I. INTRODUCTION

This paper studies the forward premium anomaly, which refers to the robust empirical finding that the forward exchange rate is not an unbiased predictor of the future spot exchange rate.1 Simply put, the most puzzling fact is the negative correlation between the forward premium and the exchange rate change. This apparent departure from uncovered interest parity suggests that a low interest rate currency tends to be depreciated rather than appreciated. To explain this puzzle, Fama (1984) shows that if the market expectation on the future spot exchange rate is rational, then the foreign exchange risk premium should be more volatile than, and negatively correlated with the expected exchange rate change. These results on the volatility relations suggest that one needs a model that is able to generate: (1) a high volatility of both exchange rate changes and marginal rates of substitution; (2) a low volatility of both expected exchange rate changes and interest rates.

The above two conditions are also consistent with two empirical regularities. On the one hand, studies initiated by Meese and Rogoff (1983) find that exchange rates follow a near-random walk.2 On the other hand, Hall (1978) provides well-known empirical evidence that marginal utility of consumption also closely follows a random walk. The key idea of this paper is to tie the near-random walk behaviors of both exchange rates and consumption to the issue of the forward premium anomaly.

For this purpose, we introduce goods market frictions into the standard macro model and examine if sticky-price general equilibrium models with complete asset markets, such as those developed in Chari, Kehoe, and McGrattan (2002) (hereafter CKM), and Obstfeld and Rogoff (2003), can generate the volatility relations described above.3 As shown in Engel and West (2005), the nominal exchange rate closely follows a random walk under certain conditions in a class of asset-pricing models.4 Since our model with interest-sensitive money demand satisfies those conditions, a near-random walk behavior of the nominal exchange rate is endogenously derived in response to monetary shocks: the change in the exchange rate is likely to display large variation but the expected exchange rate change is likely to exhibit small variation. When this channel is combined with the assumption of sticky prices, the marginal utility of consumption is determined in a similar way as the nominal exchange rate in the money market equilibrium. However, in contrast to the nominal exchange rate, some features such as staggered price setting that generate gradual price adjustment are necessary to induce a near-random walk behavior of marginal utility of consumption. Under one-period in advance synchronized price setting as in Obstfeld and Rogoff (2003) and Engel (1999), consumption does not exhibit any persistence since all future prices fully adjust after the realization of unexpected monetary shocks. On the other hand, as prices are fixed more than one period, households adjust their consumption accordingly since their future consumption will be affected by current shocks. As a result, consumption becomes more persistent as the degree of sluggish price adjustment becomes larger. Therefore, the marginal rate of substitution is likely to display large variation but the interest rate is likely to exhibit small variation.

Using our sticky-price general equilibrium model, we first show that nearrandom walk behaviors of both exchange rates and marginal utility of consumption, in response to monetary shocks, can be endogenously derived. Based on this approach, we provide quantitative results on Fama’s volatility relations described above with some success. The benchmark model generates negative correlation between the risk premium and the expected depreciation and improves its performance on the volatility of the risk premium, while matching volatilities of exchange rates and consumption and autocorrelations of forward premium. However, the variance of the risk premium is still less than that in the data and quite similar to that of the expected exchange change.

Duarte and Stockman (2005) also use a sticky-price model and study how rational speculation behavior of economic agents affects the risk premium. Their study, motivated by Flood and Rose (1995), and Obstfeld and Rogoff (2000), pays attention to channels that affect the risk premium and the nominal exchange rate without affecting other macroeconomic variables much. Our study, motivated by Engel and West (2005), focuses on the role of expectations about future fundamentals operating through nominal interest rates. Alvarez et al. (2006) present a monetary model in which asset markets are endogenously segmented and show that the risk premium can be time varying even if the distributions of the fundamentals are time invariant. While they investigate the effects of frictions in asset markets on the risk premium, we study the effects of frictions in goods markets. Engel (1999) and Obstfeld and Rogoff (2003) analytically show that the foreign exchange risk premium can arise endogenously in sticky price models with a synchronized price setting. We extend their analyses to a more general setting that incorporates staggered price setting and time-varying risk premium, and link persistence of both exchange rates and consumption to Fama's volatility relations.

1)See Lewis (1995) and Engel (1996) for surveys of empirical evidence on the forward premium anomaly.

2)See Cheung et al. (2002) for a comprehensive survey.

3)The existing approach, based on Lucas (1982), attempts to generate high volatilities of foreign exchange risk premia with risk aversion parameters and/or habit persistence in preferences. Examples include: Backus et al. (1993) for habit persistence; Bekaert et al. (1997) for first-order risk aversion; Bekaert (1996) for habit persistence, consumption duration, and transaction costs. See, also, Lewis (1995) and Engel (1996) for a comprehensive survey.

4)Engel and West (2005) show that asset prices will exhibit a near-random walk behavior in a class of present-value models if (i) fundamentals have a unit autoregressive root and (ii) the discount factor is close to unity. Our model with persistent money growth rates and interest-sensitive money demand satisfies these two conditions. Engel and West (2005) also show that the monetary models which incorporate a variant of the Taylor rule can generate a near-random walk behavior of exchange rates. See also Gali and Monacelli (2005) for the nonstationary of behavior of exchange rates in a small open economy with a variant of the Taylor rule.

II. THE MODEL

We use CKM’s two-country monetary general equilibrium model, modified by abstracting from capital accumulation but by introducing an input-output production structure in producing intermediate goods, to study if the sticky-price model can generate a high volatility of the foreign exchange risk premium. Specifically, we focus on linking persistent real effects of monetary shocks, induced by gradual price adjustment, to the volatility of the risk premium. The presentation of the benchmark model is brief since it is directly drawn from CKM.

