Dealing with double default under Basel II

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Dealing with double default under Basel II

• Erik HeitfieldBoard of Governors of the Federal
  Reserve System
• 20th and C Street, NW
• Washington, DC 20551 USA
• Erik.heitfield_at_frb.gov

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Todays Talk

• Basel IIs credit risk capital model for unhedged
  exposures
• The substitution approach for hedged exposures
• The ASRF/Merton model for hedged exposures
• Example calibrations
• A simple alternative to the ASRF/Merton Model
• Conclusions

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Basel II

• Basel II is intended to align regulatory
  capital requirements more closely with underlying
  risks, and to provide banks and their supervisors
  with several options for the assessment of
  capital adequacy.
• -- William McDonough

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Basel IIs Credit Risk Capital Rule

• Capital charges are designed to satisfy a
  portfolio-level solvency target (VaR rule)
• Charges must be assessed on a loan-by-loan basis

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The ASRF Framework

• In general, a VaR capital rule cannot be applied
  on a loan-by-loan basis because the marginal
  contribution of a single exposure to portfolio
  risk depends on its correlation with all other
  exposures
• Gordy (2003) shows that under stylized
  assumptions a simple, decentralized capital rule
  satisfy a VaR solvency target
• Collectively these assumptions are called the
  asymptotic-single-risk-factor (ASRF) framework

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ASRF Assumptions

• Cross-exposure correlations in losses are driven
  by a single systematic risk factor
• The portfolio is infinitely-fine-grained (i.e.
  idiosyncratic risk is diversified away)
• For most exposures loss rates are increasing in
  the systematic risk factor

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ASRF Capital Rule

• The ?th percentile of X is
• Set capital to the ?th percentile of L to ensure
  a portfolio solvency probability of ?
• Plug the ?th percentile of X into c(x)

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Merton Model

• Obligor i defaults if its normalized asset
  return Yi falls below the default threshold ?.
• where

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IRB Risk Weight Function

• The conditional expected loss function for
  exposure i given X is
• Plugging the 99.9th percentile of X into ci(x)
  yields the core of the Basel II capital rule

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Credit Risk Mitigation

• Basel II will provide some capital relief to
  account for the risk-mitigating effects of credit
  hedges
• Financial guarantees
• credit default swaps

Bank
Obligor
Guarantor
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Joint Default Probabilities
Joint default probability is generally much lower
than either marginal default probability
?og 60
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The Substitution Approach

• Credit risk mitigation in the form of
  derivatives and financial guarantees must not
  reflect the effects of double default. --CP3
  paragraph 270
• Under the substitution approach a bank can
  substitute the PD and LGD of the guarantor for
  those of the obligor if this would result in a
  lower risk weight

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Substitution Approach
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Substitution Approach

• Shortcomings of the substitution approach
• Provides no incentive to hedge high quality
  exposures
• Not risk sensitive for low-quality hedged
  exposures
• Lacks theoretical foundation
• Solution
• The same ASRF framework used to derive capital
  charges for unhedged loans can be used to derive
  capital charges for hedged loans

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ASRF/Merton Approach

• A Merton model describes default by both the
  obligor (o) and the guarantor (g)
• Two risk factors drive default correlations
• X affects all exposures in the portfolio
• Z affects only the obligor and the guarantor

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ASRF/Merton Approach

• Model allows for
• Guarantors with high sensitivity to systematic
  risk
• Wrong way risk between obligor and guarantor
• Three correlation parameters

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ASRF/Merton Approach

• Plugging the 99.9th percentile of X into the
  conditional expected loss function for the hedged
  exposure yields an ASRF capital rule

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Base Case

• Treats guarantors in a manner symmetric with
  corporate obligors
• Guarantor asset correlation is the same as asset
  correlation for corporate obligors
• ?g declines from 24 to 12
• No extra wrong-way risk

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Base Case
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Substitution Approach
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Base Case

