Plain Vanilla Swaps Group member: Shanwei Huang An Gong

1. History of Interest rate swaps Interest-rate swaps have grown tremendously over the last 10 years. Swaps were first created to exploit comparative advantage. This is when two companies who want to borrow money are quoted fixed and floating rates such that by exchanging pay ments between themselves they benefit, at the same time benefiting the intermediary who puts the deal together.

2. The structure of interest rate swap In an interest rate swap, each counterparty agrees to pay either a fixed or floating rate denominated in a particular currency to the other counterparty. The fixed or floating rat e is multiplied by a notional principal amount (say, USD 1 million). This notional amount is generally not exchanged between counterparties, but is used only for calculating th e size of cash flows to be exchanged. The most common i nterest rate swap is one where one counterparty A pays a fixed rate (the swap rate) to counterparty B, while receivin g a floating rate (usually pegged to a reference rate such as LIBOR). A pays fixed rate to B (A receives variable rat e); B pays variable rate to A (B receives fixed rate).

3. Plain vanilla interest-rate swap The simplest swap structure is a plain vanilla interest rate swap, in which one party receives a fixed interest rate agr eed in advance and the other party a variable reference in terest rate. This swap specifies the notional amount or fac e value of the swap, the payment frequency (quarterly, se mi-annual, etc.) the maturity, or fixed rate, and the floating rate. One party (Counterparty A) makes fixed rate paymen ts on the notional amount. The other party (Counterparty B) makes floating rate payments to Counterparty A based on the same notional amount.

4. Valuation and pricing for the interest rate swap The present value of a plain vanilla (i.e. fixed rate for floati ng rate) swap can easily be computed using standard met hods of determining the present value (PV) of the fixed leg and the floating leg. The value of the fixed leg is given by the present value of t he fixed coupon payments known at the start of the swap, i.e., where C is the swap rate, M is the number of fixed pa yments, P is the notional amount, ti is the number of days i n period i, Ti is the basis according to the day count conve ntion and dfi is the discount factor.

4. Valuation and pricing for the interest rate swap Similarly, the value of the floating leg is given by the present value of th e floating coupon payments determined at the agreed dates of each pa yment. The value of the floating leg is given by the following: Where N is the number of floating payments, fj is the forward rate, P is t he notional amount, tj is the number of days in period j, Tj is the basis a ccording to the day count convention and dfj is the discount factor. The discount factor always starts with 1. The discount factor is found as foll ows: [Discount factor in the previous period]/[1 + (Forward rate of the floating underlying asset in the previous period × Number of days in period/36 0)].

4. Valuation and pricing for the interest rate swap The fixed rate offered in the swap is the rate which values the fixed rates payments at the same PV as the variable r ate payments using today's forward rates, i.e.: Therefore, at the time the contract is entered into, there is no advantage to either party, i.e., Thus, the swap requires no upfront payment from either p arty.

5. EXCEL Application Build an application in Excel/VBA to value plain vanilla sw aps. Input: a short rate vector: r(1), r(2), …, r(n) basis points the nominal value and a vector of times where the cash flows exist t(1), t(2), …, t(n). In this model, the fixed and floating cash flows can have di fferent periods. E.g., the fixed cash flows can be paid ann ually and the floating semi-annual.