Question

What is the price of a $10,000 bond with a 4.5% coupon rate with quarterly coupon payments, and 7 years to maturity if it has a YTM of 6%?

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

Quarterly bond interest = 4.5% * $10000 * 3 /12 = $112.5

Bond interest payments will be quarterly 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 quarterly yield to maturity rate, which is 6% /4 = 1.5%, with 7*4 = 28 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 = $112.5, r is the rate of interest = 1.5% and n is the time period = 28

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

PVA = $112.5 * (1 - (1 + 1.5%)^-28 / 1.5%)

PVA = $112.5 * (1 - ( 1+ 0.015)^-28 / 0.015)

PVA = $112.5 * (1 - ( 1.015)^-28 / 0.015)

PVA = $112.5 * ((1 - 0.65909924935) / 0.015)

PVA = $112.5 * (0.34090075065 / 0.015)

PVA = $112.5 * 22.72671671

PVA = $2556.76

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 = $10000, PV = Present value, r = rate of interest = 1.5%, n= time period = 28

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

$10000 = PV * (1 + 1.5%)²⁸

$10000 = PV * (1 + 0.015)²⁸

$10000 = PV * (1.015)²⁸

$10000 = PV * 1.5172221801

PV = $10000 / 1.5172221801

PV = $6591

Now,

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

Bond price = $2556.76 + $6591 = $9147.75