Mathematical Modeling in Finance with Stochastic Processes

72 CHAPTER 1. BACKGROUND IDEAS

Problems to Work for Understanding

1. Suppose that there is a 20% decrease in the default rate from 5% to
   4%. By what factor do the default rates of the 10-tranches and the
   derived 10th CDO change?


2. For the tranches create a table of probabilities of default for tranches
   i= 5 toi= 15 for probabilities of defaultp= 0.03, 0.04, 0.05 0.06 and
   0 .07 and determine where the tranches become safer investments than
   the individual mortgages on which they are based.


3. For a base mortgage default rate of 5%, draw the graph of the default
   rate of the tranches as a function of the tranche number.


4. The text asserts that the expected payout from the collection of tranches
   will be

E[U] =

∑^100

n=0

∑n

j=0

(

100

j

)

pj(1−p)^100 −j=

∑^100

j=0

j

(

100

j

)

pj(1−p)^100 −j= 100p.

That is, the expected payout from the collection of tranches is exactly

the same the expected payout from the original collection of mortgages.

More generally, show that

∑N

n=0

∑n

j=0

aj=

∑N

j=0

j·aj.

Outside Readings and Links:

1. Wall Street Journal.com : The Making of a Mortgage CDO An an-
   imated graphic explanation from the Wall Street Journal describing
   mortgage backed debt obligations.


2. Portfolio.com: What’s a CDO Another animated graphic explanation
   from Portfolio.com describing mortgage backed debt obligations.