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# Present Value of an Annuity:

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1. Define a explain the present value of an annuity.
2. How is it calculated?
3. What are the benefits of its calculation?

Contents:

## Definition and Explanation:

The present value of an annuity is an amount of money today which is equivalent to a series of equal payments in the future. For example, you have won a lottery and lottery officials give you the choice of having a lump-sum payment today or a series of payments at the end of each of the next 5 years. The two alternatives would be considered equivalent (in a monetary sense) if by investing the lump-sum today you could generate (with accumulated interest) annual withdrawal equal to five installments offered by the lottery. An assumption is that the final withdrawal would deplete the investment completely. Consider the following example.

## Formula:

Following formula is use for the calculation of present value of an annuity: R = Amount of an annuity i = interest rate per compounding period n = Number of annuity payments (also, the number of compounding periods) Present value of the annuity

## Example:

A person recently won a state lottery. The terms of the lottery are that the winner will receive annual payments of \$20,000 at the end of this year and each of the following 3 years. If the winner could invest money today at the rate of 8 percent per year compounded annually, what is the present value of the four payments?

#### Solution:

 i = 0.08 n = 4 R = \$20,000 The present value of the four payments is calculated as follows: A = \$20,000(3.31213*) = \$66,242.6

## Determining the Size of Annuity:

There are problems in which we may be given the present value of an annuity and need to determine the size of the corresponding annuity. For example, given a loan of \$10,000 which is received today, what quarterly payments must be made to repay the loan in 5 years if interest is charged at the rate of 10 percent per year, compounded quarterly? The process of repaying loan by installment payments is referred to as amortizing a loan.

To determine the size of an annuity, the formula is solved for R: The quarterly payment necessary to repay the above motioned \$10,000 is calculated as follows:

 A = \$10,000 i = 0.10/4 = 0.025 n = 20 R = \$10,000/15.58916* = \$641.50 There will be 20 payments totaling \$12,830; thus interest will equal \$2,830 on the loan

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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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