Question

​A coupon bond which pays interest semi-annually, has a par value of $1,000, matures in 15 years, and has a yield to maturity of 4.2%. If the coupon rate is 3.7%, the market price of the bond today will be

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 = 3.7% * $1000 * 1/2 = $18.5

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 yield to maturity rate, which is 4.2% /2 = 2.1%, with 15*2 = 30 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 = $18.5, r is the rate of interest = 2.1% and n is the time period = 30

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

PVA = $18.5 * (1 - (1 + 2.1%)^-30 / 2.1%)

PVA = $18.5 * (1 - ( 1+ 0.021)^-30 / 0.021)

PVA = $18.5 * (1 - ( 1.021)^-30 / 0.021)

PVA = $18.5 * ((1 - 0.53607769477) / 0.021)

PVA = $18.5 * (0.46392230522 / 0.021)

PVA = $18.5 * 22.091538344

PVA = $408.69

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 = 2.1%, n= time period = 30

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

$1000 = PV * (1 + 2.1%)³⁰

$1000 = PV * (1 + 0.021)³⁰

$1000 = PV * (1.021)³⁰

$1000 = PV * 1.8654012464

PV = $1000 / 1.8654012464

PV = $536.077

Now,

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

Bond price = $408.69 + $536.077 = $944.77