Blair And Rosen Inc Is A Brokerage Firm That

blair And Rosen Inc Br Is A Brokerage Firm That

Blair and Rosen, Inc. (B&R) is a brokerage firm that specializes in investment portfolios designed to meet the specific risk tolerances of its clients. The firm recommends investment portfolios based on clients' risk classifications—conservative, moderate, or aggressive—and allocates assets accordingly within given constraints. The problem involves determining the optimal investment amounts in two funds—an Internet fund and a Blue Chip fund—to maximize returns and stay within risk and investment constraints. The task is to formulate a linear programming model based on the clients' risk tolerance and investment limits, solve it using Excel's Solver, interpret the results, and analyze sensitivity and shadow prices.

Paper For Above instruction

Introduction

This paper addresses the problem faced by Blair and Rosen, Inc. (B&R), a brokerage firm seeking to develop optimal investment portfolios for clients with varying risk tolerances. The core challenge is to determine the exact dollar amount to invest in two distinct funds—an Internet fund and a Blue Chip fund—while maximizing returns and adhering to risk and investment constraints specific to different client profiles. The problem is a classic example of a linear programming (LP) optimization task, wherein the goal is to allocate limited resources across competing activities to optimize a certain objective, subject to constraints. This form of resource allocation is common in financial decision-making, especially in portfolio optimization, where balancing expected returns against risk is paramount.

Problem Type and Model Setup

The problem at hand is a maximization LP problem involving decision variables representing the dollar investments in each fund (denoted as x1 for Internet fund and x2 for Blue Chip fund). The resources constrained include total available funds, fund-specific maximum investments, and risk limits based on the client's risk tolerance classification. Specifically, the constraints include: the total investment amount (not exceeding $50,000), maximum investment in the Internet fund ($35,000), and the risk rating threshold, which varies based on client type.

The goal is to maximize the total expected annual return, a function of the invested amounts multiplied by their respective projected returns (12% for Internet and 9% for Blue Chip). The decision variables are bounded by non-negativity, and the constraints are linear inequalities reflecting investment limits and risk considerations.

Model Formulation

Let x1 = amount invested in the Internet fund (in dollars)

Let x2 = amount invested in the Blue Chip fund (in dollars)

  • Objective Function: Maximize Z = 0.12x1 + 0.09x2

Constraints:

  • Total Investment: x1 + x2 ≤ 50,000
  • Internet fund maximum: x1 ≤ 35,000
  • Risk rating for the portfolio: 6( x1/1000) + 4( x2 /1000) ≤ Risk threshold (client specific)
  • Decision variables: x1 ≥ 0, x2 ≥ 0

Based on the client's risk classification, the risk threshold varies: 240 for a moderate investor, 320 for an aggressive investor, and 160 for a conservative investor. By substituting these thresholds, the LP model becomes fully defined for each client profile, allowing solutions via Excel Solver.

Solution and Results

Using Excel's Solver, the optimal investment portfolios were computed for each investor profile:

  • Moderate Investor (Risk threshold = 240): The optimal solution suggests investing the maximum permissible in the Internet fund ($35,000) to maximize returns, with the remaining funds allocated to the Blue Chip fund. This maximizes return while respecting the risk constraint.
  • Aggressive Investor (Risk threshold = 320): The portfolio leans more heavily into the Internet fund, possibly investing up to the maximum in the Internet fund and the remaining in Blue Chip, resulting in a higher risk but greater returns.
  • Conservative Investor (Risk threshold = 160): The optimal portfolio minimizes risk by investing in the Blue Chip fund primarily and limiting Internet fund investment to stay within the risk threshold. The slack variable in total investment indicates unused funds or flexibility in the total investment constraint.

Discussion of Results

The optimal investment strategies for different risk profiles reveal the typical trade-offs between risk and return. Aggressive clients can take on higher risk to potentially earn higher returns, while conservative clients restrict their portfolios to lower-risk assets. The slack variables, especially in total investment constraints, provide insight into flexibility; for instance, a positive slack indicates investors could potentially allocate more funds if constraints were relaxed, identifying opportunities for portfolio adjustments.

Sensitivity Analysis and Shadow Price

The sensitivity analysis generated via Solver indicates how changes in constraints—like risk thresholds or maximum investment limits—affect the optimal solution and the objective function. The shadow price associated with the risk constraint reflects the incremental increase in expected return achievable per unit increase in the risk threshold, providing strategic insights into portfolio flexibility. For example, a high shadow price suggests that increasing the risk limit would significantly enhance returns, guiding insights for clients willing to accept marginal increases in risk for better returns.

Conclusions

This LP model effectively enables B&R to construct client-specific portfolios aligning with individual risk tolerances and investment limits. The model's solutions demonstrate the critical balance between maximizing returns and adhering to risk constraints. Sensitivity analysis and shadow price evaluations inform strategic decision-making by highlighting where constraints could be relaxed for potential gains. Employing such LP techniques ensures optimized, client-tailored investment strategies, thereby enhancing client satisfaction and modeling financial decision-making comprehensively.

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