Linear Programming Applications in Finance

Linear Programming Applications in Finance. Michelle Permejo Nory Conanan King John Dominique Zoleta.

Linear Programming. mathematical modeling technique in which a linear function is maximized or minimized when subjected to various constraints . This technique has been useful for guiding quantitative decisions in business planning, in industrial engineering , and—to a lesser extent—in the social and physical sciences ..

Linear Programming (cont.). The solution of a linear programming problem reduces to finding the optimum value (largest or smallest, depending on the problem) of the linear expression (called the objective function ) subject to a set of constraints expressed as inequalities:.

Linear Programming in Finance Applications.

Portfolio Selection in Linear Programming.

Portfolio Selection. Portfolio selection problems involve situations in which a financial manager must select specific investments—for example, stocks and bonds—from a variety of investment alternatives. Managers of mutual funds, credit unions, insurance companies, and banks frequently encounter this type of problem. The objective function for portfolio selection problems usually is maximization of expected return or minimization of risk. The constraints usually reflect restrictions on the type of permissible investments, state laws, company policy, maximum permissible risk, and so on. Problems of this type have been formulated and solved using a variety of mathematical programming techniques. In this section we formulate and solve a portfolio selection problem as a linear program..

Portfolio Selection Example:. Consider the case of Welte Mutual Funds, Inc., located in New York City. Welte just obtained $100,000 by converting industrial bonds to cash and is now looking for other investment opportunities for these funds. Based on Welte’s current investments, the firm’s top financial analyst recommends that all new investments be made in the oil industry, in the steel industry, or in government bonds. Specifically, the analyst identified five investment opportunities and projected their annual rates of return. The investments and rates of return are shown in Table 9.3..

Portfolio Selection Example:. Management of Welte imposed the following investment guidelines: 1. Neither industry (oil or steel) should receive more than $50,000. 2. Government bonds should be at least 25% of the steel industry investments. 3. The investment in Pacific Oil, the high-return but high-risk investment, cannot be more than 60% of the total oil industry investment..

Portfolio Selection Example:. What portfolio recommendations—investments and amounts—should be made for the available $100,000? Given the objective of maximizing projected return subject to the budgetary and managerially imposed constraints, we can answer this question by formulating and solving a linear programming model of the problem. The solution will provide investment recommendations for the management of Welte Mutual Funds..

Portfolio Selection Example:. Graphical user interface, application, table Description automatically generated.

Let A = dollars invested in Atlantic Oil P = dollars invested in Pacific Oil M = dollars invested in Midwest Steel H = dollars invested in Huber Steel G = dollars invested in government bonds.

The requirement that government bonds be at least 25% of the steel industry investment is expressed as:.

FIGURE 9.3 SENSITIVITY REPORT FOR THE WELTE MUTUAL FUNDS PROBLEM Variable Cells . file WEB Model Variable A P M H G N ame Atlantic Oil Amount Invested Pacific Oil Amount Invested Midwest Steel Amount Invested Huber Steel Amount Invested Gov't Bonds Amount Invested Final Value Final Value Reduced Cost -0.011 Shadow Price 0.069 0.022 -0.024 0.030 Objective Coefficient 0.073 o. 103 0.064 0.075 0.045 Constraint R.H. Side 0.000 0.000 Allowable Increase 0.030 IE+30 0.01 1 0.0275 0.030 Allou able Increase IE+30 Allowable Decrease 0.055 0.030 IE+30 0.011 IE+30 Allowable Decrease Constraints Constraint Number 1 2 3 4 5 N ame Avl- Funds Oil Max Steel Max Gov't Bonds Pacific Oil.

Portfolio Selection Example:. The sensitivity based on Excel Solver for this linear program is shown in Figure 9.3. Table 9.4 shows how the funds are divided among the securities. Note that the optimal solution indicates that the portfolio should be diversified among all the investment opportunities except Midwest Steel. The projected annual return for this portfolio is 0.073(20000) + 0.103(30000) + 0.064(0) + 0.075(40000) + 0.045(10000) = $8000, which is an overall return of 8%..

