Valuing Bonds with Uncertainty

The assignment is about valuing bonds in an environment of uncertain interest rates.

• The interest rate for the first year is 4% (no uncertainty).
• Each year the one-year interest rate could move up or down by 1%.
• The probability that the rate moves higher is 70%; it moves lower is 30%. Note the values differ from the lecture examples.
• All bonds have a face value of $1,000.

We will complete the following table by answering the following nine questions. Some of the cells are already complete so that you can verify your calculations against mine.

Asset	Price	Yield	Discount Factor
1-year zero-coupon
face $1,000. coupon 0%
2-year zero-coupon	$921.0854	4.20%	$0.9211
face $1,000, coupon 0%
3-year zero-coupon	$879.1023	4.39%	$0.8791
face $1,000, coupon 0%
4-year zero-coupon
face $1,000, coupon 0%
4-year coupon
face $1,000, coupon 5%

Consider a one-year zero-coupon bond. Do your calculations on a calculator or with Excel. Round your final answer as in the table above. Do not round intermediate values; use all the decimal places in subsequent computations.

1. What is the price or value of the bond?

2. What is the yield to maturity of the bond?

3. What is the one-year discount factor (present value of $1 delivered in the future)?

Consider a four-year zero-coupon bond. Use a binomial tree for this calculation. Do not round intermediate values.

4. What is the price or value of the bond?

5. What is the yield to maturity of the bond?

6. What is the four-year discount factor (present value of $1 delivered in the future)?

7. Use the discount factors to value a four-year coupon bond (face value of $1,000 and a coupon rate of 5%). You do not need to use the binomial tree, but you can verify your calculation using the tree if you are unsure of the answer.

8. Calculate the yield on the four-year coupon bond using your answer to question 7. I suggest using the IRR function in Excel.

9. The airline industry is currently considered very risky. I just paid $900 for a one-year zero-coupon airline bond with a face value of $1,000. Use the 4% certain interest rate for the next year. What default rate does the price imply? Be specific about your assumption on recovery and whether you are using an approximation or an exact formula.