Investment Portfolio Optimization With Real-World Data

• 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

C1 Mining and Save Time On Research and Writing Hire a Pro to Write You a 100% Plagiarism-Free Paper. Get My Paper	Materials	Financial	Retail	Pharmaceutical Save Time On Research and Writing Hire a Pro to Write You a 100% Plagiarism-Free Paper. Get My Paper
Energy		Sector Services		s,
				Biotechnology
				& Life Sciences
APA Group	Alkane	Commonwealth	Woolworths	ACRUX Limited
(APA)	Resources	Bank of Australia	Group Limited	(ACR)
	Limited (ALK)	(CBA)	(WOW)
BHP Billiton	ABM Resources	ASX Limited	Accent Group	AUSCAN Group
Limited (BHP)	NL (ABM)	(ASX)	Limited AX1	Holdings Ltd
				(ACB)
Caltex Australia	Alicanto minerals	AMCIL Limited	AP Eagers	ALCHEMIA
Limited (CTX)	Limited (AQI)	(AMH)	Limited (APE)	Limited (ACL)

• 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

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
ALK	CTX	CBA	BHP
ABU	WOW		ASX
AQI
AMH
AX1
APE
ACR
AC8
ACL
APA

• 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,

• 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;

LP

• The amounts in $ to invest are shown below

	Amounts to Invest in $
Max	3992
L	2150
M	3100
H	2150
V	2600

• 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

V	38%	3800
L	32%	3200
M	16%	1600
H	14%	1400

• 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

	R1	R2	R3	R4	ASX
R1				–
0.0084%	0.1170%	-0.0115%	12.1000%	0.1500%
R2	0.1170%	2.3950%	7.8900%	0.0120%	0.2400%
R3	-0.0115%	7.8900%	0.0012%	0.0015%	0.1900%
R4	-12.1000%	0.0120%	0.0015%	0.0124%	0.1600%
ASX	0.1500%	0.2400%	0.1900%	0.1600%	2.1000%
			0.00000		0.00000	Variance	0.0000000
Variance Terms				Std. Dev.
0.00000%	0.00000%	%	0.00000%	%	0.00%
						Des. Ret	10.28%
Return Terms	0.00%	0.00%	3.04%	7.24%	0.00%	Return	10.28%

 

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