Sunday, February 8, 2015

Calculating Variance Explained from PCA Loadings


I have a return history for a universe of risky assets and I've run a principal component algorithm and obtained a loadings matrix (num_factors by num_assets) for the first 5 factors.


I have a portfolio as well (a subset of the above universe) with weights w for each of the assets. This portfolio has a variance \sigma^2. How do I figure out the percentage of the variance in the portfolio that comes from factor 1?



Answer



PCA gives you a decomposition of the covariance matrix of the form Σ=VΛVT

where Λ is diagonal with the eigenvalues in the diagonal. Your portfolio variance is wTΣw=(VTw)TΛ(VTw)
On the other hand if you take your return matrix R and define F=VTR
then the covariance matrix of these so called principle portfolios is Λ. You find this here by Meucci.


In fact he writes V1R for the return of principle portfolios and defines the weights w=V1w for the weight of the original portfolio on the principle portfolios.


He then defines vn=(w)2λ2n for the contribution of the n-th principle portfolio to the portfolio variance. If you relate this to the total volatility of the portfolio then you are done. Note that V is orthogonal which means that V1=VT.


I recommend to read the following white paper or this blog entry or this to get more details.


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