# Linear programming bank assets problem

• Dr Zoidburg
In summary, the bank's objective is to maximise the annual return on its investment portfolio. The LP (in standard form) that will enable the bank to meet this goal is a car loan portfolio that has a 12% ROI.

#### Dr Zoidburg

I've got this question to do:
A bank is attempting to determine where its assets should be invested during the current
year. At present \$500 million is available for investment in bonds, home loans, car loans,
and personal loans. The annual rate of return on each type of investment is known to be:
bonds, 7%; home loans, 8%; car loans, 12%; personal loans, 11%. In order to ensure that
the bank’s portfolio is not too risky, the bank’s investment manager has placed the
following three restrictions on the bank’s portfolio:
(a) The amount invested in personal loans cannot exceed the amount invested in bonds.
(b) The amount invested in home loans cannot exceed the amount invested in car loans.
(c) No more than 25% of the total amount invested may be in personal loans.
The bank’s objective is to maximize the annual return on its investment portfolio.
Formulate an LP (in standard form) that will enable the bank to meet this goal. Assume
interest is calculated annually.

Pretty straight forward I think. I did this:
Let:
B = Bonds
H = Home Loans
C = Car Loans
P = Personal Loans
Maximise
Z = 0.07B + 0.08H + 0.12C + 0.11P
Subject to:
P <=B
H <=C
P <= 125 Million
B,H,C,P >=0
B+H+C+P <= 500 Million

However, using the constraints as they are, intuitively - and supported by excel solver - the best way to maximise profits is to put everything into car loans (at 12% ROI). Am I right, or did I miss something?
I ask because it just seems too easy to lump it all into car loans, and it hardly matches the goal of minimising risk by spreading the loans.

Dr Zoidburg said:
However, using the constraints as they are, intuitively - and supported by excel solver - the best way to maximise profits is to put everything into car loans (at 12% ROI). Am I right, or did I miss something?
I ask because it just seems too easy to lump it all into car loans, and it hardly matches the goal of minimising risk by spreading the loans.

Based on the constraints, that seems like the right answer. It doesn't actually minimize risk by diversifying, but it satisfies the constraints that the problem defines as the ones chosen to minimize risk.

The problem would be more interesting if, say, personal loans had the highest yield instead of car loans. Then things would be more interesting because you have two upper bounds on P, whereas you only have the one lower bound on C.