TECHNISCHE UNIVERSITÄT DRESDEN Fakultät Wirtschaftswissenschaften

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1 TECHNISCHE UNIVERSITÄT DRESDEN Fakultät Wirtschaftswissenschaften Dresdner Beiträge zu Quantitativen Verfahren Nr. 37/04 BLUEs for Default Probabilities von Konstantin Vogl und Robert Wania Herausgeber: Die Professoren der Fachgruppe Quantitative Verfahren ISSN

2 BLUEs for Default Probabilities Konstantin Vogl and Robert Wania Technische Universität Dresden, Germany Department of Business Management and Economics This Version: December 2, 2003 Abstract Assigning default probabilities to rating classes is a prerequisite for most credit risk models. In this paper the best linear unbiased estimator for the default probabilities is obtained as Aitken-estimator for correlated defaults. The estimation requires only a minimum of assumptions. Remarkable in these findings is that the Aitken-estimator uses the number of defaults in one rating class to estimate the default probability of another rating class, provided they are correlated. 1 Introduction Default probabilities play an essential role in credit risk models. The estimation of default probabilities is therefore a crucial problem. There are different estimation methods applied in practice which usually require extensive assumptions. Maximum-Likelihood based methods need distributional assumptions, scoring systems rely on explaining variables, or other methods simply assume independence of defaults to use well known estimators. In this paper only a minimum of assumptions is made for the estimation procedure. A prerequisite is that obligors are grouped together in rating classes. The estimation is based on historical default data which is available for all rating classes. This makes the method valuable especially for the retail banking portfolio where other methods fail which rely on bond spreads or balance sheet data. To achieve sound estimates, a generalized least squares estimator is used. This estimator is the best linear unbiased estimator (BLUE). In the following sections the variables and assumptions are described, the estimator is set up and special cases for applications are discussed. 2 The Framework In this section the basic variables and model assumptions are introduced. For the estimation problem historical default data of time periods t = 1,...,T is used. Within each period the rating classes r = 1,...,R are considered. In period t 1

3 and rating class r there are n tr obligors. The default variable if obligor i in rating class r 1 defaults over time period t, Y tri := i =1,...,n tr, 0 otherwise, is assigned to the i-th obligor of rating class r in time period t. Similar obligors are grouped together in each rating class, therefore these obligors can be assumed to have the same default probability. This homogeneity leads to E(Y tri )=:π r, under the assumption that the default probability of each rating class is constant over time. The Bernoulli-distributed default variables have the variance σ 2 r := V(Y tri )=π r (1 π r ). Additionally to the time stability of the default probabilities, it is assumed that the covariance of the default variables is constant over time 3 The Estimator Cov(Y tri,y tsj )=:γ rs for i j if r = s. For the estimation it is sufficient to consider the relative default frequencies K tr := 1 n tr n tr The relative default frequencies for different rating classes are combined in a vector for each time period i=1 Y tri. K t := (K t1,...,k tr ). The relative default frequency is a straightforward unbiased estimator for the unknown default probabilities E(K t )=π, where π =(π 1,...,π R ). In the notation of linear regression the residuals are defined by e t := K t π, which yields K t = π + e t, with E(e t )=0. 2

4 The covariance matrix of the relative default frequencies equals the covariance matrix of the residuals since π is non-stochastic. It follows that the covariance matrix can be written as Σ t := E(e t e t)=γ+d t (1) with γ 11 γ 12 γ 1R γ 21 γ 22 γ 2R Γ :=.... γ R1 γ R2 γ RR and σ1 2 γ 11 n t1 0 0 σ γ 22 D t := n t σ 2 R γ RR n tr To formulate the problem of estimation combined for all time periods the vectors are aggregated. In compound vector and block matrix notation and with the R R identity matrix I R R it follows that K 1 I R R e 1 K 2 I R R e 2 K T } {{ } =: K with covariance matrix = Σ := E(ee ) = I R R } {{ } =: M π + e T } {{ } =: e Σ Σ Σ T since the independence of time periods is assumed. The block diagonal elements Σ t are defined in (1). The best linear unbiased estimator ˆπ blue for the parameter vector π is given by ) 1 T ˆπ blue := (M Σ 1 M) 1 M Σ 1 K = ( T t=1 Σ 1 t t=1 (2) (3) Σ 1 t K t. (4) The estimator (4), which is also known as Aitken-estimator 1, is the generalized least squares estimator 2 for π in the linear model (2). 1 After [Ait35]. 2 For a proof of the minimum variance property (also known as Gauss-Markov-Theorem) see [SO91, pp.712] and for the generalization see [JGH + 85, pp.26]. 3

