Question

PB Investments is investing $50,000 for a client into a bond fund paying 4.5% per year and into a stock fund paying 6% per year. PB Investments always places 5% of a portfolio into a noninterest bearing cash account for safety. The amount invested into stocks should be no more than 3 times the amount invested into bonds. At least 25% of the market investment should be in bonds. At least 40% of the return should come from bonds.

Formulate into linear programming equations to maximize return.

a) PB Investments is investing $50,000 for a client into a bond fund paying 4.5% per year and into a stock fund paying 6% per year. PB Investments always places 5% of a portfolio into a noninterest bearing cash account for safety.

b) The amount invested into stocks should be no more than 3 times the amount invested into bonds.

c) At least 25% of the market investment should be in bonds.

d)At least 40% of the Return on Investment should come from bonds.

Answer 1

Let us denote the different instruments with the following variables:

Amount invested in bond fund = x

Amount invested in stock fund = y

5% into non-interest bearing cash account = 5% of 50,000 = 2500

Balance amount = 50,000 – 2500 = 47,500

Also, as per condition given in b: y <= 3x

As per condition in c: x >= 25% of 50,000; i.e. x >= 12,500

Also the overall return from stocks and bonds = 4.5% of x + 6% of y = 0.045x + 0.06y

Also, since the balance amount is 47,500 that means x + y = 47,500

Further as per condition given in d: 40% of (0.045x + 0.06y) >= 0.045x

                                                       Which is simplified as: y >= 1.125x

So the overall Linear programming is:

Maximise 0.045x + 0.06y

Where, x,y >=0

              y>=1.125x and y <=3x

              x + y = 47,500

              x>=12,500 which implies y >= 14,062.5 and y <= 37,500