Doctoral Program: Business Administration Doctoral Thesis ...shodhganga.inflibnet.ac.in/bitstream/10603/21971/1... · 6.1 Weights of the optimum portfolios under Sharpe’s Single

THE UNIVERSITY OF BURDWAN

Doctoral Program:

Business Administration

Doctoral Thesis

SELECTION OF PORTFOLIO WITH COMPARATIVE

ANALYSIS FOR SOME LISTED COMPANIES IN NIFTY

SOMA PANJA

Thesis Advisor: Professor Dilip Roy

Department of Business Administration

October 2012

Dedicated to my Parents and my Beloved Husband

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Acknowledgements

First of all, I would like to express my deepest sense of gratitude to my guide

Professor Dilip Roy, retired Professor, Department of Business

Administration, The University of Burdwan for giving the opportunity of

developing my Ph.D. thesis under his supervision. I am grateful for his

scholarly guidance, inspiring and valuable suggestions, tirelessly monitoring

of the work and for his timely corrections with positive criticism, without which

this study would not have been completed or even started. I thank him for

believing in me, sometimes more than what I believe myself.

I am thankful to the authorities of The University of Burdwan for giving me

permission to do this research work.

I express my gratitude to all the members of the Centre for Management

Studies for creating a pleasant place to work and for providing me with all the

resources necessary to carry out this work with important suggestion.

I would like to thank to all my teachers, colleagues, co-researcher and friend

for their help and cooperation.

Finally, I would like to thank my family members, especially to my mother,

father and my husband Subroto and my son Suyash along with my in laws

and my elder sister and her family for being so supportive and

accommodative throughout the course of the study. They always supported

me and motivated and encouraged me to continue towards the completion of

this thesis. This achievement is theirs.

Date:

(Soma Panja)

ii

Table of Contents Contents Page

No:

Acknowledgements i

Table of Contents ii

Index of Tables V

Index of Figures Viii

List of Works X

List of Presentation xi

Notations xii

Chapter 1: Introduction

1.1 An Invitation to the Problem 1

1.2 Portfolio Management 2

1.3 Concept of Risk and Return 9

1.3.1 Concept of Risk 9

1.3.2 Concept of Return 11

1.4 Existing Approaches 13

1.4.1 Mathematical Programming Approach 13

1.4.2 Model Approach 20

1.5 Research gap and Research Problem 24

1.6 Preview of the Work 30

1.7 Scope of the Work 33

Chapter 2: Research Methodology

2.1 Introduction 36

2.2 Research Questions 36

2.3 Objectives 37

2.4 Research Design 38

2.4.1 Sampling Design 38

2.4.2 Statistical Design 39

2.5 Methods to be Used 59

iii

2.5.1 Optimization Method 59

2.5.2 Heuristic Method 60

2.5.3 Model Based 62

Chapter 3: Selection of Portfolio based on Minimum Return Constraint

3.1 Introduction 65

3.2 Earlier works 66

3.3 Optimization Framework 70

3.4 Heuristic Framework 72

3.5 Closeness between Heuristic and Optimization Framework 74

3.6 Data Analysis and Interpretation 74

3.7 Conclusion 99

Chapter 4: Selection of Portfolio based on Coefficient of Optimism

4.1 Introduction 100


4.3 Optimization Framework 105

4.4 Heuristic Framework 106

4.5 Closeness between Heuristic and Optimization Framework 108


4.7 Conclusion 130

Chapter 5: Value Based Selection of Portfolio: A Heuristic Approach



5.3 Choice of weights under different value systems 134


5.5 Conclusion 145

Chapter 6: Comparative Analysis on Sharpe’s Cut off Principle and a Near Optimum Principle for Portfolio Selection under Model Approach



6.3 Sharpe’s Single Index Model 152

iv

6.4 Sharpe’s cut-off Principle 152

6.5 Optimum framework under Value System Sharpe’s Single Index Model

154

6.6 A Near Optimum Approach 155

6.7 Comparative study 156

6.8 Conclusion 164

Chapter 7: End Remarks

7.1 Importance of value system 165

7.2 Closeness between Optimum Solution and Heuristic Solution

167

7.3 Limitations of the Current work 170

7.4 Future Scope 171

References 173

v

Index of Tables Table No:

Details Page No:

3.1 Weights of optimum portfolio and heuristic portfolios under -0.0057 minimum return constraint

76

3.2 Expected return and risk of three different portfolios under – 0.0057 as the minimum return constraint

77

3.3 Weights of optimum portfolio and heuristic portfolios under – 0.0047 as the minimum return constraint

79


80


82


83


84


85

3.9 Weights of optimum portfolio and heuristic portfolios under -0.0017 as the minimum return constraint

87


88

3.11 Weights of optimum portfolio and heuristic portfolios under -0.0007 as the minimum return constraint

89


90

3.13 Weights of optimum portfolio and heuristic portfolios under 0.0003 as the minimum return constraint

92

3.14 Expected return and risk of three different portfolios under 0.0003 as the minimum return constraint

93

3.15 Weights of optimum portfolio and heuristic portfolios under 0.00065 as the minimum return constraint

94

3.16 Expected return and risk of three different portfolios under 0.00065 as the minimum return constraint

95

vi

3.17 Euclidean Distance between pessimistic and optimum solutions and between optimistic and optimum solutions

97

4.1 Optimum weights of fifteen securities under different values of coefficient of optimism

109

4.2 Weights of heuristic optimistic, heuristic pessimistic, heuristic risk planner and heuristic random selector solutions

111

4.3 Expected return and risk of the optimum portfolio under different α values

112

4.4 Values of expected returns and risks of heuristic portfolios

113

4.5 City Block Distance between the optimistic & optimum choices of portfolios and pessimistic & optimum choices of portfolios

114

4.6 City Block Distance between the optimistic & optimum choices of portfolios and risk planner & optimum choices of portfolios

116

4.7 City Block Distance between the optimistic & optimum choices of portfolios and random selector & optimum choices of portfolios

118

4.8 City Block Distance between the pessimistic & optimum choices of portfolios and risk planner & optimum choices of portfolios

120

4.9 City Block Distance between the pessimistic & optimum choices of portfolios and random selector & optimum choices of portfolios

122

4.10 City Block Distance between the pessimistic & optimum choices of portfolios and random selector & optimum choices of portfolios

124

4.11 City Block Distance between optimistic and optimum solutions, between pessimistic and optimum solutions, between risk planner and optimum solutions and between random selector and optimum solutions

127

5.1 Weights of the optimum portfolios of fifteen selected securities under different α coefficients

137

5.2 Expected return and risk of the fifteen selected 138

vii

securities under different α coefficients

5.3 Weights of two heuristic portfolios taken into consideration

139

5.4 Weights of the heuristic portfolios of fifteen selected securities under different α coefficients

140

5.5 Expected return and risk of heuristic portfolios under different α coefficients

142

5.6 City Block Distances between optimum portfolios and heuristic portfolios under different α coefficients

143

6.1 Weights of the optimum portfolios under Sharpe’s Single Index Model of ten securities under different α values

157

6.2 Expected return and risk of the optimum portfolios under Sharpe’s Single Index Model under different values of α

158

6.3 Weights of the near optimum portfolios of ten securities under different α values

159

6.4 Expected return and risk of the near optimum portfolios under different values of α

160

6.5 CBD between the optimum portfolio under Sharpe’s Single Index Model and Sharpe’s cut-off principle portfolio

161

6.6 CBD between the optimum portfolios under Sharpe’s Single Index Model and the near optimum portfolios

162

viii

Index of Figures

Figure

No: Page No:

3.1 Expected return and risk under -0.0057 minimum return constraint

78


80


83


86


88


91

3.7 Expected return and risk under 0.0003 minimum return constraint

93

3.8 Expected return and risk under 0.00065 minimum return constraint

96

3.9 ED of Optimistic and Pessimistic Solution 98

4.1 City Block Distance between Optimum Portfolio and Optimistic Portfolio and Optimum Portfolio and Pessimistic Portfolio

115

4.2 City Block Distance between Optimum Portfolio and Optimistic Portfolio and Optimum Portfolio and Risk Planner's Portfolio

117

4.3 City Block Distance between Optimum Portfolio and optimistic Portfolio and Optimum Portfolio and Random Selector's Portfolio

119

4.4 City Block Distance between Optimum Portfolio and Pessimistic Portfolio and Optimum Portfolio and Risk Planner's Portfolio

121

ix

4.5 City Block Distance between Optimum Portfolio and Pessimistic Portfolio and Optimum Portfolio and Random selector Portfolio

123

4.6 City Block Distance between Optimum Portfolio and Risk Planner's Portfolio and Optimum Portfolio and Random Selector's Portfolio

125

4.7 City Block Distance between Optimum Portfolio and Heuristic Portfolios

128

5.1 CBD between Optimum and Heuristic Portfolios 144

6.1 City Block Distance between Sharpe’s Single Index Model and Cut-off principle and Sharpe’s Single Index Model and Near Optimum Portfolio

163

x

This thesis is based on the following published works and a few works

submitted for publication:

Published:

1. Roy, D & Panja,S. (2011): Heuristic selection of portfolio based

on coefficient of optimization, International Journal of Financial

Management, (ISSN: 2229-5682), Volume 1 Issue 1, 2011.

2. Roy, D, Mitra, G. & Panja,S. (2010): Closeness between Heuristic

and optimum selection of portfolio: An Empirical analysis, GITAM

Review of International Business, (ISSN No. 0974-357X), Vol 2,

Issue 2, January 2010.

Submitted for Publication

1. Panja, S. & Roy, D. (2011): A comparative study on Sharpe’s cut

off principle and a near optimum principle for portfolio selection

(Communicated for Publication to Journal of Financial and

Quantitative Analysis- Cambridge Journal).

2. Panja, S. & Roy, D. (2011): Risk based Selection of Portfolio:

Heuristic Approach (Communicated for Publication to The

Journal of Portfolio Management-Institutional Investor Journal).

xi

Conference Attended

Presented Paper titled “Closeness between Heuristic and optimum selection

of portfolio: An Empirical analysis” International conference on Advanced Data

Analysis, Business Analytics and Intelligence by IIM Ahmedabad (6-7 June

2009)

xii

Notations

ijR = Daily return of the ‘i’th security, where, i= 1,2,3,……………….,n. and

j= 1,2,3,………..,k.

σi = Standard Deviation of the ‘i’th security

i = Average return of the ‘i’th security

μmin = Minimum return of the security in the portfolio

μmax = Maximum return of the security in the portfolio

OPR = Expected return of the optimum portfolio

o

iw = Optimum weight of the ‘i’th security

Ri =Expected return of the ‘i’th security

n = Number of the securities in the portfolio

OP = Risk of the optimum portfolio

rij = Correlation coefficient between ‘i’th and ‘j’th securities

i = Standard Deviation of ‘i’th security.

HO

iw = Heuristic optimistic weight of the ‘i’th security

μi = Expected return of ‘i’th security

HOPR = Expected return of the optimistic portfolio

HOP = Standard Deviation of the heuristic optimistic portfolio.

HPiw = Heuristic pessimistic weight of ‘i’th security

HPPR = Expected return of the pessimistic portfolio

HPP = Standard Deviation of the pessimistic portfolio

xiii

α = Coefficient of optimism of the investor (0≤α≤1)

∑ = Dispersion matrix of the securities

~w = Weight vector

``R = Expected return vector

HRPiw = Heuristic risk planner weight of the ‘i’th security

HRPPR = Expected return of the heuristic risk planner portfolio

HRPP = Standard Deviation of the heuristic risk planner portfolio

HRSpR = Expected return of the heuristic random selector portfolio

HRSp = Standard Deviation of the heuristic random selector portfolio

Hw Weight of the ‘i’th security in the heuristic portfolio with coefficient of

optimism as α

HR Expected return of the general heuristic portfolio

H Standard deviation of the general heuristic portfolio

mR Rate of return on the market index, a random variable

i = Expected value of the component of return independent of the market’s

performance

i The expected change in the rate of return on stock ‘i’ associated with a

1% change in the market return

ie = Error component

xiv

2m Variance of market return

2ei = Variance of error term

iR Average return of ‘i’th security

pR Return of the portfolio

iw = Weight of ‘i’th security

p Standard deviation of the portfolio

2p = Variance of returns or portfolio risk

iC = Cut-off rate

*C = A candidate of iC

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1 | P a g e

1.1 An invitation to the problem

In financial management, risk management is an

important concept. Financial Risk Management can be defined

as an act of creating economic value of the firm by effectively

managing the exposure to the risk, with judicious use of several

financial instruments and sophisticated techniques. The different

type of exposure to risk, a firm is subjected to mainly involve

credit risk and market risk. Financial risk management can be

qualitative and quantitative in approach.

In order to determine the optimal asset allocation

strategies mathematical models are vastly used by the corporate

investors and portfolio managers respectively. The key

organizations in finance are households, business firms,

financial intermediaries and capital markets. The tradition in

neoclassical economics is to consider the existence of

households, their tastes and their endowments as exogenous to

the theory. But other economic organizations are regarded as

primary because of the functions they serve and are therefore

endogenous to the theory. Giving concentration more on

household, there are two players – consumer and investor. The

consumer chooses how much of her income and wealth to

allocate to current consumption and thereby, how to save for


2 | P a g e

future consumption. As investor, the household solves the

problem to determine the fractional allocation of her savings

among the available investment opportunities.

Risk management plays a critical role in determining the

financial dynamics of the corporate investors. However, risk

management holds equal significance when individual investors

and their investment objectives are considered. Therefore for

this purpose asset management is important for the individual

investors. If a person owns more than one asset for investment,

she has an investment portfolio. A portfolio consists of more

than one asset. The main aim of the portfolio owner is to

enhance the value of portfolio by selecting investments that yield

good returns.

1.2 Portfolio Management

Portfolio Management refers to the science of analyzing

the strengths, weaknesses, opportunities and threats for

performing wide range of activities related to the one’s portfolio

for maximizing the return at a given risk. It helps in making

selection of Debt Vs Equity, Growth Vs Safety, and various other

tradeoffs. Portfolio management involves the task of taking

decisions about investment policy and mix, matching


3 | P a g e

investments to the objectives of the investors, allocating assets

for individuals and institutions and balancing risk against

performance. Portfolio management has a great importance in

theory of finance. Managing a portfolio involves inherent risks.

Portfolio management is goal driven and target oriented.

Constructing a portfolio involves making wide range of decisions

regarding buying or selling of stocks, bonds, or other financial

instruments. In portfolio management both time and magnitude

are very important.

Portfolio optimization plays a critical role in determining

portfolio strategies for investors. In portfolio optimization

investors want to either maximize portfolio returns or minimize

portfolio risks. Since return is compensated based on risk,

investors have to balance the risk-return tradeoff for their

investments and it depends upon investors risk-return

preferences. So, a single optimized portfolio is not in a position

to satisfy all investors.

The traditional mean variance optimization approach fails

to meet the demand of investors who have multiple investment

objectives. In order to achieve the multiple objective of the

investor and to satisfy the aim of obtaining the optimal portfolio it

is necessary to understand the proportion of investment of


4 | P a g e

different assets in the portfolio. That means fixation of asset

weights correctly of a given portfolio is of prime importance from

investors point of view. The above discussed premises points at

the ultimate necessity of fixation of assets’ weight as the

decision variable in the portfolio optimization problems. The use

of portfolio weights for model generation may be especially

important in a setting where the expected return and the

variance are assumed to be correctly predicted and are

expected to show same behavior in future. The use of portfolio

weights to measure the performance of trading strategies was

pioneered by Cornell (1979). Cornell’s measure was modified

by Copeland and Mayers (1982) to analyze Value Line rankings.

The concept was applied to weight-based measure of mutual

fund performance (Grinblatt and Titman, 1993).

The weight-based performance measures of portfolio are

rather simple. For example, an investor if knows that the returns

of assets are likely to be higher or lower than expected by the

market, then, other things remaining same, the investor can

earn profit by changing his portfolio weights toward those assets

whose returns are likely to be higher than expected and away

from those assets whose returns are likely to be lower. In other

words the covariance between the change in a portfolio’s


5 | P a g e

weights and subsequent abnormal asset returns may be used to

measure performance of portfolio.

Most of the portfolio measurement techniques are return-

based, and involve regressing the return of a portfolio on some

benchmark return. The measure of performance is the intercept

in the regression. The minimal information requirement of the

return methodologies is cited as one of the strength. An investor

needs only returns on the managed portfolio and the benchmark

in portfolio optimization problem. The overemphasis on the risk

and return dimensions of the portfolio optimization often

overlooks the necessity of the potential important information

regarding the composition of the portfolio. Previously, portfolio

weights were used with unconditional moments to measure

portfolio performance. However, the inclusion of return-based

measures into a conditional framework changed the results

(Ferson and Schadt, 1996) and hence made it interesting to

consider formulation and modeling in weight-based measures of

performance.

The manner in which the choice between the different

courses of action or inaction is made plays a critical role in

deciding the effectiveness of a portfolio manager. The easiest

way to make the decision is to evaluate from the array of


6 | P a g e

predicted risk and return of the securities in the portfolio and

select the best combination. But the question which remains to

be addressed is the search and applicability of a common model

that can be used to capture the risk and returns, the different

prediction mechanisms, the role played by different information

dynamics which are supposed to influence the assessment of

the risks and returns, and so many prior expectations that may

be brought to influence the proportion of security in the design of

the optimum portfolio.

Since the middle of twentieth century, financial economist

or practitioner and statisticians had been measuring the

performance of a managed portfolio from various angles. Even

after years of research, several issues remain unsolved. The

classical mean variance model is aimed at satisfying the

optimizing needs of the risk averse investors. However, in real

world investors exhibit a multitude of risk profiles which explicitly

points to the fact and necessity of a model which can take into

account the dynamics of the different categories of investors on

the parameter of risk taking aptitude and attitude. Thus, it

becomes evident that the investors are influenced by their

nature or value system. According to Heller (1971) there are six


7 | P a g e

values of importance. Out of which, propensity to take risk is

most important value system (Jauch and Glueck, 1988).

Nature of the investors may not be same and their

investment needs depend on their nature. Some investors are

risk taker, some are risk aversive, some other investors invest

their wealth after a proper planning, who are known as risk

planners and the rests are random selectors who randomly

select the assets without giving any importance on expected

return and risk. Thus, if an investor can correctly predict the

proportion of amount to be invested in each assets of the

portfolio then right decision can be taken to earn maximum

utility.

An important unsolved issue is how to handle the

dynamic behavior of a managed portfolio. Not only because of

the existence of time-varying required returns in a portfolio but

also due to investor’s strategy or other influencing factors,

management of portfolios become difficult. Thus creation of

optimal portfolio strategy for investors of various risk profiles

becomes a lurking question in the financial risk management

scenario. An attempt has been made in this work to formulize

and propose a multi-objective approach to portfolio optimization

problems. In this work portfolios risk and return are optimized


8 | P a g e

and various portfolio optimization models are integrated.

Detailed analysis based on heuristic weight generation and

subsequent optimization and application of the model are

provided and compared to portfolio generated through the

mean-variance approach and Sharpe’s approach.

The basics and ideas of investment portfolio

management are used in various dimensions as discussed:

Application Portfolio Management: it involves

management of complete group or subset of computer software

applications in a portfolio. As the applications of software include

maintenance cost and development cost, these can be

considered as investment. The decisions regarding purchasing

new software or modifying existing software are important parts

in application portfolio management.

Product Portfolio Management: it means grouping of

major products developed or sold by businesses into (Logical)

portfolio. These products are arranged according to major line-

of- business or business segment. In this case investment

decision involves development of new product or modifying the

existing product or discontinuation any other product.


9 | P a g e

Project Portfolio management or initiative portfolio

management: it includes a specified beginning and end; precise

and limited collection of desired results or work products and

management team for taking the initiative and utilizing the

resources.

1.3 Concept of Risk and Return

1.3.1 Concept of Risk

The earliest definition of risk was given by Knight (1921).

