Geometric Mean vs Geometric Mean Return

Hello I understand to to analyze past return data of a portfolio we will always use the Geometric Mean Return but when do we ever use the general Geometric Mean formula? I am trying to understand when to use either of these and what is the difference in the application of either. If there is a negative return, mathematically I understand the regular formula wouldn’t make sense. In what cases then would we ever use the general formula and not the Geometric Mean Return? Thanks in advance!

Returns can be calculated using a geometric mean or arithmetic mean. I’m not sure if the tittle of your post is indeed correct or what you meant.

The curriculum states the cases at when geometric mean is more suitable than arithmetic mean.

For example, if you are reinvesting returns of a portfolio (like compound interest), then use geometric mean of returns to know “mean” historical returns.

For example, If you are receiving dividends from a stock position (like simple interest), then use arithmetic mean of returns to know “mean” historical returns.

they are synonymous

You can use general geometric mean formula as long as all the values (in this case returns of a portfolio) is positive under the radical sign (square root). Since its not practical that all the returns of portfolio will always be positive so we modified it a little bit.

You will have to use the modified formula i.e. Geometric Mean Return (as stated in Curriculum) as long as the returns are in negative (even if a single value).

The reason for adding 1 to each return has nothing to do with the fact that returns can be negative. Even if all of the returns were positive, you’d get the wrong answer if you fail to add 1 to each return first.

The reason for adding 1 to each return is that when returns are compounded (i.e., multiplied) you need to add 1 to each return; i.e, it is the relative value factors that are multiplied, not merely the returns. In essence, you’re solving this equation for rgeom:

(1 + rgeom)n = (1 + r1)(1 + r2) ∙ ∙ ∙ (1 + rn)

And even if they aren’t.

Thank You all. @s2000magician so we are all using the Geometric Mean Return formula ALWAYS regardless to entertain to affects of compounding. Got ya. Than you very much