Question

A $1,000 par value bond matures in 8 years. It has a 7% coupon rate, with semi-annual interest payments. The yield (the rate at which investors are using to calculate the price of the bond) is 8%. What is the fair market value of the bond?

Answer 1

Fair market value or 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 = 7% * $1000 * 1/2 = $35

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 rate, which is 8% /2 = 4%, with 8*2 = 16 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 = $35, r is the rate of interest = 4% and n is the time period = 16

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

PVA = $35 * (1 - (1 + 4%)^-16 / 4%)

PVA = $35 * (1 - ( 1+ 0.04)^-16 / 0.04)

PVA = $35 * (1 - ( 1.04)^-16 / 0.04)

PVA = $35 * ((1 - 0.53390817568) / 0.04)

PVA = $35 * (0.46609182431 / 0.04)

PVA = $35 * 11.6522956079

PVA = $407.83

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 = 4%, n= time period = 16

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

$1000 = PV * (1 + 4%)¹⁶

$1000 = PV * (1 + 0.04)¹⁶

$1000 = PV * (1.04)¹⁶

$1000 = PV * 1.87298124573

PV = $1000 / 1.87298124573

PV = $533.91

Now,

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

Bond price = $407.83 + $533.91 = $941.74

So, fair market value of the bond is $941.74.