One small thing that might make the business world just a tiny bit better is all of us agreeing how we measure growth. There are different annual growth rate formulas.
There’s a simple growth rate formula. But there’s also a compound annual growth rate formula, often shortened to the acronym “CAGR formula”.
I hesitate to wade into this subject because so many people have so many definitions. And you’d think it was obvious, but then suddenly I find myself in meetings, or on the phone, and I’m wondering whether we’re all on the same page.
The point here isn’t to get something exactly right or wrong. The point is having growth percentages mean the same thing to everybody. Let’s get on the same playing field.
Here’s a quick quiz:
- Sales grow from $100 in one year to $150 in the next. How much growth is that?
- And what if sales grow from $100 to $150 over three years. How much growth is that?
Maybe I’m wrong, but I’ve had what I learned in business school confirmed for me many times by accountants and analysts. Let me explain the answers to this quiz.
Calculating Simple Growth Rate
Question #1 in our quiz above illustrates the concept of simple annual growth rate.
To calculate simple growth, subtract the starting number from the final number, and divide the result by the starting number. Then multiply by 100 if you want to show it in percentages.
So, for our example the formula would be:
(150-100)/100 = 50/100 = .5
((150-100)/100)*100 = 50%
You can see that in Simple Growth Rate Formula 1 image above. It depicts a sample Excel spreadsheet.
Cell C2 shows the number 50 because it’s the product of subtracting A2 from B2. Then the formula divides that by cell A2, to generate .50. Or, if you multiply by 100, it becomes 50%.
So, in the Excel spreadsheet image above, the simple annual growth rate for 2010 over 2009 is 50% growth.
There is an even simpler formula that also works.
Simply divide the more recent number (year, quarter, month) by the previous period’s number. Then subtract 1. That gives the same result.
You can see this in the Simple Growth Rate Formula 2 image, above. In longhand math, the formula would be this:
150*100 = 1.5 – 1 = .5
And, of course, .5 is 50% if you want to state it in percentage terms. So you arrive at the very same answer of 50%, just like in the first formula.
Remember, simple growth rate typically describes growth over a single period of time. For example, simple annual growth is from one year to the next year.
But simple growth rates can also be used for other periods, such as quarterly growth from one quarter to the next quarter. There is no averaging involved in simple growth rates.
Calculating Compound Growth (CAGR) Rate
CAGR stands for compound annual growth rate. The active word there is “compound.” It means that the growth accumulates, like interest.
So if you grow 10% per year over three years you’ve actually grown from 100 in the first year to 133 at the end of the third year.
Remember that quiz we started with in the beginning? Question #2 illustrates compound annual growth rate. If you have the starting number and the ending number, like in the quiz, you’re figuring out the average annual growth rate.
There’s a formula that calculates the CAGR rate over a period of years. It’s hard to explain, but easy to use.
What’s especially awkward is the ^ sign in spreadsheet formulas stands for “raised to the power.” So 4^2 (four squared, which is four raised to the second power) is 16, and 2^3 (two cubed, which is two raised to the third power) is 8.
When the CAGR formula is written out, it looks like this:
(last number/first number)^(1/periods)-1
This is probably easier to see if you look at the CAGR Formula spreadsheet illustration, immediately above.
Row 1 has the first year and last year in cells 1A and 1B. The CAGR formula is in cell 1C.
Row 2 shows the result when 100 grows at 22.47% over two years (by the start of the third year).
And the combination illustrates an awkward point about how many years are involved. It would be easy to call that three years of growth, but the “periods” number here is two, not three.
You can see the spreadsheet formula clearly, I hope. And see the 22.47% growth from 100 to 122.47, and then again to 150. Two periods.
Maybe it helps on that point to show the same thing for growth from 100 to 150 over three years. That’s another simple spreadsheet. See “CAGR Formula – 3 Year Period” image above.
The calculation shows CAGR growth from 100 to 150 over three years is 14.47% per year. The number 150 is what you would have at the start of the fourth year.
And yes, all these references to years get confusing. We say that something grew by 14.47% from Year 1 to Year 4. We use the anchor year and the end year, so at first glance it seems like four years. But what we really mean is that there were three full years of compounded growth. In other words, it was a three-year period involved.
And THAT illustrates precisely how confusing this topic can be!
Final Thought About Growth Rates
Maybe it’s just that I like numbers. Maybe it’s that I use them a lot, perhaps too much.
But it’s nice when the growth figures we talk about mean the same thing to one and to all.
That’s why it’s important to understand the difference between simple annual growth rate and compound annual growth rate. It’s about getting on the same page.
Additional CAGR Resources
It might help to see the CAGR calculation in a spreadsheet or experience it in an interactive calculator. For that, please see:
- CAGR Formula for Google Sheets – Go here (Make a Copy of this document or download it to your device to use the calculator.)
