Question

7- What is the price of a bond with the following features?

• Face Value = $1,000
• Coupon Rate = 3% (stated as an ANNUAL rate)
• Semiannual coupon payments
• Maturity = 6 years
• YTM = 5.2% (Stated as an APR)

State your answer to the nearest penny (e.g., 984.25)

Answer 1

Price of the bond can be calculated by the following formula:

Bond price = Present value of interest payment + Present value of bond payment at maturity

Semi annual bond interest = 3% * $1000 * 1/2 = $15

Bond interest payments will be semi annual every year, so it is an annuity. Bond payment at maturity is a one time payment. The interest rate that will be used in calculating the required present values will be the semi annual YTM rate, which is 5.2% /2 = 2.6%, with 6*2 = 12 periods.

Now,

First we will calculate the present value of interest payments:

For calculating the present value, we will use the following formula:

PVA = P * (1 - (1 + r)^-n / r)

where, PVA = Present value of annuity, P is the periodical amount = $15, r is the rate of interest = 2.6% and n is the time period = 12

Now, putting these values in the above formula, we get,

PVA = $15 * (1 - (1 + 2.6%)^-12 / 2.6%)

PVA = $15 * (1 - ( 1+ 0.026)^-12 / 0.026)

PVA = $15 * (1 - ( 1.026)^-12 / 0.026)

PVA = $15 * ((1 - 0.73490579284) / 0.026)

PVA = $15 * (0.26509420715 / 0.026)

PVA = $15 * 10.1959310444

PVA = $152.9389

Next, we will calculate the present value of bond payment at maturity:

For calculating present value, we will use the following formula:

FV = PV * (1 + r%)ⁿ

where, FV = Future value = $1000, PV = Present value, r = rate of interest = 2.6%, n= time period = 12

now, putting theses values in the above equation, we get,

$1000 = PV * (1 + 2.6%)¹²

$1000 = PV * (1 + 0.026)¹²

$1000 = PV * (1.026)¹²

$1000 = PV * 1.36071862507

PV = $1000 / 1.36071862507

PV = $734.9057

Now,

Bond price = Present value of interest payment + Present value of bond payment at maturity

Bond price = $152.9389 + $734.9057 = $887.84