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THE EVALUATION OF MODERN PORTFOLIO THEORY SKRIPSI Presented in partial fulfillment of the requirements for The Bachelor’s Degree in Accounting By: Putu Chantika Putri Dhammayanti 008201500126 FACULTY OF BUSINESS ACCOUNTING STUDY PROGRAM PRESIDENT UNIVERSITY CIKARANG, BEKASI 2019

THE EVALUATION OF MODERN PORTFOLIO THEORY

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Page 1: THE EVALUATION OF MODERN PORTFOLIO THEORY

THE EVALUATION OF MODERN PORTFOLIO

THEORY

SKRIPSI

Presented in partial fulfillment of the requirements for

The Bachelor’s Degree in Accounting

By:

Putu Chantika Putri Dhammayanti

008201500126

FACULTY OF BUSINESS

ACCOUNTING STUDY PROGRAM

PRESIDENT UNIVERSITY

CIKARANG, BEKASI

2019

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ACKNOWLEDGMENT

First of all I would like to praise God for the guidance and strength during the process of

finishing this research and also be able to finish the research on time. The research

entitled with “ The Evaluation of Modern Portfolio Theory” is established in order to

complete the requirement for me to finish the education in President University

Accounting study program. The researcher also acknowledge all people who have

contributed in fulfilling this thesis, especially to:

1. My family who supports me financially and mentally during my university life.

2. My advisor, Dr. Josep Ginting,CFA, and also my co advisor, Maria Yus Trinity

Irsan, M.Si for being patience in giving me supports and understanding so that I

could finish this thesis.

3. The Dean faculty of Business, Head of Accounting Study Program, Accounting

Study Program Lectures and Accounting Study Program Staff of President

University who already give their best assistance and guidance to support all

accounting students finish their education in President university.

4. My best friend and colleagues in President University for listening, offering me

advice and supporting me to finish this thesis.

I realize that this research is still far from perfection. Thus any recommendations and

suggestions are very welcome in order to establish a better research in the future.

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TABLE OF CONTENTS

DECLARATION OF ORIGINALITY ................................................................ i PANEL OF EXAMINER APPROVAL .......................................................................... ii

ACKNOWLEDGEMENT ............................................................................................... iii

TABLE OF CONTENTS ................................................................................................ iv

LIST OF TABLES .......................................................................................................... vi

LIST OF FIGURES ....................................................................................................... vii

ABSTRACT .................................................................................................................. viii

INTISARI ........................................................................................................................ ix

CHAPTER I - INTRODUCTION ................................................................................... 1

1.1 – Research Background ............................................................................................. 1

1.2 – Problem Identification ............................................................................................ 5

1.3 – Research Objectives ............................................................................................... 5

1.4 – Research Scope and Limitation .............................................................................. 6

1.5 – Research Benefit .................................................................................................... 7

1.6 – Thesis Organization ............................................................................................... 7

CHAPTER II – LITERATURE REVIEW .................................................................... 9

2.1 – Basic Concepts ...................................................................................................... 9

2.2.1 – Return ............................................................................................................ 9

2.2.2 – Risk ............................................................................................................. 10

2.2.3 – Covariance and Coefficient Correlation ...................................................... 11

2.2 – Portfolio Return and Risk ..................................................................................... 14

2.3 – Modern Portfolio Theory ..................................................................................... 18

2.4 – Efficient Frontier .................................................................................................. 21

2.5 – Capital Allocation Line and Separation Theorem ................................................ 23

CHAPTER III – RESEARCH METHODOLOGY ..................................................... 26

3.1 – Research Design .................................................................................................. 26

3.2 – Sampling Design .................................................................................................. 28

3.3 – Data Collection and Procedure ............................................................................ 32

CHAPTER IV – DATA ANALYSIS AND INTERPRETATION .............................. 36

4.1– Data Description ................................................................................................... 36

4.1.1 – Analysis of Individual Return and Standard Deviation of Stocks ........... 38

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4.1.2 – Analysis of Market Return .......................................................................... 40

4.1.3 – Analysis of Risk Free Rate .......................................................................... 41

4.1.4 – Analysis of Covariance and Coefficient Correlation Matrix ....................... 42

4.1.5 – Analysis of Markowitz Portfolio ................................................................. 45

4.1.6 – Analysis Passive Strategy Capital Allocation Line ..................................... 48

4.1.7 – Analysis of Efficient Frontier Curve model by using Polynomial ............. 51

4.2– Discussion and Interpretation Data ....................................................................... 55

CHAPTER V – CONCLUSION AND RECOMMENDATION ................................. 58

5.1 – Conclusion............................................................................................................ 58

5.2 – Limitation and Recommendation ......................................................................... 60

5.3 – Implications .......................................................................................................... 61

APPENDICES ................................................................................................................. 63

REFERENCES ................................................................................................................ 85

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LIST OF TABLES

TABLE 1 – REPRESENTATIVE SAMPLE CHOSEN ....................................28 TABLE 2 – LISTED OF SAMPLE COMPANIES ..........................................34

TABLE 3 – ANALYSIS OF MARKET RETURN ..........................................38

TABLE 4 – PORTFOLIO RANK BASED ON THE EXPECTED RETURN .45

TABLE 5 – RESULT OF POLYNOMIAL INTERPOLATION ......................49

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LIST OF FIGURES

FIGURE 1 – COVARIANCE AND COEFFICIENT CORRELATION MATRIX ..... 15

FIGURE 2 – PORTFOLIO RISK AND LEVEL OF DIVERSIFICATION ............... 17

FIGURE 3 – EFFICIENT FRONTIER ........................................................................ 21

FIGURE 4 – CAPITAL ALLOCATION LINE .......................................................... 22

FIGURE 5 – AVERAGE ANNUAL RETURN AND STANDARD DEVIATION ... 37

FIGURE 6 – COVARIANCE MATRIX RESULT ...................................................... 41 FIGURE 7 – COEFFICIENT CORRELATION MATRIX RESULT ....................................... 42

FIGURE 8 – FEASIBLE SET OF PORTFOLIO ...................................................................... 44

FIGURE 9 – EFFICIENT FRONTIER AND CAPITAL ALLOCATION LINE ...................... 48

FIGURE 10 – POLYNOMIAL CURVE AND EQUATION .................................................... 49

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ABSTRACT

The objective of this thesis is to discuss the strategy of asset allocation in

perspective of quantitative approach through the practical application based on the

Modern Portfolio Theory. This discussion is required in order to learn whether the

optimal portfolios constructed on the theory formulate the best allocation. In this

paper, readers will have the overall information about the role of asset allocation

and general knowledge about Modern Portfolio Theory. The outcomes of the

model and element are discussed and evaluated and will be a good source to be

referred. The empirical result show that the portfolio model can be modified so it

is useful to be used as an investment portfolio strategy under some circumstances.

Last but not least, the author will emphasize some significant improvement

suggestion at the end of the paper.

Key words : Modern Portfolio Theory, Optimal Portfolio, Efficient Frontier,

Capital Allocation Line

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INTISARI

Tujuan dari tesis ini adalah membahas model stategi untuk alokasi asset dengan

pendekatan kuantitatif melalui aplikasi praktis mengacu pada Teori Portofolio

Modern untuk mengetahui apakah portofolio optimal yang dibangun berdasarkan

teori merumuskan alokasi terbaik. Dalam tulisan ini, pembaca akan memiliki

informasi keseluruhan tentang peran alokasi aset dan pengetahuan umum tentang

Teori Portofolio Modern. Hasil dari model dan elemen yang dibahas dan di

evaluasi akan menjadi sumber yang baik untuk dirujuk. Hasil menunjukkan

bahwa secara garis besar model portfolio ini dapat dimodifikasi sehingga

bermanfaat untuk digunakan sebagai strategi portofolio investasi dalam beberapa

keadaan. Terakhir, penulis akan menekankan beberapa saran peningkatan yang

signifikan pada akhir penelitian.

Kata Kunci : Modern Portfolio Theory, Optimal Portfolio, Efficient Frontier,

Capital Allocation Line

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CHAPTER I

INTRODUCTION

1.1. Research Background

Investment is highly related with the process of allocating financial

resources in variety of assets which aims to obtain certain benefit as a result of

investment. Thus, implementing a professional investment strategy is necessary

and it usually starts with determining the return requirements and the risk appetite

of the investor. This idea is the fundamental of strategic asset allocation and

diversification. These critical concepts play a key role in portfolio construction.

The first concept is asset allocation that can be defined as a specific

scheme to construct a single portfolio which involves specified target allocation of

various asset classes to provide the optimal balance between expected return and

risk. In other words, asset allocation is the most basic form for fund manager and

investor to decide how to weight financial instrument such as stocks, bonds,

treasury bills and other financial instruments. Too much investment in bonds or

cash will result in lower volatility than stocks, but may not produce optimal

returns. On the contrary, too much investment in stocks will result in higher return

but higher volatility as well.

The second concept is diversification. This concept was proposed by

(Markowitz H. , 1952). Diversification is actually a risk management technique

which reduce investment risk by investing in various financial instruments. It is

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possible to achieve higher return and lower overall volatility, even though there

can be no guarantee that any particular yield or return will be achieved from any

investment due to sudden changes in the financial markets. But, this is the best

element to any investment plan because investment success depends on being

prepared for and being willing to take risk. The benefits of diversification could

be achieved with a portfolio consisting of several asset that have a negative or low

correlation between them (Bodie Z. a., 1980). However, even with a large number

of assets, risk cannot be reduced to zero since portfolios are affected by

macroeconomic factors which influence the market. It is called as systematic risk.

(Bodie Z. a., 2011).

Before the existence of Modern Portfolio theory, the classic portfolio

theory states that assets selection is determined by the maximization of the

expected value. But now in the existence of the modern portfolio theory, the

investment strategy must be conducted in a risk diversification perspective.

