Question

Kathleen Allen has $100,000 to divide among several investments. The alternative investments are municipal bonds with a 5% return, certificate of deposits with a 5% return, treasury bills with a 6.5% return, and a growth stock with a 13% annual return. To minimize risk, Kathleen follows these guidelines:

1. No more than 20% of the total investment should be in municipal bonds.

2. The amount invested in certificates of deposit should not exceed the total amount invested in the other three alternatives.

3. At least 30% of the investments should be in treasury bills and certificate of deposits.

4. The amount invested in CDs and treasury bills should equal to or greater than the amount invested in municipal bonds and the growth stock fund. Kathleen wants to know how much to invest in each alternative in order to maximize the return.

Answer 1

  

ment alternative:X1=amount($) invested in municipal bondX2 =amount ($) invested in certificates of deposit

X3= amount ($) invested in treasury bills

X4= amount ($) invested in growth stock fund

The Objective Function

The objective of the investor is to maximize the total return from the

investmentin the four alternatives. The total return is the sum of the individual returns fromeach alternative. Thus, the objective function is expressed as

MAXIMIZE Z = $0.05x1+0.X2+0.65X3+0.130X4

Model Constraints

In this problem the constraints are the guidelines established for diversifying thetotal investment. Each guideline is transformed into a mathematical constraintseparately.The first guideline states that no more than 20% of the total investment should be≤in municipal bonds. The total investment is $100000; 20% of $100000 is $200000.Thus, this constraint is X1≤ $2000,000

second guideline indicates that the amount invested in certificates of depositshould not exceed the amount invested in the other three alternatives. Becausethe investment in certificates of deposit is x2

and the amount invested in the otheralternatives is

, the constraint isxX1,X2,X3,X4

X2≤ X1+X4+X3

This constraint is not in what we referred to as standard form because allvariables would be on the left-hand side of the inequality (≤), and all the numericvalues would be on the right side. We will convert the constraint to

X2-X1-X3-x4<=0

third guideline specifies that at least 30% of the investment should be intreasury bills and certificates of deposit. Because 30% of $100,000 is $21,000 and

the amount invested in certificates of deposit and treasury bills is represented by

X2+X3 the constraint isx2+X3

≥ 3000000The forth guideline states that the ratio of the amount invested in certificates ofdeposit and treasury bills to the amount invested in municipal bonds and the growthstock fund should be at least 1.2 to 1.[(x2+X3)/X1+X4]>=1.2

is constraint is not in standard linear programming form because of thefractional relationship of the decision variables, (x2+X4)/X1+X4

It is convertedasfollows:

X2+X3≥ 1.2(x1+X4)-1.2X1+X2+X3-1.2≥ 0Finally the investor wants to invest the entire $70,000 in the four alternatives.Thus, the sum of all the investments in the four alternatives must

equal

$100,000.x1+X2+X3+X4=100000