Question

Three retired couples each require an additional annual income of \(\$ 2000\) per year. As their financial consultant, you recommend that they invest some money in Treasury bills that yield \(7 \%\), some money in corporate bonds that yield \(9 \%,\) and some money in "junk bonds" that yield \(11 \%\). Prepare a table for each couple showing the various ways that their goals can be achieved: (a) If the first couple has \(\$ 20,000\) to invest. (b) If the second couple has \(\$ 25,000\) to invest. (c) If the third couple has \(\$ 30,000\) to invest. (d) What advice would you give each couple regarding the amount to invest and the choices available?

Short answer

Define investment amounts, form equations, solve for each couple’s budget, and suggest a balanced investment strategy.

Step-by-step solution

- Define Variables

Let the amount invested in Treasury bills be denoted by \( x \), in corporate bonds by \( y \), and in junk bonds by \( z \). Note that the total return from these investments must equal \$2000 per year.

- Formulate Equations

The returns on the investments can be expressed as: \[ 0.07x + 0.09y + 0.11z = 2000 \] Additionally, the total amount invested by each couple must equal their respective budgets: \[ x + y + z = 20000 \text{ for the first couple} \] \[ x + y + z = 25000 \text{ for the second couple} \] \[ x + y + z = 30000 \text{ for the third couple} \]

- Create a Table for Various Allocations for the First Couple

For each allocation, solve the system of equations: \[ 0.07x + 0.09y + 0.11z = 2000 \] \[ x + y + z = 20000 \] Example to consider: Invest \( x = 10000 \), \( y = 5000 \), \( z = 5000 \).

- Create a Table for Various Allocations for the Second Couple

For each allocation, solve the system of equations: \[ 0.07x + 0.09y + 0.11z = 2000 \] \[ x + y + z = 25000 \] Example to consider: Invest \( x = 12000 \), \( y = 8000 \), \( z = 5000 \).

- Create a Table for Various Allocations for the Third Couple

For each allocation, solve the system of equations: \[ 0.07x + 0.09y + 0.11z = 2000 \] \[ x + y + z = 30000 \] Example to consider: Invest \( x = 15000 \), \( y = 10000 \), \( z = 5000 \).

- Advise Each Couple

Advise each couple to balance between safety and yield. Suggest a greater percentage investment in safer Treasury bills, a moderate amount in corporate bonds, and a smaller amount in junk bonds. For precise allocations, personal risk tolerance and current market conditions should be considered.

Key concepts

systems of equations

A system of equations is a set of multiple equations that you solve together. In this case, we have two main equations for each couple. The first equation tells us the total return from the investments: \[ 0.07x + 0.09y + 0.11z = 2000 \] The second equation represents the total amount invested: \[ x + y + z = \text{{budget}} \] To find solutions, we need to find the values of \( x \), \( y \), and \( z \) that satisfy both equations. Using different values for the budget, we apply the same method to solve for each couple. By solving these systems, we can determine various ways to distribute their investments to achieve the desired returns.

investment strategies

Investment strategies are plans to allocate money in various financial products to achieve specific financial goals. For the retired couples, the strategy involves investing in a mix of Treasury bills, corporate bonds, and junk bonds. Here's what to consider:

• **Treasury bills:** These are low-risk investments with a yield of **7%**.
• **Corporate bonds:** Moderate-risk investments that yield **9%**.
• **Junk bonds:** Higher-risk options with an **11%** yield.

A balanced strategy means distributing investments to manage risk while achieving the required annual return of \(\$2000\). We can create multiple scenarios to see different allocations and their benefits.

financial planning

Financial planning involves setting goals and devising a plan to achieve them using various financial instruments. In our scenario, each couple's goal is to ensure an additional \(\$2000\) annual income through investments. Steps for good financial planning here include:

• **Goal Definition:** Each couple needs an extra \(\$2000\) per year.
• **Resource Assessment:** Each couple has available funds of \(\$20000\), \(\$25000\), and \(\$30000\) respectively.
• **Strategy Formulation:** Determine how much to invest in each type of bond to meet the goal.

By following these steps, the couples can create diverse portfolios that cater to their income needs and risk preferences. Ensuring a mixture of safer and riskier investments helps manage potential losses while aiming for higher returns.

yield calculation

Yield calculation is about finding the percentage return on an investment over a period. In our problem, the yield from each type of bond is: \[ \text{Treasury bills: } 0.07x \] \[ \text{Corporate bonds: } 0.09y \] \[ \text{Junk bonds: } 0.11z \] To find the total yield, we sum the returns: \[ 0.07x + 0.09y + 0.11z = 2000 \] This equation helps us identify the right amounts to invest in each type of bond to achieve the \(\$2000\) target. By doing these calculations, we can design an investment portfolio that provides the necessary return while considering the risk associated with each type of bond.