# Introduction to Interest Rate Swaps

 In a plain vanilla interest rate swap, two counterparties agree to exchange interest cash flows on a periodic basis for a predetermined length of time. Usually, one party will pay a certain fixed interest rate to the other while receiving an amount that is based on a benchmark floating interest rate. This constitutes a "fixed-for-floating" swap where the cashflows are based on the same notional principal sum and are denominated in the same currency. A simple example is as follows. Consider a two year swap that is initiated on February 1, 2000. The terms of the deal require that company A pay company B a fixed amount of 7.5% on a notional principal of \$10M, while company B pays company A on the basis of 3-month LIBOR for the same principal amount. This means that company A will pay fixed and receive floating, while company B will pay floating and receive fixed. All payments are made on a quarterly basis, meaning that there will be 8 payments in total. If we assume (unrealistically) that the length of each 3 month period is exactly 0.25 years, then the schedule of payments for the swap might follow that shown in the table below. Reset Date Payment Date LIBOR Floating Payment Fixed Payment Net CashFlow 1 Feb 2000 1 May 2000 7.35 \$183,750 \$187,500 -\$3,750 1 May 2000 1 Aug 2000 7.42 \$185,500 \$187,500 -\$2,000 1 Aug 2000 1 Nov 2000 7.45 \$186,250 \$187,500 -\$1,250 1 Nov 2000 1 Feb 2001 7.67 \$191,750 \$187,500 \$4,250 1 Feb 2001 1 May 2001 8.05 \$201,250 \$187,500 \$13,750 1 May 2001 1 Aug 2001 7.75 \$193,750 \$187,500 \$6,250 1 Aug 2001 1 Nov 2001 7.80 \$195,000 \$187,500 \$7,500 1 Nov 2001 1 Feb 2002 8.00 \$200,000 \$187,500 \$12,500 The amount of the fixed payment is of course constant for all 8 payments, and is computed as 0.25´10,000,000´0.075. However, the values of all but the first of the floating payments are unknown at the inception of the swap because they are dependent on the realizations of the LIBOR rate on each of the subsequent reset dates. The payments actually shown in the table are based on one possible path for the LIBOR rate over the next two years, and each payment is computed as 0.25´10,000,000´(LIBOR/100). Note that it is possible to forecast the level of the floating payments when the swap agreement is entered into by using the existing zero curve. In most cases an interest rate swap is structured so that both the fixed and floating payments are not actually paid. Rather, the difference between the two amounts is paid by the counterparty who faces the net shortfall at each payment date. The net cashflow figures shown in the table above are expressed from company A's point of view and indicate that company A must pay company B on each of the first 3 payment dates. Because the floating payment received by company A exceeds the fixed payment on all of the other payment dates, company A will receive a net cash inflow on the remaining 5 payment dates. One of the main reasons for a company to enter into an interest rate swap is to transform an existing liability. Suppose that company A in the example above has borrowed \$10M at LIBOR plus 100 basis points. Then the swap might be used to transform the floating-rate loan into a fixed rate commitment by combining the following three cashflows: 1. Payment of LIBOR plus 1% to lending institution2. Payment of fixed 7.5% to company B3. Receipt of LIBOR from company B These cashflows net to a fixed payment of 8.5%, and company A has now effectively borrowed \$10M at that fixed rate. The swap can be valued from company A's point of view by simply recognizing that the agreement is equivalent to taking a long position in a floating-rate bond (receive floating payments) and a short position in a fixed-rate bond (pay fixed payments). That is: where, SVa= the swap value to company A.Bfl = the value of the floating-rate bond.Bfx= the value of the fixed-rate bond. Determining these values is quite straightforward assuming that an appropriate zero curve is available. However, we can develop the necessary logic here by recognising that the value of both bonds is simply equal to the present value of the future cashflows that will accrue to the bondholders. Thus, the value of the fixed-rate bond can be determined as: where, FV = the face value (or principal) of the bondrfx = the fixed interest rateTi = the tenor of the i'th coupon payment, computed as the number of days in the period divided by the assumed number of days in a year.DFi = the discount factor with the same maturity as coupon payment iN = the total number of coupon payments outstanding Determining the current value of the floating rate bond is complicated by the fact that the amount of each coupon payment is dependent on the future level of the floating rate at each of the reset dates. Although these rates are obviously unknown on the valuation date, the floating-rate bond can be valued using our current expectation for the floating rates: where, E(ri) = the expected value of the floating rate at reset i The expected reset rates are typically estimated by deriving forward rates from the current zero curve. That is: where, Fi = the forward rate applicable to reset i