When Do You Use Arithmetic vs Geometric Means?

A description of each mean (Arithmetic and Geometric) and when to use each.

[azure-hawk]

Quick comment in case this helps someone else: the arithmetic mean is treating each year as independent from the others and the starting value at each year as the new baseline. The geometric mean instead takes into account the returns from previous years and sets the modified values after each year as the new baseline. So instead of 0.2, -0.03, and 0.08, the corrected proportions would be 0.2, -0.03(1.2), and 0.08(1.2)(0.97).

[binaandhika]

The simplest explanation about arithmetric v geometric I found on YouTube. Thanks!

[wilstra]

Hey Matt, I am from Brazil and I don't miss a single video explanation. Thank you!

[siekphried]

Hi Matt, great explanation and thank you for it
I've been struggling with a question for so long now about parametric VaR. As you may know, 1 part of this method is calculating the average return for the mean of the distribution, but I'm not quite sure if I should use the average on log returns or the average of discrete returns, I tend to associate Geometric mean with log returns and use it for Parametric VaR, but given that you said that for expected values you should use Arithmetic mean I'm so confused on which mean should I use as they may lead to different results:
Imagine you have an asset that can only have the following daily returns with 50/50 chance, +50% and -40% discrete returns (the price series would be 100, 60, 90), the log returns would be 40.5% and -51.1%. When you take the average they would be +5% for discrete and -5.27% for log returns. Also, the Geometric mean for the discrete returns would be the same as the Arithmetic mean of the discrete returns. This really confuses me on what mean method should I choose for parametric VaR expected mean.
I'll be very thankful for your help with this subject.

[alextab]

I understand both methods. But what I remain confused about is using the Arithmetic method to add percentages to find the mean. I always thought we weren't allowed to add them up and then divide by the total number like we would with regular numbers? I see so many people doing it just like you are, but I don't understand why its allowed because doesn't each percentage represents a different number? in other words were trying to average unlike numbers?