Question

An Individual Retirement Account (IRA) is an annuity that is set up to save for retirement. IRAs differ from TDAs in that an IRA allows the participant to contribute money whenever he or she wants, whereas a TDA requires the participant to have a specific amount deducted from each of his or her paychecks. When Shannon Pegnim was 14, she got an after-school job at a local pet shop. Her parents told her that if she put some of her earnings into an IRA, they would contribute an equal amount to her IRA. That year and every year thereafter, she deposited $500 into her IRA. When she became 25 years old, her parents stopped contributing, but Shannon increased her annual deposit to $1,000 and continued depositing that amount annually until she retired at age 65. Her IRA paid 8.5% interest. If Shannon Pegnim had started her IRA at age 35 rather than age 14, how big of an annual contribution would she have had to have made to have the same amount saved at age 65? (Round your answer to the nearest cent.) $

Answer 1

The future value of an annuity A over period N at interest rate of R is given by: A / R x [(1 + R)^N - 1]

A = $ 1,000 (Initially it was $ 500 each from Shannon and her parents, subsequently it was from her only; people making contribution changed, but the amount remained as same i.e. $ 1,000 per period)

R = 8.5%

N = 65 years - 14 years = 51 years

Hence, the future value of the account = A / R x [(1 + R)^N - 1] = 1,000 / 8.5% x [(1 + 8.5%)⁵¹ - 1] = $ 742,454.73

If Shannon had waited until 35

T = 65 - 35 = 30 years

Let's assume she has to contribute an amount P over T years to get the same FV then,

the future value of the account = P / R x [(1 + R)^T - 1] = P / 8.5% x [(1 + 8.5%)³⁰ - 1] =  124.21P = FV = $ 742,454.73

Hence, her annual contribution should be, P = $ 742,454.73 / 124.21 = $ 5,977.19 (please round it off as per your requirement)