Stefan Huschens

Country Default Probabilities: Assessing and Backtesting

We address the problem how to estimate default probabilities for sovereign countries based on market data of traded debt. A structural Merton-type model is applied to a sample of emerging market and transition countries. In this context, only few and heterogeneous default probabilities are derived, which is problematic for backtesting. To deal with this problem, we construct likelihood ratio test statistics and quick backtesting procedures.

Estimation of Default Probabilities and Default Correlations

This paper provides estimators for the default probability and default correlation for a portfolio of obligors. Analogously to rating classes, homogeneous groups of obligors are considered. The estimations are made in a general Bernoulli mixture model with a minimum of assumptions and in a single-factor model. The first case is treated with linear distribution-free estimators and the second case with the maximum-likelihood method. All problems are viewed from different points of origin to address a variety of practical questions.

Anmerkungen zur Value-at-Risk-Definition

In einfachen Fallen kann der Value-at-Risk (VaR) mit dem p-Quantil einer Verlustverteilung zu einer vorgegebenen Wahrscheinlichkeitpz. B. p = 99%, identifiziert werden. Fur eine allgemeine Definition ist aber zu berucksichtigen, das das p-Quantil mehrdeutig und in extremen Fallen auch negativ sein kann. Der Beitrag diskutiert alternative Varianten der VaR-Definition und schliest mit einer Definition, die auch in Fallen der Mehrdeutigkeit oder Nichtnegativitat des p-Quantils den VaR eindeutig festlegt.

Simultaneous Confidence Intervals for Default Probabilities

A single-factor portfolio model for credit risk with K rating categories and different default probabilities p k and correlations ρ k for each rating category k = 1,…,K is considered. In this framework simultaneous confidence intervals for the default probabilities based on observed relative default frequencies are derived.

Necessary sample sizes for categorial data

This paper deals with the problem how to determine the necessary sample size for the estimation of the parameter π=(π1,...,πk) (πj ≥ 0, Σjπj=1) based on the vector f=(f1,...,fk) of relative frequencies with sample size n. The vector n-f has a multinomial distribution. For a given precision c, 0≤c≤1, and a given confidence number β, 0≤β≤1, there exists a smallest positive integer N0=N0(β, c, k) with P{|fj−πj|≤c; j=1, ...,k}≥β for all sample sizes n≥N0 and for all π. As results are given in this paper exact upper bounds for N0 and an improved asymptotical upper bound for N0 which is derived from the asymptotical multinormal approximation for the distribution of f.

Confidence Intervals for Asset Correlations in the Asymptotic Single Risk Factor Model

The asymptotic single risk factor (ASRF) model, which has become a standard credit portfolio model in the banking industry, is parameterized by default probabilities and asset (return) correlations. In this model, individual and simultaneous confidence intervals for asset correlations are developed on the basis of observed default rates. Since the length of these confidence intervals depends on the confidence level chosen, they can be used to define stress scenarios for asset correlations.

Estimation of Default Probabilities in a Single-Factor Model

The default probability is a central parameter of credit risk models and can be estimated by the relative default frequency in a portfolio. The distribution of this estimator is derived in the framework of a single-factor model. For a sample portfolio consisting of 15 rating categories with different default probabilities individual and simultaneous probability intervals are given.

Konfidenzintervalle für den Value-at-Risk

Fur die parametrische Schatzung des Value-at-Risk (VaR) bei multinormalverteilten Renditen und linearer Portfoliostruktur werden exakte und asymptotische Konfidenzintervalle angegeben und verglichen.1

Nachträglich geschichtete Stichproben und partielle Information

Die Kenntnis der Verteilung eines Merkmales X in der Grundgesamtheit kann auf zwei Arten bei der Schatzung des Mittelwertes oder eines anderen Verteilungsparameters eines interessierenden zweiten Merkmales Y verwendet werden: un bereits in der Phase der Stichprobenerhebung eine Schichtung durchzufuhren oder um erst in der Phase der Schatzung durch Schichtung einen verbesserten Schatzer zu erhalten. Im zweiten Fall spricht man von Posteriori-Schichtung (post stratification) oder nachtraglicher Schichtung. Diese ermoglicht die Verwendung modifizierter Schatzer mit Effizienzgewinnen durch den Schichtungseffekt.

From Credit Scores to Stable Default Probabilities: A Model Based Approach

The default probability of borrowers is one of the crucial inputs for calculating the regulatory capital underlaying credit risk. Currently, default probabilities are estimated in rating grades by using historical default data. In this attempt, information provided by many banks is lost by the classification of credit scores into rating grades. To avoid this loss of information, a direct mapping of credit scores to default probabilities is presented in this paper. This also guarantees stable default probabilities as demanded by the Basel Committee. The model uses an increasing and convex functional relation between credit scores and default probabilities of individual borrowers. Consequently, the approach focuses on the estimation of model parameters instead of the default probability itself. Maximum-Likelihood estimators for the model parameters are derived, which account for heteroskedasticity.