The changing base is the whole idea
A fictional $1,000 investment earns 10% in its first year and 10% in its second. After year one it is worth $1,100. The second 10% adds $110, leaving $1,210. The two-year gain is 21%, because the second return applies to the first year’s gain as well as the original money.
For returns r1 and r2, the combined return is (1 + r1) × (1 + r2) − 1. That expression assumes the returns belong to consecutive periods of the same consistently measured investment. Adding new money, withdrawing capital or changing distribution treatment introduces other measurement questions.
Equal percentage gains and losses do not cancel
Start again at $1,000, gain 20% and then lose 20%. The sequence ends at $960: $1,000 × 1.20 × 0.80. The arithmetic average of the two returns is zero, but the compounded return is −4%. The loss applies to a larger base than the preceding gain.
Reversing those two returns produces the same final value if there are no intervening deposits or withdrawals. It can produce a different experience when cash flows occur between periods. A person withdrawing money after a loss has less capital participating in any later recovery.
An average annual rate needs a label
A compound annual growth rate translates a starting and ending value into a constant hypothetical annual rate. If $1,000 becomes $1,210 after two years with no external cash flows, that rate is 10%. The path could still have included an uneven first and second year. A smooth equivalent rate is not a record of smooth realized returns.
An arithmetic average answers a different question about the individual period returns. Comparing an arithmetic strategy average with a compounded benchmark can therefore mix incompatible measurements. Check how both numbers were calculated before interpreting a gap as outperformance.
A compounding illustration is not a forecast
Constant-return examples make the mechanism visible. They do not promise that markets will deliver that rate, that distributions will always be available for reinvestment or that losses can be avoided by waiting. Fees, taxes and inflation can materially change what an investor keeps and what it buys.
When reading a report, keep a daily return, a cumulative five-session return and an annualized rate distinct. A five-day gain is not five times the latest daily return. Use the actual consecutive observations and a consistent price or total-return convention.
An investment rises 10% and then falls 10%. Is it back where it started?
Show the explanation
No. Its multiplier is 1.10 × 0.90 = 0.99, leaving a 1% cumulative loss before any other effects.
Sources and editorial notes
Original explanations and hypothetical examples prepared with AI assistance. Primary references checked October 16, 2026. Numerical examples are hypothetical unless explicitly identified otherwise.
- SEC Investor.gov: Compound Interest
Growth earned on prior principal and accumulated returns; numerical return paths are original illustrations.