There are two countries in the world, home (H) and foreign (F). The population of monopolistically competitive intermediate goods producers in each country is normalized to 1. Intermediate goods producers set prices in a staggered way following a variant of the Taylor (1980) staggered nominal price contract. Markets for intermediate goods are segmented across countries so that consumers cannot engage in arbitrage activities. Intermediate goods producers must set prices in consumer’s currency in each market (local currency pricing). Under these two assumptions, intermediate goods producers can discriminate prices across countries and thus the law of one price does not hold. There is a representative household who lives infinitely in each country. Finally, we assume that there exist complete nominal bond markets across countries as well as within each country. The model is driven by exogenous shocks to the growth rates of money supply in each country. In the beginning of each period t, one of many finite states, denoted by s_t, is realized. s^t = (s^t-1, s_t) denotes the history of states up to time t and π(s^t) is the probability, as of period 0, of a history s_t. In what follows, we mainly describe the economy of the home country. Foreign quantities and prices are attached an asterisk superscript.

The representative home household has preference given by the expected infinite life-time utility function

where C denotes consumption, denotes real money balances, L is labor, β is the discount factor, σ is risk aversion, ∅ governs the interest elasticity of money demand, and γ is the labor supply elasticity.

Both home and foreign households can trade state contingent nominal bonds denominated in the home currency. Let Q(s^t+1|s^t) denote the nominal price (in home currency units) of one home state contingent bond paying one unit of home currency at s^t+1 and 0 otherwise. B(s^t+1|s^t) denotes the number of home state contingent bonds held by the home household between s^t and s^t+1. The home household's budget constraint (in home currency units) is:

where M is nominal money balances; II represents the profit of the home intermediate firms; and T denotes nominal transfer paid from the home government. is a borrowing constraint. represents a upper bound of real borrowing of the consumer. The initial conditions are given by M(s^-1) and B(s⁰).

Households are assumed to take prices of goods and labor as given. Then, the home household’s first order conditions are derived by maximizing its expected utility subject to the budget constraint and the borrowing constraint (the optimal conditions for the foreign representative household can be derived analogously)

where U_c(s^t) = C(s^t)^-σ denotes the marginal utility of consumption, denotes the marginal utility of real balances, U_l(s^t) = κ₂L(s^t)^γ denotes the marginal disutility from work, ε(s^t) denotes the nominal exchange rate of home currencies per foreign currency, and denotes the inverse of the home gross nominal interest rate. Equation (2-3) shows that money demand for the home household is sensitive to the nominal interest rate. As will be discussed in detail later, this interest-sensitive money demand derived from the utility maximization problem is one of the key mechanisms that generate near-random walk behaviors of both exchange rates and consumption. Equations (2-5) and (2-6) are related to home and foreign nominal intertemporal Euler equations expressed in the home currency for each state. The price, Q(s^t+1|s^t), of one state contingent home nominal bond should be equal to the marginal rate of substitution in home consumption between s^t and s^t+1 weighted by the change in purchasing power of the home currency. Or, it should be equal to the marginal rate of substitution in foreign consumption weighted by the change in purchasing power of the foreign currency once converted into the home currency.

In each period t, the home competitive representative firm produces a final composite good by using home intermediate goods produced in home country and foreign intermediate goods produced in foreign country according to the following technology:

where Y(s^t) denotes the home final composite good, denotes a composite good of home intermediate goods, denotes a composite good of foreign intermediate goods, Y_H(i, s^t) denotes home intermediate goods i, and Y_F(i, s^t) foreign intermediate goods i. denotes the elasticity of substitution between home and foreign composite goods, denotes the elasticity of substitution between differentiated intermediate goods within the country, and a₁ denotes a share for the domestic intermediate goods and determines the ratio of imports to output along with ρ and θ. The home final goods producer takes as given prices P(s^t), P_H(i, s^t) for i ∈ [0, 1], and P_F(i, s^t) for i ∈ [0, 1], to maximize its profit given by

subject to (2-7), where P(s^t) is the price of the final goods, P_H(i, s^t-1) is the price of home intermediate goods i, and P_F(i, s^t-1) is the price of foreign intermediate goods i at time t. Intermediate goods prices do not depend on s^t because they are set before period t shocks are realized. These prices are denominated in home currency. From this problem, we can derive input demand functions for home and foreign intermediate goods i:

where Using zero profit condition from the above profit maximization problem, the price of the final goods is defined by

The home final goods is distributed to the home representative household and to home intermediate goods producers according to

where Z(i, s^t) denotes a final good purchased by intermediate goods producer i.

The home firm that produces intermediate goods i uses the home labor service as well as the final goods according to the following production function:

where L(i, s^t) denotes labor input, Z(i, s^t) denotes the composite intermediate input, α is the cost share for the intermediate input, and Y_H(i, s^t) and denote the amounts of intermediate goods i used in the production of home and foreign final goods, respectively. Firms producing intermediate goods are assumed to take as given prices of inputs and of other intermediate goods, while they set prices of their own intermediate goods according to a variant of the Taylor staggered price contract. In each period t, a fraction of the intermediate firms choose new prices and fix them for N_p periods. Once prices are set, each intermediate goods producer must meet the forthcoming demand at the same prices. Since firms are assumed to set their prices in the consumer’s currency and intermediate goods markets are segmented across countries, the home firm i chooses P_H(i, s^t-1) in the home currency for sales to the home market and in the foreign currency for sales to the foreign market to maximize its expected profit given by

subject to P_H(i, s^t-1) = P_H(i, s^t) = ⋯ = P_H(i, s^t+N_p-1) and denote home and foreign demand for home good i at time t, respectively, and MC(i, s^t) = α^-α(1 − α)^-(1-α)W^1-α(s^t)P^α(s^t) denotes marginal cost for home firm i. All home firms have the same marginal cost since they have the same production function and face the same input prices.

