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Linear PnL is usually correctly estimated by the inner product of risks and market movements:: $$ Pnl = S \cdot \Delta r = S^T \Delta r$$ Where you apply a linear transformation to those risks to express it in some other mathematical basis (e.g. PCA respresentation), then you have some transformation matrix, $T$, and it is easier to show that the PnL is ...


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You're right - I've looked, and there are many good tutorials on IR curve PCA out there, e.g. https://mockquant.blogspot.com/2010/12/principal-component-analysis-to-yield.html , https://plus.credit-suisse.com/r/kv66a7 , but I don't see anywhere a good explanation of atributing P&L to the IR curve changes in terms of PCs. Therefore I will outline it. ...


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Back in the day, I used to do precisely this on a cross-asset basis. The critical point being that the correlations of any of your PCs to any other PC will be zero, else you will have miscalculated your PCs in the first place. This being a given, you can regress your P&L to any and every PC in isolation, safe in the knowledge that all the others are ...


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I added an update to answer your updated question below. The PCA pertains to the attribution of changes (i.e. returns) of your valuation factors (e.g. zero rates across tenors) to latent "principal components", either thru eigendecomposition or thru SVD decomposition. Let $X$ denote the $T\times k$ matrix of observed zero rates shifts across the $k$...


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