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A business needs $450,000 in five years. How much should be deposited each quarter in a sinking fund that earns 9% to have this amount in five years?

Again, suppose that x size 12{x} {} dollars are deposited each quarter in the sinking fund. After five years, the future value of the fund should be $450,000. Which suggests the following relationship:

x 1 + . 09 / 4 20 1 . 09 / 4 = $ 450 , 000 x 24 . 9115 = 450 , 000 x = 450000 24 . 9115 x = 18 , 063 . 93 size 12{ matrix { { {x left [ left (1+ "." "09"/4 right ) rSup { size 8{"20"} } - 1 right ]} over { "." "09"/4} } =$"450","000" {} ## x left ("24" "." "9115" right )="450","000" {} ##x= { {"450000"} over {"24" "." "9115"} } {} ## x="18","063" "." "93"} } {}
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If the payment is made at the beginning of each period, rather than at the end, we call it an annuity due. The formula for the annuity due can be derived in a similar manner. Reconsider [link] , with the change that the deposits be made at the beginning of each month.

If at the beginning of each month a deposit of $500 is made in an account that pays 8% compounded monthly, what will the final amount be after five years?

There are 60 deposits made in this account. The first payment stays in the account for 60 months, the second payment for 59 months, the third for 58 months, and so on.

The first payment of $500 will accumulate to an amount of $ 500 1 + . 08 / 12 60 size 12{$"500" left (1+ "." "08"/"12" right ) rSup { size 8{"60"} } } {} .

The second payment of $500 will accumulate to an amount of $ 500 1 + . 08 / 12 59 size 12{$"500" left (1+ "." "08"/"12" right ) rSup { size 8{"59"} } } {} .

The third payment will accumulate to $ 500 1 + . 08 / 12 58 size 12{$"500" left (1+ "." "08"/"12" right ) rSup { size 8{"58"} } } {} .

And so on.

The last payment is in the account for a month and accumulates to $ 500 1 + . 08 / 12 size 12{$"500" left (1+ "." "08"/"12" right )} {}

To find the total amount in five years, we need to find the sum of the following series.

$ 500 1 + . 08 / 12 60 + $ 500 1 + . 08 / 12 59 + $ 500 1 + . 08 / 12 58 + . . . + $ 500 1 + . 08 / 12 size 12{$"500" left (1+ "." "08"/"12" right ) rSup { size 8{"60"} } +$"500" left (1+ "." "08"/"12" right ) rSup { size 8{"59"} } +$"500" left (1+ "." "08"/"12" right ) rSup { size 8{"58"} } + "." "." "." +$"500" left (1+ "." "08"/"12" right )} {}

Written backwards, we have

$ 500 1 + . 08 / 12 + $ 500 1 + . 08 / 12 2 + . . . + $ 500 1 + . 08 / 12 60 size 12{$"500" left (1+ "." "08"/"12" right )+$"500" left (1+ "." "08"/"12" right ) rSup { size 8{2} } + "." "." "." +$"500" left (1+ "." "08"/"12" right ) rSup { size 8{"60"} } } {}

If we add $500 to this series, and later subtract that $500, the value will not change. We get

$ 500 + $ 500 1 + . 08 / 12 + $ 500 1 + . 08 / 12 2 + . . . + $ 500 1 + . 08 / 12 60 $ 500 size 12{$"500"+$"500" left (1+ "." "08"/"12" right )+$"500" left (1+ "." "08"/"12" right ) rSup { size 8{2} } + "." "." "." +$"500" left (1+ "." "08"/"12" right ) rSup { size 8{"60"} } - $"500"} {}

Not considering the last term, we have a geometric series with a = $ 500 size 12{a=$"500"} {} , r = 1 + . 08 / 12 size 12{r= left (1+ "." "08"/"12" right )} {} , and n = 60 size 12{n="60"} {} . Therefore the sum is

$ 500 1 + . 08 / 12 61 1 . 08 / 12 $ 500 = $ 500 74 . 9667 $ 500 = $ 37483 . 35 $ 500 = $ 36983 . 35 size 12{ matrix { { {$"500" left [ left (1+ "." "08"/"12" right ) rSup { size 8{"61"} } - 1 right ]} over { "." "08"/"12"} } - $"500" {} ## =$"500" left ("74" "." "9667" right ) - $"500" {} ##=$"37483" "." "35" - $"500" {} ## =$"36983" "." "35"} } {}
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So, in the case of an annuity due, to find the future value, we increase the number of periods n size 12{n} {} by 1, and subtract one payment.

The Future Value of an Annuity due = m 1 + r / n nt + 1 1 r / n m size 12{"The Future Value of an Annuity due"= { {m left [ left (1+r/n right ) rSup { size 8{ ital "nt"+1} } - 1 right ]} over {r/n} } - m} {}

Most of the problems we are going to do in this chapter involve ordinary annuity, therefore, we will down play the significance of the last formula. We mentioned the last formula only for completeness.

Finally, it is the author's wish that the student learn the concepts in a way that he or she will not have to memorize every formula. It is for this reason formulas are kept at a minimum. But before we conclude this section we will once again mention one single equation that will help us find the future value, as well as the sinking fund payment.

The equation to find the future value of an ordinary annuity, or the amount of periodic payment to a sinking fund

If a payment of m size 12{m} {} dollars is made in an account n size 12{n} {} times a year at an interest r size 12{r} {} , then the future value A size 12{A} {} after t size 12{t} {} years is

A = m 1 + r / n nt 1 r / n size 12{A= { {m left [ left (1+r/n right ) rSup { size 8{ ital "nt"} } - 1 right ]} over {r/n} } } {}

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Present value of an annuity and installment payment

Section overview

In this section, you will learn to:

  1. Find the present value of an annuity.
  2. Find the amount of installment payment on a loan.

In [link] , we learned to find the future value of a lump sum, and in [link] , we learned to find the future value of an annuity. With these two concepts in hand, we will now learn to amortize a loan, and to find the present value of an annuity.

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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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