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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. 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
  • 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
a metric for allocation
A metric for allocation

Exactly accounts for portfolio CES

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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