Do markowitz and sharpe models improve investment performance of an investor in vietnamese stock market

8 2.1.1 Original Idea on Portíolio Analysis...8 2.1.2 Markowitz Portfolio Selection M odel...11 2.1.3 Diversiíication in Markowitz M odel...13 2.2 Markowitz Model: Mean-Variance Portfoli

DO M ARKOW ITZ AND SHARPE MODELS IMPROVE INVESTMENT PERFORMANCE OF AN INVESTOR IN

VIETNAMESE STOCK MARKET?

G ra d u a tio n thesis D e c e m b e r 2 0 0 8

ABSTRACT

The current context o f Vietnamese market is fa r from the efficient level due lo such shortage o f /ìnancial advisors, least trading o f government securities, dominance o f some types o f market, rumour trading and so forth This síudy is to make a small contribution to the invesíing world o f Vietnam by iníroducing a powerful “tool" which may help improve the investors’ investment decisions without affecting the reíurn Within just about 10,000 words, the research paper is elaborating alỉ the theories as vvell as procedures so as to figure out optimal porựolios in which investors are recommended to invesl in Also, the paper is equippìng investors with a very popular spreadsheet to handỉe aỉl the complicated tasks - Excel The research paper is hoped to bring about a new approach to the investing society o f Vietnam.

G r a d u a tio n thesis _ _ _ _ D e c e m b e r 2008

STATEMENT OF AUTHORSH1 p

"Except where reference is made in the text of the thesis, this thesis contains no material published elsewhere or extracted in \vhole or in part from a thesis or any other degree or diploma

No other person's work has been used vvithout due acknovvledgment in the main text

o f the thesis

This thesis has not been submitted for the award o f any degree or diploma in any other tertiary institution."

Student signature

G r a d u a tio n thesis D e c e m b e r 2 0 0 8

ACKNOWLEDGEMENTS

I am indebted to Ms Phuong - my guiding teacher, for all the things that she has helped me; vvithout her help and encouragement, this paper cannot tum out to be a vvhole I also indebted to her enthusiastic lectures and concems during the past four year period which has helped me accumulate a lot o f understandings, which is the springboard for my íuture jobs

I would like to extend my greatest thanks and appreciation to all the teachers of the faculty, who have taught me during the last four years For all your enthusiasm, exciting and interesting lectures, all your considerateness, I never íorget and never feel enough to say thank you

In here, I do like to send my most whole-hearted thanks to all o f my íriends - the ones who have stood by me for the whole course sharing with me the ups and downs of tertiary studies and aỉl happiness as well as soưovv

Withouí you alỉ, I would have be none o f mine todayỉ

Thankyou, mybeloved!

G r a d u a tio n thesis D e c e m b e r 2008

TABLE OF CONTENT

ABSTRACT i

STATEMENT OF AUTHORSHIP ii

ACKNOWLEDGEMENTS iii

TABLE OF CONTENT iv

1 INTRODUCTION 1

1.1 Background Overview 1

12 Research Signiíícance 3

1.2.1 Vietnamese Stock Market Review 4

1.2.2 Research Signiíìcance 6

1.3 Organisation of the Research 7

2 Literature Review 8

2.1 Harry M Markowitz and Portíòlio Selection Model 8

2.1.1 Original Idea on Portíolio Analysis 8

2.1.2 Markowitz Portfolio Selection M odel 11

2.1.3 Diversiíication in Markowitz M odel 13

2.2 Markowitz Model: Mean-Variance Portfolio Selection (Risk and Reward)16 2.3 Sharpe Model o f Portfolio Optimisation 19

