arXiv subjects

Dirk Tasche

Publications and source records attributed to Dirk Tasche.

At least 37 records · Page 2Linked to original sources

arXiv2014

The Law of Total Odds

The law of total probability may be deployed in binary classification exercises to estimate the unconditional class probabilities if the class proportions in the training set are not representative of the population class proportions. We argue that this is not a conceptually sound approach and suggest an alternative based on the new law of total odds. We quantify the bias of the total probability estimator of the unconditional class probabilities and show that the total odds estimator is unbiased. The sample version of the total odds estimator is shown to coincide with a maximum-likelihood estimator known from the literature. The law of total odds can also be used for transforming the conditional class probabilities if independent estimates of the unconditional class probabilities of the population are available. Keywords: Total probability, likelihood ratio, Bayes' formula, binary classification, relative odds, unbiased estimator, supervised learning, dataset shift.

math.PR↗

arXiv2013

The art of probability-of-default curve calibration

PD curve calibration refers to the transformation of a set of rating grade level probabilities of default (PDs) to another average PD level that is determined by a change of the underlying portfolio-wide PD. This paper presents a framework that allows to explore a variety of calibration approaches and the conditions under which they are fit for purpose. We test the approaches discussed by applying them to publicly available datasets of agency rating and default statistics that can be considered typical for the scope of application of the approaches. We show that the popular 'scaled PDs' approach is theoretically questionable and identify an alternative calibration approach ('scaled likelihood ratio') that is both theoretically sound and performs better on the test datasets. Keywords: Probability of default, calibration, likelihood ratio, Bayes' formula, rating profile, binary classification.

q-fin.RM↗

arXiv2013

Bayesian estimation of probabilities of default for low default portfolios

The estimation of probabilities of default (PDs) for low default portfolios by means of upper confidence bounds is a well established procedure in many financial institutions. However, there are often discussions within the institutions or between institutions and supervisors about which confidence level to use for the estimation. The Bayesian estimator for the PD based on the uninformed, uniform prior distribution is an obvious alternative that avoids the choice of a confidence level. In this paper, we demonstrate that in the case of independent default events the upper confidence bounds can be represented as quantiles of a Bayesian posterior distribution based on a prior that is slightly more conservative than the uninformed prior. We then describe how to implement the uninformed and conservative Bayesian estimators in the dependent one- and multi-period default data cases and compare their estimates to the upper confidence bound estimates. The comparison leads us to suggest a constrained version of the uninformed (neutral) Bayesian estimator as an alternative to the upper confidence bound estimators.

q-fin.RM↗

arXiv2012

Bounds for rating override rates

Overrides of credit ratings are important correctives of ratings that are determined by statistical rating models. Financial institutions and banking regulators agree on this because on the one hand errors with ratings of corporates or banks can have fatal consequences for the lending institutions and on the other hand errors by statistical methods can be minimised but not completely avoided. Nonetheless, rating overrides can be misused in order to conceal the real riskiness of borrowers or even entire portfolios. That is why rating overrides usually are strictly governed and carefully recorded. It is not clear, however, which frequency of overrides is appropriate for a given rating model within a predefined time period. This paper argues that there is a natural error rate associated with a statistical rating model that may be used to inform assessment of whether or not an observed override rate is adequate. The natural error rate is closely related to the rating model's discriminatory power and can readily be calculated.

q-fin.RM↗

arXiv2012

Capital allocation for credit portfolios under normal and stressed market conditions

If the probability of default parameters (PDs) fed as input into a credit portfolio model are estimated as through-the-cycle (TTC) PDs stressed market conditions have little impact on the results of the capital calculations conducted with the model. At first glance, this is totally different if the PDs are estimated as point-in-time (PIT) PDs. However, it can be argued that the reflection of stressed market conditions in input PDs should correspond to the use of reduced correlation parameters or even the removal of correlations in the model. Additionally, the confidence levels applied for the capital calculations might be made reflective of the changing market conditions. We investigate the interplay of PIT PDs, correlations, and confidence levels in a credit portfolio model in more detail and analyse possible designs of capital-levelling policies. Our findings may of interest to banks that want to combine their approaches to capital measurement and allocation with active portfolio management that, by its nature, needs to be reflective of current market conditions.

q-fin.RM↗

arXiv2010

Estimating discriminatory power and PD curves when the number of defaults is small

