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Outline how a single factor model gauges portfolio credit risk, correlation effects included.

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

The idea

In the single factor model each firm's standardised asset return is Ri = βi × M + √(1 − βi²) × ei. Here M is a common factor such as the state of the economy, and ei is an idiosyncratic shock. Both are standard normal, and the ei are independent of M and of each other. A firm defaults when Ri falls below a threshold, and that threshold is ki = N⁻¹(PDi). Conditional on M, defaults are independent, so each firm has CondPD = N((ki − βi × M) ÷ √(1 − βi²)). Averaging over M recovers the unconditional PD. The model needs one beta per firm instead of N(N − 1) ÷ 2 pairwise correlations. A bad outcome for M raises every firm's default probability at once. Common errors are flipping the sign of βi × M and omitting the square-root denominator.

Formulas

Asset return in the single factor model

Ri = βi × M + √(1 − βi²) × ei

βi
the firm's loading on the common factor
M
the common factor, standard normal
ei
the firm's idiosyncratic shock, standard normal and independent of M and of other firms' shocks

Ri is standardised, with variance βi² from the factor and 1 − βi² from the shock. Correlation between firms runs only through M.

Conditional default probability given the factor

CondPD = N((ki − βi × M) ÷ √(1 − βi²))

N
the standard normal cumulative distribution function
ki
the default threshold, N⁻¹ of the firm's PD
βi
the firm's loading on the common factor
M
the realised value of the common factor

A negative M raises the probability; flipping the sign of βi × M lowers it instead. Dividing by 1 − βi² rather than its square root is the other slip.

A practice question

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

In the single factor model R_i = beta_i x M + sqrt(1 - beta_i²) x e_i, which assumption is made about the idiosyncratic shocks e_i?

  1. They are correlated with M by an amount equal to beta_i.Why not: Beta_i is the loading on M; the idiosyncratic shock is assumed independent of M.
  2. They are correlated across firms in the same industry in proportion to their betas.Why not: Industry effects would require extra factors; the single factor model has none.
  3. They are standard normal and independent of M and of each other, so the common factor is the only source of correlation. - correct
  4. They have variance equal to beta_i².Why not: The factor part has variance beta_i²; the idiosyncratic part has variance 1 - beta_i².

Independence of the e_i from M and from each other makes all dependence between firms run through the factor, which is what makes conditional defaults independent.

A harder one

In a single factor model, a firm has beta = 0.5 and PD = 2% (threshold -2.054). If the common factor realises M = -2.0, the firm defaults when its idiosyncratic shock e is below a value closest to:

  1. -3.53Why not: Adds beta x M instead of subtracting it, giving (-2.054 - 1.0) / 0.866.
  2. -1.05Why not: Omits the division by sqrt(1 - beta²).
  3. -1.40Why not: Divides by 1 - beta² = 0.75 instead of by its square root.
  4. -1.22 - correct

Default occurs when beta M + sqrt(1 - beta²) e < k, so e < (k - beta M) / sqrt(1 - beta²) = (-2.054 + 1.0) / 0.866 = -1.22. A bad factor draw makes default more likely.

The hardest tier

A firm has PD = 1% and beta = 0.4 in a single factor model. In a one-in-a-hundred bad economy the factor is M = -2.326. Use these table values: N(-1.40) = 0.0808, N(-1.52) = 0.0643, N(-2.13) = 0.0166, N(-3.55) = 0.0002. The firm's conditional default probability is closest to:

  1. 0.02%Why not: Gets the sign of beta x M wrong, giving z = (-2.326 - 0.930) / 0.917 = -3.55.
  2. 8.1%Why not: Omits the division by sqrt(1 - beta²), giving z = -1.40.
  3. 1.7%Why not: Uses beta² = 0.16 as the loading and sqrt(1 - 0.16) as the denominator, giving z = -2.13.
  4. 6.4% - correct

z = (N^-1(0.01) - beta M) / sqrt(1 - beta²) = (-2.326 + 0.930) / 0.9165 = -1.52. N(-1.52) = 0.0643, so the default rate rises from 1% to about 6.4% in the bad state.

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