What it means
The raw material is simple: for each period, the change in value plus any income received, divided by the value at the start. String those figures together and you have a track record covering a year, a decade or a century.
Two different averages can be taken from the same series and they answer different questions. The arithmetic mean adds the annual returns and divides by the number of years, while the geometric mean, also called the compound annual growth rate, reflects what an investor actually ended up with after the ups and downs compounded.
The geometric figure is always the lower of the two whenever returns vary, and the gap widens with volatility. That gap is not a technicality: quoting the arithmetic mean for a volatile investment systematically overstates what any real investor experienced.
Historical returns matter in business decisions well beyond portfolios. They feed the assumed return on pension assets, the discount rate in a valuation, the expected cost of equity and the hurdle a project must beat, so an over-optimistic history quietly inflates every one of those numbers.
The nuance everyone quotes and few respect is that past performance does not predict future performance. A sensible use is to understand the range of outcomes an asset has produced, including its worst stretches, rather than to project the average forward as though it were a promise.
In practice
Real-world examples.
Example
A pension trustee board reviews 20 years of returns on its growth fund and finds a geometric average of 6.1% against an arithmetic average of 7.4%. It lowers the assumed return used in the funding valuation to the geometric figure, which increases the required employer contribution.
Example
A founder building a five-year forecast uses the historical average return on invested capital in her sector, about 11%, as the hurdle for a new production line. Any project returning less becomes hard to justify against simply investing elsewhere.
Example
An investment committee compares two funds with identical five-year total returns but very different year-by-year paths. It selects the steadier one because the volatile fund's worst single year was a 34% loss, which its liquidity needs could not absorb.
Formula
Calculation
Annual return = (ending value - beginning value + income) / beginning value.
Arithmetic mean = sum of annual returns / number of years.
Geometric mean = (product of (1 + each annual return)) raised to the power of 1 divided by the number of years, minus 1.
Take five years of returns on a fund: 12%, -8%, 15%, 6% and 10%.
Arithmetic mean = (12 - 8 + 15 + 6 + 10) / 5 = 35 / 5 = 7.0% a year.
For the geometric mean, multiply the growth factors: 1.12 x 0.92 x 1.15 x 1.06 x 1.10 = 1.3817. Taking the fifth root gives 1.0668, so the geometric mean is 6.68% a year.
Check it against the money. An investment of $10,000 grows to $10,000 x 1.3817 = $13,817 over the five years. Compounding at the geometric mean of 6.68% reproduces that figure, while compounding at the arithmetic mean of 7.0% would suggest $14,026, overstating the result by roughly $209. That difference of 0.32 percentage points a year is the cost of using the wrong average, and it grows with both time and volatility.Case study
Seen in the real world.
Calder Foundation is a fictional charitable endowment used here for an illustrative case study. It holds $40 million and its board sets an annual spending rate by looking at the historical returns of its portfolio, which have averaged 7.0% a year on an arithmetic basis over the past five years.
The finance committee proposes spending 6.5% a year, reasoning that returns of 7.0% comfortably cover it. An adviser recalculates using the geometric mean and gets 6.68%, then points out that the endowment also pays 0.9% in fees and needs roughly 2.5% growth to keep pace with rising costs. On those figures the sustainable spending rate is closer to 3.3%, not 6.5%.
The board recalculates and settles on 4.0%, phased in over three years to avoid a sudden cut to the charities it supports. In this illustrative outcome, the decision reduces annual grants from $2.6 million to $1.6 million, a difficult conversation that the trustees judged far preferable to slowly consuming the capital. The point of the fictional example is that a single misread average, compounded over decades, decides whether an endowment survives.
Watch out
Common mistakes.
- Quoting the arithmetic mean as though it were the return an investor earned. For anything volatile, the geometric mean is the honest figure and the arithmetic mean flatters the record.
- Choosing a start date that makes the numbers look good. A track record beginning just after a market crash tells you about the recovery, not about the investment.
- Projecting the historical average forward as an expectation. History gives you a range of plausible outcomes, and treating the midpoint as a forecast ignores everything the range was telling you.
Questions
People also ask.
Should returns be shown before or after fees?
After fees and costs, since gross returns describe an experience no actual investor ever had.
How many years of history are enough?
Longer is better, and a period covering at least one full downturn is far more informative than a decade of steady growth.
Do historical returns predict future returns?
No, though they do give a useful sense of volatility and of how bad the worst periods have been, which is often the more valuable information.
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