Arithmetic vs. geometric returns: why a 10% average is not 10% growth

Use consecutive, equal-length periods for the same investment or portfolio. Work in one currency, with a positive starting value and no external cash flows. State whether income is reinvested and whether returns include costs.

Write each period return as a decimal r: 40% becomes 0.40 and a 20% loss becomes −0.20. Its growth factor is 1 + r, not r itself.

Arithmetic average = (r1 + r2 + ... + rn) ÷ n. This describes the average observed single-period return. It does not reconstruct the ending balance of a continuously invested account.

For consecutive periods, multiply the growth factors: cumulative return = (1 + r1) × (1 + r2) × ... × (1 + rn) − 1.

Keep price return and total return consistent throughout. Switching between them halfway through a series produces a different problem from choosing an average.

Consider two complete years and 10,000 units of any unchanged currency. The hypothetical returns are +40% in year one and −20% in year two. There are no fees, distributions, deposits, or withdrawals.

Compounding the arithmetic average would instead give 10,000 × 1.10 × 1.10 = 12,100. That extra 900 never appeared in this account; it belongs to a different return path.

The example is original arithmetic, not a backtest, a live investment result, or a forecast. The currency symbol would not change the percentages.

• Start: 10,000.
• After year one: 10,000 × 1.40 = 14,000.
• After year two: 14,000 × 0.80 = 11,200.
• Arithmetic average: (40% − 20%) ÷ 2 = 10% per year.
• Cumulative return: 11,200 ÷ 10,000 − 1 = 12% over two years.

The geometric return is the constant per-period rate g that reproduces the actual growth factor: (1 + g)^n = (1 + r1) × ... × (1 + rn).

Thus g = [(1 + r1) × ... × (1 + rn)]^(1/n) − 1. Take the nth root of the product of growth factors, not of the return percentages.

For the two-year example, g = (1.12)^(1/2) − 1 ≈ 5.8301% per year. Using the unrounded rate gives 10,000 × (1 + g)^2 = 11,200.

When the periods are years, this geometric return equals the compound annual growth rate, or CAGR. With no external flows, CAGR = (ending value ÷ starting value)^(1/t) − 1, where t is elapsed years.

A separate example grows 10,000 to 15,000 over five years. Its total gain is 50%, but CAGR is (1.5)^(1/5) − 1 ≈ 8.45%, not 50% ÷ 5 = 10%.

[FINRA's annualized-return explanation](https://syndication.finra.org/content/how-are-your-investments-doing-returns-explained) also distinguishes compound annualization from dividing a total gain by years.

Two years of +10% and +10% have the same arithmetic average as +40% and −20%. Yet 10,000 grows to 12,100 in the first case and 11,200 in the second.

For positive growth factors, the geometric mean is no greater than the arithmetic mean. They are equal when every period return is the same. Apply that statement to the same observations, not unrelated funds.

The gap is not an extra commission to subtract from the balance. The compounded ending value already reflects the returns. Nor does the comparison prove that any lower-volatility investment must outperform another.

[CFA Institute's discussion of the two means](https://rpc.cfainstitute.org/blogs/enterprising-investor/2015/the-myth-of-volatility-drag-part-1) explains why compounding is a product rather than a sum.

Reordering this fixed pair of returns leaves the product unchanged when there are no cash flows. It can still change the path and observed drawdowns.

A geometric average from monthly observations is a monthly rate. If twelve consecutive months each return exactly 1%, annual growth is (1.01)^12 − 1 ≈ 12.68%, not 12%.

That example specifies all twelve months. Annualizing a single observed 1% month would describe a hypothetical repetition, not a year's realized performance or a prediction.

For unequal observation intervals, a geometric mean per observation is not automatically an annual rate. Use actual elapsed years for a valid endpoint CAGR, and state the day-count convention for partial years.

An annualized number smooths the endpoints. It does not say the account earned that rate in every year or avoid large interim losses.

Suppose 100 is invested in asset A and 900 in asset B over the same period. A gains 50%, while B loses 10%. Their ending values are 150 and 810, totaling 960.

The portfolio return is (960 − 1,000) ÷ 1,000 = −4%. Equivalently, beginning weights give 0.10 × 50% + 0.90 × (−10%) = −4%.

The unweighted average of +50% and −10% is +20%, which ignores allocation. Multiplying 1.50 by 0.90 is also wrong here: those holdings coexist rather than represent two periods of the same capital.

External deposits and withdrawals create another boundary. A balance rising from 10,000 to 15,000 after a 5,000 deposit is not necessarily a 50% investment gain.

[GIPS cash-flow methodology](https://www.gipsstandards.org/standards/gips-standards-for-firms/gips-standards-handbook-for-firms/) distinguishes time-weighted performance from returns that reflect the timing and size of cash flows.

A percentage gain on a small option premium is not the same as a percentage gain on the whole account. Different trade sizes, overlapping positions, and idle cash prevent casual multiplication of trade-level returns.

[OIC's leverage guide](https://www.optionseducation.org/optionsoverview/leverage-risk) explains why option premiums can show magnified percentage gains and losses.

For futures, posted margin is collateral rather than the purchase price of the exposure. Changing that denominator does not create a new return on the underlying asset.

[CME's margin explanation](https://www.cmegroup.com/education/courses/introduction-to-futures/margin-know-what-is-needed) identifies what those funds represent.

Use futures return denominators for a trade comparison, and consistent whole-account valuations for portfolio compounding. Include open positions and count each cost once.

Start with the stated question: average observation, growth of one investment, or return on a portfolio of simultaneous holdings. These require different calculations.

The positive-factor formulas assume each return exceeds −100%. At a complete loss, there is no remaining capital to compound. Negative account equity requires cash-flow and liability analysis, not a normal CAGR comparison. [!TRYMARK] TryMark average-return audit At the next performance review, target the reported ending value. Rebuild it from dated account returns, then recheck the result when costs, reinvestment assumptions, or external cash flows change. [!WARNING] A historical average does not promise repeatable growth Neither the arithmetic nor geometric average establishes next year's return. A larger required gain or a smooth CAGR does not justify more leverage. These examples explain measurement, not an investment recommendation.

• Record dates, period lengths, currency, and beginning and ending values.
• Identify income, reinvestment, fees, and external deposits or withdrawals.
• Rebuild the ending value from successive growth factors before annualizing.
• Label arithmetic average, cumulative return, geometric period return, and CAGR separately.