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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.

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From what I understand about your problem, you are a EUR investor looking to hedge the downside risk of USD depreciating against EUR such that returns earned in a USD ETF are worth less in your domestic currency (EUR).

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.

Answering your question: I do not think there is a need to use ETF returns in computing the MVHR. Rather, it is about computing and matching the notionals to determine the hedge amount to use for the FX futures on USDEUR.

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  • $\begingroup$ Thanks for your comment, the ETF returns would still matter because if the ETF USD returns are negatively correlated with the FX returns, then I would need to hedge less because my FX movement would counter the ETF returns (partially) and reduce overall volatility. Conversely, if the ETF returns and FX returns are more positively correlated, closer to 1, I would need to hedge more because my FX movement would 'amplify' the ETF returns and increase overall vol $\endgroup$
    – sjedi
    Feb 7 at 9:44
  • $\begingroup$ I think you need to decide which risk you would like to hedge first, because you are talking about different types of risk in 1 point. The returns of the ETF are due to market risk, whereas the change in USDEUR is due to FX risk. If you wish to hedge them together, you need to consider what are the returns of the USD ETF at every point in time converted to EUR (as you mentioned the FX rate "hedges" the returns of the USD ETF). This is more complicated and honestly does not reflect the right application of the MVHR in my opinion. $\endgroup$
    – KaiSqDist
    Feb 7 at 10:43
  • $\begingroup$ Hi, if you feel my response has helped you, you can give an upvote (if you haven't done so) or accept it as the solution (if it answered your question). :) $\endgroup$
    – KaiSqDist
    Apr 21 at 13:07

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