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THEORY 32 VALUE-AT- METHODS AND MODELS...

VALUE-AT-RISK METHODS AND MODELS AND THEIR APPLICATION

doc. Ing. Tatiana Varcholová, CSc., Ing. Marián Rimarčík University of in Bratislava, Faculty of Business Economics in Košice

The concept of Value-at-Risk (VaR) was used for loss in the case that the would have to be the first time by large financial institutions at the end held for a certain fixed period according to past of the eighties for measuring in portfolios. This experience with a certain rate of certainty (as a rule period was characterised by huge exchange-rate 95 or 99%). In other words, if the future is like the volatility and rapid growth in the use of derivatives past, the volume of loss estimated by the risk models useful for managing currency and -rate risks. will occur every 20 days or every 100 days, depen- Modern derivatives such as forwards, future swaps ding on the degree of certainty set, naturally only if and options assist in managing exchange-rate and the company does not manage to take preventive interest-rate volatility. Since these times there has measures in order to reduce such a loss. occurred a boom in the use of VaR, which has cea- At present various methods exist for determining sed to be merely a matter of internal interest to finan- VaR, the most important of which are considered to cial institutions - regulatory authorities have begun to be: the - method (also termed the take an interest in them too. In April 1995 the Basel analytical method), the historical method Banking Supervision Committee proposed that and the Monte Carlo simulation method. in calculating their capital adequacy be ena- The variance-covariance method allows an esti- bled to apply their own VaR models for mate to be made of the potential future losses of a with the use of certain parameters determined by the portfolio through using on volatile values in Committee. Similar measures were implemented in the past and correlations between changes in their the same year also in the USA: by the Federal values. Volatilities and correlations of risk factors are Reserve Fund and the Securities and Exchange calculated for a selected period of holding the port- Commission. The European Commission directive folio and the historical period through using historical on capital adequacy, which has been in force since data. VaR derives from the distribution of the proba- 1996 has allowed for the use of VaR models for cal- bility of risk factors of change in the value of the port- culating capital adequacy for positions held in foreign folio, where the simplest models envisage a normal currencies, where this counted on a rapid expansion distribution of risk factors and their stable correlation. in the use of VaR also for calculating capital needs In using this method it is necessary to take into con- for other market risks. sideration in particular the fact that: • movements in market prices do not have always VaR methods a normal distribution -they exhibit so-called heavy tails, which means a tendency to have a relatively Financial models include a number of ways for more frequent occurrence of extreme values than expressing risk, which may be used to measure the does a normal distribution, market risk of a , an investment portfolio or • models may not appropriately depict market risk financial instruments. The most important ways inc- ensuing from extraordinary events, lude: variance of past yields, standard deviation of • the past is not always a good guide to the future, yields for a certain previous period, maximum loss in for example correlation forecasts may founder. a set of possible scenarios, the Monte Carlo compu- The historical simulation method is based on ter simulation and VaR models. Standard VaR data of losses that a bank would have suffered in a models include all types of market risks, i.e. interest, given portfolio in a past period. This method is sim- stock, commodity and currency risks. Successful pler compared to the previous, since it does not world banks presently use VaR models also for mea- require demanding work breaking down the probabi- suring . The further development of the lity of risk factors and determining correlations bet- application of these models envisages their expansi- ween risk factors. The disadvantage is the need for a on also for measuring operational and legal risks. sufficient quantity of historical . VaR models allow an estimate to be made of the The Monte Carlo method is founded on the gene- value of risk in a portfolio as the maximum volume of ration of a large number of simulations (scenarios) of