According to him, risk is measurable uncertainty. Many authors

have given various definitions of risk. Risk is a concept that

denotes a potential negative impact to an asset. However, there

was an attempt made by Holton (2004) to summarize the most

relevant definition of risk to give a more general concept of risk.

According to him, risk is the exposure to a proposition of which

one is uncertain. A finance relevant definition given by Jorion

(2000) defines risk as the volatility of the expected results on the

value of assets and liabilities of interest. Portfolio management

relevant definition given by Cool (1999) defines risk as the

absolute value of probable loss.

From the economic point of view, risk is any event or

action that may adversely affect an organization's ability to


10 | P a g e

achieve its objectives and execute its strategies. In finance,

financial risk is essentially any risk associated with any form of

financing. The existence of risk means that the investor can only

associate a single number or payoff with investment in any asset

with certain probability associated with it. A common property of

investment opportunities is that their actual returns might differ

from what has been expected; or in short: they are risky. When

the actual return is lower than the expected outcome, it is known

as downside risk where as when the deviation from the actual

return is more than the expected outcome is known as upside

risk.

Risk reflects not only the dangers associated with an

investment, but also the chances. Therefore, a risky situation is

one in which surprises and unexpected developments might

occur. Volatility of return is this type of risk measure which is

one of the foundations of portfolio theory. Risk can be measured

in different ways; viz. semi-variance which measures only the

negative deviations from the expected value. More recently,

Value at Risk (VaR) has been used by the authors. It means the

maximum loss within a certain period of time with a given

probability. A third type of risk refers to a situation of danger or

peril. In finance, this concept is applied as an aspect of risk


11 | P a g e

measure as applied in circumstances where catastrophes ought

to be prevented; investors ought to be saved from hazardous

situations. However, in finance there are two types of risk-

systematic risk or un-diversifiable risk and unsystematic risk or

diversifiable risk. Systematic risk is the market risk or the risk

that cannot be diversified away. Systematic risk affects the

whole economy of a country. A perfectly diversifiable portfolio

also carries some systematic risk which we cannot avoid.

Unsystematic risk or diversifiable risk is the risk which is

associated with individual assets and it differs from asset to

asset. It is industry or company specific risk. Unsystematic risk

can be diversified away by including a number of assets in the

portfolio. However, unsystematic risks do not present enough

information about the overall risk of the entire portfolio. The

impact of risk has two components: uncertainty and exposure.

The chance of facing risk is uncertainty and exposure is the

amount of the possible loss if the risk has been faced.

1.3.2 Concept of Return

Return in its optimistic outlook can be defined as any

potential gain which is supposed to be over and above than the

amount invested in any asset. In other words, return is the ratio

of money gained or lost (whether realized or unrealized) on an


12 | P a g e

investment relative to the amount of money invested. A general

investor always tries to earn some return from any investment.

The return can be calculated over a single period, or expressed

as an average over multiple periods of time. Return can be

calculated in various ways viz. arithmetic return, geometric

return, logarithmic return. Return may be annual return and

annualized return. An annual return is a single-period return,

while an annualized return is a multi-period or arithmetic

average return.

Personalized investment returns are of recent origin and

are in much demand amongst the investor’s community. The

demand for personalized investment returns holds the argument

that the fund returns may not be the actual account returns

which are based upon the actual investment account transaction

history. This occurs because investments may have been made

on various dates and additional purchases and withdrawals may

have incurred on various dates and the related amount is

varying and thus is unique to the particular account. The fund

returns may be more or less than the account return. More and

more fund and brokerage firms have begun providing

personalized account returns on investor's account statements

in response to this need.


13 | P a g e

1.4 Existing Approaches

1.4.1 Mathematical Programming Approach

The selection of a best element (with regard to some

criteria) from some set of available alternatives in parlance of

mathematics, computer science, or management science is

known as mathematical

optimization (alternatively, optimization or mathematical

programming).In the simplest case, an optimization

problem consists of maximizing or minimizing a real function by

systematically choosing input values from within an allowed set

and computing the value of the function. The generalization of

optimization theory and techniques to other formulations

comprises a large area of applied mathematics. More generally,

optimization includes finding "best available" values of some

objective function given a defined domain, including a variety of

different types of objective functions and different types of

domains (Wikipedia,2012).

The following are applications of mathematical

optimization and include the following:

Convex programming studies the case when the

objective function is convex (minimization)


14 | P a g e

or concave (maximization) and the constraint set is convex. This

can be viewed as a particular case of nonlinear programming or

as generalization of linear or convex quadratic programming

(Wikipedia, 2012).

Linear programming (LP), a type of convex

programming, studies the case in which the objective

function f is linear and the set of constraints is specified using

only linear equalities and inequalities. Such a set is called

a polyhedron or a poly tope if it is bounded (Wikipedia, 2012).

Second order cone programming (SOCP) is a convex

program, and includes certain types of quadratic programs

(Wikipedia, 2012).

Semidefinite programming (SDP) is a subfield of

convex optimization where the underlying variables are semi

definite matrices. It is generalization of linear and convex

quadratic programming (Wikipedia, 2012).

Conic programming is a general form of convex

programming. LP, SOCP and SDP can all be viewed as conic

programs with the appropriate type of cone (Wikipedia, 2012).


15 | P a g e

Geometric programming is a technique whereby

objective and inequality constraints expressed

as polynomials and equality constraints as monomials can be

transformed into a convex program (Wikipedia, 2012).

Integer programming studies linear programs in which

some or all variables are constrained to take on integer values.

This is not convex, and in general much more difficult than

regular linear programming (Wikipedia, 2012).

Quadratic programming allows the objective function to

have quadratic terms, while the feasible set must be specified

with linear equalities and inequalities. For specific forms of the

quadratic term, this is a type of convex programming (Wikipedia,

2012).

Fractional programming studies optimization of ratios of

two nonlinear functions. The special class of concave fractional

programs can be transformed to a convex optimization problem

(Wikipedia, 2012).

Nonlinear programming studies the general case in

which the objective function or the constraints or both contain

nonlinear parts. This may or may not be a convex program. In


16 | P a g e

general, whether the program is convex affects the difficulty of

solving it (Wikipedia, 2012).

Stochastic programming studies the case in which

some of the constraints or parameters depend on random

variables (Wikipedia, 2012).

Robust programming is, like stochastic programming,

an attempt to capture uncertainty in the data underlying the

optimization problem. This is not done through the use of

random variables, but instead, the problem is solved taking into

account inaccuracies in the input data (Wikipedia, 2012).

Combinatorial optimization is concerned with problems

where the set of feasible solutions is discrete or can be reduced

to a discrete one (Wikipedia, 2012).

Infinite-dimensional optimization studies the case

when the set of feasible solutions is a subset of an infinite-

dimensional space, such as a space of functions (Wikipedia,

2012).

Heuristics and metaheuristics make few or no

assumptions about the problem being optimized. Usually,

heuristics do not guarantee that any optimal solution need be


17 | P a g e

found. On the other hand, heuristics are used to find

approximate solutions for many complicated optimization

problems (Wikipedia, 2012).

Constraint satisfaction studies the case in which the

objective function f is constant (this is used in artificial

intelligence, particularly in automated reasoning) (Wikipedia,

2012).

Dynamic programming studies the case in which the

optimization strategy is based on splitting the problem into

smaller sub-problems. The equation that describes the

relationship between these sub-problems is called the Bellman

equation (Wikipedia, 2012).

Mathematical programming with equilibrium

constraints is where the constraints include variational

inequalities or complementarities (Wikipedia, 2012).

A subset of mathematical programming approach, the

linear programming is a class of optimization problems. The

linear programming problems have one objective function and

the set of constraints with linear equalities and inequalities.

Because of the effectiveness and robustness of linear program


18 | P a g e

solving algorithms this techniques are useful for portfolio

rebalancing problems (Wikipedia, 2012).

Chekhlov et al (2004) stated that portfolio allocation

problems can efficiently be handled with linear programming

based algorithms. The techniques are attractive to the investors

as it demonstrates the problems with thousands of instruments

and scenarios. The portfolio mean-variance optimization

techniques are a class of quadratic programming problems. The

quadratic programming Optimization models can lead to non-

convex multi extrema problems.

Modern Portfolio theory: Markowitz, Sharpe, Tobin and

more

The modern portfolio theory introduced by Harry

Markowitz (1952) was based on linear programming problem. In

the model he defined the linear programming as either by

maximizing the return subject to certain amount of risk or

minimizing the risk subject to certain amount of return. He

proposed that investors should focus on selecting portfolios

based on their joint risk-reward feature. The expected return of

any portfolio can be measured by using the historical returns of

each asset on the portfolio. Various statistical measures such as


19 | P a g e

average (return), standard deviation and linear correlation are

used to measure the volatility of the portfolio. Markowitz (1959)

used volatility and expected return as proxies for risk and

reward. Markowitz defined an optimal way of selecting a

portfolio by balancing the risk and reward features of the

portfolio. According to him a rational investor should select a

portfolio that lies on the efficient frontier. Modern Portfolio

Theory as propounded by Harry Markowitz has introduced the

concept of efficient frontier. "Efficient Frontier" can be defined as

the combination of assets, i.e. a portfolio, if it has the best

possible expected level of return for its level of risk (usually

proxied by the standard deviation of the portfolio's return).

Markowitz work was expanded by James Tobin (1958) by

adding a risk-free asset to the analysis. He showed that by using

leverage or deleverage on the portfolios on the efficient frontier it

was possible to outperform them in terms of their risk and

reward relation. To prove it he introduced the concepts of

“Capital Market Line” and “super-efficient portfolio”.

In the Capital Asset Pricing Model (CAPM) Sharpe (1964)

pointed out that the market portfolio lies on the efficient frontier

and is also actually Tobin's super-efficient portfolio. CAPM first

introduced the concept of “beta” and relates an asset's expected


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return to its beta. He showed that according to the risk apatite,

all investors should hold the market portfolio where it is

leveraged or de-leveraged with positions in the risk-free asset.

1.4.2 Model Approach

The description of a system using mathematical concepts

and language is known as a mathematical model. Mathematical

modeling is the process of developing a mathematical model. In

order to explain a system and to study the effects of different

components, and also to make predictions about behavior

mathematical models comes handy. Mathematical models can

take many forms, including but not limited to dynamical

systems, statistical models, differential equations, or game

theoretic models. A mathematical model usually describes a

system by a set of variables and a set of equations that

establish relationships among the variables. There are six basic

groups of variables namely: decision variables, input variables,

state variables, exogenous variables, random variables, and

output variables. Since there can be many variables of each

type, the variables are generally represented by vectors.

Decision variables are sometimes known as independent

variables. Exogenous variables are sometimes known as

parameters or constants. The variables are not independent of


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each other as the state variables are dependent on the decision,

input, random, and exogenous variables. Furthermore, the

output variables are dependent on the state of the system

(represented by the state variables).

Objectives and constraints of the system and its users

can be represented as functions of the output variables or state

variables. The objective functions will depend on the perspective

of the model's user. Depending on the context, an objective

function is also known as an index of performance, as it is some

measure of interest to the user. Although there is no limit to the

number of objective functions and constraints a model can have,

using or optimizing the model becomes more involved

(computationally) as the number increases (Wikipedia, 2012).

Classifying mathematical models

Many mathematical models can be classified in some of

the following ways:

Linear vs. nonlinear: Mathematical models are usually

composed by variables, which are abstractions of quantities of

interest in the described systems, and operators that act on

these variables, which can be algebraic operators, functions,

differential operators, etc. If all the operators in a mathematical


22 | P a g e

model exhibit linearity, the resulting mathematical model is

defined as linear. A model is considered to be nonlinear

otherwise (Wikipedia, 2012).

Deterministic vs. probabilistic (stochastic):

A deterministic model is one in which every set of

variable states is uniquely determined by parameters in the

model and by sets of previous states of these variables.

Therefore, deterministic models perform the same way for a

given set of initial conditions. Conversely, in a stochastic model,

randomness is present, and variable states are not described by

unique values, but rather by probability distributions (Wikipedia,

2012).

Static vs. dynamic: A static model does not account for

the element of time, while a dynamic model does. Dynamic

models typically are represented with difference equations or

differential equations (Wikipedia, 2012).

Discrete vs. Continuous: A discrete model does not

take into account the function of time and usually uses time-

advance methods, while a Continuous model does. Continuous

models typically are represented with f (t) and the changes are

reflected over continuous time intervals (Wikipedia, 2012).


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Deductive, inductive, or floating: A deductive model is

a logical structure based on a theory. An inductive model arises

from empirical findings and generalization from them. The

floating model rests on neither theory nor observation, but is

merely the invocation of expected structure. Application of

mathematics in social sciences outside of economics has been

criticized for unfounded models. Application of catastrophe

theory in science has been characterized as a floating model

(Wikipedia, 2012).

In portfolio management a lot of the research has been

done in modeling the uncertainty of the value of assets on a

portfolio and the relations between them, often heavily relying in

probability theory and statistics. These models are very often

used to simulate possible future scenarios through extensive

computer programs. These models may play a vital role in the

decision making process of the investors and they represent

solely another tool of analysis. Although the use of a model

constitutes another risk by itself, it may enable portfolio and risk

managers to explicitly take into consideration some of the

uncertainty they face and to quantify and estimate as accurately

as possible the risks they take (Wikipedia, 2012).


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1.5 Research gap and Research Problem

Markowitz (1952) is credited to proclaim the Numero Uno

position in successfully quantifying the two basic conflicting

objectives of investing in a portfolio viz. maximizing expected

return and minimizing risk. Since the formulation of the modern

portfolio theory, his work has attracted the attention of the

academic world and has been instrumental in providing the

fundamental direction to address the issue of portfolio

optimization. Unfortunately, in the real world of investment

management, the Markowitz framework has had surprisingly

little impact. The reasons are, first, investors tend to focus on

small segments of their potential investment universe. They

select the undervalued assets and finds assets with positive

momentum, or identifying relative value trades. Unfortunately,

the Markowitz model needs expected returns to be specified for

every component of the relevant universe which is unrealistic.

But in practice this is typically defined by a broad benchmark.

Secondly, the investors put emphasis on the weights in a

portfolio. They are not much involved in balancing the expected

returns against the contribution to portfolio risk which is the

relevant margin in the Markowitz framework. The application of

Markowitz model to ascertain the weights of the assets to

construct the portfolio often results in extreme values which

seems to be computationally justified but lacks logical and


25 | P a g e

intuitive appeal to the investor community. Thus, in practice the

situation demands that substantial amount of energy needs to

be invested in quest for reasonable numbers to make logical

appeal and intuitive acceptance by manipulating the original

model. The basic motivation for framing the research is driven

by the above discussed premise.

Choosing a single asset for investment is not a difficult

task for an investor. However, the investor’s decision becomes

difficult when faced with virtually innumerable investable choices

and more importantly an infinite number of combinations of

assets. The investor chooses those portfolios which lie on the

efficient frontier. The choice may be affected by investor’s

beliefs, objectives, preferences, expectations, risk aversion, time

and budget constrains, estimations among other. In addition,

external factors will also affect the choice. Due to all the external

and internal factors, the investor faces a dynamic decision

problem in selection of optimal portfolio. Also, it is quite evident

that the interactions of risk and reward are stated in portfolio

theory in a very broad framework and has deeply influenced the

way institutional portfolios are managed, and is also successful

in motivating the “passive management” investment techniques.

The mathematics of portfolio theory is widely used in risk


26 | P a g e

management and is basic for more recent risk measures. But

when the investors use the models to solve real world problem,

every assumption of the model becomes its limitation and often

become obvious and thus is expected to have deep implications

on the actual risk and reward that the portfolio’s holders will

bear. Most recent works have shown that practically one can not

select an optimal portfolio by considering the mean-variance

portfolio theory. And above it, the gains from portfolio

optimization are seen to have been nullified by the error explicit

on the most common model’s parameters estimators (Uppal,

DeMiguel, Garlappi and Nogales, 2007). Uppal, DeMiguel,

Garlappi and Nogales (2007) have put forward their argument

by demonstrating how a naively diversified portfolio with equal

weights in every asset, can out-perform out-of-sample on a risk

adjusted basis (Sharpe-ratio in this case) an “optimally”

diversified portfolio. However, some parameters of mean-

variance model have been improved in several other studies.

Sharpe’s (1963) study is one of the examples where he

observed the market portfolio in order to improve the estimations

of the expected return and covariance matrix.

Thus, though Markowitz model is the pioneer work in

portfolio management, it suffers from some other serious


27 | P a g e

limitations. To overcome the computational complexity, it has to

rely on a number of strict technical assumptions which are more

or less away from reality viz. markets are assumed to be perfect

that means there are neither taxes nor transactions costs and

assets are infinitely divisible; investors make their decisions at

exactly one point in time for a single-period horizon; and the

means, standard deviations and correlation coefficients are

sufficient to describe the assets’ returns. This theory needs a

large number of data. W. Sharpe introduced Single Index Model

to reduce huge data need. This model can be used with

relatively few data as compared to Markowitz’s mean – variance

– efficient portfolio theory. Single Index Model assumes that the

only reason of security movement is a common co-movement

with the index and the index is unrelated to a security’s unique

return. The model does not consider other factors which affect

the security return such as company performance, economy of

the concerned industry, economic condition of the company etc.

Multi Index Model was introduced to capture some of the

non-market influences that cause securities to move together.

Non-market influences means a set of economic factors or

structural groups that account for common movement in security

prices beyond that accounted for by the market index itself. It


28 | P a g e

uses extra indices in the hope of capturing additional information

which are not present in Sharpe’s Single Index Model. The

problem of introducing additional indices is that they may pick up

random noise rather than real influences. To eliminate the

problem of picking up random noise averaging techniques were

introduced. But the disadvantage of averaging technique is that

real information may be lost in the averaging process.

In spite of the classical mean – variance- optimum

portfolio theories, some other approaches are also used

frequently by the authors such as stochastic dominance,

geometric mean return and analysis in terms of characteristics

of the return distribution etc. Later many works have been

reported in the literature on portfolio management. Many authors

have suggested selection of portfolio in many ways. These have

established a close relationship with statistic of modeling.

Various stochastic formulations (see Marton,1980; Sahalia &

Brandt, 2001; Detemple, Garcia and Rindisscher, 2003;

Beliakov and Bagirov, 2006; Okhrin and Schmid, 2008 etc.)

have been used to discus the problem of selection of portfolio

and various complicated statistical tools have been used to

discus the problem of selection of weight of the portfolio (the

proportion of wealth invested in each individual asset). However,


29 | P a g e

the adoption of more sophisticated risk measures like value at

risk and constraints including restrictions on the maximum

number of different assets in a portfolio and minimum holding

size, have made it all but impossible to optimize portfolios with

classical techniques.

Earlier attempt by Elton and Gruber (1973) and recent

attempt by Ledoit and Wolf (2004) has come up with the finding

that the output of optimally portfolios can be improved by

imposing a structure to the covariance matrix as opposed to its

sample estimator. Although this recent research gives some

insight for the mean-variance approach, it fails to give a well-

known robust estimate for the expected return of most assets

and also fails to achieve the benefits promised by portfolio

optimization in its conception. Hence, there is a need to delve

deep into the nuances of mean-variance approach to design an

integrated approach to address the issue of portfolio

optimization in more simplistic terms and churn out results from

the proposed model which will be more attractive intuitively. To

address the issue, the need to frame integrated robust model

motivates the present research to integrate different

mathematical and statistical models along with heuristics which

allow us to estimate and emulate the risk features of a given


30 | P a g e

portfolio and to use simulation techniques to generate scenarios

and weights which enable us to perform portfolio optimization

taking into account explicitly the role of value system in the

decision making framework of the investor.

1.6 Preview of the work

Selection of the optimum portfolio is a complex task for

the general investors as choice of optimum weight is very

difficult to make. There may be basically two ways of arriving at

an optimum portfolio – one by minimizing the risk and the other

by maximizing the return. This doctoral work proposes to strike a

balance between these two. In chapter 3 optimum portfolios

have been constructed subject to minimum return constraints.

