Question

A brokerage firm has just been instructed by one of its clients to invest $1,000,000 of her money. The client’s goal is to maximize total projected return on investments. The analysts at the brokerage firm are considering the following options for investment:

Projected Rate

Investment Option of Return (%) Risk (%)

Municipal bonds 2.2 0.1

Company A stocks 12.1 2.5

Company B stocks 9.4 1.7

Company C stocks 7.2 0.8

The client has specified the following guidelines:

- The total risk should be less than or equal to $12,000.

- Municipal bonds should constitute at least 20% of the money invested.

- At least 60% of the funds available should be placed in a combination of company A, B, and C stocks.

- No more than 35% of the amount invested in municipal bonds should be invested in stock C.

Formulate a linear optimization model for this investment problem.

(a) Define the decision variables.

(b) Determine the objective function. What does it represent?

(c) Determine all the constraints. Briefly describe what each constraint represents.

Answer 1

Solution:

a: Let's define the variables,

x1 = Amount Invested in Municipal Bonds
x2 = Amount Invested in Stock A
x3 = Amount Invested in Stock B
x4 = Amount Invested in Stock C

b: Objective function:

Maximize Z = 0.022x1 + 0.121x2 + 0.094x3 + 0.072x4

This objective function represents the total "returns" from the given investments and we intend to maximize the returns. These are the returns over the investment amount.

c: Constraints:

x1 + x2 + x3 + x4 <= 1000000 -- (Constraint of total available funds)

0.001x1 + 0.025x2 + 0.017x3 + 0.008x4 <= 12000 --- (Constraint of maximum risk tolerance of investments)

x1 >= 0.2(x1+x2+x3+x4)
0.8x1 - 0.2x2 - 0.2x3 - 0.2x4 >= 0 --- (Constraint of at least 20% of total invested amount to be invested in municipal bonds)

x2 + x3 + x4 >= 600000 --- (Constraint of at least 60% of funds available to be invested in combination of A,B, and C)

0.35x1 - x4 >= 0 --- (Constraint of investment in Stock C to be maximum of 35% of investment in Municipal Bonds)

x1,x2,x3,x4 >= 0 -- (Non-negativity constraint.

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