Question

Financial Planning A recently retired couple needs \(\$ 12,000\) per year to supplement their Social Security. They have \(\$ 300,000\) to invest to obtain this income. They have decided on two investment options: AA bonds yielding \(5 \%\) per annum and a Bank Certificate yielding \(2.5 \%\). (a) How much should be invested in each to realize exactly \(\$ 12,000 ?\) (b) If, after 2 years, the couple requires \(\$ 14,000\) per year in income, how should they reallocate their investment to achieve the new amount?

Short answer

The couple should invest \$180,000 in AA bonds and \$120,000 in the Bank Certificate to realize \$12,000 per year. To realize \$14,000 per year after 2 years, they should invest \$260,000 in AA bonds and \$40,000 in the Bank Certificate.

Step-by-step solution

Define Variables

Let’s define the variables. Let x be the amount invested in AA bonds and y be the amount invested in the Bank Certificate. Thus, our equations are formed from the given yield rates.\[ x + y = 300,000 \] \[ 0.05x + 0.025y = 12,000 \]

Formulate Equations

We have the following system of equations from the given information:From the total investment:\[ x + y = 300,000 \]From the annual income requirement:\[ 0.05x + 0.025y = 12,000 \]

Solve the System of Equations

First, solve the first equation for y:\[ y = 300,000 - x \]Substitute this into the second equation:\[ 0.05x + 0.025(300,000 - x) = 12,000 \]Simplify and solve for x:\[ 0.05x + 7,500 - 0.025x = 12,000 \]\[ 0.025x = 4,500 \]\[ x = 180,000 \]Then, find y by substituting x back into the first equation:\[ y = 300,000 - 180,000 \]\[ y = 120,000 \]

Formulate Equations for New Requirement

After 2 years, their income requirement increases to \(14,000\). Thus, the new equation becomes:\[ 0.05x + 0.025y = 14,000 \] The total investment remains the same:\[ x + y = 300,000 \]

Solve the System of Equations for New Requirement

First, solve the first equation for y:\[ y = 300,000 - x \]Substitute this into the second equation:\[ 0.05x + 0.025(300,000 - x) = 14,000 \]Simplify and solve for x:\[ 0.05x + 7,500 - 0.025x = 14,000 \]\[ 0.025x = 6,500 \]\[ x = 260,000 \]Then, find y by substituting x back into the first equation:\[ y = 300,000 - 260,000 \]\[ y = 40,000 \]

Key concepts

Financial Planning

Financial planning is essential for managing income and investments, especially for retirement. It ensures that you allocate your resources properly to meet current and future needs. In this exercise, the retired couple needs to invest their savings wisely to supplement their yearly Social Security income. They selected two investment options with different yields, affecting how much they’ll earn annually from these investments. Proper financial planning encompasses:

• Setting clear financial goals like the couple’s required annual income.


• Understanding different investment options and their yields.


• Formulating a strategy to meet financial needs both now and in the future.

By choosing AA bonds and a Bank Certificate, the couple looked to balance their investments to meet their financial targets.

System of Equations

A system of equations is a set of two or more equations with the same variables. To solve for the variables, we find values that satisfy all equations in the system simultaneously. Here, the couple's total investment and required annual income create a system of two equations:
\[ x + y = 300,000 \ 0.05x + 0.025y = 12,000 \]
Both equations represent constraints on the investment amounts (\(x\) and \( y \)). Solving this system helps determine how much to allocate to each investment option to meet their income goal. Techniques like substitution or elimination can be used to find the solution.

Linear Equations

Linear equations are equations of the first degree, meaning they graph as straight lines. Each variable in a linear equation is raised to the power of one. In this context, the equations:
\[ x + y = 300,000 \ 0.05x + 0.025y = 12,000 \]
are linear because the variables (\(x\text{ and }y\text{)\)}) have exponents of 1. Solving these linear equations involves finding the point where their graphs intersect, representing the unique solution that satisfies both investment constraints. The simplicity of linear equations makes them ideal for modeling financial scenarios like this.

Investment Yield

Investment yield is the income return on an investment, expressed as a percentage of the invested amount. It helps investors understand how much they will earn from their investments. Here, the AA bonds yield 5%, and the Bank Certificate yields 2.5%. These percentages are crucial because:

• They determine the annual income derived from each investment.


• They influence how much needs to be invested in each to meet the income requirements.


• Reallocating investments affects future yields and income.

By solving the exercise's equations, we find the optimal investment amounts in each option to achieve the desired yield and income.