F71CM: CREDIT RISK MODELLING A summary of the syllabus
M.A. Fahrenwaldt based on A.J. Mc Risk Credit-Risky Instruments
Definition of credit risk. Key terms like obligor and default. The main features and risks of loans and bonds.
Counterparty credit risk in derivative contracts including the example of an interest- rate swap. Terms like CVA, netting and collateralization.
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Credit default swaps (CDSs): structure, payment flows and uses.
Exposure (EAD), probability of default (PD), loss-given-default (LGD) and recovery.
Measuring Credit Quality
The two main philosophies for measuring credit quality: using ratings and scores or using market prices.
Concept of a rating and the systems used by Moody’s and S&P.
The typical form of rating transition data.
Meaning of through-the-cycle (TTC) and point-in-time (PIT).
Discrete-time Markov chain models for rating transitions or migrations. The Markov assumption. Meaning of stationarity. Definition of the transition matrix.
Transition matrices for multiple time periods as products of the one-period transition matrix with itself.
How transition probabilities are estimated in a discrete-time model. (The derivation of the multinomial log-likelihood will be given in an exercise but not examined.)
Drawbacks of the discrete-time Markov chain for rating transitions.
Explanation of how the continuous-time Markov chain may be viewed as a limit as the time step in a discrete-time chain goes to zero.
Definition of generator matrix and relationship to matrices of transition probabilities via the matrix exponential function.
Construction (simulation) of a continuous time Markov chain. Estimator of the generator matrix.
Criticism of the Markov assumption for rating transitions and evidence against as- sumption.
1.3 Structural Models of Default
• The assumptions of the Merton model concerning the financing of a firm through debt and equity.
• The interpretation of equity and debt as contingent claims on the assets of the firm.
• Modelling the asset value process with geometric Brownian motion. Formulas for
SDE, solution of SDE and physical default probability.
• Relationship between the physical default probability and: (i) the drift of the asset value process; (ii) the volatility of assets; (iii) the initial assets of the firm; (iv) the face value of the debt.
• Formula for the risk-neutral default probability and relationship between physical and risk-neutral probabilities.
• Derivation of formula for value of equity using Black–Scholes formula. (The Black Scholes formula will usually be given.) Assumptions that are made in applying Black–Scholes.
• Derivation of formula for value of debt; derivation of formula for value of a zero- coupon bond issued by the firm.
• Definition of the credit spread of a bond.
• Derivation of a formula for the credit spread in Merton’s Model.
• Relationship between credit spread and: (i) leverage of firm; (ii) asset volatility; (iii) time to maturity of debt.
• Drawbacks of Merton’s model.
• The concept of EDF (expected default frequency) as applied to Merton’s model.
• The Moody’s Public-Firm EDF model and the ways in which the Merton model is adapted.
• How the concept of backing out asset values and asset volatility from quoted equity prices works.
• The concept of distance-to-default (DD). Calculation of DD when the default point is known and asset value and asset volatility are recovered from equity data.
1.4 Bond and CDS Pricing in Hazard Rate Models
• Definition of survival (or tail) function, hazard function and cumulative hazard func- tion of a random time τ. Understanding of the connections between these functions. Calculation for simple examples such as Weibull distribution.
• Definition of default indicator process and the filtration generated by default infor- mation only.
• Fundamental formula for E(I{τ>t}X | Ht) and its application to computing a condi- tional survival probability.
• • • • • • •
Knowledge of the martingale modelling philosophy of financial valuation and its pros and cons.
Pricing of a zero-coupon bond under deterministic interest rates and recoveries. Decomposition of payment streams of a bond into a survival claim and a recovery payment.
Derivation of formula for the price of a survival claim.
The RT (recovery-of-treasury) assumption.
The RF (recovery-of-face-value) assumption.
Derivation of price of recovery payment under RT.
Derivation of formula for the spread of a zero-coupon bond under RT. Derivation of price of recovery payment under RF.
Assumptions concerning the payment flows of a simple CDS without accrued premi- ums.
Derivation of formula for the value of the premium leg.
Derivation of formula for the value of the default leg. (This is more difficult and we would usually derive a formula for a recovery payment under RF as an intermediate step.)
Concept of the fair swap spread.
Calibration of the hazard-rate model using observed CDS spreads. In particular, the case where there is a single spread and the risk-neutral hazard rate is assumed to be constant.
A clear appreciation of the difference between real-world and risk-neutral probabili- ties of default and an awareness that there are empirical studies of the relationship between the two.
Shortcomings of simple hazard-rate models.
Portfolio Credit Risk Management Threshold Models
Causes of dependence between credit events and a sense of why this is important in risk management.
Bernoulli default indicator variables for fixed time periods. Default correlation.
Concept of exchangeability and notation for default probability, joint default prob- ability and default correlation.
