Article

Value-at-Risk Estimation of Foreign Exchange Rate Risk in India Onkar Shivraj Swami1 Santosh Kumar Pandey1 Puneet Pancholy1

Asia-Pacific Journal of Management Research and Innovation 12(1) 1–10 © 2016 Asia-Pacific Institute of Management SAGE Publications sagepub.in/home.nav DOI: 10.1177/2319510X16650057 http://apjmri.sagepub.com

Abstract Banks are required to maintain an appropriate level of capital which must commensurate with the riskiness of their portfolio. Recently, the Reserve Bank of India (RBI) issued a circular on Prudential Guidelines on Capital Adequacy—Implementation of Internal Models Approach (IMA) for Market Risk to select a suitable method for the banks to determine the regulatory capital requirement under the market risk exposure. Banks which adopt this approach are required to quantify market risk through their own Value-at-Risk (VaR) model. Therefore, it is a challenging task for risk managers of the bank to select an appropriate risk model which reasonably covers the risk of the bank’s portfolio. Use of wrongly calibrated risk models may lead to undercapitalised banking system. This article aims at exploring the suitable risk model for measuring foreign exchange risk in banks’ portfolio. The objective of present study is to empirically test the appropriate VaR model for foreign exchange rate risk. Value-at-Risk has been estimated for foreign exchange rate risk by using parametric variance–covariance method and non-parametric historical simulation (HS) method. Under parametric method, VaR is estimated by assuming that returns follow normal and Student’s t-distribution. Backtesting results for various VaR models have been done based on Kupiec’s proportion of failures (KPOF) test and regulatory ‘traffic light’ test. This article concludes that when returns are non-normal, VaR model based on the assumption of normality significantly underestimates the risk. Our empirical results based on backtesting show that most accurate VaR estimates are obtained from Student’s t VaR model.

Keywords Foreign exchange rate risk, internal models approach, Value-at-Risk, historical simulation, variance–covariance, VaR backtesting

Introduction According to the Basel Committee on Banking Supervision (BCBS), market risk is defined as ‘the risk of losses in on and off-balance-sheet positions arising from movements in market prices’ (BIS, 2005). Market risk is the risk of losses to the bank arising from movements in market prices as a result of changes in interest rates, foreign exchange rates and equity and commodity prices. Foreign exchange rate risk is the risk that the value of the bank’s assets or liabilities changes due to currency exchange rate fluctuations. By buying and selling the foreign exchange on behalf of their customers, banks are exposed to exchange rate risk. Generally, banks are vulnerable to three types of foreign exchange risk: transaction (commitment),

economic (operational, competitive or cash flow) and translation (accounting) (Abor, 2005). Transaction risk arises when the value of existing obligations is deteriorated by movements in foreign exchange rates (Abor, 2005). Economic risk occurs due to impact of high unexpected volatility in the exchange rate on equity/income for both domestic and foreign operations. Translation risk is associated with the assets or income derived from offshore activities (Abor, 2005). BCBS has adopted VaR as a primary measure of market risk for determining the bank capital adequacy (BCBS, 2006). Subsequently, VaR has become the standard measure for estimating the market risk in financial sector industries. Value-at-Risk is defined as ‘the worst loss over a target horizon with a given level of confidence’ (Jorion, 2007).

Disclaimer: The views expressed in this study are of the authors and not of the institution to which they belong. 1  Banking Policy Division, Department of Banking Regulation, Central Office, Reserve Bank of India, Shahid Bhagat Singh Marg, Fort, Mumbai, Maharashtra,

India. Corresponding author: Onkar Shivraj Swami, Banking Policy Division, Department of Banking Regulation, Reserve Bank of India, Shahid Bhagat Singh Marg, Fort, Mumbai 400001, India. E-mail: [email protected]

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The portfolio VaR estimates the maximum loss with the given probability an investor may suffer over a given time period (Abad, Benito & López, 2014). The objective of the present study is to empirically test the foreign exchange rate risk for the management of market risk with the use of VaR methodology. Value-atRisk has been estimated for foreign exchange rate risk by using (i) parametric method called as variance–covariance approach and (ii) non-parametric HS method. The article is organised as follows. In the next section, we review HS and parametric approach to estimate the VaR. The third section provides the regulatory framework for market risk in India. The fourth section delineates topics related to data and methodology. Empirical findings are given in the fifth section. The last section concludes the findings of the study.

