Problem 34 - Precalculus: Mathematical 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

Short-term: $10,000, Intermediate-term: $45,000, Long-term: $45,000.

Step 1: Define Variables

Let's designate variables to represent the amount invested in each bond type. Let \( x \) represent the amount invested in short-term bonds, \( y \) represent the amount invested in intermediate-term bonds, and \( z \) represent the amount invested in long-term bonds. According to the problem, \( y = z \), which means equal amounts are invested in intermediate- and long-term bonds.

Step 2: Formulate the Equations

We have three key pieces of information that translate into equations:- Total investment: \( x + y + z = 100,000 \)- Total annual return: \( 0.04x + 0.06y + 0.08z = 6700 \)Additionally, since \( y = z \), we can simplify our equations using this relationship.

Step 3: Simplify Equations

Since \( y = z \), substitute \( z \) for \( y \) in the equations. This gives us:1. \( x + 2y = 100,000 \)2. \( 0.04x + 0.06y + 0.08y = 6700 \), simplifying to \( 0.04x + 0.14y = 6700 \).

Step 4: Solve for 'x' and 'y'

We will now solve this system of equations:1. \( x + 2y = 100,000 \)2. \( 0.04x + 0.14y = 6700 \)From the first equation: \( x = 100,000 - 2y \).Substitute \( x \) in the second equation:\( 0.04(100,000 - 2y) + 0.14y = 6700 \)Simplify and solve for \( y \):\( 4000 - 0.08y + 0.14y = 6700 \)\( 0.06y = 2700 \)\( y = \frac{2700}{0.06} = 45000 \)

Step 5: Calculate 'x' and 'z'

Using \( y = 45000 \) and \( x = 100,000 - 2y \), we find:\( x = 100,000 - 2(45000) = 10000 \)Since \( z = y = 45000 \), \( z \) is also 45000.

Step 6: Conclusion

The investor should invest \( \\(10,000 \) in short-term bonds, \( \\)45,000 \) in intermediate-term bonds, and \( \$45,000 \) in long-term bonds to satisfy the conditions.

Key Concepts

Bonds

Bonds are a type of investment where an investor loans money to an entity, like a government or corporation. In return, the issuer of the bond promises to repay the loan amount on a set date, while paying interest over time. This interest is known as the coupon rate.
Bonds can be categorized based on their term:

• Short-term bonds: These typically mature in a few months to a few years. They are considered safer but usually offer lower interest rates.
• Intermediate-term bonds: These bonds mature in about 3 to 10 years. They often offer higher returns compared to short-term bonds but come with moderate risk.
• Long-term bonds: These have maturity periods longer than 10 years and can offer even higher returns, along with higher risk.

In the exercise, our investor distributes her capital among these three types of bonds to achieve a certain return rate.

Financial Mathematics

Financial mathematics involves using mathematical methods to solve financial problems, like determining optimal investment strategies. It helps us understand how interest rates, returns, and investments impact financial decisions.
Key concepts in financial mathematics used for bond investment solutions include:

• Interest rates: The percentage at which money grows over a period. Every type of bond offers a distinct interest rate, influencing the total return.
• Annual return: The total amount of money received from an investment each year, derived from interest earnings.
• Equation formulation: Using the given data to set up mathematical equations that model the investment situation.

This combination of financial and mathematical understanding is necessary to create effective investment plans and solve problems like the one in the exercise.

System of Equations

A system of equations is a set of two or more equations with a common set of variables. Solving these systems helps find the values of the variables that satisfy all equations simultaneously.
In the context of our bond investment problem, a system of equations is formed based on:

• Total investment equation: The sum of all amounts invested across different bond types should equal the total available investment, i.e., \(x + y + z = 100,000\).
• Total annual return equation: The expected return is calculated based on bond interest rates, i.e., \(0.04x + 0.06y + 0.08z = 6700\).
• Equality condition: In this case, the same amount is invested in intermediate and long-term bonds, hence, \(y = z\).

The solution involves algebraically manipulating these equations to find the amounts to invest in various bond types.

Annual Return

The annual return is the financial gain generated by an investment in a year. It’s a key metric for investors to assess the performance of their investments.
Calculating annual return involves:

• Interest income: The interest paid by bonds annually. For short-term bonds, this is \(4\%\), for intermediate-term it’s \(6\%\), and for long-term bonds, it’s \(8\%\).
• Total return: The combined interest from all bond investments, which needs to meet the investor’s target, as in the exercise with a target of \$6700.

By understanding the annual return, investors can adjust their strategy to meet specific investment goals, ensuring they achieve the desired returns across different types of bonds.