Problem 38 - Precalculus Mathematics for Calculus

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? Short-term bonds pay \(4 \%\) annually, intermediate-term bonds pay \(6 \%,\) and long-term bonds pay \(8 \% .\) The investor wishes to have a total annual return of \(\$ 6700\) on her investment, with equal amounts invested in intermediate- and long-term bonds.

Step-by-Step Solution

Answer

Invest \$ 10,000 in short-term, \$ 45,000 in intermediate-term, and \$ 45,000 in long-term bonds.

Step 1: Define Variables

Let's define the variables for the amounts invested in each type of bond. Let \( x \) be the amount invested in short-term bonds, \( y \) be the amount invested in intermediate-term bonds, and \( z \) be the amount invested in long-term bonds. According to the problem, \( y = z \). We also know that the total investment is \$ 100,000, so \( x + y + z = 100,000 \).

Step 2: Set Up the Equation for Total Return

The total annual return from the investments is obtained by adding the individual returns from each type of bond. The return from short-term bonds is \( 0.04x \), from intermediate-term bonds is \( 0.06y \), and from long-term bonds is \( 0.08z \). We want the total return to be \$ 6,700: \[ 0.04x + 0.06y + 0.08z = 6700 \]

Step 3: Use the Equality Condition for Intermediate and Long-Term Investments

We know \( y = z \). Replace \( z \) with \( y \) in both equations. This gives us two equations to solve: \( x + 2y = 100,000 \) and \( 0.04x + 0.06y + 0.08y = 6700 \). Simplifying the second equation, we have \( 0.04x + 0.14y = 6700 \).

Step 4: Solve the System of Equations

From the equation \( x + 2y = 100,000 \), we have \( x = 100,000 - 2y \). Substitute this expression for \( x \) into the second equation: \[ 0.04(100,000 - 2y) + 0.14y = 6700 \]Simplify and solve for \( y \):\[ 4000 - 0.08y + 0.14y = 6700 \]\[ 0.06y = 2700 \]\[ y = 45000 \]

Step 5: Find the Value of Short-term Investment

Using \( y = 45000 \) in the equation \( x = 100,000 - 2y \), we find:\[ x = 100,000 - 2(45000) \]\[ x = 100,000 - 90000 \]\[ x = 10000 \]

Step 6: Verify the Solution

With the values \( x = 10,000 \), \( y = 45,000 \), and \( z = 45,000 \), verify the total return: \( 0.04(10,000) + 0.06(45,000) + 0.08(45,000) = 400 + 2700 + 3600 = 6700 \), which matches the required return. Verify the total investment as well: \( 10,000 + 45,000 + 45,000 = 100,000 \). Both conditions are satisfied.

Key Concepts

System of Equations

A system of equations is used to find the values of unknown variables that satisfy the given conditions simultaneously. In finance, especially in investment allocation, this concept helps determine how much to invest in different options to meet certain goals. Here, we have three unknowns: the amounts to be invested in short-term, intermediate-term, and long-term bonds. These are represented as variables \( x \), \( y \), and \( z \) respectively.
The problem provides two key equations:

• The total amount invested: \( x + y + z = 100,000 \)
• The total annual return: \( 0.04x + 0.06y + 0.08z = 6700 \)

This forms our system of equations. Further simplification and conditions (such as \( y = z \)) allow us to solve for specific values. By treating one variable in terms of another, we narrow down the number of equations and solve them more easily.

Investment Return Calculation

Investment return calculation is essential in ensuring that the invested funds yield the expected profits each year. For bonds, the annual return is calculated by multiplying the invested amount by the bond's interest rate.
For instance, if you invest in short-term bonds with an interest rate of \(4\%\), then the return on the investment is calculated as \(0.04x\), where \(x\) is the money invested. Similarly, intermediate-term returns are \(0.06y\) and long-term are \(0.08z\).

• Short-term return: \(0.04x\)
• Intermediate-term return: \(0.06y\)
• Long-term return: \(0.08z\)

The annual target is set to a total return of \(\$6700\). By ensuring the sum of all individual returns matches this target, investors can assess their investment strategy's success. This forms part of financial planning, ensuring optimal allocation to meet financial goals.

Types of Bonds

Choosing between different types of bonds involves understanding their characteristics and associated risks and returns. Bonds typically fall into these categories:

• **Short-term Bonds:** These have maturities of up to three years and often offer lower returns because of their lower risk. They are less affected by interest rate changes.
• **Intermediate-term Bonds:** With maturities between three to ten years, these bonds offer moderate risk and moderate returns. They strike a balance between short-term flexibility and long-term growth.
• **Long-term Bonds:** These typically mature in more than ten years and often have higher yields. However, they come with higher risks due to price fluctuations over extended periods.

Investors may prefer bonds based on their investment goals. For instance, someone looking to preserve capital might lean towards short-term bonds, while someone aiming for higher returns may consider long-term bonds.
In our problem, equal investments in intermediate- and long-term bonds reflect a strategy that balances risk and return while achieving desired annual gains.