hard · FRM Part 2 Credit Risk

A credit portfolio of 100 identical names is modeled with a one-factor Gaussian copula, asset correlation ρ, and common marginal default probability p. An analyst recalibrates by doubling ρ while holding p fixed, then is surprised that the expected number of defaults is unchanged but the equity-tranche (first-loss) value rises. The most accurate explanation is:

1. Raising ρ leaves the marginal p and hence expected defaults unchanged, but fattens both tails of the loss distribution, shifting mass toward zero and many defaults — sparing the equity tranche while hurting senior tranches.
2. Raising the asset correlation lowers each obligor's marginal default probability while the joint dependence structure stays fixed, so expected defaults fall and the equity tranche gains from this reduced mean portfolio loss.
3. Raising the asset correlation raises expected defaults but simultaneously compresses the loss variance around that higher mean, so the equity tranche gains purely from the reduced dispersion of losses across the range of possible scenarios.
4. Raising the asset correlation leaves the entire loss distribution unchanged because the Gaussian copula is exchangeable across all names, so any observed tranche-value shift must instead stem entirely from a mispriced recovery-rate assumption.

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