# Kreditrisiko und Basel III

### Network modelling of credit concentration risk

*Master Thesis in "Mathematical Finance", Oxford (2016)*

Concentration risk is the risk arising from high exposure to single risk factors. Standard concentration measures rely mainly on the aggregation of the measures of each portfolio element at, for example the sectorial level (Lutkebohmert, 2008). They underestimate, however, direct economic links between the elements of the portfolio and therefore the underlying structure of the portfolio. This thesis builds on previous work on concentration risk with a network model (Sindel, 2009), which used a varying parameter for progressively eliminating interdependencies between obligors, and analysed the exposure of the component with largest exposure to measure the portfolio's concentration risk. While this method is able to measure the properties of specifc portfolios, the expected theoretical properties of the model have not been studied yet. This thesis studies the theoretically expected distribution of the largest component in the sparsified correlation matrix by making a link between the varying parameter and the distribution function of the interobligor correlations, and interpreting these on the light of the random graph model. For this, it makes the assumptions that (1) the exposure is homogeneously distributed among the obligors and that (2) the correlations between obligors are statistically independent. Under these assumptions, we find that the distribution of the expected largest component is strongly dominated by the so-called giant component and therefore strongly dependent on the assumed distribution of the correlations.

### Impact of the revised standardized approach for credit risk on capital requirements for interbank exposures

*Master Thesis in "Risk Management and Regulation", Frankfurt School of Finance & Management (2015)*

Regarding the interconnectedness of the banking industry and the crucial role the banking sector plays for a working economy, an effective banking regulation is essential. As a reaction to the financial crisis starting in 2007, several changes have been made to improve the regulatory standards, in order to prevent bank failures and bailouts that had led to economic disruptions. Many of the past and ongoing modifications are related to the capitalization of banks. In 1988, the BCBS (Basel Committee on Banking Supervision) published a set of minimum capital requirements that banks should fulfill, known as Basel I. With the second Basel Accord - Basel II, initially disclosed in 2004 - those requirements were enhanced. The three pillars of Basel II set out regulatory standards regarding capital requirements, internal risk management practices and disclosure requirements. As a result of the financial crisis, those regulatory rules were further enhanced by Basel III, with the objectives of tightening capital requirements, decreasing bank leverage and raising liquidity. The amendments initiated with Basel III are accompanied by several other regulatory changes most recently announced.

### Backtesting Derivatives Exposures

*Master Thesis in "Mathematical Finance", Oxford (2015)*

The use of over-the-counter derivatives almost certainly implies having an inherent credit risk against the counterparty of the trade. Accordingly,larger banks have developed own forecasting models to handle this risk. Since regulators require to proof a good model quality, banks are obliged to validate their models on a regular basis. However, there is no known method to rigorously validate the models mathematically. In general, banks use representative portfolios rather than the complex real portfolio and argue qualitatively why they assume this is justified. This dissertation proposes a new method to quantitatively evaluate whether those representative portfolios are properly chosen. In addition, for a specific case, the probability of making an error with this approach is estimated in dependence of the choice of the representative portfolio. It is shown that this estimation corresponds to the economic intuition and it is discussed how future model validations can take advantage of the suggested approach.

### Valuation of Contingent Convertible Bonds

*Master Thesis in "Risk Management and Regualtion", Frankfurt School of Finance & Management (2014)*

### Analytical Framework for Credit Portfolio Risk

*Master Thesis in "Mathematical Finance", Oxford (2013)*

Credit portfolio models, in particular the Merton model, are used by many banks for economic capital calculation and steering. The standard approach to Merton style portfolio models is to apply Monte Carlo simulations. However, stable estimates are computationally expensive and can only be obtained by incorporating complex Monte Carlo variance reduction techniques. Furthermore, the Monte Carlo approach is suitable for a full portfolio analysis for the purpose of economic capital estimations, but is di cult to apply to time critical questions like risk-based pricing and real-time deal decision making. Likewise, tasks such as portfolio optimisation and scenario-based stress testing require multiple re-calculation of the loss distribution in numerous di erent settings. This thesis analyses the applicability of an analytical model rst proposed by Pykhtin to accurately estimate economic capital gures both on a portfolio and on a transaction level in a multi-factor Merton framework.

