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APPLICATION OF VALUE-AT-RISK METHODS FOR MEASURING OF THE
FINANCIAL RISK WITHIN COMPANY
Ing. Jozef Glova, PhD.
Technical university of Košice
Faculty of Economics
Nemcovej 32, 042 00 Košice
e-mail: [email protected]
Abstract
Financial turmoil in the global financial
market has led to the extreme negative
consequences in almost all economics around the
globe. The global uncertainty in financial system
admittedly affects the firm environment in different
industries, where the manufacturing industry also
could not be omitted. This article describes possible
ways of managing the financial risk, especially
exchange rate risk, in regard to diminish this
exposure from the firm’s balance.
Key
words:
Uncertainty,
financial
risk
measurement, Value-at-Risk, exchange rate risk.
INTRODUCTION
The recent history of financial risk
measurement begins when basic bond duration
concept in 1938 was developed by Frederick
Macualay, for more detail see the reference in [3].
Except of this contribution in the field of fixedincome securities, the pioneering work and research
of Harry Max Markowitz [4], [5] in his article
Portfolio selection revolutionized finance and
significantly accelerated the application of
quantitative methods to financial analysis.
The table 1 shows the genesis of analytical
financial risk measurement techniques in 20th
century.
Table.1 The evolution of financial
measurement techniques, source: own
Year
1938
1952
1963
1966
1973
1988
1993
1994
1997
1998
1998
risk
Instrument
Bond duration
Markowitz mean-variance framework
Sharpe’s capital asset pricing model
Multiple factor models
Black-Scholes option pricing model
“Greeks”
Risk-weighted assets for banks
Value at Risk
Risk Metrics
Credit Metrics
Integration of credit and market risk
Risk budgeting
METHODOLOGY
The main goal of the empirical part of this
paper is to apply and calculate the VaR using
selected methods and then compare the results
achieved by using them.
First, we need to build a hypothetical
portfolio, for which VaR will be calculated and
analysed. In order to do this, we need information
about the assets of the portfolio and their weights.
We assume a hypothetical investor who holds a
hypothetical portfolio in foreign currencies. For
these foreign currencies historical information
about their past value is available. Information
about the weights of the portfolio is also available.
The weights are assumed to be relatively constant
over the analysed time frame.
For calculating the VaR the Delta-normal
method will be applied. The tool for performing
these simulations is Microsoft Excel. In an
overview, the similarities and the differences of the
outputs for the Delta-normal method will be
analysed. Finally, hypothetical financial advice will
be given to the hypothetical portfolio investor. This
implies using the indicator VaR for taking decisions
regarding the currency risk.
DATA DESCRIPTION
For the empirical part of this article we use
historical data for different currencies and time
frames. We consider data for five exchange rates to
Euro, namely United States Dollar (USD), Great
Britain Pound (GBP), Chinese Yuan (CHY), Swiss
Franc (CHF) and Polish Zloty (PLZ), collected at
multiple time periods. The daily observations in this
data set begin in first quarter of 2007, and end in
the second quarter of 2011. The data set contains
1122 observations. Summary descriptive statistics
of data sets applied is shown in Table.2.
Table.2 Summary statistics of particular exchange
rates of foreign currencies to €, source: own
calculations based on Bloomberg data
GBP
USD
CHF
PLZ
CHY
Mean
Standard
Error
0.8114
1.3899
1.5138
3.9093
9.7389
0.0025
0.0026
0.0036
0.0097
0.0231
Median
0.8358
1.3759
1.5182
3.9112
9.7532
Mode
Standard
Deviation
Sample
Variance
0.6789
1.4365
1.5230
3.8745
10.0141
0.0823
0.0887
0.1190
0.3249
0.7728
0.0068
0.0079
0.0142
0.1056
0.5972
-1.01
-0.452
-0.777
-0.051
-1.2441
Kurtosis
86
Skewness
-0.488
0.399
-0.598
0.134
-0.0332
Range
0.3220
0.4103
0.4332
1.6714
3.0435
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Minimum
0.6553
1.1913
1.2450
3.2038
8.1360
Maximum
0.9773
1.6016
1.6782
4.8752
11.1794
1122
1122
1122
1122
1122
Count
Our portfolio of mean position held in
balance sheet in form of account receivable, current
accounts and cash balance in foreign currencies
contains 100 Mio. €. Particular parts of assets
exposure with exchange rate risk are shown in
Table.3.
