A VaR Investigation of Currency Composition In

8/7/2019 A VaR Investigation of Currency Composition In

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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: [email protected]

Tel: 886-2-25381111 ext. 8927

Jui-Cheng Hung

Department of Finance and Banking

Yuanpei Institute of Science and Technology, Taiwan


Mei-Maun Hseu

Department of Finance, Chihlee Institute of Techonology, Taipei,Taiwan


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. Basedon 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 makeVaR 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 currencywhile 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. IntroductionExperiencing from the Asian financial crisis of 1997-98, most developing Asian economies haverapidly 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 inseveral 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


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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 internationalportfolio.

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 needsfor economy stability. Foreign exchanges have not only become a countrys key reserves assets but

also protect the countrys 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 majormeans; the U.S. dollar inevitably acts an essential part in a countrys foreign exchange reserves.

According to the IMFs 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 countrys foreign reserves policies are under investigation. However, a

countrys 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 thestability of foreign reserves was resulted from the commodity trading, the capital flow, and the fixedexchange 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 intoseparated Value-at-Risks in parts, while Jos, Carlos, and Juan (2001) take the conceptual Value-at-

Risk 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 banks investmentportfolio 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 reservescurrencies 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 banks 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 currenciesweighted importance, we analyze if the increasing Euro holdings was able to reduce the portfolios

risks.

2. Methodology2.1. 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 andlosses 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 IMFs annual report presents 8 currencies and 1 unspecified currency. In this study we take the Australia dollar as the unspecified

currency.


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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 marketrisk, but also summarizes the effects of leverage, diversification, and probabilities of adverse price

movements in a single dollar amount.

Lets define W0 as the initial investment at the beginning, and R*

represents the estimateexpected returns, then VaR can be written as:

meanVaR = ( )*0- W R - . (3)

If defined VaR as the absolute loss that excludes the expected returns, then VaR can berewritten as:

zeroVaR =*

0- W R . (4)

If we transform the probability distribution (W)f into a standard normal distribution (),

where ~ N(0, 1). Then the probability of possible returns which is less than W*

will be 1-C, and it can

be rewritten as:

d)(dr)r(fdw)w(fC1RW **

=== , (5)

wheret

Rt

*

= and t is the time factor. After the value of is determined, Value at Risk

can be written as: ( )ttWVaR 0zero = (6)tWVaR 0mean = . (7)

In this study we take zeroVaR 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 canbe 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

( )tttzero,1t ZWVaR = + (8)

ttmean,1t ZWVaR = + , (9)

where Wt is the assets value at time t, t is the average returns at time t, t is the standard deviation at

time t, and Z represents the critical value with a confidence level of 1-.

Under the definition of absolute return and the assumption of W t=1, Z, t and t can then beobtained. 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- 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:2

1t,i

2

1t,i

2

t,i r)1( += (10)

1t,j1t,i1t,ijt,ij rr)1( += , (11)

where ji , 2,ti and tij, are the variance of asset i and the covariance of asset i and j at time t. 1, tir

and 1, tjr are the return ratios of asset i and j at time t-1. 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 bewritten as:

TZWVaR t,ptt = , (12)


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where = wwtP ',

NN

2

t,Nt,2Nt,1N

t,N2

2

t,2t,21

t,N1t,12

2

t,1

=L

MOMM

L

K

1NN

2

1

w

w

w

w

=M

;

iw is the weighted ratio of the ith

foreign asset to the total portfolio, i,t, j,t, and ij,t are obtained by

EWMA and they are the variance and covariance of the foreign asset, respectively. tP, is the standarddeviation 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 usedto calculate the underlying VaR. The method takes the limited historical returns and go through the

iterate sampling to construct the asset portfolios 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 directlyutilizes the future returns distribution that has only one price path, while Bootstrapping makes an

iterate sampling from historical data to simulate the real returns distribution to improve the

shortcoming of a Historical Simulation.Assume it is known for each assets prices in a portfolio for the last 251 days. It would be

attainable to get next days 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.

)T(Ri , where i 1, 2...,N= T 1,2,...,10000=

Step 2Multiply 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 days 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-.

2.4. Monte Carlo Simulation

Before applying the Monte Carlo Simulation, it is necessary to assume that the underlying portfoliosreturn 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:

dZdtR

dRtt

t

t += , (13)

where Rt is the return of portfolio, t is the drift term of the portfolio at time t, t is the standard

deviation at time t, and dZ is normally distributed with 0 mean and dt variance, whereby ( )dt,0N~dZ .From (13) we found that if a portfolios return follows a multivariable normal distribution that

is ( )NN1N ,N~R , then in order to simulate the return as an N N normal distribution, a MonteCarlo process as be described as follows:

Step 1


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80 International Research Journal of Finance and Economics - Issue 21 (2008)

Take the Cholesky Decomposition to decompose an NN variance- covariance matrix as =TAAlike follows

=

NN2N1N

2221

11

aaa

0

aa

00a

A

L

OMM

ML

L

=

NN

2N22

1N2111

T

a00

aa0

aaa

A

L

MOMM

L

L

=

2

N2N1N

N2

2

221

N112

2

1

L

MOMM

L

L

.

Step 2

An 1N Z matrix can be generated, where Z is a multivariate standard normal distributed randomvariable, and ( )NI,0N~Z , where

=

100

00

001

IN O .

Step 3

Multiplying the A matrix from step 1 by the Z matrix, we can get R=AZ, which is an N multivariablenormal random variable. Here the covariance matrix of R is

( ) ( ) ( ) ( ) TTTTT AZZAEAAZZERRERVAR === = == TTN AAAAI .Step 4

Repeat step (3) 10000 times and get a sample size of 10000 next days portfolios return. Sorting the

data in increasing order, we get the distribution for the portfolios next days return. VaR can thereforebe computed by a percentile criterion on the confidence level of 1- for this simulated future returns.

2.5. VaR, Evaluation and Performance

In VaRs evaluation and its performance effectiveness, we adopt Kupiecs (1995) Proportion of Failure

Test (PF-Test). This gives the verification of whether the constructed 0 was consistent with the actual

. Here, the null hypothesis is H0 0 = , while the statistics are

)])1((ln))1((ln[2LR xn0x

0

xnx

PF

= ~ )1(2 , (14)

where 0 is the exception rate, n is the number of observations, x is the counts which shown the actual

return greater than the computed VaR ratio in the models, and nx = is the ratio of which actualreturns 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 acceptthe 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 bothrisk control and funds usage. RMSE can be written as

( )

n

VaRr

RMSE

n

1t

2

tt=

= , (15)

where rt is the actual returns and VaRt is the value computed from the model.


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Benet (1992) investigated foreign exchange futures, and suggested that if out-of-sample or ex-

ante 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 thefuture potential losses.

2.6. The Decomposition of VaR

The main reason for forming a portfolio is hoping through investing in different assets investor candiversify the markets non-systematic risk and minimize the investment risk, whereby VaR can pre-

reveal the possible loss for the portfolio. Since VaR can integrate all of the possible holding portionsfor an institution and monitor the volume of risks 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 assets VaR can be written as

iii ZwVaR = ,

where N,,1i K= , 1wN

1i

i ==

. The total risk for the portfolio is =

=N

1i

iP VaRVaR , without considering

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

assets VaR, that can be written as =

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A VaR Investigation of Currency Composition In