Forecasting Exchange Rates 2

FIGURE V.2 CHF/USD Levels and Changes (January 1971-January 2010) CHF/USD Exchange Rate 5 CHF/USD 4 3 2 10 1/8/2009 1/8/2007 1/8/2005 1/8/2003 1/8/2001 1/8/1999 1/8/1997 1/8/1995 1/8/1993 1/8/1991 1/8/1989 1/8/1987 1/8/1985 1/8/1983 1/8/1981 1/8/1979 1/8/1977 1/8/1975 1/8/1973 0 1/8/1971 1 CHF/USD Changes 8 Changes (%) 6 4 2 0 -2 -4 -6 -8 -10 Many researchers found the variance of asset returns to be predictable over time. The problem faced by researchers was to find an appropriate model that incorporated the empirical characteristic of financial assets returns. Robert F. Engle, in a paper published in Econometrica in 1982, introduced the Autoregressive Conditional Heteroskedastic (ARCH) model. This model became very popular because of its simplicity and intuitive appeal. Engle's ARCH(q) model is specified as: 2t = 0 + iq i e2t-i. (V.2) The variance at time t depends on past conditional errors. Therefore, the variance at time t is predictable with the information set available at time t-1. This information set will be labeled t-1. The ARCH(q) model has a familiar time-series structure. If we consider the squared error, e2t, as an approximation for the variance at time t, the ARCH(q) model has a similar structure than the AR(q) model (see Appendix V). V.16 Empirical Regularities and ARCH Models Rob Engle's model incorporates an empirical fact observed in returns of financial assets: large (small) changes in returns are likely to be followed by large (small) changes in returns. A large shock yesterday (et-1 is big) tends to increase the variance today, 2t. This phenomenon is called volatility clustering and it is typical of financial assets. Volatility clustering is easily observed in the second panel of Figure V.2. Another empirical fact that the ARCH(q) model incorporates is that the unconditional distribution of financial assets' returns has thicker (fatter) tails than the normal distribution. That is, in statistical language, the unconditional distribution is leptokurtic. Later, Tim Bollerslev, in a paper published in the Journal of Econometrics, in 1986, introduced the GARCH model, which generalized the ARCH model. The GARCH(q,p) model is specified as follows: 2t = 0 + iq i e2t-i + ip ßi 2t-i. (V.3) Now, in equation (V.3), the variance at time t depends on past errors and past variances. It can be shown that the GARCH(q,p) is similar to an ARMA model. Since a variance has to be positive, the restrictions 00, i0 (i=1,...,q), and ßi0 (i=1,...,p) are usually imposed. If we take expectations in equation (V.3), and set E[e2t]=E[2t]=2, we can obtain the unconditional (or average) variance, 2, that is, 2 = 0/(1 - iq i - ip ßi). For the unconditional variance to be well defined (i.e., bounded and positive), we need the following condition: iq i + ip ßi < 1. GARCH models have been successfully employed for exchange rates and other financial series at different frequencies: monthly, weekly, daily and intradaily. In several of these applications a useful regularity has emerged: in general, a GARCH(1,1) specification is able to capture the dynamics of the conditional variance: 2t = 0 + 1 e2t-1 + ß1 2t-1. (V.4) The GARCH(1,1) model says that the current variance estimate can be calculated from the variance estimated last period, 2t-1, modified by the squared error also observed last period. In the GARCH(1,1) the unconditional variance is well defined when 1+ß1<1. Let =1+ß1. For reasons that will become clear below, the parameter  is called the persistence parameter. Estimation of GARCH models is usually done using a method called maximum likelihood estimation. To use this method of estimation, we have to assume a distribution for the conditional errors, et. We will assume that et, conditional on the information set t-1, follows a normal distribution with mean V.17 zero and variance 2t. That is, et|t-1 ~ N(0,2t). Given the GARCH structure, the estimation involves nonlinear optimization techniques. Sometimes, it is difficult to obtain convergence for the GARCH model. For those familiar with nonlinear optimization, it is well-known that initial values are very important. Many statistical packages, like SAS and TSP, estimate GARCH models. Once the parameters i's and ßi's are estimated, equation (V.3) can be used to forecast the next-period variance. Example V.5: Sietes Swiss, a Swiss company, wants estimate the volatility of the weekly CHF/USD exchange rate. From their