SOLUTION: Mr Hammond is 45 years of age and has a life expectancy of 10 more years. He wishes to invest $20,000 in an annuity that will make a level payment at the end of each year until h

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Question 1183073: Mr Hammond is 45 years of age and has a life expectancy of 10 more years. He
wishes to invest $20,000 in an annuity that will make a level payment at the end of
each year until his death. If the interest rate is 8%, what income can Mr Hammond
expect to receive each year?
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.

            I will solve the problem under the assumption that Mr. Hammond withdraws a level payment
            at the beginning of each year  to provide the money for living this year.

            It looks like much more realistic scenario,  and it is  usual assumption  in this class of problems. 


Use the general formula  A = W%2Ap%2A%28%281-p%5E%28-n%29%29%2Fr%29.


In this formula,  A is the deposited value ($20,000 in this case);  

                  W  is the withdrawal annual rate W (the unknown value in this problem);  

                  the annual compounding rate is  r = 0.08;  p = 1 + r = 1 + 0.08 = 1.08;   

                  n is the number of payment periods  n= 10. 


From the formula


          W = A%2F%28%28%28p%2A%281-p%5E%28-n%29%29%29%2Fr%29%29 = 20000%2F%28%281.08%2A%28%281-1.08%5E%28-10%29%29%2F0.08%29%29%29 = 2759.80 dollars.                 ANSWER


Next, the TOTAL amount obtained from the account (2759.80 each year) during 10 years is  2759.80*10 = 27,598 dollars.


The difference  $27,598 - $20000 = $7.598  is the INTEREST earned by the account during 10 years.   


ANSWER.  The uniform level withdrawal is $2759.80 per year.

Solved.

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

    - Ordinary Annuity saving plans and geometric progressions
    - Annuity Due saving plans and geometric progressions
    - Solved problems on Ordinary Annuity saving plans
    - Withdrawing a certain amount of money periodically from a compounded saving account (*)
    - Miscellaneous problems on retirement plans

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


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The difference in answers between my post and the post by tutor  @MathTherapy is due

to the fact that we solved the problem under  DIFFERENT  ASSUMPTIONS.



Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
Mr Hammond is 45 years of age and has a life expectancy of 10 more years. He
wishes to invest $20,000 in an annuity that will make a level payment at the end of
each year until his death. If the interest rate is 8%, what income can Mr Hammond
expect to receive each year?


When either of the 3 is applied, he should be expecting annual payments of highlight_green%28%22%242%2C980.59%22%29%29, at the END of each year, for  10 years.

So, the formula for the present value of an ORDINARY ANNUITY was used. I may be WRONG. Tutor IKLEYN's answer is more practical though,

since she used the formula for payments/payouts at the beginning of the year (ANNUITY DUE) - which is when payments are normally 

made - as opposed to the end. To be 100% certain though, the problem needs to state when exactly he's expecting these payouts.

I RETRACT

I actually read the problem AGAIN, something I didn't do before, and it clearly states that he wishes to get the annual payments 

at the END of the year, so highlight_green%28%22%242%2C980.59%22%29%29, at the END of each year, for  10 years is what he should expect.