Question

Montana State Savings Bank is currently scheduling $10 million in deposits. First trust deeds yield 9%, second trust deeds 10.5%, automobile loans 12.25%, and business loans 11.75%. In addition, Montana State Savings Bank can invest in risk free securities yielding 6.75%. Regulatory commissions of the state and federal governments require the following: • At most one-third of deposits must be in risk-free securities. • Home loans (first and second trust deed) cannot exceed the amount in risk-free securities. • Business loans may not account for more than 49% of the total loans and trust deed investments (deeds and loans, but not securities). • Automobile loans may not exceed 50% of the home loans (first and second trust deeds). a) Formulate the mathematical linear programming model for this problem. Solve the problem. b) What is the total profit? c) Which one is the biggest investment? What is the value? d) Does the bank deposit in all of these possible investments? e) The Bank deposits a significant amount in the Risk Free Securities although its yield is the lowest. Explain why.

Answer 1

a) Mathematical linear programming model for this problem is following

Decision variables:

X1 = amount to be invested in first trust deeds

X2 = amount to be invested in second trust deeds

X3 = amount to be invested in automobile loans

X4 = amount to be invested in buisness loans

X5 = amount to be invested in risk free securities

Objective: Max 0.09X1+0.105X2+0.1225X3+0.1175X4+0.0675X5

s.t.

X1+X2+X3+X4+X5 = 10

X5 <= 10*1/3

X1+X2 <= X5

X4 <= 0.49*(X1+X2+X3+X4)

X3 <= 0.5*(X1+X2)

X1, X2, X3, X4, X5 >= 0

Solution of the problem using LINDO follows:

b) Total profit = $ 1.002761 million

c) The biggest investment in business loans (X4). Value = $ 3.656716 m

d) No, the bank does not deposit in first trust deeds.

e) Risk free securities investment = $ 2.537313 m is a significant amount, despite the lower yield. This is because of the constraint that home loans cannot exceeed the amount in risk free securities. The maximum yield is of automobile loans, but the constraint is that it may not exceed 50% of the home loans. All of the constraints lead to a mandatory investment in risk free securities so that investment in home loans can be made, which will make it possible to invest in automobile loans, whose yield is the highest.