. Mr. David, an individual investor, has $70,000to divide among several investments. The alternative investments are municipal bonds with an 8.5% annual return, certificates of deposit with a 5% return, treasury bills with a 6.5% return, and a growth stock fund with a 13% return. The investments are all evaluated after one year. However, each investment alternative has a different perceived risk to the investor; thus, it is advisable to diversify. David wants to know how much to invest in each alternative in order to maximize the return.
a) The following guidelines have been established for diversifying the investments and lessening the risk perceived by the investor.
b) No more than 20% of the total investment should be in municipal bonds.
c) The amount invested in certificates of deposit should not exceed the amount invested in the other three alternatives
d) At least 30% of the investment should be in treasury bills and certificates of deposit.
e) To be safe, more should be invested in certificates of deposit and treasury bills than in municipal bonds and the growth stock fund by a ratio of at least 1.2 to 1.
Required: If David wants to invest the entire $70,000, formulate the Linear Programming Model
Just write algebra for the conditions. Let
w = municipal bonds
x = CD's
y = T-Bills
z = Stocks
Then the conditions are
w <= (w+x+y+z)/5
x <= w, x <= y, x <= z
x+y >= 3/10 (w+x+y+z)
x+y >= 1.2(w+z)
Rearrange things a bit, and you want to
maximize p = .085w+.05x+.065y+.13z
subject to
4w <= x+y+z
3x <= w+y+z
7x+7y >= 3w+3z
6x+6y >= 5w+5z
the constraints could be massaged into other forms if you like.
Student
Give me answer
To formulate the Linear Programming Model for this problem, we need to define the decision variables, objective function, and constraints.
Decision Variables:
Let x1 represent the amount invested in municipal bonds.
Let x2 represent the amount invested in certificates of deposit.
Let x3 represent the amount invested in treasury bills.
Let x4 represent the amount invested in the growth stock fund.
Objective Function:
The objective is to maximize the return on investments, which can be represented as:
Maximize Z = 0.085x1 + 0.05x2 + 0.065x3 + 0.13x4
Constraints:
1) No more than 20% of the total investment should be in municipal bonds:
x1 ≤ 0.2 * ($70,000)
2) The amount invested in certificates of deposit should not exceed the amount invested in the other three alternatives:
x2 ≤ x1 + x3 + x4
3) At least 30% of the investment should be in treasury bills and certificates of deposit:
x2 + x3 ≥ 0.3 * ($70,000)
4) More should be invested in certificates of deposit and treasury bills than in municipal bonds and the growth stock fund by a ratio of at least 1.2 to 1:
x2 + x3 ≥ 1.2 * (x1 + x4)
Non-negativity Constraints:
x1, x2, x3, x4 ≥ 0
By solving this Linear Programming Model, we will find the optimal values for x1, x2, x3, x4 that will maximize the return on investments while satisfying the given constraints.