As an investor, you are considering an investment in the bonds of the Conifer Coal Company. The bonds, which pay interest semiannually, will mature in 8 years, have a coupon rate of 9.5% on a face value of $1,000. Currently, the bonds are selling for $872.
a. If you required return is 11% for bonds in this risk class, what is the highest price you would be willing to pay?
b. What is the yield to maturity on these bonds if you purchase them at the current price?
c. If the bonds can be called in three years with a call premium of 4% of the face value, what is the yield to call on these bonds?
a. To determine the highest price you would be willing to pay, you need to calculate the present value of the future cash flows from the bond.
The coupon payments for each period (semiannually) are calculated as follows:
Coupon payment = (Coupon rate x Face value) / 2
Coupon payment = (9.5% x $1,000) / 2
Coupon payment = $47.50
The number of periods (n) is calculated by multiplying the number of years to maturity by 2 (since the bond pays semiannual interest).
n = 8 years x 2 = 16 periods
The required return is 11%, or 0.11 in decimal form.
Using the present value formula:
Price = [Coupon payment x (1 - (1 + r)^-n)] / r + Face value / (1 + r)^n
where r is the required return rate, and n is the number of periods.
Price = [$47.50 x (1 - (1 + 0.11)^-16)] / 0.11 + $1,000 / (1 + 0.11)^16
Calculating this equation will give you the highest price you would be willing to pay.
b. To calculate the yield to maturity (YTM), you need to find the rate that equates the current market price ($872) with the present value of the bond's future cash flows.
You can use trial and error or financial calculators/software to find the YTM, which is the discount rate (or yield) that makes the present value of the bond's cash flows equal to its current market price.
c. To calculate the yield to call (YTC), you need to find the rate that equates the call price to the present value of the bond's future cash flows, assuming the bond is called in three years.
The call price is the face value + call premium. In this case, the call premium is 4% of the face value, so the call price would be $1,000 + (0.04 x $1,000) = $1,040.
You can then use trial and error or financial calculators/software to find the YTC, which is the discount rate (or yield) that makes the present value of the bond's cash flows equal to the call price ($1,040) when called in three years.
a. To determine the highest price you would be willing to pay for these bonds, you need to calculate the present value of the future cash flows using the required rate of return of 11%. The cash flows in this case are the semiannual coupon payments of $47.50 (9.5% of $1,000 divided by 2) and the face value of $1,000 at maturity.
The formula to calculate the present value of a bond is:
P = C * (1 - (1 + r)^(-n)) / r + F / (1 + r)^n
Where:
P = Present value of the bond
C = Coupon payment per period
r = Required rate of return
n = Number of periods
F = Face value of the bond
Using the given values:
C = $47.50
r = 11% or 0.11 (expressed as a decimal)
n = 16 periods (8 years * 2 semiannual periods per year)
F = $1,000
Plugging these values into the formula, you can calculate the present value (P):
P = $47.50 * (1 - (1 + 0.11)^(-16)) / 0.11 + $1,000 / (1 + 0.11)^16
Calculate these expressions within the brackets first, then evaluate the rest of the expression. The result will give you the highest price you would be willing to pay.
b. To calculate the yield to maturity (YTM) on the bonds, you need to solve for the discount rate (YTM) that equates the present value of the cash flows to the current price of $872.
Using the same formula as above, instead of solving for the present value (P), you'll solve for the yield to maturity (r). The formula becomes:
$872 = $47.50 * (1 - (1 + r)^(-16)) / r + $1,000 / (1 + r)^16
Rearrange the equation to solve for r. Since this is a complex equation, you can use numerical methods, such as trial and error or the Newton-Raphson method, to find the yield to maturity that satisfies this equation.
c. To calculate the yield to call (YTC) on the bonds if they can be called in three years with a call premium of 4% of the face value, you follow a similar approach as in part b.
You'll use the formula to calculate the present value of the cash flows until the call date (3 years) and then add the call premium to the final cash flow. The formula for the yield to call is a variation of the YTM formula, with the present value (P) equal to the current price ($872) and the number of periods (n) equal to 6 (3 years * 2 semiannual periods per year).
$872 = $47.50 * (1 - (1 + r)^(-6)) / r + ($1,000 + (0.04 * $1,000)) / (1 + r)^6
Solve for the yield to call (r) using numerical methods similar to those mentioned in part b.