# Calculate minimum variance hedge ratio for foreign-denominated asset hedged to domestic currency

The formula for minimum variance hedge ratio (MVHR) is conceptually the correlation multiplied by the ratios of volatilities. correl (Y,X) * (STDEV Y / STDEV X)

Suppose I am a EUR investor purchasing an S&P 500 ETF denominated in USD currency and I want to get the MVHR to determine how much to hedge from USD to EUR. To apply the above formula, is Y the unhedged S&P 500 returns in EUR or the S&P 500 returns in USD. I.e. should it be the returns in foreign currency or returns in domestic currency (unhedged). And is X using the 1M USDEUR FX Forward Rate or the USDEUR spot rate.

Therefore, in the MVHR given by $$h^* = \rho \frac{\sigma_{S}}{\sigma_{F}}$$, the "spot" asset is the USDEUR FX rate (how much is 1USD worth in EUR), which should be hedged with a "futures" asset using a short FX futures on USDEUR (if there are only FX futures on EURUSD, just take the opposite position) that hedges against a depreciation in USD. However, as the underlying of the FX futures is the same as the "spot" asset, the MVHR is just $$1$$.
Moving back to your case, the hedge ratio simplifies to $$h = \frac{N_{A}}{N_F}$$, where $$N_A$$ is the notional of your ETF and $$N_F$$ is the notional of the FX futures that should be same in amount as that of the ETF in USD.