# Basis Swaps

What is a basis? A basis is a beautiful financial concept that is used to apply a numerical value (and therefore being able to trade it) to a discrepancy or to what at first can seem an anomaly. As far as curve construction is concerned, a basis is extremely important as it appears to indicate where two floating rates, which we would have assumed are equal, are in fact different. Within the same currency, a basis is traded through tenor basis swaps and between two currencies through (cross) currency basis swaps.

## Tenor Basis Swaps

A tenor basis swap can be defined as an exchange between a longer rate and a shorter rate plus a basis *b*T, that is,

(2.7)

where the rate *L*YM accrues over a number of months *Y* multiple of *X,* the number of months over which the rate *L*XM accrues.

At the origin of tenor basis swaps–a reason for which they have become more and more common lately–is an issue of credit. In a safe environment it should be irrelevant whether we choose to borrow a certain amount over a period of, say, six months or we choose to compound it in two three- month periods. The rate we should be charged should be the same. In a less-safe environment this is no longer the case: the longer term is riskier for the lender and therefore the borrower is charged a higher rate. In liquid currencies there are several types of tenor basis swaps available, as shown in Figure 2.6 (which shows only a subsection of the USD ones): the swap is quoted, as we did in Equation 2.7, by putting the basis on top of the rate

FIGURE 2.6 A few examples of quotes for common USD tenor basis swaps. *Source:* I homson Reuters Eikon.

with shorter maturity. In Figure 2.6, for example, we see quotes for swaps exchanging USD three-month LIBOR flat versus USD one-month LIBOR plus basis; USD six-month LIBOR flat versus USD three-month LIBOR plus basis; and finally, in the third column, USD six-month LIBOR flat versus USD one-month LIBOR plus basis. One can check that, as it should be, the average between the bid and offer (the mid) in the second column should be the sum of the mids in the first and second column. In other words it should be the same to swap, in one step, a one-month rate for a six-month rate or, in two steps, to swap a one-month rate for a three-month rate and then a three-month rate for a six-month rate.