Question

nnis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client’s needs.

For a new client, Innis has been authorized to invest up to $3 million in five investment funds: stock fund A, stock fund B, Bond fund A, Bond fund B and a money market fund. Stock fund A provides an annual rate of return of 15%, stock fund B provides an annual rate of return of 12.5%, Bond fund A provides an annual rate of return of 8%, Bond fund B provides an annual return of 9% and the money market fund provides an annual rate of return of 4%. The client wants to maximize return subject to the requirement that the total annual risk index be no more than $2.65 million. According to Innis’s risk measurement system, each dollar invested in stock fund A has a risk index of 0.8, each dollar invested in stock fund B has a risk index of 0.7, each dollar invested in Bond fund A has a risk index of 0.4, each dollar invested in Bond fund B has a risk index of 0.48 and each dollar invested in the money market fund has a risk index of 0.2. The higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innis’s client also specified that at least $300,000 be invested in stock fund B. The amount invested in stock fund A must be at least 12% of the total investment. The amount invested in bond funds must be within +/- 7% of the total amount invested in stocks.

Formulate a Linear Programming model to determine optimal investment amounts.

Answer 1

Let X1,X2,X3,X4,And X5 Be The Amount Of Funds Invested In Stock Fund A, Stock Fund B, Bond Fund A, Bond Fund B, And Money Market Fund.

Objective Function : Maximize Return

Maximize Z = 0.15X1+0.125X2+0.08X3+0.09X4+0.04X5 .

Constraint Equation 1 :   Innis has been authorized to invest up to $3 million in five investment funds.

So X1+X2+X3+X4+X5 $\leq$ $30,00,000.

Constraint Equation 2   : at least $300,000 be invested in stock fund B.

   X2 $\geq$   $300,000.

Constraint Equation 3 : The amount invested in stock fund A must be at least 12% of the total investment.

X1  $\geq$ 0.12(X1+X2+X3+X4+X5)

Constraint Equation 4 : The amount invested in bond funds must be within +/- 7% of the total amount invested in stocks.

X3+X4  $\geq$ 0.07(X1+X2)

(OR)

X3+X4 $\leq$   0.07(X1+X2)

Constraint Equation 5 : Risk Index For Investment Of Each Doller.

X1 = 0.8

X2 = 0.7

X3 = 0.4

X4 = 0.48 And

X5 = 0.2

Constraint Equation 6 :  the total annual risk index be no more than $2.65 million.

0.8X1+0.7X2+0.4X3+0.48X4+0.2X5 < $ 26,50,000.