Arithmetic mean versus geometric mean return

Two investments can have the same arithmetic average return and very different ending values. The difference comes from compounding: a loss applies to the reduced balance after a prior gain, not to the original amount. Financial planning questions therefore distinguish the arithmetic mean from the geometric mean.

Arithmetic mean: the ordinary average

Add each period's return and divide by the number of periods. For returns of 8%, 4%, and −3%, the arithmetic mean is (8% + 4% − 3%) ÷ 3 = 3%. It gives the average return across the observed periods without calculating the effect of reinvesting each period's result.

The arithmetic mean is useful as a description of a typical observation and can be relevant to an expected one-period return when the problem's assumptions call for it. It is not the rate that the investment actually compounded over a multi-period holding period.

Geometric mean: the compound rate

The geometric mean incorporates the sequence of gains and losses. For n periods, multiply one plus each period's return, take the nth root, and subtract one: [(1 + R1)(1 + R2)…(1 + Rn)]^(1/n) − 1. It is the steady per-period rate that would lead from the same starting value to the same ending value over the same number of periods.

For 8%, 4%, and −3%, the growth factors are 1.08, 1.04, and 0.97. Their product is about 1.090. The cube root of that product, minus one, gives a geometric mean close to 2.9%. The exact result will generally be no greater than the arithmetic mean when returns vary, with equality when every period has the same return.

Why volatility lowers the compounded result

Suppose an account rises 50% one year and falls 50% the next. The arithmetic mean is zero: (50% − 50%) ÷ 2. But $100 becomes $150 after the gain, then $75 after the loss. The two-year compound result is a 25% decline, and the geometric mean annual return is √(0.75) − 1, or about −13.4%.

The loss percentage is applied to the larger balance after the gain, while the later gain in a reverse sequence would be applied to a smaller balance after the loss. This is why returning to a prior high requires a gain larger than the preceding percentage loss: after a 20% decline, a 25% gain is needed to recover.

Cumulative return and annualized return are different

Cumulative return is the total change from start to finish. Multiply the growth factors and subtract one. In the 50% gain followed by a 50% loss example, cumulative return is −25%. The geometric mean annualizes that two-period result; it is about −13.4% per year. The arithmetic mean of zero is neither the cumulative result nor the compound annual growth rate.

When comparing records of different lengths, check whether the problem asks for a cumulative return, an annualized compound return, or an average of periodic returns. A number labeled simply 'average return' can be ambiguous without a stated method and period.

Exam calculation steps

1. List each periodic return as a decimal or percentage, using one format consistently.
2. For the arithmetic mean, add the returns and divide by the number of periods.
3. For the geometric mean, convert every return to a growth factor, multiply the factors, take the root matching the number of periods, and subtract one.
4. For cumulative return, multiply the factors and subtract one without taking a root.
5. Check the result: with variable returns, the geometric mean should not exceed the arithmetic mean.

Common mistakes

• Do not average gains and losses arithmetically when asked for the compound growth rate.
• Do not divide cumulative return by the number of years to get an exact annualized return.
• A zero arithmetic mean does not mean the account ended unchanged.
• Match the root to the number of periods. Three annual observations call for a cube root, not a square root.
• The average of returns is not the same as the return on an average balance or the return of a dollar-weighted investor. Cash-flow timing may require a different measure.

Choosing the right measure in a client discussion

Use arithmetic averages carefully when discussing a multi-year investment experience. They may describe the center of annual observations, but they can overstate the compound pace an investor experienced when returns are volatile. Geometric returns explain historical compounding, but neither historical measure guarantees future results. A forecast should state its assumptions and distinguish expected periodic return from a realized compound return.

A worked example shows the difference

Suppose a portfolio returns +20% in year one and −20% in year two. The arithmetic mean is (20% + −20%) ÷ 2 = 0%. The investor’s $100 becomes $120, then falls by 20% of $120 to $96. The two-year cumulative return is −4%, and the geometric annual return is √(0.96) − 1, about −2.02%. The arithmetic average did not describe the actual compound growth path.

The geometric mean is the constant annual rate that would produce the observed cumulative value. Multiply each period’s gross return, take the nth root for n periods, and subtract one: [(1+r1)(1+r2)…]^(1/n) − 1. It preserves compounding and is useful for describing historical growth. It is not a guaranteed expected return or forecast.

Why volatility creates a drag

A percentage loss requires a larger percentage gain to recover. A 20% loss takes a 25% gain to return to the original value; a 50% loss requires a 100% gain. Alternating equal positive and negative returns therefore produces a decline, not a flat result. Greater variation generally lowers the geometric mean relative to the arithmetic mean when the arithmetic mean is held constant.

The arithmetic mean remains useful as an estimate of the expected return for one period under assumptions, especially in some portfolio models. The geometric mean describes realized compound growth over multiple periods. They answer different questions. A long-term projection that uses an arithmetic return as if it were a guaranteed compound rate can overstate ending wealth.

Choose the measure that fits the question

Use arithmetic mean for the average of single-period observations and many one-period expected-return calculations. Use geometric mean for annualized growth over a multiyear period. Use money-weighted return when the investor’s cash-flow timing matters; use time-weighted return to evaluate an investment manager’s performance apart from external flows. None of the means by itself describes risk or sequence.

Check the data before calculating: the periods must be equal length, returns must be converted to gross factors before multiplying, and total return should include reinvested distributions if that is the chosen measure. For an exam, state whether you are averaging periodic percentages or finding the compounded rate that links beginning and ending value. That wording usually identifies the correct mean.