Under the standardised approach, risk-weighted assets are a sumproduct of exposures and regulatory weights, but the answer turns on three details that are easy to skip: the credit conversion factor, the mortgage split and the defaulted book.
Risk-weighted assets under the standardised approach are each exposure multiplied by its regulatory risk weight, with off-balance-sheet commitments first converted into exposure by a credit conversion factor. For an illustrative bank with 41,000 million of assets and 7,000 million of undrawn lines, credit RWA come to 22,200.0 million, a density of 54.15 per cent, and the 8 per cent Pillar 1 minimum asks for 1,776.0 million of capital.
Worked in full in Financial Regulation by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Every capital ratio has RWA as its denominator, so an error here moves the CET1 ratio, the buffers, the distribution trigger and the resolution requirement at once. The arithmetic is a sumproduct. The judgement sits in choosing the right weight for each line, and in remembering the lines that are not on the balance sheet.
Weights are broadly those of the Basel III final standardised approach, implemented in the EU by CRR3, shown fully phased in. Real portfolios need the exposure class, the rating, the collateral and the currency of every exposure; this bank is simplified to eleven on-balance-sheet lines and two commitment lines.
| Exposure class | Exposure | Risk weight | RWA |
|---|---|---|---|
| Cash and central bank reserves | 4,000 | 0% | 0.0 |
| Sovereign bonds, domestic currency | 6,000 | 0% | 0.0 |
| Banks rated A | 1,500 | 30% | 450.0 |
| Corporates rated BBB | 3,000 | 75% | 2,250.0 |
| Unrated corporates | 8,000 | 100% | 8,000.0 |
| Regulatory retail | 3,000 | 75% | 2,250.0 |
| Residential mortgages, LTV 60% | 12,000 | 24.58% | 2,950.0 |
| Commercial real estate loans | 2,000 | 100% | 2,000.0 |
| Equity holdings | 300 | 250% | 750.0 |
| Defaulted loans, net of provisions | 400 | 150% | 600.0 |
| Fixed and other assets | 800 | 100% | 800.0 |
| On balance sheet | 41,000 | 20,050.0 |
Credit RWA = Σ on-balance exposure × RW + Σ off-balance nominal × CCF × RW
In Excel, with exposures in B2:B12 and weights in C2:C12, commitments in B15:B16, factors in C15:C16 and weights in D15:D16: =SUMPRODUCT(B2:B12,C2:C12)+SUMPRODUCT(B15:B16,C15:C16,D15:D16).
Step 1: the zero-weighted lines. Central bank reserves and domestic sovereign bonds in their own currency carry 0 per cent: 10,000 million of assets, a quarter of the balance sheet, add nothing.
Step 2: rated and unrated counterparties. A bank rated A takes 30 per cent, so 1,500 becomes 450.0. Corporates rated BBB take 75 per cent and unrated corporates 100 per cent: 2,250.0 plus 8,000.0, together 10,250 of RWA, almost half the total.
Step 3: the mortgages. Under the EU loan-splitting approach the part of each loan up to 55 per cent of the property value weighs 20 per cent and the rest weighs as the borrower, 75 per cent for a retail borrower. At an LTV of 60 per cent: (55 × 20% + 5 × 75%) ÷ 60 = 24.58 per cent, so 12,000 of loans produce 2,950.0 of RWA. The mortgages are 29.3 per cent of the assets and only 13.3 per cent of the RWA. Why a book's average LTV understates or overstates this weight is worked in the mortgage loan-splitting example.
Step 4: the heavy lines. Equity at 250 per cent turns 300 into 750.0. Defaulted loans with specific provisions below 20 per cent of the outstanding weigh 150 per cent: 400 becomes 600.0. Fixed and other assets weigh 100 per cent.
Step 5: off balance sheet.
| Commitment | Nominal | CCF | Exposure | Risk weight | RWA |
|---|---|---|---|---|---|
| Undrawn corporate commitments | 5,000 | 40% | 2,000 | 100% | 2,000.0 |
| Unconditionally cancellable card lines | 2,000 | 10% | 200 | 75% | 150.0 |
| Off balance sheet | 2,150.0 |
Credit RWA are 20,050.0 plus 2,150.0, that is 22,200.0 million. On the 41,000 of assets the density is 54.15 per cent; counting the converted commitments in the denominator it is 51.39 per cent. The 8 per cent Pillar 1 minimum is 1,776.0 million of total capital, of which 999.0 million must be CET1 at 4.5 per cent. Operational risk and market risk RWA are calculated separately and added before any ratio is struck; the operational risk piece is worked in how operational risk capital is calculated.
| Change | Credit RWA | Change in RWA | Capital at 8% |
|---|---|---|---|
| Base case | 22,200.0 | ||
| Undrawn commitments at a CCF of 100% | 25,200.0 | +3,000.0 | +240.0 |
| Mortgages at LTV 80% (37.19%) | 23,712.5 | +1,512.5 | +121.0 |
| Banks rated BBB (50%) | 22,500.0 | +300.0 | +24.0 |
| Defaulted loans provisioned 20% or more (100%) | 22,000.0 | −200.0 | −16.0 |
| Unrated corporates rated BBB | 20,200.0 | −2,000.0 | −160.0 |
The two largest levers are the commitment book and the share of corporates without an external rating. The unrated corporate line alone is 36.0 per cent of credit RWA.
Leaving out the off-balance-sheet lines, or weighting them at nominal. Omitted, RWA read 20,050.0, understated by 2,150.0. Taken at nominal without a conversion factor, they read 26,550.0, overstated by 4,350.0. A bank that runs its capital plan from the balance sheet alone will find the difference when its first regulatory return is filed. The second mistake is applying a flat weight to a class that is graded, mortgages and defaulted loans above all.
All figures are illustrative. Weights depend on the jurisdiction, the transitional schedule and the exposure's details; this is a method, not a regulatory return.
The book's own risk weight engine, with both ladders and the output floor across its schedule, is in the free workbooks for this case.
It depends on the book. The illustrative bank here has credit RWA of 22,200.0 million on 41,000 million of assets, a density of 54.15 per cent, because 10,000 million of cash and sovereign bonds weigh nothing while 8,000 million of unrated corporate loans weigh in full. A mortgage-heavy bank sits far lower, a corporate lender higher; the density is a description of the mix, not a target.
Multiply the undrawn amount by a credit conversion factor, then by the borrower's risk weight. Under the Basel III standardised approach most commitments take a 40 per cent factor and unconditionally cancellable ones 10 per cent. Here 5,000 million of undrawn corporate lines become 2,000 of exposure and 2,000.0 of RWA; at a 100 per cent factor they would add 3,000.0 more.
The Pillar 1 minimum is 8 per cent of RWA in total capital, of which 4.5 per cent must be CET1: 1,776.0 million and 999.0 million on 22,200.0 of RWA. Pillar 2 requirements and the combined buffer sit on top, so the CET1 a bank actually needs to avoid distribution limits is several points higher.
This article is one calculation from Financial Regulation. 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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