The home government issues the home currency. Money supplies for the home country are assumed to follow a univariate process of the form

where G(s^t) denotes stochastic home money growth rates. The stochastic process for money growth in the home country is given by

where ξ_t is a home stochastic disturbance term and E[g] denotes the unconditional mean of home money growth rates. In our model, time varying risk premium is driven by time varying distributions of home and foreign money growth rates. Based on our estimation, we consider that the conditional variances of home money growth rates are time varying and follow a univariate GARCH (1,1) process:

where ρ_h denotes the persistent coefficient of conditional variance shocks, ρ_ξ denotes the kurtosis coefficient, var(ξ) denotes the unconditional variance of stochastic disturbances, h_t+1 denotes the conditional variance of home monetary shocks at time t, and ρ_h + ρ_ξ < 1. Then, where N(0,1) is a random number drawn from the normal distribution with mean zero and variance 1. We assume that the stochastic process for money in the foreign country is the same and the cross correlation between ξ_t and is zero.

The home government runs a balanced budget in each period. So, home nominal transfers are given by

An equilibrium for this economy is a collection of allocations for the home and foreign final goods producers Y(s^t), Y^∗(s^t); allocations and prices for the home and foreign intermediate goods producers indexed by i ∈ [0,1]; allocations for home and foreign households C(s^t), L(s^t), M(s^t), B(s^t), C^∗(s^t), L^∗(s^t), M^∗(s^t), B^∗(s^t); allocations for the home and foreign governments T(s^t), T^∗(s^t); prices P(s^t), W(s^t), P^∗(s^t), W^∗(s^t), Q(s^t+1|s^t); the nominal exchange rate ε(s^t) that satisfy the following conditions: (i) Optimality of final firms' behavior: taking the prices as given, allocations for both home and foreign final goods producers solve their problems; (ii) Optimality of intermediate firms’ behavior: taking all prices except its own as given, prices for both home and foreign intermediate goods producers solve their problems; (iii) Optimality of households’ behavior: taking prices as given, allocations for both home and foreign households solve their problems; (iv) Government's budget balance: both home and foreign governments run a balanced budget; (v) All markets clear including both final and intermediate goods markets, labor markets, and bonds markets. To make the economy stationary, all nominal variables are normalized by the level of the relevant money supply.

1. The Foreign Exchange Risk Premium

In this section, we derive the foreign exchange risk premium driven by home and foreign monetary volatilities. From equations (2-5) and (2-6), we derive the following risk sharing condition under complete asset markets:

where the exchange rate is linked to foreign and home nominal marginal rates of substitution. This relation holds regardless of frictions in goods markets such as price rigidities and deviations from PPP. Previous studies based on the Lucas-type exchange economy with complete markets also use this relation for studying the behavior of the risk premium. For example, Backus et al. (1993) investigate how habit persistence affects the foreign exchange risk premium, while setting the joint stochastic process of exchange rate, inflation, and consumption growth from the data. Instead, we let our sticky-price model generate the behaviors of those variables in response to monetary shocks. Using an arbitrage condition (covered interest parity), we define the forward premium by

where F_t denotes the forward exchange rate, E_t[⋅] is a mathematical conditional expectation, and E_t[Q_t+1] and denote the inverse of the home and foreign nominal interest rates, respectively. For simplicity, we henceforth suppress notation for state.

In order to derive the foreign exchange risk premium, we take second order approximations around a zero money growth steady state, while ignoring terms higher than second order. Then, the second order approximated version of equation (2-16) is

where a hat over a small letter denotes the log deviation of the corresponding capital letter except for the nominal exchange rate: is the log deviation of the nominal exchange rate at time t. This relation apparently looks similar to the one obtained by assuming that all relevant variables follow log-normal distributions. But our analysis only concerns with deviations from the steady state values. By taking logs on both sides of equation (2-17) and taking conditional expectations given the information at time t, we derive the log deviation of the expected exchange rate change

By subtracting equation (2-20) from (2-19) we derive the following foreign exchange rate risk premium

Equation (2-19) shows that foreign exchange rate risks originate from both home and foreign nominal interest rates: the risk premium increases as relative risks of holding foreign bonds become higher. Using the relation in equation (2-17), we can rewrite the relation for the risk premium in the following way

where are related to Jensen’s inequalities and is interpreted as the true risk premium following Engel (1992). Here, we omit time t variables since their conditional variances are zero at time t. Engel (1992) shows that the true risk premium is zero in the Lucas (1982) environment with flexible prices unless monetary shocks are correlated with real shocks; the true risk premium, however, arises endogenously in our model with sticky prices because monetary shocks affect both consumption and exchange rate.