3 RESEARCH METHODOLOGY 19

3.1 Revievv o f Studies in Vietnamese Stock M arket 19

3.2 The Research’s Hypotheses 20

3.3 Sampling and Method o f Analysis 21

3.4 Procedures o f forming optimal portfolios under Markowitz model 22

3.5 Procedures o f íòrming an optimal portíblio under Sharpe model 23

3.6 Procedures o f Forming Optimal Portfolios by Using Excel 24

4 RESEARCH FINDINGS - ANALYSIS AND DISCUSSION 33

4.1 Findings under the Case o f Markowitz Model 33

4.1.1 Retum and Risk of Individual Securities and the M arket 33

4.1.2 Selectìon o f T wo-stock Portfolios 35

4.1.3 Calcuỉation o f Portfolio Weights, Retum and Standard deviation 36

4.1.4 Identiíĩcation o f Efficient Sets 36

4 1.5 Ranking o f Eíĩícient Sets 38

4.2 Findings under the Case o f Sharpe Model 39

4.2.1 The Optimal Portfolio under Sharpe Model 39

4.3 Optimal Portfolios Formed by Application o f Optimising Model in Excel 41 4.4 Discussion on Research vveaknesses 48

4.5 Research Implications 51

5 RECOMMENDED FUTURE RESEARCH OPPOTƯNITIES 53

6 CONCLUSION 55

Annex 56

References 57

G r a d u a tio n thesis D e c e m b e r 2008

USJ OI 1 K,l iu.s

Figure 1: Statistics o f Listed Stocks 4

Figure 2: Market Capitalisation per GDP 5

Figure 3: Markowitz Portfolio Selection 12

Figure 4: Relationship between expected retum and SD o f retum for vanous coưelations 15

Figure 5: Eíĩicient Frontier and Indifference Curve 17

Figure 6: Excel Add-ins 25

Figure 7: Add-in Dialogue Box 25

Figure 8: Excel Data Analysis 26

Figure 9: Data Analysis Box 26

Figure 10: Correlation Dialogue B ox 27

Figure 11: Coưeỉation Matrix 28

Figure 12: niustration o f Portíolio Variance Formula 30

Figure 13: Solver 31

Figure 14: Solver Parameters 32

Figure 15: Eíĩĩcient Frontier o f Two-stock Portfolios under Markowitz Theory 37

Figure 17: Solver Options 46

A s som eone might think o f it, Finance is a branch o f social sciences - a combination o f M athematics and Economics Yet, Finance is such an immense íĩeld that not all in-the-fíeld scholars or investors could have a deep understanding and rigid master o f it; howevei/M <LTom Smith - Professor o f Finance at the Australian National University - as quoted saying that the tíeld

o f fínance can be viewed as being m ade up o f three simple ideas1:

S The time value o f money;

s Diversification; and

s Arbitrage

ln fact, Finance is rooted in the idea o f the time o f money The fírst idea is quite plain - a dollar today is worth m ore than a dollar tomorrow It is this rule that is the basis o f all valuation Second, i f there are two ways to get the same

cash flow, they must have the sam e price This statement is knovvn as va ỉm tio n

by arbitrage The arbitrage idea is a t the heart o f the valuation o f íutures and

options W hat about the last branch o f ĩinance under Tom ’s idea - Diversiíĩcation? If investors wish to m aximize ửieir retum on investment vvithout suíĩering more risk exposure, they should diversiíy since

1 Ross & Thompson & Christensen and W estemfield 2004.‘Fundamentals o f Corporate Fi ư edn

C h a p t e r 1 - Introduction

diversiíìcation reduces risk The idea o f diversiíícation was formalized by Harry M Markowitz in 1952, which has absolutely altered the way investment decisions have been made in the investing world since then The theory by

M arkowitz vvas later named M odem Portfolio Theory (M PT) or M odem Portíolio Selection

At íírst glance, M odem Portíòlio Theory is not as m odem as its implications Just like other theorems

and m odels which w ent

M PT also has its own

story The insight for

M arkow itz received the

Nobel Prize was fírst

published in 1952 in an article nam ed “Portíòlio Selection” Later in 1959,

M arkowitz expanded the article into a book entitled “Portfolio Selection -

E íĩícient Diversifícation o f Investm ents” They were both ứie milestones

m arking a break-through alteration in the investing approach Yet, the quantitative approach did appear far back in past and it w as modelled on the investm ent trusts o f the England and Scotland, w hich began in the middle o f the nineteenth century The idea that diversification could reduce risk and that

it happened in the earlier time w as actually put into poem by Shakespeare’s wording:

H a rry M arkow itz

Kíy veníures are n o í in one botíom trusted

N or to one place; n o r is m y whole estate

ư p o n the /o rtu n e o f this presen t year;

There/ore, m y m erchandise m akes m e not sad.

