Corporation has issued​ zero-coupon corporate bonds with a 5​-year maturity​ (assume ​$100 face value​ bond). Investors believe there is a 40 % chance that Grummon will default on these bonds. If Grummon does​ default, investors expect to receive only 50 cents per dollar they are owed. If investors require a 6 % expected return on their investment in these​ bonds, what will be the price and yield to maturity on these​ bonds?
Note​: Assume annual compounding.

To solve this problem, we'll need to break it into several steps:

### 1. Understanding the Situation

- **Zero-coupon bond:** This means the bond doesn't pay interest, so the investor will only receive the face value (or a portion thereof in case of default) at maturity.
- **Maturity period:** 5 years.
- **Face value of the bond:** $100.
- **Default probability:** 40% chance Grummon will default.
- **Recovery rate in case of default:** Investors expect to recover only 50% of the face value, or $50.
- **Required return:** Investors want a 6% expected return on their investment.

### 2. Calculation of Expected Payoff

There are two possible outcomes at the end of 5 years:

1. **No default (probability = 60%):** Investors will receive the full face value of $100.
   
2. **Default (probability = 40%):** Investors will receive 50% of the face value, i.e., $50.

So, the **expected payoff** of the bond at maturity is:
\[
\text{Expected payoff} = (0.60 \times 100) + (0.40 \times 50) = 60 + 20 = 80
\]

### 3. Required Price for 6% Expected Return

The price today must be such that the investor earns a 6% return over the 5-year period. The formula for the price $ P $ of a zero-coupon bond is:
\[
P = \frac{\text{Expected Payoff}}{(1 + r)^n}
\]
Where:
- $ r = 6\% $ (the required return),
- $ n = 5 $ years (the maturity period),
- Expected payoff = 80 (calculated above).

Now we can calculate the price $ P $.

\[
P = \frac{80}{(1 + 0.06)^5}
\]

Let's calculate this value.The current price of the bond, based on a 6% expected return, would be approximately **$59.78**.

### 4. Yield to Maturity (YTM)

Next, we calculate the **Yield to Maturity (YTM)**, which is the internal rate of return of the bond based on its current price. We already know that the expected payoff at maturity is $80, and the price today is approximately $59.78. The YTM can be found using the formula:
\[
P = \frac{\text{Expected Payoff}}{(1 + YTM)^n}
\]
We can solve for $ YTM $ using this equation:
\[
YTM = \left(\frac{\text{Expected Payoff}}{P}\right)^{\frac{1}{n}} - 1
\]

Let's now calculate the YTM.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can manually calculate the Yield to Maturity (YTM) using the formula provided earlier:

\[
YTM = \left(\frac{\text{Expected Payoff}}{P}\right)^{\frac{1}{n}} - 1
\]

Where:
- Expected payoff = $80
- Price = $59.78
- $ n = 5 $ years

If you have any questions or want me to clarify further, feel free to ask!

### Related Questions:
1. How does default risk impact bond pricing in general?
2. What factors influence the recovery rate in a default scenario?
3. What is the difference between the required return and the yield to maturity?
4. How does a zero-coupon bond differ from a coupon bond in terms of risk?
5. What are the tax implications of purchasing zero-coupon bonds?

#### Tip:
When investing in bonds, always assess both the credit risk (default probability) and the recovery rate to ensure the expected returns match your risk tolerance.

Corporation has issued zero-coupon corporate bonds with a 5-year maturity (assume $100 face value bond). Investors believe there is a 40% chance that Grummon will default on these bonds. If Grummon does default, investors expect to receive only 50 cents per dollar they are owed. If investors require a 6% expected return on their investment in these bonds, what will be the price and yield to maturity on these bonds? Note: Assume annual compounding.

Learn how to calculate the price and yield to maturity of a zero-coupon bond with a 40% default probability and a recovery rate of 50%. This detailed solution explains how to apply probability, the bond pricing formula, and the yield to maturity calculation to solve the problem.

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