For crypto returns, the arithmetic mean averages the simple returns observed per period. The geometric mean return is the constant per-period growth rate that reproduces their compounded result. Calculate it by multiplying one plus each decimal return, taking the nth root, and subtracting one. Use arithmetic mean to describe average one-period outcomes and geometric mean to summarize a linked historical growth path; neither alone predicts future performance.
A report can correctly show a positive average weekly return while the account finishes below its starting value. The useful check is whether “average” describes the return observations or the growth of the money exposed to them.
Define a comparable return series first
Use consecutive, equal-length simple returns for the same investment boundary. An hourly coin-price change, a completed trade's profit divided by entry notional, and a daily whole-account return are different observations. Their percentages cannot be placed in one average just because they share a percent sign.
For a portfolio without external flows, calculate each return from consistently valued opening and closing equity. Preserve the reporting currency, idle cash, open positions and applicable costs. If the purpose is net account performance, the return series must already reflect the costs included in that policy. Compounding a price-only series cannot supply missing fees or funding afterward.
Deposits and withdrawals require a flow-aware calculation before this averaging step. The time-weighted versus money-weighted return guide explains which question each method answers. Flow-boundary subperiods can have unequal durations: geometrically linking them produces a cumulative return, but taking a root based only on their count does not produce an equal-time average.
This article stays with simple returns in a fixed observation interval. The simple versus log return guide covers conversion between return representations; the averaging distinction here does not require that conversion.
Average the percentages or average their growth factors
For n eligible periods, where n is at least one, with decimal simple returns r₁ through rₙ:
arithmetic mean a = (r₁ + r₂ + … + rₙ) / n
geometric mean g = [(1 + r₁) × (1 + r₂) × … × (1 + rₙ)]^(1/n) − 1
The second formula applies a geometric mean to the growth factors, then converts back to a return. A +20% return has factor 1.20; a −20% return has factor 0.80. Here “growth factor,” sometimes called a gross return factor, means one plus the return. It does not declare whether trading costs were deducted.
OpenStax's method for arithmetic and geometric investment returns sets out this product-and-root calculation and the distinction from an arithmetic average. With strictly positive factors, g is no greater than a; equality holds when all the period returns are identical.
The reconstruction test is particularly useful:
(1 + g)^n = product of all (1 + r)
That identity gives the geometric mean its meaning. Applying the same g each period reaches the original series' endpoint. It does not assert that the actual path was smooth.
Four hypothetical weeks: positive average, negative growth
Consider an illustrative account starting with $1,000. The following stipulated weekly returns are invented, with no external flows. The equity path has no additional fees, funding, borrowing or taxes; these large moves make the distinction visible rather than represent measured crypto performance.
| Week | Opening equity | Simple return | Growth factor | Closing equity |
|---|---|---|---|---|
| 1 | $1,000.00 | +50% | 1.50 | $1,500.00 |
| 2 | $1,500.00 | −40% | 0.60 | $900.00 |
| 3 | $900.00 | +20% | 1.20 | $1,080.00 |
| 4 | $1,080.00 | −20% | 0.80 | $864.00 |
The arithmetic mean is:
a = (0.50 − 0.40 + 0.20 − 0.20) / 4 = 0.025 = +2.5% per week
The linked growth factor is:
1.50 × 0.60 × 1.20 × 0.80 = 0.864
Therefore, the account's cumulative return is −13.6%, and its geometric mean is:
g = 0.864^(1/4) − 1 ≈ −0.035885897 = −3.5886% per week
Using the unrounded g, $1,000 × (1 + g)^4 = $864. By comparison, inserting the arithmetic mean into that same compounding expression gives $1,000 × 1.025^4 ≈ $1,103.81. That amount belongs to a different, hypothetical path earning exactly +2.5% every week.
The arithmetic mean remains a correct description of the four observed weekly percentages. The error would be presenting it as the constant growth rate that this account experienced. Likewise, −3.5886% is a per-week equivalent rate, while −13.6% is the full four-week result. These are different units of presentation.
