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Quantitative Risk Management / Portfolio Credit Risk and Credit Derivatives

Asymptotics for Large Portfolios and the Basel IRB Formula

The Vasicek distribution derived in Bernoulli Mixture Models is not just a theoretical curiosity. It is the mathematical foundation of the Basel II Internal Ratings-Based (IRB) capital formula — the formula that determines minimum regulatory capital for credit portfolios in thousands of banks worldwide. This note makes that connection rigorous.

The key result is a two-step argument:

  1. Conditional LLN (Proposition 11.17): In a Bernoulli mixture model with a common factor Ψ\Psi, the relative portfolio loss L(m)/amL^{(m)}/a_m converges almost surely to the deterministic function ℓˉ(Ψ)\bar\ell(\Psi) conditional on each Ψ=ψ\Psi = \psi. This means idiosyncratic risk is fully diversified away in large portfolios — only systematic risk remains.
  2. Quantile limit (Proposition 11.18): The α\alpha-quantile of the portfolio loss converges to ℓˉ(qα(Ψ))\bar\ell(q_\alpha(\Psi)) — the asymptotic loss function evaluated at the α\alpha-quantile of the factor. For the one-factor Gaussian model this is exactly the Vasicek formula, and setting α=99.9%\alpha = 99.9\% yields the Basel risk capital contribution per obligor.
The Basel IRB formula (QRM (11.37)). For obligor ii, the required regulatory risk capital is:
RCi=c δi ei Φ ⁣(Φ−1(pi)+βi Φ−1(0.999)1−βi).(1)\text{RC}_i = c\,\delta_i\,e_i\,\Phi\!\left(\frac{\Phi^{-1}(p_i) + \sqrt{\beta_i}\,\Phi^{-1}(0.999)}{\sqrt{1-\beta_i}}\right). \tag{1}

This lesson derives every term in (1) from first principles.

Learning objectives. By the end of this note you should be able to:
  • state Propositions 11.17–11.18 and their key assumptions;
  • explain how idiosyncratic risk diversifies in large portfolios while systematic risk does not;
  • derive the Vasicek VaR formula for the homogeneous one-factor Gaussian model;
  • trace each component of the Basel IRB formula back to the ASRF derivation;
  • state the limitations of the IRB formula: ad-hoc βi\beta_i, granularity, and one-factor restriction.

In this lesson

    1. The ASRF framework: formalising the asymptotics
    1. The one-factor Gaussian model and Vasicek VaR
    1. Derivation of the Basel IRB formula
    1. Limitations of the Basel IRB approach

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