In the first part of this lab, we will use the data file to estimate various probabilities.
In the second part, we will use these probabilities to decide whether a bank should work out
an arrangement with or foreclose on a borrower whose business is currently illiquid.
Before proceeding we define the following events:
- S is the event that a randomly chosen borrower is succesful in paying back
the loan in full.
- F is the event that a randomly chosen borrower fails to pay back the loan in full.
- Y is the event that a randomly chosen borrower has 7 years of business experience.
- T is the event that a randomly chosen borrower has a Bachelor's Degree.
- C is the event that the economy is in normal times.
Part One
- Download the file and open it with MINITAB, using the File -> Open
Worksheet... command.
- Use Stat -> Tables -> Tally... to estimate the following
probabilities:
- Pr(S)
- Pr(F)
- Pr(Y)
- Pr(T)
- Pr(C)
Note that for the last three cases, we assume that each probability can be
estimated using a restricted data set obtained by limiting ourselves to the records of
a particular bank.
- Use Stat -> Tables -> Cross Tabulation... to find the following
conditional probabilities. Note that you will have to add an asterisk (*) in the
empty cells of the last row of data, to prevent MINITAB from thinking that some of the columns
are shorter than the others.
- Pr(Y | S) (this is the probability that a borrower has 7 years of experience,
given that he/she paid back the loan in full)
- Pr(T | S)
- Pr(C | S)
- Pr(Y | F)
- Pr(T | F)
- Pr(C | F)
- Pr(S | Y)
- Pr(F | Y)
- Two events A and B are independent if and only if Pr(AB)
= Pr(A) Pr(B). A similar definition applies when the events are conditioned to another
event. Assuming that the events Y, T, and C are independent, even when they
are conditioned to S and F, estimate the following probabilities:
- Pr(Y T C | S)
- Pr(Y T C | F)
- Draw a picture of the sample space S which contains the events S, F, and
Y T C.
Use the picture to obtain a formula for Pr(Y T C) in terms of
Pr(Y T C | S), Pr(Y T C | F), P(S) and
P(F).
- Use Bayes's theorem to calculate the following probabilities:
- Pr(S | Y T C)
- Pr(F | Y T C)
Part Two
Assume that you are a loan officer and that the bank you work for has an outstanding commercial loan
with a full value of $4,000,000. If the bank decides to foreclose on the borrower, it will recover
$2,100,000. If the bank decide to work out an arrangement, either the work out will be successful,
in which case the bank will get the $4,000,000 back, or the work out will fail, in which case the bank
will only recover $250,000. You also know that the borrower has a Bachelor's degree, that he has
7 years of business experience, and that the economy is in normal times. Using this information and
the probabilities calculated in Part One, decide whether or not the bank should work out an arrangement
with this particular borrower.
Reading Assignment: Sections 2.1 through 2.3.
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