PCA as a Tool for Analyzing the Market

Efe Yigitbasi1

1Boston University

(Dated: May 5, 2015)

We have applied the method of Principal Component Analysis to two di↵erent markets, oneconsisting of stocks only, and one consisting of bonds only. We have tried to find the numberof relevant dimensions for each of those markets, and inspected the eigenvectors corresponding tothose dimensions. By using R2 factor as a measure of goodness of our approximations by dimensionreduction, we inspected the usefulness of PCA as a tool for analyzing di↵erent markets.

PACS numbers:

INTRODUCTION

A simple model for the market from the point of viewof an investor is such that the market consists of N se-curities and an investment period ⌧ . In general thosesecurities can be a combination of bonds, commodities,mutual funds, currencies etc. One can build a time seriesout of the prices of such securities sampled at specifictimes. For a market with N such securities the prices upto a time t0 will form a T ⇥ N dimensional matrix Pt0 ,where T is the length of the time series for the price forone of the securities. The question is then, to determinethe multivariate matrix Pt0+⌧ which will give the pricesof the securities at the end of the investment period.This general procedure is not a simple one. We have

to apply several approximations to be able to get a rea-sonable estimation for Pt0+⌧ . One of the approximationsthat can be applied is called Dimension Reduction. Al-though the multivariate matrix Pt0+⌧ has N columns, ingeneral those N securities are not independent. There-fore, by looking at the structure of the covariance matrixbuilt by using the matrix Pt0 we can reduce the num-ber of its dimension by projecting it onto a space whichrepresents the most of the information in our matrix.One method for applying Dimension Reduction is

called Principal Component Analysis (PCA). The ideabehind PCA is to find the subspace with most amountof information by looking at the eigenvalues of the co-variance matrix constructed from Pt0 . In the followingsections I’ll go over the general procedure of constructingthe covariance matrix from Pt0 , and then I’ll apply PCAto two di↵erent covariance matrices that represents twodi↵erent markets. Those markets are the following:

• 26 stocks from Dow-Jones Index.

• US National Treasury Bonds.

THEORY

The first step in analyzing the market is to chooseour parameters. The matrix Pt0 is a matrix of

prices, however the price of a security is not a use-ful parameter when we are trying build a model. Ifwe consider our parameters to be stochastic vari-ables, we require them to be time homogenous in-variants. With this requirement the distributionof our invariants will not depend on time. Thisis essential for making future predictions by look-ing at past data. In order to find out if a variablesatisfies the above requirement we can look at thelagged correlation of a given time series. For timehomogenous invariants that distribution should bea circular cloud. This is easy to see by looking attwo di↵erent time series for stocks. First time se-ries to consider is the price of a stock Pt, the secondone is the logarithmic (compounded) returns of thesame stock Xt,t0 . Where:

Xt,t0 = lnPt

Pt�t0(1)

It is easy to see that the price is not a time homoge-nous invariant, whereas compounded return is.

FIG. 1: Lagged correlation of the price of a stock.

The same relation applies to the price of a bondand the di↵erence of yield to maturity for the same

2

FIG. 2: Lagged correlation of the compounded returns of astock.

bond. Once again the price of the bond is not aninvariant, however the di↵erence in yield rates is.Assume that the price of a bond at time t whichwill expire at time t+ v is given by Z

t+vt , the yield

rate and the di↵erence in yield rates are given by:

Y

vt = �1

v

lnZt+vt (2)

X

vt,t0 = Y

vt � Y

vt�t0 (3)

In general, for a market with both bonds and stocksthe multivariate matrix that is important would bethe combination of the invariants defined above.Therefore, for a market with N securities we willhave a T ⇥N matrix X which includes the invari-ants for all our securities.

The next step in a general market analysis is to ana-lyze the parameterXt0 up to time t0, and make pre-dictions for a later time Xt0+⌧ . Finally one shouldconvert the parameter Xt0+⌧ to the prices of thesecurities. Here I will only focus on the analysis ofXt0 by using PCA.

First we have to construct the covariance matrixusing X. The covariance matrix for N securities isan N ⇥ N matrix given by the following equationwhere T is the number of rows in the matrix X:

C =1

T

X

TX (4)

The idea behind PCA is to find the eigenvalues andeigenvectors of C and try to extract informationfrom them. Pictorially one can plot the location-dispersion ellipsoid by using C, and the eigenvec-tors of C are the principal axes of this ellipsoid.

