Why the average overstates what you earn
Start with the classic coin flip: your portfolio gains 50% one year and loses 50% the next. The average of +50% and −50% is exactly zero — yet$10,000 becomes $15,000, then $7,500. You are down 25%, because returns multiply rather than add: 1.5 × 0.5 = 0.75. The single rate that actually reproduces that outcome — the geometric mean — is -13.40% per year, a full 13.40 percentage points below the arithmetic average. That gap is volatility drag, and every volatile return stream pays it. Losses hurt more than equal-sized gains help, so the bumpier the path, the further the compound return falls below the quoted average. We wrote a full explainer on this —Average Returns Lie: why the number your fund reports isn't what you earned— and this calculator is its companion tool.
For portfolios described by a mean and a standard deviation, the drag has a famously compact estimate: about half the variance. Thomas Messmore named the phenomenon "variance drain" in a 1995Journal of Portfolio Management paper, and the σ²/2 term is the same one that appears in the mathematics of lognormal growth. It is an approximation for real-world annual returns — exact only in the continuous/lognormal limit — but it is accurate enough to be the standard back-of-envelope conversion between the two means.
The volatility drag formulas
g ≈ μ − σ²/2
gexact = [(1+r₁)(1+r₂)…(1+rₙ)]^(1/n) − 1
drag = μ − g
where μ is the arithmetic mean return, σ the standard deviation of returns (both as decimals), andr₁…rₙ the individual yearly returns. The first line is the approximation — exact only in the continuous-time / lognormal limit; the second is the exact geometric mean computed from an actual series. The drag is the distance between the average you are quoted and the rate your money compounds at.
Worked example
Take the calculator's default portfolio: a 9% average annual return with 15% volatility, $10,000 invested for 30 years. Here is what the σ²/2 drag does to the outcome:
| Step | Amount |
|---|---|
| The quoted average return (μ)an arithmetic mean, with 15% annual volatility (σ) | 9% |
| Volatility drag = σ²/2(0.15)² ÷ 2 = 1.125 percentage points, every year | 1.125pp |
| = Approximate geometric return g ≈ μ − σ²/29% − 1.125pp — the rate your money actually compounds at | 7.875% |
| $10,000 projected at the 9% average for 30 yearswhat the brochure math implies | $132,677 |
| $10,000 compounded at 7.875% for 30 yearswhat the volatile path actually delivers | $97,191 |
| = What the average overstatesa 1.125pp annual drag compounds into real money over 30 years | $35,486 |
Computed with this calculator's default settings — open the tool above and you'll see the same numbers, then swap in your own mean and volatility, or an actual return series.
Reading fund returns honestly
The practical upshot: whenever returns are quoted, ask which mean you are looking at. Fund fact sheets that report an "annualized return" or CAGR are giving you the geometric figure — the honest one for compounding. A simple average of yearly returns, or an expected return built from a forecast, is arithmetic and will overstate long-run growth unless you subtract the drag. The same logic explains why two funds with identical average returns are not interchangeable: the one with lower volatility compounds to more money, which is the entire argument for looking at risk-adjusted measures rather than raw averages.
To go deeper, compute the honest compound rate between two balances with ourCAGR calculator, work out the money-weighted return of a portfolio with contributions using theinvestment return calculator, or put a number on return-per-unit-of-risk with theSharpe ratio calculator.
Frequently asked questions
What is volatility drag?
Volatility drag (sometimes called variance drain) is the gap between the arithmetic average of a series of returns and the geometric return the money actually compounds at. It exists because gains and losses multiply rather than add: a 50% loss needs a 100% gain to recover, so a bumpy path always ends below a smooth path with the same average. The drag is roughly half the variance of returns — σ²/2 — per year, which means a portfolio with 15% volatility gives up about 1.1 percentage points of compound return relative to its quoted average.
What is the difference between the arithmetic and geometric mean return?
The arithmetic mean is the simple average: add up each year’s percentage return and divide by the number of years. The geometric mean is the single constant rate that, compounded over the same period, reproduces the actual ending balance — it is computed by multiplying the growth factors (1 + r) together and taking the n-th root. The geometric mean is what your wealth actually grew at, and it is always less than or equal to the arithmetic mean; the two are equal only when every year’s return is identical.
Is the σ²/2 rule exact?
No — it is an approximation. The identity g = μ − σ²/2 holds exactly only in the continuous-time limit, where returns are lognormally distributed and μ and σ describe the instantaneous process. For a real series of discrete annual returns, the exact geometric mean comes from compounding the actual growth factors, and the true drag can land above or below σ²/2 — usually close, but not equal. That is why this calculator has a yearly-returns mode: enter the actual series and it computes the exact means alongside what the rule of thumb would have predicted.
Why does volatility matter more over long horizons?
Because the drag never averages away — it compounds. A 1.1-percentage-point annual gap between the quoted average and the true compound return sounds small in any single year, but it is subtracted from the exponent every year. Over one year the difference on $10,000 is about a hundred dollars; over thirty years, the default example on this page shows the arithmetic projection overstating the real outcome by tens of thousands. Time diversification reduces the odds of a losing decade, but it does not shrink volatility drag — it gives the drag more years to work.
Can volatility drag be reduced?
The drag is a mathematical property of the return path, so the only lever is the volatility itself. That is one framing of why diversification is valuable: combining imperfectly correlated assets can lower portfolio volatility more than it lowers the average return, which narrows the gap between the arithmetic and geometric means. Disciplined rebalancing is often discussed in the same breath, since it keeps a portfolio’s risk from drifting upward. None of this is a free lunch or a recommendation — it is a description of why risk-adjusted thinking, not just average returns, drives long-run compounding.
Sources
- Quanticed — Average Returns Lie: Why the Number Your Fund Reports Isn’t What You Earned
- Investor.gov (U.S. SEC) — Glossary of investing terms
The official figures this page quotes are drawn from the primary sources above — check them (or a qualified professional) before relying on a result.
Disclaimer: This calculator is foreducation and illustration only. The σ²/2 formula is an approximation, real return distributions are not lognormal, and projections at any constant rate are simplifications — no volatility figure predicts an actual future path. Nothing here is investment, financial, or tax advice.