A VaR Investigation of Currency Composition in Foreign Exchange

International Research Journal of Finance and Economics ISSN 1450-2887 Issue 21 (2008) © EuroJournals Publishing, Inc. 2008 http://www.eurojournals.com/finance.htm A VaR Investigation of Currency Composition in Foreign Exchange Reserves Jer-Shiou Chiou Department of Finance and Banking, Shih-Chien University, Taipei, Taiwan E-mail: jschiou@mail.usc.edu.tw Tel: 886-2-25381111 ext. 8927 Jui-Cheng Hung Department of Finance and Banking Yuanpei Institute of Science and Technology, Taiwan E-mail: 891490053@s91.tku.edu.tw Mei-Maun Hseu Department of Finance, Chihlee Institute of Techonology, Taipei,Taiwan E-mail: meimaun@mail.chihlee.edu.tw Tel: 886-2--2253-7240 ext. 347 Abstract In this study, Exponential Weighted Moving Average (EWMA), Bootstrapping, and Monte Carlo Simulation are used to calculate the VaRs for three groups’ foreign reserves portfolio from the year of 1995 to 2001. Empirically we find that EWMA finished relatively better than the other two. Based on developing countries’ currency in 2001 if the other currencies’ holding ratio is fixed, reducing the U.S. dollar holding ratio while increasing the Euro holding ratio will make VaR decrease in EWMA. However, if the Euro holding ratio is high, VaR increases. This implies that despite the hedging effect of Euro, as its holding ratio increases, the marginal effect decreases. The hedging effect of Euro is not persistent. In general, increase the holdings of the lowest-valued component VaR currency while decreases the holdings of the highest-valued component VaR currency can in fact reduce the risk of the portfolio. Keywords: VaR, RMSE, RAPM, Marginal VaR, Component VaR 1. Introduction Experiencing from the Asian financial crisis of 1997-98, most developing Asian economies have rapidly increased their external surpluses and accumulated foreign reserves to reduce their vulnerability to future shocks; the expansion in US current account deficit is among the most. However, the countries with largest reserves holdings were least affected by speculative pressures. But the recent increasing oil prices and the foreign direct investment and portfolio inflows have begun to fade in several Asian countries, Asian governments suffer a sharp fall in the value of their dollar holdings and, more importantly, would see their export-dependent economies hit hard by a US slowdown. In this International Research Journal of Finance and Economics - Issue 21 (2008) 77 study, we seek to examine an alternative reserves management strategy in the level of international portfolio. There are three reasons make central banks keep certain amount of foreign exchanges. Those are liquidity needs for balance of trade, steady needs for government financing and interfering needs for economy stability. Foreign exchanges have not only become a country’s key reserves assets but also protect the country’s monetary interests. When the authority alters its portfolio holdings, the context of safety, liquidity, and profitability in foreign exchange reserves have always been taken into the consideration. Due to the most international business and liability takes U.S. dollars as major means; the U.S. dollar inevitably acts an essential part in a country’s foreign exchange reserves. According to the IMF’s annual report, from 1995 to 2001, there is over 60% of developing countries’ foreign exchange reserves were U.S. dollars. On April 29, 1999, the World Bank Annual Conference, Pam Greens suggested Value-at-Risk should be adopted when a country’s foreign reserves policies are under investigation. However, a country’s actual reserves are usually confidential, not only retrieving truthful data becomes difficult, but the relative literatures are rare. Because the data retrieval difficulty, most studies focus on the determinants in foreign reserves management. Beschloss and Mendes (1999) considered liquidity as the main concern for central banks in foreign reserves management. Dooley, Lizondo, and Mathieson (1989) found the fixed exchange policy, the major international competitors, and the foreign debts are the key determinants in reserves management decision. Barry and Donald (2000) had shown that the stability of foreign reserves was resulted from the commodity trading, the capital flow, and the fixed exchange policy. While Kenen (2002) suggested that as Euro increases their influential on the members in the European Union; Euro still can hardly replace the U.S. dollar as the major international currency. In terms of risk management, Winfried (1999) decomposes the portfolio total risks into separated Value-at-Risks in parts, while José, Carlos, and Juan (2001) take the conceptual Value-atRisk in risk management practice. The study on center bank foreign reserves risk management only appears in Blejer and Schumacher (1998) who theoretically construct a center bank’s investment portfolio VaR evaluation model and then analyze the associates policies implications, but unfortunately no empirical