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A note on interest rate defense policy and exchange rate volatility
Economic Modelling 24 (2007) 768 – 777 www.elsevier.com/locate/econbase
A note on interest rate defense policy and exchange rate volatility ☆ Shiu-Sheng Chen ⁎ Department of Economics, National Taiwan University, No. 21, Hsu-Chow Road, Taipei, Taiwan Accepted 22 February 2007
Abstract In this paper, the effectiveness of an interest rate defense policy is investigated theoretically. Chen [Chen, Shiu-Sheng, 2006. Revisiting the interest rate–exchange rate nexus: a Markov switching approach. Journal of Development Economics 79 (1), 208–224] has documented an empirical regularity that higher interest rates are associated with higher exchange rate volatility. In order to account for the empirical findings, a simple theoretical model by incorporating interest rate rules in a noise trader model is proposed. © 2007 Elsevier B.V. All rights reserved. JEL classification: F30; F31; F41; G15 Keywords: Exchange rates; Interest rates; Noise traders
1. Introduction There are debates of whether a higher interest rate stabilizes exchange rates. A number of studies have empirically investigated the effectiveness of interest rate defense, but failed to find a systematic relationship between interest rates and exchange rates. Using a nonlinear Markovswitching framework, a recent paper by Chen (2006) has addressed an empirical regularity that higher interest rates are associated with higher exchange rate volatility. The paper concludes that high interest rate policy is unable to defend the exchange rate. ☆
I would like to thank Kenneth West and Charles Engel for their valuable suggestions and comments. Any remaining errors are my own responsibility. This research is supported by a grant from the Program for Globalization Studies, Institute for Advanced Studies in Humanities and Social Sciences, National Taiwan University (95R0064-AH03-03). ⁎ Tel.: +886 2 2351 9641x481. E-mail address: [email protected]. 0264-9993/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2007.02.002
S.-S. Chen / Economic Modelling 24 (2007) 768–777
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In order to explain the empirical regularity, I developed a simple theoretical model that modifies the framework of microstructural theory of exchange rates in Jeanne and Rose (2002). Their model mixes elements from two disparate branches of economic theory: the macroeconomic theory of exchange rate determination and the noise trading approach to asset price volatility. Instead of using a conventional monetary model of the exchange rate on the macroeconomic side, I assume that an interest rate rule is adopted to defend the exchange rate in my modification. To the best of my knowledge, this is the first paper to investigate theoretically the relationship between interest rate defense policy and exchange rate volatility. This modified model has the features of multiple equilibria, which consequently imply a possible switching between the regimes of high and low volatility of the exchange rates, in other words, a shift between “crisis” and “tranquil” regimes. Furthermore, incorporating an interest rate rule in the model helps me to investigate the effects of monetary policy, especially when the government intends to raise nominal interest rates to stabilize exchange rates. It will be shown that under plausible conditions, higher exchange rate volatility is associated with higher interest rates. In this simple, stylized model, a tightened monetary policy induces capital inflow as predicted by conventional wisdom. However, tightened monetary policy attracts more noise trading thus increasing exchange rate volatility when noise traders are present in the financial market. The paper is structured as follows. Section 2 presents a model of exchange rates determination. The concluding remarks are offered in Section 3. 2. A model of exchange rates determination In this section, I present a modified version of Jeanne and Rose's (2002) model which contains a microstructural theory of exchange rates. In the microstructure theory, I introduce the presence of noise trading in the foreign exchange rate market by following Jeanne and Rose (2002) closely.1 In addition, I incorporate an interest rate rule as the monetary policy of the central bank. The main results of multiple equilibria are still reached as in Jeanne and Rose (2002), however, according to an interest rate rule in the model, I can further study the effectiveness of interest rate defense. This modified model has shown the existence of multiple regimes in exchange rates with low to high volatility. Further, high interest rate policy could be destabilizing since raising interest rates to defend the currency may push the currency into a highly variable exchange rate regime. There are two countries, “domestic” and “foreign”, moreover, it is assumed that international traders locate around the world. In order to focus on the domestic country, I assume that the foreign country is in the steady state with constant price level. Therefore, anything measured in foreign currency can be interpreted as in real terms. Two kinds of bonds are issued, by domestic and foreign countries respectively. In the bond market, the international investors can hold bonds denominated in domestic currency (I will call them Peso bonds thereafter) and in foreign currency (I will call them Dollar bonds).2 Assume that the international traders care about the return of their portfolio measured in real terms, i.e., the returns are in terms of the foreign currency noting that the foreign price level is constant. As argued in Jeanne and Rose (2002), we may imagine that the foreign country is the center of the international financial system, and the domestic country is a small economy, thus the Dollar bonds is in perfectly elastic supply. Investors require a risk premium to hold bonds 1 Their model is in the spirit of the well-known model of noise trading developed by DeLong, Shleifer, Summers, and Waldmann (1990). 2 Using the names, “Peso bond” and “Dollar bond”, is just for convenience as in Jeanne and Rose (2002).
