Question

What amount must be deposited in an account that earns 10% on January 1, 2017 if an investor wants to withdraw $5,000 January 1 of each year beginning on January 1, 2018 with the last withdrawal being made on January 1, 2021? What if the withdrawals take place on December 31 of each year starting on December 31, 2017 until December 31, 2020? What if withdrawals of $2,500 occur every six months on June 30 and Dec 31 of each year starting on June 30, 2017 until December 31, 2020?

Answer 1

Solution 1:

Amount to be withdrawn from 01.01.2018 for 4 years = $5,000

Therefore required balance to be maintained in account on 01.01.2018 = Present value of annuity due for amount to be withdrawn discounted at 10% for 4 periods

$5,000 * Cumulative PV factor at 10% for annuity due for 4 periods

= $5,000 * 3.486852

= $17,434

Required balance on 01.01.2018 (Future value) = $17,434

Therefore amount to be deposited on Jan 1 2017 in order to have $17,434 after 1 year = $17,434 / 1.10 = $15,849

Solution 2:

It will not have any impact on amount to deposited on January 1, 2017 and withdrawl is only 1 day before or on the end date of every year. Therefore amount to be deposited on January 1 2017 will remain tge same i.e. $15,849

Solution 3:

Amount to be withdrawn from 30.06.2017 for 8 semiannual period = $2,500

Semiannual interest rate = 10%/2 = 5%

Therefore required balance to be maintained in account on 30.06.2017 = Present value of annuity due for amount to be withdrawn discounted at 5% for 8 periods

$2,500 * Cumulative PV factor at 5% for annuity due for 8 periods

= $2,500 * 6.786373

= $16,966

Required balance on 30.06.2017 (Future value) = $16,966

Therefore amount to be deposited on Jan 1 2017 in order to have $16,966 after 6 months = $16,966 / 1.05 = $16,158