In this paper it is shown how a Vasicek-model approach and the assumptions in Basel 2 regulatory framework can be used to develop measures of the probability of banks' failure. The Basel 2 framework is based on a Vasicek-model approach. The estimation of the propose measure of bank probability of default could be made over the capital ratio from supervisory authorities (non-public information) or over the

capital ratio from balance sheet data (public available information)

Minkova, Leda

Radkov, Petar

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Munich Personal RePEc ArchiveAssessing bank’s default probabilityusing the ASRF modelRadkov, Petar and Minkova, LedaFaculty of Mathematics and Informatics, Sofia University "St. Kl.Ohridski"25 June 2011Online at https://mpra.ub.uni-muenchen.de/60186/MPRA Paper No. 60186, posted 26 Nov 2014 06:59 UTCAssessing bank’s default probability using theASRF modelPetar Radkov Leda MinkovaFaculty of Mathematics and InformaticsSofia University ”St. Kl.Ohridski”AbstractIn this paper it is shown how a Vasicek-model approach and the assumptions inBasel 2 regulatory framework can be used to develop measures of the probability ofbanks’ failure. The Basel 2 framework is based on a Vasicek-model approach. Theestimation of the propose measure of bank probability of default could be made overthe capital ratio from supervisory authorities (non-public information) or over thecapital ratio from balance sheet data (public available information).Key words: Vasicek model, ASRF model, Basel 2, banks’ probability of defaultMathematics Subject Classification 2000: 60J05; 60J10; 60J20; 91B24.1 IntroductionThe Vasicek-model approaches and the assumptions in Basel 2 regulatory framework couldbe used to develop measures of the probability of banks’ failure. Capital requirements’set by Basel 2 are based on standardize approach or advanced internal risk based (IRB)approach. The banks’ probability of default measure could be extracted by the capital ratiofrom supervisory authorities (non-public information). The same ratio could be derived1from balance sheet data (public available information).When we know the probability ofdefault of loans portfolio according to the Vasicek-model approach (see Vasicek 2002) wecould calculate the requirement bank capital equal to 99.9The modified model could be usedto estimate the banks’ probability of default where the IRB approach set by Basel 2 is notaccepted from regulatory institutions or the banks do not use it.The remainder of this paper is organized as follows. The second section presents a briefreview .. In the third section we describe our model.2 BackgroundUsually in practice there are two alternative methodologies for estimating the probabilityof default. The first one is based on the historical data analysis. The second is based onthe conception of imply probability measure from the option pricing theory, developed byMerton (1974) and then continuing by Vasicek (1987). The last method is extended byBasel 2 as a regulatory framework to valuate the loss on loan portfolio and to calculaterequirement capitals. This paper is closely related to the second concept for quantifying thebanks probability of default.3 MethodologyBasel 2 takes the Vasicek model which is used to determine the regulatory capital in theBasel 2 framework. Under the assumptions of the model could be estimate the expected loss(EL) and unexpected loss (UL) of the banks. The expected loss is determined by three mainingredients: Probability of default (PD), Exposure at default (EAD) and Loss given default(LGD). The EAD means the amount of money owed at the moment counterparty goes intodefault. The LGD is the percentage of the EAD lost during a default. The expected loss isgiven by the following product2EL = PD ∗ EAD ∗ LGD(1)and determines the amount of provisions which should be taken by the banks as a reserves.The UL is the additional loss in nonregular circumstances, for which tier capital should beretained. In the Vasicek model approach the UL is defined by the PD in a downturn situation.The model assumes that the EAD and LGD are not affected by dire circumstances. Theboth parameters are considered constant for a company.VaR is defined as a threshold value x such that the probability of exceedance x is equalto α < 1. α is called a confidential level. Basel 2 uses VaR coefficient as a measure of risk.The VaR is obtained by algebraically inverting the distribution function F (x)P (L ≤ x) = F (x),(2)where