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S.-S. Chen / Economic Modelling 24 (2007) 768–777
denominated in Peso since they are risk averse and the exchange rate is stochastic. I further assume that the supply of Peso bonds is held at a constant level B¯. 2.1. Setups of microstructure: trading behavior Consider an overlapping-generation setting. N risk-averse traders are born with a full stomach and an endowment of W at each period. They do not consume in the first period of life and make portfolio decisions to maximize the expected utility of second-period wealth, which is used to finance their consumption in the second period. There is no cost to invest in Dollar bonds. Traders live for two periods and allocate their portfolio between Peso and Dollar one-period bonds in the first period of their life. There are two kinds of traders distinguished by the ability to trade in the Peso bond market. Some of them called “informed” traders can form correct expectations on risk and returns without any cost. Others, however, having noisy expectations and needing to pay an entry cost to invest in Peso bonds are called “noise” traders. Both informed and noise traders have same tastes and endowments of W. If we denote the entry decision by a dummy variable ηtj, where ηtj = {1,0} = {enter, not enter}, each trader's decision of whether or not entering the Peso bond market is based on: j
gt ¼
1 0
j j ðUtj jgtj ¼ 1ÞzEt−1 ðUtj jgtj ¼ 0Þ; if Et−1 otherwise:
ð1Þ
Where Utj represents j's trader at utility t, time and Etj − 1 denotes trader j's expectation conditional on the information available at time t − 1. I assume that trader's entry decision is made before the time t shocks are revealed. Assume that a trader will invest btj in Peso bonds when she enters the Peso bond market. Therefore, trader j's maximization problem can be written as: j
Utj ¼ Etj ð−e−aWtþ1 Þ; max j bt
s:t:
j Wtþ1 ¼ ð1 þ i⁎ ÞW þ gtj ðbtj qtþ1 −cj Þ:
ð2Þ
Where cj are the costs associated with entering the Peso bond market, and Wtj+ 1 is trader j's wealth in the second period. a is a positive constant. The excess return (risk premium) on Peso bonds, ρ is defined by3: qtþ1 Jit −ðetþ1 −et Þ−i⁎ ;
ð3Þ
where it is the interest rate for Peso bonds and i⁎ is the interest rate for Dollar bonds. The foreign interest rates are constant over time since I assume that the foreign country is in the steady state. There are NI informed traders ( j = 1, …, NI) and NN ( j = NI + 1, …, N ) noise traders in each generation, NI + NN = N. As mentioned above, these two different kinds of traders are distinguished When trader j enters the Peso bond market, her end-of-life wealth is indeed W˘ tþ1 uð1 þ i⁎ ÞðW −btj Þþ ˘ tj+ 1 as: W˘ j ¼ ð1 þ i⁎ ÞW þ bt ð1 þ it Þ SStþ1t −cj , where St is the exchange rate in level. However, we can rewrite W tþ1 h i i h btj ð1 þ it Þ SStþ1t −ð1 þ i⁎ Þ −cj cð1 þ i⁎ ÞW þ btj ln ð1 þ it Þ SStþ1t −1 þ 1 −i⁎ −cj ¼ ð1 þ i⁎ ÞW þ btj ½it þ ðet −etþ1 Þ−i⁎ − 3
j
j
t . Therefore, to simplify the analysis, it makes no harm to consider Wtj+ 1 rather than cj ¼ ð1 þ i⁎ ÞW þ btj qtþ1 −cj uWtþ1 j ˘ t + 1 in the utility function. W
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by their costs of processing new information and costs of entry. Assume that the informed traders can make their decisions based on rational expectations and do not bear any entry cost: 8 j < Et ðqtþ1 Þ ¼ Et ðqtþ1 Þ; 8j V NI ; vartj ðqtþ1 Þ ¼ vart ðqtþ1 Þ; : cj ¼ 0;
ð4Þ
where Etj( ρt + 1) and vartj( ρt + 1) are the expected value and conditional variance of the excess return on the Peso bonds, and Et and vart are mathematical operators conditional on information available at time t. In contrast, assume that the noise traders perceive the second moment of excess returns correctly, but their perception of first moments is affected by noise that is unrelated to economic fundamentals, i.e. 8 <
Etj ðqtþ1 Þ ¼ q ¯ þ st ; 8j N NI ; vartj ðqtþ1 Þ ¼ vart ðqtþ1 Þ; : cj z 0;
ð5Þ
where ρ¯ is the unconditional mean of the excess return (or average risk premium), and τt is a stochastic i.i.d. normal shock. The variance of τt is assumed to be proportional to the true unconditional variance of the exchange rate: var(τ) = λ(var(e)), λ N 0.4 I further assume that noise traders are ordered by increasing entry cost and the entry cost is not too small. That is: ( 8j N NI ;
dcj =dj N 0; 1 cj N lnð1 þ kÞ: 2a
ð6Þ
2.2. Monetary policy Assume that the home country adopts an interest rate rule of the form: it ¼ gðet − e˜ t Þ þ xt ;
ð7Þ
where γ is a parameter, and e˜t is the target exchange rate. It is clear that the monetary authority is assumed to lean against exchange rate depreciations (γ N 0), i.e. the government raises interest rates to defend the currency. In Eq. (7), the i.i.d. normal shock, ωt contains omitted terms which may affect monetary policy. 2.3. Main results The equilibrium will be presented by two steps. I first determine the equilibrium exchange rate, taking the number of noise traders, n as given. Next, I solve for n endogenously. The main results of the model can be summarized as follows while the details are left in Appendix A. 4 It may be implausible to assume that the unconditional variance of the noise is constant since the constancy may imply that noise traders expect the exchange rate to be stochastic when it is in fact fixed. See Jeanne and Rose (2002).
