Visualising Diversification using PCA (or "Why FRES is not a miner"!)

30th Apr '17 by Nick Ray

7 comments 5738 reads

PCA (Principal Component Analysis) is a statistical technique that can be used to identify the main factors (components) which account for the variance between a set of observations. It has been applied to human personality traits to establish the five factor model and it has been applied to political leanings where we find there is really only a single big axis (left-right) with other components very much smaller.

In MPT (Modern Portfolio Theory) we work with a covariance matrix of stock returns and use it to try to minimise the variance of the portfolio. This is the mathematical equivalent of using diversification to spread risk. So I thought it would be interesting to look at the principal components of this matrix (using the FTSE100 over a 256 day period) to show graphically how the risk gets spread and also to show where a naive sector-based approach to diversification can sometimes be misleading.

PCA is purely mathematical technique and the fun starts when you try to interpret what the components which it reveals might really mean. This short post is not intended to get to technical or academic though. So I'll have a go at some cod analysis but mainly it is intended to provide a bit of fun over the bank holiday weekend.

The first report is called a "scree plot" and it shows the magnitude of each component it has found. (These magnitudes are called the eigenvalues). For an analysis of 100 stocks there will be 100 components altogether but the plot below only shows the first ten.

You can see that they get rapidly smaller towards the right. The biggest components here have the greatest effect on how a stock varies. The first and largest component here broadly corresponds to whole-market movements (the beta for each stock) and for this quick look is not very interesting.

So let's plot the second and third components against each other (PC2 and PC3 in the plot. This is called a "biplot" in the PCA literature.) Right-click/View Image to see a bigger version.

We can see that Anglo American (LON:AAL) Antofagasta (LON:ANTO) Rio Tinto (LON:RIO) BHP Billiton…

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gus 1065

1st May '17

Thanks Nick. Interesting analytic of something I've had an intuition about for a while. I find the broad sector definition "Industrials" is especially diverse in terms of identifying the correlation between companies' share price action, which seems to be borne out by the last chart. One quick question as I think the detail may have been lost in your PCA Magnitude graphic. What are the 2 factors behind PC2 and PC3 if PC1 is general market Beta?

Best,

Gus.

Nick Ray

What are the 2 factors behind PC2 and PC3 if PC1 is general market Beta?

There might not be one!

The trouble with PCA is that it is a purely mathematical technique which tries to construct a "basis" (set of orthogonal components) by looking at the total variance in the data. This can result in the components being a blend of several underlying factors. There are more advanced techniques (under the general banner of "exploratory factor analysis") to try to "rotate" (apply an orthogonal transformation to) the vector to separate out the true factors.

There is a hint that such a rotation is required because the banks and the miners are orthogonal in the first plot, but not aligned with the axes. Anecdotally, this suggests that the basis vectors are a blend of something deeper.

However it did look as if the services/manufacturing split could be seen and possibly (but more weakly) defensives versus cyclicals. I only looked at FTSE100 stocks over 256 (working) days though. Possibly stronger patterns would emerge for more stocks over a longer time period.

As you probably know, the financial literature is awash with discussions about factor investing, and trying to identify the factors. There are lots of candidates: sectors, sector kinds, regions, market cap size, growth/value/income and so on. But it is important to note that this is not saying some factors are "better" than others. They are just different (have low covariance) so you can get a diversified portfolio by choosing from across the range of a given factor.

dmjram

Interesting Nick, many thanks.

On the face of it, looks as if Fresnillo is an alternative to gold ETF's without the counter party risks and with a dividend stream to boot. Would think Centamin would show similar characteristics as well.

Can I ask what software you used for the analysis please? I used to run PCA on GenStat many moons ago, would like to know how things have moved on.

I use R together with a few packages, mainly 'quantmod' and 'TTR' to customise it for financial work. R is not exactly a spring chicken though!

To give you an idea how it looks in R the code below will run in a fresh copy of R provided you have installed the XML, quantmod and TTR packages. (just use the packages menu). It produces essentially the same results as in the top post, except that I used a customised version of 'biplot' because the built-in one is pretty horrible!

