Chapter 11.8, Problem 23AYU

Return on Investment An investment broker is instructed by her client to invest up to $\$20,000$, some in a junk bond yielding $9\%$ per annum and some in Treasury bills yielding $7\%$per annum. The client wants to invest at least $\$8,000$ in T-bills and no more than $\$12,000$ in the junk bond.

(a) How much should the broker recommend that the client place in each investment to maximize income if the client insists that the amount invested in T-bills must equal or exceed the amount placed in the junk bond?

(b) How much should the broker recommend that the client place in each investment to maximize income if the client insists that the amount invested in T-bills must not exceed the amount placed in the junk bond?

To determine

To solve: The given linear programming problem.

Answer to Problem 23AYU

Solution:

a. The maximum return on investment is $\$1500$, when $\$10,000$ is invested in T-Bills and $\$10,000$ is invested in junk bond.

Explanation of Solution

Given:

• Total investment upto $\$20,000$.
• Minimum investment in Treasury Bills $\$8,000$.
• Maximum investment in junk bond $\$12,000$.
• Interest from Treasury Bills $=7\%$.
• Interest from junk bond $=9\%$.

Calculation:

Begin by assigning symbols for the two variables.

$x=$ Amount invested in T-Bills.

$y=$ Amount invested in junk bond.

a. If $P$ is the total return from both the investment, then

$P=0.07x+0.09y$

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

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

From the given data we get

$x+y\le 20,000$

$x\ge 8,000$

$y\le 12,000$

$x\ge y$

Therefore, the linear programming problem may be stated as

Maximize $P=0.07x+0.09y$

Subject to

$x\ge 0$

$y\ge 0$

$x+y\le 20,000$

$x\ge 8,000$

$y\le 12,000$

$x\ge y$

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

Corner points are	Value of objective function
$\left(20000,0\right)$	$\$1,400$
$\left(8000,0\right)$	$\$560$
$\left(8000,8000\right)$	$\$1,280$
$\left(10000,10000\right)$	$\$1500$

The maximum return on investment is $\$1500$, when $\$10,000$ is invested in T-Bills and $\$10,000$ is invested in junk bond.

To determine

To solve: The given linear programming problem.

Answer to Problem 23AYU

Solution:

b. The maximum return on investment is $\$1640$, when $\$8,000$ is invested in T-Bills and $\$12,000$ is invested in junk bond.

Explanation of Solution

Given:

• Total investment upto $\$20,000$.
• Minimum investment in Treasury Bills $\$8,000$.
• Maximum investment in junk bond $\$12,000$.
• Interest from Treasury Bills $=7\%$.
• Interest from junk bond $=9\%$.

Calculation:

Begin by assigning symbols for the two variables.

$x=$ Amount invested in T-Bills.

$x=$ Amount invested in junk bond.

b. If $P$ is the total return from both the investment, then

$P=0.07x+0.09y$

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

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

From the given data we get

$x+y\le 20,000$

$x\ge 8,000$

$y\le 12,000$

$x<y$

Therefore, the linear programming problem may be stated as

Maximize $P=0.07x+0.09y$

Subject to

$x\ge 0$

$y\ge 0$

$x+y\le 20,000$

$x\ge 8,000$

$y\le 12,000$

$x<y$

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

Corner points are	Value of objective function
$\left(8000,12000\right)$	$\$1640$
$\left(8000,8000\right)$	$\$1,280$
$\left(10000,10000\right)$	$\$1500$

The maximum return on investment is $\$1640$, when $\$8,000$ is invested in T-Bills and $\$12,000$ is invested in junk bond.