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INTERNI MODELI

UPRAVLJANJA KREDITNIM

RIZIKOM

Rezime

Bazelski Komitet je ustanovio predloge za pristup zasnovan na unutrašnjem rejtingu (IRB - Internal Rating Based) kapitalnih zahteva za kreditni rizik. Pristup zasnovan na unutrašnjem rangiranju ima za cilj da unapredi sigurnost i ispravnost u finansijski sistem. Takav pristup, koji se oslanja na bankarsku unutrašnju procenu drugih strana ugovora (counterparties) i izloženosti, ostvaruje dodatna osetljivost na rizike (additional risk sensitivity), u kome kapitalni zahtevi zasnovani na unutrašnjim rejtinzima mogu da budu znatno osetljiviji na pokretače (drivers) kreditnog rizika i ekonomskih gubitaka u bankarskom portfoliju. U prilog aktuelnosti teme, paket reformi Basela III obuhvata, između ostaloga, i kvalitetnije pokriće rizika uključujući i kapitalni zahtev za pokriće kreditnog rizika druge ugovorne strane.

Ključne reči: kreditni rizik, rejting, banka, obveznice, izloženost, verovatnoća neplaćanja, gubici u slučaju neplaćanja, izloženost gubicima

JEL klasifikacija: G21, G32

originalni naučni rad

UDK 005.334:336.71

Rad primljen: 03.04.2012.

Odobren za štampu: 18.06.2012.

Viktorija MišićStudent doktorskih studijaviktorija.misic@gmail.com

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Summary

Basel Committee set out proposals for the internal rating based approach - IRB to the credit risk capital requirements. This approach is based on internal rating with the objective of enhancing the financial system safety and fairness function. The approach based on the bank’s internal rating of the counterparties and exposure is providing additional risk sensitivity, where capital requirements are based on internal ratings that may be significantly more sensitive to the credit risk and economic loss drivers in the bank portfolio. In support to the relevance of this topic, Basel III reform package covers, among others, also a higher quality risk coverage including capital requirement for the counterparty credit risk.

Key words: credit risk, rating, bank, bonds, exposure, probability at default - PD, loss given default - LGD, exposure at default - EAD.

JEL Classification: G21, G32

INTERNAL MODELS FOR THE CREDIT RISK MANAGEMENT

original scientific paper

UDC 005.334:336.71

Paper received: 03.04.2012

Approved for publishing: 18.06.2012

Viktorija MišićPhD studentviktorija.misic@gmail.com

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Pojam i karakteristike rejting sistema

Kreditni rizik je primarni finansijski rizik u bankarskom sistemu. Identifikovanje i rejting kreditnog rizika je ključni prvi korak u efektivnom upravljanju kreditnim rizikom. Proces upravljanja sistemom rejtinga dozvoljava menadžmentu banke da upravlja rizicima kako bi optimizirao prihode. Interni rejtinzi su sumarni indikatori rizika svojstven individualnom kreditu analiziranom od strane banke. Rejtinzi sadrže procenu rizika gubitaka zbog neispunjenja obaveze druge strane a zasnivaju se na razmatranjima odgovarajućih kvantitativnih i kvalitativnih informacija. U bankama izloženost svakog internog stupnja se tretira prema njihovim specifičnim i merljivim karakteristikama gubitka. Verovatnoća

neplaćanja (PD - Probability at Default) i ostale dve komponente rizika LGD (Loss Given Default - gubici u slučaju neplaćanja) i EAD (Exposure At Default - izloženost gubicima,) su ključni ulazni parametri kalkulacije regulatornog kapitala. Samim tim, validacija ovih komponenata i rejting sistema je ključna komponenta procesa pregleda supervizora. Pristup zasnovan na internom rejtingu (IRB) u Novom Bazelskom Kapitalnom dogovoru dozvoljava bankama da koriste sopstvene modele rejtinga kod procene verovatnoće difolta (PD) sve dok sistemi ispunjavaju propisane minimalne zahteve. U skladu sa tim, očekivani gubici se izračunavaju:

EL=PD*LGD*EAD

Tabela 1. Zašto odabrati IRB?

STD prema IRB Standardni pristup IRB pristup

Individualna procena rizika

Samo eksterni rejtinzi

• Primenjeni PD zadat na osnovu regulatora

• Veliki deo portfolia je neocenjen (eksterno)

• Bez eksplicitne vrednosti LGD• Ograničeno prepoznavanje

kolaterala

Interni rejtinzi i PD• Ceo portfolio može biti obuhvaćen• Interni LGD• Prošireno prepoznavanje

kolaterala

Portfolio procene rizika Jednostavni stepeni• Bez ekonomskog utemeljenja

Kompleksna funkcija integracije parametara rizika • Izvedena iz najnovijeg

ekonomskog modela

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The notion and characteristics of the rating system

Credit risk is the primary risk in the banking system. Credit risk identifying and rating is the crucial first step in effective credit risk management. The process of the rating system management allows the bank management to manage risks in order to optimize profits. Internal ratings are the summary risk indicators particular to each individual credit analyzed by the bank. Ratings contain risk assessment in case of the counterparty default, and are based on the assessment of corresponding quantitative and qualitative information. In banks, the exposure on every internal level is treated according to their specific and measurable loss

characteristics. Probabilities of default - PD, and the other two risk components, of the loss given default - LGD and the exposure at default - EAD, are the key input parameters for the calculation of the regulatory capital. Hence the validation of these components and the rating system is the key component of the supervisory review process. The internal rating based approach - IRB, under the new Basel Capital Accord, allows the banks to use their own rating models in the assessment of the probability at default - PD, for as long as the systems are complying with the minimum prescribed requirements. In accordance with this, the expected losses are calculated as follows:

EL=PD*LGD*EAD

Table 1 Why chose the IRS?

STD vs IRB Standard approach IRB approach

Individual risk assessment

Only external ratings

• Implemented PD provided by the regulator

• Large section of the portfolio remains unappraised (externally)

• Without any explicit LGD value• Limited recognition of collaterals

Internal ratings and PD• The portfolio can be appraised in

its entirety• Internal LGD• Expanded recognition of

collaterals

Risk assessment portfolio

Simple grades• Without economic grounds

Complex function of risk parameters integration • Derived from the latest economic

model

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Tabela 2. Proces rejtinga rizika

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Table 2. Risk rating process

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Statistički modeli u razvoju rejting sistema

Statistička teorija nudi različite metode građenja i procene modela rejtinga. Fokus je na statističkim metodama što uključuje parametarske modele, kao što su linearna regresiona analiza, diskriminantna analiza, analiza binarnog odgovora, metode vremenski diskretnih panela, modele hazarda i neparametarske modele kao što su to neuralne mreže i stabla odluka.

Neuralne mreže su kao alternativa parametarskim modelima. One nude fleksibilniji dizajn i predstavljaju spoj između nezavisnih i zavisnih varijabli. Neuralne mreže pripadaju klasi neparametarskih metoda. One su inspirisane uticajem bioloških nervnih sistema, kao što je mozak i procesom obrade informacija. One se tipično sastoje od mnogo nodova (čvorova) koji odašilju određeni output u slučaju da prime specifični input od drugog noda u mreži sa kojim su povezani. Kao i parametarski modeli i neuralne mreže se obučavaju putem uzoraka za obučavanje koji korektno klasifikuju dužnike. Finalna mreža se ostvaruje prepodešavanjem međuveza između inputa, outputa i bilo kojih potencijalnih međuveza nodova.

