Therefore, conditional on knowing the value M, the total number of defaults is

actually binomially distributed with parameters n, the number of names in the

portfolio, the number of credits, the number of bonds and probability q M.

Why is that? Well if you think about it, conditional

on knowing M, we've already seen what's the probability of default is, it's q M.

But also conditional on knowing M, all of these default events become independent.

Because X i depends on M and Z i, and Z i, the Z i's are IID normal random

variables. So, if I condition on the value of M,

then the X i's become independent, which in particular means that the default

events of all of the names in the portfolio are also independent.

The probability of default conditional on M is q M.

So therefore, the total number of defaults in the portfolio condition on M

has a binomial distribution with parameters of M and q M.

So therefore the probability of l defaults, and note I've dropped the

dependence on t here which is fine. Because we're just considering one period

with t equals one year. So therefore the probability of l

defaults, out of the n names in the portfolio.

Or n credits in the portfolio, conditional on the value of M is

binomially distributed. And this is equal to N choose l times q M

to the power of l, times 1 minus q M to the power of N minus l.

Where phi we said in the previous slide is the standard normal CDF, and we know

this expression for q M as well. Recall the q the risk neutral probability

of a single name default within one year. So just be clear.

You shouldn't get confused between q and q M.

Q is the risk-neutral probability of a single name defaulting within one year.

And qM is the same probability that is the risk neutral probability of a single

name defaulting within one year, conditional on the value of M.

So here are two questions. The first question is the following, when

correlations on default probabilities are not identical, why is four no longer

valid? Well, that's because, if the correlations

an default probabilities are not identical, then these qM's are not the

same for all of the names in the portfolio.

You end up having a different qM, for each of the N names.

So this quantity here is no longer a binomial.

Random variable or a binomial probability.

Next question, how do we calculate P superscript N l M in that case.

Well, we'll just do it as we did in the last module, we saw that we had an

iterative algorithm, remember we had the two nested for loops.

That was the way, how we calculate P N of l in general.

In this example, we don't need to run through those two for loops, because here

we can just use the fact that the qMs are identical for all names I equals 1 to

capital N. And therefore this is a binomial

probability and we know this expression. [SOUND] So at this point, in this

example, we actually know this quantity here.

This quantity is a binomial probability. Assuming we can calculate q M, and we can

because q M can be calculated as this expression here.

We know these binomial probabilities, and so in particular we know the risk neutral

probability of l defaults occurring conditional on the random variable M.

How can we use this knowledge? Well, we can use it to compute the

expected tranche losses. And that's what we're going to do in this

slide. First of all, maybe I should go down to

equation five first. So remember, what we're going to say is

the following. The probability of l losses.