Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
FOREX risk premia and policy uncertainty:
a recursive utility analysis
Lynne Evans a,∗ , Turalay Kenc b,1
a
School of Economics Finance and Business, University of Durham, 23-26 Old Elvet, Durham DH1 3HY, UK
b Management School, Imperial College, 53 Prince’s Gate, Exhibition Road, London SW7 2PG, UK
Received 10 March 2002; accepted 17 February 2003
Abstract
We compare actual and calibrated values for the foreign exchange risk premium based on the
definition in [J. Int. Econ. 32 (1992) 305]. Calibrated values are found from within a dynamic stochastic
general equilibrium model of a small open economy consisting of risk averse optimizing agents with
unconventional preferences. We find that the equilibrium foreign exchange risk premium is a function
of exogenous shocks in the model and is sensitive to assumed attitudes towards risk. Furthermore,
various forms of policy uncertainty improve the capacity of the model to generate values closer to
those found in the data.
© 2003 Elsevier B.V. All rights reserved.
JEL classification: F31
Keywords: Foreign exchange risk premium; Stochastic general equilibrium models; Policy uncertainty;
Recursive utility
1. Introduction
While a considerable amount of research effort has been devoted to modelling foreign
exchange risk premia, models capable of producing risk premia which match the data are
proving elusive. In this paper, we contribute to the literature which models asset returns as
the outcome of a dynamic stochastic equilibrium (see for example, Sibert (1996) and Bekaert
et al. (1997)) and build a general equilibrium model which is suitable both for analyzing
∗
Corresponding author. Tel.: +44-191-374-7287; fax: +44-191-374-7289.
E-mail addresses: lynne.evans@durham.ac.uk (L. Evans), t.kenc@ic.ac.uk (T. Kenc).
1 Tel.: +44-207-594-9212; fax: +44-207-823-7685.
1042-4431/$ – see front matter © 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S1042-4431(03)00041-6
2
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
a number of potential determinants of the foreign exchange risk premium; and also for
carrying out a numerical analysis of the foreign exchange risk premium. Furthermore, we
seek to establish the potential role played by various forms of policy uncertainty.
Our approach is distinguished from many of those in the literature by a number of features. Firstly, we use the Engel (1992) definition to calculate both the observed foreign
exchange risk premium and the calibrated value. Secondly, our dynamic general equilibrium model incorporates portfolio choice-thereby giving rise to an integrated analysis of
exchange rate determination with a risk-adjusted PPP and portfolio equilibrium. Thirdly, the
model includes a recursive utility function that disentangles risk aversion from intertemporal substitution-thereby enabling an analysis of the distinct roles played by agents’ attitudes
towards risk and intertemporal substitution. Fourthly, we have an exact continuous-time
stochastic model rather than the discrete time stochastic approximations (through Markov
chains) more commonly adopted in the literature. Although continuous-time techniques are
more restrictive compared to discrete time techniques, they are largely favored in the finance
literature because the results are more transparent and typically more insightful than those
found from discrete time analysis. In addition, our allowance for different forms of policy
uncertainty is also distinctive.
Thus, the model builds on early examples of the general equilibrium stochastic growth
model developed by Eaton (1981) and Gertler and Grinols (1982). More recent examples include Turnovsky (1995), Grinols (1996), Smith (1996); and Turnovsky and Grinols
(1996). All of these models represent uses of stochastic calculus, they are all based on the
intertemporal optimizing behavior of a risk averse representative agent and they derive a
macroeconomic equilibrium in which both the means and variances of the relevant variables
are simultaneously determined. The model developed and used in this paper is extended
along the lines of Obstfeld (1994b) and Grinols (1996) to include a recursive utility function
which disentangles risk aversion from intertemporal substitution. In addition, we specifically model a small open economy which takes the world interest rate as given, in preference
to using a two-country model. Furthermore, policy uncertainty is captured through (i) different specifications for government expenditure shocks; and, more importantly, through
(ii) monetary policy uncertainty—time varying means as developed by Williams (1977)
and Stulz (1986).
The paper presents closed-form solutions for the model, amongst which is an expression
for the foreign exchange risk premium which reveals that not only is the risk premium a
function of the differential between domestic and foreign interest rates and certain variance
terms as in Grinols and Turnovsky (1994) but it is also a function of a range of exogenous
shocks and attitudes to risk as in Sibert (1996). Furthermore, various forms of policy uncertainty are found to improve the capacity of the model to generate values closer to those
found in the data. Significantly, unlike a common approach used in the literature, we adopt
a proper definition of the foreign exchange risk premium as set out in Engel (1992).
Our paper is organized as follows: Section 2 outlines our development of existing continuous-time stochastic endogenous growth models and presents the solution. Section 3 is
devoted to an analysis of the foreign exchange risk premium; Section 4 to monetary policy
uncertainty; and Section 5 presents a numerical analysis of the model and compares the
numerical estimates with the ‘true’ value of the foreign exchange risk premium. Section 6
concludes the paper.
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
3
2. The model
We use a representative agent model which has money in the utility function (MIUF)2 and
assumes a Rebelo (1991) ‘AK’ production function which leads to endogenous growth. The
stochastic nature of the model is characterized by four exogenous stochastic shocks. Two
of these are policy shocks, namely government expenditure shocks and monetary growth
shocks; the other two are productivity and foreign price shocks. Other shocks could be
included but this set is characteristic of the most important exogenous stochastic influences
on a small open economy. We allow for a full range of interactions across shocks. The economy specializes in the production of a single good which is assumed to be sufficiently small
in the world production to have no significant impact on its market. The behavioral nature
of the model is described by the utility maximizing and portfolio optimizing behavior of a
representative household. The paper deals with only the steady-state stochastic equilibrium
which is separated into deterministic and stochastic components. The remainder of this
