An Average That Nobody Receives
Two investments, both advertising an average annual return of 25%.
Investment A returns 25% every year, steadily. Investment B returns +100% one year and −50% the next, repeating.
The average of +100 and −50 is +25. Same headline number.
Put £10,000 into each for four years.
A: 10,000 → 12,500 → 15,625 → 19,531 → 24,414 B: 10,000 → 20,000 → 10,000 → 20,000 → 10,000
Investment B has a 25% average annual return and has made exactly nothing. Not a rounding difference. Not a fee. The average is simply describing something other than what happened to the money.
Arithmetic vs Geometric Mean
The number in the brochure is usually the arithmetic mean — add the annual returns, divide by the number of years. It answers: what was the typical single-year return?
What your balance follows is the geometric mean — the constant rate that would have produced the same final value. It answers: what did the money actually compound at?
geometric mean = (product of all growth factors)^(1/n) − 1
For Investment B: growth factors are 2.0 and 0.5. Their product is 1.0. Take the fourth root of 1.0 and you get 1.0. Geometric mean: 0%.
For Investment A: 1.25 four times over. Geometric mean: 25%.
Two critical properties:
The geometric mean is always less than or equal to the arithmetic mean. They're equal only when every return is identical. Any variation at all creates a gap.
The gap widens with volatility. This is not a small correction for volatile assets — it's the dominant effect.
Volatility Drag
The size of the gap has a useful approximation:
geometric ≈ arithmetic − (σ² / 2)
where σ is the standard deviation of returns, expressed as a decimal.
Run it for a few realistic cases:
| Arithmetic mean | Volatility (σ) | Drag | Approx. geometric |
|---|---|---|---|
| 8% | 10% | 0.5% | 7.5% |
| 8% | 20% | 2.0% | 6.0% |
| 8% | 30% | 4.5% | 3.5% |
| 8% | 50% | 12.5% | −4.5% |
The drag term is squared, so it grows fast. Doubling volatility quadruples the penalty.
At 50% annual volatility, an 8% average return produces a negative compound growth rate. The portfolio loses money while the average looks fine. This is the arithmetic behind why highly leveraged and highly volatile strategies can post attractive average returns while destroying capital.
Why the asymmetry exists
The underlying reason is simple and worth internalising: a percentage loss requires a larger percentage gain to recover.
- Lose 10%, need +11.1% to get back
- Lose 20%, need +25%
- Lose 33%, need +50%
- Lose 50%, need +100%
- Lose 90%, need +900%
Losses compound against a shrinking base. Gains compound onto a growing one. Averaging percentages treats a −50% and a +50% as cancelling out. In reality, £100 → £50 → £75. You've lost a quarter of your money and the arithmetic mean says you broke even.
What This Means for Comparing Investments
Look for CAGR, not average return. Compound annual growth rate is the geometric mean. Reputable fund reporting standards require it, but marketing material and informal comparisons frequently don't.
You can check any advertised figure yourself. Take the start value, the end value, and the number of years:
CAGR = (end / start)^(1/years) − 1
If a fund quotes an average return well above its calculated CAGR, the difference is volatility drag, and the CAGR is the number that reflects your experience.
Volatility isn't just discomfort — it's cost. Two portfolios with the same expected arithmetic return but different volatility do not have the same expected terminal wealth. The lower-volatility one wins. This is a substantial part of the mathematical case for diversification: it reduces σ, which raises the geometric mean, even without raising the arithmetic mean at all.
Be suspicious of long track records with big drawdowns. A strategy with several strong years and one catastrophic one can still show an appealing average. Ask for the year-by-year sequence, not the summary.
Modelling It Properly
The Compound Interest Calculator works with a steady rate, which makes it the right tool for the question "what does my money do at a constant compound rate?":
- Enter your starting principal.
- Enter the annual rate — use the geometric mean or CAGR, not the arithmetic average.
- Set the time period and compounding frequency.
- Add regular contributions if applicable.
- Read the projected final value.
Running it twice is instructive. Try it with the arithmetic mean, then with the geometric mean, and compare the two end values. For a volatile asset over 20 or 30 years, the difference is often enormous — and the second number is the honest one.
If you know an asset's historical average return and volatility, you can estimate the geometric mean with the drag formula and feed that in.
Sequence Risk: When Order Suddenly Matters
Everything above assumed no cash flows. Multiplication is commutative, so with a lump sum left untouched, the order of returns doesn't affect the final value. Good year first or bad year first, same destination.
Add regular deposits or withdrawals and that stops being true.
Consider two retirees, each starting with €500,000 and withdrawing €25,000 a year. Both experience the same set of annual returns over 25 years. One gets the bad years at the start; the other gets them at the end.
The one who hits poor returns early is selling assets at depressed prices to fund withdrawals, permanently reducing the capital base that would otherwise have recovered. The one who gets good years first builds a cushion before the bad years arrive. Same average return, same geometric mean, dramatically different outcomes.
This is sequence of returns risk, and it's why the years immediately before and after retirement are treated as a distinct risk period. It also works in the saver's favour in reverse: someone accumulating benefits from poor early returns, because they're buying more units cheaply while their balance is small.
Practical Tips
Convert everything to CAGR before comparing. It's the only figure that's comparable across investments with different volatility profiles.
Treat volatility reduction as a return enhancement. Lowering σ raises your compound growth even if it does nothing to the arithmetic average.
Look at drawdowns alongside averages. Maximum drawdown tells you what the sequence looked like in a way an average never can.
For projections, be conservative with the rate. Long-horizon compound projections are extremely sensitive to the assumed rate. A percentage point over 30 years is a very large difference.
Remember inflation. A 7% nominal CAGR with 3% inflation is roughly 4% real. Project in real terms if you want the answer in today's purchasing power.
FAQ
Is CAGR the same as the geometric mean? Yes. CAGR is the geometric mean of annual growth factors, expressed as a rate.
Why do funds quote average annual returns at all? It's a legitimate statistic — it describes the typical single year. It just isn't the number that describes what happened to invested capital, which is why performance reporting standards generally require compound figures.
Does volatility drag mean volatility is always bad? It means volatility has a mathematical cost to compound growth. Whether an asset is worth that cost depends on whether its higher expected return more than offsets it.
Does the order of returns matter? Not for a lump sum left alone. It matters a great deal when you're adding or withdrawing money along the way.
How do I calculate CAGR from a start and end value? Divide the end by the start, raise to the power of one over the number of years, subtract one.
The Takeaway
Compounding is multiplication, and averages are addition. Any time returns vary, those two operations disagree — and the gap between them is exactly the volatility drag. Ask for the compound figure, and if you're only given an average, treat it as an upper bound on what actually happened.
Model compound growth free with the Compound Interest Calculator at sadiqbd.com — no sign-up, instant results. This article is general information, not financial advice; consult a qualified adviser for decisions about your own money.