examinations were made. Departure from the existing literatures, in this study we take nine1 regularly-taken reserves currencies daily data, and the associates three major groups’ (that the IMF 2002 annual report categorized as the whole world, the industrial countries and the developing countries) reserves weighted information to study a central bank’s risk management strategy in terms of foreign reserves portfolio. We adopt models of the Exponential Weighted Moving Average (EWMA), Bootstrapping, and Monte Carlo Simulation to compute the VaR for these three groups’ foreign reserves portfolio from the year 1995 to 2001. Then, we take the best model to calculate and compare the risk-adjusted performance measurement index (RAPM) for each group. At last, by adjusting each currencies weighted importance, we analyze if the increasing Euro holdings was able to reduce the portfolio’s risks. 2. Methodology 2.1. Value at Risk In measuring variations, variance and standard error have often been taken to reveal the magnitude in the changes of future asset prices. As the fluctuation in future prices is inherent, potential gains and losses in assets holding become inevitable. But most investors seem concern losses more than gains; the variation criteria stated above is undesirable in describing this downside risks phenomenon. VaR is defined as the worst expected loss over a great horizon within a given confidence level, Jorion (1996). 1 In practice, the IMF’s annual report presents 8 currencies and 1 unspecified currency. In this study we take the Australia dollar as the unspecified currency. 78 International Research Journal of Finance and Economics - Issue 21 (2008) It not only provides an aggregate statistic of the order of magnitude of potential losses due to market risk, but also summarizes the effects of leverage, diversification, and probabilities of adverse price movements in a single dollar amount. Let’s define W0 as the initial investment at the beginning, and R* represents the estimate expected returns, then VaR can be written as: VaR mean = - W0 R * -μ . (3) ( ) If defined VaR as the absolute loss that excludes the expected returns, then VaR can be rewritten as: (4) VaR zero = - W0 R * . If we transform the probability distribution f (W) into a standard normal distribution Ф(ε), where ε ~ N(0, 1). Then the probability of possible returns which is less than W* will be 1-C, and it can be rewritten as: 1− C = ∫ W* −∞ f ( w ) dw = ∫ − R* −∞ f ( r ) dr = ∫ −α −∞ Φ (ε ) dε , (5) where − α = − ，and Δt is the time factor. After the value of α is determined, Value at Risk σ Δt can be written as: VaR zero = W0 α σ Δt − μΔt (6) μΔt − R * ( ) VaR mean = W0 α σ Δt . (7) In this study we take VaR zero as the measurement. That is, we take VaR as the absolute loss without considering the expected returns. 2.2. EWMA with Variance-Covariance consideration It had been shown in Jorion (2000) that when the returns are normally distributed, Value-at-Risk can be expressed in two different forms depending on whether absolute returns or average returns were taken. If foreign positions had been taken, the VaRs can be rewritten as (8) VaR t +1,zero = Wt (Z α σ t − μ t ) VaR t +1,mean = Wt Z α σ t , (9) where Wt is the asset’s value at time t, μt is the average returns at time t, σt is the standard deviation at time t, and Zα represents the critical value with a confidence level of 1-α. Under the definition of absolute return and the assumption of Wt=1, Zα, σt and μ t can then be obtained. If a parametric model was taken to compute the VaR for a single asset, the variances multiple by a critical value at the confidence level of 1-α is necessary. While in computing the VaR for certain portfolios, the influential of covariance has to be considered, other than the variance. We therefore take EWMA as a mean to estimate the fluctuations of the portfolio. Their variance and covariance can thus be rewritten as: σ i2, t = λσ i2, t − 1 + (1 − λ ) ri2t − 1 (10) , σ ij , t = λσ ij , t − 1 + (1 − λ ) ri , t − 1 r j , t − 1 , (11) where i ≠ j , σ i2,t and σ ij,t are the variance of asset i and the covariance of asset i and j at time t. ri ,t −1 and r j ,t −1 are the return ratios of asset i and j at time t-1. λ is the decay factor of which 0.94 for the daily data and 0.97 for the monthly data. When EWMA is in use, the VaR for the next T days can be written as: VaR t = W t Z α σ p , t T , (12) International Research Journal of Finance and Economics - Issue 21 (2008) 2 ⎡ σ1,t ⎢ σ ' = w ∑ w ， ∑ = ⎢ 21,t ⎢ M ⎢ ⎢ σ N1,t ⎣ 79 where σ P ,t σ12,t σ 2,t 2 σ N 2,t M K L O L σ1N ,t ⎤ ⎡ w1 ⎤ ⎥ ⎢w ⎥ σ 2 N ,t ⎥ ，w = ⎢ 2 ⎥ ; ⎢ M ⎥ M ⎥ ⎥ ⎢ ⎥ σ 2 ,t ⎥ N×N ⎣ w N ⎦ N×1 N ⎦ wi is the weighted ratio of the ith foreign asset to the total portfolio, σi,t, σj,t, and σij,t are obtained by EWMA and they are the variance and covariance of the foreign asset, respectively. σ P,t is the standard deviation for the portfolio at time t, while T is the holding period for the foreign asset. Under the assumption of the holding period for the foreign asset is 1 day (T=1) and the confidence level is 99%, the daily VaR is obtained. 