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.

View Explanation

Log in for full answers

Similar Questions

What is the cost of bond?

A bond’s yield to maturity represents the investor’s realized return for any holding period, regardless of whether the bond is held to maturity.

A bond with a coupon rate of 8% will also have a yield to maturity of 8%.

Suppose you buy a 10-year $1,000 face-value bond. The annual coupon rate is 7% and interest payments are paid semi-annually. If the bond price is $932, what is the YTM on this bond?

A zero-coupon bond is currently priced at $456, has a face value of $1,000, and matures in 10 years. What is the yield to maturity of this bond?

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler