Question

An S&L pays 5% per annum compounded quarterly. Bert and Bertha are now 50 years old. They will deposit $200 per quarter at the end of each quarter until they are 65 years old. How much is in their retirement account at the end of this period i.e. at 65? 3 months after their last deposit they start withdrawing equal amounts each quarter until they are 80 i.e. for the next 15 years Find the size of the withdrawals.
How much cash did they deposit?
How much cash did they withdraw?
What was the total interest?

(I need a step-by-step explanation of the formulas used. Thanks)

Answer 1

How much cash did they deposit?

We can use following formula for Future value (FV) of ordinary annuity calculation (as the deposits are made at the end of quarter)

FV = PMT * [(1+i) ^n – 1] /i

Where FV = future value of deposits when they turn 65 years =?

PMT or deposit per quarter= $200

And i= I/Y = 5% is the interest rate per annum or 5%/4 = 1.25% per quarter

The time period n = (65 years – 50 years) = 15 years * 4 = 60 quarters

Therefore,

FV = $200 * [(1+ 1.25%) ^60 -1]/ 1.25%

FV = $17,714.90

Therefore they will accumulate $17,714.90 in their account at age of 65 years.

How much cash did they withdraw?

Now we can use PV of an Annuity formula to calculate the quarterly withdrawals and amount accumulated in account (calculated in previous part) will be present value of the withdrawals

PV = PMT * [1-(1+i) ^-n)]/i

Where,

Where PV = Present value of deposits when they turn 65 years = $17,714.90

PMT or withdrawal per quarter=?

And i= I/Y = 5% is the interest rate per annum or 5%/4 = 1.25% per quarter

The time period n = (80 years – 65 years) = 15 years * 4 = 60 quarters

Therefore,

$17,714.90 = PMT* [1- (1+1.25%)^-60]/1.25%

PMT = $421.44

Therefore they can withdraw $421.44 per quarter for 15 years

What was the total interest?

Total interest = Total withdrawals – total deposits

Where total withdrawals = size of the withdrawals * number of withdrawals

= $421.44 * 60 = $25,286.18

And total deposits = size of the deposits * number of deposits

= $200 * 60 = $12,000

Therefore,

Total interest = $25,286.18 - $12,000.00 = $13,286.18