DETERMINATION OF CREDIT RISK BY THE USE OF CREDITRISK + MODEL

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1 DETERMINTION OF CREDIT RISK BY THE USE OF CREDITRISK + MODEL Kataría Kočišová, Mária Mišaková INTRODUCTION CreditRisk + is the method for the calculatig the distributio of of potetial credit losses of the portfolio which was developed ad published by Credit Suisse i This method ca be used to determiig of credit risk i retail ad also i corporate sector, it meas loas, derivatives ad also marketable bods. This method is based o portfolio approach to modellig risk of default, it is cosiderig iformatio relatig to size ad maturity of the istrumet, credit quality (credibility) ad the systematic risk of the borrower (the systematic risk is the risk arisig from overall ecoomic developmet affectig all subects. CreditRisk + has oe special problem regardig the aggregatio of portfolio risk. It is the oly model whose authors ited to avoid computer simulatios to calculate portfolio risk ad attai a aalytical solutio for the portfolio loss distributio. For this reaso, the authors choose a Poisso approximatio of the distributio of the umber of defaultig credits i a portfolio segmet. s a cosequece each segmet. s a cosequece each segmet cotais a ifiitive umber of credits [1]. This hidde assumptio may lead to a sigificat overestimatio of risk i small segmets, e.g. whe the segmet of very large exposures i a bak portfolio is cosidered that is usually quite small. Thus, CreditRisk + is particularly suited for very large ad homogeous portfolios. However, at high percetiles, the reported portfolio losses eve always exceed the total portfolio exposure [6]. I CreditRisk + systematic risk factors are modelled as hidde variables that iduce cliets s default probabilities to be gamma distributed with a give mea ad variace. I order to be able to compute the portfolio loss distributio aalytically, the authors of the model assume that systematic risk factors oly refer to the cliets i a specific sector while risk factors of differet sectors are idepedet by suppositio. Note that this presumptio implies that cliets i differet sectors are idepedet as well, a problematic cocealed structural decisio for a portfolio model. Oly cliets who are at least partially represeted i the same idustrial sectors appear to be depedet i their default behaviour [8]. CreditRisk + is statistical model of credit risk of default which does ot create assumptios about causes of default. This approach is similar as approach which is used to market risk maagemet (there is o effort to modellig of causes of chages i market prices. CreditRisk + assumes that defaults are occurred i sequece of evets so that caot be predict exact time of their appearace ad or their umber. To model the radomess of borrower default i the model are used mathematical methods that are ofte used i isurace idustry. We expect portfolio with may idividual risk with low probability of occurrece. It meas that CreditRisk + is aalytical model which allowig quick ad explicit calculatio of total portfolio loss distributio. Model is based o value of property ad bods is essetially prospective ad is determied by expected future of idividual debtor from the perspective of ivestors. It coects actual credibility of debtors ad their expected future developmet. Therefore it ca be assumed that default rate varies cotiuously. Model CreditRisk + cosiders default rate as cotiuous radom variables. Stadard deviatios ca be sigificat compared with default rates, it reflects real fluctuatio of ecoomic cycles. I practise we do ot have idividual default rates of idividual debtors, appropriate method for determie default rates is for example assigmet of default probability accordig to credit ratigs. Exteral factors for example state of the ecoomy ca caused correlatio betwee idividual defaults although there is o causal relatioship betwee them. Effects of these factors are processed ito the model CreditRisk + usig the volatilities of default rates ad aalysis of sectors istead of usig correlatio of default as a direct iput ito the model [3]. Tab. 1: Compariso of some curret model of measuremet credit risk CreditMetrics CreditRisk * KMV Model developer J.P. Morga Credit Suisse KMV Defiitio of risk Market value of assets Losses from state of Losses from state of Source of risk ssets valued o the basis of market value Probability of default ad default rates Value of assets