BIATEC, Volume XI, 10/2003 THEORY VALUE-AT-RISK METHODS AND MODELS... 33 the development of a portfolio's value. Each simula- the contract is (approx.) zero. On the day following tion is created through a combination of randomly the concluding of the contract three market factors generated values of risk factors from their probabili- (two interest rates and the koruna’s spot exchange ty distribution. The result of the simulations is the rate) change, whereby the value of the contract also generation of probability estimates of the VaR. changes. A depreciation of the koruna, a fall in the It is important to mention that all models of mea- euro interest rate and a rise in the koruna interest suring risk are based on different mathematical rate increase the contract’s value. models and statistical relationships with varying The question is, what value will the forward con- degrees of reliability. Therefore, at present there are tract have tomorrow, i.e. 12 February 2003? If the no preconditions for the definition of a single “cor- changes of the three market factors are favourable, rect” model of measuring risk. It is however important the value will increase, i.e. the firm will record a pro- to have an understanding of the overall characteris- fit, and in the opposite case the firm will record a tics and properties of the models as well as their loss. For calculating the VaR of a contract, which we demands in terms of data input, since the correct- define as the fifth (or first) of the distributi- ness of determining the VaR and quality of manage- on of possible profits and losses, we use the histori- ment decisions thus adopted depends also on the cal percentage daily changes of the three factors correctness of input, for example, on the types of recorded for the preceding 200 days. The current as well as the parameters of value of the forward contract is recalculated 200 these distributions. times using the values of the factors of 11 February The use of information technology now provides 2003 and using the daily percentage changes of the almost unlimited possibilities for programming ever factors over the period from 7 May 2002 to 11 Feb- more perfect models. Since many banks have only ruary 2003. The VaR value is then determined from recently moved toward using special programming the distribution of calculated profits and losses as systems for managing risk, an assessment has not the fifth (or first percentile). yet been made of the results of new information The VaR defined as the fifth (first) percentile cor- technology applications in this field. Those program- responds to a loss of 23 325 (39 859) koruna ming systems which allow an increase in the quality of in the framework of financial Fig. 1 Distribution of possible profits and losses on a for- ward contract institutions’ integrated information systems are beco- ming the centre of attention among managers.

20 % Application of historical simulation

We will demonstrate the use of historical simulati- 15 % on in the example of a forward contract, which a company concluded with a bank on 11 February 10 % 2003 and by which one month later its secures the transfer of EUR 100 000, undertaking to pay SKK 5 % 42.16 per EUR, i.e. SKK 4 216 000. The present value of the forward contract is calculated according 0 % Profit to the formula:

EUR 100 000 EUR 100 000 Existing VaR models are able to calculate a value PV =SSKK/EUR . –––––––––––– – 42.16 . –––––––––––– 1 + rEUR (1/12) 1 + rSKK (1/12) at risk; often however they are not able to correctly estimate the development of risk factors in particular Where: in newly-emerging markets. If financial markets do

SSKK/EUR –Spot exchange rate of the koruna not behave in accordance with modelled results, rEUR – LIBOR euro interest rate then the abnormality of the market's movements is rSKK – BRIBOR koruna interest rate not in itself a problem, the problem however is ove- restimating the reliability of models founded on the Since the forward exchange rate is set so that assumption that these markets will behave in accor- arbitrage is not possible, then on the day of conclu- dance with past experience. Examining the causes ding the contract, and in the case of the exchange of “global” losses leads us to the fact that for an effi- rates 42.03 SKK/EUR, BRIBOR equalling 6.42% cient evaluation of new, in particular qualitative, com- and LIBOR equalling 2.808%, the current value of ponents of in current conditions other

BIATEC, Volume XI, 10/2003 THEORY 34 VALUE-AT-RISK METHODS AND MODELS...

instruments are necessary. The developing appro- risks, apply modern models for measuring it and ach to risk management supplements a set of mea- effectively use modern information technology, can surable risks with a portfolio of less quantifiable dan- gain a competitive edge and thereby also profit. Risk gers, which exceeds the boundaries of VaR models. management is currently taking on new dimensions. The developing approach to risk management cros- ses the boundaries of traditional risk measurement, Conclusion therefore new more effective instruments are requi- red, which will be ever more difficult to handle. One of the paths to success for a financial institu- Objective knowledge and conclusions nevertheless tion is risk management founded upon an evaluation remain essential, primarily with regard to seeking of a set of measurable and non-measurable risks ever more modern approaches to financial risk through using modern information technology. Only management appropriate to the ongoing globalisati- those financial institutions that best classify possible on changes in society.

BIATEC, Volume XI, 10/2003