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…

Well, this ideal scenario is not really implausible. True, the year needs to be 365 or 366 days long to keep it in sync with how the earth moves, and neither number is divisible by 7, so you end up taking for granted that the first of January must shift days every year. But the multiplication table of 7 is just a tiny inconvenience in the larger scheme of things. Ways have been proposed to get past it and have one permanent calendar, once and for all.

We can intuitively predict how the hours, minutes and seconds will progress on an unknown day 20 years hence. We can quickly tell which year it will be 40 years from now. But to know the day a particular date will fall even one month later, we don’t feel sure unless we’ve referred to a calendar. Can this get more absurd?

You see, the kind of shifting calendar we currently use—and which we are used to the way we are to the air and water around us—is one of the oldest surviving ways of organising life, and it has seen little change for centuries. Rather, it has spread across the world with little resistance and evolution, as it helps unify humanity like few other things. We may have different languages and customs, but we operate with the same calendar to keep life smooth. Traditional and religious calendars may dictate local festivals, but the January 1 to December 31 cycle is so ingrained that not following it would cause chaos. The last time someone made a tweak to the calendar, countries saw riots.

So should we expect any massive change that could fundamentally change how we count our days? Well, not right now; though the world has been much closer to a calendar revolution in the 20th century (which I will come to in a bit). Let me argue why it would be nice to revive the debate.

The calendar we so intimately know is a sun-based one, called the Gregorian calendar, and has been in use since 1582, when Pope Gregory XIII decided to revamp the Julian calendar that was in existence back then. The older calendar, which Roman emperor Julius Caesar formalised more than 2,000 years ago in 45 BCE, was actually pretty much the same as what we know the calendar to be: it only counted three extra days every 400 years. In that sense, Gregory’s innovation in how we structure our calendar was marginal.

The flaw with Caesar’s calendar was that its three extra days every 400 years made it a bit too slow. Over the centuries, the equinoxes were not falling on their expected dates any longer, and if left unchanged, dates would begin to fall out of sync with the seasons. So Gregory’s innovation set it right by deciding that century years that don’t divide by 400 would no longer be leap years. It isn’t perfect yet, but now the error will not make itself felt for thousands of years.

The basic structure of Caesar’s calendar (which, too, was inspired from the structure of the ancient Egyptian calendar) still thrives simply because we’re so used to it. Who finds it intuitive to have a short February, equal lengths of July and August, and the ever-changing day for any given date? If you forget that it’s an unavoidable way of life, you might realise it’s a ridiculous way to structure your year.

The structural flaws of the Julian calendar have their roots in arbitrary choices of Roman dictators. Their sole (and crucial) priority was to average 365.25 days a year (and 365.2425 a year in the Gregorian tweak) over longer periods, and rightly so. But they also had an eye on the traditional imperative to align festivals with the lunar months, leading to irregularly-sized months. Secondly, the Julian calendar, and its Gregorian innovation, are deeply rooted in religion, hardly in science. Gregory wasn’t out to impress the Renaissance vote bank; the problem was that scholars had complained for centuries that the choice of date for the annual Easter, which was linked to the March equinox, was falling out of line with the sun’s position in the sky. Also at play was rivalry with the Jewish calendar, which didn’t face the issue in scheduling its own equinox-linked festival Passover. The Christians simply wanted a foolproof method of their own.

So essentially what we currently use to dictate our rhythm of life is a calendar that could reform itself only out of religious compulsions, where scientificity was unintentional. It led to quarrels within Christianity; Britain (which had tense relations with the Vatican) didn’t adopt Gregory’s reforms until 1752, and the Russian Orthodox Church still uses the Julian calendar and is by now 13 days out of line with the earth’s motion! This deeply unsecular calendar then travelled the world solely by virtue of European colonialism. It doesn’t really need to run the 21st-century world.

The world needs a secular calendar that can shed its religious and colonial baggage. A common calendar that pleases no religion, but works for all humans and makes date-counting—its basic purpose—easier.

The World Calendar, my favourite alternative, was proposed in the 20th century, and it made it close in debates at the United Nations. It retains the same 12 months. January 1 always falls on a Sunday. Each quarter is made up of months 31, 30 and 30 days long. That adds up to 91 days, or exactly 13 weeks, which means that each quarter, not just each year, starts and ends on the same day. The cycle of days and dates repeats every quarter, unlike every 28 years like now (our current calendar normally repeats every 28 years due to the leap year), making it easier to remember days. You wouldn’t need to change calendars every year!

But that adds up to 364 days, a day and a quarter too short. So it adds one extra date at the end of December—a global festival of sorts, Worldsday—that does not get either a day or a date. Every four years, to account for the leap year, we add a similar Worldsday at the end of June as well. Since Worldsday doesn’t get a day, the following January 1 or July 1 don’t fall out of cycle. Alongside, we get an extra day to celebrate a secular, simple calendar that governs our lives so intimately.