2 States Chage of Default Cotiuous rate of ratig/default probability Probability Ucoditioal Ucoditioal Coditioal Volatility Costat Variable Variable Correlatio of risk Calculated from mutual Calculated from Calculated from mutual factors movemet of assets process resp. state of movemet of assets Rate of recovery Radom Costat withi a Radom certai bad Model desig Simulatio/aalytical alytical alytical Source: Ow processig 1 BSIC MODEL Every model of calculatig credit risk is depedet o iput data which quality directly affects the accuracy of result. Model CreditRisk + requires followig iput data [3]: exposure, default rates of idividual debtors, volatility of idividual default rates, default rate of retur. It is assumed: for a loa, the probability of default the term is the same for ay equally time horizo, for large umber of debtors, the probability of default of idividual debtor is low ad umber of default which are appeared i a give time period is idepedet o amout of default i the past time horizos. Uder these assumptios, the probability distributio of the umber of default durig give time period well represeted by a Poisso distributio with parameter µ: e P ( ), pre 0,1,2,..., (1) where: - umber of default, µ - average umber of default for oe year period, P - where P is probability of default of debtor. Obviously, i portfolio is usually fial quatity of bods, so Poisso distributio which specifies the probability of default for ifiite amout of default, oly by approximatig the distributio of default. If umber of debtors are large eough, the probability is egligible that the umber of defaults exceed umber of debtors. If we assume Poisso distributio of default umber, we expect that stadard deviatio of default rate will be approximated by square root of the mea default rate. I fact, we ca observe that Poisso distributio uderestimates the probability of default for all ratig grades. This is due to variability of itesity of default i the time which is modelled as a fuctio of chagig the selected risk factors. If the average umber of default is stochastic ature ad has Gamma distributio with parameters µ ad ca be represeted by Poisso distributio. 1.1 DISTRIBUTION OF PORTFOLIO LOSSES Derivatio of idividual asset risk is based o the calculatio of expected loss. To derive the probability of losses well diversified portfolio, the losses are divided ito groups accordig to the size of losses. Each group cotais debtors with the same credit risk ad is cosidered as idepedet portfolio of bods with followig markig [4]: debtor, L possible loss, P probability of debtor default (), - expected loss, V possible loss i the group, - expected loss i the group,

3 - expected umber of default i the group. With portfolio of bods each group of debtors ca be worked as a uchagig portfolio. Possible loss of each debtor i the group ca be obtaied as v. L. Uder the defiitio we get: v. (2) L. P (3) Mark expected loss of debtor, it meas L the is expected loss i horizo of oe year i the group expressed by the amout of expected losses of all debtors i the group, it meas. : v v Expected umber of defaults i horizo of oe year i the group is: v v v (4) : v v : v v Derivatio of distributio the probability of losses for the whole portfolio cosists of several steps. 1. Derivatio of probabilistic geeratig fuctio for every group v (5) 0 0 G ( z) P( loss L) z P( defaults) z Because we assume that the umber of default is govered by the Poisso distributio. We ca derive: e v v (z) exp (6) 0 G z z 2. Derivatio of probabilistic geeratig fuctio for all portfolio ssumig idepedece of each group is probabilistic geeratig fuctio for whole portfolio: m m m v v G( z) exp z exp z Where deotes expected umber of default for the whole portfolio (7) m. 3. Derivatio of losses probability for all portfolio From probabilistic geeratig fuctio for whole portfolio, we ca derive distributio probability of losses as: CONCLUSION 1 d G( z) P( loss from L) for 1,2,... (8) dz z0 Model CreditRisk + is simple ad easy implemet model for calculatio of expected losses i a state of default. CreditRisk + is suitable model for calucate of credit risk of homogeeous portfolio cosistig of a large umber of debtors with low probability of default. It is based o Poisso approximatio of idividual default. Disadvatage of this model is that it does ot ivolve the risk of dowgrade. The model is i cotrast to the method CreditMetrics model aims to determie of volume of veture capital assets, estimated distributio of expected losses ad values i risk. Ulike KMV model, this method does ot cocetrate o relative risk of default to capital structure of the compay. The model does ot use Mote Carlo simulatio therefore outputs are fully coditioed to iput data. Probabilistic distributio of portfolio losses ca be derived from probabilistic geeratig fuctio with umerically stable algorithm. d advatage of CreditRisk + is that it requires a limited amout of data as iputs (basically oly idividual exposures ad default probabilities), ad the computatio of the loa loss is rather easy to 1

