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Cuadernos de economía
versión On-line ISSN 0717-6821
Cuad. econ. v.45 n.132 Santiago nov. 2008
http://dx.doi.org/10.4067/S0717-68212008000200001
Cuadernos de Economía, Vol. 45 (Noviembre), pp. 161-183, 2008
What Exactly is 'Bad News' in Foreign Exchange Markets? Evidence from Latin American Markets*
CECILIA MAYA^{1}, KAROLL GÓMEZ^{2} ^{1} Universidad EAFIT, Colombia cmaya@eafit.edu.co,
This paper asks whether the 'leverage effect -as defined by Black (1976) for stock markets- is also a characteristic of foreign exchange markets. The study focuses on five Latin American emerging markets which have adopted a floating exchange regime. It finds that the response of exchange rates to volatility shocks is characterized by long memory and symmetry in most countries. The response is asymmetric only in Brazil and Peru. A possible explanation for this asymmetry is the fear of floating' that induces side-effects on interest rates and inflation, which the market considers 'bad news'. The opposite direction of the asymmetry may be explained by the particular characteristics of each economy. Keywords: Exchange Rate Volatility, Leverage Effect, Asymmetric Volatility, GARCH, HYAPARCH. JEL: CÍO, C22, F31, G10, G15
RESUMEN El objetivo de este artículo es investigar si el efecto apalancamiento (leverage effectj encontrado en Black (1976) para los mercados accionarios también está presente en el mercado cambiarlo. El estudio se realizó para los cinco países latinoamericanos que han adoptado el régimen de flotación en sus tipos de cambio. Encontramos que las respuestas a los shocks en volatilidad se caracterizan generalmente por larga memoria y simetría, excepto en Brasil y Perú donde dichas respuestas son asimétricas. Una posible explicación a este comportamiento es el miedo a flotar (Tear or floating) que al inducir efectos colaterales sobre la tasa de interés y la inflación provoca "malas noticias" para el mercado. Las características peculiares de cada economía pueden explicar las diferentes direcciones que toma la asimetría.
1. Introduction It is a well known fact that returns in foreign exchange markets cannot be predicted. Tests on most of the different exchange rate series cannot reject the hypothesis of a zero conditional mean (Yang, 2006). Volatility dynamics, on the other hand, has attracted most of the attention due to the abundant evidence of heteroskedasticity on returns. The presence of volatility-volatility correlation^{1} or volatility clusters is a well known characteristic of financial time series and has been widely studied, yet it does not fully explain the volatility dynamics of these series (Engle, 2004). Other features -such as leverage, asymmetries, and long memory- ought to be also considered in volatility models to better match the data. In 1976, Fisher Black discussed for the first time what is known as the 'leverage effect' (Le., the increase in volatility when a stock price falls), which indicates a negative correlation between return and volatility. He argued that when the stock price falls, the leverage of the firm increases and its volatility rises due to the 'bad news'^{2}. This paper asks if the leverage effect is also a characteristic of foreign exchange markets and whether it is a result of bad news. The latter result would clearly require a different economic explanation than that given by Black. Identifying leverage when modeling volatility dynamics is important for a variety of reasons. An important one is for option pricing which rests on the correct modeling of the underlying asset. In particular, the most commonly used model for foreign exchange options is that of Garman and Kohlhagen (1983) which assumes log-normality based on constant volatility, in clear contradiction with the existing evidence. Even when non constant volatility is identiñed in the form of volatility smiles, a long-observed pattern in which at-the-money options tend to have lower implied volatilities than other options, a misconception remains since leverage may be best represented by volatility skews^{3}. However, the textbook application to foreign exchange options pricing assumes always a symmetric volatility smile (see Hull, 2006, p. 376) leading to incorrect valuations. Furthermore, modeling leverage is crucial for market risk measures, Le. Value at Risk -VaR. This is a widely used measure of market risk mandatory for financial institutions in countries that have adopted the Basilea agreement. VaR focuses on the left tail of the return distribution and computes the worst probable loss within a certain level of conñdence. Ignoring leverage may result in under-estimating that risk (Engle, 2004). Having in mind the above motivation, we search for evidence of leverage in Latin American foreign exchange markets. We find volatility asymmetries in both directions, not just left asymmetry as in the leverage effect, and also find evidence of long memory, indicating that volatility shocks persist for a period of time longer than expected. The paper proceeds straightforwardly. We review the empirical evidence on leverage in section two. Section fhree discusses the models we then apply to Latin America currencies in section four. Finally, we present some conclusions. 