III. CALIBRATION AND ESTIMATION OF THE FOREIGN EXCHANGE RISK PREMIUM

1. Calibration

The parameter values for the benchmark model are reported in Table 1. We begin by choosing parameter values for the utility function specified specified in (2-1). We set γ to 2 so that the intertemporal elasticity of substitution in labor supply is 0.5, which is within the range of estimates in the empirical labor literature. The discount factor β is set so that an annual interest rate is equal to 4%. Since preferences are separable between consumption and real money balances, both the consumption and interest elasticities of demand for money are tied with the level of risk aversion. First, we set the consumption elasticity of money demand following Mankiw and Summers (1986). Next, we set the level of risk aversion σ at 7 to match the relative standard deviation of the median nominal exchange rate change in the sample to the US consumption growth. This value seems relatively high compared to other studies in the literature. However, our numerical results on the volatilities of the risk premium and exchange rates do not depend much on this parameter value. As can be seen in detail later, the effects of monetary shocks on the marginal utility of consumption are almost irrelevant to the value of σ in our sticky-price model. Rather, the curvature parameter σ mainly affects the relative standard deviations of both the marginal rate of substitution and the real exchange rate to consumption, respectively. We set ϕ = 7, which governs the interest elasticity of money demand, as a consequence of parameterization on and σ. κ₁ is set so that the steady state velocity of money is 1.

We now consider the intermediate goods technology parameters. The cost share α of the composite intermediate goods in the production function (2-11) is calibrated in the following way. We first obtain the steady state ratio of intermediate goods to output

by combining the market clearing conditions for intermediate goods with the optimal condition obtained from the cost minimization problems of intermediate goods producers. We then set θ = 0.9 so that an annual markup is 11 %. This value is the same as the one used in CKM and Huang et al. (2004) and less than those in Christiano et al. (2005) and Bergin and Feenstra (2001) who set the markup to 46% and 50%, respectively. We finally set α = 0.7, following Huang et al. (2004) who find the ratio of intermediate input to the industrial production in the US manufacturing sector is 68 %. We set N_p = 4 so that prices are set for one year.

For the final goods technology parameters, we first set ρ so that the elasticity of substitution across countries is 1.5, following Backus et al. (1994). Estimates of this elasticity vary a lot across studies. In general, studies in the RBC literature consider estimates ranging between 1 and 2 as reliable. On the other hand, studies in the international trade literature report higher estimates ranging between 5 and 10, using micro (sectoral) data.5 However, our results remain unchanged with respect to changes in this parameter value. We now relate the home bias in the final goods production function to the share of imports and use a US import share of 0.15 to obtain values for α₁ and α₂. Since α₁ and N_p affect the degree of both international and domestic price adjustments and thus the risk premium, we report sensitivity analysis with respect to changes in these parameter values.

We now set parameter values for the money growth processes in (2-14)-(2-15). As reported in Table 2, the quarterly growth rates in M1 in the US contain strong ARCH components that support our specification for the process of time-varying conditional variances of money growth rates. For residual series in the regression of the form (2-14), we apply for ARCH LM tests for conditional homoskedasticity and reject the null hypothesis. This result is consistent with Hodrick (1989) for the monthly growth rates in M1 in the US, Canova and Marrinan (1993) for the monthly and quarterly growth rates in M1 in the US, and Bekeart (1996) for the weekly growth rates in M1 in the US. Parameter values in the AR(1)-GARCH (1,1) model of the forms (2-14) and (2-15) are jointly estimated using quarterly US data for M1 between the second quarter of 1973 and the third quarter of 2003, obtained from the Board of Governors of the Federal Reserve System Database: ρ_m = 0.66, ρ_h = 0.54, and ρ_ξ = 0.37. The unconditional variance of stochastic disturbances are set at 0.015² for all experiments.

2. Estimation of the Foreign Exchange Risk Premium

To estimate expected returns from currency speculation, we run the OLS regression of the form

following Cumby (1988), Backus et al. (1993), and Canova and Marrinan (1993). e_t denotes the log of the price of foreign currency in terms of home currency at quarter t, f_t denotes the log of quarter t home currency price of a one-quarter forward contract specifying delivery of one unit of foreign currency at quarter t + 1, and v_t+1 denotes an error term. We assume that the US is the home country. If expectations are rational and uncovered interest parity holds, then both b₁ and b₂ should be zero. Non-zero estimates of b₂ imply that the forward premium f_t − e_t may contain predictable components. We measure fitted values from regression (3-5) by the foreign exchange rate risk premium.

Data consists of quarterly spot and forward rates for the US dollar price of the Japanese yen, the British pound, the French franc, the Italian lira, and the German mark obtained from Data Resources Incorporated (DRI). The series for non-EU currencies run from the second quarter of 1973 to the third quarter of 2003 while the series for the Italian lira and the German mark end in the fourth quarter of 2001. The series for the French franc run from the first quarter of 1980 to the fourth quarter of 2001.

As reported in Panel A in Table 3, we find that the estimated slope coefficients are strictly positive but the French franc and the Italian lila are not statistically significant: estimates of the slope coefficient range from 0.89 for the French franc to 1.84 for the British pound. Previous empirical studies using monthly or weekly series have consistently documented non-zero estimates of the slope coefficient b₂ for a variety of currencies. Although magnitudes of the estimated slope coefficient are less than those in the previous studies, our results using quarterly series are consistent with them. We report properties of expected returns, interpreted as the foreign exchange risk premium in our study, in Panel B in Table 3. Expected returns are volatile and persistent: the standard deviations of these returns range from 0.008 for the French frac to 0.020 for the Japanese Yen and the autocorrelations range from 0.69 for the French franc to 0.91 for the German mark.

To derive Fama’s volatility relations, we decompose the estimated slope coefficient into two parts:

where 6 is mainly determined by the time-varying risk premium and is related to expectation errors. By assuming that expectations are rational, and the estimate is consistent, we can derive the following two necessary conditions for obtaining non zero values of the estimated slope coefficient from regression (3-2):

We call these two conditions Fama’s volatility relations and ask whether or not the benchmark model can generate these relations. The implication of the negative correlation between the expected rate of depreciation and the risk premium can be easily seen from excess return on foreign currency, e_t+1 − f_t, which is obtained by selling home currency in the forward market for foreign currency and by using that foreign currency to buy home currency at future spot rate. As the expected depreciation of the dollar becomes higher, that is, the conditional expectation of the future spot exchange rate becomes higher, the higher expected excess return should be required. And thus the risk premium should be negatively related to the expected depreciation.7

5)e.g., Harrigan (1993) and Obstfeld and Rogoff (2000).

