Question

Sof ́ıa saves money for retirement. She deposits $150 on the first day of every month (starting today) for 30 years in a saving account. Altogether, 360 investments. She plans to retire after 30 years and from that time on she does not invest money anymore, and rather she plans to withdraw a fixed amount of money $Q every month (starting on the first day of the 361st month) for 40 years. Altogether, 480 withdrawals. Assume that the annual interest rate r=4% is compounded monthly.

(a) Find the monthly effective interest rate.

(b) How much money does she have in her account immediately after the fifth deposit?

(c) How much money does she have in her account immediately after the last deposit?

(d) Find Q such that on the time of the last withdrawal her balance is zero for the first time.

Answer 1

a) Monthly effective interest rate = 4%/12 = 0.3333% = 0.003333

b) Immediately after the 5th deposit,

Balance in account = Future value of deposits

=150*(1+0.003333)^4+150*(1+0.003333)^3+150*(1+0.003333)^2+150*(1+0.003333)^1+150

= $755.02

c) Immediately after the last deposit (360 deposits)

Amount = 150*1.0033^359+150*1.0033^358+...+150*1.0033+150

=150*1.0033^359*(1-1/1.0033^360)/(1-1/1.0033)

=$104107.41

d) For balance to be zero at the end of 480 withdrawals (starting one month after the last deposit)

Present value of withdrawals = $104107.41

=> Q/0.003333*(1-1/1.003333^480) = 104107.41

=> 239.2697*Q = 104107.41

=> Q = 435.10

So, the withdrawal amount is $435.10 each month