### Counterparty Credit Risk - A Mathematical Framework for Backtesting of Internal Model Methods

*Master Thesis in "Mathematical Finance", Oxford (2012)*

The highlight of the latest financial crisis was the default of Lehmann Brother - one of the largest banks world-wide with a large portfolio of OTC contracts with other financial institutions. This default showed the existence of counterparty credit risk - the risk of a counterparty's default prior to a contract's maturity - even if contracts are trated with well known and solvent counterparties. Thus supervisors highlighted this risk lately within new regulations and demanded firms to enhance their counterparty risk controlling.

Internal Model Methods - hereafter also reffered as IMMs - are one way to calculate Counterparty Credit Risk and base on internal calculated distributions. Important aspect of an Internal Model Method is the observation of the used framework with focus on a good performance. Backtesting is a wide spreaded and generally well accepted method to measure the performance on internal models and shall thus be used for the observation of Internal Model Methods in Counterparty Credit Risk. If backtesting of Internal Model Methods states that these models are not sufficiently accurate the regulatory capital to cover Counterparty Credit Risk must be increased or even worse the usage necessary data for IMMs based on potential future exposures this work introduces a first mathematical framework for the problem of backtesting for the used counterparty credit risk model. This framework includes various test methods expanding well known and sufficiently discussed market risk methods like traffic light approaches, conditional and unconditional coverage tests and independence testing to the new challenge of counterparty credit risk. Major problems of the backtesting approach are described and solutions to minimize its impact are introduced with major focus on autocorrelation of time series and the problem of frequently changing portfolios.

### Incremental Risk Charge Modelling within a Merton-Style Factor Model

*Master Thesis in "Mathematical Finance", Oxford (2011)*

This thesis deals with the computation of the incremental risk charge (IRC) with a modified Mertonstyle portfolio model.The incremental risk charge is an additional capital buffer for a bank’s trading book and covers risks arising from rating migration and default events. Modelling the IRC is currently a top priority topic in market risk management since the charge becomes effective by 2012 and since it heavily increases the overall regulatory capital to be hold for the trading book. The IRC models that are currently developed by major banks combine both market- and credit risk components and hence IRC is a first step to integrate market and credit risk modelling.

In this thesis we firstly present the motivation, the regulatory evolution and the final regulatory requirements on IRC.

Secondly, we develop a fully-fledged IRC model based on a Merton-style credit portfolio that is compliant with the regulatory requirements.

In the third chapter, we specify three enhancements of the basic IRC model that improve its risk sensitivity, namely how to incorporate active short term management of trading products (constant level of risk assumption), the modelling of a stochastic recovery rate and the consideration of the default risk of hedge counterparties.

Fourthly, we implement the specified IRC model and conduct a wide range of quantitative studies to assess the effect of model assumptions, calibration parameters and portfolio composition of real bank portfolios with different risk profiles as well as different compositions of traded bonds and related hedge positions.

The numerical results are presented in chapter 5.

One major feature of this thesis is the consistent integration of a wide range methodologies from market- and credit risk: to model correlated migration- and default events a Merton-style portfolio model commonly used to compute economic capital for banking books is used, liquidity aspects are incorporated by a multistep extension of this model and the reevaluation of positions given rating migration is based on the standard spread curve model for general- and specific market risk. The repricing of positions with modified credit spread curves is based on the standard pricing formulae. Therefore, we elaborate in detail the pricing theory for bond- and CDS positions.

Another major feature of the thesis is the practical relevance of its quantitative results: The comprehensive test calculations have been conducted on real world (sub) portfolios with realistic parameterizations and condense the experience gained on several IRC consulting projects. Altogether, we believe that this work presents interesting results regarding the severity of the IRC, the materiality of rating migration events (non-default) for the IRC, the IRC’s sensitivities to correlation parameters, the impact of the constant-level-of-risk assumption and the effects implied by stochastic recovery and the modelling of hedge counterparty defaults.

### Counterparty Exposure for Claims with Optionality

*Master Thesis in "Mathematical Finance", Oxford (2010)*

This study analysises the impact of potential counterparty default on optimal exercise decisions. It considers financial claims with embedded optionality under a hybrid credit & interest-rate set-up to capture risk factors to actual pay-out from both credit and market risk.

First a brief review of counterparty credit risk for financial claims, and standard mitigating measures like central clearing or collateral and netting agreements is given. Netting agreements typically interconnect value of claims on a portfolio level. This makes conventional contract-wise valuation methods inapplicable. In order to isolate the optionality effect a single claim with netting to a static background portfolio is considered.

