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This is the second module on pricing defaultable bonds.

Â In this module, we're going to extend, the idea that we had developed in the

Â previous module, to pricing coupon bearing bonds, and also show you how

Â these are done in practice using several bonds and calibration on an Excel

Â spreadsheet. In this module we're going to assume that

Â the hazard rates h i j are state independent.

Â This ensures that the default probability is going to be independent of the

Â interest rate dynamics. It'll be easier for us to keep track of

Â the events by defining a quantity called q t which at the risk mutual probability

Â that the bond survives until date t. A simple recursion defines what q t is

Â going to be. So look at q t plus one, which is a

Â probability that the bond survives up to date t plus 1.

Â This is going to be the probability that the bond survives up to date t, and the

Â conditional probability that the bond survives one extra period which is 1

Â minus h t. And therefore you can write q t plus 1 as

Â just a product of k going from 0 to t, 1 minus h k.

Â Let I t denote the indicator that the bone survives up to time t.

Â So I t is going to be 1, if the bond is not in default at time t, and its going

Â to be 0 otherwise. Then the indicative variable that the

Â default occurs exactly at time t is going to be the difference between I t minus 1

Â and I t. I t minus 1 equals 1 and I t equals 0,

Â tells me that exactly that the bond defaulted at time t.

Â From the definition of I t, it immediately follows that the expected

Â value, under the risk-neutral measure of I t is exactly equal to q t.

Â Once we have this indicator val, variable I t, we can define various events using

Â this indicator value, variable. And it's going to be easier for us to

Â keep track of various events that happen to the bond.

Â We're going to assume that the random recovery rate, R tilde, is going to be

Â independent of the interest rate dynamics under the risk neutral measure Q.

Â And we're going to let R, without the tilde, denote the expected value of R

Â tilde under the risk measure, risk neutral measure Q at time 0.

Â Recall that R tilde is a fraction of the face value F 8 on default.

Â Here are the details of the pricing. We are going to assume that the current

Â date is equal to 0. t 1 to t n are the future date at which

Â the coupons are going to be paid. The coupon on date t k is paid only if i

Â t k, the indicator vari, variable that indicates whether the bond is in default

Â or not, is equal to 1, meaning that the bond is not in default.

Â Therefore the random cash flow associated with the coupon payment on date t k is c

Â times I t k. The randomness comes from the fact that

Â although the coupon payment is deterministic, the fact whether the bond

Â is in, not in default or is in default, is going to be a random quantity.

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But this random quantity is going to be paid only if the default occurs on date t

Â k, which have been indicator vari, variable I t k minus 1 minus I t k.

Â Therefore, the random cash-flow associated with the recovery on date t k

Â is going to be R tilde t k times F times I t k minus 1 minus I t k.

Â Now that we have all the random cash-flows associated with a bond, a

Â defaultable bond with some random recovery, we can now price this bond by

Â just discounting all of these random cash-flows with respect to the risk

Â neutral measure, and that's what we're going to do on the next slide.

Â Let B t denote the value of the cash account at time t, then the price at time

Â 0 of a defaultable fixed coupon bond is given by, the expected value under the

Â risk neutral measure of the random cash-flows discounted by the cash

Â account. So C times I t k is the random cash flow

Â associated with the bond, with the bond coupon payment, it has to, and it occurs

Â at time t k, therefore it has to be discounted by the cash amount at time t

Â k. F times I t n is going to be the random

Â cash flow associated with the face value. It's going to have to be discounted, so

Â there's no b t, but it's just 1. And, at the rate b t n, R tilde t k times

Â F is going to be the random cash-flow associated with the recovery at time t k,

Â so it's discounted by B t k, but this extra variables has the default, of

Â course at time t k. Now, we have, we have assumed that the

Â default is independent, of the interest rate dynamics.

Â So the first expectation, I can split it up into two expectations.

Â This expectation is the expectation of the default, the second expectation is

Â just the expectation with respect to the risk neutral dynamics of the interest

Â rate, or the short rate. And this split, happens, only because

Â I've assumed that the default and the interest rate dynamics are independent.

Â Again, I'm going to split up the next term into two terms.

