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 6 %, and on B bonds 11 %.  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 $26,000 , and the investor wants an annual return of $1,620 on the three investments. B. The values in part A are changed to $38,000 and $2,360 , respectively A. The client should invest $(?) in AAA bonds, $(?)in A bonds, and $(?)in B bonds.

Answer 1

Part A.

1) Variables:

X: amount invested in AAA bonds

Y: amount invested in A bonds

Z: amount invested in B bonds.

2) Return:

4%* X + 6% * Y + 11% * Z = 1620

3) Investment:

 X + Y + Z = 26,000

4) Ratio between AAA and B bonds.

X = 2*Z

5) Solution of the system of equations:

Replace X with 2Z in the two first equations:

2Z * 4% + Y * 6% + Z * 11% = 1620

=> 0.08Z + 0.06Y + 0.11Z = 1620

=> 0.19Z + 0.06Y = 1620

2Z + Y + Z = 26,000

=> 3Z + Y = 26,000 => Y = 26,000 - 3Z

Replace Y with 26,000 - 3Z

0.19Z + 0.06(26,000 - 3Z) = 1620

0.19Z + 1560 - 0.18Z = 1620

0.01Z = 1620 - 1560

0.01Z = 60

Z = 60 / 0.01 = 6000

=> X = 2*6000 = 12,000

=> Y = 26,000 - 12,000 - 6,000 = 8,000

Answer: 12,000 in bonds AAA, 8,000 in bonds A, and 6000 in bonds B.

Part B.

2) Return:

0.04X + 0.06 Y + 0.11Z = 2360

3) Investment:

 X + Y + Z = 38,000

4) Ratio between AAA and B bonds.

X = 2Z

5) Solution of the system of equations:

Replace X with 2Z in the two first equations:

=> 0.08Z + 0.06Y + 0.11Z = 2360

=> 0.19Z + 0.06Y = 2360

2Z + Y + Z = 38,000

=> 3Z + Y = 38,000 => Y = 38,000 - 3Z

Replace Y with 38,000 - 3Z

0.19Z + 0.06(38,000 - 3Z) = 2360

0.19Z + 2280 - 0.18Z = 2360

0.01Z = 2360 - 2280

0.01Z = 80

Z = 80 / 0.01 = 8000

=> X = 2*8000 = 16,000

=> Y = 38,000 - 16,000 - 8000 = 14,000

Answer: 16,000 in AAA bonds, 14,000 in A bonds, and 8,000 in B  bonds.