# Principal Components Analysis: Application in Value at Risk and Expected Shortfall

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Principal Component Analysis (PCA) is used in financial risk management to reduce the dimensionality of a multivariate problem, thus creating a simpler representation of the risk factors in the dataset. Only a few judiciously chosen hypothetical variables are needed to explain a large proportion of the variability in the data. These principal components are obtained through the singular value decomposition of the return series.

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Contributed by: Pichet Thiansathaporn (July 2012)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: PCA VaR stabilizes when the number of principal components exceeds three

Snapshot 2: divergence of PCA VaR (plotted points) from the "true" asymptotic VaR (dashed horizontal lines) is more pronounced: (1) toward the tail of the distribution (higher confidence level); and (2) with a small number of data points (1,000 observation days)

Snapshot 3: divergence decreases with a larger number of data points (10-fold increase from Snapshot 2)

Reference

[1] Kevin Dowd, *Measuring Market Risk*, 2nd ed., West Sussex, England: Wiley, 2005 pp. 118–125.

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