2.3. Bootstrapping It is not necessary to fully understand the distribution of the population when Bootstrapping is in used to calculate the underlying VaR. The method takes the limited historical returns and go through the iterate sampling to construct the asset portfolio’s distribution for future returns. The VaR of an asset portfolio is then obtainable, whenever the confidence level and holding period are specified. The difference between Bootstrapping and Historical Simulation is that Historical Simulation directly utilizes the future returns’ distribution that has only one price path, while Bootstrapping makes an iterate sampling from historical data to simulate the real return’s distribution to improve the shortcoming of a Historical Simulation. Assume it is known for each asset’s prices in a portfolio for the last 251 days. It would be attainable to get next day’s VaR for the portfolio by the Bootstrapping. The procedure can be described as follows: Step 1 Convert the historical 251 days prices into 250 returns and perform 10000 repeated sampling from the underlying returns. The same process can apply to any portfolio which contains N assets. R i (T ) , where i = 1, 2...,N T = 1, 2,...,10000 Step 2 Multiply the returns by the corresponding importance ratio, which is measured by any single asset to the portfolio. Sorting the data in increasing order and sum together, the distribution of next day’s return can therefore be obtained. Step 3 Using the above simulated future returns, VaR can be computed by a percentile criterion on the confidence level of 1-α. 2.4. Monte Carlo Simulation Before applying the Monte Carlo Simulation, it is necessary to assume that the underlying portfolio’s return follows a certain random process. Usually, the Geometric Brownian Motion Model (GBM) is applicable to the assets such as stocks and foreign exchange. It can be expressed as: dR t = μ t dt + σ t dZ , (13) Rt where Rt is the return of portfolio, μ t is the drift term of the portfolio at time t, σ t is the standard deviation at time t, and dZ is normally distributed with 0 mean and dt variance, whereby dZ ~ N(0, dt ) . From (13) we found that if a portfolio’s return follows a multivariable normal distribution that is R ~ N(μ N×1 , Σ N×N ) , then in order to simulate the return as an N × N normal distribution, a Monte Carlo process as be described as follows: Step 1 80 International Research Journal of Finance and Economics - Issue 21 (2008) N × N variance- covariance matrix as AA T = Σ 2 L a N1 ⎤ ⎡ σ1 ⎢ L a N2 ⎥ ⎥ ， Σ = ⎢ σ 21 ⎢ M O M ⎥ ⎢ ⎥ 0 a NN ⎦ ⎢σ N1 ⎣ Take the Cholesky Decomposition to decompose an like follows 0 L 0 ⎤ ⎡ a 11 ⎡a 11 a 21 ⎢a ⎢0 a ⎥ a 22 L M ⎥ 22 ， AT = ⎢ A = ⎢ 21 ⎢ M ⎢ M ⎥ M O 0 M ⎢ ⎢ ⎥ ⎣a N1 a N 2 L a NN ⎦ ⎣0 L σ12 σ 2 2 σN2 M L σ1N ⎤ ⎥ L σ2N ⎥ . O M ⎥ ⎥ L σ2 ⎥ N ⎦ Step 2 An N × 1 Z matrix can be generated, where Z is a multivariate standard normal distributed random variable, and Z ~ N(0, I N ) , where ⎡1 0 0 ⎤ I N = ⎢0 O 0 ⎥ . ⎢ ⎥ ⎢0 0 1 ⎥ ⎣ ⎦ Step 3 Multiplying the A matrix from step 1 by the Z matrix, we can get R=AZ, which is an N multivariable normal random variable. Here the covariance matrix of R is VAR (R ) = E RR T = E AZZT A T = AE ZZT A T = AI N A T = AA T = Σ . Step 4 Repeat step (3) 10000 times and get a sample size of 10000 next day’s portfolio’s return. Sorting the data in increasing order, we get the distribution for the portfolio’s next day’s return. VaR can therefore be computed by a percentile criterion on the confidence level of 1-α for this simulated future returns. ( ) ( ) ( ) 2.5. VaR, Evaluation and Performance In VaRs’ evaluation and its performance effectiveness, we adopt Kupiec’s (1995) Proportion of Failure Test (PF-Test). This gives the verification of whether the constructed α 0 was consistent with the actual ˆ α . Here, the null hypothesis is H0： α = α 0 , while the statistics are x ˆ ˆ LR PF = 2 [ ln ( α x ( 1 − α ) n − x ) − ln ( α 0 ( 1 − α 0 ) n − x )] ~ χ 2 ( 1 ) , (14) where α 0 is the exception rate, n is the number of observations, x is the counts which shown the actual ˆ return greater than the computed VaR ratio in the models, and α = x n is the ratio of which actual returns are greater than the VaR. After the underlying VaR models passed the PF-Test, the prediction effectiveness for the models is investigated then. Forward-testing and the efficiency of funds uses are taken to be the measurement indices. Next, we take the Root Mean Square Error (RMSE) as the criterion to determine the usage efficiency of funds in the short run. This can be seemed as the square root of the average sums of square of differences between the predicted value and actual value. This implies that when we accept the condition of a reliable VaR model, the smaller RMSE is the closer is a VaR from the actual loss. It also shows that there are no excess reserve funds to compensate for the possible loss in the short run. For a reliable VaR model with a smaller RMSE, this implies that there are certain advantages on both risk control and funds usage. RMSE can be written as RMSE = ∑ (r t =1 n t − VaR t ) n 2 , (15) where rt is the actual returns and VaRt is the value computed from the model. International Research Journal of Finance and Economics - Issue 21 (2008) 81 Benet (1992) investigated foreign exchange futures, and suggested that if out-of-sample or exante were taken to evaluate the hedging effect, external effect should be emphasized in order to be more meaningful for the investors. Similarly, forward-testing is based on ex-ante and consider the future potential losses. 