Arithmetic mean is also known as the simple average. It is the sum of all numbers divided by the number of observations. For example, the arithmetic mean of 4,5,6, and 7 is 5.5, calculated the following way:
(4+5+6+7)/4 = 5.5
The geometric mean is notably different. As it relates to investing, the geometric mean can be used to calculate a time-weighted compounded rate of return. The formula to calculate geometric mean is the following:
The difference between the two as it applies to investing is best illustrated through an example:
From year 1 to year 2 a stock increases in value by 100%.
From year 2 to year 3 the stock decreases in value by 50%.
The arithmetic return is 25%, calculated as (100%+-50%)/2
The geometric return is 0%, calculated using the formula.
This is a significant difference. If we use some specific stock prices and apply the returns, we can more clearly understand:
From year 1 to year 2 a stock at $20 increases in value by 100% to $40.
From year 2 to year 3 the stock at $40 decreases in value by 50% to $20.
Even though the arithmetic return is 25%, the stock which was worth $20 at the beginning of year 1 is still worth $20 at the end of year 2.
Another example from a different perspective:
You have $100. You lose 10% and then gain 10%. How much do you now have?
The answer is $99 because 10% of 100 is 10 but 10% of 90 is 9.
Even though the arithmetic return is 0%, the geometric return is actually negative.
This is one of the reasons why minimizing losses can be more important than maximizing gains - losses hurt more than gains help, both literally and psychologically through loss aversion.
-Joe
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