Question

Assume that seven months from today you plan to make the first of a series of semiannual deposits into an account that pays an APR of 6.5% with monthly compounding. Your first deposit will equal $300 and your final deposit will occur four years and seven months from today. Each deposit will be 0.5% larger than the previous one. Five years and three months from today, you plan to make the first of a series of annual withdrawals from an account. You will continue to make withdrawals through nine years and three months from today. Each withdrawal will be 1.5% smaller than the previous one. How large can you make your final withdrawal?

Note: You don’t need to solve anything. Just set everything up. Setting everything up means writing down all relevant equations and plugging in all possible numbers. If in some step you are using a value from a previous step, make it clear what you are doing. If you are solving for something other than the left-hand side of the equation, tell me what you are solving for.

Answer 1

The entire situation is assumed to be after seven months. Hence my first deposit will occur after seven months.
It is given that the final deposit will occur four years and seven months from today, which indicates that the deposit will be for four years of $300 semi-annual deposits larger than the previous one by 0.5%.

APR = 6.5%, so semi annual interest rate = 3.25%

Formula for Interest(I) = Beginning Balance(B) x Periodic rate of Interest (r)
Ending Balance (E) = Beginning Balance(B) + Interest(I)

Ending Balance of Year 0.5 = B + B x r/2 = E1
300.00 + 300x6.5%/2 = 309.75
Ending Balance of Year 1 = (E1+Bx(1.005)) + (E1+Bx(1.005)) x r/2 = E2
(309.75+300x1.005) + (309.75+300x1.005)x6.5%/2 = 631.12
Ending Balance of Year 1.5 = (E2+Bx(1.005)^2) + (E2+Bx(1.005)^2) x r/2 = E3
(631.12+300x1.005^2) + (631.12+300x1.005^2)x6.5%/2 = 964.48
Ending Balance of Year 2 = (E3+Bx(1.005)^3) + (E3+Bx(1.005)^3) x r/2 = E4
(964.48+300x1.005^3) + (964.48+300x1.005^3)x6.5%/2 = 1310.25
Following the same equation set up for the next 2 years, we will get the ending values for every deposits.

On solving, total deposited amount after 4 years = $2,825.71

I will start to withdraw money from this account after five years and three months from today i.e. four years and eight months after my first deposit.
So, the withdrawal will start eight months after the last deposit.

Therefore, the account will pay APR of 6.5% for eight more months,
$2,825.71 x 6.5% x 8/12 = $122.45
Hence the total amount becomes = $2,825.71 + $122.45 = $2,948.16

Now the withdrawals will continue till nine years and three months from today i.e. total of 5 withdrawals starting after five years and three months from today.

Each withdrawals will be 1.5% smaller than the previous one i.e. 98.5% of the previous withdrawals.
Let the first withdrawal be $W.
Second withdrawal will be $W x 98.5%.
Third withdrawal will be $W x 98.5%^2.
Fourth withdrawal will be $W x 98.5%^3.
Fifth withdrawal will be $W x 98.5%^4.
So,
Final Equation is
$W + $W x 98.5% + $W x 98.5%^2 + $W x 98.5%^3 + $W x 98.5%^4 = $2,948.16 (Answer)
Now solve for $W to get the final withdrawal amount.