Conditional and Unconditional Portfolio Loss in Vasicek and Gordy Models

Summary

The document asks how to distinguish the large-portfolio loss results associated with the Vasicek credit risk model and Gordy’s foundation for ratings-based bank capital rules. It focuses on whether Vasicek establishes convergence of portfolio loss to a conditional expected loss given a common risk factor, while Gordy additionally establishes convergence without conditioning on that factor.

The question states the conditional loss expression in terms of the default probability, asset correlation, and a standard normal risk factor, but does not provide an answer or supporting derivation. Its main value is identifying a subtle issue in interpreting asymptotic results and inconsistent notation across the cited papers. Readers should consult the original papers to resolve whether the unconditional convergence claim is distinct or already follows from Vasicek’s result; this document alone does not settle the point.

Key ideas

• The question distinguishes portfolio loss conditional on a common risk factor from unconditional portfolio loss.
• It presents a conditional default probability as a function of the factor, baseline default probability, and asset correlation.
• It asks whether Gordy adds an unconditional convergence result beyond the result attributed to Vasicek.
• The document gives no derivation or resolution, so the cited papers are needed to answer the question.

Tags

# Difference between Vasicek and Gordy models


# Difference between Vasicek and Gordy models












I'm trying to understand what Gordy [1] added to Vasicek [2] model (the core of the IRB formula of Basel Accords).

Is it correct to say the Vasicek shows that the portfolio loss conditional on $Y$ converges, by the law of large numbers, to its expectation $$p(Y)=\Phi\left(\frac{\Phi^{-1}(p)-\sqrt{\rho}\,Y}{\sqrt{1-\rho}}\right),$$ whereas Gordy says that, under their assumptions, even the (unconditional) portfolio loss converge to $p(Y)$?

That is, for Vasicek $$L\mid Y\to p(Y)$$ whereas in Gordy $$L\to p(Y)$$ or the latter result is already present in Vasicek?

I'm confused because the notation used in Vasicek is not perfectly consistent with what written in their text.

[1]: Gordy, Michael B. (2003). A risk-factor model foundation for ratings-based bank capital rules. Journal of financial intermediation 12(3), 199–232.

[2]: Vasicek, Oldrich A. (2002). The distribution of loan portfolio value. Risk 15(12), 160–162.

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