Question 1135731
.


Let me re-formulate the post to make the problem formulation correct,  precise and clear,  as it should be.


<pre>
    You want to be able to withdraw $45,000 from your account <U>at the beginning of each year</U> for 25 years after you retire. 

    You expect to retire in 30 years. 

    If your account earns 9% interest <U>compounded yearly</U>, how much will you need to deposit <U>at the end of</U> each year 
    until retirement to achieve your retirement goals?
</pre>


<U>Solution</U>


The solution is in 2 steps.


<U>Step 1</U>


<pre>
Step 1 is to determine what amount you need to have at your account after 30 years depositing to it

in order for to be able to withdraw $45,000 from your account <U>at the beginning of each year</U> for 25 years after you retire.


The general formula to calculate this amount is  

    X = {{{W*p*((1-p^(-n))/r)}}}.

In this case  the withdrawal semi-annual rate is W = $45000,  the annual compounding rate 
is  r = 0.09,  p = 1 + 0.09 = 1.09, the number of payment periods  is n = 25. So


    X = {{{45000*1.09*((1-1.09^(-25))/0.09)}}} = 481,797.53 dollars.     
</pre>


<U>Step 2</U>


<pre>
At this step we determine how big the annual deposit should be to provide this amount of $481,797.53 after 30 years of depositing.


This time it is classic Ordinary Annuity saving plan.  The general formula is 


    FV = {{{P*(((1+r)^n-1)/r)}}},   


where  FV is the future value of the account;  P is annual payment (deposit); r is the annual percentage yield presented as a decimal; 
n is the number of deposits (= the number of years, in this case).


From this formula, you get for the annual payment 


    P = {{{FV*(r/((1+r)^n-1))}}}.     (1)


Under the given conditions, FV = $481,797.53;  r = 0.09;  n = 30.  So, according to the formula (1), you get for the annual payment 


    P = {{{481797.53 *(0.09/((1+0.09)^30-1))}}} = $3534.64.


So, your annual deposit should be  $3534.64.     <U>ANSWER</U>


<U>ANSWER</U>.  To provide your goal, you need to deposit  $3534.64 dollars annually at the end of each year during 30 years.
</pre>


Solved.


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If you want to learn the theory of this financing and/or see other similar solved problems, &nbsp;look into my lessons in this site


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&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Withdrawing-a-certain-amount-of-money-periodically-from-a-compounded-saving-account.lesson>Withdrawing a certain amount of money periodically from a compounded saving account</A> 

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Happy learning !