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 calculate the price you should be willing to pay for the bond today, we need to determine the present value of the bond's future cash flows.
In this scenario, the bond pays $50 in interest every six months for a total of 20 years. So, over the bond's life, it pays 40 times ($50 x 2 = $100) every 6 months.
First, let's calculate the present value of each cash flow using the required annual return of 12 percent. We will use a financial calculator or formula to do this:
PV = CF / (1 + r)^n
Where PV is the present value, CF is the cash flow, r is the discount rate (annual return as a decimal), and n is the number of periods.
For the cash flows received semi-annually, we will adjust the discount rate and the number of periods accordingly.
Cash flows 1-40: $100 every 6 months.
Discount rate for the first 10 years (semi-annually): 12% / 2 = 6% as a decimal.
Number of periods for the first 10 years (semi-annually): 10 x 2 = 20.
Cash flows 41-80: $100 every 6 months.
Discount rate for the last 10 years (semi-annually): 8% / 2 = 4% as a decimal.
Number of periods for the last 10 years (semi-annually): 10 x 2 = 20.
Now, let's calculate the present value of each cash flow:
PV1-40 = $100 / (1 + 0.06)^20
PV41-80 = $100 / (1 + 0.04)^20
Next, we need to calculate the present value of the bond's final payment, which is the face value or principal value of the bond after 20 years.
The final payment's present value is calculated as follows:
PV_final_payment = Face Value / (1 + r)^n
Where the face value is the value of the bond after 20 years, r is the discount rate (8% for the last 10 years as a decimal), and n is the number of periods (20 for the last 10 years).
To determine the final payment's present value, you'll need to know the face value of the bond.
With the present values calculated for each cash flow and the present value of the final payment, sum all the present values together to get the total present value of the bond. This will be the price you should be willing to pay for the bond today.