Question

2.35 A father wants to set aside money for his 8-year-old daughter's future education, by making monthly deposits to a bank account that pays 8% per year, compounded annually. What equal monthly deposits must the father make-the first 1 month after her 9th birthday and the last on her 17th birthday-in order for her to withdraw $4000 on each of her next four birthdays (the 18th through the 21st)? Ans. $103.80

Please show step by step how to do with the formulas, thank you.

Answer 1

Annuity required after from 18th to 21st birthday 4000

Annuity period= 4 years

Interest rate = 8% or 0.08

We need to Calculate present Value of annuity to Calculate Amount that must have at the time of 17th birthday for Annuity withdrawl of $4000.

Present Value of annuity formula = Annuity *(1-(1/(1+i)^n))/i

4000*(1-(1/((1+0.08)^4)))/0.08

13248.51

So, We must have accumulated $13248.51 at the time of 17th birthday, that is Future value of annuity monthly contribution made by us.

Time in months from 9th to 17th birthday = (n) 8 years*12= 96

Interest rate is 8% comoounded annually. So effective Interest rate is 8% p.a. We have to Calculate monthly interest rate by following formula: ((1+ r))^no. of months)-1 = Effective rate per annum

((1+ r)^12)-1 = 0.08

((1+ r)^12)= 1.08

(1+ r)= (1.08)^(1/12)= 1.00643403

r or monthly interest rate= 0.006434

Future value of annuity formula = P *{ (1+r)^n - 1 } / r

13248.51 = P*(((1+0.006434)^96)-1)/0.006434

13248.51/ 132.2544132 =P

P = 100.1744365

So, We need to contribute $100.17 monthly.

This is exact answer by Calculation.