A Theory of Exchange Rates and the Term Structure of Interest Rates


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1 Review of Development Economics, 17(1), 74 87, 013 DOI: /roe.1016 A Theory of Exchange Rates an the Term Structure of Interest Rates HyoungSeok Lim an Masao Ogaki* Abstract This paper efines the concepts of inirect an irect risk premium effects an analyzes their properties in an exchange rate moel. In the moel, these effects are enogenously etermine in a rational expectations equilibrium. For the effect of an interest rate shock, they have the opposite signs an the inirect risk premium effect can ominate the irect risk premium effect uner reasonable parameters. This means that omestic shortterm bons an foreign bons are complements in the moel even though omestic longterm bons an foreign bons are substitutes. This moel, focusing on the inirect risk premium effect an on the term structure of interest rates, can be combine with a small sample bias approach to explain stylize facts about the forwar premium anomaly, which is foun for shortterm interest rates, but not for longterm interest rates. 1. Introuction Uncovere Interest Parity (UIP) states that the interest rate return on a omestic currency asset shoul equal the interest rate on each foreign currency asset, less the expecte appreciation of the omestic currency. However, contrary to what UIP woul preict, many empirical finings about shortterm interest rates have shown that currencies with high interest rates ten to appreciate. This is calle the forwar premium anomaly. On the other han, more favorable evience for UIP has been foun for longterm interest rates. This paper efines the concept of an inirect risk premium effect an analyzes properties of this effect in an exchange rate moel. To investigate the role of the inirect risk premium effect in explaining the stylize facts mentione above, we buil a partial equilibrium moel of exchange rate etermination for a small open economy. The omestic investors have a constant absolute risk aversion utility function over their wealth in the next perio. The asset returns are normally istribute conitionally on the available information. We assume that there are three assets in the moel: one risk free asset omestic shortterm bons; an, two risky assets omestic longterm bons an foreign bons. The investors are also assume to have a short investment horizon in our moel. Employers of professional traers active in foreign exchange markets are likely to assess traers base on their shorthorizon performances. Therefore, our short investment horizon assumption is justifiable. Given the conitional expectations an variances of all risky assets, we can ecompose the effect of a change in the omestic shortterm interest rate on the eman for * Lim: Korea Institute of Finance, MyungDong, ChungGu, Seoul , Korea. Tel: ; Fax: ; Ogaki: (corresponing author): Department of Economics, Keio University, Mita, Tokyo , Japan. Tel: ; Fax: ; We gratefully acknowlege the very helpful comments from Paul Evans, Lars Hansen, Poksang Lam, Nelson Mark, Masaya Sakuragawa, an two anonymous referees, as well as seminar participants at Ohio State University an the University of Rochester. Ogaki s research is supporte by GrantsinAi for Scientific Research (B)
2 EXCHANGE RATES AND THE TERM STRUCTURE OF INTEREST RATES 75 foreign bons into two components. First is the irect risk premium effect. It is the change in eman ue to changes in the risk premium for foreign bons when the risk premium for omestic longterm bons is kept constant. The other is the inirect risk premium effect. It is the change in eman ue to changes in the risk premium for omestic longterm bons when the risk premium for foreign bons is kept constant. The change in the eman for foreign bons is the sum of these irect an inirect risk premium effects. In the special case of risk neutral investors, the inirect risk premium effect oes not play any role. However, when investors are risk averse, it is necessary to evaluate both the irect an inirect risk premium effects in orer to stuy how the foreign exchange rate changes when the omestic shortterm interest rate changes. The irect an inirect risk premium effects are properties of the eman for foreign bons, given the istributions of the asset returns an wealth, which are conitional on the information available to the investors. In orer to examine how these effects work in equilibrium, we consier the term structure of interest rates, where some shocks to shortterm interest rates are not transmitte to longterm interest rates, even though they affect the risk premium of long term bons. Let us suppose that the omestic shortterm interest rates rise, but the longterm interest rates o not change. First, the risk premium for foreign bons falls an the irect risk premium effect lowers the eman for foreign bons without changing the exchange rate. Since the supply for foreign bons is essentially fixe in the short term by the cumulative current balance in the moel, the omestic currency appreciates, creating expecte future epreciation of the currency in orer to restore equilibrium. Secon, the risk premium for omestic longterm bons also falls as the omestic shortterm interest rates rise. If the conitional covariance of the two risky assets returns is positive, the inirect risk premium effect increases the eman for foreign bons. In orer to restore equilibrium, the omestic currency must epreciate this perio, creating an expecte appreciation in response to the inirect risk premium effect. The sign an magnitue of the inirect risk premium effect epens on the conitional covariance. In the partial equilibrium moel with given stochastic processes for interest rates, we enogenously erive the eman for foreign bons by solving for the rational expectations equilibrium of the conitional expectation, variance, an covariance of the exchange rate. In equilibrium, the conitional covariance of the two risky assets returns is positive, an the irect an inirect risk premium effects have opposite signs. We show that uner some reasonable parameter configurations, the inirect risk premium effect ominates the irect risk premium effect, even when the egree of risk aversion is low. As a result, the omestic currency epreciates when the omestic shortterm interest rates rise without the increase of longterm interest rates. This counterintuitive result helps in making the moel more consistent with the stylize facts regaring UIP. On the other han, when omestic longterm interest rates also rise with omestic shortterm interest rates, the risk premium for omestic longterm bons oes not change. In this case, the inirect risk premium effect oes not affect the equilibrium exchange rate. Therefore, the omestic currency appreciates when the shortterm an longterm interest rates rise together. In this way, our moel is consistent with stylize facts about exchange rates an the longterm interest rates. The intuition behin these results for irect an inirect risk premium effects can be generalize with Ogaki (1990) s concepts of irect an inirect substitution effects to the cases of other utility functions an more than three assets as explaine in section 4.
