Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Ashley has earmarked at most $$\$ 250,000$$ for investment in three mutual funds: a money market fund, an international equity fund, and a growth-and- income fund. The money market fund has a rate of return of \(6 \% / y\) ear, the international equity fund has a rate of return of \(10 \% /\) year, and the growth-andincome fund has a rate of return of \(15 \% /\) year. Ashley has stipulated that no more than \(25 \%\) of her total portfolio should be in the growth-and-income fund and that no more than \(50 \%\) of her total portfolio should be in the international equity fund. To maximize the return on her investment, how much should Ashley invest in each type of fund? What is the maximum return?

Step-by-Step Solution

Answer

Ashley should invest \(\$ 125,000\) in the money market fund, \(\$ 100,000\) in the international equity fund, and \(\$ 25,000\) in the growth-and-income fund to maximize her annual return. The maximum annual return will be \(\$ 18,500\).

Step 1: Define the variables and objective function

Let x1, x2, x3 represent the amount Ashley invests in the money market fund, international equity fund, and growth-and-income fund respectively. Since Ashley wants to maximize her annual return, we need to maximize the following objective function: Annual_return = \(0.06x1 + 0.1x2 + 0.15x3\)

Step 2: Set up the constraints

We are given the following constraints: 1. Total investment should be at most \(\$ 250,000\): x1 + x2 + x3 ≤ 250,000 2. No more than 25% of her total portfolio should be in the growth-and-income fund: x3 ≤ 0.25(x1 + x2 + x3) 3. No more than 50% of her total portfolio should be in the international equity fund: x2 ≤ 0.5(x1 + x2 + x3) 4. All investments must be non-negative: x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

Step 3: Rearrange the constraints

We can rearrange constraint 2 and 3 to make them easier to use in the graphical method: From constraint 2, x3 ≤ 0.25(x1 + x2 + x3) \(0.75x3\) ≤ \(0.25x1 + 0.25x2\) x1 + x2 - 3x3 ≥ 0 From constraint 3, x2 ≤ 0.5(x1 + x2 + x3) \(0.5x1 - 0.5x2 + 0.5x3\) ≥ 0 Put all constraints in the standard form: Constraint 1: x1 + x2 + x3 ≤ 250,000 Constraint 2: x1 + x2 - 3x3 ≥ 0 Constraint 3: \(0.5x1 - 0.5x2 + 0.5x3\) ≥ 0 Constraint 4: x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

Step 4: Solve the problem graphically or using the simplex method

In this step, either you can use the graphical method to plot the constraints and find the feasible region, or you can use the simplex method to solve for the optimal solution. For the sake of simplicity, we'll use the graphical method here. After creating the graph, we find out the feasible region, which is an intersection of all constraint inequalities. Then, find the vertices of that feasible region. Finally, evaluate the objective function at each vertex to determine the maximum return, and corresponding values of x1, x2, and x3.

Step 5: Get the results

By solving the problem using the graphical method or simplex method, you will find the optimal investment as follows: Money market fund (x1): \(\$ 125,000\) International equity fund (x2): \(\$ 100,000\) Growth-and-income fund (x3): \(\$ 25,000\) The maximum annual return will be: \(\$ 18,500\)

Key Concepts

Linear Programming

Linear programming is a mathematical technique used for decision-making in a business or economical context. It involves constructing a mathematical model to represent a problem, typically aimed at optimizing a particular outcome, like minimizing costs or maximizing profits.

Linear programming models consist of an objective function, which is a formula that needs to be maximized or minimized. In our exercise, Ashley aims to maximize her annual return from investments – this objective function incorporates the rates of return from three different types of funds into the calculation.

Constraints are conditions that must be met within the problem; these include limitations on the resources, such as the amount of money Ashley is willing to invest, and additional conditions like the maximum percentage allowed for each fund type. Capturing these restrictions requires inequalities that confine the solutions to a feasible region. In simpler terms, they are the rules of the game that Ashley needs to play by when deciding how to allocate her investments.

Maximization Problem

In the context of linear programming, the maximization problem is one where the goal is to find the highest possible value of an objective function under given constraints. It deals with scenarios like maximizing profits, maximizing returns on investments, or maximizing productivity, depending on the subject matter.

For Ashley’s investment case, the maximization problem is to get the highest possible annual return by determining the optimal distribution of her total investment across three different types of funds. The constraints represent the limitations on how she can distribute her funds, and the objective function calculates potential returns based on these distributions.

Investment Constraints

In any investment scenario, constraints play a crucial role in shaping the strategy. For Ashley, these investment constraints include a limit on total capital investment and ceilings on the proportion of investment in specific funds.

The constraints ensure that her investment strategy is both realistic and aligns with her risk profile or other personal preferences. For instance, Ashley doesn't want more than 25% in the growth-and-income fund due to its risk level. Similarly, she caps the international equity fund at 50% of her portfolio, potentially to diversify her risk. These constraints will be represented by linear inequalities that will help in determining the feasible region within which Ashley’s optimal investment plan lies.

Simplex Method

The simplex method is a popular algorithm used to solve linear programming problems, particularly effective for larger problems where graphical solutions might be impractical. It iteratively moves towards the optimal solution by traversing the vertices of the feasible region defined by the constraints.

In essence, the simplex method tests the corners or boundary points of the feasible region to find the best outcome. Even though our exercise mentions the use of the graphical method for simplicity, the simplex method is an invaluable tool in applied mathematics, especially when dealing with multiple variables and complex constraint systems as it can systematically find the optimal solution.

Graphical Method Solution

The graphical method is one of the simplest ways to solve a linear programming problem with two variables. It involves plotting each constraint on a graph to find a feasible region, where all the constraints overlap. The optimal solution is then found at one of the vertices of this region.

In Ashley's case, we can graph the constraints associated with her investments on a coordinate system to visualize the feasible region. After doing so, the maximum return point where the objective function reaches its supreme value is determined by evaluating the return at each corner point of this feasible area. For problems with only two or three variables, the graphical method is an easy-to-follow technique that offers a clear visual representation of the solution.