Chapter 11.2, Problem 85AYU

Financial Planning 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?

[Hint: Higher yields generally carry more risk.]

To determine

To find: 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.

Answer to Problem 85AYU

Amount Invested at
$7\%$	$9\%$	$11\%$
0	10000	10000
1000	8000	11000
2000	6000	12000
3000	4000	13000
4000	2000	14000
5000	0	15000

Explanation of Solution

Given:

Financial Planning 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 $11\%$.

Calculation:

Let the amount invested in treasury bills be $x$, in corporate bonds be $y$ and in junk bonds be $x$. Since the treasury bills yield $7\%$, corporate bonds yield $9\%$ and junk bonds yield $11\%$ and they together yield $\$2000$.

$0.07x+0.09y+0.11z=2000$   -----(1)

a. Since the first couple has $\$20000$ to invest, we can write the second equation.

$x+y+z=20000$   -----(2)

Let $x=0$, from (2) $z=20000-y$

$2000=0.09y+\left(20000-y\right)0.11$

$2000=2200-0.02y$

$0.02y=200$

$y=10000$

Therefore $z=10000$.

Let $x=1000$, from (2) $z=19000-y$

$0.07x=70$

$1930=0.09y+\left(19000-y\right)0.11$

$1930=2090-0.02y$

$0.02y=160$

$y=8000$

Therefore $z=11000$.

Let $x=2000$, from (2) $z=18000-y$

$0.07x=140$

$1860=0.09y+\left(18000-y\right)0.11$

$1860=1980-0.02y$

$0.02y=120$

$y=6000$

Therefore $z=12000$.

Let $x=3000$, from (2) $z=17000-y$

$0.07x=210$

$1790=0.09y+\left(17000-y\right)0.11$

$1790=1870-0.02y$

$0.02y=80$

$y=4000$

Therefore $z=13000$.

Let $x=4000$, from (2) $z=16000-y$

$0.07x=280$

$1720=0.09y+\left(16000-y\right)0.11$

$1720=1760-0.02y$

$0.02y=40$

$y=2000$

Therefore $z=14000$.

Let $x=5000$, from (2) $z=15000-y$

$0.07x=350$

$1650=0.09y+\left(15000-y\right)0.11$

$1650=1650-0.02y$

$0.02y=0$

$y=0$

Therefore $z=15000$.

Amount Invested at
$7\%$	$9\%$	$11\%$
0	10000	10000
1000	8000	11000
2000	6000	12000
3000	4000	13000
4000	2000	14000
5000	0	15000

To determine

To find: A table for each couple showing the various ways that their goals can be achieved:

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

Answer to Problem 85AYU

Amount Invested at
$7\%$	$9\%$	$11\%$
12500	12500	0
14500	8500	2000
16500	4500	4000
18750	0	6250

Explanation of Solution

Given:

Financial Planning 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 $11\%$.

Calculation:

Let the amount invested in treasury bills be $x$, in corporate bonds be $y$ and in junk bonds be $x$. Since the treasury bills yield $7\%$, corporate bonds yield $9\%$ and junk bonds yield $11\%$ and they together yield $\$2000$.

$0.07x+0.09y+0.11z=2000$   -----(1)

b. Since the second couple has $\$25000$ to invest, we can write the second equation.

$x+y+z=25000$   -----(2)

Let $z=0$, from (2) $x=25000-y$

$2000=0.09y+\left(25000-y\right)0.07$

$2000=1750+0.02y$

$0.02y=250$

$y=12500$

Therefore $x=12500$.

Let $z=2000$, from (2) $x=23000-y$

$1780=0.09y+\left(23000-y\right)0.07$

$1780=1610+0.02y$

$0.02y=170$

$y=8500$

Therefore $x=14500$.

Let $z=4000$, from (2) $x=21000-y$

$1560=0.09y+\left(21000-y\right)0.07$

$1560=1470+0.02y$

$0.02y=90$

$y=4500$

Therefore $x=16500$.

Let $z=6250$, from (2) $x=18750-y$

$1312.5=0.09y+\left(18750-y\right)0.07$

$1312.5=1312.5+0.02y$

$0.02y=00$

$y=0$

Therefore $x=18750$.

Amount Invested at
$7\%$	$9\%$	$11\%$
12500	12500	0
14500	8500	2000
16500	4500	4000
18750	0	6250

To determine

To find: A table for each couple showing the various ways that their goals can be achieved:

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

Answer to Problem 85AYU

See explanation

Explanation of Solution

Given:

Financial Planning 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 $11\%$.

Calculation:

Let the amount invested in treasury bills be $x$, in corporate bonds be $y$ and in junk bonds be $x$. Since the treasury bills yield $7\%$, corporate bonds yield $9\%$ and junk bonds yield $11\%$ and they together yield $\$2000$.

$0.07x+0.09y+0.11z=2000$   -----(1)

c. All the money invested at $7\%$ provides $\$2100$ more than what we required.

To determine

To find: A table for each couple showing the various ways that their goals can be achieved:

d. What advice would you give each couple regarding the amount to invest and the choices available?

Answer to Problem 85AYU

d. Advice: Higher interest rates come with higher risk. If you are concerned about the risk, then more money should be put into corporate bond and little or none into Junk Bonds. Therefore, minimizing risk means living with less income.

Explanation of Solution

Given:

Financial Planning 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 $11\%$.

Calculation:

Let the amount invested in treasury bills be $x$, in corporate bonds be $y$ and in junk bonds be $x$. Since the treasury bills yield $7\%$, corporate bonds yield $9\%$ and junk bonds yield $11\%$ and they together yield $\$2000$.

$0.07x+0.09y+0.11z=2000$   -----(1)

d. Advice: Higher interest rates come with higher risk. If you are concerned about the risk, then more money should be put into corporate bond and little or none into Junk Bonds. Therefore, minimizing risk means living with less income.