The Basel II IRB formula

The Basel Committee (2004) set bank capital for credit risk with Vasicek’s model: capital per unit of exposure is the 99.9% quantile of the large-pool loss rate, minus the expected loss already covered by provisions, times a maturity adjustment. The supervisory asset correlation falls from 24% to 12% as the PD rises. Gordy (2003) showed this is the only setting where each loan’s capital charge does not depend on the rest of the portfolio.

import matplotlib.pyplot as plt
import numpy as np

from financialkit.credit import basel_correlation, basel_irb_capital, basel_risk_weight, one_factor_loss_distribution

pd = np.geomspace(3e-4, 0.3, 200)
fig, ax = plt.subplots(1, 2, figsize=(12, 4))
for cls in ("corporate", "mortgage", "revolving", "retail"):
    ax[0].semilogx(pd, 100 * np.asarray(basel_risk_weight(pd, asset_class=cls)), label=cls)
    ax[1].semilogx(pd, 100 * np.asarray(basel_correlation(pd, cls)), label=cls)
ax[0].set_xlabel("PD")
ax[0].set_ylabel("risk weight (%), LGD 45%")
ax[0].legend()
ax[1].set_xlabel("PD")
ax[1].set_ylabel("asset correlation (%)")
print("illustrative corporate risk weights:", {f"{p:.2%}": f"{100 * basel_risk_weight(p):.2f}%" for p in (0.0003, 0.001, 0.01, 0.05, 0.2)})

illustrative corporate risk weights: {'0.03%': '14.44%', '0.10%': '29.65%', '1.00%': '92.32%', '5.00%': '149.85%', '20.00%': '238.23%'}

Capital is the gap between the 99.9% quantile and the expected loss

p, lgd = 0.01, 0.45
R = float(basel_correlation(p))
dist = one_factor_loss_distribution(p, R, 2000, lgd=lgd)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(dist.losses, np.cumsum(dist.pmf))
ax.axvline(dist.mean, color="C0", ls=":", label=f"expected loss {dist.mean:.2%}")
ax.axvline(dist.quantile(0.999), color="C3", ls="--", label=f"99.9% quantile {dist.quantile(0.999):.2%}")
ax.set_xlim(0, 0.08)
ax.set_xlabel("portfolio loss rate")
ax.set_ylabel("cumulative probability")
ax.set_title(f"K at M = 1: {basel_irb_capital(p, lgd, maturity=1.0):.2%} of exposure")
ax.legend()

plt.show()