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## Main Question or Discussion Point

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.

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.