The intention with this paper is to provide all the estimation concepts and techniques that are needed to implement a two-phases approach to the parametric estimation of probability of default (PD) curves. In the first phase of this approach, a raw PD curve is estimated based on parameters that reflect discriminatory power. In the second phase of the approach, the raw PD curve is calibrated to fit a target unconditional PD. The concepts and techniques presented include a discussion of different definitions of area under the curve (AUC) and accuracy ratio (AR), a simulation study on the performance of confidence interval estimators for AUC, a discussion of the one-parametric approach to the estimation of PD curves by van der Burgt (2008) and alternative approaches, as well as a simulation study on the performance of the presented PD curve estimators. The topics are treated in depth in order to provide the full rationale behind them and to produce results that can be implemented immediately.

q-fin.RM↗

arXiv2008

Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle

Despite the fact that the Euler allocation principle has been adopted by many financial institutions for their internal capital allocation process, a comprehensive description of Euler allocation seems still to be missing. We try to fill this gap by presenting the theoretical background as well as practical aspects. In particular, we discuss how Euler risk contributions can be estimated for some important risk measures. We furthermore investigate the analysis of CDO tranche expected losses by means of Euler's theorem and suggest an approach to measure the impact of risk factors on non-linear portfolios.

q-fin.PM↗

arXiv2008

Capital allocation for credit portfolios with kernel estimators

Determining contributions by sub-portfolios or single exposures to portfolio-wide economic capital for credit risk is an important risk measurement task. Often economic capital is measured as Value-at-Risk (VaR) of the portfolio loss distribution. For many of the credit portfolio risk models used in practice, the VaR contributions then have to be estimated from Monte Carlo samples. In the context of a partly continuous loss distribution (i.e. continuous except for a positive point mass on zero), we investigate how to combine kernel estimation methods with importance sampling to achieve more efficient (i.e. less volatile) estimation of VaR contributions.

math.ST↗

arXiv2007

Incorporating exchange rate risk into PDs and asset correlations

Intuitively, the default risk of a single borrower is higher when her or his assets and debt are denominated in different currencies. Additionally, the default dependence of borrowers with assets and debt in different currencies should be stronger than in the one-currency case. By combining well-known models by Merton (1974), Garman and Kohlhagen (1983), and Vasicek (2002) we develop simple representations of PDs and asset correlations that take into account exchange rate risk. From these results, consistency conditions can be derived that link the changes in PD and asset correlation and do not require knowledge of hard-to-estimate parameters like asset value volatility.

q-fin.RM↗

arXiv2006

Measuring sectoral diversification in an asymptotic multi-factor framework

We investigate a multi-factor extension of the asymptotic single risk factor (ASRF) model that underlies the capital charges of the "Basel II Accord". In this extended model, it is still possible to derive closed-form solutions for the risk contributions to Value-at-Risk and Expected Shortfall. As an application of the risk contribution formulae we introduce a new concept for a diversification measure. The use of this new measure is illustrated by an example calculated with a two-factor model. The results with this model indicate that, thanks to dependence on not fully correlated systematic sectors, there can be a substantial reduction of risk contributions by sectoral diversification effects.

physics.soc-ph↗

arXiv2006

Validation of internal rating systems and PD estimates

This paper elaborates on the validation requirements for rating systems and probabilities of default (PDs) which were introduced with the New Capital Standards (Basel II). We start in Section 2 with some introductory remarks on the topics and approaches that will be discussed later on. Then we have a view on the developments in banking regulation that have enforced the interest of the public in validation techniques. When doing so, we put the main emphasis on the issues with quantitative validation. The techniques discussed here could be used in order to meet the quantitative regulatory requirements. However, their appropriateness will depend on the specific conditions under which they are applied. In order to have a common ground for the description of the different techniques, we introduce in Section 3 a theoretical framework that will be the basis for the further considerations. Intuitively, a good rating system should show higher probabilities of default for the less creditworthy rating grades. Therefore, in Section 4, we discuss how this monotonicity property is reflected in the theoretical framework from Section 3. In Section 5, we study the meaning of discriminatory power and some tools for measuring it in some detail. We will see that there are tools that might be more appropriate than others for the purpose of regulatory validation of discriminatory power. The topic in Section 6 is calibration of rating systems. We introduce some of the tests that can be used for checking correct calibration and discuss the properties of the different tests. We then conclude in Section 7 with some comments on the question which tools might be most appropriate for quantitative validation of rating systems and probabilities of default.