6)For defining we omit the sample averages of the forward premium and the exchange rate change. This does not change the results because they are very small.

7)see, also, Hodrick and Srivastava (1986).

IV. RESULTS

The main question we ask in this paper is whether or not our sticky-price model can produce enough variation in the risk premium to explain the forward premium anomaly. In particular, we are interested if the model can generate Fama’s volatility relations. The numerical results in the benchmark economy as well as in other economies are reported in Table 4. The statistics in this table are averages of moments across 1000 simulations with a sample length of 120 periods each. The column labeled with ‘Bench’ represents the benchmark economy.

The main findings in the benchmark model are: (a) The variance of the risk premium is greater than (but close to) that of the expected rate of depreciation. The variance of the true risk premium is 0.25E-4 while that of the expected depreciation is 0.14E-4. The variances of predictable returns from currency speculation, interpreted as the risk premium, are 0.62E-4 for the French frac, which is the smallest value, 1.12E-4 for the German mark, which is the median value, and 4.08E-4 for the Japanese yen, which is the largest value in the sample. (b) The covariance of the risk premium with the expected rate of depreciation is negative. The cross correlation between these two quantities is -0.74. (c) The correlation between the forward premium and the risk premium is positive but close to zero. The cross correlation between the two quantities is 0.07. (d) The autocorrelation of the risk premium is in the range of our sample, whereas the forward premium is less persistent than those in the data. The autocorrelation of the risk premium is 0.78 in the benchmark model, whereas it is 0.68 for the Italian lira, which is the lowest value, and 0.91 for the German mark, which is the highest value. The autocorrelation of the forward premium is 0.33 in the benchmark model, whereas they are 0.73 for the Italian lira and 0.89 for the German mark, respectively. (e) The benchmark model produces volatilities and autocorrelations of both exchange rates and consumption matched with the data: for example, the standard deviations of both the nominal and real exchange rate changes are 0.061 and 0.067, respectively, while the corresponding sample median values are 0.061 and 0.062. Further, the standard deviation of consumption growth is 0.007, which is the same as that in the US consumption growth. We will discuss the autocorrelations of exchange rates and consumption below.

1. Near random walk behaviors and Fama’s volatility relations

The main mechanisms for obtaining results (a) and (b) are sticky prices and interest-sensitive money demand that generate near random walk behaviors of exchange rates and marginal utilities of consumption.

To understand near random walk behavior of the nominal exchange rate, we first substitute the home money market clearing condition into equation (2-3) and then take second order approximations:

where denotes the log-linearized home aggregate price index, denotes the log-linearized home money supply, denotes the log-linearized home marginal utility of consumption, denotes a steady state interest rate, denotes home nominal interest rate risks, and is a collection of second order terms derived from second order approximations on the home money market clearing condition. Here, we do not explain the economic interpretations of the second order terms except for the risk premium since our primary concern is the behavior of the risk premium.

The nominal exchange rate can then be derived by using both the home and foreign money market clearing conditions and the risk sharing condition (2-17) from the bonds markets:

where denotes the log-linearized real exchange rate. Equation (4-2) shows that, like as asset prices, the nominal exchange rate is determined in a present value model where the exchange rate is a discounted sum of current and expected future fundamentals. By assuming PPP holds for simplicity, we obtain a relation for the nominal exchange rate change from equation (4-2):

where One distinct feature of equation (4-3) is that dominates the effects of other terms on the exchange rate change at t + 1 since, as can be seen in calibration, that governs the discount factor in the determination of the nominal exchange rate is very small. This implies that the nominal exchange rate closely follows a random walk.8 This is notable in the sense that only matters for the determination of the risk premium. Consequently, the expected deprecation tends to be small. Note that near-random walk behavior of the nominal exchange rate arises regardless of the degree of persistence in the process of money growth rate as long as money demand is interest sensitive. As ϕ goes infinity, interest elasticity of money demand becomes zero so that the interest rate effects on the nominal exchange rate would vanish and the nominal exchange rate would no longer follow a near-random walk. This is one of the reasons why our model with interest-sensitive money demand can generate more volatile risk premia and much less variable expected exchange rate changes than previous studies that have the quantity equation with a unitary income velocity of money. Because of the same reason just mentioned, the model also generates persistence of the exchange rate change closely matched with the data: the autocorrelation of the exchange rate depreciation is -0.00 in the benchmark model, while it ranges from 0.027 for the German mark to 0.159 for the Italian lira in our sample.

Consequently, we find that variation in is large: the variance of is 0.12E-04, which is about half the variance of the true risk premium.