Theoretical Background, Concept and Definitions Definition and Concept of VaR Value-at-Risk is a statistical measure that summarises all the risks of a portfolio into a single number suitable for use in the boardroom, for reporting to regulators or for disclosure in an annual report. More precisely, according to Elliott and Miao (2009), ‘VaR is a statistical estimate of a portfolio loss with the property that, with a small probability α, the owner of the portfolio stands to incur that loss or moreover a given (typically short) holding period.’ In general, a may be fixed at 1, 2.5 and 5 per cent and the holding period may be taken as 1, 2 and 10 business days or 1 month. According to Jorion (2007), ‘VaR is the worst loss over a target horizon such that there is a low, prespecified probability that the actual loss will be larger.’ This definition consists of two parameters: the risk horizon and the confidence level. Let a be the confidence level and L be the loss, measured as a positive number. Value-at-Risk is also reported as a positive number (Jorion, 2007). Again as mentioned by Jorion (2007), a general definition of VaR is that it is the smallest loss, in absolute value, such that

P [L > VaR] # 1 – a

Take, for example, a 99 per cent confidence level (i.e., a = 0.99). Value-at-Risk then is the cut-off loss such that the probability of experiencing a greater loss is less than 1 per cent. Jorion (2007) also has suggested that the length of time required to hedge the market risk should be in alignment with the risk horizon. It is convenient to use short risk horizons than the longer one for the frequently

trading portfolios such as foreign exchange rates (Sirr, Garvey & Gallagher, 2011).

VaR Models The non-parametric HS method and parametric method for estimating the VaR, that is, variance–covariance approach, are described below: Historical Simulation Method The historical VaR model assumes that all possible future variations have been experienced in the past, and that the historically simulated distribution is identical to the returns distribution over the forward-looking risk horizon (Alexander, 2008a, 2008b). Thus, non-parametric HS has some advantages over the parametric method, as it makes no assumption about the shape of the distribution of returns. Further, HS method is very easy to implement. However, more recent studies (Abad et al., 2014; Angelidis, Benos & Degiannakis, 2007; Ashley & Randal, 2009; Trenca, 2009) have mentioned that the HS method gives poor VaR estimates. Filtered HS and conditional extreme value provide more adequate VaR estimates than HS VaR (Abad et al., 2014). Nevertheless, by far HS approach is most widely implemented by the banks for estimating the daily VaR (Pérignon & Daniel, 2010). Parametric Method Wang, Wu, Chen and Zhou (2010) mentioned that the variance–covariance method is a parametric method to calculate VaR. This parametric method needs the estimation of standard deviation of the portfolio by means of sample estimation of variance (Rossignolo, Duygun & Mohamed, 2012). The 100a per cent h-day normal linear VaR for single exchange rate is given by

VaR h, a = U -1 (1 – a) v h

where vh is the standard deviation of the sample returns and U –1 (1 – a) is the inverse of the cumulative density function of the standard normal distribution. Similarly, 100a per cent h-day normal linear VaR for portfolio of foreign exchange rates is given by

VaR h, a = U –1 (1 – a) wl Vh w – wl E (X h)

where risk factor returns are multivariate normal and independent and identically distributed (iid), w denotes the current vector of portfolio weights, E(Xh) is the n × 1 vector of the expected/mean excess h-day returns and Vh is the h-day covariance matrix of foreign exchange rates return. To estimate the VaR with high peaks and heavy tails of exchange rate returns, it is ideal to assume that U(·)

Swami et al. 3 follows Student’s t-distribution (Wang et al., 2010). When h is small, an approximate formula for the 100a per cent h-day Student’s t VaR is given by

Student t VaR h, a, y = y -1 (y - 2) ht y-1 (1 – a) v – hn

where y denotes degrees of freedom for Student’s tdistribution. Similarly, 100a per cent h-day Student’s t VaR for foreign exchange rates portfolio is given by

Student t VaR h, a, = y -1 (y - 2) ht -y 1 #  (1 – a) il X h i – il n h)

where Xh denotes the m × m covariance matrix of the risk factor and nh denotes the m × 1 vector of expected/mean excess returns over the h-day risk horizon.