2.6. The Decomposition of VaR The main reason for forming a portfolio is hoping through investing in different assets investor can diversify the market’s non-systematic risk and minimize the investment risk, whereby VaR can prereveal the possible loss for the portfolio. Since VaR can integrate all of the possible holding portions for an institution and monitor the volume of risk’s exposure. And makes the risk of the investment position fulfills the requirement for the limits of the capital possible. VaR decomposition hence reveals the importance of any single asset in an investment portfolio. Those include the individual VaR, the marginal VaR, incremental VaR, and the component VaR. A brief discussion is as follows. 2.6.1. Individual VaR Assume that there are N assets within a portfolio. Any single asset’s VaR can be written as VaR i = w i Z α σ i , where i = 1, K , N , ∑ w i = 1 . The total risk for the portfolio is VaR P = ∑ VaR i , without considering i =1 i =1 N N N the interaction among the assets. But in fact, every coefficient between any two assets should lie between –1 and 1. Therefore, theoretically total risk for the portfolio should less than the sum of every asset’s VaR, that can be written as VaRP < ∑ VaRi . i =1 2.6.2. Marginal VaR and Incremental VaR Marginal VaR represents ith asset increases by one unit how much risk of the total investment portfolio will increase. MVaR i = ∂VaR . ∂w i W Here, MVaR is the Marginal VaR, wi is a weighted ratio of the ith asset to the whole portfolio, and W is the value of the portfolio. Incremental VaR shows the amount of VaR that increases or decreases, because of the increase in an additional asset to the portfolio. That can be written as VaR a = VaR P +a − VaR P , where VaR P +a is the VaR after adding the ath additional asset, while VaR P is the original value. 2.6.3. Component VaR In order to make the risk of portfolio less than the maximum tolerance level, it becomes necessary to understand the contribution of every single asset to the whole portfolio. This gives the possibility to adjust every asset’s holding ratio to the portfolio, making the risk exposure less than the tolerance level possible. Through the help of the Component VaR, we can distinguish an asset’s positive effect from the negative effect to the portfolio. In order to lower the risk exposure to an acceptable level, decreasing the holdings of those positive effect assets and increasing the holdings of those negative effect assets may become necessary. In this study we apply the Component VaR and calculate the VaR effect of Euro to the portfolio so as to discuss if Euro could become a hedging object for foreign exchanges portfolio. That is if it was possible by variation of individual foreign exchange to lower the risk of the foreign exchange portfolio. The Component VaR can be written as: CVaR i = MVaR i × w i W , 82 International Research Journal of Finance and Economics - Issue 21 (2008) where CVaR is the Component VaR, wi is the ith weighted importance ratio to the investment portfolio, and W is the value of the portfolio. After considering the relationships among the assets within the portfolio, Component VaR can present the risks that every asset would share. Therefore, the sum of every asset’s Component VaR will equal to the risk of the portfolio. That is: ∑ CVaR i =1 N i = VaR P . 3. Empirical Results 3.1. Data Description The study includes the U.S. Dollar, Japanese Yen, British Pound, Swiss Franc, the Euro, Deutsche Mark, French Franc, Netherlands Guilder, and the Australian Dollar. They are expressed in terms of Special Drawing Right (SDR)2, and retrieved from the Pacific Exchange Rate Service. The Euro was introduced on January 1, 1999, and since its exchange rate was 1:1 with European Currency Units (ECU), we use the ECU instead of the Euro prior to 1999. If there was a missing value, we took the average of the day before and after. All of the exchange rates are converted in terms of return by the following: R i ,t = ln(Si ,t ) − ln(Si ,t −1 ) , where Ri ,t is the return of currency i at time t, and Si ,t is the exchange rate of currency i at time t. Table 1: Statistics of Three Portfolios 1995-2001 Min -0.75076 -0.38325 -0.34017 -0.45949 -0.42731 -0.33396 -0.48314 Min -0.59539 -0.40851 -0.28968 -0.40963 -0.43897 -0.36516 -0.54105 Min -0.91140 -0.35796 -0.43797 -0.49751 -0.43076 -0.31426 -0.44324 Max 0.726121 0.420626 