1 points TanFin12 2.3.041. Mr. and Mrs. Garcia have a total of $100,000 to be in

Question

1 points TanFin12 2.3.041. Mr. and Mrs. Garcia have a total of $100,000 to be invested in stocks, bonds, and a money market account. The stocks have a rate of return of 6%/year, while the bonds and the money market account pay 4%/year and 2%/year, respectively. The Garcias have stipulated that the amount invested in stocks should be equal to the sum of the amount invested in bonds and 3 times the amount invested in the money market account. How should the Garcias allocate their resources if they require an annual income of $5,000 from their investments? Give two specific options. (Let x1, Vi, and z1 refer to one option for investing money in stocks, bonds, and the money market account respectively. Let x2, Y2, and z2 refer to a second option for investing money in stocks, bonds, and the money market account respectively.) 3. / My Notes Ask Your Teacher Need Help? Readt

Explanation / Answer

Let the amounts invested in the stocks, bonds and the money market account be $ x,y and z respectively. Then, x +y +z = 100000…(1) as the sum of all investments is $ 100000.

Also, since the investment in the stocks is equal to the sum of the investment in bonds and 3 times the investment in the money market account, we have x = y+3z or, x-y-3z = 0…(2).

Also, since the total annual returns from these investments is $ 5000, we have 0.06x+ 0.04y+0.02z = 5000. On multiplying both the sides by 50, we get 3x+2y+z = 250000…(3).

The augmented matrix of the above linear system is A =

1

1

1

100000

1

-1

-3

0

3

2

1

250000

To solve the above linear system, we will reduce A to its RREF as under:

Add -1 times the 1st row to the 2nd row

Add -3 times the 1st row to the 3rd row

Multiply the 2nd row by -1/2

Add 1 times the 2nd row to the 3rd row

Add -1 times the 2nd row to the 1st row

Then the RREF of A is

1

0

-1

50000

0

1

2

50000

0

0

0

0

It implies that x-z = 50000 or, x = 50000+z and y+2z = 50000 or, y = 50000-2z. Then (x,y,z) = (50000+z,50000-2z,z).

Now, let z1 = 10000 and z2 = 20000. Then, (x1,y1,z1) =( 60000, 30000,10000) and (x2,y2,z2) =(70000,10000,20000).

1

1

1

100000

1

-1

-3

0

3

2

1

250000

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