Credit Risk and Basel II
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 creditand
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.
Analysis of the Gaussian Copula Assumption in Credit Portfolio Factor Models
Master Thesis in "Mathematical Finance", Oxford (2006)
Credit portfolio factor models have become an industry standard for the calculation
of the loss distribution and the economic capital of large loan or bond portfolios.
These models incorporate the idea that the default event of an obligor can be modeled
by its underlying asset process and a critical threshold, the default point. The
asset value processes of all obligors are simultaneously described by a small set of
factors that represent macro-economic variables like industry and country indices.
The default correlation between the obligors is expressed by the correlations of their
underlying factors. This procedure facilitates the loss calculation for large portfolios.
The models usually assume that the factors follow a standard normal distribution and
use multivariate normal distributions to describe their joint probability distribution,
implicitly applying a Gaussian copula.
Assuming that theMSCI indices can be used as a proxy for the underlying industry
and country factors of the portfolio model, we show that the Gaussian assumption
for the marginal distributions, the distribution of the asset value process, does not
hold. Instead, the log-returns show a pronounced tail structure, hinting at a Student
t distribution as a better approximation.
Based on this result, we propose the family of copulas derived from multivariate
normal mixture distributions for the description of the dependency structure of the
underlying factors. This family provides both the Gaussian and the t copula so that a
systematic comparison between both models becomes possible. In addition, a suitable
copula is fitted to the historical data of the MSCI indices.
Applying a credit portfolio factor model based on the MSCI indices to different
test portfolios, results for the Gaussian copula and different t copulas are obtained
by means of Monte Carlo simulation. By comparing the loss distribution and the
economic capital we show that the choice of the copula has a far reaching effect
on the calculated quantities. The economic capital increases significantly if the tail
dependence of the copula is increased. This effect is most pronounced for obligors of
high credit quality and for weakly correlated portfolios.
Statistical Inference of Default Probabilities for Companies
Master Thesis in "Mathematical Finance", Oxford (2004)
Default risk is the uncertainty of a company's ability to meet its financial obligations.
Prior to the observation of a default, it is not possible to discriminate unambiguously
between companies that will default and those that will not. At best predictions
can be made about the probability of default. In this thesis, a model is presented
to assesss the one-year default probability conditional on the companies' financial
ratios derived from published accounting information. The thesis discuss the model
development in detail, analyse its accuracy and provides a comparison with published
default rates corresponding to external ratings. The model validation suggests that
the introduced internal rating system provides reliable and robust estimates of the
default probability for publicly traded companies.
Credit Risk: Multiperiod Default Prediction and Estimation of Loan Spreads
Master Thesis in "Mathematical Finance", Oxford (2004)
In this thesis, a model for the risk related part of credit spreads for bank loans to corporates is
presented. The underlying condition used for the determination of the spread is that the present
value of earnings from spread is to cover the present value of the expected loss and a part of the
unexpected loss. The relation of the estimated spread on the size of a bank's loan portfolio and
the default correlation among borrowers is discussed in detail.
The model presented for the estimation of credit spreads is based on the term structure of the
probability of default for the borrower and the expected recovery rate for the loan in case of
default. The spread is shown to strongly depend on the timing and amounts for repayments and
interest payments.
The probability of default for corporates is estimated using Generalized Linear Models on the
basis of financial ratios calculated from annual reports. Model parameters are estimated using a
diversified reference sample of publicly quoted industrial companies. Missing financial ratios
due to lack of data are imputed in a separate step.
The numerical results for risk related credit spreads are compared to corresponding market
quoted spreads.
Shape Factor Models In Credit Risk
Master Thesis in "Mathematical Finance", Oxford (2004)
The shape factor model is a new type of term structure model. At first
developed for interest rate risk, it can be expanded to cover credit risk,
as well. The term structure of interest rates or credit spreads is modelled
as a linear combination of basis functions (the shape factors) weighted
by stochastic coefficients. As the comparison of a principle component
analysis of the time series for CDS spreads of different companies reveals,
the shape factor model allows to price credit risky securities with respect
to the most important underlying risk driver, specific to each counterparty.
Furthermore, it offers an attractive platform to integrate credit and interest
rate risk, and can be fitted to incorporate all relevant market data:
interest rates and spread curves, current levels and historical time series,
and finally, implied option volatilities. It is flexible and can be implemented
for two, three, four or more underlying risk factors. Moreover,
by switching between risk-neutral and physical probability measures investors
can estimate the risk premium, which facilitates the investment
process.
The Basel II IRB Approach and Internal Credit Risk Models
Master Thesis in "Mathematical Finance", Oxford (2004)
The following thesis is intended to build a link between the regulatory requirements
given by the Basel II capital accord and an internal Credit risk model.