This minimum return constraint starts from the minimum return

of the security in the portfolio and increases step by step to

maximum return of the security in the portfolio. Then a heuristic

procedure for arriving at security weights has been introduced

based on the investors’ propensity to take risk. For this purpose,

two extreme situations have been chosen – risk taker and risk

aversive investor. After constructing heuristic portfolio the extent

of closeness between the ideal portfolio constructed on the

basis of optimization method and portfolio constructed on the

basis of heuristic methods has been examined. For this purpose


31 | P a g e

Euclidian distance is considered. After detailed analysis, a point

of change have been identified beyond which the optimum

portfolio is closer to optimistic portfolio than to pessimistic

portfolio before which the optimum portfolio is closer to

pessimistic portfolio than optimistic portfolio.

In chapter 4 the optimum portfolio is obtained through a

mathematical programming framework so as to minimize the

portfolio risk subject to return constraint expressed in terms of

coefficient of optimism (α), where α varies from 0 to 1.

Simultaneously, four heuristic portfolios have been developed

for optimistic and pessimistic investors, risk planners and

random selectors. Given the optimum portfolio and a heuristic

portfolio City Block Distance has been calculated to measure the

departure of the heuristic solution from the optimum solution.

In chapter 5 coefficient of optimism has been introduced

in the weight of risk planner to observe the change of the

behavior of the heuristic portfolio. The City Block Distance is

used to calculate the distance between the optimum portfolio

and the heuristic portfolios. For moderate values of the

coefficient of optimism a heuristic investor’s decision nearly

coincides with the corresponding optimum portfolio. However,


32 | P a g e

for extreme situations i.e. optimistic and pessimistic situations

heuristic portfolio differs from optimum portfolio.

Chapter 6 states the comparison between Sharpe’s cut

off principle portfolio and proposed near optimum portfolio with

that of optimum portfolio under Sharpe’s Single Index Model.

Here also the coefficient of optimism in the decision making

process has been considered to compare Sharpe’s approach

under optimality principle and cut off principle and the proposed

near optimum portfolio, based on Single Index Model and to

examine the suitability of near optimum portfolio over Sharpe’s

cut off principle portfolio. These optimum portfolios have been

obtained through a mathematical programming framework so as

to minimize the portfolio risk subject to return constraint

expressed in terms of coefficient of optimism. To know the

similarity between the cut off principle portfolio and the near

optimum portfolio with that of optimum portfolio under Sharpe’s

Single Index model, City Block Distance has been considered.

Up to moderate value and very high value of coefficient of

optimism, near optimum portfolio shows better result. However,

for moderate to high value of coefficient of optimism, the cut off

principle portfolio shows closer result. This put forward the

admissibility of the near optimum portfolio.


33 | P a g e

Chapter 7 concludes the work and presents some

limitation of the present work. It also provides a direction for

future research which can be carried out to make the proposed

approach more robust and practically applicable.

1.7 Scope of the work

The portfolio optimization problem is mainly concerned

with selecting the optimal investment strategy of an investor. In

other words, the investor looks for an optimal decision on how

many shares of which security should be purchased to maximize

the expected utility. If the investor knows the securities that may

give maximum expected return or minimum expected risk, it is

easy to take optimal decision. But in real world it is difficult to

find out those securities due to presence of efficient market. The

statistical models used in behavioural finance are not very easily

understandable to the general investor. When a general investor

wishes to invest money in any portfolio of securities they are

more concerned about the expected return and risk of the

portfolio not about the various statistical models. The present

study mainly focuses on the weight of the securities in the

portfolio and proposes a simple heuristic tool to help those

investors so that they can get a near optimum portfolio for


34 | P a g e

investment. Heuristic method is not universally accepted but is

having intuitive appeal.

The present study aims to identify the objectives,

background, methodology, and proposes a model which

proposes to simplify the portfolio optimization problem. The

academic endeavor provides a rigorous treatment to the weight

as a decision variable in the optimization framework. The

decision variable obtained heuristically is also factored into the

optimization framework so as to provide an all inclusive

dimension to the simplified approach. The work also provides a

framework and analysis of the allocation decisions of the linear

programming model and non-linear programming model. The

interactions of the different value system with different

decision- making systems of the investors have been well

captured in the heuristic model generation process and its

applicability has been ensured by comparing with the

performance of the optimal solution generated by classical

models.

One can get number of values for weight, risk and return

in the optimization framework depending on the computational

techniques used to compute weight, risk and return. The

resultant weight values churned out form optimized solutions


35 | P a g e

provides a complex set which thereby makes it practically

impossible to determine the global optima. Hence, in order to

simplify the process, weights have been considered as the only

decision variable in the optimization framework considering the

assumption that for different set of investors the risk and return

are held constant but the proportion of total investment in

different securities in the portfolio can be manipulated and

hence controlled to reach at the optimum solution. Therefore,

according to the class of investors and their corresponding risk

appetite heuristic weights can be generated. Heuristic portfolios

are designed to compute the heuristic risk and return to

compare with the optimum portfolios’ risk and return to ascertain

their closeness and make informed decisions.

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36 | P a g e

2.1 Introduction

Empirical research can be conducted by basically

following two approaches: quantitative and qualitative approach.

The generation of data in quantitative form and its subsequent

rigorous treatment in a formal and rigid fashion defines

quantitative approach. Quantitative approach can be further sub-

divided into inferential, experimental and simulation approaches

to research. In our study we have adopted experimental

approach whereby we have exercised much control over the

research environment and some variables are manipulated to

observe their effect on other variables. We have also adopted

simulation approach wherein construction of an artificial

environment is facilitated so that relevant information and data

can be generated and its effect can be studied. Thus, our

research approach has allowed for an observation of the

dynamic behaviour of a system (or its sub-system) under

controlled conditions.

2.2 Research Questions

The computational complexity of the optimization models

coupled with different investor sentiments and the role played by

different value systems ultimately directs to address the issue


37 | P a g e

and demands for designing and selecting optimum portfolio

based on different value systems. The classification of the

investors sentiments and the role played by risk taking aptitude

in portfolio optimization calls for considering the issue of why not

to consider the concept of degree of optimism in the decision

framework of the portfolio optimization. Also it is necessary to

have a mathematical formulation to arrive at the optimum

decision rule given a value for degree of optimism of the

investor. Thus we note that there is a need to suggest a

heuristic framework and subsequently examine the closeness

between optimum solution and heuristic solution. Of the

classical models of portfolio optimization Sharpe’s model is most

accepted. The basic task thus is to reconsider Sharpe’s model,

and subsequently come up with modified decision rule. Lastly it

also becomes pertinent to examine the closeness between

Sharpe’s model and modified decision rule.

2.3 Objectives

The main objective of this study is to select optimum

portfolio based on different value systems. Specifically, the

study has the following objectives:


38 | P a g e

1. To introduce the concept of co-efficient of optimism

in the decision framework of the portfolio

optimization.

2. To present the mathematical formulation to arrive at

the optimum decision rule given a value for

coefficient of optimism.

3. To suggest a heuristic framework and examine the

closeness between the optimum solution and

heuristic solution.

4. To reexamine Sharpe’s model.

5. To suggest a modified decision rule, based on

Sharpe’s concept.

6. To make a comparative study between Sharpe’s

model and modified decision rule.

2.4 Research Design

2.4.1 Sampling Design

For empirical study, we have considered daily security

wise data of fifteen companies listed in Nifty for ten years (2000

to 2009). Data have been collected from www.nseindia.com.

Then, random selection technique has been considered to


39 | P a g e

select fifteen companies out of fifty companies listed in the Nifty.

To select the companies 11th June, 2010 trading day has been

randomly considered. Jindal Steel, Ranbaxy, Reliance, HDFC,

Tata Motors, Hindalco, Wipro, Ambuja Cement, Siemens, Tata

Steel, ACC, Reliance Infrastructure, Axis Bank, BPCL and Sun

Pharmaceutical – these fifteen companies have been

considered for empirical analysis based on simple random

sampling.

As per the specifications of the model requirement and

for the purpose of model validation as discussed in chapter six

different data set has been used. The example given by Elton,

Gruber and Padberg (1978) has been considered for arriving at

the optimal portfolio as per Sharpe’s Single Index Model.

2.4.2 Statistical Design

To undertake comparative study we need to find out the

deviation from the optimum decision rule. This calls for suitable

choice of a measure of distance. In statistics, various types of

distance measures are found. The main distance measures are

discussed one by one.


40 | P a g e

City block distance

The City block distance between two points,

a( ~a a1,a2,a3,…….ak.) and b (

~b b1,b2,b3,……bk.), with k

dimensions is calculated as:

k

jjj ba

1 , j=1,2,3,……,k

The City block distance is always greater than or equal to

zero. The measurement would be zero for identical points and

high for points that show little similarity. It is being used in plant

location and other areas (Wikipedia, 2012).

Euclidean distance

Euclidean distance or Euclidean metric is the "ordinary"

distance between two points that one would measure with a

ruler, and is given by the Pythagorean formula. By using this

formula as distance, Euclidean space becomes a metric space.

The associated norm is called the Euclidean norm. Older

literature refers to the metric as Pythagorean metric.


41 | P a g e

Definition

The Euclidean distance between points p and q is the

length of the line segment connecting them ( ).

In Cartesian coordinates, if p = (p1, p2,..., pn) and

q = (q1, q2,..., qn) are two points in Euclidean n-space, then the

distance from p to q, or from q to p is given by:

(1)

Euclidean vector is the position of a point in a Euclidean

n-space. So, p and q are Euclidean vectors, starting from the

origin of the space, and their tips indicate two points. The

Euclidean norm, or Euclidean length, or magnitude of a vector

measures the length of the vector:

where the last equation involves the dot product.

A vector is a directed line segment from the origin of the

Euclidean space (vector tail), to a point in that space (vector tip).

If it is considered that the length of the vector is actually the

distance from its tail to its tip, it becomes clear that the


42 | P a g e

Euclidean norm of a vector is just a special case of Euclidean

distance: the Euclidean distance between its tail and its tip.

The distance between points p and q may have a

direction (e.g. from p to q), so it may be represented by another

vector, given by

The Euclidean distance between p and q is just the

Euclidean length of this distance or displacement vector (when p

and q represent two positions of the same point at two

successive instants of time):

2)

Which is equivalent to equation 1, and also to:

One dimension

In one dimension, the distance between two points on the

real line is the absolute value of their numerical difference. Thus


43 | P a g e

if x and y are two points on the real line, then the distance

between them is given by:

In one dimension, there is a single homogeneous,

translation-invariant metric up to a scale factor of length, which

is the Euclidean distance. In higher dimensions there are other

possible norms.

Two dimensions

In the Euclidean plane, if p = (p1, p2) and q = (q1, q2) then

the distance is given by

This is equivalent to the Pythagorean theorem.

N dimensions

In this way the distance for an n-dimensional space is

(Wikipedia, 2012)


44 | P a g e

Mahalanobis distance

Mahalanobis distance is a distance measure introduced

by P. C. Mahalanobis in 1936. According to this distance

different patterns can be identified and analyzed with the help of

correlation between the variables. It describes the similarity of

an unknown sample set to a known one. It takes into account

the correlations of the data set and is scale-invariant which are

not considered in case of Euclidean distance. In other words, it

is a multivariate measure

Definition

Formally, the Mahalanobis distance of a multivariate

vector from a group of values with

mean is defined as:

where, = variance-covariance matrix

Mahalanobis distance can also be defined as a

dissimilarity measure between two random vectors and of

the same distribution with a common covariance matrix as


45 | P a g e

The Mahalanobis distance reduces to the Euclidean

distance when the covariance matrix is the identity matrix. If the

covariance matrix is diagonal, then the resulting distance

measure is called the normalized Euclidean distance:

Where, is the standard deviation of the and over

the sample set.

The Mahalanobis distance can be used to classify a test

point as belonging to one of k classes. One first estimates the

common covariance matrix of all the classes, usually based on

samples known to belong to each class. Then the Mahalanobis

distance to each class can be computed with a given test

sample, and can be classified as belonging to that class for

which the Mahalanobis distance is minimal.

Applications

Mahalanobis' discovery was prompted by the problem of

identifying the similarities of skulls based on measurements in

1927. It is widely used in cluster analysis and classification


46 | P a g e

techniques. It is closely related to Hotelling's T-square

distribution used for multivariate statistical testing and Fisher's

Linear Discriminant Analysis that is used for supervised

classification. This distance can also be used in the

development of the linear regression model (Wikipedia, 2012).

Chebyshev distance

Chebyshev distance or Tchebychev distance is

developed by Pafnuty Chebyshev. It is a metric defined on a

vector space where the distance between two vectors is the

greatest of their differences along any coordinate dimension.

The distance is used in chess. The Chebyshev distance

between two spaces on a chess board gives the minimum

number of moves a king requires to move between them. For

this reason it is also known as chessboard distance.

Definition

The Chebyshev distance between two vectors or points p

and q, with standard coordinates and , respectively, is


47 | P a g e

hence it is also known as the L∞ metric (Wikipedia, 2012).

Minkowski distance

The Minkowski distance is a metric on Euclidean space

which can be considered as a generalization of both the

Euclidean distance and the Manhattan distance.

Definition

The Minkowski distance of order p between two points

is defined as:

The Minkowski distance is a metric as a result of the

Minkowski inequality (Wikipedia, 2012).

Reasons for considering Euclidean Distance and City Block

Distance in the Study

However, in this doctoral thesis only Euclidean Distance

(ED) and City Block Distance (CBD) have been considered as

two standard measures of distance for a formal look at the


48 | P a g e

behavior of heuristic solutions in respect of the optimum

solutions. In other words, with these distance measures the

similarity between the optimum choice of portfolios and the

heuristic choice of portfolios is computed. As distance between

two points has been measured in this thesis, ED and CBD are

the appropriate measures. The Mahalonobis distance is used in

cluster analysis and classification techniques mainly. Chebyshev

distance is mainly used in chess. For these reasons these

distance measures have not been used.

Decision theory

Decision theory is concerned with identifying the values,

uncertainties and other issues relevant in a given decision, its

rationality, and the resulting optimal decision. It is closely related

to the field of game theory as to interactions of agents with at

least partially conflicting interests whose decisions affect each

other.

Most of decision theory is normative or prescriptive.

Normative decision theory is concerned with identifying the

optimum decision to take, assuming an ideal decision maker

who is fully informed, able to compute with perfect accuracy,

and fully rational. When people violates the optimality rules i.e.;


49 | P a g e

people do not behave in ways consistent with axiomatic rules,

often their own, there is a related area of study, called a positive

or descriptive discipline. This type of decision theory is

attempting to describe what people will actually do. Since the

normative, optimal decision often creates hypotheses for testing

against actual behaviour, the two fields are closely linked.

In recent decades, a new term ‘behavioural decision

theory’ is emerging and this has contributed to a re-evaluation of

what rational decision-making requires.

Some decisions are difficult because of the need to take

into account how other people in a given situation will respond to

the decision that is taken. Game theory is used to analyse such

social decisions. From the standpoint of game theory most of

the problems treated in decision theory are one-player games

(or the one player is viewed as playing against an impersonal

background situation).

Other areas of decision theory are concerned with

decisions that are difficult simply because of their complexity, or

the complexity of the organization that has to make them. In

such cases the difficulty arises in determining the optimal

behaviour. The Club of Rome, for example, developed a model


50 | P a g e

of economic growth and resource usage that helps politicians

make real-life decisions in complex situations.

Alternatives to decision theory

The probability theory can be used as an alternative to

decision theory. Though, it is a highly controversial issue. The

proponents of fuzzy logic, possibility theory, Dempster–Shafer

theory and info-gap decision theory state that probability is only

one of many alternatives and point to many examples where

non-standard alternatives have been implemented with apparent

success. Non- probabilistic rules such as minimax and maximax

principles are also used in making decision. The rules are

robust. Hurwitz developed the maximin and maximax rules to

take decision (Wikipedia, 2012).

Hurwitz Criterion

A compromise between the maximax and maximin criteria

is the Hurwitz criterion. Maximax criterion assumes that the

decision maker is not optimistic. According to this criterion the

maximum payoff for each alternative is located first and then the

alternative with the maximum number will be selected. This

criterion decision locates the alternative with the highest

possible gain. The maximin criterion, on the other hand,


51 | P a g e

assumes that the decision maker is not totally pessimistic and in

maximin criterion, the minimum payoff for each alternative is

located first and then the alternative with the maximum number

will be selected .

Thus, the Hurwitz principle assumes that the decision

maker is neither totally optimistic nor totally pessimistic. With the

Hurwicz Criterion, the decision payoffs are weighted by a

Coefficient of Optimism, a measure of the decision maker’s

degree of optimism. The coefficient of optimism, defined as “α”,

lies between 0 and 1 (i.e. 0≤α≤1.0).

α=1.0 represents that the decision maker is completely

optimistic; α=0 represents that the decision maker is completely

pessimistic. (Given this definition, 1-α is the Coefficient of

Pessimism). When the value of α=0.5, the decision maker is

neutral. The advantage of this approach is that it allows the

decision maker to build in personal feelings about relative

optimism and pessimism.

The formula is

Criterion of Realism = α (maximum payoff for an

alternative) + (1-α) (minimum payoff for an alternative)


52 | P a g e

As Hurwitz principle talks about optimistic, pessimistic

and neutral investors (to be referred as risk planner in the

present study) this decision criterion is applicable in the present

study (Wikipedia, 2012).

Linear programming

Linear programming or linear optimization is a

mathematical method for determining a way to achieve the best

outcome (such as maximum profit or minimum cost) in a given

mathematical model for some list of requirements represented

as linear relationships. Linear programming is a specific case of

mathematical programming or mathematical optimization.

In other words linear programming is a technique for the

optimization of a linear objective function, subject to certain

linear constraints. The constraints may be linear equality or

linear inequality type. Its feasible region is a convex polyhedron.

That means the feasible region is a set defined as the

intersection of finitely many half spaces, each of which is

defined by a linear inequality. Its objective function is a real-

valued affine function defined on this feasible region. A linear

programming algorithm finds a point in the feasible region where


53 | P a g e

this function has the smallest (or largest) value if such point

exists.

Linear programs are problems that can be expressed in

canonical form:

where x represents the vector of variables (to be

determined), c and b are vectors of known coefficients, A is a

known matrix of coefficients, and is the matrix transpose.

The expression to be maximized or minimized is called the

objective function (cTx in this case). The inequalities Ax ≤ b and

X >= are the constraints which specify a convex polytope over

which the objective function is to be optimized. In this context it

can be stated that when two vectors have the same dimensions,

they can be comparable. If every entry in the first is less-than or

equal-to the corresponding entry in the second then the first

vector is less-than or equal-to the second vector.

Linear programming can be applied to various fields of

study. It is used in business, economics, finance and in some

engineering problems. Industries that use linear programming


54 | P a g e

models include transportation, energy, telecommunications, and

manufacturing. It is also useful in modeling diverse types of

problems in planning, routing, scheduling, assignment, and

design.

Existence of optimal solutions

Geometrically, the linear constraints define the feasible

region, which is a convex polyhedron. When a linear function is

a convex function, it implies that every local minimum is a global

minimum; similarly, when a linear function is a concave function,

it represents that every local maximum is a global maximum.

Optimal solution need not exist, for two reasons. First, no

feasible solution exists if two constraints are inconsistent.

Second, when the feasible region is unbounded in the direction

of the gradient of the objective function, which is the vector of

the coefficients of the objective function, then no optimal value is

attained.

Otherwise, in the presence of a feasible solution and

bounded objective function, the optimum value is always

attained on the boundary of optimal level-set, by the maximum

principle for convex functions (alternatively, by the minimum

principle for concave functions).However, some problems have


55 | P a g e

distinct optimal solutions. For example, when the objective

function is the zero function, the problem of finding a feasible

solution to a system of linear inequalities is a linear

programming problem. For this feasibility problem with the zero-

function for its objective-function, if there are two distinct

solutions, then every convex combination of the solutions is a

solution.

Algorithms

A series of linear constraints on two variables produces a

region of possible values for those variables. Solvable problems

will have a feasible region in the shape of a simple polygon. The

vertices of the polytope are also called basic feasible solutions

(Wikipedia, 2012).

