Midland Oil has $1,000 par value bonds outstanding at 8 percent interest. The bonds will mature in 25 years. Compute the current price of the bonds if the present yield to maturity is:
A. 7 percent.
B. 10 percent.
C. 13 percent.
To compute the current price of the bonds, we need to calculate the present value of the bond's cash flows, which includes the periodic interest payments (coupons) and the face value (par value) at maturity.
The bond pays annual interest of 8% of the $1,000 par value, which is equal to $80 per year. Given the current yields to maturity, we can calculate the present value of these coupon payments using the following formula:
Present Value of Coupons = C * (1 - (1 + YTM) ^ -N) / YTM,
where C is the coupon payment, YTM is the yield to maturity, and N is the number of years to maturity.
We also need to calculate the present value of the face value at maturity using the following formula:
Present Value of Face Value = M / (1 + YTM) ^ N,
where M is the face value (par value) of the bonds.
Finally, we add these two present values together to determine the current bond price.
Here are the calculations for each yield to maturity:
A. 7% yield to maturity:
Present Value of Coupons = $80 * (1 - (1 + 0.07) ^ -25) / 0.07 = $832.41
Present Value of Face Value = $1,000 / (1 + 0.07) ^ 25 = $349.41
Current Bond Price = $832.41 + $349.41 = $1,181.82
B. 10% yield to maturity:
Present Value of Coupons = $80 * (1 - (1 + 0.10) ^ -25) / 0.10 = $611.66
Present Value of Face Value = $1,000 / (1 + 0.10) ^ 25 = $228.30
Current Bond Price = $611.66 + $228.30 = $839.96
C. 13% yield to maturity:
Present Value of Coupons = $80 * (1 - (1 + 0.13) ^ -25) / 0.13 = $441.41
Present Value of Face Value = $1,000 / (1 + 0.13) ^ 25 = $111.27
Current Bond Price = $441.41 + $111.27 = $552.68
So, the current bond prices for different yields to maturity are:
A. 7% yield to maturity: $1,181.82
B. 10% yield to maturity: $839.96
C. 13% yield to maturity: $552.68
To compute the current price of the bonds, we can use the present value formula. The formula for calculating the present value of a bond is as follows:
P = (C / (1 + r)^1) + (C / (1 + r)^2) + ... + (C / (1 + r)^n) + (M / (1 + r)^n)
Where:
P = Present value of the bond
C = Annual coupon payment
r = Yield to maturity
n = Number of periods
M = Face value of the bond
Let's compute the current price of the bonds for each given yield to maturity:
A. 7 percent yield to maturity:
P = (80 / (1 + 0.07)^1) + (80 / (1 + 0.07)^2) + ... + (80 / (1 + 0.07)^25) + (1000 / (1 + 0.07)^25)
B. 10 percent yield to maturity:
P = (80 / (1 + 0.10)^1) + (80 / (1 + 0.10)^2) + ... + (80 / (1 + 0.10)^25) + (1000 / (1 + 0.10)^25)
C. 13 percent yield to maturity:
P = (80 / (1 + 0.13)^1) + (80 / (1 + 0.13)^2) + ... + (80 / (1 + 0.13)^25) + (1000 / (1 + 0.13)^25)
Let's calculate the values for each yield to maturity:
To compute the current price of the bonds, we need to use the present value formula for bond valuation. The formula is as follows:
PV = C * (1 - (1 + r)^(-n)) / r + F / (1 + r)^n
where:
PV = present value (current price) of the bonds
C = coupon payment (interest payment) per period
r = yield to maturity (interest rate)
n = number of periods (number of years until maturity)
F = face value (par value) of the bond
In this case, the par value (face value) of the bond is $1,000, the coupon payment is 8% of the par value ($1,000 * 0.08 = $80), and the maturity is 25 years.
A. For a yield to maturity of 7%:
r = 0.07
n = 25
C = $80
F = $1,000
Substituting the values into the formula:
PV = $80 * (1 - (1 + 0.07)^(-25)) / 0.07 + $1,000 / (1 + 0.07)^25
B. For a yield to maturity of 10%:
r = 0.10
n = 25
C = $80
F = $1,000
Substituting the values into the formula:
PV = $80 * (1 - (1 + 0.10)^(-25)) / 0.10 + $1,000 / (1 + 0.10)^25
C. For a yield to maturity of 13%:
r = 0.13
n = 25
C = $80
F = $1,000
Substituting the values into the formula:
PV = $80 * (1 - (1 + 0.13)^(-25)) / 0.13 + $1,000 / (1 + 0.13)^25
By plugging in the respective values, you can use a calculator or spreadsheet software to calculate the current price (present value) of the bonds for each yield to maturity scenario.