Question

A young couple has \(\$ 25,000\) to invest. As their financial consultant, you recommend that they invest some money in Treasury bills that yield \(7 \%,\) some money in corporate bonds that yield \(9 \%,\) and some money in junk bonds that yield \(11 \% .\) Prepare a table showing the various ways that this couple can achieve the following goals: (a) \(\$ 1500\) per year in income (b) \(\$ 2000\) per year in income (c) \(\$ 2500\) per year in income (d) What advice would you give this couple regarding the income that they require and the choices available?

Short answer

Set up and solve \(x + y + z = 25,000\) and \(0.07x + 0.09y + 0.11z = required\text{ }income\) for each goal.

Step-by-step solution

Define Variables

Let \(x\) be the amount of money invested in Treasury bills, \(y\) be the amount invested in corporate bonds, and \(z\) be the amount invested in junk bonds.

Set Up Equations

Since the couple has \(\text{\textdollar} 25,000\) to invest, the first equation is: \( x + y + z = 25,000 \). Moreover, the income from investments is given by the equation: \(0.07x + 0.09y + 0.11z\).

Achieve Income Goals

Using the equations: \( x + y + z = 25,000 \) and \(0.07x + 0.09y + 0.11z\), let's solve for the various income goals.

Goal (a) - \(\text{\textdollar}1500\)

We need: \(0.07x + 0.09y + 0.11z = 1500\). Using the amount equation: \(x + y + z = 25,000\). Use matrices or substitution method to solve these equations.

Goal (b) - \(\text{\textdollar}2000\)

We need:\(0.07x + 0.09y + 0.11z = 2000\). Using the amount equation: \(x + y + z = 25,000\). Use matrices or substitution method to solve these equations.

Goal (c) - \(\text{\textdollar}2500\)

We need:\(0.07x + 0.09y + 0.11z = 2500\). Using the amount equation: \(x + y + z = 25,000\). Use matrices or substitution method to solve these equations.

Contextual Solution

Solve each system of equations to find the values of \(x, y,\) and \(z\) for each income goal.

Provide Advice

Analyze the solutions and give the couple advice about how feasible it is to achieve each income goal based on the distribution of investments.

Key concepts

systems of equations

When dealing with investment allocation, we often need to solve systems of equations to determine how much money should be invested in different options. A system of equations is a set of multiple equations that share the same variables. For our problem, we have defined three variables - let's call them x (amount invested in Treasury bills), y (amount invested in corporate bonds), and z (amount invested in junk bonds).

The first important equation we use is the total investment amount: \[ x + y + z = 25,000 \] This equation ensures that our total investment does not exceed \(25,000.

The second equation is used to define how much income we want to achieve from these investments. For example, if we aim for \)1500 in annual income, we set up the income equation: \[ 0.07x + 0.09y + 0.11z = 1500 \]

By solving these systems of equations, we can find out the exact amounts to invest in each type of bond to meet the desired income goal. Utilizing methods like substitution or matrix methods makes solving these equations more manageable.

simple interest calculations

Simple interest calculations are fundamental when it comes to calculating returns on investments such as Treasury bills, corporate bonds, and junk bonds. Simple interest is calculated using the formula: \[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]

In our exercise, the interest rates for different investment options are given - 7% for Treasury bills, 9% for corporate bonds, and 11% for junk bonds. The principal is the amount invested in each type of bond.

For example, if you invest \(10,000 in corporate bonds with an annual interest rate of 9%, the simple interest earned in one year would be: \[ 10,000 \times 0.09 = 900 \text{ dollars} \]

When you sum up the interest from all three types of investments, you get the total income from your portfolio. This sum needs to match the income goals specified, like \)1500, \(2000, or \)2500.

matrix methods

Matrix methods are extremely helpful in solving systems of linear equations, especially when dealing with multiple variables as in our investment problem. A matrix is a rectangular array of numbers arranged in rows and columns.

For our system of equations, we can express the problem in matrix form. Let's consider the goal of achieving $1500 income as an example. Our system of equations would be: \[ \begin{bmatrix} 1 & 1 & 1 \ 0.07 & 0.09 & 0.11 \ \text{Solution} \end{bmatrix} \begin{bmatrix} x \ y \ z \ \text{Solution} \end{bmatrix} = \begin{bmatrix} 25,000 \ 1500 \ \text{Solution} \ \text{ \ } \end{bmatrix} \]

To find the values of x, y, and z, we can use matrix operations such as Gaussian elimination or matrix inversion. These methods systematically simplify the matrix to find the solution for the variables. Understanding matrix methods is crucial for efficiently and accurately solving complex systems of equations that arise in investment allocation scenarios.