The price sits between the risk-free value and the recovery, and where it sits is the default probability, once you fix what default pays.
Set the bond price equal to the present value of two outcomes, the coupons and principal if the issuer survives and the recovery if it defaults, and solve for the default probability that makes them match. An illustrative three-year 8 per cent bond at 62, with a 35 recovery and a 4 per cent risk-free rate, implies an annual default probability of 31.30 per cent, 67.57 per cent over three years. The figure is only as good as the recovery you assume: at 45 the same price implies 77.0 per cent over three years.
Worked in full in The Distressed Debt Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A distressed bond trades between two anchors. If default were impossible it would be worth its cash flows discounted at the risk-free rate, 111.10 here. If default were certain within a year it would be worth the recovery discounted for a year, 33.65. The price of 62 sits between them, and the implied default probability is simply where it sits, once you have fixed what default pays.
The result is a risk-neutral probability. It contains the market's estimate of default plus the premium investors demand for bearing it, so it is higher than a real-world forecast. That makes it useful for comparing bonds and for asking whether your own view is more or less pessimistic than the price, not for booking an expected loss.
| Input | Value |
|---|---|
| Price, per 100 of face | 62.00 |
| Annual coupon | 8.00 |
| Years to maturity | 3 |
| Recovery if default, per 100 of face, paid at the end of the year of default | 35.00 |
| Risk-free rate | 4.00% |
| Default probability | Constant each year, conditional on survival |
With one period left the algebra is a single line, and it shows the whole logic.
Price × (1 + r) = (1 − q) × (100 + coupon) + q × recovery
q = (100 + coupon − price × (1 + r)) ÷ (100 + coupon − recovery)
Excel: =(100+C-P*(1+r))/(100+C-R)
Price 62 grown at 4 per cent is 64.48. The gap between full payment, 108.00, and that figure is 43.52. The gap between full payment and recovery is 73.00. So q = 43.52 ÷ 73.00 = 59.62 per cent. A bond at 62 with a year to run is pricing default as more likely than not.
With more than one period, each year's cash flow is weighted by the probability of surviving to it, and each year's recovery by the probability of defaulting in that year. With a constant annual hazard q:
Price = Σ [ St−1 × (1 − q) × CFt + St−1 × q × R ] ÷ (1 + r)t, with St = (1 − q)t
Solve for q with Goal Seek or a bisection: set the price cell to 62 by changing q.
The solution is q = 31.30 per cent. The table proves it by rebuilding the price.
| Year | Alive at start | Defaults in the year | PV if it survives | PV of recovery |
|---|---|---|---|---|
| 1 | 1.0000 | 31.30% | 5.28 | 10.53 |
| 2 | 0.6870 | 21.50% | 3.49 | 6.96 |
| 3 | 0.4720 | 14.77% | 31.14 | 4.60 |
| Total | 67.57% | 39.91 | 22.09 |
The two columns sum to 62.00. The cumulative default probability is 1 − (1 − 0.3130)3 = 67.57 per cent, so the market gives the issuer about a one in three chance, 32.43 per cent, of paying in full. Multiply the cumulative probability by the loss given default and the price embeds a risk-neutral principal loss of 43.9 points of face, undiscounted and before lost coupons.
The recovery assumption moves the answer more than most analysts expect, because every point of recovery added makes default less costly, so more of it is needed to explain the same discount.
| Recovery | Price 50 | Price 62 | Price 75 |
|---|---|---|---|
| 25 | 36.4% / 74.2% | 26.4% / 60.1% | 17.8% / 44.4% |
| 35 (case) | 44.8% / 83.1% | 31.3% / 67.6% | 20.7% / 50.1% |
| 45 | 60.5% / 93.8% | 38.8% / 77.0% | 24.7% / 57.3% |
Push the recovery towards the price and the implied probability runs to certainty: at a recovery of 55 the price of 62 implies 89.0 per cent over three years, at 60 it implies 95.8 per cent. A distressed bond trading close to your recovery estimate is not a bond the market expects to survive. That is why, at these prices, the work moves from default probability to recovery, and from there to the enterprise value the price implies, which is the subject of what enterprise value a distressed bond price implies.
Using the credit triangle, hazard = spread ÷ (1 − recovery). It is a fair shortcut for a bond near par. For a distressed bond it breaks. At 62 the bond's yield to maturity is 28.48 per cent, a spread of 2448 bp over the risk-free rate. Divided by 0.65 that gives an annual hazard of 37.66 per cent and 75.77 per cent over three years, 6.36 points a year too high. The triangle assumes default costs a fraction of the current value, when in fact recovery is paid on face: a holder who paid 62 and recovers 35 loses far less than 65 per cent of what it owns. Price the cash flows directly.
The second mistake is reading the result as a forecast. A risk-neutral 67.57 per cent includes the compensation buyers want for holding a binary, illiquid position. If your own work says the issuer has a better chance than one in three, the bond is cheap on your numbers; the price alone cannot tell you which view is right.
The Distressed Debt Investor works recovery, the fulcrum and the return on a distressed position; the free workbook for this case carries the recovery and return models. To turn a recovery view into a buying price, see what recovery a distressed bond needs to break even.
It depends on recovery and maturity. For an illustrative three-year 8 per cent bond with a 4 per cent risk-free rate and a 35 recovery, 31.30 per cent a year and 67.57 per cent over three years. With one year to run, the same price implies 59.62 per cent, because there is less time for the coupons to compensate.
Because default becomes less costly, so more of it is needed to explain the same discount to the risk-free value. At a price of 62 on the illustrative bond, a 25 recovery implies 60.1 per cent over three years, 35 implies 67.6 per cent and 45 implies 77.0 per cent.
No. It is risk-neutral: it includes the premium investors demand for bearing default risk and illiquidity, so it exceeds a real-world estimate. Use it to compare bonds and to test your own view against the price. A 67.57 per cent implied figure means the price is cheap if you think default odds are lower.
Not well. The credit triangle assumes a near-par bond. At 62 the illustrative bond yields 28.48 per cent, a 2448 bp spread, and the shortcut gives 37.66 per cent a year against 31.30 per cent from pricing the cash flows, because recovery is paid on face, not on the current price.
This article is one calculation from The Distressed Debt 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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