# How can I get an average of percentages?

I can't seem to find an answer to my exact question.

I want find how much on average a stock or the stock market goes up during an up week (or month) and down during a down week (or month). Let's say it goes up 5% the first week up 10% the second week down 20% the third down 10% up 15% up 20% down 5%.

Can I simply average the percentages? In this case the average for up weeks is (5+10+15+20)/4 = 12.5% and down weeks is (20+10+5)/3 = 11.67%. Or would I need to do something else?

I found a similar question here

Can percentages be averaged?

but no one really addressed it.

• First of all, you should rather build some kind of model for this data to have a meaningful estimate rather then blindly average them. As about "averaging" see also: stats.stackexchange.com/questions/155817/… – Tim May 15 '17 at 19:24
• Take a look at my answer here. – Dimitriy V. Masterov May 15 '17 at 19:28
• You can average anything as part of operational semantics. The question is whether you will be able to interpret the result in a way that will be useful to you or not. So first of all you need to define what is it that you want to achieve, after that we can answer whether arithmetic mean is the operation that will achieve that or not. – Cowboy Trader May 15 '17 at 19:59

## 1 Answer

When dealing with percentages like this, you wouldn't simply take an "arithmetic" average as the result wouldn't be meaningful. Instead, you'll do different kind of averaging. In your example, the arithmetic (regular) mean over all 7 weeks is 1.05 (assuming you add a one to each value). But if you had gained 5% each week, then after 7 weeks, you'd be left with a total return of 40.7%. In reality though, you end up with a total return of 33.2%. So that begs the question, "what average weekly gain would result in a total return of 33.2%?" The answer is 4.19%. The way you figure out this number is simple. You take the "geometric mean" which is defined as $(x_1 \times x_2 \times ... \times x_n)^{1/n}$. In your case, $x_1=1.05$, $x_2=1.1$, etc.

• I am not looking for total return or average weekly gain. What I would like to know is how much can I expect a stock to go up for an up week, and how much to go down for a down week. – Eliot May 15 '17 at 20:19
• Eliot, it isn't apparent that averaging would tell you that. For instance, if seven weeks ago the market changed by $-99\%$ and then by $+100\%$ each of the next six weeks, it would still be down by $36\%$--but there are few averaging methods indeed that would tell you the average of the changes $-99, 100, 100, 100, 100, 100, 100$ is $-36$! This is what lies behind the comment made by @Cagdas. – whuber May 15 '17 at 20:55
• Ok I didn't catch that when I first read your question. I think the best way to do this is with a simple modification similar to what you did but instead using the harmonic mean. The average "up-week" would then be $(1.05 \times 1.10 \times 1.15 \times 1.20)^{1/4} = 1.124$. You'd have to really stretch to give this a good interpretation though. – jjet May 15 '17 at 20:58
• Thanks. It could have practical application. Let's say historically a stock has 33% down weeks and 67% up weeks. Let's also say you decide to buy call options on the stock. Knowing how much you can expect it to go up or down would give you an indication on how much to bet each week (with all the caveats about the past not predicting the future, etc). – Eliot May 15 '17 at 21:17
• @jjet What you have there is the geometric mean. – Matthew Gunn May 16 '17 at 14:24