Problem 34 - 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? $$ \begin{array}{l}{\text { Short-term bonds pay } 4 \% \text { annually, intermediate-term bonds }} \\ {\text { pay } 6 \%, \text { and long-term bonds pay } 8 \% . \text { The investor wishes }} \\ {\text { to have a total annual return of } \$ 6700 \text { on her investment, }} \\ {\text { with equal amounts invested in intermediate- and long-term }} \\ {\text { bonds. }}\end{array} $$

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 denote the amounts invested in short-term, intermediate-term, and long-term bonds by \( S \), \( I \), and \( L \) respectively.

Step 2: Establish Equations

Based on the problem, the first equation relates to the total investment: \[ S + I + L = 100,000 \]

Step 3: Set Up Earnings Equation

The second equation relates to the total annual return of $6,700: \[ 0.04S + 0.06I + 0.08L = 6,700 \]

Step 4: Equal Investment in Intermediate and Long-term Bonds

The problem states equal investment in intermediate- and long-term bonds, which gives us the third equation: \[ I = L \]

Step 5: Simplify Using Equal Investments

Substitute \( L = I \) into both equations: 1. \( S + 2I = 100,000 \) 2. \( 0.04S + 0.06I + 0.08I = 6,700 \)

Step 6: Solve the System of Equations

Substitute \( I \) and simplify:From the first equation: \( S = 100,000 - 2I \).Substitute \( S \) into the earnings equation:\[ 0.04(100,000 - 2I) + 0.14I = 6,700 \] \[ 4,000 - 0.08I + 0.14I = 6,700 \] \[ 0.06I = 2,700 \] \[ I = 45,000 \]

Step 7: Determine Remaining Investments

With \( I = L = 45,000 \), find \( S \): \[ S = 100,000 - 2 imes 45,000 \] \[ S = 10,000 \]

Step 8: Verify the Solution

Verify that the total amount invested and returns meet the conditions:Total invested: \( 10,000 + 45,000 + 45,000 = 100,000 \).Total return: \( 0.04 \times 10,000 + 0.06 \times 45,000 + 0.08 \times 45,000 = 400 + 2,700 + 3,600 = 6,700 \). Both conditions are satisfied.

Key Concepts

Bonds

Bonds are a type of fixed-income investment where you lend money to an entity for a defined period at a fixed or variable interest rate. When you invest in bonds, essentially you’re acting as a creditor, lending your money to the bond issuer, which can be a corporation or government entity. They promise to return your principal when the bond matures and pay regular interest, known as the coupon rate, during the bond's lifetime.

• Short-term bonds: Typically mature in 1 to 3 years and tend to offer lower yields due to their reduced risk and shorter investment period.
• Intermediate-term bonds: These mature in 3 to 10 years and generally offer better yields than short-term bonds.
• Long-term bonds: With maturities over 10 years, they usually offer higher yields due to increased exposure to interest rate changes and inflation.

Investors choose bonds to diversify their portfolio, manage risk, and secure reliable income through interest payments.

System of Equations

A system of equations is a set of multiple equations, all using the same variables, seeking a common solution. Solving a system of equations means finding values for the variables that satisfy all equations simultaneously. It's a foundational concept in algebra and crucial in solving real-world problems efficiently.
To illustrate in this context, the investor's scenario involves three equations:

• The sum of investments in short-term, intermediate-term, and long-term bonds equals the total available funds, represented by: \[ S + I + L = 100,000 \]
• The total expected return, which combines returns from all types of bonds, is represented by: \[ 0.04S + 0.06I + 0.08L = 6,700 \]
• And the condition of equal investment in both intermediate and long-term bonds is given by: \[ I = L \]

Using these equations, one can resolve the uncertainties to find precise investment figures allocating the total sum optimally, while satisfying all given conditions of yield and equality among bond types.

Annual Return

The annual return is the percentage change in an investment's value over a year, taking into account all capital appreciation plus distributions such as dividends, interest, and other income. Understanding how to calculate and interpret this return is pivotal for investors wanting to measure how much they gain over time.
In this investment problem, the annual return is established from the set return rates of different bonds:

• Short-term bond: Returns 4% annually.
• Intermediate-term bond: Provides a 6% return.
• Long-term bond: Offers a higher yield at 8% due to longer commitment and higher risk.

Thus, to achieve a calculated total annual return of $6,700, the investor mixes these bond types strategically, ensuring that the returns align with her investment requirements while considering potential risks involved in longer-term bonds.