estimating value at risk on the belgrade stock exchange

8/11/2019 Estimating Value at Risk on the Belgrade Stock Exchange

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ECONOMIC ANNALS, Volume LIV, No. 183 / October December 2009.UDC: 3.33 ISSN: 0013-3264

COMMUNICATIONS

* Novi Sad and Te College o ourism, Belgrade

JEL CLASSIFICATION:C51, C52, G21

ABSTRACT: Tis paper presents market

risk evaluation for a portfolio consistingof shares that are continuously tradedon the Belgrade Stock Exchange, byapplying the Value-at-Risk model the analytical method. It describes themanner of analytical method applicationand compares the results obtained byimplementing this method at different

confidence levels. Method verification was

carried out on the basis of the failure ratethat demonstrated the confidence level forwhich this method was acceptable in viewof the given conditions.

KEY WORDS: market risk, Value-at-risk(VaR) model, analytical method, financialmarket

Milica D. Obadovi and DOI:10.2298/EKA0983119O

Mirjana M. Obadovi*AN ANALYTICAL METHOD OF

ESTIMATING VALUE-AT-RISK ON THE

BELGRADE STOCK EXCHANGE


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Economic Annals, Volume LIV, No. 183 / October December 2009.

1. INTRODUCTION

Although in modern parlance the term risk has come to mean danger o loss,finance theory defines risk as the dispersion o unexpected outcomes due tomovements in financial variables (J. P. Morgan1). A risk describes selection osuch alternatives that do not have a predetermined yield, but or which there is arecognized comparison o alternative yields with their occurrence probabilities.Tereore, a risky investment is one or which the distribution is known with alesser or higher accuracy.

Te total risk o investing in securities is the aggregate o systematic (market risk)

and non-systematic risk (specific risk). Systematic risk is a financial risk originatingas a consequence o an institutionally and structurally organized market, whereinall the participants have to ollow a certain business code. Te main elements osystematic risks are the ollowing: significant (sudden) drop in securities price,rapid expansion o securities to other markets, payment delay, banking crisis,payment system crisis etc. Systemic risk is ollowed and expressed by means omarket indexes, as indicators o average market trends and profitability. Non-systemic risk is linked to the operations o the securities issuer and depends on:the issuers financial status, their market position, attractiveness o their product

range, liquidity, solvency, etc.

VaR measures the worst expected loss over a given time interval under normalmarket conditions at a given confidence level. Based on firm scientific oundations,VaR provides users with a summary measure o market risk. VaR is a methodo assessing risk that uses standard statistical techniques routinely used in othertechnical fields(Jorion 1997: xiv). For instance, a bank might say that the dailyVaR o it trading portolio is $40 million at the 99 percent confidence level. Inother words, there is only 1 percent probability rom 100 percent, under normalmarket conditions, or a loss greater than $40 million to occur. VaR measuresrisk using the same measurement units as banks e.g. dollars. Shareholders andmanagers may thus decide whether they consider a given risk level appropriate.In case they are not comortable with the proffered risk level, the very processleading to VaR calculations may be utilized to make a decision on risk mitigation.Although it virtually always represents a loss, VaR is conventionally reported as apositive number. A negative VaR would imply the portolio has a high probabilityo making a profit, or example a one-day 5% VaR o negative $1 million implies

1 Jorion (1997: 63)


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the portolio has a 95% chance o making $1 million or more over the next day(Crouhy 2001).

It is important to emphasize the ollowing characteristics o VaR. First o all, VaRis an estimated and not a uniquely defined value. VaR evaluations will depend onthe stochastic processes that instigate random realizations o market data. Tisrequires the rearranging o past data and determining the size o the historic sampleto be used. Tere is also the question o whether more value should be attached tomore recent events than to those that occur urther in the past. Basically, the goalis to achieve the best possible evaluation o the stochastic process that generatesmarket data during the particular calendar period to which VaR evaluation is

applied. VaR does not reer to the distribution o potential losses in those rarecases when the VaR evaluation is surpassed. It is incorrect to view VaR evaluationsas worst-case scenario losses. Analysis o rare but extreme losses has to includealternative tools, such as extreme value theory or simulations made in accordancewith the worst-case scenarios o market trends.

VaR has multiple roles:

Informative role - VaR may be used to inorm the upper management on the

existing risk in the market and investment transactions.

Resource allocation role VaR may be used to set position limits or traders and tomake a decision on where to allocate limited capital resources. Te advantage oVAR is that it creates a common denominator, with which it is possible to comparerisky activities in different markets.

