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.
