ooley, Inc., has outstanding $100 million (par value) bonds that pay an annual coupon rate of interest of 10.5 percent. Par value of each bond is $1,000. The bonds are scheduled to mature in 20 years. Because of Dooley’s increased risk, investors now require a 14 percent rate of return on bonds of similar quality with 20 years remaining until maturity. The bonds are callable at 110 percent of par at the end of 10 years.
a. What price would the bonds sell for assuming investors do not expect them to be called?
b. What price would the bonds sell for assuming investors expect them to be called at the end of 10 years?
To determine the price of the bonds, we need to calculate the present value of the future cash flows associated with the bonds. The cash flows include the annual coupon payments and the maturity value at the end of the bond term.
a. To calculate the price of the bonds assuming they will not be called, we will use the current required rate of return of 14 percent.
Step 1: Calculate the present value of the coupon payments:
The bonds have a $100 million (par value) with an annual coupon rate of 10.5 percent. Since the par value of each bond is $1,000, the annual coupon payment will be $1,000 * 10.5% = $105.
To find the present value of the coupon payments, we use the formula for the present value of an ordinary annuity:
PVA = C * [(1 - (1 + r)^(-n)) / r]
Where:
PVA is the present value of the annuity (coupon payments)
C is the coupon payment per period
r is the discount rate per period
n is the number of periods
Using the given values:
C = $105
r = 14% = 0.14
n = 20 years
PVA = $105 * [(1 - (1 + 0.14)^(-20)) / 0.14]
PVA = $105 * [(1 - 0.029283) / 0.14]
PVA = $105 * (0.970717 / 0.14)
PVA = $725.52
Step 2: Calculate the present value of the maturity value:
The maturity value of the bond is $1,000. We need to find the present value of this amount, which is simply $1,000.
Step 3: Calculate the total present value of the bond:
The total present value of the bond is the sum of the present value of the coupon payments and the present value of the maturity value.
Total Present Value = PVA + Present Value of the Maturity Value
Total Present Value = $725.52 + $1,000
Total Present Value = $1,725.52
Therefore, the bonds would sell for $1,725.52 assuming investors do not expect them to be called.
b. To calculate the price of the bonds assuming investors expect them to be called at the end of 10 years, we need to consider the cash flows until the bonds are called.
Step 1: Calculate the present value of the coupon payments and the maturity value for the first 10 years (before the bond is callable):
Using the same method as in part a, calculate the present value of the coupon payments for the first 10 years (n = 10) and the present value of the maturity value at the end of 10 years.
Step 2: Calculate the present value of the maturity value when the bonds are called:
The bonds are callable at 110 percent of par value, which is $1,000 * 1.10 = $1,100. We need to find the present value of this amount at the end of 10 years.
Step 3: Calculate the total present value of the bond:
The total present value of the bond is the sum of the present value of the coupon payments and the present value of the maturity value.
Total Present Value = Present Value of Coupon Payments (first 10 years) + Present Value of Maturity Value (at the end of 10 years)
Therefore, the bonds would sell for the total present value calculated in step 3, assuming investors expect them to be called at the end of 10 years.