Question

Assume that four years and one month from today you plan to make the first ofseveralannual withdrawals from an account. Your first withdrawal will equal $1000. You plan for these withdrawals to continue through ten years and one month from today and for each withdrawal to be 0.5% larger than the previous withdrawal. You plan to fund these withdrawals by making a series of equal semiannual deposits into an account earning an APR of 8% with semiannual compounding. Your first deposit will occur one year and four months from today and your final deposit will occur three years andten months from today. How large must you make each deposit? Formula is required.

Answer 1

Total no of withdrawals starting from four years and one month from today till ten years and one month from today (both inclusive) =6*12+1=73

Monthly interest rate = (1+8%/2)^(1/6)-1 = 0.0065582

So, value of these withdrawals four years from today

= 1000/1.0065582+1000*1.005/1.0065582^2+.....+1000*1.005^72/1.0065582^73

From Sum of GP formula

=1000/1.0065582*(1-(1.005/1.0065582)^73)/(1-1.005/1.0065582)

=$68626.77

Six monthly rate = 8%/2 = 4% or 0.04

No of equal semiannual deposits = 6

So, size of semiannual deposit (A) is given by

Future value three years and ten months from today = A*1.04^5+....+A

=A/0.04*(1.04^6-1)

So, Value of these deposits 4 years from today = A/0.04*(1.04^6-1) *1.0065582^2

value of withdrawals four years from today =Value of deposits 4 years from today

=> A/0.04*(1.04^6-1) *1.0065582^2 = 68626.77

=> A* 6.720261465 = 68626.77

=> A = $10211.92

So, each deposit has to be $10211.92