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# Future Value of An Annuity:

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1. What is an annuity?
2. How the future value of an annuity is calculated?
3. Examples

Contents:

## Definition and Explanation:

An annuity is a series of periodic payments. Examples of annuities include regular deposits to a saving account, monthly car, mortgage, or insurance payments, and periodic payments to a person from a retirement fund. Although an annuity may vary in dollar amount, we will assume that an annuity involves a series of equal payments. We will also assume that the payments are all made at the end of a compounding period. One may certainly argue that end of one period coincides with the beginning of the next period. The important point is that payment does not qualify for interest in the previous period but will earn full interest during the next period.

Following is the illustration of a series of payments R, each of which equals \$1,000. These might represent year-end deposits in a savings account or quarterly tax payments by a self-employed person.

Annuity Future value of a lump sum investment is explained on the future value of a single sum page. In this article future value or sum of an annuity is determined.

## Formula:

The following formula is used to calculate future value of an annuity: R = Amount an annuity i = Interest rate per period n = Number of annuity payments (also the number of compounding periods) Sn = Sum (future value) of the annuity after n periods (payments)

## Examples:

### Example:

A person plans to deposit \$1,000 in a tax-exempt savings plan at the end of this year and an equal sum at the end of each following year. If interest is expected to be earned at the rate of 6 percent per year compounded annually, to what sum will the investment grow at the time of the fourth deposit?

#### Solution: ### Example:

A teenager plans to deposit \$50 in savings account at the end of each quarter for the next 6 years. Interest is earned at a rate of 8 percent per year compounded quarterly. What should her account balance be 6 years from now? How much interest will she earn?

#### Solution:

 In this example: R = \$50 i = 0.08/4 = 0.02 n = (6 years × 4 quarters per year) = 24 S4 = \$50(30.42186*) = \$1,521.09 Over 6-year period she will make 24 deposits of \$50 for a total of \$1,200. Interest for the period will be \$1,521.09 - \$1,200.00 = \$321.09

## Determining the Size of An Annuity:

The above formula can be solved for any of the four parameters, given values for the other three. For example, we might have a goal of accumulating a particular sum of money by some future time. If the rate of interest which can be earned is known, the question becomes, what amount should be deposited each period in order to reach the goal? In other words formula can also be used to determine the size of an annuity. For this purpose, formula can be solved for R: Suppose a corporation wants to establish a sinking fund beginning at the end of this year. Annual deposits will be made at the end of this year and for the following 9 years. If deposits earn interest at the rate of 8 percent per year compounded annually, how much money must be deposited each year on order to have \$12 million at the time of the 10 deposit? How much interest will be earned?

#### Solution:

 n = 10 i = 0.08 Sn = \$12,000,000 R = ? The money to be deposited each year (R) is calculates as follows: R = \$12,000,000 / 14.48656* R = 828354Since 10 deposits of \$828,354 will be made during this period, total deposits will equal \$8,283,540. Because these deposits plus accumulated interest will equal \$12 million, interest of \$12,000,000 - \$8,283,600 = \$3,716,400 will be earned.
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## More study material from this topic: Methods for the evaluation of capital investment analysis Average rate of return or accounting rate of return method Cash payback method Net present value method Internal rate of return method Simple interest Future value of a single sum Future value of an annuity Present value of a single sum Present value of an annuity Qualitative consideration in capital investment analysis Capital investment analysis and unequal proposal lives Capital rationing decision process Difference between simple interest and compound interest Difference between nominal and effective interest rate Future value of \$1 table Present value of \$1 table Present value of ordinary annuity table Future value of ordinary annuity table

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