Answer:
To calculate the purchase price of the bond, we need to determine the present value of all the future cash flows associated with the bond.
First, let's calculate the number of semi-annual periods between April 15, 2006, and January 1, 2021. Since each year has two semi-annual periods, there are 15 years * 2 = 30 semi-annual periods in total.
Next, let's calculate the coupon payments. The bond has a face value of $1,000 and a coupon rate of 4% payable semi-annually. Each period, you will receive 4% * $1,000 / 2 = $20.
Now, let's calculate the present value of the coupon payments. Since the market interest rate is 4.5% compounded semi-annually, we'll use this rate as the discount rate for our present value calculations.
To calculate the present value of each individual coupon payment, we divide the coupon payment by the semi-annual interest rate and then discount it back to the purchase date. Since the bond was purchased on April 15, 2006, and there are 30 semi-annual periods, we can discount each coupon payment back 30 periods.
The present value of a coupon payment is given by the formula:
PV = C / (1+r)^n
where PV is the present value, C is the coupon payment, r is the interest rate per period, and n is the number of periods.
Using this formula, we can calculate the present value of the coupon payments by discounting each individual payment:
PV_coupon = $20 / (1 + 0.045/2)^30
Next, let's calculate the present value of the face value, which is the final payment you will receive when the bond matures on January 1, 2021. Similar to the coupon payments, we'll discount the face value back to the purchase date:
PV_face_value = $1,000 / (1 + 0.045/2)^30
Finally, the purchase price of the bond is the sum of the present values of the coupon payments and the face value:
Purchase price = PV_coupon + PV_face_value
Calculating these values, the purchase price of the bond is the sum of both present values:
PV_coupon = $20 / (1 + 0.045/2)^30 ≈ $356.01
PV_face_value = $1,000 / (1 + 0.045/2)^30 ≈ $624.02
Purchase price = $356.01 + $624.02 ≈ $980.03
Therefore, the approximate purchase price of the bond on April 15, 2006, is $980.03.