Question

The City of Allentown is issuing a 30-year bond with a face value of $80,000,000 and a stated annual interest rate of 5 percent. The town will make interest payments twice a year.

1. Calculate the semiannual interest payment.

2. Calculate how much Allentown will receive from the bond offering under the following conditions:

a. Market interest rates remain unchanged at the time of the offering.

b. Market interest rates increase to 6 percent at the time of the offering

Answer 1

1. Calculate the semiannual interest payment.

M = value at maturity, or face value = $ 80,000,000

C = coupon payment or annual interest payment = 5% per annum but it makes coupon payments on semiannual basis therefore coupon payment semiannual interest payment

= 5%/2 of face value = 5%/2 * $80,000,000 = $2,000,000

2. Calculate how much Allentown will receive from the bond offering under the following conditions:

a. Market interest rates remain unchanged at the time of the offering.

Let’s calculate the bond price with the help of following formula

Bond price P0 = C* [1- 1/ (1+i) ^n] /i + M / (1+i) ^n

Where (for 30-year bond with 5% coupons and a yield of 5%)

Market price of the bond, P0 =?

M = value at maturity, or face value = $ 80,000,000

C = coupon payment or annual interest payment = 5% per annum but it makes coupon payments on semiannual basis therefore coupon payment = 5%/2 of $80,000,000 = $2,000,000

n = number of payments or time remaining for the maturity of bond = 60 (30*2 for semiannual payments)

i = yield to maturity (YTM) = 5% per annum or 2.5% per semiannual

Therefore,

Price of bond P0 = $2,000,000 * [1 – 1 / (1+2.5%) ^60] /2.5% + $80,000,000 / (1+2.5%) ^60

= $61,817,312.97 + $18,182,687.03

= $80,000,000.00

Allentown will receive $80,000,000 from the bond offering

b. Market interest rates increase to 6 percent at the time of the offering

Now for 30-year bond with 5% coupons and a yield of 6%

M = value at maturity, or face value = $80,000,000

C = coupon payment or annual interest payment = 5% per annum but it makes coupon payments on semiannual basis therefore coupon payment = 5%/2 of $80,000,000 = $2,000,000

n = number of payments or time remaining for the maturity of bond = 60 (30*2 for semiannual payments)

i = yield to maturity (YTM) = 6% per annum or 3% per semiannual

Therefore,

Price of bond P0 = $2,000,000 * [1 – 1 / (1+3%) ^60] /3% + $80,000,000 / (1+3%) ^60

= $55,351,127.33 + $13,578,647.20

= $68,929,774.53

Allentown will receive $68,929,774.53 from the bond offering