Future Value of a Single Amount?

Future value of a particular quantity is its value at a specified interest rate in the future.

If interest is compounded, “future value” is the investment’s future growth. A lump sum invested at the start of a period (e.g., year 1) and kept intact is the single amount.

## Explanation

Let’s use the table below to explain the future value of a specific quantity.

This table shows the future value of $10,000 invested at 12% yearly interest for three years under a specified compounding pattern. This is a single-amount future value calculation.

No further investments or interest withdrawals. Future value or compound interest projections are crucial to many personal and commercial financial choices.

## Example

A corporation may want to know the value of a $50,000 investment after 5 years whether interest is compounded semi-annually versus quarterly, or what rate of return is needed on a $10,000 investment to get $18,000 in 7 years.

Both challenges require determining the future worth of a single amount.

These challenges can be solved with tables like the one above. This method is slow and inflexible.

You can also utilize mathematical formulas. The formula in the next section is used to calculate the accumulated amount of a single amount at different compounded rates in the tables above.

Formula for Accumulated Amount at Different Compounded Rates

## This formula defines variables as:

The principal amount

I = Interest rate

number of compounding periods

The following formula yields $14,049.28 from $10,000 compounded annually for 3 years at 12%:

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

= $14,049.28

One simple technique is to utilize tables that show $1’s future value at different interest rates and timeframes.

For $1 principal, these tables interpret the preceding mathematical formula for different interest rates and compounding periods.

Multiplying the future $1 amount by the required principal amount makes it easy to calculate any principal. Function keys on many hand calculators can solve these problems.

The table below shows $1’s future value over 10 periods with interest rates from 2% to 15%.

Continued from above, we wish to calculate $10,000’s future value after 3 years if interest is compounded annually at 12%.

Check the 12% column in the table till we reach 3 interest periods. The table factor is 1.40493, meaning $1 invested at 12% will grow to $1.405 in 3 years.

We multiply 1.40493 by $10,000 to calculate the principal’s future value as we want $10,000, not $1.

Despite a rounding error, the amount is $14,049.30, matching the table.

Formula for generalizing future value table use:

Amount accumulated = Table Factor x Principal

= 1.40493 x $10,000

= $14,049.30

This formula solves several similar problems.

As mentioned above, you may want to know what interest rate to get on a $10,000 investment to make $18,000 in 7 years.

Or you may want to know how long an investment must last to grow. In all these circumstances, we have two of the three formula elements and can solve for the third.

Interest compounded more than annually

Most interest is compounded more than once a year. We change the interest rate and interest intervals in certain cases.

To calculate $10,000’s value after 3 years assuming interest is compounded quarterly at 12%, check the 3% column until we reach 12 periods (see Table 1.1).

The factor is 1.42576. The general formula yields $14,257.60:

The total = Factor x Principal

= 1.42576 x $10,000

= $14,257.60

Setting the Periods or Interest Rate

In many cases, the unknown variable is the number of interest periods or interest rate that must be earned.

Consider investing $5,000 in a savings and loan organization that pays compounded interest yearly.

A project requires $8,857.80.

The savings and loan organization must hold the investment for how many years?

According to the general formula, 6 years is the answer:

The total = Factor x Principal

Amount accumulated / Principal

= $8,857.80 / $5,000.00

= 1.77156

The sixth-period row of Table 1.1’s 10% column has the factor 1.77156. Interest compounds annually, therefore the sixth period is 6 years.

In this case, the factor is a round number of periods. Without it, interpolation is needed. The examples, exercises, and problems in this article don’t need interpolation.

The required interest rate can be calculated using the same way.

Say you invest $10,000 for 8 years. What annual compounded interest rate must you earn to accumulate $30,590.23?

## The general formula yields 15%. Determined as follows:

The total = Factor x Principal

Amount accumulated / Principal

= $30,590.23 / $10,000.00

= 3.05902

The 15% column of the eighth-period row has a factor of 3.05902.