Question

4) Nandana invests $500 at the start of each year for 20 years in a bank account paying interest at the effective annual rate i. She takes the interest paid at the end of each year and invests it in a different account paying an effective annual rate i/2. The effective annual rate she earns on her combined investments is 6% a) How much money does she have at the end of 20 years? (Total of both accounts.) b) What is i?

Answer 1

a) As her investments earn an effective annual return of 6%, on her combined investments, the amount of money that she will have at the end of the 20 years is the same amount if her investment were earning 6% and she had not taken out any money

So, Amount at end of 20 years = FV of all the installments paid

=500 * (1.06)²⁰ + 500 * (1.06)¹⁹+500 * (1.06)¹⁸+....+500 * (1.06)¹

= 500 *1.06 * (1.06²⁰ -1) /(1.06-1)

= $19496.36

b) The interest amount at the end of the first year is 500*i  (as this interest is withdrawn there will be no interest on the same)

The interest amount at the end of the second year is 1000*i

and so on till

The interest amount at the end of the 20th year is 10000*i

The amount in the bank account in which installments are paid at the end of 20 years will only be $500 deposited each year for 20 years as only principal will remain in this account

=500 +500 + ... upto 20 times

= 500*20 = $10000

So , remaining amount of $9496.36 should have come from the Second account in which interest was reinvested at the rate of i/2

Therefore, amount in the 2nd account will be the future value of all the interest investments 500*i , 1000*i, .... , 10000* i

Now, 9496.36 = 500 * i * (1+i/2)¹⁹ + 1000 *i * (1+i/2)¹⁸+...... +10000 * i

Solving for i by Hit and Trial Method, we get

i = 7.1373% = 7.14%