Based on model assumptions for the internal and external rating structure, the
regulatory internal ratings based approach and the CreditMetricsTM methodology
are applied for a bond portfolio. The characteristic risk parameters of
the internal model are extracted from the loss distribution which results from
Monte Carlo simulations of correlated scenarios. On the level of single positions
the relationship between these results and the risk values following from the
analytic formulas of the capital requirement can be described with the use of a
power law. This enables to connect both approaches and to account for portfolio
effects within the regulatory framework. Furthermore the introduced method
represents a framework of potential practical relevance that could be used for
the internal estimation of the probability of default.
LGD-Rating for a Portfolio of Retail Loans
Diploma Thesis in "Mathematical Finance", Oxford (2004)
The thesis describes a method to develop a rating system for LGD in the case of portfolio of retail loans. The method is based on a LGD-score and uses a calibration technique – the power curve calibration technique – well known from the calculation of probability of default. We show that the method developed is powerful enough to calculate statistically significant values for LGD in a general case without making assumptions about the functional form of the loss given default rate the last chapter of the thesis presents a kind of excurse and deals with the important problem of the plausiblisation [SP?] of the loss given default rates calculated using statistical models.
Pricing and Risk Measurement of Collateralized Mortage Obligations Using Monte Carlo Methods
Master Thesis in "Mathematical Finance", Oxford (2004)
The largest sector of the debt market in the world is the mortgage market. Mortgage loans are widely used as collateral for the creation of securities. Some of these simply pass through the cash flows, like standard mortgage-backed securities (MBS). Others, like collateralized mortgage obligations (CMO), redirect the underlying cash flow to a number of pieces, so-called tranches, in order to meet the investors' needs.
Using four different Monte Carlo methods, the values and risk profiles of a pass-through MBS and of four different CMO containing 12 tranches were determined. A simple lognormal interest rate model and an arctangent prepayment model were applied. The code used for this task was developed in Visual Basic for Applications (VBA).
In a methodological part, four Monte Carlo methods were compared: Firstly, two kinds of sequences were used, namely pseudo-random sequences (MC) and quasi-random, low discrepancy sequences (QMC). Secondly, these sequences were used in the standard way or by applying the negative values of each sequence along with the original ones (antithetic method). It was shown that QMC is more efficient (provides smaller empirical errors) than MC and that the antithetic use of the sequences is more efficient than the standard one. In most cases, antithetic MC did better than standard QMC. However, the convergence speed was larger with QMC than MC. This held especially if the QMC sequence was not used in the antithetic way.
The payment rules underlying the CMO were shown to provide very specific risk profiles, especially regarding the prepayment sensitivity. The prepayment risk can be largely concentrated in one tranche, the Z tranche, as often intended by the issuers of CMO. The sensitivities to interest rate market parameters are changed by the payment rules, too, but they display a superposition of the direct sensitivity to interest rates and the indirect dependence on interest rates via prepayment rates.
Credit Risk: Design and Validation of Rating models
Master Thesis in "Mathematical Finance", Oxford (2004)
We describe the techniques we assume to be most useful in the construction and validation of corporate default prediction models. This approach uses a combination of statistical and computational methods to address the data problems that appear in model setup and validation. We show in detail how a score is calculated from a set of financial ratios using logistic regression or a hazard rate model. The score is then used to estimate the probability of default. We show that hazard rate models contain more information than the single period logit model. Because simple statistics (such as the number of defaults correctly predicted) are insufficient and often inappropriate in the domain of rating models, we describe the theory and the application of metrics for evaluating model performance: Gini curves (or cumulative accuracy profiles (CAP) plots), Gini coefficients (or accuracy ratios (AR)), receiver operational characteristic (ROC) curves and coefficient of concordance (CoC). We use a bootstrap procedure to account for all typical error sources in the calculation of the prediction uncertainties. This procedure enables us to make statements about the quality of the rating model. We apply these techniques to a set of 25 financial ratios, which we group into seven different classes, for non-financial corporates of the Bureau van Dijk's OSRIS databases from June 2002 and 2003 in order set up and validate a powerful rating model.
Calculating Basel II Risk Parameters for a Portfolio of Retail Loans
Master Thesis in "Mathematical Finance", Oxford (2003)
Under the Basel II regime, banks can choose among different approaches to
measure the regulatory capital to underpin their risky assets. From the point
of view of the amount of capital required, the Retail IRB Approach can be very
advantageous. To satisfy its requirements, banks have to estimate sensible values
for the risk parameters Probability of Default (PD) and Loss Given Default (LGD)
on the basis of their own default and loss data. In part due to the segmentation
rules particular to the Retail IRB Approach, this is very difficult, and the simple
calculation of relative frequencies will not do in general – the sample data do not
allow one to make a sensible distinction between the structure of the default and
loss densities and the randomness of the sample data, as we see in this thesis;
all methods we derive for computing risk parameters are developed using real
bank data.
