Sizing a Currency Hedge for a Foreign-Denominated Equity ETF
Summary
The document considers a euro-based investor holding a US dollar S&P 500 ETF and asks how to calculate a minimum-variance hedge ratio against dollar depreciation. Although the question starts from the usual correlation-and-volatility formula, the answer reframes the exposure: the currency risk is the value of the dollar in euros, so the relevant hedge is a short position in USDEUR FX futures, or the equivalent opposite position if contracts are quoted as EURUSD.
Because the futures hedge and the spot currency exposure refer to the same exchange rate, the answer says the ratio is one in this simplified setup. The practical sizing rule is therefore to match the ETF’s dollar notional with the FX futures notional, rather than calculate a ratio from ETF returns and currency returns. This explanation assumes the hedge instrument tracks the same currency exposure and focuses on hedging currency risk; it does not address basis risk, contract specifications, changing exposures, transaction costs, or broader portfolio risk.
Key ideas
- The currency exposure is the value of USD in the investor’s domestic currency, EUR.
- A short USDEUR FX futures position can hedge against USD depreciation, with the inverse position used for EURUSD quoting.
- When futures and spot represent the same exchange-rate exposure, the answer gives a hedge ratio of one.
- The suggested position size matches the ETF’s USD notional to the FX futures notional.
- The explanation does not cover basis risk, contract details, or other portfolio exposures.
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Full text
# Calculate minimum variance hedge ratio for foreign-denominated asset hedged to domestic currency
# 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.
## Answer by KaiSqDist (score 1)
https://quant.stackexchange.com/a/78243
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.