3 76 HyoungSeok Lim an Masao Ogaki The moel in this paper has policy implications. It implies that the effectiveness of central bank attempts to affect exchange rates through the control of shortterm interest rates epens on the responsiveness of longterm interest rates to changes in shortterm interest rates. The rest of the paper is organize as follows: section provies a literature review, while section 3 efines the irect an inirect risk premium effects. Section 4 presents the moel an section 5 erives the rational expectations equilibrium. An finally, section 6 investigates the moel s implications on the relationship between the exchange rate an the term structure of interest rates, then conclues with irections for future work.. Literature Review For shortterm interest rates an forwar exchange rates, UIP is typically rejecte (Engel, 1996). As Fama (1984) fins, one reason many papers reject UIP is that the regression of future epreciation on the current forwar premium yiels negative estimates of the slope coefficient. This is calle the forwar premium anomaly. For longterm interest rates, more favorable evience for UIP has been foun (Chinn, 006). Direct evience is given by recent papers, such as Alexius (001) an Chinn an Mereith (005). They fin that regressions of future epreciation over a long horizon on the current longterm interest rate ifferential typically yiel significantly positive estimates of the slope coefficient. Inirect evience has been foun in the stanar exchange rate moels, such as Meese an Rogoff (1988) an Baxter (1994). They show that the longterm interest rate ifferential is more consistent with the assumptions of UIP an long term PPP than the shortterm rate ifferential. Similarly, implications of stanar exchange rate moels hol better for longhorizon ata than for shorthorizon ata. It is challenging to fin an economic explanation for the forwar premium anomaly for shortterm interest rates. Neither the stanar consumptionbase asset pricing moel with riskaverse investors (Mark an Wu, 1998) nor the ynamic term structure moel (Wu, 005) can explain it. Several explanations have attempte to meet this challenge. Alvarez et al. (00) construct a moel of segmente asset markets which can be consistent with the forwar premium anomaly. McCallum (1994) an Chinn an Mereith (004) provie an explanation for the forwar premium anomaly base on policy reactions. However, in their moels, an unspecifie error term is necessary for the uncovere interest parity relationship. Bacchetta an van Wincoop (010) show that infrequent foreign currency portfolio ecisions can explain the anomaly. Fisher (006) explains the forwar premium anomaly in terms of a moel where agents have iverse prior beliefs about omestic an foreign inflation. If some agents have iffuse priors about a country s inflation process, then the one month forwar rate will be negatively correlate with realize epreciations. Except for Chinn an Mereith (004), these explanations are focuse on the anomaly for shortterm interest rates an o not show formal analysis of whether or not their explanations are also consistent with more favorable evience for UIP for longterm interest rates. Another way to explain the forwar premium anomaly emphasizes small sample econometric problems. Baillie an Bollerslev (000), for example, argue that the forwar premium anomaly can be viewe mainly as a statistical phenomenon cause by small sample sizes an persistent autocorrelation in the forwar premium (Maynar an Phillips, 001).