section sets out the key features of the model and its solution. Further details may be found
in Evans and Kenc (2003).
2.1. Household optimization
At each point in time the representative household chooses its consumption C(t) and
allocates its portfolio of wealth, W(t), across four assets: money M, government bonds B,
capital K, and foreign bonds B∗ . Two of these assets (capital and bonds) are internationally
traded.3 The only source of income for the representative household is the capital income
received from holding these assets.
The representative agent’s intertemporal utility is given by:
U(t) = e−δt [ũ(t)ζ + e−δdt V(t + dt)ζ ]1/ζ
V(t + dt) = [Et U(t + dt)1−γ ]1/(1−γ)
M(t) 1−θ
θ
ũ(t) = C(t)
P(t)
ζ < 1, = 0; 0 < γ, = 1
(1)
1≥θ≥0
where E is the expectation operator, δ the preference rate, (1 − ζ)−1 the usual intertemporal
substitution elasticity parameter and γ the degree of relative risk aversion. This class of
2 There are alternative ways of incorporating money: the most well-known alternative is transaction cost
technologies such as a cash-in-advance constraint (Lucas, 1982; Svensson, 1985; Rebelo and Xie, 1999). There
are difficulties in imposing a cash-in-advance constraint in continuous-time. As pointed out by Rebelo and Xie
(1999) the cash-in-advance constraint in a general equilibrium model requires a simplified framework. However,
the introduction of money in the utility function is widely used (see for example Stulz, 1986; Turnovsky, 1993, and
Basak and Gallmeyer, 1999). The appeal lies not only in its tractability but also in its ability to capture money’s
role as a store of value and a medium of exchange (see Feenstra, 1986 and Danthine et al., 1987). The paper deals
with only the steady-state stochastic equilibrium which is separated into deterministic and stochastic components.
3 The assumption that some assets are nontraded does not pose any problem as long as the risk characteristics
of these nontraded assets can be replicated with those of the traded assets, i.e. nontraded assets are spanned. In
other words, markets are complete in the sense that the number of stochastic processes equals the number of traded
assets.
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L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
preferences states that lifetime utility U(t) at time t is a function of known consumption
of composite good ũ, where ũ is composed of the consumption good C and real money
balances M/P, and a discounted certainty equivalent of random utility in the next instant,
V(t, dt). Attitude towards risk is captured by this V -function and intertemporal substitution
by the functional relationship between ũ and V . This structure implies that utility satisfies
intertemporal consistency of preferences and removes the restriction that ζ + γ = 1.4 The
relative importance of money in composite consumption is measured by θ.
Utility is maximized subject to the following wealth W constraint, which in real terms is
M
B EB∗
+ +
+K
P
P
P
where E is the exchange rate and P equals the price level
and to the stochastic wealth accumulation equation:
W=
dW = W[nM dRM + nB dRB + nF dRF + nK dRK ] − C(t) dt − dT
(2)
where ni refers to share of portfolio held in asset i. More specifically, nM = (M/P)/W,
nB = (B/P)/W, nK = K/W and nF = (E B∗ /P)/W. dRi refers to the total (sum of
deterministic and stochastic) rates of return on asset i. dT is the taxation paid on holdings
of wealth. The representative consumer constructs an optimal portfolio of his total wealth
subject to the adding up condition for portfolio shares:
1 = nM + nB + nK + nF
(3)
Consumers are assumed to purchase output over the instant dt at the nonstochastic rate
C(t) dt using the capital income generated from holding assets. To define each asset return,
dRi , requires a description of the dynamics which generate asset prices and asset returns.
2.1.1. Prices and asset returns
There are three commodity prices in the model: the domestic price of the traded good
(P); the foreign price level of the traded good (F ); and the exchange rate (E). F is assumed
to be exogenous and other two prices P and E are endogenously determined. The exchange
rate (E) is measured in units of domestic currency per unit of foreign currency. Prices and
returns are both generated by geometric Brownian motion (Wiener) processes. Each of the
prices P, F and E evolves according to
dx
= (drift term) dt + (diffusion term) dZ
(4)
x
where x is either P, F or E; and π (σP ), πF (σF ) and (σE ) are respective drift(diffusion)
terms of these price processes. Thus, for example, π dt is the expected mean rate of change of
4
The utility function employed here disentangles risk aversion from intertemporal substitution as proposed by
Epstein and Zin (1989) and Weil (1990). There are two reasons for choosing a utility function with this property:
(i) it has been shown that allowing for unconventional preferences may help explain the foreign exchange risk
premium, Sibert (1996); and (ii) one would like to answer questions about how preference parameters influence the
numerical solutions for the FOREX risk premium. There is a range of possible formulations for the recursive utility
function (see Dolmas, 1998): the particular formulation used in this paper was chosen by reference to a criterion
of requiring tractability of the model (and differs slightly from that used in Svensson (1989)). Our formulation
resembles that of Grinols (1996).
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
5
P and σP dt is the volatility of this rate of change. Zj is a Wiener process for j = P, F, E.
We use ρ to denote the “instantaneous” correlation coefficient between any two Wiener
processes: ρij = dZi dZj .
This small open economy is linked with the rest of the world through the law of one price.
Formally, it means that the exchange rate E relates foreign prices F to domestic prices of
traded goods P, which is referred to as the purchasing power parity (PPP) relationship. The
domestic price of the imported good is then given by
P = EF
(5)
which yields the following price process
dP
= (πF + + ρFE σF σE ) dt + σF dZF + de
P
(6)
where de is a random variable (an endogenous shock) with mean zero and variance σE2 dt,
ρFE is the correlation coefficient between foreign prices and the exchange rate with σE
being its diffusion term.
Incorporating money into the model gives rise to a separation of real returns from nominal
returns. Return to the productive asset, capital, will be described below but returns to other
assets can be described in terms of the interest rates they pay. Domestic and foreign bonds
pay nominal rates of interest, i and iF , respectively. Applying stochastic calculus, we obtain
the real rates of return to domestic holders of money, domestic bonds, and foreign bonds as
follows:
dRM = rM dt − dp
rM = −π + σP2
(7a)
dRB = rB dt − dp
rB = i − π + σP2
(7b)
dRF = rF dt − dZF
rF = iF − πF + σF2
(7c)
where dp is a temporally independent, normally distributed, random variable with mean