C h a p tc r 1 - Introduction

Harry M Markowitz is, indeed, credited with presenting new concepts o f risk measurement and their application to the portíolio selection He started with the idea o f risk aversion o f average investors and their desire to maximise the expected retum with the least risk Thereíbre, M arkowitz model acts as the theoretica! background for analysis o f risk and retum and their inter- relationships Under the modei, risk is measured by the statistical parameters and assets in a portíolio are selected in an efficient m anner through

m athematical methods The íramevvork led to the concept o f efficient portíòlios

- the ones that maximize retum at a given level o f risk or minimize risk at a given level o f retum , which can be íormed by com bining securities having less than perfect positive correlations in their retums

M arkowitz model w as theoretically elegant and conceptually sound Yet, its serious limitation w as the volume o f work well beyond the capacity o f all except a few analysts To resolve the problem, WiHiam F Sharpe developed a

simplifíed variant o f the M arkowitz model that reduces substantially its data and computational requirements (Sharpe, 1963)

As per Sharpe’s model, the construction o f an optimal portfolio is simpliíied if a single number measures the desirability o f including a stock in the optimal portfolio If accept his model, such a number exists In this case, the desirability o f any

VVilliam F S harpe

stock is directly related to its excess retum-to-beta ratio If the stocks are ranked from the highest to lowest order by excess retum to beta, that represents the desirability o f any stock’s inclusion in a portíolio The number o f stocks selected depends on a unique cutoíĩ rate such that all stocks vvith higher ratios vvill be included and all stocks with lower ratios be excluded

1.2 Research Signitìcance

C h a p te r 1 - In tro d u c tio n

This study is to shed light on the investment vvorld o f Vietnam - applying the Markowitz and Sharpe model - with the help o f Excel - so as to construct eữícient portfolios, the portfolios which performed best in the period 2005- 2007

1.2.1 V ie tn a m e s e S to c k M a rk e t R e v ie w

The stock m arket o f Vietnam emerged ữ o m July 2000 with only two stocks It was marked w ith a formalized m ilestone at early 2006 w hen a num ber o f stocks were listed To date, after nearly 8 years in operation, the num ber o f listed stocks has reached 156 on H o Chi M inh Stock Exchange (HOSE); Hanoi Securities Trading Center (HASTC) also records a signiíícant boom ing during the past three years - from 9 listed stocks in 2005 up to 148 in middle 2008 The grovvth o f H O SE and HASTC can be seen from the follow ing íígure

C hapter 1 - ln tro d u c tio n

F ig u r e 2 : M â r k e t C a p ita lis a tio n p e r G D P

Source: VDSC

Although this w as a quite dramatic grovvth, it does not mean that the stock

m arket o f V ietnam has reached an efficient level due to the limited number o f listed securities (156 securities) by the end o f July 2008, dom inance o f some types o f securities o f market portfolio, namely, ones o f Finance & Banking (36.5% ) (vneconom y, 2007), least trading o f govem m ent securities, absence o f proíessional investm ent advisors, very low level o f inform ation disclosure and trading driven by rum ors than system atic analysis

At present, Capital m arket is only proíítable to the investors w ho can overlook the rule o f gam e, i.e accessing relevant information in advance Thereíore, this poses a g reat challenge for the rational investors o f Vietnam who utilizes a system atic approach in investment decision

The stock m arket o f Vietnam is currently in the infant-transitional stage; hovvever, all types o f investors can fare from the sound and in-depth knowledge

o f portíòlio analysis which would help them diversiíy their investment risk The system atic analysis o f attainable portfolios and thereby, selection o f the optim al portíòlio w ould help diversiíy risk without adversely decreasing the

C h a p te r 1 - Introduclion

retum Furthermore, it also facilitates the mobilization o f resources in all sector

o f economy by inducing investors to invest in stocks o f different industrial categories and thereby fostering th e econom y grovvth o f the country

1.2.2 R e s e a rc h S ig n itic a n c e

This research is set fire to solve the optim ization problem raised by M arkow itz

56 years ago It is aim ed at ííguring out the efficient frontier, thereby arriving at efficient portíòlios - by applying M arkow itz m odem portíòlio theory into the case o f two-stock portíblio Furtherm ore, an optimal portíolio w hich may include a num ber o f securities will also be w orked out in the case o f Sharpe model All the procedures will only be processed in a single environment - EXCEL The m ain focus o f this study is “The Third M odel”- Application o f the optim izing model in Excel to establishing optimal portfolios with minimum variance subject to a certain level o f required rate o f retum Since the period o f concem is 2005 - 2006 - 2007; thereíòre, only securities listed before 1/1/2005 are selected as inputs for the study Since HA STC began its official operation

at 8/3/2005, only 26 securities were listed on H O SE at 31/12/2004 (at this time, there w as another investment fimd on H O SE - Vietíìm d M anagement 1 Yet this research only targets com m on stocks; hence, investm ent fund is out o f analysis range)

