Question 1173176
.
A person wants to buy life insurance policy which would yield a large enough sum of money to provide for 20 annual payments 
of $50000 to surviving members of the family. The payment should begin 1 year from the time of death. It is assumed that interest 
could earned on the sum received from the policy at the rate of 8% compounded annually.
a) What amount of insurance should be taken out so as to insure the desired annuity?
b) How much interest will be earned on the policy benefits over the 20 year period?
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Let me re-formulate the problem to make it crystally clear.


<pre>
    There is an account with starting amount X dollars; the value of X is unknown now.


    We want to withdraw $50000 from this account during 20 years, at the beginning of each year 

    (starting 1 year after the death: so, if the person died 01/01/2030, for example, then the account starts working
     on 01/01/2031 morning with the amount of X dollars, and first withdraw goes on 01/01/2031 afternoon - and then every year after that).

    The account is compounded annually at the annual rate of  8%  at Dec., 31.


    Find the starting value X at the account (at 01/01/2031), which provides uninterrupted functioning of the account during 20 years.
</pre>


<U>Solution</U>


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


In this formula,  W  the withdrawal annual rate W = $50000;  the annual compounding rate 

is  r = 0.08;  p = 1 + r = 1 + 0.08 = 1.08;   n is the number of payment periods  n= 20. 


From the formula


          X = {{{50000*1.08*((1-1.08^(-20))/0.08)}}} = 530,180 dollars.                 <U>ANSWER to question (a)</U>


Next, the amount obtained from the account ($50000 each year) is  50000*20 = 1,000,000 dollars.


The difference  $1,000,000 - $530,180 = $469820  is the INTEREST earned by the account during 20 years.     <U>ANSWER to question (b)</U>


<U>ANSWER</U>.  To provide the  goal, the starting amount should be  $530180  at the account.

         The earned interest is $469820.
</pre>

Solved.


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See my lessons in this site associated with annuity saving plans and retirement plans 


&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/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> (*)

&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> 


and especially lesson marked &nbsp;(*) &nbsp;in the list as the most relevant to the given problem.