Q. 23 - Precalculus Enhanced with Graphing Utilities

Question

Return on Investment An investment broker is instructed by her client to invest up to $\backslash (20,000$, 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 $\backslash )8000$ in T-bills and no more than $\$12000$ 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 junk bonds?
(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 junk bonds? 

Step-by-Step Solution

Answer

The maximum returns are attained when the investment in the junk bond is $x=\$10000$ and the investment in the Treasury bills is $y=\$10000$. The maximum return from the investment is S1600. 

Step 1. Given information

Let the amount invested in the junk bond and Treasury bills be x and y.
The broker decides to deposit atleast $\$8000$ in the T-bills and no more than $\$12000$ in the junk bond.

$$\begin{gathered} x\le 12,000 \\ y\ge 8000 \end{gathered}$$

Step 2. Finding the value of R

The client has a total of $\$20,000$ to be invested in the bonds. Also they instruct their broker that the amount invested in T-bills must equal or exceed the amount placed in junk bonds.

$$\begin{gathered} x+y\le 20,000 \\ y\ge x \end{gathered}$$

The interest obtained from the junk bond and T-bills are $9\%$ and $7\%$ respectively. The return function can be written as follows:

$R=0.09x+0.07y$

Step 3. Finding the maximum return from the investment

The above obtained inequalities can be graphed to obtain the vertices in order to obtain a maximum of the return from the amount invested given by  $P=0.08x+0.04y$

The return from the investment has to be evaluated on the vertices of the feasible region obtained in the above graph. We have tabulated the returns from the total investment in the table given below. 

Vertex	Value of the return function $R=0.09x+0.07y$
$(0,8000)$	$R=0.09\left(0\right)+0.07\left(8000\right)=560$
$(8000,8000)$	$R=0.09\left(8000\right)+0.07\left(8000\right)=1280$
$(10000,10000)$	$R=0.09\left(10000\right)+0.07\left(10000\right)=1600$
$(0,20000)$	$R=0.09\left(0\right)+0.07\left(20000\right)=1400$

The maximum returns are attained when the investment in the junk bond is $x=\$10000$ and the investment in the Treasury bills is $y=\$10000$. The maximum return from the investment is  $\$1600$