Principal
Components Analysis (PCA) is a dimension reduction technique that aims at
identifying communalities between the initial number of variables and scale
down to the number to few meaningful components that are able to explain
certain amount of variance. In other words, the underlying idea behind the PCA
is to extract common features of the shares’ returns, i.e. instead of looking
at all shares the PCA concentrates on communalities and extracts several components
that are able to explain certain amount of the variability of all data. What
remains unexplained by the factors is the specific risk.
PCA extracts the factors but does not name these factors, i.e. it is
only doing a partial job. But nevertheless, it is useful to identify the
specific and systematic risks of the listed shares.
There are
several key features that make PCA quite a sensitive technique (but as a
bottom-line this is valid to all kind of techniques that deal with identifying
key meaningful factors):
(1)
Identifying the number of
components – a number of stopping rules apply – as the graphical analysis - scree
plot (the point at which the line levels out), Kaiser’s stopping rule (only components
with eigenvalue over 1.0 should be taken into consideration), etc.
(2)
Defining the options for the
PCA – namely analysis of correlation or covariance and rotation method. Generally, the covariance matrix is preferred
when the variables have similar scales (for instance logarithmic stock returns)
or when the variables have been transformed, while the correlation matrix is
used when variables are on different scales (and correlation standardized the
data). On rotation method, seems more widely accepted to use direct oblimin that
allows components to be correlated (non-orthogonal solution), the alternative solution
is varimax rotation (orthogonal solution – components are not correlated).
The PCA analysis of 14 Romanian listed shares for the period Sept 3,
2012 – Dec 31, 2014 (excluding several blue chips that arrived on the market
after Sept 2012 – Romgaz, Electrica, Nuclearelectrica) is based on 581 daily
log-return series for each company. Before conducting the PCA analysis two preliminary
tests were taken into consideration – Kaiser-Meyer-Olkin Measure of Sampling
Adequacy (KMO) that has a value of 0.892 (threshold value of 0.6 is considered
as a minimum; closer to 1.0 the better) and Bartlett's Test of Sphericity.
The results:
(1)
Percent of variance of
returns series of the 14 Romanian shares explained by the first component – 30.3%
and by the second component – 7.8%; so cumulatively the first two components
are able to explain as much as 38.1% of the variance of the returns. And
together with the third component the % of the variance explained is 44.2%. The first component can be interpreted as “market
wide” component.
(2)
Proportion of variance of the
14 Romanian shares that can be explained by the 2 components (can be
defined as the sum of the squared factor loadings) - high for SIF3, SIF1 and
Transelectrica but rather low for the pharma shares – Biofarm (BIO),
Antibiotice (ATB) and Fondul Proprietatea (FP).
(3)
How the companies can be
combined into groups with respect to how they respond to the 2 components?
Answer to this question provides the “pattern matrix”. So, in the first group
we have: Transelectrica, Bucharest Stock Exchange, BRD, Banca Transilvania,
Transgaz and Petrom. Almost all the large caps fall in one group (intuitive, isn’t;
but this also means in case the first component is the general market we cannot
rely on having sort of safe havens and really defensive stocks). Also
interesting is the second group – all the SIFs (negative loadings to the second
component). The SIFs have low loadings with respect to the first component
(market component) – so there is opportunity for diversification between the 2
group of stocks identified. The correlation between the 2 components is -0.519.
This publication is for information purposes only and should not be
construed as a solicitation or an offer to buy or sell any securities.
