Ways of quantifying market risk used in financial practice.

Armeanu, Daniel ; Vintila, Georgeta ; Nedelescu, Mihai 等

1. INTRODUCTION

Value at Risk measures the largest loss an institution can expect in an established time interval in normal market conditions with a given level of trust. This risk is estimated with the help of statistic and simulations methods designed with the scope of acquiring the volatility of assets in the company's portfolio.

From the specialty literature 5 methods of calculus arise: the delta-normal method (known under the name of parametric method, due to the work hypothesis of normal distribution or the variance-covariance method), delta-gamma method (Greek method), historical simulation method, testing external conditions method (scenario analysis) and the Monte Carlo simulations method (Berkowitz & Brien, 2001).

In this study, using the variance-covariance method, we follow the application of VaR method for measuring the risk inherent to a currency portfolio owned by a bank.

2. MARKET RISK IN ROMANIAN BANKING SISTEM

The currency risk represents the risk of loss due to an adverse change of exchange rate compared to the base currency. It includes the risk of term transactions--comes from the alteration of exchange rate of term contracts and volatility risk--appears due to the alteration of exchange rate's volatility. The instruments used regarding currency risk management are Currency position and VaR (Jorion 2001).

"Position risk" is known as a market risk or price risk. This represents risk as position opened to generate loss due to the market variation in the negative sense. So long as a position stays open, there is the risk of market prices fluctuating in the negative sense and transforming a profitable position in a loss or a loss in an even bigger loss (Hull 2006).

The bank's currency position is the net cash-flow (long or short) denominated in currency, without taking into consideration claim day. The currency net position includes both spot transactions which have not yet matured the net currency position of term transactions, if there is any as well as their cash in currency.

Value at risk for n positions (for the entire portfolio) is (Penza & Bansal, 2000):

The portfolio's VaR represents the maximum loss determined by the market variation. The period of detaining represents the time interval in which a position can be liquidated and multiplies the value of VaR with "own period" which means that, the longer the period the higher the risk. Trust interval used is 99% and a period of 10 days is considered. Thus, the size of the risk the bank takes through the respective exposing cannot be covered in less than 10 days of transactions. We will use a history of the market evolution for a period of 24 months in order to estimate similar evolutions of markets in the future.

The long and short positions for each currency owned in a Romanian commercial bank's portfolio on the 30th of June 2009 are presented in the table 1.

We take into consideration the analysis period of 2nd of July 2007--29th of June 2009 in order to construct the data base which contains the daily exchange rates of each currency owned in the bank's portfolio in order to compute the historical volatility and the correlation coefficient between exchange rates. We use this 24 months time frame in order to highlight the scenarios regarding possible modifications of exchange rates and in order to compute daily VaR for the given portfolio. The daily profitability at moment t(Rit) is expressed in continuous form (logarithmic) following the formula:

Rit = LN [C.sub.t]/[C.sub.t-1] (2)

We compute the daily profitability's in a continuous time and for each of the 1 currencies owned in the portfolio we will have a number of T=506 profitability's because the number of working days in the analyzed period is 507 days and the calculus of daily profitability begins with the 2nd day from the time interval chosen, more specific the 3rd of July 2007. The value of each position expressed in currency it is transformed to the RON equivalent at NBR exchange rate for the day we compute the VaR and the values obtained are presented in table 2.

The net position in RON on the total portfolio is computed as a difference between total long position and total short position, this being-7.620.098. The calculus of volatility for each currency in the portfolio is done with the help of the relation which explains the average square deviation:

[[sigma].sub.i] = [square root of [T.summation over (t=1)] [([R.sub.i,t] - [[bar.R].sub.i]).sup.2]/T - 1 (3)

[[bar.R].sub.i] = [T.summation over (t-1)] [R.sub.it]/T (4)

The correlation coefficients for each two currencies in the portfolio and then building the correlation matrix for all currencies in the portfolio are based on the following formulas:

[[rho].sub.ij] = [[sigma].sub.ij]/[[sigma].sub.i] x [[sigma].sub.j] (5)

[[sigma].sub.ij] = [T.summation over (t=1)]([R.sub.it] - [[bar.R].sub.i]([R.sub.jt] - [[bar.R].sub.j])/T - 1 (6)

Daily VaR for each currency in the portfolio is computed using the formula: (Armeanu & Balu 2007)

[VaR.sub.i] = -[W.sub.i,0] x [alpha] x [[sigma].sub.i] (7)

Where: [W.sub.i0] = the net currency position for the currency taken into consideration in RON equivalent, [alpha] = trust coefficient ([alpha] = 2.33 for a probability of 99%), [[sigma].sub.i] = daily volatility of currency i (table 3).

