DSA: Daily Stock Activity Part II ... a continuation of Part I
Motivated by mental frailty

Remember when I was trying to identify stocks whose daily price movements were dramatic. Stocks that opened, then rose and/or fell ...

>So you could buy and sell on the same day?
Exactly! So I thought I'd measure this daily stock activity by calculating DSA = (High - Low)/Open, expressed as a percentage.
If it was 2.0%, it meant the spread between High and Low was 2.0% of the Opening price ... and that'd be significant.

>And DSA is the Daily Stock Activity, right?
Aha! You do remember.
 Okay, here's the problem. When I look around for stocks with large DSAs I often find stocks that are dropping like a rock. They open, then drop dramatically to their daily low. >And you can make money with that stock? No, I can't ... so I realize that DSA ain't what I'm after. What I want is to identify stocks that open, then go up and down from that open. In fact, I'd like to have the Open somewhere in the middle of Low-to-High range ... sorta like this
In fact, I'd like to have the Open halfway between the Low and the High.
>Huh? Why not 1/3 or 3/4 of the way between Low and High?
Okay, here's my argument:

• If I buy at the Open and sell at the High, I measure my gain as: A = (High - Open).
• Or, I may sell at the Open and buy at the Low. I measure my gain as: B = (Open - Low).
• Suppose I measure my total gain as the product: P = A*B = (High - Open)(Open - Low)
• Suppose that Open is somewhere between Low and High, say Open = Low + x(High - Low) where 0 < x < 1.
• Then P = x(1-x)(High - Low)2 which has its maximum value when x = 1/2 ... meaning that Open is halfway between Low and High.
The maximum value of x(1-x) is 1/4 ... which occurs when x = 1/2.
So here's what we do.
• Each day we measure x(1-x) = P / (High - Low)2 = (High - Open)(Open - Low) / (High - Low)2.
• We hope to see lots of days when this is close to its maximum value of 0.5 ... meaning that Open is often halfway between Low and High.
• We look for stocks where the average value of (High - Open)(Open - Low) / (High - Low)2, over the past umpteen days, is as large as possible ... meaning that it's close to 0.5.
Okay, so we're now looking at:
 DSA = (High - Open)(Open - Low) / (High - Low)2
Note that the formula doesn't distinguish between prices measured in Dollars, Yen or British pounds.
It also doesn't give extra weight to stock with BIG stock prices.
That's important.

Anyway, I looked at various stocks and ...

>Don't tell me! You have a spreadsheet.
Of course.
It looks like this:

• You type in Yahoo symbols for 30 stocks (in Column K).
• You click a button (called "get Stock DSA2s").
• A years worth of daily stock prices is downloaded and DSA is calculated each day for each stock ... and plotted.
• The list of stocks is sorted according to their average DSA ... and plotted.
• When the downloading is completed, you can move the slider to stare in awe and wonder at each of the 30 stocks.
In the picture of the spreadsheet, it's stock #26: the DOW, with an average DSA of 0.71
>What!? The DOW is the best? But it's dropped over the past year!
Always.

>That's it?
That's it.

Unfortunately, the above DSA prescription gives more weight to stocks with small (High - Low) variation.
We'd really like to identify stocks that have large (High - Low) variation, so we'll multiply the above DSA formula by (High - Low)2.
Alas, that'll give DSA-values that depend upon the stock price, so we also divide by Open2.
 DSA = (High - Open)(Open - Low) / Open2

That's an interesting one, 'cause look what it picks out:

>DRYS? Never heard of it?
Now you have!

>But what if I don't like that DSA formula? What is I'd rather have ... ?
Uh ... did I forget to mention that you can stick in your own formula? Like so:

For example, you might try any or all of the following:

•   (High - Low) / Open   ... which is the "original" DSA of Part I
•   (High/Open)*(Close/Low)   ... the product of two imaginary gains: (High/Open) and (Close/Low)
>Huh?
It'd be the gain if you bought at the Open and sold at the High, then bought at the Low and sold at the Close.
>Yeah, very funny.
Continuing ... before I was interrupted:
•   (High - Open)*(Open - Low) / Open^2   ... that's the one that picked our DRYS
•   ((High - Open) + (Close - Low))/Open   ... the sum of two imaginary gains, (High - Open) and (Close - Low), relative to the Open
•   (Open + High + Low + Close)/4
>Why that last one?
I thought I'd toss that in, just for fun.
>So what's best?
How would I know? However, consider this:
Knowing only the Open, High, Low and Close we can't tell which of these occurred:

>You don't know which came first, the High or the Low, right?
Exactly, so we might consider:
• For the first chart, we'd look at (High - Open)*(Close - Low)
• For the second chart, we'd look at (Open - Low)*(High - Close)
In any case, we'd want a zero value if Open or Close equals the High or the Low.
>Why not multiply them all together?
Uh ... yeah, except we'd have to "normalize" so \$100 stock prices don't override \$10 stock prices.
>Then divide by Open or Close.
Better to divide by Open2*Close2.
But then we're talking about the product of daily changes, like:
[ (High - Open)/Open ] * [ (Close - Low)/Close ] * [ (Open - Low)/Open ] * [ (High - Close)/Close ]
These may be something like 0.5% and the product of four of these is gonna be awfully small.
We should maybe take the 4th root of DSA = (High - Open)*(Close - Low)*(Open - Low)*(High - Close) / Open2*Close2 ... that'd be the geometric mean, eh?
Okay, let's consider this one:
 DSA = ( (High - Open)*(Close - Low)*(Open - Low)*(High - Close) )(1/4) / ( Open*Close )(1/2) ... expressed as a percentage
That last fella, the divisor, is to "normalize" ... eliminating the effect of widely varying (i.e. big) stock prices and/or currency values.
It also gives equal weight to the Open and Close.
It also ...
>So how does it work? Does it identify stocks that have the property you want?
Here's an example:

>Is that DRYS ag'in !?
Looks like it, don't it?

Here's another ingteresting formula:
Let's pretend you always buy a stock at the Open and sell at the High. Your gain would be High / Open - 1.
>Good luck!

Nevertheless, it's interesting to know what the Open-to-High return is, on average.
It's assumed you never own the stock by the end of the day. You always buy then you sell.
Of course, maybe you misjudge when the High occurs and sell too soon or too late.
Suppose you only get 50% of the Open-to-High gain. You can play with that, too:

On the other hand, suppose you always own the stock at the end of the day.
You buy at the Low and sell at the Close. Here are the results:

I should point out that we're looking at the last 100 market days.
>What about looking at the last 200 days ... or maybe 50 days or ...