2. Empirical Evidence on Le ver age Evidence of leverage in stock markets is common in empirical studies. Johnson and Soriano (2003) perform a study on 39 countries for the 1990-2002 period and conclude that the EGARCH(1,1) and TGARCH(1,1) models^{4} fit adequately the daily return data on those markets. Bouchaud et al. (2001) find correlation between future volatility and past returns in a sample of seven major indices^{5} and 437 stocks on the S&P index, whereindices exhibit a higher leverage than stocks. Shively (2007) applies a structural bivariate threshold model^{6} and finds that temporary innovations account for 68% of the innovations after negative returns and less than 4% after the positive ones, concluding that the negative-return regime is the high-volatility one. For foreign exchange returns, on the other hand, symmetry has been the textbook typical assumption (Hull, 2006). Bollerslev et al. review a large amount of empirical evidence and conclude that "whereas stock returns have been found to exhibit some degree of asymmetry in their conditional variances, the two-sided nature of foreign exchange markets makes such asymmetries less likely" (Bollerslev, Chou and Kroner, 1992, p. 38). Hsieh (1988), Diebold and Nerlove (1989), and Taylor (1986) among others, support this assertion by finding the symmetric GARCH(1,1) to be the best fit ting model for exchange rate returns (US$ against other major currencies), making it the natural choice for modeling their volatility. Andersen et al. (2001) also find symmetry in the exchange rate volatility dynamics. On the contrary, recent studies find evidence of asymmetry for some exchange rates. A study by Oh and Lee (2004) on the Korean Won/USD and Won/ Yen find an asymmetric response to devaluation or revaluation. They model the return series using the EGARCH(1,1) and the TGARCH(1,1) model of Glosten, Jaganatfhan and Runkle's (1993), Yang (2006) applies the latter, the GARCH(1,1) model, and a semiparametric GARCH^{7} model on the Mark and the British Pound to the US dollar, finding that the last model performs better than the symmetric GARCH and the TGARCH. Additional evidence on asymmetric volatility in foreign exchange markets is found in Hsieh (1989), Tse and Tsui (1997), McKenzie and Mitchell (2002), and Adler and Qi (2003). In Latin America, Fernández (2003) studies the Chilean Peso to the US Dollar exchange rate after the adoption of a floating exchange regime and finds the asymmetric GARCH^{8}, the exponential smooth transition GARCH^{9} and the EGARCH to be the best fitting models from a wide selection of twelve ARCH type models. Domac and Mendoza (2003) find evidence in favor of an EGARCH(1,1) model for the returns of the Mexican peso to the US Donar from August 1996 to June 2001. For the Brazilian Real to the US Dollar returns from January 1999 to May 2004, Vilela and Holland (2004) fit a GARCH(1,1) process. In conclusión, the empirical evidence on leverage for foreign exchange returns is mixed. We find both symmetry and asymmetry, although the latter is more common in emerging markets. On the other hand, volatility is sometimes higher for negative returns as defined by the leverage effect, yet this is not always the case. Clearly, the leverage effect -defined by Black (1976) as an increase in the debt-to-equity ratio on individual companies- is not an adequate explanation for the asymmetric volatility found in the analysis of some foreign exchange return series. Furthermore, it is possible that the two-sided nature of these series men-tioned by Bollerslev et al. (1992), instead of generating a symmetric effect on volatility, causes volatility to be higher on one side, not necessarily the negative one. This means that some markets may be more sensitive to devaluation while others would show a stronger reaction to revaluation. This fact makes it difficult to define bad news in foreign exchange markets. In contrast to stock markets where bad news are equated to negative returns (Nelson, 1991)^{10}, in foreign exchange markets they could go in either direction. In the next two sections we define and apply the main volatility models which incorpórate asymmetries in order to draw some conclusions about what exactly is bad news in foreign exchange markets in Latin America. 3. Asymmetric Volatility Models One of the most important characteristics of Autoregressive Conditional Heteroskedasticity -ARCH- models is their ability to capture many of the empirical regularities present in fmancial return series, such as time varying variances and clusters of volatility or the abovementioned volatility-volatility correlation. These models nave evolved since they were initially suggested by Engel (1982) to include asymmetries and some other characteristic features of fmancial time series. From the large number of ARCH models that can be found in the literature, we present a selection of the most useful specifications to detect asymmetries, departing from the classical symmetric GARCH proposed by Bollerslev (1986). Bollerslev (1986) generalized Engel's ARCH model with a more parsimonious specification for the conditional variance equation including lags of the variance. In this GARCK(p,q) model, the conditional variance σ^{2}follows an ARMA (p, q) and The model must satisfy the following restrictions in order to guarantee a non-negative