Modern Portfolio Theory had become one of the most significant and influence

theories in investment and finance. The outgrowth of Modern Portfolio Theory

(MPT) is Markowitz’s model in Portfolio selection. In his paper, he developed the

model of Efficient Frontier Curve (see figure 3), in which the development of a

plotted graph represents the set of efficient portfolio that have the maximum

expected return for any given level of risk, or the minimum level of risk for

several of risk assets combinations. Then, selecting the best portfolio from the set.

The rational investors will prefer to invest in the efficient portfolio.

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In Modern Portfolio Theory, there is one portfolio out of the set of

efficient portfolio which have the highest returns per additional unit of risk. An

optimal portfolio is achieved where the Capital Allocation line intersects the

Efficient Frontier Curve (see figure 4). The theoretical concept of Capital

Allocation Line was introduced by (Tobin, 1958). In his paper of separation

theorem, by combining this risk-free asset with risky asset can construct portfolio

with better outcomes for risk averse investor rather than only use Efficient

Frontier Curve model.

Given the fact that such a large proposition of portfolio is explained by the

asset allocation decision, it could be argued that the asset allocation decision is the

most important decision in performance of portfolio. However, the process of

asset allocation has been largely split into two opposing groups: qualitative

approach and quantitative approach. In qualitative approach, fund manager prefer

to get to know a company’s management including the education and professional

background (Drachter, 2007), fund manager conducted private meeting with

companies to find the informational advantages (Barker R. H., 2012), fund

manager do the asset allocation process depends on the fund manager that had

more experienced and stock-picking abilities because the result of asset allocation

is more conservative (Lord, 2014). This could be happen because fund manager

tend to maximize their own utility of their compensation and firm’s interest rather

than that of investors (Athanassakos, 1992). Another reason, fund manager can

also find the theory is impractical to apply (Holland, 2006); (Wang, 2012);

(Coleman, 2014).

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Though some researchers believe that process of asset allocation based on

qualitative analysis is more practical than quantitative analysis, it should be noted

that the qualitative analysis involved widespread conflict of interest and

behavioral errors within the finance industry which can be damaging to investor.

In regards with the limitation of qualitative approach, some researchers expect

that quantitative approach might also do better than qualitative approach due to

the set mathematical models that fund manager has employed to do in processing

of asset allocation. The popularity of the quantitative approach is attributed to the

belief that it has potential to be less susceptible to cognitive errors and biases.

Only a few studies address this aspect, which remains considerably

understudied. (Wermers, 2007) directly examine the differences in the

performance between fund managers that employ qualitative approaches and fund

managers that use quantitative approach in their selection process. They find that

employing quantitative models that are largely designed to take advantage of

known market which would impact to the performance of portfolio. (Jongwook

Won, et. al., 2014) shows that the quantitative approach can measuring expected

market returns with corresponding risk level.

The researcher believes that each of existing approach provided its

advantages. This raised a debate among fund manager to emphasize one approach

among quantitative approach or qualitative approach for the asset allocation

process. Based on this background, researcher have interest to examine the

process of asset allocation from the perspective of quantitative approach using

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Markowitz’s model and whether there should be any modification that can be

done to increase the accuracy and more relevance to use in the practical world

1.2. Problem Identification and Statement

There has always been ongoing debate regarding quantitative approach

and qualitative approach for fund managers to make decision in doing asset

allocation to construct the portfolio. For quantitative oriented, fund manager will

design the asset allocation model according to the historical data of return and risk

in order to meet the overall return objectives with acceptable risk into their

quantitative model and derive a new portfolio construction whose value

characteristics make sense in the current economic (Cowell, 2002). For qualitative

oriented, the aspect of decision making to the process will use subjective in nature

and experience. Therefore, the researcher attempts to do the process of asset

allocation from the perspective of quantitative approach. Based on this research,

the question that will be answered are:

1. How do the process of asset allocation according to Markowitz ?

2. How to improve the Markowitz efficient frontier model ?

1.3.Research Objectives

Based on the research problems mentioned in the previous section, the

objective of this research is to examine the asset allocation in constructing

portfolio from the perspective of quantitative approach and whether there should

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be any modification that can be done to increase the accuracy and relevancy to use

in the practical world. This study also provide a future recommendation for

portfolio manager in choosing several scenario within certain return and risk. To

appraise the return and risk, relevant information will be extracted from historical

company’s share price concerning the years 2015 until 2018. This research also

applies prediction, assumption, and approaches method to give a precise output.

1.4. Research Scope and Limitation

To have a common perception between reader and researcher, there are

some points should be clarified. The data used are secondary data where the data

is coming from Indonesia Stock Exchange. In terms of the accuracy of the

secondary data is beyond the control of the researcher. In this study, the selected

stocks investment is LQ-45 index without concerning the types of shares traded

throughout the research period. Three years data has been considered for the study

due to time constraints. The data used in secondary data which only included

active stocks as a representative of risky asset and Bank Certificate Indonesia

(SBI) for non-risky asset. The theories that will be examined include the portfolio

theory. The portfolio theory will be analyzed based on a literature review, such as

published books, journal and articles. The assumptions for doing the process of

asset allocation are the same assumption as (Hull, 2015) made which are no

transaction cost for trading, tax and short selling allowed. Indeed, this

assumptions not the best representation of reality, but allows to do valuable

analysis. The extent of this study is limited to the portfolio construction.

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1.5. Significance of the Study

The result of this research is expected to give benefits to several parties, they

are :

1. Investors : the result of this research are expected to provide to the

framework of asset allocation as fundamental of investing principles

and the study will inform investors the benefit of diversification

strategy.

2. Capital Market Academician : the result of this research are expected

to enrich the field of science, particularly in the field of finance

economics. In addition, this research is also expected to provide new

insights about the portfolio construction from the perspective of

quantitative approach.

3. Researcher : this research could exercise the researcher’s skill in doing

allocation of asset to construct feasible set of efficient portfolio based

on the Markowitz modern portfolio theory. It also gains experience in

academic life, especially in surveying the case or problem in the real

world.

1.6. Thesis Organization

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Writing system in this research are:

Chapter 1 Introduction

Discussion of research background, research question, research

objectives, significance of study, and writing systematic discuss in

introduction section.

Chapter 2 Literature Review

Discussion of the theories and concept that support research,

previous research, literature review, hypothesis development, and research

model

Chapter 3 Methodology

Discussion of the research method, variable used in the research,

research design, population and sample, instruments, and tools used in the

research, sources, and data collection method, analytical method.

Chapter 4 Analysis of data and result

Discussion of research analysis and interpretation of the result of

analysis

Chapter 5 Conclusions and Recommendations

Discussion of conclusion of the research, research limitations, and

suggestion

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CHAPTER II

LITERATURE REVIEW

2.1 Basic Concepts

2.2.1 Return

The ultimate goals for any investors are achieving the return when they

engage in investment activity. In most cases, the investor can evaluate how

successful an investment has been by recognizing the benefit that they received in

terms of a capital gain or loss and some form of income it may generate

(dividend). This return represents how much an asset has increased or decreased

in value over a period of time.

The simple net return on share(s) will be estimated:

𝑅(𝑡0, 𝑡1) = 𝑃𝑡1−𝑃𝑡0𝑃𝑡0

or 𝑅(𝑡0, 𝑡1) = 𝑙𝑛(𝑃𝑡1𝑃𝑡0

) (2.1)

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Here : - R is the stock return on period of time

- Pt0 is the price of stock at the beginning of the holding period

- Pt1 is the price of stock at the end of the holding period

The stock return, calculated in formula (2.1) between t0 and t1 is called

holding period return. This formula is used, if assume there are no cash-flow

(e.g., dividend) during time interval denoted by t. The holding period can be used

for any amount of time such as daily, weekly, monthly and more. Another

formula that primarily used to calculate asset price return is log returns (ln), the

concept between those formulas are similar.

2.2.2 Risk

Investment risk comes in many forms, this risk happened when there is a

difference between the expected return and the actual return. Of course, investor

will concerned if the actual return is less than the expected return. (Bodie Z. a.,

2011) talks about the characteristics of risk related of stock investment is volatile

and unstable. Obviously, it is likely to happen because our equity market is

dynamic. (Markowitz H. , 1959) introduced variance as regards to the statistical

measure to quantify risk of an asset’s expected return and cannot eliminate the

entire variance through diversification. Risk is related to the dispersion within the

uncertainty outcome and dispersion refers to variability. So, the overall risks of

investments are often measured with such common absolute measures utilized in

statistics as variance and standard deviation.

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Variance can be estimated as the dispersion of the favorable outcome from the

expected of each investment return:

𝑉𝑎𝑟 [𝑅�𝑖] ∶= 𝐸(𝑅𝑖 − 𝑅�𝑖)2 = 𝐸[𝑅𝑖2] − [𝑅�𝑖]2 (2.2)

The variance in formula (2.2) is one measure to assess the volatility of

individual asset return. Investors have a tendency to choose the small value of

variance. Markowitz emphasized that the variance is comparable to the riskiness

of investment so that decreasing desired level of asset return. (Fabozzi FJ., 2002)

argued the limitation of variance due to this measure is the only one to describe

how the returns differ from the expected return.

Since the variance is denoted by Var [R�i] or 𝜎2 which can be interpreted

as squared units of the standard deviation, hence we can state that the standard

deviation is the square root of the variance :

𝑆𝑡𝑑 𝐷𝑒𝑣 (𝑅𝑖) = �𝑉𝑎𝑟(𝑅𝑖) (2.3)

This, imply the standard deviation can determine the likelihood of asset

return will reach on specific range. The two ideas are linear, meaning that is the

greater the investment risk indicate the larger the variance or standard deviation.

