Problem 1
We consider in this problem a mixed binomial setting for modeling the losses
of a loan portfolio. We are interested in comparing two different mixed
binomial models. In model 1, the random default probability of each loan
is given as a random variable ˜p1 which can take values 0, 0.01, 0.02, . . . , 0.10
with probabilities given as
P(˜p1 = 0.01i) =
10
i
!
0.6
i
· 0.4
10−i
for i = 0, 1, . . . , 10.
Another way of expressing ˜p1’s distribution is to say that it is equal to that
of 1
100B˜
1 where B˜
1 is a binomial distribution with N = 10 and p = 0.6.
Similarly, in model 2 we assume that the random default probability ˜p2 is
given as another scaled binomial distribution:
P(˜p2 = 0.01i) =
20
i
!
0.3
i
· 0.7
20−i
for i = 0, 1, . . . , 20,
which we recognize as being equal to that of 1
100B˜
2 where B˜
2 is a binomial
distribution with N = 20 and p = 0.3.
1. Compute the mean and variance of ˜p1 and ˜p2.
2. In a portfolio of 50 loans, compute the expected number of defaults and
the variance of the number of defaults under both models.

Now assume that the loan portfolio considered is very large, i.e., consists of
a very large number of small loans of equal size, such that both in model
1 and in model 2 the loss fraction ( DN
N
in the notation of the notes) is well
approximated with the distribution of ˜p1 and ˜p2, respectively.
3. Under the two models, what is the probability that the fraction of loans
that default is smaller than 1.5%?

4. Under the two models, what is the probability that a fraction larger than
9.5% defaults?
5. Assume that we construct a CDO with the portfolio of loans as collateral,
and we design a senior tranche which repays in full as long as the loss
fraction is below or equal to 7%. but which takes losses thereafter. Which
model would you expect to assign the highest expected pay-off to the
senior tranche? Explain your answer.

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20210315160736problemset2__finance