5 4 Discussing the Estimator In the following the Aitken-estimator from (4) is discussed in some special cases. Single Class, Multiple Periods If there is only one single rating class r with historical data of several time periods t =1,...,T the Aitken-estimator simplifies to 3 ˆπ blue (r) := T t=1 T n tr 1+(n tr 1)ϱ rr n sr K tr, (5) s=1 1+(n sr 1)ϱ rr where ϱ rr denotes the default correlation between the default variables. Proofof(5) Corr(Y tri,y trj )=ϱ rr := γ rr, for i j σr 2 In the single class case the covariance-matrix Σ from (3) reduces to ( 1 Diag(V(K 1r ),...,V(K Tr )) = γ rr I T T +(σr 2 γ rr )Diag,..., n 1r and with M = (1,...,1) of dimension T and K =(K 1r,...,K Tr ) the equation (5) is obtained from (4). It can be remarked that even though independence of time periods is assumed, the estimator 1 T T t=1 K tr for π r is not optimal in case n tr varies over time. Multiple Classes, Single Period Considering a single time period t and R rating classes, the best linear unbiased estimator for π is simply given by 1 n Tr ˆπ blue (t) = K t, (6) the vector of the R relative default frequencies in time t. This follows from (4) with M = I R R and K = K t. 3 Please note, the estimator ˆπ (r) blue in the single class case is not the r-th element ˆπ r blue vector ˆπ blue from (4), where all R classes are considered. ) of the 4

6 5 Conclusion The Aitken-estimator provides an approach to the problem of estimating default probabilities for rating classes. Though accounting for default correlation the estimator presented needs only the assumption of time-stable default probabilities and covariances. Proposed for the case of multiple rating classes and multiple time periods, the estimator can easily be customized for various data. Applying the theoretical findings on real data shows that the Aitken-estimator can apply negative weights on observed relative default frequencies. Further, the observed relative default frequencies from all rating classes and all time periods contribute to the estimate of the default probability of each specific rating class. References [Ait35] A.C. Aitken, On least squares and linear combination of observations, Proceedings of the Royal Society of Edinburgh A55(1935), 42. [JGH + 85] G.G. Judge, W.E. Griffiths, R.C. Hill, H. Lütkepohl, and T.-C. Lee, The theory and practice of econometrics, 2 ed., John Wiley & Sons, New York, [SO91] A. Stuart and J.K. Ord, Kendall s advanced theory of statistics, 5ed., vol. 2, Edward Arnold, London,

7 Dresdner Beiträge zu Quantitativen Verfahren (ISSN ) Ältere Ausgaben (1994 7/97): 8/97 S. Huschens: Genauigkeit von Schätzungen des Risikopotentials 9/97 S. Huschens: Confidence intervals for the Value-at-Risk 10/97 S. Huschens: Konfidenzintervalle für den Value-at-Risk 11/97 S. Huschens: Risikoabschätzung durch historische Simulation 12/97 A. Henking: Some approaches in order to model bivariate densities with fixed marginals 13/97 R. Roth: Die Bestimmung des At-the-money-Punktes europäischer Optionen - Implikationen für die Einführung neuer Basispreise an der DTB 14/97 R. Roth: Die Eignung eines Futures auf implizite Forwardvolatilitäten zum Handeln des Vega- Risikos von Optionen 15/97 M. Brechtmann: Wochentagseffekte am deutschen Aktienmarkt unter Berücksichtigung von ARCH-Effekten 16/98 R. Roth: Der VOLAX-Future - Ein Derivat zum Handeln des Vega-Risikos von Optionen 17/98 S. Huschens: Messung des besonderen Kursrisikos durch Varianzzerlegung 18/98 S. Huschens (Hrsg.): Value-at-Risk-Schlaglichter, Ausgabe 1/98 19/98 S. Huschens: Historische Simulation 20/98 J. F. Kiviet, G. D. A. Phillips, B. Schipp: Alternative bias approximations in first order reduced form models 21/98 R. Roth: Die Bewertung des VOLAX-Futures mit dem Arbitrageansatz 22/98 H. W. Brachinger, U. Steinhauser: Konzepte zur Messung von Risiko - vom intuitiven Risikobegriff zum Value at Risk 23/98 S. Huschens (Hrsg.): Value-at-Risk-Schlaglichter, Ausgabe 2/98 24/98 S. Huschens, J.-R. Kim: Measuring risk in Value-at-Risk based on Student s t-distribution 25/98 S. Huschens, J.-R. Kim: Measuring Risk in Value-at-Risk in the Presence of Infinite Variance 26/99 S. Huschens, J.-R. Kim: Blue for β in CAPM with Infinite Variance 27/99 R. Prinzler: Reliability of neural network-based Value-at-Risk estimates 28/99 S. Huschens: Anmerkungen zur Value-at-Risk-Definition 29/99 S. Huschens: Verfahren zur Value-at-Risk-Berechnung 30/00 S. Huschens: Value-at-Risk-Berechnung durch historische Simulation 31/00 S. Huschens: Von der Markt- zur Kreditrisikomessung 32/02 S. Höse, S. Huschens: Sind interne Ratingsysteme im Rahmen von Basel II evaluierbar? Zur Schätzung von Ausfallwahrscheinlichkeiten durch Ausfallquoten 33/03 S. Höse, S. Huschens: From Credit Scores to Stable Default Probabilities: A Model Based Approach 34/03 S. Höse, S. Huschens: Estimation of Default Probabilities in a Single-Factor Model 35/03 S. Höse, S. Huschens: Simultaneous Confidence Intervals for Default Probabilities 36/03 S. Huschens, K. Vogl, R. Wania: Estimation of Default Probabilities and Default Correlations 37/04 K. Vogl, R. Wania: BLUEs for Default Probabilities