Nonlinear programming

Nonlinear programming (NLP) is defined as the process

of solving a system of equalities and inequalities (collectively

termed constraints) over a set of unknown real variables, along

with an objective function to be maximized or minimized, where

at least one of the constraints and the objective function is

nonlinear.


56 | P a g e

Applicability

A typical non-convex problem is that of optimising

transportation costs with various connectivities and capacity

constraints. The transportation costs can be selected from a set

of transportation methods, one or more of which exhibit

economies of scale. An example would be petroleum product

transport given a selection or combination of pipeline, rail tanker,

road tanker, river barge, or coastal tank ship. Owing to

economic batch size the cost functions may have discontinuities

in addition to smooth changes.

Mathematical formulation of the problem

The problem can be stated simply as:

to maximize some variable such as product

throughput

or

to minimize a cost function

where


57 | P a g e

s.t. (subject to)

Methods for solving the problem

If the objective function f is linear and the constrained

space is a polytope, the problem is a linear programming

problem and can be solved by using linear programming

solutions.

If the objective function is concave or maximization type

problem, or convex or minimization type problem and the

constraint set is convex, then the program is called convex and

then convex optimization method can be used to solve the

problem.

When the objective function is a ratio of a concave and a

convex function (in the maximization case) and the constraints

are convex, then fractional programming techniques can be

used by transforming the problem to a convex optimization

problem.


58 | P a g e

To solve non-convex problems several methods are

available. One approach is to use special formulations of linear

programming problems. Another method involves the use of

branch and bound techniques. In this method the program is

divided into subclasses to be solved with convex (minimization

problem) or linear approximations that form a lower bound on

the overall cost within the subdivision. With subsequent

divisions, at some point an actual solution will be obtained

whose cost is equal to the best lower bound obtained for any of

the approximate solutions. This solution is optimal, although

possibly not unique. The best possible solution is within a

tolerance from the best point found, known as ε-optimal.

Terminating to ε-optimal points is typically necessary to ensure

finite termination. This ε-optimal solution is useful for large,

difficult problems and problems with uncertain costs or values

where the uncertainty can be estimated with an appropriate

reliability estimation. Under differentiability and constraint

qualifications, the Karush–Kuhn–Tucker (KKT) conditions give

necessary conditions for a solution to be optimal. These

conditions are also sufficient under convexity (Wikipedia, 2012).


59 | P a g e

2.5 Methods to be used

2.5.1 Optimization Method

Every investor fundamentally wants to select an optimal

portfolio. But construction of optimal portfolio is very difficult. The

basic of portfolio optimization method has given by Markowitz in

1952. Optimization of portfolio can be done in two ways – (1)

maximizes return subject to a certain level of risk and (2)

minimize risk subject to a certain level of minimum return. An

investor always wants to go for either of these above two

optimizations.

The present thesis has considered Markowitz

optimization model. The objective function is considered to

minimize the risk subject to a step by step increase in the return.

The two optimization approaches can be balanced by

minimizing the risk subject to a step-by-step increase in the level

of minimum return. For the lower values of minimum return the

risk can be minimized into lesser restrictions. For higher values

of the minimum return the minimization problem becomes more

restrictive giving thereby more importance on return.

In the true sense, choice of portfolio depends on the

value system of the investors and arriving at value based


60 | P a g e

optimum solution may not be an easy task. To handle this

problem, the thesis proposes to incorporate in the optimization

framework the coefficient of optimism (α) introduced by Hurwitcz

(see Taha) in the context of decision theory.

Depending on the choice of an optimum solution can be

obtained in a non linear programming approach. One can get

the optimum portfolio corresponding to the minimum return of

the security by considering = 0 and the optimum portfolio

corresponding to the maximum return of the security by

considering =1. Here, = 0 represents pessimistic investors

and = 1 represents optimistic investors. For random selector

and risk planner’s the value of lies between 0 and 1. In each

case, one can get different optimum weights and with those

weights return and risk of the optimum portfolio can be

calculated.

2.5.2 Heuristic Method

Classical optimization models may not be easy to handle

in the real world. Sometimes, heuristic methods are the only

way out. Mansini and Speranza (1999), Chang et al. (2000)

suggested the use of heuristic optimization techniques to

portfolio selection. Heuristic defines as experience-based


61 | P a g e

techniques for problem solving, learning, and discovery.

Heuristic methods are used to speed up the process of finding

an acceptable solution where an in-depth search is not possible.

In other words, heuristic methods are the strategies which are

readily accessible but loosely applicable. A rule of thumb, an

educated guess, an intuitive judgment, or common sense etc. is

used as the heuristic methods.

In the present thesis the security weights of the investors

can also be generated heuristically based on the value system

of the investors. Four types of investors are considered

according to their propensity to take risk – risk taker or optimistic

investor, risk averse or pessimistic investor, risk planner and

random selector. As their risk apatite is different, their security

weights are different.

Optimistic investor’s put emphasis on the expected

return. So when the weight of optimistic investor is to be

generated emphasis has been given to the security returns and

the security weights are considered as directly proportional to

the expected return of the security. For heuristic pessimistic

portfolio, security weights are to be considered as inversely

proportional to standard deviation of the security. Here an

assumption is to be made that a pessimistic investor gives more


62 | P a g e

importance on risk of the portfolio ignoring the expected return

part and more is the risk less is the chance of selection. On the

other hand, as the risk planner considers both risk and return in

choosing the securities of a portfolio, the formula of security

weight for them is to be constructed by considering both return

and risk of the securities. Lastly for random selector the

assumption is to be taken that they randomly select securities

for the portfolio instead of giving importance to either return or

risk. Thus, they put equal weight to each security in the portfolio.

2.5.3 Model based

Prior to Markowitz's work, the assessment of the risks

and rewards of portfolios was carried out through the analysis of

individual securities independently. Markowitz was the person

who first introduced the concept of diversification of securities.

He proposed that investors should focus on joint risk-reward

features of the securities while selecting portfolios. He argued

that a portfolio consisted with more than one security would

have less risk than that of an individual security.

Markowitz assumed that, a rational investor would

choose the portfolio with the highest expected return, for a given


63 | P a g e

level of risk and similarly, for a given level of expected return, a

rational investor would choose the portfolio with the lowest risk.

The theory used historical returns of each asset of the

portfolio and average return is used to estimate expected return.

Standard deviation or variance of return is used to calculate risk

and linear correlation is also used for this purpose. Markowitz

used volatility and expected return as proxies for risk and

reward.

Markowitz defined an “optimal” way of selecting a

portfolio by balancing the risk and reward features of the

portfolio within the infinite possible alternatives of portfolio. In

other words, a portfolio is said to be optimal if there is no

portfolio having the same risk with a greater expected return,

and there is no portfolio having the same expected return with a

lower risk. The set of portfolios constructed in this optimal

manner conform what he called the efficient frontier. He

concludes that an investor should select a portfolio that lies on

the efficient frontier.

The classical Markowitz portfolio selection is based on

the assumption of multinormally distribution of returns. Since the

multinormal model is inadequate for risk estimation, the


64 | P a g e

Markowitz optimal portfolio might not be optimal any more. The

mean-variance theory is applicable where the asset class is

different and it is giving an open form of solution.

The Single Index Model is based on the observation that

securities move together with the market. That means when a

market represented by a market index moves up or down,

security prices will also move accordingly. Therefore, there exist

a positive correlation between the security prices and the market

index. In fact, the Single Index Model assumes that the only

relevant cause of correlation between two assets is their

common dependency on market movements. The other

assumption of the model is that a security’s return does not

affect the market index. That means, the index is unrelated to

unique return.

The mathematical model of Single Index Model is a two

variable linear regression equation with the return on each

security as dependent variable and the return on a market index

as independent variable. Like Markowitz, Sharpe also used

variance as the measure of risk. But he introduced covariance

term to calculate risk of a portfolio.

V{tÑàxÜ F

fxÄxvà|ÉÇ Éy cÉÜàyÉÄ|É Utáxw ÉÇ `|Ç|ÅâÅ exàâÜÇ VÉÇáàÜt|Çà


65 | P a g e

3.1 Introduction

A portfolio means a combination of two or more assets.

From a given set of assets a large number of portfolios can be

selected. Each portfolio has its own risk-return characteristics. In

view of the innumerable parameters involved in selection of a

portfolio the concerned decision making becomes a complicated

task. Selection of the optimum portfolio is a complex task for the

general investors as choice of optimum weight is very difficult to

make. In most of the cases it is difficult to arrive at an optimal

solution. One of the key problems of portfolio optimization is to

obtain correct and precise portfolio weights (the function of

wealth invested in individual asset). Precise estimation of

weights reduces unnecessary transaction costs and helps to

construct the optimum portfolio. The problem of optimum

portfolio selection has been discussed by various researchers

from various angles. Some authors preferred classical Mean –

Variance technique to select optimal portfolio, whereas other

group proposed new methods of optimal portfolio selection.

Arriving at a heuristic solution may be possible at a given point

of time and at a given set of conditions but the same may not be

a uniformly optimal one.


66 | P a g e

When Markowitz (1952) developed a Mean – Variance

framework of modern portfolio optimization more that 50 years

ago, simplicity was the key part of its appeal. However, the

adoption of more sophisticated risk measures like value at risk

and constraints including restrictions on the maximum number of

different assets in a portfolio and minimum holding size, have

made it all but impossible to optimize portfolios with classical

techniques. Financial analysts have been talking about

downside risk for years but few have attempted to use it for

portfolio selection because of the resultant optimization problem

and its difficulty to solve. Since the crucial work of Markowitz

and Sharpe Mean – Variance have been the central focuses of

portfolio analysis, most authors follow this Mean – Variance

technique to select an optimum portfolio. The present study has

tried to show that one investor heuristically can reach near to the

optimum solutions.

3.2 Earlier works

Ross (1976) developed an alternative asset pricing

theorem called Arbitrage Pricing Theory (APT) which is based

on less restrictive assumptions. Under no arbitrage conditions,

the theory states that the return on any stock is linearly related


67 | P a g e

to asset of systematic factor or risk factors. It means that two

portfolios that have the same risk cannot offer different expected

returns. However, APT fails to prescribe the nature and number

of factors.

Most asset allocation analysis used the Mean – Variance

approaches for analyzing the trade off between risk and

expected return. Merton (1980) described the difficulty of

precisely estimating the expected portfolio return. In the

classical Merton problem the investor can allocate her/ his

money into a risk less account governed by a deterministic

interest rate and different risky stock. Merton was able to reduce

the portfolio problem to a stochastic optimum control problem,

which can be solved by using stochastic control methodology.

He considered New York stock exchange data from 1926 to

1978 for the analysis.

Jauch and Glueck (1988) have pointed out seven values.

The most prominent is risk value that affects the business

decision-making process.

Sahalia & Brandt (2001) studied the asset allocation

problem when the conditional moments of return are partly

predictable. They examined how the optimal portfolio weights


68 | P a g e

depend on the predictive variables. To know the nature of

dependency of the optimal portfolio weights on the predictive

variables they combined the predictor in to a single index that

best captures time variations in investment opportunities.

Schaerf (2002) discussed the problem of a portfolio

selection, which provide a suitable balance of expected risk and

return to the investors. He considered additional constraints on

the cardinality of the portfolio and on the quantity of individual

assets over the seminal mean – variance model of Markowitz.

The Detemple, Garcia and Rindisscher (2003) proposed

a new simulation based approach for the optimal portfolio

allocation in realistic environment with complex dynamics for the

state variables and large numbers of factors and assets. They

also analyzed settings where stock returns are predicted by

dividend yields and where investors have wealth dependent

relative risk aversion. They used Nasdaq index, and S& P 500

index as empirical evidence.

Beliakov and Bagirov (2006) examined numerical

performance of various methods of calculations of the

conditional values at risk (CVaR) and carried out portfolio

optimization with respect to this risk measure. They found that


69 | P a g e

nonsmooth optimization can be used efficiently for large portfolio

optimization and also examined parallel execution of this

method on computer cluster.

Brandt and Clara (2006) presented a novel approach to

dynamic portfolio selection that is as easy to implement as the

static Markowitz paradigm. They expanded the set of asset and

statistically optimized the portfolio in that extent asset space.

They considered conditional portfolio and timing portfolios. They

concluded that the static choice of this managed portfolios

represents a dynamic strategy that closely approximates the

optimal dynamic strategy for horizons up to five years.

A multivariate shrinkage estimator was proposed by

Okhrin and Schmid (2008) to calculate the optimal portfolio

weights. They discussed the estimated shrinkage weights

analytically and also used Monte Carlo simulation technique to

calculate the estimated shrinkage weight of the portfolio. They

have shown how the shrinkage estimator gives the most

accurate estimation of the portfolio weights. They also carried

out an empirical analysis in support of their observation.


70 | P a g e

3.3 Optimization Framework

In portfolio theory expected return and risk play vital role

in determining the choice set. The formula, we have used to

calculate return is given below:

ijR (Closing price of the ith security for the jth period –

Opening price of the ith security for the jth period) / (Opening

price of the ith security for the jth period) (1)

The risk of a security is the standard deviation of the

observed returns and is given by the formula:

k

jiiji R

1

2 (2)

Average return and risk for the securities so selected

have been calculated for subsequent analysis. An investor

always wants to go for either maximizing return subject to

certain level of risk or minimizing risk subject to certain level of

return. A balance between these two approaches has been

made by minimizing the risk subject to a step-by-step increase

in the level of minimum return. The optimization problem

considered for the current work is of the type:


71 | P a g e

Min ww ˆˆ

Subject to oiRw ˆ

μmin μ0 μmax

The risk and return of the optimum portfolio have been

calculated by using the following equations.

i

n

i

OP RwR O

i

1

(3)

OP =

n

i

n

j

ijjiji rww OO

1 1

(4)

For the numerical study the minimum return of the

security in the portfolio is considered as the starting point and in

each step the value of minimum return requirement has been

step by step increased by 0.001 point. In this process finally, the

maximum return of the security in the portfolio has been

reached. In each case, we have got different optimum weights.

With those weights return and risk of the optimum portfolio have

been calculated. Even if multiple assets are there in the

portfolio, the final return cannot be less than the minimum return

and more than the maximum return. As a result, there is no trial

and error in this approach. It is a systematic process followed


72 | P a g e

from minimum possibility to maximum possibility. The increment

of 0.001 is of course a subjective choice.

3.4 Heuristic Framework

In this phase a heuristic solution have been introduced

with the help of which one may get an alternative choice of

portfolio based on propensity to take risk. In this chapter, two

types of portfolios have been constructed one for optimistic

investor and the other for pessimistic investor.

In case of heuristic optimistic portfolio, we have

considered security weights directly proportional to expected

return of the security based on the assumption that an optimistic

investor gives more importance on expected return irrespective

of the risk. To ensure non-negativity in the system a locational

shift has been undertaken with point of origin as μmin. Thus, for

heuristic optimistic solution following formula has been used for

calculating the weights of the securities of the portfolio.

min1

min

i

n

i

iHOiw (5)

Given this weight function we have calculated the

expected return and risk as follows:


73 | P a g e

i

n

i

HOP RwR HO

i

1

(6)

HOP =

n

i

n

j

ijjiji rww HOHO

1 1

(7)

For heuristic pessimistic portfolio, security weights have

been considered as inversely proportional to standard deviation

of the security. Here we have made an assumption that a

pessimistic investor gives more importance on risk of the

portfolio ignoring the expected return part and more is the risk

less is the chance of selection. For pessimistic solution following

formula has been used for calculating the weights of the

securities of the portfolio. For heuristic pessimistic solution

following formula has been used for calculating the weights of

the securities of the portfolio.

i

n

i

iHPiw

61

61

1

(8)

Given this weight function the expected return and risk

are calculated as follows:

i

n

i

HPP RwR HP

i

1

(9)


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HPP =

n

i

n

j

ijjiji rww HPHP

1 1

(10)

3.5 Closeness between Heuristic and Optimization

framework

The closeness between the optimum portfolio and the

heuristic portfolio is measured with the help of Euclidean

Distance, a distance between two points. The closeness is

measured to observe that whether the heuristic choice of

portfolio is near to the optimum portfolio.

3.6 Data Analysis and Interpretation

The objective is to see the closeness between the

optimum solution and the two risk-based heuristic solutions. This

has been carried out graphically to examine the closeness

between the optimum solution and heuristic solutions. For this

purpose, different tables have been framed and corresponding

graphs have been drawn for depicting the closeness between

the optimum solution and heuristic solutions. Another graph has

been drawn to find out the threshold point beyond which the

optimum portfolio is closer to optimistic portfolio than to

pessimistic portfolio and before which the optimum portfolio is


75 | P a g e

closer to pessimistic portfolio than optimistic portfolio. This point

has been determined by identifying the point of intersection

between lines obtained by joining the Euclidian Distance (ED)

between the optimum solution and heuristic solutions. However,

ED has been considered as a standard measure of distance for

a formal look at the behavior of heuristic solutions in respect of

the optimum solutions.

Equations 3 and 4 are used to calculate the return and

risk of the optimum portfolio respectively. Equations 5 and 8 are

used to calculate the weights of the optimistic portfolio and

pessimistic portfolio respectively. The return and risk of the

heuristic optimistic portfolio are calculated by using Equations 6

and 7 respectively. On the other hand Equations 9 and 10 are

used to determine the return and risk of heuristic pessimistic

portfolios respectively. The weights of the optimum portfolios

and heuristic portfolios under different values of minimum return

constraints, expected return and risk of the optimum portfolio,

heuristic optimistic portfolio and heuristic pessimistic portfolio

and ED between the optimum portfolios and the heuristic

portfolios have been shown separately in different tables from

Table 3.1 to Table 3.17. Regarding the graphical presentation of

closeness between optimum solutions and heuristic solutions, in


76 | P a g e

the horizontal axis we have measured the expected return and

in the vertical axis we have measured the expected risk. From

graph no: 3.1 to 3.8, point O represents the optimum choice of

the portfolio, point A represents the pessimistic choice of the

portfolio and point B represents the optimistic choice of the

portfolio.

The first minimum return constraint considered is -0.0057.

Table 3.1 and Table 3.2 describe the optimum weights, heuristic

optimistic weights, heuristic pessimistic weights and the returns

and risks of the three portfolios under the three different value

systems respectively. The expected return and risk of the three

portfolios are graphically described in Figure 3.1.

Table 3.1: Weights of optimum portfolio and heuristic

portfolios under -0.0057 minimum return constraint

Securities Optimum weights

Heuristic Optimistic Weights

Heuristic Pessimistic

Weights JINDALSTEL

0 0.090668 0.053313

RANBAXY

0.1936191 0.066351 0.074502

RELIANCE

0.06280846 0.077863 0.076569

HDFC

0.1424088 0.101525 0.075374

TATAMOTORS

0.005869277 0.059956 0.065455

HINDALCO

0.05411821 0.072906 0.065483


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WIPRO

0 0.064353 0.059075

AMBUJACEM

0.1068151 0.051962 0.077734

SIEMENS

0.07751241 0.073386 0.067679

TATASTEEL

0 0.056439 0.064709

ACC

0.02198061 0.061715 0.071019

RELINFRA

0 0.064433 0.059468

AXISBANK

0.04030778 0 0.054746

BPCL

0.1195407 0.071308 0.064254

SUNPHARMA

0.1750195 0.087136 0.07062

Table 3.2: Expected return and risk of three different

portfolios under – 0.0057 as the minimum return constraint

Portfolio Expected return Expected risk

Optimum portfolio -0.001142735 0.024383995

Heuristic Optimistic portfolio

-0.001074532 0.016490497

Heuristic pessimistic portfolio

-0.001478186 0.016463641


78 | P a g e

Figure 3.1: Expected return and risk under -0.0057 minimum

return constraint

From Figure 3.1, it is clear that the optimum portfolio (O) and

optimistic portfolio (B) are closer to each other as compared to

the pessimistic portfolio (A). Expected returns of the optimum

portfolio and optimistic portfolio are very close. Though, the

expected risk of the optimum portfolio is higher than the risk of

optimistic portfolio. Whereas, the risks of pessimistic and the

optimum portfolio are more or less same but expected return of

optimistic portfolio is higher than that of pessimistic portfolio.

