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Instructions: Please read carefully!
The assignment consists of two questions. Read each question carefully and perform the following tasks.
• Modelling Tasks: For each problem, you need
o to provide the complete mathematical programming formulation in a compact form in terms of all sets of decision variables, the objective function, constraints, and parameters you use to write down the problem formulation.
o to use AMPL to solve the underlying optimisation model with appropriate solver and data.
o to provide a brief explanation of main observations if needed.
• Writing Format: Handwritten solutions are not allowed! Write your answers clearly using MS Word or LaTeX with the font size 11. The main body of the assignment should NOT exceed 5
pages (including the cover page).
o Enter your ID number at the beginning of your work. Make sure that each page (in the
main document) has your ID number and the question number.
o Your AMPL codes must be named as “QuestionNumber”. For example, AMPL codes of
Question 1 part (b) should be called as ‘Q1b.mod’, ‘Q1b.dat’, and ‘Q1b.run’.
o Do not include your AMPL codes into the main document as the answer to any question.
• Submission and Deadline: A pdf version of Word or Latex document should be submitted to the ‘Individual Assignment (15 CATS)’ assessment area on my.wbs. Your AMPL files should be submitted in a zip file to the ‘Individual Assignment – Zip File for Codes’ area. Submission is to be made electronically, following the electronic submission guidelines, on or before the date displayed against this assessment. Late submissions are automatically marked down. Ensure your submission will print clearly in black and white.
Finally, problem formulations, AMPL models as well as relevant explanations have to be your own work; any similarity between submissions (solution, writing and construction) shall be dealt with accordingly.
Questions 1 (60% of marks)
CASE A:
The portfolio manager of a telecommunication company is currently considering different fixed income securities such as government and commercial bonds (labelled as ? = 1,2, … , ?) to pay off a series of future cash obligations over a four-year planning horizon. They now need to decide the number of securities to purchase today so that the future cash requirements ?? (£) in year ? for ? = 1, . . . , 4 are met. The portfolio manager assumes that these securities are widely available in the market and can be purchased in any quantities at the given price. Moreover, he considers only securities with 1, 2, 3, and 4-year maturities.
Let ??(£) denote the current market price of security ?. Each security ? yields annual coupon payment of ??(£) up to its maturity. The principal ??(£) of security ? is paid out at maturity. After an initial investment on bonds is made at t = 0, they can also apply for a one-year loan or borrow a certain amount of money at any time if they need, but do not wish to consider another reinvestment opportunities. An amount of money can be borrowed from year ? to year ?+1 with annual rate of ??(%) at each time-period. Similarly, a loan to be received at each time-period will be paid-off at the next time-period with annual interest rate ??(%) from year ? to the next year ?+1. Moreover, there is no cash reinvestment, loan or borrowing at the end of planning horizon. The portfolio manager would like to determine an optimal portfolio dedication strategy that maximizes the final cash on hand at the end of investment horizon and minimizes the total cost of investment.
a) Assume that all model parameters are known, and the fixed rates remain the same over a year. Formulate (but do not solve) a deterministic linear programming model of the portfolio dedication problem.
(10 marks)
b) Now, ignore the optimization model developed in part (a). They assume that both rates ̃?? ??? ?
̃ ? are uncertain. Thus, they generate a scenario tree, that is showing a probabilistic representation of random rates of one-year loan and annual borrowing over the five-year period. They observe either one or two different events, representing realizations of random rates at each node of scenario tree with certain probability, over the investment horizon. Modify the linear program developed in part (a) and formulate (but do not solve) a scenario based linear programming model that maximizes the total expected cash on hand at the final time-period and minimizes the total cost of investment. Briefly explain what additional variables/constraints you need to add to the model developed in part (a).
(20 marks)
c) Consider an instance of the firm’s financing problem consisting of up to 10 securities. For the scenario based stochastic programming formulation developed in part (b), generate an appropriate sample data set as input to the optimisation model. Find the optimal investment strategy by using the numerical data (to be generated). Briefly summarize your observations. (30 marks)
Question 2 (40% of marks)
CASE B:
Janet has been working as a chief data scientist in a retailer for the last three years. She has recently been concerned with her personal finance as she would like to buy her first house in Warwickshire. Currently, she has a capital of £70k and considers investing today in stocks of three British companies (namely, BT, Shell and Sainsbury) from FTSE 100. She would like to have at least £120k after nine years (starting from today) to be able to pay a deposit for her first house. If she does not have £120k after nine years, she plans to borrow an amount (that she needs) at a cost of 5%. Once she makes the initial investment decisions today, she can review and change these decisions once every 3 years. She would like to develop a stochastic programming model and considers a scenario tree to model an evaluation of asset returns over time where at most two different realisations (of asset returns at each node of the scenario tree) with a certain branching probability are observed.
• Generate a sample scenario tree with an appropriate structure as described. Plot the scenario tree and clearly display decision stages, branching structure with probabilities, and return realisations. Briefly describe first-stage and second-stage decision variables and clearly show them on the scenario tree.
• Formulate and solve the problem as a stochastic linear optimisation model that maximizes the expected value of lump sum cash, she has left, at the end of planning horizon by taking into account expected cost of borrowing. Briefly summarize your observations.
