Cross currency basis swaps, like the tenor kind, also highlight the difference between two floating rates, but they do so with rates of two difference currencies. Although simple in principle they raise a lot of questions and not everyone agrees about the answers.

A cross currency basis swap is defined as an exchange between the floating rate of one currency (usually USD but sometimes EUR) without basis (flat) and the floating rate plus basis bC of another currency. Also, crucially, the two parties exchange the principals at the beginning and at time T at the end of the transaction.^{[1]} We can write this as

(2.8)

where N is the principal amount. Let us try to understand the concept of principal exchange. Before doing so let us state a very important fact:

A stream of floating payments (leg) where the length of the rate coincides with the payment frequency and where the principal is paid at maturity should always price more or less at par.

Let us try to understand this simple and yet fundamental concept (about which more or less volumes could be written). First of all, to price at par, which we introduced in Section 2.2.4, means here the same as it did previously: it means that there is no gain for either side. When there are two legs, that is, two sides as in a swap, it means that both legs are worth the same; when there is only one leg it means that it is worth 0. (To be more precise, 0 or 1 (or 100), depending on whether we consider the principal itself.) A simple way to illustrate this is by using some of the tools and instruments previously shown.

Let us imagine that we have a floating leg that pays once in six months and a second time plus principal in one year. We know, and we consider these the only instruments traded in the market, the six-month cash deposit R6 and the six-month rate setting in six months L6,6. The present value of our floating leg will be given by

We have already found in Section 2.2.4 the values of D6 and D12, so by substituting we obtain

(2.9)

which, after simplification, leads to V = 1, that is, par. The above can be obviously extended to any maturity, provided that we pay every six months a rate of type Li,6 and that we pay the principal at maturity. Since we have made no mention of currency we must assume the above to be true for any currency.

If we now return to the definition of cross currency basis swap we begin to understand the role of the principal exchange. It is a way of saying: “Let us set the stage for the accepted situation, that is, that a floating leg should price at par in any currency, at which point we introduce the basis to show that this is not always the case.” This also follows our first definition of basis as a way of trading an anomaly, in this case the anomaly on the intuitive idea that a floating leg should price at par.^{[2]}

There is no single accepted reason for this anomaly and this anomaly has not always been acknowledged. Before 2005, even sophisticated market participants accepted the idea that all floating legs should price at par. The next section, dedicated to constructing curves with multiple inputs, will try to shed some light on the issues involved in the use of currency basis swaps.

[1] There are also rarer cases in which the principal is exchanged at intermediate stages of the swap's life at an exchange rate fixed at the moment of each initial exchange. These types of swaps can be seen as a series of forward starting, standard currency basis swaps.

[2] While this is true mathematically and it is useful in our subsequent discussion, a trader would not see the principal exchange in the same light. The principal exchange is primarily seen as a way of eliminating currency risk from the trade and simply concentrating on the interest rate sensitivity linked to the floating rate payments.

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