Allocation of Economic Capital

1
Allocation of Economic Capital

• May 2008

Preben Munch Thomsen pt_at_danskebank.dk Credit Risk
Modelling Risk Management
2
Motivation and Summary

• As the Basel II framework defines the standard
  for capital adequacy, the most valuable outcome
  of an internal credit risk model is actually not
  the EC level, but rather the capital allocation
  key.
• In the light of portfolio optimization it is
  very important that capital allocation is
  calculated as the marginal contribution to the
  total risk and thereby catching concentration
  and diversification.
• To serve all applications, it is furthermore
  essential that EC is allocated to the lowest
  possible level facility level.

3
Use of allocated credit EC and allocation model

• Performance (BU-, branch-, and customer level)
• Risk based pricing Corporate loans and
  derivative contracts
• ICAAP
• Pillar II Stresstests
• Industry analysis
• Risk management reporting
• Monthly BU reports
• Large counterparty watch list

4
Calculation of Credit Risk EC / principle
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Capital allocation is a 2-step process

• The credit risk model allocates EC to
  individually simulated obligors and pools
• Capital allocation model allocates EC to facility
  level
• Developed internally by Danske Bank
• Consist of a capital allocation function build
  upon typical risk parameters PD, EaD, LGD, M, a
  concentration term and some regression parameters
  
• Regression parameters are estimated from data on
  large obligors including allocated EC from the
  Credit Risk Model.

6
Allocation inside the Portfolio Credit Risk Model

• Allocation to
• Large individual obligors 70 of portfolio
  EaD-wise
• Pools of facilities for small and medium-size
  companies and households
• Coherent capital allocation with a coherent risk
  measure
• Ensures full allocation
• Additive (Eulers theorem)
• Allocate diversification benefits to obligors
  that contributes to portfolio diversification

7
Expected Shortfall vs. VaR
ES is the common risk measure for allocation in a
credit risk model

• Coherent opposite to VaR.
• Main difference is that ES (as opposed to VaR) is
  sub additive
• which reflects that risk can be reduced
  by diversification
• Average measure Increased stability at obligor
  level (compared to VaR with equal number of
  simulations)

VaR_at_99.97
EL
EC_at_99.97
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Expected Shortfall choice of quantile

• Theoretically, use ES_at_99.85 as it approximately
  match the VaR 99.97 loss at portfolio level.
• However, level problematic with respect to
  stability of the regression parameters Less
  statistical significant and too sensitive to
  portfolio changes.
• ES_at_95 chosen as compromise
• The 95 quantile is far from the tail of the
  loss distribution.

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Capital allocation function
Scaling
Single-name
Maturity
Rating
Loss amount
Concentration
- a ensures that aggregation of all facilities
EC match the portfolio EC - Increased maturity
increases migration risk and hence EC - 9 country
weights - based on obligors turnover
distribution - Allocation function modelled at
obligor level and applied at facility level
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Regression process- Model issues / Log
transformation

• Log-transform is good
• Ensures usable fit
• Optimization easy (linear model)
• Transformation of data of unlike characteristic
• Large and small EC values ends up on same scale
• Heavy-tailed data countered
• Log-transform is bad
• Error minimized in wrong space (OLS), i.e.
  log(EC)-space. Error tolerence in EC-space very
  un-even for large and small EC values.

Fixed tolerencein log(EC)-space
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Regression process- Model issues / Weighted
Least Squares (WLS)

• WLS optimization- data points treated with
  individual weight in error function as opposed to
  OLS.

The weights are related to noise variance for the
given data point

• Strong EC weighting
• Proportional EC weighting
• (compromise for better numerical properties)

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Regression process- Model performance

• Model fit is satisfactory and optimized in the
  space of interest by means of WLS

• All variables are reliable

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Regression process- Model performance

• Predicted EC values versus the true EC values in
  both the log-transformed space and the direct EC
  space
• The residuals must ideally be without pattern.

14
Application of the allocation function and
Single-name concentration

• Application of the allocation function at
  facility level requires special treatment of
  single-name concentration

• Calculate multiplier for single-name
  concentration (SNC) for each obligor
• Use allocation function on facility level
  corrected by each obligors SNC multiplier

• Allocation model characteristics
• EC increases more than linear with LGD times
  exposure.

15
Allocation model characteristics - Rating
dependency

• EC for a B3 rated customer is 400 hundred times
  higher than for an A1

• EC depends less on PD than EL does

16
Allocation model characteristics - Geographic
concentration

• The dependency between geographic concentration
  and EC

• Corporate customers are sensitive to geography
• Retail customers diversify the portfolio
  independent of geography

17
Allocation model characteristics - Maturity as
proxy for migration risk

• Migration loss counts for 22 of the total loss
• Maturity therefore contributes significantly to
  explaining EC
• The difference between the shortest possible
  maturity (1 year) and the longest (5 years), is
  45 percent with respect to EC

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Summary and Conclusion

• To reap the gain of a portfolio credit risk model
  and use it actively for risk management, it is
  essential that EC is allocated marginally to each
  loan, customer, and business unit.
• The allocation model described in this
  presentation estimates the marginal loss at
  facility level from the tail of the loss
  distribution of a banks credit portfolio
  generated by simulating a large number of
  scenarios in a credit risk model.
• The allocation process consists of two steps
• Allocating to large obligor level with ES_at_95 and
• Allocating to facility level with a regression
  based allocation function, having PD, LGD, EaD,
  M, and a country based concentration term as
  drivers.
• Questions and comments.
• How do other banks allocate capital?