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• Step 1: When the number of noise traders is given, I can then solve for the average risk premium and the variance of the exchange rate as: q ¯¼
a B¯ varðeÞ; NI þ n
varðeÞ ¼
ð8Þ
g2 varð˜eÞ þ varðxÞ 2 : ð1 þ gÞ2 −k NnI
ð9Þ
Some remarks to address: First of all, note that increase in the fundamental variance (var(e˜ ) or var(ω)) result in higher exchange rate variance. Secondly, an exogenous increase in the number of noise traders, n, makes the variance of exchange rate increase as well. Finally, the effects of increasing n on average risk premium are non-monotonic. On the one hand, a higher n lowers ρ¯ directly in Eq. (8), on the other hand, it makes var(e) increase, and therefore, raises ρ¯. In other words, noise traders play two roles in this model: they create risk as well as share risk at the same time. • Step 2: I now consider the entry decisions of both informed and noise traders in the Peso bond market and present the analysis when the number of noise traders is endogenous. 1. The first simple result, which can be easily observed, is that the informed traders always enter the Peso bond market in equilibrium since the informed traders bear no entry costs. 2. Noise trader j enters the Peso bond market in equilibrium only if the gross benefit of entry, (GB), exceeds her cost of entry. That is GBð q; ¯ varðeÞÞ ¼
1 1 q ¯2 þ lnð1 þ kÞzcj : 2að1 þ kÞ varðeÞ 2a
ð10Þ
See Proposition 1 in Appendix A. 3. For some levels of fundamental variance and entry costs, there exist multiple equilibria with a range of entry (from low to high) by noise traders. That is, according to the traders' entry decision, we will obtain multiple equilibria in the model. In order to illustrate the multiple equilibria property in my model, I plot the gross benefit and entry cost in Fig. 1.5 The interceptions of gross benefit and cost of entry are net benefit of entry for the marginal noise trader. Investigating first the benchmark case with γ = 0.5 (GB0 in Fig. 1), we can clearly find out that points E1 and E3 are two stable equilibria and point E2 is an unstable equilibrium. Since the variance of the exchange rate is a monotonic function of the number of noise traders, point E1 represents an equilibrium with low exchange rate volatility while a low risk premium cannot attract many noise traders to enter. By contrast, point E3 corresponds to a high volatility equilibrium since more noise traders are attracted to the Peso bond market by the high risk premium endogenously generated by their own entry (see Proposition 2 in Appendix A). 4. For sufficiently high levels of fundamental variances (var(e˜ ) or var(ω)), there is one unique equilibrium with all noise traders entering the Peso bond market. For sufficiently low levels of fundamental variances, there is an equilibrium with no noise traders in the Peso bond market. For intermediate levels of fundamental variances, there exist multiple ¯ = 20, and cj = 0.2775 for all The values of the parameters in Fig. 1 were set as follows: λ = 7, a = 4, NI = 60, NN = B noise traders. The variances of fundamentals are: var(e˜ ) = 3, and var(ω) = 1. γ = 0.5 for GB0 and γ = 1.1 for GB1. 5
S.-S. Chen / Economic Modelling 24 (2007) 768–777
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Fig. 1. Benefit and cost of entry for the marginal noise trader.
equilibria with low and high entry by noise traders (or low and high exchange rate volatility). Thus the emergence of multiple equilibria for intermediate levels of fundamental variances in this model has some interesting implications. First of all, there is no simple relationship between the volatility of fundamentals and the exchange rate. Two countries may exhibit different levels of exchange rate volatility even though they have similar fundamentals. Secondly, the state of an economy may jump between multiple equilibria due to changes of sunspot variables or parameters such as unrelated policy actions (see Proposition 3 in Appendix A). 5. In this model, I consider γ in Eq. (7) as a policy parameter indicating how the interest rate policy responses to the exchange rate. If γ N 0, it means that the monetary authority leans against exchange rate depreciations. It is of interest to ask whether or not an interest rate defense can stabilize the exchange rate, that is, whether or not an interest rate defense leads to an equilibrium with lower exchange rate volatility. Under some conditions, given γ N 0, higher values of γ leads to higher exchange rate volatility, and may end up with a unique equilibrium in which all noise traders enter the Peso bond market and lead to higher exchange rate volatility. This can be illustrated in Fig. 1. A higher value of γ (from 0.5 to 1.1) shifts the gross benefit curve up from GB0 to GB1, and induces a high volatility equilibrium, point E4. Therefore, the model predicts that under some conditions, given that a policymaker employs an interest rate defense policy, the more aggressive she responds to the exchange rate depreciations (larger the value of γ), the higher the exchange rate volatility (see Proposition 4 in Appendix A). Raising interest rate to defend the currency may push the currency into a highly variable exchange rate regime, i.e. an interest rate defense policy may destabilize rather than stabilize the exchange rate. The intuition is that higher interest rates induce higher risk premia, which raises the
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gross benefit of entry for noise traders, attracts more noise traders to enter the market, and then makes the risk premium even much higher. Finally, it may end up with a high exchange rate volatility with high risk premium equilibrium that induces all noise traders to enter the Peso bond market. In sum, a tightened monetary policy induces capital inflow as conventional uncovered interest rate parity framework predicted. However, if both rational and noise traders coexist in the financial market, a higher interest rate raises the excess return, which attracts more noise trading and hence exaggerates the exchange rate volatility. 