Cosmic Communications Inc.is planning two new issues of 25-year bonds.Bond Par will be sold at its $1,000 par value, and it will have a 10% semiannual coupon.Bond OID will be an Original Issue Discount bond, and it will also have a 25-year maturity and a $1,000 par value, but its semiannual coupon will be only 6.25%.If both bonds are to provide investors with the same effective yield, how many of the OID bonds must Cosmic issue to raise $3,000,000? Disregard flotation costs, and round your final answer up to a whole number of bonds.

Question

Quizplus · Accepted Answer

First, we need to calculate the effective yield of the bond with a 10% coupon:

- Each coupon payment will be $50 (=$1,000 par value x 10% coupon rate x 6/12)
- There will be 50 coupon payments over 25 years, so the total coupon payments will be $2,500 (=50 x $50)
- At maturity, the bondholder will also receive the par value of $1,000
- Therefore, the total future cash flows for this bond are $3,500 (= $2,500 + $1,000)

Using a financial calculator or spreadsheet function, we can solve for the yield to maturity (YTM) that makes the present value of these future cash flows equal to the price of the bond:

PV = $1,000 (for a par bond)
PMT = $50
N = 50 x 2 = 100 (since there are two payments per year)
FV = $1,000
Solving for I/Y gives a YTM of 7.22%

Now, we need to find the number of OID bonds Cosmic Communications must issue to raise $3,000,000 while providing investors with the same effective yield as the bond with a 10% coupon.

Let's assume they issue X number of OID bonds. The total dollar amount raised by the OID bonds will be:

$1,000 per bond x (1 - price of OID bond) x X bonds = $3,000,000

Since the OID bond has a 6.25% coupon rate, the effective yield will be lower than 7.22%. We need to find the price of the OID bond that will give us the same effective yield as the bond with a 10% coupon and use it to solve for X.

Using the same method as before, the YTM of the OID bond that makes the present value of its future cash flows equal to its price can be found:

PV = price of OID bond
PMT = $31.25 (= $1,000 x 6.25% / 2)
N = 50 x 2 = 100
FV = $1,000
Solving for I/Y gives a YTM of 9.479%

To make the effective yields of the two bonds equal, we need to find the price of the OID bond that gives a YTM of 7.22%:

PV = $31.25 / (1 + 0.0361)^1 + $31.25 / (1 + 0.0361)^2 + ... + $31.25 / (1 + 0.0361)^100 + $1,000 / (1 + 0.0361)^100 = $608.01

Therefore, the number of OID bonds needed to raise $3,000,000 is:

X = $3,000,000 / [$1,000 x (1 - $608.01 / $1,000)] = 4,562.16

Rounding up to the nearest whole number gives us 4,562 bonds. Therefore, the best answer is D) 4,562.