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

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Conditional Expected Loss; an allocation metric. Calculation using Bayes' Theorem ... We can easily calculate this by removing issuer i from the final portfolio and ... – PowerPoint PPT presentation

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Title: Modelling Default Risk in the Trading Book: Accurate Allocation of Incremental Default Risk Charge J


1
Modelling Default Risk in the Trading Book
Accurate Allocation of Incremental Default Risk
ChargeJan Kwiatkowski
2
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

3
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.

4
Trading book default risk models(calibrated to a
given time-horizon)
  • Many models use systematic risk factors and
    correlations

5
Conditional PDs
  • Require an algorithm for computing distribution
    of portfolio loss conditional on any X
  • Integrate over X (e.g. Monte Carlo or quadrature)

6
Algorithms for conditional losses
  • Monte Carlo
  • Transforms
  • ASB
  • Must keep in mind the need for allocation

7
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

8
The ASB algorithm
  • Discretise LGDs as multiples of a fixed Loss
    Unit
  • ui loss units for issuer i
  • Let qi PD for issuer i (conditional on given X )

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

9
ASB implementation
10
A metric for allocation
  • Exactly accounts for portfolio CES

11
Bayes Theorem
12
Allocation methodology
  • We can easily calculate this by removing issuer i
    from the final portfolio and adding its LGD, ui,
    to the resulting portfolio distribution.
  • We use the reversal of the ASB algorithm to
    remove issuer i

13
Reversal of ASB
  • We illustrate this for a long position
    (uigt0) this is easily adaptable to short
    positions.

14
Warning
  • This becomes unstable for qigt0 close to 1.
  • Can be mitigated (Parcell 2006)

15
Summary of method Phase 1
  •  
  •  
  •  
  •  
  • For various systematic effects, X, use ASB to
    find the conditional distribution.
  • Integrate over X
  • Compute La, the required portfolio CES

16
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 La is
    exceeded
  • Multiply by uiqi/(1-a)
  • Integrate over X

17
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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