Question

Assume that two months from today you plan to make the first of a series of quarterly deposits into an account that pays an APR of 6.5% with monthly compounding. Your first deposit will equal $100 and your final deposit will occur two years and five months from today. Each deposit will be 1.5% smaller than the previous one. Three years and seven months from today, you plan to make the first of a series of semiannual withdrawals from an account. You will continue to make withdrawals through five years and one month from today. Each withdrawal will be 2.5% larger than the previous one. How large can you make your final withdrawal?

Answer 1

The Annual Percentage Rate (APR) for the deposit = 6.5%

Monthly Interest Rate for the deposit = APR/12 = 0.5417%

$100 is deposited after 2 months, deposits reduce by 1.5% every quarter and deposits continue till 2 years and 5 months from today (29 months from today)

The schedule for deposits is given below:

The deposit for second month is calculated as 100X(1-1.5%) = 98.5

Deposits for subsequent quarters are similarly calculated

Growing annuity formula can be used to calculate the future value of these deposits after 29 months

Future Value = P X [ (1+r)ⁿ-(1+g)ⁿ]/(r-g)

Where P is the first payment, r is interest rate, g is growth rate and n is number of periods

Here, P =$100

r = quarterly interest rate = [(1+monthly interest rate)^3]-1 = [(1+0.5417%)^3]-1 = 1.6338%

g = quarterly growth rate = -1.5% (here growth rate will be negative as the deposits actually reduce rather than grow)

n = number of quarters for which deposits have been made = 10

Future Value of Deposits after 29 months = 100 X [ (1+1.6338%)¹⁰-(1-1.5%)¹⁰]/(1.6338% - (-1.5%))

= 1008.999546

No further deposits are made and these deposits grow for another 14 months (till 3 years and 7 months or 43 months) till the first withdrawal is made

Future value of deposits after 43 months = Future Value of Deposits after 29 months X (1+monthly interest rate)¹⁴

= 1008.999546 X (1+0.5417%)^{14 =} 1088.268588

Now let us assume that the first withdrawal made is w

The subsequent withdrawals will be as follows

second withdrawal = first withdrawal X (1+2.5%) and so on

The present value of these withdrawals at month 43 can be calculated as shown below

Discount Factor = 1/ (1+monthly interest rate)ⁿ

where n is the number of months from 43 month to the withdrawal month

Discounted Withdrawal = Withdrawal X Discount Factor

Present value of all withdrawals at 43 month, PV = sum of discounted withdrawals

= 3.954096w

The present value of withdrawals at 43 month and the future value of deposits at 43 months will be equal if after final withdrawal there is no amount remaining in the deposit

PV of all withdrawals = FV of all deposits

3.954096w = 1088.268588

w = 275.2256148

first withdrawal = 275.2256148

final withdrawal = 1.076890625w = 296.3878843

So the maximum final withdrawal that can be made is 296.3879