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A portfolio is a collection of financial instruments. We often collect instruments together to represent the complete holdings of an investor and to analyze the overall risk (which may be lower due to diversification, i.e the portfolio holding multiple instruments).
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Rebucketing Risk using PCA/other methods
Sounds like PCA is not the approach you're looking for. If you're looking to transform a risk vector in terms of securities V into a risk vector in terms of securities W, then the basic approach would …