Further, the unconditional mean of is close to the unconditional variance of This is natural because the exchange rate is highly volatile and follows a near-random walk. But variation in is zero because prices do not respond to current monetary shocks in the benchmark model. Our results are consistent with previous studies about Jensen's equalities since those studies mainly focus on the behavior of 9

To discuss the role of staggered price setting in the determination of the risk premium, we solve equation (4-1) forward for p_t and rearrange it for the home marginal utility of consumption:

Here, consumption must respond to current monetary shocks to clear the money market because the aggregate price index does not change with respect to them as well as changes in the nominal exchange rate.10 As a result, sticky prices together with interest-sensitive money demand induce the marginal utility of consumption to be apparently determined in a similar way as the nominal exchange rate: the marginal utility of consumption is mainly driven by the discounted sum of current and expected future money supplies. However, in contrast to the determination of the nominal exchange rate, the effects of monetary shocks are also significantly affected by the degree of the price adjustment, which are summarized in the discounted sum of expected future marginal utilities of consumption and current price. For example, when N_p = 1, all firms set their prices at the beginning of each period before monetary shocks are realized and fix them only one period. In this case, consumption does not exhibit any persistence even if monetary shocks are persistent because future prices are fully adjusted right after the realization of monetary shocks. On the other hand, if prices are fixed for a certain periods in a staggered way, households would consider the effects of current monetary shocks on their future consumption since some fraction of the firms will not change their future prices in response to them. Therefore, the marginal utility of consumption becomes more volatile as price adjustments become slower.

To see this more precisely, we calculate the marginal rate of substitution in the two extreme cases: the marginal utility of consumption does not exhibit any persistence in the first case and follows a random walk in the second case. In the first case, the marginal rate of substitution is

where in equation (4-4). In the second case, the marginal rate of substitution is

where 11 As can be seen in equations (4-5) and (4-6), the marginal rate of substitution can be largely amplified when consumption follows a random walk: the effect of ξ_t+1 on the marginal rate of substitution is about ϕ times greater than when consumption exhibits no persistence.12 This example shows that a mechanism that increases persistent real effects of monetary shocks on consumption can also play a significant role in increasing the volatility of the risk premium. In our numerical experiments, when prices are fixed for one year in a staggered way, consumption exhibits a near-random walk: the autocorrelation of consumption growth rates in the benchmark model is -0.01, whereas it is -0.48 in the model with N_p = 1.13 Consequently, variation in the nominal interest rate significantly decreases with respect to the length of contract periods: the variance of the forward premium in the model with N_p = 1 is 0.12E-4, whereas it is 0.02E-4 in the benchmark model. The volatility of the marginal rate of substitution, on the other hand, increases with respect to the contract periods: the variance of the true risk premium in the model with N_p = 1 is 0.16E-4, while it is 0.25E-4 in the benchmark model. Although staggered price setting increases variation in the risk premium, its quantitative effects are not so impressive. This may be because staggered price setting, alone, may not be able to generate large persistence in consumption as in CKM (2000) and Christiano et al. (2005). Hence, in the next section, we consider nominal and real features that prove to lead to longer periods of endogenous price stickiness and thus persistence in real variables in response to monetary shocks.

We now discuss how the benchmark model is likely to produce the negative correlation between the expected depreciation and the risk premium. By taking conditional expectation on equation (4-3), we derive the expected rate of depreciation

where For deriving this relation, we use the condition that the risk premium is a function of time-varying conditional variances of home and foreign money growth rates and assume that the real exchange rate is zero for simplicity. Equation (4-7) illustrates the negative relation between the expected depreciation and the risk premium, holding other things constant. Using the forward premium can be derived:

Note that equations (4-7) and (4-8) would not be equal due to the presence of the risk premium. Hence, the forward premium anomaly may be reconciled with uncovered interest parity as long as the risk premium is highly volatile as Fama suggests. We obtain this result because exchange rate risks in the nominal exchange rate are transmitted from the home and foreign nominal interest rates via the intertemporal link of interest-sensitive money demand as shown in equation (4-3). This link is absent in a simple cash-in-advance-constraint model in which money demand is independent of interest rates.

2. Deviations from PPP

To study how much real exchange rate risks affect the risk premium, we compare the benchmark economy to an economy in which PPP holds. For this, we modify the assumptions of currency pricing and home bias in the final goods production function in the benchmark economy. When prices are preset in the consumer’s currency, the law of one price does not hold because there is no pass-through of the exchange rate to import prices. Hence, home monetary shocks mostly affect the home marginal utility of consumption even in the presence of complete asset markets.14 On the other hand, when intermediate goods prices are set by producers’ currency, import prices completely absorb changes in the nominal exchange rate. That is, the relative price between home and foreign goods fluctuates even if prices are unchanged in terms of domestic currencies. As a result, each country’s aggregate consumption is internationally diversified. Our experiments show that real exchange risks significantly increase variation in the risk premium. The column labeled with ‘PPP’ reports statistics from the economy in which prices are set in producer’s currency and α₁ = 0.5: the variance of the true risk premium is 0.09E-4, which is about twice less than that of the true risk premium in the benchmark economy.

There are two elements in the benchmark model that cause deviations from PPP: one is the segmentation of international goods markets combined with local currency pricing and the other is home bias in the final goods production function. We conduct some experiments to see which of these two elements more significantly affect the volatility of the risk premium. First, we modify the degree of home bias in the final goods production by setting α₁ = 0.5, while keeping the assumptions of segmentation of international goods markets and local currency pricing. The results in this economy are very similar to those in the benchmark economy: the variances of the true risk premium and the expected exchange rate change are 0.23E-4 and 0.12E-4, respectively. Second, we modify the assumption of currency pricing in the benchmark model so that the law of one price holds, while keeping the assumption of home bias in the final goods production function. The results are reported in the column labeled with ‘PCPH’ in Table 4. The modified model also produces similar results as the benchmark model. However, the results are sensitive to the degree of openness. The variance of the true risk premium is 0.16E-4 when α₁ = 0.85, which is a bit lower than that in the benchmark model. But the variance of the true risk premium is 0.09E-4 when α₁ = 0.5.

