A yield to maturity is the return on a promise kept, stated as though the keeping were certain; the expected yield prices the chance that it is not.
A bond’s loss-adjusted (expected) yield is its yield to maturity minus its expected loss, where expected loss equals the annual probability of default multiplied by one minus the recovery rate. For a CCC bond quoting 10.06 per cent, with a 15.00 per cent annual default probability and 40 per cent recovery, the expected loss is 9.00 per cent and the expected yield is 1.06 per cent: below the 4.00 per cent Treasury bill it was meant to beat.
Worked in full in The Credit Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Two quantities decide how much a lender loses on average. The probability of default, or PD, is the chance that the borrower fails to pay within a year. The recovery is the share of the claim the lender eventually receives after a default. One minus the recovery is the loss given default.
Expected loss = PD × (1 − recovery). The expected yield is the displayed yield less that figure. Neither input can be observed in advance for one borrower, so both are assumptions. A rating is an opinion about credit quality, not a probability. The conversion below is made by assumption, with a recovery of 40 per cent for senior unsecured bonds and 65 per cent for senior secured loans.
| Rating | Annual PD | Expected loss, bond | Cumulative PD, 5 years |
|---|---|---|---|
| A | 0.05% | 0.03% | 0.2% |
| BBB | 0.20% | 0.12% | 1.0% |
| BB | 1.00% | 0.60% | 4.9% |
| B | 3.00% | 1.80% | 14.1% |
| CCC | 15.00% | 9.00% | 55.6% |
The cumulative column is computed by survival. A B bond survives each year with a probability of 0.97, and five years of that leave about 0.859, so the chance of default at some point is 14.1 per cent. For CCC the yearly survival is 0.85, five years leave about 0.444, and the cumulative probability of default is 55.6 per cent. Under these assumptions a five-year CCC bond is more likely than not to default before it matures. The step from B to CCC multiplies the annual PD by five.
Harrowgate Pneumatics, rated CCC, was bought at 94.00 with a coupon of 8.50 per cent. Its yield to maturity was 10.06 per cent and its spread over the 4.00 per cent bill was 6.06 per cent. Its expected loss is 15.00 × 0.60, or 9.00 per cent a year. The expected yield is 10.06 minus 9.00: 1.06 per cent.
The spread makes the point sharper. The spread is what the investor is paid, over the risk-free rate, for bearing the chance of loss. Here the payment is smaller than the thing it pays for, by 2.94 points (9.00 minus 6.06). The expected yield therefore sits 2.94 points below the bill.
The useful inversion is the break-even PD: the default probability at which the expected yield exactly equals the bill. It is the spread divided by the loss given default. For Harrowgate, 6.06 divided by 0.60 is, on the unrounded spread, 10.09 per cent a year. The assumption for CCC is 15.00 per cent, so the bond was priced as if its PD were about two-thirds of the assumed one. A B bond yielding 8.75 per cent, by contrast, has a spread of 4.75 and a break-even PD of about 7.9 per cent, against an assumed 3.00. Its expected yield, 8.75 minus 1.80, is 6.95 per cent, and it comfortably beats the bill.
| Line | Rating | Displayed | Expected loss | Expected |
|---|---|---|---|---|
| Aldermoor Utilities | A | 4.80% | 0.03% | 4.77% |
| Hesketh Foods | BBB | 5.15% | 0.12% | 5.03% |
| Blakeney Packaging | BB | 6.99% | 0.60% | 6.39% |
| Marlow Home Brands | B | 8.75% | 1.80% | 6.95% |
| Harrowgate Pneumatics | CCC | 10.06% | 9.00% | 1.06% |
| Brindle Floating Rate Fund | n/a | 6.65% | 1.05% | 5.60% |
For investment grade the adjustment is almost invisible: 5.13 per cent displayed, 5.05 expected. For the high-yield sleeve it is large: 8.15 displayed, 5.60 expected, a gap of 2.55 points, most of it from one bond. The loan fund’s 1.05 per cent is the same arithmetic with a 65 per cent recovery: 3.00 × 0.35. Across the whole portfolio, a displayed 7.46 per cent becomes an expected 6.31 per cent.
An expected loss is an average across many bonds. A single bond never loses 9.00 per cent a year. Either it pays and earns roughly the displayed 10.06 per cent, or it defaults and loses most of the principal at once. The 1.06 per cent is a weighted average of two outcomes, neither of which the holder can actually receive.
Harrowgate defaulted. It missed its coupon at quarter 6 and paid a 40 per cent recovery at quarter 10: $5,000 back on $12,500 of face, against $11,750 paid. Across a six-bond high-yield sleeve, the realised IRR over three years moves fast with each default.
| Defaults | Recovery 25% | Recovery 40% | Recovery 55% |
|---|---|---|---|
| None | 8.32% | 8.32% | 8.32% |
| One | 3.32% | 4.22% | 5.10% |
| Two | −2.27% | −0.33% | 1.57% |
One default at 40 per cent recovery left the sleeve at 4.22 per cent, just above the bill. Two take it to minus 0.33 per cent. Recovery decides whether a default is survivable.
Every figure here is reproduced by a live formula in the companion workbook. The full model, with the PD and recovery inputs open to change, is on The Credit Investor page.
Loss-adjusted, or expected, yield is the yield to maturity minus the expected annual credit loss, where expected loss is the probability of default times one minus the recovery rate. A B-rated bond yielding 8.75 per cent with a 3.00 per cent PD and 40 per cent recovery loses 1.80 per cent a year in expectation, leaving an expected yield of 6.95 per cent.
The break-even default rate is the annual probability of default at which a bond’s expected yield equals the risk-free rate. It equals the spread divided by the loss given default. A CCC bond with a 6.06 per cent spread over the bill and 40 per cent recovery breaks even at 10.09 per cent a year; above that PD, Treasury bills are expected to do better.
With an assumed annual default probability of 15.00 per cent, a CCC bond survives each year with probability 0.85. Five years of survival leave about 0.444, so the cumulative probability of default is 55.6 per cent. A five-year CCC bond is more likely than not to default before maturity, against 14.1 per cent for a B bond.
This article is one calculation from The Credit Investor. 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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