zero and variance σP2 dt.
The real rate of return to equity holders is calculated from the flow of new output dY
per capital K. We assume that output is produced from capital by means of the stochastic
constant returns to scale technology; and the economy-wide capital stock is assumed to
have a positive external effect on the individual factor capital. We therefore write the aggregate production function as an AK function of the kind discussed by Rebelo (1991)with
a stochastic linear coefficient
dY(t) = α[dt + σY dZY (t)]K(t)
(8)
where α is the marginal physical product of capital and σY dZY (t) a productivity shock
with σY being the volatility of the shock. Technically, dZY represents increments to a
Brownian motion with zero drift and variance unity. Thus the return to capital before and
after separating its deterministic and stochastic terms is, respectively5
dRK = α dt + ασY dZY ,
5
rK = α dt,
duK = ασY dZY .
To derive Eq. (9), a linear investment technology for capital is assumed.
(9)
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L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
Taxes are endogenously determined to satisfy the government budget constraint, see
Section 2.2 below, and include a stochastic component reflecting the changing need for
taxes. Because, in a growing economy, taxes and other real variables grow with the size of
the economy, measured here by real wealth, we relate total taxes to wealth according to
dT = τ W dt + W dv.
(10)
where τ is the tax rate and dv a random variable (an endogenous shock) with mean zero
and variance σV2 dt.
As we will see below, the tax rate on the deterministic component of total wealth, τ, is
nondistortionary: it operates essentially as a lump-sum tax. However, this is not true of the
stochastic component, which will have real effects through the portfolio decision.
The wealth accumulation of the representative consumer can be redefined to separate
deterministic and stochastic components as follows:
dW
= ψ dt + dw
W
ψ = nM rM + nB rB + nK rK + nF rF − τ −
(11a)
C(t)
W(t)
dw = −nM dp − nB dp + nK ασY dZY − nF σP dZF − dv
(11b)
(11c)
The maximization of (1) subject to the stochastic differential Eq. (11a) and the adding
up condition (3) represents a continuous-time stochastic dynamic optimization problem of
the type pioneered by Merton (1969). The solution strategy is based on the dynamic programming approach of Grinols (1996) which is similar to the solution method of Svensson
(1989) and Obstfeld (1994b).6 The household maximizes utility by choosing the optimal
full (composite) consumption-wealth ratio and the optimal portfolio shares of assets, taking
the rates of return on assets, and the relevant variances and covariances as given7 .
2.2. Government policy
The government engages in four activities, (i) choosing its expenditure and financing it
by (ii) taxation (iii) printing money and (iv) issuing bonds. Government expenditure G is
determined in the following stochastic way:
dG = gαK dt + αKσG dZG .
(12)
This states that on average the level of public spending is a fraction g of the level of
economy output. The stochastic term σG dZG reflects the inability of the government in
setting government expenditures with certainty with σG being the volatility of government
expenditures and ZG being a Wiener process generating the randomness. The diffusion is
therefore different from production risk, σY dZY .
6
It is perhaps worth noting that there is current interest in alternative characterizations of the problem such as
approaches based on stochastic differential utility, stochastic variational utility and utility gradient.
7 However, the general equilibrium conditions, i.e. market-clearing conditions, of the model will determine
these rates of return, variances and covariances.
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
7
We also consider a different form of government expenditure uncertainty in which the
stochastic term is proportional to the production shocks. It follows that setting σG dZG =
gσY dZY will eliminate the uncertainty stemming from government actions. Formally, this
can be represented as:
dG = g[αK dt + αKσY dZY ].
(13)
The government pursues the following monetary and borrowing policy rules:
dM
= φ dt + σX dZX
M
B
=λ
M
(14)
(15)
where φ is the mean monetary growth rate and λ a policy parameter, reflecting the choice
between monetary expansion and borrowing, set by the government. The stochastic term
σX dZX may reflect exogenous stochastic failures to meet the monetary growth target set
by the monetary authority. Correlation between monetary growth shock and other shocks
is important. For example, correlation between dZX and dZF may reflect stochastic adjustments in the money supply as the authorities respond to exogenous stochastic movements
in the intermediate target, the exchange rate.
Finally, the government budget constraint is
M
B
M
B
d
+d
(16)
=
dRB +
dRM + dG − dT.
P
P
P
P
Given the policy rules for monetary growth and borrowing, the stochastic component of
taxation must adjust to maintain the government budget constraint.
2.2.1. Goods market equilibrium and balance of payments
In our small open economy net exports in real terms are given by
net exports = dY − dC − dG − dK.
The balance-of-payments equilibrium condition in real terms is
∗ ∗
EB
EB
=
dRF + net exports
d
P
P
(17)
Substituting and simplifying we derive the following expression for the rate of growth of
the capital stock:
1 C
dK
= ω α(1 − g) −
+ (1 − ω)rF dt
K
nK W
+ ωα(σY dZY − σG dZG ) − (1 − ω)σF dZF .
(18)
2.3. Macroeconomic equilibrium
The assumption of constant drift (mean) and diffusion (variance) parameters in the geometric Brownian motion which describes the model variables ensures that risks and returns
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L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
on assets are unchanging through time. This feature of the model, together with the constant
elasticity utility function, generates a recurring equilibrium, implying that the consumer
chooses the same portfolio shares nM , nB nF , nK and consumption-wealth ratio, C/W,
at each instant of time. Moreover, the multiplicity of all shocks (meaning that stochastic
disturbances are proportional to the current state variables such as the capital stock and
wealth), leads to an equilibrium in which means and variances of the relevant endogenous
variables are jointly and consistently determined—a mean-variance equilibrium.
The exogenous factors, apart from the four stochastic shocks, dy, dz, dx and dpF , explained above, include (i) the preference and technology parameters γ, ζ, δ, θ, α, ν (ii) the
policy parameters φ (monetary), g (government spending), λ (government borrowing), and
(iii) the mean foreign inflation rate πF . The endogenous variables include (i) the stochastic
adjustments in the economy dp (the stochastic adjustment in the domestic price level), de
(the stochastic PPP relationship), dv (the stochastic adjustment in taxes), dw (the stochastic