The next minimum return constraint considered is -

0.0047. Table 3.3 and Table 3.4 describe the optimum weights,

heuristic optimistic weights, heuristic pessimistic weights and the

returns and risks of the three portfolios under the three different


79 | P a g e

value systems respectively. The expected return and risk of the

three portfolios are graphically described in Figure 3.2.






Weights JINDALSTEL

0 0.090668 0.053313

RANBAXY

0.1936189 0.066351 0.074502

RELIANCE

0.06280945 0.077863 0.076569

HDFC

0.1424077 0.101525 0.075374

TATAMOTORS

0.005868386 0.059956 0.065455

HINDALCO

0.05411870 0.072906 0.065483

WIPRO

0 0.064353 0.059075

AMBUJACEM

0.1068156 0.051962 0.077734

SIEMENS

0.07751275 0.073386 0.067679

TATASTEEL

0 0.056439 0.064709

ACC

0.02197968 0.061715 0.071019

RELINFRA

0 0.064433 0.059468

AXISBANK

0.04030873 0 0.054746

BPCL

0.1195404 0.071308 0.064254

SUNPHARMA

0.1750197 0.087136 0.07062


80 | P a g e



Portfolios Expected Return Expected Risk


Heuristic Optimistic

portfolio

-0.001074532 0.016490497

Heuristic

pessimistic

portfolio

-0.001478186 0.016463641


return constraint


81 | P a g e

From Figure 3.2, it is evident that the optimistic portfolio

(B) and pessimistic portfolio (A) are placed more or less same

distance to the optimum portfolio (O). The expected risk of the

optimum portfolio is much lower than that of optimistic and

pessimistic portfolios. Whereas, expected return of optimistic

portfolio is slightly higher than that of the optimum portfolio.

Optimistic portfolio’s expected return is lowest in this case. On

the other hand, expected risks of pessimistic and optimistic

portfolios are more or less same under this minimum return

constraint.

The third minimum return constraint used is -0.0037 and

Table 3.5 and Table 3.6 represent the optimum weights,

heuristic optimistic weights, heuristic pessimistic weights and the

returns and risks of the three portfolios under the three different

value systems respectively. The expected return and risk of the

three portfolios are described in Figure 3.3.


82 | P a g e






Weights JINDALSTEL

0 0.090668 0.053313

RANBAXY

0.1936181 0.066351 0.074502

RELIANCE

0.06281429 0.077863 0.076569

HDFC

0.1424053 0.101525 0.075374

TATAMOTORS

0.005865420 0.059956 0.065455

HINDALCO

0.05412018 0.072906 0.065483

WIPRO

0 0.064353 0.059075

AMBUJACEM

0.1068162 0.051962 0.077734

SIEMENS

0.07751338 0.073386 0.067679

TATASTEEL

0 0.056439 0.064709

ACC

0.02197784 0.061715 0.071019

RELINFRA

0 0.064433 0.059468

AXISBANK

0.04031004 0 0.054746

BPCL

0.1195388 0.071308 0.064254

SUNPHARMA

0.1750204 0.087136 0.07062


83 | P a g e

Table3.6: Expected return and risk of three different


Portfolios Expected Return Expected Risk



portfolio

-0.001074532 0.016490497

Heuristic

pessimistic portfolio

-0.001478186 0.016463641


return constraint


84 | P a g e

In Figure 3.3 again the optimum solution (O), the

pessimistic solution (A) and the optimistic solution (B) are

situated in the same place as under -0.0047 minimum return

constraint. Pessimistic portfolio shows the worst result in this

figure. Both expected return and risk of pessimistic portfolio are

more than that of optimistic portfolio.

The next minimum return constraint is -0.0027 and the

weights of the three different portfolios along with the expected

returns and risks of them are shown in Table 3.7 and Table 3.8

with corresponding graph (Figure 3.4) representing the position

of the three portfolios with their expected return and risk.






Weights JINDALSTEEL

0 0.090668 0.053313

RANBAXY

0.1936165 0.066351 0.074502

RELIANCE

0.06282562 0.077863 0.076569

HDFC

0.1423984 0.101525 0.075374

TATAMOTORS

0.005858875 0.059956 0.065455

HINDALCO

0.05412454 0.072906 0.065483

WIPRO 0 0.064353 0.059075


85 | P a g e

AMBUJACEM

0.1068160 0.051962 0.077734

SIEMENS

0.07751463 0.073386 0.067679

TATASTEEL

0 0.056439 0.064709

ACC

0.02197602 0.061715 0.071019

RELINFRA

0 0.064433 0.059468

AXISBANK

0.04031180 0 0.054746

BPCL

0.1195347 0.071308 0.064254

SUNPHARMA

0.1750229 0.087136 0.07062

Table 3.8: Expected return and risk of three different portfolios under – 0.0027 as the minimum return constraint

Portfolios Expected return Expected risk



portfolio

-0.001074532 0.016490497

Heuristic


-0.001478186 0.016463641


86 | P a g e


return constraint

Figure 3.4 projects that the expected return of the

pessimistic solution (A) is less than that of the optimum solution

(O) and risk of pessimistic solution (A) and optimistic solution

(B), both is more than that of the optimum solution (O). But the

expected return of the optimistic solution (B) is slightly more

than that of the optimum solution (O). The behaviour of the

optimum portfolio, optimistic portfolio and pessimistic portfolio

are almost same like under the minimum return constraint -

0.0037.

Now consider the minimum return constraint -0.0017. The

weights of the three different portfolios and returns and risks of

them are shown in Table 3.9 and Table 3.10 with corresponding


87 | P a g e

graph, Figure 3.5, representing the position of the three

portfolios under minimum return constraint -0.0017.


portfolios under -0.0017 as the minimum return constraint




Weights JINDALSTEL

0 0.090668 0.053313

RANBAXY

0.1935911 0.066351 0.074502

RELIANCE

0.06289563 0.077863 0.076569

HDFC

0.1423614 0.101525 0.075374

TATAMOTORS

0.005813745 0.059956 0.065455

HINDALCO

0.05418887 0.072906 0.065483

WIPRO

0 0.064353 0.059075

AMBUJACEM

0.1068153 0.051962 0.077734

SIEMENS

0.07754785 0.073386 0.067679

TATASTEEL

0 0.056439 0.064709

ACC

0.02203039 0.061715 0.071019

RELINFRA

0 0.064433 0.059468

AXISBANK

0.04028770 0 0.054746

BPCL

0.1195005 0.071308 0.064254

SUNPHARMA

0.1749674 0.087136 0.07062


88 | P a g e






portfolio

-0.001074532 0.016490497

Heuristic


-0.001478186 0.016463641


return constraint

Under -0.0017 minimum return constraint the positions of

the optimum solution (O), pessimistic solution (A) and optimistic


89 | P a g e

solution (B) are same as under -0.0027 minimum return

constraint. The pessimistic portfolio shows the worst result.

The Table 3.11 and Table 3.12 under -0.0007 minimum

return constraint depicts the optimum weights, heuristic



systems. The expected return and risk of the three portfolios are

described in Figure 3.6.

Table 3.11: weights of optimum portfolio and heuristic

portfolios under -0.0007 as the minimum return constraint




Weights JINDALSTEL

0 0.090668 0.053313

RANBAXY

0.1712044 0.066351 0.074502

RELIANCE

0.08403390 0.077863 0.076569

HDFC

0.2177282 0.101525 0.075374

TATAMOTORS

0 0.059956 0.065455

HINDALCO

0.05253162 0.072906 0.065483

WIPRO

0 0.064353 0.059075

AMBUJACEM

0.04933059 0.051962 0.077734

SIEMENS

0.08757217 0.073386 0.067679

TATASTEEL 0 0.056439 0.064709


90 | P a g e

ACC

0.01605094 0.061715 0.071019

RELINFRA

0 0.064433 0.059468

AXISBANK

0 0 0.054746

BPCL

0.1197338 0.071308 0.064254

SUNPHARMA

0.2018143 0.087136 0.07062





Heuristic Optimistic portfolio

-0.001074532 0.016490497

Heuristic pessimistic portfolio

-0.001478186 0.016463641


91 | P a g e


return constraint

In Figure 3.6 the optimum solution (O) is going towards

the origin. That means its expected return is increasing and risk

is more or less same as under -0.0017 minimum return

constraint. From the figure it can be obtained that distance

between the optimum portfolio and the pessimistic portfolio is

more than that of the optimum portfolio and the optimistic

portfolio.

The Table 3.13 and Table 3.14 under 0.0003 minimum

return constraint depicts the optimum weights, heuristic




92 | P a g e

systems respectively. The expected return and risk of the three

portfolios are described in Figure 3.7.


portfolios under 0.0003 as the minimum return constraint




Weights JINDALSTEL

0.1072901 0.090668 0.053313

RANBAXY

0 0.066351 0.074502

RELIANCE

0.02598263 0.077863 0.076569

HDFC

0.60152040 0.101525 0.075374

TATAMOTORS

0 0.059956 0.065455

HINDALCO

0 0.072906 0.065483

WIPRO

0 0.064353 0.059075

AMBUJACEM

0 0.051962 0.077734

SIEMENS

0 0.073386 0.067679

TATASTEEL

0 0.056439 0.064709

ACC

0 0.061715 0.071019

RELINFRA

0 0.064433 0.059468

AXISBANK

0 0 0.054746

BPCL

0 0.071308 0.064254

SUNPHARMA

0.2652069 0.087136 0.07062


93 | P a g e




Optimum portfolio 0.0003 0.020345017


portfolio

-0.001074532 0.016490497

Heuristic pessimistic

portfolio

-0.001478186 0.016463641

Figure 3.7: Expected return and risk under 0.0003 minimum

return constraint

Here, the optimum portfolio has sifted to the first

quadrant. It is very clear from the above figure that the optimistic

portfolio (B) is situated near to the optimum portfolio (O) as


94 | P a g e

compared to the pessimistic portfolio (A). The expected return of

the optimistic portfolio is near to that of the optimum portfolio.

Expected return of the pessimistic portfolio is far from that of the

optimum portfolio. But the risks of both optimistic and

pessimistic portfolio are same.

The last minimum return constraint considered is

0.00065. The weights of fifteen selected companies for three

different portfolios are given in Table 3.15 and the expected

return and risk of the concerned portfolios have been shown in

Table 3.16. The expected return and risk of the three portfolios

are shown graphically in Figure 3.8 under 0.00065 minimum

return constraint.






Weights JINDALSTEL

0 0.090668 0.053313

RANBAXY

0 0.066351 0.074502

RELIANCE

0 0.077863 0.076569

HDFC

1 0.101525 0.075374

TATAMOTORS

0 0.059956 0.065455

HINDALCO

0 0.072906 0.065483

WIPRO 0 0.064353 0.059075


95 | P a g e

AMBUJACEM

0 0.051962 0.077734

SIEMENS

0 0.073386 0.067679

TATASTEEL

0 0.056439 0.064709

ACC

0 0.061715 0.071019

RELINFRA

0 0.064433 0.059468

AXISBANK

0 0 0.054746

BPCL

0 0.071308 0.064254

SUNPHARMA

0 0.087136 0.07062




Optimum portfolio 0.00065 0.027169


portfolio

-0.001074532 0.016490497

Heuristic pessimistic

portfolio

-0.001478186 0.016463641


96 | P a g e

Figure 3.8: Expected return and risk under 0.00065

minimum return constraint

Under 0.00065 minimum return constraint, again the

optimum solution is going far from the other two heuristic

solutions. Though from the above figure it is clear that the

optimistic solution (B) is situated near to the optimum solution

(O) but the pessimistic solution (A) is situated far from the

optimum solution (O). The expected return of optimistic portfolio

is more than that of pessimistic portfolio. But the expected risks

of the optimistic portfolio and pessimistic portfolio are more or

less same.

Now the ED values under all minimum return constraints

are shown in Table 3.17 to graphically present the curves drawn

by considering the coordinates between the minimum return


97 | P a g e

constraints and the ED between the optimum solution and the

heuristic solutions. Figure 3.9 shows the said curves. To present

Euclidian Distance (ED) graphically the horizontal axis has

measured the minimum return constraints and the vertical axis

has measured the ED values between the optimum and

pessimistic solutions and the ED values between the optimum

and optimistic solutions.

Table 3.17: Euclidian Distance between pessimistic and

optimum solutions and between optimistic and optimum

solutions

Minimum Return

Constraints

Ed between

optimum and

Pessimistic

solutions

ED between

optimum and

Optimistic solutions

-0.0057 0.0079274545032 0.0078937946135

-0.0047 0.0012753798258 0.0012591483233

-0.0037 0.0012753798258 0.0012591483233

-0.0027 0.0012753823348 0.0012573369874

-0.0017 0.001275685698 0.0012594901746

-0.0007 0.0013331751198 0.0011708631432

0.0003 0.0042693120289 0.0040922686373

0.00065 0.0109148470901 0.0108168589202


98 | P a g e

Figure 3.9: ED of optimistic and pessimistic solutions

In Figure 3.9, the point T1 and T2 are the two threshold points.

This identification of point of change has been undertaken for

the two risk situations – risk taker or optimistic investor and risk

aversive or pessimistic investor. Before the point T1 the ED

between the optimum solution and the pessimistic solution and

the ED between the optimum solution and optimistic solution are

same. They are overlapping before point T1. But beyond the

point T1 the ED between the optimum solution and the optimistic

solution is less than the ED between the optimum solution and

pessimistic solution. In other words, before the threshold point

T1, both the pessimistic approach and the optimistic approach

are closer to the optimization approach. The situation has got

change beyond the point T1. We have got the same result

beyond the second threshold point T2. Beyond the point T2 the


99 | P a g e

pessimistic solution and the optimistic solution are overlapping.

In between the points T1 and T2, an optimistic investor can get

the benefit of the optimum solution.

3.7 Conclusion

After detailed analysis, we have identified an ideal point

of change to be taken into consideration while examining the

behaviour of an investor, following heuristic approach in

investment planning. We have observed that in between the

said threshold points the decision of the optimistic investor is

very close to the decision of the optimum investor. Before the

first threshold point and beyond the second threshold point the

behaviour of the optimistic investor and the pessimistic investor

is same as the behaviour of the optimum investor. Thus, it can

be claimed that heuristically one can reach the optimum

solution. For those investors who find it difficult to

mathematically/statistically arrive at sophisticated optimum

solution, they can heuristically obtain a portfolio similar to the

optimum portfolio. The value system of the investor under

consideration has been numerically demonstrated given the

minimum return constraints. The heuristic method is very simple

to implement. However, a heuristic solution can be at best at par

with the optimum solution but can never be better than the

optimum solution.

V{tÑàxÜ G

fxÄxvà|ÉÇ Éy cÉÜàyÉÄ|É Utáxw ÉÇ VÉxyy|v|xÇà Éy bÑà|Å|áÅ


100 | P a g e

4.1 Introduction

In asset allocation analysis one mostly uses the mean –

variance approach for analyzing the trade off between risk and

expected return. Since the seminal work of Markowitz (1952,

1959) the computational aspects of finding efficient portfolios

have been a concern of the finance profession. In Markowitz

portfolio theory, the portfolio of financial assets is characterized

by a desired property, the ‘reward’, and something undesirable,

the ‘risk’. Balancing financial aspects against statistical and,

particularly, computational limitations, Markowitz identified

reward and risk with the expectation and the variance of returns

respectively.

Since 1952, many works have been done to select an

optimum portfolio. But in most of the cases it is difficult to arrive

at an optimal solution. The main problem of selection of

optimum portfolio is to obtain correct and precise portfolio

weights (the proportion of wealth invested in each individual

asset). If weights can be precisely estimated, it can reduce

unnecessary transaction costs and help to construct the

optimum portfolio. From a theoretical point of view, one should

optimize expected utility for which the mean-variance approach

can be an option. Some other techniques are there to optimize


101 | P a g e

the portfolio. Heuristically one can reach to the optimum or near

optimum solution. At a given point of time and a given set of

conditions, heuristic solution may give reasonable result.

The portfolio optimization problem is mainly concerned

with selecting the optimal investment strategy of an investor

without considering the value system. In other words, the

investor looks for an optimal decision on how many shares of

which security she should purchase to maximize her expected

utility. If the investor knows the securities that may give her

maximum expected return or minimum expected risk, it is easy

for her to take optimal decision. But in real world it is difficult to

find out those securities due to presence of efficient market. In

this thesis we have tried to find out a heuristic solution with four

different value systems which can help an investor to take

decision which is near to the optimum decision. In this thesis we

have considered the choice of the securities, construction of

distance metric and value system of the investor.

4.2 Earlier works

Li, Yang and Deng (2007) selected optimal portfolio with

earning – at – risk (EaR). They used EaR of terminal wealth as

a measure of risk. They obtained closed form expressions for

the best constantly – rebalanced portfolio investment portfolio


102 | P a g e

strategy and the efficient frontier of the mean – EaR analysis to

the classical mean – variance analysis and to the mean – CaR

(capital - at - risk) analysis.

Doganoglu, Hartz and Mittnik (2007) proposed a practical

approach to portfolio selection which considers conditionally

varying volatility and fat – tailedness of risk factors explicitly into

account, while retaining analytical tractability and ease of

implementation. They used nine German DAX stocks to

illustrate their model and they found that the model is strongly

favoured by the data and it is practically implemental.

Aydin Ulucan (2007) investigated optimal holding period

for the classical mean-variance portfolio optimization problem.

He took 648 cases of Istanbul Stock Exchange ISE-100 index

and Athens Stock Exchange FTSE-40 index data. He used

three different target return levels, three different risk measures

and 36 different time periods with rolling data. He found that

portfolio returns with varying holding periods have a convex

structure with an optimal holding period.

Golosnoy and Okhrin (2007) published a paper which

proposes a multivariate shrinkage estimator for the optimal

portfolio weights. The estimated classical Markowitz weights are

shrunk to the deterministic target portfolio weights. To derive the


103 | P a g e

explicit solutions for the optimal shrinkage factors, they

assumed that log asset returns to be i.i.d. Gaussian. They also

empirically studied the competing portfolio selection

approaches. They prescribed that both simulation and empirical

studies show the robustness of the shrinkage estimator and that

provide significant gains to the investor compared to benchmark

procedures.

Ma and Pohlman (2008) discussed about the general

interpretation of quantile regression in the financial market. After

exploring the distributional impact of factors on returns of

securities they have found that factor effects on return depends

on the quantiles of return. They have proposed two methods for

the return forecasting and portfolio construction. Their models

show that under mild conditions more accurate forecasts can be

got and one can get more value added portfolios than the

classical conditional mean method.

Danielsson, Jorgenson, Vries and Yang (2008) proposed

optimal portfolio allocation subject to a budget constraint and a

probabilistic VaR (value – at - risk) constraint in complete

markets environments with a finite number of states. Their

method restored monotonicity of the optimal portfolio allocation

in the state-price density and reduced computational complexity.


104 | P a g e

Huang (2008) discussed portfolio selection problem in

combined uncertain environment of randomness and fuzziness

in his paper. In this paper, security returns were regarded as

random fuzzy variables, i.e. random returns with fuzzy expected

values. He proposed new mean – variance model and a hybrid

intelligent algorithm to solve the new model problem. He

reached his objectives with the empirical analysis.

Shaw, Liu and Kopman (2008) studied a portfolio

selection problem subject to a cardinality constraint. They

developed a dedicated Lagrangian relaxation method to

optimize the portfolio. They empirically tested the algorithm to

S&P 500 index, S&P 100 index, FTSE 100 index and FTSE 250

index.

Markus and Thorsten (2008) suggested that a time –

varying estimation of return correlations fits the data

considerably better than time – invariant estimation and thus,

increases the efficiency of risk estimation and portfolio selection.

According to them for the selection of optimal portfolios and for

risk assessment the dynamic and distributional behaviour of risk

factors are very vital.