3. Concluding remarks In this paper the effectiveness of interest rate defense policy is investigated theoretically. Chen (2006) has documented an empirical regularity that higher interest rates are associated with a longer length of spells of high exchange rate volatility and a shorter length of spells of low exchange rate volatility. Hence, the paper suggests that any structural model of the interest rate– exchange rate nexus ought to predict this relationship between interest rates and exchange rate volatility. Clearly, the simple, stylized model proposed in this paper is consistent with the empirical regularity and provides one of cause and effect. Appendix A In this appendix, I will show how to obtain the main results of the model outlined in Section 2. First of all, let me define an equilibrium in this model. Definition 1. (Equilibrium) An equilibrium consists of a path of stochastic processes for the exchange rate {et}, the risk premium {ρt}, and traders' optimal decisions ηtj and btj, such that (a) ηtj satisfies the entry condition (1); (b) btj solves for the maximization problem (2); P (c) Market clearing condition for Peso bonds is: B¯ ¼ Nj¼1 gtj btj . As in Jeanne and Rose (2002), I solve the model by the strategy of “guess-and-verify”. Since all the shocks are assumed to be independently and identically distributed xt f i:i:d: N ð0; varðxÞÞ; e˜ f i:i:d: N ðEð˜eÞ; varð˜eÞÞ, I take a guess that the solution will be: (a) et f i:i:d: N ðe¯; varðeÞÞ, and qt f i:i:d: N ðq ¯ ; varðqÞÞ; (b) all informed traders, and a constant number of noise traders, n ≤ NN, enter the Peso bond market at each period. The equilibrium will be presented by two steps. I first determine the equilibrium exchange rate, taking the number n as given. Next, I use the entry condition to solve for n. A.1. Analysis when the number of noise traders is given If the risk premium ρt is normally distributed, the maximization problem (2) can be rewritten as a j j max Utj ¼ Etj ðWtþ1 Þ− vartj ðWtþ1 Þ; j 2 bt
j s:t: Wtþ1 ¼ ð1 þ i⁎ ÞW þ gtj ðbtj qtþ1 − cj Þ:
ð11Þ
S.-S. Chen / Economic Modelling 24 (2007) 768–777
Thus, I can get the demand for Peso bonds by an individual trader as follows: ! 1 Etj ðqtþ1 Þ j bt ¼ : a vartj ðqtþ1 Þ
775
ð12Þ
Then using market clearing condition in the Peso bond market, I have ¯¼ B
NI a
Et ðqtþ1 Þ n q¯ þ st þ vart ðqtþ1 Þ a vart ðqtþ1 Þ
1 NI Et ðqtþ1 Þ þ nð q¯ þ st Þ ¼ ; a varðeÞ
ð13Þ
ð14Þ
where I use the fact that in equilibrium, vart ( ρt + 1) = vart (et + 1) = var (e) from Eq. (3). I can then solve for the average risk premium from Eq. (14), and solve for the equilibrium exchange rate and the variance of the exchange rate as follows,6 a B¯ varðeÞ; NI þ n 1 n gð˜e−Eð˜eÞÞ−xt − st ; et − e¯ ¼ 1þg NI
q¯ ¼
varðeÞ ¼
g2 varð˜eÞ þ varðxÞ 2 : ð1 þ gÞ2 −k NnI
ð15Þ ð16Þ ð17Þ
1 Where ¯e ¼ ðq ¯ þ i⁎ Þ. g A.2. Analysis when the number of noise traders is endogenous I now consider the entry decisions of both informed and noise traders in the Peso bond market. The first simple result, which can be easily observed, is that the informed traders always enter the Peso bond market in equilibrium since the informed traders bear no entry costs. Some other important results can be summarized in following propositions: Proposition 1. Noise trader j enters the Peso bond market in equilibrium only if the gross benefit of entry, (GB), exceeds her cost of entry. That is GBðq ¯ ; varðeÞÞzcj ;
ð18Þ
where GBð q; ¯ varðeÞÞ ¼
1 1 q¯2 þ lnð1 þ kÞ: 2að1 þ kÞ varðeÞ 2a
Proof. See Jeanne and Rose (2002). 6
ð19Þ □
To get Eq. (16), we need to combine 2 Eqs. (3), (7), (14), and (15). Eq. (17) can be derived from Eq. (16). Note that we impose the condition ð1 þ gÞ2 −k NNNI N0 to ensure the variance of exchange rate is positive for all n ≤ NN.
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S.-S. Chen / Economic Modelling 24 (2007) 768–777
According to the traders' entry decision, I can get the following proposition relating to multiple equilibria: Proposition 2. For some levels of fundamental variance and entry costs, there exist multiple equilibria with a range of entry (from low to high) by noise traders. Proof. Combining Eqs. (15), (17), and (19), I can get
GB ¼
2 2 1 a B¯ g varð˜eÞ þ varðxÞ 1 2 þ lnð1 þ kÞ; 2að1 þ kÞ NI þ n 2a 2 n |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ð1 þ gÞ −k NI A when n z |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð20Þ
z when n z
that is, generally speaking, GB is a non-monotonic function in n. Thus, there will be more than one interception of the gross benefit curve, GB, and the entry cost curve, cj. □ Proposition 3. For sufficiently high levels of fundamental variances (var(e˜ ) or var(ω)), there is one unique equilibrium with all noise traders entering the Peso bond market. For sufficiently low levels of fundamental variances, there is an equilibrium with no noise traders in the Peso bond market. For intermediate levels of fundamental variances, there exist multiple equilibria with low and high entry by noise traders (or low and high exchange rate volatility). Proof. Since increase of the fundamental variances shifts up the GB curve, the GB curve is everywhere above the entry cost curve when the fundamental variances are sufficiently high. This results an unique equilibrium with all noise traders entering the Peso bond market. Moreover, there can be another unique equilibrium with no noise traders in the Peso bond market when the fundamental variances are sufficiently low, which makes the GB curve to be everywhere below the entry cost curve. Finally, Proposition 2 is applied for intermediate levels of fundamental variances. □ Proposition 4. Given γ N 0, higher values of γ leads to higher exchange rate volatility, and may end up with a unique equilibrium in which all noise traders enter the Peso bond market 1 eÞvarðxÞÞ2 ˜ ∀n ≤ NN , where g˜ ¼ AþðDþ4varð˜ and lead to higher exchange rate volatility if γ N γ ; 2varð˜ eÞ 2 A ¼ −varð˜eÞ þ Nn kvarð˜eÞ þ varðxÞ ; and D ¼ A2 . I
Proof. First, I have the result that 0
1 2 2 n 2 2gvarð˜eÞ ð1 þ gÞ − NI k −2½g varð˜eÞ þ varðxÞð1 þ gÞC ¯ 2 B AGB 1 aB B C ¼ B C: 2 2 @ A Ag 2að1 þ kÞ NI þ n 2 n ð1 þ gÞ − NI k
ð21Þ
2
Thus, AGB eÞg2 þ varð˜eÞ− NnI kvarð˜eÞ−varðxÞ g−varðxÞN0, (i.e. γ N γ˜ ∀n ≤ NN while I Ag N0 if varð˜ consider the positive value of γ only). Since the GB curve shifts up, it may end up that the GB curve is everywhere above the curve of entry cost, so that there is one unique equilibrium in which □ all noise traders enter the market.
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References Chen, Shiu-Sheng, 2006. Revisiting the interest rate–exchange rate nexus: a Markov switching approach. Journal of Development Economics 79 (1), 208–224. DeLong, J. Bradford, Shleifer, Andrei, Summers, Lawrence H., Waldmann, Robert J., 1990. Noise trader risk in financial markets. Journal of Political Economy 98 (4), 703–738. Jeanne, Olivier, Rose, Andrew K., 2002. Noise trading and exchange rate regimes. Quarterly Journal of Economics 117 (2), 537–569.