Portfolio Selection Example.

Financial Planning in Linear Programming.

Financial Planning. Hewlitt Corporation established an early retirement program as part of its corporate restructuring. At the close of the voluntary sign-up period, 68 employees had elected early retirement. As a result of these early retirements, the company incurs the following obligations over the next eight years: The cash requirements (in thousands of dollars) are due at the beginning of each year..

Financial Planning. The corporate treasurer must determine how much money must be set aside today to meet the eight yearly financial obligations as they come due. The financing plan for the retirement program includes investments in government bonds as well as savings. The investments in government bonds are limited to three choices: The government bonds have a par value of $1000, which means that even with different prices each bond pays $1000 at maturity. The rates shown are based on the par value. For purposes of planning, the treasurer assumed that any funds not invested in bonds will be placed in savings and earn interest at an annual rate of 4%..

We define the decision variables as follows: F = total dollars required to meet the retirement plan’s eight-year obligation B1 = units of bond 1 purchased at the beginning of year 1 B2 = units of bond 2 purchased at the beginning of year 1 B3 = units of bond 3 purchased at the beginning of year 1 Si = amount placed in savings at the beginning of year i for i 1, . . . , 8.

The funds available at the beginning of year 1 are given by F. With a current price of $1150 for bond 1 and investments expressed in thousands of dollars, the total investment for B1 units of bond 1 would be 1.15B1. Similarly, the total investment in bonds 2 and 3 would be 1B2 and 1.35B3, respectively. The investment in savings for year 1 is S1. Using these results and the first-year obligation of 430, we obtain the constraint for year 1:.

Similarly, the constraints for years 3 to 8 are:.

file WEB Hewlitt I URE Variable Cells Model Variable B2 Constraints Constraint Number 2 3 4 5 6 7 8 SENSITIVITY REPORT FOR THE HEWLITT CORPORATION CASH REQUIREMENTS PROBLEM N ame Dollars Needed Bond I - Year I Bond 2 - Year 2 Bond 3 - Year 3 Savings Year I Savings Year 2 Savings Year 3 Savings Year 4 Savings Year 5 Savings Year 6 Savings Year 7 Savings Year 8 Name Year I Flow Year 2 Flow Year 3 Flow Year 4 Flow Year 5 Flow Year 6 Flow Year 7 Flow Year 8 Flow Final Value 1728.794 144.988 187.856 228.188 636.148 501.606 349.682 182.681 0.000 0.000 0.000 0.000 Final Value 430.000 210.000 222.000 231.000 240.000 195.000 225.000 255.000 Reduced Cost 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.064 0.013 0.021 0.671 Shadow Price 1.000 0.962 0.925 0.889 0.855 0.760 0.719 0.671 ()bjeetive Coefficient 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Constraint R.H. Side 430.000 210.000 222.000 231.000 240.000 195.000 225.000 255.000 Allowable Increase IE+30 0.067 0.013 0.023 0.110 0.143 0.211 0.414 IE+30 IE+30 IE+30 IE+30 Allowable Increase IE+30 IE+30 IE+30 IE+30 IE+30 2149.928 3027.962 1583.882 Allowable Decrease 0.013 0.020 0.750 0.055 0.057 0.059 0.061 0.064 0.013 0.021 0.671 Allowable Decrease 1728.794 661.594 521.670 363.669 189.988 157.856 198.188 255.000.

Financial Planning Example. The optimal solution and sensitivity report based on Excel Solver is shown in Figure 9.4. With an objective function value of F 1,728.794, the total investment required to meet the retirement plan’s eight-year obligation is $1,728,794. Using the current prices of $1,150, $1000, and $1350 for each of the bonds, respectively, we can summarize the initial investments in the three bonds as follows:.

Product Mix in Linear Programming.

Transportation Problem in Linear Programming.