We have proposed in chapter 3 heuristic solutions to

reach near the optimum portfolio, considering two extreme value


105 | P a g e

systems of investors covering optimistic and pessimistic

investors. We have examined closeness between these

heuristic solutions and the optimum solution with the help of a

distance measure. Euclidean Distance technique has been used

by them to show the similarity between the optimum portfolio

and the heuristic portfolios. For the purpose of empirical

analysis, we have selected ten companies of Nifty and the data

range they have taken from 2006 to 2007.

4.3 Optimization Framework

Following the existing literature, there are two ways

through which a portfolio can be optimally constructed. One

method is to minimize risk subject to a minimum return

constraint. The other way is to maximize the expected return

subject to a certain level of risk. We propose to strike a balance

between these two approaches by allowing either the minimum

return or the maximum risk to vary. This variation is to be

incorporated in the mathematical programming framework in

terms of coefficient of optimism (α) in the constraint set. The

optimization problem considered for the current work is of the

following type:


106 | P a g e

Min ````ww

Subject to minmax````

)1( Rw

Depending on the choice of we get an optimum solution

of this problem. Thus, weight vector will be a function of . One

can get the optimum portfolio corresponding to the minimum

return of the security by considering = 0 and the optimum

portfolio corresponding to the maximum return of the security by

considering =1. However, for the purpose of empirical study,

we have varied from 0 to 1 with step 0.125 resulting in nine

steps. Here, = 0 represents pessimistic investors and = 1

represents optimistic investors. For random selector and risk

planner’s the value of lies between 0 and 1. In each case, one

can get different optimum weights and with those weights return

and risk of the optimum portfolio can be calculated using

equations (3) and (4) as shown in chapter 3.

4.4 Heuristic Framework

In the next phase we have developed a heuristic solution

with the help of which we may get an alternative choice of

portfolio based on propensity to take risk. In fact, we have


107 | P a g e

constructed four types of heuristic portfolios covering optimistic

investor, pessimistic investor, risk planner and random selector.

In case of heuristic optimistic portfolio and heuristic

pessimistic portfolio the formulae of weights, risk and return

developed in chapter 3 is used here. In case the investor is a

risk planner, we have assumed that he/she will give importance

on both risk and return. Here we have considered both risk and

return to calculate the weights of the securities. As risk planner

investor considers both return and risk to select portfolio.

For risk planner following formula has been used for

calculating the weights of the securities of the portfolio where

both return and risk have been considered. Basically this is a

combination of the two extreme situations mentioned earlier.

)(

)( w

HRP

i

i

i

i

i

i

i

i

i

Min

Min

(11)

Given this weight function we have calculated the expected

return and risk as follows:

n

i i

i

i

i

n

ii

i

i

i

i

HRPP

R

R

1

1

)min(

)min(

(12)


108 | P a g e

n

i

n

jijji

j

j

j

j

i

i

i

in

i i

i

i

i

HRPP r

1 1

1

)min)(min()min(

1

(13)

For random selector, we have considered equal weight to

each security as they do not put emphasis either on return or on

risk. The weight for random selector is 1/n. Given this equal

weights we have calculated the expected return and risk for the

random selector as follows:

n

ii

HRSp R

nR

1

1 (14)

n

i

n

jijji

HRSp r

n 1 1

1 (15)

4.5 Closeness between Heuristic and Optimization framework

The closeness between the optimum framework and heuristic

framework is measured by applying the City Block Distance. The

closeness is observed to know whether an investor heuristically

can reach to near optimum solution.


109 | P a g e



have been calculated with the help of equation 1 and 2

respectively to arrive at the objective. Classical optimum

technique has been used to optimize the portfolio. Optimum

weights of the ten securities have been calculated under

different minimum return constraints. The results are shown

below:

Table 4.1: Table showing optimum weights of fifteen

securities under different values of coefficient of optimism

Securities α values

0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1

JINDALSTEL

0 0 0 0 0 0 0 0.04870561

0

RANBAXY

0.1936191

0.1936190

0.1936186

0.1936176

0.1936158

0.1935900

0.1936190

0.07933459

0

RELIANCE

0.06280846

0.06280902

0.06281166

0.06281766

0.06282900

0.06289579

0.06280901

0.07960539

0

HDFC

0.1424088

0.1424080

0.1424066

0.1424035

0.1423969

0.1423651

0.1424096

0.4121370

1

TATAMOTORS

0.005869277

0.005868693

0.005866985

0.005863447

0.005856850

0.005814507

0.005869393

0 0

HINDALCO

0.05411821

0.05411855

0.5411938

0.05412130

0.05412621

0.05418588

0.05411801

0.01009264

0

WIPRO

0 0 0 0 0 0 0 0 0

AMBUJACEM 0.1068 0.1068 0.1068 0.1068 0.1068 0.1068 0.1068 0 0


110 | P a g e

151 155 160 164 158 162 148

SIEMENS

0.07751241

0.07751266

0.07751307

0.07751375

0.07751504

0.07754704

0.07751214

0.05838301

0

TATASTEEL

0 0 0 0 0 0 0 0 0

ACC

0.02198061

0.02197995

0.02197871

0.02197696

0.02197645

0.02202873

0.02198111

0 0

RELINFRA

0 0 0 0 0 0 0 0 0

AXISBANK

0.4030778

0.04030851

0.04030944

0.04031073

0.04031108

0.04028933

0.04030694

0 0

BPCL

0.1195407

0.1195405

0.1195397

0.1195376

0.1195332

0.1195004

0.1195407

0.06761105

0

SUNPHARMA

0.1750195

0.1750197

0.1750200

0.1750210

0.1750237

0.1749671

0.1750193

0.2441307

0

The weights of optimistic, pessimistic and risk planner solutions

have been calculated using the equations (5), (8) and (11)

respectively. Table 4.2 shows the weights of the optimistic,

pessimistic, risk planner and random selector solutions.


111 | P a g e

Table 4.2: Weights of heuristic optimistic, heuristic

pessimistic, heuristic risk planner and random selector

solutions

Securities Heuristic Optimistic Weights


Weights

Heuristic Risk

Planner Weights

Heuristic Random Selector Weights

JINDALSTEL

0.090668 0.053313 0.096953 0.066667

RANBAXY

0.066351 0.074502 0.06138 0.066667

RELIANCE

0.077863 0.076569 0.077593 0.066667

HDFC

0.101525 0.075374 0.112735 0.066667

TATAMOTORS

0.059956 0.065455 0.057583 0.066667

HINDALCO

0.072906 0.065483 0.074128 0.066667

WIPRO

0.064353 0.059075 0.066541 0.066667

AMBUJACEM

0.051962 0.077734 0.03797 0.066667

SIEMENS

0.073386 0.067679 0.07398 0.066667

TATASTEEL

0.056439 0.064709 0.053592 0.066667

ACC

0.061715 0.071019 0.056634 0.066667

RELINFRA

0.064433 0.059468 0.066428 0.066667

AXISBANK

0 0.054746 0 0.066667

BPCL

0.071308 0.064254 0.072559 0.066667

SUNPHARMA

0.087136 0.07062 0.091924 0.066667


112 | P a g e

Our objective is to examine the closeness between the

optimum solution and heuristic solutions. For this purpose firstly,

we have calculated the expected return and risk of the optimum

portfolio by using equation 3 and 4 as shown in chapter 3. To

calculate these expected return and risk we have used the

weights shown in Table 4.1. These expected return and risks of

the optimum portfolio under different values of the coefficient of

optimism (α) are shown in Table 4.3.

Table 4.3: Expected return and risk of the optimum portfolio

under different α values

Values of α

Minimum Return

Constraints

Expected Returns

Expected Risk

0 -0.0057 -0.0032105 0.024384

0.125 -0.00490625 -0.00114 0.015233

0.250 -0.0041125 -0.00169801 0.025905757

0.375 -0.00331875 -0.001142747 0.015233

0.500 -0.002525 -0.0011427 0.015233

0.625 -0.00173125 -0.00114272 0.015232856

0.750 -0.0009375 -0.00114273 0.0152332

0.875 -0.00014375 -0.00014375 0.017076738

1 0.00065 0.00065 0.027169


113 | P a g e

To study the closeness between the optimum choice of portfolio

and the heuristic choice of portfolio, we have next calculated

expected return and risk of the heuristic portfolios using

equations 6 & 7, 9 & 10 shown in chapter 3, and equations 12 &

13 and 14 & 15. Weights of the heuristic portfolios shown in

Table 4.2 are used to calculate these expected return and risk.

Table 4.4 shows the values of expected returns and risks of the

heuristic portfolios.

Table 4.4: Values of expected returns and risks of heuristic

portfolios

Heuristic portfolios Expected Return Expected Risk

Optimistic -0.001074532 0.016490497

Pessimistic -0.001478186 0.016463641

Risk Planner -0.001016865 0.016534909

Random Selector -0.001530278 0.016738164

However, for a formal look at the behaviour of heuristic

solutions in respect of the optimum solutions, we would like to

take into consideration a standard measure of distance. For this

purpose City Block Distance has been considered. With the

expected returns and risks of the optimum solution and heuristic

solutions we have computed City Block Distances (CBD) among


114 | P a g e

the optimum portfolio and the heuristic portfolios. With that City

Block Distance we want to know the similarity between the

optimum choice of portfolios and the heuristic choice of

portfolios. For this purpose, we have graphically shown the City

Block Distances between the optimum choice of portfolio and

the heuristic choices of portfolios. Given four heuristic solutions

we can pair wise compare them resulting in six combinations.

We have drawn six graphs. Each graph shows the City Block

Distance between the optimum solution and two heuristic

solutions. In the horizontal axis we have measured the values of

minimum return constraints and in the vertical axis we have

measured the values of City Block Distance. Graphs are shown

below:

Table 4.5: City Block Distance between the optimistic &

optimum choices of portfolios and pessimistic & optimum

choices of portfolios

Values of α Minimum Return

Constraints

Optimistic Pessimistic

0 -0.0057 0.001325534 0.001565927

0.125 -0.00490625 0.001325535 0.00156592

0.250 -0.0041125 0.001325538 0.001565915

0.375 -0.00331875 0.001325548 0.001565917


115 | P a g e

0.500 -0.002525 0.001325556 0.001565924

0.625 -0.00173125 0.001325831 0.001566249

0.750 -0.0009375 0.001417119 0.001793917

0.875 -0.00014375 0.001517023 0.001947533

1 0.00065 0.012403035 0.012833545

Figure 4.1

In Figure 4.1, there is one prominent threshold point T1

where the value of α is 0.875. Before the point T1 the optimistic

choice of portfolio placed near to the optimum portfolio than that

of the pessimistic choice of portfolio. Beyond the threshold point


116 | P a g e

T1, the two CBD curves overlapped each other. That means

beyond the point T1 the decisions of both the optimistic and

pessimistic investor are same. Thus, it can be stated that from 0

to 0.875 value of α (coefficient of optimism), the optimistic

investor’s decision is near to the optimum decision as compared

to the pessimistic decision and from 0.875 to 1 value of α the

optimistic and pessimistic investors’ decisions are same.


optimum choices of portfolios and risk planner & optimum



Constraints

Optimistic Risk Planner

0 -0.0057 0.001325534 0.001427613

0.125 -0.00490625 0.001325535 0.001427614

0.250 -0.0041125 0.001325538 0.001427617

0.375 -0.00331875 0.001325548 0.001427627

0.500 -0.002525 0.001325556 0.001427635

0.625 -0.00173125 0.001325831 0.00142791

0.750 -0.0009375 0.001417119 0.001403864

0.875 -0.00014375 0.001517023 0.001414944

1 0.00065 0.012403035 0.012300956


117 | P a g e

Figure 4.2

In Figure 4.2, T2 and T3 are the threshold points. To the

left of the threshold point T2 where the value of α is 0.736, the

optimistic investor’s decision is closer than that of the risk

planner decision, though the distances are very less. It means

that the optimistic investor’s solution is closer to the optimum

solution than that of the risk planner’s solution. In between T2

and T3 (value of coefficient of optimism α is 0.875) the risk


118 | P a g e

planner’s decision is closer to the optimum decision than that of

the optimistic investor’s decision. On the other hand, in the right

side of the point T3, the CBD between optimum and optimistic

portfolios is overlapping with the CBD between optimum and risk

planner portfolios. That means the decisions of the optimistic

and the risk planner is same beyond 0.875 value of α.


optimum choices of portfolios and random selector &

optimum choices of portfolios


Constraints

Optimistic Random Selector

0 -0.0057 0.001325534 0.001892542

0.125 -0.00490625 0.001325535 0.001892535

0.250 -0.0041125 0.001325538 0.00189253

0.375 -0.00331875 0.001325548 0.001892532

0.500 -0.002525 0.001325556 0.001892539

0.625 -0.00173125 0.001325831 0.001892864

0.750 -0.0009375 0.001417119 0.002120532

0.875 -0.00014375 0.001517023 0.001725102

1 0.00065 0.012403035 0.012611114


119 | P a g e

Figure 4.3

From the above Figure, it is observed that to the left of

the threshold point T4, where the value of coefficient of optimism

is 0.757, optimistic investor’s decision is closer to the optimum

solution than that of random selector’s decision. In other words,

from 0 to 0.757 values of coefficient of optimism the optimistic

investor’s decision is better than that of random selector’s

decision. But beyond the point T4 (or from the value of α 0.757

to 1) both optimistic and random selector shows the same

decision.


120 | P a g e

Table 4.8: City Block Distance between the pessimistic &

optimum choices of portfolios and risk planner & optimum



Constraints

Pessimistic Risk Planner

0 -0.0057 0.001565927 0.001427613

0.125 -0.00490625 0.00156592 0.001427614

0.250 -0.0041125 0.001565915 0.001427617

0.375 -0.00331875 0.001565917 0.001427627

0.500 -0.002525 0.001565924 0.001427635

0.625 -0.00173125 0.001566249 0.00142791

0.750 -0.0009375 0.001793917 0.001403864

0.875 -0.00014375 0.001947533 0.001414944

1 0.00065 0.012833545 0.012300956


121 | P a g e

Figure 4.4

From the above Figure 4.4 it can be stated that the CBD

between the optimum portfolio and the risk planner’s portfolio

gives better result than the CBD between the optimum portfolio

and the pessimistic portfolio. What is important to note is that no

threshold point can be identified. The risk planner gives better

decision than that of the pessimistic investor through out the

whole range of coefficient of optimism.


122 | P a g e





Constraints

Pessimistic Random Selector

0 -0.0057 0.001565927 0.001892542

0.125 -0.00490625 0.00156592 0.001892535

0.250 -0.0041125 0.001565915 0.00189253

0.375 -0.00331875 0.001565917 0.001892532

0.500 -0.002525 0.001565924 0.001892539

0.625 -0.00173125 0.001566249 0.001892864

0.750 -0.0009375 0.001793917 0.002120532

0.875 -0.00014375 0.001947533 0.001725102

1 0.00065 0.012833545 0.012611114


123 | P a g e

Figure 4.5

When pessimistic investor and random selector are

chosen for comparison it is observed that the decision of both

the pessimistic investor and random selector is very close to the

optimum solution beyond the threshold point T6 (value of α is

0.875). Between the points T5 (value of α 0.824) and T6 random

selector’s decision is closer to the optimum solution than that of

the pessimistic investor’s decision. But before the threshold

point T5 the pessimistic investor’s decision gives better result as


124 | P a g e

compared to the random selector’s decision, as the CBD

between the optimum portfolio and the pessimistic portfolio is

closer than that of the optimum portfolio and random selector’s

portfolio. Thus, it can be stated that in between 0<α<0.824 the

pessimistic solution is closer to the optimum solution than that of

the random selector’s solution and in between 0.824 and 0.875

the random selector shows the better result as compared to

pessimistic investor. Beyond the value of α 0.875 both heuristic

portfolios show same result.





Constraints

Risk Planner Random Selector

0 -0.0057 0.001427613 0.001892542

0.125 -0.00490625 0.001427614 0.001892535

0.250 -0.0041125 0.001427617 0.00189253

0.375 -0.00331875 0.001427627 0.001892532

0.500 -0.002525 0.001427635 0.001892539

0.625 -0.00173125 0.00142791 0.001892864

0.750 -0.0009375 0.001403864 0.002120532

0.875 -0.00014375 0.001414944 0.001725102

1 0.00065 0.012300956 0.012611114


125 | P a g e

Figure 4.6

By putting the CBD values related to the risk planner and

random selector, T7 is marked as the threshold point where the

value of α is 0.97 in Figure 4.6. When the decision between risk

planner and random selector is compared from the above figure,

it is found that to the left hand side of the threshold point T7 the

risk planner’s solution is giving better result than that of the

random selector’s solution. But beyond point T7 (or the value of


126 | P a g e

α 0.97) the risk planner’s decision is same as the random

selector’s decision, because the CBD between the optimum

portfolio and the risk planner’s portfolio is same as the CBD

between the optimum portfolio and random selector’s portfolio.

After completing this pair-wise comparison, a combined

graph is drawn which shows all the six combinations so that the

distances of all the heuristic solutions with the optimum solution

can be examined together. Like the above graphs in Figure 4.7,

on the horizontal axis minimum return constraints are measured

and on the vertical axis CBD values are measured.


127 | P a g e

Table 4.11: City Block Distance between optimistic and

optimum solutions, between pessimistic and optimum

solutions, between risk planner and optimum solutions and

between random selector and optimum solutions

Values of α

Minimum Return

Constraints

Optimistic Pessimistic Risk Planner

Random Selector

0 -0.0057 0.001325534

0.001565927

0.001427613

0.001892542

0.125 -0.00490625 0.001325535

0.00156592

0.001427614

0.001892535

0.250 -0.0041125 0.001325538

0.001565915

0.001427617

0.00189253

0.375 -0.00331875 0.001325548

0.001565917

0.001427627

0.001892532

0.500 -0.002525 0.001325556

0.001565924

0.001427635

0.001892539

0.625 -0.00173125 0.001325831

0.001566249

0.00142791

0.001892864

0.750 -0.0009375 0.001417119

0.001793917

0.001403864

0.002120532

0.875 -0.00014375 0.001517023

0.001947533

0.001414944

0.001725102

1 0.00065 0.012403035

0.012833545

0.012300956

0.012611114


128 | P a g e

Figure 4.7

In the above Figure 4.7 it is observed that before the

threshold point T8 (where the value of coefficient of optimism α

is 0.736) optimistic solution is giving better result than that of all


129 | P a g e

other heuristic solutions. Then the risk planner’s solution is

showing better result than that of the pessimistic solution and

random selector’s solution. The random selector’s solution is

worst before the point T8. Between the threshold point T8 and

T10 (value of α is 0.875) the risk planner’s decision is better

than the optimistic investor’s decision, as the CBD between the

optimum portfolio and optimistic portfolio is situated far than the

CBD between the optimum portfolio and the risk planner’s

portfolio. At the threshold point T9 where the value of α is 0.824

a little change is occurred. Beyond the point T9 the random

selector’s decision is placing in the third position irrespective of

the pessimistic investor. In between the points T9 and T11 the

decision of the pessimistic investor is worst. But beyond the

threshold point T11 all the heuristic solutions are showing same

result. In other words, it can be stated that beyond the threshold

point T11 the CBD between the optimum portfolio and the four

heuristic portfolios are overlapping. Thus, beyond the point T11

all the heuristic portfolios give the same decision.


130 | P a g e

4.7 Conclusion

After detailed analysis, a mixed result has been obtained.

When the value of coefficient of optimism lies between 0 and

0.750 the decision of the optimistic investor is best than the

other heuristic investors though the risk planner’s decision is

very close to the optimistic investor’s decision. Only at 0.750

and 0.850 values of coefficient of optimism the risk planner’s

decision is best as compared to the other heuristic solutions. But

at the coefficient of optimism 1, all the four heuristic solutions

show the same result. 0=<α>=0.824 the random selector shows

the worst result. In this range the pessimistic investor is having

the third best position. Thus, we can say that heuristically the

optimum solution can be reached. In other words, it can be said

that closeness is there between the optimum portfolio

constructed on the basis of existing techniques and portfolio

selected by investors who follow heuristic approach.

