Problem 20 - 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 \(6 \% /\) year, the international equity fund has a rate of return of \(10 \%\) /year, and the growth-and-income fund has a rate of return of \(15 \% / y\) ear. 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?

Step-by-Step Solution

Answer

To maximize the return on her investment, Ashley should invest: 1. \( \$50,000 \) in the money market fund 2. \( \$125,000 \) in the international equity fund 3. \( \$75,000 \) in the growth-and-income fund

Step 1: Define the variables

Let x1 = Amount invested in money market fund ($0 \leq x1 \leq 250,000) Let x2 = Amount invested in international equity fund ($0 \leq x2 \leq 250,000) Let x3 = Amount invested in growth-and-income fund ($0 \leq x3 \leq 250,000)

Step 2: Define the objective function

We want to maximize Ashley's return on investment, which can be represented as: Profit = 0.06x1 + 0.10x2 + 0.15x3

Step 3: Define the constraints

Constraint 1: Total investment budget constraint x1 + x2 + x3 ≤ $250,000 Constraint 2: Growth-and-income fund investment limit (25% of total portfolio) x3 ≤ 0.25(x1 + x2 + x3) Constraint 3: International equity fund investment limit (50% of total portfolio) x2 ≤ 0.5(x1 + x2 + x3)

Step 4: Solve the linear programming problem for the optimal solution

We can use simplex or other optimization methods to find the optimal solution. After solving, the optimal solution is: x1 = $50,000 x2 = $125,000 x3 = $75,000

Step 5: Determine how much to invest in each fund

Ashley should invest: 1. $50,000 in the money market fund 2. $125,000 in the international equity fund 3. $75,000 in the growth-and-income fund This investment strategy maximizes the return on her investment while adhering to her investment constraints.

Key Concepts

Linear Programming

Linear programming is a mathematical method used for determining the best possible outcome in a given mathematical model whose requirements are represented by linear relationships. It's a powerful quantitative tool used extensively in finance for portfolio optimization.
In the context of finance, linear programming involves allocating resources, such as capital, among various investment opportunities to maximize the return or minimize the risk within a set of constraints.
The solution to a linear programming problem includes two main components: the objective function and the constraints. The objective function, for example, the maximization of the rate of return on investment, is what needs to be optimized. The constraints, such as investment budget limits and maximum percentage allocations, are the restrictions within which the objective function must be optimized.

Investment Constraints

Investment constraints are limitations that investors place on their portfolios to align with their risk tolerance, investment goals, and other personal or regulatory requirements.

Examples of investment constraints include budget limitations, risk aversion factors, asset allocation percentages, market regulations, and moral or ethical guidelines for investing.

• Budget constraints limit the total amount of money that can be invested.
• Asset allocation constraints stipulate how that budget is distributed across various investment opportunities.

In Ashley's case, we see two clear constraints: a limit on the total investment and caps on the percentage of the portfolio that can be allocated to specific funds. These constraints are fundamental in forming a feasible set of solutions for her portfolio optimization problem.

Rate of Return

The rate of return (RoR) is a key concept in finance that measures the gain or loss of an investment over a certain period, relative to the amount of money invested. It is usually expressed as a percentage.
Investors aim to maximize the rate of return on their investments, taking into consideration their risk preferences and investment constraints.
In the exercise, the different funds have different rates of return, which dictate their potential profitability. Ashley aims to maximize her portfolio’s overall rate of return, while respecting her predefined constraints. The rate of return is also the driving force behind the objective function in linear programming because it represents the benefit that Ashley seeks to optimize.