experience, forecasters at Sietes Swiss know that changes in exchange rates are very difficult to forecast. Therefore, they use a very simple AR(1) process for the mean return. The variance is modeled using the popular GARCH(1,1) specification: et|t-1 ~ N(0,2t). st = [log(St) - log(St-1)]x100 = a0 + a1st-1 + eFt, 2t = 0 + 1 e2t-1 + ß1 2t-1. Sietes Swiss using SAS estimates the above model using weekly data provided by the Federal Reserve of Chicago (http://www.frbchi.org). The sample starts in January 8, 1971 and ends in January 21, 2010, for a total of 2038 observations. The estimated model for st is given by: st = -0.055 + (0.32) 2t = 0.236 + (1.71) 0.051 st-1, (1.66) 0.092 e2t-1 + 0.822 2t-1, (4.00) (11.83) where t-statistics are in parenthesis. In Figure V.3 we plot the conditional volatility (squared root of the variance) over time for the CHF/USD exchange rate, as calculated by the estimated GARCH(1,1) model. Consistent with the visual intuition in Figure V.1, volatility changes over time. We observe the high volatility periods associated with the 1978-1980 FIGURE V.3 GARCH(1,1) CHF/USD Volatility Estimates (1/1971-1/2010) 14 12 8 6 4 2 V.18 2/5/2009 2/5/2007 2/5/2005 2/5/2003 2/5/2001 2/5/1999 2/5/1997 2/5/1995 2/5/1993 2/5/1991 2/5/1989 2/5/1987 2/5/1985 2/5/1983 2/5/1981 2/5/1979 2/5/1977 2/5/1975 2/5/1973 0 2/5/1971 Variance 10 Note that Sietes Swiss finds that =.955 < 1. That is, the unconditional variance, 2, is well defined. Given that e1048 = -0.551, and 22038 = 2.554, Sietes Swiss is able to forecast the variance for the week ending on January 28, 2010 --22039. That is, 21039 = 0.140 + 0.079 (-0.551)2 + 0.876 (2.554) = 2.401. ¶ Multi-period variance forecasts can be easily calculated using a GARCH(1,1) model. It can be shown that the conditional expectation of the variance k periods ahead is given by Et[2t+k] = (1 + ß1)k [2t - 0/(1-1-ß1)] + 0/(1-1-ß1) = k [2t - 2] + 2 Note that if =(1+ß1)<1, as k, k0. That is, as we forecast more into the future, i.e., k, the forecasts converges to the unconditional variance, 2. Note that the parameter  represents the rate of convergence (decay) of the conditional variance towards 2. Example V.6: Suppose Cannigia Co. estimates a GARCH(1,1) model using 300 daily observations for the GBP/USD exchange rate. Cannigia Co. uses percentage changes of the USD/GBP exchange rates to estimate the GARCH model. Cannigia Co. obtains the following estimates for the variance parameters :0 = .175, 1 = .123, and ß1 = .847. The estimate for the conditional variance of the last observation is 2300 = 1.959. Cannigia wants to forecast the variance 10 days from now. The unconditional variance is given by 2 = .175/(1 - .123 - .847) = 5.833. Then, the 10-day-ahead variance forecast is: E300[2310] = (.97)10 [ 1.959 - 5.833] + 5.833 = 2.976. That is, the volatility forecast 10 days ahead is 310 = 1.725%. ¶ Many empirical studies also find that =1+ß1 is close to one. For instance, in Example V.2, we find that =.955. When =1, the GARCH(1,1) model is called integrated GARCH model, or IGARCH. An IGARCH model does not have a well-defined 2. This formulation reduces the number of parameters of the conditional variance equation (V.4) to two. Therefore, (V.4) becomes 2t = 0 + 1 e2t-1 + (1-1) 2t-1. When a series is described by an IGARCH process, the multi-period variance forecast becomes: Et[2t+k] = 2t + k 0. The variance at time t affects the conditional variance forecasts into the indefinite future. Thus, today's shocks to the conditional variance persists forever in conditional variance forecasts. When  is close to one, shocks to the variance today show a high persistence in the conditional variance forecasts. That is, a big surprise today (big et-1) has an influence on the variance for several periods. V.19 2.A J.P. Morgan's RiskMetrics Approach In October 1994, J.P. Morgan unveiled and made freely available its risk management software RiskMetrics (http://www.jpmorgan.com/RiskManagemet/RiskMetrics). RiskMetrics is a method that estimates the market risk of a portfolio, using the Value-at-Risk (VAR) approach. RiskMetrics uses a simple specification to model a time-varying variance: 2t = (1 - ) e2t-1 +  2t-1. That is, next period's variance is a weighted-average of this period's squared forecast error and this period's variance. The weights are given by the parameter , which is referred to as the decay factor and, then, should be less than one. You should note the similarity of the RiskMetrics