Calculus help?

As soon as you start your job after graduation, you plan to start saving for retirement. You figure you will work for 40 years, and then live for another 30 years after you stop working. You want to have enough in your retirement account to be able to withdraw $3200 per month for the entire time of your retirement, but you don't plan to leave any money behind for your children. With luck, you will be able to find an investment that earns 0.55% per month.





How much money will you need 40 years from now in order to accomplish this financial objective? Remember that your account continues to earn interest, even though you are no longer making contributions (in fact you're taking money out).


$_____________





How much must you deposit each month over the next 40 years so that you will have the required amount for retirement?


$___________|||First question: What is the present value, at .55% per mo, of an annuity certain of 360 monthly payments of one dollar, paid at the end of the month?





The formula for this is


a = (1 鈥?1.0055^-360) / .0055.





Doing the calculation, you get a = 156.578 124 623. Then, we conclude, to provide 3200 per month, requires capital, at retiral, of 3200a = 501 049.998 79, or $ 501,050, to the nearest dollar.





Second question: What is the accumulated value, with interest at 0.55% per month, of 480 monthly payments, of one dollar, paid in at the end of each month?





The formula for this is


S = (1.0055^480 鈥?1) / .0055.


Doing the calculation gets you S = 2347.673 631 936.





Divide this last result into the amount of capital required at retrial, to get the monthly amount needed to be saved each month, in order to reach that capital requirement.


That is, a/S = 501050 / 2347 = 213.49..673


Therefore, you need to save $213.50, at the end of every month.





Explanation: Both of these well-known compound-interest formulae are based on the standard mathematical formula for summing a geometric series:


1 + x + x^2 + x^3 + 鈥?+ x^( n 鈥?1 ) = ( x^n 鈥?1 ) / ( x 鈥?1 ).


For the accumulation formula, substitute 1.0055 for x;


for the annuity formula, substitute 1.0055 for x, and


then divide the numerator by (1.0055)^n. In the latter case, you are taking the present value, rather than their value at the end of the accumulation process.