Portfolio Credit Risk and the Vasicek Single-Factor Model

Two banks each hold $1billionofcorporateloanswithidenticalPD,LGD,andaverageexposure.Bank{A}^{\prime }sportfoliois1,000loansof$1 million; Bank B's is ten loans of $100 million. Same expected loss, very different unexpected loss. Granularity changes everything once you move past the average, and the Vasicek formula tells you exactly how much.

If defaults across loans were independent, portfolio UL would scale with √N: diversification would be powerful, and even concentrated portfolios with many loans would have small relative UL. Real defaults cluster. When the economy weakens, many firms default at once because they share exposure to the same macro factor. That clustering shows up as default correlation, and it pulls portfolio UL above the independence benchmark.

For two loans with single-loan ULs of ${\text{UL}}_1$ and ${\text{UL}}_2$ and default correlation ${\rho }_d$:

$${\text{UL}}_p^2={\text{UL}}_1^2+{\text{UL}}_2^2+2{\rho }_d\cdot {\text{UL}}_1\cdot {\text{UL}}_2$$

Generalized to N homogeneous loans, portfolio UL grows with the square root of $N(1-{\rho }_d)+N^2{\rho }_d$. At ${\rho }_d=0$ it scales with √N (full diversification). At ${\rho }_d=1$ it scales with N (no diversification).

Common mistakes

• Confusing asset correlation with default correlation. The single-factor model uses asset correlation ρ; the resulting default correlation is much smaller for investment-grade obligors. A choice that uses asset correlation as default correlation overstates UL by a factor of three or more.
• Treating the Vasicek formula as exact for finite portfolios. Vasicek assumes infinite homogeneous portfolios. For 50-name portfolios, idiosyncratic risk adds materially on top. Choices that report the Vasicek number as final capital for small portfolios understate capital. Trap: a 50-loan question expects you to add a granularity adjustment.
• Using Vasicek for non-homogeneous portfolios. The closed form requires identical PD, LGD, EAD across obligors. Mixed portfolios require simulation or capital pooling by sub-portfolio. Trap: a question describes two sub-portfolios (high PD and low PD) with different EAD distributions and asks for one Vasicek number. The right approach is sub-portfolio Vasicek and aggregate.

Bottom line

• Default correlation drives portfolio UL above the independence benchmark; the asset-return correlation ρ from the single-factor model is much higher than the resulting default correlation.
• Basel IRB asset correlation is regulator-set at 0.12 to 0.24 by borrower type, while default correlations run only 0.02 to 0.10 for investment grade.
• Single-factor model: $R_i=\sqrt{\rho }\cdot M+\sqrt{1-\rho }\cdot {\epsilon }_i$. Common factor M creates correlation; idiosyncratic ε is independent across firms, and asset correlation between any two firms is exactly ρ.
• Vasicek conditional default rate at confidence c: $\Phi \left(\frac{{\Phi }^{-1}(\text{PD})+\sqrt{\rho }\cdot {\Phi }^{-1}(c)}{\sqrt{1-\rho }}\right)$, the basis of the Basel IRB capital formula. Higher PD or higher ρ both raise it.

Exam shortcut

When a question gives you PD, LGD, EAD, and ρ for a homogeneous portfolio and asks for credit VaR, plug into Vasicek immediately. That is the test. Compute ${\Phi }^{-1}(\text{PD})$, compute ${\Phi }^{-1}(c)$, apply the formula. The conditional default rate at 99.9% is always above PD; the gap grows with ρ.

The full lesson (about 3,082 words, 21 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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