In this module we're going to introduce you to swap contracts, introduce you to the ideas why these swap contracts are constructed, what is the advantage of swap contracts. And also show you how to price a very simple interest rate swap using a no-arbitrage principle. Swaps are contracts that transform one kind of cash flow into another. A plain vanilla swap transforms a fixed interest rate cash flow into a floating interest rate cash flow. A commodity swap, swaps or exchanges floating price of, for the commodity for a fixed price for the same commodity. Examples include gold swaps, oil swaps, and so on. Currency swaps allows you to swap a cash flow in one currency for a cash flow in another currency. So why do companies construct or entities construct swaps? They do because you want to change the nature of cash flows. The example is fixed interest rate versus floating interest Another possibility is to construct swaps to leverage strengths in different markets. Here's an example. Company A, were it to borrow in the fixed interest rate market would be charged 4% per annum. If it were to borrow in the floating rate market, it would have to pay LIBOR plus 0.3%. LIBOR stands for the London Interbank Offer Rate, and that's the rate which is used as the base for floating, floating interest rates. Company B, were it to borrow in the fixed interest-rate market, would be charged 5.2 percent, and in the floating interest-rate market, it could borrow at LIBOR plus one percent. So company A is clearly superior to company B, both in the fixed interest rate market as well as the floating interest rate market. However, company B is relatively stronger in the floating rate market. The difference between the rate for company A and B is only 0.7% in the floating rate market. Whereas it's one point two percent in the fixed rate market. So these companies, could take advantage of the difference of their relative strengths in the true markets to create an, an instrument which lets them borrow at the better rate than they could have individually borrowed in either of the two markets. Company A, which is stronger in the fixed-rate market, borrows in the fixed-rate market, company B, which is stronger in the floating-rate market, borrows in the floating-rate market, and then they construct a swap in order to make an additional product or a derivative product. Which is going to be better than each of these individual deals that are available to this company. So here's company A, it borrows at 4%, which means that it's going to pay out 4%. Here's company B, which borrows at libor plus 1%. And then we construct a swap, we have company A, B is company B LIBOR and company B pays company A 3.95%. And if you see what the net effect of this swap is to each of these companies. Company A now ends up paying LIBOR plus 0.05%, which is better than what it could have borrowed in the floating rate market, and company B ends up paying 4.95%. Which is better than what it could have gotten in the fixed-rate market. So by constructing this swap, these two companies are able to leverage their relative strength to get a deal which is better than what they could have achieved in, either the fixed-rate market or the floating-rate market. Both of them end up gaining. The details of how to the 3.95% gets set depends on supply and demand. But there is an inplicit assumption that is being made in this particular example, and that's that company A and company B continue to exist. That neither of them is going to default. If one of these companies were to default then this entire superstructure breaks down and company A or company B, depending upon which one has a default, will be exposed to a big risk. So most of the times when swaps are constructed. You don't make a swap with a counterparty directly because you don't want to be exposed to the counterparty default risk. You would rather make it with an intermediary, a financial intermediary, that is able to take on the counterparty risk, and guarantees you, that you will get the swap cash flow that you're expecting to get from the counterparty. So here's how, these swaps get, set up. The same two companies, A and B, and now there's a financial intermediate that contracts the swap. Company A borrows in the fixed market at 4%, swaps with an intermediary and pays LIBOR and receives 3.93%. So, 0.02% less. Company B borrows in the floating rate market at LIBOR plus 1%. Constructors swap with an intermediary, receives LIBOR and pays, 3.97%. So 0.02% more, or this is the same thing as saying two basis points less, two basis points more. The financial intermediary in the middle makes this 0.04%. It receives two basis points from company A, and receives two basis points from company B. Why does it get that? This is the compensation for taking on the counter party risk. If either of these two companies default, the financial intermediary's on the hook, to provide the cash flow necessary for the surviving party. And it also constructs a service in the sense that, typically, in the market, company A and company B don't know that they exist, and their relative strengths are different so, by creating a swap, they would be able to better position themselves. So financial intermediary is able to bring these two parties to the table and construct a swap that is going to mutually beneficial, and ends up getting paid for providing this service. So let's see how an interest rate swap is priced. Let rt denote the floating unknown interest rate of time t. And let's consider a swap whose cash flows at time little t equal to 1 through capital T is given as false. Company A, which takes on the long position in the swap, receives a notional principal n, times the random interest rate prevailing at time t minus 1. So the cash flow that it receives at time t is given by the notional principal n. Times the interest rate at time T minus 1. And it pays the same notion of principal n times a fixed interest rate x. Company B, which nominally takes on a short position on the swap, receives the fixed interest rate payment n times capital X, and pays the floating payment n times rt minus 1. Now what we want to do is compute the value of the swap to company A. So there are two pieces to the cash flow to company A. It receives the cash flow, the principle, n times r 0, r 1, r 2 and so on up to r T minus 1 at times 1, 2, 3, 4 up to capital T minus 1. This is precisely the cash flow associated with the floating weight bond, minus the face value. So in a floating weight bond, at the expiration capital T, in addition to the coupon payment, you have received the notional principle back. This time you do not get the notional principle. And therefore, the value of the swap to company A consists of two elements. This is the value, of the floating rate, payments received, this is the value of fixed rate payments, paid to company B. We know that the value of a floating rate bond is directly equal to the principle. But we don't get the face value back, so I have to subtract from that value n times d, 0, t, which is the discounted value of the principal which was received at time capital T. What happens to the value of the fixed-rate payment? Every period, I get n times x. I have to discount that from time t equal to 1 through capital T. So this is simply the discount values. How is this x set? This is this x set, in a similar manner as we had done for former contracts. We set at, at a value such that the va is exactly equal to 0 at time t equal to 0. If you set it up, p equal to 0 implies that x must be equal to 1 minus d, 0, t divided by the sum of little t going from 1 to capital T to 0 T. This is the interest rate that you would have to set up, so that the value of the swap is exactly equal to company, zero for company A. And it's also equal to zero for company B. So, the two companies going into this swap, are eq, are, indifferent between taking a long position or a short position.