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52

Principal Component Analysis

Many financial markets are characterized by a high degree of collinearity between returns. Term structures such as yields of different maturities on similar bonds exhibit a very high level of correlation, that is, they are highly collinear. Variables are highly collinear when there are only a few important sources of information in the data that are common to many variables. This chapter is about a standard method for extracting the most important uncorrelated sources of variation in a multivariate system, which is called principal component analysis (PCA). Not only term structures but also implied volatilities of different options on the same underlying asset, and futures of different maturities on the same underlying, lend themselves to PCA. It also has very useful applications to modelling equity markets, or currency markets, or indeed any market where there is a reasonably high level of correlation between the returns.

It is a common problem in risk management that risk measures and pricing models are applied to a very large set of scenarios based on movements in all possible risk factors. The dimensions are so large that the computations become extremely slow and cumbersome, so it is quite common that over-simplistic assumptions are made. For example, smile surfaces may be assumed constant, or at least only parallel shifts in the smile will be considered. Yield curves might be interpolated between just a few points along the curve, rather than taking scenarios for all maturities in the term structure.

This chapter describes a modelling tool for correlated financial systems that is based on only a few key market risk factors. Large systems for the pricing and risk management of financial assets use hundreds of different risk factors. One of the aims of principal component analysis is to reduce dimensions so that only the most important sources of information are used. The two main advantages of this approach are:

>- The computational efficiency that results from the lack of correlation between the principal components and the dimension reduction from taking just a few of them.

>- That it provides a tractable and intuitive framework that will often aid understanding of the dynamics of market behaviour.

In many cases the objective of PCA is to reduce dimensionality by taking only the first m principal components in the representation (6.2) below. This is certainly

Variables are highly collinear when there are only a few important sources of information in the data that are common to many variables



useful in highly correlated systems because there will only be a few independent sources of variation and most of it can be explained by just a few principal components. The reduction in dimensionality achieved by a principal component representation can greatly facilitate calculations. Transformations to the principal components may be applied, and then the factor weights are used to relate these transformations to the original system. This has enormous advantages in scenario analysis (§9.6). The significant reduction in computation time makes principal components a very useful tool for risk measurement in large portfolios.

Another very useful application of principal component analysis is the construction of large positive definite covariance matrices

Another very useful application of principal component analysis is the construction of large positive definite covariance matrices. In this case the advantage of PCA is not so much in the reduction of dimensionality; rather it is due to the orthogonalization of variables.1 Since principal components are orthogonal their unconditional covariance matrix is diagonal. The variances of the principal components can be quickly transformed into a covariance matrix of the original system using the factor weights. Only the m principal component variances need to be calculated, instead of the k(k + l)/2 different elements of the x covariance matrix of the original system. Full details of the method are given in §7.4.

A linear algebraic approach to PCA is taken in §6.1. Section 6.2 will help the reader to gain an intuitive grasp of principal components, and their factor weights, through empirical examples of the application of PCA to term structures (term structures lend themselves to PCA and many financial analysts are familiar with PCA in this context). Section 6.3 describes some of the latest research on the application of PCA to modelling volatility smiles and skews (Alexander, 2000b, 2001a). This section shows how non-linear movements of smiles and skews, corresponding to a quadratic parameterization of the volatility surface, may be modelled using PCA.

PCA can be used to fill in missing data points for new issues or illiquid assets

The final section of this chapter explains how PCA may be used to overcome various problems with data. A common problem that arises with many models, such as multi-factor models and benchmark tracking models that are based on regression analysis, is that explanatory variables are often highly collinear. This multicollinearity can cause difficulties with model estimation (§A.4.1). Section 6.4.1 shows how PCA provides a robust and efficient method for estimating parameters in these models. The second data problem concerns missing data on relatively new assets that do, however, have a reasonably high correlation with other assets in the system. Section 6.4.2 explains how PCA can be used to fill in missing data points for such assets: an empirical example that simulates an artificial price history over more than 2 years from just a few months of real

If two random variables have zero unconditional correlation they are called orthogonal.

2 Such an approach presents a useful background for the applications of PCA that will be explored in this and subsequent chapters. The CD contains spreadsheets that illustrate the optimization approach. Jolliffe (1986) gives more details on the theoretical background.



data is given. It has obvious applications to new issues, and to financial assets that are not heavily traded.

6.1 Mathematical background

The data input to PCA must be stationary (§11.2). Prices, rates or yields are generally non-stationary and so they will have to be transformed, commonly into returns, before PCA is applied. These returns will also need to be normalized before the analysis, otherwise the first principal component will be dominated by the input variable with the greatest volatility. We therefore assume that each column in the T x stationary data matrix X has mean 0 and variance 1, having previously subtracted the sample mean and divided by the sample standard deviation.3

PCA is based on an eigenvalue and eigenvector analysis of V = XX/J, the x symmetric matrix of correlations between the variables in X. Each principal component is a linear combination of these columns, where the weights are chosen in such a way that:

>- the first principal component explains the greatest amount of the total variation in X, the second component explains the greatest amount of the remaining variation, and so on;

the principal components are uncorrelated with each other.

It is now shown that this can be achieved by choosing the weights from the set of eigenvectors of the correlation matrix. Denote by W the x matrix of eigenvectors of V. Thus

VW = WA,

where A is the x diagonal matrix of eigenvalues of V. Order the columns of W according to size of corresponding eigenvalue. Thus if W = (n>,:/) for /, 7=1,..., k, then the wth column of W, denoted w,„ = (wlm, . . ., wkm)\ is the x 1 eigenvector corresponding to the eigenvalue Xm and the column labelling has been chosen so that X{ > X2 > ... > Xk.

Define the wth principal component of the system by

Pm = WimXi + w2mX2 + + wkmXk,

where X, denotes the ith column of X, that is, the standardized historical input data on the rth variable in the system. In matrix notation the above definition becomes

3 Other forms of standardization are occasionally applied, which explains why different statistical packages ma> give different results. Note that if, after normalizing the data in X, the observation at time t is weighted by an exponential smoothing constant \T~, then V= (1 - ?)XX will be the exponentially weighted correlation matrix of the input data. An example of this is given on the CD, Normalization of the eigenvectors also plays a role, as the reader will discover when using the different spreadsheets for PCA on the CD. A full explanation of differences due to normalization is given in their help and tutorial files.



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