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A swap is an agreement between two counterparties to exchange something (one "leg" of the swap) for something else (the other "leg"). These things can be anything that has a financial value. Typically they are quantities determined by some form of interest rate, in an interest rate swap or derivative.
Interest rate swaps take many forms. Typically they consist of a number of component swaps on a frequent basis according to a predetermined payment schedule.
Usually, one leg involves quantities that are known in advance, known as the "fixed leg", the other involves quantities that are not known in advance, known as the "floating leg". The floating leg must therefore be reset against an agreed reference rate, which will become known at some point before the payment or settlement takes place. Ideally, the determination of the reference rate must be outside the control of the counterparties, otherwise a conflict of interest will arise. However, many financial products in the retail market (such as capped mortgages) involve reference to a managed interest rate which is actually controlled by the mortgage provider. Typically, the reference rate is some figure made publicly available by a third party information vendor, or by government agencies. For example, BBA LIBOR.
Once a component of the floating leg is fixed (or "reset"), the fixed and floating components can be swapped or settled (typically one or two days after the fixing date).
Interest rate swaps allow parties to re-allocate their exposure to interest-rate fluctuations, typically by exchanging fixed-rate obligations for floating rate obligations.
Party A may hold a fixed-rate loan, party B a variable-rate loan. In a swap, A will make the payments on B's loan and vice versa. There is no change in the balance sheets of either party, because the principal, i.e. the underlying 'notional' amounts, stay where they were. In other words, what is called a $1 billion swap actually involves amounts much smaller than $1 billion.
An interest-rate swap is one of the more common forms of over-the-counter derivative security transactions.
The present value of a vanilla swap can easily be computed using standard methods of determining the present value of the components.