Â One that corresponds to the default and one the other one that corresponds to the

Â interest rate dynamics. And finally with the same, same thing

Â with the last term, this corresponds to default.

Â And this other one corresponds to the interest rate dynamics.

Â [SOUND]. Now we know from the definition of the

Â indicator function that I expectation under Q sub 0 of I t k is nothing but Q t

Â k. In the expectation under Q sub 0 of I t n

Â is nothing but Q t n. Similarly, this quantity expectation

Â 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

Â Q t k. So the, all of these are just coming from

Â the definition of the indicator function of the expectations of that indicator

Â function under the risk neutral measure. What happens to these quantities?

Â These are nothing but the prices of the zero coupon bond.

Â So that particular quantity is nothing but the price at time 0 of a 0 coupon

Â bond that pays $1 at time t k. This quantity over here, is the price of

Â the 0 coupon bond at ti, that pays $1 at time t n.

Â And similarly again, this is the same repeat, it's a 0 coupon bond paying $1 at

Â time t k. We can further simplify this exp,

Â expression. This quantity is nothing but the discount

Â rate up to time t k, discount rate up to time t n, and discount rate up to time t

Â k. Here's a formula that tells me what the

Â price is. The only thing that is going into this

Â formula are the discount rates, which are determined by the short rates, and the

Â probabilities of default that are going to be determined by hazard rates.

Â So in principle, if I had the prices of lots of defaultable bonds with fixed

Â coupon payments, I can infer from there what the hazard rates are going to be.

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So my story is going to be, I'm going to assume that the interest rate is dynam,

Â is deterministic and known. We could have calibrated this and you

Â have done this in another module. To keep the story simple and focus on the

Â hazard rates, we are going to assume that the interest rate dynamics are going to

Â be deterministic. If that is deterministic, then I can

Â write, the model price of a defaultable bond as a function of the hazard rates.

Â I'm going to assume that I have some observed prices of the market bonds and I

Â can use those prices, and compare it with the model price, get an error.

Â So here's the market price for the Ith bond.

Â Here's the model price for the Ith bond. I compare the two, take the square.

Â That is going to be the error that I'm going to be making on the Ith bond.

Â And I take the sum over all possible bonds, that's the total error, and the

Â calibration problem that I'm going to be facing, is to minimize over H F H.

Â And I'm going to show you this, using a numerical example in the associated

Â spreadsheet. So here's a simple spreadsheet that I'm

Â going to be working with. In this sheet, I'm just, I'm going to

Â assume, that the discount rate is given to me.

Â So the discount rate is assumed to be 5% per annul, deterministic.

Â And from that interest rate, I can compute out what the discount rates are

Â going to be. For the six month discount rate, it's

Â just going to be half of this. So if you look at the formula, it's just

Â going to be 1 plus R F divided by 2 times the number of half year periods that have

Â elapsed. So this is simple, this is something that

Â we have done before. Now, in this particular worksheet, I'm

Â going to assume that the hazard rate, which is going to be the six month hazard

Â rate, is fixed at 0.02. What does that mean?

Â So right now, it's time 0. So the survival probability is 1 because

Â the bond exists. Now from this survival probability, I

Â want to compute the survival probability and the default probability in six months

Â from now. The hazard rate of time, 0 is 0.02, so

Â the probability that I default, in the next six months, is simply going to be

Â the probability, survival probability times the default probability.

Â The conditional probability, of default is the hazard rate.

Â The probability that I have survived right now is 1.

Â I take the product of that, that gives me the default probability.

Â What is the probability that I survive, it's the probability that I'm surviving

Â right now times 1 minus H, where H is the hazard rate.

Â That gives the survival probability in six months.

Â What is the survival probability in 12 months?

Â It's, again the same thing, which is 0.98, which is the survival probability

Â at six months times 1 minus the hazard rate.

Â What is the default probabilitiy? It's the surval probability times the

Â hazard rate. So all of this table has been computed

Â using, the survival probabilities and default probabilities form the hazard

Â rate. Now let's see what happens to a coupon.

Â So here is a bond, which is a one year bond so it expires in one year.

Â It me, it has two coupon payments and the face value payment.

Â I'm going to assume here that the face value is 100, and the coupon is 5%,

Â therefore the coupon is 5. The recovery rate is 10%.

