Lesson Withdrawing a certain amount of money periodically from a compounded saving account

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<H2>Withdrawing a certain amount of money periodically from a compounded saving account</H2>

<H3>Problem 1</H3>There is an annually compounded saving account with initial amount of &nbsp;X &nbsp;dollars.
A person withdraws certain constant amount &nbsp;W = 24000 &nbsp;dollars at the beginning of each year.
The account is compounded annually at &nbsp;r = 6% annual rate at the end of each year.  
What the initial value &nbsp;X&nbsp; of the account should be in order for to feed the person for &nbsp;20&nbsp; years 
in accordance with this scheme ?

<B>Solution</B>

<pre>
Let  r = 6% = 0.06  and let  p = {{{1 + r}}}.  Let us move in time back from the 20-th year to the beginning.

20)  At the beginning of the 20-th year, before withdrawing, the account must have exactly W = 24000 dollars in order for the person 
     could withdraw it and complete/(zeroed) the account.

It means that at the end of the 19-th year, after compounding, the account should have exactly W = 24000 dollars.
     In turn, it means that at the end of the 19-th year, <U>before</U> compounding, the account should be exactly  {{{W/p}}} dollars.

Hence, at the beginning of the 19-th year, after withdrawing of W = $24000, the account must be exactly  the same  {{{W/p}}} dollars.

19)  It means that at the beginning of the 19-th year, <U>before</U> withdrawing, the account must be  {{{W/p + W}}} dollars.

Hence, at the end of the 18-th year, after compounding, the account should be the same {{{(W/p+W)}}} dollars.
     In turn, it means that at the end of the 18-th year, <U>before</U> compounding, the account shoud be exactly  {{{(1/p)*(W/p + W)}}} dollars.

Hence, at the beginning of the 18-th year, after withdrawing of W = $24000, the account must be exactly the same  {{{(1/p)*(W/p + W)}}} dollars.

18)  It means that at the beginning of the 18-th year, <U>before</U> withdrawing, the account must be  {{{(1/p)*(W/p + W) + W}}} dollars.

Hence, at the end of the 17-th year, after compounding, the account should be the same  {{{(1/p)*(W/p + W) + W}}} dollars.
     In turn, it means that at the end of the 17-th year, <U>before</U> compounding, the account should be exactly  {{{(1/p)*((1/p)*(W/p + W) + W)}}} dollars.

Hence, at the beginning of the 18-th year, after withdrawing of W = $24000, the account must be exactly the same  {{{(1/p)*((1/p)*(W/p + W) + W)}}} dollars.

17) - 16) - 15) - . . . . . . . . - 2)

1)    It means that at the beginning of the 1-st year, <U>before</U> withdrawing, the account must be  {{{W/p^19}}} + {{{W/p^18}}} + . . .  + {{{W/p}}} + {{{W}}} dollars,
      and it is exactly the starting amount X, which is under the problem question.

Hence,  X = {{{W/p^19}}} + {{{W/p^18}}} + . . .  + {{{W/p}}} + {{{W}}},  which is equal, by applying the formula of the sum of geometric progression

X = W*(((1/p^20)-1) / ((1/p)-1)) = {{{W*p*((1-p^(-20))/((r/100)))}}}.

<U>ANSWER</U>.  The initial value X of the account to feed the person with W dollars per year for 20 years should be  X = {{{W*p*((1-p^(-20))/((r/100)))}}}.

The initial value X of the account to feed the person with W dollars per year for n years should be  X = {{{W*p*((1-p^(-n))/((r/100)))}}}.

With the given data  W = 24000 dollars,  r = 6% = 0.06,  p = 1 + 0.06,  n = 20 years  the initial amount at the account should be

X = {{{24000*1.06*((1-1.06^(-20))/0.06)}}} = 291794.80 dollars.

Notice that the person get from the fund  $24000*20 = 480,000 dollars in all, in 20 years.
</pre>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In other problems the withdrawal period &nbsp;(which is usually the same as the compounding period) &nbsp;can be different from one year.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It can be one quarter, &nbsp;or one month, &nbsp;or one week. &nbsp;But the rate of the account is usually comes as the annual rate, &nbsp;as you will see

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in problems that follow. &nbsp;The scheme of deriving the final formulas is the same, &nbsp;so I do not repeat deriving and give the final formulas

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and the way on how to use them in the problems that follow.

<H3>Problem 2</H3>There is a quarterly compounded saving account with initial amount of &nbsp;X&nbsp; dollars.
A person withdraws certain constant amount &nbsp;W = 6000 &nbsp;dollars at the beginning of each quarter.
The account is compounded quarterly at &nbsp;r = 6% &nbsp;annual rate at the end of each quarter.   
What the initial value &nbsp;X&nbsp; of the account should be in order for to feed the person for &nbsp;20 &nbsp;years 
in accordance with this scheme ?

