Compound Growth: Definition & Formula

Instructor: Sharon Linde

Sharon has a Masters of Science in Mathematics

Are you looking at a series of volatile annual sales numbers for a company and trying to find a way to make sense of it all? Using the formula for compound annual growth rate can help you answer these and other questions.

Compound Growth in Real Life

Suppose you just got an entry-level position in financial analysis. During your first week on the job, your boss asks you to look at a particular company and provide her with some annual growth numbers for the last four years. As the result of your research, you come up with the following figures. These represent an average annual growth rate of 23.8%.

Sales in Millions) Annual Growth %
$10 N/A
$12 20%
$9 -25%
$15 67%
$20 33%

After looking over your figures, your boss starts asking about the ending value of an initial $10M investment and an annual return of 23.8% in each year. To answer her question, you use the compound interest formula, which gives you the following:

10M x (1 + 0.238)^4 = $23.45M

However, your data shows the ending value to be $20M; where did you go wrong? And more importantly, how can you provide your boss with the correct answer to this not so entry-level question.

Definition of Compound Growth

We can define compound growth as the average rate of growth experienced by an investment over a multi-year period. One way to think about the compound growth rate is that it takes all the hills and valleys into account when considering the investment landscape. As we saw in our opening example, averaging year-end growth rates cannot provide us with an accurate measure of compound growth over several years. So, how do we calculate this number?

Well, we let the compound growth be equal to whatever growth rate would give us the same beginning and ending values for the same length of time. We do this by using the following formula:

compound growth

Since the initial value of the investment is $10M, the final value is $20M, and the elapsed time is four years. Yes, there are five years of data. However, since they're reported at the end of the period, there are only four years between the first and last data points. Plugging these known quantities into our equation then yields:

CG = ($20M / $10M)^(1/4) - 1

CG = 2^(1/4) - 1

CG = 1.189 - 1

CG = .189 or 18.9%

Does this new number give us what we want?

Final value = initial value x (1+1.189)^4

Final value = $10M x 2.00 = $20M, which matches our original data.

Another thing to note here is that an 18.9% growth rate is significantly lower than the 23.8% growth rate you got by averaging the year-end growth rates. It's not always the case that the two methods differ by this much. When growth is fairly steady over time, then both methods give you very similar numbers. The larger and more chaotic the changes are from year to year, the better the compound growth method works.

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