Analysis of portfolio optimization with and without shortselling based on diagonal model: evidence from indonesian stock market

ANALYSIS OF PORTFOLIO OPTIMIZATION WITH AND
WITHOUT SHORTSELLING BASED ON DIAGONAL MODEL:
EVIDENCE FROM INDONESIAN STOCK MARKET

KALEEM SALEEM

GRADUATE SCHOOL
BOGOR AGRICULTURAL UNIVERSITY
BOGOR
2013

DECLARATION
I do hereby declare that the thesis entitled "ANALYSIS OF PORTFOLIO
OPTIMIZATION WITH AND WITHOUT SHORTSELLING BASED ON
DIAGONAL MODEL: EVIDENCE FROM INDONESIAN STOCK MARKET" is
my original work produced through the guidance of my academic advisors and that to
the best of my knowledge it has not been presented for the award of any degree in any
educational institution. All of the incorporated material originated from other
published as well as unpublished papers are clearly stated in the text as well as in the
references.
Hereby I delegate the copy rights of this work to the Bogor Agricultural University.

Bogor, October 2013
Kaleem Saleem
NRP H251118101

SUMMARY
KALEEM SALEEM. Analysis of Portfolio Optimization with and without
Shortselling based on Diagonal Model: Evidence from Indonesian Stock Market.
Supervised by ABDUL KOHAR IRWANTO dan ENDAR HASAFAH
NUGRAHANI.
Markowitz (1952, 1959) and Roy (1952) proposed portfolio theory which narrate
that risk of a portfolio is the variance of individual securities and covariances among
those securities comprising the portfolio. The objectives of this research is to construct
an optimized portfolio using Sharpe's diagonal model with and without short selling
respectively, and to analyze the performance of less diversified optimal portfolios by
investing in a single industry versus well diversified portfolio by investing across all
industries. Finally to analysis the statistical properties of the estimates of diagonal
model.
Markowitz mean-variance portfolio optimization theory is implemented for all
stocks listed in 9 sectors of Indonesian stock market during 2007-2011. However our
model used 269 stocks which are not contradicting to the basic assumptions of the

diagonal model. Diagonal model used in this research is a linear model and data used
are time series secondary data. Firstly 9 optimal portfolios are computed for individual
sectors with the assumption that short selling not permitted and then computation is
done with assumption that short selling is permitted. Similarly 2 optimal portfolios are
computed consisting of all stocks with assumptions mentioned above.
Results showed that both well diversified short and long portfolios provide higher
returns at a specified risk level compare to portfolios consist of individual sectors
stocks. Largest weight in short portfolio is from infrastructure, utilities and
transportation sector equaling 17.2 percent. Largest weight in long only portfolio are
from trade, investment and services sector equaling 23.2 percent. The more diversified
and larger portfolios provide better tangency portfolio. All stocks short and long
portfolios performed better than any other individual sector portfolio. Standard error
for β and residual standard error for other portfolios are large compare to individual
stocks indicating risk was not diversified away. Least square parameters estimates
were not quite promising such as value of portfolio beta happened to be less than
individual stocks as well as value of its coefficient of determination shows most of the
systematic risk could not get eliminated as low values of R2 for portfolios shows that
most of the variance in stock return as well as in portfolio return is not explained by
market variance therefore unsystematic risk could not get eliminated despite of
forming a portfolio encompassing all equities listed in Indonesian Stock Exchange as

suggested by results of regression analysis.
Keywords: Diagonal Model, Diversification, Portfolio Optimization, Short Sales

RINGKASAN
KALEEM SALEEM. Analisis Optimisasi Portofolio dengan dan tanpa Shortselling
bardasarkan Diagonal Model: Bukti dari Pasar Saham Indonesia. Dibimbing oleh
ABDUL KOHAR IRWANTO dan ENDAR HASAFAH NUGRAHANI.
Markowitz (1952 , 1959) dan Roy (1952) mengusulkan teori portofolio yang
menceritakan bahwa risiko portofolio adalah varians dari sekuritas individual dan
covariances antara mereka. Tujuan dari penelitian ini adalah untuk membangun
sebuah portofolio optimal menggunakan model diagonal Sharpe dengan and tanpa
short selling berturut-turut, dan untuk menganalisis kinerja portofolio optimal yang
kurang terdiversifikasi dengan berinvestasi di portfolio industri individu dibandingkan
portofolio yang terdiversifikasi dengan berinvestasi di semua industri. Akhirnya
menganalisis sifat statistik perkiraan model diagonal .
Markowitz teori mean-variance diterapkan untuk semua saham yang tercatat di 9
sektor pasar saham Indonesia selama 2007-2011. Namun model kami menggunakan
269 saham yang tidak bertentangan dengan asumsi dasar dari model diagonal. Model
diagonal digunakan dalam penelitian ini adalah model linier dan data yang digunakan
adalah data time series sekunder. Pertama 9 portofolio optimal dihitung untuk

masing-masing sektor dengan asumsi bahwa short selling tidak diperbolehkan dan
kemudian dengan asumsi bahwa short selling diperbolehkan. Demikian 2 portofolio
optimal dihitung terdiri dari semua saham dengan asumsi tersebut di atas.
Hasil penelitian menunjukkan bahwa kedua terdiversifikasi portofolio short dan
long memberikan hasil yang lebih tinggi pada tingkat risiko tertentu dibandingkan
dengan portofolio terdiri dari saktor saham individu. Bobot terbesar di portofolio short
adalah dari infrastruktur , utilitas dan sektor transportasi setara 17,2 persen. Bobot
terbesar di portofolio long adalah dari sektor perdagangan, investasi dan jasa setara
23,2 persen. Portofolio lebih beragam dan lebih besar menyediakan portofolio
singgung yang lebih baik. Semua saham portofolio short dan long dilakukan lebih
baik daripada portofolio sektor individu lain. Standar error untuk β dan residual
standar error untuk portofolio lainnya besar dibandingkan dengan saham individu
yang menunjukkan risiko tidak didiversifikasi. Least square perkiraan parameter tidak
cukup menjanjikan seperti nilai portofolio β kebetulan kurang dari saham individu
serta nilai koefisien determinasi menunjukkan sebagian besar risiko sistematis tidak
bisa mendapatkan dieliminasi sebagai nilai-nilai rendah R2 untuk portofolio
menunjukkan bahwa sebagian besar dari varians dalam return saham serta return
portofolio tidak dijelaskan oleh varians pasar sehingga risiko tidak sistematis tidak
bisa mendapatkan dihilangkan meskipun membentuk portofolio meliputi semua
saham yang terdaftar di Bursa Efek Indonesia seperti yang disarankan oleh hasil

