IEOR242 Applications in Data Analysis
EOR 242A: Machine Learning and Data Analytics I
Problem 1: (30 points) Let us consider an extension of the lending decision problem from class.
The lender is now deciding between three options: (i) funding the loan with low interest, (ii) funding the loan with high interest, and (iii) not funding the loan. The amount of the principal
of the loan is still $4000; so if the borrower defaults, then the lender loses $4000. Again, let p denote the probability that the borrower defaults. Assuming that the borrower does not default,
the low-interest option would yield a total profit of $1000 for the lender and the high-interest option would yield a total profit of $1500. If the lender chooses the high-interest option, then the borrower will agree to the terms of the loan with probability q (this event happens independently of the defaulting event). If the lender chooses the low-interest option, then the borrower is guaranteed to agree to the terms of the loan.
Please answer the following questions.
a) (8 points) Create a decision tree diagram to model the previously described scenario. Use squares to denote decision nodes and circles to denote chance nodes representing random
events. Each terminal node of the tree should have a corresponding profit value.
b) (8 points) Derive formulas for the expected profit under each of the three possible decisions for the lender: (i) fund with low interest, (ii) fund with high interest, and (iii) do not fund.
Your formulas should depend on the parameters p and q.
c) (8 points) Suppose that q = 1/2. Segment the range of possible values of p, i.e., the interval [0, 1] into three subintervals corresponding to ranges of values where each of the three options
are optimal decisions in order to maximize expected profit. Create a graph to visually display your answer.
d) (6 points) Briefly discuss how one might estimate the parameters p and q in practice, in a personalized way depending on features associated with the borrower. Your discussion should
include what type of dataset(s) would need to be collected and what model(s) you would fit.
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