If all of our N securities are perfectly independentof each other, the location-dispersion ellipsoid willbe spherical. However if there are some dependentsecurities, the ellipsoid might have a pancake-likedistribution. For such a market we can apply PCAto find the irrelevant directions in our ellipsoid, andreduce the dimension of the ellipsoid by projectingit out of the irrelevant directions.

FIG. 3: Location-dispersion ellipsoid and the principal axes.

This approximation is only a good one if the princi-pal axes that we are projecting onto correspond tonearly the entire variation in our system. A goodway to measure this is comparing the relative size ofthe first k eigenvalues starting from the largest one,to the sum of all the eigenvalues. The parameterR

2 is defined as following:

R

2 =

Pki=1 �iPNi=1 �i

(5)

If R2 ⇡ 1 for k < N we can project the ellipsoidonto the subspace given by the first k eigenvectorswithout much error.

In the following sections, I will try to look at amarket that consists of stocks, and a market thatconsists of bonds to see if we can get some dimen-sion reduction by using PCA.

STOCK MARKET

Assume that our market consists of 26 stocks whichare all in Dow-Jones Industrial Average Index. Wetake daily close price data from: 1/1/1990 up to:4/20/2015. We calculate the compounded returnsand the covariance matrix of compounded returns,by using the definitions that are given above. The

3

eigenvalues and the R

2 value that we obtain byusing the first k eigenvalues is given in Fig. 4 andFig. 5.

FIG. 4: Eigenvalues of the stock market.

FIG. 5: R2 if we include the first k eigenvalues.

As we can see, apart from the first one or two eigen-values, all the eigenvalues have similar magnitudes.Therefore the value of R

2 does not become closeto unity before including nearly all the eigenval-ues. This means that the location-dispersion ellip-soid does not have any irrelevant directions, andwe cannot reduce the number of dimensions of thismarket without introducing significant errors.

However, we might be able to get some informationby looking at the eigenvectors that correspond tothe first few eigenvalues. If we plot the contributionfrom all the stocks to the first eigenvector (Fig. 6)we see that it has similar contributions from allthe stocks in our market. This principal axis isusually called the market mode. The market, andthe prices of all the stocks in it tend to change in thesame way, and this is by far the biggest contributionin the change of price of any individual stock.

Another information that we can get from the

FIG. 6: Contribution of companies to the eigenvector withthe largest eigenvalue.

eigenvectors is the relations between di↵erentstocks. These relations will usually be betweencompanies in the same industry, or similar sectors.If we look at the eigenvalue that corresponds to thethird largest eigenvalue (Fig. 7), we can see thatthe contributions from all the stocks are close tozero except for two huge contributions from stockslabeled as 11 and 18. Those two companies are In-tel, and Microsoft respectively, which are in similarsectors. The other contributions to this eigenvec-tor mainly come from company 1 which is Apple,company 17 which is Merck & Co.,and company 20which is Pfizer where the last two companies areboth pharmaceutical companies.

FIG. 7: Contribution of companies to the eigenvector withthe third largest eigenvalue.

BOND MARKET

In this section we will look into a bond market.Assume that our market consists of several bondswith di↵erent maturities. We take the zero coupon

4

bond prices for each maturity, and calculate thedi↵erence in yield rates. Then we construct thecovariance matrix as explained above. The covari-ance matrix between two bonds with maturity v

and v + p is in the form:

C(v, p) = Cov(Xv, X

v+p) (6)

In general for bond markets, this matrix has thefollowing properties:

C(v + dv, p) ⇡ C(v, p+ dv) (7)

C(v, 0) ⇡ C(v + ⌧, 0) (8)

C(v, p) ⇡ C(v + ⌧, p) (9)

The above equations mean that our covariance ma-trix is only a function of one parameter, which isthe di↵erence in maturities of the two bonds. It isalso a real, symmetric matrix which is smooth inthis one argument, mostly diagonal, and constantalong its diagonal. These special matrices are calleda Toeplitz Matrix.

The infinite dimensional case of a Toeplitz Matrixis called a Toeplitz Operator and the solution forits eigenvalues and eigenvectors is given in Fig. 8.