4 EXCHANGE RATES AND THE TERM STRUCTURE OF INTEREST RATES 77 In this paper, we show that the inirect risk premium effect alone oes not solve the anomaly for shortterm interest rates. However, the effect can be complementary to a solution for the anomaly in builing a moel that is consistent with both the anomaly an more favorable evience for UIP for longterm interest rates. We also show that our moel complements the explanation that emphasizes the small sample econometric issue. We observe these patterns with high probability in small samples. However, the inirect risk premium effect coul also be introuce to any economic moel that solves the anomaly for shortterm interest rates. 3. Inirect Risk Premium Effect This section efines the concept of the inirect risk premium effect. Consier an economic agent who consiers allocating his wealth between N ifferent assets in perio t. Given the istributions of all returns in the future an the variance, covariance an higher moments of returns from perio t to perio t + 1 conitional on the information set available for perio t, we can write the eman function for an asset i as A i,t (r 1, t,...,r i, t,...,r N,,t, r t, W t), where r j, t is the risk premium for asset j (j = 1,...,N), r t is the riskfree shortterm interest rate, an W t is the wealth. In this paper, we are intereste in the effect of a change in r t on the eman, given the expecte returns of the assets. Because a rise in r t ecreases each risk premium, the effect is given by: Ait, Ajt, A + ρ ρ r it, j i jt, it, t (1) If the agent is risk neutral, then the first term is infinite an ominates the other two terms. We efine the irect risk premium effect to be this term, A it,. We also ρ it, efine the inirect risk premium effect, 1 as A jt, j i ρ. jt, Our intuition tens to focus on the irect risk premium effect, but we show that the inirect risk premium effect can ominate the irect risk premium effect, even when the agent s risk aversion is fairly small. In our moel, we assume that the agent s absolute risk aversion is constant, an the returns are normally istribute, conitional on the information set. Uner these assumptions, the thir term in equation (1) is zero. 4. The Moel This paper aopts a simple partial equilibrium exchange rate moel following Driskill an McCafferty (1980). We enogenously erive the eman for foreign bons by solving for the rational expectation of the covariance, so that the covariance assume by agents is consistent with the one implie by the eman function. It is technically ifficult to solve for the rational expectation of the covariance in complicate asset pricing moels. For this reason, we employ three asset moels. Consier a partial equilibrium moel of exchange rate etermination. For simplicity, the overall price level is assume to be constant. Alternatively, all variables can be consiere to be measure in real terms. Investors live for two perios, an the same number of investors is born every perio. There are three assets: omestic shortterm bons (B S, t), omestic longterm bons (B L, t), an foreign bons (B F, t). Since the foreign interest rate is assume to be constant, the foreign short an longterm bons
5 78 HyoungSeok Lim an Masao Ogaki are perfect substitutes an o not nee to be istinguishe. The omestic short an longterm bons are iscount bons paying one unit of the omestic currency after one perio an two perios, respectively. The foreign bons behave in the same manner. At time t, a representative investor allocates his/her initial wealth W t among the three assets an he/she collects the payoffs pai by the assets he hols at the beginning of time t + 1. Let q t be the price of omestic longterm bons at time t. Let r t be the omestic shortterm interest rate, an let R t be the omestic longterm interest rate. Then, the rate of return on holing omestic longterm bons for one 1 1 perio (r L, t) is q 1 t. Since qt =, we have r q + r Lt, = ( 1+ Rt) ( 1+ R ) t 1 t Rt rt + rt ( + Rt) The risk premium for omestic longterm bons (r L, t) is efine to be the ifference between the expecte rate of return on holing longterm bons for one perio an the rate of return for shortterm bons. Thus, we have: ( ) 1 { ( )} 1, () ρ Lt, = E t r Lt, r t R t r t + E t r t + where E t is the expectation operator conitional on the information set in perio t, W t. We assume that W t inclues the current an past values of r t, R t, r t *, R t *, an s t, where r t *, an R t * are the foreign short an longterm interest rates, respectively, an s t is the natural log of the exchange rate expresse in terms of the omestic currency. The rate of return on holing foreign bons for one perio in terms of the omestic currency (r F,t) is rt* + st+1 st. Let the risk premium for foreign bons (r F,t) enote the ifference between the expecte rate of return on holing foreign bons for one perio an the rate of return for shortterm bons. Then, ρ Ft, = E t ( r F, t) r t = r t * + E t ( s t +1 ) s t r t. The moel assumes