The kingdom did not grow 50 percent because one pauper found a dollar.
The spreadsheet says 50 percent. The treasury says one dollar. Both are right, but only one matters.
The pauper starts with $1, finds another, and celebrates a 100 percent gain. The king starts with $999,999 and gains nothing. Average the two rates and you get 50 percent. Add the starting wealth, though, and the kingdom had $1,000,000. It now has $1,000,001. Its aggregate growth is 0.0001 percent.
No, the math isn’t broken. The question changed halfway through.
Averaging percentages is how a dollar becomes an economic miracle. The spreadsheet will average anything you let it. Dashboards repeat the mistake when they average desk returns, fill rates, or error rates without checking what sits under each ratio.
If you want the growth of the whole, do this:
- Add the starting amounts first.
- Add the ending amounts.
- Calculate the percentage change from those totals.
- Or weight each row’s growth rate by its starting amount when those rates describe the same period and measure.
A percentage point is not a dollar, despite what the slide deck implies. Yes, both answers can be mathematically correct. Only one answers this question.
Keep each numerator with its denominator. The denominator is not a decorative accessory. A ratio without its denominator is just a number in a nice suit.
And yes, a zero starting amount has no ordinary percentage growth rate. Cash flows or changing exposures can call for a different return calculation. Do not fix one bad average by confidently choosing the wrong weights.
Ask what grew before announcing how much.