Regulatory Framework To allocate adequate regulatory capital under the market risk, Basel II framework on capital adequacy provides two methods, namely, standardised measurement method (SMM) and Internal Models Approach (IMA). Since 31 March 2005, banks in India have been measuring market risk under Basel II accord by using the SMM. Compared to the SMM, the IMA is considered to be more risk sensitive and aligns the capital charge for market risk more closely to the actual losses likely to be incurred by banks due to movements in the market risk factors. Therefore, the Reserve Bank of India (RBI) issued circular on Prudential Guidelines on Capital Adequacy—Implementation of Internal Models (IMA) Approach for Market Risk on 7 April 2010 to provide a wider choice to banks in selecting a method for determining the regulatory capital requirement for their market risk exposure (RBI, 2010). As of now, none of the banks in India are determining the regulatory capital requirement for their market risk exposure by using IMA.

VaR As per the RBI’s circular on Prudential Guidelines on Capital Adequacy—Implementation of Internal Models (IMA) Approach for Market Risk issued on 7 April 2010, the capital requirement is a function of three components: ‘(1) Normal VaR Measure (for general market risk and specific risk) (2) Stressed VaR Measure (for general market risk and specific risk) and (3) Incremental Risk Charge (IRC) (for positions subject to interest rate specific-risk capital charge)’. Further, according to the same circular on IMA dated 7 April 2010, mandatory parameters for VaR models are given below:

1. Value-at-Risk must be computed on a daily basis. 2. In calculating VaR, a 99th percentile, one-tailed confidence interval is to be used. 3. In calculating VaR, an instantaneous price shock equivalent to a 10-movement in prices is to be used, that is, the minimum ‘holding period’ will be 10 trading days. Banks may use VaR numbers calculated according to shorter holding periods scaled up to 10 days in an appropriate manner. Banks should use the ‘square root of time rule’ only for linear portfolios with identically and independently normally distributed returns. A bank using this approach, for portfolios other than linear portfolios with identically and independently normally distributed returns, must periodically justify the reasonableness of its approach to the satisfaction of the RBI. For calculating VaR, it is essential to have a minimum, past 1 year daily returns data (i.e., 250 trading days). Banks in India are permitted to use any VaR model subject to the condition that it is satisfies the above requirements, that is, variance–covariance matrices, HSs, etc., for calculating the VaR under the market risk exposure.

Regulatory Backtesting Rossignolo et al. (2012) mentioned that ‘backtesting constitutes a statistical technique to assess the quality of the risk measurement specifications which involves the comparison between the daily VaR forecast with the actual losses’. Backtesting involves comparing the number of times actual losses exceed VaR estimates in approximately 250 trading days (Rossignolo et al., 2012). In accordance with Basel II requirements, the RBI under IMA guideline (2010) has stated that ‘backtesting will be based on 1 percent daily VaR estimate i.e. holding period is assumed as one day and it will cover a period of 250 days’. For backtesting, most recent 12 months of data (i.e., 250 daily observations) are used. Apart from backtesting VaR model based on 1 per cent VaR, banks can validate model assumptions at 2, 5 and 10 per cent VaR. In general, to have one exception in a month (1 in 20 trading days) for backtesting, 95 per cent confidence level is considered to be more appropriate. As per RBI’s circular on IMA dated 7 April 2010, backtesting outcomes can be classified into three zones depending on the number of exceptions coming out of the backtesting, often called the ‘traffic light’ approach. Based on the number of exceptions, backtesting outcomes will fall in either green zone, yellow zone or red zone and the corresponding zone-wise increase in the multiplication factors for both VaR and stressed VaR are given in Table 1.

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Table 1. Zone-wise Classification Backtesting Outcomes Zone

Number of Exception

Green zone

Yellow zone

Red zone

Increase in Multiplication Factor

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.40

6

0.50

7

0.65

8

0.75

9

0.85

10 or more

1.00

Source: Reserve Bank of India.