8)We obtain this result since the discount factor is close to one in our present value model. See, Engel and West (2005) for the sufficient conditions that generate the results.

9)For example, see Engel (1984) and Backus et al. (1993).

10)This transmission mechanism would disappear if prices are flexible because they will immediately adjust in response to monetary shocks. Further, this mechanism will be weakened in the model where PPP holds because the pass-through from nominal exchange rate movements to import prices is significant.

11)If consumption follows a random walk, should be equal to once second order terms are ignored.

12)The analogous number used in CKM is about 2.5. And their estimate of interest elasticity of money demand is similar to that of Mankiw and Summers (1986) and smaller than that of Stock and Watson (1993).

13)The autocorrelation of the US consumption growth is 0.23.

14)This may be one of the reasons that both complete and incomplete asset markets have very similar results in CKM.

V. FURTHER ANALYSIS

In the benchmark model, we link persistence of the marginal utility of consumption to the volatility of the risk premium. For example, staggered pricing setting increases variation in the marginal rate of substitution because of gradual price adjustments. However, its quantitative effects on the risk premium are not enough to match with the data. Hence, we consider some mechanisms from the monetary business cycle literature that make price adjustments further slower: sticky wages and capital utilization. In addition, we consider habit persistence in consumption that has been widely used for increasing variation in the marginal rate of substitution in both the risk premium and equity premium literatures.

1. Habit Persistence in Consumption

We first investigate quantitative implications of habit persistence for the risk premium. Previous studies find that introducing a non-linear preference specification to an otherwise standard general equilibrium model tends to increase variation in the risk premium because it allows moderate consumption fluctuations to have large impacts on the marginal utility of consumption.

To introduce habit persistence in consumption into the benchmark model, we follow Christiano et al. (2005). Preference for the home representative household is given by the following expected utility function:

where b indicates habit persistence or consumption durability. If b = 0, then the preferences are time additive, if b > 0 then consumption exhibits habit persistence, and if b < 0 then consumption is durable. In our quantitative study, we set b equal to 0.7, following Christiano et al. (2005).

Our benchmark model with habit persistence generates unrealistically high values of the relative standard deviation of the real exchange rate to consumption, although the absolute volatilities of marginal utility of consumption and exchange rates do not depend much on the risk aversion and habit persistent parameters. Hence, we set σ = 2 in order to reduce the relative standard deviation of the real exchange rate. Further, we set ϕ = 10.26 following Christiano et al. (2005). The results from these modifications are reported in the column labeled with ‘Habit’ in Table 4. Our numerical results show that, in contrast to previous studies that abstract from production, habit persistence does not play a significant role in the determination of the risk premium. Variations of the risk premium, the forward premium, and the expected depreciation in this modified model do not much increase: The variance of the true risk premium is 0.26E-4, while those of the forward premium and the expected depreciation are 0.05E-4 and 0.14E-4.

The reason why there is not much difference between the two models with and without habit persistence can be easily seen when the money demand function is static

Since prices are fixed before monetary shocks are realized, the marginal utility of consumption should change one-for-one with changes in nominal money balances in order to clear money markets regardless of whether or not consumption exhibits habit persistence. This implies that the conditional volatility of the marginal rate of substitution is independent of the risk aversion as well as the habit persistence parameters. That is, in contrast to endowment economies, the effect of habit persistence (or the degree of risk aversion) on the marginal utility of consumption is exactly offset by that of the elasticity of intertemporal substitution.15 Since only this conditional volatility matters for the determination of the risk premium, the introduction of habit persistence does not much improve the result on the variation of the risk premium in the benchmark model. Similarly, although consumption is more persistent in the habit persistence model than in the benchmark model by construction, it would not help to increase the volatility of marginal rate of substitution in our framework. By inserting equation (5-2) and the foreign counterpart into the risk sharing condition (2-15), one can easily see that the conditional volatilities of changes in both the nominal and real exchange rates do not depend on these two parameters either.

2. Sticky Wages

Huang and Liu (2002) and Christiano et al. (2005) find that staggered wage setting can generate more persistent aggregate quantities than staggered price setting. To study this effect on the variation of the marginal rate of substitution, we extend the benchmark model by assuming that labor inputs are differentiated and households set wages according to a variant of the Taylor staggered wage contract.

In the presence of sticky wages, the household’s problem is changed, while the problems of the final goods producers and intermediate goods producers remain the same as before. Following Christiano et al. (2005), we introduce the home competitive representative firm that produces aggregate labor L(s^t) by combing a continuum of differentiated labor inputs, indexed by n = [0,1], using the technology:

where L(n, s^t) denotes home household n’s labor service and ϑ denotes substitutability between differentiated labor inputs. The home competitive firm takes as given wages W(s^t), W(n, s^t-1) for n = [0,1] to maximize its profits given by

subject to (5-3). W(s^t) is the aggregate wage rate and W(n, s^t-1) is the price of home labor input n at time t. Prices of differentiated labor input services do not depend on s^t because they are set before period t shocks are realized. Wages in home country are denominated in home currency. From this problem, the demand function for labor input n is defined by

Using the zero profit condition, the price of the composite labor service is defined by and home final labor service is distributed to the home intermediate goods producers according to Here, L(i, s^t) denotes the home composite labor service purchased by home intermediate goods producer i.

In the beginning of each period t, a fraction of home households, set wage W(n, s^t-1) and fix it for the subsequent N_w periods. Household n maximizes his expected utility given in (2-1) subject to the budget constraint and demand for labor input n

Following CKM, we choose initial bond holdings so that each household has the same present discounted value of income. Then, the optimal wage condition for home household n is

Equations (2-3) and (2-5) and initial bond conditions guarantee that U_c(n, s^τ) and U_m(n, s^τ) are equal across households. For calibration, we set ϑ = 0.87 and N_w = 4 from CKM.

