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.

Requests for specific extensions (of up to 15 days) which are typically for longer and more serious   concerns must be submitted via my.wbs ideally 72 hours BEFORE the deadline. Extensions can only be approved if you clearly detail your circumstances and provide supporting documentation (or a reason as to why you cannot provide the supporting documentation at the time) asset out in the Mitigating Circumstances Policy.

Self-certification is a university-wide policy whereby you are permitted an automatic extension of 5 working days on eligible written assessed work without the need for evidence. WBS permits self-certification for all types of written, assessed works such as essays and dissertations. It is not permitted for exams, course tests, or presentations.

You can self-certify twice within each year of study, starting from the anniversary of your course start date. This will coverall eligible written assessments that fall within the self-certification period, as long as they have not previously had an extension applied. To find out further details about the self-certification policy please see:https://my.wbs.ac.uk/-/academic/20778/item/id/1244460/ . If you wish to self-certify for an extension of 5 working days, please select 'Self-certification' in the Extension Type field. If you wish to request a longer extension than 5 working days, please leave the Extension Type as 'Standard'.

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.