PCA for stand alone equity VaR

Question

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.

Answer 1

If I have understood your question correctly, no adjustment is necessary if you are using log-returns. Returns of a stock in a differing numeraire is simply a sum of the currency spot process and the locally denominated process. Since you are trying to find the principal components, well, a linear term like this doesn't matter, since that variation will be captured by the components that capture the variation of the local currency (supposing that the relevant currency is one of the pairs you are considering! ).

Answer 2

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.