Question

A corporation creates a sinking fund in order to have $360,000 to replave in some machinery in 10 years. How much should be placed in this account at the end of each quarter if the annual interest rate is 8.6% compounded quarterly? (Round your answer to the nearest cent.)

2.) How much interest would they earn over the life of the account?

3.) Determine the value of the fund aftwr 2, 4, 6 years.

4.) How much interest was earned during the third quarter of the 5th year?

Answer 1

Approximately $3,139.95 should be placed in the sinking fund at the end of each quarter.

Step-by-step explanation:

To calculate the amount that should be placed in the sinking fund at the end of each quarter, we can use the formula for the future value of an annuity:

FV = PMT x (((1 + r/n)^(n*t)) - 1) / (r/n)

Where:
FV = future value of the annuity (360,000)
PMT = payment per period
r = annual interest rate (0.086)
n = number of compounding periods per year (4 quarters)
t = number of years (10)

By plugging in the values, we get:
360,000 = PMT x (((1 + 0.086/4)^(4*10)) - 1) / (0.086/4)

Solving for PMT, we find that approximately $3,139.95 should be placed in the account at the end of each quarter.

Answer 2

To create a sinking fund that amasses $360,000 with 8.6% annual interest compounded quarterly, a specific quarterly payment is required. The total interest earned is the difference between the final amount and the sum of all quarterly payments. The value of the fund after 2, 4, and 6 years is calculated by using the future value formula for each period, and the interest earned for any quarter is the change in fund value for that quarter.

Step-by-step explanation:

To solve how much should be placed in the sinking fund quarterly, we can use the future value of an annuity formula for compound interest. Since the annual interest rate is 8.6%, compounded quarterly, the quarterly interest rate is 8.6% / 4 = 2.15%.

We need to find the regular payment (R) for a future value (FV) of $360,000, the number of periods (n) which is 10 years x 4 quarters/year = 40 quarters, and the interest rate per period (i) which is 2.15% or 0.0215 per quarter.

The future value of an ordinary annuity formula is FV = R * [(1 + i)^n - 1] / i. Plugging in the values, we solve for R as follows:

R = $360,000 / [(1 + 0.0215)^40 - 1] / 0.0215

After calculation, the quarterly payment will be a certain amount, rounded to the nearest cent.

Calculating the total interest earned over the life of the account involves subtracting the total amount deposited (R * n) from the future value ($360,000).

To determine the value of the fund after 2, 4, and 6 years, we would use the same future value formula but with n reflecting 8, 16, and 24 quarters, respectively. Calculate separately for each period.

For the interest earned during the third quarter of the 5th year, we would need to calculate the value of the fund at the beginning and end of that quarter and then subtract the fund's value at the beginning of the quarter from its value at the end of the quarter, subtracting the quarterly payment as well.