Stabla odluke su poznata i kao stabla klasifikacije. Stabla predstavljaju modele koji se sastoje iz seta ako-onda uslova deljenja kod slučajeva klasifikacije na dva ili više različitih grupa. Pod ovakvim metodama osnovni uzorak se deli na grupe u skladu sa kovarijatima. U slučaju binarne klasifikacije, na primer, svaki nod stabla je dodeljen pravilu odlučivanja koje opisuje uzorak u skladu i deli ga na dve podgrupe. Sada se proces posmatranja razvija naniže preko drveta u skladu sa pravilom donošenja odluka sve do krajnjeg noda u razgranatoj šemi stabla, koji tada predstavlja klasifikaciju ovog posmatranja.

Jedan od najupečatljivih razlika parametarskih modela je ta da su kovarijati tretirani kao kategorijske varijable. Nadalje, u slučaju kada specifična varijabla ili kategorija postaje relevantna u zavisnosti od kategorije ili varijable iz prethodnog nivoa. Na primer, varijabla “godine u biznisu” je relevantna za kompanije koje posluju u građevinskom

sektoru. Ova vrsta zavisnosti između varijabli se naziva interakcijom.

Statistički rejting sistemi primarno uključuju pretragu zavisnih varijabli koje daju jasne i pouzdane procene narušavanja situacije dužnika kao moguće. Statistički modeli se mogu opisati kao polazna tačka svakog modela koji koristi karakteristike dužnika kao indikatore i eventualno makroekonomske varijable koje su sakupljene kroz istorijske podatke i nalaze se na raspolaganju kod opisivanja dužnika u difoltu. Ako se definišu karakteristike dužnika kao vektor od n različitih varijabli (koje takođe nazivamo kovarijantama) tako da je x = x1, ..., xn posmatranih u vremenu t-L. Stanje difolta se označava binarnom varijablom x koju posmatramo u vremenu t. Varijabla y se definiše kao y=1 u slučaju difolta i y=0 kada nema difolta.

U zavisnosti od statističke primene ovih podataka različite metode se mogu koristiti radi predviđanja osobina. Zajednička osobina ovih metoda leži u činjenici da one procenjuju korelaciju između karakteristika dužnika i stanja difolta u prošlosti i koriste ovu informaciju radi građenja modela predviđanja. Model predviđanja se projektuje da procenjuje bonitet dužnika sa poznatim osobinama. Ovo se postiže uvođenjem karakteristika x u model gde je izlazna veličina modela procenjeno ponašanje. Vremenski pomak L između x i y određuje horizont predviđanja.

Regresiona analiza

Regresioni model uspostavlja linearni odnos između karakteristika dužnika i varijable difolta prema:

yi=β’ x xi + ui. (1)

Ovde yi označava da li se dužnik i nalazi u difoltu (kada je yi = 1) ili ne nalazi u difoltu (pa je yi=0). U vremenskom periodu t, xi predstavlja vektor karakteristika dužnika posmatranih u periodu t-L dok veličina β predstavlja vektor parametara koje zahvataju učinak promene karakteristika na varijablu difolta. Rezidualna varijabla je ui koja sadrži varijacije koje nisu obuhvaćene karakteristikama xi. Standardna procedura podrazumeva procenu (formula 1)

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Statistical models in the rating systems development

Statistical theory offers different models for the building up and assessment of the ratings model. The focus is on the statistical methods that include parametric models, such as the linear regression analysis, discrimination analysis, binary response analysis, methods of time discreet panels, hazard models, and non-parametric models such as the neural networks and the decision trees.

Neural networks serve as an alternative to the parametric models. They offer a more flexible design and are an interconnection between independent and dependent variables. Neural networks belong in the class of non-parametric methods. They are inspired by the impact of the biological neural systems, such as the brain, and the information processing procedure. They are typically composed of many nodes that are sending the given output in case of receiving a given input from another node in the net where they are connected. Not unlike parametric models, neural networks are also having a learning process from the instruction examples which are correctly classifying obligors. The final network is achieved through the readjustment of the input - output interconnection, and any other potential nodes interconnection.

The Decision trees are also known as Classification trees. Decision trees are the models which consist of a set of “if-and-then” attributes for the division in case of classification into two or more different groups. The main example in such methods is being divided into groups in accordance with the co-variants. In case of a binary classification, for example, each node of the Tree is attributed to the Rule of induction which cuts the space of the example parallel to the two sub-groups. The observation process now develops downwards, down the Decision tree according to the Rule induction system, all the way down to the final node in the branching scheme of the Decision tree, which then stands for the classification of this observation.

One of the most outstanding differences between parametric models is that the co-variants are treated as variable categories. In addition, this is also the case when the

specific variable or category becomes relevant depending on the category or the variable from the previous level. For example, the “years in business” variable for companies operating in construction works is relevant. This type of interdependency between variables is called interaction.

Statistical rating systems primarily cover the search for interdependent variables which give clear and reliable assessment that the deteriorating situation of the obligor is possible. Statistical models may be described as the starting point for every model which is using obligor’s characteristics as indicators, and eventually macro-economic variables gathered through the historical data available when describing the obligor in default. If the obligor characteristics are defined as vector n of different variables (which we also call co-variances), the x = x1,…,xn in the observed time t-L. The default situation is designated with the binary variable x which we observe in time t. Variable y is defined as y = 1 in case of default, and as y = 0 without default.

Depending on the statistical application of these data, different methods may be used for the forecast of characteristics. The common feature of these methods is the fact that they are assessing the correlation between the obligor’s characteristics and the history of default in the past, and are using this information for purpose of building up the forecasting model. Forecasting model is projected so as to evaluate credit rating of the obligor with the known characteristics. This is achieved by introducing characteristic x into the model where the output value of the model is the estimated conduct. The time shift L between the x and y will determine the forecasting horizon.

Regression analysis

Regression model establishes a linear relation between the obligor’s characteristics and the default variable, as follows:

yi=β’ x xi + ui. (1)

Where yi is designating whether the obligor i is in default (when yi = 1) or he is not in default (when yi = 0). Over the time period t, the xi is the

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prema prostim najmanjim kvadratima veličine β koji su dodeljeni na osnovu veličine b.

Procenjeni rezultat daje skor dužnika Si koji se obračunava kao:

Si=E (yi / xi) = b‘ x xi. (2)

Ova jednačina iskazuje stav da skor dužnika predstavlja očekivanu vrednost osobina varijable kada su njegove karakteristike poznate. Ovaj skor može da se obračuna uvođenjem vrednosti karakteristika dužnika u linearnu funkciju datu formulom 2. Skor Si je kontinuelna veličina (dok je yi binarna varijabla), pa će zbog toga izlaz modela dati rezultat unutar intervala od 0 do 1. Predikcija može da poprimi vrednosti veće od 1 ili manje od 0. Ovo ima za posledicu da rezultat proizašao iz modela ne može da se interpretira kao verovatnoća. Međutim, skor Si može biti uporebljen u svrhe poređenja između različitih dužnika gde je veća vrednost skora Si je u korelaciji sa povećanim rizicima difolta. Koristi i mane ovog modela formule 1 i 2 su sledeće:1. Procenitelji po osnovu prostih najmanjih

kvadrata su dobro znani i lako primenljivi;2. Model predviđanja je linearni model

pa stoga jednostavan za izračunavanje i razumenje;

3. Slučajna varijabla ui je heteroskedastične prirode tj. varijansa ui nije konstantna za sve vrednosti i pošto:

VaR (ui) = VaR (yi) =E (yi / xi) x [1 - E (yi / xi) ] = b‘ x xi (1 - b‘ x xi) (3)

Zbog toga je procena β nedovoljna i standardna greška koeficijenta procene b je sa predrasudama. Jedan efikasan način procene b predstavlja primena ponderisanih najmanjih kvadrata.

