Question

What is the present value of the following bond offering. Face value of $1000, maturity in 10 years, the coupon rate is 8%, the yield to maturity is also 8%, coupon is paid semi-annually.

a. $1,050

b. $950

c. 825

d. 1,231

e. 1,000

Answer 1

Option (e) is correct

Present value of the bond with coupon rate and yield to maturity rate same, will be equal to its face value. So, present value will be $1000, which is the face value also.

The above can be explained / calculated as per below:

Present 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 = 8% * $1000 * 1/2 = $40

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 8% /2 = 4%, with 10*2 = 20 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 = $40, r is the rate of interest = 4% and n is the time period = 20

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

PVA = $40 * (1 - (1 + 4%)^-20 / 4%)

PVA = $40 * (1 - ( 1+ 0.04)^-20 / 0.04)

PVA = $40 * (1 - ( 1.04)^-20 / 0.04)

PVA = $40 * ((1 - 0.4563869462) / 0.04)

PVA = $40 * (0.54361305379 / 0.04)

PVA = $40 * 13.590326345

PVA = $543.61

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 = 20

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

$1000 = PV * (1 + 4%)²⁰

$1000 = PV * (1 + 0.04)²⁰

$1000 = PV * (1.04)²⁰

$1000 = PV * 2.19112314303

PV = $1000 / 2.19112314303

PV = $456.39

Now,

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

Bond price = $543.61 + $456.39 = $1000