The daily VaR for the entire currency portfolio is computed using a formula similar to the one of risk for a portfolio, the difference being the fact that this time the average square deviation indicator ([sigma]) is substituted through the VaR indicator. The formula of VaR for a portfolio is (the weights are taken into consideration in the calculus of individual VaR for each currency position):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

VaRp = [V.sup.t] x [PSI] X V (9)

Where: V = the vector of VaR individual values of the currencies in the portfolio, [OMEGA] = the matrix of correlation coefficients between daily profit abilities of currencies in the portfolio.

For our portfolio the VaR will be: VaRp = 117.857 RON Value of VaR for the chosen portfolio in a chosen time horizon "h" is:

[VaR.sub.P ,h] = [VaR.sub.p] [square root of (h)] (10)

where [VaR.sub.p] = is VaR computed for a time horizon of 1 day, for example, for an interval of 10 days the value of VaR will be [VaR.sub.p], 10 = 372.696 RON.

Because of national currency's exchange rate variation, the value of the portfolio differs from one day to another registering a loss or profit and through VaR methodology it was determined what the maximum loss will be for a portfolio with a probability of 99%. After we have computed the intrinsic VaR for each currency in the portfolio we obtained the VaR of 117.857 RON of the portfolio of 11 currencies during one day, with a probability of 99%, this meaning the in the next day (30th of June 2009) the value of the portfolio is going to diminish with 117.857 at most (maximum loss) compared to its current value (29th of June 2009) with chances of 99%.

For the established time frame, respectively 10 days we have a VaR of 372.696 RON which means that this is the size of the risk the bank takes through that expose which cannot be covered in less than 10 days of transactions for a net value of the portfolio of 7.620.098 RON.

3. CONCLUSION

The VaR methodology is especially important both for banking institutions as well as for the other investors because it allows the identification of maximum loss registered by the value of the portfolio of financial assets, which can appear in the following period with a certain pre-established probability. In times of economic crisis this way of measuring an investor's exposure on the financial market can be adjusted in the sense that the correlation coefficients between assts are no longer computed and it is considered that their value is 1, so that the biggest possible loss of the investor at a certain level of trust can be identified. This calculus variant for the VaR presented in the study done on the currency market can be also used for the portfolio of titles constituted on the capital market.

4. REFERENCES

Armeanu, D. & Balu, F.O. (2007). VaR Methodology Application for Banking Currency Portfolios, Theoretical and Applied Economic, No. 2(507)/2007, pp 83-93, ISSN 1841-8678

Berkowitz, J. & Brien, J. (2001). How Accurate are Value-at-Risk Models at Commercial Banks?, Graduate School of Management Division of Research and Statistics University of California, Irvine Federal Reserve Board, 2001

Hull, J. (2006). Risk Management and Financial Institutions, John Wiley & Sons

Jordon, P. (2001). Value-at-Risk: the New Benchmark for Controlling Market Risk, McGraw-Hill

Penza, P. & Bansal, V. (2000). Measuring Market Risk with Value at Risk, John Wiley & Sons

Tab. 1. Portfolio of currency owned by the bank

Currency Open position NBR exchange rate

 EUR 875.693,30 4,2067
 USD 161.509,77 2,9792
 GBP 201.375,65 4,9485
 SEK 542.136,36 0,388
 CHF 292.781,21 2,7579
 DKK 522.775,18 0,5649
 JPY 3.211.241,00 0,031095
 AUD 138.975,12 2,4208
 CAD 293.417,55 2,5844
 NOK 188.259,72 0,4653
 HUF -8.857.695,93 0,01543

Tab. 2. Currency positions in RON

Currency Open position NBR exchange Long Short
 rate position position
 in RON in RON

 EUR 875.693,3 4,20670 0 3.683.779
 USD 161.509,7 2,97920 0 481.170
 GBP 201.375,6 4,94850 0 996.507
 SEK 542.136,3 0,38800 0 210.349
 CHF 292.781,2 2,75790 0 807.461
 DKK 522.775,1 0,56490 0 295.316
 JPY 3.211.241,0 0,03110 0 99.854
 AUD 138.975,1 2,42080 0 336.431
 CAD 293.417,5 2,58440 0 758.308
 NOK 188.259,7 0,46530 0 87.597
 HUF -8.857.695,9 0,01543 136.674 0
 TOTAL 136.674 7.756.772

Tab. 3. Maximum individual losses

Currency Open net Average Daily VaR
 position (RON square 99%
 equivalent) deviation

 CAD -758.308,32 0,97% -17.146,38
 GBP -996.507,40 0,98% -22.689,76
 JPY -99.853,54 1,59% -3.697,80
 USD -481.169,91 1,11% -12.456,30
 EUR -3.683.779,01 0,62% -53.371,24
 AUD -336.430,97 1,11% -8.647,81
 DKK -295.315,70 0,62% -4261,728482
 NOK -87.597,25 0,84% -1713,866843
 SEK -210.348,91 0,77% -3.777,27
 HUF 136.674,25 0,91% 2907,122558
 CHF -807.461,30 0,93% -17.559,51