FIG. 8: Eigenvalues, R2, and first three eigenvectors of aToeplitz Operator.

Apparently our stock market example did not havethe above given properties, and we did not see asimilar pattern for the stock market. However ourbond market example might have a similar eigen-value and eigenvector composition even thoughthey are finite dimensional. The first thing thatwe can check is the representation of the covari-ance matrix of a bond market to a stock market.As we can see in Fig. 9 the covariance matrix of

our stock market is not a smooth function at all.However, it can be seen in Fig. 10 that for the USTreasury Bond Market the covariance matrix lookslike a smooth two dimensional function. The otherproperties other than smoothness are not really vis-ible, therefore we have to find the eigenvalues forourselves and see if they have the same form as aToeplitz Operator.

FIG. 9: Representation of the covariance matrix of the stockmarket.

FIG. 10: Representation of the covariance matrix of the bondmarket.

US Treasury Bond Market

In this bond market example we will look into thebond market of US Treasury Bonds. We take thedaily prices from: 2/9/2006 up to: 4/20/2015, cal-culate the di↵erence in yields and the covariancematrix. The eigenvalues and the R

2 value that we

5

obtain by using the first k eigenvalues is given inFig. 11 and Fig. 12.

FIG. 11: Eigenvalues of US bond market.

FIG. 12: R2 if we include the first k eigenvalues.

We can see that the eigenvalues are completelydominated by the first four eigenvalues, and thatthe R

2 goes above 0.95 if we include the first foureigenvalues. This means that the US Treasurybond market is not really 11 dimensional as our in-variants suggest, but it is in fact close to being fourdimensional. Looking at the contributions to thefirst four eigenvectors, we can see that the eigen-vectors resemble the eigenvectors of the ToeplitzMatrix solution (Fig 13). However due to the co-variance matrix being not an exact Toeplitz Matrixwe can see that the eigenvectors are not exactly theoscillatory solutions we see in the exact case.

Now the last step is to actually reduce the numberof dimensions by projecting our invariants in thedirection of our eigenvectors. By doing this we arebasically mixing our bonds in such a way that ourinvariant matrix now consists of our eigenvectorsinstead of our bonds. By doing this we can seethe di↵erent things happening in our market more

FIG. 13: Contributions to the first four eigenvectors.

easily than the previous case. By looking at thetime series of the first three eigenvectors, we cansee that there are some points where the valueschange dramatically (Fig. 14, Fig. 15, Fig. 16). Inthis case the location of these jumps (around day500 in our time series) happens to be around theyear 2008. We can see that the market has someunusual price changes around that period.

FIG. 14: Time series of the first eigenvector.

CONCLUSION

I have applied the method of Principal ComponentAnalysis to an example of a stock market and anexample of a bond market. For both cases themethod is useful in having a better understandingof the market. However the information gatheredfrom PCA is di↵erent for the two examples that aregiven here.

For the stock market PCA does not help in reducingthe number of relevant dimensions in our market.

6

FIG. 15: Time series of the second eigenvector.

FIG. 16: Time series of the third eigenvector.

It is possible that for markets with greater num-ber of stock than 26 that I have used here, PCAmight give some dimension reduction, however itwas not the case here. The usefulness of PCA instock market example was to get an understand-ing of underlying connections between the prices ofstocks of di↵erent companies.

For the bond market, PCA helped us identify therelevant directions in the location-dispersion ellip-soid. For the US Treasury Bond Market, the num-ber of relevant dimensions was 4 out of a totalnumber of 11 di↵erent bonds. This is a huge di-mension reduction. Also the plot of eigenvectorsshows that the changes in yield rates of bonds canbe analyzed with a few simple functions. By far thegreatest contributors to the R

2 factor is the firsttwo eigenvalues. The e↵ect of the first two eigen-vectors can be understood to be a constant shift inthe bond yield rates (from first eigenvector), and askew (from the second eigenvector). These contri-butions are easy to understand, visualize and usewhen trying to understand the behavior of the mar-ket. The e↵ect of a financial crisis is clearly visibleon the time series of the most important eigenvec-tors. This information can also be useful in under-standing the overall behavior of the bond marketduring financially unstable periods.

[1] Risk and Asset Allocation, Meucci, Attilio