In this second module on credit default swaps we're going to introduce you to a very simple pricing formula for credit default swaps, and then using this simple formula I'm going to show you how you can price CDSs on a spreadsheet. In this module, we are going to start with a very simple CDS pricing formula. The value of the CDS to a buyer is going to be the risk neutral value of the protection minus the risk neutral value of the premiums. We're going to assume that the default event is uniformly distributed over the premium interval delta. So let's walk through each of the pieces that go into constructing the value for the CDS. The risk-neutral value for single-premium payment on date t k, is going to be delta times S. Delta is the period, times S, which is the spread, times N, which is a notional amount, times the expected value under the risk-neutral measure at time zero, of I t k divided by B t k. Recall. I t k is the prop, is the indicative function that the entity does not default, is not in default. And B t k is a cash account at time t k. So this going to be here simply is a discounted value of the event that there is no default up to time t k, discounted back to time 0. Using the cash account B t k. Again we're going to assume as we have done in the modules corresponding to defaultable bonds that I t k and B t k are independently distributed, and therefore I can take the expectation separately. If I take the expectation separately I'll end up getting Q t k which is the time zero probability. That the bond doesn't default up to time t k. This quantity which is Z t k 0 is simply the expectation or under the risk neutral measure of 1 over B t k, which is nothing but the price of a zero-coupon bond which pays $1 at time t k. This can further be simplified as the discount rate. Up to time t k. So what's the risk neutral value of all the premium payments? Just sum this overall times k, that's k going from one through n, delta times S time N, which is the coupon payment, this is the probability that this coupon has to be paid, this is the discounted value of that quantity. So the expected discounted value of the coupons is what the coupon payments are going to be. What about the accrued interest. We've got to assume, that if t k is here, and this dot, some default time tau happens, it's uniformly distributed over this. So the value of coupon that you're going to be paying, would be approximately half. So that's what this quantity here is. About, this is about, approximately half the coupon. And this comes from the fact that we are assuming that the default is uniformly distributed over the interval t k minus 1 to t k. Now, what is the probability that the default occurs at time t k? It's I t k minus 1 minus I t k, divided by B t k to discount things over, again. Just taking the expectation and using the fact that the default is going to be independent of the interest rate dynamics. We end up getting that this quantity is going to be delta S N divided by 2 q t k minus 1 minus q t k. This is the probabilities of the two indicators times Z t k 0 which is the price of a zero-coupon bond. Which pays $1 at time t k. We can dis, we can simplify this further and write this quantity as a discount phase. If you add up the two quantities, this tell you the risk-neutral value of the premium and the accut, accrued interests can be approximated and the approximation here comes from the fact that we're approximating this uniform distribution. Is also another subtle approximation which I don't want to go in too much detail and that comes from the fact. That if you look at the, hazard rates, those are going to be going monotonically down, so even if you assume that the period, that it's going to be in the interval is going to be uniform, the probabilities are going to be slightly different. We're going to approximate all of that and assume that it's sort of a flat probability, in that interval. Okay, to sum that up, you get an expression, and we're going to be using this expression in the next page to try to figure out what the par value is going to be. What is the risk-neutral present value of the protection or the contingent payment? It's 1 minus R times N. This is going to be the amount of protection that you have to be paid. It's another subtle amount here. We're trying to price the CDS. We're going, but in the pricing we're assuming that the R is known. But really R gets known only on default. So in some sense we're making an implicit assumption that these CDSs have been around, and so we have a good idea what the expected recovery is going to be on particular CDS that we are going to be pricing. What is I t k minus 1 minus I t k, this is the event that a default happens at time t k, and, times B t k which is a discount that I have to do in order to bring the quantities back to time zero. Again I've gone through two steps. We can first write it as the price of zero-coupon bond or you can directly go to the discount and I'm directly going to the discount here. That is the quantity that is going to be the contingent payment or the protection payment. So S par which is the par spread is defined to be the spread that makes the value of the contract exactly equal to 0, you compare this term with that term. This term involves S and solve for S and you end up getting to the solve for, the value, the notional amount goes away. It's 1 minus R. This is protection. Down here it's premium plus accrued interest. If you assume that the default probabilities remain sort of flat over the entire premium interval so q t k is going to be 1 minus some hazard rate h times q t k minus 1. Then you can approximate