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Describe how the Gaussian copula and the CreditMetrics approach are each used to estimate credit VaR.

FRM® Part II · Credit Risk Measurement and Management · Credit Risk · FRM2.CR.12.g

The idea

Both methods turn default correlation into a loss distribution and read credit VaR as a percentile. The Gaussian copula (Vasicek) approach suits a large portfolio of similar loans. Let ρ be the copula correlation, X the confidence level and WCDR the worst-case default rate at that level. With these definitions, the worst-case default rate at confidence level X for a portfolio of similar loans is: WCDR = N((N⁻¹(PD) + √ρ × N⁻¹(X)) ÷ √(1 − ρ)). The loss at that level is EAD × LGD × WCDR. This approach is default-only, and credit VaR subtracts the expected loss, so the unexpected loss that it measures is: UL = EAD × LGD × (WCDR − PD). CreditMetrics instead simulates correlated normal variables for each borrower and maps them through transition-matrix thresholds to a new rating at the horizon, usually one year. It revalues each position at that rating, or at recovery on default, and reads the percentile of portfolio value below its mean. Exam point: CreditMetrics captures migration losses. The large-portfolio formula assumes granularity and a single factor.

Formulas

Worst-case default rate at confidence level X

WCDR = N[(N⁻¹(PD) + √ρ × N⁻¹(X)) ÷ √(1 − ρ)]

N
the standard normal cumulative distribution function, with N⁻¹ its inverse
PD
the one-year default probability of each loan
ρ
the copula correlation
X
the confidence level

A large, granular, single-factor portfolio. Adding the factor term gives the worst case; subtracting it gives a benign rate.

Credit VaR of a large homogeneous portfolio

credit VaR = EAD × LGD × (WCDR − PD)

EAD
the total exposure
LGD
loss given default
WCDR
the worst-case default rate at the confidence level
PD
the probability of default

Default-only, so migration losses are missing. Omitting PD reports the worst-case loss rather than the unexpected loss.

A practice question

From the question bank, with the reason each wrong option is wrong.

The main output of a CreditMetrics analysis is:

  1. a distribution of default times for each borrower in the portfolio.Why not: Describes a time-to-default copula model, not the horizon valuation approach of CreditMetrics.
  2. a single worst-case default rate for a large homogeneous portfolio.Why not: Describes the Vasicek large-portfolio result, which is default-only.
  3. a distribution of portfolio value at the horizon, reflecting rating migrations and defaults. - correct
  4. a set of risk-neutral default probabilities extracted from bond spreads and used to price credit derivatives.Why not: That is an input-type calculation, not what CreditMetrics produces.

CreditMetrics simulates rating changes over the horizon, revalues positions at the new ratings, and builds the distribution of portfolio value from which credit VaR is read.

A harder one

A homogeneous portfolio has total exposure of USD 500 million, LGD of 55%, a one-year default probability of 2% and a copula correlation of 0.15. Credit VaR is defined as the loss at the worst-case default rate (99.9%) minus the expected loss. Credit VaR is closest to:

  1. USD 43.0 million. - correct
  2. USD 78.2 million.Why not: Omits the LGD factor, treating the whole exposure as lost on default.
  3. USD 48.5 million.Why not: Reports the worst-case loss without subtracting the expected loss.
  4. USD 0.1 million.Why not: Subtracts the factor term instead of adding it, giving a low default rate rather than the worst case.

WCDR = N[(N^-1(0.02) + sqrt(0.15) x 3.090)/sqrt(0.85)] = 17.63%. Credit VaR = 500 x 0.55 x (17.63% - 2.00%) = 43.0 million.

The hardest tier

A bond's value in one year in CreditMetrics depends on its rating: upgrade to A, USD 101.5 (probability 5.0%); stays BBB, USD 100.0 (86.0%); BB, USD 96.0 (6.0%); B, USD 88.0 (2.1%); CCC, USD 75.0 (0.5%); default, USD 50.0 (0.4%). Credit VaR at 99% is defined as the expected value minus the 1st percentile value. It is closest to:

  1. USD 24.3.Why not: Takes the CCC value as the 1st percentile, although the cumulative probability from below is only 0.9% at CCC.
  2. USD 49.3.Why not: Uses the default outcome, which is the worst case, not the 1st percentile.
  3. USD 11.3. - correct
  4. USD 12.0.Why not: Measures the loss from the current value of 100 rather than from the expected value.

Expected value = 99.26. Cumulative probability from the bottom: default 0.4%, CCC 0.9%, B 3.0%, so the 1st percentile is the B value, 88.0. Credit VaR = 99.26 - 88.0 = 11.3.

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