that, at time t, a representative investor with a constant absolute risk aversion utility function maximizes his/her expecte utility of wealth at the kw e t +1 beginning of the time t + 1(= W t+1) subject to a buget constraint; max Et k, s.t. Wt = BS, t + BLt, + BFt,, where k is the coefficient of absolute risk aversion, an enotes eman, so that omestic currency amounts investe in omestic short, longterm, an foreign bons are B St,, B Lt, an B Ft,, respectively. W t is the initial wealth at time t, an the value of an investor s assets at the beginning of time t + 1, W t+1, satisfies: Wt+ 1 = BS, t( 1+ rt) + BL, t( 1+ rlt, ) + BF, t( 1+ rft, ). In the partial equilibrium moel, the stochastic processes for the interest rates are exogenously given, an the utility function is parameterize. The equilibrium s exchange rate satisfies the foreign bons market clearing conition, BFt, = BFt,, where s B Ft, is the supply of foreign bons to the omestic resients. It is assume to be equal to the cumulative current account balance an to follow the ynamic equation: s s BFt, = BFt, 1 + Ct. An, C t is the current account balance in perio t that satisfies: Ct = a+ b St + ut, where b is a positive number, an u t is a trae shock which is assume to be white noise with variance σ u. Suppose that W t+1 is normally istribute conitional on W t an that the measure of the absolute risk aversion, k, is a positive constant. Uner these assumptions, a representative investor s optimization problem is equivalent to maximizing: t
6 EXCHANGE RATES AND THE TERM STRUCTURE OF INTEREST RATES 79 k max ( ) var ( ) { B,, B E, } t Wt+ 1 t Wt+ 1, where Et( Wt+ 1) = Wt( 1+ rt) + BL, t( ρlt, ) + BF, t( ρft, ) Ft Lt an var t( Wt ) ( BL, t) var t( rt ) ( BF, t) var t( st ) ( BL + 1 = , t)( BFt, )cov t( rt+ 1, st+ 1 ). Firstorer conitions with respect to B Ft, an B Lt, are ρ Ft, kb ( Ft, )var t( st+1) + kb ( Lt, )cov t( rt+ 1, st+ 1)= 0 an ρ Lt, kb ( Lt, )var t( rt+ 1) + kb ( F, t)cov t( rt+ 1, st+ 1) = 0, respectively. Solving for B Ft, an B Lt, yiels the eman functions for foreign bons an omestic longterm bons, respectively. In particular, the eman for the foreign bon is: B Ft, ρft, ρlt, ψ ρft, ψ φ ρlt, [, ] =. (3) An, the eman for the omestic longterm bon is: s B Lt, ρft, ρlt, ψ φ ρft, ψ σ = + Lt, σ ρ r [, ]. (4) Here, ψ = 1 k σ 1 s ( cor ), φ = 1 cov, σ σ = s E t [ s t + 1 E t ( s t + 1)], σ r = r Et[ rt+ 1 Et( tt+ 1)] 1, cov, = E t[{s t+1  E t(s t+1)}{r t+1  E t(r t+1)}], an cor = cov, σ s σr respectively. The eman function for foreign bons, equation (3), epens on both the conitional covariance between the exchange rate an the shortterm interest rate (cov) an the conitional variance of the exchange rate ( σ s ). At the same time, the stochastic processes of the exchange rate an cov also rely on the eman function for foreign bons. Therefore, in orer to solve for a rational expectations equilibrium, the values of cov an σ s must be consistent with the stochastic process of the exchange rate implie by the eman function for foreign bons. When the shortterm interest rate rises, there are two opposite effects on the eman for foreign bons given the secon moments of the exchange rate an the shortterm interest rate. These effects are the irect an inirect risk premium effects we efine in section 3. The irect risk premium effect is shown in the first term of equation (3). This effect is the change in the eman for foreign bons when the shortterm interest rates rise, holing the risk premium for longterm bons constant. This effect is equal to y an is negative. The inirect risk premium effect is the secon term of equation (3). This effect is the change in the eman for foreign bons when the shortterm interest rate rises, holing the risk premium for foreign bons constant. This effect is equal to yf. In the rational expectations equilibrium erive in the next section, cov is negative, which implies that the inirect risk premium effect is positive. An intuitive explanation of the inirect risk premium effect is as follows: If the shortterm interest rate unexpectely rises, the price of a longterm bon falls an longterm bon holers suffer an unexpecte capital loss. When cov is negative, the exchange rate tens to appreciate causing investors to suffer an aitional unexpecte loss if they hol foreign bons. Therefore, as long as an increase in the shortterm interest rate is associate with an appreciation of the omestic currency, risk averse agents will want to avoi holing both longterm bons an foreign bons. When this association is stronger, investors are more willing to substitute between omestic longterm bons an foreign bons. In particular, when an increase in shortterm interest rates reuces the risk premium for longterm bons, risk averse investors