Performance evaluation role VaR may be used to adjust risk perormance. Tisis essential or the market environment where traders have a natural tendency to

take extra risk. For example, in its 1994 annual report, J.P. Morgan revealed that thedaily trading VaR had an average o 15 million dollars at the 95% confidence level.Tus, investors may decide whether they consider this level o risk acceptable.Prior to the publishing o such figures, investors only had a vague idea on thescope o trading activities undertaken by banks.

During the eighties, large financial institutions (Bankers rust, Chase ManhattanBank, Citibank and others) began publishing the application o VaR in riskmanagement systems. o implement this concept, a large amount o mutuallyinterchangeable data was required, which was a huge problem until the appearance


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o RiskMetrics2. It consists o detailed technical documentation, as well as thecovariance matrix or several hundred key points, which were updated daily.Since J.P. Morgans Riskmetrics were published in 1994, there has been a swifexpansion o research into VaR methodology. Although the zone o evaluating andanalyzing market risk exposure remained the main field o VaR implementation,applications were expanded to other types o risk as well. In the last ew years,many financial institutions have accepted VaR as an instrument or evaluatinginormation on their portolio positions. Te regulatory authorities o mostcountries have recognized and acknowledge the VaR approach as one o severalmethods or measuring the market risk o financial institutions. Apart rom itsconceptual appeal, its popularity was promoted by the Basel Committee, which

allowed banks to calculate their capital requirements or market risks using VaRmethodology. In order to calculate VaR, banks may choose between the historicalsimulation method, analytical method and Monte- Carlo simulation.

Tere are several ways o expressing VaR:

1. VaR may be expressed as an absolute amount or as a percentage o marketvalue. For example, VaR is 7 million dollars or VaR is 3.5% o the portoliovalue.

2. VaR may be given at an aggregate level, and may be broken down into businessunits, by risk type, by instrument type, or as a combination o the two.

3. VaR may be given at different confidence levels. As an example, J. P. MorgansRiskMetrics is used at the 95% confidence level. Te Basel Committee selectedthe 99% confidence level, which reflects the regulators wish to provide a secureand healthy financial system. Generally speaking, the levels applied in practiceare 95%, 97.5%, 99% and 99.9%. Te most used confidence levels are 97.5% or99%. When comparing VaR reports o two institutions, it is necessary to settheir confidence levels to be equal.

Te first step towards VaR measurement is to select two quantitative actors: theholding periodand the confidence level.

Te usual holding period is a day or a month, but institutions may choose otherperiods (e.g. a quarter or more), depending on their investments and reportingperiods. Te holding period may also depend on the liquidity o the markets inwhich an institution operates. All other things being equal, the ideal holding period

2 RiskMetrics is a ree service offered by JP Morgan in 1994, to promote VaR as a riskmeasurement tool


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in any market is the time required to ensure uniorm liquidation o the marketpositions. Te holding period may also be specified by regulations. Accordingto the Basel Agreement, the rules on capital adequacy demand that the internalevaluation models used to determine the minimum regulatory capital or marketrisk, must reflect a time horizon o two weeks (i.e. 10 working days). Selectiono the holding period may also depend on the ollowing actors: the assumptionthat the portolio does not change over the holding period is easier to upholdwith a shorter holding period; or model validation, so-called back testing, a shortholding period is more desirable. Reliable validation requires a large data set anda large data set requires a short holding period. Te commercial banks currentlyhave daily VaR reports due to the ast turnover o their portolios. In contrast,

investment portolios such as pension unds slowly adjust their risk exposure, andhence a one-month period is usually chosen or their investment purposes.

Te selection o confidence level depends mostly on the purpose behind the riskmeasurement. Tereore, a very high confidence level, ofen as great at 99.97%, isappropriate i used or risk measurement in order to set capital requirements orin order to achieve a low insolvency probability or high credit rating. Tis choiceshould reflect the level o a companys aversion towards risk and expenses dueto losses caused by exceeding the VaR. Te larger the aversion towards risk or

expense, the greater is the amount o capital required to cover potential losses,which leads to a higher level o confidence. On the other hand, to validate a model, itis desirable to have relatively low confidence levels, in order to obtain a reasonableproportion o observed losses. In contrast, i VaR is only used by companies tocompare the risks in various markets, then the selection o confidence level isirrelevant.

Tus, amongst other reasons, the best choice o these parameters dependson the context. It is important to make clear choices in each context and or

those choices to be completely clear throughout an institution, so that settingconstraints and making other decisions connected to risk may be made in light othis understanding.

