This is the second module on pricing defaultable bonds. In this module, we're going to extend, the idea that we had developed in the previous module, to pricing coupon bearing bonds, and also show you how these are done in practice using several bonds and calibration on an Excel spreadsheet. In this module we're going to assume that the hazard rates h i j are state independent. This ensures that the default probability is going to be independent of the interest rate dynamics. It'll be easier for us to keep track of the events by defining a quantity called q t which at the risk mutual probability that the bond survives until date t. A simple recursion defines what q t is going to be. So look at q t plus one, which is a probability that the bond survives up to date t plus 1. This is going to be the probability that the bond survives up to date t, and the conditional probability that the bond survives one extra period which is 1 minus h t. And therefore you can write q t plus 1 as just a product of k going from 0 to t, 1 minus h k. Let I t denote the indicator that the bone survives up to time t. So I t is going to be 1, if the bond is not in default at time t, and its going to be 0 otherwise. Then the indicative variable that the default occurs exactly at time t is going to be the difference between I t minus 1 and I t. I t minus 1 equals 1 and I t equals 0, tells me that exactly that the bond defaulted at time t. From the definition of I t, it immediately follows that the expected value, under the risk-neutral measure of I t is exactly equal to q t. Once we have this indicator val, variable I t, we can define various events using this indicator value, variable. And it's going to be easier for us to keep track of various events that happen to the bond. We're going to assume that the random recovery rate, R tilde, is going to be independent of the interest rate dynamics under the risk neutral measure Q. And we're going to let R, without the tilde, denote the expected value of R tilde under the risk measure, risk neutral measure Q at time 0. Recall that R tilde is a fraction of the face value F 8 on default. Here are the details of the pricing. We are going to assume that the current date is equal to 0. t 1 to t n are the future date at which the coupons are going to be paid. The coupon on date t k is paid only if i t k, the indicator vari, variable that indicates whether the bond is in default or not, is equal to 1, meaning that the bond is not in default. Therefore the random cash flow associated with the coupon payment on date t k is c times I t k. The randomness comes from the fact that although the coupon payment is deterministic, the fact whether the bond is in, not in default or is in default, is going to be a random quantity. The face value F, is going to be paid on date t n only if I t n is equal to 1. Therefore, the random cash flow associated with the face value payment on date t n is going to be a deterministic quantity F times the random quantity I t n. The recovery itself is random, so recovery at date t k is going to be R tilde t k. The random recovery rate times the face value F. But this random quantity is going to be paid only if the default occurs on date t k, which have been indicator vari, variable I t k minus 1 minus I t k. Therefore, the random cash-flow associated with the recovery on date t k is going to be R tilde t k times F times I t k minus 1 minus I t k. Now that we have all the random cash-flows associated with a bond, a defaultable bond with some random recovery, we can now price this bond by just discounting all of these random cash-flows with respect to the risk neutral measure, and that's what we're going to do on the next slide. Let B t denote the value of the cash account at time t, then the price at time 0 of a defaultable fixed coupon bond is given by, the expected value under the risk neutral measure of the random cash-flows discounted by the cash account. So C times I t k is the random cash flow associated with the bond, with the bond coupon payment, it has to, and it occurs at time t k, therefore it has to be discounted by the cash amount at time t k. F times I t n is going to be the random cash flow associated with the face value. It's going to have to be discounted, so there's no b t, but it's just 1. And, at the rate b t n, R tilde t k times F is going to be the random cash-flow associated with the recovery at time t k, so it's discounted by B t k, but this extra variables has the default, of course at time t k. Now, we have, we have assumed that the default is independent, of the interest rate dynamics. So the first expectation, I can split it up into two expectations. This expectation is the expectation of the default, the second expectation is just the expectation with respect to the risk neutral dynamics of the interest rate, or the short rate. And this split, happens, only because I've assumed that the default and the interest rate dynamics are independent. Again, I'm going to split up the next term into two terms. One that corresponds to the default and one the other one that corresponds to the interest rate dynamics. And finally with the same, same thing with the last term, this corresponds to default. And this other one corresponds to the interest rate dynamics. [SOUND]. Now we know from the definition of the indicator function that I expectation under Q sub 0 of I t k is nothing but Q t k. In the expectation under Q sub 0 of I t n is nothing but Q t n. Similarly, this quantity expectation under Q sub 0 of I t k minus 1 is Q t k minus 1, and E sub 0 super Q of I t k is Q t k. So the, all of these are just coming from the definition of the indicator function of the expectations of that indicator function under the risk neutral measure. What happens to these quantities? These are nothing but the prices of the zero coupon bond. So that particular quantity is nothing but the price at time 0 of a 0 coupon bond that pays $1 at time t k. This quantity over here, is the price of the 0 coupon bond at ti, that pays $1 at time t n. And similarly again, this is the same repeat, it's a 0 coupon bond paying $1 at time t k. We can further simplify this exp, expression. This quantity is nothing but the discount rate up to time t k, discount rate up to time t n, and discount rate up to time t k. Here's a formula that tells me what the price is. The only thing that is going into this formula are the discount rates, which are determined by the short rates, and the probabilities of default that are going to be determined by hazard rates. So in principle, if I had the prices of lots of defaultable bonds with fixed coupon payments, I can infer from there what the hazard rates are going to be. So my story is going to be, I'm going to assume that the interest rate is dynam, is deterministic and known. We could have calibrated this and you have done this in another module. To keep the story simple and focus on the hazard rates, we are going to assume that the interest rate dynamics are going to be deterministic. If that is deterministic, then I can write, the model price of a defaultable bond as a function of the hazard rates. I'm going to assume that I have some observed