Friday, March 8, 2013

Is X better than Y if X is perfectly correlated with Y?

The Dow Jones Industrial Average hit its first record in a few years the other day and I was bombarded with mathematical innumeracy stampeding out of the speakers of my car like a herd of ... well, innumerate journalists.

DAVIDSON: Here's the thing. For reasons I cannot understand, nobody adjusts for inflation when they're talking about the Dow. .... And anyway, even if it did reach a record, this is not the measure we should be paying attention to. 
BLOCK: OK. Well, if it's not the measure we should be paying attention to, what is? 
DAVIDSON: There are, as I mentioned, a handful of indexes that do a much better job, like the S&P 500 ...

Don't worry too much right now about the fact that the S&P 500 isn't adjusted for inflation either. More came from Marketplace minutes later:

“[The Dow] a rough indicator of the health of the market,” says Kelly School of Business professor Scott Smart, “but there are some problems with the Dow as such an indicator." 
For one, the Dow is a very, very small sample ... “since it only looks at 30 stocks, there are obviously big portions of the market that the Dow doesn't monitor or doesn't capture.” ... 
... “it is weighted in a very unusual way.” ... 
That’s the Dow. No complicated formula. No algorithms. No wonder more serious investors prefer the S&P 500 ...

So while the Dow is a giant turd on the world of financial journalism, the S&P 500 is, like, totally the awesomest.

Except they have a correlation of 0.96 ...

The Dow is in blue and the S&P 500 is in gray. Or, wait ...

Here is another view. Basically Dow = a*S&P 500 (where a is ~ 9.2) ... and you will only be off by a couple percent.

The simplicity of the formula, its "unusual" weighting based on price not market cap, its small sample size: none of these things matter. Why? Because most companies large enough to be listed in an index are highly correlated with each other. Here, for example, are Boeing and GE:

It doesn't matter how you weight the companies since these weights have no effect on correlation  ... if the correlation of X and Y is c, then the correlation of a*X and b*Y is also c.

And even if individual companies weren't very correlated with each other, creating indices that lump individual companies together tends to destroy the information about their individual performance so you end up with an index that shows an average trend (this is the idea behind these indices in the first place). All that matters is how highly correlated the stocks are in the first place whether it takes 30 companies, 500 or 5000 to get there. (The answer is 30.)

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