Table.3 Total sum of particular currencies in firm
portfolio, source: own
Local
currency
In Mio.
Exchange
rate to €
USD
GBP
CHY
CHF
PLZ
20
36
21
60
196
1.42
0.87
9.32
1.28
3.98
The most important property of the used
data set for our analysis is the value of the daily
return of the currencies. This is easily calculated by
using information about the actual currency value
and its value on the previous days. Therefore, the
daily returns of the currencies are the basis for
every calculation. For describing the relationships
between daily return of the currencies the Pearson
correlation coefficient will be applied for further
calculations of the risk exposure.
DETERMINATION OF VaR USING THE
DELTA-NORMAL METHOD
For calculating the VaR using the Deltanormal method we follow the next five steps [2]:
•
Step 1: Calculate the daily returns (or value
changes) of the portfolio currencies;
•
Step 2: Calculate the covariance matrix of the
currencies returns;
•
Step 3: Calculate the weights of the portfolio
assets;
Step 4: Calculate the portfolio variance;
•
•
Step 5: Calculate VaR.
In the next paragraphs we will show how
the calculation is done step by step. For simplicity
reasons we well show how to do the calculations
only for one day horizon.
Step 1: Calculation of the daily returns (or
value changes) of the portfolio currencies. The
calculation of the daily returns of all the assets in
the portfolio for the analysed time frame is being
done by subtracting the price of the foreign
from its price at the
currency at the time
. The result will be divided by the value
time
of the foreign currency at the time t.
t is the time frame, t={01.01.2007,…,19.04.2011},
c is the currency, c={USD, GBP, CHY, CHF,
PLZ},
i is the day horizon, i={1, 2, 3, 4, 5, 10}.
The calculation will be done for all the
portfolio currencies, this means for all five of them.
In addition, this calculation must be performed for
different time horizons, since it is calculated not
only on daily basis, but also on two, three, four, five
and ten days horizon. In this table we can see a
sample of the daily returns values for each
currency. The calculation was based on one day
horizon.
Step 2: Calculate the covariance matrix of
the currencies returns. This is done using the Excel
function. The necessary inputs for the calculation
are the historical daily returns which have been
calculated in the previous step using historical data.
Since we want to calculate the VaR for six different
day horizons, we need six different correlation
matrixes.
GBP
GBP
0.00004
USD
0.00002
USD
CHF
PLZ
CHY
0.00005
CHF
0.00000
0.00001
0.00002
PLZ
0.00001
-0.00002
-0.00001
0.00006
CHY
0.00001
0.00003
0.00001
-0.00001
0.00005
Figure.1 Covariance matrix of the currencies
returns for one day horizon, source: own calculation
Step 3: Calculate the weights of the
portfolio assets. For calculating these, we need
information about the amount of € hold in each
foreign currency. In addition, we divide the amount
of assets hold in that currency by the total value of
the portfolio. This means calculating the weights
for the six currencies. In this case, the total value of
the portfolio is 100 Mio. €. The proportion of assets
hold in foreign currencies is 25% GBP, 23% USD,
21% CHF, 16% PLZ, and 15% CHY. We assume
that the portfolio weights are constant for each
period.
Table.4 Weights of the portfolio, source: own
UK
25%
USA
23%
Swiss
21%
Zloty
16%
Yuan
15%
Step 4: Calculate the portfolio variance
and the standard deviation. The calculation of the
portfolio variance is done by multiplying the
portfolio weights by the covariance matrix and by
the transpose matrix of the portfolio weights. In
addition, the calculation of the standard deviation is
done by calculating the square root from the
variance. Formally, the calculation for the portfolio
deviation is done as follows:
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Table.5 Calculation of the variance and standard
deviation for the portfolio, source: own
Variance
0,000012626
Standard deviation
0,003553275
Step 5: Calculate VaR. For this, we need to
multiply the equivalent parameter for the desired
confidence level by the portfolio standard deviation
and by the portfolio value. Formally, the calculation
is done as follows [1]:
V is 100 Mio. €
is 1,28; 1,64 and 2,32 for 90%, 95%, and 99%
confidence level.
