Arithmetic and geometric mean
Why does averaging weekly returns overstate the rate at which capital actually grows?
In plain words
The arithmetic mean adds up a series of returns and divides by the count, ignoring that losses shrink the base for later gains. The geometric mean chains each return into one compounded growth rate, which is the rate that turns the starting capital into the ending capital.
Why it matters
Sizing and performance targets built on arithmetic averages assume returns compound from a fixed base that does not exist. The larger the swings, the wider the gap between the two means, so in volatile markets the arithmetic figure flatters what capital actually earned.
In a real week: ETH, week of 23 April 2018
ETH's four prior weekly returns were -27.75%, 5.76%, 33.34% and 16.3%, and the current week added 11.31%. The arithmetic mean of those returns sits above the geometric mean that actually compounds capital, because a -27.75% week shrinks the base later gains build on. The week ranged 20.07% from a high of 712.99 to a low of 593.83, with realised volatility of 116.6% annualised, a best day of 8.84% and a worst day of -11.86%.
A common mistake
Using the arithmetic mean as the expected growth rate of a position. It is the right average for a single period's expected return, but the wrong rate for projecting capital forward through compounding.