11. The IRB RWA Formula
The IRB Formula: how a PD becomes a risk weight
RWA, AIRB, IRB, Capital
Two banks hold the same loan to the same company. One reads a 100% risk weight off the standardised table. The other runs its own formula and gets 92%. Both are following the rules.
Risk-weighted assets (RWA), are the denominator of every capital ratio. Under the internal ratings-based approach (IRB), a bank computes each risk weight itself, one loan at a time, from three estimates:
1) PD, the probability the borrower defaults within 1 year.
2) LGD, loss given default, the share you do not recover after collateral.
3) EAD, exposure at default, what is still outstanding the day of the default.
Those three feed one supervisory formula:
➡️ Capital K = LGD × (stressed PD − PD), then RWA = K × 12.5 × EAD.
The formula charges capital only for the loss you do not expect. Expected loss is PD × LGD, and provisions cover it as part of loan pricing.
Capital is for the bad year, so the formula subtracts the average and keeps the unexpected.
The model assumes one force, the economy, pulls down on every borrower at once: The economy goes to its worst level in 1000 years and the default rate in such case is the stressed PD.
One input does most of the work: the asset correlation. It measures how tightly a borrower follows the economy. Basel sets it between 0.24 and 0.12 for companies, and it falls as PD rises, because a very weak name defaults for its own reasons more than because the cycle turned.
A worked example. A corporate loan, PD 1%, LGD 45%, maturity 2.5 years. The formula returns a 92% risk weight. Now double the PD to 2%. If capital scaled with risk, the weight would be 184%. But it lands at 115%, roughly a quarter more for twice the default risk, because the correlation drops as the borrower weakens.
That is why the IRB curve bends.
The regulator built it to be portfolio-invariant: the capital on one loan does not depend on the rest of the book, so you compute it name by name and add it up. The cost is that concentration in a few large borrowers stays invisible to the formula, and supervisors handle it separately under Pillar 2.
Full breakdown in the slides below.
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