In the first part the modelling of counterparty valuation in a consistent replication approach within a simple one interest-rate index economy with bilateral default risk is analysed. In the second part a numerical experiment is used to quantify impact on the approximations to the Credit Valuation Adjustment (CVA) and the exposure profile for typical examples of swaps, European Swaptions and Bermudan Swaptions. Generally, the difference to the standard, non-exercise shifting calculation results is found to be not significant for typical market positions in comparison to other effects. The numerical schemes for CVA and exposure are much more complex than their pricing counterparts, making practical applications unlikely, and increase the uncertainy in numerical values considerably. The difference in exposure profiles could potentially impact risk management of a portfolio, though, and further study on the inclusions of suitable approximations in large-scale portfolio simulations currently being rolled out in banks is potentially warranted.

### Comparison of Credit Risk Models by a Historical Analysis of Portfolio Hedges

*Master Thesis in "Mathematical Finance", Oxford (2009)*

This thesis contains an analysis of credit risk models in which the quality of hedges of credit derivatives was investigated over a time period in the past. The credit derivatives under consideration were index credit default swaps and collaterlized debt obligations. We consider two market models for pricing these derivatives, the uniform market model and the Gaussian copula model. For a time series of daily price quotes the model parameters are determined such that the market prices are reproduced. The obtained model calibrations were used to determine hedges that reduce the dependence on risky market parameters. From the history of profits and losses of the resulting hedged portfolios the standard deviations were taken as a (reciprocal) measure of the hedge quality.

The results indicate that the models are similarly successful. Hedging away the hazard rate risk reduces the standard deviation of profits and losses in most cases by about 40%{60%. If the calibration is working well and profits and losses can be explained in terms of the greeks, the hedge can be improved by additionally hedging the correlation risk with the equity tranche. However, the improvement is limited if only parallel shifts of the correlations are considered. It turns out that the choice of the hedge has more infuence than the pricing model. If a fixed amount of a certain tranche is held, the uniform market model performs slightly better. If, on the other hand, one is interested in a fixed coupon leg size, then the Gaussian Copula model seems to be advantageous. However, the small sample sizes of about 200 days make a definite statement diffcult.

### A new Methodology for the Assessment of Concentration Risk

*Master Thesis in "Mathematical Finance", Oxford (2009)*

One important goal of the Basel II framework is to provide a universal and simple formula for the calculation of the regulatory capital of a bank. Due to this constraints the Concentration Risk included in any real-world portfolio is not taken into account within this framework. Motivated by this drawback we develop a universal methodology for the assessment of the Concentration Risk of an arbitrary portfolio in this thesis. In contrast to traditional methods we use both the correlation- and exposure-structure of the portfolio as an input. We employ graph theory for the measurement of this quantity. We resolve the internal structure of a portfolio by gradually increasing a so-called ramping parameter and monitoring the effective "exposure-weighted" graph for each value of the sequence. This procedure provides the functional dependence between the biggest exposure carried by a single connected component and the employed ramping parameter. We weight this curve with an economically motivated weight function and measure the Concentration Risk by computing the area under the resulting curve. We employ our methodology on various test portfolios and demonstrate its adequacy. Our methodology provides a universal framework for the assessment of Concentration Risk: instead of describing the considered portfolio by its correlation-matrix (which is usually difficult to obtain) one can also use any other quantity describing the internal structure of the portfolio (e.g. an experts judgement).

### Stability of Credit Portfolio Models

*Master Thesis in "Mathematical Finance", Oxford (2008)*

In this thesis we examined the stability of credit portfolio models which are at the heart of modern credit risk management. Therefore, we investigated the impact of changes in certain model parameters or the portfolio setup on the loss distribution as well as the resulting economic capital. The parameters of interest for which we analysed the model sensitivities covered both, obligor specific (probability of default and loss given default) as well as portfolio specific quantities (correlation and concentration). For our numerical simulations we focused on CreditRisk+ and CreditMetrics since these models also are widely used within the banking industry. In order to quantify economic capital we used risk measures based on Value at Risk as well as Expected Shortfall, which have been controversially discussed in this context. Our main findings are that both models proved to be relatively stable over a wide parameter range of practical interest. However, even small parameter shifts as they can occur, e.g., in a period of economic downturn or simply as a result of parameter misspecification may lead to a quite dramatic increase of the predicted economic capital. Further, we found that both high portfolio concentration as well as strong correlations may lead to a breakdown of the standard behavior thereby indicating a model instability.