Question

Grandma decides to put 1000 dollars every month into an account for you. She makes 25 monthly deposits, the last coming September 1, 2019 - the day you start college. She wants you to be able to withdraw money from this account at the beginning of each month, with the first withdrawal coming September 1, 2019 and the last coming June 1, 2024, (when you'll graduate). (Note: that makes 58 withdrawals total.) How much will you be able to withdraw each month if the account is earning a nominal interest rate of 9.3 percent convertible monthly?

Answer 1

Amount deposited by Grandma each month = X = 1000

Step 1

Number of Deposits = n = 25

Rate of return = r = 9.3%

Hence, Future Value on Sep 1 2019 = X(1+r)^n-1 +....+ X(1+r)² + X(1+r) + X = X[(1+r)ⁿ -1]/r = 1000[(1+0.093)²⁵ -1]/0.093 = $88563.74

Step 2

Let the amount withdrawn each month be P

Number of withdrawals = t = 58

Rate of return = r = 9.3%

The Present Value of these withdrawals will be equal to the Value calculated in Step 1 above

=> PV = 88563.74 = P + P/(1+r) +....+ P/(1+r)^n-1 =P + P[1- (1+r)^-(n-1)]/r = P + P[1- (1+0.093)^-57]/0.093

=> 88563.74 = 11.685P

=> P = $7579.27

Hence, she can withdraw 7579.27 from Sep 1 2019 to June 1 2024