Analiza diskriminante

Analiza diskriminante je tehnika klasifikacije koja se primenjuje na korporativne bankrote. Linearna analiza diskriminante se zasniva na proceni linearne funkcije diskriminante u cilju odvajanja individualnih grupa (u ovom slučaju grupa dužnika u difoltu i dužnika koji nisu u difoltu) a na osnovu specifičnih karakteristika. Funkcija diskriminante glasi:

Si= β‘ x xi (4)

Skor Si je varijabla diskriminante. Procena funkcija diskriminante se temelji na sledećem principu: Maksimizacija spreda između grupa (dobrih i loših dužnika) i minimizacija spreda unutar svake pojedine grupe. Maksimizacija određuje optimalne proporcije između koeficijenata vektora β. Koeficijenti su normalizovani po osnovu odabrane varijanse unutar grupe tako da poprimaju vrednost 1. To ima za posledicu da je veličina apsolutnog nivoa varijable diskriminante Si proizvoljna i ne može da se intepretira na izdvojen način. Kao i kod linearne regresione analize Si može da se koristi radi poređenja predikcije za različite dužnike (viši skor, viši rizik).

Logit i Probit modeli

Logit i Probit modeli predstavljaju ekonometrijske tehnike osmišljene u cilju analize binarnih zavisnih varijabli. Pristup sa latentnim varijablama podrazumeva neprimetnu (latentnu) varijablu y* koja se odnosi na karakteristike dužnika po sledećem obrascu:

y i* = β‘x xi +ui (5)

Ovde su veličine β, xi i ui već definisane a varijabla y i* je metrički skalirana i poprima vrednost binarne varijable difolta yi tako da je:

To znači da difolt nastaje onda kada latentna varijabla pređe vrednost praga od nule. Stoga, verovatnoća dešavanja difolta je jednaka:

P (yi = 1) = P (ui > – β‘ x xi) = 1 - F (-β‘ x xi) = F (β‘ x xi) (7)

Ovde veličina F(.) označava nepoznatu distributivnu funkciju. Poslednji korak u prethodnoj formuli podrazumeva da je distributivna funkcija simetričke gustine oko nule. Izbor distributivne funkcije F (.) zavisi od distribucionih pretpostavki učinjenih oko reziduala (ui). Ako se pretpostavi normalna

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vector of the obligor’s characteristics observed over the period t-L, while the value β is the vector of parameters which cover the impact of changes in the characteristics on the default variable. Residual variable is the ui which contains variations which are not covered by the characteristics xi. The standard procedure makes an assessment (equation 1) according to the ordinary least squares of the value β which are allocated on the basis of the value b.

The estimated result offers the score for the obligor Si which is calculated as follows:

Si=E (yi / xi) = b‘ x xi. (2)

This equation expresses the view that the obligor’s score represented the anticipated value of the characteristics variable when his characteristics are known. This score may be computed by the introduction of the obligor’s characteristics value in the linear function given in the equation 2. The score Si is a continuous value (while the yi remains a binary variable), hence the model output will give the result within an interval from 0 to 1. Prediction may acquire values higher than 1 or lower than 0. This will have as a consequence the fact that the result obtained from this model can not be interpreted as a probability. However, the Si score may be used for purpose of comparison between different obligors where the higher value of the score Si is in correlation with the higher risk of default. Advantages and disadvantages of the formula 1 and 2 models are the following:1. Estimates based on the ordinary least squares

are well known and easily applicable2. Forecasting model is the linear model

and thus simple for computation and understanding

3. Random variable ui is of a heteroscedastic nature, i.e. it is not constant across all the values, as:

VaR (ui) = VaR (yi) =E (yi / xi) x [1 - E (yi / xi) ] = b‘ x xi (1 - b‘ x xi) (3)

Hence the estimate of β is insufficient, and the standard error of the estimate coefficient b is prejudiced. An efficient way of estimating b is the use of weighted least squares method.

Discriminant analysis

Discriminant analysis is the method of classification used for predicting corporate bankruptcies. Linear discriminant analysis is based on the assessment of the linear discriminate function used to separate two or more individual classes (in this case classes of obligors in default, and obligors who are not in default), based on specific characteristics. Discriminant function is as follows:

Si= β‘ x xi (4)

The Si score is the discriminant variable. Discriminant function estimate is based on the following principle: Maximisation of spread between classes (good and bad obligors) and minimisation of spread within each particular class. Maximisation determines optimum proportions between coefficients of the vector β. Coefficients are normalised on the basis of the selected variance within the class so as to acquire value 1. This results in the fact that the value of the absolute level of the discriminant variable Si is random and can not be interpreted in a separate way. Not unlike the case of the linear regression analysis, Si may be used for comparison of prediction for different obligors (higher score, higher risk).

Logit and Probit models

Logit and Probit models are the econometric techniques designed for purpose of analysing binary dependent variables. The approach to the latent variables requires a latent variable y* which pertains to the obligor’s characteristics according to the following equation:

y i* = β‘x xi +ui (5)

The values β, xi, and ui are here already defined, and the variable yi* is metrically scaled and acquires the value of the binary default variable yi, and gives the following:

else

if

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distribucija tada važi Probit model gde je distributivna funkcija definisana kao:

Ako umesto reziduala pretpostavimo da sledimo logaritamsku distribuciju dobijamo rezultat iskazan kao Logit model funkcije distribucije koji glasi:

Polazeći od nastojanja da izvršimo procenu verovatnoća difolta u Logit i Probit modelima za jednog jedinog dužnika dolazi se do toga da verovatnoće difolta ne mogu biti posmatrane kao ostvarenje verovatnoće difolta. Međutm, kod grupa dužnika učestanosti osmotrenih događaja difolta se mogu interpretirati kao verovatnoća difolta. Polaznu osnovu je procena najmanjih kvadrata u sledećoj regresiji:

pi = b’ x xi + ui (10)

U ovoj jednačini indeks i označava grupu koju čini broj individua, pi je frekvencija difolta u posmatranoj grupi i, dok je veličina xi karakteristika posmatrane grupe i. Ovaj model međutim nije odgovarajući. Potrebno je razmotriti ishod koji nije ograničen na veličine između nule i jedinice pa stoga i ne može da se interpretira kao verovatnoća (a to je funkcija E (yi / xi) =b‘ x xi).

Pošto je uopšteno uzevši neverovatno pretpostaviti da verovatnoća može biti obračunata kao linearna funkcija linearni izraz b’ x xi treba transformisati kao nelinearnu funkciju (link funkciju F):

pi=F (b’ x xi) (11)

Ova odgovarajuća link funkcija transfomiše veličinu b’ x xi u srazmeri unutar zatvorenog interval [0,1]. Ovo se ostvaruje na osnovu bilo koje distributivne funkcije. Izbor odgovarajuće link funkcije određuje tip modela: sa logističkom funkcijom linka ova jednačina 11 postaje Logit model dok sa normalnom distribucijom (jednačina 11) dobijamo rezultate iskazane kroz Probit model. Međutim, kod procenjivanja pomoću najmanjih kvadrata (jednačina 10),

koeficijenti će biti heteroskedastični stoga što je VaR (ui) = VaR (pi) = p (xi) x (1 - p (xi) ). U statistici niz slučajnih varijabli predstavlja heteroskedastični skup ako i samo ako slučajne varijable imaju različite varijanse. Niz slučajnih varijabli koji ima konstanstnu varijansu naziva se homoskedastičan.