A note on interest rate defense policy and exchange rate volatility ☆ Shiu-Sheng Chen ⁎ Department of Economics, National Taiwan University, No. 21, Hsu-Chow Road, Taipei, Taiwan Accepted 22 February 2007
Abstract In this paper, the effectiveness of an interest rate defense policy is investigated theoretically. Chen [Chen, Shiu-Sheng, 2006. Revisiting the interest rate–exchange rate nexus: a Markov switching approach. Journal of Development Economics 79 (1), 208–224] has documented an empirical regularity that higher interest rates are associated with higher exchange rate volatility. In order to account for the empirical findings, a simple theoretical model by incorporating interest rate rules in a noise trader model is proposed. © 2007 Elsevier B.V. All rights reserved. JEL classification: F30; F31; F41; G15 Keywords: Exchange rates; Interest rates; Noise traders
1. Introduction There are debates of whether a higher interest rate stabilizes exchange rates. A number of studies have empirically investigated the effectiveness of interest rate defense, but failed to find a systematic relationship between interest rates and exchange rates. Using a nonlinear Markovswitching framework, a recent paper by Chen (2006) has addressed an empirical regularity that higher interest rates are associated with higher exchange rate volatility. The paper concludes that high interest rate policy is unable to defend the exchange rate. ☆
I would like to thank Kenneth West and Charles Engel for their valuable suggestions and comments. Any remaining errors are my own responsibility. This research is supported by a grant from the Program for Globalization Studies, Institute for Advanced Studies in Humanities and Social Sciences, National Taiwan University (95R0064-AH03-03). ⁎ Tel.: +886 2 2351 9641x481. E-mail address: [email protected]. 0264-9993/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2007.02.002
S.-S. Chen / Economic Modelling 24 (2007) 768–777
769
In order to explain the empirical regularity, I developed a simple theoretical model that modifies the framework of microstructural theory of exchange rates in Jeanne and Rose (2002). Their model mixes elements from two disparate branches of economic theory: the macroeconomic theory of exchange rate determination and the noise trading approach to asset price volatility. Instead of using a conventional monetary model of the exchange rate on the macroeconomic side, I assume that an interest rate rule is adopted to defend the exchange rate in my modification. To the best of my knowledge, this is the first paper to investigate theoretically the relationship between interest rate defense policy and exchange rate volatility. This modified model has the features of multiple equilibria, which consequently imply a possible switching between the regimes of high and low volatility of the exchange rates, in other words, a shift between “crisis” and “tranquil” regimes. Furthermore, incorporating an interest rate rule in the model helps me to investigate the effects of monetary policy, especially when the government intends to raise nominal interest rates to stabilize exchange rates. It will be shown that under plausible conditions, higher exchange rate volatility is associated with higher interest rates. In this simple, stylized model, a tightened monetary policy induces capital inflow as predicted by conventional wisdom. However, tightened monetary policy attracts more noise trading thus increasing exchange rate volatility when noise traders are present in the financial market. The paper is structured as follows. Section 2 presents a model of exchange rates determination. The concluding remarks are offered in Section 3. 2. A model of exchange rates determination In this section, I present a modified version of Jeanne and Rose's (2002) model which contains a microstructural theory of exchange rates. In the microstructure theory, I introduce the presence of noise trading in the foreign exchange rate market by following Jeanne and Rose (2002) closely.1 In addition, I incorporate an interest rate rule as the monetary policy of the central bank. The main results of multiple equilibria are still reached as in Jeanne and Rose (2002), however, according to an interest rate rule in the model, I can further study the effectiveness of interest rate defense. This modified model has shown the existence of multiple regimes in exchange rates with low to high volatility. Further, high interest rate policy could be destabilizing since raising interest rates to defend the currency may push the currency into a highly variable exchange rate regime. There are two countries, “domestic” and “foreign”, moreover, it is assumed that international traders locate around the world. In order to focus on the domestic country, I assume that the foreign country is in the steady state with constant price level. Therefore, anything measured in foreign currency can be interpreted as in real terms. Two kinds of bonds are issued, by domestic and foreign countries respectively. In the bond market, the international investors can hold bonds denominated in domestic currency (I will call them Peso bonds thereafter) and in foreign currency (I will call them Dollar bonds).2 Assume that the international traders care about the return of their portfolio measured in real terms, i.e., the returns are in terms of the foreign currency noting that the foreign price level is constant. As argued in Jeanne and Rose (2002), we may imagine that the foreign country is the center of the international financial system, and the domestic country is a small economy, thus the Dollar bonds is in perfectly elastic supply. Investors require a risk premium to hold bonds 1 Their model is in the spirit of the well-known model of noise trading developed by DeLong, Shleifer, Summers, and Waldmann (1990). 2 Using the names, “Peso bond” and “Dollar bond”, is just for convenience as in Jeanne and Rose (2002).
