Problem 66 - 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. \(\textit{Total Investment}\) \(\$10,000\) \(\textit{Annual Return}\) \(\$760\)

Step-by-Step Solution

Answer

To get the final amount of money invested in the AAA, A, and B type bonds respectively, first calculate the inverse of the given coefficient matrix and then multiply it with the given RHS matrix. The resulting values indicate the amount invested in each type of bond.

Step 1: Writing the Coefficient Matrix and the RHS Vector

Here are the two matrices formed from the system of equations:Coefficient Matrix\[ A = \left[ \begin{array}{rrr} 1 & 1 & 1 \ 0.065 & 0.07 & 0.09 \ 0 & 2 & -1 \end{array}\right] \]RHS Vector\[ B = \left[ \begin{array}{r} 10000 \ 760 \ 0 \end{array}\right] \]

Step 2: Calculating the Inverse of Coefficient Matrix

The inverse of matrix A, labelled as \(A^{-1}\), can be calculated using algebraic methods (find the determinant, then adjoint, and divide each element by the determinant), or computing technologies (like a graphing calculator or a computer software). Please compute the inverse of A, \(A^{-1}\).

Step 3: Multiplying the Inverse of A with Matrix B

In order to find the solution for \(x\), \(y\), and \(z\) (representing the amount invested in AAA, A, and B-rated bonds), multiply the inverse of A, \(A^{-1}\), with the RHS vector, \(B\). The result will be a vector containing the three values. Please multiply \(A^{-1}\) with \(B\).

Step 4: Interpreting the Solution

The resulting column vector from the multiplication contains three elements each representing the amount invested in AAA, A and B type bonds respectively. Please interpret the result to find the amount of money invested in each type.

Key Concepts

Investment Portfolio

When people think about an investment portfolio, they envision a collection of assets such as stocks, bonds, and more. In this scenario, we are examining a portfolio comprised solely of bonds: AAA-rated, A-rated, and B-rated. Each bond has a different yield:

• AAA-rated bonds: 6.5% yield
• A-rated bonds: 7% yield
• B-rated bonds: 9% yield

An essential part of managing an investment portfolio is determining how much money should be allocated to each type of bond to achieve a specific return. In our problem, we are given a total investment amount of $10,000 and an expected annual return of $760. Additionally, there is a condition linking A and B bonds - the investment in B bonds is twice that of A bonds.
Considering such conditions, mathematical tools, especially matrices, become handy to find how much was invested in each type.

Linear Equations

Linear equations form the backbone of these calculations because they allow us to represent the constraints and returns involved in this investment scenario. We use three linear equations:

• Firstly, the sum of the investments in AAA-rated, A-rated, and B-rated bonds should be equal to the total investment of $10,000.
• Secondly, we calculate the annual return from these investments, ensuring it matches the expected return of $760.
• Lastly, the equation that describes the relationship between A and B bonds is included, indicating that the amount in B bonds is twice that in A bonds.

Each equation corresponds to a line in our investment scenario, and together they form a system that can be solved to give the amounts of money (x, y, z) invested in each type of bond. Solving these equations typically involves organizing them into matrices, which streamlines the calculation process significantly.

Matrix Multiplication

Matrix multiplication is a central mathematical technique utilized in solving systems of equations, such as those found in this portfolio problem. By representing the system of linear equations using a coefficient matrix and a result matrix, we can find variables through matrix multiplication.
In this exercise, the coefficient matrix reflects the equations' constants, and the right-hand-side (RHS) vector contains the totals. The equation is simplified to \[ A \cdot X = B \]Where:

• \( A \) is the coefficient matrix,
• \( X \) is the column vector \([x, y, z]^T\), and
• \( B \) is the RHS matrix.

To find \( X \), we compute the inverse of \( A \) and multiply it by \( B \), yielding \( X = A^{-1} \cdot B \). This results in a new matrix giving the investment amounts in each bond type. This multiplication is crucial as it gives a straightforward solution to complex investment scenarios, enabling investors to determine how to allocate their resources optimally.