<B>Solution</B>

<pre>
Use the general formula  X = {{{W*p*((1-p^(-n))/((r/100)))}}}.

In this case  W = $6000,  the quarterly rate is  r = 0.06/4 = 0.015,  p = 1 + 0.015 = 1.015, the number of payment periods (= the number of quarters) 
is n = 20*4 = 80, so

X = {{{6000*1.015*((1-1.015^(-80))/0.015)}}} = 282620.60 dollars.     <U>ANSWER</U>

Notice that the person get from the fund  $6000*20*4 = 480,000 dollars in all, in 20 years.
</pre>

<H3>Problem 3</H3>There is a monthly compounded saving account with initial amount of &nbsp;X &nbsp;dollars.
A person withdraws certain constant amount &nbsp;W= 2000 &nbsp;dollars at the beginning of each month.
The account is compounded monthly at &nbsp;r = 6% &nbsp;annual rate at the end of each month.  
What the initial value &nbsp;X&nbsp; of the account should be in order for to feed the person for &nbsp;20 &nbsp;years 
in accordance with this scheme ?

<B>Solution</B>

<pre>
Use the general formula  X = {{{W*p*((1-p^(-n))/((r/100)))}}}.

In this case  W = $2000,  the monthly rate is  r = 0.06/12 = 0.005,  p = 1 + 0.005 = 1.005, the number of payment periods (= the number of months) 
is n = 20*12 = 240, so

X = {{{2000*1.005*((1-1.005^(-240))/0.005)}}} = 280557.40 dollars.     <U>ANSWER</U>

Notice that the person get from the fund  $2000*20*12 = 480,000 dollars in all, in 20 years.
</pre>

<H3>Problem 4</H3>You have $300,000 saved for retirement. Your account earns 10% interest. 
How much will you be able to pull out at the beginning of each month, if you want to be able to take withdrawals for 25 years?

<B>Solution</B>

<pre>
Use the general formula  A = {{{W*p*((1-p^(-n))/r)}}}.

Here A is the initial amount at the account; W is the monthly withdrawn value; r is the nominal monthly percentage r = 0.1/12; 
presented as a decimal;  p = 1 + r  and n is the number of withdrawing periods (months, in this case).

In this problem,  W is the unknown;  the monthly rate is  r = 0.10/12 = 0.00833,  p = 1 + 0.00833 = 1.00833, the number of payment 
periods (= the number of months) is n = 25*12 = 300.  So

300000 = {{{W*1.00833*((1-1.00833^(-300))/0.00833)}}}.

The factor  {{{1.00833*((1-1.00833^(-300))/0.00833)}}} is equal to 110;  therefore

W = {{{300000/110}}} = 2727 dollars  (rounded to the closest lesser dollar).

Thus you will be able to withdraw about $2727 every month during 25 years.    <U>ANSWER</U>
</pre>

<H3>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>The conclusion and the lesson to learn</U></H3>

If you are given a problem like this

<pre>
    There is a saving account with initial amount of &nbsp;X&nbsp; dollars, compounded periodically (yearly, or quarterly, or monthly).
    A person withdraws certain constant amount &nbsp;W&nbsp; from the account at the beginning of each period.
    The account is compounded at the end of each period at the annual rate  r.  
    What the initial value &nbsp;X&nbsp; of the account should be in order for to feed the person for &nbsp;n &nbsp;years 
    in accordance with this scheme ? 
</pre>

then the solution is as follows:

<pre>
    a)  Use the decimal rate for the withdrawal (compounding) period  r (for the yearly period ),  or  r/4 (for the quarterly period)  or  r/12 (for the monthly period).

Then compute p = 1+r  and the number of the withdrawal periods m = n, or m = 4n  or  m = 12n, correspondingly;

b)  then use the formula  X = {{{W*p*((1-p^(-m))/r)}}}  to determine the initial amount &nbsp;X&nbsp; at the account.
</pre>

My other lessons in this site associated with annuity saving plans and retirement plans are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Geometric-progressions.lesson>Geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-geometric-progressions.lesson>The proofs of the formulas for geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Ordinary-Annuity-saving-plans-and-geometric-progressions.lesson>Ordinary Annuity saving plans and geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Annuity-due-saving-plans-and-geometric-progressions.lesson>Annuity Due saving plans and geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problem-on-Ordinary-Annuity-saving-plans.lesson>Solved problems on Ordinary Annuity saving plans</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Finding-present-value-of-an-annuity.lesson>Finding present value of an annuity, or an equivalent amount in today's dollars</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-Annuity-saving-plans.lesson>Miscellaneous problems on retirement plans</A>

OVERVIEW of my lessons on geometric progressions with short annotations is in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Review-of-lessons-on-geometric-progressions.lesson>OVERVIEW of lessons on geometric progressions</A>.

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-II.