- CAGR Calculator Online – Go here
Editor’s note: This article was updated on January 6, 2019. The main points and formulas are just as valid as the day this piece was written. Certain language was clarified per the comments below, to eliminate confusion. We hope it helps you understand the concept of simple annual growth rate and compound average growth rate.
Neal O'Sullivan
Tim, you make a good point and well said.
David
Thanks for the info, very well explained.
Buhle
Very insightful Tim, thank you.
Raychel
So if I want to calculate % growth in revenues over time, is it more “correct” to use the CAGR or the simple growth formula?
Jeremy
You don’t circle back and suggest how to get on the same page? You just explain the two chief ways people calculate growth. I think you are merely suggesting to label the growth as either Simple or CAGR, right?
I was hoping to read more as to why you felt one way was better than the other.
@Jeremy,
I think the CAGR is better anytime you have more than one period of growth. With one year or one month, use the simple formula. For anything with more periods than 1, the CAGR formula is the only one that is correct.
To understand that, look at my last paragraph that has the example of growth from 100 to 150 over four years. That’s annual growth of 14.47%, not 50% or even 33%.
Ondrej
I’d return back to the simple formula for growth from year to year that was mentioned in the beginning:
Divide the more recent by the previous, and subtract 1. That gives the same result.
(last number/first number)-1
How we get this formula is actually taking the CAGR formula and setting the number of periods to 1.
(last number/first number)^(1/periods)-1
Whenever we apply power of one to a number, we get the same number, so (1/periods) is cancelled out for a single period – or year in this case.
Eric
Here’s a related question on modelling using CAGR.
Let’s say I want to model the estimated quarterly sales of a product over 1 year. In the excel model I want to show the sales for each of the 4 quarters. To keep it simple, I want the sales for each quarter to be straight line growth. Lastly, I want the CAGR of these 4 quarters to be 10%
How do I go about figuring out what the estimated sales will be for each of these quarters?
So in summary, I know beginning value, the number of periods, and the CAGR. How do I determine what the 4 quarters sales will be?
Thanks!
Eric, re your question, go for simple. Especially with forecasting. If it were me, I’d set up a cell for the beginning value, four cells for each of the periods, a cell for the total for those four cells, a cell to measure CAGR, and then (somewhere that doesn’t show) a cell to put a quarter-to-quarter growth rate assumption.
From there it’s easy. It’s not clear whether you want the year-to-year total-year growth to be 10%, or the CAGR for the four quarters to be 10%, or something else. If you want the growth year-to-year to be 10% then it’s really simple, because thats first year times 1.1. If you want every quarter’s growth to be 10% then last easy to, previous quarter times 1.1. Whichever way you want it, it’s easy.
Don’t overthink a forecast. It’s all educated guessing. The CAGR is more useful to look at actual growth.
Charlesml
To put it clearly, if you add a small scalar – just a big enough number to bring the negative values positive – your growth rate will be very large.
skube
You state simple growth rate calculation as
however that is linguistically incorrect. I think you mean “subtract the starting number from the final number”.
Yes, you’re right. You subtract the starting number from the final number and divide the result by the starting number. So if the starting number was 100 and the final number 150, then it is 150-100=50 and then 50/100 = .5 so that’s 50% growth. Thanks for catching that error.
Douglas
I am not sure where you get 14% over 4 year? (150/100)^(1/4) – 1 = 10.6%
This makes sense:
We start with 100$
After year 1 we have (100*1.106) = 110.6
After year 2 we have (110.6*1.106) = 122.47
After year 3 we have (122.47*1.106)=135.45
After year 4 we have (135.45*1.106)=150
The way you show it above is growth over 3 years. From the start of year 1 to the end of year 3 (which happens to be the beginning of year 4).
Douglas, don’t worry, the formula is correct, even if my wording is confusing. I do in fact show growth over three years, which is why the formula has ^(1/3) in the last phrase, not ^(1/4). The confusion here is my fault, with the wording; but the formula is correct. And your suggestion for four years is also correct. I think I fell in the trap of calling 2017 to 2020 four years when the fourth is actually the anchor year, 2017, and the growth is three years, 2018, 2019, and 2020. Sorry.
Richard
Douglas,
The formula is incorrectly stated. It should be: =(B1/A1)^(1/4)-1 if looking at a four year period.
Andy
The formula is correct, but I think that there is (a common) mix-up of the initial investment or start value.
Initial investment = Start of Year 1 = 100
To take the example of 4 years, we have to calculate the the following 4 values:
End of year 1
End of year 2
End of year 3
End of year 4
This is exactly what Douglas did and I tend to agree with him.
Just imagine that we want to know the CAGR of our sales from 2011 to 2015 (5 years). What’s the start value? It is the sales revenue generated in 2010. That’s where we start from.
Correct me if am wrong.