7 80 HyoungSeok Lim an Masao Ogaki want to ajust their portfolios towar holing more foreign bons an fewer longterm bons. This inirect risk premium effect allows the eman for foreign bons to increase when the shortterm interest rate rises. The existence of two opposite effects on the eman for foreign bons implies that the impact of a rise in the shortterm interest rate on the eman for foreign bons epens on the relative strengths of these two effects. The inirect risk premium effect ominates the irect risk premium effect if an only if f > 1. Therefore, f coul be thought of as the relative magnitue of the inirect risk premium effect. In the next section, it will be shown that f is greater than 1 uner reasonable parameter configurations. The intuition for inirect an irect risk premium effects can be generalize with Ogaki (1990) s concepts of irect an inirect substitution effects to the cases of other utility functions an more than three assets. Given the secon moments, equations (3) an (4) give the eman functions for foreign bons an longterm bons, respectively, as functions of expecte returns. Hence it is possible to efine substitution an income effects for changes in expecte returns as in Blanchar an Plantes (1977) an Royama an Hamaa (1967). Because absolute risk aversion is assume to be constant, income effects o not appear in the eman for risky assets, notably foreign bons an longterm bons. Therefore, the price effects which appear in equations (3) an (4) are also the substitution effects. We can ecompose the substitution effect into irect an inirect substitution effects. Even though the inirect substitution effect is not equal to the inirect risk premium effect, the inirect risk premium effect ominates the irect risk premium effect if an only if the inirect substitution effect ominates the irect substitution effect. As we show in the next section, the inirect risk premium effect ominates with reasonable parameter values in our moel. In such cases, omestic shortterm bons an foreign bons are complements. For the general utility function with any number of assets, a substitute of a substitute is always an inirect complement. In our moel, foreign bons an omestic longterm bons are substitutes, an both omestic short an longterm bons are substitutes. This means that omestic shortterm bons an foreign bons are inirect complements. 5. The Rational Expectations Equilibrium We use the moel presente in section 4 to erive the rational expectations equilibrium. The stochastic processes of interest rates are assume to be as follows: rt = μ+ et + ε, t (5) Rt = 1 + μ 1 + ( + c ) et, (6) 1 an r t * = μ, R * 1 t = + μ, where e t is a persistent interest rate shock an e t is a temporary interest rate shock. It is assume that e t follows an AR(1) process: e = ce + v, c <, (7) t t 1 t 1 an that it is inepenent of u t. It is also assume that e t, an v t are white noise with variance σ ε an σ v, respectively, an that they are inepenent of both each other an u t. 3 Finally, an m are positive numbers.
8 EXCHANGE RATES AND THE TERM STRUCTURE OF INTEREST RATES 81 It follows that: Et( rt+ 1) = μ + cet, (8) an from equation () an equation (8), ρlt, = εt. (9) For our purposes, we assume that the risk premium for longterm bons is nonzero. As shown in equation (9), the assumption employe here is that only e t is transmitte to the longterm interest rate, 4 so the risk premium is equal to the mean of the longterm interest rate plus a temporary interest rate shock. Define η σ ε =, which may be interprete as the measure of substitution between σ e both short an longterm bons. If h = 0, then the risk premium for longterm bons will be the mean of the longterm interest rate, implying that the short an the longterm bon will become more substitutable. The greater the magnitue of h, the smaller the egree of the substitution. Let L be the lag operator. Then, the equilibrium conition in perio t is, E[ A ( L) s ] = a+ D, t 0 t 0 (10) where A 0(L) =yl 1 s + (b + y) an D0 = ut BF, t 1 ψφ ψet + ψ( φ 1) ε t. Taking conitional expectations with respect to W t on both sies yiels the equilibrium conition for perio t + 1: Et[ A( L) st+ 1] = a+ D1, (11) where A(L) =yl 1 + (b + y)  yl an D 1 = y(1  c)e t  y(f  1)e t. Analogously, taking expectations conitional on W t on both sies yiels the equilibrium conition for any perio t = t, where t : Et[ A( L) st+ τ ] = a+ D, (1) where D = y(1  c)e tc t1. Solving equations (10), (11) an (1) as a ifference equation system of E t(s t+t) with respect to t provies the unique sale point solution for s t: s = s ( ) ( ) ( ) + 1 λ 1 λ s λ u B, 1 b b c e t λφ ( 1) εt, (13) 1 λ t t F t ( ) a where s 1 λ = φψ is the longrun equilibrium exchange rate that clears the b b current account, an ψ λ = 1+ b b 1+ 4 ψ ψ b. (14) It can be shown that 0 < l < 1, l/ y > 0, lim y 0l = 0, an lim y l = 1. Equation (13) shows that the investor s expecte values of cov an σ s affect the exchange rate ynamics through l an f. On the other han, the exchange rate