Table.6 Overview of the portfolio variance and
standard deviation for different time horizons,
source: own calculations
1 day
2 days
3 days
4 days
5 days
Variance
0.00001
0.00003
0.00004
0.00006
0.00008
Standard
deviation
0.00355
0.00531
0.00662
0.00772
0.00875
Table.7 Results of the VaR for different confidence
levels, source: own calculations
Confidence level
0,90%
0,95%
0,99%
VaR (Mio. €)
0,45537049
0,584461696
0,826615331
CONCLUSION
After a short definition and description of
financial risk measurement techniques developed in
the last decades, a hypothetical portfolio consisting
of assets in different currencies has been built and
demonstrated. The type of risk, which is relevant
for this application, is the exchange rate risk. This
means that changes of the exchange rates can affect
the future value of the portfolio in a negative way.
Based on historical data for the exchange rates time
series, important statistical key indicators have been
calculated. In addition, plausible distribution
functions for simulations have been assumed. Using
them, we were able to perform calculation using the
described VaR method; Delta-Normal method was
suggested and applied.
In conclusion, the choice of the VaR
method and its calculation depends on many
factors. On the one hand, there are legal factors,
according to them the method and the used
methodology are prescribed especially for
10 days
institutions and other companies. On the other
0.00017
hand, the calculation of the VaR is not mandatory
for many companies. Therefore, depending on the
0.01298
technical abilities of the risk managers or of the
managers, the interest and the risk policy of the
stakeholders and of the availability of the necessary
data, different VaR methods can be chosen. A
trade-off between the accuracy of the VaR
calculation and the put in effort must also be taken
in consideration.
References
After calculating the VaR for different
confidence levels and for different days horizon, we
can put all the result in one table in order to have an
overview of our work. In the following table we can
see the results of the calculations in absolute values
(Mio. €) or relative values (%).
Table.8 Overview of the absolute VaR results using
the Delta-Normal method (in per cent), source: own
calculation
Confidence
level 1 day
90% 0,46
95% 0,58
99% 0,83
Time horizon
2
3
4
days days days
0,36
0,45
1,08
0,47
0,58
1,39
0,66
0,82
1,97
5
days
0,58
0,75
1,06
Table.9 Overview of the relative VaR results using
the Delta-Normal method
Confidence level
90%
95%
99%
1 day
0,46%
0,58%
0,83%
2 days
0,36%
0,47%
0,66%
Source: Own calculations
88
Time horizon
3 days 4 days
0,45% 1,08%
0,58% 1,39%
0,82% 1,97%
5 days
0,58%
0,75%
1,06%
[1] Alexander, C.: Market Risk Analysis Volume
IV. West Sussex: John Wiley&Sons, 2008. 449 pp.
ISBN 978-0-470-99788-8.
[2] Jorion, P.: Value at risk: the new benchmark
for managing financial risk. New York : McGrawHill, 2007. 531 pp. ISBN 0-07-135502-2.
[3] Macaulay, F.: Some Theoretical Problems
Suggested by the Movements of Interest Rates,
Bond Yields and Stock Prices in the United States
since 1856. New York : NBER, 1938. 625 pp.
ISBN 0-87014-032-9. Also available on
http://www.nber.org/books/maca38-1
[4] Markowitz, H. M.: Portfolio Selection. New
York : The Journal of Finance, 1952. 7(1), pp.
77-91. ISSN
[5] Markowitz, H. M.: Portfolio Selection:
Efficient Diversification of Investments. New York
: John Wiley & Sons, 1959. 356 pp.
Contribution has been supported by
project VEGA No. 1/0897/10 "Measuring and
managing
interest
rate
risk
(IntraRiskMetrics)".