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Converting USD Equity Returns into EUR Returns

Article Quant Q&A · Author: user8170
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Summary

The document explains how to express returns on a US equity index in euros using the EURUSD exchange rate. It distinguishes currency conversion from currency hedging: converting the index measures its value in euros, so the index return combines the local equity return with the change in the euro value of one dollar. With EURUSD quoted as dollars per euro, the USD index level is divided by EURUSD to obtain its euro-denominated value.

It presents both arithmetic and logarithmic one-day return formulations. In the arithmetic calculation, the index price ratio is multiplied by the inverse EURUSD ratio; in log returns, the two component returns add, with the exchange-rate term reversing sign. A separate answer notes that a fully hedged USD return would, before hedging costs, match the underlying USD return. The document gives no discussion of hedge instruments, rebalancing, transaction costs, or practical hedging error.

Key ideas

  • Convert the index level into euros by dividing its USD value by EURUSD when the quote is dollars per euro.
  • The euro return combines the USD index return with the change in the euro value of a dollar.
  • Arithmetic returns combine multiplicatively, while logarithmic returns combine additively.
  • A fully currency-hedged return is distinct from an unhedged return merely converted into euros.
  • Hedging costs and implementation details are outside the document’s treatment.

Tags

EquitiesForexUS markets
Full text
# hedge a USD index into EUR


# hedge a USD index into EUR












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?

## Answer by Richi Wa (score 2, accepted)

https://quant.stackexchange.com/a/25399

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: $$ 1+ r_{ \text{ S&P in EUR}} = (1+r_{\text{ S&P in USD}})(1+ r_{\text{1 USD = x EUR}}). $$

## Answer by nbbo2 (score 1)

https://quant.stackexchange.com/a/25402

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 $\ln(sp_t)-\ln(sp_{t-1})+\ln(eu_{t-1})-\ln(eu_t)$

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.