Problem 33 - Algebra and Trigonometry

Question

Finance An investor has \(\$ 100,000\) to invest in three types of bonds: short- term, intermediate-term, and long-term. How much should she invest in each type to satisfy the given conditions? $$ \begin{array}{l}{\text { Short-term bonds pay } 4 \% \text { annually, intermediate-term bonds }} \\ {\text { pay } 5 \% \text { , and long-term bonds pay } 6 \% . \text { The investor wishes }} \\ {\text { to realize a total annual income of } 5.1 \%, \text { with equal }} \\ {\text { amounts invested in short- and intermediate-term bonds. }}\end{array} $$

Step-by-Step Solution

Answer

Invest $30,000 in short-term, $30,000 in intermediate-term, and $40,000 in long-term bonds.

Step 1: Define the Variables

Let \( x \) be the amount invested in short-term bonds and \( y \) be the amount invested in intermediate-term bonds. Since the amount invested in short-term and intermediate-term bonds should be equal, we can say \( x = y \). Let \( z \) be the amount invested in long-term bonds.

Step 2: Set Up the Equations

The total investment is $100,000, so we have the equation: \( x + y + z = 100,000 \). Since \( x = y \), this equation becomes \( 2x + z = 100,000 \).

Step 3: Determine the Income Equation

The annual incomes from the bonds are: short-term: \( 0.04x \), intermediate-term: \( 0.05y \), long-term: \( 0.06z \). The total annual income needed is 5.1% of \(100,000, which is \)5,100. Thus, we have: \( 0.04x + 0.05y + 0.06z = 5,100 \). Since \( x = y \), we can rewrite it as: \( 0.04x + 0.05x + 0.06z = 5,100 \).

Step 4: Simplify the Equations

From Step 3, simplify the income equation to: \( 0.09x + 0.06z = 5,100 \). We now have two equations: \( 2x + z = 100,000 \) and \( 0.09x + 0.06z = 5,100 \).

Step 5: Solve the System of Equations

Solve the first equation for \( z \): \( z = 100,000 - 2x \). Substitute \( z \) in the second equation: \( 0.09x + 0.06(100,000 - 2x) = 5,100 \). Simplify and solve for \( x \):\[ 0.09x + 6,000 - 0.12x = 5,100 \]\[ -0.03x + 6,000 = 5,100 \]\[ -0.03x = -900 \]\[ x = 30,000 \].Since \( x = y \), \( y = 30,000 \). Using \( z = 100,000 - 2x \), find \( z \): \( z = 40,000 \).

Step 6: Verify the Solution

The amounts should be: Short-term bonds: \\(30,000, Intermediate-term bonds: \\)30,000, Long-term bonds: \\(40,000. The total annual income should be calculated as:\[ 0.04(30,000) + 0.05(30,000) + 0.06(40,000) = 1,200 + 1,500 + 2,400 = 5,100 \].This confirms the total income is indeed \)5,100.

Key Concepts

Understanding Bonds

Bonds are a type of investment where the investor loans money to an entity, typically a corporation or government, which borrows the funds for a period at a fixed interest rate. They are often viewed as a relatively safe investment, especially when compared to stocks. Bonds pay interest annually or semi-annually until they reach maturity, at which point the original investment is returned to the investor.
Bonds are generally categorized into three types based on their terms:

• Short-term bonds: These typically mature in 1 to 3 years and offer lower interest rates than longer-term bonds because they're considered lower risk.
• Intermediate-term bonds: These have maturities ranging from 4 to 10 years. They strike a balance between risk and return, offering more interest than short-term bonds.
• Long-term bonds: With maturities over 10 years, these bonds often provide the highest interest rates, reflecting the higher risk taken by investors.

When investing in bonds, one must consider the bond's issuer, the credit rating, the current interest rates, and how interest rate changes could affect bond prices.

Calculating Annual Income from Investments

Annual income from investments is essentially the money that the investments generate over a year. For bonds, this income comes in the form of interest payments.
The formula to calculate the annual income from an investment in bonds is:

• Multiply the amount invested by the interest rate.

For example, if an investor puts \(10,000 into a bond with a 5% interest rate, the annual income from this investment would be calculated as:\[ \text{Annual Income} = \text{Amount Invested} \times \text{Interest Rate} \]\[ \text{Annual Income} = 10,000 \times 0.05 = 500 \]In the context of the original exercise, the investor seeks a total of \)5,100 in annual income from differing interest bonds by balancing the allocation among them to achieve this target. The understanding is key to meeting financial goals through strategic investment allocation.

Solving Systems of Equations

A system of equations is a set of equations with the same variables that are solved together. Solving these systems allows us to find the values that satisfy all equations simultaneously.
In finance, systems of equations often help in determining how to allocate assets to achieve a financial target, like in this exercise.
Here’s a step-by-step rundown of solving a system like the one in our exercise:

• Set up the equations: Determine equations based on the problem's conditions. For example, total investment must equal the available funds, and desired income equals the income from all investments.
• Simplify and organize: Sometimes, combining similar terms or substituting values can make solving easier.
• Solve: Use substitution or elimination methods to find solutions for the variables.

For this specific problem, notice how the problem is modeled by two main equations resulting from setup steps: the total amount to be invested and the required income. Substitution is used to solve for one variable in terms of others, easing the solving process. Once you find a solution, verify by plugging back into the original equations to ensure no errors are present.