Question

Jeffrey deposited $18,000 into a fund at the beginning of every quarter for 13 years. She then stopped making deposits into the fund and allowed the investment to grow for 4 more years. The fund was growing at 4.90% compounded monthly.

a. What was the accumulated value of the fund at the end of year 13?
(Round to the nearest cent)

b. What was the accumulated value of the fund at the end of year 17?
(Round to the nearest cent)

c. What is the total amount of interest earned over the 17-year period?
(Round to the nearest cent)

Answer 1

Explanation:

To solve this problem, we can use the formula for the future value of an annuity:

FV = PMT * [(1 + r/n)^(nt) - 1] / (r/n)

where FV is the future value, PMT is the periodic payment, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.

a. At the end of year 13, Jeffrey made 13 * 4 = 52 quarterly deposits of $18,000 each. The annual interest rate is 4.90%, so the monthly interest rate is 4.90% / 12 = 0.4083%. The number of compounding periods per quarter is 3, so we have:

FV = 18000 * [(1 + 0.0490/3)^(3*13) - 1] / (0.0490/3)

= $856,167.46

Therefore, the accumulated value of the fund at the end of year 13 is $856,167.46.

b. At the end of year 17, Jeffrey made 17 * 4 = 68 quarterly deposits of $18,000 each. We can use the same formula, but with t = 4 and the new future value as the starting point:

FV = 856167.46 * (1 + 0.0490/12)^(12*4)

= $1,217,856.61

Therefore, the accumulated value of the fund at the end of year 17 is $1,217,856.61.

c. The total amount of interest earned over the 17-year period is the difference between the accumulated value at the end of year 17 and the total amount of deposits made:

Total interest = 18000 * 68 * 4 - 1217856.61

= $253,143.39

Therefore, the total amount of interest earned over the 17-year period is $253,143.39.