physics.soc-ph↗

arXiv2005

Estimating Probabilities of Default for Low Default Portfolios

For credit risk management purposes in general, and for allocation of regulatory capital by banks in particular (Basel II), numerical assessments of the credit-worthiness of borrowers are indispensable. These assessments are expressed in terms of probabilities of default (PD) that should incorporate a certain degree of conservatism in order to reflect the prudential risk management style banks are required to apply. In case of credit portfolios that did not at all suffer defaults, or very few defaults only over years, the resulting naive zero or close to zero estimates would clearly not involve such a sufficient conservatism. As an attempt to overcome this issue, we suggest the "most prudent estimation" principle. This means to estimate the PDs by upper confidence bounds while guaranteeing at the same time a PD ordering that respects the differences in credit quality indicated by the rating grades. The methodology is most easily applied under an assumption of independent default events but can be adapted to the case of correlated defaults.

cond-mat.other↗

arXiv2005

Calculating credit risk capital charges with the one-factor model

Even in the simple one-factor credit portfolio model that underlies the Basel II regulatory capital rules coming into force in 2007, the exact contributions to credit value-at-risk can only be calculated with Monte-Carlo simulation or with approximation algorithms that often involve numerical integration. As this may require a lot of computational time, there is a need for approximate analytical formulae. In this note, we develop formulae according to two different approaches: the granularity adjustment approach initiated by M. Gordy and T. Wilde, and a semi-asymptotic approach. The application of the formulae is illustrated with a numerical example. Keywords: One-factor model, capital charge, granularity adjustment, quantile derivative.

cond-mat.other↗

arXiv2004

The single risk factor approach to capital charges in case of correlated loss given default rates

A new methodology for incorporating LGD correlation effects into the Basel II risk weight functions is introduced. This methodology is based on modelling of LGD and default event with a single loss variable. The resulting formulas for capital charges are numerically compared to the current proposals by the Basel Committee on Banking Supervision. Keywords: Regulatory capital charge, loss given default (LGD).

cond-mat.other↗

arXiv2004

Calculating Concentration-Sensitive Capital Charges with Conditional Value-at-Risk

By mid 2004, the Basel Committee on Banking Supervision (BCBS) is epected to launch its final recommendations on minimum capital requirements in the banking industry. Although there is the intention to arrive at capital charges which concur with economic intuition, the risk weight formulas proposed by the committee will lack an adequate treatment of concentration risks in credit portfolios. The question arises whether this problem can be solved without recourse to fully-fledged portfolio models. Since recent practical experience shows that the risk measure Conditional Value-at-Risk (CVaR) is particularly well suited for detecting concentrations, we develop the semi-asymptotic approach by Emmer and Tasche in the CVaR context and compare it with the capital charges recently suggested by the Basel Committee. Both approaches are based on the same Vasicek one-factor model.

cond-mat.other↗

arXiv2003

A traffic lights approach to PD validation

As a consequence of the dependence experienced in loan portfolios, the standard binomial test which is based on the assumption of independence does not appear appropriate for validating probabilities of default (PDs). The model underlying the new rules for minimum capital requirements (Basle II) is taken as a point of departure for deriving two parametric test procedures that incorporate dependence effects. The first one makes use of the so-called granularity adjustment approach while the the second one is based on moment matching.

cond-mat↗

arXiv2002

Credit Risk Contributions to Value-at-Risk and Expected Shortfall

This paper presents analytical solutions to the problem of how to calculate sensible VaR (Value-at-Risk) and ES (Expected Shortfall) contributions in the CreditRisk+ methodology. Via the ES contributions, ES itself can be exactly computed in finitely many steps. The methods are illustrated by numerical examples.

cond-mat↗

arXiv2002

Expected Shortfall and Beyond

Financial institutions have to allocate so-called "economic capital" in order to guarantee solvency to their clients and counter parties. Mathematically speaking, any methodology of allocating capital is a "risk measure", i.e. a function mapping random variables to the real numbers. Nowadays "value-at-risk", which is defined as a fixed level quantile of the random variable under consideration, is the most popular risk measure. Unfortunately, it fails to reward diversification, as it is not "subadditive". In the search for a suitable alternative to value-at-risk, "Expected Shortfall" (or "conditional value-at-risk" or "tail value-at-risk") has been characterized as the smallest "coherent" and "law invariant" risk measure to dominate value-at-risk. We discuss these and some other properties of Expected Shortfall as well as its generalization to a class of coherent risk measures which can incorporate higher moment effects. Moreover, we suggest a general method on how to attribute Expected Shortfall "risk contributions" to portfolio components. Key words: Expected Shortfall; Value-at-Risk; Spectral Risk Measure; coherence; risk contribution.

cond-mat↗