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Show why a securitization's credit risk depends on the default probabilities and default correlations of its collateral.

FRM® Part II · Credit Risk Measurement and Management · Structured Credit Risk · FRM2.CR.22.i

The idea

A tranche's credit risk depends on how likely the collateral is to default and on how defaults cluster. The expected pool loss depends on default probabilities and loss given default only, so correlation does not change it. Correlation changes how that loss is distributed. With low correlation, defaults are spread evenly, so the pool loss is close to its mean, which lets equity absorb it and leaves senior nearly safe. With high correlation, outcomes are all-or-nothing: either few defaults occur or many occur together. This lowers the expected loss of equity and raises that of senior, with mezzanine somewhere in between and sometimes either way. Let p1 and p2 be the default probabilities of two assets and ρ their default correlation. For two assets, Pjoint = p1 × p2 + ρ × √(p1 × (1 − p1) × p2 × (1 − p2)). Exam point: correlation moves risk between tranches without changing the pool's expected loss.

Formulas

Joint default probability of two assets

P(both) = p₁ × p₂ + ρ × √(p₁(1 − p₁)p₂(1 − p₂))

p₁, p₂
the two assets' default probabilities
ρ
the default correlation, the correlation of the default indicators

At ρ = 0 it reduces to the independent product. The chance of at least one default is p₁ + p₂ minus this, so correlation helps a first-loss position.

Variance of the number of defaults in a homogeneous pool

variance = n × p × (1 − p) × [1 + (n − 1) × ρ]

n
the number of names
p
each name's default probability
ρ
the pairwise default correlation

Correlation widens the distribution without changing the mean, n × p. Take the square root of the whole product for the standard deviation.

A practice question

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

If default correlation among the collateral rises while default probabilities and loss given default stay constant, the expected loss of the equity tranche will generally:

  1. fall, and that of the senior tranche will rise. - correct
  2. rise, and that of the senior tranche will fall.Why not: Reverses the effect: with high correlation, defaults cluster and the equity is less likely to be hit in moderate scenarios while senior is hit in extreme ones.
  3. rise, and that of the senior tranche will rise.Why not: Assumes all tranches get riskier; correlation redistributes losses without changing the pool's expected loss.
  4. stay unchanged, as will that of every tranche.Why not: Only the pool's expected loss is unchanged by correlation; the tranche losses change.

Higher correlation makes the pool loss distribution wider: more probability of very low losses (good for equity) and of very high losses (bad for senior). Since the total expected loss is fixed, equity gains and senior loses.

A harder one

Two loans each have a one-year default probability of 5 percent, and the default correlation (correlation of default indicators) is 0.20. Closest to the probability that both default?

  1. 0.25 percentWhy not: Multiplies the probabilities as if defaults were independent.
  2. 5.0 percentWhy not: Assumes perfect dependence, in which the second loan always defaults with the first.
  3. 1.2 percent - correct
  4. 0.95 percentWhy not: Adds only the correlation term rho x p(1-p) and leaves out the independent term p².

Joint default probability = p x p + rho x p(1-p) = 0.0025 + 0.20 x 0.05 x 0.95 = 0.0025 + 0.0095 = 1.2 percent, nearly five times the independent value.

The hardest tier

A pool has 10 names, each with default probability 4 percent, and a pairwise default correlation of 0.15. The variance of the number of defaults is np(1-p)[1 + (n-1)rho]. Closest to the standard deviation of the number of defaults?

  1. 0.95 - correct
  2. 0.62Why not: Ignores correlation and uses the binomial variance np(1-p) alone.
  3. 0.90Why not: Reports the variance (0.902) as if it were the standard deviation.
  4. 1.46Why not: Takes the square root of the binomial variance only and multiplies by (1 + (n-1)rho), rather than taking the square root of the product.

Variance = 10 x 0.04 x 0.96 x (1 + 9 x 0.15) = 0.384 x 2.35 = 0.902. Standard deviation = 0.95, compared with 0.62 without correlation, which shows the widening of the loss distribution.

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