x is set to be equal to the maximum losses (Value at Risk). Basel 2 sets the probability1− α equal to 0.999 (BIS 2005) and V aR99.9 is equal to x = F−1(0.999).The unexpected loss is calculated as followingUL = V aR99.9 − PD ∗ LGD(3)More specifically, the new capital accord uses the Vasicek result to compute the Value atRisk (VaR) for each obligor, which is the maximum loss amount that is not expected to beexceeded with a given level and within a defined time horizon. In general the banks areexpected to cover their expected loss on an ongoing basis by provisions and write-offswhile Basel 2 requires banks to hold capital only to protect against unexpected losses.Our task is to assess banks’ default probability from the banks’ required capital or capitalfrom the balance sheets. That is an inverting problem of Vasicek model. VaR is computedfor each obligor and found out the total required capital when the probability of default3of each obligor is known. The estimation of the proposed measure of bank probability ofdefault could be made over the capital ratio from supervisory authorities (non-public infor-mation) or over the capital ratio from balance sheet data (public available information). Theassumptions under which we calculate the implied probability of default are the following:• Banks portfolios are exposed only to the credit risk, Banks’ credit portfolio is homo-geneous and the portfolio is infinitely divisible,• The maximum loss (VaR) is set by Basel 2 to be equal to 99.9• Required capital determinate by standardized approach or by internal rating basedapproach IRB (foundation and advanced approaches) both of them base on Basel 2 isa sufficient to cover the unexpected loss in 0.999.The basis of the Basel 2 IRB approach is the Vasicek portfolio credit loss model (Vasicek,1987, Vasicek, 2002). Vasicek offers one factor model which was firstly introduce by Mertonin 1974 . Consider a portfolio consisting of n loans. Dynamic of the value of the assets of aborrower i for i = 1, 2, ...n can be described by the geometrical Brownian motion:dAi = µAi(t)dt+ σAi(t)dWi(t)(4)where W (t) is a Wiener process and W (t) N(0, t).Thus the value of assets at time T can be expressed as:Ai(T ) = Ai(0)e(µ−σ22)T+σ√TW (T )(5)where W(T) is a normal distributed random variable.4Basel’s 2 formula uses Merton’s model interpretation of default probability of the bor-rower i, so an obligor will default if the value of his assets falls below the value of his debt:p = P (Ai(T ) < Bi) = P (Wi < ci) = N(ci) = N(−di)(6)ci =ln(Bi/Ai)− (µ+ σ22)Tσ√T(7)where N(−d2) is the standard normal distribution function giving the default probabilitywhich is equal to p. Then p = N(ci) and therefore ci = N−1(p). For fixed t we assume thatWi has the following expression:Wi = by + aεi(8)where y represents systemic risk affecting the entire portfolio and is the specific risk forborrower i. It is assumed that both of them follow standard normal distributions. Thevalues of b and a are given by:b =√ρ a =√1− ρ(9)where ρ measures the asset correlation between borrowers’ assets. It could be describedas the dependence of the assets of a borrower on the general state of the economy; soall borrowers are linked to each other by the single risk factor, etc the common pairwisecorrelation between obligors (see Vasicek 1991).Under those conditions, default probability of any loan, conditional on the single riskfactor y, can be written as:P (x|y) = P (by + aεi < ci) = P(εi <ci − bya)P (x|y) = P(εi <ci − bya)= P(εi <N−1(p)−√ρy√1− ρ)(10)5We can establish that the percentage of default on total portfolio equals the number ofindividuals that default divided by total number of obligators. If n is large enough, by thelaw of the large numbers it can be stated that the fraction of clients x that default is equalto the default probability conditional on y (4).Under certain homogeneity assumptions, the Vasicek model leads us to very simple an-alytic asymptotic approximation of loss distribution due to each obligor (see Vasicek 2002).The cumulative distribution function of loan losses L on a very large portfolio is given bythe limit:P (L ≤ x) = F (x) = N[√1− ρN−1(x)−N−1(PD)√ρ](11)where N is the standard normal distribution function. We use the notation PD for theprobability of default, previously noted by p.According to the Basel 2, if the LGD is equal to 1, 0.999 VaR (BIS 2005) has the followingrepresentation:V aR99.9 = N[√11− ρN−1(PD) +√ρ1− ρN−1(0.999)](12)the loss could be recovered, i.e. the LGD