0:02

In this module,

we're going to continue on with the example we introduced in the last example.

That was the simple model, it was a one period model with identical risk mutual

default probabilities for each bond in the reference portfolio, and so on.

So what we're going to do here is we're going to look at some of the tranche

losses or the expected tranche losses.

And we're going to look at the characteristics of these expected tranche

losses as a function of the correlation parameter rho and so on.

We're also going to see how the total expected losses in a portfolio

does not depend on the correlation parameter rho.

0:34

So recall the details for our simple example.

We want to find the expected losses in a CDO, or in CDO tranches,

with the following characteristics.

The maturity is 1 year, and

it's just a 1-Period CDO, there are 125 bonds in the reference portfolio.

Each bond pays a coupon of one unit after one year, if it has not defaulted.

And the recovery rate on each bond is zero.

So in particular, if a bond has defaulted before the one year is up,

then that bond will pay a coupon of zero.

There are three tranches of interest, the equity, mezzanine, and senior tranches.

1:09

The equity tranche is exposed to defaults 0 to 3,

the mezzanine tranche is exposed to defaults 4 to 6, and

the senior tranche is exposed to defaults 7 to 9.

We saw in the last module that we could

actually compute the expected tranche losses.

We saw our expressions for the expected loss in the equity tranche, the expected

loss in the mezzanine tranche, and the expected loss in the senior tranche.

So, on this slide, we're actually going to see what these expected losses are as

a function of rho, which is the common pairwise correlation between the various

name in the portfolio, and the individual risk mutual default probability.

So this is the q = 1%.

So this corresponds to the case where the probability of each individual

name defaulting is 1%.

So there's 125 bonds in the portfolio,

we assume each of them defaults would probability equal to 1% here.

Over here, each of them default to probability q=2%,

down here, 4%, and over here, 3%.

And on the y-axis we have the expected losses.

Note, we have 125 names in the portfolio.

Each name will pay one unit, if it hasn't defaulted after one year.

So therefore, the total portfolio notional is 125.

The equity tranche is on the hook for the first three losses, so

the equity tranche can lose a maximum of three.

The mezzanine tranche is on the hook for losses 4, 5, and 6.

So the maximum the mezzanine tranche can lose is also three.

Likewise, the maximum the senior tranche can lose is also a three.

It's on the hook for losses 7, 8, and 9.

So the maximum loss in any of these tranches is three, and that's why you

don't see in any of these cases anything greater than three on the y axis.

Across the x -xis, we're actually plotting

these expected losses as a function of the correlation parameter rho.

So this is that value rho we saw in the last module,

which defines the normalized asset value, xi.

3:22

So there are some important observations we should make here.

The first observation is the following.

Regardless of the individual default probability, q, and correlation parameter,

rho, we see that the expected equity tranche loss is greater than or

equal to the expected mezzanine tranche loss, is greater than or

equal to the expected senior tranche loss.

Now this only holds when each tranche has the same notion exposure,

in this case three units.

So we see it here.

We see the blue line,

which is the expected tranche losses in the equity tranche, is always greater or

equal to the red line, which corresponds to the mezzanine tranche,

which in turn is always greater or equal to the senior tranche, expected losses.

And that makes sense.

After all, we can see that the equity tranche is the riskiest.

You can only lose, the mezzanine tranche can only lose,

if the equity tranche has already lost everything.

In other words, if you have four defaults, the mezzanine tranche loses one unit,

then that means we've had more than three losses, and

the equity tranche has lost everything.

4:22

Similarly, if the senior tranche is to lose anything, than that can only be

the case if we have seven or more losses, which means the equity and

mezzanine tranche have already lost everything as well.

And so we should be absolutely certain and

understand why the expected losses in the equity tranche must be greater than or

equal to the expected losses in the mezzanine tranche, must be greater than or

equal to the expected losses in the senior tranche.

4:48

Another important observation is that the expected losses in the equity

tranche are always decreasing in this correlation parameter, rho.

So we see it here.

So let's pick q=2% for example.

If we look a q=2%,

we can see that the equity tranche losses is a decreasing function of rho.

And in fact it's true in all four graphs, it's always a decreasing function of rho.

Why might this be the case?

Well, one easy way to see this, perhaps, is the following.

Imagine two possibly different values of rhos,

so imagine rho being equal to 0, or rho being equal to 1.

Well, if rho is equal to 1, well, then what happens, either all of the names,

all of the credits default together, or none of them default together.

5:42

So in that case, the equity tranche will actually be as risky as the senior

tranche, because either all of the names default or none of them default.

The probability of all of them defaulting will therefore just be

q = 1% in the 1% case, or 2 in the 2% case, and so on.

So in other words, let's considered this example here.

So the q=1% case, for numerical reasons we didn't take the value of rho all the way

equal to 1, but we can look at the value rho = 0.99 to see what's going on.

