Question

a) You invest 10000$ in a savings account today at j1 = 5%. At the end of each year you withdraw exactly 500$ from the account. What is the account balance at the end of year 15?

b) Let us assume that instead of withdrawing 500$ at the end of each year, you withdraw 500$ at the end of year 1, 550$ at the end of year 2, 600$ at the end of year 3, and so on (increasing the size of the annual withdrawal by 50$ each year with respect to the previous year). What is the account balance at the end of year 10?

c) In which year does the account balance reach zero? Use linear interpolation to solve, using 15years & 20years to set up the interpolation.

Answer 1

a) The balance at the end of 15th year can be calculated from the difference between future value of savings account balance $10000 which is invested for 15 years and an annuity of $500 annual withdrawals

Future value of $10000 = 10000 * 1.05¹⁵ =$20789.28

Future value of annuity of $500 = 500/0.05 * (1.05¹⁵-1) = $10789.28

So, account balance at the end of 15th year = $20789.28 - $10789.28 = $10000

This is because only the interest is withdrawn at the end of each year leaving the principal intact

b) In this case,

Future value of $10000 after 10 years = 10000 * 1.05¹⁰ =$16288.95

Future value of growing annuity of $500,$550 and so on

=  500* 1.05⁹ + 550 *1.05⁸ + .... 950*1.05⁰

= $8866.84

So, account balance at the end of 10th year = $16288.95 - $8866.84 = $7422.107

c) Similarly,we can calculate account balance at end of 15 years and 20 years

Future value of $10000 after 15 years = 10000 * 1.05¹⁵ =$20789.28

Future value of growing annuity of $500,$550 and so on for 15 years

=  500* 1.05¹⁴ + 550 *1.05¹³ + .... 1200*1.05⁰

= $17367.85

So, account balance at the end of 15th year = $20789.28 - $17367.85 = $3421.44

Future value of $10000 after 20 years = 10000 * 1.05²⁰ =$26532.98

Future value of growing annuity of $500,$550 and so on for 20 years

=  500* 1.05¹⁹ + 550 *1.05¹⁸ + .... 1450*1.05⁰

= $29598.93

So, account balance at the end of 20th year = $26532.98 - $29598.93 =   - $3065.95

By Linear Interpolation Year in which account balance reaches zero

= 15 + ( 3421.44-0)/(3421.44- (-3065.95)) * (20-15)

=15+ 2.64

=17.64

So, the account balance reaches Zero in the 18th year.