2. GENERAL SAMPLE INFORMATION

In this section, the estimation o market risk through application o the analyticalValue-at-Risk calculation method is presented. A portolio was created, consistingo shares o 27 companies that are continuously traded at the Belgrade Stock


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Exchange. Te data analyzed belong to a yearlong period, between 22 May 2006and 21 May 2007, or shares rom the ollowing companies:

1. AGBC -Agrobaka a.d. Baka opolaBELEX2. AIKB -AIK banka a.d. Ni BELEX3. ALFA -Alfa plam a.d. Vranje BELEX4. BMBI -Bambi Banat a.d. Beograd BELEX5. BNNI - Banini a.d. Kikinda BELEX6. CCNB -aanska banka a.d. aak BELEX7. DNVG- Dunav Grocka a.d. Grocka BELEX8. ENHL- Energoprojekt holding a.d. Beograd BELEX

9. FIO - Galenika Fitofarmacija a.d. Zemun BELEX10. IMLK - Imlek a.d. Beograd, simbol BELEX11. MLNS - Novosadska mlekara a.d. Novi Sad BELEX12. MLC - Metalac a.d. Gornji Milanovac BELEX13. NPRD - Napred GP a.d. N. Beograd BELEX14. PLNM - Planum GP a.d. Beograd BELEX15. PRGS - Progres a.d. Beograd BELEX16. PLK - Pupin elecom a.d. Zemun BELEX17. PUUE - Putevi a.d. Uice BELEX

18. RDJZ - Radijator a.d. Zrenjanin BELEX19. RMBG - Ratko Mitrovi a.d. Beograd BELEX20. SJP - Soja protein a.d. Beej BELEX21. SRBL - Srbolek a.d. Beograd BELEX22. GAS - Messer ehnogas a.d. Beograd BELEX23. IGR - igar a.d. Pirot BELEX24. LKB - elefonkabl a.d. Beograd BELEX25. UNBN - Univerzal banka a.d. Beograd BELEX26. UNVR - Univerzal - holding a.d. Beograd BELEX

27. ZOPH - Zorka Pharma a.d. abac BELEXTe initial assumption was that on 22. May 2006, 10,000,000 dinars were investedin a portolio consisting o these companies shares. It was additionally assumedthat the same amount was invested into each company.

Let Vo be the initial 10,000,000 dinar investment. Since the same amount wasinvested into each o the 27 companies, it means that invested into every companywas:

10.000.000 / 27 = 370.370,370 dinars. (1)


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I p1,1

, p1,2

, ...., p1,27

are the share prices as o 22 May 2006, and q1, q

2, ...., q

27are the

numbers o purchased shares, then:

p1,1q

1,1= p

1,2q

1,2= .... = p

1,27q

1,27= 370.370,370 dinara, (2)

and qn,n

= 370.370,370 / pn,n

. (3)

Te number o purchased shares in this example does not change, i.e. it remainsconstant during the observed analysis period, so only market price fluctuationsand their impact on the portolio value are being ollowed.

When the market prices o the shares p2,1, p2,2, ....., p2,27 change, in this case, theollowing day the portolio value was reduced to 9,930,981.340 dinars. Te relativeportolio value change amounts to (9,930,981.340 10,000,000) / 10,000,000 =-0.00690 = -0.690 %, which means that a loss o 69,018.66 dinars was made.

A year later, in this case 249 working days, the portolio value amounted to24,521,567.951 dinars, i.e. a gain was made o 14,521,567.951 dinars. In this casethe investment paid off because a profit o 145.22% was made.

3. CALCULATING VAR VALUES BY APPLYING THE ANALYTICAL METHOD

Te analytical method is an approach that assumes market variables have a normaldistribution o probabilities and then uses the eatures o the normal probabilitydistribution to determine VaR. Normal (Gaussian) distribution is given by theollowing unction:

(4)

where: x independent variable, 3,14159, e 2,71828, - mean (expected)distribution value and - standard deviation.

