PCA for stand alone equity VaR

I am trying to compute equity VaR, forex VaR and total VaR on an international portfolio (10 stocks x 4 countries). Since I am not interested in the risk disaggregation among diffrent countries I was thinking to apply PCA directly on $\sigma_E$, $\sigma_X$ and $\sigma$ respectively; where $\sigma_E$ is the covariance matrix of the stocks log-returns in local currencies, $\sigma_X$ is the covariance matrix of log-returns on exchange rates and $\sigma$ the covariance matrix of all log-returns (stocks and exchange rates).

While there shouldn't be any problem for the forex VaR, I am not quite sure I can use PCA on log-returns denominated in different currencies in order to find the equity VaR. My main concern is on how to find and interpret the principal components coefficients. Let’s say that I decide to use 5 PCs that will replace my 40 stocks log-returns, how do I find the coefficients?

Would it be possible to create an a-doc portfolio from the log returns denominated in different currencies (without converting them):

$r_p = w_1 r_1^€+ … + w_{10} r_{10}^€ + w_{11} r_1^{DKK}​ + … + w_{20} r_{10}^{DKK}+ ...$

And then regress it on the principal component factors like this:

$r_p = α + β_1 PC_1 + ... + β_5 PC_5 + ϵ$

The reason I am trying to do this is because I would prefer avoid having PCA (or a foundamental factor model) for each country, otherwise I would still have an equity and total variance-covariance matrix with nonzero covariance’s and I would then have to use a multivariate GARCH.

• NB: I have already posted this question in another forum without any luck last week. However I decided tho ask here as well because its seems more appropriate for this topic. stats.stackexchange.com/questions/223323/… – Marco Jul 19 '16 at 17:09
• do you think you could simplify your question ? short questions attract more answers most of the time – MJ73550 Jul 20 '16 at 11:44
• Thank you for the advice, I did simplify it a bit. Next time I will kip that in mind. Please let me know if something is still not clear. – Marco Jul 21 '16 at 2:00
• Have a look a covariance regularization as well to understand what the options are. There is also something called LKJ Covariance, Hierarcical risk parity, C-vines blah blah. Not necessarily worth using but conceptually is a rabbit hole you can know about and avoid for now. But honestly some simple shrinkage and hVar (non-parametric) will give you enough. Compute the Var deltas to portfolio changes to look for any weirdness. – mathtick Apr 3 '20 at 12:19

PCA itself just help you find the correlated movement. Since you equity variance is cross multiple countries. So the actual return is definitely related to the fx changes. So my suggestion is to add PCA to the $\sigma$ directly.