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Why is it so popular to use market capitalization weighted indices instead of taking the first principle component that explains the most variation of the constituents? I haven't yet seen an academic study that does the latter. What's the reason for this?

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a) because it does not matter how you weigh each constituents as long as the methodology is publicly accessible and as long as it more or less reflects the original intent. That is why there are market cap weighted indexes but also why there are indexes that apply different weighting methodologies.

b) because PCA is computationally way more expensive. Why would you chose to do things the most complicated way if results can be had more efficiently. The point with such indexes is not to derive the most accurate results in terms of attribution but to set a standard on which options can be written, portfolios bench-marked against, and relative performance evaluated.

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The first principal component of a large covariance matrix is extremely expensive to replicate in a real portfolio.

While it is true principal components provide true (ex post) orthogonal factors, this is not necessarily relevant to the business of risk management. The market index is what most investors are benchmarked by, and is furthermore often available to trade through cheap technology like futures, swaps, or ETFs.

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