Problem 68 - Precalculus with Limits

Question

INVESTMENT PORTFOLIO In Exercises 65-68, consider a person who invests in AAA- rated bonds, A-rated bonds, and B-rated bonds. The average yields are 6.5% on AAA bonds, 7%on A bonds, and 9% on B bonds. The person invests twice as much in B bonds as in A bonds. Let \(x\), \(y\), and \(z\) represent the amounts invested in AAA, A, and B bonds, respectively. \(A = \left[ \begin{array}{rcl} x + y + z&=&\textrm{(total investment)} \\\ 0.065x + 0.07y + 0.09z&=&\textrm{(annual return)} \\ 2y - z&=&0 \end{array} \right]\) Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond. \(\textit{Total Investment}\) \(\$500,000\) \(\textit{Annual Return}\) \(\$38,000\)

Step-by-Step Solution

Answer

Without a specific numerical output from the inverse calculation, a short answer cannot be given. Please use a matrix calculation tool to determine the inverse of matrix \(A\) and multiply it with \(B\) as instructed in Step 3 to find the respective investment amounts for AAA, A, and B bonds.

Step 1: Represent the system of equations as a matrix

Create a matrix \(A\) where each row represents a coefficient of \(x\), \(y\), and \(z\) from the respective equation in the system. Also create a column vector \(B\) that represents the constant on the right side of each equation. As such, we have:\[A = \begin{bmatrix} 1 & 1 & 1 \ 0.065 & 0.07 & 0.09 \ 0 & 2 & -1 \end{bmatrix} , B = \begin{bmatrix} 500000 \ 38000 \ 0 \end{bmatrix}\]

Step 2: Compute the inverse of matrix A

This step involves calculating the inverse of the matrix \(A\), denoted as \(A^{-1}\). The inverse of a matrix is computed using a number of methods, including the adjoint method or the row operations method. Compute this using a calculator.

Step 3: Solve for vector x

After computing \(A^{-1}\), solve for the vector \(x\) (which represents the amounts invested in each type of bond) by multiplying \(A^{-1}\) with vector \(B\). That is, \(x = A^{-1}B\). The result will be a column vector whose elements represent the solutions for \(x\), \(y\), and \(z\) respectively (amounts to be invested in AAA, A, and B bonds).

Step 4: Interpret the solution

Each entry in the solution vector \(x\) corresponds to the amount of money to be invested in each type of bond. The first entry represents the investment in AAA bonds (\(x\)), the second represents the investment in A bonds (\(y\)), and the third represents the investment in B bonds (\(z\)).

Key Concepts

System of Equations

A system of equations consists of two or more equations that share variables. In this investment scenario, you have three equations involving three types of bonds: AAA, A-rated, and B-rated. Each equation describes a different aspect of the investment.

- The first equation shows that the total investment is equal to the sum of the amounts invested in each type of bond. Think of this equation as your budgetary constraint. - The second one outlines the expected annual return from these investments, calculated as a weighted sum of the yields from each bond category. Here, percentages become coefficients as they relate to the principal, or in this case, the amount invested. - The third equation establishes a relationship between the investments in A-rated and B-rated bonds. Specifically, it states that twice as much is invested in B-rated bonds as in A-rated bonds.

Systems of equations like these can be solved using a variety of techniques, including substitution, elimination, and matrix methods.

Investment Returns

Investment returns refer to the gain or loss on an investment over a specified period, expressed as a percentage of the original investment. In this problem, the returns are based on the interest or yield from different bond investments.

Here's how it breaks down:

- **AAA-rated bonds** offer a return of 6.5%. - **A-rated bonds** provide a 7% yield. - **B-rated bonds** offer a 9% return.

These percentages represent the income from the bonds relative to the money invested. Essentially, they are profits derived annually. Calculating expected returns is essential for making informed investment decisions. Knowing how your money grows allows for strategic financial planning, aligning with your financial goals.

In this exercise, we derive the total expected return using the second equation in the system. By solving it using matrix algebra, we determine the optimal investment in each bond type to meet the desired annual return of $38,000.

Bond Investment

Bonds are essentially loans made by investors to either corporate entities or the government. In exchange for this loan, investors receive interest payments, known as yields, until the bond's maturity. The exercise focuses on three types of bonds: AAA, A-rated, and B-rated, each with varying risks and returns.

Here's a quick overview of these bond types:

- **AAA-rated bonds** are high-grade and deemed to have the lowest risk. Hence, they offer lower returns compared to other bonds. - **A-rated bonds** have a moderate risk level with slightly better returns than AAA bonds. - **B-rated bonds** pose a higher risk, but they compensate for this with higher potential returns.

In this example, a strategy is developed where the person invests twice as much in B-rated bonds as in A-rated bonds. This strategy impacts potential returns significantly. Understanding bond investment, the associated risks and the structured return, is crucial for a balanced investment portfolio.

Matrix Representation

A matrix is a rectangular array of numbers that can be used to represent and solve systems of linear equations. In this problem, we use matrices to efficiently handle multiple equations and solve for the unknowns.

Here's a step-by-step to how matrix representation applies to our exercise:

- The **coefficient matrix** has elements corresponding to the coefficients of the variables in the equations.- The **column vector** represents the constants from the equations (e.g., total investment, annual return).- The complete matrix equation looks like this: \ \[ A \cdot x = B \] \ Where \( A \) is the coefficient matrix, \( x \) is a column vector of the variables, and \( B \) is the constant matrix.
Once the matrix is set up, we calculate the inverse of matrix \( A \) to solve for \( x \). The product of the inverse matrix and the constant vector gives us the amounts invested in each type of bond. Using matrices simplifies the process of solving complex systems of equations and is a powerful tool in financial computations.