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2.2.3 Covariance and Coefficient Correlation

Covariance is statistical measure that indicate the movement of two

variables. Unlike the other statistical result that talk about how big or small the

value, but this more to emphasize the positive or negative value. There is no

limited extent of covariance, it has value from 1 to infinity. The way asset return

associated with one and another will have an impact on the portfolio risk. The

covariance between two assets X and Y may be calculated as under :

𝐶𝑜𝑣𝑥,𝑦 = ∑ [𝑅𝑥− 𝑅�𝑥] �𝑅𝑦− 𝑅�𝑦�𝑁𝑖=1

𝑁 (2.4)

Where : - 𝐶𝑜𝑣xy is covariance between x and y

- 𝑅𝑥 is return of asset x

- 𝑅�𝑥 is expected or mean return of asset x

- 𝑅𝑦 is return of asset y

- 𝑅�𝑦 is expected or mean return of asset y

- N is number of observations

Covariance does not give any detailed regarding the relationship between

the return on two assets, either both of the assets have strong or weak relation. By

using covariance, there will be three possible outcomes in analyzing the relation

of two asset returns such as positive covariance, negative covariance, and zero

covariance. Positive values of covariance have meaning that asset X and Y has

same direction. If the return of asset X is above its mean of return that represent

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positive value as well as asset Y. Negative value of covariance indicates that if the

return of asset X is below its mean of return that represent negative value as well

as asset Y will tend to have the negative value. Otherwise, If covariance indicates

zero value is identified both of assets are independent. But in the practice, the case

of zero covariance is rare.

The coefficient correlation is beneficial to simplify the interpretation of

covariance result, since the value of covariance can never be greater than +1 or

less than -1. As mentioned earlier, covariance value can take any value; therefore

it would be harder to explain about the degree of the relationship of two variables.

Further, coefficient correlation formula is shown in equation (2.5)

𝜎𝑥,𝑦 = 𝜌𝑥,𝑦𝜎𝑥𝜎𝑦 and 𝜌𝑥,𝑦 = 𝜎𝑥,𝑦

𝜎𝑥𝜎𝑦 (2.5)

Here : - 𝜌𝑥,𝑦 is coefficient correlation between asset X and asset Y

- 𝜎𝑥,𝑦 is covariance between asset X and asset Y

- 𝜎𝑥 is standard deviation of asset X

- 𝜎𝑦 is standard deviation of asset Y

In the coefficient correlation framework, correlation may be positive,

negative, or zero. Positive correlation, +1.0 denoting the strong movement that

one assets going up or down, the other asset will have tendency to move in the

same direction, vice versa, Negative correlation, -1.0 denoting the strong

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movement in opposite direction. For zero correlation, there will be no correlation

at all. However, very strong correlation being rare in the investment practices.

Most correlation fall on scale -1.0 to +1.0 The discussion of coefficient

correlation deserve to have an attention that the relationship between two

variables does not imply causation, this means that they simply are related or

associate with one and another variable.

2.2 Portfolio Return and Risk

Diversified investment portfolio is part of risk management strategy to

implement risk reduction of an investment by combining variety of assets. (Lee,

2016) assumes that investor would have a more desirable risk/return trade-off

through combined assets over a wide range of industries whose uncorrelated

return rather than investing in any single asset. According to the paper Asset

Allocation Models using the Markowitz approach (Kaplan, 2015), the investment

decision is the matter of how to allocate the funds into various of asset, not merely

which securities to own in order to achieve desired outcome.

Diversification is a well-known concept and being fundamental of portfolio

management since Markowitz introducing Modern Portfolio Theory (Markowitz

H. , 1952). In Modern portfolio theory, we assume those investors are risk averse.

In other words, if the investor faced with two scheme of investment with similar

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return, they likely to choose return which have lower risk. There are some key

consideration to measure the expected return and risk of a portfolio.

Expected return which can defined as probability of possible return

obtained by investor. As we noted earlier, this is only consider as prediction or

estimation. Therefore, the actual return may be higher or lower than as it expected

before. The portfolio expected return is computed as the weighted average of

return on the assets held in the portfolio. The weight represent the proportion of

assets that need to be invested in portfolio and it estimated by the sample of

historical average. in spite of the fact that their estimation probably won't be the

best, it align with our purpose. The general formula of expected return on a

portfolio is :

𝐸�𝑅𝑝� = 𝑊1.𝐸(𝑅1) +𝑊2.𝐸(𝑅2) + 𝑊3.𝐸(𝑅3) + ⋯+𝑊𝑛.𝐸(𝑅𝑛) (2.6)

∑ 𝑊𝑛𝑛𝑖=1 = 1 (2.7)

Let 𝐸(𝑅1), denote the return on asset 1. Assuming the total of weight of asset in

the portfolio allocation is 1 or in term of percentage is 100% as shown in equation

(2.7). In the formula (2.6), we will allocate 𝑊𝑛 to asset 𝐸(𝑅𝑛). This formula can

be extended based on the number of assets in portfolio.

From Portfolio risk standpoint, risk can be expressed as the opportunity

for investor to carries the probability of loss. When an asset has larger

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probabilities of loss, the asset is classified as the risky asset. The probability of

loss is occurred due to the existence of uncertainty. Mostly people who has role as

a buyer of stocks are genuinely wish trying to get as much as gain without endure

the risk even it is likely impossible. Uncertainty can be implying as the

knowledge and information were practically non-existence. Therefore, the suitable

portfolio seems become major concern for people when they get into investment

world.

The proper strategy could ensure share-owner against the risk and

meanwhile obtain the highest return, but it also requires estimating risk and return

of portfolio more accurately. Having knowledge about portfolio risk or portfolio

variance such an important matter before we do the selection of portfolio as the

main point of Markowitz study. We need to do the examination about the

potential performance (e.g., expected return) and potential magnitude of risk (e.g.,

variance) of portfolio.

The author composing portfolio of multiple assets. Therefore, the

calculation is slightly difference from the portfolio consisting of two assets. Even

so, the basic principle remains the same. Covariance and coefficient correlation

are the essential constituents of the variance of the portfolio. It is more difficult to

calculate covariance and coefficient correlation when the number of assets in

portfolio becomes large. Therefore, (Markowitz H. , 1952) has presented in the

classical mean-variance optimization framework about covariance and coefficient

correlation matrix. This method provided a convenient for us to calculate multiple

assets and depressed the error estimation (K. Chopra, 2006) and is essential to

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deal with multi asset allocation. (John, 2017) argued that there is the downside of

the mean-variance analysis such as it is mostly identified with its affectability to

the inaccuracy estimation of the means and covariance matrix estimation of asset

return. Applying matrix notation n x n covariance matrix of returns, 𝐶𝑜𝑣𝑥,𝑦 =

𝐶𝑜𝑣(𝑅𝑥 ,𝑅𝑦) and coefficient correlation denoted by 𝝆𝒙,𝒚 as follows by figure 1

Figure (1) Covariance and Coefficient and Correlation Matrix

Thus far we have given the background and key parts to get the portfolio

variance, which are covariance and coefficient portfolio. This all elements above

are to compile the portfolio variance. The historical simulation method is one of

the most frequently method for forecasting the risk. The equations for portfolio

variances for multiple assets are the extension of portfolio risk consisting of two

assets. Further, portfolio variances is shown in equation (2.8) and (2.9)

𝜎𝑃2 = 𝑤12𝜎12 + 𝑤22𝜎22 + 𝑤32𝜎32 + 2𝑤1𝑤2𝜎12 + 2𝑤1𝑤3𝜎13 + 2𝑤2𝑤3𝜎23 (2.8)

𝜎𝑃2 = 𝑤12𝜎12 + 𝑤22𝜎22 + 𝑤32𝜎32 + 2𝑤1𝑤2𝜌12 + 2𝑤1𝑤3𝜌13 + 2𝑤2𝑤3𝜌23 (2.9)

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In words, equation (2.8) is used covariance as the component to get

portfolio variance, on the contrary for equation (2.9) is used coefficient

correlation. But, at last those equations states the weighted total of individual

variances of the assets in the portfolio plus the weighted total of the extent to

which the asset differ from each other.

Risk related to any material loss attached to the performance of asset that

may affect the return of the asset. The types of risk can be classified under two

main groups: systematic and non-systematic risk. So, the total risk are containing

the systematic risk and unsystematic risk, see figure (2).

Figure (2) Portfolio Risk and level of diversification

The systematic risk or also known as undiversifiable risk the risk that

occur by the macroeconomic factors that affect for all risky assets, for instance

inflation risk, interest rate risk, market risk and more. The unsystematic risk is the

part of total risk that called as diversifiable risk. Systematic risk arises due to the

unique event that happen in the firm or in the industry and not have big impact for

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entire firm and market e.g. business risk, financial risk and liquidity risk. This

type of risk can be reduce through diversification.

2.3 Modern Portfolio Theory

Modern Portfolio Theory was introduced by Markowitz (1952), the well-

known reputable ones. In its simplest form, the modern portfolio theory is about

constructing the portfolio with target to maximize the return in any given level of

risk. This theory or it can refer as mean-variance analysis figure prominently the

relationship between risk and return. Prior to Modern Portfolio Theory, it was

determined by the optimization of the real value of the possible value on the

assets. In other words, investors were just concentrating on the return and risk of

individual’s assets. His concept has been a foundation for the development of

financial economics and corporate finance. Basically, this theory focuses on the

risk of portfolio, not only risk generated by individual securities. The concept of

diversification assets or investing in much kind of assets plays a very crucial part

in the portfolio theory. Markowitz’s model concentrates on extremely complicated

statistical mathematical modelling and formulas that support the theoretical

assumption.

(Fabozzi FJ., 2002) mentioned portfolio theory is portfolio construction

framework for investment practitioners to selective in choosing portfolio depend

on the performance of expected return and level of risk. Markowitz’s model

portrays a principle of conduct about how investor should engage in building a

portfolio ( in a rational way of investing) , but in fact the majority of investors did

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in the opposite way. Markowitz portfolio model or mean-variance formulation is

not only highly concern about the number of stock in the portfolio but finding the

small or constant variance with maximize expected return or vice versa, in the

practice this is an important point to the coefficient correlation approach.

