SOLUTION: Ishan and Hazel plan to retire at age 60 with a retirement income of $48,000 a year from their savings. Rather than pay themselves the whole amount at the beginning of each year,

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Question 1065758: Ishan and Hazel plan to retire at age 60 with a retirement income of $48,000 a year from their savings. Rather than pay themselves the whole amount at the beginning of each year, they have decided that payment at the beginning of each quarter of $12,000 gives them the right balance of flexibility and maximized interest earnings. They feel they can safely earn an interest rate of 4%, compounded quarterly, on their money and they are budgeting based on the prediction that they will live until they are 90 years old. How much money will they have to have saved by the time they are 60 in order to reach their retirement goal?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
they plan to retire at 60 and expect to live until 90.
that means they need enough money for 30 years.

they believe they can earn 4% per year.
divide that by 4 and it becomes 1% per quarter.

they will pay themselves 12,000 at the beginning of each quarter.

thy will need 844,770.327 in their account at the beginning of their retirement period.

i used an online time value of money calculator to come up with this amount.

the online calculator i used is at http://arachnoid.com/finance/

my inputs and outputs are shown below:

first the inputs.

$$$

next the outputs.

$$$

the inputs were:

present value = 0
future value = 0
number of periods = 30 years * 4 quarters per year = 120 quarters.
payment amount = 12,000 per qusarter.
interest rate per period = 4% per year divided by 4 quarters per year = 1% per quarter.
payment is at the beginning of each quarter.

the outputs are the present value of 844,770.33 which gets displayed after i select the compute pv button.

this can also be done manually using financial formulas.

i have a list of these formulas.

they can be found at https://www.algebra.com/algebra/homework/Finance/THEO-2016-04-29.lesson#formulas

the formula you want would be:

PRESENT VALUE OF AN ANNUITY WITH BEGINNING OF TIME PERIOD PAYMENTS

that formula will be:

PRESENT VALUE OF AN ANNUITY WITH BEGINNING OF TIME PERIOD PAYMENTS
p = ((a*(1-1/(1+r)^n))/r)*(1+r)
p is the present value of the annuity.
a is the annuity.
r is the interest rate per time period.
n is the number of time periods.

a is 12,000.
r is .01
n is 120

you would get p = ((12000*(1-1/(1+.01)^120))/.01)*(1+.01)

you should get p = 844770.327 rounded to 3 decimal places.