Question

Following her 18th birthday, Madison began investing $41 at the end of each week in an account earning 5% per year. She plans to continue making weekly investments until she turns 68. Instead, if she hadn't started investing until she turned 56, how much would she have had to invest each week in order to have the same retirement nest egg at age 68

Answer 1

Madison will invest during 40 years, since we are not told if the account capitalizes interest monthly, weekly or annually, I will assume the interests are capitalized weekly:

total number of payments = 40 x 52 = 2,080

weekly interest rate =5% / 52 = 0.09615%

future value annuity factor = [(1 + 0.09615%)²⁰⁸⁰ - 1 ] / 0.09615% = 6,637.2376

Madison will have $41 x 6,637.2376 = $272,126.74 by the time she is 68.

If instead she starts to save when she is 56, she will only make 12 x 52 = 624 deposits

we know the future value = $272,126.74

the future value annuity factor = [(1 + 0.09615%)⁶²⁴ - 1 ] / 0.09615% = 854.4573

weekly deposit = $272,126.74 / 854.4573 = $318.48