Problem 37 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Mr. and Mrs. Garcia have a total of $$\$ 100,000$$ to be invested in stocks, bonds, and a money market account. The stocks have a rate of return of \(12 \%\) lyear, while the bonds and the money market account pay \(8 \% /\) year and \(4 \%\) year, respectively. The Garcias have stipulated that the amount invested in stocks should be equal to the sum of the amount invested in bonds and 3 times the amount invested in the money market account. How should the Garcias allocate their resources if they require an annual income of \(\$ 10,000\) from their investments? Give two specific options.

Step-by-Step Solution

Answer

Based on our calculations, we found two specific investment options for Mr. and Mrs. Garcia: Option 1: - Invest $60,000 in stocks - Invest $30,000 in bonds - Invest $10,000 in the money market account Option 2: - Invest $70,000 in stocks - Invest $10,000 in bonds - Invest $20,000 in the money market account Both options will satisfy their requirements for the investment allocations and provide the desired annual income of $10,000.

Step 1: Write the system of equations.

We can write the following equations: 1. \(x + y + z = 100,000\): The total investment budget is $100,000. 2. \(x = y + 3z\): They want the investment in stocks to be equal to the sum of the amount invested in bonds and 3 times the amount invested in the money market account. 3. \(0.12x + 0.08y + 0.04z = 10,000\): The total annual return on investments should be $10,000.

Step 2: Solve the system of equations.

First, we can use equation 2 to get \(y\) in terms of \(x\) and \(z\): \(y = x - 3z\). Now we will substitute this expression for \(y\) in equation 1 and equation 3: Equation 1: \(x + (x - 3z) + z = 100,000\) Equation 3: \(0.12x + 0.08(x - 3z) + 0.04z = 10,000\) Now, we have two equations with two variables: Equation 1: \(2x - 2z = 100,000\) Divide by 2: \(x - z = 50,000\) Equation 3: \(0.12x + 0.08x - 0.24z + 0.04z = 10,000\) Combine: \(0.20x - 0.20z = 10,000\) Divide by 0.20: \(x - z = 50,000\) We find that both equations are the same, so we only have one solution for two variables. This means we have infinitely many solutions. To find two specific options, we will choose arbitrary values for one variable and solve for the other variable.

Step 3: Choose arbitrary values for z and find the corresponding x and y values

Let's choose z = 10,000 for Option 1 and z = 20,000 for Option 2: Option 1: If z = 10,000, then from \(x - z = 50,000\), we have x = 60,000. And from \(y = x - 3z\), we have y = 60,000 - 3(10,000) = 30,000. Option 2: If z = 20,000, then from \(x - z = 50,000\), we have x = 70,000. And from \(y = x - 3z\), we have y = 70,000 - 3(20,000) = 10,000.

Step 4: Present the two specific investment options

Based on our calculations, we found two specific investment options for Mr. and Mrs. Garcia: Option 1: - Invest $60,000 in stocks - Invest $30,000 in bonds - Invest $10,000 in the money market account Option 2: - Invest $70,000 in stocks - Invest $10,000 in bonds - Invest $20,000 in the money market account Both options will satisfy their requirements for the investment allocations and provide the desired annual income of $10,000.

Key Concepts

Systems of Linear Equations

Understanding systems of linear equations is fundamental not just in mathematics, but also in analyzing real-world scenarios like investment allocation. Essentially, a linear equation represents two values that are proportional to each other. When we talk about a 'system' of these equations, we mean a set of two or more equations with the same variables.

In the context of the Garcias' investment, the system was created by expressing the total amount of money to be invested as a sum of different assets - stocks, bonds, and a money market account. Each asset category had an equation associated with it based on the total amount they had for investment and their desired returns. To solve such systems, we usually look for values of our variables (in this case, the amounts to invest in each asset) that satisfy all the equations simultaneously.

Sometimes, like in the Garcias' case, we get infinitely many solutions, which is a result of having less unique equations than variables. This situation allows for flexibility in choosing the investment amounts. The key is to select the amount for one variable and calculate the others, ensuring all original conditions are still met.

Rate of Return

The rate of return is a measure of the gain or loss on an investment over a specified period, expressed as a percentage of the investment's initial cost. This metric is vital for investors as it helps them gauge the efficiency of their investments.

For Mr. and Mrs. Garcia, understanding the rate of return was crucial to ensuring that their investments yielded a specific annual income. In the exercise, each type of investment had a different rate of return, influencing how much of their total funds they would need to allocate to each type in order to achieve their target income.

Applying these rates to the corresponding investment amounts via a linear equation allowed them to determine the combination of investments that would reach the desired outcome. In real-life scenarios, investors would look for a balance that maximizes return while minimizing risk.

Investment Strategies Mathematics

Applying mathematics to investment strategies enables individuals to make informed decisions based on logical, quantitative assessments. It involves using equations and formulas to predict future performance, assess risks, and balance portfolios.

In this exercise, we applied a mathematical approach to find viable investment strategies for the Garcias. By setting up and solving a system of linear equations, we could determine the allocation of funds into various assets to meet their financial goals. This problem-solving method illustrates how mathematics guides the decision-making process in creating an effective investment plan.

Mathematical modeling, like the one used for the Garcias, is a powerful tool for investors as it can filter through multiple variables and scenarios, assessing each against the investor’s objectives. With disciplined use of mathematics, investors can systematically tackle complex financial decisions and craft strategies that align with their financial targets and risk tolerance.