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[BLANK_AUDIO] . In this short module, we're going to

discuss the mechanics of a synthetic CDO tranche.

While we haven't actually described for you what a synthetic CDO tranche is yet,

we will actually do that in a later module.

In particular, we will distinguish between a cash CDO and a synthetic CDO.

But for now, we're just going to go into the details of the mechanics, of how a

synthetic CDO tranche actually works. So now let's discuss the mechanics of the

synthetic CDO tranche. I'm going to describe or explain for you,

the distinction between a synthetic and a cash CDO in a later module.

For now we're just going to discuss the details of a synthetic tranche.

As I said, I will distinguished between the synthetic and a cache CDO, in the

later model. So recall there are N credits in the

reference portfolio. Each credit has the same notional amount,

A. If the ith credit defaults, then the

portolio incurs a loss of A times 1 minus r.

Where R is the recovery rate which is assumed fixed, known, and constant across

all credits. A tranche is defined by the lower and

upper attachment points, L and U respectively.

So, we've already seen examples of L and U, L and U, 0 to 3 and so on.

3 to 6, 6 to 9. Usually L and U are given as percentages

of the total portfolio notional amount. In our simple example on the previous two

modules, L and U were given as the number of losses.

0, 1, 2, or 3, 4, 5, or 6, 7, 8 or 9, or so on.

But, typically in practice, they're given as percentages.

The tranche loss function, TL for tranche loss, superscript l and u, to denote the

lower and upper attachment points are parameters of this function.

So its a function of the number of losses in the portfolio L, is a function given

as follows. First of all we take the minimum of LA1

minus R and U. So this, here, is actually the total

portfolio loss. So if the total portfolio loss exceeds U,

then the tranche loss is given by U. After all, the tranche cannot lose more

than U. U is the upper attachment point, it

cannot lose more than U. So if the total portfolio loss exceeds U,

then this minimum is given to us by U. Otherwise the minimum is given by the

total portfolio loss. Now the lower attachment point is L, so

we then have to subtract L from this minimum here.

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And finally we take the maximum of that last quantity and 0, and that gives us

our tranche loss. It tells us for a given number of

defaults, what loss is suffered by the tranche.

So for example, suppose L is 3% and U is 7%, so we've got some sort of CDO as

follows, maybe there's an equity piece here, which is the 0 to 3%, we've got a

mezzanine tranche, which is maybe 3 to 7, and so on.

Well, suppose the losses in the portfolio add up to 5%.

Well, in that case, this piece represents the losses in the portfolio.

The first 3% of losses are incurred by the equity tranche.

But the next 2% of losses fall in here, and they are incurred by this tranche

here, with lower attachment point l equals 3%, and upper attachment point U

equals 7%. So, therefore, the tranche loss is 2%,

and actually, that's 50% of the tranche notional.

The tranche notional is 7% minus 3%, which is 4%.

So, we've incurred 2% losses out of a total maximum loss of 4%.

So, therefore, in this example we have lost 50% of the tranche notional.

When an investor sells protection on the tranche, she's guaranteeing to reimburse

any realized losses on the tranche to the protection buyer.

In return, the protection seller receives a premium, at regular intervals from the

protection buyer. These payments ticky, typically take

place every three months. So, when you see the word protection

here, what you might want to do is think of it as being insurance.

So protection, think of this as insurance.

And what's going on here, is that one person is selling insurance, and the

other person is buying insurance. So the person who's buying insurance, is

insuring against losses, in the underlying portfolio that impact their

given tranche. In return for providing insurance, the

insurance seller receives an insurance premium.

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And they receive that premium at regular intervals, from the protection or

insurance buyer. In some cases, the protection bar may

also make what is called an upfront payment, we're not going to be concerned

about this. But sometimes they might make enough

front payment, in addition to, or instead of the regular payment which might take

place every three months. This is often the case, for example, in

the case of equity tranches which, as I said earlier have a lower attachment

point of zero. The fair value of the CDO tranche, is

that value of the premium plus upfront payment, if applicable, for which the

expected value of the premium leg equals the expected value of the default leg.

So, just like a swap, the initial value of a CDO tranche position is 0.

Now, if this doesn't seem very clear to you yet, that's fine, we're going to see

a diagram of the next page, which will make it clearer still.

And then in the next module, we're going to go through the premium leg and the

default leg in more detail, so that hopefully the two legs of a CDO tranche

position, should become clear to you, and you understand exactly what is going on,

with a CDO. Another point I want to make is the

following. We have already seen the tranche loss

function, we're going to need to compute the expected tranche loss function, nor

did it compute the value of the CDO. We actually already computed this, in our

earlier example, our one period example. We computed the expected tranche loss in

an equity tranche, a mezzanine tranche, and the senior tranche.

Well, the expression we used for that was just this ex, this expression here.

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So, it's equal to the sum from L equals zero to capital N, T L T of L times the

probability of L losses in the portfolio. And remember, we can compute this

probability by conditioning on the random variable M, which makes the default

events of each name independent. So, we're, we were then able to compute

this quantity, either using the iterative.

In that case for a simple example, we saw that this was just a binomial

probability. But either way, we can compute this.

This is the standard normal PDF, so we can compute this quantity using a

numerical integration. So this gives us our expected tranche

loss function. So now let's see what happens visually

with a CDO. We've got a number of periods, so we've

got a period here, here, here and so on. This is the premium leg.

So these are the payments made by the person, or investor, who buys insurance,

or buys protection. S is the annual premium, or spread per

unit of notional. These here, are default events.

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We will see in the next module, that the tranche notional decreases after each

default. So, we, these vertical arrows represent

the size of the payment. Because S is an annual spread, we have to

multiply be delta T, which is the length of an interval.

Typically approximately one quarter, representing payments every three months.

So the, the, the premium per quarter will be delta TS.

It's paid on the notional of the tranche, and in fact it's paid on the outstanding

notional on the beginning of each period. So at this point, there haven't been any

defaults yet, any default's that have impacted to tranche yet.

At this point we have a loss, and this loss impacts the tranche, so this

actually decreases the notional of the tranche, by an amount of 1 minus RA.

and so we get decreased payments, after each default event.

So a second default event occurs here, this default event also impacts the

tranche, and that lowers the outstanding notional in the tranche.

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And so the insurance payments, or the premiums, are payed only on the

outstanding notional in the tranche. As I've said in the previous slide, this

is just a schematic, describing basically what goes on in the CDO.

We have two legs. We have the premium leg, and we have the

default leg. If you like, you can think of this like

being a swap, where we've go the fixed payments and we have the floating

payments. The fixed payments correspond to the

premium leg, they're not always fixed. The rate is fixed, the delta TS, but

they're paid on the outstanding notional. The default lagger like the floating

payments, you're never sure what you're going to get in each period.

Most of the time you'll get nothing, if there hasn't been any default event in

the tranche. But if there has been a default event

that impacts the tranche, you will receive a payment.

You will receive an insurance payment, for that event of 1 minus R times A.

And we will see in the next module, that the way the CDO is priced, or in other

words, the way the S value is calculated. It is calculated by equating the value of

the premium leg with the default leg. And so the initial value of an investment

in a CDO tranche would be zero, and we're going to find what value of S makes that

value equal to zero.