# Get a list of the FTSE-100 companies
# e.g. by scraping a suitable web page
require(XML)
pagescrape = readHTMLTable('http://shareprices.com/ftse100')
n.rows = unlist(lapply(pagescrape,function(tab)if(!is.null(dim(tab)[1]))dim(tab)[1] else 0))
stocktable = pagescrape[[which.max(n.rows)]] # grab biggest table on the page

stocktable$TIDM = gsub('\\.','-',stocktable$TIDM) # cope with BT.A or similar
ftse100 = paste0(stocktable$TIDM,'.L') # add .L to all stock names

# Now grab the historical data for the ftse100 stocks
require(quantmod)
getSymbols(ftse100,auto.assign=TRUE,src='yahoo',env=.GlobalEnv,from='2015-01-01') # get historical stock data from yahoo

# Build the list of daily returns
require(TTR)
roclist = do.call(cbind,lapply(ftse100,function(x){ROC(Cl(get(x)))}))
names(roclist) = ftse100 # these names will be used in the plots
roclist = tail(roclist, 256) # only keep last 256 days of data
roclist = roclist[, colSums(is.na(roclist)) < nrow(roclist)*0.1] # remove stocks with insufficient data
roclist = na.omit(roclist) # remove any days with missing data

# Do the PCA analysis and show the results
pca.analysis = prcomp(roclist)
dev.new(); plot(pca.analysis) # scree plot
dev.new(); biplot(pca.analysis, choices=c(2,3), col=c('grey80','blue'), var.axes=T)

(NB: the "code" formatting does not seem to work so I've used "quote" which is not great but better than nothing.)

colin1862

1st Oct '20

Hi Nick, thanks for taking the time for analysis - very interesting. It shows clearly that some stocks are misclassified and this weakens the value of sector analysis. I'm trialing a short/long pairing strategy. I identify strongest and weakest sectors (and largest Industry Groups) using momentum screen then go long/short on the highest/lowest QM stocks within best/worse sector. The strength of this approach is that it protects against large market moves as, if the market tanks, the loss on the longs is cancelled by the rise on the longs - this lets me sleep at night during these turbulent times......The key to success is for the longs to rise faster than shorts in bull phase and vice versa forbear phase. Surprisingly it has worked well over the last couple of months with the shorts making most money but t needs to experience all market conditions. It does rely on Stocko's QM ranking to be robust.to have chance of success over long term but by investing small amounts in large no of stocks I shold get reasonable exposure. Any ideas you might have would be welcome

I think long-short portfolios are more Gromley's speciality!

The big problem with them (in my opinion) is that they are designed to cancel out the beta of the market. This works well in a year like 2020 but it's not so great in a year like 2017 where the market as a whole rose significantly and long-short portfolios performed a lot worse than long only portfolios.

The goal of market-neutral approaches is to reduce volatility but it is very difficult to do by subtracting (going short) unless you can finely balance the betas of the stocks. And even then there is a danger that you are left with a portfolio which is barely moving in any direction.

As I mentioned in the article, I tend to try to minimise the variance of the portfolio by combining stocks with low covariance. But I am also slowly moving towards looking at longer and longer periods over which to assess a stock's (or a portfolio's) variance. So minimising the Sharpe ratio over five years or even ten years rather than one year for example.

However, perhaps others who do use long-short portfolios can chip in on what works for them.

2nd Oct '20

thanks Nick, I take your points re shorts stifling bull market returns but I'll have to see what happens when a bull market returns......Instead of sectors it might be better group stocks by scenarios i.e. if war breaks out which stocks group together long or short. You could then adjust your portfolio according to which scenario occurs. After SARS in 2000 it was obvious that another similar outbreak would occur - I should have planned for it. Will try and track Gromley down

Visualising Diversification using PCA (or "Why FRES is not a miner"!)

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Nick Ray

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