5. What are the proceeds If a company issues 10 year bonds with a face value of $10,000,000 in bonds with a coupon rate of 7% and a market rate of 7.5% paying interest payments semi annually.
Will they sell at a premium or a discount?
To determine whether the bonds will sell at a premium or a discount, we need to compare the coupon rate with the market rate. Here are the steps to calculate the proceeds and determine whether the bonds will sell at a premium or a discount:
Step 1: Calculate the coupon payment
The annual coupon payment can be calculated using the formula: Coupon Payment = Coupon Rate x Face Value.
Coupon Payment = 7% x $10,000,000 = $700,000
Since interest payments are made semi-annually, the semi-annual coupon payment will be $700,000 / 2 = $350,000.
Step 2: Determine the market interest rate per payment period.
The stated market rate is given as an annual rate, so we need to divide it by the number of compounding periods in a year to get the semi-annual market interest rate.
Market Interest Rate (per payment period) = 7.5% / 2 = 3.75%
Step 3: Calculate the present value of the bond.
To calculate the present value of the bond, we need to discount each semi-annual coupon payment and the face value using the market interest rate.
Using the present value of an ordinary annuity formula, the present value of the coupon payments (PVCP) can be calculated:
PVCP = Coupon Payment x [(1 - (1 + Market Interest Rate)^-number of payments) / Market Interest Rate]
In this case, the number of payments is twice the number of years, so 10 years would have 20 semi-annual payments.
PVCP = $350,000 x [(1 - (1 + 3.75%)^-20) / 3.75%]
Using a financial calculator or software, you can find that PVCP is approximately $6,393,625.
The present value of the face value (PVFV) can be calculated using the formula:
PVFV = Face Value / (1 + Market Interest Rate)^number of payments
PVFV = $10,000,000 / (1 + 3.75%)^20
Again, using a financial calculator or software, you can find that PVFV is approximately $4,120,633.
Step 4: Calculate the total proceeds.
The total proceeds from selling the bonds can be calculated by adding the present value of the coupon payments and the present value of the face value.
Total Proceeds = PVCP + PVFV
Total Proceeds = $6,393,625 + $4,120,633 = $10,514,258
Step 5: Determine whether the bonds will sell at a premium or a discount.
If the total proceeds are higher than the face value of the bonds ($10,000,000 in this case), the bonds will sell at a premium. If the total proceeds are lower than the face value, the bonds will sell at a discount.
In this case, the total proceeds ($10,514,258) are higher than the face value ($10,000,000), so the bonds will sell at a premium.