Computing the expectation and variance of the portfolio loss in a model with default correlation.
• Definition of a threshold model and notation.
• Asset correlation.
• Copulas. Definition, Sklar’s theorem.
• Gaussian, t, Gumbel and Clayton copulas.
• Equivalence of threshold models when the critical variables share a copula and the default probabilities are the same.
• Derivation of formula linking joint default probability to marginal default probabil- ities via the copula of the critical variables.
• Calculating examples of this formula for simple copulas, such as Archimedean copula.
• Multivariate Merton model as a Gaussian threshold model.
• The general form of the factor model that is used in Gaussian threshold models. Statement of assumptions in factor model. Formula for asset correlation.
• The one-factor special case of the Gaussian threshold model.
• How the Gaussian threshold model might be generalized to obtain a t copula thresh-
old model.
2.2 Mixture Models
• General definition of a Bernoulli mixture model of default.
• Derivation of expressions for joint conditional probability mass function and joint
unconditional probability mass function.
• Calculation of the Laplace-Stieltjes transform or moment generating function. An insight into when this can be useful.
• The exchangeable one-factor Bernoulli mixture model. Derivation of formulas for default and joint default probabilities as well as default correlation. Distribution of number of defaults.
• Calculation of all these quantities in the special case of a beta mixing distribution.
• The probit-normal and logit-normal distributions as alternatives to beta.
• The mixture distribution corresponding to a threshold model with Clayton copula. The distribution is obtained as a transformation of a gamma distribution and the transformation and the gamma density would usually be given.
• Relationship between default probability and default correlation and the two param- eters of a typical mixing distribution.
• One-factor Bernoulli mixture model with covariates, particularly the special case with exchangeable group structure.
• The idea that systematic recoveries can be modelled as a function of the factor(s) in a Bernoulli mixture model.
• Representation of a Gaussian threshold model as a Bernoulli mixture model; obser- vation that the conditional default probability has a probit-normal distribution.
• Poisson approximation of a Bernoulli distribution.
• Definition of the CreditRisk+ model.
• Proof that a gamma mixture of Poisson variables has a negative binomial distribu- tion; students would generally be expected to know the pmf of a Poisson distribution but would be helped with the density of gamma and the pmf of negative binomial.
• Proof that the number of defaults in CreditRisk+ is a sum of independent negative binomial random variables.
2.3 Asymptotics for Large Portfolios
• How the strong law of large numbers may be applied conditionally to an exchange- able Bernoulli mixture model; insight that the loss distribution is driven by the distribution of the mixing variable.
• Computations for large portfolios in the probit-normal mixture model (Gaussian threshold model with one factor).
• Insight into tail risk in exchangeable models.
• Definition of the aymptotic relative loss function.
• Statement of general result for the distribution of the relative loss (relative to total exposure) in a Bernoulli mixture.
• Statement of general result for quantiles of the relative loss in a one-factor Bernoulli mixture model.
• Application of these results to a one-factor Bernoulli mixture model with exchange- able groups.
• Application to a model incorporating stochastic LGDs that depend on the factor. depends
• Concept of risk-weighted assets and the standardized approach in Basel capital framework.
• Formulas for the risk weights and capital contributions in the IRB (internal-ratings- based) approach to credit risk.
• Proof that these formulas correspond to contributions to the portfolio Value-at-Risk if we assume that credit losses are governed by a one-factor model and the portfolio has an exchangeable group structure.
• Discussion of the Basel formula.
2.4 Monte Carlo Methods
• Algorithm for naive Monte Carlo and justification by SLLN.
• Algorithm for importance sampling and justification.
• Calculation of variance of estimators and implication for choosing importance sam- pling density when estimating a rare event probability.
• Illustration of technique of exponential tilting for normal density (or other simple continuous densty like gamma).
• Presention of importance sampling and exponential tilting as a change of measure technique.
• General formula for changing measure from P to Qt using exponential tilting based on a random variable L; choice of t.
• Illustration for independent Bernoulli (or independent Poisson) risks where L is a weighted sum.
• Explanation of how importance sampling ideas can be applied to a Bernoulli mixture model through ‘inner’ and ‘outer’ sampling.
2.5 Statistical Inference in Portfolio Credit Risk Models
• Definition of the Gaussian threshold model (Gauss copula model) with factor struc- ture that is commonly used in industry.
• Discussion of the calibration of the thresholds using information about default prob- abilities (based on ratings or EDFs).
• Detailed assumptions for the factor model for the critical variables.
• Step by step method for calibrating the factor model to asset return data or a proxy
for asset returns like equity returns.
• Derivation of estimators for default probability and default correlation based on data on defaults for a homogeneous population of obligors that can be assumed to follow an exchangeable default model in each time period.
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