Snaga stabilnosti Logit i Probit modela može sumarno da se iskaže kao:• Metodi su zasnovani na teoriji,• Dobijeni rezultati se mogu direktno

interpretirati kao verovatnoća difolta,• Značajnost modela i individualni koeficijenti

se mogu testirati.

Statistički pristup PD validacije (probability of default - verovatnoća neplaćanja)

Verovatnoća difolta je rizik da dužnik neće biti sposoban ili je nevoljan da isplati svoj dug u potpunosti ili na vreme. Rizik difolta proizilazi iz analiziranja kapaciteta dužnika da isplati dug u saglasnosti sa uslovima ugovora. Verovatnoća difolta je opšte povezana sa finansijskim karakteristikama kao što je neadekvatan cash flow radi servisiranja duga, opadanje prihoda ili operativnih margina, visok leveridž, opadanje likvidnosti, ili nesposobnost uspešnog implementiranja poslovnog plana. U dodatku ovih merljivih faktora, dužnikova volja da isplati dug mora takođe biti procenjena. Statističke analize rejting sistema i funkcija skora su zasnovane na pretpostavki da postoje dve kategorije dužnika u banci:• Dužnici koji će biti u difoltu u nekom

definisanom vremenskom horizontu i• Dužnici koji neće biti u difoltu u nekom

vremenskom periodu.Obično nije poznato da li jedan dužnik

pripada prvoj ili drugoj kategoriji. Banke se suočavaju sa dihotomnom (ili binarnom) klasifikacijom problema pošto one treba da procene budući status jednog dužnika kroz izvor njegovih trenutnih dostupnih karakteristika.

Diskriminacija je procedura primene alata za klasifikaciju jednog dužnika, za procenu njegovog budućeg statusa. Diskriminatorna snaga rejting sistema obeležava svoju sposobnost diskriminisanja ex-ante između dužnika koji su

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This means that the default occurs when the latent variable crosses the threshold value of zero. Hence the probability of default is equal to the following:

P (yi = 1) = P (ui > – β‘ x xi) = 1 - F (-β‘ x xi) = F (β‘ x xi) (7)

The value F(.) here designates the unknown distribution function. The last step in the previous equation implies that the distribution function of symmetric density is around zero. The choice of distribution function F(.) depends on the distribution assumptions made around the residual (ui). If the assumption is a normal distribution, then the Probit model applies where the distribution function is defined as follows:

If we are to assume, instead of a residual, to be following the logarithmic distribution, the result obtained as a Logit model of the distribution function will be as follows:

If we are to engage in making the prediction of the probability of default, in both the Logit and the Probit models, for solely one single obligor, what results is that the probabilities of default can not be observed as the materialisation of the probability of default. However, in case of a group of obligors, the frequency of the observed default events may be interpreted as the probability of default. The starting point is the estimate of the least squares in the following regression:

pi = b’ x xi + ui (10)

In this equation index i designates the group composed of a number of individuals, where pi is the frequency of default in the observed group i, and the value xi is the characteristic of the observed group i. This model, however, is not adequate. It is necessary to observe the outcome which is not limited to the values between zero and one, and thus can not be interpreted as a probability (and this is the function E (yi/xi)=b’ x xi).

Generally speaking, as it is improbable to assume that the probability can be calculated as a linear function, the linear expression b’ x

xi should be transformed into a non-linear function (link function F):

pi=F (b’ x xi) (11)

This corresponding link function transforms the value b’ x xi in proportion within the closed interval [0,1]. This is achieved on the basis of any distribution function. The choice of the corresponding link function shall determine the type of the model: with the logistic link function this equation 11 becomes Logit model, while with the normal distribution (equation 11) we obtain the results expressed through the Probit model. However, when estimating by means of the least squares (equation 10), coefficients will be heteroscedastic because VaR(ui) = VaR(pi) = p(xi) x (1 - p(xi)0. In statistics, a series of random variables represents a heteroscedastic set if and only if the random variables are having different variances. The series of random variables which is having a constant variance is called homoscedasticity.

The power of stability of the Logit and Probit models may be summarily expressed as the following:• Methods are based on theory;• Results obtained may be directly interpreted

as the probability of default;• Significance of the model and individual

coefficients may be tested.

Statistical approach to the probability of default - PD validation

Probability of default is a risk that the obligor will not be able or willing to repay his debt either fully or in time. The risk of default derives from the analysis of the obligor’s capability to discharge debt liabilities under contractual terms. Probability of default is generally connected with the financial characteristics such as an inadequate cash flow for debt servicing or operative margin high leverage, fall in liquidity, or inability to implement business plan successfully. In addition to these measurable factors, what remains is obligor’s willingness to repay his debt which must also

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u difoltu i onih koji to nisu. Različite statističke metodologije procene diskriminatorne snage popularne u bankarskom sektoru su:• Profil kumulativne preciznosti (Cumulative

Accuracy Profile - CAP) i njegov sumarni indeks, racio preciznosti (Accuracy Ratio);

• Operativne karakteristike primaoca (Receiver operating characteristic - ROC) i njegovi sumarni indeksi: Oblast ispod ROC (AUROC-Area Under the ROC) i Pietra koeficijent;

• Stopa Bajesinove greške;• Uslovna entropija, Kulbek-Lejbrelovo

rastojanje, Uslovna informativna entropija;• Vrednosti informacije (divergencija, indeks

stabilnosti);• Kendelov‘t i Somerov’D;• Brijerov skor.

Entropija je koncept iz teorije informacija koji se odnosi na isključenje nesigurnosti u eksperimentu. Posmatranje dužnika u vremenu sa namerom da se donese odluka o njegovom solventnom statusu može da se interpretira kao jedan eksperiment.

Brijerov skor je procenitelj uzorka razlike srednjih kvadrata varijabli indikatora difolta (na primer, vrednost 1 se dodeljuje u slučaju difolta i vrednost nula u slučaju preživljavanja) u portfoliju i predvidjanja vrednosti verovatnoce difolta za rejting kategorije ili vrednosti skora. Brierov skor nije mera sa kalibracionom preciznošću.

U praksi bančina procena vrednosti verovatnoće difolta će se razlikovati od posmatrane stope difolta. Ključno pitanje je da li je ova devijacija čisto slučajna ili se dešava sistematski. Kvalitet procena verovatnoće difolta treba sagledati u svetlu sledećih metodologija:1. Binominalni test,2. Hi kvadrat test,3. Normalni test,4. Pristup saobraćajnog semafora.

Normalni test predstavlja multiperiodni test korektnosti predviđanja verovatnoće neplaćanja kod jedne jedine rejting kategorije. On se primenjuje pod pretpostavkom da srednja stopa difolta ne varira previše u vremenu i da je nastajanje difolta u različitim godinama nezavisno. Normalni test je motivisan centralnom graničnom teoremom i zasnovan je na normalnoj aproksimaciji distribucije vremenski usrednjenih stopa difolta. Snaga

testa je umerena i uglavnom primenljiva na kratkovremenske serije (na primer do 5 godina).

