Question

A company is issuing a 6-year maturity bond that is expected to pay $40 annually and a lump sum of $1000 at the end of six-year from now. If bond investors require 9% annual nominal interest rate (APR) but the interest rate is compounded quarterly, how much is the price of this bond today?

Please show your formula in your answer and explain step-by-step calculation to arrive to your final answer.

Answer 1

First, we compute the effective annual rate (EAR) of 9% compounded quarterly.

EAR = (1 + (r/n))ⁿ - 1

where r = annual nominal interest rate

n = number of compounding periods per year

EAR = (1 + (9%/4))⁴ - 1

EAR = 9.3083%

Price of a bond is the present value of its cash flows. The cash flows are the coupon payments and the face value receivable on maturity.

Present value of lump-sum = future value / (1 + R)ⁿ

where R = interest rate

n = number of years

PV of face value = $1,000 / (1 + 9.3083%)⁶ = $586.25

PV of annuity = P * [1 - (1 + r)^-n] / r,

where P = periodic payment. This is $40.

r = interest rate per period. This is 9.3083%

n = number of periods. This is 6

PV of annuity = $40 * [1 - (1 + 9.3083%)^-6] / 9.3083%

PV of annuity = $177.80

PV of coupon payments =  $177.80

Price of bond = PV of face value + PV of coupon payments

Price of bond = $586.25 + $177.80

Price of bond = $764.05