Diversification will provides favorable set of portfolio if the portfolio consist of

companies in different industries because it can reduce the portfolio variance.

Normally, it have small covariance, which mean between one companies and

another do not affect each other.

In mean-variance approach, mean returns is a key parameters for expected

return and variance of return is a proxy for determine the risk of asset. This basic

concept were encapsulated in his journal and widely used up to now. According to

(Markowitz H. , 1952) portfolio theory was paying attention more to second stage

of the process of selection portfolio start with the relevant way of thinking about

the future performance, and continue with choosing the suitable portfolio.

Portfolio Theory explains that through diversification or holding the

enormous of assets can reduce the risk of portfolio. Therefore, (Markowitz H. ,

1987) in his model depends on the several assumptions and basics that are the

main concept whereupon it has been developed, like: the probability normal

distribution of individual asset returns, only just used the portfolio standard

deviation to measure risk, following the belief that investor are rational and risk

averse ( investor prefer for maximum risk for a given level of risk and vice versa),

there are no transaction cost for buying and selling securities, taxes is not put into

consideration while investor making an investment decision.

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There are several assumption that he developed in his model but it raised

question and criticize by researcher. (Maehl, 2008) mentioned that investor have

strong urge to have a profit maximization, therefore it can prompt investor to do

speculative decision based on rumor and intuition (Morien, 2005). Another point

of criticism come from (Fabozzi FJ., 2002) pointed about all relevant parameters

for calculate portfolio such as predicting return, standard deviation, coefficient

correlation based on historical data are not plausible analysis to represent future

performance due to the fact that neither expected return nor risk are directly

observable. One of the principle issues when computing these estimates is

selecting a representative set of historical simulation data. The chosen set ought to

as accurately as possible represent the time horizon to be predicted. This

theoretical model about the inter-relationship between risk and return of financial

assets also led to the concept of efficient portfolio which contains all feasible

optimal portfolio.

2.4 Efficient Frontier

Investor are faced with many options about the proper asset allocation.

Generally speaking, portfolio can be referred as the variety of asset to be allocated

into a spread of investment product with purpose to fulfil the primary objective of

investment holder. What is the best one combination of asset is one of the

question which triggered the concept of asset allocation appears in the first place.

Asset allocation is a broad subject especially to be implemented this subject in the

dynamic investment world. We marking boundary just to make it clear to do the

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selection of portfolio because asset allocation consist of any number with different

asset proportion.

Assuming economic rationality, Investor are concerned with the efficient

portfolio. Having the set of efficient portfolio lead investor gain the best expected

on a given level of risk, or minimum risk for a given expected return and provides

the most satisfaction and be preferred by all investor (Bringham, 2010).

In the context of Markowitz efficient frontier curve model, when it is not

meet the criteria, then we can called as inefficient portfolio. Inefficient portfolio

has lower expected return for the amount risk taken on or they have to sacrificing

the return for reducing the risk. Therefore, this portfolio does not lie on the

efficient frontier. Meaning that, we should not invest it because the portfolio

requires too much risk and inferior. An efficient frontier is the best combination

of asset allocation that offers the highest possible return for a specific level of risk

and vice versa. Efficient frontier shows the graph of expected return (vertical axis)

and expected risk ( horizontal axis) see Figure (3).

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Figure (3) Efficient Frontier

When choosing a portfolio, investor will choose that suit their risk

preference and the level of risk tolerance. (Girard, 2005) states that investor’s

utility functions toward are taken into consideration in the development of

efficient portfolio. The purpose of the utility theory depicts the preference by

estimating the level of gratification of decision maker. Utility approach give us an

approach to measure financial agent’s preferences for wealth and the number of

risk they will attempt in the desire for achieving greater wealth. This makes it

attainable for development portfolio theory.

2.5 Capital Allocation Line and Separation Theorem

In his essay, “ Liquidity Preference as Behavior Toward Risk”, (Tobin,

1958) extended Markowitz model efficient frontier adding the concept of risk-free

asset. This assets is one kind of asset that can be allocated in the portfolio. The

concept is also known as Tobin’s separation theorem. It clearly seen that if we

put our money to invest in stocks ( risky asset ), the size of potential loss is greater

than if we put our money to invest in separation asset i.e. risky asset and risk free

asset. In simply word, invest in one safe asset and large number of risky asset

much preferred by risk averse investor (Buitler, 2003). So, the expected risk can

be managed effectively. Forming a combination risky asset will construct efficient

frontier model that derived from mean-variance optimization, in purpose to

achieve the highest attainable return based on set of feasible portfolio. According

to (Tobin, 1958) argued that investor can build an ideal portfolio which exceed the

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performance of portfolio on efficient frontier by adding risk-free assets and it will

create the Capital Allocation Line (CAL). Tobin was introduced the risk free asset

and this development of Tobin Separation Theorem (TSB) underpins the

framework of Markowitz’s work (Figure 4)

Figure (4) Capital Market Line (CML)

Furthermore, the tangency point where capital market line intersects with

efficient frontier indicates the best possible capital market line or as the optimal

risk portfolio on efficient frontier consisted by risky asset. Tangency Portfolio

represent the maximum Sharpe ratio. see formula (2.10)

𝑆 = ERp − Rf 𝜎𝑝

(2.10)

Where : - ERp is expected return of portfolio

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- Rf is risk free rate of investment

- 𝜎𝑝 is standard deviation of portfolio

Since this point has the property that is has the most astounding possible

mean-standard deviation proportion, (Engels, 2014). That is the reason we called

this the tangency portfolio. It occurred because risk-free investment is present

when constructing the optimal portfolio. According to (Bodie Z. a., 2011), the

composition of optimal portfolio will be as follows:

With a proportion, x, in the risky portfolio, and 1-x in the risk free asset,

the rate return on the complete portfolio, denoted 𝐸𝑅𝑐

𝐸𝑅𝑐 = 𝑥 𝐸𝑅𝑝 + (1 − 𝑥) 𝑅𝑓 (2.11)

Where : - ERc is expected return of optimal portfolio

- Rf is risk free rate of investment

- ERp is expected return of portfolio

- x is the proportion of risky asset

- 1-x is the proportion of risk free asset

For the standard deviation of optimal portfolio will be denoted by 𝜎𝑐, the formula

is depicted as follow :

𝜎𝑐 = 𝑥 𝜎𝑝 (2.12)

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By varying the proportion of the non-risky asset and risky asset is the beneficial

way to adjust the riskiness of portfolio.

CHAPTER III

METHODOLOGY

3.1 Research Design

In conducting the research, researchers have three types of method that can be

used: a qualitative, a quantitative and a mixed method ( combining between

qualitative and quantitative method). All of them have same purpose to capture

the wide range of phenomena and find out things of the certain area, in order to

increasing the knowledge. The distinction between quantitative and qualitative in

nature, A quantitative method is used to analyzed the phenomena, human problem

or testing a hypothesis. It contains a systematic or mathematic process which are

estimated numbers as an outcome (Creswell, 1994); (Gay, 2002). With this

approach, the researcher gather information from example database.

A qualitative method is subjective is subjective interpretation in different set

of paradigms (Hitchcock, 1995) and the type of research that develop findings that

cannot be obtained by using statistic procedure. In spite of the fact that qualitative

method is hard to define but some of issues and situation could be more

understandable when used qualitative method, also the instrument use typically

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provides flexibility does not use rigid style. For this method, researcher gather

data in form of observation, in-depth interviews, focus group which are the data

format is more textual. Which method to use depends on the purpose of the study.

If in the end of the study, researcher want to deeper understanding find unique

details upon the phenomenon studied, then qualitative method seems a suitable

method to use. But, if researcher want to wide and direct result, then the analysis

can use quantitative method as its method.

The researcher’s input is numerical data extracted from historical data weekly

stocks price of 45 most liquid stocks included in LQ-45 index and BI rate for risk

free instrument, the analyzing process in this research also includes of various

mathematical formulas and output is a number showing the proportion of asset

allocation. Accordingly, it employs the quantitative research method as the

priority. The most valid method to use in this study is the quantitative method,

considering that the implementation of efficient frontier model and capital

allocation will be engaged in the simulation techniques and ending with

examining the model performance. In measuring the performance of the establish

portfolio, this study also requires comparative data as benchmark, therefore we

use LQ45 index.

The purpose of this study is to implement the process of asset allocation in

constructing portfolio from the perspective of quantitative approach and whether

there should be any modification that can be done to increase the accuracy and

relevancy to use in the practical world, by using the model of Markowitz because

the model is well-known especially in the investment community as a

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fundamental for portfolio management. In fact, there are some findings that we

have known about the main characteristics and most relevant aspect of efficient

frontier, but researcher assumes more essential information is needed for

generating the theory to convince us that theory is applicable even in the changing

business environment.

3.2 Sampling Design

In this research, the author have construct a portfolio consisting stocks that

ever listed in LQ-45 index. The historical data LQ-45 performance during time

period cover years from July 2015 to July 2018. The LQ-45 index is a stock

market index for Indonesia Stock Exchange (Jakarta Stock Exchange). The

Liquid-45 index (LQ-45) is the national benchmark index that was established in

February 1997 that represents 45 of the most liquid stocks , this index is one of

indicator of stocks in the capital market in Indonesia. The LQ-45 consist of 45

common stocks that have been chosen through the following criteria that have to

be fulfilled by firms which are being among the Top 60 common stocks with the

highest transaction value over the past 12 months , being among Top 60 common

stocks with the highest market capitalization in regular market over the past 12

months, having listed on the Indonesia Stock Exchange for at least 3 months,

having good financial condition and the prospect of future growth.

