How much will each withdrawal be in the following case:
Manu's family has decided that in three months, they will start depositing $ 160 every three months for three years. The account will earn interest 5.7% per annum, compounded quarterly. Then, three months after the last deposit, they plan to withdraw money every three months for equal 24 payments for music lessons
Manu's family plans to deposit $160 every three months for three years starting after 3 months. The account in which the deposit is made gives a return of 5.7% per annum compounded quarterly.
The total amount in the account after 3 years can be determined using the formula for compound interest with the interest rate for each term equal to `(5.7%)/4 = 0.01425` . There are 11 deposits in all after 3 months each.
After 3 years the amount in the account is equal to:
`160*(1.01425^10 + 1.01425^9 +...1) `
= `160*1*(1.01425^11 - 1)/(1.01425 - 1)`
= `160*(1.01425^11 - 1)/0.01425`
After 3 months, this increases to $1917.86.
Now, the funds are withdrawn in 24 equal payments each after 3 months. Let the payment be denoted by P. As the account continues to earn 5.7% p.a compounded quarterly, the withdrawals have to be discounted at the same rate. This gives:
`P + P/1.01425 + ... + P/1.01425^23 = 1917.86`
=> `P*(1 - 1/1.01425^24)/(1 - 1/1.01425) = 1917.86`
=> `P ~~ 93.58`
Each of the 24 withdrawals is equal to $93.58