In other words, the study answ ers the question: Do Markowitz and Sharpe models improve investment performance o f an investor in Vietnamese stock

In the context o f Vietnamese stock m arket reviewed above - coupled with the

m arket cu ư en t situation, the study hopes to bring about a new & m ore

“coníídent” view to the investment vvorld o f Vietnam Even though the current

“ investing society” o f Vietnam still lack a great num ber o f íínancial experts and analysts, individual investors can take utility o f available resources like

Excel in order to com e up with better investment decisions In addition, it is so useíul & effective for the academic research at tertiary schools Students can use Excel to strengthen their understanding about investment, thereby bridging the distance betvveen theory and empirical practice.

Above all, the research is to empirically test vvhether the models developed by Markowitz and Sharpe five decades before, which have been proved to be true

in several other markets, can tum out to be “applicable and íeasible” in Vietnamese stock market

1.3 Organisation ofthe Research

The Research includes íòur main chapters Literature Revievv is the next chapter w here th e m ajor insights o f M arkowitz & Sharpe model are discussed Following is about Research M ethodology presenting sampling m ethod and procedures putting M arkowitz - Sharpe models into practice as well as steps using the add-in “Solver” to solve the same optimization problem The subsequent chapter discusses the fíndings, limitations and implications o f the research The last but not least chapter o f this paper concludes with íiiture research opportunities

Chapter 1 - I n t r o d u c t i o n

C h a p t e r 2 - Literature R e v i e u

2 Literature Review

This part o f the paper is supposed to review tw o m odels introduced by Harry Markowitz (Portíolio Selection M odel) and W illiam Sharpe (Sharpe Model o f Portfolio Optimisation) Speciíícally, M arkowitz Portíolio Selection M odel and

M ean-Variance A nalysis are discussed Afterw ards, given those understandings, a sim plifíed variant o f M arkow itz model - presented by Sharpe

- will be operationalised A review o f m athem atics w ill also be brieíed

2.1 Harry M Markowitz and Portíoĩio Selection Model

2.1.1 Original Idea on Porttolio Analysỉs

It is w orth to cast a b rief overview to w hat the portíòlio analysis is Throughout this paper, “portfolio selection” will be used instead o f “ security selection” Regarding a portíòlio, M arkow itz states that it is more than a long list p f good stocks and bonds; it is a balanced w hole, providing the investor vvith protections and opportunities w ith respect to a w ide range o f contingencies And the investor should build tovvard an integrated portfolio which best suits his needs Also according to M arkow itz’s school o f thought, the process o f selecting a portfolio - portíòlio analysis, may be divided into tw o stages It fírst starts with information concem ing individual securities Then it ends w ith conclusions concem ing portíolios as a vvhole Thus, th e analysis’s purpose is to fínd portíolios which best meet the objectives o f the investor

A variety o f iníorm ation concem ing securities can be used as the raw material

o f a portfolio analysis Typically, the past perform ance o f individual securities

is one source The second one is the belieís o f one o r more securities analysts concem ing íìiture performances Accordingly, w hen past perform ances o f securities are used as inputs, the outputs o f the analysis are portíolios w hich

C h a p t c r 2 l.ilcraturc Rcviev.

períormed particularly well in the past Meanvvhile, when beliets o f security analysts are used as inputs, the analysis’s outcom es are ìmplications o f these beliets for better and vvorse portíolios

Additionally, it is also worth o f taking into consideration broad principles upon

vvhich the techniques o f portíolio analysis are based Markowitz argued that

Uncertainty is a salient /eature o f security im e stm en ẽ Economic íòrces are

not understood well enough for predictions to be beyond doubt or errors Even

if the consequences o f economic conditions were perfectly made sense of, non- economic iníluences can change the course o f general prosperity, the level o f the market, or the success o f a particular security For instance, changes in International tensions, increases or decreases in govem m ent spending, an exừem e ílooding or dry summer, the success o f an invention and so íòrth - all can have an impact on the Capital gains or dividends or one o r many securities

M arkowitz hum orously said that only the clairvoyant could hope to predict with certainty A nd surely they are not the target reader o f any analysis technique But the existence o f uncertainty does not m ean that careíiil security anaỉyses are valueless The security analyst may be expected to arrive at reasonable opinions to the effect that:

growữi of security E is more ccrtain but has less potential than that o f security

F ; only ỉf the demand for theừ industiy’8 prodúct continues to expand (as ít

is ỉỉkely, but not certain, to do) will the retum on sccurities G and H be satisfactory.

