Why is the geometric mean required for calculating equivalent annual growth rates over multiple periods?

Conditions

• Dealing with multiplicative growth or decay
• Equal time periods involved

Reasoning, step by step

1. Convert percentage changes into growth factors (e.g., $1+r$).
2. Multiply the growth factors together.
3. Take the $n$-th root of the product (for $n$ periods).
4. Subtract 1 to convert back to a rate if necessary.

Example

For an item increasing by 10% then 20%, you multiply 1.1 and 1.2, take the square root, yielding ~1.1489. The true annual rate is ~14.89%, not the arithmetic average of 15%.

Common misconceptions

• Simply averaging the percentage rates arithmetically (e.g., claiming 15% for 10% and 20%).
• Geometrically averaging the percentage rates themselves instead of the growth factors.

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