Question

Huey McDuck has received $70,000 from his rich uncle and wants to determine how much to invest in each of four different investments: municipal bonds expected to return 8.5%, CDs expected to return 5%, T-bills expected to return 6.5% and a stock fund expected to return 13%.

The following guidelines have been established to ensure that the portfolio is diversified for reduced risk.

1. No more than 20% of the total investment should be in municipal bonds
2. At least 30% of the investment should be in T-bills

Write the LP formulation

Set the problem up in Excel and solve using Excel Solver

Report the optimal solution and total expected return

Answer 1

Let the amount of money invested in municipal bonds , CDs, T-bills and stock funds be x1,x2,x3, and x4 respectively.

So, the profit function which we need to maximize is Z = 1.085x1 + 1.05x2+1.065x3 + 1.13x4, this is because if x1 amount is invested in municipal bonds which have a return of 8.5% , then the amount at the end of the period would become = x1+ 8.5% * x1 = 1.085 x1 and so on.

Now we need to factor in the constraints:-

Constraint 1 : Total amount Available constraint

Since the total amount of money that can be invested is $ 70000, therefore; C1: x1+x2+x3+x4 <= 70000

Constraint 2 : Max. cap on municipal bond investment

C2: x1 < = 20%*70000

=> x1 < = 14000

Constraint 3 : Min amount to be invested in T bills

C3: x3 > 30% * 70000

=> x3>= 21000

After all the constraints and problem is defined the same can be modeled on excel as shown below:-

Formula Sheet for the above excel is :-

Type of Investment		Min Amount Allowed		Actual Amount Invested		Max Amount Allowed	Rate of Return
Municipal Bonds	x1	0	<=	0	<=	14000	0.085
CDs	x2	0	<=	0	<=		0.05
T Bills	x3	21000	<=	0	<=		0.065
Stock Fund	x4	0	<=	0	<=		0.13
Total		0	<=	=SUM(E2:E5)	<=	70000
Profit Function		=E2*(1+H2)+E3*(1+H3)+E4*(1+H4)+E5*(1+H5)

Adding the constraints in solver we have:-

Solving the problem we get :-

Type of Investment		Min Amount Allowed		Actual Amount Invested		Max Amount Allowed	Rate of Return
Muncipal Bonds	x1	0	<=	0	<=	14000	8.50%
CDs	x2	0	<=	0	<=		5.00%
T Bills	x3	21000	<=	21000	<=		6.50%
Stock Fund	x4	0	<=	49000	<=		13.00%
Total		0	<=	70000	<=	70000
Profit Function		77735

Total expected Returns = 77735

Optimal Solution is summarized below:-

Municipal Bonds	x1	=	0
CDs	x2	=	0
T Bills	x3	=	21000
Stock Fund	x4	=	49000
Total		=	70000

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