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# Future Value of A Single Sum:

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1. Define and explain the future value.
2. How is it calculated?

Contents:

Future value concept into two types. These are: (1) future value of a single sum and (2) future value of an annuity. In this article future value of a single sum is explained. To understand the concept of the future value of an annuity read future value of an annuity article.

## Definition and Explanation:

To understand the concept of future value we need to understand compound interest first.  Under the procedure of compounding, the interest is reinvested. The interest earned each period is added to the principal for the purpose of compounding interest for the next period. The amount of interest computed using this procedure is called the compound interest. The principle plus any interest earned during a period is called Compound amount or future value.

Suppose that we have deposited \$8,000 in a credit union which pays interest of 8 percent per year compounded quarterly. We want to determine the amount of money we will have on deposit at the end of 1 year if all interest is left in the savings account. At the end of the first quarter, interest is computed as follows:

 I1 = (\$8,000)(0.08)(0.25*) = \$160 *1/4 - First quarter

With the interest left in the account, the principal on which interest in the second quarter is the original principal plus the \$160 in interest earned during the first quarter or \$8,000 + \$160 = \$8,160. Interest for the second quarter is computed as follows:

 I2 = (\$8,160)(0.08)(0.25*) = \$163.20 *1/4 - Second quarter

The calculation for the four quarters is as follows:

 Quarter (P) Principle (I) Interest (S = P + I) Compound Amount 1 \$8,000.00 \$160.00 \$8,000.00 + \$160.00 = \$8,160.00 2 \$8,160.00 \$163.00 \$8,160.00 + \$163.00 = \$8,323.20 3 \$8,323.20 \$166.46 \$8,323.20 + \$166.46 = \$8,489.66 4 \$8,489.66 \$169.79 \$8,489.66 + \$169.79 = \$8,659.46

In this example, compound interest for one year is \$659.46 (\$160.00 + 163.20 + 166.46 + 169.79). The total amount at the end of the year (principal plus interest earned) is \$8,659.46. This is compound amount or future value of the original amount (principal) of \$8,000.

## Compound Amount or Future Vale Formula:

We have already discussed that the interest earned plus principal is equal to compound amount or future value. The above method of computing compound amount is time consuming. To save time and make the procedure simple, we can use the following formula:

 S = P(1 + i)n Where; P = Principal, dollars i = Interest rate per compounding period n = Number of compounding periods (number of periods in which the principal has earned interest) S = Compound amount

## Examples:

### Example 1:

Suppose that \$1,000 is invested in savings bank which earns interest at a rate of 8 percent per year compounded annually. If all interest is left in the account, what will the account balance be after 10 years.

#### Solution:

 S = P(1 + i)n S = \$1,000(1 + 0.08)10 S = (\$1,000)(2.15892*) = \$2,158.92 The \$1,000 investment will grow to \$2,158.92, meaning that interest of (\$2,158.92 - \$1,000) \$1,158.92 will be earned. In other words the \$1,000 will have a value equal to \$2,158.92 after 10 years.

### Example 2:

A long term investment has been made by a small company. The interest rate is 12% per year, and interest is compounded semiannually. If all interest is reinvested at same rate of interest, what will the value of the investment be after 8 years?

#### Solution:

 S = P(1 + i)n S = \$250,000(1 + 0.06)16 S = \$250,000(2.54035*) = \$635,087.5 Compounding occurs twice a year (semiannually). The interest rate for six months is the annual interest rate divided by the number of compounding periods per year i.e: 0.12/2 = 0.06 The number of compounding periods over the 8-year period is 8 × 2 = 16
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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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