approach with the IGARCH model, with 0=0. In this case,  = (1-1) = ß1. Therefore, shocks to exchange rates persist forever in variance forecasting. Based on convenience and experience, J.P. Morgan selected to set  equal to .94 for daily data. Ideally, this parameter should be estimated for every series, every day. Given the large number of series that J.P. Morgan uses (more than 500), RiskMetrics uses the same  for all the series. For monthly volatility, RiskMetrics sets =.97. Example V.7: Suppose Cannigia Co. estimates a time-varying variance using the RiskMetrics model using 300 daily observations for the GBP/USD exchange rate. Cannigia Co. uses percentage changes of the USD/GBP exchange rates to estimate volatility. The estimate for the conditional variance of the last observation is 2300 = 1.959 and e300 = 0.663. Cannigia wants to forecast the variance tomorrow. Using RiskMetriks, Cannigia forecasts: E300[2301] = (.94) 1.959 + .06 (.633)2 = 1.8655 That is, the volatility forecast for tomorrow is 310 = 1.366%. ¶ 2.B Using Forecasts of the Variance: Value-at-Risk (VaR) Value-at-Risk (VaR) provides a number, which measures the market risk exposure of a portfolio of a firm over a given length of time. VaR measures the maximum expected loss in a given time interval, within a confidence interval. Note that to calculate a VaR of a portfolio, we need to specify a time interval and the significance level for the confidence interval. Suppose we specify a day as the time interval and 5% as the significance level to calculate the VaR of a FX portfolio. Suppose the VaR of the FX portfolio equals to USD 10,000. This VaR amount (USD 10,000) represents the potential loss of the FX portfolio in about one every twenty days. Example V.8: Microsoft uses a VaR computation, within a 97.5% confidence interval, to estimate the maximum potential 20-day loss in the fair value of its foreign currency denominated investments and account receivables, interest-sensitive investments and equity securities. At the end of June 2001, Microsoft calculated a VaR of negligible for foreign currency instruments, USD 363 million for interest sensitive V.20 instruments, and USD 520 million for equity investments. ¶ 2.B.1 Calculating a VaR Suppose that a firm has a portfolio of foreign currencies, with a value W. Let the returns, r, of this portfolio follow a probability distribution, with mean return of  and volatility equal to . The initial value of the portfolio is W0. The value of the portfolio at the end of the period is W1= W0 (1 + r). Now, let W* be the minimum value of the portfolio within a confidence interval c. That is, W*= W0(1 + r*), where r* represents the minimum return of the portfolio within a confidence interval. Now, we can define a measure for the mean exposure: VaR (mean) = E(W1) -W* = W0(1+) - W0(1+r*) = W0 ( - r*) Note that computing a VaR is equivalent to computing r*. We can derive the VaR using the distribution of the future values of the portfolio, f(w). Within a confidence interval c, we can find W* such that c  W* f ( w ) d ( w) A usual assumption is to convert f(w) into a standard Normal distribution. Let’s calculate the VaR (mean) using a standard Normal distribution. Recall that a standard normal distribution has =0 and =1. First, we need to standardize the data. That is, Z = (r-)/ . Our goal is to find W*. That is, we want to find r*, which is equivalent to find - = (r*-)/. Note that >0. W*  1  c   * f ( w)d ( w)    ( )d ( )  N (  ) For a one-tail 95% confidence interval,  = 1.65. For a one-tail 97.5% confidence interval,  = 1.96. Now, we can easily find r*, for a given confidence interval: r* = - +  Therefore, the VaR (mean) is equal to: VaR (mean) = W0(-r*) = W0() In general,  and  are annualized. We have to adjust these numbers to the VaR’s time interval. This can be done by multiplying the numbers by t, which represent the VaR’s frequency in years. V.21 That is, VaR can be calculated by VaR (mean)  W0 t. You should notice now the usefulness of a good volatility forecast. The only variables needed to calculate VaRs are variances. All the other parameters are given to us. Example V.9: Suppose we want to calculate the VAR of a portfolio of foreign currencies. The value of the portfolio is USD 220 million. Using a GARCH model, we estimated the annualized volatility of changes in exchange rates as 12%. The time interval is ten days (t=10/252=.03968). We want to use a 95% confidence interval. VaR (mean) = W0t. = USD 220Mx1.65x0.12x.03968 = USD 8,677,082. That is, the maximum one-day loss of this portfolio is USD 8,677,082. ¶ It is possible to calculate a VaR without calculating variances. One way to do this is to use the empirical distribution. That is, we can find W*, such that, p = P(wW*) = 1-c. Thus, we can calculate W* by looking at the histogram (empirical distribution). We can locate W* from the past history. For example, if we use a 95% confidence interval, W* is the value in the histogram that leaves 5% of the observations to the left. Example V.10: Suppose we have managed the same portfolio for two years and we have recorded the value of the portfolio at the end of each day. That is, we have roughly 500 observations. If we use a 95% confidence interval, W* is calculated by looking at the 25th (500x.05) ordered observation, which is equal to USD -6.1M. The mean of the daily value of the portfolio is USD 3.2M. Thus, the daily VAR is: VaR (mean) = E(W1) -W* = USD 3.2M - (-USD 6.1M) = USD 9.3M. ¶ Interesting readings: Parts of Chapter V were based on the following books: International Financial Markets, by J. Orlin Grabbe, published by McGraw-Hill. International Investments, by Bruno Solnik, published by Addison Wesley. V.22 APPENDIX V-A BASIC FORECASTING MODELS A.I Forecasting from Econometric Models The econometric approach to forecasting consists first of formulating an econometric model that relates a dependent variable to a number of independent variables that are expected to affect it. The model is then estimated and used to obtain conditional or unconditional forecasts of the dependent variable. The models are generally formulated using economic theory and the statistical properties of the variables included in the model. Example A.V.1: In Example V.3, a company believes that monthly changes in the MYR/USD exchange rate are related to the interest rates differential between Malaysia and the U.S. (INTt) and income growth rates differential (INCt) between Malaysia and the U.S. That is, the econometric model is given by: sMYR/USD,t,one-month = a0 +  INTt +  INCt + t, (A.1) where t is a prediction error assumed to follow a normal distribution with zero mean and constant variance, 2. The IFE predicts that INTt should have a positive coefficient. That is, if Malayan interst rates increase relative to U.S. interest rates, then the MYR should depreciate with respect to the USD (i.e.,  should be positive). Similarly, the Asset Approach predicts that INCt should have a negative coefficient. That is, if income grows in Malaysia at a faster rate than in the U.S., the MYR should appreciate with respect to the USD (i.e.,  should be negative). ¶ Several economic series seem to show seasonal effects. For example, many researchers have found a Monday effect in the U.S. stock market. Since these seasonal effects are predictable, many forecasters include seasonal variables in an econometric model like equation (A.1). Example A.V.2: In Example A.V.1 a forecaster might like to introduce monthly seasonal variables to predict the monthly change in the MYR/USD. In this case, equation (A.1) would include eleven monthly dummy variables. sMYR/USD,t,one-month = a0 +  INTt +  INCt + Jan DJan + ... + Nov DNov +t, where Di =1 =0 if t=i, i= Jan,..., Nov. otherwise. ¶ A.II Forecasting from Time Series Models Econometric models are generally based on some underlying economic model. A popular alternative to econometric models, especially for short-run forecasting is known as time series models. These models typically relate a dependent variable to its past and to random errors that may be serially correlated. V.23 Time series models are generally not based on any underlying economic behavior. A powerful time series model is the ARMA (Autoregressive Moving Average) process. The basic idea is that the series st at time t is affected by past values of st in a predictable manner. A general ARMA(p,q) can be written as: st = 0 + 1 st-1 + ... + p st-q + ß1 t-1 + ... + ßp t-q + t, (A.2) where t is the prediction error at time t assumed to have a constant variance 2. The terms with the 's coefficients are the moving average terms. The terms with the ß's coefficients are the moving average terms. In order for the ARMA model in (A.2) to have nice properties -i.e., to be stationary-, we need to check that the roots of the polinomial 1 - (1 z + 2 z2 + ... + p zp) = 0 lie outside the unit circle. In general, this requires that |I| < 