Te normal distribution unction may also be presented graphicallywith a bell-shaped curve. Te main eature o the normal distribution is that it is symmetricaland that it may be completely determined with only two parameters, as ollows: (mean value) and (standard deviation). I these two parameters are known,

then the normal distribution is ully defined. Normal distribution is denoted with


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N(,2). Te normal distribution N(0,l) is called the standard normal distribution.Te standard normal distribution is obtained by standardizing deviations orandom variable xirom its mean value, i.e. by showing each value o the random

variable through its distance rom the mean value expressed in standard deviations.Te standard deviation o random variable x

irom its expected or mean value is

marked with zand calculated on the basis o the ormula:

(5)

where < z< +. Each value o the random variable can be standardized usingthe previous ormula, obtaining the specific value z

i. For each z

iit is possible to

calculate the probability o random variable xi taking a value that is smaller orlarger than this value afer standardization. Te probability o z

itaking a value that

is smaller thanymay be represented in the ollowing manner:

P(zi


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Table 1.Calculation of VaR for different confidence levels

Confidence levels VaR 90% -1,28**V95% -1,65**V98% -2,05**V99% -2,33**V

Te largest problem in applying this method is calculating the standard deviationo the portolio value change. Once this value is calculated, VaR is obtained as theproduct o standard deviation , the specific value o parameter zand portolio

value V.

Te standard deviation is calculated using a covariance matrix, obtained on thebasis o ndays rate o assets return, in the ollowing manner:

, (8)

where w = (w1, w

2, ... , w

n) is the weighted portolio vector (nominal amounts

invested into each asset), and V is covariance matrix, obtained on the basis o

ndays rate o assets return. In this example the same amount is invested intoeach asset, so there is 3.7% initial portolio value invested into each asset. Tecovariance matrix Vlooks as ollows:

, (9)

where Var(Rn)is variance o assets return rate n, and Covar(R

n,R

m)is covariance

between assets return rates n and m, with Covar(Rn,R

m)= Covar(R

m,R

n).

Subsequently, the standard portolio deviation is calculated as the square root oportolio variance, in the ollowing manner:


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, (10)

where:

. (11)

o calculate the covariance matrix in this example, the method o equally weightedhistorical data was applied, as well as the EWMA (exponentially weighted movingaverage) method, and a decay parameter = 0.94 (which suggests how muchthe value o each observation decreases day by day). According to the equallyweighted historical data method, volatility is calculated as ollows:

, (12)

whereas the relevant ormula or volatility calculation according to EWMAmethod is:

, (13)

where rtand r

mare individual asset return rates and mean value o assets return

rate, respectively. is the number o observed daily rates o portolio value change( = 249).

In this example VaR is calculated or confidence levels o 90%, 95% and 99%, aswell as or one-day and ten-days prediction periods (able 2, Figure 1, Figure 2,Figure 3), by applying the basic analytical method and the ollowing results wereobtained:


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Table 2.VaR values calculatedfor the local market using the basic method

1-day 10-daysVaR (90%) 1.0338% 3.27%VaR (95%) 1.4346% 4.54%VaR (99%) 2.1866% 6.91%

Te broken line in the figures 1 - 6 represents data on 88 consecutive VaR values,estimated on the basis o data on 249 value changes or the portolio consisting o27 local shares.

Te black line in the figures 1 - 6 represents data on 88 consecutive actual portolioreturns to which VaR evaluations are compared.

Figure 1.Graph o VaR values obtained by the analytical method, or a local sharesportolio with the error risk o 0.01 and = 1


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In the ollowing example, VaR values are calculated or confidence levels o 90%,95% and 99%, or one-day and ten-day prediction periods (able 3, Figure 4,Figure 5, Figure 6), by applying the analytical EWMA method and the ollowingresults were obtained:

Table 3.VaR values calculatedfor the local market using EWMA method

1-day 10-daysVaR (90%) 1.9833% 6.27%VaR (95%) 2.7439% 8.68%VaR (99%) 4.1707% 13.19%

Figure 4.Graph o VaR values obtained by the analytical method, or a local sharesportolio with the error risk o 0.01 and = 0.94


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MODEL VERIFICATION BASED ON THE FAILURE RATE

Te above demonstrates how to evaluate the basic parameters or VaRmeasurement, mean value, standard deviation and quintile, based on actual data.Tese evaluations should not be taken or granted and the accuracy o a modelshould be verified.

Te simplest way o veriying model accuracy is to record the failure rate, whichgives the proportion o how many times VaR has exceeded the expectations in agiven sample.

In order to determine whether these models predict risks well, VaR is evaluated or88 consecutive days, or a local shares portolio. Te ollowing tables demonstratehow many times the loss was larger than predicted by VaR.

Table 4.Review of the calculated values how many times the actual loss exceededthe previous day VaR for a portfolio consisting of local shares using theanalytical method ( = 88)

analytical method = 1 = 0.94p = 0.01 N = 4 N = 2

p = 0.05 N = 5 N = 6p = 0.1 N = 7 N = 9

where total number o consecutive VaR predictions, and N number showinghow many times the actual loss exceeded the VaR rom the previous day.

