Question

A trust officer at the Blackburg National Bank needs to determine how to invest $100,000 in the following collection of bonds to maximize the annual return. The information is given in the following table:    

Bond   Annual Return(%)       Maturity     Risk   Tax Free                  

A     9.5   Long        High    Yes

B 8.0 Short   Low     Yes

C 9.0 Long    Low     No

D 9.0 Long High    Yes

E    9.0 Short    High    No              

The officer wants to invest at least 50% of the money in short-term issues and no more than 50% in high risk issues. At least 30% of the funds should go in tax-free investments and at least 40% of the total return should be tax free.

i) Formulate as an LP.

ii) Solve this using Excel’s Solver.

Answer 1

We tabulate the data as shown below:

Let the amount invested in Bond A be Xa, Bond B be Xb, Bond C be Xc, Bond D be Xd, Bond E be Xe

Total Annual return = 9.5% * Xa + 8.0% * Xb + 9.0% * Xc + 9.0% * Xd + 9.0% * Xe = 0.095Xa + 0.08Xb + 0.09Xc + 0.09Xd + 0.09Xe

We have to maximize this return. Hence, we get objective function as:

Maximize Return R = 0.095Xa + 0.08Xb + 0.09Xc + 0.09Xd + 0.09Xe

Total amount invested = $100,000

Hence,

Xa + Xb + Xc + Xd + Xe = 100,000

As given, at least 50% is invested in short term issues. Hence,

Xb + Xe >= 50%*100,000 = 50,000

No more than 50% in high risk issues. Hence,

Xa + Xd + Xe <= 50%*100,000 = 50,000

At least 30% should go in tax-free. Hence,

Xa + Xb + Xd >= 30%*100,000 = 30,000

At least 40% of the total return should be tax free. Hence,

0.095Xa + 0.08Xb + 0.09Xe >= 40% * (0.095Xa + 0.08Xb + 0.09Xc + 0.09Xd + 0.09Xe)

Hence, 0.095Xa + 0.08Xb + 0.09Xe >= 40% of Objective function

Hence, we get LPP formilation as:

Maximize Total Return R = 0.095Xa + 0.08Xb + 0.09Xc + 0.09Xd + 0.09Xe

Subject to constraints

Xa + Xb + Xc + Xd + Xe = 100,000

Xb + Xe >= 50,000

Xa + Xd + Xe <= 50,000

Xa + Xb + Xd >= 30,000

0.095Xa + 0.08Xb + 0.09Xe >= 40% of Objective function

Xa, Xb, Xc, Xd, Xe >=0.....non-negative constraint

Solving the above LPP, we get

The above table in form of formulas along with excel solver extract is shown below for better understanding and reference: