## The Problem with Percentages

I read a lot of economic analysis - obviously a complete waste of time from a system trading point of view! Dispersion charts are very common, even with "average percent changes" displayed on them; consider this speech given by Mishkin. It contains a whole panel of such charts with averages.

I like percentages. I think they are useful in system design and analysis because they lead you to expectations for the future rather than fictional back-tested equity curves. But percentages, in their usual form, have a problem that anyone in trading is familiar with: if you lose 10% you need to make 11.1(1) % to break even.

It's not symmetrical and it annoys me!

This is an issue because dispersion charts are misleading as a result. Consider some economic data: 100, 95, 90.25, 95, 100. A dispersion chart of this data would read: -5%, -5%, +5.26%, +5.26%. So, if we looked at the dispersion chart alone and made the mistake of drawing in the average growth rate, we would see +0.13%. In what world does average +0.13% growth result in zero growth?

A simple little trick (anyone familiar with financial mathematics will know it) is to use continuous compounding. That is, $\Delta = \ln \left ( \frac{P_{0}}{P_{-1}} \right )$.

So our dispersion chart would appear as follows: -5.13%, -5.13%, +5.13%, +5.13%. Average growth is now zero, so we EXPECT the factor under discussion to end where it started, as it does. Now, isn't that better? Symmetrical and common-sensical!

There are additional advantages to doing your percentages this way - you can simply add them up when they occur in series. You can divide and multiply them to move from one time frame to another (e.g. 12% pa is the same as 1% per month for 12 months instead of 0.95%). And in the end, you can always easily convert back to real values or old-school percentages.

The one draw back is that you have to get used to the concept of negative hundreds of percent - e.g. 100 -> 30.1 over 12 months is a decline of 120% pa continuously compounded (cc)! But you need a 120% cc return to get back to where you started, so it's a wash.

Ah, but over what time-frame? It doesn't matter! It could be a year or a decade, it's the same total 120% cc return that is needed (12 mo @ 10% pcm cc, or 10 years @ 12% pa cc, or 5 years @ 24% pa cc it all adds up to 120% continuously compounded).

I don't think I have ever seen a dispersion chart done this way, but I finally understand why I find them un-helpful!