Question

1. A customer has approached a bank for a $100,000 one-year loan at 8% interest rate. If the bank does not approve this loan application, the $100,000 will be invested in bonds that earn a 4% annual return. Without additional information, the bank believes that there is a 5% chance that this customer will default on the loan, assuming that the loan is approved. If the customer defaults on the loan, the bank will lose $100,000 entirely. At a cost of $1000, the bank can thoroughly investigate the customer’s credit record and supply a favorable or unfavorable recommendation. Past experience indicates that in cases where the customer did not default on the loan, the probability of receiving a favorable recommendation on the basis of the credit investigation was 80%. Furthermore, in cases where the customer defaulted on the loam, the probability of receiving a favorable recommendation on the basis of the credit investigation was 15%.

Construct the decision tree and explain what strategy should the bank follow to maximize its expected profit.

Answer 1

First, define the following events nomenclature:

D = Default
ND = Not default
F = Favorable
UF = Unfavorable.

The following probabilities are given:

P(D) = 0.05
P(F | ND) = 0.80
P(F | D) = 0.15

Use the Baye's theorem, to compute the posterior probabilities from the priors and conditionals as follows:

F
Prior		Conditional		Joint		Posterior
P(D)	0.05	P(F | D)	0.15	P(D) x P(F | D)	0.008	P(D | F)	0.0098
P(ND)	0.95	P(F | ND)	0.80	P(ND) x P(F | ND)	0.760	P(ND | F)	0.9902
Total	1.0			P(F) =	0.7675		1.000
UF
Prior		Conditional		Joint		Posterior
P(D)	0.05	P(UF | D)	0.85	P(D) x P(UF | D)	0.043	P(D | UF)	0.1828
P(ND)	0.95	P(UF | ND)	0.20	P(ND) x P(UF | ND)	0.190	P(ND | UF)	0.8172
Total	1.0			P(UF) =	0.2325		1.000

Decision Tree:

So,

the optimal strategy is to first investigate the credit and if found favorable, accord the load, otherwise, invest in the bond.