Week 8 Project: You Are A Portfolio Manager For Xyz Invest

Week 8 Projectyou Are A Portfolio Manager For The Xyz Investment Fund

Week 8 Project 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 widely 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? Portfolio Portfolio Manager Portfolio Stocks Real Estate Bonds CD Profit Percentage: 0.1 0.07 0.04 0.01 Resources Resources: Investments Constraint Available Left over Total Investments = $50,000.00 Real Estate minimum holding -- 1 -- -- >= $50,000.00 Bond minimum holding -- -- 1 -- >= $50,000.00 CD minimum holding -- -- -- 1 >= $50,000.00 CD & Bonds for liquidity -- -- 1 1 >= $200,000.00 Liquidity constraint on Real Estate holding -- 1 -- --

Paper For Above instruction

The optimal management of an investment portfolio involves balancing expected returns with risk constraints while adhering to diversification and liquidity requirements. This paper develops a linear programming model for the XYZ investment fund, which aims to maximize returns from four asset classes: stocks, real estate, bonds, and certificates of deposit (CD). The formulation considers historical return data, risk limitations measured via Value at Risk (VaR), and specific investment constraints to ensure diversification and liquidity adequacy.

Objective Function:

The primary goal is to maximize the total expected return of the portfolio. If \( x_1, x_2, x_3, x_4 \) represent the dollar amounts invested in stocks, real estate, bonds, and CDs respectively, then the objective function is:

\[ \text{Maximize } Z = 0.10x_1 + 0.07x_2 + 0.04x_3 + 0.01x_4 \]

subject to the constraints detailed below.

Constraints:

1. Total Investment Constraint:

\[ x_1 + x_2 + x_3 + x_4 \leq 1,000,000 \]

2. Risk Constraints via VaR:

- Stocks: \( 0.06x_1 \leq 25,000 \Rightarrow x_1 \leq \frac{25,000}{0.06} \approx 416,666.67 \)

- Real estate: \( 0.02x_2 \leq 15,000 \Rightarrow x_2 \leq \frac{15,000}{0.02} = 750,000 \)

- Bonds: \( 0.01x_3 \leq 2,500 \Rightarrow x_3 \leq 250,000 \)

- CD: \( 0 \leq 0 \Rightarrow x_4 \) has no VaR constraint directly.

3. Minimum Investment Constraints:

\[ x_1 \geq 50,000 \]

\[ x_2 \geq 50,000 \]

\[ x_3 \geq 50,000 \]

\[ x_4 \geq 50,000 \]

4. Diversification and Liquidity Constraints:

- The combined investment in bonds and CD should be at least $200,000:

\[ x_3 + x_4 \geq 200,000 \]

- Real estate holdings should not exceed 30% of total assets:

\[ x_2 \leq 0.3 \times (x_1 + x_2 + x_3 + x_4) \]

- Liquidity requirement: combined bonds and CD holdings at least $200,000:

\[ x_3 + x_4 \geq 200,000 \]

- Additional constraints regarding the maximum holdings in real estate and other assets depend on specific policies; for illustrative purposes, the model may restrict real estate to not exceed $300,000.

Solution Approach:

Using linear programming techniques, such as the simplex method or solvers in Excel/solver software, the optimal investment allocation can be computed. Key considerations involve ensuring the constraints are satisfied while maximizing returns.

Expected Results:

Based on the parameters, the model determines optimal allocations:

- Stocks: approximately $416,666 to utilize maximum VaR limit.

- Real estate: near the upper bound of $750,000 but constrained by diversification limits.

- Bonds and CD: allocated to satisfy liquidity constraints and risk limits while maximizing returns.

The solution balances the trade-off between higher expected returns from stocks and risk limitations.

Effect of Additional Capital and Changing Yields:

- With an additional $500,000 investment capacity, the model reallocates funds, possibly increasing the holdings in higher-return assets within the new total.

- If stock yields decrease to 6%, the new objective function's coefficients change, leading to adjusted allocations favoring safer or higher-yield assets, and the re-distribution minimizes the impact of the declining stock returns by increasing investments in assets with relatively stable or higher returns.

Conclusion:

Formulating this portfolio optimization problem with linear programming provides a strategic framework for balancing risk and return within specified constraints. It highlights the importance of quantitative decision-making tools in effective portfolio management, especially when navigating multiple constraints like risk limits, diversification, and liquidity. Regular re-evaluation of parameters, such as asset yields and risk tolerances, is vital for maintaining optimal investment performance over time.

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