Hypothetical worked example and primary educational references; not observed market data.
An average is not an account balance
The arithmetic mean adds periodic returns and divides by the number of periods. Compound growth multiplies the corresponding growth factors. Investor.gov cautions that historical fund results do not predict future performance; even a correct summary of the past needs a defined method. An average of returns can be useful for describing a series, but it does not directly show what happened to invested capital. An investor reading an average should ask whether it is arithmetic, geometric, or a different presentation. Those measures can differ substantially even when they use exactly the same underlying observations.
Supporting references: 1. Investor.gov: Mutual Funds, Past Performance
Follow the changing capital base
Start a hypothetical account with USD 1,000. A first-year gain of 20% raises it to 1,200. A second-year loss of 20% reduces that larger balance to 960. The arithmetic average of the two returns is zero, while cumulative return is minus 4%. The annual geometric return is approximately minus 2.02%, because multiplying that annual growth factor twice reproduces 960 from 1,000. This path is invented for arithmetic demonstration, not a forecast. Equal positive and negative percentage changes do not cancel because each percentage applies to the capital available at a different point in time.
Hypothetical worked example
Same arithmetic mean, different ending wealth
Model setup
Read these assumptions before comparing the scenarios.
- All monetary amounts in this example are hypothetical U.S. dollars (USD), not current prices, quoted account terms or observed market data. Percentages, counts, time periods and index levels retain their stated units.
- Both accounts begin with USD 1,000.
- No flows, fees or taxes; two annual periods.
Geometric annual return (%)
- 1. +20%, -20%
- -2.02
- 2. 0%, 0%
- 0.00
Shared zero-based value scale. Range: -2.02 to 0.00, in the units above. Case numbers match the labels; no probability or time trend is implied.
Hypothetical model, not observed market data or a forecast. Assumptions above; original values and calculations below.
| Annual returns | Arithmetic mean (%) | Ending value (USD) | Geometric annual return (%) |
|---|---|---|---|
| +20%, -20% | 0.00 | 960.00 | -2.02 |
| 0%, 0% | 0.00 | 1,000.00 | 0.00 |
Calculation details
- Volatile ending value
- 960.00 USD
((1000 * 1.2) * 0.80000000000000004) - Volatile geometric rate
- -2.02 % per year
(((0.95999999999999996 ^ 0.5) - 1) * 100) - Flat ending value
- 1,000.00 USD
((1000 * 1) * 1) - Arithmetic mean
- 0.00 %
((20 + -20) / 2)
Compare a stable path using the same convention
A second hypothetical account earns zero in each year, ending at 1,000. Both paths have an arithmetic average of zero, but their compounded outcomes differ. The comparison shows that the dispersion of returns matters when converting an average into growth. It does not establish a permanent mathematical penalty independent of the actual return distribution or a reason to reject volatile investments categorically. The investor still needs to consider price, risk, horizon, and possible outcomes. The essential correction is narrower: do not use an arithmetic average as though it were a constant annual rate that reproduces terminal wealth.
Geometric growth needs a defined interval
To calculate geometric annual growth for a multi-year period, divide ending wealth by starting wealth and take the root corresponding to the number of years, then subtract one. This works cleanly when there are no external contributions or withdrawals and the return convention includes relevant distributions. Additional flows require more careful performance measurement. A deposit changes wealth without being investment profit. A withdrawal changes the compounding base. The simple endpoint formula should therefore not be applied indiscriminately to a household account merely because its first and last balances are readily available.
A smooth summary conceals the path
The geometric rate reproduces the endpoints but does not imply that the investment grew smoothly each year. Two paths can have the same compound rate while differing in drawdowns and in the timing of income. An investor who needed to sell during the weak period could experience a different practical outcome from an investor who held throughout. Investor.gov's report guidance encourages reading performance with broader fund information, which is useful here. A return summary should be paired with evidence about the path and exposures. Endpoint consistency does not eliminate risk encountered between those endpoints.
Supporting references: 2. Investor.gov: How to Read a Mutual Fund or ETF Shareholder Report
Use the measure that matches the question
For a no-flow account's realized growth over time, a geometric measure links directly to the wealth ratio. For certain statistical descriptions of individual period returns, an arithmetic mean answers another question. Keep the labels and units explicit, and avoid converting a hypothetical average into an expected future rate without additional assumptions. If a model uses a constant growth rate for illustration, say that it is a mathematical input rather than a prediction. The distinction helps investors evaluate performance claims, compare scenarios fairly, and recognize why a reassuring average can coexist with a disappointing account result.
Test another situationCompare dispersion using a different pair of gains and losses
A separate hypothetical case. Its inputs differ from the main worked example above.
Compare dispersion using a different pair of gains and losses
Start an independent hypothetical investment with USD 2,000 and no external flows. Let the first annual return be positive 50% and the second negative 40%. At the end of the second year, the balance is exactly USD 1,800. A positive arithmetic average would therefore coexist with a loss of capital on this path. The illustration does not estimate any asset's return distribution. Smoothing the displayed average cannot change the actual multiplication of wealth. Lower fluctuations could produce another outcome, but deciding whether to accept risk also requires the possible return levels and the goal's timing rather than a preference for smoothness alone.
Choose averages according to what is being aggregated
Investor.gov cautions that prior fund performance is not a reliable prediction. Even descriptive summaries need the correct unit of observation. An equal-weight average across different funds in one period answers a cross-sectional question; compounding one fund across successive periods answers a time-growth question. Do not replace one with the other because both tables contain annual percentages. Similarly, a geometric summary of a no-flow path cannot be applied directly to balances changed by deposits. Identify whether the analysis follows invested dollars, a set of opportunities or periodic observations before selecting the averaging method and drawing conclusions from it.
Supporting references: 1. Investor.gov: Mutual Funds, Past Performance
Check this case’s calculations
- Two-year wealth under unequal opposite returns
- 1,800.00 USD
((2000 * 1.5) * 0.59999999999999998)
Limits of this additional case
- The chosen path illustrates averaging and does not estimate an expected return, volatility penalty or future outcome.
Limits of this analysis
- Endpoint growth formulas require adjustment for external flows.
- A geometric summary hides interim drawdowns.
- Neither historical mean nor hypothetical constant growth forecasts future returns.
Sources and scope
- Investor.gov: Mutual Funds, Past Performance
Past fund performance does not necessarily predict future results.
Scope: U.S. investor education; supports the stated mechanism, not the hypothetical scenario or a forecast.. Reference accessed 2026-10-06.
- Investor.gov: How to Read a Mutual Fund or ETF Shareholder Report
Reports provide performance, benchmark and portfolio context; past returns are not predictions.
Scope: U.S. investor education; supports the stated mechanism, not the hypothetical scenario or a forecast.. Reference accessed 2026-10-06.
Illustrative calculations are not forecasts or personal recommendations. Editorial standards