and non-persistent conditional variance: c > 0, α_{j} , ß _{j} ≥ 0 and , respectively. Additionally, it requires p ≥ 0 and q > 0 for all j =1, ...,p and i= 1, ..., q. However, both ARCH and symmetric GARCH models have some limita-tions. The most important one has to do with the non negativity constraints for parameters in equation (1) to guarantee a nonnegative variance for all t with probability one. This constraint makes the estimation difficult and implies that increasing , at time t increases , excluding the possibility of a future oscillatory behavior (Nelson, 1991). Nelson (1991) also points out that the GARCH model assumes that posi-tive and negative shocks have a similar impact on the conditional variance since is a function of the square of the error term u_{t}. He argües that in practice the effect may not always be symmetric as it is the case for stock returns where bad news increases volatility due to the leverage effect. Consequently, he proposes the exponential GARCH -or EGARCH- model which can capture those leverage effects. If returns follow an EGARCH process, the conditional variance depends on both the magnitude and the sign of lagged residuals which allow asymmetric responses to negative and positive shocks. The equation for the conditional variance is: where and δ and γ are real constants. The 'news impact curve', g(v_{t}), represents the asymmetric response to news and is a function of the standardized residuals v_{t-i} = u_{t-i} /σ_{t-i} that can be written as For negative values of the innovation v_{t}, the function g(v_{t}) is linear with trend (δ + γ), while for positive ones the trend is (δ - γ). The second term in equation (2) represents the magnitude of ARCH effects determined by the difference between the actual value of innovation and its expected value. The non-negativity condition for the conditional variance is guaranteed by its logarithmic specifica-tion, and will be stationary if ß_{j}, < 1. Along the same line, Ding, Granger, and Engle (1993) proposed the Asymmetric Power GARCH (APGARCH) which allows for non linearities and asymmetry in any direction. Its specification is as follows: with and = 1,...,q. The APGARCH nests many other models like ARCH, GARCH, Taylor's GARCH^{11} (1986), and the Threshold GARCH (Glosten, Jaganatthan and Runkle, 1993; Zakoian, 1994). It becomes an ARCH when p = 2, ß = 0 and γ = 0; a GARCH in the case of p = 2, ß > 0 and γ = 0; finally, it would be a TGARCH if ρ = 2, ß > 0 and 0 < γ < 1 or ρ = 1, ß > 0 and -1 < γ < 1, where this last specification corresponds to the Zakoian (1994) TGARCH model. A more general model which nests this wide range of GARCH models (see Table A1 in the Appendix) is the Hyperbolic Asymmetric Power ARCH -HYAPARCH- proposed by Dark (2006), which accounts for both long memory^{12 }and volatility asymmetries. HYAPARCH is generally preferred over some other specifications allowing for long memory like HYGARCH (Davidson, 2004) which assumes symmetric responses and FIGARCH (Baillie, Bollerslev and Mikkelsen, 1996), FEGARCH (Bollerslev and Mikkelsen, 1996) and FIAPARCH (Tse, 1998) since all these fractionally integrated GARCH models lead to infinite variances preventing their use for most financial time series. Instead, the HYAPARCH is covariance stationary, it models amplitude and memory separately^{13}, and it also allows the power of the heteroskedastic equation to be estimated from the data. The HYAPARCH model may be expressed as where and are the conditional variance polynomials with all roots outside the unit circle and without common factors. represent amplitude, memory, power, and asymmetry, respectively, and (1 - L)^{d} is the lag operator of the fractional differencing parameter. The conditions for weak stationarity are (Schoffer, 2003; Davidson, 2004). The estimation method is máximum likelihood under different assumptions for the error distribution^{14}. 4. Latin American Exchange Rates: Volatility dynamics^{15} This study focuses on the dynamics of the foreign exchange rate of five Latin American currencies expressed as units of local currency to the US DoUar and the Euro. The group under study includes only emerging markets^{16} which had adopted floating exchange rate regimes by 2000: Brazil, Chile, Colombia, México, and Perú. The data^{17} is the daily exchange rate for the period August 1, 2000-July 31, 2007 and returns are computed as logarithmic variations of these daily exchange rates (see Figure A1 in the Appendix). A descriptive statistics summary for each of the series is shown in Table 1. Standard deviations show the Brazilian Real to the US Dollar and the Euro as the most volatile return series. The Jarque Bera test rejects normality for all series which are also characterized by leptokurtosis, especially for the exchange rates in terms of dollars. The Brazilian Real and the Peruvian New Sol exhibit negative skewness as well. The Ljung Box statistic shows evidence of serial correlation significant to the tenth lag in squared returns. However, in levels, this statistic shows serial correlation for all currencies to the Dollar -except the New Sol-, but not to the Euro -with the exception of the Real. Figure 1 shows the histogram, Kernel density and the Q-Q plot for each of the series. They confirm non normality and fat tails, more