IB3K20
Financial Optimisation
Individual Assignment, 2023-24
Assignment Instructions
All assignments must be submitted ONLINE via my.wbs by 12pm (midday) UK time on the date displayed against this assessment.
Please ensure that you have inserted a completedassignment coversheet, which must be included as the first page of your script. This should include your Student ID number, but not your name.
Word Limit
Maximum 5 pages, including the cover page (an equivalent of 1500 words).
Word Count Policy
WBS has a school-wide policy on word counts. This is strictly enforced to ensure consistency across modules and programme. You can find more information about this policy in the Undergraduate
Student Handbook under Academic Practice -7i. Word count policy.
This is a strict limit not a guideline: any piece submitted with more words than the limit will result in the excess not being marked.
Academic Practice
Please ensure you read the full guidelines forAcademic Practicein the Undergraduate Student Handbook and ensure you understand it. If in doubt, please seek clarification in advance of your submission. This includes important information on:
• Cheating, plagiarism and collusion
• Correct referencing
• Using internet sources in assessments
• Academic writing
• English Language support
• Word count policy
When you submit this assignment online, you will be required to tick a declaration box indicating that the work involved is entirely your own. Each assignment will be put through plagiarism software to identify any collusion or inadequate referencing of materials used from different sources. Please do not submit images of your typed work unless you have been specifically requested to do so.
We would consider taking action if your work:
1. is too reliant on the words of particular authors (rather than presenting your ideas in your own words), if the essay uses the ideas or words of an author without referencing them or putting their words into quotations (plagiarism).
2. suggests that you have worked very closely with another student or students (unless explicitly asked to do so by your Module Leader/Tutor) (collusion).
3. includes unreferenced work that you have previously submitted for any accredited course of study (unless explicitly asked to do so by your Module Leader/Tutor) (self-plagiarism).
The Use of Artificial Intelligence (AI)
The University recognises an increasing number of technologies such as Artificial Intelligence and that they maybe applicable in your completing this assessment. The assessment brief sets out specific requirements or restrictions, and theUndergraduate Student Handbookhas further guidance and advice.
You are reminded that the inappropriate use of such a technology may constitute a breach of University policy, such as theProofreading PolicyorRegulation 11 (Academic Integrity). If you breach these policies, it may have significant consequences for your studies. Please make sure you read and understand the assessment brief and how AI mayor may not be used.
If a generative AI or similar is permitted and has been used you MUST make clear why you used such a tool or service, what you used it for and you will be obliged to confirm that you take sole intellectual ownership of any submitted work. As appendices, and as part of your submitted work, you must provide screenshots of the question and the AI-generated response, alongside an explanation of how the content has been utilised. You should note the relevant reference alongside each screenshot.
When you submit you must complete (physically or electronically) a declaration. This requires you to explain the use of any AI. Failure to disclose at the point of submission maybe prejudicial in any later investigations should they arise.
For this assessment the use of AI is:
- Prohibited
Where AI is prohibited:
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Extensions and Self-certification
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Your assignment instructions begin below.
Instructions: Please read carefully!
This assignment consists of 3 questions related to a mini case. The marks for parts a, band c are 25%, 35% and 40%, respectively. Read each question carefully and perform the following tasks.
• Modelling Tasks
For each problem, you need 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.
Use AMPL to solve the underlying optimisation model with appropriate solver and data.
Provide abrief 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).
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.
Your AMPL codes must be named by ‘your ID Number’ . For example,AMPL codes should be called as ‘ IDnumber.mod’, ‘ IDnumber.dat’, and ‘ IDnumber.run’ .
。 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.
The assignment submission is to be made electronically, following the electronic submission guidelines, on or before Thursday, 21 March 2024. 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.
Case: John Harrison is a portfolio manager of an IT development company and currently holds a portfolio consisting of n risky assets (represented by i = 1, ... , n) and cash investment (denoted by i = 0). He would like to determine holdings (defined as the contents of an investment portfolio held by the investor) over a financial planning horizon of T years. The portfolio can be restructured at discrete time periods t = 1, ... , T in terms of holdings; in particular, t = 0 represents today. Let pit denote price of a share of asset i at time t. It is assumed that cash account (i = 0) earns the annual interest rate of rt(%) at each time t and is also used to pay out the total transaction costs (£). In addition, the short-sale and borrowing are not allowed. Let hit represent holdings as the number of shares of asseti in the portfolio and h0t denote cash holdings at time t. Let bit and s it denote the number of shares of asset i to be bought and sold, respectively, at the beginning of each time t. Transaction costs (denoted as c it) based on the purchase and sale of asseti at time t are paid out from the cash account. Given the current portfolio position hi0for each asseti = 1, ... , nat the beginning of investment horizon t = 0, John needs to determine how to adjust the portfolio at discrete time periods so that the total wealth at the end of planning horizon is maximised.
a) John assumes that the annual interest rate and asset prices at each time period are known. Formulate (but do not solve) the deterministic portfolio selection problem.
b) Now he ignores the optimisation model developed in part (a) and would like to take into account uncertainty. He assumes that annual interest rate rt and price pit of asseti at time t are uncertain. Thus, they generate a scenario tree, that is showing a probabilistic representation of random interest rates and asset prices over the investment horizon. They observe that the finite number of events, representing realisations of random rates and asset prices at each node of the 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 maximises the total expected wealth at the final time-period. Briefly explain what additional variables/constraints you need to add to the optimisation model developed in part (a).
c) Consider an instance of the multi-stage portfolio management problem consisting of 5 risky securities and a cash investment over the 4-year investment horizon. Generate an appropriate sample data set as input to the optimisation models developed in parts (a) and (b). You can consider a scenario tree structure with finite number of realisations at each node of the scenario tree. As an example, at each node of the scenario tree you may consider at most three realisations such as increasing and/or decreasing by a certain rate or remaining at the same level as in the previous time period. Find the optimal strategyy obtained by solving the multi-stage optimisation models with the numerical data to be generated. Briefly summarise your observations.
Hints:
- Write down ageneric formulation of the deterministic linear optimisation and the scenario-based linear optimisation problems in parts (a) and (b),respectively.
- In part (c), as the AMPL code involves the model using the selected data, you should
present only how the data and scenario tree is generated and the solution in the report.
- The scenario tree maybe generated in anyway you wish, but an example is given on part (c).