analisis regresi.
Kata kunci : Model Diagonal , Diversifikasi , Optimasi Portofolio , Shortselling

Copyright © 2013, Bogor Agricultural University.
All rights reserved.
No part or all of this work may be excerpted without inclusion or mentioning the
sources. Exception only for research and education use, writing for scientific papers,
reporting, critical writing or reviewing of a problem; and this exception does not
inflict a financial loss in the proper interest of Bogor Agricultural University.
No part or this entire work may be transmitted and reproduced in any form without
written permission from Bogor Agricultural University.

ANALYSIS OF PORTFOLIO OPTIMIZATION WITH AND
WITHOUT SHORTSELLING BASED ON DIAGONAL MODEL:
EVIDENCE FROM INDONESIAN STOCK MARKET

KALEEM SALEEM
A Thesis submitted in partial fulfillment of the requirements for the award of the degree
Master of Science
in

Management

GRADUATE SCHOOL
BOGOR AGRICULTURAL UNIVERSITY
BOGOR
2013

External Examiner: Dr. Muhammad Najib, S.TP., MM

-

Title
セ。ュ・@

:.mP

: Analysis of Portfolio Optimization with and without Shortselling based
on Diagonal Model: Evidence from Indonesian Stock Market
: Kaleem Saleem
: H251118101


Approved by
Advisory Committee

Dr Ir Abdul Kohar Irwanto, MSc
Supervisor

Dr Ir Endar Hasafah Nugrahani, MS
Co-supervisor

Agreed by

Program Coordinator
Management Science

Dr Ir Abdul Kohar Irwanto, MSc

Examination Date: 29 July 2013

Dr Ir Dahrul Syah, MScAgr


Submission Date:

2 3 0CT 20 13

Title

: Analysis of Portfolio Optimization with and without Shortselling based
on Diagonal Model: Evidence from Indonesian Stock Market
Name : Kaleem Saleem
NRP
: H251118101

Approved by
Advisory Committee

Dr Ir Abdul Kohar Irwanto, MSc
.
Supervisor


Dr Ir Endar Hasafah Nugrahani, MS
Co-supervisor

Agreed by
Program Coordinator
Management Science

Dean of Graduate School

Dr Ir Abdul Kohar Irwanto, MSc

Dr Ir Dahrul Syah, MScAgr

Examination Date: 29 July 2013

Submission Date:

FOREWORD
I would borrow few sentences to praise God Almighty Allah SWT, and Our Lord
Mohammad SAW from Richard Burton's translation of 'The Arabian Nights', "Praise

be to Allah, The Beneficient King, The Creator of The Universe, Lord of the Three
Worlds, who set up the firmament without pillars in its stead, and who stretched out the
earth even as a bed, and grace and prayer-blessing be upon our Lord Mohammed, Lord
of Apostolic Men, and upon His family and companion".
I would like to express my deepest gratitude and thanks to the Indonesian
taxpayers for financing my postgraduate studies at Bogor Agricultural University in
Indonesia and was deeply honored to receive the Indonesian Government Postgraduate
Scholarship for Developing Countries which enabled me to pursue advance studies in
management science. This research is also made possible by funding from Bureau of
Planning and International Cooperation, Ministry of National Education of The
Republic of Indonesia and author is extremely grateful to the ministry for this
generosity.
This work could not be made possible without the supervision, support and
constant guidance of my supervisors Dr Ir Abdul Kohar Irwanto MSc and Dr Ir Endar
Hasafah Nugrahani MS. I am highly grateful to them for this kindness. I am also
thankful to Dr Mukhlis Ansori MSc for his lessons of Bahasa Indonesia. Without the
firm support and prayers of my beloved parents Dr Muhammad Saleem Baloch and
Zubaidah Baloch and my beloved family Parigul, Shahgulee, Fauziah, Faheem,
Saimah, Waseem and Hassam this milestone could never be accomplished. I am also
grateful to my classmates Jonathan, Roto, Didu, Budi, Mita, Nunung, Hamadani,

Ajen, Dani, Yanti, Jay, Dewi, Arfan and to all of them whose names are not mentioned
here. Finally I am extremely indebted to Jazirotul and Adek Aruna, Dr Adil,
Wahidullah, Dr Walter, Petlane, Princy, Constantine, Hagim, Hirmawan, Ujang,
Pungki, Pak Soleh Hidayat, Gani, Sinath, Faye, Andrew, Chris, Fredrik, Aziz, Sari,
Isma, Viladmir, Musa and whole KNB family for their enormous support during all
these years in Indonesia. Last but not the least I am extremely thankful to Indonesia
itself, I have experienced living in one of the most beautiful places on planet, with
surprisingly wonderful and friendly people. Indonesia is no doubt a piece of paradise
on earth.
Bogor, October 2013
Kaleem Saleem