Pristup saobraćajnog semafora je (za razliku od normalnog testa) potpuno nezavisan od bilo kakvih pretpostavki o konstantnosti veličine verovatnoće difolta u vremenu. Distribucija broja difolta u toku jedne godine približno podleže normalnoj distribuciji.

Modeliranje LGD u bankarskom sektoru (Loss Given Default - gubici u slučaju neplaćanja)

LGD (gubici u slučaju neplaćanja) je važan element pristupa zasnovan na unutrašnjem rejtingu merenja kapitala (IRB pristup). LGD je finansijski gubitak banke koji se desio kada dužnik ne može ili ne želi da isplati svoj dug. Validacija internog merenja LGD-a je krucijalna za validaciju pogodnosti merenja kapitala. LGD je posebno važan jer su zahtevi minimalnog regulatornog kapitala visoko osetljivi na gubitke u slučaju neplaćanja. Ključne teme koje dotiču LGD su:• Šta LGD znači i koja je njegova uloga u

pristupu zanovanom na unutrašnjem rejtingu?

• Kako je LGD definisan i na koji se način meri?• Šta pokreće različitosti vrednosti LGD?• Koji pristupi mogu da se uzmu radi

modeliranja ili procenjivanja LGD?Faktori koji igraju ključnu ulogu kod svake

procene bančinog LGD-a i upotrebljenog modela su kapitalna struktura, prisustvo i kvalitet kolaterala, industrijski sektor i tajming poslovnih ciklusa

Po definiciji dugovni instrument može iskusiti gubitak samo ako je postojao difolt. Svakako ne postoji standardna definicija difolta. Različite definicije se mogu koristiti za različite svrhe. Tipično se difolt dešava kada se dešavaju sledeći slučajevi:• Zajam je postavljen kao neizvršen;• Charge off se već desio.• Dužnik je više od 90 dana u docnji;• Dužnik je zapao u bankrotstvo.

Pre početka bilo koje LGD procedure procene neophodno je definisani gubitak. Prvo, ekonomski gubici nisu isto što i računovodstveni gubici: ”definicija gubitaka koja se koristi u proceni LGD-a je ekonomski

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be assessed. Statistical analysis of the rating system and the score function are based on the assumption that there are two categories of the bank’s obligors:• Obligors who will be in default over some

defined time horizon, and• Obligors who will not be in default over

some time period.Usually it is not known whether an obligor

belongs in the first or in the second category. Banks are faced with the dichotomous (or binary) classification of the problem as they are the ones to predict the future status of one obligor through the source of his currently available characteristics.

Discrimination is a procedure of the classification tools application on one obligor, for purpose of assessing his future status. Discriminatory power of the rating system marks its discriminatory capability ex ante between obligors who are in default and those who are not. Different statistical methodologies of discriminatory power assessment, popular in the banking sector, are the following:• Cumulative Accuracy Profile - CAP and its

summery index, the Accuracy Ratio;• Receiver Operating Characteristics - ROC

and its summary indices: Area Under the ROC - AUROC, and the Pietra Ratio;

• Bayesian Rate Error;• Conditional entropy, Kullback-Leibler

Divergence, Conditional Informative Entropy;

• Information value (divergence, stability index) ;

• Kendall’s tau and Somer’s D ;• Brier Score.

Entropy is the concept from the information theory that pertains to the elimination of uncertainty in an experiment. Observation of an obligor over a time period with the intent of deciding on his solvency status may be interpreted as one such experiment.

The Brier Score measures average squared deviations between default indicator variables (for example, value 1 is allocated in case of a default, and the value zero in case of a survival) in a portfolio and the predicted value of the probability of default for rating categories, or for the score value. The Brier Score is not a measure with calibrated precision.

In practice, the bank’s prediction of the probability of default value will differ from the observed default rate. The key question is whether this deviation is purely random, or it happens systematically. The quality of the probability of default prediction should be assessed in the light of the following methodologies:1. Binomial test;2. Hi square test;3. Normal test;4. The Traffic Light Approach.

The Normal Test is a multi-periodical testing of correct prediction for the probability of default in solely one single rating category. It is being applied under the assumption that the average default rate is not showing excessive variation over time, and that the default events during different years occur independently. Normal Test is motivated by the central limiting theorem and is based on the normal approximation of distribution of the time averaged default rates. The testing power is moderate and mainly applicable on short time series (for example, up to 5 years).

The Traffic Lights Approach (as an opposite of the normal test) is completely independent from any assumptions on the constancy of the probability of default value over time. Distribution of the number of defaults over one year approximately has a normal distribution.

Loss Given Default - LGD modeling in the banking sector

Loss given default - LGD is an important element in the internal rating based approach - IRB to the capital measurement. LGD is the financial loss of the bank that occurs when an obligor can not or will not repay his debt. The LGD internal rating validation is of crucial importance for the validation of the capital measurement facility. LGD is especially important as the minimum requirements for the regulatory capital are highly sensitive to the loss given default. The key topics impacting the LGD are the following:• What is the meaning of the LGD and what is

its role in the internal rating based approach?• How is the LGD defined and how is it

measured?

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gubitak što uključuje materijalne efekte diskonta i materijalne direktne i indirektne troškove povezane sa prikupljanjem (collecting) po izloženosti”. Ekonomski gubitak može biti određen korišćenjem različitih metoda koje se klasifikuju bilo u eksplicitnim ili implicitnim metodama. Fokusiranje na eksplicitne metode, kada je povezanost jednog ekonomskog gubitka sa svakim elementom uključenih u referentne setove podataka, dva različita pristupa se mogu koristiti: tržišni LGD i workout LGD. Tržišni LGD zavisi od tržišnih cena difolt usluga odmah nakon datuma difolta (obično oko 30 dana). Većina rejting agencija u svojim studijima o stopi oporavka koriste ovaj pristup. Ovaj metod je koristan pošto cene odražavaju investitorske procene diskontnih vrednosti oporavka (Eskontovani neto-novčani tok, tehnika procene investicije koja uzima u obzir različite vrednosti budućih prihoda u zavisnosti od toga kada će biti ostvareni). Ako je tržište likvidno ili ako proizilazi iz šokova koji nisu povezani sa očekivanim oporavkom, ova mera neće biti pogodna. Ovo je od posebnog interesa za relativno nova kreditna tržišta. Druga mera LGD, workout LGD, koristi informacije o vrednosti gubitka iz workout-a. Gubici povezani sa difolt facility (uslugama) su izračunati kroz discounting cash flows, uključujući troškove, koji su rezultat workout-a od datuma difolta do kraja procesa oporavka. Gubici se u tom slučaju mere kao procenat od izloženosti difolta. Vremenski cash flow, tržišni i workout LGD i diskontna stopa su krucijalni u ovom pristupu.

