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State what Credit VaR is and compute it.

FRM® Part II · Credit Risk Measurement and Management · Credit Value at Risk · FRM2.CR.10.b

The idea

Credit VaR is the loss that will not be exceeded at a chosen confidence level over a chosen horizon, measured relative to expected loss. Subtracting the mean from the quantile gives the unexpected loss, which is the basis for economic capital. In symbols, the credit measure at a given confidence level is VaR = loss quantile − expected loss. Expected loss is covered by pricing and provisions, so capital is held only for the excess. For a single exposure, expected loss is the product of default probability, loss given default and exposure at default, written EL = PD × LGD × EAD. To compute credit VaR, build the portfolio loss distribution, read the quantile where the cumulative probability first reaches the confidence level, then subtract the mean. The usual mistakes are quoting the quantile without deducting expected loss, reading the wrong point of a discrete distribution, and forgetting LGD. On a discrete distribution, take the first loss whose cumulative probability is at least the confidence level.

Formulas

Expected loss on a single exposure

EL = PD × LGD × EAD

PD
the probability of default over the horizon
LGD
loss given default, as a share of the exposure
EAD
the exposure at default

The mean of the loss distribution, covered by pricing and provisions, not by capital. Using the recovery rate in place of LGD is the common slip.

Credit VaR

credit VaR = loss quantile − EL

loss quantile
the first loss whose cumulative probability reaches the confidence level
EL
the expected loss, the mean of the distribution

Over a stated horizon and confidence level. Reporting the quantile alone overstates it by the expected loss.

A practice question

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

Which of the following best defines credit VaR?

  1. The expected loss, calculated as PD x LGD x EAD.Why not: Gives expected loss, the mean of the distribution, not a high quantile measured from it.
  2. The average loss in the tail beyond the VaR quantile.Why not: Describes expected shortfall, not VaR; VaR is the quantile itself.
  3. The loss if the largest single borrower defaults with no recovery.Why not: A single-name concentration scenario; credit VaR is a statistical quantile of the whole portfolio loss distribution.
  4. The loss at a chosen confidence level over a set horizon, measured in excess of expected loss. - correct

Credit VaR = loss quantile at the confidence level minus expected loss, i.e. unexpected loss, which is the amount that economic capital must cover.

A harder one

A portfolio's one-year loss distribution (USD million) is: loss 0 with probability 90%; 2 with 5%; 5 with 3%; 10 with 1.5%; 20 with 0.5%. The 99% credit VaR, defined relative to expected loss, is closest to:

  1. USD 9.50 million - correct
  2. USD 19.50 millionWhy not: Takes the maximum loss (20) as the quantile; the 99th percentile is 10 because cumulative probability reaches 99.5% there.
  3. USD 10.00 millionWhy not: Reads the correct quantile but does not deduct the expected loss of 0.5.
  4. USD 4.50 millionWhy not: Stops at 5, where cumulative probability is only 98%, before the 99% level is reached, then deducts 0.5.

Cumulative probabilities: 90, 95, 98, 99.5. The 99% quantile is 10. EL = 0.1 + 0.15 + 0.15 + 0.1 = 0.5. Credit VaR = 10 - 0.5 = 9.5 million.

The hardest tier

A portfolio has three independent loans, each with EAD of USD 10 million, a one-year PD of 4% and LGD of 100%. The 99.9% one-year credit VaR (quantile minus expected loss) is closest to:

  1. USD 20.0 millionWhy not: Reads the correct quantile (two defaults) but does not subtract the expected loss of 1.2.
  2. USD 8.8 millionWhy not: Uses the 99% quantile (one default, cumulative 99.53%) instead of 99.9%, then subtracts 1.2.
  3. USD 28.8 millionWhy not: Takes the worst case of three defaults (30) as the quantile, which cumulates only the last 0.0064%, then subtracts 1.2.
  4. USD 18.8 million - correct

P(0)=0.8847, P(1)=0.1106, P(2)=0.0046, P(3)=0.00006, so cumulative reaches 99.99% at two defaults: the 99.9% quantile is 20. EL = 3 x 0.04 x 10 = 1.2. Credit VaR = 20 - 1.2 = 18.8 million.

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