Chapter 11.8, Problem 28AYU

Financial Planning A retired couple have up to $\$50,000$ 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 decide to place at most $\$20,000$ in the AAA bond and at least $\$15,000$ 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?

To determine

To solve: The given linear programming problem.

Answer to Problem 28AYU

Solution:

The couple have to invest $\$20,000$ in AAA bonds and $\$30,000$ in CD.

Explanation of Solution

Given:

• Total amount to be invested is upto $\$50,000$.
• AAA bond and CD yields $8\%\text{and}4\%$ per annum.
• The amount that can be invested in AAA bond can be at most $\$20,000$.
• The amount that can be invested in CD has to be atleast 15,000.
• To place investment of at least as much in the CD as in the AAA bond.

Calculation:

Begin by assigning symbols for the two variables.

$x$ be the amount invested in AAA bonds.

$y$ be the amount invested in certificate of Deposit.

If $P$ be the total return on investment after one year,

$P=0.08x+0.04y$

The goal is to maximize $P$ subject to certain constraints on $x\text{and}y$. Because $x\text{and}y$ represents amount to be invested, the only meaningful values of $x\text{and}y$ are non-negative.

Therefore, $x\ge 0,y\ge 0$.

From the given data we get,

$x+y\le 50000$

$x\le 20000$

$y\ge 15000$

$y\ge x$

Therefore, the linear programming problem may be stated as,

Maximize, $P=0.08x+0.04y$.

Subject to,

$x\ge 0$

$y\ge 0$

$x+y\le 50000$

$x\le 20000$

$y\ge 15000$

$y\ge x$

The graph of the constraints is illustrated in the figure below.

The corner points are as follows:

Corner points are	Value of objective function
(0, 50000)	$\$2000$
(0, 15000)	$\$600$
(15000, 15000)	$\$1800$
(20000, 20000)	$\$2400$
(20000, 30000)	$\$2800$

Therefore, the couple have to invest $\$20,000$ in AAA bonds and $\$30,000$ in CD.