Question

Mary has created a college trust fund for her son when he turned 8 by making equal deposits every 6 months for 10 years and leaving money in the account afterwards. At age 21, her son started receiving $4000 at the beginning of every quarter for 4 years, leaving 0 balance in the account after the last payment.
a. find the amount of semi annual deposits of money earning 6.5% interest compounded monthly
b. what was the total interest accumulated during the 17 years?

Answer 1

Answer a

During Deposits:

No. of payments = 10*2=20

Amount = A (say)

Rate=6.5% compounded monthly

Annual Effective Interest Rate = (1+6.5%/2)^2-1 = (1+3.25%)^2-1=1.0325^2-1 = 1.0661-1=0.0661 or 6.61%

Semi annual effective Rate =6.61%/2 = 3.3%

Hence FV of deposits at end of deposits (i.e at age 18) = A*((1+r)^n-1)/r

=A*((1+3.3%)^20-1)/3.3%

=A*((1.033)^20-1)/0.033

=A*(1.9153-1)/0.033

=A*0.9153/0.033

=$27.74A

FV of deposits at Age 21 (i.e after 3 years) = 27.74A*(1+6.61%)^3 = 27.74A*1.0661^3=27.74*1.2117=$33.61A

During Withdrawals

Amount=$4000

No of withdrawal=4*4=16

Effective quarterly Rate = 6.61%/4 = 1.6525%

PV of withdrawals = 4000+PV of next 15 withdrawals

PV of last 15 withdrawal will be A*(1-(1+r)^-n)/r

=4000*(1-(1+1.6525%)^-15)/1.6525%

=4000*(1-1.016525^-15)/0.016526

=4000*(1-0.7829)/0.016525

=4000*0.2171/0.016525

=$52758.82

Hence PV of withdrawal = 4000+52758.81 = $56758.81

Since PV=FV for account balance to be zero

33.61A =55758.81

or, A = 54758.81/33.61 = $1688.75

Hence Seminannual deposits = $1688.75

Answer b

Total Deposits = 1688.75*20 =$33775

Total Withdrawal = 4000*16 = $64000

Interest accumulated = 64000-33775 = $30225