Subsequently, at the given confidence level, we need to know whether N is toosmall or too large, under the null hypothesis that p is a true probability. Once theailure rate is calculated as N/ and compared to the lef tail probability, e.g. p =

0.01, which is used to determine VaR evaluation, i they match the VaR evaluationwas correct, and i they differ significantly the model has to be rejected.

For the sake o illustration, the confidence level is set at 95 percent. Tis numberdoes not reer to the quantitative levelpthat was selected as the VaR, which mightbep= 0.01 or instance. Tis confidence level reers to the decision on whether toreject the model or not. It is generally set at 95 percent because this correspondsto two standard deviations under a normal distribution.


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Kupiec (1995) developed the confidence regions or such a test. Tese regions aredefined by the tail points o likelihood ratio:

LR= -2ln [(1 - p) - NpN] + 2ln [(1 (N/) - N(N/)N], (14)

which is distributed through 2test with one degree o reedom under the nullhypothesis that p is a true probability.

For example, with data or the portolio consisting o local shares ( = 88) it couldbe expected that N = p = 10% 88 = 8.8 deviations will be observed. However,the regulator will not be able to reject the null hypothesis as long as N is within

the confidence interval [2 < N < 15]. Values o N greater or equal to 15 suggestthat VaR model represents the probability o large losses lower than they actuallyare; values o N that are less than or equal to 2 suggest the VaR model is tooconservative. Te ollowing table shows the regions in which the model is notrejected on the basis o significance = 0.05.

Table 5. Model verification: Regions in which the model is not rejected at thesignificance level of 0.05

Portolio consisting o local shares ( = 88)

p = 0.01 N


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Te ollowing table presents whether a model is acceptable or not, on the basis oKupiec likelihood ratio.

Table 6.Model verification for a portfolio consisting of local shares using theanalytical method ( = 88)

analytical method = 1 = 0.94p = 0.01 rejected acceptedp = 0.05 accepted acceptedp = 0.1 accepted accepted

5. CONCLUSIONS

Analytical approaches provide the simplest and most easily implementedmethods to estimate VaR. Tey rely on parameter estimates based on market datahistories that can be obtained rom commercial suppliers or gathered internallyas part o the daily mark-to-market process. For active markets, vendors such asRiskMetrics supply updated estimates o the volatility and correlation parametersthemselves. But while simple and practical as rough approximations, analytic

VaR estimates also have shortcomings. Perhaps the most important o these isthat many parametric VaR applications are based on the assumption that marketdata changes are normally distributed, and this assumption is seldom correct inpractice. Assuming normality when our data are heavy-tailed can lead to majorerrors in our estimates o VaR. VaR will be underestimated at relatively highconfidence levels and overestimated at relatively low confidence levels.

Market value sensitivities ofen are not stable as market conditions change.Since VaR is ofen based on airly rare, and hence airly large, changes in marketconditions, even modest instability o the value sensitivities can result in major

distortions in the VaR estimate. Such distortions are magnified when optionsare a significant component o the positions being evaluated, since market valuesensitivities are especially unstable in this situation.

Analytic VaR is particularly inappropriate when there are discontinuous payoffsin the portolio. Tis is typical o transactions like range floaters and certain typeso barrier options.

In summary, analytic approaches provide a reasonable starting point or deriving

VaR estimates, but should not be pushed too hard. Tey may be acceptable on a


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long-term basis i the risks involved are small relative to a firms total capital oraggregate risk appetite, but as the magnitude o risk increases, and as positionsbecome more complex, and especially more nonlinear, more sophisticatedapproaches are necessary to provide reliable VaR estimates (Koenig 2004: 81).

Using the analytical method on the portolio consisting o shares traded in thelocal market, it has been shown under null hypothesis that p = 0.05 is an accurateprobability, based on the Kupiec likelihood ratio distributed through 2 testwith one degree o reedom, so this method is accepted in all cases except whenapplying the basic analytical method to the portolio at the confidence level o99%, wherein this model shows the probability o larger losses to be smaller than

it actually is.

Applying this method in local market conditions is risky at a confidence level o99%, because that level corresponds to extremely rare events. Tis explains whysome banks preer lower confidence levels, in order to be able to observe a sufficientnumber o deviations or the purpose o model validation. A multiplication actoris thus applied to transorm VaR into secure capital to be set aside or protectionrom market risk.

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Bluhm, C., L. Overbeck and C. Wagner (2001),An introduction to credit risk modeling, Chapman &

Hall, Boca Raton, Florida.

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estimating value at risk on the belgrade stock exchange

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