These stocks will be evaluated and monitored by Indonesia Stock

Exchange every six months, included in the calculation LQ-45 index. The stock

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will be replaced on the next cycle of stock selection, if a stock does not fulfil the

criteria that have been set. This replacement occurs every six months, and

effective in early February and August. The stock used in LQ45 index is common

stocks, meaning that the investor who hold this stock will have voting rights and

right to receive dividends under certain condition, if the corporation able to make

profit. Company usually choose this strategy by issuing common stocks with

purpose to seek funds in the equity market (e.g., Indonesia Stock Exchange).

Stock that listed on LQ-45 are attractive stocks for investor to buy and sell due to

their high liquidity assets criteria. However, it does not mean that the performance

of LQ-45 index is the best performing major index because this index also heavily

reliance on the global market.

This research is limited using the LQ45 index because researcher were

trying to capture the phenomena that efficient frontier curve model that have been

made based on modern portfolio theory. MPT have an assumptions that investor

are risk averse, they prefer to hold stocks which gives maximizes return and

minimizes risk. Theoretically, all set of portfolio that lies on the curve are

efficient portfolio indicates the portfolio offer the minimum risk and greater

return. According to the purpose of construct portfolio based on EFC model,

researcher assumes that LQ-45 index is suitable to be used as a sample in this

study because stocks that listed on LQ-45 index are traded actively in Indonesia

Stock Exchange. In this regards, these stock have been proven to have a strong

fundamental analysis for investor to follow the rules of selecting stocks. With a

shortlist of companies, an investor might to examine the capabilities of the

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company and potential for future growth. In simple word, investor have to choose

which company is good to be an investment choice.

The concept of Capital Allocation Line is involving the combination

between risky assets and non- risky asset. Another instrument was involved in

forming portfolio is non-risky asset e.g. Bank Indonesia Certificate (BI Rate). The

data used in this study is weekly historical data for three years period, so the

interest rate taken is the interest rate during the study period. For the comparison

performance portfolio and marked index, we used historical weekly data of LQ-45

index as our benchmark.

In constructing optimal portfolio of stocks, The selection of data samples

is done by purposive sampling, which is the selection of data based on specific

criteria. The criteria of shares listed is actively traded companies on the Indonesia

Stock Exchange which consistently appears in the index with a certain period.

This limitation is implemented to avoid the extreme changes in stock prices

during the observation period, average prices and incomplete data. This criterion

is used in order to prevent any research bias caused by the inclusion of

inconsistent stocks in the LQ45 index. Based on screening and the

implementation of Modern portfolio theory, there are nine stocks were selected as

candidates for building portfolio. Below is the list, appeared alphabetically: (table

1)

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Code Company

1. AKRA AKR Corporation, Tbk.

2. BSDE Bumi Serpong Damai, Tbk.

3. JSMR Jasa Marga ( Persero ), Tbk.

4. KLBF Kalbe Farma, Tbk.

5. LPKR Lippo Karawaci, Tbk.

6. LPPF Matahari Department Store, Tbk.

7. PGAS Perusahaan Gas Negara ( Persero), Tbk.

8. PTPP PP ( Persero ) Tbk

9. WIKA Wijaya Karya ( Persero ) Tbk

Table 1 Representative Sample Chosen

Source : data processed, 2018

The estimation period in this study is three years. Researcher assumes that

three years is adequate to do this research because of the consideration about

macroeconomics in Indonesia. As part of emerging market, Indonesia has faced

challenging times for the world ’s developing economies. In recent weeks, money

has been flowing out of emerging markets and into the US as international

investors repositioned their portfolios in anticipation of a fed rate hike by the

Federal Reserve. This indicates that the impact of the macroeconomic situation

reflects the changes in the stock market.

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The condition of the capital market has strongly influenced by

macroeconomic variables such as rupiah exchange rate, inflation, interest rate,

gross domestic product. From the point of view of our current economic

condition, it can be ascertained that Indonesia cannot be separated from the

positive and negative sentiments brought by developed countries. For instance, a

significant slowdown in economic activity resulting a decrease in GDP. The

strong decline of the Rupiah, the interest rate hikes, these will burden demand

sides and the economy. Economic actors may decide to invest or sell an

investments. If we see the GDP growth rate from 2015 until 2018 is relatively

stable rather than GDP growth rate in 2013 and 2014 during Jokowi’s era, see

figure (A.1). For this reason, researcher have an intention to limit the time

period with assumption the data taken is stable and not heavily rely on economic

condition.

3.3 Data Collection and Procedure

Sources of information are divided into two types: primary sources and

secondary sources. Secondary data are data gathered and recorded by someone

else prior to the current needs of the researcher. Secondary data are usually

historical, already assembled, and do not require access to respondents or subjects

and researchers are able to build on past research a body of business knowledge.

The primary advantage of secondary data is the obtaining secondary data is almost

always less expensive than acquiring primary data. In addition, secondary data

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can generally be obtained rapidly and may include information that not available

in primary data.

Researcher using weekly stock price data of listed company in LQ-45

index over period 2015 – 2018. Historical data of stocks and LQ-45 index were

taken by from yahoo finance, from starting period of July 30th, 2015 until the last

week of July 2018. Meanwhile, to obtain Bank Indonesia (BI) rate as a proxy to

non-risky rate, researcher gather data through the website of Bank Indonesia (The

type of data used in this study can be categorized as secondary data. As far as we

all know, there is no study that allows us to understand what is the best period of

time horizon to get reliable input data ( average return, standard deviation and

covariance ). Thus, we have a consideration a period between three years may

sufficient to produce final result. Researcher also need to employ Microsoft excel

for data processing because this software has the function and features needed for

analysis the data. The advantages of using Microsoft excel as tools to do the

portfolio calculation based on Markowitz approach is because this software is

easy to use and very popular in our community.

In order to attain the final result of this analysis, there are some steps

appear in this research conducted by researcher and according to reference on

chapter 2: (1) Calculate the return, standard deviation, and variance of each stocks

listed in LQ-45 index. The price that is used to calculate those variables is

adjusted closing price of each week. This is useful for researcher to do screening

with an intention to select stock that have higher mean and lower standard

deviation. This screening refers to mean-variance framework The first step of this

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process generate nine stocks that match with the criteria (2) Determine the

proportion of possible funds from each share; This research used Excel software

and its random number. In this study, random number used as representative of

probability. Basically, we apply the additional requirement, which is the

proportion of each asset is more than zero and with the constraints 𝑤𝑖 = 100%, the

consideration of this assumption is the total portion of asset must be equal to

100%. In addition, this indicates that researcher will allocate fund for all asset.

(3) Calculating the covariance and correlation matrix of portfolio which consist of

nine stocks. Due to the difficulty of calculate the covariance and correlation more

than two assets. Thus, researcher use this matrix in order to make it in easier way.

(4) Determining the efficient portfolio return and risk based on the scheme

proportion of possible funds. Each of respective portfolio will present different

level of return and risk. (5) Making chart and comparing all the efficient portfolio

which has been formed and constructing the efficient frontier of all the

probability. From this steps, as a result of many probabilities. Researcher obtain

the result that some of portfolio have different return but the same level of risk or

have the same return but different level of risk. From that finding, researcher will

choose that given a higher return or lower risk. This assumption is supported by

Modern Portfolio Theory (MPT). (6) Establish a Capital Allocation Line (CAL).

From an efficient set of portfolios, an optimal portfolio can be formed with the

help of the Capital Allocation Line (CAL). To make Capital Allocation Line, we

need put another kind of asset, which is non-risky assets. Therefore, researcher

used Bank Indonesia Certificate ( BI rate ) that represent non-risky asset. (6)

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From the steps before, the optimal portfolio will be obtain from the measurement

index that has the highest value, which is the maximum Sharpe Ratio. The

tangency portfolio or the optimal portfolio can be measure using Sharpe Ratio.

This measurement is used to help investor understand about risk adjusted return.

This result will support the final result to generalize the conclusion of this

research.

Validity relates to the thesis’s ability to look at what's meant to be

researched, notice association between theory and empirical findings (Kumar,

2011). (Brown, 2003) states that the information collected should be precise and

correct with the set purpose, this to form the correct interpretations and to provide

a plausible analysis. Researcher follow the steps that often used and recommended

by some previous researcher and the efficient frontier model has examined by

mathematical programming model using Matlab software.

In order to verify the efficient frontier model is coded by Matlab 16

software. To identify whether the model can be used to accurately predict the

standard deviation where the value of expected return has been known.

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CHAPTER IV

DATA ANALYSIS AND INTERPRETATION

4.1 Data Description

The research used the secondary data taken from LQ-45 index of the

companies listed in Indonesia Stock Exchange from the year of 2015 – 2018. LQ-

45 consist of 45 most liquid companies. There are several considerations that

underlying the selection of stocks, the criteria by using purposive sampling as

follows: first, the companies have already been listed on Indonesia Stock

Exchange for period July 2015 – July 2018. Second, the companies that have been

listed on LQ-45 index during the research period consistently and stocks are

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qualified active. This research will take companies that listed in LQ-45 index for 3

consecutive years and fulfilled the criteria that set by researcher.

The table below shows the list of companies that used as samples :

Code Company Mean Std. Dev Variance

1. AKRA AKR Corporation 5,37% 31,86% 10,2%

2. BSDE Bumi Serpong Damai 4,88% 31,84% 10,1%

3. JSMR Jasa Marga 0,39% 29,11% 8,5%

4. KLBF Kalbe Farma 6,03% 28,61% 8,2%

5. LPKR Lippo Karawaci 38,78% 33,94% 11,5%

6. LPPF Matahari Department

Store

24,30% 41,09% 16,9%

7. PGAS Perusahaan Gas Negara 18,39% 46,73% 21,8%

8. PTPP Perusahaan

Pembangunan

18,15% 36,60% 13,4%

9. WIKA Wijaya Karya 12,78% 33,77% 11,4%

Table 2 Listed of Sample Companies

Source : Processed by researcher (2018)

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Based on the criteria above, only thirty one companies that fulfilled the

criteria. In addition, researcher attempts to add additional requirement, which is

screening the stock selection based on MPT framework. Thus, this process

generate nine stocks that match with the all the criteria, see table (2). From the list

of LQ-45 shares taken from historical data, researcher has to calculate average

rate of return, standard deviation and variance from each of shares. Calculation of

returns and standard deviation are calculated using weekly observation from

closing price. Thus, researcher calculates the performance of the stock. The most

effective way of expressing investment returns is on an annual basis. For a weekly

investment return, the researcher will be presented in value per annum.

