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. 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 ChargeJan Kwiatkowski

Group Risk Management


Summary
Summary Allocation of Incremental Default Risk Charge

  • 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 Allocation of Incremental Default Risk Charge

  • 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 Allocation of Incremental Default Risk Charge(calibrated to a given time-horizon)

  • Many models use systematic risk factors and correlations


Conditional pds
Conditional PDs Allocation of Incremental Default Risk Charge

  • 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 Allocation of Incremental Default Risk Charge

  • Monte Carlo

  • Transforms

  • ASB

  • Must keep in mind the need for allocation


Allocation of idrc
Allocation of IDRC Allocation of Incremental Default Risk Charge

  • 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 Allocation of Incremental Default Risk Charge

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 Allocation of Incremental Default Risk Charge


A metric for allocation
A metric for allocation Allocation of Incremental Default Risk Charge

Exactly accounts for portfolio CES


Bayes theorem
Bayes’ Theorem Allocation of Incremental Default Risk Charge


Allocation methodology
Allocation methodology Allocation of Incremental Default Risk Charge

  • 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 Allocation of Incremental Default Risk Charge

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


Warning
Warning Allocation of Incremental Default Risk Charge

  • This becomes unstable forqi>0 close to 1.

  • Can be mitigated (Parcell 2006)


Summary of method phase 1
Summary of method – Phase 1 Allocation of Incremental Default Risk Charge

  • 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 Allocation of Incremental Default Risk Charge

  • 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 Allocation of Incremental Default Risk Charge

  • 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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