component of wealth), (ii) the tax rate τ, (iii) the optimal consumption-wealth ratio and optimal portfolio shares, (iv) the equilibrium prices π (the expected domestic inflation rate),
i (the nominal domestic interest rate), (the expected exchange rate depreciation), and (v)
the equilibrium growth rate ψ.
The determination of endogenous variables involves several stages. By using the constancy assumption of portfolio shares we first solve the model for the price level and thereby
π and dp. The next stage is to determine stochastic adjustments. With the stochastic adjustments obtained, one can then calculate the endogenous variances and covariances that
appear in the optimality conditions for the consumption-wealth ratio, portfolio shares etc.
The final stage is to substitute these variances and covariances into the deterministic components of the equilibrium.
2.3.1. Solution
The solution is given in the Appendix A to this paper: key results are reproduced here
for convenience.
The solution of consumption is obtained from the first-order optimality condition as
C
θ
1 2
,
(19a)
=
δ − ζ β − γσW
W
1 − ζθ
2
2 as the risk-adjusted
where β = nM rM + nB rB + nK rK + nF rF − τ. We define β − (1/2)γσW
rate of return. Expression (19a) reveals that the optimal consumption and saving decision
depends on both the intertemporal elasticity of substitution and the coefficient of risk aversion.
The first-order optimality condition for money holdings yields an expression for its optimal portfolio share nM :
θ
C/W
nM =
.
(19b)
1−θ
i
The remaining two first-order optimality conditions for equities and bonds give CAPM type
expressions:
(rK − rB ) dt = γcov(dw, α dy + dp),
(19c)
(rF − rB ) dt = γcov(dw, −df + dp).
(19d)
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
9
These two equations describe the differential real rates of return on the assets in terms of
their respective real risk differentials, as measured by the covariances with the overall market
return. The right-hand-side of the equations represent the real risk premium on domestic
capital over domestic bonds and that on foreign bonds over domestic bonds, respectively,
both being ‘priced’ at γ. To solve these FOCs and the remaining equations we first need to
obtain expressions for the variance–covariance terms. They are given by:
dw = αω(σY dZY − σG dZG ) − (1 − ω)σF dZF
(19e)
dp = σX dZX − αω(σY dZY − σG dZG ) + (1 − ω)σF dZF
(19f)
de = σX dZX − αω(σY dZY − σG dZG ) − ωσF dZF
(19g)
where ω = (nK )/(nK + nF ) and is solved as
ω=
α − (iF − πF + σF2 ) αρFY σF σY + σF2
+
γΛ
Λ
(19h)
Λ = α2 [σY2 − ρGY σY σG ] + α[2ρFY σY σF − ρFG σG σF ] + σF2
Solving these equations in conjunction with the adding up condition for portfolio shares
(3), determines the optimal portfolio share of each of the remaining three assets (equity,
foreign bonds and domestic bonds), nK , nF and nB
nK = ω[1 − (1 + λ)nM ]
nF =
1−ω
nK
ω
nB = λnM
(19i)
(19j)
(19k)
The solution for the exchange rate is
= π − πF − cov(de, df )
(20)
which is referred to as the “risk-adjusted” PPP equation. The solution for the inflation rate,
π, is
π = φ − ψ − ρXW σX σW + σW
(21)
The solution for the interest rate, iD , is
i = α + π − σp2 − γ[ρWY σW ασY + ρWP σW σP ]
The solution for the growth rate, ψ, is
1 C
ψ = ω α(1 − g) −
+ (1 − ω)(iF − πF + σF2 )
nK W
(22)
(23)
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L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
3. Foreign exchange risk premium
The model developed here, with its mean-variance equilibrium and general equilibrium
features, offers a convenient vehicle for examining the determinants of the foreign exchange
risk premium which is implicit in the forward exchange rate. In addition, our unconventional
preferences assumption gives scope for a quantitative analysis of the foreign exchange risk
premium. In general, the literature distinguishes between real and nominal measures and
across several definitions. We shall focus on the real measure as being of greater relevance
and use the proper definition of Engel (1992) as used by Grinols and Turnovsky (1994).
Our expression for the real foreign exchange risk premium is therefore:
Definition. The real foreign exchange risk premium over the period (t, T ) is the difference
between the risk-neutral forward rate F RN (t, T ) and the forward rate F(t, T ), divided by
the risk-neutral forward rate (the home currency cost of buying a unit of foreign exchange
one period forward)
Θ(t, T ) ≡ 1 −
F(t, T )
F RN (t, T )
(24)
where
F RN (t, T ) =
Et [E(T )/P(T )]
Et [1/P(T )]
(25)
F(t, T ) is the forward exchange rate at time t of the spot exchange rate E(T) at time T .
To derive an expression for Θ(t, T ) we exploit covered interest parity (CIP), which stems
from the equilibrium (no-arbitrage) condition of the forward foreign currency market:
F(t, T )B(t, T ) = E(t)B∗ (t, T )
where B(t, T ) is the time t price of a domestic zero-coupon bond paying a certain pound at
date T and B∗ (t, T ) is the price of a foreign zero-coupon bond. The price of zero coupon is
given by
B(t, T ) = exp [−i(T − t)]
Substituting and simplifying this into the above equation we obtain8
F(t, T )
exp [iF (T − t)] = exp [iD (T − t)]
E(t)
To determine Et [E(T)/P(T)], we apply stochastic calculus to
2
dE dP
dE dP
dP
d[E/P]
=
−
−
+
E/P
E
P
E P
P
8
In economics the discrete-time version of this expression is widely used (see Obstfeld and Rogoff, 1996):
1 + i = (1 + iF )
F(t, t + 1)
.
E(t)
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
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Taking expectations gives
E(T )
E(t)
Et
=
exp [( − π − πF + + ρEP σE σP + σP2 )(T − t)]
P(T )
P(t)
Similarly calculating Et [1/P(T)] we obtain
F RN (t, T ) =
Et [E(T)/P(T)]
= E(t) exp [( − σEP )(T − t)]
Et [1/P(T)]
Combining these terms into the foreign exchange risk premium expression yields
Θ(t, T ) = 1 − exp [(i − iF − + σEP )(T − t)]
(26)
It is evident that the risk premium is a function of the differential between domestic and
foreign interest rates and of certain variance and covariance terms. However because key
variables are determined in the general equilibrium, it is also a function of all four exogenous
shocks (see, for example Eq. (19f)); and a function of attitudes towards risk amongst other
things. Thus our model sheds new light on the determinants of the foreign exchange risk
premium.
4. Modelling monetary policy uncertainty
The assumption that the drift and diffusion terms of all stochastic process are constant
implies that investors’ probability beliefs converge instantaneously to true probabilities.
In other words, there is no policy uncertainty such as monetary policy uncertainty. This
assumption was relaxed by Stulz (1986) building on Williams (1977). He develops a model
in which optimizing households are uncertain about the distribution of monetary growth
and learn about it over time. This monetary policy uncertainty implies that the households’
predictive distribution over money growth has a higher variance than the variance of money
