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Say I have the daily index levels for the S&P. It is then very simple to calculate the daily returns.

I however want to calculate the hedged returns into Euros. I have the daily exchange rates for EURUSD however I'm not sure how to hedge the returns into EURs?

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    $\begingroup$ I hope I'm not being petty here, but there is a significant difference between hedging and translating. Translating returns into EUR would be to take the FX risk, while hedging would try to eliminate the FX risk by using products such as forwards. I'm assuming you just mean translate and I think that is what your responses infer. $\endgroup$
    – horseless
    Apr 13, 2016 at 16:26

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if you hedge it means that your USD return equals (neglecting hedging cost) your EUR return. You just change the name. If you want to know what the return measured in EUR is, then you either calculate the price of S&P in EUR and then take returns or equivalently you calculate the product of the local return and the return of the USD in EUR in the following sense: $$ 1+ r_{ \text{ S&P in EUR}} = (1+r_{\text{ S&P in USD}})(1+ r_{\text{1 USD = x EUR}}). $$

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The EUR is normally quoted as EURUSD, i.e. the value of one euro measured in dollars, currently about 1.1281.

If the S&P index is $sp_t$ and the EURUSD rate is $eu_t$ then the S&P converted into Euros is $sp(t)/eu(t)$. The arithmetic 1 day return on this is $-1+\frac{sp_t}{sp_{t-1}}\frac{eu_{t-1}}{eu_t}$. The logarithmic return is $\ln(sp_t)-\ln(sp_{t-1})+\ln(eu_{t-1})-\ln(eu_t)$

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