TABLE OF CONTENTS

TABLE OF CONTENTS

vii

LIST OF FIGURES

x

LIST OF TABLES

xii

LIST OF APPENDICES
1
INTRODUCTION
1.1
Background
1.2
Problem Formulation
1.3
Research Objectives
1.4
Research hypothesis
1.5
Research Benefit
2
LITERATURE REVIEW
2.1
Emergence of Modern Portfolio Theory
2.2
Overview of Indonesian Stock Market
2.3
Choices Under Certainty
2.4
Mean and Variance
2.5
Mean, Variance and Covariance of Portfolio
2.6
Diversification
2.7
Portfolio Possibilities Curve
2.8
One Fund Theorem
2.9
Portfolio Optimization with Quadratic Programming
2.10 Single Index Model
2.11 Previous Researches
3
METHODOLOGY
3.1
Research Framework
3.2
Diagonal Model
3.3
Research Method
3.4
Research Limitations
3.5
Data
3.6
Portfolio Optimization with R
4
RESULTS & DISCUSSION
4.1
Agriculture Sector
4.2
Mining Sector Portfolio
4.3
Basic Industry & Chemicals Sector Portfolio
4.4
Miscellaneous Industry Sector Portfolio
4.5
Consumer Goods Industry Portfolio
4.6
Property, Real Estate and Building Construction Portfolio
4.7
Infrastructure, Utilities and Transportation Portfolio
4.8
Finance Portfolio

xiv
1
1
2
2
3
3
4
4
5
6
7
8
9
10
13
14
17
20
23
23
23
24
25
26
26
29
29
33
37
41
45
49
53
57

5
6

4.9
Trade, Services & Investment Portfolio
4.10 All Stocks Short and Long Portfolios
CONCLUSION
RECOMMENDATIONS

60
64
73
73

REFERENCES

74

APPENDICES

78

LIST OF FIGURES

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33

Consumption in period 1 and period 2 for a single asset with certain
outcomes
Consumptions in period 1 and 2 for multiple assets with certain outcomes
Perfectly positive covariance between asset A and asset B
Perfectly negative covariance between asset A and asset B
Zero covariance between asset A and asset B
Combination of riskless asset with risky assets

< 0 at maximum feasible value of θ
Maximum value of θ when dX
i
when X i = 0

Maximum value of θ when dX
= 0 at maximum feasible value of θ
i
when X i = 0
β is shown by the slope of line
Return Plot during 2007-2011
Efficient Frontier Optimal Portfolio Agriculture Sector
Performance Chart Agriculture Portfolio
Efficient Frontier Optimal Portfolio Mining Sector Sector
Performance Chart Mining Portfolios
Efficient Frontier Optimal Portfolio Basic Industry & Chemicals Sector
Performance Chart Basic Industry & Chemicals Portfolios
Efficient Frontier Optimal Portfolio Miscellaneous Industry Sector
Performance Chart Miscellaneous Industry Portfolios
Efficient Frontier Optimal Portfolio Consumer Goods Industry Sector
Performance Chart Consumer Goods Portfolios
Efficient Frontier Optimal Portfolio Property, Real Estate & Building
Construction Industry Sector
Performance Chart Property, Real Estate & Building Construction
Portfolios
Efficient Frontier Optimal Portfolio Infrastructure, Utilities and
Transportation Sector
Performance Chart Infrastructure, Utilities and Transportation Short &
Long Portfolios
Efficient Frontier Optimal Portfolio Finance Sector
Performance Chart Finance Short & Long Portfolios
Efficient Frontier Optimal Portfolio Trade, Services & Investment Sector
Performance Chart Trade, Services & Investment Short & Long Portfolios
Efficient Frontier Optimal Portfolio All Stocks
Performance Chart All Stocks Short Portfolio
Performance Chart All Stocks Long Portfolio
Circular Map of Optimal Portfolio Allocation with Short Selling
Circular Map of Optimal Portfolio Allocation with no Short Selling

6
7
10
11
12
14
16
17
19
29
30
33
34
37
38
41
42
45
46
48
49
52
53
56
57
60
61
63
65
67
68
69
70

LIST OF TABLES

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

Highlights of Equities 2007-2011 in IDX
Prices of JCI 2007-2011 in IDX
Sectors Number of Stocks consistently Listed 2007-2011
Regression Agriculture Long Portfolio against Market Return
Regression Agriculture Short Portfolio against Market Return
Regression Mining Short Portfolio
Regression Long Mining Portfolio against Market
Regression Basic Industry & Chemicals Long Portfolio against Market
Regression Basic Industry & Chemicals Short Portfolio against Market
Regression Miscellaneous Industries Long Portfolio against Market
Regression Miscellaneous Industry Short Portfolio against Market
Regression Consumer Goods Industry Long Portfolio against Market
Regression Consumer Goods Industry Short Portfolio against Market
Regression Property, Real Estate & Building Construction Long
Portfolio against Market
Regression Property, Real Estate & Building Construction Short
Portfolio against Market
Regression Infrastructure, Utilities and Transportation Long Portfolio
against Market
Regression Infrastructure, Utilities and Transportation Short Portfolio
against Market
Regression Finance Long Portfolio against Market
Regression Finance Short Portfolio against Market
Regression Trade, Services & Investment Long Portfolio against Market
Regression Trade, Services & Investment Short Portfolio against Market
Regression All Stocks Long Portfolio
Regression All Stocks Short Portfolio
Performance Measurement each Sector Portfolio & Portfolio of All Stocks

1
1
26
31
32
35
36
39
40
43
44
47
48
50
51
54
55
58
59
62
63
66
67
71

LIST OF APPENDICES
1
2
3
4

Parameter Estimates for individual Stocks and for Portfolios
Return, Risk, Short & Long Weights of Each Stock
Return & Risk of Each Portfolio and Market
Descriptive Statistics