significant in the cases of the Real to both the US Dollar and the Euro, and the New Sol to the US Dollar. There is also evidence of long memory in the series, a stylized fact for financial series largely documented in the literature (Lo (1991), Ding, Granger y Engle (1993), Baillie (1996) y Granger y Ding (1996)). The sample correlogram up to the 100^{th} lag is shown in Figure 2. Squared returns provide evidence of long memory, exhibiting hyperbolic decay characteristic of volatility persistence, more pronounced in the case of the Brazilian Real and the Colombian Peso, both versus the Dollar and the Euro. Given the serial correlation and long memory evidence f ound in some returns and square returns, to model the conditional mean and conditional variance we proceed to estímate ARFIMA-HYAPARCH model. We checked different conditional distributions for the error term including student's t, skewed student's t and GED, given the rejection of normality. Results are reported in Tables 2 and 3. The appropriate specification in each case was selected following nested models strategy^{18} and based on the Akaike Information Criterion and Schwarz Bayesian Criterion. To check for serial correlation in the residuals we applied the Ljung Box test on the standardized residuals and the squared residuals. In both cases, the nuil hypothesis was rejected for all cases, indicating that the absence of time dependency in the residuals or additional GARCH effects. Therefore, the adjustment of the models is deemed satisfactory. Finally, we found evidence in favor of the generalized error distribution in all cases. Most of the exchange rates display evidence of volatility persistence. In the case of the Colombian Peso to the US Dollar, we obtained the same result found by Castaño, Gómez and Gallón (2007), Le., a long memory model for the mean process and an integrated process in variance. The mean process for the Chilean Peso to the US Dollar follows an ARMA(1,1). For the rest, including all exchange rates to the Euro, we can not reject the nuil hypothesis of zero mean, a common result in the literature. For the variance, with the exception of Colombia to the Dollar and Chile to the Euro which are integrated processes, and Perú to the Dollar which follows an EGARCH, all the other processes are hyperbolic or fractionally integrated GARCH (see Tables 2 and 3). These results transíate into a very slow decay in shocks to the variance of most of the exchange rates examined in this study. There is also evidence of skewness in the cases of the Real to the Dollar and to the Euro, and the New Sol to the Dollar, given that γ_{1} is different from zero at a level of 1%. For the Real to both the Dollar and the Euro, the negative sign associated to this parameter indicates that volatility is higher when shocks are negative, that is, when the local currency appreciates. For the New Sol to the Dollar there is also evidence of asymmetry, but in this case volatility is higher when shocks are positive, meaning the response is stronger to devaluation of the local currency. There is no evidence of asymmetry in the other exchange rates. Additionally, we conclude that the most appropriate way to model volatility in most of the returns is in terms of the conditional variance, with the exception of the Chilean Peso to the Euro, where 8 takes a value around one, suggesting that this exchange rate should be modeled in terms of the conditional standard deviation. We also explore coefficient stability in these models. When the timing of potential structural shifts is uncertain, a useful approach is to apply a general test of coefficient stability. Nyblom (1989) and Hansen (1992) develope stability tests that do not require specification of the timing of shifts. The Nyblom-Hansen tests evalúate the nuil hypothesis of coefficient stability against the alternative hypothesis that at least one coefficient does shift at some unspeciñed breakpoints. The tests are not designed for determining the timing of structural breaks; they merely test the nuil of parameter constancy. The resulting Nyblom-Hansen statistics for the joint stability of the coefficients in the different models are not large enough to reject the nuil hypothesis of coefficient stability at conventional size levels. As a final check for the robustness of our results, we analyze weekly returns. We obtain the same results for the exchange rates to the US Dollar. For the Euro, the results are similar to those for daily exchange rates except for the fact that GARCH effects disappear for the Mexican Peso and the Peruvian New Sol (see Tables A2 y A3 in the Appendix). Although it is frequently argued that the inclusión of exogenous variables -in the form of dummies or other measured variables- is required to explain these dynamics, Davidson shows that, while any of these may be the true explanations, there is no need to introduce them since "the behavior of currency markets [...] can be well described by a very simple endogenous mechanism, driven solely by the information contained in the shock process itself' (Davidson, 2004, p.15)^{19}. Given that we observe different responses to exchange rate shocks, our original question of what exactly is bad news in foreign exchange markets, leads us to issues of foreign exchange management by the Central