I want to calculate the EUR equivalent yield from the USD yield curve. For example : how to translate the USD libor curve into an EUR equivalent yield curve ?

Do I need to use FX forwards or is the X-CCY basis enough ?


I'm not sure exactly what you mean by the "EUR equivalent yield", but I think that what you are asking is "How do I calculate the effective cost of funding EUR cashflows at USD Libor?".

If that's the case then, fundamentally, what you need are FX forward rates. For a given forward date you can imagine funding EUR at USD Libor by switching some amount of EUR into USD at the spot FX rate, accruing it at USD Libor, and then switching it back into EUR at the forward date using the forward FX rate. You then find something like

$$ P^\mathrm{EUR}_\mathrm{USD\ Libor\ 3m}(T) = P^\mathrm{USD}_\mathrm{USD\ Libor\ 3m}(T) \frac{fx(T)}{fx(0)}$$

which defines the discount factors of your "equivalent curve" in terms of those of the native USD Libor 3m curve (take care to get the FX in the correct direction, depending on the convention for the relevant pair).

The question then is, whence do you source FX forward rates? Well, there is of course an observable FX forward market, but they are typically only observable for the relatively short term. What about longer terms? This is where the cross-currency basis spreads come in: the value of a cross-currency basis swap depends on forward FX rates, so it follows that observed cross-currency basis spreads can be used to imply forward FX rates. Cross-currency basis swaps are typically quoted to much longer maturities than are seen in the FX forward market (as far out as 50Y+ in major currencies), and so these tend to be the best source of information on long-term FX forward expectations.

Ultimately then, the answer to your question is that you can use either or both FX forward rates and cross-currency basis spreads, since they both serve as a source for FX foward rates. Which you should prefer will depend on the relative liquidity of the respective products at each point in the term structure, and this depends on the particular currency pair you're dealing with. For EUR and USD, however, in my experience, the cross-currency basis swap market alone is deemed to allow the full FX forward term structure to be calibrated sufficiently well for the purpose of determining cross-currency funding costs.


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