# How to compute the foreign exchange volatility within a portfolio

Suppose I have a portfolio of 5 assets. Assets 1 and 2 have foreign exchange exposures and therefore foreign exchange volatility. How can I calculate the marginal contribution to the total portfolio volatility from the individual foreign exchange exposures?

How about letting the FX rates remain fixed, and recalculate the portfolio volatility. That seems very obvious - am i missing something?

• That will calculate the total contribution, not the marginal contribution. Your answer was accepted as correct by two other readers already - they are probably also confused by the term "marginal." – mathguy Apr 24 '16 at 12:22

If a USD based investor owns shares of Toyota Motor in Japan, the variance of USD based returns is approximately equal to the variance of Toyota in yen, plus the variance of USDJPY plus twice the covariance between Toyota and the exchange rate. The last term could be positive or negative; if I had to guess for a big exporter like Toyota it is probably slightly negative.

• Not really - that would be correct if the entire portfolio was unhedged, not only part of it. In addition, your answer, like dm63's, looks at total contribution, not marginal. And why "approximately" equal and not "exactly" equal? – mathguy Apr 24 '16 at 12:24

If $R$ and $r$ are the return on the portfolio after currency hedging and on the currency, if I write $V(\cdot)$ for variance, and a fraction of $t$ of the portfolio is exposed to currency risk, then the return of the unhedged portfolio is $R+tr$. Then: $$V(R+tr) = V(R) + 2t\mathrm{Cov}(R,r) + t^2 V(r)$$ so the marginal contribution (derivative with respect to t) is $$\frac {\text d} {\text d t}V(R+tr) = 2\mathrm{Cov}(R,r) + 2tV^2(r).$$