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


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Modelling Default Risk in the Trading Book: Accurate Allocation of Incremental Default Risk Charge Jan Kwiatkowski. Summary. Background on IDRC Trading book default risk models The need for accurate allocation Andersen, Sidenius & Basu (ASB) algorithm

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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 charge jan kwiatkowski

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

Group Risk Management


Summary

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

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

Trading book default risk models(calibrated to a given time-horizon)

  • Many models use systematic risk factors and correlations


Conditional pds

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

Algorithms for conditional losses

  • Monte Carlo

  • Transforms

  • ASB

  • Must keep in mind the need for allocation


Allocation of idrc

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

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


Asb implementation

ASB implementation


A metric for allocation

A metric for allocation

Exactly accounts for portfolio CES


Bayes theorem

Bayes’ Theorem


Allocation methodology

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

Reversal of ASB

We illustrate this for a long position (ui>0); this is easily adaptable to short positions.


Warning

Warning

  • This becomes unstable forqi>0 close to 1.

  • Can be mitigated (Parcell 2006)


Summary of method phase 1

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

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

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


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