Question

Following her 18th birthday, Madison began investing $20 at the end of each week in an account earning 7% per year. She plans to continue making weekly investments until she turns 68. If she hadn't started investing until she turned 45, how much would she have to invest each week in order to have the same retirement nest egg at age 68? Round to the nearest cent. [Hint: Find the size of the retirement nest egg under the first long horizon scenario, then use that number to solve for CF under the short investment horizon scenario.]

Answer 1

FV of Annuity :
Annuity is series of cash flows that are deposited at regular intervals for specific period of time.

FV of Annuity = CF [ (1+r)^n - 1 ] / r
r - Int rate per period
n - No. of periods

Assuming 52 weeks per anum.

Particulars	Amount
Cash Flow	$           20.00
Int Rate	0.135%
Periods	2600

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 20 * [ [ ( 1 + 0.00135 ) ^ 2600 ] - 1 ] / 0.00135
= $ 20 * [ [ ( 1.001346 ) ^ 2600 ] - 1 ] / 0.001346
= $ 20 * [ [33.0376] - 1 ] / 0.001346
= $ 20 * [32.0376] /0.001346
= $ 475987.22

Weekly deposit if she starts from 45th Bday:

Particulars	Amount
FV of Annuity	$   4,75,987.22
Int Rate	0.1346%
Periods	1196

FV of Annuity = Cash Flow * [ [(1+r)^n ] - 1 ] /r
$475987.22 = Cash Flow * [ [ ( 1 + 0.0013 ) ^ 1196 ] - 1 ] / 0.0013
$475987.22 = Cash Flow * [ [ ( 1.0013 ) ^ 1196 ] - 1 ] / 0.0013
$475987.22 = Cash Flow * [ [ ( 4.9974 ] - 1 ] / 0.0013
$475987.22 = Cash Flow * [ 3.9974 ] / 0.0013
Cash Flow = $ 475987.22 * 0.0013 / 3.9974
Cash Flow = $ 160.29

Weekly deposit shall be made is $ 160.29