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Apply the gain to its starting amount
The first period starts at 100 and earns 20%, increasing wealth by 20 to 120. No cash is added or withdrawn. The second period starts from that new amount, not the original 100.
Apply the loss to the enlarged amount
A 20% loss on 120 is 24. Wealth ends at 96. Linked cumulative return is 1.2×0.8−1 = −4%, while the arithmetic mean of the two periodic returns is zero.
| Point | Wealth | Change from preceding point |
|---|---|---|
| Initial | 100 | — |
| After gain | 120 | +20 |
| After loss | 96 | −24 |
Find an endpoint-equivalent annual rate
If both periods are years, the geometric annual rate is √0.96−1 = -2.02041%. Applying that same rate twice to 100 also ends at 96. It does not reproduce the intermediate peak at 120.
Keep the means tied to their questions
The arithmetic mean summarizes the two observed rate numbers with equal weight. The geometric mean matches cumulative wealth growth. Reporting one as the other can misstate performance even when the input observations are correct.