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Judge the consequences of correlation for a credit portfolio and the Credit VaR figure attached to it.

FRM® Part II · Credit Risk Measurement and Management · Portfolio Credit Risk · FRM2.CR.11.c

The idea

Correlation changes the shape of the loss distribution, not its mean. Expected loss is the sum of PD × LGD × exposure over the loans, so it does not depend on correlation at all. Unexpected loss does depend on it. Let σ1 and σ2 be the stand-alone loss standard deviations of two loans, and ρ their default correlation. Putting the two loans together, the standard deviation of the portfolio loss, which is the unexpected loss, is: Sigma = √(σ1² + σ2² + 2 × ρ × σ1 × σ2). This grows with ρ, because higher default correlation makes joint defaults more likely, fattens the right tail and shrinks the diversification benefit. Credit VaR is the loss quantile at a confidence level less the expected loss, so it normally rises with correlation and reaches the undiversified sum at perfect correlation. Be careful with small, lumpy portfolios, where the quantile is a step function and can even fall when correlation rises. Always quote Credit VaR with its confidence level and horizon.

Formulas

Standard deviation of loss on two loans

σP = √(s1² + s2² + 2 × ρ × s1 × s2)

s1
the stand-alone loss standard deviation of the first loan
s2
that of the second loan
ρ
the default correlation between them

Grows with ρ while expected loss stays put. Each stand-alone figure is exposure × LGD × √(PD × (1 − PD)); subtracting the covariance term is the sign slip.

Stand-alone loss standard deviation of one loan

s = exposure × LGD × √(PD × (1 − PD))

exposure
the amount lent
LGD
loss given default
PD
the probability of default

Treats the loss as exposure × LGD on default and zero otherwise, a Bernoulli outcome.

A practice question

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

Holding PDs, LGDs and exposures fixed, a rise in the default correlation between the loans of a portfolio will:

  1. Raise expected loss and Credit VaR in the same proportion.Why not: Expected loss is a sum of PD x LGD x exposure and does not depend on correlation.
  2. Leave expected loss unchanged and raise Credit VaR. - correct
  3. Lower both expected loss and Credit VaR, because correlated loans offset each other.Why not: Positive correlation makes defaults cluster; it does not create offsetting losses.
  4. Leave Credit VaR unchanged, because only the PDs determine the loss distribution.Why not: Ignores that the joint distribution, not just the marginal PDs, determines the tail quantile.

Correlation moves probability into both tails but leaves the mean loss unchanged. More clustered defaults raise the high-quantile loss, so Credit VaR increases.

A harder one

A portfolio has two loans of USD 10 million each, with LGD of 50%, a one-year PD of 2% and a default correlation of 0.30 between them. Treating each loan's loss as exposure x LGD on default, the standard deviation of portfolio loss is closest to (USD million):

  1. 1.40Why not: Adds the two stand-alone standard deviations, which assumes perfect correlation.
  2. 0.99Why not: Takes the square root of the sum of the variances, which assumes zero correlation.
  3. 1.13 - correct
  4. 0.83Why not: Subtracts the covariance term instead of adding it.

Stand-alone sigma = 10 x 0.5 x sqrt(0.02 x 0.98) = 0.7. Portfolio sigma = sqrt(0.49 + 0.49 + 2 x 0.3 x 0.49) = sqrt(1.274) = 1.13.

The hardest tier

Two loans of USD 10 million each have LGD of 100% and a one-year PD of 5%. Their default correlation implies a probability of 2.0% that both default (it would be 0.25% under independence). Credit VaR is defined as the 99% loss quantile less the expected loss. Under the stated correlation, Credit VaR is closest to (USD million):

  1. 19 - correct
  2. 20Why not: Reports the 99% loss quantile and forgets to subtract the expected loss of USD 1 million.
  3. 10Why not: Uses the independent-case 99% quantile, as if correlation did not alter the tail, and does not deduct expected loss.
  4. 9Why not: Reports the independent-case Credit VaR (quantile of 10 less expected loss of 1).

With a 2.0% joint probability: P(loss = 20) = 2%, P(loss = 10) = 2 x (5% - 2%) = 6%, P(0) = 92%. P(loss <= 10) = 98% < 99%, so the quantile is 20. Expected loss = 1, so Credit VaR = 19.

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