## What is the future value of a single amount? – Definition

The value of a current single amount taken to a future date at a specified interest rate is called the **future value of a single amount.**

or

Future value means the amount to which the investment will grow at a future date if interest is compounded. The single amount means that a lump sum was invested at the beginning of year 1 and was left intact for all the periods.

## Explanation

To explain the concept of the future value of a single amount, let’s assume the following data:

**Table 1.1**

In the above data, we are really trying to determine what the future amount of $10,000 invested at 12% for three years would be, given a certain compounding pattern. This is an example of determining the future value of a single amount. There were no additional investments or interest withdrawals. These future value or compound interest calculations are important in many personal and business financial decisions.

## Example

For example, an individual may be interested in determining how much an investment of $50,000 will amount to in 5 years if interest is compounded semiannually versus quarterly, or what rate of return must be earned on a $10,000 investment if $18,000 is needed in 7 years.

All of these situations relate to determining the future value of a single amount. One way to solve problems of this type is to construct tables similar to the one above. However, this method is time-consuming and not very flexible. Mathematical formulas can also be used. For example, the tables used above to determine the accumulated amount of a single amount at different compounded rates are based on the following formula:

### Formula to determine the accumulated amount at different compounded rates

**Where:**

p = Principal amount

i = Interest rate

n = Number of compounding periods

That is, in the example of the $10,000 compounded annually for 3 years at 12%, the $14,049.28 can be determined by the following calculation:

= $10,000(1 + .12)^{3}

= $14,049.28

However, one of the simplest methods is to use tables that give the future value of $1 at different interest rates and for different periods. Essentially these tables interpret the above mathematical formula for various interest rates and compounding periods for a principal amount of $1. Once the amount for $1 is known, it is easy to determine the amount for any principal amount by multiplying the future amount for $1 by the required principal amount. Many hand calculators also have function keys that can be used to solve these types of problems.

To illustrate, the following example (table 1.2) shows the future value of $1 for 10 interset periods for interest rates ranging from 2% to 15%. Suppose that we want to determine the future value of $10,000 at the end of 3 years if interest is compounded annually at 12% (the example previously used).

**Table 1.2**

In order to solve this, we look down the 12% column in the table until we come to 3 interest periods. The factor from the table is 1.40493, which means that $1 invested today at 12% will accumulate to $1.405 at the end of 3 years. Because we are interested in $10,000 rather than $1, we just multiply the factor of 1.40493 by $10,000 to determine the future value of the principal amount. The amount is $14,049.30, which, except for a slight rounding error, is the same as we determined from table 1.1.

We can generalize the use of the future value table by using the following formula:

Accumulated amount = Factor (from the table) x Principal

= 1.40493 x $10,000

= $14,049.30

This formula can be used to solve a variety of related problems. For example, as we noted above, you may be interested in determining what rate of interest must be earned on a $10,000 investment if you want to accumulate $18,000 at the end of 7 years. Or you may want to know the number of years an amount must be invested in order to grow to a certain amount. In all these cases, we have two of the three items in the formula, and we can solve for the third.

### Interest Compounded More Often Than Annually

As we stated, interest usually is compounded more often than annually. In these situations, we simply adjust the number of interest periods and the interest rate. If we want to know what $10,000 will accumulate to at the end of 3 years if interest is compounded quarterly at an annual rate of 12%, we just look down the 3% column until we reach 12 periods (see Table 1.3).

**Table 1.3**

The factor is 1.42576, and employing our general formula, the accumulated amount is $14,257.60, determined as follows:

Accumulated amount = Factor x Principal

= 1.42576 x $10,000

= $14,257.60

### Determining the Number of Periods or the Interest Rate

There are many situations in which the unknown variable is the number of interest periods that the dollars must remain invested or the rate of return (interest rate) that must be earned. For example, assume that you invest $5,000 today in a savings and loan association that will pay interest at compounded annually. You need to accumulate $8,857.80 for a certain project. How many years does the investment have to remain in the savings and loan association? Using the general formula, the answer is 6 years, determined as follows:

Accumulated amount = Factor x Principal

Factor = Accumulated amount / Principal

= $8,857.80 / $5,000.00

= 1.77156

Looking down the 10% column in Table 1.3, the factor of 1.77156 appears at the sixth-period row, Because the interest is compounded annually, the sixth period is interpreted as 6 years. This example was constructed so that the factor equals a round number of periods. If it does not, interpolation is necessary. The examples, exercises, and problems in this book will not require interpolation.

We can use the same method to determine the required interest rate. For example, assume that you invest $10,000 for 8 years. what rate of return or interest rate compounded annually must you earn if you want to accumulate $30,590.23? Using the general formula, the answer is 15%, determined as follows:

Accumulated amount = Factor x Principal

Factor = Accumulated amount / Principal

= $30,590.23 / $10,000.00

= 3.05902

Looking across the eighth-period row, we find the factor of 3.05902 at the 15% column.