Editing Principal component analysis (section)

=== Quantitative finance ===
In [[quantitative finance]], PCA is used<ref name="Miller">See Ch. 9 in Michael B. Miller (2013). ''Mathematics and Statistics for Financial Risk Management'', 2nd Edition. Wiley {{ISBN|978-1-118-75029-2}}</ref>
in [[financial risk management]], and has been applied to [[Financial modeling#Quantitative finance|other problems]] such as [[portfolio optimization]].

PCA is commonly used in problems involving [[fixed income]] securities and [[Bond fund|portfolios]], and [[interest rate derivative]]s. 
Valuations here depend on the entire [[yield curve]], comprising numerous highly correlated instruments, and PCA is used to define a set of components or factors that explain rate movements,<ref name="Hull"/>
thereby facilitating the modelling.
One common risk management application is to [[Value at risk#Computation methods|calculating value at risk]], VaR, applying PCA to the [[Monte Carlo methods in finance|Monte Carlo simulation]].
<ref>§III.A.3.7.2 in Carol Alexander and Elizabeth Sheedy, eds. (2004). ''The Professional Risk Managers’ Handbook''. [[PRMIA]]. {{isbn|978-0976609704}}</ref>
Here, for each simulation-sample, the components are stressed, and rates, and [[Monte Carlo methods for option pricing#Methodology|in turn option values]], are then reconstructed; 
with VaR calculated, finally, over the entire run. 
PCA is also used in [[hedge (finance)|hedging]] exposure to [[interest rate risk]], given [[Key rate duration|partial duration]]s and other sensitivities.
<ref name="Hull">§9.7 in [[John C. Hull (economist)|John Hull]] (2018). ''Risk Management and Financial Institutions,'' 5th Edition. Wiley. {{isbn|1119448115}}</ref>
Under both, the first three, typically, principal components of the system are of interest ([[Fixed-income attribution#Modeling the yield curve|representing]] "shift", "twist", and "curvature").
These principal components are derived from an eigen-decomposition of the [[covariance matrix]] of [[yield curve|yield]] at predefined maturities; 
<ref>[https://www-2.rotman.utoronto.ca/~hull/RMFI/PCA_6thEdition_Example.xls example decomposition], [[John C. Hull (economist)|John Hull]]</ref>
and where the [[variance]] of each component is its [[eigenvalue]] (and as the components are [[orthogonal]], no correlation need be incorporated in subsequent modelling).

For [[equity (finance)|equity]], an optimal portfolio is one where the [[expected return]] is maximized for a given level of risk, or alternatively, where risk is minimized for a given return; see [[Markowitz model]] for discussion.
Thus, one approach is to reduce portfolio risk, where [[asset allocation|allocation strategies]] are applied to the "principal portfolios" instead of the underlying [[Capital stock|stock]]s.
A second approach is to enhance portfolio return, using the principal components to select companies' stocks with upside potential.
<ref>Libin Yang. [https://ir.canterbury.ac.nz/bitstream/handle/10092/10293/thesis.pdf?sequence=1 ''An Application of Principal Component Analysis to Stock Portfolio Management'']. Department of Economics and Finance, [[University of Canterbury]], January 2015.</ref> 
<ref>Giorgia Pasini (2017); [https://ijpam.eu/contents/2017-115-1/12/12.pdf Principal Component Analysis for Stock Portfolio Management]. ''International Journal of Pure and Applied Mathematics''. Volume 115 No. 1 2017, 153–167</ref>
PCA has also been used to understand relationships <ref name="Miller"/> between international [[equity market]]s, and within markets between groups of companies in industries or [[Stock market index#Types of indices by coverage|sectors]].

PCA may also be applied to [[Stress test (financial)|stress testing]],<ref name="IMF">See Ch. 25 § "Scenario testing using principal component analysis" in Li Ong (2014). [https://www.elibrary.imf.org/display/book/9781484368589/9781484368589.xml "A Guide to IMF Stress Testing Methods and Models"], [[International Monetary Fund]]</ref> essentially an analysis of a bank's ability to endure [[List of bank stress tests|a hypothetical adverse economic scenario]]. Its utility is in "distilling the information contained in [several] [[Macroeconomic model|macroeconomic variables]] into a more manageable data set, which can then [be used] for analysis."<ref name="IMF"/> Here, the resulting factors are linked to e.g. interest rates – based on the largest elements of the factor's [[eigenvector]] – and it is then observed how a "shock" to each of the factors affects the implied assets of each of the banks.