Michael Kalkbrener

Risk management of non-maturing liabilities

Abstract Risk management of non-maturing liabilities is a relatively unstudied issue of significant practical importance. Non-maturing liabilities include most of the traditional deposit accounts like demand deposits, savings accounts and short time deposits and form the basis of the funding of depository institutions. Therefore, the asset and liability management of depository institutions depends crucially on an accurate understanding of the liquidity risk and interest rate risk profile of these deposits. In this paper we propose a stochastic three-factor model as general quantitative framework for liquidity risk and interest rate risk management for non-maturing liabilities. It consists of three building blocks: market rates, deposit rates and deposit volumes. We give a detailed model specification and present algorithms for simulation and calibration. Our approach to liquidity risk management is based on the term structure of liquidity, a concept which forecasts for a specified period and probability what amount of cash is available for investment. For interest rate risk management we compute the value, the risk profile and the replicating bond portfolio of non-maturing liabilities using arbitrage-free pricing under a variance-minimizing measure. The proposed methodology is demonstrated by means of a case study: the risk management of savings accounts.

On the complexity of Gröbner bases conversion

In this paper, the complexity of the conversion problem for Grobner bases is investigated. It is shown that for adjacent Grobner bases F and G, the maximal degree of the polynomials in G, denoted by deg(G), is bounded by a quadratic polynomial in deg(F). For non-adjacent Grobner bases, however, the growth of degrees can be doubly exponential.

Correlation Under Stress in Normal Variance Mixture Models

We investigate correlations of asset returns in stress scenarios where a common risk factor is truncated. Our analysis is performed in the class of normal variance mixture (NVM) models, which encompasses many distributions commonly used in financial modelling. For the special cases of jointly normally and t-distributed asset returns we derive closed formulas for the correlation under stress. For the NVM distribution, we calculate the asymptotic limit of the correlation under stress, which depends on whether the variables are in the maximum domain of attraction of the Frechet or Gumbel distribution. It turns out that correlations in heavy-tailed NVM models are less sensitive to stress than in medium- or light-tailed models. Our analysis sheds light on the suitability of this model class to serve as a quantitative framework for stress testing, and as such provides valuable information for risk and capital management in financial institutions, where NVM models are frequently used for assessing capital adequacy.

An axiomatic characterization of capital allocations of coherent risk measures

An axiomatic definition of coherent capital allocations is given. It is shown that coherent capital allocations defined by the proposed axiom system are closely linked to coherent risk measures. More precisely, the associated risk measure of a coherent capital allocation is coherent and, conversely, for every coherent risk measure there exists a coherent capital allocation.

Asymptotic behaviour of multivariate default probabilities and default correlations under stress

We investigate default probabilities and default correlations of Merton-type credit portfolio models in stress scenarios where a common risk factor is truncated. For elliptically distributed asset variables, the asymptotic limits of default probabilities and default correlations depend on the max-domain of attraction of the asset variables. In the regularly varying case, we derive an integral representation for multivariate default probabilities, which turn out to be strictly smaller than 1. Default correlations are in (0, 1). In the rapidly varying case, asymptotic multivariate default probabilities are 1 and asymptotic default correlations are 0.

CORRELATION UNDER STRESS IN NORMAL VARIANCE MIXTURE MODELS: correlation under stress

We investigate correlations of asset returns in stress scenarios where a common risk factor is truncated. Our analysis is performed in the class of normal variance mixture (NVM) models, which encompasses many distributions commonly used in financial modeling. For the special cases of jointly normally and t�?distributed asset returns we derive closed formulas for the correlation under stress. For the NVM distribution, we calculate the asymptotic limit of the correlation under stress, which depends on whether the variables are in the maximum domain of attraction of the Fréchet or Gumbel distribution. It turns out that correlations in heavy�?tailed NVM models are less sensitive to stress than in medium�? or light�?tailed models. Our analysis sheds light on the suitability of this model class to serve as a quantitative framework for stress testing, and as such provides valuable information for risk and capital management in financial institutions, where NVM models are frequently used for assessing capital adequacy. We also demonstrate how our results can be applied for more prudent stress testing.

Stressed Testing in Credit Portfolio Models

As, in light of the recent financial crises, stress tests have become an integral part of risk management and banking supervision, the analysis and understanding of risk model behaviour under stress has become ever more important. In this paper, we present a general approach to implementing stress scenarios in a multi-factor credit portfolio model and analyse asset correlations, default probabilities and default correlations under stress. We use our results to study the implications for credit reserves and capital requirements and illustrate the proposed methodology by stressing a large investment banking portfolio. Although our stress testing approach is developed in a particular credit portfolio model, the main concept - stressing risk factors through a truncation of their distributions - is independent of the model specification and can be applied to other risk types as well.

Default probabilities and default correlations under stress

We investigate default probabilities and default correlations of Merton-type credit portfolio models in stress scenarios where a common risk factor is truncated. The analysis is performed in the class of elliptical distributions, a family of light-tailed to heavy-tailed distributions encompassing many distributions commonly found in financial modelling. It turns out that the asymptotic limit of default probabilities and default correlations depend on the max-domain of the elliptical distributions mixing variable. In case the mixing variable is regularly varying, default probabilities are strictly smaller than 1 and default correlations are in (0; 1). Both can be expressed in terms of the Student t-distribution function. In the rapidly varying case, default probabilities are 1 and default correlations are 0. We compare our results to the tail dependence function and discuss implications for credit portfolio modelling.