Why does the average return overstate reality?
Two funds both advertise "10% average annual return." One made its holders rich; the other went sideways. Both are telling the truth. Welcome to the most legal lie in finance.
The two averages
- Arithmetic average: add the yearly returns, divide by the years. Simple — and systematically flattering.
- Geometric average (CAGR): the single steady rate that would actually turn the starting money into the ending money. Formula for the curious: (end ÷ start)^(1/years) − 1. This is the one your bank balance obeys.
Both averages assume nothing was added or withdrawn. Once money moves in or out, end divided by start measures your saving habit as much as the market, and an account funded every month shows a "return" that is mostly the deposits. Two honest fixes exist and a broker statement usually prints both. The time-weighted return chains the return of each period between cash flows, so the size and timing of your deposits drop out; it is the number to set against a benchmark. The money-weighted return keeps the deposits in and answers what your euros actually earned; it is the number that describes you. Read the label before the figure, and never run end over start on an account you kept paying into.
Watching them diverge
Fund A: +50% then −50%. Arithmetic average: (50 − 50) ÷ 2 = 0% per year. Actual money: €1,000 → €1,500 → €750 — a −13.4% CAGR. The arithmetic mean says "flat"; your account says "quarter gone."
The gap between the two averages grows with volatility — professionals call it volatility drag. Wild rides don't just feel bad (Course 1's loss-aversion lesson); they mathematically subtract from compound growth. Two funds with the same arithmetic average but different turbulence end at different destinations — the calmer one richer. Keep this drag in your pocket: it returns in the risk unit as one of the honest arguments for smoother rides.
How big is the drag, in numbers?
There is a rule of thumb, and it is worth carrying: compound return ≈ arithmetic average − half the volatility squared. An index with the ~18% annual volatility you will meet in the risk unit loses about 0.5 × 0.18² = 1.6 percentage points a year to the drag. A fund advertising a 10% arithmetic average at that turbulence compounds at roughly 8.4%. Over thirty years that gap is €1,000 growing to about €11,200 instead of about €17,400. Nothing was hidden and nothing was stolen; the average was simply the wrong average.
The practitioner's rule
When you see "average annual return," always ask: arithmetic or compound? Marketing prefers arithmetic (it's never smaller, usually bigger). Auditors, regulators and your own balance live on CAGR. If a claim doesn't say which — assume the flattering one was chosen, and ask for the CAGR.
Try it now
- Compute both averages for two years of +100% then −50% (start €1,000). Arithmetic says +25%/yr; where does the money actually end?
- One sentence: why does higher volatility widen the gap between the two averages?
- Do it on something real. The chart below holds two decades; switch it to Monthly and Measure each calendar year to build a column of annual returns. Take their arithmetic mean. Then measure the whole span in one drag and read the percentage that gives you the compound answer. Write down the gap between the two figures in percentage points, because that gap is the volatility drag, measured rather than described.
- Add "arithmetic or CAGR?" to your three honest questions from last lesson — that's four now.