Compound Growth Rate

The compound growth rate for a pair of values, V1 and V2, and a number of time periods, n, is the constant growth rate, g, that will cause V1 to grow to V2 over n time periods. In equation form, g must satisfy

If you know V1, V2, and n, then you can solve for g:

The following examples can help explain the uses of the compound growth rate formulas.

Suppose sales increased from $100 million in 1999 to $180 million in 2003. What is the average annual sales compound growth rate? We know sales increased by a total of 80% over these four years. However, this does not represent an annual sales growth rate of 80 ÷ 4 = 20%. Rather, the annual sales growth rate is

The growth rate is less than 20% because of compounding. At a growth rate of 15.86%, sales will grow as follows:

In 2000, sales increased by 15.83% of $100, or $15.83. In 2001, sales increased by another $15.83 plus 15.83% of the additional $15.83 sales in 2000. Thus not only does the initial $100 in sales grow by 15.83% each year, but the additional sales in each year will also grow by 15.83%.

The same calculations apply to interest earnings. If you invest $100 in a savings account at 6%, then in one year you will have $106. In two years you will have $106 × (1.06) = $112.36. In three years you will have $112.36 × (1.06) = $119.10. In the second and third year you earned interest on the $100, plus interest on the interest you had earned earlier. This is called compound interest. See the entry for Time Value of Money for a discussion about compound interest.

Compounding Intervals

The compound growth rate depends on the beginning and ending values and on the number of compounding intervals. For example, if you started with $100 in a stock account and five years later you had $146.93, then the annual compound growth rate, which is also equal to the annual compound rate of return, would be

However, if you wanted to know the monthly compound rate you earned, the calculation would be

This is not equal to 8% ÷ 12 = 0.667%.