9 8 HyoungSeok Lim an Masao Ogaki ynamics in equation (13) imply certain values of cov an σ s, which nee to be consistent with the investor s expecte values in the rational expectations equilibrium. To analyze the equilibrium, we first solve for the rational expectation of cov. We then, show the uniqueness an existence of the equilibrium 5 by solving for the rational expectation of σ s. Before solving for the equilibrium, note the nature of equation (13). It explains the iscrepancy between actual an longterm equilibrium exchange rates through four factors: the trae shock at perio t, the cumulative current account balance, an, the persistent an temporary interest rate shocks. All factors, except the temporary interest rate shock, prouce effects consistent with the expecte irections. However, the temporary interest rate shock (e t) has a positive effect if the relative magnitue of the inirect risk premium effect, f, is greater than one. Calculating cov = E t[{s t+1  E t(s t+1)}{r t+1  E t(r t+1)}] from equations (5) an (13) yiels: λ cov = ( c e c) ( 1 ) σ + λ( φ 1) σ ε. (15) 1 λ Substituting the efinition of f into equation (15), an solving for cov gives the rational expectations equilibrium values for cov an f: λ( 1 c ) + λη( 1 λc) 1 c + η cov = e c c + + < 1 λ 1 η( 1 λ) σ λ( 1 c ) + λη( 1 λc) 1 φ = 0. 1 λc 1 c + η( 1+ λ) > 0, (16) In the rational expectations equilibrium, cov is negative, an f is positive. This implies that the inirect risk premium effect is positive as shown in the previous section. The main issue for the purpose of this paper is whether f is greater or less than one. In orer to etermine this, we will investigate the sign of: ( 1 c ){ λ( 1+ c) 1} η( 1 λc) φ 1 =. ( 1 λc){ 1 c + η( 1+ λ)} (17) Equation (17) shows that f can be either greater or less than one, epening on parameter values. One interesting case arises when the investor is close to being risk neutral. For a very small k, an approximate formula for equation (17) with 1 is: ( 1+ cc ) η φ 1 =. 1 c + η (18) We investigate the conitions require to exhibit the forwar premium anomaly uner low egrees of risk aversion. The forwar premium regression for a shortterm interest rate ifferential is st+ 1 st = α + β( rt rt*) + error term. Let ˆβ be the estimate of b. The probability limit of the estimator is: rt rt st st rt st st plim ˆ cov( *, + 1 ) cov(, + 1 ) β =. var( r r*) var( rt ) t t
10 EXCHANGE RATES AND THE TERM STRUCTURE OF INTEREST RATES 83 For this to be negative, we nee cov(r t, s t+1  s t) < 0, which implies: cov(, rt st+ 1) < cov(, rt st) (19) 1< ( φ 1) η However, substituting equation (18) into (19) oes not prouce a positive value of h that satisfies equation (19). Thus, the population limit of ˆβ is always positive in this moel. The persistent shock creates a positive slope coefficient because both shortan longterm interest rates move together in response to this shock in the moel. The persistent shock ominates the temporary shock in the limit; however, given that the temporary shock can cause the slope to be negative ue to the ominant inirect risk premium effect, there is likely to be a small sample bias when the exchange rate is persistent. We conuct a MonteCarlo simulation 6 uner these parameterizations to investigate the possibility of the small sample bias. Because the moel is highly stylize so that we can analytically solve for the rational expectation of the covariance, we o not try to calibrate the ata in this paper. 7 Suppose that the AR(1) coefficient of the persistent interest rate shock, c, in equation (7) is close to one (for example, c = 0.9). Then, f in equation (18) becomes greater than one as long as h < When the investor is close to being riskneutral, the egree of substitution between short an longterm bons must be high, an consequently, h shoul be very small. Uner these parameter configurations, our moel presente in the previous section preicts that when the measure of the relative magnitue of the inirect risk premium effect, f, is greater than one, the eman from foreign bons increases as the shortterm interest rate rises, resulting in the epreciation of omestic currency to cause an expecte future appreciation of omestic currency. A MonteCarlo simulation base on these parameter configurations consistently generates a negative slope coefficient to show the forwar premium anomaly. As Table 1 shows, the stronger the inirect risk premium effect, the more statistically significant the negative slope coefficient. Table 1. A Monte Carlo Simulation of the Slope Coefficient for the ShortTerm Regression (a) ( st+ 1 st) = α + β( rt rt*) + error term H 0 : b = 0 f (h = 0.7) (h = 0.3) (h = 0.) (h = 0.1) (h = 0.1) p lim ˆβ Mean of cov ˆ (b) (, rt st+ 1 st) Mean of ˆβ Negative frequency (c) % significance level () (10% significance level) () (3.4) (14.7) (7.4) (60.7) (96.5) Notes: (a) Sample size is 10 an c = 0.9. (b) Sample covariance. (c) Percentage of negative coefficients among total iteration (1,000). () Percentage of total iteration (1,000) that reject H 0 at a 5% significance level. Numbers in parentheses are that of 10% significance level.