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S.-S. Chen / Economic Modelling 24 (2007) 768–777
denominated in Peso since they are risk averse and the exchange rate is stochastic. I further assume that the supply of Peso bonds is held at a constant level B¯. 2.1. Setups of microstructure: trading behavior Consider an overlapping-generation setting. N risk-averse traders are born with a full stomach and an endowment of W at each period. They do not consume in the first period of life and make portfolio decisions to maximize the expected utility of second-period wealth, which is used to finance their consumption in the second period. There is no cost to invest in Dollar bonds. Traders live for two periods and allocate their portfolio between Peso and Dollar one-period bonds in the first period of their life. There are two kinds of traders distinguished by the ability to trade in the Peso bond market. Some of them called “informed” traders can form correct expectations on risk and returns without any cost. Others, however, having noisy expectations and needing to pay an entry cost to invest in Peso bonds are called “noise” traders. Both informed and noise traders have same tastes and endowments of W. If we denote the entry decision by a dummy variable ηtj, where ηtj = {1,0} = {enter, not enter}, each trader's decision of whether or not entering the Peso bond market is based on: j
gt ¼
1 0
j j ðUtj jgtj ¼ 1ÞzEt−1 ðUtj jgtj ¼ 0Þ; if Et−1 otherwise:
ð1Þ
Where Utj represents j's trader at utility t, time and Etj − 1 denotes trader j's expectation conditional on the information available at time t − 1. I assume that trader's entry decision is made before the time t shocks are revealed. Assume that a trader will invest btj in Peso bonds when she enters the Peso bond market. Therefore, trader j's maximization problem can be written as: j
Utj ¼ Etj ð−e−aWtþ1 Þ; max j bt
s:t:
j Wtþ1 ¼ ð1 þ i⁎ ÞW þ gtj ðbtj qtþ1 −cj Þ:
ð2Þ
Where cj are the costs associated with entering the Peso bond market, and Wtj+ 1 is trader j's wealth in the second period. a is a positive constant. The excess return (risk premium) on Peso bonds, ρ is defined by3: qtþ1 Jit −ðetþ1 −et Þ−i⁎ ;
ð3Þ
where it is the interest rate for Peso bonds and i⁎ is the interest rate for Dollar bonds. The foreign interest rates are constant over time since I assume that the foreign country is in the steady state. There are NI informed traders ( j = 1, …, NI) and NN ( j = NI + 1, …, N ) noise traders in each generation, NI + NN = N. As mentioned above, these two different kinds of traders are distinguished When trader j enters the Peso bond market, her end-of-life wealth is indeed W˘ tþ1 uð1 þ i⁎ ÞðW −btj Þþ ˘ tj+ 1 as: W˘ j ¼ ð1 þ i⁎ ÞW þ bt ð1 þ it Þ SStþ1t −cj , where St is the exchange rate in level. However, we can rewrite W tþ1 h i i h btj ð1 þ it Þ SStþ1t −ð1 þ i⁎ Þ −cj cð1 þ i⁎ ÞW þ btj ln ð1 þ it Þ SStþ1t −1 þ 1 −i⁎ −cj ¼ ð1 þ i⁎ ÞW þ btj ½it þ ðet −etþ1 Þ−i⁎ − 3
j
j
t . Therefore, to simplify the analysis, it makes no harm to consider Wtj+ 1 rather than cj ¼ ð1 þ i⁎ ÞW þ btj qtþ1 −cj uWtþ1 j ˘ t + 1 in the utility function. W
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by their costs of processing new information and costs of entry. Assume that the informed traders can make their decisions based on rational expectations and do not bear any entry cost: 8 j < Et ðqtþ1 Þ ¼ Et ðqtþ1 Þ; 8j V NI ; vartj ðqtþ1 Þ ¼ vart ðqtþ1 Þ; : cj ¼ 0;
ð4Þ
where Etj( ρt + 1) and vartj( ρt + 1) are the expected value and conditional variance of the excess return on the Peso bonds, and Et and vart are mathematical operators conditional on information available at time t. In contrast, assume that the noise traders perceive the second moment of excess returns correctly, but their perception of first moments is affected by noise that is unrelated to economic fundamentals, i.e. 8 <
Etj ðqtþ1 Þ ¼ q ¯ þ st ; 8j N NI ; vartj ðqtþ1 Þ ¼ vart ðqtþ1 Þ; : cj z 0;
ð5Þ
where ρ¯ is the unconditional mean of the excess return (or average risk premium), and τt is a stochastic i.i.d. normal shock. The variance of τt is assumed to be proportional to the true unconditional variance of the exchange rate: var(τ) = λ(var(e)), λ N 0.4 I further assume that noise traders are ordered by increasing entry cost and the entry cost is not too small. That is: ( 8j N NI ;
dcj =dj N 0; 1 cj N lnð1 þ kÞ: 2a
ð6Þ
2.2. Monetary policy Assume that the home country adopts an interest rate rule of the form: it ¼ gðet − e˜ t Þ þ xt ;
ð7Þ
where γ is a parameter, and e˜t is the target exchange rate. It is clear that the monetary authority is assumed to lean against exchange rate depreciations (γ N 0), i.e. the government raises interest rates to defend the currency. In Eq. (7), the i.i.d. normal shock, ωt contains omitted terms which may affect monetary policy. 2.3. Main results The equilibrium will be presented by two steps. I first determine the equilibrium exchange rate, taking the number of noise traders, n as given. Next, I solve for n endogenously. The main results of the model can be summarized as follows while the details are left in Appendix A. 4 It may be implausible to assume that the unconditional variance of the noise is constant since the constancy may imply that noise traders expect the exchange rate to be stochastic when it is in fact fixed. See Jeanne and Rose (2002).
