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5 MATHEMATICS OF FINANCE

Copyright © Cengage Learning. All rights reserved.

5.2 Annuities

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Future Value of an Annuity

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Future Value of an AnnuityAn annuity is a sequence of payments made at regular time intervals.

The time period in which these payments are made is called the term of the annuity.

Depending on whether the term is given by a fixed time interval, a time interval that begins at a definite date but extends indefinitely, or one that is not fixed in advance, an annuity is called an annuity certain, a perpetuity, or a contingent annuity, respectively.

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Future Value of an AnnuityIn general, the payments in an annuity need not be equal, but in many important applications they are equal. Annuities are also classified by payment dates.

An annuity in which the payments are made at the end of each payment period is called an ordinary annuity, whereas an annuity in which the payments are made at the beginning of each period is called an annuity due.

Furthermore, an annuity in which the payment period coincides with the interest conversion period is called a simple annuity, whereas an annuity in which the payment period differs from the interest conversion period is called a complex annuity.

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Future Value of an AnnuityIn other words, we study annuities that are subject to the following conditions:

1. The terms are given by fixed time intervals.

2. The periodic payments are equal in size.

3. The payments are made at the end of the payment periods.

4. The payment periods coincide with the interest conversion periods.

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Future Value of an AnnuityTo find a general formula for the accumulated amount S of an annuity, suppose that a sum of $R is paid into an account at the end of each period for n periods and that the account earns interest at the rate of i per period.

Then, proceeding as we did with the numerical example, we obtain

S = R + R(1 + ) + R(1 + )2 + · · · + R(1 + )(n – 1)

(9)

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Future Value of an Annuity

𝑆=𝑅 [ (1+ 𝑟𝑚 )

𝑛−1

𝑟𝑚 ]

𝑟𝑚

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Example 1Find the amount of an ordinary annuity consisting of 12 monthly payments of $100 that earn interest at 12% per year compounded monthly.

Solution:Since i is the interest rate per period and since interest is compounded monthly in this case, we have = = 0.01.

Using Formula (9) with R = 100, n = 12, and = 0.01, we have

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Example 1 – Solution 1268.25

or $1268.25.

Use a calculator.

cont’d

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Practicep. 291 Self-Check Exercises #1