Važna su tri pitanja u vezi LGD procene: koje facilities (usluge) treba da se uključe u referentne setove podataka korišćenih u procenu procene; kada je proces oporavka gotov; i kako procene LGD koji koristi specifične definicije difolta mogu biti transformisani na procene pod drugim definicijama difolta. Za određen portfolio jedni interni ili spoljni referentni set podataka je zahtevan radi procene parametara rizika (PD, LGD i EAD) koji su potrebni za unutrašnje korišćenje i izračunavanje kapitalnih zahteva za IRB pristup. Ovi referentni setovi podataka bi trebali:• Pokriti najmanje kompletan poslovni ciklus;• Obuhvatiti sve difoltove koji su se desili u

toku određenog vremenskog okvira;

• Uključiti sve relevantne informacije radi procene parametara rizika i

• Uključiti podatke na relevantnim drajverima gubitaka (pokretačima gubitaka).LGD se definiše kao odnos gubitka naspram

izloženosti u slučaju difolta. U slučaju nastanka difolta LGD sadrži sledeće tipove gubitaka:• Gubitak glavnice;• Carrying costs (finansijski troškovi, troškovi

kamata po osnovu obveznica, troškovi kamata po osnovu zajmova radi plaćanja hartija od vrednosti, ekonomski troškovi; troškovi koji se povećavaju sa povećanjem nivoa investicija u obrtni kapital; troškovi držanja zaliha od datuma nabavke do datuma prodaje) po neostvarenim zajmovima na primer: prihod po kamatama;

• Workout troškovi (naplate, pravni troškovi...).Kod obveznica i zajmova koji su ušli u

difolt, a sa kojima se trguje na tržištu moguće je posmatrati cene direktno sve dok se trgovina stvarno i odvija. Studije oporavka koje su radile rejting agencije su se zasnivale na ovakvom pristupu. Stvarne cene su bile bazirane na paritetu 100 (centa na dolar) pa su naknadno lako bile prevedene u procente oporavka (ili LGD kao 100% minus procenat oporavka). Ove cene predstavljaju rezultat tržišnih transakcija.

Komponente workout LGD-a

Tri glavne komponente za izračunavanje workout gubitaka su: oporavljanja (keš ili ne keš), troškovi (direktni i indirektni) i diskontni faktor (eskontni faktor, stopa koja se koristi za dobijanje neto-sadašnje vrednosti novca koji će biti isplaćen u budućnosti) koji su osnova za građenje svih cash flow-a u pojmu monetarne jedinice u datumu difolta.

Ako su svi cash flow-i povezani sa default facility (uslugama) od datuma difolta do kraja procesa oporavka su poznati (postoji kompletna informacija):

Realizovani LGD = [1-(∑i Ri (r) - ∑j Pj (r) / EAD)]

Gde Ri je svaki od i diskontni oporavak difolt facility (usluga), pj je svaki od j diskontnog plaćanja ili troškova za vreme perioda oporavka i r predstavlja diskontnu stopu (diskontna/

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• What is triggering differences in the LGD values?

• What are the approaches that may be applied for modeling or evaluating the LGD?Factors playing the crucial role in every

evaluation of the bank’s LGD and the model applied are the capital structure, presence and quality of collateral, industrial sector, and the business cycle timing.

Per definition, the debt instrument may suffer losses only in the presence of a default. There is certainly no standard definition of a default. Different definitions may be used for different purposes. Typically, default occurs in the case of the following events:• Loan remains unpaid;• Charge off has already occurred.• Obligor is more than 90 days in default;• Obligor has suffered bankruptcy.

Before starting on any of the LGD procedures it is necessary to define loss. Firstly, economic losses are not the same as the accounting losses: “definition of losses which is applied in the LGD estimation is the economic loss which includes material, tangible effects of the discount, and the tangible direct and indirect costs related to the collecting according to exposure”. Economic loss may be determined through the use of different methods which are classified either as the explicit or the implicit methods. Focusing on the explicit methods, when the connection of one economic loss is set with every element included in the reference data sets, offers two different approaches that may be used: the marketing LGD, and the workout LGD. The marketing LGD depends on the market prices of default services immediately after the date of default (usually some 30 days). Most of the rating agencies in their studies on the recovery rate are using this approach. This method is useful as the prices are reflecting investors’ estimates of the recovery discount values (discounted net cash flows, investment assessment techniques which are taking into consideration different values of future income depending on when they will be materialized). If the market is liquid, or if it is coming out of shocks that are not related to the expected recovery, this measure will not be suitable. This is especially interesting for the relatively new credit markets. The other LGD measure,

the workout LGD, is using information on the workout loss value. Losses connected with the default facility are calculated through the discounting cash flows, including costs, which are the result of the workout from the default date and up to the end of the recovery process. Losses in this case are measured as a percentage of the default exposure. Time cash flows, market and workout LGD, and the discount rate, are crucial in this approach.

There are three important questions arising in connection with the LGD estimates: what are the facilities to be included in the reference data sets used in the estimation process; when is the recovery process finished; and how can the LGD estimate, using specific default definition, be transformed into estimates under different default definitions. For a certain portfolio what is required is one internal or external reference data set for purpose of evaluating the risk parameter (PD, LGD, and EAD) which are necessary for the internal use and calculation of the capital requirements for the IRB approach. These reference data sets should allow for the following:• Cover the least complete business cycle;• Include all the defaults occurring during the

given time frame;• Include all the information relevant for the

risk parameter evaluation; and• Include data on the relevant loss drivers.

LGD is defined as a ratio between loss and the default given exposure. In case of a default, the LGD contains the following loss types:• Loss of principal;• Carrying costs (financial costs, interest costs

on bonds, interest costs on loan for securities payment, economic costs, costs accruing with the higher level of investment into working capital, costs of holding stocks from the supply date to the selling date) on unrealized loans, for example: interest income;

• Workout costs (collections, legal costs).In case of defaulting in bonds and loans

traded on the market it is possible to observe prices directly for as long as trading is being done in real time. Recovery studies made by the rating agencies were based on such an approach. Real prices were based on a 100 (cent to dollar) parity, and were subsequently easily translated into the recovery percentage (or LGD

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eskontna stopa, kamatna stopa koju banke zaračunavaju za obezbeđivanje gotovine za menice). Kada gubitak je izračunat kroz uspostavljanje svih negativnih observacija gubitaka nula, kao prikaz u formuli 12:

Realizovani LGD = Max[1-(∑i Ri (r) - ∑j Pj (r) / EAD), 0] (12)

Izračunavanje ekonomskih gubitaka difolt usluga (facilities), korišćenje posmatranih oporavka i troškova, neophodno je diskontovati ih korišćenjem neke diskontne stope. Uticaj izabrane diskontne stope na procene LGD-a je posebno značajno u portfolijima gde je period oporavka dug i ima nizak nivo rizika. S obzirom da je EAD direktno spregnut sa LGD neophodno je istovremeno uzeti u obzir obe veličine.

Primer 1.• Difolt nastaje po osnovu usluge vredne 1

milion €,• U trenutku nastanka defaulta povučeno je

svega 500.000 €,• Postoji cena pravnih troškova od 1.000 € po

godini, nakon difolta,• Nakon 2 godine objavljen je bankrot i banka

je povratila nazad 200.000 €,• Diskontna stopa iznosi 5%.