The results from this modification are reported in the column labeled with ‘Sticky Wages’ in Table 4. The modified model improves the benchmark model’s performance on the variation of the true risk premium slightly: the variance of the true risk premium is 0.31E-4, while it is 0.25E-4 in the benchmark model.

3. Capital Utilization

In this section, we consider another mechanism that increases persistent real effects of monetary shocks and thus might increase the conditional variation of the marginal rate of substitution: variable capital utilization. For this purpose, we extend the benchmark model by introducing variable capital utilization and investment adjustment costs from Christiano et al. (2005).

We assume that households own capital and decide how many units of capital services to supply. Accordingly, the home representative household’s budget constraint is modified in the following way:

where I(s^t) denotes the purchase of investment goods, denotes the physical stock of capital at the end of time t − 1, v(s^t) denotes the utilization rate of capital, a(v) denotes the capital utilization function, R^k(s^t) denotes the price of capital service, denotes the stock of installed capital at time t, and denotes capital service at time t. The household’s capital stock evolves according to:

where δ denotes the depreciation rate of physical capital and F(I(s^t), denotes investment adjustment costs. We assume the same properties of functions α(⋅) and F(⋅,⋅) as Christiano et al. (2005) for our calibration. In terms of the capital utilization function α(⋅), we assume that v = 1 and α(1) = 0 in the steady state. And for function S(⋅), we assume S(1) = S′(1) = 0 and S′′(1) > 1.

The production function for intermediate goods i is modified by:

where α_κ is the cost share for capital service and set at Accordingly, the resource constraint is modified in the following way:

The results from these modifications are reported in the column labeled with “Capital Util” in Table 4. The quantitative performance of the model with capital utilization and investment adjustment costs on the volatilities of the true risk premium, the forward premium, and the expected depreciations is very similar to that of the benchmark model: the variance of the true risk premium is 0.24E-4, while those of the forward premium and the expected depreciation are 0.03E-4 and 0.14E-4, respectively. Finally, we add to the benchmark model all real and nominal frictions that we have considered: habit persistence in consumption, sticky wages, capital utilization, and investment adjustment costs. The results from these modifications are reported in the column labeled with “All” in Table 4. Again, we find that the model with all these features improves its quantitative performance slightly: the variance of the true risk premium is 0.30E-4, while those of the forward premium and the expected depreciation are 0.04E-4 and 0.23E-4, respectively.

VI. CONCLUDING REMARKS

Studies such as CKM (2000) and Christiano et al. (2005) in the monetary business cycle literature focus on developing mechanisms that lead to endogenous price stickiness and thus persistent output movements. Based on their frameworks, in the present paper, we focus on quantitative implications of persistent real effects of monetary shocks for the volatility of the risk premium in foreign exchange markets. In particular, our study links random walk behaviors of both exchange rates and consumption to variation in the risk premium and to Fama’s volatility relations in order to account for the forward premium anomaly. In the benchmark model, elastic money demand and persistent money growth produce a near-random walk behavior of the nominal exchange rate. Further, when they interact with the frictions in goods markets that affect the degree of price adjustments, the model can also produce a near-random walk behavior of the marginal utility of consumption. With these features, the benchmark model generates Fama’s volatility relations since both the exchange rate and the marginal rate of substitution display large variation, while both the expected depreciation and interest rates exhibit small variation.

However, the risk premium in the benchmark model is less volatile than in the data: the variance of the true risk premium is similar to that of the expected depreciation. We interpret this as staggered price setting, by itself, may not produce enough persistence in the marginal utility of consumption to generate the volatility of the risk premium observed in the data. To improve this, we conduct several experiments using various nominal and real frictions that produce the right persistence in real variables in the monetary business cycle literature. The models with these features improve on the variation of the risk premium although their quantitative effects are not so large. But we do not view this as discouraging. The volatility of price changes in these models is much larger than in the data. This suggests that there is still room for making price adjustments even slower and thus increasing persistence in the marginal utility of consumption. We leave this for future study.

In addition to these features, we find that the risk premium is determined quite differently between an endowment economy and a production economy. For example, studies that introduce habit persistence in consumption into the Lucas model succeed in increasing variation in the marginal utility of consumption. In these stylized frameworks, the marginal rate of substitution depends mainly on the risk aversion and habit persistent parameters since the equilibrium consumption process is exogenously given. However, raising the degree of risk aversion and/or introducing habit persistence do not help to increase the volatility of the marginal utility of consumption in our sticky-price model with production because a rise in risk aversion is offset by a fall in the elasticity of intertemporal substitution. This result is consistent with those in the equity-premium studies with production economies.16

15)See, also, equation (4-5) and (4-6). We also conduct some experiments by varying σ from 1 to 100 to study how the degree of risk aversion affects our results: variances of the true risk premium are 0.24E-4 for σ = 1 and 0.25E-4 for σ = 100, while those of the expected depreciation are 0.16E-4 for σ = 1 and 0.14E-4 for σ = 100. These results show that rasing the degree of risk aversion does not help to increase the volatilities of the marginal utility of consumption and exchange rates. Rather, it increases the relative standard deviation of the real exchange rate to consumption because households’ incentives for smoothing consumption increase as the elasticity of intertemporal substitution becomes lower: the relative standard deviation of the real exchange rate to consumption is 1.31 for σ = 1 and 128.56 for σ = 100.

16)For example, see, Rouwenhorst (1995).