11 84 HyoungSeok Lim an Masao Ogaki Table. A Monte Carlo Simulation of the Slope Coefficient for the LongTerm Regression (a) ( st+ st) = α + β( Rt Rt*) + error term H 0 : b = 0 f (h = 0.7) (h = 0.3) (h = 0.) (h = 0.1) (h = 0.01) p lim ˆβ Mean of cov ˆ (b) ( Rt, st+ st) Mean of ˆβ Positive frequency (c) % significance level () (10% significance level) () (40.4) (40.) (39.4) (36.0) (8.0) Notes: (a) Sample size is 10 an c = 0.9. (b) Sample covariance. (c) Percentage of positive coefficients among total iteration (1,000). () Percentage of total iteration (1,000) that reject H 0 at a 5% significance level. Numbers in parentheses are that of 10% significance level. On the contrary, a MonteCarlo simulation for the longterm interest ifferential still generates, as Table shows, a positive slope coefficient uner the same parameter configurations as the stanar exchange rate moel preicts. 6. Conclusion We erive the eman function for foreign bons enogenously by solving for the rational expectations equilibrium an investigate how a rise in the shortterm interest rate affects the eman for foreign bons. It generates two opposite effects on the eman for foreign bons. The irect risk premium effect comes from the fact that risk averse agents with short investment time horizons want to reuce the eman for foreign bons to increase the amount investe in risk free assets. On the other han, investors have another incentive: the inirect risk premium effect, to increase the eman for foreign bons to minimize potential capital losses resulting from holing both foreign bons an longterm omestic bons. We show that uner reasonable parameter configurations, the inirect risk premium effect is quantitatively important. In fact, the inirect risk premium effect can ominate the irect risk premium effect, causing eman for foreign bons to increase. Combining the ominance of the inirect risk premium effect with a smallsample bias solution causes our moel to be consistent with the stylize facts regaring the forwar premium anomaly. A MonteCarlo simulation for shortterm regressions base on this parameter specification consistently shows that the ominant inirect risk premium effect is likely to cause ownwar small sample bias when the exchange rate is persistent. The stronger the inirect risk premium effect, the more statistically significant the negative slope coefficient. In this case, the forwar premium anomaly about the shortterm interest rate can be explaine; the omestic currency epreciates now, creating expecte future appreciation of the currency. For the longterm interest rate ifferential, this moel still shows the same preiction on the exchange rate as stanar exchange rate moels. Byeon an Ogaki (1999) fin such results for many of
12 EXCHANGE RATES AND THE TERM STRUCTURE OF INTEREST RATES 85 the G7 countries with cointegrating regressions of real exchange rates onto both short an longterm interest rate ifferentials. Ogaki an Santaella (000) obtain similar results for Mexico. If the inirect risk premium effect is quantitatively significant, then the effectiveness of central bank attempts to affect the exchange rate by controlling the shortterm interest rate epens on whether the longterm interest rate respons to changes in the shortterm interest rate. Anecotal evience suggests that further empirical investigation is warrante. For example, from the mile of March 198 to the en of November 198, the Bank of Japan aopte a policy of increasing the omestic shortterm interest rate in orer to cause an appreciation of the yen (Komiya an Sua, 1983, p ). The shortterm interest rate in Japan increase, but the yen tene to epreciate rather than appreciate against the US ollar uring this perio. One remarkable fact was that the longterm interest rate i not increase when the Bank of Japan began to increase the shortterm interest rate (Komiya an Sua, 1983, p. 349). In typical VAR with the recursive ientification assumption, the elaye overshooting puzzle has been foun as for example in Eichenbaum an Evans (1995). One aspect of the elaye overshooting puzzle is that UIP is severely violate. Therefore, the inirect risk premium effect in principle can help in resolving the puzzle. Incorporating the inirect risk premium effect into an open economy general equilibrium moel is of interest to see whether or not the elaye overshooting arises in such a moel. In this paper, we evelop a highly stylize partial equilibrium moel to obtain the rational expectation of the covariance between the exchange rate an the shortterm interest rate, which is a key parameter for the inirect risk premium effect. It is of interest to stuy whether the qualitative implications of the moel still hol in more realistic moels. There has been relatively little empirical work on the interaction between exchange rate an the term structure of interest rates. Further empirical investigation is warrante on this complicate interaction. References Alexius, A., Uncovere Interest Parity Revisite, Review of International Economics 9 (001): Alvarez, F., A. Atkeson, an P.J. Kehoe, Money, Interest Rates, an Exchange Rates with Enogenously Segmente Markets, The Journal of Political Economy 110 (00): Bacchetta, P. an E. Wincoop, Infrequent Portfolio Decisions: A Solution to the Forwar Discount Puzzle, American Economic Review 100 (010): Baillie, R. an T. Bollerslev, The Forwar Premium Anomaly Is Not As Ba As You Think, Journal of International Money an Finance 19 (000): Baillie, R. an R. Kilic, Do asymmetric an Nonlinear Ajustments Explain the Forwar Premium Anomaly? Journal of International Money an Finance 5 (006): 47. Baxter, M., Real Exchange Rates an Real Interest Differentials: Have We Misse the BusinessCycle Relationship? Journal of Monetary Economics 33 (1994):5 37. Blanchar, O. an M. K. Plantes, A Note on Gross Substitutability of Financial Assets, Econometrica 45 (1977): Byeon, Y. an M. Ogaki, An Empirical Investigation of Exchange Rates an the Term Structure of Interest Rates, Working Paper 0, Ohio State University (1999). Chinn, M.D., The (Partial) Rehabilitation of Interest Rate Parity in the Floating Rate Era: Longer Horizons, Alternative Expectations, an Emerging Markets, Journal of International Money an Finance 5 (006):7 1.
