# Calculating marginal risk contribution of FX for foreign asset portfolio

I am a European investor investing in US equities. My US equities portfolio returns in EUR can be broken down into (1) equities returns in USD terms, and (2) USDEUR spot currency returns.

Using the time series of US equities returns in USD and USD EUR spot currency returns, how do I calculate the marginal risk contribution of FX for the unhedged US equities portfolio, 50% hedged and 100% hedged to EUR portfolio. I am able to calculate standard deviation, covariance etc. from the time series.

I understand that for multi-asset portfolios e.g. bonds and equities. I would multiply the % weights of bonds and equities by the covariance matrix to obtain the marginal risk contribution, i.e. mmult([%allocation],[cov matrix])

How can I do the same for the FX impact on the portfolio (USD to EUR translation effect), as there are no 'actual' weights given for FX? Hedging would also entail a short USDEUR position, but against a long USD asset, so how would the hedged weight (hedging ratio) be accounted for?

I am not sure what you mean by Marginal Risk Contribution in this case. A European investor in American stocks experiences a different variance of returns than a US investor. Define the following:

RLSP = the Logarithmic Return on a portfolio of US stocks, which could be the S&P or any other portfolio

RLEUR = the Logarithmic Return of the EUR exchange rate EURUSD

RLEINV = the Logarithmc Return of a European INVestor

Then we have the following relationship

RLEINV = RLSP - RLEUR

The variance of RLEINV can therefore be found as

Var(RLEINV) = Var(RLSP) -2 Covar(RLSP, RLEUR) + Var(RLEUR)

In the simplest case, where stock prices are not related to exchange rates, the covariance term is zero. IN THIS CASE THE VARIANCE OF THE EXCHANGE RATE IS SIMPLY ADDED TO THE VARIANCE OF THE AMERICAN STOCKS. In general though the covariance matters, could be of either sign and could even be changing over time (as american companies expand or contract their foreign activities, etc.).

When a fraction $$\alpha$$ of the foreign exchange is hedged (where $$\alpha=0$$ means no hedging), the equation becomes

Var(RLEINV) = Var(RLSP) -2 (1-$$\alpha$$) Covar(RLSP, RLEUR) + (1-$$\alpha$$)^2 Var(RLEUR)