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Use the single factor model to estimate the joint default probability and the default correlation for any two credits.

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

The idea

To get default quantities, first compute the asset return correlation from the betas. Rho = βi × βj. Next find each default threshold, which is the inverse standard normal of the firm's PD. The joint default probability is the bivariate normal probability that both asset returns fall below their thresholds. It exceeds PDi × PDj when rho is positive. Equivalently, it is the average over the factor of the product of the two conditional default probabilities. Then the default correlation, written Corr, comes from the joint default probability. Corr = (JDP − PDi × PDj) ÷ √(PDi(1 − PDi) × PDj(1 − PDj)). It is usually far smaller than rho. Given that firm i defaulted, the probability that firm j defaults is JDP ÷ PDi. A common mistake is reporting the asset correlation as the default correlation.

Formulas

Default correlation from the single factor joint default probability

Corr = (JDP − PDi × PDj) ÷ √(PDi × (1 − PDi) × PDj × (1 − PDj))

JDP
the bivariate normal probability that both firms default
PDi
firm i's default probability
PDj
firm j's default probability

JDP comes from the asset correlation βi × βj. Reporting that asset correlation as the default correlation is the common mistake.

Probability that one firm defaults given that the other has

P(j | i) = JDP ÷ PDi

JDP
the probability that both firms default
PDi
the default probability of the firm already known to have defaulted

A conditional probability, not a correlation. Under independence it collapses to PDj.

A practice question

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

In the single factor model, the joint default probability of firms i and j is:

  1. PD_i x PD_j, because the single factor makes defaults of different firms independent.Why not: Defaults are independent only conditional on the factor, not unconditionally.
  2. The bivariate normal probability that both asset returns fall below their default thresholds, with correlation beta_i x beta_j. - correct
  3. The smaller of PD_i and PD_j, because the firms share the same factor.Why not: This is the upper bound reached only at perfect dependence.
  4. PD_i x PD_j x (1 + beta_i x beta_j), a linear uplift for the shared factor.Why not: Not a valid formula; the joint probability needs the bivariate normal distribution.

Both default when R_i < k_i and R_j < k_j. R_i and R_j are standard normal with correlation beta_i beta_j, so the probability is the bivariate normal M2(k_i, k_j; beta_i beta_j).

A harder one

Two firms each have a PD of 3% and an asset return correlation of 0.30 in a single factor model. The bivariate normal calculation gives a joint default probability of 0.317% (0.00317). The default correlation is closest to:

  1. 0.109Why not: Omits the subtraction of PD x PD: 0.00317 / (0.03 x 0.97).
  2. 0.30Why not: Reports the asset correlation as the default correlation.
  3. 0.0023Why not: Reports the covariance of the default indicators, 0.00317 - 0.0009, without dividing by the variance.
  4. 0.078 - correct

rho = (JDP - p²) / (p(1-p)) = (0.00317 - 0.0009) / 0.0291 = 0.078, far smaller than the asset correlation of 0.30.

The hardest tier

In a single factor model firm A has beta 0.6 and PD 1%, and firm B has beta 0.5 and PD 4%. The bivariate normal probability that both asset returns fall below their thresholds is tabulated by asset correlation: 0.25 gives 0.00132, 0.30 gives 0.00161, 0.55 gives 0.00368. The default correlation is closest to:

  1. 0.168Why not: Averages the betas (0.55) to get the asset correlation and reads the wrong row.
  2. 0.083Why not: Omits the subtraction of PD_A x PD_B: 0.00161 / sqrt(0.01 x 0.99 x 0.04 x 0.96).
  3. 0.062 - correct
  4. 0.047Why not: Uses beta_B² (0.25) as the asset correlation and reads the wrong row.

Asset correlation = 0.6 x 0.5 = 0.30, so JDP = 0.00161. Default correlation = (0.00161 - 0.0004) / sqrt(0.0099 x 0.0384) = 0.00121 / 0.0195 = 0.062.

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