Methodology The main data contain univariate price series of three foreign exchange currencies, that is, Indian rupee (INR)/ US Dollar (USD), INR/POUND and INR/EUR. We have taken a hypothetical portfolio of 200 million INR. Of these, 100 million INR foreign exchange business of a bank is in USD, 50 million INR business is in pound sterling and euro each (i.e., INR/USD has 50 per cent weight of total portfolio, and INR/POUND and INR/EUR each have 25 per cent weight in total portfolio). The data are retrieved from RBI’s database on Indian economy for the period from 4 January 1999 to 31 December 2013. The daily logarithmic return Rt is defined as Pt n . For each exchange rate, we have 3,400 Rt = ln d Pt–1 daily logarithmic returns. As per regulatory requirements,

this study calculates VaR with 99 per cent confidence level for 1-day horizon. First, we estimate 1-day VaR for 3 January 2000, that is, 251st day by using the entire returns of the year 1999 (i.e., around 250 days observations). Second, we calculate VaR for 4 January 2000, that is, 252nd day using returns from 5 January 1999 to 3 January 2000. In a similar way, VaR is estimated up to the last data point of the present study (i.e., 31 December 2013). We have used 1-year data, that is, about 250 days data rolling window, for our VaR analysis and backtesting. Although it is computationally convenient to assume normal distribution of the returns when calculating the VaR, it always does not hold true. Estimation based on normal distribution produces significant errors when dealing with skewed data. We have used Q–Q plots and Jarque–Berra test for detecting the non-normality. Value-at-Risk violation occurs when returns exceed the estimated VaR. Backtesting is a statistical procedure in which actual returns are systematically compared to corresponding VaR estimates. The backtesting of estimated VaR is carried out for each year from 2000 to 2013 by using the latest preceding 1-year data (i.e., about 250 days). In addition to regulatory backtesting, we have also performed backtesting using KPOF test (Kupiec, 1995). Kupiec’s test measures whether the number of exceptions is consistent with the confidence level (Nieppola, 2009).

Results Descriptive Statistics The descriptive statistics of daily logarithmic returns of the three series and portfolio are given in Table 2. For all the three series, skewnesses of daily logarithmic returns are not equal to zero and kurtosises are greater than three. Further, portfolio reruns series also has skewness greater

Table 2. Descriptive Statistics of Daily Logarithmic Return from Three Exchange Rates of INR Parameters

INR/USD

INR/POUND

INR/EUR

Portfolio

Mean

0.00010

0.00010

0.00014

0.00011

Median

0.00000

0.00020

0.00020

0.00010

Minimum

–0.03010

–0.05700

–0.03890

–0.03180

Maximum

0.04020

0.03680

0.04150

0.03970

Standard deviation

0.00432

0.00632

0.00680

0.00429

Skewness

0.26790

–0.41229

–0.02087

0.22510

Kurtosis

11.39130

7.86258

5.09643

9.50848

10,723

3,689

667

6,455

0.00000

0.00000

0.00000

0.00000

3,640

3,640

3,640

3,640

Jarque–Bera Probability N (sample size) Source: Authors’ calculation.

Swami et al. 5 than zero and kurtosises greater than three (Table 2). This suggests that these exchange rate series and portfolio returns do not follow the normal distribution. These findings are also supported by the Jarque–Bera statistics for daily logarithmic returns of the three series and portfolio. Figure 1 depicts the histogram of daily logarithmic return

for three exchange rates of INR and portfolio returns. From these histograms, it appears that all three series and portfolio returns have high peak than normal distribution. In general, Q–Q plot is used to identify the distribution of sample in the study. Figure 2 represents the Q–Q plot of daily logarithmic return for three exchange rates of

Figure 1. Histogram of Daily Logarithmic Return from Three Exchange Rates of INR and Portfolio Returns Source: Authors’ calculation.

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Figure 2. Q–Q Plot of Daily Logarithmic Return from Three Exchange Rates of INR and Portfolio Returns Source: Authors’ calculation.

INR and portfolio returns. The Q–Q plot compares the distribution for each exchange rate returns including portfolio returns with the normal distribution and indicates that all three exchange rates of INR and portfolio returns deviate from the normal distribution.