81
88
94
95

1

INTRODUCTION

Economics defined by Jim Tobin in one word as 'incentives'(Aumann 2006). The
incentive for investors to invest in stocks rather than in risk free assets like government
treasury bills is the relatively higher returns gain from stocks. Indonesian stock market
has played a significant role during post 1998 crisis period as it emerged as a vital and
efficient source of international as well as domestic funds inflow for businesses, and
most importantly equity financing enabled business to be less worried about
repayment to investors. In addition to this a significant portion of funds from pension
funds, insurance companies and other institutional and individual investors are
invested in stock market therefore its performance does have wide consequences (Suta
2000).
During period of 2007-2011 number of listed companies increased by 14.88%,
meanwhile 94.83% increase occured in the number of listed shares (Tabel 1). Total
capitalization increased by 77.90% in the same period (IDX Fact Book 2012).
Table 1 Highlights of Equities 2007-2011 in IDX
Period
2007
2008
2009
2010
2011
Listed Companies
383
396
398
420
440
Listed Shares (Million Shares)
1,128,174 1,374,412 1,465,655 1,894,828 2,198,133
Market Capitalization (Rp Billion) 1,988,326 1,076,491 2,019,375 3,247,097 3,537,294
Source:

IDX Fact Book 2012

High, low and closing prices of Jakarta Composite Index experienced an steady
increase during 2007-2011 period except in the year of 2008 when closing price
decreased by 49.36% compare to the price of 2007 (Tabel 2). This decrease was
caused by financial crisis that hit USA in the later half of the 2008. In 2011 Jakarta
Composite Index grew by 3.20% which was the second-best figure in Southeast Asia
(IDX Fact Book 2012).
Table 2 Prices of JCI 2007-2011 in IDX
Jakarta Composite Index
High
Low
Close
Source:

2007
2,810.96
1,678.04
2,745.83

2008
2,830.26
1,111.39
1,355.41

2009
2,534.36
1,256.11
2,534.36

2010
3,786.10
2,475.57
3,703.51

2011
4,193.44
3,269.45
3,821.99

IDX Fact Book 2012

Background
According to Dimson et al. (1999) from the time of Bernoulli (1700-1782) it was
known that individuals favour maximizing their wealth and minimizing the risk when

2
pursuing any potential gain.
However before 1952 there did not exist any
mathematical treatment regarding distribution of funds among securities in order to
obtain a diversified optimal portfolio. von Neumann (1953) narrated that "in
economic theory certain results....may be known already. Yet it is of interest to derive
them again from an exact theory". The problem of mathematical formulation of a
theory for portfolio risk and return was solved by Markowtiz (1952) and Roy (1952).
Markowitz narrates that "portfolio with maximum expected return is not
necessarily the one with minimum variance. There is a rate at which the investor can
gain expected return by taking on variance, or reduce variance by giving up expected
return". Markowtiz argued for what he called not only diversification but 'right kind'of
diversification for the 'right reason'such that investors should diversify across
industries and should only invest in securities having low covariances with each other.
Since 1952 the most part of research in investment theory has focused on how to
implement the Markowitz portfolio theory in order to obtain better estimates of risk
and returns. Markowitz proposed critical line method for its solution and for any other
quadratic programming problem. Two main differences between Markowitz and Roy
are that Markowitz impose a non-negativity constraint on the amount of investment
while Roy imposed no such constraint and secondly Markowitz allowed investor to
select any portfolio that lies on efficient frontier while Roy specify a limit where
investor should not select any portfolio beyond a certain level (Markowitz 1999).
Sharpe (1963) diagonal model made it computationally convenient to implement
portfolio theory by assuming "that single index model adequately describes the
variance-covariance structure" (Elton et al 1976). Further attempts to simplify
implementation of portfolio theory were made by Elton et al. (1976). Single index
model paved the way for development of capital asset pricing model theory (CAPM)
where Sharpe (1964) "construct a market equilibrium theory under conditions of
risk". There also exist critical literature regarding financial theories such as McGoun
(2003) who declares financial economics a failure, it is a science which lacks positive
models to describes the phenomenon of financial markets. Keasey et al. (2007)
criticized that "finance keeps itself artificially alive by taking data from the outside
world, often ignoring the rich complexities of the context which has given rise to the
data".

Problem Formulation
1. Portfolio optimization problem confronted by investors when selecting
securities in stock market using Sharpe's diagonal model.
2. Performance of less diversified optimal portfolios by investing in a single
industry versus well diversified portfolio by investing across all industries.
3. Precision and accuracy of the expected return and risk estimates.

3

Research Objectives
The objectives of this research are
1. To construct an optimized portfolio using Sharpe's diagonal model with short
selling permitted and with short selling not permitted respectively.
2. To analyze the performance of less diversified optimal portfolios by investing in
a single industry versus well diversified portfolio by investing across all
industries.
3. To analysis the statistical properties of the estimates of diagonal model.

Research hypothesis
1. Portfolio β equal market β or closer to market β than any individual stock.
2. Standard Error for portfolio β is smaller than individual stocks.
3. Increase in the number of securities would make error term irrelevant.
4. Variance in portfolio return is explained by market variance.
5. Portfolio return is maximum for any given level of risk or portfolio risk is
minimum for any given of return than any individual stock.

Research Benefit
The research determined optimal portfolio of each single sector in Bursa Efek
Indonesia (Indonesian Stock Exchange, BEI), as well as an optimal portfolio
comprised of all stocks in BEI. It would assist to identify means and variances of each
stock as well as of sectoral portfolios and of all stocks portfolios. Furthermore it help
investors to decide whether they should diversify across all industries or should invest
only in a particular industry based on past observations. This research also facilitate
decision makers to choose the right portfolio by providing an evaluation of portfolios
performance employing various performance measures.