is not equal to 1 thenV aR99.9 = LGD ∗N[√11− ρN−1(PD) +√ρ1− ρN−1(0.999)](13)The VaR quantity represents the basis for the computation of the bank capital require-ment.Then from (3) and (6) we get the formula for calculating capital requirements, known asa capital allocation parameter:C =[LGD ∗N[√11− ρN−1(PD) +√ρ1− ρN−1(0.999)]− PD ∗ LGD](14)6According to the Basel 2 the formula (7) is adjusted to the following resultC =[LGD ∗N[√11−ρN−1(PD) +√ρ1−ρN−1(0.999)]− PD ∗ LGD](1− 1.5B)−1[1 + (M − 2.5)B]1.06,where M is the time to maturity and B is a function of PD computed asB = [0.11852− 0.05478 ln(PD)]2The equation (8) represents the adjusted capital in order to take into account the expectedloss and the maturity adjustment factors.According to the Basel 2 the correlation coefficient ρ is given by:ρ = 0.121−e−50PD1−e−50+ 0.24[1− 1−e−50PD1−e−50]− 0.04[1− S−545],where S is the size of the firm.Given the assumption of the infinite divisibility, the overall capital requirement for abank is computed as a sum of the capital requirements of each loan exposure:K =∑iCi(PDi)Ai(15)that is the sum of the capital allocation parameter Ci of each exposure i multiplied by itsamount Ai. If the credit portfolio is homogenous then there is one implied probability ofdefault which could be extracted from capital adequacy ratio:K∑i Ai= C (PD∗)(16)We can derive the average PD∗ of a banks’ credit portfolio as a function of capital require-ment K for the bank and the sum of risk weight assets:7PD∗ = f(∑Ai, K)(17)estimate the implied probability of default by equation (13) under the previous assumptions.The capital ratio could be taken from supervisory authorities as a non-public information(capital adequate ratio) or from balance sheet data as a public available information.4 Numerical examplesComing after the described method for assessing the implied probability, we implement it inMatLab. We find out the implied probability of default as a function of capital ratio, givenin Figure 1. The function of implied probability of default vs. capital ratio is an increasingfunction which describes the fact that more risk taken from the bank costs more capital toit.The loss distribution as a function of default probability is given in Figure 2. The lossdistribution has been found with 1 million Monte Carlo simulations.5 ConclusionThis paper describes the derivation of implied default probabilities from the Vasicek modeland Basel 2 framework. Our method for measuring probability of default is very simple andeasy to implement in the both cases when we use the capital ratio from balance sheet data(public available information) and the capital ratio from supervisory authorities (non-publicinformation).Once we know the implied probability of default we could get the following: 1. Findout the loss distribution via Monte Carlo simulations (Figure2); 2. Find out the expectedloss which should be coved by provisions and write-offs. 3. Make simulations and stresstests what will happen if the probability of default rose sharply for one bank or for the totalbanking system.8Figure 1: The relation between implied probability of default as a function of capital ratioFigure 2: The Loss distribution as a function of default probability found with 1 millionMonte Carlo simulations.9AcknowledgmentsThe authors were partially supported by the Bulgarian NSRF grant DTK 02/71, 2011.References[1] BIS (1999), Credit Risk Modeling: current practices and applications - April[2] BIS (2003), Overview of the new Basel capital accord, Consultative document- April[3] BIS (2004), Modifications to the capital treatment for expected and unexpected creditlosses- January[4] BIS, (2004), International Convergence of Capital Measurements and Capital Standards:A Revised Framework, Basel Committee on Banking Supervision (referred to as BaselII), June.[5] BIS (2005), An explanatory note on the Basel II IRB Risk Weight Functions - July[6] Merton, R., (1974), On the Pricing of Corporate Debt: The Risk Structure of InterestRates, Journal of Finance, 29, pp. 449-470[7] Vasicek O, (2002), Loan portfolio value, Risk December, 160-162.[8] Vasicek O, (1991), Limiting Loan Loss Probability Distribution, KMV Corporation[9] Vasicek O, (1987), Probability of Loss on Loan Portfolio, KMV Corporation10

Assessing bank's default probability using the ASRF model

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Assessing bank's default probability using the ASRF model

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