Question

James invests 2,000 at an effective annual interest rate of 17 percent for 10 years. Interest is payable annually and is reinvested at an effective annual rate of 11 percent. At the end of 10 years, James’ accumulated interest is 5,685.48 Peter invests 150 at the end of each year for 20 years at an effective annual interest rate of 14 percent. Interest is payable annually and is reinvested at an effective annual rate of 11 percent. Calculate Peter’s accumulated interest at the end of 20 years.

Answer 1

To find Peter's accumulated interest after 20 years, we calculate the future value of an annuity using the annual payment, the interest rate, and the number of years. Subtracting the total contributions from this amount yields the accumulated interest. Compound interest greatly increases the amount saved over time.

Step-by-step explanation:

To calculate Peter's accumulated interest at the end of 20 years, we use the formula for the future value of an annuity compounded annually. Peter invests $150 at an effective annual interest rate of 14 percent, which is reinvested at 11 percent annually. The formula for the future value of an annuity (FVA) is FVA = Pmt * [((1 + r)^n - 1) / r], where Pmt is the annual payment, r is the annual interest rate, and n is the number of years.

In this case, Pmt is $150, r is 0.14 initially and after reinvestment is 0.11, and n is 20. After calculating the FVA, we subtract the total amount of payments made over the 20 years to find the accumulated interest.

Let's break down the calculation step by step:

1. Calculate the FVA using the initial interest rate: FVAinitial = 150 * [((1 + 0.14)^20 - 1) / 0.14].
2. Calculate the total payments made: Total payments = 150 * 20 = $3000.
3. Subtract the total payments from the FVA to find the accumulated interest: Accumulated interest = FVA - Total payments.

Compound interest can have a significant impact on savings over time, as demonstrated in the examples given.