Question

What is the price of a bond with a $1000 face value, an 9.5% coupon rate, semiannual coupons, and 7 years to maturity if it has a yield to maturity of 13%?

answer choices:

$930.24

$428.16

$842.26

$414.10

Answer 1

Option (c) is correct

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 = 9.5% * $1000 * 1/2 = $47.5

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 market rate, which is 13% /2 = 6.5%, with 7*2 = 14 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 = $47.5, r is the rate of interest = 6.5% and n is the time period = 14

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

PVA = $47.5 * (1 - (1 + 6.5%)^-14 / 6.5%)

PVA = $47.5 * (1 - ( 1+ 0.065)^-14 / 0.065)

PVA = $47.5 * (1 - ( 1.065)^-14 / 0.065)

PVA = $47.5 * ((1 - 0.41410024853) / 0.065)

PVA = $47.5 * (0.58589975146 / 0.065)

PVA = $47.5 * 9.013844233029

PVA = $428.16

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 = 6.5%, n= time period = 14

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

$1000 = PV * (1 + 6.5%)¹⁴

$1000 = PV * (1 + 0.065)¹⁴

$1000 = PV * (1.065)¹⁴

$1000 = PV * 2.41487418457

PV = $1000 / 2.19163693671

PV = $414.1

Now,

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

Bond price = $428.16 + $414.1 = $842.26