Problem 59 - College Algebra and Calculus: An Applied Approach

Question

You have a total of $$\$ 500,000$$ that is to be invested in (1) certificates of deposit, (2) municipal bonds, (3) blue-chip stocks, and (4) growth or speculative stocks. How much should be put in each type of investment? The certificates of deposit pay \(2.5 \%\) simple annual interest, and the municipal bonds pay \(10 \%\) simple annual interest. Over a five-year period, you expect the blue-chip stocks to return \(12 \%\) simple annual interest and the growth stocks to return \(18 \%\) simple annual interest. You want a combined annual return of \(10 \%\) and you also want to have only one-fourth of the portfolio invested in stocks.

Step-by-Step Solution

Answer

The final solution would come out by equating the above equations and solving them. However, as there isn't a universal strategy for stock allocation, amounts for \( x_3 \) and \( x_4 \) may vary depending on your chosen strategy.

Step 1: Define the Variables

Let \( x_1 \) be the amount invested in certificates of deposit, \( x_2 \) the amount in municipal bonds, \( x_3 \) the amount in blue-chip stocks and \( x_4 \) the amount in growth stocks.

Step 2: Build the Equations

The total sum invested is $500,000, therefore we have: \[ x_1 + x_2 + x_3 + x_4 = 500,000 \] The combined annual return should be $50,000 which makes an average of 10% of total investment, this gives us: \[ 0.025x_1 + 0.1x_2 + 0.12x_3 + 0.18x_4 = 50,000 \] One-fourth of the portfolio should be in stocks, this leads to: \[ x_3 + x_4 = 500,000/4 \]

Step 3: Solve the Equations

Now, we have a system of three linear equations with four unknowns. To solve this, some methods such as substitution or elimination can be used. This exercise doesn't stipulate how exactly to distribute between the different types of stocks, hence we will need to choose a strategy for allocation of stocks to get our fourth equation.

Step 4: Choose Strategy for Allocation of Stocks

Assume we'll invest same amount into blue-chip stocks and growth stocks. Hence, we get our fourth equation: \[ x_3 = x_4 \] Now with these 4 equations, we can solve to find out the values of \( x_1, x_2, x_3, x_4 \).

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra, widely utilized in solving real-world problems involving relationships between quantities. In this exercise, we solve a system of linear equations to determine the optimal investment allocation.

The linear equations here are based on the conditions provided in the problem:

• The total amount of money invested must equal \( \\(500,000 \).
• The desired combined annual return is \( \\)50,000 \), representing a rate of \( 10\% \) on the total investment.
• A quarter of the portfolio should be allocated to stocks.
• The specific choice of investing equally in blue-chip and growth stocks.

Each condition provides a piece of the puzzle, converted into an equation involving investments in certificates of deposit, municipal bonds, blue-chip stocks, and growth stocks. By organizing these relationships as linear equations, you can systematically solve for the amounts allocated to each investment type.

Simple Interest

Simple interest is a straightforward way of calculating the interest earned or paid on an investment or loan. It is determined by multiplying the principal amount, the rate of interest, and the time period.

In this problem, each investment category generates simple interest based on a specified annual rate. For example, certificates of deposit have a \( 2.5\% \) annual rate, while growth stocks offer \( 18\% \), reflecting different risk and return characteristics. The interest equation is:
\[\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time}\]
For investment allocation, calculating the simple interest helps to estimate how much return each asset type contributes toward the overall financial goals. Understanding these returns ensures that the investment strategy aligns with desired financial outcomes.

Investment Portfolio

An investment portfolio is a collection of financial assets that can include stocks, bonds, and other securities. This diversification helps balance risk and return. In this exercise's context, we craft a portfolio that includes certificates of deposit, municipal bonds, and two kinds of stocks.

Building a portfolio involves strategic decision-making. Here, constraints like a total investment of \( \$500,000 \), a targeted \( 10\% \) return, and a limit on stock investment, guide the portfolio construction. These parameters ensure the portfolio is aligned with both risk tolerance and return expectations.

• Certificates of deposit are considered safer, offering lower returns.
• Municipal bonds provide a middle ground with moderate risk and returns.
• Blue-chip and growth stocks offer higher potential returns but come with increased risk.

This combination is key to effectively manage growth potential while controlling risk.

College Algebra

College Algebra offers the tools needed to solve practical problems like investment allocation, leveraging concepts such as linear equations and interest calculation.

The skills developed in algebra include understanding variables, building equations from real-world scenarios, and solving these equations. College Algebra often involves simplifying complex problems into manageable mathematical models, as evidenced by the system of equations in this exercise.

Mastery in College Algebra equips learners with methods to tackle financial problems efficiently and effectively, illustrating its relevance beyond the classroom. Algebra's application extends to various fields, including finance, where it plays a crucial role in decision-making and strategic planning.