2

LITERATURE REVIEW

'Treviso Arithmetic'(1478) considered to be the foremost business textbook put in
writing in Europe to help young merchants for their mercantile trade (Miller 2001).
Later on the Smith's 'Wealth of Nations'(1776) paved the way for advancement in
economic theory however the financial economics could not catch up with the
academic growth of economics (Miller 2001). Modern finance traces its roots from the
theory of interest, which have been in practice throughout the known history despite
being opposed from religious authorities. The Jews in the medieval Europe adapted a
literal meaning of biblical scripture and earned interest from the principle amount lend
to the people outside family. Early practices to make fortune from security pricing
goes back to Swiss bankers who formed an investment trust named Trente Demoiselles
de Geneve where they pooled tontines (French government debt instruments paid
annuity till death of bondholder) of 30 young women from wealthy families. Bonds
prices were undervalued for younger ladies of affluent background due to lower risk of
death compare to risk on the tontines making this anomaly exploitable for investment
trust (Miller 2001).
From this brief historical review we came to know that practices of reducing risk
through diversification and hedging existed long before the construction of modern
financial theories in the post second world war era. The contributions of modern
financial theory is that it attempted to quantify the ideas of hedging and diversification.

Emergence of Modern Portfolio Theory
The problem of portfolio optimization was formally formulated and solved by
Markowtiz (1952 and 1959). Solution of optimization problem was facilitated due to
the development of operation research during Second World War in order to solve
complex military problems. In the year 2006 it came into knowledge of English
speaking world that de Finetti, an Italian mathematician, in 1940 proposed mean
variance solution to solve the problem of reinsurance. However the problem
Markowitz tried to solve was related to investment while di Finetti was dealt the
problem of insurance and apart from that there is no evidence that Markowitz or
anyone even in Italy ever knew about the work of de Finetti (Bernstein 2007).
Perold (2004) has mentioned that even though the stock and option markets
existed since 1602 and insurance markets since 1700s the understanding regarding risk
was too little before 1960s. Risk as a concept based on the theory of probability. The
concept of probability have existed long before the formation of stock markets.
Gambling has been practiced throughout the documented history of human being.
First systematic work on probability, Liber de Ludo Aleae (The Book of Games of
Chance), appeared in 1663 in Basle first written by Girolamo Cardano in 1525 and
revised in 1565 which paved the way for the concept of law of large numbers and he
also defined how to express probability as a fraction (Bernstein 1995). However the
theory of probability had contributed two of the most widely applied concepts in

5
almost every science during 19th century, the idea of normal distribution and the idea
of regression which were developed by Gauss and Galton during 1820s and 1880s
respectively (Bernstein 1995). The advances in the discovery of probability theory
raised the confidence that future is predictable and thereby controllable, but the First
World War shacked this belief and Keynes in his 1921 book on Treatise on Probability
stated that most of our positive actions can only be taken as a result of animal spirits...
and not as the outcome of a weighted average of quantitative benefits multiplied by
quantitative probabilities (Bernstein 1995).
The 1952 article on portfolio selection by 25 years old Markowitz considered as
the birth of modern financial theory. Markowitz (1999) later generously regarded Roy
(1952) too as the father of portfolio theory along himself, but he narrated that Roy's
disappearance from academic publishing might be the reason he was not chosen for
1990 noble prize in economics. However this led to the emergence of a series of
theories which laid the foundation of modern finance as a distinct academic field.
Since 1952 many attempts has been made to implement the Markowitz portfolio
theory. We will briefly highlight few researches conducted using data from Indonesian
stock market:

Overview of Indonesian Stock Market
Indonesia along China and India were the only countries in G20 economies that
showed economic growth during financial crisis in 2009. It was the 16th largest
economy in 2012 on the basis of gross domestic product (The World Bank 2011).
Stock market is considered an important variable to determine long term growth of
Indonesia (Cooray, 2010). The stock market is relatively new in modern Indonesia,
even though the Dutch colonial empire established a stock exchange at Batavia, the
present day Jakarta, in 1912 and it continued to operate, except during period of World
War I and World War II, till the formal abolishment of the Dutch rule at the end of first
half of 20st century. After the independence due to the policy of nationalization the
only product traded at stock exchange was the government bond called Surat Utang
Negara (SUN).
On July 10, 1977 the PT Semen Cibinong became the first go public company and
it was the rebirth of the stock exchange, however for the next 10 years only 24
companies were listed at Jakarta Stock Exchange (Suta, 2000). It was in the December
of 1987 when the stock exchange was made active by issuance of a new regulation
called Paket Kebijaksanaan Desember 1987 (Pakdes 1987) which paved the way for
many companies to go public and also opened the doors for foreign investment in
stock market, it permitted foreign investors to buy up to 49% of the shares issued by a
domestic company except banking sector (Rosul, 2002).
The importance of stock markets for Indonesia is due to the fact that it came up as
a vital and efficient source of funds inflow to businesses, furthermore as a large
portion of funds from pension funds, insurance companies and other institutional and
individual investors are invested in stock market therefore its performance does have

6
wide consequences. In 1998, Indonesia went into a deep financial crisis, as Sharma
(2001) quoted world bank saying that " no country in recent history, let alone one the
size of Indonesia, has ever suffered such a dramatic reversal of fortune". Over
guaranteed but under capitalized and under regulated banking sector and poor
macroeconomic management were regard as fundamental causes for this crisis
(Sharma, 2001).
Stock market as an alternative source of funding emerged in the post 1998 crisis era,
as heavy reliance on banking sector for funding was a vital factor for the occurrence
of the crisis (Suta, 2000). Stock market played a vital role for inflow of foreign capital
into Indonesian economy in the coming decade, there are quite a few researches which
makes it evident that financial markets has been important to the economic growth
of Indonesia. During late 1980s financial development was a passive response to the
developments in other sectors of economy but from 1990s and particularly after 1998
crisis onward the financial development 'emerged as a reason by its own to spur the
pace of economic growth in Indonesia'. JCI has shown a tremendous increase from
2003 onward excluding a short interval in the later half of 2008 and first half of 2009.

