# Calculating CAGR Manually

CAGR - Compound Annual Growth Rate
Edited on Mar 28, 2022
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How would someone go about calculating CAGR manually? During a prior interview I was asked to find revenue CAGR without the use of a calculator which I wasn't able to complete. If I had Y0 revenue of 171M and Y5 revenue of 232M - can someone provide a detailed walkthrough of how to calculate the CAGR over the 5-year period - as I don't want to repeat this mistake in future interviews. I know the formula would be (232/171)^(1/5) but I fail to see how this could easily be calculated manually.

I didn't find the PrepLounge article to be helpful for performing the raw calculations. Alternatively, if I'm wasting my time prepping for a similar question, i'd also be interested knowing that.

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Hi there,

this is a tricky one indeed! At least to my knowledge, there is no simple way to calculate this with precision, unless the applied root happens to yield an integer (here not the case). But it also doesn't matter.

While I'm of course not aware of the exact context, I can share what typically works in such situation.

Firstly, (any reasonable) MBB interviewer would not expect you to calculate an exact CAGR over 5 years manually in a live interview scenario, given the complexity of the computation without access to a calculator.

That said, you can always give an rough approximate. Management consulting is not an exact science, which is why an 80/20 type of approach is typically enough (esp. in a back-of-the-envelope), as long as the answer / recommendation does not change. The steps to do this can be as follows:

1. Check with the interviewer if he/she needs exact solution and propose a quick & dirty approximation instead (you'll actually score plus points for practicality + problem-solving right there!)
2. Calculate the approximation:
1. Get overall percentage growth 232/171 = 35%
2. Divide by no. of years (5) to get linear growth = ~7%
3. Apply a discount of 0.5-1% to account for the compounding effect
3. Give range as answer to appreciate the inaccuracy of the heuristic (~6-7%) and qualify again (say you ofc could verify exact # with a calculator for exact result, but it won't change the overall answer)

In 99% of the cases, an interviewer will be more than happy with this approach and answer and appreciate the practicality of the approach. I certainly would have been.

Hope this helps. Do feel free to send me a PM, if you have further questions.

Regards, Andi

(edited)

Completely right to ask and prepare for this type of question. Altough unlikely to have a math calculation as difficult as this one (usually you get round numbers), this might happen again and you don't want to be caught off guard again. Plus you will learn in the answers some good tips valid for easier math questions.

Here is how I will solve - detailing orally all steps to make sure my interviewer follows:

I ask how precise the CAGR needs to be - This is crucial, you don't want to waste time on math question especially if the interviewer intended to go through it quickly!

if he says 1pt precision is fine you can refer to the answer below (as Andi said, enough 99% of the time), if he asks for something more precise (+/-0.1 pt - no point in that but happened to me once) here is what I will do:

Note: It would'nt be fair to expect from a candidate with no strong scientific background the precise calculation below. So if you are not an engineer don't even bother reading further.

1. CAGR = (232/171)^(1/5)-1
2. 232/171= 1+(232-171)/171=1+62/171
3. 62/171=620/171/10; 620=600+20=4*150+4*5=4*155 so 620/171 sould be around 3. something. 620=3*171 (513) +107
4. 232/171=1.3 + 107/171/10
5. You redo the same with 107/171: 1070=6*171 (1026, easy because you just calculate 3*171 in step 3; so just need to do 513*2) + 44
6. So, 232/171=1.36 + 44/171/100 - we can forget about the 44/171/100 and round down (I would have round up if it was above 1/2/100)
7. Now for 1.36^(1/5) you need to use taylor expansion formula if you want to maintain your precision. it says basically that (1+x)^n~1+nx+n(n-1)x²/2
8. so CAGR=(1+0.36)^0.2-1~1+0.2x0.36+0.2*(-0.8)*0.36²/2-1
9. CAGR ~ 7.2% - 0.16 x ~0.13 (no need to have more digits) / 2
10. 7.2% - 8%x0.13 = (7.2 - 0.8 - 0.24)% ~ 6.2%

Again, it would be very unusual to expect from a candidate to do a taylor expansion. The interviewer who asked me this graduated from the same engineering school as me, maybe it was a way for him to connect over our common math-focused educational background.

You should be aware of two things.
One is the 72 rule. If your revenues double in n years, your CAGR is 72/n%. It is very precise.
The second one is a raw estimate, that has to be taken with caution. If your increase in revenues has been by x%, your CAGR can be estimated as x/number of years minus something. This "minus something" if x is less than 50% can be about 1%.
So in this case 232/171=1.36, which means a 36% increase, which means a 6.3% CAGR that you can calculate more or less with 36%/5 (years)=7.2% to which you deduct 1%.

You have to deduct something because CAGR is a geometric average while dividing by 5 is an arithmetic average and the first compounds faster.

You don't need an exact number, you need an approximation.

So 232-171 = 61 is the absolute difference after 5 years

61 / 171 ~ 35% is the relative difference after the 5 years (60 is ⅓ of 180, so 61/171 should be slightly higher → 35%)

35% / 5 ~ 7% is the average growth, but compound growth should be slightly lower… more likely to be ~6%.

(edited)

Hello!

It would be very very very very weird that they ask you something like this in a strategy consulting interview, so I wouldn't worry about it.

Hope it helps!

Cheers,

Clara

Hi there,

First, ask the interviewer if they require a precise answer. I would be shocked if they require you to calculate to the 1/5 power.

My guess is they wanted you to calculate the total % change over 5 years and divide by 5. They would have then expected you to flag that this is a rough estimation, not exact.