A manufacturing company issues a bond with a 100,000 face value and a coupon rate of 7%. If the bond has a life of 30 years, pays annual coupons, and the yield to maturity is 6.8%, what percentage of the bond's price is the present value of the face value?
To find the percentage of the bond's price that is the present value of the face value, we need to calculate the present value of the future cash flows in order to determine the price of the bond.
The bond pays annual coupons, which means it will pay interest equal to the coupon rate of 7% multiplied by the face value of $100,000 every year for 30 years.
To calculate the present value of the coupon payments, we can use the formula for the present value of an ordinary annuity:
PV = C * [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present value
C = Coupon payment
r = Yield to maturity
n = Number of periods
In this case:
C = Coupon payment = 7% of $100,000 = $7,000
r = Yield to maturity = 6.8% = 0.068
n = Number of periods = 30
Using the formula, we can calculate the present value of the coupon payments:
PV_coupon = $7,000 * [(1 - (1 + 0.068)^(-30)) / 0.068]
Next, we need to calculate the present value of the face value of the bond. Since the bond's face value is already the future value of the bond, we can directly calculate the present value using the formula for the present value of a single future cash flow:
PV_face_value = FV / (1 + r)^n
Where:
PV_face_value = Present value of face value
FV = Face value = $100,000
r = Yield to maturity = 6.8% = 0.068
n = Number of periods = 30
Now, we can calculate the present value of the face value:
PV_face_value = $100,000 / (1 + 0.068)^30
Finally, to find the percentage of the bond's price that is the present value of the face value, we divide the present value of the face value by the total price of the bond and multiply by 100:
Percentage = (PV_face_value / (PV_coupon + PV_face_value)) * 100
Substituting the calculated values into the formula, we can determine the answer.