Choices Under Certainty
Indifference curves were firstly drawn by Pareto in 1906 (Pareto et al. 1971).
Following figure present a case where returns on a single asset are known with
certainty and investor have to select among choices on the amount to consume in
period 1 and amount to save in period 1. Investor can not choose I 0 as no investment
opportunity available on that curve and investor will not select I 2 as it lies below the
opportunity set, therefore the better off position is I 1 as on the point D indifference
curve is tangent to opportunity set. If optimal point lie between A and B then investor
would lend the portion of its period 1 income, if the optimal point lie at point B then
neither lending nor borrowing would occur and finally if optimal point occur at point
C then investor would be better off to borrow from the future income of period 2 for
consumption in period 1. The above mentioned case becomes more complex when
investor selects multiple assets or a portfolio of assets. In the following figure an asset
with higher yield is added to the previous case to see the complexity that arises from
such situation.
In the Figure 2 investor is better off by lending one asset at higher yield and
borrowing the other one on lower interest, which gives the feasible region of A'B C.
However it is not possible to have two interest rates simultaneously as if it would have
been the case, then all investors would have purchase the asset that yield higher
interest rate. It shows that either one interest rate is prevailing or future returns are not
certain.

Consumption in P eriod 2

7

A
a′
b′

D
B

I0
I1
I3

a b
Consumption in P eriod 1

Consumption in P eriod 2

Figure 1 Consumptions in period 1 and 2 for a single asset with certain
outcomes

A′
A
B

C
C′
Consumption in P eriod 1
Figure 2 Consumption in period 1 and period 2 for multiple assets with certain
outcomes

Mean and Variance
Bakker et al. (2006) quoted Hacking (1975) that "in an ancient Indian story,
Rtuparna estimated the number of leaves and fruit on two great branches of a
spreading tree. He estimated the number on the basis of one single twig, which he
multiplied by the estimated number of twigs on the branches and found a number,
which after a night of counting turned out to be very close to the real number".

8
Bakker et al. (2006) further wrote that "the Belgian statistician Quetelet (1796-1874),
famous as the inventor of l'homme moyen, the average man, was one of the first
scientists to use the mean as the representative value for an aspect of a population". If
the probability of occurrence is same for all returns then the mean or average of an
asset i can be computed as
M
X
Rij
R̄i =
(1)
M
j=1
Here is the mean return on asset i which is calculated by dividing jth return on asset
i over M equally likely returns. In the case where returns are not equally likely the
formula become as follows
M
X
R̄i =
Pij Rij
(2)
j=1

Here represent the probability of the jth return on the ith asset which get multiply with
jth return on ith asset. Fisher (1918) coined the term variance when he stated that it is
"desirable in analysing the causes of variability to deal with the square of the standard
deviation as the measure of variability. We shall term this quantity the Variance of the
normal population to which it refers". Similarly variance for returns on asset i with
equally likely outcomes are computed as follows
σi2 =

M
X
(Rij − R̄i)2
M
j=1

(3)

Here R̄i shows the mean return on asset i and is jth return on ith asset. M is the total
number of equally likely returns. If returns over asset i are not equally likely then the
variance can be found by multiplying the jth probability on asset i with the squared
deviation of the return on asset i from the mean of asset i. It will become as follows

M 
X
2
2
(4)
σi =
Pij (Rij − R̄i)
j=1

Mean, Variance and Covariance of Portfolio
Return on a portfolio is the weighted average of the returns on individual assets.
Return on portfolio can be found by multiplying return on each individual asset with
the fraction of fund allocated for that asset. It can be written as
Rpj =

N
X

(Xi Rij )

(5)

j=1

Here represent jth return over portfolio, while is the proportion invested in ith asset
and is the jth return on asset i. N shows the number of observations. Variance for the

9
portfolio of any number of assets can be found by multiplying the sum of variances
over individual assets with the squared proportion of investment in each asset, which is
N
X

(Xi2 σi2 )

(6)

i=1

Stanton (2001) narrates that "it was the imagination of Sir Francis Galton that
originally conceived modern notions of correlation". Correlation or covariance for
portfolio of any number of assets can be found by taking the sum of multiplication of
covariance among assets j and k with the proportions of investment in the jth and kth
asset, which is as follows
N
N X
X
(Xj Xk σjk )
(7)
j=1 k=1
k6=1

Now by combining equation (6) and equation (7) we get the expression for variance of
a portfolio
N
N X
N
X
X
2
2 2
σp =
(Xj σj ) +
(Xj Xk σjk )
(8)
j=1

j=1 k=1
k6=1

Diversification
In this section we will present two cases to describe diversification using equation
(8). First case is related to when assets are independent of each other. Therefore
covariance among them is zero. Thus
N
X

(Xj2 σj2 )

(9)

j=1

As the σjk = 0 then we are only left with the first term of equation (8) which is the
portfolio variance. To illustrate the diversification, let assume that an investor allocate
equal investment in each asset. We get following equation
σp2

 N

N
X
1 2 2
1 X σj2
=
( ) σj =
N
N j=1 N
j=1

(10)

It can be seen from the equation that expression in brackets is the average of variance.
We can replace this average with σ̄j2 and equation (10) would become
σj2 = (

1 2
)σ̄
N j

(11)

If we keep putting a larger N in equation (11) the portfolio variance would come closer
to zero which is less than the variance of an individual asset. This case tells us that by

10
adding more and more independent assets the portfolio variance would become zero,
which makes it less risky. Thus the diversification reduces the risk of portfolio of
assets. In case of positive covariance we would have the equation (8), if we assume
that investor allocate equal funds to each asset, then it becomes
σp2



N 
N 
N
1 X σj2
σjk
(N − 1) X X
+
=( )
N j=1 N
N
N (N − 1)
j=1 k=1

(12)

k6=1

Here N represent j and N-1 represent k, and we have averages for both variance and
covariance expressions. Therefore average variance can be replace by σ̄j2 and average
covariance is replaced by σ̄kj and with rearrangement we get
σp2 = (

1
)(σ̄j2 − σ̄kj ) + σ̄kj
N

(13)

If we add more and more assets then we would have a large value for N which would
consequently reduce the portfolio variance. We would have a lower portfolio variance
than either of the assets may have. So it is shown that combination of assets would give
us a less risky portfolio.