0.523735 0.703617 0.554663 0.604887 0.403988 Max 0.579129 0.368508 0.402476 0.641309 0.565154 0.579782 0.445394 Max 0.878335 0.547174 0.627236 0.751983 0.546793 0.620111 0.375269 Mean 0.004008 -0.00643 -0.00518 0.014859 -0.00699 -0.00654 -0.00526 Mean 0.002047 -0.00499 -0.00227 0.013217 -0.00879 -0.00885 -0.00641 Mean 0.00603 -0.00786 -0.00759 0.01613 -0.00563 -0.00486 -0.00447 Std 0.213582 0.107816 0.138057 0.165938 0.165831 0.135744 0.143348 Std 0.173706 0.092155 0.110089 0.155033 0.174197 0.151135 0.171471 Std 0.258647 0.126545 0.167862 0.174998 0.161533 0.128726 0.126532 Skewness 0.20217 0.66633 0.52187 0.65771 0.24224 0.54679 -0.16383 0.012 0.32431 0.41454 0.52969 0.71552 0.40682 0.49722 -0.18485 Skewness 0.19158 0.38201 0.49692 0.61642 0.21453 0.33990 -0.13301 Kurtosis 1.83215 3.16717 1.02812 1.92237 0.49653 1.70586 0.20372 Kurtosis 1.65534 3.64531 1.21589 1.82881 0.49474 0.95167 0.02663 Kurtosis 1.89041 3.08053 0.82084 1.98853 0.52243 2.55359 0.37943 JB-test 37.40288 124.9567 22.35828 56.06711 5.01325 42.42726 1.55060 JB-test 31.05372 147.9086 27.09056 55.72155 6.06118 19.57732 1.43106 JB-test 39.52981 126.7672 17.30712 56.56603 4.76073 84.14456 2.23672 All countries 1995 1996 1997 1998 1999 2000 2001 Industrial countries 1995 1996 1997 1998 1999 2000 2001 Developing countries 1995 1996 1997 1998 1999 2000 2001 Note: ⎡ ⎤ 1. Skewness = E ⎣(x − μ)3 ⎦ = μ3 / σ3 . 2. Kurtosis = E ⎡(x − μ) 4 ⎤ = μ 4 / σ 4 . ⎣ ⎦ 3. JB-test represents the statistics for the Jarque-Bera Normality test, where alpha=0.05 and the critical value for the JB-test is 5.9915. 2 SDR is the booking unit of IMF members. International Research Journal of Finance and Economics - Issue 21 (2008) 83 Table 1 shows that the returns for every currency are small, actually almost equal to 0, and their fluctuations are minor too. By Skewness and Kurtosis tests, we find that each groups’ portfolios are significantly different from a normal distribution. In addition, by Jarque-Bera statistics, industrial countries are significantly different from a normal distribution in all years except 2001, while developing countries are significantly different from a normal distribution in all years except 1999 and 2001. 3.2. Evaluation of the VaR The foreign reserves consist of nine primary currencies in this study. EWMA, Bootstrapping, and Monte Carlo Simulation are adopted to compute each year’s VaR for every group’s foreign reserves portfolio. The differences between daily estimated VaR and actual portfolio gain-and-loss are calculated, the actual gain-and-loss overpass the estimated VaR are counted. That is called the number of exceptions. Divided the exceptions by 250, we get the rate of exceptions. According to Table 2, both the rate of exception and LR test shows that Bootstrapping is less accurate than EWMA and Monte Carlo Simulation did. In tables 3 and 4, industrial and developing countries’ VaR shows that EWMA is better than the other two. Therefore, we take EWMA as the means to perform the following examination. 84 Table 2: International Research Journal of Finance and Economics - Issue 21 (2008) VaR of all countries 1995-2001 Number of Exceptions Rate of Exception Zone (Basel Rule)3 LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE EWMA 3 0.012 Green 0.095 53.07% 3 0.012 Green 0.095 27.77% 2 0.008 Green 0.108 34.96% 1 0.004 Green 1.176 41.91% 4 0.016 Green 0.769 41.92% 2 0.008 Green 0.108435 34.46% 5 0.02 Yellow 1.957 35.70% BootStrap 8 0.032 Yellow 7.734*** 46.08% 1 0.004 Green 1.176 42.23% 3 0.012 Green 0.095 32.43% 4 0.016 Green 0.769 40.68% 2 0.008 Green 0.108 43.47% 0 0 Green 5.025** 38.20% 4 0.016 Green 0.769 33.99% Monte Carlo 3 0.012 Green 0.095 54.09% 4 0.016 Green 0.769 26.74% 2 0.008 Green 0.108 34.37% 1 0.004 Green 1.176 42.37% 3 0.012 Green 0.095 42.37% 2 0.008 Green 0.108435 34.35% 5 0.02 Yellow 1.957 35.38% 1995 1996 1997 1998 1999 2000 2001 Note: 1. Monte Carlo Simulation times=10000, time horizon(T)=250, lambda=0.94 ,confidence level = 0.99 (alpha=0.01) 2. *, **, and *** show the significant level at 10%, 5%, and 1%, respectively. 3. RMSE shows the average differences between VaR and actual loss, less the value less the differences. 3 For the Basel Penalty Zones, see Jorion (2000). International Research Journal of Finance and Economics - Issue 21 (2008) Table 3: VaR of industrial countries 1995-2001 Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE EWMA 4 0.016 Green 0.769 43.04% 2 0.008 Green 0.108 24.08% 3 0.012 Green 0.095 27.67% 1 0.004 Green 1.176 39.19% 3 0.012 Green 0.095 43.92% 2 0.008 Green 0.108 38.54% 3 0.012 Green 0.095 42.89% BootStrap 11 0.044 Red 15.890*** 37.13% 1 0.004 Green 1.176 36.38% 5 0.02 Yellow 1.957 25.65% 4 0.016 Green 0.769 38.16% 1 0.004 Green 1.176 45.72% 0 0 Green 5.025** 41.60% 4 0.016 Green 0.769 40.86% Monte Carlo 3 0.013 Green 0.095 43.79% 4 0.016 Green 0.769 23.13% 4 0.016 Green 0.769 27.41% 1 0.004 Green 1.176 39.60% 3 0.012 Green 0.095 44.65% 2 0.008 Green 0.108 38.08% 5 0.02 Yellow 1.957 42.47% 85 1995 1996 1997 1998 1999 2000 2001 Note: Same as Table 2. 