Statističko modeliranje LGD procene

Set podataka uglavnom koristi podatke o obveznicama. Stope oporavka (recovery rate) će biti obračunate kao tržišne vrednosti obveznica jedan mesec nakon nastanka difolta. Veza između LGD-a i stope oporavka može biti iskazana kao:

LGDt(i) =1 - Rt(i) (13)

LGDt(i) i Rt(i) označeno kao LGD i stopa oporavka obveznice i-te koja je ušla u difolt u godini t, i=1, ..., nt. Broj obveznica koje su ušle

u difolt u godini t gde se t kreće u intervalu t=1, ..., T označen je sa nt. Rezultirajuće stope oporavka i stope gubitaka normalno se nalaze u opsegu između 0 i 1 iako postoje određeni izuzeci. Stope oporavka koje su veće od 1 su veoma neuobičajene. U ovim slučajevima sa obveznicama se trguje po ceni iznad nominalne ako je emitent ušao u difolt. Ove vrednosti se onda isključuju iz seta podataka u empirijskim istraživanjima. Najpre se transformišu vrednosti LGD:

Yt (i) =log LGD t (i) / (1 - LGDt (i) ) (14)

Preveden na periode stope oporavka dobija se sledeća relacija:

Yt (i) =log (1-Rt (i) ) /Rt (i) ) = -logRt (i) / (1-Rt (i) ) (15)

LGD se može izračunati kao:

LGDt (i) = exp (yt (i) ) /1+exp (yt (i) ) (16)

Analogno, model koji se koristi u Bazelu 2 za transformisanu vrednost yt (i) je specificiran kao

Yt(i) = µ + σ x (ω)1/2 x ft + σ x (1-ω)1/2 x εt(i) (17)

Slučajne varijable ft i εt(i) su normalno distribuirane, dok za sve varijable pretpostavljamo da su nezavisne. Parameter Σ je nenegativan, dok su vrednosti varijable ω ograničene na interval [0,1].

U sledećem koraku model opisan jednačinom 18 se proširuje uključenjem čvrstih i vremenski specifičnih faktora rizika koji su podložni posmatranju. Zavisnost po osnovu ovih faktora rizika se specificira prema sledećoj linearnoj aproksimaciji:+γx

μt (i) = β 0+ β'x xt-(i) + γ' x zt-1 (18)

Gde je malo i =1, …, nt, t =1, …, T.Ovde se veličina xt-1 (i) karakteriše kao vektor

emitenta i faktora specifičnih za obveznicu koja je posmatrana u prethodnom periodu. Primeri za ovakve emitente i obveznice predstavljaju rejtinzi emitenta u prethodnoj godini ili seniority. Veličinom zt-1 označava se vektor makroekonomskih varijabli koji predstavlja potencijalne sistemske izvore rizika.

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as a 100% minus recovery percentage). Such prices are the result of market transactions.

LGD workout components

The three main components for the loss workout computation are the following: recovery (cash or no cash), costs (direct and indirect), and the discount factor (discount factor, rate used for acquiring net present value of money that will be paid in future), which are the basis for the building up of all the cash flows in the notion of a monetary unit at the date of default.

If all the cash flows connected with the default facility, from the date of default and up to the end of the recovery process, are known (if there is a complete set of information available):

Realized LGD = [1-(∑i Ri (r) - ∑j Pj (r) / EAD)]

Where Ri is every one of the i discount recovery of the default facility, pj is every one of the j discount payments or costs during the recovery period, and r is a discount rate (discount rate, interest rate that banks are charging on providing cash for draft notes). When the loss calculated through the establishment of all the negative observations is zero, it is presented in the form of the equation 12:

Realized LGD = Max[1-(∑i Ri (r) - ∑j Pj (r) / EAD), 0] (12)

When calculating economic losses of the default facilities, with the application of the observed recovery and costs, it is necessary to discount them through some discount rate. The impact of the selected discount rate on the LGD assessment is especially important in the portfolios where the recovery period is long and where the risk level is low. In view of the fact that the EAD is directly connected to the LGD it is necessary to take into account both of these values simultaneously.

Example 1

• Default based on the facility worth 1 million EUR;

• At the time of default only 500,000 EUR were withdrawn;

• There is a price of legal costs of 1,000 EUR per year, after default;

• After 2 years, bankruptcy was announced and the bank recovered 200,000 EUR;

• Discount rate is 5%.

Statistical modeling of the LGD prediction

Data set mainly uses the bonds data. Recovery rate will be computed as the market value of bonds one month after the event of default. The connection between the LGD and the recovery rate may be expressed as follows:

LGDt(i) =1 - Rt(i) (13)

Where LGDt(i) is designated as the LGD and the recovery rate of the bond i-te which has defaulted in the year t, with i=1,…,nt. The number of bonds that have defaulted in the year t where t runs in the interval t=1,…, T is marked with nt. The resulting recovery rates and the loss rates normally are to be found in the range between 0 and 1, although there are certain exceptions. Recovery rates which are higher than 1 are very unusual indeed. In such cases, bonds are traded at the price above the nominal if the issuer has defaulted. These values are then being excluded from the data set in the empirical research. The first to transform are the LGD values:

Yt (i) =log LGD t (i) / (1 - LGDt (i) ) (14)

Translated into the periods of the recovery rate, the following relation is obtained:

Yt (i) =log (1-Rt (i) ) /Rt (i) ) = -logRt (i) / (1-Rt (i) ) (15)

LGD may be also calculated as follows:

LGDt (i) = exp (yt (i) ) /1+exp (yt (i) ) (16)

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Koncept procene EAD (Exposure At Default - izloženost difoltu)

Izloženost kod difolta (EAD) predstavlja treći ulazni parametar kod proračuna IRB kapitala. Banke koje žele da koriste pristup zasnovan na unutrašnjem rangiranju moraju da koriste sopstvene procene vrednosti ovog parametra. EAD predstavlja ukupan iznos gubitka sa kojim se zajmodavac susreće kada dužnik upadne u difolt po zajmu. Postoje različiti načini izračunavanja EAD u zavisnosti od osnovnog ili naprednog pristupa. Dok kod standardnog pristupa (IRB) obračunavanje EAD je usklađeno do strane regulatora kod naprednog pristupa banke uživaju veću fleksibilnost kod obračuna EAD-a. Ove vrednosti ne uzimaju u obzir garancije, kolaterale ili hartije od vrednosti.

CCF (Credit Conversion Factors) - faktori kreditne konverzije mora da bude procenjen za van bilansne transakcije i kreditna odobrenja. Oni opisuju procenat stope nepovučenog dela kreditne linije ULC (Undrawn Credit Lines) koje nisu isplaćene, ali koje moraju biti iskorišćenje od strane dužnika dok se default ne dogodi. U

skladu sa tim, EAD za nepovučeni (undrown) iznos je definisan kao: EAD = CCF x ULC

Opšte uzevši varijacije u EAD neće biti potcenjene razmatrajući izlozenosti bilansa stanja. Ovo može biti ilustrovano kroz sledeći primer. Dva kredita se koriste na nivou od 1.000 EUR i tako da nose kamatnu stopu od 6%. Za kredit A se odnosi godišnja dogovorena otplata od 12 % a za kredit B mesečna otplata od 1%. Ako se pretpostavi da oba dužnika prestanu da plaćaju svoje mesečne otplate posle 11 meseci, što zapravo znači da je 90 dana difolta u skladu sa Basel II difolt definicijom. Za kredit A ovo znači da je u vreme difolta ukupan iznos potraživanja od 1.000 EUR dato.

