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Foundations of Credit Risk Modelling

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Default Risk
Risk that counterparty doesn't honour obligation (payment obligation, supplier doesn't deliver promised goods, contractor doesn't render promised service, etc.). Context is wide. No tx free of default risk
Default Risk for FI
Largest component of default risk is payment obligation (loans, bonds & OTC derivatives). Risk of payment default, particularly for loans & bonds, is Credit Risk
Definitions
Default : Obligation not honoured
Backruptcy Code & Default
Obligor may default on one obligation & honour others. However, creditor of defaulted obligation may go to court to force payment or institute legal proceedings. Existence of bankruptcy code gives rise to default ris…
Exposure, Default & Recovery
N¡(t) = Default indicator process (binary). 0 or 1 at time t for obligor i
Probability of Default
p¡(T) = probability of default of obligor i up to time horizon T. Eg. p¡(1) = 0.01% means 1/10,000 chance of default at some point over next year
Default Loss Computation
Default Loss = Default Arrival × EAD × LGD
Correlation of Credit Losses
If credit losses are correlated, default of one obligor makes default of other obligors more likely (concentration risk). Portfolio loss D(T) at time T is sum of individual credit losses D¡(T) for all obligors:
Credit Loss Distribution 1
Full porftolio loss D(T) is a random variable. Portfolio's credit loss distribution is probability distribution of this random variable:
Credit Loss Distribution 2
Key ingredient of portfolio loss D(T) is dependency, ρ, between individual losses:
Expected Loss (Individual Obligor)
E [D¡(T)] = p¡(T) × E¡ × L¡ (≠ 0!)
Expected Loss (Concept)
EL important for performance measurement, esp. RAROC. When loan's expected gain (in excess of fund…
Unexpected loss
More important measure of risk. Represents volatility of actual losses.
Unexpected Loss : Diversification Effect
Even at high VaR quantile of 99.9%, the UL is still much lower than maximum possible loss if total portfolio defaults with 0 recovery. This effect stems from partial diversification effect in the portfolio…
Capital Reserves & Unexpected Losses
UL is used to determine capital reserves against credit risk. Not economically viable to hold reserves against total loss, but reserving against UL at sufficiently high VaR quantile is viable if done centrally, exploiting diversification effects
Recovery Rate (Definition 1)
Recovery rate is market value per unit of legal claim amount of defaulted debt, a short time after default. Generally applies to larger obligors (eg. those rated by public rating agencies)
Recovery Rate (Definition 2)
Recovery rate is value of default settlement per unit of legal claim, discounted back to date of default after deducting legal & admin costs. For smaller obligors, no market price for distressed debt. So …
Other Factors - Industry Group of Obligor
FIs tend to have significantly higher recovery rates than industrial obligors. Less capital-intensive obligors have less substance to liquidate, eg. dotcom companies have recoveries ≈ 0
Other Factors - Obligor's Rating Prior to Default
Obligor who has spent much time close to default usually has fewer assets to liquidate than obligor who defaults suddenly from high rating class
Other Factors - Average Rating of Other Obligors in Industry Group & Business Cycle
Recoveries tend to be lower in recessions & in industry groups in cyclical downswings or with large overcapacities
Recovery Rates (Empirical Findings)
Despite empirical findings, impossible to predict recovery rates with much certainty. Dependence of recoveries on business cycle means they will not diversify away. Double whammy in recession : more defaults & lower recovery rates
Beta Distribution 1
Popular model for random recovery rates with values in [0,1]:
Beta Distribution 2
a = µ²(1−µ)/σ²