Annualized return is obtained by:

𝑅𝑦 = (𝑅𝑤 + 1)52 – 1. (4.1)

Where :

-𝑅𝑦 is the annual return

- 𝑅𝑤 is the weekly return.

For standard deviation of the average weekly return, multiply it by the square root

of the number of weeks in a year. so: 𝜎𝑤 × √52. We got the result of the average

annual return and standard deviation for the research considered periods (table

4.1)

4.1.1 Analysis of individual Return and Standard Deviation of Stocks

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Allocation for Institutional portfolios, postulated by (Kritzman, 1990),

analysis of investment return start from calculation of return for each asset. The

return of individual shares is the amount of real profits received by investors when

investing in stocks . Stock returns can be calculated by comparing the closing

price of the stock this week which is denoted by the week t minus the closing

price of the stock last week denoted by the week t-1 then divided by the closing

price of the stock week t-1. The calculation return of individual shares from nine

shares of the LQ-45 Index which are used as research samples, (can be seen in

full in appendix 3).

Processing data in this study using an average formula, where the

calculation is the cells that contain the weekly return of each share. As for the

calculation of the risk or standard deviation of each stock, it uses the formula

standard deviation where the calculation is the cells that contain weekly return

data during the analysis period. The larger standard deviation for an expected

return, the larger the dispersion of expected returns and the greater the risk of the

investment. Stock price can wildly fluctuate every day because of supply and

demand for various industries making the author difficult to estimate the

probability distribution of each stock. Therefore, for calculation expected return or

mean return, researcher assumed that the data has normal distribution of return,

which mean the denominator is the total data of return samples is 156 data

adjusting closing price per week on each stocks during the study period.

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Figure 5 Average annual return and standard deviation on each stock

Source : Processed by researcher (2018)

The figure shows it can be seen that shares that have the highest average

return are shares of Lippo Karawaci, Tbk (LPKR) with average returns of 38,78%

and risk of 33,94%, whereas, the lowest return is shares of Jasa Marga, Tbk

(JSMR) with average return of 0,39% and standard deviation of 29,11%. The

LPKR is the highest return among other stocks sampled. This is also estimated

that LPKR is able to provides the largest return in the constructed portfolio, but it

does not mean LPKR provides the highest risk too.

4.1.2 Analysis of Market Return

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Stock market index return provides a great outlook for the movement

stock price due to the volatility. It is calculated from the prices of several stocks as

guidance for investor and financial market to predict the condition of market and

also as a benchmark towards the return on particular stocks. To determine its price

usually an index used weighted average.

In this study, LQ-45 is an index to represent the stock market return. This

indicates that if the movement of stock market is positive and investor do the

passive strategy, it is more likely the return is equivalent to the index followed.

Therefore, for investor, the movement of stock market return is taking into

consideration. Market return is calculated by measuring the difference in the

Market Index (Rmt) current week with the previous week (Rmt-1) then divided

with the market index the previous week. The results of the calculation of market

return can be seen in the table 3

E(Rm) 0,087420988

𝝈m 0,178960976

Table 3 Analysis of market return and standard deviation Index LQ-45

Source : Processed by researcher (2018)

The table shows that average return market (Rm) has a positive value,

which is 0,087420988 (8,74%) per annum, standard deviation of 0,178960976

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( 17,90%), Risk free rate is 6,19% per annum. Aligned with the theoretical

concept of CAPM, market returns return can also be used as a basis for measuring

stock investment performance. If the market return is greater than the risk-free

rate of return, we conclude that investment performance is good.

4.1.3 Analysis of Risk-free rate

After calculating the return and risk each stock, the next step is calculation

of risk free rate, it can be defined as the return on investment with standard

deviation of zero (Hull, 2015). Generally, return and risk is two distinction

concept that correlate with each other. Therefore, as an investor have required to

be aware about the risk return trade off because the potential of return, the greater

probability of risk. But, this kind of asset is offering no loss because a couple of

reasons. First, the instruments is issued by government that has never defaulted on

its debt obligation. Second, this instrument has short term maturities in one year

or less, thereby this instruments is more safety, very liquid and practically no risk

attached such as interest rate risk.

In this study, we use BI 7 day repo rate (BI 7 RR) for analysis risk free

rate. We assume that Bank Indonesia certificates are type of investment without

risk for investor. BI rate 7 day repo rate is used as an interest rate for Bank

Indonesia Certificate ( SBI ) within a period of one year. The average of risk free

rate return for period August 2015 – August 2018 is 6,19% per year. From the

appendix A.4 shows that for period 31 July 2015 until 8 January 2016 is 7,5% per

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year. Then, for period 15 January 2016 until 12 February 2016 had decreased 25

bps from 7,5% become 7,25%. After that, for period 19 February 2016 until 11

March 2016 is 7,00%. Then, it had decreased 25 bps from 7,00% become 6,75%

for period 18 March 2016 until 15 April 2016. During the period above, Bank

Indonesia using BI rate as their benchmark interest rate.

Bank Indonesia introduced new policy rate which is BI 7 days repo rate

and effective from 19th august 2016 in order to strengthened monetary operations.

Therefore, starting period 19 August 2016 until the latest period 15 August 2018,

the range of BI 7 days repo rate, which is 4,25% until 5,50%. ( see appendix A.4)

4.1.4 Analysis of Covariance and Coefficient Correlation Matrix

From the discussion above, we first select nine stocks from the 45

component of stocks from Index-LQ45. Before we construct the efficient portfolio

for the case of nine stocks, it is necessary to calculate covariance and coefficient

correlation between the chosen individual assets because the portfolio risk will be

influenced by the relationship of the assets in the portfolio. However, the

covariance only determine whether the assets are positively related or inversely

related, while the coefficient correlation explained more profoundly about the

degree to which the assets move together. Covariance and coefficient correlation

between two assets are calculated with the statistical function covar and

correl .The result of covariance can be seen in the figure 6

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Figure 6 Covariance Matrix Result

Source : Processed by researcher (2018)

Covariance can be positive, negative, or zero value. Positive covariance value

means the tendency of two securities to move in the same direction, this indicates

that if stock returns rise then the market return will also rise and vice versa.

Covariance value of zero indicates that the movement of two securities is

independent of one another which shows stock returns and market returns do not

move towards the same or opposite.

From the covariance matrix (figure 6) in this study, the covariance of each

stock has a positive value, meaning that each of stocks are positively correlated to

each other and tend to move together in the same direction. By knowing

covariance and coefficient correlation among each of asset, so investors can find

out the composition of several assets to get an optimal portfolio with minimum

risk and maximum returns.

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Figure 7 Coefficient Correlation Matrix Result

Source : Processed by researcher (2018)

The next step is to calculate the correlation between each stock, the

researcher build [9x9] correlation matrix representing the correlation between the

nine stocks. The figure 7 shows that the correlation between each stock result has

positive value and weak correlation since the amounts is relatively small, that is

almost near to zero. It implies that each stock has small affects to one and another,

or in other word, the responses of the investor concerning to these stocks are not

impacted by one and another.

According to the result of coefficient correlation matrix, we did not obtain

the negative value. The biggest coefficient correlation value is shares between

PTPP and WIKA, which is 0,56 and for the coefficient correlation between shares

of LPKR and AKRA is relatively small with 0,08. Investor gain diversification

benefit of effective risk reduction when the portfolio consists of stocks whose

correlation is getting smaller or the same as negative one. In Modern Portfolio

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Theory, this is the character of coefficient correlation. So, between each stock is

independent and does not affected each other.

4.1.5 Analysis of Markowitz Portfolio

After we calculate the annual returns and standard deviation of individual

stock, the next step is calculating the expected return, covariance, and standard

deviation from each stock to construct Markowitz Portfolio. Forming the optimal

portfolio can be interpreted as set up the component of assets that produce the

most favorable expected return in any given level of risk. Therefore the expected

return and variance of portfolio are the areas that investor need to take into

consideration before make investing decisions.

The expected return can be defined as weighted average of expected

individual stock return. Asset allocation is the main concept to generate the

expected return of portfolio. We made numerous probability to produce the

optimum asset allocation ( five hundred in the example, Appendix 5). However,

we put some constraints for asset allocation, the total weight must also be equal to

1 and no negative or zero weightage. So, there would be no asset without fund

allocation. Then, to have the value of the expected return of portfolio, it simply by

multiplying the equally weighted asset with the average return each stock. (more

detail in the appendix A.5)

Portfolio Variance is to measure the risk of portfolio in any given level of

return. The role coefficient correlation and covariance are crucial to calculate the

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portfolio variance. The more portfolio’s return vary from the average portfolio

returns, the volatile of portfolio is greater. Portfolio that have combination of

small correlation among all assets will have minimum variance portfolio.

Therefore, we use this strategy to combine risky assets (stocks) and non-risky

asset (BI rate) to accomplish the principle of diversification.

The quadratic function is used to generate for five hundred feasible set of

portfolios, the function produces the expected return, variance and composition of

each asset. Each X-point produced with a combination of standard deviations and

expected returns is a point ((𝜎𝑥,𝐸(𝑅)𝑥) The appendix A.5 clearly explains the

result of portfolio return and variance.

