Q. 85 - Precalculus Enhanced with Graphing Utilities

Question

Three retired couples each require an additional annual income of \(2000 per year. As their financial consultant, you recommend that they invest some money in Treasury bills that yield 7%, some money in corporate bonds that yield 9%, and some money in junk bonds that yield 11%. Prepare a table for each couple showing the various ways that their goals can be achieved:

(a) If the first couple has \)20,000 to invest.

(b) If the second couple has \(25,000 to invest.

(c) If the third couple has \)30,000 to invest.

(d) What advice would you give each couple regarding the

amount to invest and the choices available?

Step-by-Step Solution

Answer

(a) Table of investment for the first couple is 

(b) Table of investment for the second couple

(c) All amounts can be invested in treasury bills.

(d) All couples can get more than $$2000$ if they invest the whole amount on junk bonds but a higher yield generally carry more risk so it is better to invest in all three such that they can get more than $$2000$

Step 1. Given information

require an additional annual income is $$2000$

investment in Treasury bills is $7\%$

Investment in corporate bonds $9$%

Investment in junk bonds is $11$%

 the first couple has $$20000$ to invest.

the second couple has $$25000$ to invest.

the third couple has $$30000$ to invest.

Step 2. Part (a)

Consider investment amount in Treasury bills is x, a corporate bond is y, and junk bonds are z.

so $0.07x+0.09y+0.11z=2000 \cdots \left(i\right)$

 the first couple has $20,000 to invest.

so $x+y+z=2000 \cdots \left(ii\right)$

Step 2. expression for y and z

Multiply equation ii by $0.09$ and subtract from equation i

$$\begin{gathered} 0.09x+0.09y+0.09z=1800 \\ \underline{-(0.07x+0.09y+0.11z=2000)} \\ 0.02x-0.02z=-200 \end{gathered}$$

so $z=x+10000$

Multiply equation ii by $0.11$ and subtract from equation i 

$$\begin{gathered} 0.11x+0.11y+0.11z=1800 \\ \underline{-(0.07x+0.09y+0.11z=2000)} \\ 0.04x+0.02y=-200 \end{gathered}$$

so$y=10000-2x$

Step 4. Table of values

Substitute different values of x and determine respective values of y and z

Step 5. Part (b)

the second couple has $$25000$ to invest.

so $x+y+z=25000 \cdots \left(iii\right)$

Multiply equation iii by $0.09$ and subtract from equation i

$$\begin{gathered} 0.09x+0.09y+0.09z=2250 \\ \underline{-(0.07x+0.09y+0.11z=2000)} \\ 0.02x-0.02z=250 \end{gathered}$$

so $z=x-12500$

Multiply equation iii by $0.11$ and subtract from equation i 

$$\begin{gathered} 0.11x+0.11y+0.11z=2750 \\ \underline{-(0.07x+0.09y+0.11z=2000)} \\ 0.04x+0.02y=750 \end{gathered}$$

so$y=37500-2x$

Step 6. Table of values

Step 7. Part (c)

the second couple has $$30000$ to invest.

so $x+y+z=30000 \cdots \left(iv\right)$

Multiply equation iv by $0.09$ and subtract from equation i

$$\begin{gathered} 0.09x+0.09y+0.09z=2700 \\ \underline{-(0.07x+0.09y+0.11z=2000)} \\ 0.02x-0.02z=700 \end{gathered}$$

so $z=x-35000$

Multiply equation iv by $0.11$ and subtract from equation i 

$$\begin{gathered} 0.11x+0.11y+0.11z=3300 \\ \underline{-(0.07x+0.09y+0.11z=2000)} \\ 0.04x+0.02y=1300 \end{gathered}$$

so$y=65000-2x$

All amounts can be invested on treasury bill 

$0.07\times 30000=2100$

Becouse $7$% of $$30000$ is $$2100$ which is more than required income $$2000$

Step 8. Part (d)

All couples can get more than $$2000$ if they invest the whole amount on junk bonds but a higher yield generally carry more risk so it is better to invest in all three such that they can get more than $$2000$