One small thing that might make the business world just a tiny bit better is all of us agreeing how we measure growth.

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. And the point here isn’t exactly getting something right or wrong, but 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.

**Calculating Simple Growth**

To calculate simple growth, subtract the final number from the starting number and divide the result by the starting number. Then multiply by 100 if you want to show it in percentage. So, for the example above:

(150-100)/100 = 50/100 = .5

((150-100)/100)*100 = 50%

And you can see that as a spreadsheet here to the right. C2 shows 50 because it’s the product of subtracting A2 from B2. Then the formula divides that by A2, to generate .50. Or, if you multiply by 100, 50%.

There is also a simpler formula that also works. Divide the more recent by the previous, and subtract 1. That gives the same result.

You can see that in the second illustration here.

**Calculating Compound Growth (CAGR)**

CAGR stands for compound average growth rate. The active word there is “compound.” It means that the growth accumulates, like interest. So if you grow 10% per year over four three years you’ve actually grown from 100 in the first year to 133 in the fourth.

There’s a formula that calculates the CAGR over a period of years (or months). 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’s:

**(last number/first number)^(1/periods)-1**

Which is probably easier to see if you look at the spreadsheet illustration here to the left. The first row has the first year and last year plus the CAGR formula. The second row shows the result when 100 grows at 22.47% over three years. And the combination illustrates and awkward point about how many years are involved: it would be easy to call that two years of growth, but the “periods” number here is three, not two. And you can see the spreadsheet formula clearly here, I hope. And the 22.47% growth from 100 to 122.47, and then again to 150.

Maybe it helps on that point to show the same thing for growth from 100 to 150 over four years. That’s another simple spreadsheet, and the calculation shows that the CAGR for growth from 100 to 150 over four years as 14.47% per year.

**Conclusion:** maybe it’s just that I like numbers, maybe 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.

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.

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.

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.