LINEAR PROGRAMMING APPLICATIONS IN MARKETING, FINANCE, AND OPERATIONS MANAGEMENT (2/3) Chapter 4 MANGT 521 (B): Quantitative Management

2 Chapter 4 (2/3)-2 Financial Applications n LP can be used in financial decision-making that involves capital budgeting, make-or-buy, asset allocation, portfolio selection, financial planning, and more. n Portfolio selection problems involve choosing specific investments – for example, stocks and bonds – from a variety of investment alternatives. n This type of problem is faced by managers of banks, mutual funds, and insurance companies. n The objective function usually is maximization of expected return or minimization of risk.

3 Chapter 4 (2/3)-3 Financial Applications #1: Portfolio Selection Bane Savings has $20 million available for investment. It wishes to invest over the next four months in such a way that it will maximize the total interest earned over the four month period as well as have at least $10 million available at the start of the fifth month for a high rise building venture in which it will be participating.

4 Chapter 4 (2/3)-4 Financial Applications #1: Portfolio Selection For the time being, Bane wishes to invest only in 2-month government bonds (earning 2% over the 2-month period) and 3-month construction loans (earning 6% over the 3-month period). Each of these is available each month for investment. Funds not invested in these two investments are liquid and earn 3/4 of 1% per month when invested locally.

5 Chapter 4 (2/3)-5 Financial Applications #1: Portfolio Selection Formulate a linear program that will help Bane Savings determine how to invest over the next four months if at no time does it wish to have more than $8 million in either government bonds or construction loans.

6 Chapter 4 (2/3)-6 Portfolio Selection n Define the Decision Variables G i = amount of new investment in government G i = amount of new investment in government bonds in month i (for i = 1, 2, 3, 4) bonds in month i (for i = 1, 2, 3, 4) C i = amount of new investment in construction C i = amount of new investment in construction loans in month i (for i = 1, 2, 3, 4) loans in month i (for i = 1, 2, 3, 4) L i = amount invested locally in month i, L i = amount invested locally in month i, (for i = 1, 2, 3, 4) (for i = 1, 2, 3, 4)

7 Chapter 4 (2/3)-7 Portfolio Selection n Define the Objective Function Maximize total interest earned in the 4-month period: Maximize total interest earned in the 4-month period: Max (interest rate on investment) X (amount invested) Max (interest rate on investment) X (amount invested) Max.02G G G G 4 Max.02G G G G C C C C C C C C L L L L L L L L 4

8 Chapter 4 (2/3)-8 Portfolio Selection n Define the Constraints Month 1's total investment limited to $20 million: Month 1's total investment limited to $20 million: (1) G 1 + C 1 + L 1 = 20,000,000 (1) G 1 + C 1 + L 1 = 20,000,000 Month 2's total investment limited to principle and interest invested locally in Month 1: Month 2's total investment limited to principle and interest invested locally in Month 1: (2) G 2 + C 2 + L 2 = L 1 (2) G 2 + C 2 + L 2 = L 1 or G 2 + C L 1 + L 2 = 0 or G 2 + C L 1 + L 2 = 0

9 Chapter 4 (2/3)-9 Portfolio Selection n Define the Constraints (continued) Month 3's total investment amount limited to principle and interest invested in government bonds in Month 1 and locally invested in Month 2: (3) G 3 + C 3 + L 3 = 1.02 G L 2 (3) G 3 + C 3 + L 3 = 1.02 G L 2 or G 1 + G 3 + C L 2 + L 3 = 0 or G 1 + G 3 + C L 2 + L 3 = 0

10 Chapter 4 (2/3)-10 Portfolio Selection n Define the Constraints (continued) Month 4's total investment limited to principle and interest invested in construction loans in Month 1, government bonds in Month 2, and locally invested in Month 3: (4) G 4 + C 4 + L 4 = 1.06 C G L 3 (4) G 4 + C 4 + L 4 = 1.06 C G L 3 or G 2 + G C 1 + C L 3 + L 4 = 0 or G 2 + G C 1 + C L 3 + L 4 = 0 $10 million must be available at start of Month 5: (5) 1.06 C G L 4 > 10,000,000 (5) 1.06 C G L 4 > 10,000,000

11 Chapter 4 (2/3)-11 Portfolio Selection n Define the Constraints (continued) No more than $8 million in government bonds at any time: (6) G 1 < 8,000,000 (6) G 1 < 8,000,000 (7) G 1 + G 2 < 8,000,000 (7) G 1 + G 2 < 8,000,000 (8) G 2 + G 3 < 8,000,000 (8) G 2 + G 3 < 8,000,000 (9) G 3 + G 4 < 8,000,000 (9) G 3 + G 4 < 8,000,000

