Question

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.

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 . You can attach a picture of your handwritten equations here.

Answer 1

The objective is to maximize interest income. Now, lets say that the amount of home mortgages is X, Personal Loans amount is Y and hence loan to buy cars is 1000000-(X+Y). The objective function is

.03X+.12Y+.09(1000000-X-Y)

=.03X+.12Y+90000-.09X-.09Y

=.03Y-.06X+90000

Objective function= Maximize .03Y-.06X+90000

where X is amount of home mortgages , Y is Personal Loans amount

Now lets look at constraints

X>=400000 (at least 40% of lending in mortgages)
Y<=.6(1000000-X-Y) (Amount allocated to PL cant exceed 60% of amounts allocated to car loans). Rearranging it, we get
Y<=600000-.6X-.6Y
1.6Y<=600000-.6X
1.6Y+.6X<=600000

So, there are two constraints,

X>=400000
1.6Y+.6X<=600000