What return actually means
Return is income relative to what you put in and how long you held it. Three questions matter: per year or total, before or after costs, before or after inflation.
If someone says they made fifty percent, three pieces of information are missing. Over what period? After which costs? And was inflation accounted for?
Fifty percent in one year is extraordinary. Fifty percent over ten years is about four percent a year, and unremarkable. Without a time frame, a return figure is worthless.
Always calculate returns per year. That's the only way to compare two investments. And subtract what actually leaves your pocket: fees, taxes, inflation. What's left after that is what you actually gained.
Be careful with averages. Someone who loses 50 percent in year one and gains 50 percent in year two has averaged zero, but is sitting on 75 percent of their original money. The simple average always lies upward for a fluctuating investment.
A distinction has to be made between the arithmetic and the geometric mean. The arithmetic mean of annual returns describes the expected return of a single period; the geometric mean describes the growth rate actually achieved over the entire period. The latter is the annualized return, CAGR = (final value / starting value)^(1/n) − 1, and for a fluctuating series it's always smaller than the arithmetic mean.
The gap between the two grows with volatility and can be roughly approximated as σ²/2. An investment with an expected annual return of 8 percent and a standard deviation of 20 percent delivers a geometric return of around 6 percent. That gap is why volatility isn't just unpleasant, it's mathematically expensive.
With irregular deposits and withdrawals, a further distinction is needed between time-weighted and money-weighted return. Time-weighted return measures an investment's performance independent of cash flows and is used for fund comparisons. Money-weighted return, essentially the internal rate of return, measures what the investor actually earned. The two figures can diverge sharply when a lot of money was added right after strong gains.
Summary
- A return figure with no time frame isn't information.
- The geometric mean is what counts, not the simple average.
- Volatility lowers the growth rate you actually achieve.
Did you get it?
Down 50 percent, then up 50 percent. Where do you stand?
At 75 percent of the starting value. The simple average of zero doesn't hold.
Which average describes your actual result over several years?
The geometric mean, meaning the annualized return.
Why is volatility mathematically expensive?
Because the geometric return falls below the expected return as volatility rises, by roughly σ²/2.
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