growth. Households can compute their predictive distribution available to them. This predictive distribution can be derived explicitly for the case in which households have a diffuse
prior before sampling and their only relevant information is the time series of changes in
the money supply. In this case, the predictive distribution is normal with mean
φe (t) =
1 2
1 M(t)
σ + ln
2 X t M(0)
(27)
2 = Ω2 σ 2 per unit of time. t is the time elapsed
per unit of time and variance ((t + 1)/t)σX
X
since the monetary policy was introduced. For households to be uncertain about the mean
growth rate of the monetary stock, it is required that t < ∞. In the following, Ω is used
to measure the degree of monetary policy uncertainty. When Ω = 1, there is no monetary
policy uncertainty and households know the true dynamics of the money stock. Finally, φe
follows
dφe (t) =
1 dM
1
− φe (t) dt
t M
t
(28)
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In this model, by construction, there is no uncertainty about the instantaneous variance of
the growth rate of the money supply because households sample continuously.9
5. Numerical analysis
In this section we undertake a full numerical analysis of the model and carry out a comparison of the numerical estimate with the ‘true’ value of the foreign exchange risk premium
(see Eqs. (24) and (25)). This ‘true’ value is obtained using simulations to generate expected values of the expected spot exchange rate in Eq. (25)—a different approach from
using historical observations as a proxy. The values so generated for the foreign exchange
risk premium will typically be lower under this approach.10 The generation of numerical estimates requires the specification of a number of baseline parameters and variables. Tables 1
and 2 set out the values used in the numerical exercises carried out here.
In an attempt to utilize plausible values, a number of the parameter values have been
set by reference to quarterly data relevant for the UK economy from the period 1979Q1 to
1999Q2. All data were obtained from Datastream. For the UK, we have used series averages
and standard deviations for: industrial production, government expenditure, the M1 money
supply, and the inflation rate derived from the consumer price index. To capture ‘foreign’
price and interest rate variables we have used US data, in particular: series averages and
standard deviations for the consumer price index and the 3-month Treasury bill rate.
The variance and covariance parameters were calculated using the discrete time version
of geometric Brownian motion
:ln X(ts ) = η:t + Γ :Z(ts )
(29)
where X(ts ) = (x1 (ts ), ... , x4 (ts )) is the (four) exogenous stochastic shock vector, Z(ts )
the independent normal variates vector N(0, I:t), η ≡ (µ1 −(1/2)σ12 ), . . . , (µ4 −(1/2)σ42 )
is the mean vector, and the variance–covariance matrix Σ ≡ Γ Γ with Γ = (σ1 , . . . , σ4 ).
Finally, time evolves ts+1 ≡ ts + :t, s = 0, ... , S − 1 with tS = T . Because the regressors
in (29) are identical equation by equation, the least squares estimator η̂ of η becomes
η̂ =
1
[ ln X(T ) − X(0)]
T
for S :t. Similarly, the sampling estimator Σ̂ of Σ is
Σ̂ =
S−1
1
S
:ln X(ts ) − ( ln X(T) − X(0))
S−1
S
s=0
1
× :ln X(ts ) − ( ln X(T) − X(0))
.
S
1
T
(30)
9 It should perhaps be noted that there is contemporary interest in modelling uncertainty about such variance
terms, see for example Basak (2000).
10 This is a point recently explored by Fama and French (2002).
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
13
Table 1
Baseline parameters and variables: UK
Variable
Symbol
Parameters
Marginal product of capital
Risk aversion parameter
Intertemporal substitution elasticity
Rate of time preference
Debt policy parameter
Government size
Foreign interest rate
Foreign inflation rate
α
γ
1/(1 − ζ)
δ
λ
g
i∗
π∗
Variables
Consumption–wealth ratio
Beta
Mean equilibrium growth rate
Variance of growth rate
Variance of price level
Inflation rate
Interest rate
Exchange rate
Rate of return on money
Rate of return on government bonds
Rate of return on capital
Rate of return on foreign bond
Risk-adjusted rate of return
Portfolio share of equity in tradeable
Portfolio share of money
Portfolio share of bonds
Portfolio share of equity
Portfolio share of foreign bonds
FOREX risk premium
C/W
β
ψ
σw2
σp2
π
i
rM
rB
rK
rF
β − (1/2)γσw2
ω
nM
nB
nK
nF
Θ
Value
0.08500
4.00000
0.50000
0.02500
0.25000
0.20326
0.08640
0.04140
0.00012
0.03026
0.00350
0.00009
0.00024
0.04541
0.14534
0.00411
−0.04517
0.10017
0.10000
0.10046
0.03008
0.50475
0.55240
0.13810
0.15622
0.15328
0.00729
Particular mention should be made of the values assigned to the parameters for risk
aversion and the intertemporal substitution elasticity: the values 4 and 0.5 are those used in
Obstfeld (1994b). The former is the mid point of the range of conventional estimates (2–6)
referred to in Obstfeld (1994a) although we are mindful that some authors suggest that values
of unity or values as high as 30 cannot be ruled out (see Epstein and Zin (1991); and Kandel
and Stambough (1991), respectively). The intertemporal substitution elasticity set to 0.5 is
consistent with what Epstein and Zin (1991) describe as ‘a reasonable inference’. However,
smaller values cannot be ruled out, for example Hall (1988) and Campbell and Mankiw
(1989)suggest an intertemporal substitution elasticity of 0.10; and Ogaki and Reinhart
(1998) refer to the range 0.32–0.45.
Later in this paper we explore the sensitivity of our results to different values for the key
risk aversion parameter and for a range of correlation coefficients. However, first we take
these plausible values and, it can be seen (from the bottom row of Table 1) that the model
generates a value for the foreign exchange risk premium of around 0.7. A positive real risk
14
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
Table 2
Descriptive statistics and variance–covariance matrix: UK
Variables
Panel A: Moments
Mean
Stdandard deviation
Variance
Production
(Y )
0.01225
0.02652
0.00070
Domestic inflation
rate (iD )
0.05367
0.02334
0.00054
Government
expenditure (G)
Money supply
(M1) (M)
Foreign
inflation
rate (πW )
0.20326
0.04724
0.00223
0.04885
0.01435
0.00021
0.04140
0.01487
0.00022
Panel B: Covariances (normal data) and correlations (italicized data)
Y
0.00070
−0.19624
−0.28288
iD
−0.00012
0.00054
0.22908
G
−0.00035
0.00025
0.00223
M
0.00003
−0.00003
0.00002
πW
−0.00015
−0.00001
0.00020
0.07911
−0.08268
0.02834
0.00021
−0.00015
−0.37929
0.66892
0.28209
−0.06886
0.00022
The data are quarterly observations from 1979:3 to 1999:2. All observations are taken from Datastream. The
reported estimates are based on changes in natural logarithms of the series. The variance of each series is reported
along the diagonal of the variance–covariance matrix.
Table 3
Foreign exchange risk premium
UK
‘True’ value
Mean
Standard deviation
0.04784
0.0313
Model calculated mean
Independent: dZG free to vary
Proportional: dZG = g dZY