Na drugoj strani, 6% kamate = 6 EUR za jednu godinu i kamata za period od 90 dana oko 1.6 EUR nisu bili plaćeni. Ovo se dodaje ukupnom iznosu potraživanja (EAD) od 107.6 EUR. Za kredit B, ovo znači da 11 otplata i plaćanja kamate biće pravilni, i nadalje, ostatak iznosa potraživanja je 89 EUR. Plaćanje kamata za 12-ti mesec, i za period preko, od 1.75 EUR mora biti dodat. Ukupan iznos zahtevanih potraživanja od strane klijenta (kupca) dodaje se do iznosa oko 90.75 EUR. Ovo ograničenje je takođe od priroritetnog kod LGD procene. Pretpostavimo da klijent B vraća ukupan iznos potraživanja od 90.75 EUR i sve povezane administrativne troškove. U ovom slučaju banka neće navući na sebe jedan ekonomski ili bilansni gubitak. Ali ako se pretpostavi regulatorni EAD-definicija za bilansna potraživanja, EAD se dodaje do 100 EUR i gubici (LGD) od oko 10% ce se desiti.

Razvoji EAD modela, uopšteno uzevši, zaostaju iza modela PD i LGD. Odsustvo spoljnih mera ili modela predstavlja jednu direktnu posledicu nedostatka uključivanja akademskih eksperata ili konsultanata u proces procene EAD.

Zaključak

Orijentacija pristupa zasnovanom na unutrašnjem rangiranju je u skladu sa razvojem sistema menadžmenta rizika kao pristupa internog profila kreditnog rizika i kapitalne adekvatnosti banke. Bankarsko poslovanje je usmereno ka poboljšanju značajnosti i kvantifikovanosti procene jedne od najosnovnijih pokretača kreditnog rizika - rizik od neizvršenja

BIS Bazel

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In an analogous mode, the model which is being applied in Basel 2 Accord for the transformed value yt(i) is specified as follows:

Yt(i) = µ + σ x (ω)1/2 x ft + σ x (1-ω)1/2 x εt(i) (17)

Random variables ft and ετ(i) are normally distributed while for all the variables we are assuming that they are independent. Parameter Σ is non-negative, while the values of the variable ω are limited to the interval [0,1].

In the next step, the model described in the equation 18 is being expanded by the inclusion of solid and time-specific risk factors which are subject to observation. Dependency under these risk factors is specified according to the following linear approximation:

μt (i) = β 0+ β'x xt-(i) + γ' x zt-1 (18)

where is small i=1, …, nτ t =1, …, T.The value xt-1(i) is characterized as the vector

of the issuer and the factors specific for the bond which was observed in the previous period. The examples for such issuers and bonds are the ratings of the issuers over the previous year or seniority. The value zt-1 designates the vector of macro-economic variables which represents the potential sources of systemic risk.

The concept of the exposure at default - EAD prediction

Exposure at default - EAD is the third input parameter in the calculation of the IRB capital. Banks wishing to use the internal rating based approach must use their own calculation of value of this parameter. EAD is the total amount of losses encountered by the borrower when the obligor defaults in his loan repayment. There are different ways to compute the EAD depending on whether the standard or the advanced approach is applied. In case of the standard IRB approach, the EAD calculation is harmonized by the regulator, while in case of the advanced IRB approach the banks are enjoying higher flexibility in calculating the EAD. These values are not taking into account guarantees, collaterals, or securities.

The CCF - Credit Conversion Factors, must be estimated for the off-balance sheet

transactions and credit approvals. They describe the percentage of the undrawn credit line rate which is not disbursed, but which must be drawn by the obligor until the default occurs. Concordantly, the EAD for the undrawn amount is defined as follows: EAD = CCF x ULC

Generally speaking, EAD variations will not be underestimated in examining the balance sheet exposure. This may be illustrated through the following example: Two credits are deployed on the level of 1000 EUR with an interest rate of 6%. For credit A the annual agreed repayment is 12%, and for credit B monthly repayment is 1%. Assuming that both obligors are to stop their monthly repayments after 11 months, this will actually mean that there are 90 days in default in accordance with the Basel 2 Accord default definition. For credit A this means that at the time of default the total amount of 1000 EUR claim was paid.

On the other hand, 6% interest rate = 6 EUR, for one year and the interest for the period of 90 days some 1.6 EUR, have not been paid. This is added to the total amount of claim (EAD) of 107.6 EUR. For credit B, this means that the 11 repayments and payment of the interest will be correct, and furthermore, that the remaining amount of claim is 89 EUR. Payment of interest for the 12th month, and for the overdraft period, of 1.75 EUR must be added. The total amount of required claims from the client (customer) is being added up to the amount of some 90.75 EUR. This limitation is also a priority importance in the LGD estimate. Assuming that the client B is repaying the total amount of his debt of 90.75 EUR, and all the related administrative costs, in this case the bank will not draw upon itself an economic or a balance sheet loss. But if we assume that the regulatory EAD-definition is for the balance sheet receivables-claims, the EAD is added up to 100 EUR and the loss (LGD) of some 10% will occur.

The development of the EAD models, generally speaking, is lagging behind the PD and the LGD models. The absence of exterior measures or models is a direct consequence of the reluctance of academic experts or consultants to venture into the process of the EAD assessment.

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obaveza dužnika. U skladu sa tim, mnoge banke su poslednjih godina napravile značajan napredak u poboljšanju tradicionalne, kvalitativno - orijentisane unutrašnje procene kreditnog rizika kroz širenje sopstvenih sposobnosti za kvantifikaciju kreditnog rizika povezanim sa bankarskim izloženostima.

Literatura / References

1. Alastair L. Day, Mastering Risk Modeling, First Edition, England, 2003.

2. Bessis, J., Risk Management in Banking, John Weley&Sons, Inc., New York, 2002.

3. Basel Committee on Banking Supervision, Validation of low-default portfolios in the Basel II Framework, Basel Committee Newsletter, No. 6, 2005.

4. Basel Committee on Banking Supervision, The New Basel Capital Accord, January 2001.

5. BIS, Credit risk transfer, January 2003.6. Basel Committee on Banking Supervision,

Range of Practice in Banks Internal Ratings System, January, 2000.

7. Basel Committee on Banking Supervision, Implementation of Basel II: Practical Considerations, July 2004.

8. Basel Committee on Banking Supervision, International Convergence of Capital Measurement and Capital Standards - A Revised Framework, June 2004.

9. Basel Committee on Banking Supervision, Studies on the Validation of Internal Rating Systems, Revised version, May 2005.

10. Engelmann B. And Robert Rauhmeier, The Basel II Risk Parameters: Estimation, Validation and Stress Testing, Dresdner Bank, Berlin, 2006.

11. Greuning van H. and S. Brajovic-Bratanovic, Analyzing and managing banking risk, Second Edition, The World Bank, 2003.

12. Servigny de A. and O. Reanult, Measuring and Managing Credit Risk, McGraw-Hill, New York, 2004.

13. Roy van, P.: Credit ratings and the standardised approach to credit risk in Basel II, ECB, Working Paper, No. 517, August 2005.

14. Til Schuermann, What Do We Know About Loss Given Default, The Wharton Financial Institutions Center, New York, February 2004.

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Conclusion

The focus of the Internal Rating Based approach is concordant with the development of the risk management system as an approach to the internal credit risk and the bank capital adequacy profiling management. Banking business is streamlined towards enhancing the importance and quantification of assessment of

one of the most fundamental credit risk drivers - the obligors defaulting risk. To that end, many banks over the last several years have made significant progress in the enhancement of the traditional, qualitatively- oriented internal credit risk rating, through the expansion of their own capabilities for quantification of the credit risk connected to the banking exposures.