Scope and assumptions
Single change uses Old as base; CAGR annualizes across t periods. Illustrative.
A 44% gain over two years is not "22% a year" — the compounding in between makes the true yearly rate a smoother 20%. The trick is knowing which question you're asking: the headline percent for the whole stretch, or the annualized rate that lets you compare decades honestly.
Two clocks: the headline and the smoothed rate
Total change % = (New − Old) ÷ Old × 100 CAGR = (End ÷ Start)^(1/t) − 1 (t = number of periods)
The first answers "how much did it move overall." The second answers "what steady yearly rate would land me at the same end point."
| Old | New | Periods | Total % | CAGR |
|---|---|---|---|---|
| 100 | 144 | 2 | 44% | 20% |
| 50 | 60 | 1 | 20% | 20% |
| 200 | 242 | 2 | 21% | 10% |
Why "divide by years" overshoots
Splitting 44% evenly as 22% + 22% ignores compounding. CAGR gives the steady yearly rate that actually lands you at 144: 100 × 1.20² = 144. The naive "22%" would overshoot to 100 × 1.22² = 148.84. The gap widens as the stretch grows, so the shortcut gets worse, not better, over long horizons.
The base is always the older number
Both formulas use Old / Start as the base. A rise from 100 to 144 is 44% of the start; measured against the end it's only 30.6%. The older value is the "before" you measure against — never flip it.
Change-Over-Time Figures That Mislead
- Dividing total % by years — 44% over 2 years is 20% CAGR, not 22%.
- Using New as the base — always use Old / Start.
- Forgetting the exponent in CAGR — it's (End/Start)^(1/t), not a straight average.
- Comparing total % to CAGR — different clocks; never stack them.
- Ignoring the path — CAGR hides volatility between start and end, so a bumpy ride and a smooth one can share one CAGR.
- Dropping the period unit — state t in years (or quarters) or the rate is meaningless.
Related growth & decay guides
For the rest of the growth-and-decay cluster, start at the hub: Population Growth Percentage. Adjacent: Percentage Decay Calculation and Percentage Return on Investment.
FAQ
How do I annualize a gain measured in months or quarters? CAGR works for any period t as long as the rate label matches it. A 44% gain over 8 quarters gives CAGR = (144/100)^(1/8) − 1 ≈ 4.7% per quarter, or (1.047)^4 − 1 ≈ 20% a year. Keep t and the rate label on the same clock.
What if the value went down — can CAGR be negative? Yes. A fall from 100 to 64 over 2 years gives CAGR = (64/100)^(1/2) − 1 = −20% a year. Don't mistake that for "lost 20% total" — the total loss is 36%.
CAGR versus average annual return — aren't they the same? No. CAGR is the single smooth rate connecting start to end; average annual return averages each year's return and reacts to volatility. A path 100 → 120 → 100 has 0% CAGR over two years but a positive average annual return.
Can I use this for things that aren't money? Yes — subscribers, population, website traffic, CO₂ levels. Any quantity with a start, an end, and a time span, as long as it changes multiplicatively rather than in one sudden jump.
Sources: Investopedia, CAGR; Khan Academy, Percent change.
Disclaimer: Illustrative. CAGR hides volatility within the period. Informational only.
Sources and references
- Investopedia — CAGRhttps://www.investopedia.com/terms/c/cagr.asp
- Khan Academy — Percent changehttps://www.khanacademy.org/math
Maintenance and corrections
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