Find the amount of each payment to be made into a sinking fund when $75000 @ 6% compounded semiannually for 4.5 years.
To find the amount of each payment to be made into a sinking fund, we can use the formula for calculating the future value of an annuity.
The formula for calculating the future value of an annuity is:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future Value of the annuity
P = Payment amount
r = Interest rate per period
n = Number of periods
In this case, we need to solve for P, so we rearrange the formula:
P = FV * (r / ((1 + r)^n - 1))
Now let's plug in the values given in the question:
Principal (P) = $75000
Interest rate (r) = 6% (or 0.06) compounded semiannually
Number of periods (n) = 4.5 years
First, we need to convert the interest rate and periods to match the compounding frequency. Since the interest is compounded semiannually, we divide the interest rate by 2, and since the number of periods is given in years, we multiply it by 2.
So, the adjusted interest rate (r) = 0.06 / 2 = 0.03
And the adjusted number of periods (n) = 4.5 * 2 = 9
Now we can substitute the values into the formula:
P = $75000 * (0.03 / ((1 + 0.03)^9 - 1))
Calculating the expression inside the parentheses:
P = $75000 * (0.03 / (1.03^9 - 1))
Using a calculator to find the value of 1.03^9 - 1 ≈ 0.2958:
P = $75000 * (0.03 / 0.2958)
Simplifying the expression further:
P = $75000 * 0.10155
Calculating the value:
P ≈ $7,616.25
Therefore, the amount of each payment to be made into the sinking fund is approximately $7,616.25.