13 86 HyoungSeok Lim an Masao Ogaki Chinn, M.D. an G. Mereith, Monetary Policy an LongHorizon Uncovere Interest Parity, IMF Staff Papers, 51 (004). Chinn. M. D. an G. Mereith, Testing Uncovere Interest Parity at Short an Long Horizons uring the PostBretton Woos Era, NBER Working Paper 11077, Cambrige, MA: National Bureau of Economic Research (005). Driskill, A.R. an S. McCafferty, Speculation, Rational Expectations, an Stability of the Foreign Exchange Market, Journal of International Economics 10 (1980):9 10. Eichenbaum, M. an C.L. Evans, Some Empirical Evience on the Effects of Shocks to Monetary Policy on Exchange Rates, Quarterly Journal of Economics 110 (1995): Ellingsen, T. an U. Söerström, Monetary Policy an Market Interest Rates, American Economic Review 91 (001): Engel, C., The Forwar Discount Anomaly an the Risk Premium: A Survey of Recent Evience, Journal of Empirical Finance 3 (1996):13 9. Fama, E., Forwar an Spot Exchange Rates, Journal of Monetary Economics 14 (1984): Fisher, E., The Forwar Premium in a Moel with Heterogeneous Prior Beliefs, Journal of International Money an Finance 5 (006): Kobayashi, T., On the Relationship Between Short an Longterm Interest Rates, International Finance 7 (004): Komiya, R. an M. Sua, Genai Kokusai Kinyuron: RekishiSeisaku Hen, Tokyo: Nihon Keizai Shinbun (1983). Maynar, A. an P. Phillips, Rethinking an Ol Empirical Puzzle: Econometric Evience on the Forwar Discount Anomaly, Journal of Applie Econometrics 16 (001): Mark, N. an Y. Wu, Rethinking Deviations from Uncovere Interest Parity: The Role of Covariance Risk an Noise, Economic Journal 108 (1998): McCallum, B., A Reconsieration of the Uncovere Interest Parity Relationship, Journal of Monetary Economics 33 (1994): Meese, R. an K. Rogoff, Was it Real? The Exchange RateInterest Rate Differential Relation Over the Moern FloatingRate Perio, Journal of Finance 43 (1988): Ogaki, M., A Theory of Exchange Rates an the Term Structure of Interest Rates, Working Paper 19, Ohio State University (1999). Ogaki, M., The Inirect An Direct Substitution Effects, American Economic Review 80 (1990): Ogaki, M. an J. A. Santaella, The Exchange Rate an the Term Structure of Interest Rates in Mexico, Journal of Development Economics 63 (000): Royama, S. an K. Hamaa, Substitution an Complementarity in the Choice of Risky Assets, in Hester, D.D. an J. Tobin (es), Risk Aversion an Portfolio Choice, New York: John Wiley (1967). Wu, S., Interest Rate Risk an the Forwar Premium Anomaly in Foreign Exchange Markets, Journal of Money, Creit an Banking 39 (005):43 4. Notes 1. The concepts of irect an inirect risk premium effects are closely relate, but iffer from irect an inirect substitution effects efine by Ogaki (1990).. See Ogaki (1999), an earlier version of the present paper, for etails. 3. Whether v t an u t are inepenent with each other epens on the monetary policy regime as the central bank may change the interest rate in response to a trae shock. An example of a monetary policy regime in which these two shocks are inepenent is a forwarlooking type of Taylor rule responing to both expecte inflation an omestic expecte output gap. 4. Ellingsen an Söerström (001) suggest that the comovement of term structure of interest rates epens on market participants interpretation of the policy move. They show that if a change in shortterm rate is regare as being cause by an unexpecte shift in policy preference, the feeral funs rate an the long term interest rates will move in opposite irections.
14 EXCHANGE RATES AND THE TERM STRUCTURE OF INTEREST RATES 87 Further, Kobayashi (004) emphasizes that the simultaneous occurrence of economic shocks which have ifferent signs an urations can break own the comovement of term structure of interest rates. 5. Due to the page constraint, we o not report these results here. However, the results are available on request. 6. Gauss for Winows SP Version was use to conuct the simulation for this paper. 7. Baillie an Kilic (006) employ the logistic smooth transition ynamic regression (LSTR) moel to investigate some nonlinear an asymmetric aspects of the relationship between the exchange rate an the shortterm interest rate ifferential. They show that the stylize facts of the forwar premium anomaly can be obtaine from calibrating a ata generating process from the estimate LSTR moel as long as transaction costs from closing arbitrage conitions in financial market are large relative to potential gains.
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