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. Summary. Background on IDRC Trading book default risk models The need for accurate allocation Andersen, Sidenius &amp; 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
• 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.
• 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