Question

Renfro Rentals has issued bonds that have a 9% coupon rate, payable semiannually. The bonds mature in 8 years, have a face value of $1,000, and a yield to maturity of 7.5%. What is the price of the bonds? Round your answer to the nearest cent.

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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 = 9% * $1000 * 1/2 = $45

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 7.5% /2 = 3.75%, 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 = $45, r is the rate of interest = 3.75% and n is the time period = 16

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

PVA = $45 * (1 - (1 + 3.75%)^-16 / 3.75%)

PVA = $45 * (1 - ( 1+ 0.0375)^-16 / 0.0375)

PVA = $45 * (1 - ( 1.0375)^-16 / 0.0375)

PVA = $45 * ((1 - 0.55486881088) / 0.0375)

PVA = $45 * (0.44513111891 / 0.0375)

PVA = $45 * 11.870163171

PVA = $534.1573

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 = 3.75%, n= time period = 16

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

$1000 = PV * (1 + 3.75%)¹⁶

$1000 = PV * (1 + 0.0375)¹⁶

$1000 = PV * (1.0375)¹⁶

$1000 = PV * 1.80222780662

PV = $1000 / 1.80222780662

PV = $554.87

Now,

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

Bond price = $534.1573 + $554.87 = $1089.03