Waikato Savings Bank (WSB) has $1 million of new funds from depositors that they need to loan out (so that there is some interest earned, in order to pay interest to the depositors). They have three loan products: home mortgages, personal loans, and loans to buy cars. The annual rate of return to WSB from each of these three types of loans are 3% for mortgages, 12% for personal loans, and 9% for car loans.

The lending rules that WSB operate under mandate that at least 40% of new lending has to go into mortgages. They are also constrained by a rule that the amount allocated to personal loans cannot exceed 60% of the amount allocated to car loans.

a) Using mathematical notation, write out the objective function and the constraint functions, for an objective of maximizing the returns for WSB subject to the constraints (6 marks). You can attach a picture of your handwritten equations here.

b) Using Excel, set up the objective function, the decision variables and the coefficient table in a new sheet and apply the Excel Solver to calculate the optimal portfolio. Paste an image of your worksheet and the answer report into the space below and use it to answer the following questions (3 marks each, for 12 marks total):

What is the optimal allocation to: mortgages ___________, personal loans ___________, car loans ____________

What is WSBs annual return, in $, at the optimal loan mix? ________________

c) Report the shadow price of the mortgage allocation constraint and discuss its interpretation (3 marks)

d) The management of the WSB is concerned that they are providing loans to too many small borrowers, who are expensive to service. They therefore institute a change that all loans have to be in unit sizes of $100,000 (in other words, each customer can only borrow set amounts like $100,000, $200,000, $300,000 and so on, and cannot borrow any amount below $100,000 nor can they borrow any value between $100,000 and $200,000 or between $200,000 and $300,000 and so on). With the same constraints holding as in Part A (40% of lending has to go into mortgages, the amount allocated to personal loans cannot exceed 60% of the amount allocated to car loans) what is WSBs optimal portfolio when then have $1 million of new funds to lend and what is their annual return, in $, with this additional restriction that loans have to be in unit sizes of $100,000? (2 marks each, for 8 marks total)

Allocation to mortgages ________________

Allocation to personal loans ________________

Allocation to car loans ________________

Annual return, in $, at the optimal loan mix? ________________

e) Give a name for the specific type of linear programming problem described in part (d) and provide a sketch (either hand drawn and then scanned/ or drawing using shapes in Word) to show how this type of problem differs from the usual linear programming problem. Note that your sketch is to portray a general situation and does not have to portray specific details of the WSB lending problem (6 marks)