Bank such as the 'fear of floating' hypothesis discussed by Calvo and Reinhart (2002). Although many countries may declare to have a floating exchange rate regime, in practice they intervene frequently to stabilize them. By doing so, however, they affect other variables -like inflation and interest rates-, which the market may, in fact, interpret as bad news. According to our previous results, the only two countries that exhibit asym-metric response to volatility shocks are Brazil and Perú, precisely the two countries that also show evidence of fear of floating in recent studies on this subject. On the basis of three variables (volatility of the nominal exchange rate, volatility of its change, and volatility of international reserves), Levy-Yeyati and Sturzenegger (2002), reclassify Brazil as having a ñxed exchange regime and Perú as having and intermedíate one. By their standards, Chile, Colombia and México have a floating regime by 2000, the end of their study. More evidence is found by Walker (2006), who classifies Brazil as the country with higher interventions when measured by 60-month rolling estimates of reserve monthly variation volatility. Perú classifies as a low reserve volatility country along with Chile, Colombia and México. However, a better measure of the 'fear of floating' proposed by Ibarra (2007) is the relative volatility of exchange rates to the reserve volatility^{20}, not the absolute volatility as in Walker. Although his study is restricted to Chile, Colombia and México, we extend it to Brazil and Perú. In the floating regime period up to September 2005, the relative volatility to reserves reported by Ibarra for Chile is 1.04, 0.79 for Colombia and 0.71 for México. On the basis of these results he concludes that these three countries adjust in fact to a floating regime. Our results for Brazil and Perú for this ratio are 0.14 and 0.29, respectively^{21}. This low ratio confirms that these two countries display evidence of fear of floating. Exchange rate management may transíate into asymmetric responses to volatility shocks since interventions may affect important variables such as interest rates and inflation, which the market considers truly bad news. Now, the different direction of the asymmetry depends on the particular characteristics of each economy^{22}. In the case of Brazil, which exhibits left asymmetry, a revalua-tion means large losses both to the US Dollar (as it enjoys a large positive trade balance of about US$ 40 billion) and to the Euro (since trade with the European Unión is also significant)^{23}. On the contrary, in Perú, a country that suffered hyperinflation and a deep economic crisis in the eighties, a devaluation may make the market much more nervous only to the US Dollar since this is the main currency of denomination for trade and capital flows. For Chile, Colombia and México, countries that do not suffer 'fear of floating', the response to volatility shocks is symmetric since the market expects these shocks to be absorbed mainly through the exchange rate itself. 5. CONCLUSIONS The response of exchange rates to volatility shocks varíes from one country to the other. In this study we find evidence of volatility-volatility correlation or volatility clusters frequently found in financial series, evidence on negative return-volatility correlation known as 'leverage', and also on positive return-volatility correlation, not so common in the literature. All these may be properly modeled by GARCH processes, which involve asymmetries and long memory, like the Hyperbolic Asymmetric Power ARCH (HYAPARCH) and the models nested by it. All exchange rates show evidence of long memory with the exception of the New Sol to the US Dollar. There is also evidence of skewness in the cases of the Real to the Dollar and to the Euro, and the New Sol to the Dollar. For the Real to both the Dollar and the Euro, volatility is higher when shocks are negative, that is, when the local currency appreciates. For the New Sol to the Dollar there is also evidence of asymmetry, but in this case volatility is higher when shocks are positive, meaning the response is stronger to devaluation of the local currency. There is no evidence of asymmetry in the other exchange rates. Only those countries that exhibit 'fear of floating' also show asymmetric responses to volatility shocks. From this, we conclude that exchange rate manage-ment may be a possible explanation for asymmetries since interventions may affect important variables such as interest rates and inflation, which the market considers truly 'bad news'. Now, the different direction of the asymmetry depends on the particular characteristics of each economy. In the case of Brazil, which exhibits left asymmetry, being a country with a large positive trade balance, revaluation means large losses. On the contrary, Perú, a country that suffered hyperinflation and a deep economic crisis in the eighties, devaluation may make the market much more nervous, in this case only to the US Dollar since this is the main currency of denomination for trade and capital flows. For Chile, Colombia and México, countries that do not suffer 'fear of floating', the response to volatility shocks is symmetric since the market expects these shocks to be absorbed mainly through the exchange rate itself.
