Updated for the 2026-2027 CFA® Level I curriculum.
Arithmetic and geometric returns often appear side by side because both summarize periodic investment performance. The formulas are straightforward, and the key exam skill is interpreting what each result represents.
For CFA Level I, you should be able to calculate both measures and choose the one that fits the question.
Quick Answer
The arithmetic mean is the simple average of a set of periodic returns. The geometric mean is the constant per-period return that would produce the same compound growth over the full investment period.
Use the arithmetic mean for an expected one-period return. Use the geometric mean for realized performance across consecutive periods.
Key Takeaways
The arithmetic mean treats each periodic return as a separate observation.
The geometric mean follows the compound growth of wealth through time.
The geometric return is less than or equal to the arithmetic return for the same series of valid periodic returns.
The two measures are equal when every periodic return is identical.
Greater variation between returns usually creates a wider gap between the two means.
The wording of the question determines which measure you should use.
What You Need to Know for CFA Level I
Focus on calculation and interpretation together:
Calculate the arithmetic mean by adding the periodic returns and dividing by the number of observations.
Calculate the geometric mean by multiplying growth factors, taking the appropriate root, and subtracting 1.
Convert percentage returns to decimals before using either formula.
Use the arithmetic mean for a one-period expected return based on equally weighted observations.
Use the geometric mean for compound performance across several consecutive periods.
Recognize that equal percentage gains and losses have different effects on ending wealth.
Arithmetic vs Geometric Returns: Main Difference
The arithmetic mean summarizes the returns as separate observations. The geometric mean treats the returns as one connected investment path.
Area | Arithmetic Return | Geometric Return |
|---|---|---|
What it measures | Simple average return per observation | Constant compound return per period |
Calculation | Add the returns and divide by the number of returns | Multiply the growth factors and take the relevant root |
Typical use | Expected return for one period | Realized performance across several periods |
Treatment of volatility | Keeps each observation separate | Reflects the effect of return variability on compound growth |
Relationship | Greater than or equal to the geometric return | Less than or equal to the arithmetic return |
A useful way to remember the difference is to ask two separate questions:
Arithmetic mean: What was the average return across the observations?
Geometric mean: What steady return would have produced the same ending value?
The geometric mean is appropriate for realized compound performance. The arithmetic mean is useful when estimating a one-period expected return from equally weighted observations.
Arithmetic and Geometric Return Formulas
Arithmetic Mean Return Formula
Where:
Inline equation code = return in period
Inline equation code = number of periods or observations
Inline equation code = arithmetic mean return
The arithmetic mean gives every observation the same weight. For example, returns of 8%, 4%, and 12% produce an arithmetic mean of 8%.
Geometric Mean Return Formula
Notion block equation code:
Where:
Inline equation code = return in period
Inline equation code = number of consecutive periods
Inline equation code = geometric mean return
Inline equation code = the growth factor for period
Convert each return into a growth factor before multiplying. A return of 20% becomes 1.20, while a return of -10% becomes 0.90. After multiplying the growth factors, take the nth root and subtract 1.
For the same series of returns:
Equality holds when every periodic return is the same.
When Should You Use the Arithmetic Mean?
Use the arithmetic mean when a question asks for the expected return over one future period and treats the observed returns as equally likely outcomes.
Suppose a security produced annual returns of 6%, 10%, and 14%. If each observation receives equal weight, the arithmetic mean estimates the expected return for a single period:
The result is an estimate based on the observed sample. Measuring compound performance across the three years requires the geometric mean.
When Should You Use the Geometric Mean?
Use the geometric mean when the returns occur in sequence and you want to measure how wealth grew across the full period.
The result converts the total compound change into one equivalent periodic rate. This makes the geometric mean useful for reporting historical performance and comparing investments across periods of different lengths.
When a question gives you the starting and ending values rather than each periodic return, use the following formula:
Where:
Inline equation code = beginning investment value
Inline equation code = ending investment value
Inline equation code = number of periods
This version calculates the constant periodic return that connects the beginning value to the ending value.
Why Does Volatility Create a Gap Between the Two?
Return variability reduces compound growth because each loss leaves a smaller base for the next gain.
Consider an investment that loses 10% and then gains 10%. Its arithmetic mean is 0%:
The investment still finishes below its starting value:
Its geometric mean is therefore negative:
A 10% loss reduces 100 to 90. Returning from 90 to 100 requires an 11.11% gain.
This asymmetry explains why a more volatile return series usually has a larger difference between its arithmetic and geometric means.
Worked Example
An investment earns the following annual returns:
Year 1: +20%
Year 2: -10%
Year 3: +15%
Step 1: Calculate the Arithmetic Mean
The simple average annual return is 8.33%.
Step 2: Calculate the Geometric Mean
Convert the three returns into growth factors:
Then multiply the growth factors and take the cube root:
The investment earned an equivalent compound return of 7.49% per year.
Step 3: Check the Ending Value
Starting with 100:
A constant annual return of approximately 7.49% produces the same ending value:
The geometric mean connects the starting value of 100 to the ending value of 124.20. The arithmetic mean remains higher because it averages the three annual observations without following the changing investment base.
How to Identify the Correct Measure in CFA Questions
CFA questions often signal the required measure through a few key phrases.
“Expected return for one period” usually points to the arithmetic mean.
“Average compound return” points to the geometric mean.
“Beginning value and ending value” points to the geometric mean.
“Annualized historical performance” usually points to the geometric mean.
“Simple average of periodic observations” points to the arithmetic mean.
When the wording feels less direct, decide whether the returns are being treated as separate possible outcomes or as consecutive periods in one investment history.
Common Exam Traps
Using the arithmetic mean to report compound performance across several periods.
Multiplying the raw returns instead of the growth factors represented by inline equation code .
Forgetting to subtract 1 after taking the geometric root.
Entering percentages into the formula without converting them to decimals.
Assuming a positive arithmetic mean guarantees an increase in ending wealth.
Choosing the geometric mean automatically without checking what the question asks.
Practice Question
A portfolio returns +50% in Year 1 and -20% in Year 2. What are the arithmetic and geometric mean returns?
Arithmetic mean = 15.00%; geometric mean = 9.54%
Arithmetic mean = 9.54%; geometric mean = 15.00%
Arithmetic mean = 15.00%; geometric mean = 12.25%
Correct Answer: A
The arithmetic mean is:
The geometric mean is:
Option B reverses the two measures.
Option C uses the wrong compound calculation.
The 9.54% geometric mean is the constant annual rate that links the portfolio’s starting value to its ending value over the two-year period.
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FAQs About Arithmetic vs Geometric Returns
What Is the Difference Between Arithmetic and Geometric Returns?
The arithmetic return is the simple average of periodic returns. The geometric return is the constant compound rate that produces the same ending value across consecutive periods.
Can the Geometric Return Be Higher Than the Arithmetic Return?
For the same series of valid periodic returns, the geometric return cannot exceed the arithmetic return. The two are equal only when every periodic return is identical.
Which Return Should I Use for Expected Return?
Use the arithmetic mean when estimating an expected return for one period from equally weighted observations. It treats each observation as a separate possible outcome.
Which Return Should I Use for Historical Performance?
Use the geometric mean for historical performance across several consecutive periods. It reflects compounding and connects the investment’s beginning value to its ending value.
Why Is the Geometric Return Usually Lower?
The geometric return reflects the effect of return variability on compound wealth. Losses reduce the investment base, so later gains compound from a lower amount.
Greater variability usually creates a wider gap between the geometric and arithmetic means.