An average period and a compounded path answer different questions
The arithmetic mean adds equal-period returns and divides by their count. The geometric growth rate instead finds the constant per-period factor that reproduces their compounded product for untouched money. Both can be useful summaries, but they are not interchangeable. Substituting an arithmetic average into a compound-growth projection can overstate the ending value of a particular volatile path.
This article uses invented returns for teaching rather than historical market claims or expected portfolio forecasts. All periods in each example have equal length and returns use the same fee convention. Adding deposits, withdrawals or leverage creates further measurement questions that a pair of average rates alone cannot settle. No average removes the uncertainty in future investment outcomes.
Calculate the arithmetic mean without changing the denominator
For two annual returns of +20 percent and −20 percent, the arithmetic mean is (20 + (−20))/2 = 0 percent per year. That says the two reported percentage changes average to zero. It does not say that an investor finishes with the opening balance, because the second percentage is applied to a different balance after the first year's change.
Starting with $1,000 and making no external cash flows, a 20 percent gain produces $1,200. A subsequent 20 percent loss removes $240 and leaves $960. The dollar loss in the second year exceeds the earlier $200 gain. The apparent contradiction disappears once the denominators for the two percentage changes are made explicit.
Build the geometric rate from growth factors
Convert each decimal return r_k into a growth factor 1 + r_k. For N equal periods, geometric growth rate is [(1 + r_1) × ... × (1 + r_N)]^(1/N) − 1, when the positive-factor model applies. The two-year example has product 1.2 × 0.8 = 0.96, so the rate is sqrt(0.96) − 1, approximately −2.02041 percent per year.
Compounding that constant geometric rate for two years reproduces $960 from $1,000. The cumulative return is −4 percent, which is different from both the zero arithmetic average and the approximately −2.02 percent annual geometric rate. Label cumulative and per-period outputs separately so that a summary table does not compare unlike quantities.
Use another path as an independent check
For a constructed three-year sequence of +20 percent, −10 percent and +10 percent, the arithmetic mean is approximately 6.66667 percent. The compounded factor is 1.2 × 0.9 × 1.1 = 1.188. Untouched $1,000 becomes $1,188, while the constant geometric rate is 1.188^(1/3) − 1, approximately 5.91045 percent annually.
The arithmetic mean describes the average of the three returns; it does not reconstruct the $1,188 endpoint when compounded. Under positive growth factors, the geometric mean does not exceed the arithmetic mean, with equality when period returns are identical. This mathematical comparison is not a claim that one summary is always the correct forecast for every investment question.
Keep period length and total-loss boundaries explicit
Monthly and annual observations cannot be averaged together as though each represents the same length of time. An average of one month and one year gives those unequal intervals the same weight unless a defined adjustment is made. First identify the periods and the measurement method. Do not annualize a monthly figure by relabeling its percentage as annual.
A complete loss gives a growth factor of zero, so a logarithmic geometric-mean calculation cannot simply take its logarithm. A subsequent positive percentage return cannot recover untouched money from a zero balance. Returns below −100 percent also lie outside the ordinary positive-factor unlevered model used here. Those cases need appropriate separate treatment rather than silently discarding the inconvenient period.
Reordering returns changes cash-flow outcomes even when the product matches
Without cash flows, +20 percent followed by −20 percent gives the same 0.96 product as the reversed order. Now add a hypothetical $500 at each year end, starting with $1,000. The gain-then-loss path ends at ($1,000 × 1.2 + $500) × 0.8 + $500 = $1,860. The loss-then-gain path ends at ($1,000 × 0.8 + $500) × 1.2 + $500 = $2,060.
Both paths contribute $2,000 in total and share the same arithmetic and geometric summaries of their two annual asset returns. Their investor dollar outcomes differ because the contributions encounter different later returns. A geometric average that reproduces untouched growth is not enough to reproduce a contribution plan under an arbitrary sequence. Withdrawals introduce a similar need to retain timing.
Interpret NumUtility's scenarios as constant-rate illustrations
The Investment Calculator takes lower, base and higher constant annual return assumptions, smooths them into monthly factors and includes its specified fee model. It does not accept a list of historical returns or calculate either average from such a list. Entering a historical arithmetic mean does not turn its output into a backtest of that history.
For a no-contribution check of a known geometric rate, a matching constant-rate run can reproduce the endpoint when rate and fee conventions agree. With contributions, it can differ from the actual variable path just illustrated. Use a separate sequential ledger for that path, and keep the tool's illustrative lower and higher cases distinct from statistical confidence intervals or guaranteed bounds. If supplying a geometric assumption from another calculation, confirm that it is before or after fees consistently; charging an already deducted fee again changes the path for a different reason.
Report the measure, cash flows and exclusions together
A clear return summary states period length, arithmetic average, geometric growth rate, cumulative factor and whether external flows were absent. Add the currency gain or loss only with its funding record. For an actual performance calculation, appropriate time-weighted or money-weighted methods may be needed; this article's two averages do not replace those methodologies.
This explanation is educational and is not investment advice. It does not recommend a return assumption, identify an optimal portfolio or forecast the next period. Its purpose is to prevent the same word average from hiding different calculations. Choose the measure that answers the stated question and preserve enough information to check the compounded path independently.