Quiz

1) In how many different orders can five runners finish a race?
A: 5! = 120

2) How many possibilities are there for the win, place, and show (first,
second and third) positions in a race with 12 horses?
A: 12*11*10 = 1320

3) A pair of dice is loaded. The probability that a 4 appears on the first die
is 2/7 and the probability that a 3 appears on the second die is 2/7. Other
outcomes for each die appear with probability 1/7. What is the probability of
7 appearing as the sum of the two dice?
A: p(d1+d2=7) = p(d1=1)*p(d2=6) + ...+ p(d1=6)*p(d2=1) = 5*1/7*1/7 + 2/7*2/7 = 9/49

4) Write the equation for the straight line passing through the points (5,7)
and (7,15).
A: y = ax + b ==> 7 = 5a + b and 15 = 7a + b ==> a=4 and b=-13
so the line equation is y = 4x -13.

5) One of the products our company offers is a suite of prepayment models for
various mortgage backed securities. A mortgage backed security is a bond
composed of many mortgages. Every month investors in these bonds receive the
monthly payments of the underlying mortgages. Investors like to be able to
predict the monthly payments they will receive in the future. This would be
fairly straight-forward, as home-owners generally pay the same amount every
month on their mortgages, except home-owners have an option to prepay their
mortgage at any time. A prepayment occurs when a homeowner pays the remaining
principle (each monthly payment is composed of two parts: principle and
interest) of their mortgage early. This payment, like all others, gets passed
through to the mortgage backed security抯 investors. This results in the
investors receiving a larger payment for this month, but no more payments from
this home-owner in the future. The job of a prepayment model is to take
information about a loan or a pool of loans as well as various economic
conditions and use this information to predict how fast the loan or loans will
prepay for various points in the future.

Describe the information you might need to know about the loan, home-owners,
and current and future economic conditions in order to give a general
prediction of how likely a given homeowner is to prepay their mortgage. Tell
why you would need each piece of information.

A: The prepayment will mainly come from following two reasons: 1. refinancing. 2. selling of the property, including mortgage default.
for 1. we mainly need to know the mortgage rate of the loan and tracking the mortgage rate as time goes
for 2. we might need to know more about the loan amount, remaining principle, the market value of the property, the house market trend, the family member's information for possibility of downsizing or upgrading the property, household income information, credit record, spending behavior and etc.