4 perform. limitatio of the model is that a lot of ambiguity surrouds the specificatio of the default rates for idividual obligors, which are actually basic iputs of the method. I CreditRisk +, obligors are ot assiged to ratig classes, ad their characteristics do ot determie these default rates. It is implicitly assumed that baks kow these probabilities ad their volatilities, but a cocrete method to derive them is ot offered. other limitatio is that the model does ot assume market risks [5]. There are also some limitatios to CreditRisk +. O a fier scale tha default or survival, a chage i the credit quality of a obligor that is captured as a trasitio of its iteral or exteral ratig is ot reflected. Further, we metio the determiistic descriptio of recoveries ad the fact that large loss probabilities may lead to a distortio of the loss distributio due to multiple defaults arisig from the Poisso approximatio. O the other had, however, more sophisticated models typically require more statistical iput iformatio, which i practice is ofte hard to idetify. REFERENCES [1] CISKO, Š., KLIEŠTIK, T. Fiačý maažmet podiku I. Žilia: EDIS, ISBN [2] CISKO, Š., KLIEŠTIK, T. Fiačý maažmet podiku II. Žilia: EDIS, ISBN [3] CREDIT SUISSE FIRST BOSTON INTERNTIONL. CreditRisk + [olie]. Lodo, vailable from: [4] CROUHY, M., GLI, D., MRK, R. Comparative alysis of Curret Credit Risk Models. Joural of Bakig & Fiace 24, 2000, pp ISSN [5] DERVIŠ,., KDLČÁKOVÁ, N. Methodological problems of quatitative credit risk modelig i the czech ecoomy. 2001, Prague: Czech Natioal Bak Workig Paper Series, No. 39. [6] GORDY, M. comparative aatomy of credit risk models. Joural of Bakig & Fiace 24, 2000, pp ISSN [7] HF, H., REISS, O., SCHOENMKERS, J. Numerically stable computatio of CreditRisk +, CreditRisk + i the Bakig Idustry, pp [8] RUDIGER, F, MCNEIL,., NYFELER, M. Modellig depedet defaults, Workig Papper. Swiss Bakig Istitute ad ETH Zurich [9] SMEJKL, V., RIS, K. Řízeí rizik. Praha: Grada Publishig, ISBN uthors address: Kataría, Kočišová, Ig. Žiliská uiverzita v Žilia, Fakulta prevádzky a ekoomiky dopravy a spoov, Katedra ekoomiky kataria.kocisova@fpedas.uiza.sk Mária, Mišaková, Ig. Žiliská uiverzita v Žilia, Fakulta prevádzky a ekoomiky dopravy a spoov, Katedra ekoomiky maria.misakova@fpedas.uiza.sk

5 DETERMINTION OF CREDIT RISK BY THE USE OF CREDITRISK + MODEL bstract Risk express ucertaity associated with expected yield. Credit risk makes, that the issuer of bod may be ot able to repay his debt ad iterests. It express credibility, reliability, the ability of issuers of securities to meet their commitmets. Nowadays has become the issue of credit risk a importat part of the life of each compay, which has some claims agaist other istitutios. Compaies should ot oly measure credit risk but also try to calculate ad predict potetial default of the compay i the future. This article deals with model CreditRisk +, which is based o typical isurace mathematics approach ad therefore, also ofte called a acturial model. CreditRisk + have become ifluetial bechmarks for iteral credit risk models. Practitioers ad policy makers have ivested i implemetig ad explorig each of the models idividually, but have made less progress with comparative aalyses. Key words CreditRisk +, calculatio, portfolio, loss. JEL Classificatio G32