Question

An investment firm can invest in four financial instruments for a client: Municipal Bonds, Corporate Bonds, Stocks, and Construction loans. The income from each investment will be 5%, 7%, 9%, and 10% respectively. The firm has $4 million to invest and they do not want to invest more than 40% of the fund in any single instrument. Furthermore, the firm wants at least 50% investment in Bonds and no less than 30% in Stocks. The client also wants at least 20% of total income to be tax free (Note that income from Municipal bonds is tax free).

Formulate this LP problem.

What is the solution to this LP problem?

Answer 1

This problem can be formulated as a linear programming (LP) problem to maximize the income while satisfying certain constraints. The objective function is to maximize the income, and the constraints include limits on the total investment, individual instrument investment, and minimum percentages of investment in bonds and stocks. The LP problem can be solved to find the optimal solution.

Step-by-step explanation:

This problem can be formulated as a linear programming (LP) problem to maximize the income while satisfying certain constraints. Let's define the decision variables:

• X1: Amount to invest in Municipal Bonds
• X2: Amount to invest in Corporate Bonds
• X3: Amount to invest in Stocks
• X4: Amount to invest in Construction loans

The objective function is to maximize the income, which can be represented as:

Income = 0.05X1 + 0.07X2 + 0.09X3 + 0.1X4

The constraints are as follows:

1. Total investment should not exceed $4 million: X1 + X2 + X3 + X4 ≤ 4000000
2. Not more than 40% of the fund should be invested in any single instrument: X1 ≤ 0.4(4000000), X2 ≤ 0.4(4000000), X3 ≤ 0.4(4000000), X4 ≤ 0.4(4000000)
3. At least 50% investment in Bonds: (X1 + X2) ≥ 0.5(X1 + X2 + X3 + X4)
4. No less than 30% investment in Stocks: X3 ≥ 0.3(X1 + X2 + X3 + X4)
5. At least 20% of the income should be tax-free: 0.05X1 ≥ 0.2(0.05X1 + 0.07X2 + 0.09X3 + 0.1X4)

This LP problem can be solved using optimization techniques to find the optimal solution, which will determine the amounts to invest in each financial instrument.