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Show what correlation risk contributes to both market risk and credit risk.

FRM® Part II · Market Risk Measurement and Management · Correlation Basics: Definitions, Applications, and Terminology · FRM2.MR.8.f

The idea

Correlation risk affects both main risk types. In market risk it is the co-movement of risk factors. Portfolio VaR, diversification, hedge effectiveness and the value of multi-asset derivatives all depend on correlations that change, usually upwards in stress. In credit risk it appears as default correlation, the tendency of obligors to default together. Take two obligors with default probabilities pA and pB and default correlation ρ. Their joint default probability is given by Joint = pA × pB + ρ × √(pA(1 − pA) × pB(1 − pB)). Correlation therefore raises the chance of joint defaults and fattens the tail of portfolio credit losses without changing expected loss. It also drives CDO tranche values, first-to-default baskets and wrong-way risk, where exposure to a counterparty grows as its credit quality falls. Exam point: correlation changes the shape of the loss distribution, not the expected loss.

Formulas

Joint default probability of two obligors

P(AB) = pA × pB + ρ × √(pA × (1 − pA) × pB × (1 − pB))

pA
the default probability of obligor A
pB
the default probability of obligor B
ρ
the default correlation between them

At zero correlation it collapses to pA × pB. Applying ρ as a multiplier of pA × pB, or dropping the pA × pB term, are the usual slips.

A practice question

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

In credit risk, correlation risk arises mainly through:

  1. The relationship between a bond's price and its own yield, which determines its duration.Why not: That relationship is a deterministic price-yield link for one bond, not a co-movement between obligors.
  2. Default correlation: the tendency of obligors to default together, fattening the loss tail. - correct
  3. The correlation of interest rates across currencies, which is the only driver of credit portfolio losses.Why not: Rates matter for valuation, but credit portfolio losses are driven by joint defaults.
  4. The fixed recovery rate assumed for every defaulted obligor.Why not: Recovery assumptions are a separate input and say nothing about how defaults cluster.

Default correlation raises the probability of joint defaults and so the tail of portfolio credit losses, while expected loss is unchanged.

A harder one

Two firms have one-year default probabilities of 2% and 3%, and the default correlation between them is 0.25. Using P(AB) = pA x pB + rho x sqrt(pA(1-pA) x pB(1-pB)), the joint default probability is closest to:

  1. 0.06%.Why not: Multiplies the probabilities, which assumes independence.
  2. 0.60%.Why not: Includes only the covariance term and omits the pA x pB term.
  3. 0.075%.Why not: Applies the correlation as a multiplier of pA x pB instead of using the covariance term.
  4. 0.66%. - correct

pA x pB = 0.0006 and rho x sqrt(0.02 x 0.98 x 0.03 x 0.97) = 0.25 x 0.02388 = 0.00597, so the joint probability is 0.0066, or 0.66%.

The hardest tier

A first-to-default basket on two names pays if at least one default occurs within the year. The default probabilities are 5% and 8% and the default correlation is 0.4. The probability that at least one name defaults is closest to:

  1. 12.6%.Why not: Assumes independence (0.05 + 0.08 - 0.05 x 0.08) and ignores the 0.4 correlation.
  2. 13.0%.Why not: Sums the two probabilities without subtracting the probability of both defaulting.
  3. 15.8%.Why not: Adds the joint default probability instead of subtracting it.
  4. 10.2%. - correct

P(AB) = 0.05 x 0.08 + 0.4 x sqrt(0.05 x 0.95 x 0.08 x 0.92) = 0.0277. P(at least one) = 0.05 + 0.08 - 0.0277 = 0.1023. Higher correlation lowers the first-to-default probability.

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