John decides to invest 140 000 in an account earning 13.5% interest

Question

John decides to invest 140 000 into an account earning 13.5% interest per year compounded quaterly. This new account allows him to withdraw an amount every quarter for 10 years after which time the account will be exhausted. The amount of money that John can withdraw every quarter equals? 1) 1704.28 2) 3500 3) 6429.28 4) 8594.82 5) None of the above

Answer 1

13.5% annually is 13.5/4=3.375% quarterly, so r is 0.03375. The compounding factor is 1.03375, (1+r). The period, T, is 10 years or 40 quarters. The principal is 140000, P.

If the quarterly withdrawal=Q, then after three months P, P grows to P(1+r) and Q is taken out leaving P(1+r)-Q. After 6 months, The amount grows to (P(1+r)-Q)(1+r) and Q is taken out, leaving P(1+r)^2-Q(1+r)-Q. After 9 months the amount after withdrawal is P(1+r)^3-Q(1+r)^2-Q(1+r)-Q, and after 4T periods, the end of the investment period, the amount after withdrawal is zero:

P(1+r)^(4T)-Q(1+r)^(4T-1)-Q(1+r)^(4T-2)-...-Q(1+r)-Q=0.

So P(1+r)^(4T)-Q(1+r)^(4T-1)+...+1)=0, and Q=P(1+r)^(4T)/S, where S is the sum of the series.

(1+r)S-S=((1+r)^(4T)-1) and S=((1+r)^(4T)-1)/r=(1.03375^40-1)/0.03375=82.146.

Q=140000*3.7724275/82.146=6429.28 (answer 3).