Of course, religion came in the way. The Monday to Sunday cycle is essential to some religions, and a date that doesn’t fall on any of these seven days would disturb that rhythm. Also, something that wasn’t perhaps a problem in the mid-20th century, the foundation of our current computer technology is entirely based on the Gregorian calendar, and a disruption to accommodate an unusual concept of Worldsday—a day that’s somewhat not a day—would bring chaos to digital systems.

Is there a way out? If you had to build a calendar from scratch, what would it look like? Do you have a solution, or would you rather continue with the current system? In terms of simplicity, it must win the day—but also the year (remember, that first and foremost, a solar calendar must stay in line with the seasons, with a year of accurate length).

Think about it, and please do write back. For now, it’s time to say goodbye, with the promise to write more about this topic, and to explore other alternative calendar systems, in my coming posts.

It was only in recent years that I began to realise that this quirk wasn’t as trivial or childish as I thought. As it turns out, the history of the world itself intersects surprisingly often with the history of the calendar and timekeeping. Think about it: when there was nothing, there was time. It’s always moving, only in one direction, leaving you with no option but to measure it, somehow. Early humans didn’t have any option either: and they scrambled hard to figure it all out.

I just finished reading two breathtaking books: Chad Orzel’s A Brief History of Timekeeping (BenBella Books, 2022) and David Ewing Duncan’s Calendar (Avon Books, 1998), and I’m truly spellbound at how the quest for accurate timekeeping has shaped humanity at all its stages—even when our prehistoric ancestors were largely clueless about the world. In this post, I share what I’m thinking right now as I chew on these books.

Even when maths was scarce and the number system was in dinosaur age, there were the sun and the moon, mysteriously rising out of nowhere and turning off every day, acting as natural clocks, demanding to be tracked, somehow. And track we did—in ways small and big. It gave birth to a remarkable timeline of events, driven in turns by agriculture (think of the havoc farmers would be in if the calendar was linked to the moon and not the seasons), religion (think of the tensions it would create if two priests began arguing about when a festival should be held), and medieval imperialism (think of why the British Empire would announce a massive prize for anyone who could figure out how to measure longitudes—and hence time—at the sea, a real pain for scientists at that time).

The process ended up shaping everything from mathematics, physics and engineering to astronomy, geopolitics, and our GPS and satellite systems today. Innovations in each pushed all others ahead: humanity’s puzzles, across disciplines, almost always got their answers when we ‘timed’ it right.

War—and love**—led to the basis for the calendar we use today. A major tweak to how we count days once led to a riot^. How to decide the date of Easter (and the imperative to avoid a method that mimicked the rival Jewish calendar) was the subject of hot debate and mathematical innovation for centuries. It’s truly revelatory to think of world history from the perspective of the calendar.

While I strongly recommend you read the books, here’s my list of three characters who impressed me the most (among many) along the journey:

Su Song: A Chinese polymath who designed the world’s first hydro-mechanical astronomical clock tower in the 11th century. Till then, something or the other was not quite working out as people everywhere tried to measure time. Su Song solved several issues with a clock that likely weighed in tonnes.

Ruth Belville (aka Greenwich Time Lady): A British woman who visited the Royal Observatory in Greenwich to reset her watch every week and then went door-to-door to meet (and chat with) all her subscribers so that they could sync their devices with hers. (Yes, clocks would need to be synced that often even in the 20th century—and it could be a bona fide business idea!)

Roger Bacon: The 13th-century British polymath who no one listened to when he insisted that the Julian calendar had an error, because of which it no longer aligned with the seasons. A pope who finally showed interest ended up dying before Bacon’s paperwork and calculations reached him in Rome. The remedy that Bacon had sought only came by in 1582. (In fact, the dating of Easter ultimately played a role in the correction of an imperfect calendar that had dictated the world for over 1,600 years. Had it not been for religion, would the leaders of those times have found it a matter critical enough that their calendar was sliding away against the seasons?)

These are just a few names; of course scientists and mathematicians during the middle ages played integral roles in solving the biggest questions about our earth’s motions. Some of them fought with each other for credit and acceptance when they lived, but the truth that matters 400 years later is simple: they all built upon years of human thinking and philosophy, benefited greatly from each other at a busy time for scientific discovery, and together, set the path for future innovations that would shape the world we inhabit.

*I didn’t, because by then, I had learnt that I could change the time in the device myself whenever I wanted for any experiment.

**The reference is to Julius Caesar, his entry into Egypt, and his subsequent and tireless infatuation with Cleopatra.

^The reference is to the shift from the Julian to the Gregorian calendar in the UK in 1752, due to which 11 days just vanished from the calendar. (If it happened today, how much rent would you pay your landlord for the affected month?)

All information shared here is my interpretation of my reading of the two books, and my own research. I learned about Su Song and Ruth Belville in Orzel’s book, and Roger Bacon found mention in both books.