How do I calculate single-name credit VaR for an FRM Part II question?

Question

AcadiFi · Accepted Answer

Single-name credit VaR measures the potential loss on one obligor at a chosen confidence level. Consider a five-year $12 million loan to Vermilion Industries with EAD = $12M, LGD = 45%, and one-year PD = 2.3%. Expected loss EL = EAD * LGD * PD = $124,200.

Two approaches:

(1) Mode-of-loss method: unexpected loss depends on loss distribution. Assuming default is Bernoulli with stochastic LGD following a beta distribution (mean 45%, standard deviation 20%), the variance of loss is EAD^2 * [PD*(1-PD)*LGD_mean^2 + PD*Var(LGD)]. With those inputs, variance = $12M^2 * [0.023*0.977*0.2025 + 0.023*0.04] = $810k^2. Standard deviation of loss is $810k.

(2) Empirical quantile: for a single name, the 99% VaR either equals zero (if 1 - PD > 0.99) or equals the full loss. Here PD = 2.3% > 1%, so at 99% confidence the worst 1% includes default states. 99% credit VaR = EAD * LGD * 1 = $5.4M in the default branch.

Unexpected loss (UL) is 99% VaR minus EL = $5.4M - $124k = $5.28M, the capital cushion needed above reserves. Note the contrast with market VaR: credit loss distributions are highly skewed and discrete, so VaR at common confidence levels often jumps from zero to full LGD.

How do I calculate single-name credit VaR for an FRM Part II question?

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