How to handle Percent Change and CAGR for negative numbers

by fortmarinus

Sometimes finance deals with negative quantities that become less negative over time. For example, consider a profit/(loss) of ($50M) in year 1 that becomes a profit/(loss) of only ($1M) in year 4. If we apply the traditional formulas for Percent Change and Compound Annual Growth Rate (CAGR), we find that the results do not align with common-sense interpretation. Beginning with % Change, the usual formula is:

\%\triangle=\frac{F-I}{I}=\frac{F}{I}-1, plugging in our example values: \%\triangle=\frac{-1--50}{-50} = -98\%

Common-sense says our profit is increasing, therefore we expect +98%. Using an absolute value in the denominator adjusts the formula in such a way that is consistent with the common-sense interpretation.

\%\triangle_{ADJ} = \frac{F-I}{abs(I)}, plugging in our example values: \%\triangle_{ADJ} = \frac{-1--50}{abs(-50)}=+98\%


Now, the usual formula for CAGR is:

CAGR=\left (\frac{F}{I} \right )^{\frac{1}{time}}-1, plugging in our example values: CAGR=\left (\frac{-1}{-50} \right )^{\frac{1}{3}}-1=-73\%

Again, the common sense interpretation expects a positive growth rate since profit is increasing. We can not, however, simply reverse the sign as with % change. Let us re-write CAGR to illustrate the solution. This form is identical to the usual formula. Re-arranging it in this way allows us to see that % Change is embedded in the formula:

CAGR=\left (\frac{F}{I}-1+1 \right )^{\frac{1}{time}}-1=\left (\%\triangle+1 \right )^{\frac{1}{time}}-1

If we replace % Change with Adjusted % Change, we will have an Adjusted CAGR that yields the growth rate consistent with the common-sense interpretation:

CAGR_{ADJ}=\left (\%\triangle_{ADJ}+1 \right )^{\frac{1}{t}}-1=\left (\frac{F-I}{abs(I)}+1\right )^{\frac{1}{t}} -1=\left (\frac{F-I+abs(I))}{abs(I)} \right )^{\frac{1}{t}}-1


CAGR_{ADJ}=\left (\frac{F-I+abs(I))}{abs(I)} \right )^{\frac{1}{time}}-1, plugging in our example values: CAGR_{ADJ}=\left (\frac{-1--50+abs(-50))}{abs(-50)} \right )^{\frac{1}{3}}-1=+26\%