V{tÑàxÜ H

itÄâx Utáxw fxÄxvà|ÉÇ Éy cÉÜàyÉÄ|ÉM T [xâÜ|áà|v

TÑÑÜÉtv{

Chapter 5: Value based Selection of Portfolio: A Heuristic Approach

131 | P a g e

5.1 Introduction

The portfolio optimization problem aims to find out the

optimal investment strategy of an investor. In other words, the

investor looks for an optimal decision on how many shares of

which security he/she should invest to maximize his/her

expected return or minimize his/her risk. Markowitz (1959) was

the first person who had shown that through mean – variance

analysis an investor can get optimum portfolio. According to

Markowitz model, we need large information to calculate an

efficient portfolio. Sharpe (1964) worked on the same problem

and developed Capital Asset Pricing Model (CAPM) which can

be used for multiple securities to construct an efficient portfolio

and can be used with relatively less amount of information.

Inspired by their seminal works many researchers have been

doing work on optimum choice of the portfolio. Amongst them

Merton (1969, 1971, 1973, and 1980) is the lead researcher who

discussed the continuous time portfolio optimization problem in

the utility framework. Brown (1978) studied the problem when an

investor has only two options - a single risky asset and an asset

with a constant and known rate of return. He considered Optimal

Baye’s Portfolio and Certainty Equivalence Strategy to calculate


132 | P a g e

the efficiency of the asset. He found that the optimal Baye’s

portfolio invests less in risky securities.

5.2 Earlier works

While addressing the issue whether the systematic risk

plays any role to choose a portfolio of international equities, Das

and Uppal (2004) developed two models- one that incorporates

systemic risk and the other that ignores it. The models of asset

returns help to capture the jumps, which reflects the returns on

international equities, occurring at the same time across

countries. They also showed how an investor would choose an

optimal portfolio when returns have these features. They tested

their method to determine the weights for a portfolio allocated

over a risk less asset. However, they have got the result that “for

investors with low risk aversion who desire levered positions, the

cost of ignoring systemic risk is larger, and in the case of a

highly levered portfolio, there is a positive probability of losing

one’s entire wealth if there is a large negative systemic shock”.

So to analyse a portfolio, the systematic risk plays an important

role.

On the other hand, Jones (1999) showed that for

empirical analysis of classical mean-variance problem,

regression method is a simple tool to portfolio analysis. By using


133 | P a g e

GRS F- test (formulated by Gibbons, Ross, and Shanken

(1989)) of portfolio efficiency he showed that a test of the

restriction, the weights of the ‘tangency’ portfolio equal the

weights of the test portfolio. He also proposed the

implementation of GRS F-test for portfolio efficiency using linear

restrictions on a single linear OLS regression. He analyzed the

weights of an international efficient portfolio. He used monthly

data from Morgan Stanley Capital International (MSCI) for the

20-years for the equity markets of 11 developed countries.

Thus, over the years, many authors have used various

techniques to find out an optimum portfolio that can help an

investor to take the optimum decision. Some of them gave

emphasis on the number of assets in the portfolio; others

assigned importance on the weights of the assets of the

portfolio. Recently works are on the suitability of heuristic

approach for solving this complex problem. In this thesis we

have considered two types of portfolios – optimum portfolio and

heuristic portfolio and determined weight of the portfolio, i.e. rate

of investment in each asset of the portfolio. Calculation of

assets’ weight is the most crucial task of the investors. If

investor can correctly predict the proportion of amount to be

invested in each assets of the portfolio, he/she can take right


134 | P a g e

decision to earn maximum utility. But in real world in an efficient

market, it is very difficult to predict assets’ weights accurately.

The problem of arriving at weights of the assets in the portfolio

has been discussed from various angles by different authors.

Though optimum portfolio is conceptually the best choice

yet investors opt for conditional portfolios given their own value

systems. This calls for detailed study on effects on value

systems on the optimum choice of the portfolio. In this thesis we

have tried to examine those effects. In chapter 3 we have

considered two human value systems in terms of propensity to

take risk. These are optimistic investor and pessimistic investor.

But these are two extreme states of an investor’s value. In this

chapter we would like to generalize this concept in terms of

dependency on coefficient of optimism of the investor. We would

like to compare this heuristic formulation with the optimum

portfolio.

5.3 Choice of weights under different value systems

In the next stage, we propose a heuristic solution to get

an alternative choice of portfolio based on the same propensity

to take risk. For this purpose, we have considered three types of

investor viz. optimistic investor, pessimistic investor and risk


135 | P a g e

planner. Since in chapter 3 we have considered optimistic

portfolio and pessimistic portfolio and calculated their closeness

from the optimum portfolio we propose to strike a balance

between these two polar opposite states to describe all possible

situations and present a general heuristic portfolio with α as the

coefficient of optimism.

n

i

HPi

HOi

HPi

HOi

H

ww

www

1

)2/1( (16)

We would like to observe the closeness between the

optimum portfolios and the heuristic portfolios with an effect of

coefficient of optimism. Whether coefficient of optimism has any

effect on choice of a heuristic portfolio? To find out the answer

of the question, we have considered the following weight

generated with the help of equation 17. We assume that a

general investor gives emphasis both on the return and risk.

Weights of optimistic investor and pessimistic investor have

been considered together with the value of coefficient of

optimism to calculate the weight of the heuristic portfolio. The

formula is given below:


136 | P a g e

n

i

HPi

HOi

HPi

HOi

H

ww

www

1

)1(

)1(

)(

(17)

Given the above weight function one may calculate the

expected return and risk of the general heuristic portfolio with

the following formula.

n

i

n

j

HPi

HOi

n

ii

HPi

HOi

H

ww

RwwR

1 1

1

1

1

(18)

n

i

n

jijji

HPj

HOj

HPi

HOi

HPi

HOi

H rwwwwww 1 1

11

1

))((1

(19)

It is not possible to analytically examine the closeness of

the heuristic portfolio with the optimum portfolio; empirical

analysis is the only way out to undertake a comparative study



have been calculated by applying equations 1 and 2 shown in

chapter 3. For nine different values of α coefficient optimum

weights of the fifteen securities have been calculated. These

optimum weights are given in Table 5.1:


137 | P a g e

Table 5.1: Weights of the optimum portfolios of fifteen

selected securities under different α coefficients

Securities

α values

0 0.125 0.250 0.375 0.500 0.625 0.750 0.875

1.000

JINDALSTEL

0 0 0 0 0 0 0 0.048705

61

0

RANBAXY

0.1936191

0.1936190

0.1936186

0.1936176

0.1936158

0.1935900

0.1936190

0.079334

59

0

RELIANCE

0.0628084

6

0.0628090

2

0.0628116

6

0.0628176

6

0.0628290

0

0.0628957

9

0.0628090

1

0.079605

39

0

HDFC

0.1424088

0.1424080

0.1424066

0.1424035

0.1423969

0.1423651

0.1424096

0.412137

0

1

TATAMOTO

RS

0.0058692

77

0.0058686

93

0.0058669

85

0.0058634

47

0.0058568

50

0.0058145

07

0.0058693

93

0 0

HINDALCO

0.0541182

1

0.0541185

5

0.5411938

0.0541213

0

0.0541262

1

0.0541858

8

0.0541180

1

0.010092

64

0

WIPRO

0 0 0 0 0 0 0 0 0

AMBUJACE

M

0.1068151

0.1068155

0.1068160

0.1068164

0.1068158

0.1068162

0.1068148

0 0

SIEMENS

0.0775124

1

0.0775126

6

0.0775130

7

0.0775137

5

0.0775150

4

0.0775470

4

0.0775121

4

0.058383

01

0

TATASTEEL

0 0 0 0 0 0 0 0 0

ACC 0.0219806

1

0.0219799

5

0.0219787

1

0.0219769

6

0.0219764

5

0.0220287

3

0.0219811

1

0 0

RELINFRA

0 0 0 0 0 0 0 0 0

AXISBANK

0.4030778

0.0403085

0.0403094

0.0403107

0.0403110

0.0402893

0.0403069

0 0


138 | P a g e

1 4 3 8 3 4

BPCL 0.1195407

0.1195405

0.1195397

0.1195376

0.1195332

0.1195004

0.1195407

0.067611

05

0

SUNPHARM

A

0.1750195

0.1750197

0.1750200

0.1750210

0.1750237

0.1749671

0.1750193

0.244130

7

0

These optimum weights under different α coefficients

have been used to calculate the expected return and risk of the

optimum portfolios. In this manner we have nine optimum

portfolios to be considered. Equations 3 and 4 as shown in

chapter 3 have been used to calculate the expected risk and

return of the respective optimum portfolios. In the following table

the expected return and risk of the optimum securities under

different values of coefficient of optimism are given in Table 5.2:

Table 5.2: Expected return and risk of the fifteen selected securities under different α coefficients

α coefficient Expected Return Expected Risk

0 -0.0032105 0.024384

0.125 -0.00114 0.015233

0.250 -0.00169801 0.025905757

0.375 -0.001142747 0.015233

0.500 -0.0011427 0.015233

0.625 -0.00114272 0.015232856

0.750 -0.00114273 0.0152332

0.875 -0.00014375 0.017076738

1.000 0.00065 0.027169


139 | P a g e

In the next phase we have considered the two heuristic

portfolios for optimistic investor and pessimistic investor. The

weights of the heuristic portfolios have been calculated by using

equations 5 and 8 respectively. Table 5.3 represents the weights

of these two heuristic portfolios.

Table 5.3: Weights of two heuristic portfolios taken into

consideration

Securities Optimistic Portfolio Pessimistic Portfolio

JINDALSTEL 0.090668 0.053313

RANBAXY 0.066351 0.074502

RELIANCE 0.077863 0.076569

HDFC 0.101525 0.075374

TATAMOTORS 0.059956 0.065455

HINDALCO 0.072906 0.065483

WIPRO 0.064353 0.059075

AMBUJACEM 0.051962 0.077734

SIEMENS 0.073386 0.067679

TATASTEEL 0.056439 0.064709

ACC 0.061715 0.071019

RELINFRA 0.064433 0.059468

AXISBANK 0 0.054746

BPCL 0.071308 0.064254

SUNPHARMA 0.087136 0.07062


140 | P a g e

After calculating the weights of the optimistic and

pessimistic investors we have calculated the weights of the

selected securities of the general heuristic portfolio. For this

purpose equation 17 has been used. The weights of the

securities of the general heuristic portfolios under different

coefficient of optimism values are given in Table 5.4:

Table 5.4: Weights of the heuristic portfolios of fifteen selected securities under different α coefficients

Securities

α values

0 0.125

0.250

0.375

0.500

0.625

0.750

0.875

1.000

JINDALST

EL

0.053313

0.060000264

0.063783791

0.067757147

0.071926093

0.076296261

0.080873

13

0.085661989

0.090668

RANBAXY

0.074502

0.077333969

0.075824542

0.074291121

0.072736197

0.071162256

0.069571766

0.067967182

0.066351

RELIANC

E

0.076569

0.080808152

0.080555369

0.080245638

0.079879428

0.079457315

0.078979983

0.078448221

0.077863

HDFC

0.075374

0.082391597

0.085070973

0.087774326

0.090498248

0.093239143

0.095993226

0.098756526

0.101525

TATAMOTORS

0.065455

0.068182275

0.067086828

0.065961521

0.064808

3

0.063629136

0.062426019

0.061200949

0.059956

HINDALCO

0.065

0.069895

0.070470

0.070999

0.071480

0.071912

0.072295

0.072626

0.0729


141 | P a g e

48 679 803 572 702 985 296 593 06

WIPRO

0.059075

0.062884237

0.063229096

0.063530135

0.063786556

0.063997

64

0.064162

75

0.064281336

0.0643533

AMBUJAC

EM

0.077734

0.077846268

0.073637952

0.069607

05

0.065749329

0.062060441

0.058535939

0.055171303

0.051962

SIEMENS

0.067679

0.072001455

0.072354

53

0.072657059

0.072908215

0.073107267

0.073253577

0.073346609

0.073386

TATASTEE

L

0.064709

0.066993697

0.065514936

0.064022749

0.062519413

0.061007171

0.059488237

0.057964783

0.056439

ACC

0.071019

0.073492666

0.071837395

0.070168916

0.068489743

0.066802349

0.065109166

0.063412583

0.061715

RELINFRA

0.059468

0.063259956

0.063564051

0.063823683

0.064038144

0.064206801

0.064329109

0.064404611

0.064433

AXISBANK

0.054746

0 0 0 0 0 0 0 0

BPCL

0.064254

0.068556297

0.069092621

0.069583073

0.070026449

0.070421623

0.070767553

0.071063276

0.071308

SUNPHARMA

0.07062

0.076353488

0.077977116

0.079578

01

0.081153183

0.082699611

0.084214

25

0.085694

04

0.087136


142 | P a g e

With the weights given in Table 5.4 we have calculated

the expected return and risk of the general heuristic portfolio.

Equations 18 and 19 have been used to find out the expected

return and risk of the general heuristic portfolios respectively.

Expected return and risk of the general heuristic portfolios under

different values of coefficient of optimism is show in the following

Table 5.5.

Table 5.5: Expected return and risk of heuristic portfolios

under different α coefficients

α coefficient Expected Return Expected Risk

0 -0.00147818 0.016463595

0.125 -0.001214085 0.016412405

0.250 -0.00119441 0.016414064

0.375 -0.001174638 0.01641861

0.500 -0.001154781 0.016426197

0.625 -0.001134838 0.016436991

0.750 -0.001114814 0.016451174

0.875 -0.001094711 0.016468941

1.000 -0.001074536 0.016490524

Once we have obtained the expected return and risk of the

general heuristic portfolios under different values of coefficient of

optimism, we have considered City Block Distance to measure

the closeness of the heuristic portfolio with the optimum


143 | P a g e

portfolio. City Block Distance is a standard measure of distance

to calculate the similarities or dissimilarities between two points.

Since we would like to find out the similarities or dissimilarities

between the optimum portfolios and the heuristic portfolios, this

City Block Distance can be applied on a two dimensional set up

with the expected return and risk of the optimum portfolios and

the heuristic portfolios. City Block Distance between the

optimum portfolios and the heuristic portfolios under different

coefficient of optimism values are given in Table 5.6:

Table 5.6: City Block Distances between optimum portfolios

and heuristic portfolios under different α coefficients

α Coefficient City Block Distance values

0 0.009652725

0.125 0.00125349

0.250 0.009995293

0.375 0.001217501

0.500 0.001205278

0.625 0.001212017

0.750 0.00124589

0.875 0.001557958

1.000 0.012403012


144 | P a g e

CBD between Optimum and Heuristic Portfolios

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 1.125

Values of α coefficient

CB

D v

alues

CBD between optimum and heuristic portfolios

Figure 5.1

These City Block Distances between the optimum portfolios and

heuristic portfolios are presented in a graph (see figure 5.1). In the

said graph we have presented the values of coefficient of optimism

(α coefficient) along the horizontal axis and the values of City Block

Distance along the vertical axis.

Form the above figure it is clear that when the value of

the coefficient of optimism (α coefficient) lies between 0 or 0.25

the decision of the heuristic investor does not match with the

decision of the optimum investor. It means that for 0≤ α ≤0.25,

there is less similarities between the optimum solution and the

heuristic solution. However, at the value of α at 0.125 the


145 | P a g e

decision of the heuristic investor is closer to the optimum

solution. The decision of the heuristic investor again does not

match when the value of α coefficient is nearly 1. For 0.375≤ α

≤0.875, the heuristic portfolio is very close to the optimum

portfolio. So, there is a marked similarity between the optimum

portfolio and the heuristic portfolio when the value of coefficient

of optimism is in between 0.375 to 0.875. Thus, we can say that

risk planners’ decision is near to optimum decision.

5.5 Conclusion

After thorough empirical analysis along with comparative

studies, it has been noted that heuristically one can get a set of

weights that provide a result very close to the optimum result.

The investor has to be aware of the coefficient of optimism while

making investment in any security. It has been observed that the

decision of the pessimistic investor does not match with the

optimum solution because when the value of coefficient of

optimism α is equal to 0 (represents the weight of the

pessimistic investor) the distance between the optimum solution

and the heuristic solution is very large. When the value of the

coefficient of optimism is 1 then also the decision of the

optimistic investor does not match with the optimum decision

because the CBD between the optimum solution and optimistic


146 | P a g e

solution is large. But when we have considered the value of

coefficient of optimism between 0.375 to 0.875 covering the risk

planner’s decision, the heuristic solution is very close to the

optimum decision. So the moderate risk taker may reach nearly

optimum portfolio if they considered the heuristic approach. And

mostly an investor falls in this moderate range. Thus, an investor

can easily arrive at near optimum solution heuristically.

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Chapter 6: Comparative Analysis on Sharpe’s Cut off Principle and a Near Optimum Principle for Portfolio Selection under

Model Approach

147 | P a g e

6.1 Introduction

Although the pioneer work done in portfolio management

by Markowitz gained much importance in portfolio optimization

but was not without limitations. However, the next pioneering

work was done by W. Sharpe (1971) in order to overcome the

limitations of the mean-variance model of portfolio optimization.

He developed a model, also known as, Single Index Model

which describes risk and return of a portfolio with relatively less

calculations. His work was influenced by a prior work known as

Capital Asset Pricing Model (1964). In Single Index Model,

requirement of data is comparatively less than the Mean –

Variance approach. In this model Sharpe assumes that ‘the co-

movement of the stocks is due to a single common influence or

index’. That is why this model is known as Single Index Model.

Later many works have been reported in the literature on

portfolio management. Many authors have suggested selection

of portfolio in many ways. These have established a close

relationship with statistic of modeling. With various stochastic

formulations (see Marton,1980; Sahalia & Brandt, 2001;

Detemple, Garcia and Rindisscher, 2003; Beliakov and Bagirov,

2006; Okhrin and Schmid, 2008 etc.) these authors discussed


Model Approach

148 | P a g e

the problem of selection of portfolio from optimum to heuristic

point of view.

6.2 Earlier works

The relationship between the optimal portfolio obtained

by using Elton, Gruber and Padberg’s algorithm under the

Single Index Model and the optimal portfolio under the Single

Index Model with the introduction of a market security was

discussed by Stucchi (1991). The author empirically showed the

relationship and got that there is an exact relationship between

the optimal portfolio obtained by using Elton, Gruber and

Padberg’s algorithm under the Single Index Model and the

optimal portfolio under the Single Index Model with the

introduction of a market security.

Dutt (2003) constructed an optimal portfolio using

Sharpe’s Single Index Model. For analysis purpose he took data

of BSE 100 as market index from October 2002 to April 2003.

He observed that all the selected stock turn out to be bank

stocks. Antomil, J. et al (2004) discussed the application of

Fuzzy Compromise Programming to Portfolio Selection using

Sharpe’s single index model. They introduced the fuzzy ideal

solution concept based on soft preference and indifference

relationships and they canonically represented the fuzzy


Model Approach

149 | P a g e

numbers by means of their α –cuts. The fuzzy parameters are

used to evaluate the accuracy between the ideal solution and

the objective values through their expected intervals. Their

model work with data that contain more information than any

classical model and the model can easily be dealt with.

Computational effort is less in their model. Their model allows

the preference of the decision makers to select an optimal

portfolio.

Bilbao, A. et al (2006) presented Sharpe's single-index

model and on Fuzzy Sets Theory to deal with the problem of

portfolio selection. Their model has three basic concepts like

value, ambiguity and fuzziness. They have introduced ‘Expert

Betas’ which are nothing but the expert estimations about future

Betas. Each financial asset’s ‘Extra Beta’ is modelled as

trapezoidal fuzzy numbers. They have also proposed Goal

Programming model to select an optimal portfolio which can

include imprecise investor's aspirations considering both, high-

and low-risky assets. To illustrate the proposed model they have

considered a real portfolio selection problem.