1. The prediction error, t, is just the difference between the realization of st and the prediction of st using the ARMA(p,q) model. Example A.V.3: Suppose we estimate equation (A.2) and we obtain spt = a0 + a1 st-1 + ... + ap st-q + b1 t-1 + ... + bp t-q, where spt is the predicted change in st, the ai's are the estimated i's coefficients, and the bi's are the estimated ßi's coefficients. Then, t = st - spt. ¶ Note: Suppose that st represents changes in the MYR/USD exchange rate. According to (A.2), the past p changes in the MYR/USD exchange rate affect today's change in the MRY/USD exchange rate. Also, the past q prediction errors affect today's change in the MYR/USD exchange rate. The key component of the ARMA model is to determine q and p. Several statistical packages provide identification tools to determine q and p. Many forecasters prefer to work with simpler AR(p) models. In this case, to determine p, a simple rule of thumb can be followed: start with an AR(1) model and add terms until the added terms are not statistically significant. A.III Forecasting Using a Combination of Methods Many forecasters use a combination of the methods described in A.I and A.II. The dependent variable might depend on theoretical grounds on a set of independent variables. On empirical grounds it has V.24 been found that the dependent variable shows a high degree of autocorrelation. Although this autocorrelation is not present in the economic model, an economist might combine an economic model with an ARMA model to produce a better forecast. Example A.V.4: Suppose a forecaster believes that changes in the monthly MYR/USD exchange rate are determined by the IFE. She also has found that an ARMA(1,1) helps to predict future changes in exchange rates. She decides to use the following forecasting model: sMYR/USD,t,one-month = 0 +  INTt + 1 st-1 + ß1 t-1 + t, where t is a prediction error with a constant variance, 2. ¶ A.IV Stationarity and Trends in Macroeconomic and Financial Data In the previous sections, we have implicitly assumed that the dependent variable and independent variables are stationary. Roughly speaking, stationarity implies that the unconditional moments of a time series are independent of time. That is, they are constant. Example A.V.5: the process for Yt is said to be weakly stationary if: E[Yt] =  E[(Yt-) (Yt-j-)] = 2 =0 for all t for j = 0 for j  0. ¶ The assumption of stationarity might not be appropriate for many of the economic and financial series used in practice. Several economic and financial series show clear trends: GDP, Consumption, CPI prices, stock prices, exchange rates, etc. For example, in Figure V.2, the CHF/USD shows a clear, predictable positive trend. This trend should be incorporated into any forecasting model. There are two ways to achieve stationarity for these non-stationary series. The idea is to incorporate this trend in the model: (1) a deterministic time trend and (2) stochastic trend. The first model, also referred as trend-stationary, includes a deterministic time trend. The second model, also referred as a unit root process, uses first differences instead of levels. Example A.V.6: Suppose yt is a non-stationary series. (A) Trend-stationary process. yt =  +  t + t, where t is a stationary error. (B) Unit Root process. yt - yt-1 =  + t, where t is a stationary error. This simple process is called a random walk with drift . We should note that this unit root process can be written in an AR(1) form: V.25 yt = 1 yt-1 +  + t, where 1=1. ¶ Both processes have different implications. If the series yt follows a trend-stationary process, a shock has a temporary effect on the series, and the series eventually catches up with its trend. On the other hand, if yt follows a unit root process, a shock might have permanent consequences for the level of future yt's. There are several tests to check if a series has a unit root. These tests usually find a unit root on all major macroeconomic data. Of particular interest to us, exchange rates, GNP, money supply, and price levels have unit roots. Therefore, it is highly advisable to estimate models for these series in first differences. It is common to take logs of the data before using it (see the Appendix of the Review Chapter). For small changes, the first difference of the log of a variable is approximately the same as the percentage change in the variable: log(yt) - log(yt-1) = log(yt/yt-1) = log[1 + (yt-yt-1)/yt]  (yt-yt-1)/yt, where