This module is the first of two modules on credit default swaps. In this module, we are going to introduce to you to the details of what a credit default swap is. Talk about, on an intuitive level, of how credit default swaps give you some information about the default probability. And how these quantities can be used for hedging, can be used for investment, can be used for speculation. And how credit default swaps have played an important role in the financial crisis, and also the sovereign debt crisis that is currently going on in Europe. The seller of a credit default swap, CDS, agrees to compensate the buyer in the event of a loan default or some other credit event on a reference entity. The reference entity could be a corporation, could be a sovereign, that's a country, in return for periodic premium payments. So the dynamics are as follows, this is what happens to the buyer. So the buyer pays periodic premiums, and what is a premium? It's some coupon rate S which is also called the spread. D is in years, so it's a fraction of a year, so d times S is the total coupon that has been accumulated over d years. N is a notional principal, so there's a certain amount of principal that has been protected. In order to protect that principal, you have to pay a spread or a coupon S times N, it has to be paid every d fraction of the year. So every coupon payment is going to be d times S times N. And this keeps on going until some kind of a credit event happens. Maybe the corporation defaults on the bond, maybe the company is not able to make an interest payment, and so on. We're going to talk about credit events later on, and talk about the fact that defining whether the current event has happened or not, in itself, a difficult problem. So once that credit event happens, two things happen. This credit event has happened somewhere in between two coupon payments. There should've been a coupon payment there, there should have been a coupon payment there. So the credit event happened right in between these two coupon payments. So at the next coupon payment date, the buyer has to pay the accrued interest over this interval. And I'm showing this with a smaller arrow to suggest that the accrued interest is actually less than the full coupon payment. On the other hand, the seller, the one who decided to sell the protection on this underlying credit event. After the credit event has happened, at the next coupon date, the seller has to pay 1-R, where R is the recovery rate, times the notional payment N. Because the buyer pays premiums, the premium payments are sometimes called the premium leg of the CDS. And because the seller always pays the amount only on default, this is also called the protection leg. Here's a simple numerical example. So consider a hypothetical two-year CDS on a notional principle N equal to $1 million. And a spread S equal to 160 basis points, so just about 1.6%. And let's assume that the payments are quarterly. Suppose a default occurs in month 16 of a 24-month protection period, and the recovery rate at that time is 45%. And now let's understand what happens to the payments of the buyers and the sellers. The buyer pays premiums, so he pays premiums at month 3, 6, 9, 12, and 15. And this is going to be S, the spread, times a notional principal of $1 million divided by 4, why? Because these are quarterly payments, so it's one-fourth of a year. So the payments in all of these months is going to be $4,000. Now, here's month 15, the next coupon is going to be on 18. But in month 16, the default has happened. So this is the period over which the interest has accrued. So the accrued interest that the buyer has to pay in month 18 is just one-third of 4,000, so it's $1,333.33. What about the payments from the protection seller? Nothing paid up to month 15, because the default has not happened. Default happens in month 16, in month 18, which is the next coupon date, the seller has to pay (1-R) times N. R was 45%, therefore (1-R) = 55% of N, which is $550,000. This is the total protection payment. Some other names for these payments, we've called them premium payments. We have called them, another name for the same payment is fixed leg or the fixed payments. Because the premiums are fixed except for the accrued interest amount. The name for the protection seller's payments is also sometimes called a contingent leg or the contingent payment. Because it's contingent on a default happened. So the basic model for the CDS cash flow that we're going to be using in this module is what we saw in this example. There will be a fraction delta, which is a fraction of a year, times k would be the times at which the coupon payments are going to happen. Delta typically is one quarter, that its quarterly payments. And the dates of the payments are also set on March 20th, June 20th, September 20th, and December 20th. If the reference entity is not in default at time tk, the buyer pays the premium delta. Which is the fraction times S, which is the spread, times N, which is the notional principal. If the referencing entity defaults at some time tau between tk-1 and tk, the contract terminates at time tk. So in the example, the default time was month 16. This was between the two coupon payments at month 15 and month 18, the contract terminates at time 18. The buyer pays the accrued interest over whatever fraction is left over. So one month was what we did in the numerical example. And the buyer receives, or equivalently the seller pays (1-R) times N, where R denotes the recovery rate of the underlying. We are going to be working with this basic model to price and understand what is going to happen to the CDS sensitivities. But the details behind CDSs are enormous. They have been standardized by the International Swaps and Derivative Association, ISDA, in 1999. There were changes made in 2003, then again changes were made in 2009. And may yet again changes be made if CDSs become exchange traded. And the reason there are so many different details in a CDS contract is there are many difficult issues. How does one define that a credit event has occurred? Was the interest payment just late, did it not occur at all, it's a problem. How does one determine the recovery rate? There's often litigation, there's delays, and so on. So we're not going to be worrying about that in this module, we are going to assume that the recovery rate is somehow known. And we're going to price assuming that this recovery rate is known. And also many, many details, how is the spread set? How is it set for junk bonds versus investment grade bonds? What about countries, how is the spread said for countries? When is the coupon payment done, in advance or in arrears? How is the spread quoted? Is the spread quoted in terms of power spread, meaning the value that makes the net value of the CDS equal to zero, or some other standardized spread? All these details are important when you talk about particular CDS contracts. But in order to understand the basic mechanisms of how CDSs work, the basic model that we have introduced is sufficient, and it highlights all the main features. So we're going to focus on the basic model to illustrate the details of pricing and the sensitivity to hazard rates, which are the probabilities of default.