Portfolio Possibilities Curve
Shape of portfolio curve depends upon the correlation among the assets in portfolio.
Correlation is measured with correlation coefficient that ranges from -1 to +1, where -1
describes perfect negative correlation and +1 represent perfect positive correlation and
0 means assets are uncorrelated. Let see how efficient frontier look like in the cases
of different correlations. If the covariance among two assets is perfectly positive, then
in the mean variance space we would have a straight line connecting the asset with
minimum variance to the asset with maximum return showing the linear relationship
among the assets in portfolio. Figure 3 illustrate it when ρ = +1 In the Figure 3 A and

B stocks represent minimum variance and maximum return respectively. The above
illustration shows case of a portfolio consisted of two assets. According to Elton et al.
(2007) the fraction of fund to be invested in each stock when covariance among two
stocks is perfectly positive and where portfolio standard deviation for two stocks A and
B is given as
σp = XA σA + (1 − XA )σB
(14)
Here X A and X B represent the fractions of fund invested in the securities A and B in
the portfolio and and are the variances of securities A and B. The equation for return on
portfolio of two stocks A and B is as follows
R̄p = XA R̄A + (1 − XA )R̄B

(15)

11

Rp

B

A

σp
Figure 3 Perfectly positive covariance between asset A and asset B
Now the fraction of fund to be invested in security A can be find as follows
XA =

σp − σB
σA − σB

(16)

By substituting value of X A into equation (15) we get


σp − σB
σp − σB
R̄A + 1 −
R̄B
R̄p =
σA − σB
σA − σB

(17)

We can rewrite above equation as


σp − σB
R̄p = R̄B + (R̄A − R̄B )
σA − σB



(18)

In the case of perfect negative covariance among any two assets we would an efficient
frontier which looks like Figure 4. This is the case where ρ = +1 In Figure 4 we can

see two lines representing two assets A and B which are moving exactly in opposite
direction when correlation among them in perfectly negative. As covariance among
two stocks is negative therefore portfolio standard deviation would be
σp = XA σA − (1 − XA )σB

(19)

Here fraction of fund invested in security B times its standard deviation is negative
which shows the negative covariance among two stocks, however it can also be as
follows
σp = −XA σA + (1 − XA )σB
(20)

12

Rp

B

A

σp
Figure 4 Perfectly negative covariance between asset A and asset B
As we get two different equations for standard deviation of portfolio therefore we would
have two values for X A which are
XA =

σp + σB
σA + σB

(21)

or it can also be

σp + σB
(22)
−σA + −σB
Zero covariance among assets in a portfolio means returns on assets are independent
of each other, there would still occur reduction in risk which make portfolio less risky
than the risk of individual securities. Figure 5 illustrate the case when ρ = 0 In this
XA =

case value of X A can be computed by taking the derivative of the portfolio variance
with respect to X A and then equaling it to zero. Variance of portfolio is
σp =



XA2 σA2 + (1 − XA )2 σB + 2XA (1 − XA )σA σB ρAB

 12

The derivative with respect to X A is


2
2
2
2XA σA − 2σB + 2XA σB + 2σA σB ρAB − 4XA σA σB ρAB
∂σp
1
=
 21
∂XA
2 
2 2
2
XA σA + (1 − XA ) σB + 2XA (1 − XA )σA σB ρAB

(23)

(24)

13

Rp

B

A

σp
Figure 5 Zero covariance between asset A and asset B
Equation (24) can be simplified by dividing numerator with
equals 0 and multiplying denominator with 0 we get

1
2

and setting equation

XA σA2 − σB2 + XA σB2 + σA σB ρAB − 2XA σA σB ρAB = 0
σB2 − σA σB ρAB
XA = 2
σA + σB2 − 2σA σB ρAB
As covariance among A and B equals zero therefore equation (26) becomes
XA =

σB2
σA2 + σB2

(25)
(26)

(27)

One Fund Theorem
The assumption introduced in this section is that investors can lend and borrow
unlimited funds to invest. Riskless lending and borrowing means future returns on
security are known with certainty, here lending can be considered as purchasing short
term government treasury bills and borrowing is short selling of these securities. Return
and variance on a combination C consisted of riskless asset RF and risky asset A is as
follows
R̄C = (1 − XA )2 RF + XA R̄A
(28)


(29)
σC = (1 − XA )2 σA2 + 2X(1 − X)σA σF ρF A
Future outcome of riskless asset are certain therefore its standard deviation equals zero,
then the equation (29) would become
σC = XA σA

(30)

14
Fraction of fund X A to be invested in this combination C is
XA =

σC
σA

(31)

Substituting value of XA into equation (28) and by its rearrangement would give us
R̄C = RF +




R̄A − RF
σC
σA

(32)

Equation (32) is an equation of a straight line, so all combinations of riskless and risky
asset lies on this line are illustrated in Figure (6).
Combination of RF and A gives us a combination of riskless and risky asset.
However this combination is not desirable due to availability of portfolios as we move
clockwise in the efficient frontier. Combination of portfolio C with RF is the point
where the straight line representing return on combination is tangent to the efficient
frontier, this line is also called as capital market line as it represent the return on
efficient portfolio. However as investor is allowed to borrow unlimited amount
therefore an risk lover investor can choose a combination between C and D. Similarly
a risk averse investor may select a combination of RF and C. One fund theorem said
that an efficient portfolio can be constructed by combining fund of risky asset C with a
riskless asset RF . In the next section we would see how we can find such optimized
combination.