Monetary policy uncertainty
0.00729
0.02452
0.02574
The first two rows show the mean and standard deviation of the observed FOREX risk premium. This is calculated as
the percentage difference between the observed 3-month forward exchange rate and the corresponding risk-neutral
forward rate using Monte-Carlo methods. The remaining rows present the corresponding mean for the FOREX
risk premia obtained from the stochastic dynamic general equilibrium model. These means are obtained using
different specifications for uncertainty.
premium means that the foreign currency is regarded as riskier than the home currency. This
is not surprising: the UK inflation rate has both a higher mean and a higher standard deviation
than that experienced in the rest of the world. This generates a high equilibrium interest
rate and a low exchange rate resulting in a positive risk premium.11 This figure is less than
the ‘true’ mean value which is shown in Table 3. However, it is also clear from Table 3 that
allowance for other forms of policy uncertainty in the model gives rise to calculated values
of the foreign exchange risk premium which are considerably closer to the ‘true’ value. The
11
This is the mean expected risk premium. We did not analyze its standard deviation. This can be done by
calculating the standard deviations of the components of the risk premium expression (26).
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
15
presence of uncertainty about monetary policy leads to an appreciation in the exchange rate
(see Stulz (1987)) and thereby increases the risk premium which can be seen from (26).
This suggests that this model, when extended to accommodate policy uncertainty, may
be capturing some important influences on the foreign exchange risk premium which are
missing from other models: It does seem that it may be important to focus attention on
policy uncertainty.
Fig. 1. FOREX risk premium: UK.
16
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
However, it may be premature to conclude this before seeing the results of a sensitivity
analysis in which the risk aversion parameter and correlation coefficients are allowed to
vary in the model. Fig. 1 presents results of allowing the risk aversion parameter to vary
across the positive range up to 20; and correlation coefficients to vary between −1 and 1.
Fig. 2. Correlation coefficient sensitivity.
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
17
One observation (bottom right) is that the calibrated foreign exchange risk premium, for
the conventional range of risk aversion parameters, remains below the ‘true’ value-but the
premium is very sensitive to the risk aversion parameter.12 As this parameter increases, the
foreign exchange risk premium rises markedly.
The other elements of Fig. 1 reveal something of how the key variables in the model are
related to the foreign exchange risk premium. In particular, there is a positive association of
the FOREX premium with the portfolio share of money—not so evident from the solution
of the model, but it does show how important a role portfolio adjustment may have within
this model.
Turning to Fig. 2 it is evident that it is only with high values for the risk aversion parameter that the foreign exchange risk premium shows sensitivity to different values of the
respective correlation coefficients. In fact, there is no sensitivity to the correlation coefficient
between foreign price variability and productivity shocks (top left). Consider for example
the correlation coefficient between money growth variability and production shocks (bottom left). At low values of risk aversion, the correlation coefficient has no impact on the risk
premium. However, with more risk averse agents, the risk premium is considerably higher
as the correlation coefficient rises. At negative values a positive monetary growth shock
is associated with negative effects on output and thus the inflationary impulse reduces the
demand for money, domestic bonds and equity at home. At positive values, a positive monetary growth shock is associated with expansionary output and an increase in the demand for
equity at home. In this latter environment, domestic households require a higher return to
hold foreign currency denominated assets—the more so because they are highly risk averse.
A similar result is obtained for the correlation coefficient between government expenditure
shocks and foreign price shocks: this is because government expenditure shocks increase
the riskiness of domestic bonds. A positive monetary growth shock coupled with a negative
foreign price shock leads to the highest values for the risk premium (bottom right). Unsurprisingly, with a positive correlation the degree of risk aversion has a negligible effect on
the risk premium: the shocks at home and abroad balance each other out.
6. Conclusion
This paper has developed a stochastic general equilibrium model of exchange rate determination incorporating portfolio choice and unconventional preferences. These features
of the model give rise to an integrated analysis of exchange rate determination with a
risk-adjusted PPP and portfolio equilibrium; and enable us to analyze the distinct roles
played by agents’ attitudes towards risk and intertemporal substitution. The model is calibrated to examine the FOREX risk premium with different specifications of policy uncertainty. We find that while the foreign exchange risk premium is a function of the differential
between domestic and foreign interest rates and certain variance terms, it is also a function
of attitudes towards risk. Moreover, numerical analysis yields a range of parameter values
12 The result is insensitive to the other preference parameter, intertemporal elasticity of substitution, because
the model used in this paper is a representative agent model in which there is no saving effect and it is not an
overlapping generations model as used by Sibert (1996).
18
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
that are broadly consistent with ‘true’ values of the foreign exchange risk premium, particularly when the model is extended to allow for policy uncertainty. Given the widespread
difficulty of generating values for the FOREX risk premium which are at all close to those
of the real world, the approach used here seems to shed new and important light on the
determinants of the foreign exchange risk premium.
Acknowledgements
We are grateful for comments from an anonymous referee and the participants at conferences of the European Financial Management Association (EFMA), Lugano 2001 and
Money Macro and Finance research group (MMF), Belfast 2001; and seminar participants
at the University of Manchester and the Judge Institute of Management Studies, University
of Cambridge. Lynne Evans acknowledges support from the Economic and Social Research
Council (Award Reference: L138 25 1026).
Appendix A. Solution to the consumer’s optimization problem
The representative consumer’s optimization problem is to find the solution to

