Q. 28 - Precalculus Enhanced with Graphing Utilities

Question

Financial Planning A retired couple has up to \($50000$ to place in fixed-income securities. Their financial adviser suggests two securities to them: one is an AAA bond that yields $8$% per annum; the other is a certificate of deposit (CD) that yields $4$%. After careful consideration of the alternatives, the couple decides to place at most \)$20000$ in the AAA bond and at least $$15000$ in the CD. They also instruct the financial adviser to place at least as much in the CD as in the AAA bond. How should the financial adviser proceed to maximize the return on their investment? 

Step-by-Step Solution

Answer

To maximize the return $$20,000$ should be invested in AAA bond and $$30,000$ should be invested in CD bond.

Step 1. Given Information

Total money to be invested in two securities is $$50,000$.

Two securities are there: one is AAA bond which yields $8$% per annum and the other is CD which yeilds $4$%.

At most $$20,000$ is placed in the AAA bond and at least $$15,000$ in the CD.

They also decided to place at least as much in the CD as in the AAA bond. 

Step 2. Form the inequalities

Let the amount invested in the AAA bond and CD bond be $x$ and $y$ dollars respectively.

The couple has a total of $\$50,000$ to be invested. So

$x+y\le 50000$

The couple decides to place at most $$20000$ in the AAA bond and at least $$15000$ in the CD, so two inequalities can be written as

$$\begin{gathered} x\le 20000 \\ y\ge 15000 \end{gathered}$$

The couple instructed to place at least as much in the CD as in the AAA bond, so the fourth inequality is

$y\ge x$

The amount invested cannot be negative so

$$\begin{gathered} x\ge 0 \\ y\ge 0 \end{gathered}$$

Step 3. Form the expression that needs to be maximized

The AAA bond that yields $8$% per annum and the CD bond yields $4$% .

So if $x$ dollars is invested in AAA bond and $y$ dollars is invested in CD bond. The return is given as $0.08x+0.04y$.

So we need to maximize the expression $R=0.08x+0.04y$

Step 4. Graph the inequalities

The inequalities are graphed and the feasible region has been shaded as

Step 5. Find the value of return at each boundary point

The feasible region is marked by five boundary points and they are

$(0,15000),(15000,15000),(20000,20000),(20000,30000),(0,50000)$

Find the return for each point and fill the table we get

$\left(x,y\right)$	$R=0.08x+0.04y$
$(0,15000)$	$0.08\left(0\right)+0.04\left(15000\right)=600$
$(15000,15000)$	$0.08\left(15000\right)+0.04\left(15000\right)=1800$
$(20000,20000)$	$0.08\left(20000\right)+0.04\left(20000\right)=2400$
$(20000,30000)$	$0.08\left(20000\right)+0.04\left(30000\right)=2800$
$(0,50000)$	$0.08\left(0\right)+0.04\left(50000\right)=2000$

So from the table, it can be seen that the maximum return possible is $2800$ dollars and it is attained when $$20000$ is invested in AAA bond and $ $15000$ in CD bond.