Why your portfolio's average return overstates what you'll get

"The market returns 10% a year" is the most common way to overstate a compound path without noticing. An average of annual returns is not the rate you can compound. The rate you can compound is the geometric mean, or, on a sample of lives, something closer to the median terminal. This page measures the gap on a frozen no-spend companion of the B4 retiree mix: $1,000,000, 80/15/5, 30 years, no withdrawals, seed 20260622, 1,000 paths.

We used a no-spend companion so the annual change in wealth is a return, not a return mixed with a $40,000 withdrawal. The spending cone lives on the sibling pages. This identity is cleaner without it. The treatment is the summary statistic: arithmetic mean of annual simple returns versus the geometric mean implied by the median terminal.

Why your portfolio's average return overstates what you get

Arithmetic mean on this freeze: 5.0%. Geometric mean from the median ending: 3.3%. The gap is 1.74 points a year. That does not sound like much. Compounded for 30 years it is $4,332,063 versus $2,624,399. Almost $1,707,664 of paper wealth that the average promised and the median life did not collect.

The identity is old. For a simple two-year example, +50% then −50% averages 0% and compounds to −25%. The average is not a lie about the years. It is a lie about the pile. Volatility is not a feeling. It is a leak in the compound. This freeze is that leak with 1,000 years-of-years instead of two.

People still quote arithmetic means because they are larger and because they match the way a calendar is averaged. Fund fact sheets, back-of-envelope retirement math, and "stocks return 7% real" slides are usually arithmetic or worse, a rounded headline. If you then raise that number to the 30th power, you have built the dashed line on this chart.

Median life versus mean path

The solid line is the year-by-year median of the no-spend fan. It ends at $2,624,223. The dashed line compounds 5.0% from the same $1,000,000. They start together and they do not finish together. The dashed line is not a life. It is what you get if you give every year the average year. No year is the average year. The median life is a sequence of un-average years that happened to sort to the middle at the end.

Skew makes this worse. A few excellent sequences pull the arithmetic mean of annual returns up. The median terminal does not care about those sequences except as other people's luck. Planning as if you will receive the mean of other people's luck is the same error as planning to the p90 on a cone.

Inflation is in the engine. The terminals here are the real fan when it exists. If you quote a 10% nominal average and then forget inflation, you have stacked two overstatements. The real return calculator is the matching identity for that second mistake. This page is the first.

Scope

Q1 run on the no-spend companion (same seed and mix as B4_RETIREE, contributions and withdrawals off). Q2 not run: the gap is variance drag inside this calibration. Q3 not run: this is not the staying-invested historical table, which is a different question about missing months. Q4: 5 bps, tax off. Teaching identity. No forecast of next decade returns.

What would falsify: geometric mean greater than or equal to arithmetic mean on these 1,000 annual-return samples, or the arithmetic-compounded 30-year wealth falling at or below simulated p50. Arithmetic stayed above geometric. The dashed line finished above the median.

Annual returns were taken from year-over-year changes in each no-spend wealth path. That is a wealth-relative annual change, named here so it is not confused with a time-weighted return after deposits. There were no deposits on this companion.

The dashed line, year by year

At year 10 the no-spend median is $1,397,462. The arithmetic compound is about $1,630,447. At year 20 the median is $1,837,962 and the arithmetic compound is about $2,658,356. The leak is not a last-year surprise. It accumulates. That is why compounding a headline is worse than quoting the headline. The headline is only wrong by a couple of points. The power is wrong by a million dollars on this freeze.

Fund fact sheets that show "average annual return" without a geometric twin are doing this. So are back-of-envelope models that take 7% or 10% and raise it to 30. The real return calculator on this site will at least strip inflation out of a nominal headline. It will not, by itself, turn an average into a compound. For that you need a distribution, which is the cone.

The no-spend median at year 30 is $2,624,223. The arithmetic-compounded 30-year value is $4,332,063. Ratio: the average path is claiming about 1.65 times the median life. If a slide used that average as "what you will have," it overstated the typical life by that factor, on this specimen, before spending even started.

