scan4llm/3_surya.md

2. Literature review

2.1. Highlights of the IRB set-up and evolution

1987 Vasicek model. IRB owes to the Nobel laureate in economics Robert Merton and Oldrich von Vasicek for its birth. As early as in Merton (1974), there came an idea to model a hypothetical borrower’s finance, its assets more specifically, as a Normally-distributed random walk. The default event happens when the asset pattern breaches the fixed thresholds signaling for the artificial liabilities. Hence, the probability of such a breach given the known (assumed) nature of the probabilistic distribution allows us to proceed with the probability of default ($PD$) at the individual borrower’s level.

More than a decade later, the concept to consider a bunch of such individual borrowers was developed in Vasicek (1987). The starting point was to decompose the individual borrowers’ pattern into specific (idiosyncratic) and common (systemic) components as in Eq. (1).

$$ A_i = Z \cdot r + u_i \cdot \sqrt{1 - R} \quad (1) $$

where $Z \sim N(0, 1)$ - common factor; $u_i \sim N(0, 1)$ - specific factor; $r = \text{corr}(A_i; Z)$ - correlation of factors (conventional mathematical correlation) as it originally appeared without a squared power in the works by Vasicek (1987); Gordy (2000) (let us call it default correlation);

$R = r^2$ - asset (value) correlation using BCBS notations (mathematically speaking, it is the square of the conventional mathematical correlation between $A_i$ and $Z$).

Due to the nice properties of the Gaussian distribution, Vasicek uses the Value-at-Risk (VaR) risk-measure to slice the default rate (DR) distribution of the loan portfolio and arrives at the worst feasible default rate realization given the chosen significance level of $\alpha$ in Eq. (2).

$$ VaR = N \left( \frac{N^{-1}(PD) + N^{-1}(1 - \alpha) \cdot \sqrt{R}}{\sqrt{1 - R}} \right). \quad (2) $$

Important to note that Vasicek did not impose any restrictions over $r$, over its feasible values. The only thing he notes is that in case $r > 50\%$, the DR distribution becomes U-shaped (bimodal). Remembering the notations that $R = r^2$, or $r = \sqrt{R}$, we should take away that Vasicek was speaking of DR distribution bimodality when the asset (value) correlation exceeds 25% ($R > 25\%$).

2006 BCBS amendment to the Vasicek model. The idea to internationally introduce model-based (IRB) credit risk regulation seems to have become popular after the success of such regulation introduction for the market risk in the Basel I amendment BCBS (1996). Thus, the very first draft of the future Basel II Accord incorporating IRB appeared on the edge of the new millennium, see Penikas (2020a), while it took another six years to polish it to the very final (comprehensive) version of BCBS (2006).

When designing IRB, the Basel Committee implemented five conceptual amendments, originally not previewed in the Vasicek model:

1. BCBS chose the IRB significance level of $\alpha = 0.1\%$ (inversely, confidence level of 99.9%) what corresponds to the risk-measure breach (bank failure) once in one hundred years given the annual horizon for $PD$, see BCBS (2005a).2. BCBS added other credit risk parameters like loss given default ($LGD$), exposure at default ($EAD$), maturity ($M$) etc. For instance, $VaR$ from Eq. (2) is multiplied by $LGD$ and $EAD$.

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