George Recently Inherited A Large Sum Of Money

George Recently Inhereted a Large Sum Of Money He Wants to Use a Port

George recently inherited a large sum of money. He wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: 1 a bond fund and 2 a stock fund. The projected returns over the life of the investments are 6% for the bond fund and 10% for the stock fund. Whatever portion of the inheritance he finally decided to commit to the trust fund, he wants to invest at least 30% of that amount in the bond fund. Also he wants to select a mixture with a total return of at least 7.5%. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each investment idea.

Paper For Above instruction

In the realm of investment planning, constructing an optimal asset allocation strategy is crucial for maximizing returns while adhering to specific constraints. The scenario presented involves allocating a certain portion of George's inheritance into two investment vehicles: a bond fund and a stock fund, each with defined projected returns. The goal is to determine the percentage allocations to each fund that satisfy the constraints and optimize the expected return.

Let's define the decision variables first. Let:

  • \( x \) = the percentage of the total inheritance allocated to the bond fund
  • \( y \) = the percentage of the total inheritance allocated to the stock fund

Since George only invests a portion of the inheritance, and the proportions are relative, these variables can be expressed as fractions or percentages of the total amount he allocates to the trust fund. For simplicity, we can model these as proportions of the total trust fund amount he commits.

The primary objective is to determine the allocation percentages, but since the problem emphasizes constraints rather than explicit maximization or minimization, we'll focus on formulating the constraints that must be satisfied, which include minimum allocations and return requirements.

Objective Function

Although the problem does not specify maximizing a particular quantity, in many such cases, the goal is to maximize expected return. Therefore, the objective function can be expressed as:

\[ \text{Maximize } R = 6\% \times x + 10\% \times y \]

Where \( R \) is the expected return of the selected investment mixture.

Constraints

  • Allocation constraints:
  • Since the total allocation must sum to 100% of the invested portion:
  • \[ x + y = 1 \]
  • Minimum bond investment:
  • At least 30% of the invested amount must go into bonds:
  • \[ x \geq 0.3 \]
  • Expected return constraint:
  • The overall expected return should be at least 7.5%:
  • \[ 6\% \times x + 10\% \times y \geq 7.5\% \]
  • Non-negativity constraints:
  • Allocations cannot be negative:
  • \[ x \geq 0 \]
  • \[ y \geq 0 \]
  • Additional implicit constraint:
  • Since total allocation sums to the entire invested amount, i.e., \( x + y = 1 \); this serves as a normalization constraint.

Complete Linear Programming Model

\[

\begin{cases}

\text{Maximize } R = 0.06x + 0.10y \\

\text{subject to} \\

x + y = 1 \\

x \geq 0.3 \\

0.06x + 0.10y \geq 0.075 \\

x \geq 0 \\

y \geq 0

\end{cases}

\]

This linear programming model captures the essence of George's investment decision-making, balancing the goal of maximizing expected returns with the constraints on minimum bond allocation and combined return rates. Solving this LP will give the optimal proportions \( x \) and \( y \) that George should allocate to each fund.

Utilizing simplex or other LP-solving techniques would allow George to determine the precise percentage investment in each fund to meet his objectives effectively. This model exemplifies a straightforward yet powerful method for strategic financial planning involving multiple constraints.

References

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