Question

A bond has a maturity of 10 years, has 8% coupon rate(paid annually) and the yield to maturity is 10%. WHat is the price of the bond? (face value is 1000)

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

Annual bond interest = 8% * $1000 = $80

Bond interest payments will be 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 yield to maturity rate, which is 10%, with 10 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 = $80, r is the rate of interest = 10% and n is the time period = 10

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

PVA = $80 * (1 - (1 + 10%)^-10 / 10%)

PVA = $80 * (1 - ( 1+ 0.10)^-10 / 0.10)

PVA = $80 * (1 - ( 1.10)^-10 / 0.10)

PVA = $80 * ((1 - 0.38554328943) / 0.10)

PVA = $80 * (0.61445671057 / 0.10)

PVA = $80 * 6.1445671057

PVA = $491.5653

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 = 10%, n= time period = 10

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

$1000 = PV * (1 + 10%)¹⁰

$1000 = PV * (1 + 0.10)¹⁰

$1000 = PV * (1.10)¹⁰

$1000 = PV * 2.5937424601

PV = $1000 / 2.5937424601

PV = $385.54

Now,

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

Bond price = $491.5653 + $385.54 = $877.11