Margin, fees, funding, costs, expected loss and the capital benefit in one ratio, then solved backwards for the minimum margin.
RAROC is the loan's risk-adjusted profit after tax divided by the capital held against it: margin and fees, less funding and operating costs, less expected loss, plus the return on the capital itself. On an illustrative EUR 120 million term loan at a 4.00 per cent margin, with an expected loss of 1.06 per cent and 12.60m of capital, RAROC is 17.5 per cent against a 12 per cent hurdle, and the minimum margin that clears the hurdle is 3.23 per cent. Move the exit multiple in default from 4.0x to 3.0x and it falls to 12.2 per cent.
Worked in full in Credit Analysis by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A margin tells you what the loan pays. It does not tell you whether that is enough, because two loans at the same margin can carry very different losses and need very different amounts of capital. RAROC puts all three in one ratio, and the ratio can be compared with the return the lender's shareholders require. It also runs backwards: fix the hurdle and solve for the margin, which is how a credit committee should test a pricing proposal before it is offered.
| Input | Value |
|---|---|
| Loan | EUR 120.0m |
| Base rate / margin | 3.40% / 4.00% |
| Upfront fee, spread over a 4-year expected life | 1.00%, so 0.25% a year |
| Lender's funding cost over the base rate | 0.75% |
| Operating cost | 0.35% |
| Probability of default, a year | 4% |
| EBITDA in the downside / exit multiple in default / enforcement costs | 23.45m / 4.0x / 6% |
| Risk weight / capital held against risk-weighted assets | 100% / 10.5% |
| Tax rate / hurdle rate after tax | 25% / 12% |
RAROC = (margin + fees − funding − operating cost − expected loss + capital × base rate) × (1 − tax) / capital
Expected loss = PD × LGD × exposure; capital = exposure × risk weight × capital ratio
Excel: =((Margin+Fee/Life-Fund-Opex-PD*LGD)*Loan+Cap*Base)*(1-Tax)/Cap
Step 1, the loss given default. Value the business where defaults happen: 23.45m of downside EBITDA at 4.0x is 93.80m, 88.17m after 6 per cent of enforcement costs. That recovers 73.48 per cent of the loan, so LGD is 26.52 per cent.
Step 2, expected loss. 4 per cent times 26.52 per cent is 1.06 per cent a year, 1.27m.
Step 3, capital. 120.0m at a 100 per cent risk weight and 10.5 per cent is 12.60m. That capital is equity, not borrowed money, so the lender saves the base rate on it: 0.43m a year, the capital benefit.
Step 4, risk-adjusted profit.
| Line | Amount |
|---|---|
| Margin, 4.00% of 120.0 | 4.80 |
| Upfront fee, a year | 0.30 |
| Less funding cost over base | −0.90 |
| Less operating cost | −0.42 |
| Less expected loss | −1.27 |
| Plus capital benefit | 0.43 |
| Risk-adjusted profit before tax | 2.94 |
| After 25% tax | 2.20 |
Step 5, the ratio. 2.20 over 12.60 is 17.5 per cent. Against a 12 per cent hurdle the loan creates 0.69m a year of value over the cost of its capital.
Step 6, the minimum margin. At 12 per cent the capital needs 1.512m after tax, 2.016m before. Solving for the margin that delivers it gives 3.23 per cent, 77 bp below the proposed 4.00.
| Exit multiple | LGD | PD 2% | PD 4% | PD 6% | Minimum margin at 4% PD |
|---|---|---|---|---|---|
| 3.0x | 44.89% | 18.6% | 12.2% | 5.8% | 3.97% |
| 3.5x | 35.71% | 19.9% | 14.8% | 9.7% | 3.60% |
| 4.0x | 26.52% | 21.3% | 17.5% | 13.7% | 3.23% |
| 4.5x | 17.34% | 22.6% | 20.1% | 17.6% | 2.87% |
At a 3.0x exit, expected loss rises to 1.80 per cent and the loan only just clears the hurdle at 12.2 per cent; its minimum margin is 3.97 per cent, three basis points under the price. Each point of default probability costs 1.89 points of RAROC at 4.0x. A credit committee that debates the PD and accepts the recovery has argued about the smaller input. The same sensitivity, without capital or costs, is in what probability of default a loan margin breaks even at.
Leaving out expected loss. Margin less costs over capital reads 25.0 per cent. Expected loss is a cost of doing the business, like funding; capital is held for the unexpected loss on top of it. Without it the loan looks 7.6 points better than it is.
Measuring recovery on the EBITDA the loan was sized on. At 4.0x of the lender's 28.46m, LGD is 10.83 per cent and RAROC 22.0 per cent. The business that defaults is the downside business, not the one that borrowed.
Forgetting the capital benefit. Without it RAROC is 14.9 per cent. That understates the loan, but it is the right figure if the lender's transfer pricing already credits the equity elsewhere: decide once and apply it to every loan.
The 100 per cent risk weight is a standardised simplification. A bank on internal ratings would compute the weight from PD and LGD, which here would move capital and therefore RAROC; the method is in how to calculate an IRB risk weight.
Credit Analysis sets a margin against expected loss for one borrower and runs the recovery on the EBITDA that exists in default; the free workbook for this case carries the pricing sheet.
One above the lender's hurdle rate, the after-tax return its shareholders require on the capital. With an illustrative 12 per cent hurdle, a loan earning 17.5 per cent creates 0.69m a year of value on 12.60m of capital. The same loan at a 3.0x exit multiple earns 12.2 per cent and barely clears it.
Yes. Expected loss is a cost of lending, deducted from revenue like funding; capital covers the unexpected loss above it. Leaving out the 1.27m of expected loss on the illustrative EUR 120 million loan lifts RAROC from 17.5 per cent to 25.0 per cent, which overstates the loan by more than seven points.
Set RAROC equal to the hurdle and solve for the margin. At 12 per cent after tax on 12.60m of capital the loan must make 1.512m after tax, 2.016m before; adding back funding, costs and expected loss and deducting the fee and capital benefit gives a minimum margin of 3.23 per cent, or 3.97 per cent if recovery is valued at 3.0x.
This article is one calculation from Credit Analysis. 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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