Annualised return

An annualised return takes a gain earned over any stretch of time and asks what constant yearly rate would have got you to the same place. That is what lets a seven month holding sit next to a three year one. With no money paid in or taken out along the way, it is the compound annual growth rate: divide the ending value by the beginning value, raise that to the power of one over the number of years, and subtract one.

$$r_{annual} = \left(\frac{V_{end}}{V_{begin}}\right)^{\frac{1}{n}} - 1$$

Annualising works through compounding rather than multiplication, the same operation behind an effective annual rate. A 1 percent month is not 12 percent a year, it is $(1.01)^{12} - 1$, or 12.68 percent. Going the other way, an 8 percent year is not 0.667 percent a month, it is 0.643 percent. The CAGR calculator does the conversion in both directions.

Two errors show up again and again. Annualising a very short period produces a figure that describes what happened and not what to expect: a 4 percent gain in a single month annualises to about 60.1 percent, which nobody should read as a forecast. And averaging yearly returns arithmetically overstates the outcome, because a loss and a gain of the same size do not cancel. Up 50 percent then down 50 percent averages to zero but leaves 75 percent of what you started with, an annualised loss of about 13.4 percent. Whatever the period, the annualised figure is only as complete as the total return it came from, and it is still a nominal one, so convert it to a real rate before comparing stretches of time when prices were rising at different speeds.