Traditional IRA ADVANCED ALGEBRA

Question

Yon made contributions to a Traditional IRA over the course of 35 working years. Her contributions averaged $2,000 annually. Yon was in the 28% tax bracket during her working years. The average annual rate of return on the account was 6%. Upon retirement, Yon stopped working and making IRA contribution. Instead, she started living on withdrawals from the retirement account. At this point, Yon dropped into the 15% tax bracket. Factoring in taxes, what is the effective value of Yon’s Traditional IRA at retirement? Assume annual compounding.

Answer 1

Josephine, here's how to tackle this.

Note that the question asks for the amount of Yon's account at retirement, so any information about the time after Yon retires is IRRELEVANT!

We need the future value (FV) at retirement of all payments, made over 35 years, net of tax.

The rate of return on the IRA is 6% py, less 28% tax => the real rate of return is 6% x (1 - 0.28) = 4.32%. This is the annual rate, i, to use.

The question does not state whether Yon made payment at the beginning or end of each year. We assume the beginning.

Amount of IRA after the 35th payment = 2000((1+i)¹ + (1+i)² + (1+i)³ + . . . + (1+i)³⁵)

Inside the parentheses is a geometric series with first term and common ratio both = (1+i)

∴ Amount at the end = 2000 (1+i) ((1+i)³⁵ - 1) / ((1+i) - 1)

= 2000 x 1.0432 (1.0432³⁵ - 1) / 0.0432

= $164,776.23

This is Yon's retirement "nest egg."