Question

An investment firm recommends that a client invest in bonds rated​ AAA, A, and B. The average yield on AAA bonds is 4​%, on A bonds 5​%, and on B bonds 8​%. The client wants to invest twice as much in AAA bonds as in B bonds. How much should be invested in each type of bond under the following​ conditions? A. The total investment is ​$28 comma 000​, and the investor wants an annual return of ​$1 comma 460 on the three investments. B. The values in part A are changed to ​$38 comma 000 and ​$1 comma 990​, respectively.

Answer 1

let A = AAA bonds that yield 4%

let a = A bonds that yield 5%

let B = B bonds that yield 8%

A = 2B

A)

A + a + B = 28,000 (replace A with 2B)

0.04A + 0.05a + 0.08B = 1,460 (replace A with 2B)

2B + a + B = a + 3B = 28,000 ⇒ a = 28,000 - 3B

0.08B + 0.05a + 0.08B = 0.05a + 0.16B = 1,460 (replace a with 28,000 - 3B)

0.05(28,000 - 3B) + 0.16B = 1,400 - 0.15B + 0.16B =1,460

1,400 + 0.01B = 1,460

0.01B = 1,460 - 1,400 = 60

B = 60 / 0.01 = $6,000

A = $12,000

a = $28,000 - $6,000 - $12,000 = $10,000

AAA bonds = $12,000

A bonds = $10,000

B bonds = $6,000

B)

A + a + B = 38,000 (replace A with 2B)

0.04A + 0.05a + 0.08B = 1,990 (replace A with 2B)

2B + a + B = a + 3B = 38,000 ⇒ a = 38,000 - 3B

0.08B + 0.05a + 0.08B = 0.05a + 0.16B = 1,990 (replace a with 28,000 - 3B)

0.05(38,000 - 3B) + 0.16B = 1,900 - 0.15B + 0.16B =1,990

1,900 + 0.01B = 1,990

0.01B = 1,990 - 1,900 = 90

B = 90 / 0.01 = $9,000

A = $18,000

a = $38,000 - $9,000 - $18,000 = $11,000

AAA bonds = $18,000

A bonds = $11,000

B bonds = $9,000