1/ζ
1−θ ζ


M(t)
lim max e−δt
C(t)θ
dt + e−δdt [Et U(t + dt)1−γ ]ζ/(1−γ)


P(t)
dt→0+ {C,n}
(A.1a)
subject to
dW
= ψ dt + n du − dv,
W
(A.1b)
n 1 = 1,
(A.1c)
where
ψ = n r − C/W − v,
and vectors, denoted by x, are:

 

1
nM
1
n 
 
 B 
n=
, 1 =  ,
1
 nK 
nF
1

rM
r
 B
r=
 rK




,

 −dp

du = 
 α dZY
rF
[n du − dv]2 is calculated as
2
dt = [n Σ n − 2n ΣUV + σV2 ] dt
[n du − dv]2 = σW
−dp
−dZQ





L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
19
where Σ U is the (instantaneous) correlation matrix of ZU whose elements are:
ρij σi σj = cov(dZi , dZj ).
Similarly, Σ UV is the correlation vector of dZU and dZV . The correlation coefficient, ρij ,
is given by
ρij =
cov(dZi , dZj )
σi · σj It follows that
∂ψ
=r
∂n
and
2
∂σW
= 2ΣU n − 2ΣUV .
∂n
Exploiting the special time dependence of intertemporal utility function and guess that the
indirect utility function takes the form
J(W(t), t) = e−δt I(W(t)).
Given the homogeneity of the budget constraint and the chosen functional forms, one may
guess that the indirect utility and consumption functions are linear in wealth,
I(W(t)) = AW,
A > 0;
C(W) = ZW,
Z > 0;
M(t)
= nM W(t).
P(t)
(A.2)
The indirect utility function is then defined recursively as
d−δt AW(t)
= lim max e−δt
dt→0+ {C,n}



Cθ
M
P
1−θ
ζ
dt + e−δdt Aζ [Et W(t + dt)1−γ ]ζ/(1−γ)
1/ζ


(A.3)
Applying Ito’s Lemma yields
2
[Et W(t + dt)1−γ ]ζ/(1−γ) = W(t)ζ exp[ζ{(1 − γ)ψ − 21 γ(1 − γ)σW
}dt]
(A.4)
Substituting Eqs. (A.2) and (A.4) into (A.3) and following Grinols (1996) we write the
following simpler problem


 (Zθ n1−θ ) ζ

1
2
M
0 ≡ max
−δ
(A.5)
+ ζ ψ − γσW

A
2
{C,n} 
First-order conditions are
ζ
[Zθ n1−θ
∂
M ]
: ζθ
− ζ = 0,
∂Z
[ZAζ ]
(A.6)
20
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
and

ζ
[Zθ n1−θ
M ]
 ζ(1 − θ)

