Question at position 10 (6 marks, difficulty level: Medium) The Dunley Corp. plans to issue one-year zero-coupon bonds. It believes the bonds will have a BBB rating, and the average debt beta for BBB rating is 0.12. Suppose AAA bonds with the same maturity have a 2.5% yield. If the market risk premium is 5%, use the annual default rates by debt rating here and calculate the yield to maturity of Dunley's one-year bond, assuming an expected 70% loss rate in the event of default during average economic times. [table]
Rating: | AAA | AA | A | BBB | BB | B | CCC | CC-C Default Rate: | | | | | | | | Average | 0.0% | 0.1% | 0.2% | 0.5% | 2.2% | 5.5% | 12.2% | 14.1% In Recessions | 0.0% | 1.0% | 3.0% | 3.0% | 8.0% | 16.0% | 48.0% | 79.0%
[/table] 5.070%4.065%3.450%2.900%

Question

Question at position 10 (6 marks, difficulty level: Medium) The Dunley Corp. plans to issue​ one-year zero-coupon bonds. It believes the bonds will have a BBB rating, and the average debt beta for BBB rating is 0.12. Suppose AAA bonds with the same maturity have a 2.5% yield. If the market risk premium is 5%​, use the annual default rates by debt rating​ here and calculate the yield to maturity of Dunley's one-year bond, assuming an expected 70% loss rate in the event of default during average economic times. [table]
Rating: | AAA | AA | A | BBB | BB | B | CCC | CC-C Default Rate: |  |  |  |  |  |  |  |  Average | 0.0% | 0.1% | 0.2% | 0.5% | 2.2% | 5.5% | 12.2% | 14.1% In Recessions | 0.0% | 1.0% | 3.0% | 3.0% | 8.0% | 16.0% | 48.0% | 79.0%
[/table] 5.070%4.065%3.450%2.900%

BlackTom AI · Accepted Answer

We start by identifying the key inputs for this one-year, zero-coupon bond analysis. The issuer is rated BBB, and the average default rate for BBB is 0.5% per year. The problem states an expected loss given default of 70% during average economic times, so when default occurs, the bondholder recovers 30% of par value.

Option-by-option analysis:

Option A: 5.070%
- What this implies: The yield to maturity would be about 5.07% for a one-year bond.
- Why this is unlikely: A yield this high would require a significantly larger expected loss or a much lower risk-free rate, neither of which is supported by the given data. The BBB default rate is only 0.5% on average, and the assumed loss given default is modest (70%), so the expected loss is small and does not justify such a high yield.

Option B: 4.065%
- What this implies: A yield around 4.1% would still indicate a higher risk premium than the numbers suggest for BBB in average times.
- Why this is unlikely: The small default probability combined with the modest recovery would not produce a yield this high when the risk-free benchmark (AAA yield) is 2.5% and the debt beta is only 0.12, indicating a relatively modest risk premium.

Option C: 3.450%
- What this implies: A yield near 3.45% sits between the risk-free rate and the higher yields above; it could be plausible if the market priced in a moderate risk premium.
- Why this is not correct: While closer to a reasonable estimate, the calculation that uses a risk-free rate of 2.5% and the 0.5% default probability with 70% loss should land closer to around 3.0%, not as high as 3.45%, given the very small probability of default and the limited loss severity.

Option D: 2.900%
- What this implies: A yield of about 2.90% is only slightly above the risk-free rate from AAA (2.5%), which aligns with the very modest expected loss from BBB default
- Why this is correct: Compute the risk-neutral expected payoff for the one-year horizon. The probability of no default is 1 - d = 0.995. In the event of default (probability 0.005), the recovery is 30% of par, so the payoff is 0.3. Therefore, E[Payoff] = 0.995 * 1 + 0.005 * 0.3 = 0.9965. Discount this expected payoff at the risk-free rate of 2.5%: Price P = 0.9965 / 1.025 ≈ 0.9712. The yield y on a one-year zero-coupon that yields P is found from 1/(1+y) = P, so y ≈ 1/0.9712 − 1 ≈ 0.029 or about 2.9%. This matches the provided option 2.900%.

Additional cross-checks:
- Using a CAPM-like approach with risk-free rate 2.5% and BBB debt beta 0.12 with a market risk premium of 5% would give a rough required yield of 2.5% + 0.12×5% = 3.1%, which does not account for the low default probability and the modest expected loss. The explicit default-and-recovery calculation above is more accurate for this one-year instrument and yields about 2.9%, consistent with the option.
- The stationary AAA baseline of 2.5% as risk-free provides the proper discounting for risk-neutral valuation in this setting; elevating the yield to 3.45% or higher would require larger expected losses than the given 0.5% default rate and 70% loss rate.

Summary of reasoning for each option:
- 5.070%: too high given tiny default risk and modest recovery; not supported by the low d and loss rate. 
- 4.065%: still too high for BBB in average times with the stated parameters. 
- 3.450%: closer to plausible but still higher than the default-adjusted calculation yields. 
- 2.900%: consistent with the risk-neutral valuation using d = 0.5%, recovery = 0.3, and risk-free rate 2.5%; aligns with the expected payoff and the resulting yield.

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