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• Step 1: When the number of noise traders is given, I can then solve for the average risk premium and the variance of the exchange rate as: q ¯¼
a B¯ varðeÞ; NI þ n
varðeÞ ¼
ð8Þ
g2 varð˜eÞ þ varðxÞ 2 : ð1 þ gÞ2 −k NnI
ð9Þ
Some remarks to address: First of all, note that increase in the fundamental variance (var(e˜ ) or var(ω)) result in higher exchange rate variance. Secondly, an exogenous increase in the number of noise traders, n, makes the variance of exchange rate increase as well. Finally, the effects of increasing n on average risk premium are non-monotonic. On the one hand, a higher n lowers ρ¯ directly in Eq. (8), on the other hand, it makes var(e) increase, and therefore, raises ρ¯. In other words, noise traders play two roles in this model: they create risk as well as share risk at the same time. • Step 2: I now consider the entry decisions of both informed and noise traders in the Peso bond market and present the analysis when the number of noise traders is endogenous. 1. The first simple result, which can be easily observed, is that the informed traders always enter the Peso bond market in equilibrium since the informed traders bear no entry costs. 2. Noise trader j enters the Peso bond market in equilibrium only if the gross benefit of entry, (GB), exceeds her cost of entry. That is GBð q; ¯ varðeÞÞ ¼
1 1 q ¯2 þ lnð1 þ kÞzcj : 2að1 þ kÞ varðeÞ 2a
ð10Þ
See Proposition 1 in Appendix A. 3. For some levels of fundamental variance and entry costs, there exist multiple equilibria with a range of entry (from low to high) by noise traders. That is, according to the traders' entry decision, we will obtain multiple equilibria in the model. In order to illustrate the multiple equilibria property in my model, I plot the gross benefit and entry cost in Fig. 1.5 The interceptions of gross benefit and cost of entry are net benefit of entry for the marginal noise trader. Investigating first the benchmark case with γ = 0.5 (GB0 in Fig. 1), we can clearly find out that points E1 and E3 are two stable equilibria and point E2 is an unstable equilibrium. Since the variance of the exchange rate is a monotonic function of the number of noise traders, point E1 represents an equilibrium with low exchange rate volatility while a low risk premium cannot attract many noise traders to enter. By contrast, point E3 corresponds to a high volatility equilibrium since more noise traders are attracted to the Peso bond market by the high risk premium endogenously generated by their own entry (see Proposition 2 in Appendix A). 4. For sufficiently high levels of fundamental variances (var(e˜ ) or var(ω)), there is one unique equilibrium with all noise traders entering the Peso bond market. For sufficiently low levels of fundamental variances, there is an equilibrium with no noise traders in the Peso bond market. For intermediate levels of fundamental variances, there exist multiple ¯ = 20, and cj = 0.2775 for all The values of the parameters in Fig. 1 were set as follows: λ = 7, a = 4, NI = 60, NN = B noise traders. The variances of fundamentals are: var(e˜ ) = 3, and var(ω) = 1. γ = 0.5 for GB0 and γ = 1.1 for GB1. 5
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Fig. 1. Benefit and cost of entry for the marginal noise trader.
equilibria with low and high entry by noise traders (or low and high exchange rate volatility). Thus the emergence of multiple equilibria for intermediate levels of fundamental variances in this model has some interesting implications. First of all, there is no simple relationship between the volatility of fundamentals and the exchange rate. Two countries may exhibit different levels of exchange rate volatility even though they have similar fundamentals. Secondly, the state of an economy may jump between multiple equilibria due to changes of sunspot variables or parameters such as unrelated policy actions (see Proposition 3 in Appendix A). 5. In this model, I consider γ in Eq. (7) as a policy parameter indicating how the interest rate policy responses to the exchange rate. If γ N 0, it means that the monetary authority leans against exchange rate depreciations. It is of interest to ask whether or not an interest rate defense can stabilize the exchange rate, that is, whether or not an interest rate defense leads to an equilibrium with lower exchange rate volatility. Under some conditions, given γ N 0, higher values of γ leads to higher exchange rate volatility, and may end up with a unique equilibrium in which all noise traders enter the Peso bond market and lead to higher exchange rate volatility. This can be illustrated in Fig. 1. A higher value of γ (from 0.5 to 1.1) shifts the gross benefit curve up from GB0 to GB1, and induces a high volatility equilibrium, point E4. Therefore, the model predicts that under some conditions, given that a policymaker employs an interest rate defense policy, the more aggressive she responds to the exchange rate depreciations (larger the value of γ), the higher the exchange rate volatility (see Proposition 4 in Appendix A). Raising interest rate to defend the currency may push the currency into a highly variable exchange rate regime, i.e. an interest rate defense policy may destabilize rather than stabilize the exchange rate. The intuition is that higher interest rates induce higher risk premia, which raises the
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gross benefit of entry for noise traders, attracts more noise traders to enter the market, and then makes the risk premium even much higher. Finally, it may end up with a high exchange rate volatility with high risk premium equilibrium that induces all noise traders to enter the Peso bond market. In sum, a tightened monetary policy induces capital inflow as conventional uncovered interest rate parity framework predicted. However, if both rational and noise traders coexist in the financial market, a higher interest rate raises the excess return, which attracts more noise trading and hence exaggerates the exchange rate volatility. 