Vasicek (2002) Credit Portfolio & Basel IRB Capital Lab

Simulate asymptotic single risk factor (ASRF) credit loss distributions, 99.9% Value at Risk (VaR), Expected Shortfall (CVaR), and Basel II/III Internal Ratings-Based (IRB) regulatory capital requirements.

Banking Presets:

Credit Portfolio Parameters

Basel Asset Category

Probability of Default PD (%):

1-year unconditional default probability.

Loss Given Default LGD (%):

Economic downturn loss percentage.

Exposure at Default EAD ($):

Effective Maturity M (Years):

Portfolio Obligor Count N:

Evaluates Gordy (2003) Granularity Adjustment.

Institutional Capital & Risk Metrics

Asset Correlation (ρ)

99.9% Default VaR

Basel Capital K (%)

RWA ($)

Expected Loss (EL)

Unexpected Loss (UL)

Maturity Adj (MA)

Granularity Adj

Visual Distribution & Term Structure

Multi-Horizon Quantile & Tail Risk Breakdown

Metric	95.0% Confidence	99.0% Confidence	99.9% Regulatory (Basel)	99.9% Expected Shortfall (CVaR)	8% Min Regulatory Capital
Portfolio Default Rate	--	--	--	--	--

Sensitivity: Asset Correlation (ρ) vs Default Probability (PD) on 99.9% VaR

ρ \ PD	0.10%	0.50%	1.50%	3.00%	6.00%

Evaluates tail sensitivity to systemic factor loading.

Sensitivity: Loss Given Default (LGD) vs Maturity (M) on Basel Capital Requirement K (%)

LGD \ M	1.0Y	2.0Y	3.0Y	5.0Y	7.0Y

Displays capital requirement multiplier scaling with loan tenure.

Mathematical Foundations: Oldrich Vasicek (2002) & Basel IRB Capital Framework

1. Single-Factor Asset Return & Default Threshold

Under Robert C. Merton's (1974) structural credit framework, an obligor defaults when its asset value falls below a debt threshold. Oldrich Vasicek (1987, 2002) decomposed firm $i$'s asset return $X_i$ into a single common macroeconomic factor $Y \sim \mathcal{N}(0, 1)$ and an idiosyncratic firm-specific shock $Z_i \sim \mathcal{N}(0, 1)$:

$$X_i = \sqrt{\rho} Y + \sqrt{1 - \rho} Z_i$$

Firm $i$ defaults if $X_i \le c_i = \Phi^{-1}(\text{PD}_i)$. Conditional on the macroeconomic state $Y = y$, the default probability is:

$$p(y) = \mathbb{P}(X_i \le c_i \mid Y = y) = \Phi\left( \frac{\Phi^{-1}(\text{PD}) - \sqrt{\rho} y}{\sqrt{1 - \rho}} \right)$$

By the Law of Large Numbers, in an infinitely granular portfolio (ASRF), the idiosyncratic risks $Z_i$ diversify to zero, and the portfolio loss rate becomes exactly $p(y)$.

2. Vasicek Loss Distribution & Regulatory Capital

The portfolio loss Cumulative Distribution Function (CDF) and 99.9% Value at Risk ($\text{VaR}_{0.999}$) are derived in closed form:

$$F(x) = \Phi\left( \frac{\sqrt{1 - \rho}\Phi^{-1}(x) - \Phi^{-1}(\text{PD})}{\sqrt{\rho}} \right)$$ $$\text{VaR}_{0.999} = \Phi\left( \frac{\Phi^{-1}(\text{PD}) + \sqrt{\rho}\Phi^{-1}(0.999)}{\sqrt{1 - \rho}} \right)$$

The Basel Committee on Banking Supervision (BCBS) specifies the regulatory capital requirement $K$ by scaling unexpected loss by downturn LGD and a maturity adjustment factor $\text{MA}(M, \text{PD})$:

$$K = \text{LGD} \times \left[ \text{VaR}_{0.999} - \text{PD} \right] \times \frac{1 + (M - 2.5)b(\text{PD})}{1 - 1.5 b(\text{PD})}$$ $$\text{RWA} = K \times 12.5 \times \text{EAD}, \quad \text{Reg Capital} = 8\% \times \text{RWA} = K \times \text{EAD}$$

Interactive Credit Risk & Basel IRB Mastery Quiz

Test your comprehension of asymptotic single risk factor theory, portfolio invariance, regulatory capital formulas, and granularity adjustments.

Frequently Asked Questions

The Vasicek (1987, 1991, 2002) Asymptotic Single Risk Factor model is a structural portfolio credit risk framework. Each obligor's asset return is driven by a single common systematic macroeconomic factor and an idiosyncratic shock. Under the Law of Large Numbers in an infinitely granular portfolio, idiosyncratic risk is fully diversified away, leaving portfolio default losses as a deterministic function of the systematic factor.

The Basel Committee on Banking Supervision adopted the Vasicek ASRF model as the foundation of the Internal Ratings-Based (IRB) approach. Banks estimate their own Probability of Default (PD), Loss Given Default (LGD), Exposure at Default (EAD), and Maturity (M). The Vasicek formula evaluates the 99.9% confidence level portfolio default rate to compute Unexpected Loss and Risk-Weighted Assets (RWA = 12.5 × Capital).

In the Basel corporate asset correlation formula, rho decreases from 24% for high-grade credits down to 12% for high-PD distressed credits. Empirical credit research shows that high-grade firms default almost exclusively during severe macroeconomic downturns (high systematic correlation), whereas distressed firms often default due to firm-specific idiosyncratic problems (lower systematic correlation).

Expected Loss (EL = PD × LGD × EAD) is the statistical average loss anticipated in normal business operations and is absorbed through loan pricing and accounting loan loss provisions (IFRS 9 / CECL). Unexpected Loss (UL = VaR_99.9% - EL) represents extreme tail volatility beyond average losses and must be cushioned by equity capital.

The Vasicek ASRF theorem assumes an infinite number of infinitesimal loans where idiosyncratic risk is completely eliminated. In real-world concentrated portfolios with a finite number of obligors, residual idiosyncratic risk creates excess tail loss volatility. The Granularity Adjustment (Gordy 2003, Wilde 2001) quantifies this additional capital buffer required for finite portfolios.

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