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Review other methods for estimating default correlation.

FRM® Part II · Credit Risk Measurement and Management · Credit Derivatives · FRM2.CR.13.h

The idea

Default correlation measures how closely two names default together. Here P1 and P2 are the individual default probabilities, and P12 is the joint default probability. In symbols, the default correlation is Corr = (P12 − P1 × P2) ÷ √(P1 × (1 − P1) × P2 × (1 − P2)). It is usually much lower than asset or equity correlation, and it is hard to estimate because defaults are rare. Alternatives to implied correlation include three methods. One uses historical data, correlating annual default rates across sectors or ratings over many years. Another uses equity return correlations as a proxy for asset correlation in a structural (Merton-type) model. A third fits a factor model in which each firm loads on a common factor, so the pairwise correlation equals the product of the two loadings, the square roots of their R² values. Each method is a proxy, and none matches the market-implied approach in being forward looking.

Formulas

Default correlation from joint and individual default probabilities

Corr = (P12 − P1 × P2) ÷ √(P1 × (1 − P1) × P2 × (1 − P2))

P12
the probability that both names default
P1
the first name's default probability
P2
the second name's default probability

Without the square root the result is not scaled by the standard deviations. It is usually far below the asset correlation.

Pairwise correlation in a one-factor model

ρ = a₁ × a₂

a₁, a₂
each firm's loading on the common factor, the square root of its R² against that factor

Assumes one factor explains all common movement. Multiplying the R² values rather than their square roots is the slip.

A practice question

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

What does the default correlation between two borrowers measure?

  1. The correlation between the credit spreads of the two borrowers' bonds measured over the previous year.Why not: Spread correlation is a market proxy, not the correlation of default events themselves.
  2. The correlation between the recovery rates of the two borrowers after a default, measured on defaulted bonds.Why not: Recovery correlation concerns loss severity, not whether defaults occur together.
  3. The correlation of the two borrowers' default indicators, built from joint and individual default probabilities. - correct
  4. The probability that both borrowers default in the same calendar year.Why not: This is the joint default probability, one input to the correlation; the correlation also scales by the individual probabilities.

Default correlation = (P12 - P1 x P2) / sqrt(P1(1 - P1) x P2(1 - P2)): the covariance of the two default indicators scaled by their standard deviations.

A harder one

Two firms have one-year default probabilities of 2 percent and 3 percent, and the probability that both default in the year is 0.15 percent. The default correlation is closest to:

  1. 2.50Why not: Divides joint probability by the product of the individual probabilities, a ratio of 2.5 rather than a correlation.
  2. 1.58Why not: Leaves out the square root in the denominator, so the covariance is not scaled by standard deviations.
  3. 0.0009Why not: Reports the covariance of the default indicators and does not scale it into a correlation.
  4. 0.038 - correct

Covariance = 0.0015 - 0.02 x 0.03 = 0.0009. Denominator = sqrt(0.02 x 0.98 x 0.03 x 0.97) = 0.02388. Correlation = 0.038.

The hardest tier

In a one-factor model, the R² of two firms' equity returns against a common market factor are 0.25 and 0.64. Assuming the single factor explains all the common movement, the implied pairwise correlation between the firms is closest to:

  1. 0.16Why not: Multiplies the R² values; the correlation is the product of the loadings (square roots of the R² values).
  2. 0.45Why not: Averages the two R² values, which is not how factor loadings combine.
  3. 0.40 - correct
  4. 0.65Why not: Averages the two loadings instead of multiplying them.

Loadings are sqrt(0.25) = 0.5 and sqrt(0.64) = 0.8. In a one-factor model pairwise correlation = loading1 x loading2 = 0.5 x 0.8 = 0.40.

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