1. Dooley, 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?
a. 7
b. 6
a. To determine the price of the bonds assuming investors do not expect them to be called, we need to calculate the present value of the bond's future cash flows.
The annual coupon payment can be calculated as 10.5% of the par value, which is $1,000, so the coupon payment is $1,000 x 10.5% = $105.
The bond's cash flows consist of the annual coupon payment of $105 for 20 years and the final payment of the bond's face value ($1,000).
To calculate the present value of these cash flows, we need to discount them using the required rate of return. The required rate of return for similar quality bonds with 20 years remaining until maturity is 14%.
Using the Present Value formula:
Present Value = (Coupon Payment / (1 + Required Rate of Return)^n) + (Face Value / (1 + Required Rate of Return)^n)
Where n is the number of years remaining until maturity.
Let's calculate the present value:
Present Value = ($105 / (1 + 0.14)^1) + ($105 / (1 + 0.14)^2) + ... + ($105 / (1 + 0.14)^20) + ($1,000 / (1 + 0.14)^20)
By calculating this formula, we can find the price at which the bonds would sell assuming investors do not expect them to be called.
b. To determine the price of the bonds assuming investors expect them to be called at the end of 10 years, we need to calculate the present value of the bond's cash flows up to the call date (10 years) and the potential cash flows after the call date.
The cash flows up to the call date are the same as in part a (annual coupon payments of $105 for 10 years and the face value payment of $1,000).
After the call date, the bond will be called at 110% of the par value, which is $1,000, so the bond will be called at $1,000 x 110% = $1,100. All future cash flows after the call date are eliminated because the bond will be retired.
To calculate the present value, we need to discount the cash flows up to the call date and subtract the potential call price.
Present Value = ($105 / (1 + 0.14)^1) + ($105 / (1 + 0.14)^2) + ... + ($105 / (1 + 0.14)^10) + ($1,000 / (1 + 0.14)^10) - ($1,100 / (1 + 0.14)^10)
By calculating this formula, we can find the price at which the bonds would sell assuming investors expect them to be called at the end of 10 years.
To calculate the price of the bonds, we need to use the present value formula. The formula for the present value of a bond is:
PV = C * (1 - (1 + r/k)^(-nt)) / (r/k) + F / (1 + r/k)^nt
where:
PV = Present value of the bond
C = Coupon payment per period (annual interest payment)
r = Required rate of return (yield or discount rate)
k = Number of coupon payments per year (annual coupon payments)
n = Number of years until maturity
t = Number of years until the bond is called
F = Face value or par value of the bond
Let's calculate the price of the bonds in both scenarios:
a. Assuming investors do not expect the bonds to be called:
Rate of return (r) = 14%
Coupon payment (C) = $1,000 * 10.5% = $105
Par value (F) = $1,000
Number of coupon payments per year (k) = 1
Number of years until maturity (n) = 20
Using these values in the present value formula:
PV = $105 * (1 - (1 + 14%/1)^(-1*1*20)) / (14%/1) + $1,000 / (1 + 14%/1)^1*20
PV = $105 * (1 - (1 + 0.14)^(-20)) / 0.14 + $1,000 / (1 + 0.14)^20
PV = $105 * (1 - 0.093288) / 0.14 + $1,000 / 1.506852
PV = $9.78 + $663.60
PV = $673.38
Therefore, the price of the bonds assuming investors do not expect them to be called is approximately $673.38.
b. Assuming investors expect the bonds to be called at the end of 10 years:
We need to calculate the present value of the bond until the call date, then calculate the present value after the call date and add them together.
Present value until the call date:
Rate of return (r) = 14%
Coupon payment (C) = $105
Par value (F) = $1,000
Number of coupon payments per year (k) = 1
Number of years until maturity (n) = 10
PV1 = $105 * (1 - (1 + 14%/1)^(-1*1*10)) / (14%/1) + $1,000 / (1 + 14%/1)^1*10
PV1 = $105 * (1 - (1 + 0.14)^(-10)) / 0.14 + $1,000 / (1 + 0.14)^10
PV1 = $105 * (1 - 0.25654) / 0.14 + $1,000 / 4.317166
PV1 = $20.76 + $231.59
PV1 = $252.35
Present value after the call date:
Rate of return (r) = 14%
Coupon payment (C) = $105
Par value (F) = $1,100 (110% of $1,000)
Number of coupon payments per year (k) = 1
Number of years until maturity (n) = 10
PV2 = $105 * (1 - (1 + 14%/1)^(-1*1*10)) / (14%/1) + $1,100 / (1 + 14%/1)^1*10
PV2 = $105 * (1 - (1 + 0.14)^(-10)) / 0.14 + $1,100 / (1 + 0.14)^10
PV2 = $105 * (1 - 0.25654) / 0.14 + $1,100 / 4.317166
PV2 = $20.76 + $254.89
PV2 = $275.65
Total present value of the bonds assuming they are called at the end of 10 years:
PV_total = PV1 + PV2
PV_total = $252.35 + $275.65
PV_total = $528
Therefore, the price of the bonds assuming investors expect them to be called at the end of 10 years is approximately $528.