Portfolio Credit Risk and the Vasicek Single-Factor Model

Two banks each hold $1 billion of corporate loans with identical probability of default (PD), loss given default (LGD), and average exposure. Bank A's portfolio is 1,000 loans of $1 million; Bank B's is ten loans of $100 million. Same expected loss (EL), very different unexpected loss (UL). The Vasicek formula supplies an infinitely granular benchmark; finite portfolios require a separate treatment of residual single-name risk.

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. The resulting error depends on the PDs and model; there is no fixed overstatement factor.
• 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.

Bottom line

• Default correlation drives portfolio UL above the independence benchmark; asset-return and default-indicator correlation are different quantities, with the conversion depending on marginal PDs and the model.
• Basel IRB correlation functions depend on exposure class; asset correlation has no universal conversion to default correlation.
• Single-factor model: $R_i = \sqrt{\rho} \cdot M + \sqrt{1-\rho} \cdot \varepsilon_i$. 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 raises it; the correlation effect depends on the inputs and confidence level.

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. For typical low-PD portfolios at a high confidence level, systematic stress raises the conditional default rate above PD. Check the supplied inputs and finite-portfolio assumptions.

The full lesson (about 4,019 words, 27 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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