# PCA on portfolio depending on multiple time series

There is extensive documentation about PCA on specific time series (for example the UK yield curve). When you have a portfolio which only depends on the change of the UK yield curve then a PCA on the UK Yield Curve will be enough the calculate the risk of such a portfolio.

Now I wonder how one can calculate the risk of a portfolio which depends on multiple time series (for example the UK and the EUR Yield Curve) or a portfolio consisting of multiple different commodities. When running a PCA on the time series separately we will miss the correlation between these time series. A solution would be to run a single PCA on a combination of these time series but I wonder if this will work and if this will be statistically correct.

My main question: How can one calculate the risk of the portfolio based on PCA. Also how can I calculate the risk contribution per asset. Because normally PC1 corresponds with a parallel shift in the yield curve while you also want to know to which yield curve/time series.

• Typically you would want 3x components per ccy. Say you have two ccys, GBP and EUR yield curves. PCA1 is parallel shifts up for both (say), PCA2 is one up the other down, PCA3 is both steeper, PCA4 is one steeper one shallower, etc. You will retain the correlation between tenor points on the recovered curves (albeit imperfectly if the number of total components is fewer than total number of tenor points). Hope this is helpful. Feb 21 at 16:35
• Thanks James. Preferably I want to see the shift, twist and butterfly PC’s per currency and the contribution of these PC’s to the overall risk of the portfolio. I think I need to conduct separate PCA per curve and then combine them. Combining them would be the difficulty because I need to take correlations into account. Not sure how to do that Feb 23 at 14:16
• But as you say, these curve moves are indeed correlated so in order to decompose their portfolio risk contribution, you first need to orthogonalise the movements via PCA then calc the contributions to portfolio risk as you include more of the components. Try doing what I laid out on a 2-ccy and then a 3-ccy setting, it should then become a bit clearer. If you have 20 tenor points per ccy, obv a 2-ccy situation will be a 40x40 covar matrix as input Feb 23 at 19:29