Question

Problem 1: Consider a bank issuing loans to a population of qualified borrowers. Let Y 1 if a borrower defaults on the loan and Y = 0 otherwise. Let p denote the probability of default. Consider event B='the individual has a college degree. Assume P[B[F] = 0.9-0.7Y. (a) Find P[Y = 1[B] as a function of p. (b) Deduce the condition on p such that less than 10 percent of college educated borrowers default

Answer 1

(a)

Given, P(Y = 1) = p

=> P(Y = 0) = 1 - p

P[B | Y] = 0.9 - 0.7Y

Therefore,

P[B | Y = 0] = 0.9 - 0.7 * 0 = 0.9

P[B | Y = 1] = 0.9 - 0.7 = 0.2

By law of total probability,

P[B] = P[Y = 0] P[B | Y = 0] + P[Y = 1] P[B | Y = 1]

= (1 - p) 0.9 + p * 0.2 = 0.9 - 0.7p

By Bayes theorem,

P(Y = 1 | B) = P(B | Y = 1) * P(Y = 1) / P(B)

= 0.2p / (0.9 - 0.7p)  

(b)

For less than 10% of college educated borrowers default,

P[ Y = 1 | B] < 0.1

0.2p / (0.9 - 0.7p) < 0.1

0.2p < 0.09 - 0.07p

0.27p < 0.09

p < 0.09/0.27

p < 1/3

Thus, p should be less than 1/3.