In these case, either all the names default together or none of them default.

In that case, the three tranches are all equally risky, and

so the probability of any one defaulting is 1%.

So the expected losses in the tranche is going to be 1% times 3, which is 0.03.

So this value here is roughly 0.03.

On the other hand, if rho equals zero, well because there's no correlation

among default events, we'll always expect there to be maybe just one default or

one or two defaults or zero defaults.

But what it means is that most of the time we're actually going to see a default,

maybe one, maybe zero, but sometimes two or three.

And so with such a low correlation,

we'll always expect to see some credit event happening.

And because it's the equity tranche that is on the hook for that first credit

event, we expect the equity tranche to incur losses most of the time.

And that's why we see this number being much higher for a low value of rho.

And in fact we'll see this behavior for all values of q.

Down here for example, we see q = 4%.

And we see the expected tranche loss is now almost 3,

and that's because we expect with q = 4%,

we expect there to be 4% of 125, which is 5.

So we expect to see on average five losses in the portfolio.

Because correlation is very low down here,

we're always going to expect to see almost five losses in the portfolio,

which means that most of the time the equity tranche will be wiped out.

We're going to see more than three losses.

Most of them we're going to see four or five losses maybe.

7:59

So, that's why it will be wiped out.

Down here, for example, when rho equals 0.99,

sure, we do on average expect to see 4% losses,

which corresponds to 5 port-names defaulting.

But they're all going to default together or not at all.

Which means 96% of the time, in fact, the equity tranche won't be hit at all,

and only 4% of the time will it see a loss of 3.

4% of 3 is 0.12, so this number here is roughly at the 0.12 level.

So that's the first two observations.

Another observation to note is the following,

mezzanine tranches are often relatively insensitive to rho.

We can see that, for example, perhaps most easily in this plot, when q = 1%.

Which in many cases in practice might be the most relevant example,

because in practice, depending of course on the nature of the names in

the portfolio, you will often see a q of approximately 1%.

So you see here that actually certainly maybe in this range

here that I am circling,

9:22

Also, it actually has implications when it comes to model calibration.

Now we're not going to get into model calibration for

this Gaussian Copula model, I might say a few words about it in a later module,

but generally we won't have time to go into it in any detail.

So that's the third observation here.

So I have two more observations I want to make, but

I'm going to make those observations when we change the problem slightly.

In the remainder of this module, we're going to assume that the senior

tranche is now on the hook for all losses between 7 and 125.

So, it's not between 7 and 9, it's all losses between 7 and 125.

What we will see then, and we'll see it on the next slide,

is that the expected loss in this senior tranche,

sometimes called a super senior tranche, will see that it's increasing in rho.

And sure enough we can see that.

So this is the green here.

We see it is always increasing in rho, regardless of the value of q.

And actually the reason for this is the same reason why

the equity tranche expected losses are always decreasing in rho.

So the same intuition I gave will actually also apply in the opposite direction

to the expected losses in the senior tranche.

10:50

But the total expected losses on the three tranches,

that is, the expected losses on the index.

Remember, all of the index, now the index consists of all 125 names.

And the three tranches are equity, 0 to 3,

the mezzanine tranche, 3 to 6, and now the senior or

super senior tranche is 7 to 125.

So now these three tranches now all add up to entire index, all 125 names.

And the point I'm making, and you can see it on the next slide,

is that the total expected losses on the three tranches is independent of rho.

This is not an accident.

So what am I getting at here?

Here's what I'm getting at.

Let's pick this figure here q=3.

If I fix a value of rho, say rho equals 0.1,

and sum of the losses for the three tranches at rho equals 0.1,

I'll be summing up maybe this number plus this number, plus this number.

And that will equal to the losses at any other value of rho.

So if I take rho equals 0.7, then this, this and this number here,

the sum of these three, will be equal to the sum of these three numbers over here.

12:18

We can show that this is no accident as follows.

The expected losses, the expected losses using the risk mutual probabilities,

so the expected total loss, Is

equal to the expected value of the sum of the losses

from i equals 1 to 125 of the indicator

function that the ith bonder name defaults.

I'll just use the capital letter D and E F to denote default.

Now, I would normally multiply this by Ai(1-Ri), but,

14:04

And, the important thing to notice here, is that this,

the probability of the ith bond defaulting, well, this is equal to our q,

but this certainly does not depend, On rho.

14:38

However, it is worth pointing out that the allocation of these losses to

the three tranches, the three separate tranches, does indeed depend on rho.

We can see, as before, that the expected tranche losses in the equity piece

are decreasing in rho, this will always be the case for

an equity tranche with lower attachment point equal to zero.

Similarly, we can see that the expected tranche losses

in the senior tranche are always increasing in rho.

And this is true for any senior tranche with an upward attachment point of 100%,

or in this case 125 units.