# Is there a trick to quickly estimate the average annual growth percentage / to or total compounded growth, without a calculator? i.e. X will grow with 5% for the coming 10 years.

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Edited on May 06, 2021
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Hi,

You can use the rule of 72.

For example, if you know the revenue will grow from 100M to 400M in 12 years and you need to know the CAGR:

1. 100M will double and grow to 200M in 6 years
2. CAGR = 72 / 6 / 100 = 12%
3. Note that you can take 70-72 depending on what is easier to calculate

Best

great thank you! And if you would want to turn it around? For example a 100 million company is expected to grow for 5% per year for the coming 7 years?

(edited)

Sure, you need to re-build the formula: CAGR in percent = 72 / years to double, so that is years to double = 72 / CAGR in percent. Now you calculate growth rate over considered period = considered period / years to double. In your example this would be years to double = 72 / 5 = 14.4, growth rate over considered period = 7 / 14.4 = 0.49. Please remember that these are rules of thumb and the bigger the percentages and the considered period get the bigger the error will be. In your example the correct answer would be 1.05^10 -1 = 0.41.

### Case study - Tips for calculating with an annual growth rate

I would answer it like this:

3 * 5% = 15% + a margin for compounding of approx. 1% so the number will grow by 16%.

### Math prep : CAGR without excel - any tips?

Hi Anonymous,

basically here you have to compute

(1+x)^a=V

where you don’t know x and where a=5 and V=1,5.

For relative low interest rates and periods, the simpler process for precise results is the following:

1) Calculate what would be the solution with simple interest – in this case, 10%; the real result will always be lower than that

2) Decrease by 1 percentage point – in this case, 9%

3) Apply Taylor series till the third element. To do so, you would have to learn by heart the following formula:

1+ax+1/2*a*(a-1)*x^2

Given point 2, x=9%

1+0,45+1/2*5*4*0,09^2=

=1,45+10*0,0081=

=1,531

4) Repeat decreasing till when you find a value below V=1.5

Given x=8%

1+0,4+1/2*5*4*0,08^2=

=1,4+10*0,064=

=1,464

Thus the result is between 8% and 9%.

Hope this helps,

Francesco

(edited)

### cagr calculation (estimate)

You should be careful with rule of 72. With big growth rates it is not working. If growth rate is more than 15% results can be drastically different.
In PST sample "A" there is a question with CAGR (which, I suppose, is a reson of your question) but a specific way how you can fastly and correctly calculate an answer there is open question for me also.

The “rule of 72” will help you know when, given an average yearly return, the principal will double (72 / avg yearly return = # of years to double principal). However, to have a precise

answer the calculation will be required.

Hope this helps,

Andrea

### Math prep : CAGR without excel - any tips?

Hello,

You can use binomial appriximation but may have to iterate depending; works best with low value of r as noted below.

150/100 = (1+r)^5

Binomial approximation: means, set the right side to 1+ 5*r .... and ignore higher power terms in r for now..

Solving will give you r=0.1. Now this is equivanent to 10% simple ineterest. So compound rate will be smaller than 10%.

Also plug back and check how far off are we. (1.1)^5 = [(1.1*1.1)^2 ] *1.1 = (1.21^2) *1.1

~ 1.44*1.1 = 1.44+.14 = 1.58. Now 158 is pretty close to 150. You can stop here.

OR you can try 9% and see if that works. Now calculate 1.09^5 in bite size steps.

1.09^2 = 1+ 0.18+ 0.0081 ~1.19 ----- use (a+b)^2

Repeat again with (1.19)^2 = 1.38 + 0.19^2 ~ 1.38 + 0.04 ~ 1.40. ------ By now you have 1.09^4 approxiately

Last step 1.4*1.1 = 1.54.

So still higher than 1.5. Repeat with 8%

Regards,

SR

### Math prep : CAGR without excel - any tips?

Easiest is the rule of 72 (if you divide 72 by the interest rate in question, you'll get the number of years it will take your money to double at that interest rate). So if interest rate were 12%, it'll take 5yrs to double 100 to 200. You are at 150 after 5 years, so, you know that it will take more than 5yrs to double, so the interest rate has to be <12%. At 6%, it'll take 12yrs to double, which means at year 6 it is at least 150, so 6% is too low. You can likely guess it's not exactly 9% because compound interest grows faster with each year, so you can estimate somewhere closer to 8%. If you practice more, you can get better at this.

I also have this same question....

is there update

(edited)

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