VaR Figure 3 shows the time series of portfolio of foreign exchange rate returns. The logarithmic returns and VaR based on HS and normal and Student’s t-distribution are

Swami et al. 7

Figure 3. Graph of Daily Portfolio Returns and VaR Estimated by Using Various Models Source: Authors’ calculation.

estimated on a daily basis. The time series begins on 3 January 2000 as the previous 1-year data (i.e., year 1999) are needed to estimate the initial parameters. The chart shows the significant increase in volatility of foreign exchange rate portfolio returns from the second half of 2008 and as a result, VaR estimates also worsen. Daily estimated HS_VaR has been highest between the period December 2008 and November 2009. In fact, HS_ VaR estimates are higher than VaR estimates based on normal and Student’s t-distribution for the same period. This is the period when the Global Financial Crisis (GFC) began. As the foreign exchange rate returns are nonnormal, Normal_VaR estimates underestimate the risk throughout the study period. Moreover, as GFC progressed further, Normal_VaR has performed even more poorly. This confirms the belief that as return distributions do not follow normal distribution, risk estimates based on assumption of normality will not provide robust risk assessments. It is therefore prudent for banks and bank supervisors to look into the assumptions of return distribution while computing regulatory capital based on their internal risk models. When significant positive excess kurtosis is found in empirical return distribution, the Student’s t-distribution is likely to produce VaR estimates that are more close to historical behaviour than normal distribution (Elliott & Miao, 2009; Lin & Shen, 2006). The Student’s t-distribution is more adequate to deal with the fat-tailed and leptokurtic features. As our foreign exchange rate returns are having leptokurtic feature, the VaR estimates based on Student’s t-distribution appear to be more appropriate. From Figure 3, in general, it seems that VaR estimates based on Student’s t-distribution lie between the estimates based on HS_VaR

and Normal_VaR. During the GFC onset and the later period, the Student’s t VaR has performed better than Normal_VaR.

Backtesting of VaR Models Value-at-Risk violation occurs when actual portfolio loss exceeds the estimated VaR. Tables 3–5 represent the backtesting results for various VaR models based on KPOF test and regulatory ‘traffic light’ test. We have observed 59 exceptions/violations for HS_VaR model over the study period, that is, from 2000 to 2013. For HS_VaR model, the maximum number of violations occurred in the year 2008 followed by the year 2007 and 2013. Backtesting based on KPOF test for HS_VaR model is accepted for 11 years (p ≤ 0.05) and is rejected for the years 2007, 2008 and 2013 (Table 3). Backtesting based on Regulatory test for HS_ VaR model has indicated a result in the red zone for the year 2008 and yellow zone for the year 2003, 2007, 2012 and 2013. For Normal_VaR model, we have observed 68 exceptions/violations over the study period. For Normal_VaR model, the maximum number of violations occurred during the years 2007 and 2008. Form Figure 3, it is clear that throughout the study period, VaR number based on Normal_VaR model is underestimated. Backtesting based on KPOF test for Normal_VaR model is accepted for 11 years (p ≤ 0.05) and is rejected for the years 2004, 2007and 2008. Backtesting based on regulatory test for Normal_ VaR model has indicated a result in the red zone for the years 2007 and 2008 and yellow zone for the years 2000, 2004 and 2013.

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Table 3. Backtesting of HS_VaR Model Kupiec’s Proportion of Failures Test

Year

No. of Exceptions

Percentage of No. of Exceptions

Likelihood Ratio

2000

4

1.6

0.77

Accept/Reject

Regulatory Test

Accept

Green zone

2001

2

0.8

0.11

Accept

Green zone

2002

1

0.4

1.18

Accept

Green zone

2003

5

2

1.96

Accept

Yellow zone

2004

4

1.6

0.77

Accept

Green zone

2005

2

0.8

0.11

Accept

Green zone

2006

2

0.8

0.11

Accept

Green zone

2007

9

3.6

10.23

Reject

Yellow zone

2008

10

4.0

12.96

Reject

Red zone

2009

1

0.4

1.18

Accept

Green zone

2010

3

1.2

0.09

Accept

Green zone

2011

4

1.6

0.77

Accept

Green zone

2012

5

2

1.96

Accept

Yellow zone

2013

7

2.8

5.50

Reject

Yellow zone

Source: Authors’ calculation.