12 Chapter 4 (2/3)-12 Portfolio Selection n Define the Constraints (continued) No more than $8 million in construction loans at any time: (10) C 1 < 8,000,000 (10) C 1 < 8,000,000 (11) C 1 + C 2 < 8,000,000 (11) C 1 + C 2 < 8,000,000 (12) C 1 + C 2 + C 3 < 8,000,000 (12) C 1 + C 2 + C 3 < 8,000,000 (13) C 2 + C 3 + C 4 < 8,000,000 (13) C 2 + C 3 + C 4 < 8,000,000Non-negativity: G i, C i, L i > 0 for i = 1, 2, 3, 4 G i, C i, L i > 0 for i = 1, 2, 3, 4

13 Chapter 4 (2/3)-13 Portfolio Selection n Computer Solution Output Objective Function Value = Objective Function Value = Variable Value Reduced Costs Variable Value Reduced Costs G G G G G G G G C C C C C C C C L L L L L L L L Value of the optimal solution Optimal solution The change in the optimal value of the solution per unit increase in the RHS of the constraint.

14 Chapter 4 (2/3)-14 Financial Applications #2: Financial Planning Chen 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: Year Year Cash Cash Required Required The cash requirements (in thousands of dollars) are due at the beginning of each year.

15 Chapter 4 (2/3)-15 Financial Applications #2: 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 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: Years to Years to Bond PriceRate (%)Maturity Bond PriceRate (%)Maturity 1$ $

16 Chapter 4 (2/3)-16 Financial Applications #2: Financial Planning 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%. 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%.

17 Chapter 4 (2/3)-17 Financial Planning n Define the Decision Variables F = total dollars required to meet the retirement F = total dollars required to meet the retirement plan’s eight-year obligation plan’s eight-year obligation B 1 = units of bond 1 purchased at the beginning of B 1 = units of bond 1 purchased at the beginning of year 1 year 1 B 2 = units of bond 2 purchased at the beginning of B 2 = units of bond 2 purchased at the beginning of year 1 year 1 B 3 = units of bond 3 purchased at the beginning of B 3 = units of bond 3 purchased at the beginning of year 1 year 1 S = amount placed in savings at the beginning of S = amount placed in savings at the beginning of year i for i = 1,..., 8 year i for i = 1,..., 8

18 Chapter 4 (2/3)-18 Financial Planning n Define the Objective Function The objective function is to minimize the total dollars needed to meet the retirement plan’s eight-year obligation: Min F The objective function is to minimize the total dollars needed to meet the retirement plan’s eight-year obligation: Min F n Define the Constraints A key feature of this type of financial planning problem is that a constraint must be formulated for each year of the planning horizon. Its form is: A key feature of this type of financial planning problem is that a constraint must be formulated for each year of the planning horizon. Its form is: (Funds available at the beginning of the year) (Funds available at the beginning of the year) - (Funds invested in bonds and placed in savings) - (Funds invested in bonds and placed in savings) = (Cash obligation for the current year) = (Cash obligation for the current year)

19 Chapter 4 (2/3)-19 Financial Planning n Define the Constraints A constraint must be formulated for each year of the planning horizon in the following form: A constraint must be formulated for each year of the planning horizon in the following form: Year 1: F – 1.15 B 1 – 1 B 2 – 1.35 B 3 – S 1 = 430 Year 2: B B B 3 – 1.04 S 1 - S 2 = 210 Year 3: B B B 3 – 1.04 S 2 – S 3 = 222 Year 4: B B B 3 – 1.04 S 3 – S 4 = 231 Year 5: B B B 3 – 1.04 S 4 – S 5 = 240 Year 6: B B B 3 – 1.04 S 5 – S 6 = 195 Year 7: B B 3 – 1.04 S 6 – S 7 = 225 Year 8: B 3 – 1.04 S 7 – S 8 = 255

20 Chapter 4 (2/3)-20 Financial Planning n Optimal solution to the 12-variable, 8-constraint LP problem: Minimum total obligation = $1,728,794 Minimum total obligation = $1,728,794 Bond Units Purchased Investment Amount 1 B1 = $1150( ) = $166,736 1 B1 = $1150( ) = $166,736 2 B2 = $1000( ) = $187,856 2 B2 = $1000( ) = $187,856 3 B3 = $1350( ) = $308,054 3 B3 = $1350( ) = $308,054