that power spread to be 1 minus R times h divided by 1 minus h over 2. And, if you recall, back in the module, the first module in CDS we had said that this is approximately equal to 1 minus R times h, and this is typically because h is assumed to be pretty close to zero. When h starts becoming pretty close to 1, the approximation is not valid and one has to use a better approximation. As you would intuitively see this is increasing in the hazard rate h and decreasing in the recovery rate R. In the rest of this module, I'm going to show you this pricing using a spreadsheet. So in this spreadsheet we're going to price the hypothetical two year CDS that we had introduces in the module. The principal amount for the hypothetical CDS was $1 million. The recovery was set as 45%. So 1 minus R which is the amount that we are going to have to pay is 55% Arbitrarily I'm just setting the interest rate here to be 1% per annum. So using that interest rate, I can computer out the discount value, which is right here. So it's just going to be, the quarterly discount is going to be R divided by 4 times the month count divided by 3. So that, this interest rate directly gives me the discount rate. The hazard rate. I took it from the calibration worksheet that we worked with for the bonds. So, there, we computed the hazard rate for a six month default probability. Here, we are talking about three month default probability. So I just took that value, and divided it by 2. How did I compute the survival probabilities? I took the same formula. It's going to be the survival probability 1 times step before times 1 minus the hazard rate. The only difference is before I was using survival probabilities in absolute numbers, here I'm looking at them at percentages. What about the fixed payments, which are the fixed coupon payments that. The buyer has to make. This is simply going to be the quantity of the spread divided by 4, and I've left off the notional amount, and I'm going to look at the notional amount in a moment. So that's the fixed payment that's going to happen. Now in this column, row F, what we have done is we have taken the fixed payments and found an expected value, the expected value which takes into account the probability of default. So that's e7 times d7, divided by 100. And this 100 just comes because the probability of survival is written as percentages. So, the values in this column simply reflect the fact that fixed coupon payment has to be paid only if the reference entity survives up to that time. This is the present value. We have taken the expected value, multiplied it by the notional amount, and because, these spreads are in basis points, I multiplied it by 0.0001 to covert into absolute numbers. And multiplied it by b7, which is the discount value to get the present value. Of the coupon payments. Done this for all time periods. That tells me what the total fixed coupon payments are. But we still have to figure out what the accrued interest is. And, in order to compute the accrued interest, we need the default probabilities. So here's the default probability. This default probability is just a survival probability. Times the hazard rate, and I put that along this column all the way through. What is the accrued interest? If you click on that, it's we've assumed it to be half of what the coupon payment is going to be, because we have assumed that this is going to be half way between. Times the default probability again divided by 100 to convert the default probability, which is in percentages, into absolute numbers. What is the present value? You take the notional amount, N, which is of $1,000,000 times the accrued interest, times the discount times again 0.0001 to convert the basis points into. absolute numbers. So this is going to be the present value of the accrued interest, in different periods. Finally, what about the protection. What is the expected value of the protection, it's the default probability times 1 minus R divided by 100. The 100 again to take the default probabilities and convert them into absolute numbers. What is the present value of the protection? You multiply by the notional amount, multiply by the discount rate, that tells me what is the present value of the protection for the different time periods. So the premium leg now is going to be the sum of the present value of all the premiums and sum of all the present value of the accrued interest. And that tells you what the buyer has to pay. What the buyer will receive is just the sum of all the expected present values of the protection payments. This is the net value of the swap at time zero. The difference between the premium leg and the protection leg. So, since the value of the CDS is positive. It, it suggests that this spread is set too low. it's a good deal for the buyer. The buyer is getting protection at too lower rate. If you increase that spread, to say from 218 to 220 basis points, then you end up getting that that spread is too high, you end up getting that the difference between the premium leg and the protection leg is negative. Which means that it's not been, it's not a good deal for the buyer. This spread has been set too high. You can compute out what the correct spread is going to be. So we're going to set the objective which is the value and try to solve for the value equal to zero by changing the variable cells S, it's just a reference cell. And I'm assuming that it's going to be positives. If you solve that and you wait for Solver you end up getting that the spread is going to 218.89 which is slightly smaller then then spread that we had started with. And this brings us to the end of this module.
