Asymmetry between gains and losses: Deeper principle at work?

Question

This asymmetry between gains and losses is well known: For example, if I buy £1000 of shares in Company XYZ, and they fall to £500, a 50% loss, I need to make a 100% gain to break even.

A table shows that the gain needed to break even after loss = (1 / 1 - % Loss) - 1 https://www.wolframalpha.com/input?i=Table%5B%7B1%2F%281-+x%29-1%7D%2C+%7Bx%2C+0+%2C+0.9%2C+0.1%7D%5D

Metaphorically speaking, the deeper the hole, the harder it is to climb out.

This question has been bugging me for a while: Is there a deeper mathematical principle that explains (or generalises) this asymmetry?

Some concepts come to mind, for instance, arrow of time, path dependence and irreversibility but I don’t know if I’m barking up the wrong tree.

I’m not a mathematician so any pointers appreciated!

Stephen

Answer 1

I think the most basic mathematical result related to this is that the geometric mean is always less than or equal to the arithmetic mean.

So the arithmetic mean of $1.5$ and $0.5$ is $1$ and the geometric mean is $\sqrt {0.75}\lt 1$

This arises because taking successive percentage increases/decreases involves multiplication.

The AM/GM result tells you that if you have any sequence of percentage increases/decreases whose arithmetic mean is zero increase or decrease (ie 1 or 100%), you will always end up with a lesser value (except in the trivial case where all the percentage increases/decreases are zero).

Answer 2

In my opinion, the asymmetry is a consequence merely of us trying to explain something linearly that is inherently geometric in nature. As pointed out by Mark Bennet, the AM/GM inequality naturally creates this asymmetry.

However, why is that relevant here? Because when we do percentage increase/decrease, we're subtracting the true change factor from 100% in order to convert a multiplicative to a linear change. E.g. we could describe a 50% decrease followed by a 100% increase as such, or we could use multiplicative factors all the way through: We'd have a 0.5x change followed by a 2x change. Those are reciprocals of one another. They fundamentally don't add together, they multiply.

Personally, I think if history had gone a bit differently, it might be more elegant to describe all percentage changes as multiplicative factors instead. For instance, a 60% decrease and it's opposite 150% increase could be more clearly described as a 0.4x multiple and 2.5x multiple. You could even go further and describe them as /2.5 and 2.5x, respectively. Then you just multiply successive changes together to get final results. Of course it's a bit late to get this adopted in everyday finance, etc. But it is cool to think about.

Overall it's not as much about a fundamental asymmetry in percentages as much as it is about us trying to make something linear that just inherently isn't. Using multipliers and dividers makes the symmetry clear.