)Zl, + (~,,+BtYzt)Zzt.
Maximization is with respect to private consumption and choices of
money holdings and holdings of bonds and equities. The actual transition
probability is assumed to be known. If A, is the multiplier for the
period-t budget constraint facing the consumer, ult is the multiplier for
the period-t currency one CIA constraint, and v,, is the multiplier
for the currency two CIA constraint, then the first-order conditions are
described by (A3) through
(A10):
(A31 U1, = At + vlt,
(A41
u,,
=
0,+
~,,)9,,
(A51 XtIIlt = BE, [ (A,+1
+ v1,+1)n1,+,
I
9
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(A61 A,$,
=
BE,[(X,+,
+ ~,,+,)~,,+,l'
(A7) A,$,,
=
BE,
+ Y1,+,)
(A81
=
BE, [ ($2t+l
(A9) X,nl,nl (x,+,
9
x,)
X , I I ~ , ~ , ( X ~ x,)
+~
9
+
=
I'
et+ly2t+l)Xt+ll '
Bx,+lnl,+lF(x,+ll x,) , v
=
PX,+ln,,+lF(x,+,
ix,)
9
X,+l,
v X,+l.
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Appendix B
The t h e o r e t i c a l values of t h e c o e f f i c i e n t s i n expression (16) a r e
a~~
(1-pa)C2
=
-
(l-p7)C1 + ( 1 - 6 ) ( 1 - ~ 2 ) ~-2 ( I - 7 ) (l-p7)y1 + (l-p3Iwl
-
(1-~4)w29
aSl - as2.= aS3= as4 = 1,
a,,
=
(1-7h1,
ass = ( 1 - 6 ) ~ ~ '
as7
ass
-P39
=
Pq 9
=
(1-~>~/2,
aslo= (1-612/2,
asll
as13
-
-
Q,12 =
1/29
p7'
Pa'
as14
aSlS= aSl6= 1/6.
The t h e o r e t i c a l values of the c o e f f i c i e n t s f o r expression (20) a r e
-1nS +
ail,
=
aill
= Pi12 =
ails
=
-
wl
1
- -
~
9
(1-7)P1 ( ~ ~ - 1 )
9
~
- - (l-d3)h3/2
d
~
-/ P ~ ( ~ - P , )- { (1-d7)(h713/6,
~
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The theoretical values of the coefficients in expression (21) are
arl
=
(1-7)2PI2/2,
ar2 = (1-7)2P22/2,
Qr3
=.
( 1 + ~/2,
~~)
a*,, = ( 1 + ~ ~ ~ ) / 2 ,
~ , 5=
%e = 1, ar7 = p7/6,
Pr,
pg/6 -
=
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