Annuity word problem

Question

 Carol wants to retire at age 63. She would like to draw $3,200 each month and estimates that she will live to be 88 years old. Carol can earn nine percent interest on her investment before retirement  and will switch them over to a six percent investment after she retires. If Carol is currently 28 years old how much does she have to pay away each month 

Answer 1

Probably most important math problem one will

ever solve!!!!!! Granted, there is calculators

and/or software to do it , but here goes:

 

In order to live off of the interest
without touching the principal, Carol
needs $3200 per month of interest as

stated in the problem.

She will be investing at 6% annual interest
upon and after retirement, which 0.5% monthly

0.005*P = 3200

P = 640000

Therefore,
Carol needs $640000 by retirement.
She has 63-25 = 38 years to get it.
She has 9% annual interest which is
0.75% monthly interest at her disposal.
38 years times 12 months per year is 456 months.

For this part, the formula is needed for compounding
interest with monthly contributions. It is

P(1 + r)^t + c[ ((1 + r)^t - 1) / r ]

where P is the starting principal amount
that is being asked, r=0.5%, t is the number
of times or amount of time that the interest
compounds, c is the monthly contribution amount.

In this case, P=c is the unknown amount.
Will call this amount x, so x=P=c.
r = 0.5% and t = 456.

X(1+0.005)^456 + x[ (( 1 + 0.005)^456 - 1)/ 0.005] = 640000

X(1.005)^456 + x[ ((1.005)^456 - 1)/ 0.005] = 640000

x(1.005)^456 + 1744.26 x = 640000

9.7213X + 1744.26x = 640000

1753.98X = 640000

X=364.91

Therefore, she needs to contribute 364.91 or more
per month.


Here's the check:

364.91(1.005)^456 = 3538.55


364.91* [ ( (1.005)^456 - 1)/0.005] = 636497.61

The sum of these is 640036.16 which includes rounding error

At 0.5% monthly, the interest earned on this principal is 3200.18.

 

Thank you for this challenging and very important problem.