You are contemplating the purchase of a 20-year bond that pays $50 in interest each six months. You plan to hold this bond for only 10 years, at which time you will sell it in the marketplace. You require a 12 percent annual return, but you believe the market will require only an 8 percent return when you sell the bond 10 years hence. Assuming you are a rational investor, how much should you be willing to pay for the bond today?
To determine how much you should be willing to pay for the bond today, you need to calculate the present value of the future cash flows associated with the bond. Here's how you can do it step by step:
1. Identify the cash flows: In this case, the bond pays $50 in interest every six months for a total of 20 years, which means there will be 40 interest payments.
2. Determine the discount rate: Since you require a 12 percent annual return, this will be your discount rate for the first 10 years. However, for the remaining 10 years, you believe the market will only require an 8 percent return, so this will be your discount rate after the 10th year.
3. Calculate the present value: To find the present value, you need to discount each cash flow using the appropriate discount rate. Since the bond pays interest semi-annually, you need to divide the annual discount rate by 2 to get the semi-annual discount rate.
For the first 10 years:
- Calculate the present value of each semi-annual interest payment at a 12 percent semi-annual discount rate.
- Sum up the discounted cash flows for the first 10 years.
For the remaining 10 years:
- Calculate the present value of each semi-annual interest payment at an 8 percent semi-annual discount rate.
- Sum up the discounted cash flows for the remaining 10 years.
4. Calculate the present value of the terminal value: At the end of the 20-year period, you will also receive the principal value of the bond. Calculate the present value of this terminal value using the discount rate for the remaining 10 years.
5. Add up all the present values calculated in steps 3 and 4 to get the total present value.
6. The amount you should be willing to pay for the bond today is equal to the total present value calculated in step 5.
It's important to note that the formula used to calculate the present value of cash flows is:
PV = CF / (1 + r)^n
Where PV is the present value, CF is the cash flow, r is the discount rate, and n is the number of periods.
By following these steps and performing the necessary calculations, you should be able to determine the price you should be willing to pay for the bond today.