we have used the fact that for z close to zero, log(1+z)  z. It is usually convenient to multiply log(yt) by 100. Thus, the changes are measured in units of percentage change. We should notice, however, that several economists claim that unit root tests are not very revealing. These economists claim that in finite samples -like the ones available to us- it is very difficult to distinguish between models with a unit root -i.e., 1=1- and stationary models with 1 very close to 1. Interesting readings: Appendix V is based on Introductory Econometrics with Applications, by Ramu Ramanathan, published by Harcourt Brace Jovanovich. V.26 Appendix V-B: Taylor Rules According to the Taylor rule, the CB raises the target for the short-term interest rate, it, if: (1) inflation, It, raises above its desired level (2) output, yt, is above “potential” output The target level of inflation is positive (deflation is thought to be worse than positive inflation for the economy) The target level of the output deviation is 0, since output cannot permanently exceed “potential output.” John Taylor (1993) assumed the following reaction function by the CB: it = It + φ ( It - It* )+ γ y-gapt + r* (Equation BC.1) where y-gapt is the output gap –a percent deviation of actual real GDP from an estimate of its potential level-, and r* is the equilibrium level or the real interest rate, which Taylor assumes equal to 2%. The coefficients φ and γ are weights, which can be estimated (though, Taylor assumes them equal to .5). Let It* and r* in equation BC.1 be combined into one constant term, μ = r* - φ It*. Then, it = μ + λ It + γ y-gapt, where λ = 1 + φ. For many countries, whose CB monitors St closely, the Taylor rule is expanded to include the real exchange rate, Rt: it = μ + λ It + γ y-gapt + δ Rt Estimating this equation for the US and a foreign country can give us a forecast for the interest rate differential, which can be used to forecast exchange rates. V.27 Exercises: 1.- Go back to Example V.2. i. ii. iii. Take the autoregressive forecasting model, estimated above. What is SF1997:4? Calculate the same forecast using the "naive" model. Compare both forecasts with the in-sample PPP forecast. 2.- You work for a Tunisian investment bank. You have available the following quarterly interest rate series in the U.S., iUSD, and in Tunisia, iTND, from 1998:4 to 1999:3 (TND=Tunisian Dinar). The TND/USD in 1998:4 is equal to 1.1646. Your job is to do quarterly out-of-sample forecasts of TND/USD exchange rate for the period 1999:2 1999:3, using the linear approximation to the International Fisher Effect (IFE). Date Tunisia U.S. Quarterly Forecast (SF) 1998:4 .0590 .0621 1.1646 1999:1 .0593 .0635 1999:2 .0595 .0680 1999:3 .0599 .0714 i. Generate one-step-ahead forecasts –that is, as new information arrives, a new next period forecast is generated- for the period 1999:1-1999:4. ii. Your firm uses the following forecasting regression model to forecast interest rates. Use a regression analysis. iUSD,t = .0075 + .93 iUSD,t-1 + t. iTND,t = .0060 + .97 iTND,t-1 + t. Generate out-of-sample forecasts for the period 1999:1-1999:4. 3.- Given that firms cannot forecast exchange rates, should they worry about currency risk? 4.- J. Cruyff, a Dutch designer company, wants estimate the monthly volatility of the weekly EUR/USD exchange rate. They use the following AR(1)-GARCH(1,1) model: et|t-1 ~ N(0,2t). st = [log(St) - log(St-1)]x100 = a0 + a1st-1 + eFt, 2t = 0 + 1 e2t-1 + ß1 2t-1 + e2t-1 Dt, where Dt is the following dummy variable: Dt = 0 if et-1 ≥ 0 V.28 Dt = 1 if et-1 < 0. This GARCH model is an asymmetric model. Negative shocks increase the variance more than positive shocks. The persistence parameter should be redefined. That is, =[1+ß1+(1/2)]. Using data from January 1974 till August 1997, the "quants" at J. Cruyff estimated the model for st: st = 0.178 + 0.064 st-1, (0.90) (1.51) 2  t = 0.222 + 0.035 e2t-1 + 0.860 2t-1 + (2.09) (2.48) (12.44) 0.123e2t-1 Dt. (0.04) (a) Find  and calculate the unconditional variance, 2. Is it well defined? (b) Given that eAug 97 = -1.073, and 2Aug 97 = 7.436, forecast the variance for September 1997. (c) Forecast the variance for August 1998. 5.- You want to calculate the VAR of a position in EUR. The value of your position is USD 50 million. You estimated the volatility of changes in the USD/EUR exchange rates as 22%. The time interval is seven days. You use a 99% confidence interval to calculate VAR. V.29