### 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:

, plugging in our example values:

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.

, plugging in our example values:

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Now, the usual formula for CAGR is:

, plugging in our example values:

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:

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

Therefore,

, plugging in our example values:

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