86 Table 4: International Research Journal of Finance and Economics - Issue 21 (2008) VaR of developing countries 1995-2001 Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE Number of Exceptions Rate of Exception Zone (Basel Rule) LR test RMSE EWMA 4 0.016 Green 0.769 64.42% 2 0.008 Green 0.108 32.21% 1 0.004 Green 1.176 42.70% 2 0.008 Green 0.108 44.17% 3 0.012 Green 0.095 40.91% 2 0.008 Green 0.108 32.51% 3 0.012 Green 0.095 31.45% BootStrap 9 0.036 Yellow 10.229*** 56.75% 1 0.004 Green 1.176 48.50% 3 0.012 Green 0.095 39.43% 4 0.016 Green 0.769 42.79% 3 0.012 Green 0.095 42.54% 0 0 Green 5.025** 36.32% 4 0.004 Green 0.769 30.38% Monte Carlo 4 0.016 Green 0.769 65.73% 3 0.012 Green 0.095 31.08% 3 0.012 Green 0.095 41.83% 1 0.004 Green 1.176 44.63% 3 0.012 Green 0.095 42.05% 3 0.012 Green 0.012 32.61% 5 0.02 Yellow 1.957 31.17% 1995 1996 1997 1998 1999 2000 2001 Note: Same as Table 2. Following Jorion (2000), the Risk Adjusted Performance Measurement (RAPM) of the return that is generated by per unit of risk can be written as: RAPM = ROR P , VaR P ⋅ 365 where ROR P represents the rate of the weighted average return of underlying foreign reserves portfolio. In this study, the yields of government bond are used for all the countries, and denoted as the rate of return of the portfolio. The data of the yields of government bond are retrieved from IMF Financial Statistics (IFS) and it can also be seen in Appendix 2. Because we assume the unspecified currency was Australia dollar, we included the yields of Australian government bond as well. The weighted average yields of bond for industrial and developing countries are listed in table 5. In order to make the incidence consistent with daily VaR, we divide it by 365 , VaR P represents the average VaR that we got from EWMA, Bootstrapping, and Monte Carlo Simulation. The greater the RAPM is, the higher is the return that the VaR generates. Table 6 lists the RAPM of industrial and developing International Research Journal of Finance and Economics - Issue 21 (2008) 87 countries and the differences between them. By Table 6, we found that the performance of developing countries is worse than industrial countries did at the beginning, but became better from 1999 to 2001. The result remains consistent for all approaches. The reason is that after 1999, industrial countries held less Euro currency than they did before, while developing countries have begun to hold more and more year after year. It seems that an increasing holding of Euro may reduce the risk in foreign reserves portfolio. Table 5: Industrial Developing Weighted government bond yield of industrial countries and developing countries, 1995-2001 1995 6.8184% 6.6258% 1996 6.3176% 6.2906% 1997 5.8722% 6.0402% 1998 4.8619% 4.9826% 1999 5.2899% 5.2968% 2000 5.6724% 5.6753% 2001 4.8582% 4.8837% Source: IMF annual report and IMF International Financial Statistics Database Table 6: RAPM of industrial and developing countries, 1995-2001 EWMA 0.009394 0.014755 0.012136 0.007514 0.006797 0.008281 0.006386 0.006124 0.010866 0.007949 0.006871 0.007354 0.00998 0.008766 0.00327 0.003889 0.004187 0.000643 -0.00056 -0.0017 -0.00238 BootStrap 0.011008 0.009508 0.013193 0.007577 0.006443 0.007532 0.006694 0.007013 0.007103 0.008725 0.006925 0.006976 0.008674 0.008986 0.003995 0.002404 0.004467 0.000652 -0.00053 -0.00114 -0.00229 Monte Carlo 0.009217 0.015484 0.012294 0.007444 0.006699 0.008404 0.006471 0.005987 0.011339 0.008149 0.006801 0.007175 0.009965 0.008872 0.00323 0.004145 0.004145 0.000643 -0.00048 -0.00156 -0.0024 Industrial countries 1995 1996 1997 1998 1999 2000 2001 Developing countries 1995 1996 1997 1998 1999 2000 2001 Industrial - Developing 1995 1996 1997 1998 1999 2000 2001 3.3. The Effect of Euro In order to verify if increased the holding ratio of the Euro could reduce the risk in a foreign reserves portfolio. We take the year of 2001 for developing countries’ currencies as the base. And assume the weights of other currency holdings are fixed and then adjust the weights of the U.S. dollar and Euro only. In this study we only listed 4 different combinations. Point A is the highest weighted return of all compositions. Point B represents the original portfolio for developing countries in 2001. Point C represents the combination of highest RAPM and point D is the lowest VaR of EWMA, just like Table 7 shows. 