# USDインデックスをユーロにヘッジ

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

しかし、私はヘッジされたリターンをユーロに換算したい。私はEURUSDの日々の為替レートを持っていますが、EURへのリターンをどのようにヘッジするのか分かりません。

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## 2 答え

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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EURは通常EURUSD、すなわち1ユーロ（ドルで換算）の値で現在は約1.1281となっています。

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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