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2 Pages

### annuity

Course: MAT 142, Fall 2009
School: ASU
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Word Count: 454

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the Why Annuity Formula Is What It Is. An annuity is a fund in which you make regular deposits, and the money grows with interest. It is as though each deposit is like a &quot;new&quot; account. The value of the annuity is the sum of all these separate little accounts. Let pymt = your regular deposit (&quot;payment&quot;) into a savings fund. Let r = the annual percentage rate, c = compounding...

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the Why Annuity Formula Is What It Is. An annuity is a fund in which you make regular deposits, and the money grows with interest. It is as though each deposit is like a "new" account. The value of the annuity is the sum of all these separate little accounts. Let pymt = your regular deposit ("payment") into a savings fund. Let r = the annual percentage rate, c = compounding frequency and t = years. The first payment is made. Then the next month, the second payment is made. Meanwhile, the first 1 payment has accrued one month's worth of interest (so t = 12 ): First months' deposit: Second month's total value: pymt Total value: A = pymt 1 A = pymt + pymt (1 + r ( 12 )) For the third month, you make another pymt. The first payment has now grown for two months, and the second payment has now grown for one month: Third month's total value: 1 1 A = pymt + pymt (1 + r ( 12 )) + pymt (1 + r ( 12 )) 2 Hopefully you see a trend here. After n months, the fund's total value is 1 1 1 1 A = pymt + pymt (1 + r ( 12 )) + pymt (1 + r ( 12 )) 2 + pymt (1 + r ( 12 )) 3 + ... + pymt (1 + r ( 12 )) n -1 This type of equation is called a geometric series. The sum of a geometric series can be calculated using a nifty formula. The derivation of the formula follows: a Consider simple series that starts with some value a and each successive term is a multiplied by r raised to increasing powers. Therefore, the sum of the first n terms of a series is: S = a + ar + ar 2 + ar 3 + ... + ar n-1 A trick is to multiply the above equation by an extra r value. This increases all exponents by one: rS = r (a + ar + ar 2 + ar 3 + ... + ar n-1 ) rS = ar + ar 2 + ar 3 + ... + ar n Then subtract: rS - S = (ar + ar 2 + ar 3 + ... + ar n ) - (a + ar + ar 2 + ... + ar n-1 ) Notice that after you distribute the negative in the second set of parentheses, everything cancels except the ar n and the a terms. You get: rS - S = ar n - a So now we solve for S, the sum: S (r - 1) = a (r n - 1) S= a (r n - 1) r -1 The result is a general formu...

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