Arithmetic vs Geometric Mean Return: A Visual CFA® Level I Guide

The difference in one sentence

For equal-length periods, arithmetic mean return averages the periodic returns; geometric mean return gives the constant compounded return per period that reproduces their cumulative growth. Neither replaces the other: they describe different things.

For example, +20% and −20% have an arithmetic mean of 0%, a two-period cumulative return of −4%, and a geometric mean of approximately −2.02% per period.

Throughout the main example, each period is one year. There are no external deposits or withdrawals, and the stated returns include any investment income, which is reinvested. This lets us connect the compounded return directly to the ending investment value.

Follow $100 through two years

Imagine investing $100 at the start of Year 1. It gains 20% in Year 1 and loses 20% in Year 2.

The percentages look as if they cancel, but they apply to different amounts. You gain $20 in the first year, then lose $24 in the second. The overall loss is $4, or 4% of your original investment.

Try changing the size of the gain and loss. The arithmetic mean remains zero; the ending value changes.

The solid line shows the two actual returns. The dashed line shows a constant annual return with the same starting and ending values. At +20% and −20%, that constant annual return is approximately −2.02%.

Calculate the three numbers

1. Arithmetic mean: average the periodic returns

Add the returns and divide by the number of periods:

Arithmetic mean = (r₁ + r₂ + … + rₙ) / n

In our example:

[20% + (−20%)] / 2 = 0%

This correctly describes the average of the two annual returns. It does not say the investment ended where it started.

2. Cumulative return: multiply the growth factors

Cumulative return is the holding-period return over the full span. A return of +20% has a growth factor of 1.20. A return of −20% has a growth factor of 0.80. Multiply those factors to follow the changing investment value:

Cumulative return = (1 + r₁)(1 + r₂)…(1 + rₙ) − 1
1.20 × 0.80 − 1 = −4%

That is the return over the full two years, not the return per year.

3. Geometric mean: find the equivalent constant rate

We want a constant annual return, g, that takes the same $100 to the same $96 over two years:

100 × (1 + g)² = 96
g = √(96 / 100) − 1 ≈ −2.02%

For n equal-length periods:

Geometric mean = [(1 + r₁)(1 + r₂)…(1 + rₙ)]^1/n − 1

Use returns in decimal form inside the growth factors: 20% becomes 0.20. Keep intermediate values unrounded, then round the final answer.

Which return should you use?

Start with what the question is asking you to describe.

The arithmetic mean is also used when estimating a one-period expected return from a representative sample. That is an estimation problem, with assumptions about the return process; it is different from reconstructing a known historical investment path.

Likewise, the historical geometric mean describes the compound growth that actually occurred. It is not automatically the right forecast for a future investment. Do not turn “geometric mean captures compounding” into “always use geometric mean.”

Why equal gains and losses leave a shortfall

For a gain of x followed by a loss of x, written as decimals:

(1 + x)(1 − x) = 1 − x²

With x = 0.20, the combined growth factor is 1 − 0.04 = 0.96. A larger swing in this particular two-period example produces a larger shortfall, even though the arithmetic mean stays at zero.

For returns with positive growth factors, the geometric mean return cannot exceed the arithmetic mean return. They are equal when every periodic return is the same. For example, +5% followed by +5% has both an arithmetic and a geometric mean of 5%.

A related trap: after a 20% loss, a 20% gain does not restore the original value. $100 falls to $80, so you need a $20 gain on an $80 base: 25%.

Check your understanding

For a second check, replace the returns with +30% and −30%. Before calculating, predict whether the geometric mean will be positive, zero or negative.

Reveal the second answer

$100 × 1.30 × 0.70 = $91. The total return is −9%; the arithmetic mean is 0%; the geometric mean is √0.91 − 1, approximately −4.61% per year. The unchanged arithmetic mean does not imply unchanged compound growth.

Four mistakes to avoid

1. Multiplying the returns instead of the growth factors. Use 1.20 × 0.80, not 0.20 × −0.20.
2. Reporting total return as an annual rate. −4% over two years is not −4% per year.
3. Forgetting the period length. With monthly returns, the nth root gives a monthly geometric mean. To annualize it, use (1 + g_monthly)¹² − 1. The annualized equivalent is a summary rate, not a forecast.
4. Ignoring external cash flows. If you add or withdraw money during the investment, the change in your balance alone is not a clean measure of investment performance. Cash-flow timing matters; time-weighted and money-weighted returns address different performance questions.

Common questions

Can the geometric mean include a negative return?

Yes. A −20% return contributes a positive growth factor of 0.80, so the calculation works normally. The examples here assume each return is greater than −100%, so all growth factors are positive.

Is geometric mean return the same as CAGR?

For an investment measured across a number of years, with no external cash flows and investment income reinvested, the geometric mean of its annual total returns equals its compound annual growth rate (CAGR): (ending value / beginning value)^1/years − 1. If your periods are months or quarters, the geometric mean for those periods must first be annualized before you call it CAGR.

Does the order of these two returns matter?

With no external cash flows, +20% then −20% and −20% then +20% both end at $96 because 1.20 × 0.80 = 0.80 × 1.20. Their paths differ: the first reaches $120 after Year 1; the second reaches $80. If money is added or withdrawn between periods, the order can affect the investor’s ending wealth.

Take the idea into your next lesson

Ask two questions before choosing an average: Am I averaging separate period returns, or finding a rate that reproduces compound growth? What is the time period of the answer?

In Quizara, this topic sits in Quantitative Methods → Returns of Financial Assets and Instruments → Financial Returns. Continue with the Arithmetic and Geometric Mean Returns lesson, practise the topic, and ask about the step you find difficult.

Curriculum context and further reading

• CFA Institute: 2027 Level I topic outlines. The Quantitative Methods objectives cover comparing, calculating and interpreting return measures. This guide is an independent explanation, not an official curriculum extract.
• CFA Institute: Rates and Returns. Public learning-outcome context for return measures and their uses.
• Jacquier, Kane and Marcus: Geometric or Arithmetic Mean—A Reconsideration. Further reading on why forecasting future wealth is more subtle than reproducing a historical compound return.

All numerical examples and questions on this page were created for this guide.