See a real rate, not a headline

Use the real rate of return calculator when the headline is nominal. Use a cone when the headline is an average you were about to raise to the 30th power. This freeze's arithmetic 5.0% compounded for 30 years is $4,332,063. The median life is $2,624,399. If you take only one habit from the page, take this: when someone says average return, ask whether they mean the number they will raise to a power. If yes, they are drawing the dashed line. The dashed line is legal. It is not the pile. The pile is the compound, and the compound on this specimen is closer to 3.3% than to 5.0%.

Variance drag is not a vibe. It is why those two percentages are not equal. This engine had variance. The identity held. A later freeze with zero volatility would have been a different, boring paper, and geometric would have matched arithmetic. We did not get that paper. We got this gap.

What this page refuses to say

It refuses to forecast next decade's market return. 5.0% is an average of annual wealth changes on this freeze, not an ERP estimate for 2026–2036. It refuses to say geometric means are "true" and arithmetic means are "false." Both are true of different objects. It refuses to compound 10% for 30 years in the caption and then wave at this chart. This chart is 5.0% versus 3.3% on a no-spend companion. Different headline, same crime.

It refuses to mix spending into the annual returns. That is why the companion has no withdrawal. The retiree cone is the spending story. This page is the compound story. It refuses to cite the staying-invested article as a replication. Missing the best months is a timing question. This is an averaging question. Both can overstate a pile. They overstate it for different reasons.

It refuses a fund-fact-sheet ranking. Some sheets print geometric, some arithmetic, some a blended marketing number. This freeze does not audit them. It shows the gap on one engine so a reader can ask which number they were just handed. Ask. If the answer is "average," ask "compounded?" If the answer is silence, assume the dashed line.

Dollar gap $1,707,664 is the size of the overstatement at 30 years on this specimen. It is not a loss you can collect from a broker. It is a caption you can refuse.

Replication spec

No-spend companion: $1,000,000, 80/15/5, 30 years, buy_and_hold, seed 20260622, 1,000 paths, real fan when present. Annual simple returns from year-over-year wealth changes on each sample path. Arithmetic mean 5.0% across those annuals. Median terminal $2,624,399. Geometric mean 3.27% defined as (medianEnd/start)^(1/30)−1. Arithmetic compound 1,000,000×(1+arith)^30 = $4,332,063. n of paths used: 1000.

No-spend median yearly: year 5 $1,165,947, year 10 $1,397,462, year 15 $1,600,292, year 20 $1,837,962, year 25 $2,175,365, year 30 $2,624,223. The dashed line is not stored as an array in the freeze. It is recomputed as 1,000,000×(1+0.0501)^t so a test can rebuild a plotted point from the same inputs the SVG used.

Gap: 1.74 points a year. Dollar gap at 30 years: $1,707,664. CSV: /data/studies/average-return-overstates-compound.csv. This is not the staying-invested historical miss-the-best-months table. Different question, different producer.

Q2 not run. Q3 not run. Q4: 5 bps, tax off. Teaching identity. No forecast. If a later freeze prints geometric greater than arithmetic, the article is void. Variance was not zero here, and the identity held.

Last recap: no-spend companion, 30 years, 80/15/5, 1,000 lives. Arithmetic 5.0%, geometric 3.3%, compounded average $4,332,063, median pile $2,624,399. The average overstates the compound by $1,707,664 at the horizon on this specimen. That is variance drag plus mean-versus-median, not a forecast and not a fee. Ask which number someone is about to raise to a power. If it is the average, they are drawing the dashed line. The dashed line is not the pile.

Seed 20260622. No deposits, no withdrawals, so a wealth change is a return. Mix spending back in and you are on the cone pages, not this one. Year-10 median $1,397,462versus the arithmetic compound at 10 years already disagrees. The leak does not wait for year 30. That is why compounding a headline is worse than quoting the headline: the power does the damage early, then keeps doing it. Keep the CSV next to the caption. Do not type a rounder median. Do not raise 10 percent to the 30th power and cite this page.

Notes. No-spend companion of B4_RETIREE mix, $1,000,000, 80/15/5, 30 years, seed 20260622, 1,000 paths, buy_and_hold. Arithmetic mean of annual wealth-relative returns versus geometric mean implied by median terminal. Q1 run. Q2/Q3 not run. Q4: 5 bps, tax off. CSV: /data/studies/average-return-overstates-compound.csv. Card dated 2026-08-28. This article is educational analysis, not investment advice, and does not recommend any security or strategy.