# Compounded Annual Growth Rate with Negatives

I was trying to figure out the CAGR of EBIT of certain companies that filed for bankruptcy. Needless to say, most of these show a negative trend in their EBIT values, often starting from a positive in the past and ending up with a negative value. However, the CAGR often does not work when one of the values is negative. I did look this up online and found a couple of formulae, but I never understood the math behind those formulae. Can someone please help me figure out the logic we need to compute CAGR when either the beginning or ending value is negative? Also, how do we check if the CAGR figure we get is actually right? I mean, if we have 2 positive values and the percentage change in them, we can always substitute them back like

y=x(1+r)^n

So, is there any way to cross-check in the case of a negative beginning or ending value? Please help! Thank you in advance!

• Since you assume familiarity with the acronyms CAGR and EBIT the question is probably more suitable for money.stackexchgange or economics.stackexchange . Commented May 13, 2022 at 17:19
• @KurtG. Alright, thank you! I wanted to know the mathematical process behind it all, so thought this was the right place. Commented May 13, 2022 at 17:39
• To this question I' ve posted several answers. See here, here and here. Commented May 13, 2022 at 23:35

Essentially the mathematics of this is that if you multiply a real number by itself, you get a non-negative real number. Consider the following examples:

• If you go from $$5$$ to $$8$$ over $$10$$ years, this involves multiplying $$5$$ by $$\sqrt[10]{\frac85} \approx 1.048122$$ ten times, and we call this a compound average growth rate of $$4.8112\%$$ (look at $$1.048122-1$$).

• If you go from $$5$$ to $$2$$ over $$10$$ years, this involves multiplying $$5$$ by $$\sqrt[10]{\frac25} \approx 0.9124435$$ ten times, and we call this a compound average growth rate of $$-8.75565\%$$ (look at $$0.9124435-1$$). It is actually growing smaller so has negative growth.

• But if you go from $$5$$ to $$-3$$ over $$10$$ years, you might want to multiply $$5$$ by $$\sqrt[10]{\frac{-3}5}$$ ten times. The problem is that is no real number which when raised to the tenth power (or even squared) is equal to $${\frac{-3}5}$$.

My advice is not to look at percentage changes if the sign changes and do not deal with percentage changes more negative than $$-100\%$$. Doing so leads to other traps: for example a $$-300\%$$ change might be seen as multiplying by $$-2$$, so doing it twice is the same as multiplying by $$4$$, in which case a quadrupling over two years is both a $$100\%$$ change each year and a $$-300\%$$ change each year, which does not aid understanding.

I did this in past, it works meaningfully:

$$\left(\frac{Last Value - First Value + \lvert First Value\rvert}{\lvert First Value\rvert}\right)^{\frac{1}{n}} -1$$

• You can edit if you want Commented Jul 28, 2023 at 9:52
• I gave my contribution, who wants can add his own Commented Jul 28, 2023 at 10:10