Problem 37 - 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 \(5 \%,\) and long-term bonds pay \(6 \% .\) The investor wishes to realize a total annual income of \(5.1 \%\), with equal amounts invested in short- and intermediate- term bonds.

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: Set Up Variables

Let's define the variables for the amount the investor invests 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. From the problem statement, we know that \( x = y \). We will represent the total investment by the equation \( x + y + z = 100,000 \).

Step 2: Analyze Desired Income

The investor wants a total annual income of \( 5.1\% \), which is \( 0.051 \times 100,000 = 5,100 \) dollars. Short-term bonds contribute \( 0.04x \), intermediate-term bonds contribute \( 0.05y \), and long-term bonds contribute \( 0.06z \) to the income. Set up the income equation as: \( 0.04x + 0.05y + 0.06z = 5,100 \).

Step 3: Substitute and Simplify Equations

Since \( x = y \), substitute \( y \) with \( x \) in the equations. The total investment equation becomes \( x + x + z = 100,000 \) or \( 2x + z = 100,000 \). The income equation becomes \( 0.04x + 0.05x + 0.06z = 5,100 \) or \( 0.09x + 0.06z = 5,100 \).

Step 4: Solve the System of Equations

We have the system: 1. \( 2x + z = 100,000 \) 2. \( 0.09x + 0.06z = 5,100 \).First, solve the first equation for \( z \): \( z = 100,000 - 2x \). Next, substitute \( z \) from this into the second equation:\( 0.09x + 0.06(100,000 - 2x) = 5,100 \). Simplifying,\( 0.09x + 6,000 - 0.12x = 5,100 \) gives \( -0.03x = -900 \). Thus, \( x = 30,000 \).

Step 5: Determine Investment in Each Bond

Now that we know \( x = 30,000 \), and since \( x = y \), it follows \( y = 30,000 \). Use \( z = 100,000 - 2x \) to find \( z \): \( z = 100,000 - 2(30,000) = 100,000 - 60,000 = 40,000 \).Thus, the investments are \( x = 30,000 \), \( y = 30,000 \), and \( z = 40,000 \).

Key Concepts

Bond Investment

Bond investments are financial instruments where you can lend money to an entity (like a government or corporation) and earn interest over time. Bonds come in different types, depending on their time to maturity. In this specific example, we are examining three types:

• Short-term bonds: These generally have a maturity period of around 1-3 years and in this exercise, they offer a 4% annual return.
• Intermediate-term bonds: These have slightly longer maturity, usually from 3 to 10 years, and provide a 5% annual return.
• Long-term bonds: With maturities over 10 years, these bonds provide a 6% annual return.

When investing in bonds, it's essential to balance potential returns with risk and the need for income. In this scenario, the investor wants to achieve a 5.1% return on total investment.
Deciding how much to invest in each type of bond could influence overall income and can be planned by setting specific constraints and achieving a desired blend of investment.

Finance Mathematics

Finance mathematics is a branch of applied mathematics concerned with financial markets. It helps in making decisions for investments like determining how much to allocate to different types of bonds. In this context, it involves solving problems based on interest rates and return calculations.

• To earn returns, investments need to be strategically allocated considering the interest rates offered by each bond type.
• Knowing the relationships, like the one given: amount in short-term equals amount in intermediate-term bonds, simplifies the financial calculations.
• The desired 5.1% average return translates to maximizing income from the total $100,000 investment while considering varying bond yields.

By understanding the interconnection between percentages, yield, and invested amounts, finance mathematics provides a framework to study and optimize investment strategies.

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. Solving the system involves finding values for these unknowns that satisfy all the equations simultaneously. In the case of investment distributions:

• Equation 1 (total investment): \( x + y + z = 100,000 \) represents the entire investment across the three bonds.
• Equation 2 (income goal): \( 0.04x + 0.05y + 0.06z = 5,100 \) ensures that total interest from the funds meets the desired 5.1% outcome.
• Given \( x = y \), simplify: substitute \( y \) with \( x \), simplifying the system to work with only two variables.

The process involves solving for one variable and substituting it into other equations to find remaining unknowns. For example, solving \( z = 100,000 - 2x \) and substituting in the income equation helps achieve the final answer. It's an efficient method to determine the distribution of investments ensuring the investor's goals are met.