# Math 540

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You are a portfolio manager for the XYZ investment fund.  The objective for the fund is to maximize your portfolio returns from the investments on four alternatives.  The investments include (1) stocks, (2) real estate, (3) bonds, and (4) certificate of deposit (CD).  Your total investment portfolio is \$1,000,000.

Investment Returns

Based on the returns from the past five years, you concluded that the investment annual returns on stocks are 10%, on real estates are 7% on bonds are 4% and on CD is 1%.

Risk Constraints

However, you also have to analyze the risks associate with each investment category.  A wildly used risk measurement parameter is called Value at Risk (VaR).  (Note: VaR measures the risk of loss on a specific portfolio of financial assets.)  For example, given a million dollar stock investment, if a portfolio of stocks has a one-day 4% VaR, there is a 5% probability that the stock portfolio will fall in value by more than 1,000,000 * 0.04 = \$40,000 over a one day period.  In the portfolio, the VaR for stock investments is 6%.  Similarly, the VaR for real estate investment is 2% and the VaR for bond investment is 1% and the VaR for investment in CD is 0%.  To manage the portfolio, you decided that at 5% probability, your VaR for stocks cannot exceed \$25,000, VaR for real estate cannot exceed \$15,000, VaR for bonds cannot exceed \$2,500 and the VaR for CD investment is \$0.

Diversification and Liquidity Constraints

As a diversified investment portfolio, you also decided that each investment category must hold at least \$50,000 of the total investment assets.  In addition, you must hold combined CD and bond investment no less than \$200,000 in order to meet liquidity requirement.

The total amount of real estate holding shall not exceed 30% of the portfolio assets.

A. As a portfolio manager, please formulate and solve the investment portfolio problem using linear programming technique.  What are the amounts invest in (1) stocks, (2) real estate, (3) bonds and (4) CD?

B. If \$500,000 additional investments are available to you in your portfolio, how would you invest the capital?

C. Would you maintain the portfolio investment if stock yields lowered to 6%?  How would you re-distribute your investment portfolio?

******Your writeup should introduce your solution to the project by describing the problem. Correctly identify what type of problem this is. For example, you should note if the problem is a maximization or minimization problem, as well as identify the resources that constrain the solution. Identify each variable and explain the criteria involved in setting up the model. This should be encapsulated in one (1) or two (2) succinct paragraphs.

After the introductory paragraph, write out the L.P. model for the problem. Include the objective function and all constraints, including any non-negativity constraints. Then, you should present the optimal solution, based on your work in Excel. Explain what the results mean.

Finally, write a paragraph addressing the part of the problem pertaining to sensitivity analysis and shadow price.

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