[nM Aζ/1−γ ]
∂ 

0
:
∂n 

0

0



2

 + ζ ∂ψ − 1 γ ∂σW
− ζξi = 0

∂n
2
∂n


(A.7)
where ξ is the Lagrange multiplier for (A.1c). Eq. (A.6) implies that
(Zθ n1−θ
M )
A
ζ
=
Z
θ
(A.8)
which, after substituting into (A.5) and simplifying, yields:
Z=
θ
1 2
)],
[δ − ζ(β − γσW
1 − ζθ
2
(A.9)
and







(1 − θ)Z
θnM
0
0
0



 rM
j [ρPj σP σj ] − ρPV σP σV

 r 
 [ρPj σP σj ] − ρPV σP σV
  B 

j
+
−γ   rK 

[ρ
Yj ασY σj ] − ρYV ασY σV

j
rF
j [ρPF j σPF σj ] − ρPF V σPF σV

 
ξ
 ξ
  
 −   = 0,
 ξ
ξ
(A.10)
where β = nM rM + nB rB + nK rK + nF rF − τ.
Subtracting the second relation in Eq. (A.10) from the first one an expression for nM is
obtained as follows:
θ
C/W
nM =
.
(A.11a)
1−θ
iD
Similarly, subtracting from the remaining rows (3 and 4) yields


(rK − rB ) = γ 
σj [ρYj ασY − ρPj σP ] + σV [ρPV σP − ρYV ασY ]
(A.11b)
j
(rK − rB ) = γ[ρWY ασW σY + ρWP σW σP ]
(A.11c)


(rF − rB ) = γ 
σj [ρPF j ασPF − ρPj σP ] + σV [ρPV σP − ρPF V ασPF ] (A.11d)
j
(rF − rB ) = γ[ρWP σW σP − ρWY σW σQ ]
(A.11e)
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
21
A.1. Derivation of the rate of growth of the capital stock
Changes in net exports equal the excess of changes in production over domestic absorbtion
dY − dC − dK − dG.
Balance-of-payments equilibrium requires the following to hold:
∗ ∗
EB
EB
d
−
dRF = dY − dC − dK − dG.
P
P
(A.12)
Substituting for dRF , dY and dG in (17) and noting that dC = C dt yields:
∗ ∗ ∗
B
B
B
dK + d
= α(1 − g)K − C + rF
dt + αK(dy − dz) −
dq
Q
Q
Q
(A.13)
Dividing (17) by K + B∗ /Q and noting that ω = (nK )/(nK + nF ) yields:
dK
d(B∗ /Q)
1 C
ω
+ (1 − ω) ∗
= ω α(1 − g) −
+ (1 − ω)rF dt
K
B /Q
nK W
+ ωα(dy − dz) − (1 − ω) dq
(A.14)
Using the asset market equilibrium condition (d(B∗ /Q))/(B∗ /Q) = dK/K, Eq. (A.14) is
expressed as the rate of growth of the capital stock by the relationship
dK
1 C
= ω α(1 − g) −
+ (1 − ω)rF dt + ωα(dy − dz) − (1 − ω) dq.
K
nK W
(A.15)
A.2. Derivation of the price level
From the constant portfolio shares assumption we can write
M/P
nM
.
=
K + B∗ /Q
nK + n F
The price level can then be written as
nK + nF
K + B∗ /Q
p=
nM
M
(A.16)
Taking the stochastic differential of the above expression (A.16) (noting that portfolio shares
are constant through time), leads to
dP
= π dt + dp
P
d[K + B∗ /Q]
dM
d[K + B∗ /Q]
d[K + B∗ /Q] 2
dM
.
−
−
+
=
M
K + B∗ /Q
M
K + B∗ /Q
K + B∗ /Q
(A.17)
22
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
Using (14) and (A.13), noting that the variances are of order dt, the right-hand-side of this
equation can be expressed as
1 C
φ − ω α(1 − g) −
+ (1 − ω)rF + α2 ω2 (σy2 + σz2 ) + (1 − ω)2 σq2
nK W
(A.18)
− αω(σxy − σxz ) + (1 − ω)σxq dt + dx − αω(dy − dz) + (1 − ω) dq.
Equating the deterministic and stochastic components of (A.17) implies
1 C
π = φ − ω α(1 − g) −
+ (1 − ω)rF + α2 ω2 (σy2 + σz2 )
nK W
+ (1 − ω)2 σq2 − αω(σxy − σxz ) + (1 − ω)σxq
dp = dx − αω(dy − dz) + (1 − ω) dq.
(A.19)
(A.20)
A.3. Determination of tax adjustments
To determine the tax adjustments, we use the following government budget constraint
M
B
B
M
d
+d
=
dRB +
(A.21)
dRM + dG − dT.
P
P
P
P
Dividing both sides by W, we may rewrite this equation as
nM
d(dM/P)
d(dB/P)
dG − dT
+ nB
=
+ nM dRM + nB dRB
(dM/P)
(dB/P)
W
Substituting for debt policy (15) into the above equation this equation becomes
nM
d(dM/P)
d(λ dM/P)
dG − dT
+ nB
=
+ nM dRM + nB dRB
(dM/P)
(dM/P)
W
Substituting for government expenditure policy (12), monetary policy (14), tax collection
(10) and the price evolution (A.16) into the above equation, while noting the stochastic
derivatives of d(M/P) and d(B/P) this equation becomes
(nM + nB )(φ − π − σxp + σp2 ) dt + (nM + nB )(dx − dp)
= [αnK g − τ + nM (−π + σp2 ) + nB (i − π + σp2 )] dt + (nM + nB ) dp
+αnK dz − dv
(A.22)
Equating deterministic and stochastic parts of this equation leads to the relationship
τ = αnK g − (nM + nB )φ + nB i + (nM + nB )σxp
(A.23)
dv = αnK dz − (nM + nB ) dx
(A.24)
L. Evans, T. Kenc / Int. Fin. Markets, Inst. and Money 14 (2004) 1–24
23
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