Using PCA for VaR on International Equity and Currency Returns
Summary
The document considers principal component analysis for estimating risk in an international equity portfolio whose stocks are denominated in several local currencies. It asks whether PCA can be applied to local stock returns separately from exchange-rate returns, and how component coefficients might be interpreted for a portfolio mixing currencies.
The responses explain that local-currency stock returns and currency movements combine in the investor’s realized returns. A PCA on the full covariance structure can capture both sources of variation, while applying PCA only to local stock returns may omit currency effects relevant to the portfolio’s actual risk. The discussion is conceptual: it provides no worked calculation, VaR procedure, or empirical comparison. The appropriate return series and covariance matrix depend on the investor’s reporting currency and risk objective.
Key ideas
- International equity returns depend on both local stock movements and exchange-rate changes.
- PCA identifies correlated movement across the return series supplied to it.
- A portfolio’s VaR should reflect the currency exposure relevant to its reporting basis.
- The document gives conceptual guidance but no numerical VaR implementation.
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Full text
# PCA for stand alone equity VaR
# 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.
## Answer by deftfyodor (score 1)
https://quant.stackexchange.com/a/28212
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 by XYQ (score 0)
https://quant.stackexchange.com/a/39793
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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