Figure 8 Feasible Set of Portfolio (EFC model)

Source : Processed by researcher (2018)

𝐸(𝑅)

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Efficient Frontier Curve model is the best measurement to ensure these

combination of nine assets are presenting the optimal combination. From Figure

8 shows that only several set of portfolios whose succeeded in forming the

efficient frontier curve. The point in the graph represent the set of portfolio, thus

we only choose the point at the surface of the efficient frontier curve because

these represent the most optimal combination among the other portfolios. In this

study, we build five hundred the combination of asset, but we only select eight set

of portfolio from the total number of portfolio that we have constructed. The eight

set of portfolio can be seen from this table below.

Table 4 Portfolio Rank Based on The Expected Return

Source : Processed by researcher (2018)

The weight of asset is taking part to maximize the expected return of

portfolio. From table 4 shows that Portfolio 1 represent the portfolio with the

highest expected return, while Portfolio 6 provides the lowest risk. The result of

portfolio 1 and portfolio 6, we can concluded that the risk return trade-off still

remain in the optimal portfolio. It can likewise be seen that the more assets

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comprised in a portfolio, the more expanded the risk are, yet the lower expected

return on portfolio.

4.1.6 Analysis Passive Strategy Capital Allocation Line

Previously, in determining portfolio, all instrument used for build portfolio

is part of risky asset. If we put one kind of asset such as risk free asset e.g. Bank

Indonesia Certificate ( SBI – Surat Berharga Indonesia), then it will obtain the

new portfolio allocation. To find the optimal portfolio, we forming capital

allocation line by connecting the point on risk free asset with the point that lies on

the efficient frontier curve and determine the optimal portfolio, the straight line is

denoted by CAL(P).

The point on risk free asset (Rf) is represent the instrument with a

combination of standard deviation and expected return of an investment with zero

risk, it can be interpreted that the value of expected return and actual return are

likely to be about the same. Bank Indonesia Certificate is considered as the

instrument for risk free asset, we obtained the average of risk free rate return for

period August 2015 – August 2018 is 6,19% per year. As a result, for risk free

instruments, we get the points at coordinates (0;6,19%).

Therefore, CAL (P) is in the form of an equation of a straight line where

the capital allocation line is tangent to the efficient frontier curve. This point will

represents the optimal portfolio indicate this portfolio provides the highest returns

per additional unit of risk. In the figure below shows that the efficient frontier

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curve intersect with capital allocation line (CAL) at the third portfolio point

indicates the optimal portfolio. The optimal portfolio has an ERp (Expected Return

portfolio) of 18,2% with the 𝜎 ( Standard deviation) of 20,34% and Rf ( Risk Free

Rate) of 6,19%. Thus, the risk premium on the risky asset is ERp - Rf = 12,01%.

With a proportion, x, in the risky portfolio, and 1-x in the risk free asset,

the rate return on the complete portfolio, denoted ERc , where

ERc = x ERp + (1- x) Rf

= Rf + x [ ERp - Rf ] = 6,19 + x(18,2 – 6,19)

Because the standard deviation of the risky portfolio is 𝜎𝑝 = 20,34%

𝝈𝒄 = 𝒙𝝈𝒑 = 𝟐𝟎,𝟑𝟒𝒙

In sum, the rate of return of the complete portfolio will have expected value ERc =

6,19 + 12,01x and standard deviation 𝜎𝑐 = 20,34𝑥

However, in relation with the type of investment and allocation of investor’s fund,

there are some possible risk and excepted return that will be obtained in capital

allocation line, which are:

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1. For investor who invest all their funds into risk-free assets, then the yield

of optimal portfolio that investor obtained will equally with the point of

Rf.

2. For investor who invest all their funds into risky asset, then the yield and

risk level of optimal portfolio that investor obtained will equally with the

point of P.

3. For Investor who invest in some percentage of their fund on risk free asset

and (1-x)% of their funds on risky assets, then then the yield and risk level

of optimal portfolio that investor obtained will graph on the straight line

connecting points Rf and P.

Thus the expected return of the complete portfolio as a function of its standard

deviation is a straight line, with intercept Rf and slope.

S = ERp − Rf 𝜎𝑝

= 12,0120,34

= 0,59

The slope of the CAL, denoted S, equals the increase in the expected return of the

complete portfolio per unit of additional standard deviation.

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𝒇(𝒙) = 𝟏,𝟏𝟑 ∙ 𝟏𝟎𝟖𝒙𝟕 − 𝟏,𝟏𝟗 ∙ 𝟏𝟎𝟖𝒙𝟔 + 𝟓,𝟑𝟎 ∙ 𝟏𝟎𝟕𝒙𝟓 − 𝟏,𝟐𝟕 ∙ 𝟏𝟎𝟕𝒙𝟒 + 𝟏,𝟖𝟎 ∙ 𝟏𝟎𝟔𝒙𝟑 − 𝟏,𝟒𝟗

∙ 𝟏𝟎𝟓𝒙𝟐 − 𝟔𝟕𝟓𝟔,𝟖𝟕𝒙 + 𝟏𝟐𝟕,𝟐𝟕

Figure 9 Efficient Frontier and Capital Allocation Line

Source : Processed by researcher (2018)

4.1.7 Analysis of Efficient Frontier Curve model by using Polynomial

Equation

We make polynomial equation derived from the Efficient Frontier Curve model

using mathematical support as we see in the figure 10. From this figure, we can

state that the standard deviation is an independent variable and the expected return

𝐸(𝑅)

𝜎

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53

is an dependent variable. In mathematics, we generally found y = f(x) with x as

dependent variable and y as the independent variable. Therefore, we can construct

a curve with polynomial interpolation function (Friedman, 1994). The purpose of

interpolation is to figure out the value of missing data either within the range of

existing data or outside the experienced of data. We forming the polynomial

equation as we seen below and an expected return is denoted by x :

𝝈 = 𝒇(𝒙) = 𝟏,𝟏𝟑 ∙ 𝟏𝟎𝟖𝒙𝟕 − 𝟏,𝟏𝟗 ∙ 𝟏𝟎𝟖𝒙𝟔 + 𝟓,𝟑𝟎 ∙ 𝟏𝟎𝟕𝒙𝟓 − 𝟏,𝟐𝟕 ∙ 𝟏𝟎𝟕𝒙𝟒 + 𝟏,𝟖𝟎 ∙ 𝟏𝟎𝟔𝒙𝟑

− 𝟏,𝟒𝟗 ∙ 𝟏𝟎𝟓𝒙𝟐 − 𝟔𝟕𝟓𝟔,𝟖𝟕𝒙 + 𝟏𝟐𝟕,𝟐𝟕

Figure 10 Polynomial Curve and Equation

Source : Processed by researcher (2018)

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The polynomial equation generates the function that has degree of seven which

indicates the best fit of the series data point. From figure 10, we conclude that the

equation can be employed to predict the value because all the data used to form

the equation lies on red line. The limitation of the equation is we have to do the

long computation. This equation is practical to determine the value and forecast

the value of expected return of portfolio equal to the level of the risk.

Polynomial Interpolation : to determine the value of portfolio risk within

the range of 0,0829 ≤ x≤ 0,2335 . We obtained the range value based on the

numerous probability of the asset allocation. This indicates that the portfolio has

minimum potential to deliver the return is 0,0829 (8,29%) and the maximum

return is 0,2174 (21,74%).

Polynomial interpolation can be used in cases, for example:

- Suppose fund manager has required to estimate the potential of loss if

investors are seeking the return in the range of 10% until 15%.

𝝈 = 𝒇(𝒙) = 𝟏,𝟏𝟑 ∙ 𝟏𝟎𝟖𝒙𝟕 − 𝟏,𝟏𝟗 ∙ 𝟏𝟎𝟖𝒙𝟔 + 𝟓,𝟑𝟎 ∙ 𝟏𝟎𝟕𝒙𝟓 − 𝟏,𝟐𝟕 ∙ 𝟏𝟎𝟕𝒙𝟒 + 𝟏,𝟖𝟎 ∙ 𝟏𝟎𝟔𝒙𝟑

− 𝟏,𝟒𝟗 ∙ 𝟏𝟎𝟓𝒙𝟐 − 𝟔𝟕𝟓𝟔,𝟖𝟕𝒙 + 𝟏𝟐𝟕,𝟐𝟕

By substituting the value of x based on investor’s expected return , we can

obtained 𝝈 data using the polynomial equation as shown in the table 5:

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Actual data Forecast

Expected

Return (x)

Std.Dev (𝝈) Expected

Return (x)

Std.Dev (𝝈)

10,64% 20,42% 10,6% 19,73%

11,24% 20,64% 11,2% 20,63%

12,80% 20,87% 12,8% 23,83%

13,50% 20,60% 13,5% 24,12%

14,70% 22,45% 14,7% 22,49%

Table 5 Result of Polynomial Interpolation

Source : Processed by researcher (2018)

We can conclude that this polynomial interpolation can be used to estimate the

value of standard deviation because three of five experimental result have precise

results. In addition, investor will have the potential loss in the range of minimum

20,23% until maximum 20,96% , with the level of expected return around 10%

until 15% for period 2015 until 2018.

Polynomial extrapolation : to approximating the future value of expected

return within short term period by relying on the existing scenario data of asset

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allocation. The differences is x value is not among the range Min (0,0829) ≤ x≤

Max (0,2174).

Polynomial interpolation can be used in cases, for example:

- During this period, fund manager has achieved the best of level portfolio

return around 21,74%. Then, investors request to fund manager to generate

the return of portfolio around 22,5% and forecast the approximately value of

potential loss with given level of return in the next three years.

𝝈 = 𝒇(𝒙) = 𝟏,𝟏𝟑 ∙ 𝟏𝟎𝟖𝒙𝟕 − 𝟏,𝟏𝟗 ∙ 𝟏𝟎𝟖𝒙𝟔 + 𝟓,𝟑𝟎 ∙ 𝟏𝟎𝟕𝒙𝟓 − 𝟏,𝟐𝟕 ∙ 𝟏𝟎𝟕𝒙𝟒 + 𝟏,𝟖𝟎 ∙ 𝟏𝟎𝟔𝒙𝟑

− 𝟏,𝟒𝟗 ∙ 𝟏𝟎𝟓𝒙𝟐 − 𝟔𝟕𝟓𝟔,𝟖𝟕𝒙 + 𝟏𝟐𝟕,𝟐𝟕

By substituting the value of x based on investor’s expected return , we can

obtained 𝝈 data using the polynomial equation. If the fund manager build a

scenario in order to generate the return portfolio around 22,5%, then the potential

loss that investor have to compensate is around 33,01%.