Sharpe’s Single Index Model is therefore an interesting

area of research. Many authors have been done their research

work in this specific field. Chitnis (2010) constructed two


Model Approach

150 | P a g e

optimum portfolios using Sharpe’s Single Index Model and

compared them by Sharpe’s Ratio. The author used two

samples to construct the two portfolios. She took Nifty as the

market index and found that individual securities have risk return

characteristic of their own. According to her, portfolio spreads

risk over many securities and the greater the portfolio’s Sharpe’s

ratio, the better is its performance.

Mishra, A. K. (2011) studied the Sharpe’s Single Index

model to illustrate a real portfolio selection problem. For

empirical study he has considered thirty stocks listed in BSE

Sensex. He has given idea to the researcher that optimum

portfolio can be obtained through Sharp’s Single Index model in

real world.

Rani, M.,Bahl, S.(2012) used the Sharpe’s Single Index

model to construct an optimal portfolio with the procedure of

short sales and without the procedure of short sales. They

constructed an optimal portfolio and represented the optimal

portfolio with the percentage invested in each stock. They have

selected optimum portfolio with eleven stocks out of thirty stocks

in BSE Sensex. They have concluded that Sharpe’s Single

Index Model is of great importance and the framework Model for

optimal portfolio construction is very simple and useful. Any


Model Approach

151 | P a g e

investor can easily use this method to construct the optimum

portfolio. They have considered data from April 2006 to March

2011of BSE Sensex.

As discussed in chapter 4 and 5, the suggested heuristic

selection of portfolio is based on risk taking propensity. As

selection of optimum portfolio is a numerically involved task,

they examined the closeness between the heuristic portfolio and

the optimum portfolio. Basically, they have considered the

concept coefficient of optimism popular in decision theory to

calculate the optimum weights of different optimum portfolios

and studied the variation in closeness between the heuristic

portfolio and optimum portfolios with the variation in coefficient

of optimism. They noted that when the value of the coefficient of

optimism is 1 the optimistic investor’s decision is away from the

optimum decision and when the value of coefficient of optimism

is equal to 0 the pessimistic investor’s decision is not close to

the optimum decision. But when the value of coefficient of

optimism is moderate, i.e; in between 0.375 to 0.875, the

heuristic solution is very close to the optimum decision. These

mean the moderate risk taker may reach nearly optimum

portfolio if they consider the heuristic approach.


Model Approach

152 | P a g e

6.3 Sharpe’s Single Index Model

Sharpe made an attempt to predict the return on a

security based on a market return index. The basic equation

underlying the Single Index Model is

imiii eRR

The assumptions of Sharpe’s model are:

1. The expected value of ei is zero, i.e; E(ei) = 0,

2. The only reason of stock movement is a common co-

movement with the index, i.e. E(ei,ej) = 0 and

3. The index is unrelated to unique return, i.e,

0)]([ mmi RReE .

To select the securities for the Sharpe’s optimal portfolio, we

can consider those securities whose returns should be more

than the risk free return and the beta value for the securities

should be positive.

6.4Sharpe’s cut off principle

In the next phase, we have to rank the securities based

on [ ifi RR /)( ] to construct the optimal portfolio, where fR


Model Approach

153 | P a g e

Return of a risk free asset. The size of a security’s excess return

to beta ratio of [ ifi RR /)( ] decides whether a security is

included or excluded in the portfolio. Depending on a unique cut

off rate C *, all the securities which have higher ratio of

[ ifi RR /)( ] are included in the portfolio and those securities

are excluded from the portfolio which have lower ratio of

[ ifi RR /)( ].

Securities are ranked in descending order of magnitude

according to their excess return-to-beta ratio. Then, we have to

consider a cut-off rate C * to select the portfolio. The formula

developed by Sharpe to calculate the cut-off rate is given below:

n

j ei

im

n

j ej

ifim

i

RR

C

12

22

12

2

1

)(

C * is a candidate of Ci. The value of C * is that value of Ci ,

before which all the values of Ci are increasing and beyond

which all the values of Ci are decreasing. That means when the

values of Cis change its pattern we have to consider that value

as C *.


Model Approach

154 | P a g e

After selecting the securities we have to calculate the

expected return and risk of the portfolio using the following

equation 20 and 21 respectively.

n

iiip RwR

1

(20)

n

ieiim

n

iiip ww

1

222

2

1

(21)

6.5 Optimum framework under value system and Sharpe’s Single Index Model

Following the existing literature, a portfolio can be

optimized in two ways. One method is to minimize risk subject to

a minimum return constraint. The other way is to maximize the

expected return subject to a certain level of risk. We propose to

combine these two and minimize the portfolio risk subject to a

minimum return constraint. Under a given coefficient of optimism

under Sharpe’s Single Index Model, a mathematical

programming framework with minimum return constraint

expressed in terms of coefficient of optimism (α), is given below:

Min

n

imiip w

1

222 )( +

n

ieiiw

1

22 (22)

Subject to minmax1

)1()(

n

imiii Rw ,


Model Approach

155 | P a g e

As already stated, following the decision theory

framework we have allowed the values of coefficient of optimism

ranging to vary from 0 to 1 to get different sets of optimum

weights of the portfolio for different choices of the coefficient of

optimism (). These optimum solutions are different risk-based

optimum solutions obtained by raising the value of step by

step. With these optimum weights, the expected return and risk

of the optimum portfolio can be calculated. The risk and return of

the optimum portfolio for a given α have been calculated by

using the equations 20 and 21.

6.6 A Near Optimum Approach

We like to propose an alternative approach to be referred

as near optimum approach. In the proposed approach we

withdraw a term from equation 22. Since, the dropped term may

not have significant contribution we expect that our proposed

principle will offer a near optimum solution. For near optimum

portfolio we have used the following formula to reduce the

original non-linear programming problem stated at equation 22

to a linear programming problem.


Model Approach

156 | P a g e

Min iiw (23)

Subject to minmax1

1)(

n

imiii Rw

Since linear programming problem can be easily handled

one gets computational edge over the other two approaches.

We like to compare Sharpe’s cut off principle portfolio and the

proposed near optimum portfolio with that of optimum portfolio

under Sharpe’s Single Index Model and examine the suitability

of near optimum portfolio over Sharpe’s cut off principle

portfolio. The expected return and risk of the near optimum

portfolio have been calculated by applying equations 20 and 21.

6.7 Comparative study

For comparative analysis purpose 10 securities are

considered viz. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The average returns

of the 10 securities are 15, 17, 12, 17, 11, 11, 11, 7, 7, and 5.6

respectively. The optimal portfolio consisted of 5 securities viz.

security 1, 2, 3, 4 and 5. Their Ci’s are more than the cut-off rate

which is 5.45 for the example. We have calculated the optimum

weights for nine different values of coefficient of optimism. The

optimum weights are given below in Table 6.1:


Model Approach

157 | P a g e

Table 6.1: Weights of the optimum portfolios under

Sharpe’s Single Index Model of ten securities under

different α values

Securities

α values

0 0.125

0.250

0.375

0.500

0.625

0.750

0.875

1

1 0.023774

15

0.040958

19

0.0912

8

0.132748

0.1629226

0.189429

0.215936

2

0.238697

3

0

2 0 0 0.040052

0.098201

0.1355401

0.166186

0.196831

5

0.259457

9

0.3809524

3 0.059435

36

0.083917

33

0.152155

0.205959

0.2383973

0.264528

0.290658

9

0.171594

3

0

4 0 0 0 0 0.0444953

0.103612

0.162728

5

0.315311

5

0.6190476

5 0.029717

68

0.038878

97

0.063403

0.081995

0.0910471

0.097423

0.103799

2

0.014938

98

0

6 0 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0 0

8 0.161961

4

0.151647

9

0.131163

0.1048

4

0.0705187

0.035432

0.034452

56

0 0

9 0.059435

36

0.053120

39

0.025412

0 0 0 0 0 0

10 0.665676

1

0.626977

2

0.496534

0.376258

0.2570788

0.1433

9

0.029701

1

0 0


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158 | P a g e

We have calculated expected return and risk of the optimum

portfolios under different α values. The value of expected return

and risk of the nine different optimum portfolios are show below:

Table 6.2: Expected return and risk of the optimum

portfolios under Sharpe’s Single Index Model under

different values of α

α values Minimum Return

Constraints

Expected Return

Risk

0 5.6 6.67429453 2.265304766 0.125 7.025 6.99349983 2.181982719 0.250 8.45 8.45000015 2.065744605 0.375 9.875 9.87500041 2.207721953 0.500 11.3 11.29999927 2.388543915 0.625 12.725 12.72499895 2.686114145 0.750 14.15 14.38875508 3.106669977 0.875 15.575 15.57499968 3.673084269

1 17 17 5.313312737

We have calculated the weights of near optimum

portfolios under different values of coefficient of optimism by

applying equation 23. The weights are given in the following

Table 6.3.


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159 | P a g e

Table 6.3: Weights of the near optimum portfolios of ten

securities under different α values

Securities

α values

0 0.125 0.250

0.375

0.500

0.625

0.750 0.875

1

1 0 0.1515975

0.303192

0.454787

0.606383

0.757979

0.9095745

0.7125

0

2 0 0 0 0 0 0 0 0.2875

1

3 0 0 0 0 0 0 0 0 0

4 0 0 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0 0

8 0 0 0 0 0 0 0 0 0

9 0 0 0 0 0 0 0 0 0

10 1 0.8484043

0.696809

0.545213

0.393617

0.242021

0.0904255

0 0

The expected return and risk of the near optimum portfolios

under different co-efficient of optimism are shown in the

following Table 6.4:


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160 | P a g e

Table 6.4: Expected return and risk of the near optimum

portfolios under different values of α

α values Minimum Return

Constraints

Expected Return

Risk

0 5.6 5.6 3.098386677

0.125 7.025 7.02502658 2.879043372

0.250 8.45 8.4500001 3.190095479

0.375 9.875 9.87499968 3.906859482

0.500 11.3 11.3000002 4.852771399

0.625 12.725 12.72499978 5.918970852

0.750 14.15 14.15000047 7.051102277

0.875 15.575 15.575 5.968700916

1 17 17 7.90569415

To observe the closeness between the Sharpe’s cut off

principle and the optimum portfolio under Sharpe’s Single Index

Model and the near optimum portfolio and the optimum portfolio

under Sharpe’s Single Index Model, we have considered City

Block Distance (CBD) to combine the two dimensions of

studying risk and return. The CBD between the optimum

portfolio under Sharpe’s Single Index Model and the Sharpe’s

cut off principle portfolio is given below in Table 6.5.


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161 | P a g e

Table 6.5: CBD between the optimum portfolio under

Sharpe’s Single Index Model and Sharpe’s cut off principle

portfolio

α values Minimum Return Constraints

CBD values

0 5.6 9.900315476

0.125 7.025 9.664432223

0.250 8.45 8.324170016

0.375 9.875 6.757192409

0.500 11.3 5.151371587

0.625 12.725 3.428801677

0.750 14.15 1.344489715

0.875 15.575 0.408169178

1 17 3.473397966

The CBD between the optimum portfolios under Sharpe’s Single

Index Model and the near optimum portfolios are given in Table

6.6.


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162 | P a g e

Table 6.6: CBD between the optimum portfolios under

Sharpe’s Single Index Model and the near optimum

portfolios

α values Minimum Return Constraints

CBD values

0 5.6 1.907376441

0.125 7.025 0.728587404

0.250 8.45 1.124350923

0.375 9.875 1.699138259

0.500 11.3 2.464228414

0.625 12.725 3.232857538

0.750 14.15 4.183186912

0.875 15.575 2.295616967

1 17 2.592381413

The CBD values between the optimum portfolio under


based portfolio and between the optimum portfolio under

Sharpe’s Single Index Model and the near optimum portfolio are

graphically presented in the following Figure 6.1. Along the

horizontal axis we present the minimum return constraint and

along the vertical axis we present the CBD values.


Model Approach

163 | P a g e

Figure 6.1

From the above Figure 6.1 it is clear that to the left hand

side of the threshold point T1 (where the value of coefficient of

optimism α is 0.622), the distance between the near optimum

portfolio and the optimum portfolio under Sharpe’s Single Index

Model is less than that of Sharpe’s cut off principle portfolio. But

beyond the point T1 and up to point T2 (where the value of

coefficient of optimism is 0.962), Sharpe’s cut off principle

portfolio is closer to the optimum portfolio under Sharpe’s Single

Index Model. Again beyond point T2 the near optimum portfolio


Model Approach

164 | P a g e

is closer to the optimum portfolio under Sharpe’s Single Index

Model than that of the Sharpe’s cut off principle portfolio.

6.8 Conclusion

Based on the detailed empirical analysis, it may be

claimed that the closeness between the optimum portfolio under


portfolio and the closeness between the optimum portfolio under

Sharpe’s Single Index Model and the near optimum portfolio

depends on the value of coefficient of optimism. When the value

of coefficient of optimism or α is in between 0 and 0.6, the near

optimum portfolio is closer to optimum portfolio under Sharpe’s

Single Index Model. But when the value of coefficient of

optimism (α) is in between point 0.6 and 0.95, the Sharpe’s Cut-

off principle portfolio is closer to the optimum portfolio under

Sharpe’s Single Index Model and when the value of coefficient

of optimism (α )is more than point 0.95 the near optimum

portfolio is again closer to the optimum portfolio under Sharpe’s

Single Index Model than that of Sharpe’s cut off principle

portfolio. Thus, we can say that up to moderate value and very

high value of coefficient of optimism, near optimum portfolio

shows better result. However, for moderate to high value of

coefficient of optimism, the Cut-off principle portfolio shows

closer result. This put forward the admissibility of the near

optimum portfolio.

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165 | P a g e

7.1 Importance of value system

Portfolio optimization is a balancing act between risk and

return, with each investor having unique requirements, as well

as unique financial outlook. However, the constant factor is the

delivery of an investment strategy which should not only be

specific to an investor's personal needs but also capable to work

well and provide financial security for the future. Within these

constraints the need arises to classify investors according to

their risk and return characteristics so that portfolio optimization

can be performed in order to suit individual investors. Every

investor is different, with different financial goals, different

tolerances to risk, different personal situations and different

desires.

The psychological dimension as provided in the realms of

behavioral finance shows that investors perceive gains and

losses asymmetrically, which boils down to the fact that

downside and upside movement by identical amounts does not

guarantee equal pleasure and pain. Further to it, investors tend

to modify their behavior according to their personal bias as well

as crowd pressure, thus making it difficult to predict the decision

making characteristics.


166 | P a g e

Investors are not the same. Several dimensions including

personal and financial goals, current life situation and risk

tolerance tend to influence the differences among individual

investors. Investments that not only generate good returns but

also fulfill personal needs and provide financial security for the

future time horizon are usually preferred by the investors’

community at large. The investment decisions of the investors

are greatly influenced by the type of investor an individual is

(investment style). Value system plays a critical role in strategic

decision making. The concept has been extended to introduce

value system in order to classify investors and highlight their

performance in the portfolio optimization exercise. The following

four categories of investors are considered in this work

according to their value system.

Optimistic Investors, Risk Planner, Random Selector and

Pessimistic Investors

Optimistic investors have a propensity to take risk for

higher returns. They usually prefer risky investments and their

objective is to maximize returns in the long run. However, it is

observed that this type of investors seeks above average

returns by focusing investment in stocks and certain types of

mutual funds. Risk Planners seek to balance risk with assets


167 | P a g e

that preserve the principal investment. However, they are not

totally risk averse as they are willing to take on some amount of

risk. Their investment strategy involves investing in portfolio of

securities which promises a constant income stream over the

investment horizon. The return objective of random selector is to

optimize return of the portfolio. As far as risk appetite is

concerned they tend to demonstrate a tolerable level as deemed

fit as per the need. The investment basket of random selectors

usually consists of equities or mutual funds and a mix of

moderate investments such as unit trusts and bonds. The

Pessimistic investors are categorized to be risk averse. The

investment principle of this type of investors is to preserve their

principal investment by avoiding risky investments. Thus,

government securities and bonds that promise a constant

income stream find prominent position in their investment

basket.

7.2 Closeness between optimum solution and

heuristic solution

As mathematical are used to generate optimum portfolio

are not easy to handle, heuristic solution is the other way to get

a near optimum portfolio. For those investors who find it difficult


168 | P a g e

to mathematically or statistically arrive at sophisticated optimum

solution, they can heuristically obtain a portfolio similar to the

optimum portfolio. It means that one can reach to the near

optimum solution with the help of heuristic solution. In this

doctoral thesis, the heuristic solutions are already discussed

with empirical evidence. Detailed analysis revealed an ideal

point of change to be taken into consideration while examining

the behaviour of an investor, following heuristic approach in

investment planning. A mathematical formulation is presented to

design the optimum portfolio given a value for coefficient of

optimism.

After detailed discussion it is observed that heuristically

one can reach the optimum solution. In this doctoral thesis, the

concept of co-efficient of optimism is introduced in the decision

framework of the portfolio optimization and it is observed that

coefficient of optimism plays a crucial role in selecting portfolio

by considering human value system. Thus, investors have to be

aware of their coefficient of optimism.

When we introduced investor’s value system (according

to propensity to take risk) and coefficient of optimism in

choosing optimum portfolio heuristically, it is observed that the

optimistic investors and the risk planners are giving best


169 | P a g e

decision. That means the risk planner’s decision is very close to

the optimum decision. For lower to moderate value of

coefficient of optimism risk planners’ decision and optimistic

investor’s decision is best as compared to the other heuristic

solutions. On the other hand, at the higher value of coefficient of

optimism all the heuristic solutions showed more or less same

result. For the lower to the moderate value of coefficient of

optimism random selectors showed worst result as compared to

the other heuristic solutions.

Next in this doctoral thesis, the Sharpe’s Single Index

model is reexamined and it is presented as a non-linear

mathematical model. A comparatively easy analytical framework

is achieved by modifying the Sharpe’s model which is called

near optimum portfolio. The modified decision rule is compared

with the Sharpe’s model Based on the detailed empirical

analysis, it may be claimed that the closeness between the

optimum portfolio under Sharpe’s Single Index Model and

Sharpe’s cut off principle portfolio and the closeness between

the optimum portfolio under Sharpe’s Single Index Model and

the near optimum portfolio depends on the value of coefficient of

optimism. Also by validating the models by with the selected

data set it is observed that from moderate value of coefficient of


170 | P a g e

optimism to moderately high value of coefficient of optimism the

Sharpe’s Cut-off principle method gives the best result. For

lower value of coefficient of optimism to moderate value of

coefficient of optimism the near optimum portfolio gives the best

result. On the other hand, for a very high value of the coefficient

of optimism near optimum solution is also performing well. That

means an investor can reach near to the optimum portfolio

heuristically. This put forward the admissibility of the near

optimum portfolio.

7.3 Limitations of the Current work

The present work has aimed to reexamine the classical

model of portfolio optimization and has proposed new models of

portfolio optimization which when validated with the chosen data

set have demonstrated encouraging results. However the work

cannot be claimed to be robust in nature and readily usable for

commercial purpose due to the following limitations of the work.

1. Due to lack of time it was not possible to use

another data set in the work. So the work can be

defined as static in nature.

2. Further work can be done by considering the data of

other stock exchanges.


171 | P a g e

3. Only risk parameter is considered in the work. Thus,

the scope of considering return parameter is there in

portfolio selection.

4. Only two portfolio theories have been considered in

the work. There is the scope of using other portfolio

theories.

7.4 Future Scope

Statistical models are not very easily understandable to

the general investor. When a general investor wishes to invest

money in any portfolio of securities they are more concerned

about the expected return and risk of the portfolio and not about

the various statistical models. Thus, the present study can be

further validated with other data set so as to make it robust and

thus ultimately lead to a simple heuristic tool to help investors so

that they can get a near optimum portfolio for investment.

Though heuristic method is not universally accepted but is

logically correct and thus the initiation in the present study can

be further made to be logically accurate with more insightful

research in the same dimension. The present study mainly

focuses on the weight of the securities in the portfolio; future


172 | P a g e

studies can consider other dimensions in the portfolio

optimization framework. In real world, in an efficient market, it is

very difficult to predict assets’ weights accurately, thus, the

proposed model in the present study can be further explored

with more accurately predicted weights.

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References

173 | P a g e

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