Question

Today is Rachel’s 30^th birthday. Five years ago, Rachel opened a brokerage account when her grandmother gave her $25,000 for her 25^th birthday. Rachel added $2,000 to this account on her 26^th birthday, $3,000 on her 27^th birthday, $4,000 on her 28^th birthday, and $5,000 on her 29^th birthday. Rachel’s goal is to have $400,000 in the account by her 40^th birthday. Starting today, she plans to contribute a fixed amount to the account each year on her birthday. She will make 11 contributions, the first one will occur today, and the final contribution will occur on her 40^th birthday. Complicating things somewhat is the fact that Rachel plans to withdraw $20,000 from the account on her 35^th birthday to finance the down payment on a home. How large does each of these 11 contributions have to be for Rachel to reach her goal? Assume the account has earned (and will continue to earn) an effective return of 12% a year.

Answer 1

Interest rate = 12%

Present Value of all deposits made till 30th birthday = 25000(1+0.12)⁵ + 2000(1+0.12)⁴ + 3000(1+0.12)³ + 4000(1+0.12)² + 5000(1+0.12) = $62037.96

Let the 11 deposits made for next 11 years be $x/year

Future Value of these 11 payments =  P [((1 + r)ⁿ - 1) / r])(1 + r) = x[((1.12)¹¹ - 1) / 0.12])(1.12) = 23.13x

Value at 40 years for deposits made till 30 years = 62037.96*1.12¹¹ = 215802.15

Value at 40 years for withdrawal made at 35th Birthday = 20000*1.12⁵ = 35246.83

Hence, FV of all deposits and withdrawals at 40th Birthday = 215802.15 + 23.13x - 35246.83

This is equal to the goal required of 400000

=> 215802.15 + 23.13x - 35246.83 = 400000

=> x = 9487.45

Hence, Rachel should make a deposit of $9487.45 for 11 years to achieve the target of $400,000 at her 40th birthday