Problem 52 - Precalculus with Limits

Question

Investment Portfolio A total of \(\$ 32,000\) is invested in two municipal bonds that pay 5.75\(\%\) and 6.25\(\%\) simple interest. The investor wants an annual interest income of \(\$ 1900\) from the investments. What amount should be invested in the 5.75\(\%\) bond?

Step-by-Step Solution

Answer

The investor should place approximately \$11818.18 in the bond that gives a 5.75% return in order to achieve a total return of \$1900.

Step 1: Define Variables

Let \( x \) represent the amount to be invested at 5.75%, therefore, \( \$32000 - x \) will represent the amount to be invested at 6.25%.

Step 2: Set Up Equation

According to the question, the sum of the interest gained from both investments is \$1900. This can be represented as follows: \( 0.0575x + 0.0625(\$ 32000-x) = \$1900 \)

Step 3: Simplify and Solve the Equation

Simplify the equation and solve for \( x \), the amount investment at 5.75%. This equates to \( x \approx \$11818.18 \) after performing the algebraic operations.

Key Concepts

Investment Portfolio

Constructing a balanced investment portfolio is crucial for achieving specific financial goals. In the context of our exercise, the investor is presented with an opportunity to allocate funds between two different municipal bonds offering distinct interest rates of 5.75% and 6.25%. This scenario exemplifies the decisions investors make to optimize returns based on their desired outcome—in this case, a targeted annual interest income of $1900.

The concept of an investment portfolio goes beyond just bonds and may include a range of financial assets such as stocks, bonds, real estate, mutual funds, and more. Diversification, or spreading investments across various asset types and sectors, can reduce risk and enhance the potential for returns. Investors balance their portfolios according to their risk tolerance, investment horizon, and financial targets. Furthermore, understanding the underlying principles of simple interest, as applied in this scenario, is essential for predicting returns and making informed investment decisions.

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. They are fundamental tools for solving a myriad of problems, including those in financial mathematics. In our exercise, we set up an algebraic equation to represent the relationship between the amounts invested in the two bonds and the desired interest.

The equation, 0.0575x + 0.0625(32000 - x) = 1900, allows us to calculate the variable x, which symbolizes the amount of money to be invested in the 5.75% bond. The remaining amount, 32000 - x, automatically adjusts to fit the investment in the 6.25% bond. Through systematic operations such as distribution, combination of like terms, and solving for the variable, we find the solution to our problem. Algebraic equations are the backbone of such calculations, well-illustrated in financial and investment planning.

Financial Mathematics

Financial mathematics is a field that applies mathematical methods to solve problems in finance. It typically involves concepts such as simple and compound interest, present and future value, and amortization, to name a few. In our exercise, we utilize simple interest, a fundamental concept where the interest payment for each period is fixed and based only on the original principal.

Simple interest is calculated using the formula I = Prt, where I is the interest earned or paid, P is the principal amount invested or borrowed, r is the interest rate, and t is the time in years. The straightforward nature of simple interest makes it easier to compute than compound interest, making it a suitable choice for short-term investments. Understanding simple interest is an essential part of financial mathematics and is pivotal for effective financial planning and analysis. By mastering these principles, students can gain deeper insights into how money grows over time, which is vital for managing personal finances and building investment strategies.