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MIS775 Decision Models For Business Analytics

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The assignment is in three parts, 1,2 and 3. The requirements of each part are detailed below.

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: under the “Quote lookup” search. In this assignment you will use asset return data from a period of 4 years to identify the optimum portfolio using a variety of different optimisation methods and according to each of these methods.

Preliminary Work

The first stage is to identify a set of 15 investment items 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:

Each must have at least 48 months (March 2014 ? March 2018) of monthly data available, up to and including March 2018.

They should be selected from 5 different sectors/categories (C1, C2,…, C5) 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. 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 to 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 each asset lies in one of the Rs and in one of the Cs.

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.
  1. Choosing according to portfolio size restrictions and risk appetite.
  1. 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: In this approach, the aim is to achieve the maximum overall return, subject to specified requirementson 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.

  1. ILP model: In this approach, we assume that a balanced portfolio of exactly 8 stocks is to be chosen. The 5 assetcategories (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 these specified requirements.
  1. NLP model: In this approach, the aim is to optimise without category constraint using the methods of Module 1,topic 3 – i.e. considering the overall portfolio risk/return profile. There are three sub?problems here:
  1. Achieve the maximum overall return, subject to an upper limit on portfolio risk (your choice of limit).
  2. Achieve the minimum portfolio risk, subject to a requirement to achieve at least a specified return (your choice of required return).
  3. Achieve the maximum of risk adjusted return (Sharpe ratio).

Summarise your report (all of above parts) in a Powerpoint, present all your results comparatively in a coherent and compelling manner, and then, based on your assessment of the various approaches, explain briefly about a strategy that you might prefer to use for portfolio optimisation. For each optimisation model, explain about the optimisation approach taken, the mathematical formulation and identify the Excel Solver to be used (explaining any particular constraints used – e.g. that a variable needs to be an integer, or binary).

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.




  • 15 stocks from the ASX (Australian Stock Exchange) are evaluated using the LP model, the ILP model, and the NLP model as approaches in optimizing the portfolio
  • The securities are chosen according to restrictions of asset classes and individual risk appetites
  • The securities are also chosen according to the portfolio size restrictions and risk appetite, as well as based on portfolio risk and the required return
  • Preliminary work done by choosing and classifying securities into industries and according to risk
  • Stocks chosen from mining and energy (C1), Materials (C2), Financial Sector (C3), Retail (C4), and Pharamaceuticals, Biotechnology and Life Sciences (C5)
  • The Stocks are then classified based on Risk

Chosen Stocks

C1 Mining and







Sector Services












& Life Sciences

APA Group




ACRUX Limited



Bank of Australia

Group Limited



Limited (ALK)




BHP Billiton

ABM Resources

ASX Limited

Accent Group


Limited (BHP)



Limited AX1

Holdings Ltd






Caltex Australia

Alicanto minerals

AMCIL Limited

AP Eagers


Limited (CTX)

Limited (AQI)


Limited (APE)

Limited (ACL)


Classification Based on Risk

  • The risk levels for each was is determined by historical performance data
  • Data was collected on a monthly basis (average monthly stock prices) for the past 48 months
  • The volatility of the stock determined its risk
  • Volatility computed as a function of standard deviation of the stock performance
  • Standard deviation for each stock computed using data from past 48 months


  • Solver used with the standard deviation formula in a spreadsheet
  • The goal of the investment is to balance between risk and returns
  • The portfolio to be made up of 50% low risk assets and 50% high risk assets
  • The high risk assets are likely to result in higher returns but at a higher risk

Classification Based On Risk

  • Computing standard deviations gave a range of between 1 and 10.34
  • This was used to create percentiles for risk by dividing the range into 4 percentiles from low risk to high risk
  • The lower the standard deviation, the lower the risk but also the lower the expected returns
  • Risk classes are R1, R2, R3, and R4, in increasing order of risk

Risk Classification

Low Risk (R1)

Medium Risk (R2)

High Risk (R3)

Very High Risk (R4)

Volatility between 0

Volatility between

Volatility between

Volatility over 7.5

and 2.5

2.6 and 5.0

5.1 and 7.5










































Details and Assumptions on Investment

  • For the case, we assume there is $ 10000 to invest
  • The goal is to maximize returns at the lowest risk
  • Three approaches are used; Linear programming function,

Part 1: LP

  • Done in a spreadsheet using solver
  • The first step entailed giving each of the stocks values based on risk profile (volatility)
  • Low risk are denoted L, medium risk are denoted M, high risk are denoted H, and very high risk are denoted V
  • An objective function is then created based on the expected returns and the risk appetite


  • The objective function is subject to some constraints
  • That L+M+H+V must be less than or equal to $ 10000
  • The target of investment is to spread out risk but have a chance for highest returns
  • Each asset risk class will have no more than $ 2500 invested
  • The other condition therefore is that L+M+H+V must be equal to or less than 2500
  • The objective is to maximize revenue
  • The trivial constraints are that L+M+H+V must be greater than or equal to 0
  • The target yield is one that is above 4.3%, which is the average annual yield of the ASX based on 48 months yield data
  • The average yield of the Treasury Bills (the risk free rate) must also be exceeded by the expected yield from the stocks
  • The 10 year Australian bong yield has averaged 3.4% in the past 48 months

The targeted goal is to maximize the return

Using solver from the data ribbon (analysis)

The parameters are entered

The highest return, after solving for maximizing returns, the maximum return is found to be $ 3992 The maximum return is obtained from the figures shown in the table below;


  • The amounts in $ to invest are shown below



Amounts to Invest in $











Interval Linear Programming (ILP)

  • The aim was to maximize returns
  • The constraints are as follows;
  • At least 2 stocks from R1 (the least risky) must be in the portfolio
  • At least 2 stocks from R4 (the most risky) must be in the portfolio
  • The target is to have a portfolio of 8 stocks
  • These were also solved using Solver in a spreadsheet after creating equations to satisfy the criteria
  • Applying solver to maximize the portfolio, the results below were obtained













Non Linear Programming (NLP)

  • Entails solving optimization problem using a system of constraints consisting of both equalities and inequalities
  • The results obtained are shown in the table below


Variance/Covariance Matrix
































































Variance Terms




Std. Dev.














Des. Ret


Return Terms










The standard LP has the formula of the type minx cT x

Ax = b

X ≥ 0

The variables are split into independent and dependent The independent is set to zero and induced independent values are obtained

This method has the advantage of obtaining the optimum using the simplex interactions It is also easy to use and apply

The NLP method was more flexible, consistent with findings from research

However, developing the equations was a little of a challenge

Overall, it was a good method to use

The ILP model was by far the most challenging to use

It has internal consistencies

In choosing the best method for portfolio optimization, though, I would prefer the use of the linear model

While it is not very flexible, it is easy to obtain the optimum when using basic equations

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