88 Table 7: Point A Point B Point C Point D Point E International Research Journal of Finance and Economics - Issue 21 (2008) 2001 The effect of the Euro by EWMA approach, developing countries USD 78.40% 64.10% 52.40% 51.40% 39.40% JPY 4.50% 4.50% 4.50% 4.50% 4.50% GBP 5.50% 5.50% 5.50% 5.50% 5.50% CHF 0.90% 0.90% 0.90% 0.90% 0.90% EURO 1% 15.3% 27% 28% 40% AUD 9.60% 9.60% 9.60% 9.60% 9.60% weighted ROR 0.04882535 0.04883679 0.04884615 0.04884625 0.04885655 EWMA 0.483971 0.288596 0.205474 0.205471 0.289651 RAPM 0.005281 0.008944 0.012443 0.012441 0.008673 Table 8: A 2001 VaR decomposition, developing countries Currency USD JPY GBP CHF Euro AUD Total Currency USD JPY GBP CHF Euro AUD Total Currency USD JPY GBP CHF Euro AUD Total Currency USD JPY GBP CHF Euro AUD Total Currency USD JPY GBP CHF Euro AUD Total Weight 78.40 4.50 5.50 0.90 1 9.60 99.9 Weight 64.10 4.50 5.50 0.90 15.30 9.60 99.9 Weight 52.40 4.50 5.50 0.90 27 9.60 99.9 Weight 51.40 4.50 5.50 0.90 28 9.60 99.9 Weight 39.40 4.50 5.50 0.90 40 9.60 99.9 Individual VaR 0.5142 0.05404 0.04334 0.00989 0.01049 0.15055 0.78251 Individual VaR 0.42041 0.05404 0.04334 0.00989 0.16046 0.15055 0.83869 Individual VaR 0.34367 0.05404 0.04334 0.00989 0.28316 0.15055 0.88465 Individual VaR 0.33711 0.05404 0.04334 0.00989 0.29365 0.15055 0.88858 Individual VaR 0.25841 0.05404 0.04334 0.00989 0.41950 0.15055 0.93573 Marginal VaR 0.62416 0.0209 -0.23446 -0.75931 -0.86699 0.27091 Marginal VaR 0.60839 0.03391 -0.22462 -0.74254 -0.84822 0.38115 Marginal VaR 0.19628 -0.68917 -0.08765 0.05082 0.13762 1.24388 Marginal VaR 0.15124 -0.71287 -0.07237 0.11424 0.21145 1.26963 Marginal VaR -0.29614 -0.72792 0.08371 0.65806 0.81583 1.12822 Component VaR 0.48934144 0.0009405 -0.0128953 -0.00683379 -0.0086699 0.02600736 0.483971 Component VaR 0.38997799 0.00152595 -0.0123541 -0.00668286 -0.12977766 0.0365904 0.288596 Component VaR 0.10285072 -0.03101265 -0.00482075 0.00045738 0.0371574 0.11941248 0.205474 Component VaR 0.07773736 -0.03207915 -0.00398035 0.00102816 0.059206 0.12188448 0.205471 Component VaR -0.11668 -0.03276 0.004604 0.005923 0.326332 0.108309 0.295732 Diversified Effect 0.02485856 0.0530995 0.0562353 0.01672379 0.0191599 0.12454264 0.29461969 Diversified Effect 0.03043201 0.05251405 0.0556941 0.01657286 0.29023766 0.1139596 0.55941028 Diversified Effect 0.24081928 0.08505265 0.04816075 0.00943262 0.2460026 0.03113752 0.66060542 Diversified Effect 0.25937264 0.08611915 0.04732035 0.00886184 0.234444 0.02866552 0.6647835 Diversified Effect 0.25937264 0.08611915 0.04732035 0.00886184 0.234444 0.02866552 0.639998 B C D E International Research Journal of Finance and Economics - Issue 21 (2008) Figure 1: Different Weighted Portfolio VaR, RAPM, and Weighted ROR 89 From Figure 1, we find that when the holding ratio of Euro increases, VaR decreases, such as from point B to point D. When the ratio reaches 28%(point D), the VaR of the foreign exchange reserves portfolio reaches its minimum, while when the ratio is higher than 28%, the value at risk inverses. The value of RAPM reaches its maximum when the Euro ratio reaches 27%, as point C on the RAPM curve shows, but it begins to decline as the ratio becomes higher than 27%. All of above shows that the Euro has a hedge effect at the beginning, but its marginal effect in reducing the total risk will dissipate as the holding ratio increases. Despite the existence of the hedging effect of Euro, the effect is not persistent. According to this result, several implications can be derived. First, the Euro could be a hedge currency for central banks. Because increases its holding ratio would reduce the VaR of the portfolio. Once the holding ratio of the Euro exceeds a certain amount, its diversification effect will dissipate. Its hedging effect is not persistent. Second, for those who care more about return relative to risk, they can tag along the index of RAPM as their risk management instruction. Because RAPM illustrates the performance of the portfolio, administrators can decide their optimal currency composition through this indicator. Third, if administrators tend to be more conservative, which means they place more importance on risk, then they can consider the index of VaR. We therefore suggest that any financial institution can deal with their risk management by using the risk-return analysis above. 3.4. Risk Management In order to understand the contribution of every single currency in the foreign exchange reserves portfolio, we analyze the decomposition of the VaR. Taking point B as example, the marginal VaR for Euro is –0.84822 which is the lowest among all the currencies, while it was 0.60839 for U.S. dollars which is the highest among the currencies. And the component VaR is –0.12977766 for Euro which is the lowest among all currencies. This implies that whenever increases the holding ratio of Euro, the risk of the whole investment portfolio decreases most. On the other hand, component VaR is 0.38997799 for U.S. dollars which is the highest among all currencies. This implies that whenever increases its holding ratio of U.S. dollar, the risk of the whole investment portfolio will increase most. As a result, increase the holdings of the lowest-valued component VaR currency while decrease the holdings of the highest-valued component VaR currency can in fact reduce the risk of whole investment portfolio. 90 International Research Journal of Finance and Economics - Issue 21 (2008) The sum of the individual VaRs represents an un-diversified VaR, while the sum of the component VaR represents a diversified VaR. And the difference between these two is the diversified effect. 