We describe a method to estimate PD using the construction of a Lorenz curve
based on scoring results. While Lorenz curves usually are means to compute
efficiency ratios, we show how a Lorenz curve can serve as a vehicle to define the
borderline between structure and randomness. Values for PD can be obtained
from it in a direct way. What makes it specifically suitable for this purpose
are some invariancy properties; we show this in general and by way of sample
data of a real retail portfolio. We further compare this method to multivariate
methods, and propose a multi-component system to balance the complementary
advantages and disadvantages of both approaches.
Very often, there is no "LGD rating system" analogous to the PD rating, and so we
derive values for LGD by observing so-called special provision cohorts over time.
Making a special provision is part of the default definition, and by exponentially
modelling the time behaviour of special provision volumes one can estimate valid
LGD numbers, as we show in the relevant Chapter.
The last Chapter goes beyond Basel II. We assume the validity of the "Loss of
Memory Property" for a typical retail portfolio, and show that borrower default
under this assumption can be compared to radioactive decay. The mathematical
modelling of decaying nuclei is transported to defaulting borrowers, from where
some explicit formulae for Unexpected Loss are derived. As all terms of these
formulae can be estimated from our sample portfolio data, this model can serve
as a validation tool for an internal portfolio model.
Credit Value-at-Risk of Interest Rate Swaps
Master Thesis in "Mathematical Finance", Oxford (2003)
The thesis outlines the framework of the CreditMetrics methodology for the estimation of credit risk. The building blocks of this quantitative portfolio approach are explained in detail, such as rating migrations, obligor correlations and calculation of exposures. To quantify a portfolio's credit risk, the exposure distribution at a risk horizon has to be calculated. It is explained how Monte Carlo simulation leads to such a distribution and how the important quantities like expected loss, unexpected loss, (credit) value-at-risk, marginal value-at-risk and economic capital can be determined. Finally, emphasis is put on the exposure calculation of interest rate swaps. For an example swap, two methods are compared and the important assumptions are critically reviewed.
Statistical methods to estimate the probability of default
Master Thesis in "Mathematical Finance", Oxford 2003
Under the Basel II regime, banks can choose from different approaches to estimate
the regulatory capital needed to meet unexpected losses. Within the so called internal
rating based approaches (IRB) banks have to estimate sensible values for the risk
parameters "Probability of Default" (PD) and "Loss Given Default" (LGD) on the
basis of their own default and loss data. This thesis is dedicated to the analysis of
statistical methods to estimate the probability of default and the underlying techniques
of discrimination between the groups of defaulted and non-defaulted clients.
We begin the analysis by reviewing Bayesian statistics in the framework of PD estimation.
Applying the Bayes theorem directly leads to the likelihood ratio, which
plays a central role for discrimination of different groups. Certain model assumptions
about the parent populations of indicator variables of the group of good respectively
bad clients will be discussed. It will be shown how such model assumptions lead to
special likelihood ratios, known as linear and quadratic discriminant functions. In
addition, the linear logit model will be derived from the Bayes formula. The starting
point for the analysis will be a realistic parameterization of the parent populations of
good and bad clients taking a retail portfolio of a medium sized german bank. If the
initial parameters of these distributions are known, a systematic analysis will show
the impact of model assumptions under conditions of uncertainty on the exactness of
PD-estimation. The accuracy ratio (AR) will be introduced as a quantitative measure
of discrimination power. This accuracy ratio – also known as Gini coefficient –
is presently a kind of market standard for validating a rating system. The accuracy
ratio will be further discussed as a random variable. It will be shown how the discrimination
analysis of small statistical data samples lead to finite size effects caused by
random noise. We conclude the analysis with a discussion of important issues such as
significance, bias effects and the construction of confidence intervals of the observed
accuracy ratio.
Credit Risk in a Market Model
Master Thesis in "Mathematical Finance", Oxford (2002)This thesis deals with aspects of the Libor Market Model with and without credit risk.Starting with the pricing of default-free forward rate agreements we then study numericallythe distributional properties of forward rates with different tenors in the default-free LiborMarket Model and discuss implications on the prices of caplets. Finally we incorporatecredit risk by means of an intensity based credit risk model with both deterministic andstochastic hazard rates and different recovery schemes. We develop a pricing formula whichwe apply to several financial instruments like a zero bond, a floating rate note and a creditdefault swap. For the default-free forward rates the standard set-up of a Libor MarketModel is used. For a credit agreement and a forward rate agreement with counterpartycredit risk we reproduce and extend the results of Lotz and Schl¨ogl (J. Bank. Finance 24,301-327 (2000)). We determine the present value and the fair spread of those contracts anddiscuss our results.