4.2. Discussion and Interpretation Data

There are several empirical findings and result obtained through the

research of this study:

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First, by composing portfolio, it makes investor bear a small risk rather

than invest in a single stocks. It is the main reason why investor needs to do

understand diversification concept.

Second, in order to obtain the efficient portfolio, we apply portfolio

optimization method based on Markowitz Modern Portfolio Theory in which

selecting stocks that have minimum risk (standard deviation) and maximum

return, also considering non-correlated asset.

Third, find the optimal portfolio is part of investor’s goal but still

considering investor risk preferences. We construct the set of portfolio consisted

risky asset using efficient frontier curve model for investor risk taker, it means

they are willing to have higher return with higher risk. Then, continued by using

concept of capital allocation line, where the portfolio does not only contain risky

asset but also consist of risk free asset.

Fourth, to build an optimal portfolio, BI rate have been chosen as the

proxy for risk free investment rate and the benchmark rate that reflect the

macroeconomic condition in a country. In period 2015 -2018, the average BI rate

was relatively stable but low around 6% indicates that the research was conducted

in the relatively stable macroeconomic condition. Indonesian Bank use BI rate as

an instrument to control the level of inflation so that it will be consistent with the

inflation target established by government. Therefore, in case of the country fails

to meet economic stability, it will affect the BI rate and inflation, respectively. In

this study, one of the variable for constructing optimal portfolio is BI rate.

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Although, we find the optimal portfolio (expected return of 18,2% with the risk

level of 20,34%) from the set of efficient portfolio, it may change under these

circumstances.

Fifth, after we build the efficient frontier curve, we examined the model

and converted into mathematical function, namely polynomial equation. We

called this model as the modification of Efficient Frontier Markowitz. The

advantages using the model are fund manager can determine the value of potential

loss with a given level of expected return within certain period and fund manager

also can forecast the future value of potential loss with a given level of expected

return within next short term period. While the Markowitz portfolio model has

focused on constructing the feasible set of efficient portfolio, this model can be

used to measure how much the rate of change in expected return can affect the

rate of change in portfolio risk.

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CHAPTER V

CONCLUSION

5.1. Conclusion

The concept of asset allocation plays a key role in portfolio management.

Through clear understanding of asset allocation, investor can develop appropriate

investment strategy in selecting several investment assets into portfolio with the

purpose of seeking profit and limiting the odds of devastating loss. It has

meaningful contribution in term of literal study.

The main goal of this research is to implement the asset allocation using

Markowitz portfolio model and whether there should be any modification that can

be done to increase the accuracy and relevancy to use in the practical world. We

used the Markowitz Efficient Frontier model consist of risky asset that

represented by common stocks. The common stock have been chosen due to the

fact that common stocks are more desirable investments than other kind of assets.

Aligned with the concept of diversification. The result of coefficient correlation

and covariance among these assets has small value indicates that the assets are

independent and does not affect one and another.

In order to construct the optimal portfolio, the alteration in weight

allocation of each asset in portfolio have considered in this study. We believe that

the numerous probability that we had would produce the optimum asset

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allocation. From the result of efficient frontier curve, we have eight set of

portfolio that precisely lies on the surface of efficient frontier indicates the

efficient portfolio offers the highest expected return for a given level of risk, or

one with the lowest level of risk for a given expected return. For the portfolios

that lies below efficient frontier model is an in-efficient portfolio because they

don’t provide enough return for given level of risk.

However, the EFC model is not good enough for investor categorized as

risk-aversion who have tendency to avoid risky asset. Therefore, we have to find

the optimal one from all the set of the optimal portfolio. By using the Capital

Allocation Model, the optimal one is known as the tangency portfolio, it is an

intercept point of efficient frontier and capital allocation line. The CAL model is

required us to put one kind of non-risky asset. In this study, The optimal portfolio

has expected return of 18,2% and the risk level of 20,34%. After that, we examine

the performance of portfolio by using Shape ratio.

Moreover, we also presented the model derived from Markowitz curve

that converted into mathematical function as polynomial equation. We called this

model as the modification of Efficient Frontier Markowitz. The advantages using

the model are fund manager can determine the value of potential loss with a given

level of expected return within certain period and fund manager also can forecast

the future value of potential loss with a given level of expected return within next

short term period. While the Markowitz portfolio model has focused on construct

the efficient portfolio, this model can be used to measure how much the rate of

change in expected return can affect the rate of change in portfolio risk.

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Therefore, the modification of efficient frontier Markowitz is adding the

relevancy in the practical world.

We also conclude that applying quantitative approach provide more

robust portfolios in terms of return and relative to the risk incurred than

qualitative approach because these processes involves use of mathematical models

that aiding investors to achieve more efficient and appropriate investment

outcomes.

5.2. Limitation and Recommendation

In this study, the researcher meets some limitations about the time and

information gathered. First, the research period for observation is quite short, so it

cannot show the all the possibilities that would happen in case of longer period.

The limitation in the number of stocks selected in portfolio due to the short time

period. Second, we limit the data sample on purpose to period of stable economic

condition in order to avoid unprecedented interventions related to the frequency of

the data. If the conditions allowed, the result generating from doing stock listed on

Indonesia stock exchange from various industries and 260 historical periods

instead of stocks listed on LQ-45 index and 156 historical period will be more

convinced. Third, the methodology is not intended to fully capture about the

systematic risk that influence on stock price leading to the formation of optimal

portfolio. Fourth, the formulation has an inherent instability once the mean and

variance are replaced by their sample counterparts.

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According to the result of calculation and conclusions, some suggestion

are needed for future research:

- For further research, we suggest using another kind of asset that

represented of market such as instrument of future, options, bonds and

longer time intervals, so the result of the study are more accurate.

- Another suggestion is determine the accuracy of the Polynomial

interpolation in estimating the data in comparison to other models to

find out which produce better results.

5.3. Implications

This research contains the information of stocks performance listed on

LQ-45 index for period 2015 – 2018. This research is expected to be used as

consideration for investor and market participants to invest in financial asset

through the diversification strategy based on the model has explained in this

research. The stocks listed in LQ-45 can be selected as an alternative investment

option. The result of this study also test of Markowitz modern portfolio theory to

individual stock performance and portfolio performance with the result that this

research not only determines the set of efficient portfolio, but also select the

optimal portfolio of the efficient portfolio and modified the Markowitz model of

modern portfolio theory.

In addition, Investor should pay attention regarding the level of risk

preferences such as risk averse, risk neutral and risk seeker. This research also

stated the possible proportion of funds to invest through these stocks in order to

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achieve the optimal portfolio. For the modification of Markowitz Efficient

Frontier model, the model offers the advantages for fund manager in the real life

applications. So, investor can find out the percentage of stock funds that will be

invested.

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APPENDICES

LIST OF FIGURE

Figure 1 Covariance and Coefficient Correlation Matrix

Figure 2 Portfolio Risk and Level of diversification

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Figure 3 Efficient Frontier

Figure 4 Capital Allocation Line

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Figure 5 Average annual return and standard deviation on each stocks

Figure 6 Covariance Matrix Result

Figure 7 Coefficient Correlation matrix

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Figure 8 Feasible of Portfolio ( Efficient Frontier Curve Model )

Figure 9 Efficient Frontier and Capital Allocation Line

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Figure 10 The Polynomial Curve

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LIST OF TABLES

Table 1 Representative Sample Chosen

Table 2 Listed of Sample Companies ( Return and Risk )

Code Company Mean Std. Dev Variance

10. AKRA AKR Corporation 5,37% 31,86% 10,2%

11. BSDE Bumi Serpong Damai 4,88% 31,84% 10,1%

12. JSMR Jasa Marga 0,39% 29,11% 8,5%

13. KLBF Kalbe Farma 6,03% 28,61% 8,2%

Code Company

10. AKRA AKR Corporation, Tbk.

11. BSDE Bumi Serpong Damai, Tbk.

12. JSMR Jasa Marga ( Persero ), Tbk.

13. KLBF Kalbe Farma, Tbk.

14. LPKR Lippo Karawaci, Tbk.

15. LPPF Matahari Department Store, Tbk.

16. PGAS Perusahaan Gas Negara ( Persero), Tbk.

17. PTPP PP ( Persero ) Tbk

18. WIKA Wijaya Karya ( Persero ) Tbk

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14. LPKR Lippo Karawaci 38,78% 33,94% 11,5%

15. LPPF Matahari Department

Store

24,30% 41,09% 16,9%

16. PGAS Perusahaan Gas Negara 18,39% 46,73% 21,8%

17. PTPP Perusahaan

Pembangunan

18,15% 36,60% 13,4%

18. WIKA Wijaya Karya 12,78% 33,77% 11,4%

Table 3 Analysis of market return and standard deviation Index LQ-45

E(Rm) 0,087420988

𝝈m 0,178960976

Table 4 Portfolio Rank Based on The Expected Return

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Table 5 Result of Polynomial Interpolation

Expected Return (x) Standard Deviation (𝝈)

10,6% 19,73%

11,2% 20,63%

12,8% 23,83%

13,5% 24,12%

14,7% 22,49%

Appendix 1 Indonesia Quarterly GDP Growth for period 2009 until 2018

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Appendix 2 Company have been listed on LQ-45 index consistently for period 2015 – 2018

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Appendix 3 Historical Data Stock Return LQ-45 for Research Sample

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Appendix 4 BI 7 days RR as Risk Free Rate

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Appendix 5 Feasible set of Markowitz Portfolio

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