# Modelling Default Risk in the Trading Book: Accurate Allocation of Incremental Default Risk Charge J - PowerPoint PPT Presentation

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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