By Yuriy Smirnov Ph.D.

The concept of time value of money is based on the idea that $1 now is worth more than $1 in the future. The basis of this idea is rather straightforward. If you have $1 now, you can invest it and get more value in the future. Thus, the future value (FV) of money is a value at a specific date in the future based on the present value (PV) and on the interest rate.

Note that the process of transforming present value to future value is called compounding.

The formula to calculate the future value at the end of period N using simple interest is as follows:

FV_{N} = PV × (1 + r)^{N}

Here PV is a present value, r represents an interest rate earned per period, and N is a number of periods.

Simple interest, however, is rarely used because it does not take into account the possibility that interest earned can be reinvested. In other words, there is no compounding in such a case.

The formula to calculate the future value at the end of period N using compound interest is as follows:

FV_{N} = PV + PV × (1 + r) × N

Here PV is a present value, r represents an interest rate earned per period, and N is a number of periods.

As was mentioned above, simple interest is rarely used in financial calculations because it does not consider the impact of reinvesting interest earned. For example, if $1 is invested today at an annual rate of 12.5%, its future value after 15 years will amount to $2.88 using simple interest and $5.85 using compound interest.

The difference of more than two times is the result of compounding. Actually, the worth of $1 increases linearly using simple interest and exponentially if compound interest is applied. The effect of exponential growth is graphically presented in the figure below.

So, the greater the interest rate and the longer the investment, the faster the future value of $1 will grow.

Let’s assume that $100 is invested today for 5 years at an annual interest rate of 7.5%. The step-by-step compounding is shown in the figure below.

We start with $100 invested today and get $107.50 at the end of the first year. Now $107.50 is invested again and at the end of second year will amount to $115.56. This process continues the following years until the future value will grow to $143.56 at the end of the fifth year.

FV _{End of 1st year} = $100 × (1 + 0.075) = $107.50

FV _{End of 2nd year} = $107.50 × (1 + 0.075) = $115.56

FV _{End of 3rd year} = $115.56 × (1 + 0.075) = $124.23

FV _{End of 4th year} = $124.23 × (1 + 0.075) = $133.55

FV _{End of 5th year} = $133.55 × (1 + 0.075) = $143.56

The compounding effect occurs after the first year when the interest of $7.50 is also reinvested at 7.5%. At the end of the second year, it will earn additional interest of $0.56 ($7.5 × 0.075) to the interest of $7.50 earned on the principal amount of $100. The process of compounding continues to the end of the investment life span.

We can also get the same result using the formula mentioned above.

FV _{End of 5th year} = $100 × (1 + 0.075)^{5} = $143.56

If simple interest were applied, the future value of $100 would amount to $137.50.

FV _{End of 5th year} = $100 + $100 × 0.075 × 5 = $137.50

You can also calculate the future value of money using our online calculator or tables.

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