3. Concluding remarks In this paper the effectiveness of interest rate defense policy is investigated theoretically. Chen (2006) has documented an empirical regularity that higher interest rates are associated with a longer length of spells of high exchange rate volatility and a shorter length of spells of low exchange rate volatility. Hence, the paper suggests that any structural model of the interest rate– exchange rate nexus ought to predict this relationship between interest rates and exchange rate volatility. Clearly, the simple, stylized model proposed in this paper is consistent with the empirical regularity and provides one of cause and effect. Appendix A In this appendix, I will show how to obtain the main results of the model outlined in Section 2. First of all, let me define an equilibrium in this model. Definition 1. (Equilibrium) An equilibrium consists of a path of stochastic processes for the exchange rate {et}, the risk premium {ρt}, and traders' optimal decisions ηtj and btj, such that (a) ηtj satisfies the entry condition (1); (b) btj solves for the maximization problem (2); P (c) Market clearing condition for Peso bonds is: B¯ ¼ Nj¼1 gtj btj . As in Jeanne and Rose (2002), I solve the model by the strategy of “guess-and-verify”. Since all the shocks are assumed to be independently and identically distributed xt f i:i:d: N ð0; varðxÞÞ; e˜ f i:i:d: N ðEð˜eÞ; varð˜eÞÞ, I take a guess that the solution will be: (a) et f i:i:d: N ðe¯; varðeÞÞ, and qt f i:i:d: N ðq ¯ ; varðqÞÞ; (b) all informed traders, and a constant number of noise traders, n ≤ NN, enter the Peso bond market at each period. The equilibrium will be presented by two steps. I first determine the equilibrium exchange rate, taking the number n as given. Next, I use the entry condition to solve for n. A.1. Analysis when the number of noise traders is given If the risk premium ρt is normally distributed, the maximization problem (2) can be rewritten as a j j max Utj ¼ Etj ðWtþ1 Þ− vartj ðWtþ1 Þ; j 2 bt
j s:t: Wtþ1 ¼ ð1 þ i⁎ ÞW þ gtj ðbtj qtþ1 − cj Þ:
ð11Þ
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Thus, I can get the demand for Peso bonds by an individual trader as follows: ! 1 Etj ðqtþ1 Þ j bt ¼ : a vartj ðqtþ1 Þ
775
ð12Þ
Then using market clearing condition in the Peso bond market, I have ¯¼ B
NI a
Et ðqtþ1 Þ n q¯ þ st þ vart ðqtþ1 Þ a vart ðqtþ1 Þ
1 NI Et ðqtþ1 Þ þ nð q¯ þ st Þ ¼ ; a varðeÞ
ð13Þ
ð14Þ
where I use the fact that in equilibrium, vart ( ρt + 1) = vart (et + 1) = var (e) from Eq. (3). I can then solve for the average risk premium from Eq. (14), and solve for the equilibrium exchange rate and the variance of the exchange rate as follows,6 a B¯ varðeÞ; NI þ n 1 n gð˜e−Eð˜eÞÞ−xt − st ; et − e¯ ¼ 1þg NI
q¯ ¼
varðeÞ ¼
g2 varð˜eÞ þ varðxÞ 2 : ð1 þ gÞ2 −k NnI
ð15Þ ð16Þ ð17Þ
1 Where ¯e ¼ ðq ¯ þ i⁎ Þ. g A.2. Analysis when the number of noise traders is endogenous I now consider the entry decisions of both informed and noise traders in the Peso bond market. The first simple result, which can be easily observed, is that the informed traders always enter the Peso bond market in equilibrium since the informed traders bear no entry costs. Some other important results can be summarized in following propositions: Proposition 1. Noise trader j enters the Peso bond market in equilibrium only if the gross benefit of entry, (GB), exceeds her cost of entry. That is GBðq ¯ ; varðeÞÞzcj ;
ð18Þ
where GBð q; ¯ varðeÞÞ ¼
1 1 q¯2 þ lnð1 þ kÞ: 2að1 þ kÞ varðeÞ 2a
Proof. See Jeanne and Rose (2002). 6
ð19Þ □
To get Eq. (16), we need to combine 2 Eqs. (3), (7), (14), and (15). Eq. (17) can be derived from Eq. (16). Note that we impose the condition ð1 þ gÞ2 −k NNNI N0 to ensure the variance of exchange rate is positive for all n ≤ NN.
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According to the traders' entry decision, I can get the following proposition relating to multiple equilibria: Proposition 2. For some levels of fundamental variance and entry costs, there exist multiple equilibria with a range of entry (from low to high) by noise traders. Proof. Combining Eqs. (15), (17), and (19), I can get
GB ¼
2 2 1 a B¯ g varð˜eÞ þ varðxÞ 1 2 þ lnð1 þ kÞ; 2að1 þ kÞ NI þ n 2a 2 n |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ð1 þ gÞ −k NI A when n z |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð20Þ
z when n z
that is, generally speaking, GB is a non-monotonic function in n. Thus, there will be more than one interception of the gross benefit curve, GB, and the entry cost curve, cj. □ Proposition 3. For sufficiently high levels of fundamental variances (var(e˜ ) or var(ω)), there is one unique equilibrium with all noise traders entering the Peso bond market. For sufficiently low levels of fundamental variances, there is an equilibrium with no noise traders in the Peso bond market. For intermediate levels of fundamental variances, there exist multiple equilibria with low and high entry by noise traders (or low and high exchange rate volatility). Proof. Since increase of the fundamental variances shifts up the GB curve, the GB curve is everywhere above the entry cost curve when the fundamental variances are sufficiently high. This results an unique equilibrium with all noise traders entering the Peso bond market. Moreover, there can be another unique equilibrium with no noise traders in the Peso bond market when the fundamental variances are sufficiently low, which makes the GB curve to be everywhere below the entry cost curve. Finally, Proposition 2 is applied for intermediate levels of fundamental variances. □ Proposition 4. Given γ N 0, higher values of γ leads to higher exchange rate volatility, and may end up with a unique equilibrium in which all noise traders enter the Peso bond market 1 eÞvarðxÞÞ2 ˜ ∀n ≤ NN , where g˜ ¼ AþðDþ4varð˜ and lead to higher exchange rate volatility if γ N γ ; 2varð˜ eÞ 2 A ¼ −varð˜eÞ þ Nn kvarð˜eÞ þ varðxÞ ; and D ¼ A2 . I
Proof. First, I have the result that 0
1 2 2 n 2 2gvarð˜eÞ ð1 þ gÞ − NI k −2½g varð˜eÞ þ varðxÞð1 þ gÞC ¯ 2 B AGB 1 aB B C ¼ B C: 2 2 @ A Ag 2að1 þ kÞ NI þ n 2 n ð1 þ gÞ − NI k
ð21Þ
2
Thus, AGB eÞg2 þ varð˜eÞ− NnI kvarð˜eÞ−varðxÞ g−varðxÞN0, (i.e. γ N γ˜ ∀n ≤ NN while I Ag N0 if varð˜ consider the positive value of γ only). Since the GB curve shifts up, it may end up that the GB curve is everywhere above the curve of entry cost, so that there is one unique equilibrium in which □ all noise traders enter the market.
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References Chen, Shiu-Sheng, 2006. Revisiting the interest rate–exchange rate nexus: a Markov switching approach. Journal of Development Economics 79 (1), 208–224. DeLong, J. Bradford, Shleifer, Andrei, Summers, Lawrence H., Waldmann, Robert J., 1990. Noise trader risk in financial markets. Journal of Political Economy 98 (4), 703–738. Jeanne, Olivier, Rose, Andrew K., 2002. Noise trading and exchange rate regimes. Quarterly Journal of Economics 117 (2), 537–569.