prices of the market bonds and I can use those prices, and compare it with the model price, get an error. So here's the market price for the Ith bond. Here's the model price for the Ith bond. I compare the two, take the square. That is going to be the error that I'm going to be making on the Ith bond. And I take the sum over all possible bonds, that's the total error, and the calibration problem that I'm going to be facing, is to minimize over H F H. And I'm going to show you this, using a numerical example in the associated spreadsheet. So here's a simple spreadsheet that I'm going to be working with. In this sheet, I'm just, I'm going to assume, that the discount rate is given to me. So the discount rate is assumed to be 5% per annul, deterministic. And from that interest rate, I can compute out what the discount rates are going to be. For the six month discount rate, it's just going to be half of this. So if you look at the formula, it's just going to be 1 plus R F divided by 2 times the number of half year periods that have elapsed. So this is simple, this is something that we have done before. Now, in this particular worksheet, I'm going to assume that the hazard rate, which is going to be the six month hazard rate, is fixed at 0.02. What does that mean? So right now, it's time 0. So the survival probability is 1 because the bond exists. Now from this survival probability, I want to compute the survival probability and the default probability in six months from now. The hazard rate of time, 0 is 0.02, so the probability that I default, in the next six months, is simply going to be the probability, survival probability times the default probability. The conditional probability, of default is the hazard rate. The probability that I have survived right now is 1. I take the product of that, that gives me the default probability. What is the probability that I survive, it's the probability that I'm surviving right now times 1 minus H, where H is the hazard rate. That gives the survival probability in six months. What is the survival probability in 12 months? It's, again the same thing, which is 0.98, which is the survival probability at six months times 1 minus the hazard rate. What is the default probabilitiy? It's the surval probability times the hazard rate. So all of this table has been computed using, the survival probabilities and default probabilities form the hazard rate. Now let's see what happens to a coupon. So here is a bond, which is a one year bond so it expires in one year. It me, it has two coupon payments and the face value payment. I'm going to assume here that the face value is 100, and the coupon is 5%, therefore the coupon is 5. The recovery rate is 10%. So what happens? So the coupon and face value payments are going to be 5 in six months and 105 in one year. The recovery is going to be 10% of the face value, so it's going to be $10 if a default occurs in six months. It's going to be another $10 if a default occurs in 12 months or a year. How do I compute what is the expected value of the payments? So if you look at this formula, what's going to happen? If the bond survives in six months, you're going to get the coupon payment. So it's going to be 5 times the survival probability, which is 0.98, plus 10, which is the recovery times the default probability 0.02, and this happens in six months, and therefore you have to discount it back using the discount rate, which is 0.98. Similarly, if you look at this one, it's the same formula again. It's the coupon plus 105, which is going to be paid only if you are going to be surviving at time 12 months or in 1 year. So it's H8 times C8. And, in the case that you default, it's going to be I8 times D8, which is the default probability times the recovery. It has to be discounted back. So now you're going to use the discount rate of 0.5. Sum all of that, you end up getting what the price of this particular bond is going to be. Similarly, if you look at, here is another example of a bond, it's a two year bond with 8% recovery, same story. [SOUND]. This should be 2%, so that coupon payment is 2%. So therefore, here are the coupon payments. Here are the recovery rates. If you look at the formula, it's exactly the same. Coupon payment times the probability of survival, recovery times the probability of default, discounted back to time 0. Sum it all up, and you end up getting what the price of this bond is going to be. Great. So we know how to price bonds, given the calibrate, given the hazard rates. Now I'm going to show you the next spreadsheet, what happens when you calibrate. So here, what I've done, is I have created for you, same bonds as was there on the last sheet. I took the true price of the bond, and to it, I added a small random quantity. I took the true price that was there and then I added about $0.10 of randomness. You can play with this and see what happens when you add more randomness. Here, I'm assuming, just to keep things simple, that the hazard rate, the six month hazard rate of default is going to be constant for a year. So what I've done is that, in the first year, the six month hazard rate is an unknown quantity, but in the next six months it's exactly equal. So if you click on this, I've just made it equal to A6. Similarly, over here I've made it equal to A6, A6 and so on. Once I know the hazard rate, I can compute the survival probability to the default probability, and I can compute the model price. This is the model price that has been computed using whatever these hazard rates are going to be. Now, what I'm going to do, is I'm going to compare the model price with the true price, compute the error. This is nothing but the model price minus the true price squared. I have five different bonds. I'm going to sum up all of those errors. And this is going to be the sum of the 5 errors. Add them up, and then I'm going to minimize it. So, if I use solver, what I'm trying to do is, before I go to solver, let me randomly create some instances here. [SOUND]. So there's this a random instance of what happens to the boat. Now I'm going to go to Solver. And if you look at Solver, all I'm trying to do is, minimize the error of J21. Which is this error quantity here. By changing the variable cells, A6, A8, A10, A12, and A4, and the reason, I left off A16, is because this is the hazard rate that's going to be in the future and it's not going to matter. The A7, A9, A11, A13, and A15 have been left off because, I'm just assume that this is going to be equal over the entire year. And the reason I made that assumption is because I only have bonds that I, that are expiring in the years, and not at six months. And therefore I will not be able to calibrate the next six months of asset rate. So we minimize that. We hit solve. And, it found a solution, and the minimum error it found was 0.01. And, the prices that you end up getting is pretty to what you started of with. It's slightly different, it's 0.0201 instead of 0.002. Here it's just 0.19 and so on. And that happens just because the error is small. Some of the bonds, it's able to compute it correctly, some of the bonds it's it has small errors. And so the overall error turns out to be 0.01.
