You plan to work for 40 years and then retire using a 25-year annuity. You want to arrange a retirement income of $4500 per month. You have access to an account that pays an APR of 7.2% compounded monthly. What size nest egg do you need to achieve the desired monthly yield?

What montly deposits are required to achieve the desired monthly yield at retirement?

first lets consider the first part of the problem and is how mucho do i need to accumulate for having an annuity for 25 years. this problem can be solved applying the concept of annuity, keep in mind that an annuity is a formula which allows you to calculate the present value of future payments affected by an interest rate.by definition the present value of an annuity is given by:

[tex]a_{n} =P*\frac{1-(1+i)^{-n} }{i}[/tex]

where [tex]a_{n}[/tex] is the present value of the annuity, [tex]i[/tex] is the interest rate for every period payment, n is the number of payments, and P is the regular amount paid. so applying to this particular problem, we have:

[tex]a_{25*12} =4,500*\frac{1-(1+0.006)^{-25*12} }{0.006}[/tex]

look at the value 25*12 because the problem tells us is during 25 years but the payment is monthly, and look at the 0.006 and it is comming from the APR/12 and we must do that because this rate is componded Monthly:

[tex]a_{25*12} =623,459.79[/tex]

so for the second part we must calculate the second part we must calculate the acumulated value at 40 years of work:

[tex]s_{n} =P*\frac{(1+i)^{n}-1 }{i}[/tex]

where [tex]s_{n}[/tex] is the future value of the annuity, [tex]i[/tex] is the interest rate for every period payment, n is the number of payments, and P is the regular amount paid. so applying to this particular problem, we have:

[tex]623,459.79 =P*\frac{(1+0.006)^{40*12}-1 }{0.006}[/tex]

solving for P we have:

P=224.51