An equity call swap is similar in concept to a basis swap as both involve two variable legs. The former has two equity legs (two different stock indexes), whilst the latter is an interest rate swap in which both legs are floating rate (such as a six-month LIBOR versus six-month T-bill rate, or six-month LIBOR versus three-month LIBOR). In an equity call swap (a.k.a a relative performance swap), a counterparty pays the total return on one index and receives the total return on another stock index. In this sense, this swap can be used in asset allocation between two countries or geographical areas (and investor holding an equity portfolio in one country can convert that portfolio, entirely or partially, to an equity portfolio in another country.
A basis swap (a.k.a a floating-for-floating swap) typically embarks on the spread between two interest rate indexes (also two commodity indexes, etc). Generally speaking, this swap involves exchanging cash flows in one floating rate against cash flows in another floating rate, in the same currency. Therefore, it helps convert one floating rate to the other, i.e., it provides a means to mitigate the basis risk. For example, a government agency may need to borrow from international money markets in USD LIBOR, and then lend the money to local banks (as a small difference could exist between the interest rate being paid and that being received).
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