Tag Archives: stock market

The S&P 500 and the Dow Jones

Adam Davidson in last Sunday’s New York Times Magazine, on the drawbacks of the Dow:

None of these criticisms will come as news to finance professionals, most of whom use far more precise measures — like the S&P 500 or the Wilshire 5000, which cover more companies more precisely — when making investment decisions.

The S&P 500 certainly covers a wider range of companies.  But in a typical 5-day window its correlation with the Dow is more than 95%.  How much more precise could it be?

It’s good to have fine-grained measures, but it’s also good to know at what point extra granularity stops addding new content.

 

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Efficient markets and the digits of pi

John Quiggin had a good post yesterday on Crooked Timber about the various flavors of the Efficient Market Hypothesis, which according to Quiggin is a piece of shuffling zombie social science that won’t die no matter how many flaming sticks you jam through its skull.

The weakest form of the EMH is the so-called “random walk hypothesis” that the future behavior of a stock price is independent from its prior behavior.  If that’s the case, no amount of staring at charts is going to help you beat the market.  The random walk hypothesis is pretty well-supported by the data we have; a really nice popular account is Burton Malkiel’s A Random Walk Down Wall Street, one of the mathiest bestsellers I know of.  It makes a great present for anyone in need of a good rationale for not paying attention to their investments.

Quiggin writes

The success of the random walk hypothesis showed that the existence of predictable price patterns in markets with rational and well-informed traders was logically self-contradictory.

But this doesn’t seem quite right.  The EMF, even in its weakest form, holds that the current price of a stock is a best estimate arrived at by an aggregate of profit-maximizing investors with knowledge of the stock’s previous price:  if there were a reliable way to use previous prices to determine that tomorrow’s price would be Y, then the investors would figure that out, and today’s going price would be Y as well.

But the random walk hypothesis is much weaker.  It makes no claim about the etiology of stock prices.  It’s compatible with EMF, but it’s compatible with plenty of other models too — for instance, the one in which stock prices really are a random walk, where the price at time t+1 is the price at time t plus, let’s say, a normally distributed random variable X.  Such a market would be as impossible to beat as a roulette wheel — but evidently not because stock prices are best estimates of the future price of the stock, or of the underlying value of the company, or of anything at all.  (Fellow mathematician David Speyer makes a similar point in the comments at CT.)

You can dress this up a bit:  suppose price(t+1) – price(t) is the random variable X + [(.001) x t’th digit of pi] – (.045).   Unless something very weird is going on with the digits of pi, this version of the stock market would also satisfy the random walk hypothesis.  But unlike the pure random walk it’s a market where you can make some money; if I’m lucky enough to have access to the “digits of pi” rule, I can make a small average profit.  So the random walk hypothesis can hold for markets that are neither efficient nor unbeatable.

Stronger versions of the EMH hold that the market price already takes into account, not only the previous prices of the stock, but also publicly available information about the stock.  So what would happen if I let my secret investing strategy slip out?  This isn’t a rhetorical question; I’m authentically curious about what the rational-investor model would say about a market in which prices are publically revealed to have been governed, up until now, by some completely deterministic but “random-looking” sequence like the digits of pi.

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