Table 4. Backtesting of Normal VaR Model Kupiec’s Proportion of Failures Test

Year

No. of Exceptions

Percentage of No. of Exceptions

Likelihood Ratio

Accept/Reject

Regulatory Test

2000

5

2.0

1.96

Accept

Yellow zone

2001

1

0.4

1.18

Accept

Green zone

2002

1

0.4

1.18

Accept

Green zone

2003

3

1.2

0.09

Accept

Green zone

2004

8

3.2

7.73

Reject

Yellow zone

2005

2

0.8

0.11

Accept

Green zone

2006

2

0.8

0.11

Accept

Green zone

2007

14

5.6

25.78

Reject

Red zone

2008

14

5.6

25.78

Reject

Red zone

2009

1

0.4

1.18

Accept

Green zone

2010

3

1.2

0.09

Accept

Green zone

2011

4

1.6

0.77

Accept

Green zone

2012

4

1.6

0.77

Accept

Green zone

2013

6

2.4

3.56

Accept

Yellow zone

Source: Authors’ calculation.

We have observed 43 exceptions/violations for t_VaR model over the study period. For t_VaR model, the maximum number of violations occurred during the year 2007. Backtesting based on KPOF test for t_VaR model is accepted for all years (p ≤ 0.05) except for the year 2007. Backtesting based on regulatory test for t_VaR model has

indicated a result in the red zone for the year 2007 and yellow zone for the years 2004 and 2008. In general, VaR estimates based on normal distribution have underestimated the VaR numbers. Among the three models, Student’s t VaR model has performed better. Intuitively, the parametric standard deviation-based

Swami et al. 9 Table 5. Backtesting of Student’s t VaR Model

Year

No. of Exceptions

Kupiec’s Proportion of Failures Test

Percentage of No. of Exceptions

Likelihood Ratio

Accept/Reject

Regulatory Test

2000

1

0.4

1.1765

Accept

Green zone

2001

1

0.4

1.1765

Accept

Green zone

2002

1

0.4

1.1765

Accept

Green zone

2003

3

1.2

0.0949

Accept

Green zone

2004

6

2.4

3.5554

Accept

Yellow zone

2005

2

0.8

0.1084

Accept

Green zone

2006

1

0.4

1.1765

Accept

Green zone

2007

12

4.8

19.0162

Reject

Red zone

2008

6

2.4

3.5554

Accept

Yellow zone

2009

1

0.4

1.1765

Accept

Green zone

2010

2

0.8

0.1084

Accept

Green zone

2011

4

1.6

0.7691

Accept

Green zone

2012

1

0.4

1.1765

Accept

Green zone

2013

2

0.8

0.1084

Accept

Green zone

Source: Authors’ calculation.

approach should be more precise. Indeed, estimates of standard deviation use information about whole distribution, whereas quantile uses only the ranking of observations and the two observations around the estimated value, that is, parametric methods are inherently more precise because the sample standard deviation contains far more information than sample quantiles (Jorion, 2007).

Conclusion In this empirical study, we have applied various models/ methods for estimating VaR for the portfolio of foreign exchange currencies. Value-at-Risk has been estimated for foreign exchange rate risk by using variance–covariance approach and non-parametric HS method. For estimating the VaR based on various models, we have followed the regulatory framework by RBI on Prudential Guidelines on Capital Adequacy—Implementation of Internal Models Approach for Market Risk in India (RBI, 2007). Estimation of VaR would be made simply by using mean and standard deviation of the return distribution if returns are normally distributed. But the financial market returns usually do not follow normal distribution. The returns in our study are leptokurtic in nature. This observed non-normality of returns should be handled suitably while estimating VaR. Accordingly, we employed non-normal VaR models, such as HS and Student’s t VaR model. Our empirical results based on backtesting show that the VaR estimates based on the conventional ‘normal’ method are usually underestimated (lower than actual).

Interestingly, HS method has performed better than Normal_VaR model. However, most accurate VaR estimates are obtained from Student’s t VaR model. Acknowledgements We are extremely grateful to Sudarshan Sen, Principal Chief General Manager, Department of Banking Regulation, Reserve Bank of India, Mumbai, for his constant guidance and support in the preparation of this article. Further, helpful discussion and suggestions from Madhusmita Dutta, Manager, Foreign Exchange Department, Reserve Bank of India, Mumbai, for improving the article are highly acknowledged.

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