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
In fact he writes V−1R for the return of principle portfolios and defines the weights w∗=V−1w 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 V−1=VT.
I recommend to read the following white paper or this blog entry or this to get more details.
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