A worked loan: recovery on the downside EBITDA, expected loss against the margin, and the exit multiple that decides the price.
The break-even probability of default is the margin divided by the loss given default: the annual default rate at which expected loss consumes the whole spread. On the fictional Halstead Packaging loan, a 400 basis point margin against a loss given default of 26.52 per cent breaks even at a PD of 15.08 per cent. The figure looks comfortable, but it rests almost entirely on the exit multiple: at 3.0x instead of 4.0x it falls to 8.91 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 →
In Credit Analysis, Halstead borrows 120 million at 7.40 per cent, a base rate of 3.40 per cent plus a margin of 4.00 per cent, which is 4.80 million a year. Recovery is computed where defaults happen: on the downside EBITDA, after an 8 per cent revenue fall carried through operating leverage of 2.2. All figures are illustrative, and the probability of default is an assumption, stated as one.
| Input | Value |
|---|---|
| Senior net debt | 120 m |
| Margin over base rate | 4.00% |
| EBITDA, lender's adjusted | 28.46 m |
| EBITDA in the downside | 23.45 m |
| Reference exit multiple in default | 4.0x |
| Enforcement costs | 6% |
| Assumed annual probability of default | 4% |
Recovery. Value the business at the exit multiple on the EBITDA it would have when it defaults, take off the cost of enforcement, and divide by the debt.
Enterprise value = 23.45 × 4.0 = 93.80 m; net of 6% enforcement = 88.18 m
Recovery = 88.18 / 120 = 73.48%; loss given default = 26.52%
Expected loss. Probability of default times loss given default, on the day-one exposure.
EL = 4% × 26.52% = 1.06% a year, or 1.27 m
Share of the margin consumed = 1.06 / 4.00 = 26.52%; residual 294 bp, or 3.53 m, for funding, capital and profit
Break-even. Reverse the question. What default rate would make the expected loss equal to the whole margin?
Break-even PD = margin / LGD = 4.00% / 26.52% = 15.08%
Excel: =Margin/(1-MIN(1,EBITDA_dn*Multiple*(1-Enf)/Debt))
That is 3.8 times the assumed default rate. Sustained over four years it would mean 48.0 per cent of such borrowers defaulting. On that reading the margin looks generous.
The default rate is the input everyone argues about. The exit multiple is the one that actually drives the answer.
| Exit multiple | Net recovery value | Recovery | LGD | EL at 4% PD | Break-even PD |
|---|---|---|---|---|---|
| 3.0x | 66.13 | 55.11% | 44.89% | 1.80% | 8.91% |
| 3.5x | 77.15 | 64.29% | 35.71% | 1.43% | 11.20% |
| 4.0x | 88.18 | 73.48% | 26.52% | 1.06% | 15.08% |
| 4.5x | 99.20 | 82.66% | 17.34% | 0.69% | 23.07% |
| 5.0x | 110.22 | 91.85% | 8.15% | 0.33% | 49.08% |
Half a turn of multiple, well inside anyone's estimating error for a distressed packaging business, moves the break-even PD from 15.08 to 11.20 per cent. Doubling the assumed default rate to 8 per cent takes expected loss to 2.12 per cent, 53.0 per cent of the margin. The price is a recovery view more than a default view.
The multiple needed to be repaid in full in default is 5.44x. The loan was written at 4.22x on the lender's EBITDA and 3.19x on management's. A lender who lends at 4.22x and needs 5.44x to recover par has a gap of more than a turn, and that gap is the credit decision.
The usual error is to compute recovery on the EBITDA the loan was sized on rather than on the EBITDA at default. At 4.0x on the lender's 28.46 million, recovery is 89.17 per cent, LGD 10.83 per cent and the break-even PD a reassuring 36.95 per cent. The recovery is overstated by 15.69 points because the business that defaults is not the business that borrowed. On management's adjusted EBITDA the recovery would be 117.81 per cent of the debt, which is to say no loss at all.
The second error goes the other way: pricing on the amortising balance. Because the loan amortises, expected loss falls each year: averaged over four years it is 0.63 per cent, not 1.06. Both are correct calculations. The day-one figure prices the largest exposure the lender carries, and the average assumes the borrower survives long enough to amortise, inside a calculation about default. Use the day-one figure for pricing, and show the other.
| Year | Balance | Recovery | LGD | EL |
|---|---|---|---|---|
| 1 | 114 | 77.35% | 22.65% | 0.91% |
| 2 | 108 | 81.64% | 18.36% | 0.73% |
| 3 | 102 | 86.45% | 13.55% | 0.54% |
| 4 | 96 | 91.85% | 8.15% | 0.33% |
The pricing sheet, the reverse test and the amortising schedule are live formulas in the free workbook for this case. How the downside EBITDA is reached from an 8 per cent revenue fall is worked in how operating leverage amplifies a lender's downside case, and the bond-market version of the same arithmetic is in a high-yield bond's yield after expected default losses.
Expected loss is probability of default times loss given default times exposure. Loss given default is one minus recovery, and recovery is the enterprise value at default, net of enforcement costs, over the debt. On Halstead, a 4 per cent PD and a 26.52 per cent LGD give expected loss of 1.06 per cent a year, 1.27 million on 120 million.
The EBITDA the borrower would have when it defaults, not the figure the loan was sized on. On Halstead, 4.0x the downside EBITDA of 23.45 million gives a recovery of 73.48 per cent; 4.0x the lender's 28.46 million gives 89.17 per cent, overstating recovery by 15.69 points and lifting the break-even PD from 15.08 to 36.95 per cent.
Price on the day-one balance and show the other. Halstead's day-one expected loss is 1.06 per cent; averaged over four years of 5 per cent amortisation it is 0.63 per cent. The average assumes the borrower survives to amortise, which is an odd assumption inside a calculation about default.
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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