Â So what happens? So the coupon and face value payments are

Â going to be 5 in six months and 105 in one year.

Â The recovery is going to be 10% of the face value, so it's going to be $10 if a

Â default occurs in six months. It's going to be another $10 if a default

Â occurs in 12 months or a year. How do I compute what is the expected

Â value of the payments? So if you look at this formula, what's

Â going to happen? If the bond survives in six months,

Â you're going to get the coupon payment. So it's going to be 5 times the survival

Â probability, which is 0.98, plus 10, which is the recovery times the default

Â probability 0.02, and this happens in six months, and therefore you have to

Â discount it back using the discount rate, which is 0.98.

Â Similarly, if you look at this one, it's the same formula again.

Â It's the coupon plus 105, which is going to be paid only if you are going to be

Â surviving at time 12 months or in 1 year. So it's H8 times C8.

Â And, in the case that you default, it's going to be I8 times D8, which is the

Â default probability times the recovery. It has to be discounted back.

Â So now you're going to use the discount rate of 0.5.

Â Sum all of that, you end up getting what the price of this particular bond is

Â going to be. Similarly, if you look at, here is

Â another example of a bond, it's a two year bond with 8% recovery, same story.

Â [SOUND]. This should be 2%, so that coupon payment

Â is 2%. So therefore, here are the coupon

Â payments. Here are the recovery rates.

Â If you look at the formula, it's exactly the same.

Â Coupon payment times the probability of survival, recovery times the probability

Â of default, discounted back to time 0. Sum it all up, and you end up getting

Â what the price of this bond is going to be.

Â Great. So we know how to price bonds, given the

Â calibrate, given the hazard rates. Now I'm going to show you the next

Â spreadsheet, what happens when you calibrate.

Â So here, what I've done, is I have created for you, same bonds as was there

Â on the last sheet. I took the true price of the bond, and to

Â it, I added a small random quantity. I took the true price that was there and

Â then I added about $0.10 of randomness. You can play with this and see what

Â happens when you add more randomness. Here, I'm assuming, just to keep things

Â simple, that the hazard rate, the six month hazard rate of default is going to

Â be constant for a year. So what I've done is that, in the first

Â year, the six month hazard rate is an unknown quantity, but in the next six

Â months it's exactly equal. So if you click on this, I've just made

Â it equal to A6. Similarly, over here I've made it equal

Â to A6, A6 and so on. Once I know the hazard rate, I can

Â compute the survival probability to the default probability, and I can compute

Â the model price. This is the model price that has been

Â computed using whatever these hazard rates are going to be.

Â Now, what I'm going to do, is I'm going to compare the model price with the true

Â price, compute the error. This is nothing but the model price minus

Â the true price squared. I have five different bonds.

Â I'm going to sum up all of those errors. And this is going to be the sum of the 5

Â errors. Add them up, and then I'm going to

Â minimize it. So, if I use solver, what I'm trying to

Â do is, before I go to solver, let me randomly create some instances here.

Â [SOUND]. So there's this a random instance of what

Â happens to the boat. Now I'm going to go to Solver.

Â And if you look at Solver, all I'm trying to do is, minimize the error of J21.

Â Which is this error quantity here. By changing the variable cells, A6, A8,

Â A10, A12, and A4, and the reason, I left off A16, is because this is the hazard

Â rate that's going to be in the future and it's not going to matter.

Â The A7, A9, A11, A13, and A15 have been left off because, I'm just assume that

Â this is going to be equal over the entire year.

Â And the reason I made that assumption is because I only have bonds that I, that

Â are expiring in the years, and not at six months.

Â And therefore I will not be able to calibrate the next six months of asset

Â rate. So we minimize that.

Â We hit solve. And, it found a solution, and the minimum

Â error it found was 0.01. And, the prices that you end up getting

Â is pretty to what you started of with. It's slightly different, it's 0.0201

Â instead of 0.002. Here it's just 0.19 and so on.

Â And that happens just because the error is small.

Â Some of the bonds, it's able to compute it correctly, some of the bonds it's it

Â has small errors. And so the overall error turns out to be

Â 0.01.

Â