What it means
The everyday average, properly called the arithmetic mean, adds the numbers and divides by how many there are. That works for things that simply pile up, such as monthly sales in dollars, but it overstates performance for anything that multiplies from one period to the next.
The reason is that percentage changes compound on a shrinking or growing base. Losing 50% and then gaining 50% leaves you 25% down, yet the arithmetic mean of those two returns is zero, which describes a result nobody actually experienced.
The geometric mean fixes this by multiplying the growth factors together and taking the appropriate root. A growth factor is just one plus the return expressed as a decimal, so a 12% gain is 1.12 and an 8% loss is 0.92.
The geometric mean is always less than or equal to the arithmetic mean, and the gap widens as the returns become more volatile. That gap is a useful signal in itself: two funds can advertise the same average annual return while one of them left investors with far less money.
In practice you will meet the same idea under the label compound annual growth rate, which is the geometric mean of a series of annual growth factors. It is also the correct way to combine multiplicative factors such as a chain of index numbers, and it is used in some index construction where an ordinary average would overstate the index level.
In practice
Real-world examples.
Example
A wealth manager reports a client portfolio as returning 9.35% a year rather than 11.25%, because the geometric figure is the number that reconciles to the actual closing balance. The client's statement and the performance report agree, which removes an annual argument about why the money grew less than the reported average suggested.
Example
A retailer's revenue rises from $18,000,000 to $32,000,000 over five years. Rather than averaging the five yearly growth percentages, the finance team calculates the compound annual growth rate as the geometric mean of the growth factors, giving a figure of about 12.2% that can be used to project the next three years.
Example
A manufacturer estimates the cumulative effect of cost inflation across three inputs that each rose at different rates. Because the inputs multiply through the cost model rather than add, the analyst uses the geometric mean of the three factors to produce a single blended inflation figure for the budget.
Formula
Calculation
Geometric mean return = (product of all growth factors) raised to the power of 1/n, minus 1
where n is the number of periods and each growth factor is 1 + the period return.
A fund returns 30% in year one, -20% in year two, 25% in year three and 10% in year four. The growth factors are 1.30, 0.80, 1.25 and 1.10.
Product = 1.30 x 0.80 = 1.04, then 1.04 x 1.25 = 1.30, then 1.30 x 1.10 = 1.43.
Geometric mean = 1.43 raised to the power of 0.25, minus 1 = 1.0935 - 1 = 9.35% a year.
The arithmetic mean is (30% - 20% + 25% + 10%) / 4 = 45% / 4 = 11.25% a year, which is 1.9 percentage points higher.
Check it against real money. An investment of $100,000 grows to $100,000 x 1.43 = $143,000 over the four years. Compounding at the geometric mean gives $100,000 x 1.0935 x 1.0935 x 1.0935 x 1.0935 = $142,980, which matches within rounding. Compounding at the arithmetic mean would give about $153,200, overstating the investor's wealth by roughly $10,200.Case study
Seen in the real world.
This is an illustrative, fictional case. Bramwell Asset Partners, an invented boutique fund manager, marketed one of its funds on an average annual return of 11.25% over four years. The figure was not fabricated, it was simply the arithmetic mean of returns of 30%, -20%, 25% and 10%.
A prospective institutional client asked a simple question: if an investor had put in $5,000,000 at the start, what would the account be worth now? The answer was $5,000,000 x 1.43 = $7,150,000. Compounding $5,000,000 at the advertised 11.25% would have produced roughly $7,660,000, a difference of about $510,000 that existed only on the marketing sheet.
The fictional manager rewrote its materials to lead with the geometric mean of 9.35% and to show the volatility alongside it. Sales conversations became slightly harder and due diligence became much easier, and the invented firm found that institutional buyers, who recalculate the numbers anyway, trusted the second version far more than the first.
Watch out
Common mistakes.
- Using the arithmetic mean for a series of investment returns, which overstates the growth actually delivered whenever returns vary from year to year.
- Trying to calculate a geometric mean when one of the growth factors is zero or negative, which happens if a value falls to nothing and makes the calculation meaningless.
- Averaging the growth factors instead of multiplying them, so 1.30 and 0.80 are treated as 1.05 rather than a product of 1.04.
Questions
People also ask.
When should I use the arithmetic mean instead?
Use it for quantities that add rather than compound, such as average monthly units sold, and use it when estimating the single most likely return for one future period rather than the realised return over many.
Is the compound annual growth rate the same thing?
Yes in substance, since a compound annual growth rate is the geometric mean of annual growth factors expressed as a percentage.
Why is the geometric mean always lower?
Because volatility itself destroys compound growth, so the more the returns scatter around their average, the further the compounded outcome falls below the simple average.
From the founder's library
Take it further with the book.
Build your financial confidence beyond this definition. Shihan's full-length guide, Accounting Fundamentals, takes the same plain-English approach and turns it into a complete, practical playbook for non-finance managers, business owners and students - with chapter-end quiz answers and presentation slides included.
25% off with code MMHQ25, applied at checkout. Priced in USD - checkout may show the equivalent in your local currency.
View the book and save 25%