Question

6.16 Bond price: QBE Insurance Group Ltd has outstanding bonds with a face value of $1000 that will mature in 6 years and pay an 8 per cent coupon, interest being paid semiannually. If you paid $1036.65 today and your required rate of return was 6.6 per cent, did you pay the right price for the bond?

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 = 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 required rate, which is 6.6% /2 = 3.3%, with 6*2 = 12 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 = 3.3% and n is the time period = 12

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

PVA = $40 * (1 - (1 + 3.3%)^-12 / 3.3%)

PVA = $40 * (1 - ( 1+ 0.033)^-12 / 0.033)

PVA = $40 * (1 - ( 1.033)^-12 / 0.033)

PVA = $40 * ((1 - 0.67732349669) / 0.033)

PVA = $40 * (0.3226765033 / 0.033)

PVA = $40 * 9.77807585766

PVA = $391.12

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 = 3.3%, n= time period = 12

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

$1000 = PV * (1 + 3.3%)¹²

$1000 = PV * (1 + 0.033)¹²

$1000 = PV * (1.033)¹²

$1000 = PV * 1.4763993939

PV = $1000 / 1.4763993939

PV = $677.32

Now,

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

Bond price = $391.12 + $677.32 = $1069.24

Actual bond price should be $1069.24. We paid $1036.65, which is less than this price. So, we paid less for the price, which is our benefit.