How to calculate CAGR without wanting to throw your laptop
Your annoyance with CAGR may start withering away if you just think of it as an “average” annual growth rate.
If you do numbers regularly in the course of your work, chances are that you’re frustratingly familiar with a term called CAGR—or compound annual growth rate. Some acronymise it to make it sound like “Kagger”. It’s the mystical, little-understood, annoying cousin of the regular averages that we learnt about in the fifth grade, which were a whole lot easier. No one really taught us CAGR at school. No wonder, everyone in my profession, barring some, seems to find a CAGR calculation to be an inconvenience in the day’s task.
Why does CAGR exist in the first place? Wasn’t it difficult enough already to calculate percentage rises and declines, many journalists would wonder. Just the other day, a newsroom colleague defended a CAGR figure they used in a story by saying they got it from an AI tool. My initial reaction was one of exasperation: a financial journalist needing a Q&A bot to figure CAGR wasn’t exactly ideal. My follow-up reaction was mellower: it was that I needed to write this piece that you’re now reading.
Your annoyance with CAGR may start withering away if you just think of it as an “average” annual growth rate. The adjective “compound” gives us the impression that a choice is being made—like in the case of whether to charge someone simple interest or compound interest. So when your boss tells you to use CAGR, you think you’re being asked to go the complex way when a simpler one exists.
But don’t hate that boss. Wherever CAGR is relevant, it is actually the only thing relevant (i.e., there’s no other kind of average annual growth rate). Let me explain why, and how to recall the verbose formula to calculate CAGR easily.
Let’s start with the basic averages we understand instinctively. Five friends contribute ₹100, ₹120, ₹80, ₹70, ₹130 to buy a goodie worth ₹500. Their average contribution? The sum (₹500) divided by the number of friends (5), i.e. ₹100. What this essentially means, and watch my words, is that if each contribution in the set were to be replaced by the average, the final reality wouldn’t change: ₹100 + ₹100 + ₹100 + ₹100 + ₹100 would also be ₹500.
So an average typically collapses a set of numbers into a single number. If you replace each of the original numbers of the set with this average, the result doesn’t change.
Now let’s apply this idea to growth rates, our topic of interest today. Say the economy grew 4%, 7%, and 10% in successive years. The average, by the above logic, seems to be (4+7+10)%/3, or 7%. Let’s test this out: if the economy’s size in the year of reference was ₹100, then:
Size in year 1 = ₹100 + (4% of ₹100) = ₹104
Size in year 2 = ₹104 (i.e., the new size, that of year 1) + (7% of ₹104) = ₹111.28
Size in year 3 = ₹111.28 + (10% of ₹111.28) = ₹122.41
Now let’s perform the test I described above. Replace each year’s growth rate with the average we just assumed (i.e. 7%), and repeat this exercise:
Size in year 1 = ₹100 + (7% of ₹100) = ₹107
Size in year 2 = ₹107 + (7% of ₹107) = ₹114.49
Size in year 3 = ₹114.49 + (7% of ₹114.49) = ₹122.50
We see that the size in year 3 (and in fact in all years in between) in this “check” is different from the actual ones, once we replace the yearly growth rates with the average growth rate. This means the check has failed; something seems amiss.
What went wrong?
The problem here is that in our example of friends buying a goodie, the numbers added up on top of each other: you simply added each friend’s contribution to pool the full ₹500 budget. But in our example of an economy’s size, each year, the percentage gets applied on the new size of the economy, meaning the annual additions are not identical each year. The two situations are fundamentally different: the first one is additive; the second one has a multiplicative nature to it. So the two need different kinds of averaging.
CAGR is the correct (and the only correct) average for the second situation: if you use the AI tool that my colleague did, you’ll find that in the above example, the CAGR would be 6.972%. If you used that, the replacement technique would work:
Size in year 1 = ₹100 + (6.972% of ₹100) = ₹106.972
Size in year 2 = ₹106.972 + (6.972% of ₹106.972) = ₹114.43
Size in year 3 = ₹114.43 + (6.972% of ₹114.43) = ₹122.41
That’s the number we were looking for, proving that 6.972% is the true average, which satisfies the essential property of being a hypothetical number that can replace each element of a set without any damage.
Since each year, this average growth is compounding (as the percentage is being applied to a bigger number each year), we call it the compound annual growth rate, but as you can see, this is the only average growth rate, so better forget the word “compound” for simplicity.
Where a CAGR is relevant (because of the multiplicative nature of changes), a simple average growth rate (the 7% figure in the above example) doesn’t really satisfy the basic conditions of being an average. So you need to find a special name if you want to use the latter for any reason (out of laziness, or not knowing the CAGR, or to get a rough estimate, or sometimes, poor availability of data), since that’s not the natural method. Let’s call it the “rough average growth rate”! It’s not CAGR that needs an elaborate name and abbreviation!
What about the horrific formula?
Now, the formula using which I came up with the 6.972% figure is the other annoying part for those of us who found high school maths frustrating. The formula goes thus:
The figure n here is the number of years.
An easy way to remember this formula will reveal itself if you go back to the most basic mathematical formula that most professions use (of course, after addition, subtraction, multiplication and division): calculating percentage changes. We typically learn it as the following famous phrasing:
But if you look at it closely, this condenses down to:
Which is a remarkably easier way to put it: for example, you don’t have to bother about putting the brackets right when doing this on a calculator!
Look carefully again: this is just a special case of the CAGR formula above, where the number of years is 1. It’s the “average annual growth rate” for one year, or in other words, the growth rate for one year.
Memorising the basic percentage change formula as (Final/Initial – 1) does two things. One, it makes it easier, quicker and less clunky to get percentage changes on a calculator or in a spreadsheet. Two, it suddenly makes the CAGR formula easier to recall: if you’re confused about any part of the formula, just apply the special case of n = 1 and see if it yields the percentage change formula that you intimately know of.
Here’s the third and last trouble that many face while conjuring the CAGR formula. What is the “number of years”? If we are calculating growth between 2016 and 2026, is it 10 or 11? (11 being the number of years being covered.)
The answer, if you have this question, is to think of it as the number of “jumps” or “increments”. To reach 2026 from 2016, you jump 10 times, so the number n is 10. Or apply the special case of 1 and see how it works (this trick is always the best): to be able to use n = 11, you’d need to have n = 1 to cover only 2016, which means you’d be saying the number grew a certain percentage in zero years, which is impossible. So 2016 must correspond to n = 0, and hence 2026 to n = 10.
Hope this demystifies the peculiar concept of CAGR. If this is too basic for you, you may want to read up about log-based CAGR, and you might say, “Now we’re talking!” But that’s not relevant for most everyday uses, so I’m keeping it to this much.
Do you have any terrifying CAGR stories to share from the workplace? Do send them my way! Until next time…