Interpret the gap without inventing another cost
The example's arithmetic–geometric gap is approximately 6.0886 percentage points per week. It arises because percentage changes apply to changing capital bases. The 40% second-week loss removes $600; its preceding 50% gain added $500. Opposite percentage signs do not make those dollar effects cancel.
The gap is often discussed as volatility drag. For an observed path, it is a comparison between two summaries of that path. It is not another exchange charge to subtract from the $864 endpoint. Fees already included in the periodic returns must not be deducted again under that label.
Sampling also changes the question. Combine weeks 1–2 and weeks 3–4 into two consecutive two-week observations. Their returns are −10% and −4%, because 1.50 × 0.60 − 1 = −0.10 and 1.20 × 0.80 − 1 = −0.04.
| Observation interval | Number of returns | Arithmetic mean per interval | Geometric mean per interval | Cumulative return |
|---|---|---|---|---|
| One week | 4 | +2.5% | −3.5886% | −13.6% |
| Two weeks | 2 | −7.0% | −7.0484% | −13.6% |
The cumulative product is preserved, but the arithmetic mean even changes sign. The two-week geometric rate equals (1 + weekly g)^2 − 1. Attach an interval to every mean before comparing reports. Moving to a yearly scale introduces a separate clock decision covered by the annualized return and CAGR guide.
Catch calculation failures before publishing an average
For an empty eligible series, both means are undefined. With observations available, several errors can produce a plausible-looking number:
- Taking a geometric mean of signed net returns. Multiplying 0.50, −0.40, 0.20 and −0.20 uses the wrong inputs. The required factors are 1.50, 0.60, 1.20 and 0.80. Taking absolute values would discard the losses.
- Counting valuations rather than intervals. Five consecutive weekly endpoint valuations generate four weekly returns. The exponent is 1/4, not 1/5.
- Treating a missing week as zero. Zero means unchanged investment value under the return definition. A missing observation says the evidence is incomplete. Reconstruct the interval or disclose the changed sampling; do not silently dilute either mean.
- Averaging only weeks with trades. An account can change value while holding positions, or remain in cash. Removing calendar intervals changes the sample to an activity-conditioned statistic.
- Forcing a wipeout through an ordinary continuing series. With a positive opening value and a −100% return, the product becomes zero and the finite-horizon equivalent rate is −100%. Subsequent ordinary returns from zero capital are undefined. A later deposit needs separate flow treatment. Negative factors from returns below −100% require a different equity and liability model; a software root is not a meaningful investment growth rate.
Software labels also matter. PerformanceAnalytics documents separate geometric and arithmetic aggregation conventions, alongside an explicit periods-per-year scale. Inspect those options when reproducing a report; a function called “annualized return” does not identify every convention from its name.
Separate historical growth from forecasts and risk
A historical arithmetic mean can be used as an estimator of a one-period expected return only with assumptions about how the sample represents the future. A geometric mean describes the equivalent rate of its observed history. Neither turns a short or selected sample into an established forecast.
Brian McCulloch's Treasury working paper on geometric return and portfolio analysis distinguishes historical geometric summaries from expected-return modelling and discusses the time-series assumptions behind projections. Its examples are general investment mathematics, not evidence that crypto returns satisfy those assumptions.
Both means also discard ordering. Reordering the four stipulated returns preserves their sum and product. Yet the original equity path has a maximum drawdown of 42.4%; the order −40%, −20%, +50%, +20% takes $1,000 down to $480 before recovering to the same $864, a 52% maximum drawdown. These are checked properties of the constructed paths, assuming no additional events between valuations. The maximum drawdown and recovery guide explains that separate path measure.
A benchmark comparison needs another layer: information ratio and tracking error examine average active returns relative to their dispersion. For shortfalls below a chosen target, downside deviation and target return changes the risk question again. Neither risk measure can be inferred from the arithmetic–geometric gap alone.
Before accepting an “average return,” save the eligible return series, interval, costs, flow policy and observation count. Compute both means, retain the cumulative product, and check that the unrounded geometric mean reconstructs it. Publish the label that answers the intended question, with the actual endpoint and loss path beside it.