IMPLIED PROBABILITY OF DEFAULT ON COUPON BONDS
In Chapter 2, we worked through the calculation of the implied probability of default on a zero-coupon corporate bond. We assumed that the amount the investor would be able to recover is a constant fraction of the risk-free value at the time of default. For a coupon bond, it is more common to set the recovery rate to be a constant fraction of the par value. But these are all arbitrary assumptions that can be changed to fit the circumstances; what matters for us is the approach to the bond math problem.
Suppose that the appropriate risk-free yield curve is flat at 3.50%. Then our 4-year, 4% annual payment corporate bond would be worth 101.837 (percent of par value) if there was no risk of default.
As the bond is priced at 99.342, the investor's compensation for bearing the credit risk of the corporate issuer is the difference between the prices: 101.837 – 99.342 = 2.495 (percent of par value).
Table 3.2 displays the present values of the loss after recovery for each year. The annual (unconditional) probability of default is Q, and we assume default occurs only on a coupon payment date. Consider an event of default at the end of the second year just before the coupon payment. The bond would be worth 104.950 (percent of par value) at that time if it were default free.
The loss before recovery would be the value of the remaining two years discounted at the risk-free rate plus the coupon payment due on that date.
TABLE 3.2 Calculating the Probability of Default on the 4-Year, 4% Coupon Bond
Year |
Probability of Default |
Risk-Free Value |
Recovery |
Default Loss |
Present Value of Default Loss |
Expected Loss |
1 |
Q |
105.401 |
40 |
65.401 |
63.189 |
63.189*Q |
2 |
Q |
104.950 |
40 |
64.950 |
60.632 |
60.632*Q |
3 |
Q |
104.483 |
40 |
64.483 |
58.160 |
58.160*Q |
4 |
Q |
104.000 |
40 |
64.000 |
55.772 |
55.772* Q |
237.75 3* Q |
After (instantaneous) recovery of 40, the loss is 64.950. Discounted back at 3.50% to time zero, the present value of the loss is 60.632 [= 64.950/ (1.0350)2].
The sum of the expected losses for each year totals 237.753 * Q. Equating that to the compensation for default risk, 2.495, gives the result that Q = 0.010494. Our conclusion is that the market is factoring into the price of the corporate bond an annual default probability of 1.05%. That is a useful result but obviously one that is dependent on the various assumptions – the key one being the 40% recovery rate. You should appreciate the value of programming this calculation onto a spreadsheet (as I have done) so you, too, can make clever statements, such as: If the assumed recovery rate is only 10%, the implied default probability is just 0.72% per year; but if the recovery rate is much higher at 80% of par value, the annual probability of default is estimated to be 2.75%.