Bonds with maturity beyond a year typically pay coupons. In case these coupons are fixed we speak of fixed coupon bonds. In case these coupons are floating we speak of floating rate notes. An interest rate swap exchanges the cash flows, of fixed coupon bonds and a floating rate note. A fixed coupon bond or simply coupon bond is specified by coupon dates T1, T2 up to Tn where Tn is also called the maturity of the bond. It is also specified by fixed coupon payments, C1, C2, and so on... and so on... Being paid at the coupon dates, Ti and the bonds also comes with a principal value N that is paid at the maturity. Given what we know about discount bonds, we can now determine the price of these coupon bonds at a calendar date "t". It is given by multiplying these cash flows by the respective discount bond and summing up. We follow here the convention, that if the calendar date T coincides with one of the coupon dates, then the coupon itself is not full part of the price anymore. In other words the price is ex-dividend. A floating rate note is specified by a sequence reset and settlement dates Tnot, T1 up to Tn. Where again, Tn is the maturity of the note. While Tnot is the first reset date. Again, the principal is paid at the maturity. The coupon payments now are floating. That means (INAUDIBLE) T1, there is a coupon paid which is fixed at Tnot The same at T2, there is a coupon paid which is fixed at T1. So on and so forth. Until the last coupon has been paid. So, notice that at Tnot there is not cash flow, Tnot is simply a reset date. The price of the floating rate note, at any calendar date small "t" prior to Tnot, is surprisingly simple. It is given by the Tnot bond price times the principal N. We proof this price formula by constructing a self-financing strategy, Which has the initial cost equal to the price, and which yields the same cash flow as the floating rate note. Absence of arbitrage, that implies that this initial cost< must indeed be equal to the price. The initial cost is N x Tnot bond price. With this amount of money we can buy N Tnot bonds. We hold them until Tnot, and each of the bonds wil pay us one unit of currency. So, in total we get N We then use this amount and invest it in T1 bonds How many can we buy? Well, the amount is given by the money we have divided by the price of a T1 bond. So we pay N At the time, T1 this T1 bonds mature and for each of the bonds we receive one. We hold that many bonds so we receive this amount of money. Which we can write as sum of the floating coupon, plus the notional. So we get the notional, and we get the floating coupon. Now we repeat and use the money. N units of currency and buy T2 bonds Called into the scheme, this leads us Two sequence of cash flows, where notional income cancels notional payment, until the maturity. Where we receive the notional plus the last floating coupon. Now you see that the netcare flows of these strategies, is exactly matching the cash flows of the floating rate note, and that proves that the initial cost of this strategy is indeed equal to the price that we saw on the previous slide. When T is equal to the first reset date, the price of the floating rate note is simply equal to the notional. So we can state inverts that paying the floating coupon Ci at the coupon dates Ti and the principal N at the maturity Tn is equivalent to paying the principal N at the first reset date Tnot. Now suppose you have issued a floating rate note so that you have to pay floating rates. But you would like to pay fixed rates instead. Then you can enter an interest rate swap. An interest rate swap exchanges fixed and floating coupon payments. Formally a payer interest rate swap settles in the rears, is specified by a sequence of reset and settlment dates Tnot up to Tn as we have seen for the floating rate note, Tnot is the first reset date Tn is the maturity, and Tn minus Tnot is called the length of the swap. Further ingredient is a fixed rate K, a notional N, and for simplicity here, we assume that these time points are equally distant with constant difference Delta The holder of this payer interest rates swap, will pay a fixed rate K at the payment dates T1, T2 up to TN, and in turn receives the floating rates on there dates. The net cash flows are exactly the same as you would get from holding a floating rate note, and being short a fixed coupon bond, so the value of the payer interest rates swap at date small "t" is given by the difference of the value of the floating rate note, minus the value of the coupon bond. Is an easy exercise using the definition of forward rates expressed in terms of cero coupn bonds to show that this can be rewritten in this form. Notice that when N is one that is when there is only one cash flow date, this is exactly the same as a forward rate agreement. Now, the value of a receiver interest rate swap, it is just the counter part disposition to a payer interest rate swap. Is as given by negative of the value of the payer interest rates swap. Now again there is a unique fixed rate K, which makes the value of the payer and receiver interest rate swaps equal to cero. This fixed rate is called The Swap Rate. Prevailing at co-lender date T. Now if T is strictly smaller than Tnot, we speak of forward swap rate. If T is equal to Tnot, we speak of the spot swap rate. In either case, the expression is given by this formula. I leave it as an exercise to you. To derive these equivalent alternative expression, with weight W high. Another simple exercise for you is to proof this Lemma Which states that a coupon bond has par value at Tnot. If and only these coupon rates are equal to the corresponding swap rate. Formally speaking, this identity. Interest rate swaps trade from different starting dates Tnot. And for various lengths, this tamble gives an example of market quotes for swap rates. These are Euro swap rates, expressed in percentage points, from end of August 2013. The Euros zone swap pay annual coupons that is Delta is 1. We see here in the first column, the spot swap rates as function of the length of the swap. We get the term structure of swap rates. Similarly, we get the term structure of swap rates, for swaps starting in one year from now. And so on and so forth. Now, interest rates swap markets are over the counter. They are not exchanged traded. But the swap contracts exist in stand that is form. For example, by the International Swaps and Deivatives Association. Swap markets, therefore are extremely liquid, and the outstading notional is very large. Tha maturiries or the length of the swap, range from 1 to 30 years, which is standard, and quotes of sweap rates are even available up to 60 years. Member swaps give market participants such as institutional investors, as life insurers. The opportunity to creat synthetically long dated investments.