NOTAS * Email: cmaya@eafit.edu.co, kgomezp@unalmed.edu.co. ^{1} Time-dependence in volatility meaning that a large return is followed by another large return and a small return is followed by another small return. ^{2} Causality is not clear though since we may ask: is it increased volatility what causes the drop in pnces or is it the fact that pnces fall what increases volatility? (Bouchaud, Matacz, and Potters, 2001). Although this negative return-volatility correlation is still known as the leverage effect, other explanations like time varying risk premia and volatility feedback (Campbell and Hentschel, 1992) have been suggested in the literature (García, Luner, and Renault, 2001). ^{3} A volatility smile is a plot of the implied volatility as a function of the exercise price of the option. When the figure is asymmetric, it is called volatility skew. The implied volatility is the volatility implied by current prices when the option is valued using Black and Scholes (1973) formula. ^{4} Both belong to the class of Autoregressive Conditional Heteroskedastic models introduced by Engle (1982) and Bollerslev (1986). The Exponential GARCH (EGARCH) and the Threshold GARCH (TGARCH) go further than the previous ones by introducing asymmetries in addition to volatility clustering and excess kurtosis. ^{5} The indices are S&P 500, NASDAQ, CAC 40, FTSE, DAX, NIKKEI and Hang Seng. ^{6} This model belong to the class of Regime Switching models which define different states or regimes of the world allowing the dynamic behaviour of variables to depend on the regime that occurs any given point in time. It assumes that the regime can be defined based on observable data in a way that both, the past and the present regime, are known with certainty. ^{7} This semiparametric GARCH model also accounts for asymmetry. ^{8} See Engel (2000). ^{9} González-Rivera (1998) ^{10} A possible exception would be a market predominantly short which may occur only temporarily. ^{11} Taylor models the conditional standard deviation instead of the conditional variance dynamics as it does the widely used symmetric GARCH by Bollerslev (1986). ^{12} Engle and Bollerslev (1986) found long memory or volatility persistence in some financial series and suggested an integrated GARCH (IGARCH) process where the sum of GARCH coefficients α + ß equals one; therefore, a volatility shock does not vanish. Later approaches to long memory include FIGARCH and HYGARCH models where shocks do vanish at a very slow hyperbolic rate. ^{13} Amplitude refers to "how large the variations in the conditional variance can be while memory determines how long shocks to the volatility take to dissipate" (Davidson, 2004, p. 3). ^{14} Frequently normal, student-t, skewed student-t, and Generalized Error Distribution GED are used for this purpose. ^{15} The estimation has been carried out using the package Time Series Modelling versión 4.25 by Davidson (2003). ^{16} All of them are included in both MSCI and S&P/IFC emerging market indexes. ^{17} Source: Bloomberg Professional® Service. ^{18} The strategy of nested models consists in contrasting results under different restrictions in the general model until the best fitting model is identified. ^{19} We introduced dummy variables to capture the effect of Central Bank interventions but they were insignificant, a result in line with Davidson's argument. ^{20} This is the ratio of rolling exchange rate monthly changes volatility to international reserves monthly changes volatility. ^{21} Calculations by the authors are available upon request. ^{22} Another explanation may be that Central Banks interventions usually go in one direction, making volatility on that side higher (Wang and Yang, 2006) which may be another way of saying the same thing. ^{23} Around 20% by 2005 (Biannual Report 2004-2005, Central Bank, Brazil).
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