DETERMINISTIC DECISION MODELS
Requirements:
This assignment is to be completed in groups. A cover sheet declaring that the assignment was completed without collusion or plagiarism should also be submitted. The modelling work should be submitted as a single MS Excel file with the required information in clearly labelled separate sheets. You are also required to submit a MS Powerpoint file that summarises your results. In summary, three files should be submitted - Cover Sheet, Excel model and Powerpoint file. The assignment is in four parts, 1,2,3 and 4. The requirements of each part are detailed below. The breakdown of marks (total is 180) to be awarded is given on the last page of this document. The assignment contributes 30% towards the total assessment for this unit.
Assignment Details:
This assignment is designed to let you explore and evaluate a number of approaches to investment portfolio optimisation, using live real-world data. The relevant URL for finding stock prices is:
https://au.finance.yahoo.com/q.
In this assignment you will use asset return data from a period of 4 years to identify, using a variety of different optimisation methods, the optimum portfolio according to each of these methods. You will then compare the performance of these portfolios on a further 2 years' of data, so that you may observe just how well the optima generated from the first four years of data performed in that subsequent period, by comparison with the optimum portfolios for that period. This will allow you to make some assessment about the validity for future investment decisions of the optimisation methods which just use past data.
Preliminary Work
The first stage is to identify a set of 15 investment vehicles from which you will subsequently determine optimum portfolios, subject to various optimisation models. You may select any global assets (including indices) whose data is provided on the Yahoo finance website. The 15 assets chosen must satisfy the following general constraints:
- They should be selected from 5 different categories (e.g. banking, pharmaceuticals, media, technology, government bonds, property trusts, etc. - your choice), with at least 2 assets in each category.
- They should span a reasonable range of volatilities / risk (below you will be asked to split your assets into a set of 4 risk groups).
- Each must have at least 72 months (Oct 2010-Oct 2016) of monthly data available, up to and including Oct 2016.
The data needs to be divided into two sets:
Modelling data: For your portfolio optimisations, you should use the first 48 months of data (ie, from Oct 2010 to Oct 2014). Perform parts 1, 2, 3a, 3b and 3c on the modelling data (or Training set)
Holdout sample: The collected data from Nov 2014 to Oct 2016. This is to assess model performance as a means of model validation. Perform all these parts again using Holdout sample. This is called validation task. In part 4, you should include the results of this comparison in a table.
Optimisation parts
The assignment requires you to consider three different approaches to portfolio optimisation:
1. Choosing according to asset class restrictions, and individual asset risk appetite.
2. Choosing according to portfolio size restrictions and risk appetite.
3. Choosing according to portfolio risk and return requirements.
These three approaches allow exploration of three different optimisation techniques: linear programming, integer programming and non-linear programming:
1. LP model: Classify the assets into 4 groups according to (ascending) risk (R1, R2, R3, R4). It is up to you to determine the basis for the classification, but ensure there are at least 3 assets in each of R1 and R4 - this is a slightly arbitrary requirement! A simple, and acceptable approach would be to divide the range of risks into 4 quartiles, as long as the previous requirement is met. Recall that you also have assets classified into 5 categories (C1, C2, ..., C5), and so for each asset, it lies in one of the Rs and in one of the Cs.
In this approach, the aim is to achieve the maximum overall return, subject to specified requirements on risk mix (percentages in R1 to R4) and category mix (percentages in C1 to C5). (These requirements may be simple - such as "no more than 10% in R1), or more complex such as "there should be as much invested in R1 as there is in R4".
Other restrictions might be of the form - "at least 25% should be in the banking sector, and no more than 20% in energy".)
It is up to you to determine the restrictions that you wish to impose. I expect these to be "sensible", respecting a sense of diversity in the portfolio, and a defendable risk acceptance approach. The only requirement is that they should respect the learning aims of this assignment and therefore they should not in any way trivialise the problem. (As an example, there should be realistic range requirements for each of R1 to R4, and C1 to C5. To require all assets in the portfolio to be in risk category R1, for example, would be to trivialise the problem.)
Use a sensitivity analysis report to comment on how changes to the risk and category constraints might affect the optimum portfolio.
2. ILP model: In this approach, we assume that a balanced portfolio of exactly 10 stocks is to be chosen, with each weighted at 10%. At least 3 asset categories (the C classification) have to be included. In addition, at most 2 of the assets can be in the riskiest group R4, and at least 2 must be in the least risky group R1. The goal is to achieve the maximum overall return, subject to the specified requirements.
3. NLP model: In this approach, the aim is to optimise without category constraint using the methods of Module 2topic 7 - i.e. considering the overall portfolio risk/return profile. There are three sub-problems here:
a. Achieve the maximum overall return, subject to an upper limit on portfolio risk (your choice of limit).
b. Achieve the minimum portfolio risk, subject to a requirement to achieve at least a specified return (your choice of required return).
c. Achieve the maximum of risk adjusted return (Sharpe ratio).
4. Determine the average return and average risk achieved in the last 24 months from Nov 2014 to Oct 2016 (Holdout sample). Then calculate the optimum portfolios using the above optimisation criteria in part 1 to 3 for that 24 month-period. In that way, you can compare the various portfolios' performances in the last 24 months. This is one common way to validate an optimisation approach which is geared to decision-making for the future. Include a summary table that includes details of each chosen portfolio and the basis of choice, with percentages of assets, return and risk for the 4 years' of data used to choose the portfolio, and return and risk in the last two years (the hold-out data). Compare each of your chosen portfolios' performance in the last 2 years with the optimum portfolio for these 2 years chosen according to the same objective function, and tabulate for each of your methods the ratio of your portfolio's performance in the 2 years to the optimum performance in these 2 years.
Assignments will be marked based on the methodologies adopted, and the quality of work. Given the vast range of assets to select from on the yahoo site it is highly unlikely that you will choose the same portfolio of stocks as another student.
Assignment - Some clarifications
In assignment, you are required to develop three different optimisation techniques (LP, ILP and NLP) on your modelling and hold-out data.
For each modelling part, you are also required to formulate your model by writing decision variables, objective functions and constraints.
Preliminary work:
• Category mix: collect data from 5 different categories (C1:C5) as described in the assignment.
• from monthly stock value, calculate Returns for each portfolio. The formula is provided under week 7 lecture note and Tutorial materials.
• Risk groups (risk mix): A recommendation is based on quartiles, for example:
R1 = min to Q1
R2 = Q1 to Q2
R3 = Q2 to Q3
R4 = Q3 to max.
LP:
• This is an example of portfolio optimisation where we want to maximise overall return.
• A relevant example is provide under under M2-Topic 5 : "Exercise 2: Investment portfolio".
• The constraints should be built up based on Risk mix (4 risk groups) and category mix (5 groups) in your discretion. for example, "no more than 10% in R1, "there should be as much invested in R1 as there is in R4" or "at least 25% should be in the banking sector, and no more than 20% in energy"
ILP:
• we want to maximise overall return subject to some constraints. Constraints are provided in assignment details.
• Here, we want to decide if we take part in any of the investment options -- go all the way or not at all.
• A relevant example is provide under under M2-Topic 6 : "Exercise 2: Capital Budgeting"
NLP
• The materials from week 7 including Tutorial worksheet is useful for this part.