D
C

Rp

B
A
Rf

σp
Figure 6 Combination of riskless asset with risky assets

15

Portfolio Optimization with Quadratic Programming
Markowitz mean-variance portfolio theory is the first formulation of uncertainty
problem in economics as a mathematical programming. Markowitz formulated a
maximization problem where investor wants to maximize expected return while
considering the minimization of risk. The problem can be solved either way and it
result with same conclusions. Here we will present a maximization problem as
described by Elton et al. (2007) where optimal point can be achieved by maximizing
the slope that of the line connecting riskless asset with risky portfolio. Objective
function to be maximize is as follows
θ=

M ax

R̄P − RF
σP

(33)

The objective function θ is subject to the constraint that proportions of fund invested
by investor equals 1, which is given below.
N
X

Subject to

Xi = 1

(34)

i=1

The portfolio standard deviation σp term in objective function include quadratic terms
therefore it becomes a problem of quadratic programming. Markowitz solved the mean
variance quadratic programming problem by critical line method. However following
Elton at el. (2007) we will present its solution using Kuhn Tucker conditions. We can
write RF as follows
RF = 1RF
(35)
where

1RF =

N
X

(Xi RF )

(36)

i=1

Now rewriting the objective function by putting values of RF and σp , we get
N
X

Xi (R̄i − RF )

i=1

θ=
X
N
i=1

Xi2 σi2 +

N X
X
i=1 j=1
j6=1

Xi Xj σij

 12

(37)

Mean variance portfolio optimization can be tested with different assumptions, if we
assume that short sales are allowed then the maximum point can be achieved by setting

=0
Xi

(38)

16
Now the maximization can be achieved either X i > 0 or X i < 0. By taking first order
conditions and going through its mathematics Elton et al. (2007) derived the following
expression


N
X

2
(39)
= R̄i − RF − λ Xi σi +
Xj σij = 0
Xi
i=1
If we assume short sales is permitted and the only constraint to the objective function is
Equation (34) then the formula to solve the portfolio optimization problem is as follows


N
X

2
= R̄i − RF − λXk σk +
λXj σkj = 0
Xi
j=1

(40)

j6=k

However as in the real world there are many restrictions imposed on short sales and
no short sales case is more realistic compare to short sales, therefore in the case where
short sales is not allowed this assumption will lead to addition of the non negativity
constraint in the quadratic programming which is given below.
Xi ≥ 0

Subject to

f or all i

(41)

Due to non negativity constraint maximization at X i < 0 is not a feasible solution. This
will lead us to two cases where each one of them have a solution. First case is where
maximization occurs at Xi < 0 so the graph would look like as With X i < 0 the
θ
E
E′

Xi
Figure 7 Maximum value of θ when
when X i = 0


dXi

< 0 at maximum feasible value of θ

17
maximization occur at E which is not feasible here due to non negativity constraint.
The maximum feasible value occur at E' which is the maximum point where value of

< 0. We can summarize this with by stating
X i = 0 with downward slope that is dX
i
that in a case like Figure. 7 the maximum value can be attain where

≤0
dXi

(42)

Xi = 0

when

(43)

Second case is shown in Figure 8 where the local maximization point E is attained at

X i > 0 with first derivative is dX
< 0.
i
θ
E

Xi
Figure 8 Maximum value of θ when
when X i = 0


dXi

= 0 at maximum feasible value of θ


=0
dXi
when
Xi > 0

(44)
(45)

The problem we are trying to solve is to know which case among the above two cases
will maximize the objective function θ in Equation (37). This problem can be solved
by satisfying Kuhn Tucker conditions which we will write here more compactly. We
may introduce U i in Equation (41) to make it an equality

+ Ui = 0
dXi

(46)

18
Above equation is the first Kuhn Tucker condition. Other three conditions are as follows
Xi Ui = 0, Xi ≤ 0, Ui ≤ 0
We can add U i to the Equation (39) so it become


N
X

2
= R̄i − RF − λ Xi σi +
Xj σij + Ui = 0
Xi
i=1

(47)

(48)

Once Kuhn Tucker conditions are met then we can use the values of X i to be the optimal
fractions of fund in order to maximize the ratio of excess return to portfolio standard
deviation which is our objective function.

Single Index Model
Inputs needed to solve problem of portfolio optimization are quite large in number
if we solve Markowitz mean variance approach using quadratic programming, the
number of inputs needed would be N(N-1), where N is the number of securities. Elton
et al. (2007) stated that by using single index model for any portfolio optimization
problem we only need to compute return on individual securities, variance of
individual securities, beta for each security, market return and market variance. Later
two are constants for all securities, so the number of inputs needed to calculate
portfolio optimization for any number of securities would become 3N+2, where N
represent the number of securities. Rationale behind Single Index Model is that all
securities listed in stock exchange would move up or down along with the stock
market index which comprise all stocks listed in the stock exchange. Therefore this
single index is the only common factor among varying stocks that covary with each
other with reference to a common factor which is stock market index. Sharpe who was
also a student of Markowitz presented this simplification in 1963. Single Index Model
is a linear equation represent a straight line as follows
Ri = αi + βi Rm + ei

(49)

Where α is the expected value of i and ei is the uncertainty in the asset i, Ri is the return
on asset i and Rm is the market return. β represent the change in Ri given a change in
Rm . If we substitute the value of Ri into Equation (5) and knowing that expected value
of ei equals zero we get
R̄P =

N
X

Xi αi +

i=1

σi2

2
βi2 σm

N
X

Xi βi R̄m

(50)

i=1

2
σei

2
Substituting values for
=
+
and σij = βi βj σm
into Equation (8) and
constructing a portfolio by allocating equal amounts of funds into each N stock would
give us

 N
1 X 1 2
2
2 2
(51)
σ
σP = βP σm +
N i=1 N ei

19
Second expression in the right hand side of Equation (50) represents the average
residual risk which approaches value of zero with increase in the number of stocks N.
Theref