How does the Vasicek single-factor model work for credit portfolio loss estimation, and what is the granularity adjustment?

The Vasicek single-factor model is the theoretical foundation for the Basel IRB capital formula. It models credit losses in a portfolio where each obligor's default depends on a common systematic factor (the economy) and an idiosyncratic factor.

Model Setup

Each obligor i defaults if its asset value falls below a threshold:

> Ai = sqrt(rho) Z + sqrt(1 - rho) ei

Where Z is the common systematic factor (standard normal), ei is the idiosyncratic factor (standard normal, independent), and rho is the asset correlation. Obligor i defaults if Ai < N_inv(PDi), where PDi is its probability of default.

Conditional Default Probability

Conditional on a realization of the systematic factor Z = z:

> PD(z) = N[(N_inv(PD) - sqrt(rho) * z) / sqrt(1 - rho)]

This is the probability that any single obligor defaults given the state of the economy. In a bad economy (z very negative), PD(z) rises dramatically — this is how the model generates correlated defaults.

Example: Silverlake Bank's Corporate Portfolio

Silverlake Bank has a portfolio of 500 corporate loans, each with PD = 2% and asset correlation rho = 0.20. Under the Vasicek assumption of infinite granularity, the portfolio loss at the 99.9th percentile is:

> Loss(99.9%) = LGD N[(N_inv(0.02) + sqrt(0.20) N_inv(0.999)) / sqrt(1 - 0.20)]

Computing step by step:

• N_inv(0.02) = -2.054
• N_inv(0.999) = 3.090
• sqrt(0.20) = 0.4472
• sqrt(0.80) = 0.8944

Loss(99.9%) = LGD N[(-2.054 + 0.4472 3.090) / 0.8944]

= LGD * N[(-2.054 + 1.382) / 0.8944]

= LGD * N[-0.751]

= LGD * 0.2263

With LGD = 45%, the 99.9% portfolio loss = 0.45 * 0.2263 = 10.18% of exposure.

The Granularity Adjustment

The Vasicek formula assumes the portfolio is infinitely granular (each loan is infinitesimally small). Real portfolios have concentration: a few large exposures can dominate. The granularity adjustment adds a correction:

> GA = (1/2) HHI [VaR_adjusted_term]

Where HHI is the Herfindahl-Hirschman Index of exposure concentration. A portfolio where one loan is 10% of total exposure has much higher GA than one where each loan is 0.2%.

FRM exam tip: Know the conditional default probability formula and be able to plug in numbers. Understand that higher asset correlation means fatter tails in the loss distribution (more correlated defaults in downturns). The granularity adjustment is tested conceptually — know that concentrated portfolios need more capital than the base IRB formula suggests.

For more credit risk modeling practice, explore our FRM Part II materials.