Problem 1) Application of Linear Programming in Finance: Investment Portfolio Pr
ID: 354597 • Letter: P
Question
Problem 1) Application of Linear Programming in Finance: Investment Portfolio Problem A wealthy investor has a total of $500,000 available for investment. Three investment opportunities are available at the beginning of each of the next 4 years (Year 1, Year 2, Year 3, and Year 4) as shown in the table below. All available money at the beginning of each year must be invested. Immediate reinvestment is possible for each alternative. The investor wants to determine a plan that will maximize the amount of money that she will have at the beginning of year 5. Your Linear Programming expertise is needed here! Investment Max | Return % | Availability Timing of Invest. (beginning o) Return A Unlimited3 Years 1,2,3,4 1 year later B Unlimited6.2 Years 1,2,3 2 years later Years 1,2 3 years later $200,000 10 (a) Define all decision variables of an LP problem that solves the above investment portfolio problem as carefully and completely as you can. Hint: You will need different variables for each of the three investment options in each of the four years (when/if applicabale) (b) Write down all constraints of the problem. Hint: The main constraint in each year guarantees the sum of available money in the beginning of that year equals to the total to be invested then. (c) What is the objective function of the model? (d) MPL Part: Enter the complete model in MPL (with the exception of nonnegativity constraints), and solve it. Please print and attach your solution file only (file name" sol) to your project report. (e) Based on the answer you found in MPL, how much funds should be invested in each in each of the four years? What is maximum cash available to the investor in the option beginning of year 5? () What is the marginal return on the portfolio? Hint: The answer can be found under the 'shadow price 1. Make sure to fully explain the interpretation of the marginal return here. "of the main constraint of YearExplanation / Answer
A)
Decision Variable:
Let
Ai = amount to be invested in investment A for years i = 1, 2, 3, and 4
Bi = amount to be invested in investment B for years i = 1, 2, and 3
Ci = amount to be invested in investment C for years i = 1 and 2
B)
Constraints:
Beginning of Year 1, $500,000 are available
A1 + B1 + C1 = 500,000
Beginning of Year 2, investment from A in year 1 is available
A2 + B2 + C2 = 0.03A1
A2 + B2 + C2 – 0.03A1 = 0
Beginning of Year 3, investment from A and B in year 2 and 1are available
A3 + B3 = 0.03A2 + 0.062B1
A3 + B3 – 0.03A2 – 0.062B1 = 0
Beginning of Year 4, investment from A, B, and C in years 3, 2, and 1 is available
A4 = 0.03A3 + 0.062B2 + 0.1C1
A4 – 0.03A3 – 0.062B2 – 0.1C1 = 0
Maximum Investment in C is $200,000
C1 <= 200,000
C2 <= 200,000
Non-negative constraint
Ai, Bi, Ci >= 0
C)
Objective Function:
Investor’s goal is to maximize the returns at the end of fourth year. In fourth year returns from A, B, and C in years 4, 3, and 2 respectively is available.
Max Z = 0.03A4 + 0.062B3 + 0.1C2
Formulation:
Max Z = 0.03A4 + 0.062B3 + 0.1C2
Subject To:
A1 + B1 + C1 = 500,000
A2 + B2 + C2 – 0.03A1 = 0
A3 + B3 – 0.03A2 – 0.062B1 = 0
A4 – 0.03A3 – 0.062B2 – 0.1C1 = 0
C1 <= 200,000
C2 <= 200,000
Ai, Bi, Ci >= 0
Beginning of Year 1, $500,000 are available
A1 + B1 + C1 = 500,000
Beginning of Year 2, investment from A in year 1 is available
A2 + B2 + C2 = 0.03A1
A2 + B2 + C2 – 0.03A1 = 0
Beginning of Year 3, investment from A and B in year 2 and 1are available
A3 + B3 = 0.03A2 + 0.062B1
A3 + B3 – 0.03A2 – 0.062B1 = 0
Beginning of Year 4, investment from A, B, and C in years 3, 2, and 1 is available
A4 = 0.03A3 + 0.062B2 + 0.1C1
A4 – 0.03A3 – 0.062B2 – 0.1C1 = 0
Maximum Investment in C is $200,000
C1 <= 200,000
C2 <= 200,000
Non-negative constraint
Ai, Bi, Ci >= 0
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