• The base case represents the least conservative
  possible calibration of the ASRF/Merton model
• Generates capital charges that are significantly
  lower than the substitution approach for any
  combination of obligor and guarantor PDs

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Wrong-way Risk

• Guarantor asset correlation is the same as asset
  correlation for corporate obligors
• ?g declines from 24 to 12
• Obligor and guarantor are exposed to common
  shocks beyond those associated with systematic
  risk
• ?og 50

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Wrong-way Risk Case
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Substitution Approach
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Wrong-way Risk

• Produces capital charges that lie between the
  base case and the substitution approach
• When ?o ?g, as ?og approaches 100 the
  ASRF/Merton capital charges approach those of the
  substitution approach

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Guarantor Systematic Risk

• Guarantor exposure to systematic risk is greater
  than for typical corporate obligors
• ?g 50
• No extra wrong-way risk

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Guarantor Systematic Risk Case
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Substitution Approach
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Guarantor Systematic Risk

• For exposures to high-PD obligors ASRF/Merton
  capital charges may exceed those of the
  substitution approach
• For exposures to low-PD obligors ASRF/Merton
  capital charges remain well below those of the
  substitution approach
• As ?g approaches 100 ASRF/Merton capital charges
  approach capital charges for unhedged exposures

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Conservative Case

• Imposes conservative assumptions about
  correlation parameters
• Guarantors are more sensitive to systematic risk
  than corporate obligors
• ?g 50
• There is significant wrong-way risk
• ?og 50

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Conservative Case
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Substitution Approach
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Conservative Case

• Capital charges rise quickly with the obligor and
  guarantor PDs
• ASRF/Merton capital charges may be either higher
  or lower than under the substitution approach
• ASRF/Merton capital charges for exposures to
  low-PD obligors remain much lower than those
  generated by the substitution approach

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A Simpler Approach

• Applying the ASRF approach directly would add
  complexity to the Accord
• Uses a bivariate normal CDF
• Relies on three asset correlation parameters
• A simpler alternative would be to rescale the
  unhedged risk weight function
• The scaling function would depend on the PD of
  the guarantor
• The scaling function would be fit to the
  ASRF/Merton model

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A Simpler Approach
CP3 Risk Weight Function
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Conclusions
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Drawbacks of theSubstitution Approach

• The substitution approach is not risk sensitive
• Provides little incentive to hedge low PD credits
• Not sensitive to PDs of high PD credits
• The substitution approach has been widely
  criticized because it lacks a theoretical
  foundation

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ASRF vs. Substitution

• ASRF provides incentive to hedge risk for all
  types of obligors
• ASRF is more risk-sensitive for both high and low
  quality obligors and guarantors
• ASRF may or may not generate lower capital
  charges than substitution

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Advantages of the ASRF Approach

• The ASRF approach is more risk sensitive than the
  substitution approach
• The ASRF approach is derived from the same model
  used to construct risk-weight function for
  unhedged exposures
• The simplified ASRF approach would not add much
  additional complexity to the Accord

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References

• Available on the Federal Reserve Boards
  web-sitehttp//www.federalreserve.gov/generalinfo
  /basel2/default.htm
• Third consultative paper on the New Basel
  Capital Accord
• Risk Based Capital Guidelines Implementation of
  the New Basel Capital Accord Advance Notice of
  Proposed Rulemaking
• Treatment of Double-Default and Double-Recovery
  Effects under Pillar I of the New Basel Capital
  Accord Federal Reserve White Paper
• Additional References
• Gordy, M. (2003), A risk-factor model foundation
  for ratings-based bank capital rules, Journal of
  Financial Intermediation 12(3), pp. 199-232
• Heitfield, E. (2003), Using guarantees and
  credit derivatives to reduce credit risk capital
  requirements under the new Basel Capital Accord,
  in Credit Derivatives the Definitive Guide, J.
  Gregory (Ed.), Risk Books