Financial Price Change Distributions 2

One index I never trade in is probably the most widely known index, the Dow Jones Industrial Average (^DJI) AKA the Dow 30 or just 'Wall Street'.

The reason I avoid it is because it is an average over the share price of each composite company. Each company's share of the index is not weighted by market capitalisation as is the norm for most other indexes. This means that a relatively small company's price can go up say 10% but cause the index to go up by a disproportionate amount. E.g. say it causes the index to rise by 2% but that the company is miniscule compared to the other 29 in the index. The overall market cap of those 30 companies has not risen by anything like 2%.

This oddity has always led me to believe there will always be a greater degree of randomness in the Dow than most other indices, and possibly some weird dynamics unique to the Dow (hmm, maybe there could be something to profit from there?) and thus to just avoid it in favour of some other large cap index such as the S&P 500. If the PE is high in the S&P 500 I feel a lot safer shorting it than I would the Dow, which I feel is more likely to spring a surprise on me, e.g. a small company (with a larger share than it's market cap warrants) posting strong results.

Following on from yeserday's post then, I was wondering if this oddball index exhibits a distribution curve significantly different to market cap weighted indices. Here is a graph comparing the daily percentage price change distributions for the Russell 2000, The Dow and the S&P500. For comparison the frequencies for the Russell 2k and the S&P are scaled up to match the DOW at its peak.





And let's have a peek at one of the tails.




Note that the blip in the Dow at about -22% is from the crash of 1929.


I did wonder if the extra random element (the arbitrary weighting) in the Dow would make it's curve a bit more gaussian, apparently not. Of course there is potentially a significant effect from the buying and selling of the Dow directly, through various derivatives. If we were to take 30 random companies from the S&P 500 and construct our own unweighted index using historic data would that index still have a Levy Skew alpha-stable distribution?

Financial Price Change Distributions

I thought I'd try my hand at generating data that has some of the properties of real world financial data. My main focus right now is the Russell 2000 index, and so to get the ball rolling I thought I'd take a look at the distribution of daily price changes for that index.




Having read The Black Swan I was expecting that the distribution wouldn't be normal (the red line, using sigma calculated from the data). A fat tailed distribution makes a better fit, here I used the Cauchy Distribution for which there exists an analytically expressible probability density function, although Benoit Mandelbrot identified that changes in cotton prices fit a Levy skew alpha-stable distribution (of which the Gaussian and Cauchy distributions are specialisations) with alpha=1.7.

The key area of the graph is out on the extreme ends of the tails, so lets zoom in on the left hand tail.



We can see Mandelbrot's main issue with the normal distribution. The bumps to the far left in the raw data (the black line) represent highly improbable events if we are taking the data to be gaussian. Although the Cauchy distribution shown here is not a perfect fit (it is too high), it is a far far better model for those rare and extreme price movements. The normal distribution basically leads us to believe they will never happen in the lifetime of the universe, the Cauchy tells us to expect them once every few years. If you're betting your life savings on those rare events not occuring that's one hell of a difference.

The Numberwang Code

Mainstream culture is such a load of crap sometimes it can be pretty depressing. As a solution to this problem I can fully recommend poking fun at it at every available opportunity (and there are many). Fortunately Mitchell & Webb's new series this week got off to a good start in this area...






And of course Charlie Brooker continues to pump out satire from his south London sofa seemingly without pause...

Trading Options in the Russell 2000

I'd be interested to here from anyone in the UK that trades options in the Russell 2000(^RUT). Currently I trade using using IGIndex but their list of options is limited and doesn't include the ^RUT.

There are a couple of good reasons for buying put options on the ^RUT that I can see.

Firstly the ^RUTs P/E ratio is currently higher than most other indices. It's hard to find good quality figures for the PE but there are some (conflicting) figures out there...

The Wall Street Journal gives a figure of 40 based on trailing earnings.
http://online.wsj.com/mdc/public/page/2_3021-peyield.html

The home site of the Russell indices gives PE ratios ex-negative earnings. Which IMHO is a pretty meaningless figure, but at least we know the true PE is no lower than this figure, currently quoted at 17.59.
http://www.russell.com/Indexes/characteristics_fact_sheets/us/Russell_2000_Index.asp

The IWM is an ETF based on the ^RUT, and google finance gives us PE ratios for the ETF of 5.59
http://finance.google.com/finance?client=ob&q=IWM

Forbes though quote 86.3
http://finapps.forbes.com/finapps/jsp/finance/compinfo/CIAtAGlance.jsp?tkr=IWM

And here's yet another figure from July 2007 (40.82):
http://www.proshares.com/funds/rwm.html?Index



Quite a mess huh. Another handle on this is to look at the yield, since dividends are (mostly) paid out of earnings. Figures for the yield are a little more in agreement and I take the figure from russell.com to be about right, that being 1.35%. If all earnings were paid out as a dividend that would give a PE ratio of 74. Ok, so we can safely say that the true PE (based on trailing earnings) is between 17.59 and 74.

My own belief is that the true figure for trailing earnings is around 40. Jeremy Grantham at GMO has briefly mentioned shorting the Russell a couple of times in his quarterly "Letters to the Investment Comittee" that can be downloaded from www.gmo.com. His argument has been that the ^RUT is a good proxy for small to midcap stocks, which have outperformed large cap stocks since 2003. In times of falling profits it will probably be the blue chips that do well owing to their greater exposure to international trade and existing capital investment and infrastructure. A flight from riskier (lower or no profit) stocks to less risky stocks (good profit margins) would suggest a shift away from the Russell 200o to larger companies and thus indices such as the S&P500, Russell 1000 or the Dow. Indeed the recent falls have been more pronounced in the ^RUT than those other indices.

Essentially there is further down to fall for the ^RUT so we can expect more profts from buying puts or shorting as the PE reverts to mean. The next question then would be - What is the mean PE for the ^RUT?