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State what default correlation is for credit portfolios and compute it.

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

The idea

Default correlation is the correlation between the default indicators of two obligors over a chosen horizon. Each indicator equals 1 if the obligor defaults and 0 otherwise. Let p1 and p2 be the default probabilities and JDP the joint default probability. With those definitions, the default correlation, written Corr, is given by Corr = (JDP − p1 × p2) ÷ √(p1(1 − p1) × p2(1 − p2)). If defaults are independent, the numerator is zero, so the correlation is zero and the joint default probability is JDP = p1 × p2. Rearranging the formula, with ρ the default correlation, gives JDP = p1 × p2 + ρ × √(p1(1 − p1) × p2(1 − p2)). Default correlations are small, often a few percent, and well below the asset return correlation that drives them, because a binary event discards information. They also depend on the PDs and the horizon, and cannot reach 1 when the PDs differ. The usual errors are dropping the p1 × p2 term or the square root in the denominator.

Formulas

Default correlation of two obligors

Corr = (JDP − p1 × p2) ÷ √(p1 × (1 − p1) × p2 × (1 − p2))

JDP
the probability that both obligors default within the horizon
p1
the first obligor's default probability
p2
the second obligor's default probability

The correlation of two Bernoulli default indicators over one horizon. Independence makes the numerator zero; with unequal PDs the maximum is well below 1.

Joint default probability from the default correlation

JDP = p1 × p2 + ρ × √(p1 × (1 − p1) × p2 × (1 − p2))

p1
the first obligor's default probability
p2
the second obligor's default probability
ρ
the default correlation, not the asset correlation

The default correlation formula rearranged. Forgetting the p1 × p2 term reports only the correlation part.

A practice question

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

Which definition describes the default correlation of two obligors over a given horizon?

  1. The correlation between the two obligors' default indicators, each equal to 1 if the obligor defaults within the horizon and 0 otherwise. - correct
  2. The correlation between the two obligors' asset returns, whose fall below a threshold triggers default.Why not: Confuses default correlation with the asset return correlation that sits underneath it.
  3. The probability that the second obligor defaults given that the first has defaulted.Why not: A conditional default probability is not a correlation coefficient; it equals JDP / PD of the first obligor.
  4. The correlation between changes in the two obligors' credit spreads over the horizon.Why not: Describes a market-based spread correlation, not the correlation of default events.

Default correlation is the correlation of the Bernoulli default indicators. Asset return correlation, conditional probability and spread correlation are related ideas but are different quantities.

A harder one

Two speculative-grade borrowers have one-year default probabilities of 10% and 20%, and the probability that both default within the year is 3.5%. The default correlation is closest to:

  1. 0.292Why not: Omits the subtraction of p1 x p2 (0.02) from the joint default probability: 0.035 / 0.12.
  2. 0.125 - correct
  3. 0.106Why not: Uses sqrt(p1 x p2) as the denominator, ignoring the (1 - p) terms: 0.015 / 0.1414.
  4. 0.750Why not: Divides the covariance by the product of the PDs (0.015 / 0.02), which is not a correlation.

rho = (JDP - p1 p2) / sqrt(p1(1-p1) p2(1-p2)) = (0.035 - 0.02) / sqrt(0.09 x 0.16) = 0.015 / 0.12 = 0.125.

The hardest tier

Two obligors have one-year PDs of 1% and 10%. The highest default correlation they could possibly have is closest to:

  1. 1.00Why not: Assumes a correlation of 1 is always attainable; with different PDs the joint default probability cannot exceed the smaller PD.
  2. 0.10Why not: Uses the ratio of the PDs (1% / 10%) as the maximum correlation.
  3. 0.30 - correct
  4. 0.34Why not: Divides the maximum JDP of 1% by the standard deviations without subtracting p1 x p2 (0.01 / 0.0299).

The joint default probability is at most min(p1, p2) = 1%. Then rho_max = (0.01 - 0.001) / sqrt(0.0099 x 0.09) = 0.009 / 0.02985 = 0.30, well below 1.

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