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Modelling Default Risk in the Trading Book: Accurate Allocation of Incremental Default Risk Charge Jan Kwiatkowski

Modelling Default Risk in the Trading Book: Accurate Allocation of Incremental Default Risk Charge Jan Kwiatkowski

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## Modelling Default Risk in the Trading Book: Accurate Allocation of Incremental Default Risk Charge Jan Kwiatkowski

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**Modelling Default Risk in the Trading Book: Accurate**Allocation of Incremental Default Risk ChargeJan Kwiatkowski Group Risk Management**Summary**• Background on IDRC • Trading book default risk models • The need for accurate allocation • Andersen, Sidenius & Basu (ASB) algorithm • Conditional Expected Loss; an allocation metric • Calculation using Bayes’ Theorem • Extensions**Background on IDRC**• Required to find high percentile of portfolio default losses over a given time horizon. • We prefer Conditional Expected Shortfall (CES) at equivalent percentile: • The book tends to be ‘lumpy’; therefore we must include idiosyncratic effects.**Trading book default risk models(calibrated to a given**time-horizon) • Many models use systematic risk factors and correlations**Conditional PDs**• Require an algorithm for computing distribution of portfolio loss conditional on any X • Integrate over X (e.g. Monte Carlo or quadrature)**Algorithms for conditional losses**• Monte Carlo • Transforms • ASB • Must keep in mind the need for allocation**Allocation of IDRC**• Total IDRC is allocated/attributed to contributors (down to position level), and aggregated up the organisational hierarchy. • Allocation must be ‘fair’ and consistent • Especially, desks with identical positions should get the same allocation. • Using Monte Carlo for high percentiles , we are at the mercy of relatively few random numbers • Transforms not convenient for allocation**The ASB algorithm**Discretise LGD’s as multiples of a fixed ‘Loss Unit’ ui= loss units for issuer i Let qi= PD for issuer i (conditional on givenX ) Recursively compute the distribution of the losses for portfolios consisting of the first i exposures only, for i =0, 1, 2, …., N • The method is exact modulo discretisation • Parcell (2006) shows how effects of discretisation may be mitigated • Easily extended to multiple outcomes**A metric for allocation**Exactly accounts for portfolio CES**Allocation methodology**• We can easily calculate this by removing issuer i fromthe final portfolio and adding its LGD, ui, to the resulting portfolio distribution. • We use the reversal of the ASB algorithm to remove issuer i**Reversal of ASB**We illustrate this for a long position (ui>0); this is easily adaptable to short positions.**Warning**• This becomes unstable forqi>0 close to 1. • Can be mitigated (Parcell 2006)**Summary of method – Phase 1**• For various systematic effects, X, use ASB to find the conditional distribution. • Integrate over X • Compute Lα, the required portfolio CES**Summary of method – Phase 2**• For each issuer, i: • for each X • Use reverse ASB to find the distribution with i defaulted. • Compute the corresponding probability that Lα is exceeded • Multiply by uiqi/(1-α) • Integrate over X**Possible Extensions**• The method for VaR (rather than CES) is even simpler • Multiple outcomes: • Stochastic LGDs • Rating Downgrades • Also upgrades, but requires matrix inversion • Structured products – cascade structure