A company requires the amount of $1,000,000 in 25 years to retire a bond issue. Assume they can earn 5 3/4% interest compounded daily. What amount would they have to pay quarterly to be able to retire this debt in 25 years?
To calculate the amount the company would have to pay quarterly to retire the bond issue in 25 years, we can use the formula for calculating the future value of an ordinary annuity.
The formula for the future value of an ordinary annuity is:
FV = P * [(1 + r/n)^(n*t) - 1] / (r/n)
Where:
FV = Future Value
P = Payment amount per period
r = Interest rate per period
n = Number of compounding periods per year
t = Number of years
In this case, the future value (FV) is $1,000,000, the interest rate (r) is 5 3/4% or 0.0575, the number of compounding periods per year (n) is 365 (since it's compounded daily), and the number of years (t) is 25.
We need to solve for the payment amount per period (P). Rearranging the formula, we have:
P = FV * (r/n) / [(1 + r/n)^(n*t) - 1]
Now let's substitute the values into the formula:
P = $1,000,000 * (0.0575/365) / [(1 + 0.0575/365)^(365*25) - 1]
Let's solve this equation step by step:
Step 1: Calculate the value inside the square brackets [(1 + 0.0575/365)^(365*25) - 1]
To simplify the calculation, let's use a calculator or a spreadsheet program. The result is approximately 230.0426.
P = $1,000,000 * (0.0575/365) / 230.0426
Step 2: Simplify the expression (0.0575/365)
0.0575/365 ≈ 0.0001575
P = $1,000,000 * 0.0001575 / 230.0426
Step 3: Calculate the value of P.
Let's calculate the numerator of the expression:
$1,000,000 * 0.0001575 = $157.50
Finally, divide the numerator by the denominator to get the value of P:
P ≈ $157.50 / 230.0426
P ≈ $0.6841
Therefore, the company would have to pay approximately $0.6841 million quarterly to be able to retire this debt in 25 years.