4. Conclusion In this study we categorized nine major countries’ daily exchange rate data and the IMF 2002 annual report of weighted reserves currency data into three groups that are the world, the industrial countries and the developing countries. Then utilized the Exponential Weighted Moving Average (EWMA), Bootstrapping, and Monte Carlo Simulation to compute the VaR for those three groups’ foreign reserves portfolio from the year of 1995 to 2001. Empirically we find that EWMA finished relatively better than the other two. For the RAPM index which was computed by EWMA’s VaR, we found that RAPM in industrial and developing countries decreased from 1997 except the year 2000. Prior to 1999, RAPM in industrial countries was better than that of in developing countries. After 1999, it reversed itself. Based on developing countries’ currency in 2001 if the other currencies’ holding ratio is fixed, reducing the U.S. dollar holding ratio while increasing the Euro holding ratio will make VaR decrease in EWMA. However, if the Euro holding ratio is high, VaR increases. This implies that despite the hedging effect of Euro, as its holding ratio increases, the marginal effect decreases. The hedging effect of Euro is not persistent. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] Barry Eichengreen and Donald J Mathieson (2000), “The Currency Composition of Foreign Exchange Reserves: Retrospect and Prospect.” IMF Working Paper, WP/00/131. (Washington: International Monetary Fund). Basel Committee on Banking Supervision (1996), “Supervisory Framework for the use of ‘Backtesting’ in Conjunction with the Internal Models Approach to Market Risk Capital Requirement,” BIS. Beschloss, Afsaneh, Wendy Mendes (1999-2000), “Reserve Management Policies and Practices.” Central Banking, Vol Ⅹ, Number 4, pp. 88-96. Dooley, Michael P., J. Saul Lizondo and Donald J. Mathieson (1989), “The Currency Composition of Foreign Exchange Reserves. ”IMF Staff Papers 36, pp. 285-434. Jorion, P. (2000), Value at Risks: The new benchmark for controlling market risk, 2nd ed. McGraw-Hill, New York . José R. Aragonés, Carlos Blanco, and Juan Mascareñas (2001), “Active Management of Equity Investment Portfolios.” The Journal of Portfolio Management, Vol 27 Iss3, pp.29-pp.46. Kupiec, P. H., (1995), Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 3: 73-84. Mario I. Blejer and Liliana Schumacher (1998), “Central Bank Vulnerability and the Credibility of Commitments: A Value-at-risk Approach to Currency Crises.” IMF Working Paper, WP/98/65. Kenen, P. B., (2002), “The Euro versus the dollar: Will there be a struggle for dominance?” Journal of Policy Modeling, New York, Vol 24 Iss 4, pp.347-pp.354. Winfried G. Hallerbach (1999), “Decomposing Portfolio Value-at-Risk: A General Analysis” Tinbergen Institute Graduate School of Economics. International Research Journal of Finance and Economics - Issue 21 (2008) 91 Appendix Appendix 1: Weights of the nine currencies, 1995-2001 All countries USD JPY GBP CHF ECU/Euro DM FRF NLG AUD Industrial countries USD JPY GBP CHF ECU/Euro DM FRF NLG AUD Developing countries USD JPY GBP CHF ECU/Euro DM FRF NLG AUD Source: IMF annual report 2002 1995 57 6.8 3.2 0.8 6.8 13.7 2.3 0.4 8.9 51.8 6.6 2.1 0.1 13.4 16.4 2.3 0.2 7 62.4 7 4.3 1.5 -11 2.3 0.6 10.9 1996 60.3 6 3.4 0.8 5.9 13.1 1.9 0.3 8.3 56.1 5.6 2 0.1 12 15.6 1.7 0.2 6.7 64.3 6.5 4.8 1.4 -10.6 2 0.5 9.9 1997 62.4 5.2 3.7 0.7 5 12.9 1.4 0.4 8.4 57.9 5.8 1.9 0.1 10.9 15.9 0.9 0.2 6.4 66.2 4.7 5.1 1.1 -10.3 1.8 0.6 10.2 1998 65.9 5.4 3.9 0.7 0.8 12.2 1.4 0.4 9.3 66.7 6.6 2.2 0.2 1.9 13.4 1.3 0.2 7.4 65.3 4.5 5.2 1.1 -11.3 1.5 0.5 10.8 1999 68.4 5.5 4 0.7 12.7 ---8.8 73.5 6.5 2.3 0.1 10.7 ---6.9 64.6 4.7 5.3 1.1 14.2 ---10.2 2000 68.1 5.2 3.9 0.7 13 ---9.1 73.3 6.3 2 0.2 10.4 ---7.6 64.2 4.4 5.2 1 15 ---10.1 2001 68.3 4.9 4 0.7 13 ---9 74.5 5.5 1.8 0.4 9.7 ---8.1 64.1 4.5 5.5 0.9 15.3 ---9.6 Appendix 2: Government Bond Yield (%) 1995 6.58 2.532 8.255 3.73 6.495 7.59 7.195 8.729 9.17 1996 6.44 2.225 8.101 3.63 5.626 6.385 6.493 7.233 8.17 1997 6.35 1.688 7.087 3.08 5.077 5.627 5.805 5.958 6.89 1998 5.26 1.097 5.448 2.71 4.391 4.718 4.874 4.703 5.5 1999 5.64 1.771 4.697 3.62 4.263 4.688 4.92 4.656 6.08 2000 6.03 1.748 4.681 3.55 5.239 5.452 5.509 5.44 6.26 2001 5.02 1.334 4.783 3.56 4.699 5.047 5.169 5.028 5.64 COUNTRY UNITED STATES JAPAN UNITED KINGDOM SWITZERLAND GERMANY FRANCE NETHERLANDS EURO AREA AUSTRALIA Source: IMF Financial Statistics Appendix 3: Here, we illustrate how the sum of component VaR is the diversified portfolio VaR. N N ∂VaR P CVaR i = ∑ ⋅ w i ⋅ VP ∑ i =1 i =1 ∂w i VP N ∂ (VP ⋅ Z α ⋅ σ P ) =∑ ⋅ wi ∂w i i =1 92 = ∑ VP ⋅ Z α ⋅ i =1 N International Research Journal of Finance and Economics - Issue 21 (2008) N ∂σ P ⋅ wi ∂w i = ∑ VP ⋅ Z α i =1 N σ iP ⋅ wi σP w i σ iP σ2 P w i σ iP σ2 P = ∑ (VP Z α σ P ) i =1 = VP Z α σ P ⋅ ∑ i =1 N ⎧⎡ ⎤ 1 ⎫ = VaR P ⎨⎢∑ w i ⋅ Cov(R i ,i+1, R P )⎥ ⋅ 2 ⎬ ⎦ σP ⎭ ⎩⎣ i=1 ⎧⎡ N ⎤ 1 ⎫ = VaR P ⎨⎢∑ Cov(w i R i ,i+1, R P )⎥ ⋅ 2 ⎬ ⎦ σP ⎭ ⎩⎣ i=1 ⎡ ⎛ N ⎞ 1 ⎤ = VaR P ⎢Cov⎜ ∑ w i R i ,i+1 , R P ⎟ ⋅ 2 ⎥ ⎝ i=1 ⎠ σP ⎦ ⎣ N = VaR P ⋅ σ 2 ⋅ P = VaPP 1 σ2 P