Why Average Returns Lie: Geometric vs Arithmetic

Your financial advisor says your portfolio returned an average of 8% per year over the last decade. That sounds great. But your account balance grew by less than 8% annualized. Sometimes much less.

This is not a fee issue or a tax issue (though those matter too). It is a math issue. The "average return" your advisor quotes is the arithmetic mean. The return you actually experienced is the geometric mean. And they are never the same number when returns vary from year to year.

Understanding this gap is not optional for retirement planning. It is the difference between a plan that works on paper and one that works in reality.

Two Kinds of Average

Arithmetic mean: Add up all the annual returns and divide by the number of years.

Geometric mean: The single constant annual return that would produce the same ending balance from the same starting balance.

A simple example makes the difference obvious.

You invest 100,000. In year one, you gain 50%. In year two, you lose 50%.

Arithmetic mean: (50% + -50%) / 2 = 0%. Sounds like you broke even.

Ending balance: 100,000 x 1.50 x 0.50 = 75,000. You lost 25%.

The geometric mean here is about -13.4% per year. That is the real compound growth rate. The arithmetic mean of 0% is a fiction.

This is not a trick or an edge case. It happens in every portfolio, every year, to some degree. The only time the arithmetic and geometric means are equal is when returns are identical every year. As soon as there is any volatility, the geometric mean is lower than the arithmetic mean. Always.

Variance Drain: The Mathematical Penalty

The gap between the two means has a name: variance drain (also called volatility drag). There is an approximation formula for it:

Geometric mean is roughly equal to Arithmetic mean minus (Variance / 2)

For a portfolio with an arithmetic mean return of 10% and a standard deviation of 15%, the variance drain is approximately (0.15 squared) / 2 = 1.125%. So the geometric mean is roughly 8.875%.

That 1.1% annual difference compounds over decades. Over 30 years, growing at 10% instead of 8.875% turns 1,000,000 into about 17,449,000 instead of about 12,819,000 - a gap of more than 4.6 million, caused entirely by volatility eating into compound growth.

Higher volatility means more drain. A portfolio with 20% standard deviation loses roughly 2% per year to variance drain. At 25%, it is over 3%. Try your own numbers:

Why This Matters for Retirement Projections

When a retirement calculator asks you to input your "expected return," which number should you use?

If you enter the arithmetic mean (say 10% for equities), the calculator will overestimate your ending balance. Every year, it compounds at 10%, as if volatility did not exist. The projection will be too optimistic.

If you enter the geometric mean (say 8.5%), the calculator accounts for the drag effect, but only in aggregate. It still shows a smooth growth path with no variation, which misses the sequence-of-returns risk problem entirely.

This is one of the reasons a Monte Carlo retirement calculator is superior to a fixed-return projection. Monte Carlo does not need you to choose between arithmetic and geometric means. It generates random returns from a distribution defined by the arithmetic mean and standard deviation, then compounds them year by year. The variance drain emerges naturally from the simulation. You do not need to calculate it or adjust for it manually.

The Deception in Marketing Materials

Fund reports usually show annualized (compound) returns, but headlines, forums and back-of-the-envelope plans often quote the simple average of yearly returns. It is the bigger number, and it overstates what an investor actually earns over many years.

From 1928 to 2025, U.S. stocks averaged 11.9% a year, but compounded at 10.0% (both before inflation). Whenever someone quotes an "average return," it is worth asking which one.

For retirement planning, you care about compound growth. You care about how much money you will actually have. The arithmetic average tells you something useful about any single year's expected return, but it overpromises on multi-year outcomes. Over a 30-year retirement, the gap between what the arithmetic average implies and what the geometric average delivers can be a large share of your savings.

The Gap in Real Market Data

The S&P 500 from 1928 through 2025 has an arithmetic mean return of 11.9% per year and a geometric mean of 10.0% (Damodaran data). That gap of almost 2 points is variance drain, every single year, for nearly a century.

But century-long averages hide the worst stretches. The 2000-2009 decade tells the story more clearly:

Arithmetic mean: +1.2% per year. A surface-level reading says the market was slightly positive.

Geometric mean: -0.95% per year. An investor who held the full decade lost purchasing power.

The arithmetic average says "roughly flat." The geometric reality says "you lost money." Two crashes sandwiching a recovery created enough volatility to flip the sign of the decade's returns. Someone who plugged +1.2% into a fixed-return retirement calculator would have gotten a mildly optimistic projection. Someone living through the actual sequence - especially while withdrawing from the portfolio - would have had a very different experience.

How to Calculate Each: A Walkthrough

Using the first three years of that decade (-9.1%, -11.9%, -22.1%):

Arithmetic mean: (-9.1 + -11.9 + -22.1) / 3 = -14.4%

Geometric mean: Multiply the growth factors: 0.909 x 0.881 x 0.779 = 0.6238. Then take the cube root: 0.6238^(1/3) = 0.8545. Subtract 1: -14.55%.

In this case the two are close because the returns are all negative (low variance around the mean). But run the same calculation across all ten years - where returns swing from -37% to +28.7% - and the gap between arithmetic (+1.2%) and geometric (-0.95%) becomes large enough to change the conclusion entirely.

The Interaction With Fat Tails

Variance drain gets worse under fat-tailed distributions. Why? Because fat tails mean more extreme returns in both directions, which increases variance. Higher variance means more drag.

With 16% volatility, variance drain is about 1.3% per year. If a fat-tailed model adds variance on top, the drag grows with it - and the extra losses tend to come in the worst years, which is exactly where they hurt a retiree most.

What This Means for Your Plan

Three practical implications:

Do not trust simple projections using arithmetic returns. If a calculator tells you that at 10% annual growth you will have X by age 80, the real number will be meaningfully lower. Use a Monte Carlo simulator that handles the math correctly.

Higher-volatility portfolios pay a steeper penalty. An all-equity portfolio has a higher arithmetic mean than a balanced one, but the variance drain is also larger. After adjusting for drag, the expected compound growth gap between a 100% equity portfolio and a 70/30 portfolio is smaller than the arithmetic means suggest.

This is one reason diversification works. A diversified portfolio typically has lower volatility than its individual components (especially when assets have low correlation). Lower volatility means less variance drain, which means a higher geometric return relative to the arithmetic return. Diversification does not just reduce risk. It improves compound growth.