The IRB formula looks forbidding and is five cells in Excel; worked once on a single loan, it explains why a bank's capital sits where it does and what a risk transfer can release.
The IRB risk weight of a corporate loan is 12.5 times K, where K is the loss given default times the probability of default conditional on a 99.9 per cent systematic shock, less expected loss, multiplied by a maturity adjustment. An illustrative loan with a 0.67 per cent PD, a 42 per cent LGD and a 4.3-year maturity has a conditional PD of 11.47 per cent, K of 7.54 per cent and a risk weight of 94.28 per cent. Run the same formula over a whole portfolio and it gives KIRB, the number every synthetic risk transfer is structured around.
Worked in full in Significant Risk Transfer by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The formula is in the regulation, but it is rarely worked by hand, and so its two strongest levers, the maturity adjustment and the correlation, are rarely questioned. Below it is done in five steps on one loan, then rolled up to the reference portfolio of a significant risk transfer.
| Input | Value |
|---|---|
| Probability of default (PD), one year | 0.67% |
| Loss given default (LGD) | 42% |
| Effective maturity (M), years | 4.3 |
| Exposure at default (EAD) | €10,000,000 |
| Borrower's sales | Between 50 and 500 million: no SME adjustment, own LGD allowed |
R = 0.12 × w + 0.24 × (1 − w), where w = (1 − e−50 PD) ÷ (1 − e−50)
b = (0.11852 − 0.05478 × ln PD)2
K = [LGD × N((G(PD) + √R × G(0.999)) ÷ √(1 − R)) − PD × LGD] × (1 + (M − 2.5) b) ÷ (1 − 1.5 b)
Risk weight = 12.5 × K
N is the standard normal distribution and G its inverse. In Excel the conditional PD is =NORM.S.DIST((NORM.S.INV(PD)+SQRT(R)*NORM.S.INV(0.999))/SQRT(1-R),TRUE).
Step 1, correlation. The weight w is 0.2847, so R = 0.12 × 0.2847 + 0.24 × 0.7153 = 0.2058. Low PDs get correlations near 0.24, high PDs near 0.12: the regulation assumes safer companies fail mainly when the whole economy does.
Step 2, conditional PD. G(0.67 per cent) is −2.4730 and G(0.999) is 3.0902. With √R = 0.4537 the argument is (−2.4730 + 0.4537 × 3.0902) ÷ √(1 − 0.2058) = −1.2017, and N of that is 11.47 per cent. In a one-in-a-thousand year, a borrower that defaults 0.67 per cent of the time defaults 11.47 per cent of the time.
Step 3, unexpected loss. 42 per cent of 11.47 is 4.82 per cent. Less expected loss of 0.67 × 42 = 0.281 per cent leaves 4.54 per cent. Capital covers only the unexpected part.
Step 4, maturity. b = (0.11852 − 0.05478 × ln 0.0067)2 = 0.1542. The adjustment is (1 + 1.8 × 0.1542) ÷ (1 − 1.5 × 0.1542) = 1.6622.
Step 5, K and the weight. K = 4.54 per cent × 1.6622 = 7.54 per cent, and the risk weight is 12.5 times that, 94.28 per cent.
| Line | Amount |
|---|---|
| Risk-weighted assets at 94.28% | 9,427,508 |
| Capital at the 8% minimum (K × EAD) | 754,201 |
| Capital at a 13.5% CET1 target | 1,272,714 |
| Expected loss, for provisioning and pricing | 28,140 |
The maturity adjustment does more work than its name suggests. At 4.3 years it multiplies capital by 1.6622; at the 2.5-year reference it would be 1.3010 and the risk weight 73.79 per cent. More than a fifth of this loan's capital, 21.7 per cent, comes from maturity beyond the 2.5-year reference.
| PD | LGD 25%, M 2.5 | LGD 45%, M 2.5 | LGD 45%, M 5 |
|---|---|---|---|
| 0.05% | 10.9% | 19.7% | 33.7% |
| 0.10% | 16.5% | 29.7% | 48.0% |
| 0.67% | 43.9% | 79.1% | 109.5% |
| 1.00% | 51.3% | 92.3% | 124.0% |
| 3.00% | 71.4% | 128.4% | 159.4% |
| 10.00% | 107.3% | 193.1% | 222.0% |
LGD is linear: the weight scales with it exactly. PD is concave, rising fast at the bottom and slowly at the top, because correlation falls as PD rises. The SME adjustment pulls correlation down for borrowers with sales between 5 and 50 million: a BB loan at a 2.24 per cent PD, 45 per cent LGD and 3.7 years weighs 133.70 per cent for a large corporate and 115.68 per cent for a company with 24 million of sales, its correlation cut from 0.1592 to 0.1360.
Run the formula grade by grade over an illustrative €4,000m reference portfolio of seven grades, from a 0.09 per cent PD weighing 26.62 per cent to a 4.10 per cent PD weighing 120.42 per cent, and the RWA total €3,077.1m, a density of 76.93 per cent. Expected loss is 14,034,720, or 0.351 per cent. KIRB, RWA times 8 per cent plus expected loss over exposure, is 6.15 plus 0.351, or 6.51 per cent. That is the attachment every tranche in the structure is measured against; the next step is in how to calculate the SEC-IRBA risk weight of a tranche.
Two rule changes catch out older models. The final Basel III reforms, CRR3 in the EU, removed the 1.06 scaling factor: a model that keeps it shows 99.93 per cent for this loan, 6 per cent too much. They also raised the corporate PD floor to 0.05 per cent, so the first row of the table is the lowest PD a corporate can now carry. And for corporates with revenue above 500 million, CRR3 withdraws own LGD estimates: a senior unsecured loan to such a borrower takes the foundation LGD of 40 per cent.
Correlation, conditional PD, unexpected loss, maturity adjustment, times 12.5: 94.28 per cent for this loan, and 6.51 per cent of KIRB for the portfolio it sits in. Significant Risk Transfer takes one bank and one credit fund from that capital calculation to the price of the tranche between them, and the free workbook for this case derives KIRB from the seven buckets as live formulas.
It is the sensitivity of the borrower to a single systematic factor, set by the regulation from PD: between 0.12 for high PDs and 0.24 for low ones, with up to 0.04 deducted for SMEs with sales under 50 million. An illustrative loan with a 0.67 per cent PD has a correlation of 0.2058; at 2.24 per cent it is 0.1592, or 0.1360 for a company with 24 million of sales.
Because capital covers unexpected loss only; expected loss is meant to be covered by provisions and pricing. On an illustrative loan the loss at the 99.9 per cent conditional PD is 4.82 per cent of exposure and expected loss 0.281 per cent, so the capital charge before the maturity adjustment is 4.54 per cent.
KIRB is the capital the bank would hold on the underlying pool under IRB, as a share of the pool: risk-weighted assets times 8 per cent, plus expected loss. On an illustrative €4,000m corporate portfolio with RWA of €3,077.1m and expected loss of 14,034,720, KIRB is 6.51 per cent. It anchors the SEC-IRBA tranche formula.
This article is one calculation from Significant Risk Transfer. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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