3

Such analysis illustrates that careíìil and expert judgem ents concem ing the potentialities and vveaknesses o f securities w ould still build up the best background for portfolio analysis

2 Harry Markowitz 1959 Portfolio Selection: ERicient Diversiíication o f Investment, Journal o f Finance

3 The same source as above.

“perfect” , i.e i f retum s on all securities m oved up and dow n together in períect unison, diversification could do nothing to elim inate risk; the fact that security retum s are highly correlated, but not perfectly correlated, implies that diversification can reduce risk but not totally elim inate it And since the correlation am ong retum s is not the sam e for all securities, it is essential to avoid a portfolio whose securities are all highly correlated w ith each other in order to reduce risk.

Above are tw o broad principles w hich act as the framework for Markowitz techniques o f portíolio analysis So, what are the objectives o f portfolio analysis? Because to derive all possible conclusions concem ing portfolios is impossible, a portíòlio analysis m ust be based on criteria which serve as a guide to the important and unim portant, the relevant and irrelevant The proper choice o f criteria depends on the nature o f th e investor For some investors, taxes are prime consideration; for others, such as non-profít corporations, they are irrelevant Hence, for each type o f investor, the details o f the portíblio analysis must be suitably selected, Yet, tw o objectives are com m on to all investors for w hich the m odel is designed:

s They want “retum ” to be high The appropriate deĩinition o f “retum ” might vary from investor to investor B ut they prefer more to less o f it regardless o f whatever appropriate sense it is

s They w ant this retum to be dependable, stable, not subject to

uncertainty The portfolio selection model is for the investor who, ceteris paribus, prefers certainty to uncertainty

C h a p l c r 2 Literaturc K c v ie u

2.1.2 M a rk o w itz P o rtto lio S e le c tio n M o del

Prior to M arkow itz’s work, investors focused on assessing the risks and revvards o f in d iv id u al securities in constructing their portfolios Standard investment advice vvas to identity those securities that oíĩered the best opportunities for gain (expected retum s) vvith the least risk - as measured by its risk S ta n d a r d deviation from the mean o f expected retum s and then constructed

a portíolio from these Following this advice, an investor m ight conclude that bank stocks all offered good risk-reward characteristics and compiled a portfolio from these Intuiíively, this w ould be íoolish

Markowitz form alized this intuition M arkowitz began a revolution by suggesting that the value o f a security to an investor m ight best be evaluated by its mean, its Standard deviation, and its correlation to other securities in the portfolio This “ venture” suggestion am ounted to ignoring a lot o f iníbrm ation about the firm - its eam ings, dividend policy, Capital structure, m arket and competitor; and calculating a few statistics Elaborating maứiematics for

diversiíícation, he proposed that investors /o c u s on selecting portfolios based

on their overall risk-reward properties in lieu o f m ereỉy compiỉing portfoỉios J'rom securities thai each indixidnally h a s attractive risk rew ard properties In

brief, investors should select portíolios, not individual securities

portĩolio at the beginning o f ứie period The investor’s objective is to maximize the portfolio’s expected retum subject to an acceptable level o f risk or

m inim ize risk subject to an acceptable expected retum The assumption o f a single period in couple with assum ptions about the investor’s attitude toward risk, allows risk to be measured by the variance or Standard deviation o f ứie portfolio’s return Thus, as indicated by the arrow in Figure 1, the investor is trying as far to go northwest as possible:

C h a p t e r 2 L ite ratu re Revievv

Standard Devỉation

Figure 3: M arkow itz P ortíolio Sclection

As securities are added to a portíòlio, the expected retum and Standard deviation change in very speciĩic ways, based on the w ay in w hich the added securities co-vary w ith the others in the portfolio The best that an investor can

do (i.e the íurthest northwest a portfolio can be) is bounded by a curve that is the upper h alf o f a hyperbola, as show n in the íígure above This curve is known as the eữ ícien t frontier A ccording to M arkow itz m odel, investors select portíòlios along this curve given their risk tolerance An investor w ho can live with lots o f risk m ight choose portfolio A, w hile a m ore risk-averse investor would be more likely to choose portíolio B O ne o f M arkow itz theory’s m ajor insights is that it is a security’s expected retum in combination with how it co- varies with other securities which determ ines its candidacy to the investor’s portfolios

M arkow itz’s prim ary contribution consisted o f developing a strictly íormulated, operational theory for portíòlio selection under uncertainty Due to the possibility o f reducing the risk through diversiĩication, the risk o f the portfolio - m easured as its variance or S t a n d a r d deviation, vvill depend not only

C h a p t c r 2 Litcrature Reviev.

on the individual variances o f the retum on ditĩerent assets but also on the pair-

wise covariance o f all assets Thus, the essentiaỉ respect relating to the risk o f

an asset ĩs noi the risk o f each asset in isolation but the contribution o f each asset to the risk o f the aggregate portfolio Yet, the law o f large numbers is not

totally applicable to the diversiíícation o f risks in portíblio choice since the retum s on different assets are correlated in practice Hence, generally risk cannot be absolutely eliminated regardless oí" how many types o f securities included in a portíòlio

2.1.3 Diversification in Markowitz Model

M arkowitz model suggests that it is possible to reduce the Ievel o f risk belovv the non-diversiíĩable risk M arkowitz diversiíication can be categorized on five basic interrelated concepts:

i The W eig h ts Sum to One: The fírst concept requires that the weights o f the assets in the portfolio sum to 100% Sim ply the investment vveights are a decision variable, which is the main task for portfolio manager to determine them

W here X represents weights or participation level o f asset i in a portfolio

th at contains N assets W hen th e portíolio involves short sales, weights can be negative; hovvever, they should not violate this concept A portíolio w hich has negative vveights for som e assets is called leveraged portfolio or borrowing portfolio

ii A Portfolio’s Expected Return: It is the weighted average o f the

expected retum s o f the assets Uiat make up the portfolio, ứie portfolio’s expected rate o f retum for N-assets portfolio is,

.V

C h a p t c r 2 U te r a t u r e R e v ic u

.V

E i R p ) = £ v , £ ( / ? , )

/=1

Where E(Rj) is the security analysts íorecast for expected rate of retum

from the /th asset

iii T h e O bjective: Investment weights chosen by portíolio managers should add up to an efficient portíolio w hich is:

• The m aximum e x p e c t e d retum in its risk-class, or

• The minimum risk at its level o f expected retum

The set o f all efficient portíolios is called eíficient frontier T his is the

m aximum retum at each level o f risk The eữícient frontier dominates all other investment opportunities

iv P o rtío lio R isk: In contrast w ith expected retum o f a portfolio which isbased on íòrecast, the risk o f a portíolio is caiculated from historicaldata available to the asset manager The risk o f the portíolio, or its variance should be broken into tw o parts, the variance which represents the individual risks and interaction betvveen N candidate assets This equation (double sum m ation) represents the variance-covariance matrix and can be expanded and vvritten in m atrix form

VAR(Rp) = ± ± x ixJa iJ

ì- 1 1

ơ j — Ơ Ị Ơ p ị ị p

assets i and j In order to have a portfolio well-diversifíed according to

M arkowitz, the assets included in the portíòlio should have low enough correlations between their rates o f retum s As show n in the íígure below, a portfolio with correlation coefficient equal to zero gives ứie sam e level o f return - but w ith a low er risk level than a portíòlio which the assets including it have a correlation coefficient o f one If an investment or portíolio m anager achieves to include securities whose

C h a p t c r 2 I Ileraturc R c v ie u

rates of retum have low enough correlation, accordmg to Markowitz, he

or she can reduce a portfolio's risk below the non-diversifiable level

V T h e C a p ita l A llocatỉon Line: The last concept to consider on diversiíícation by M arkowitz is The Capital A llocation Line (CAL) This concept discusses the possibility o f lending and borrowing at a risk free rate o f interest provided by M arkow itz model An example can be a govem m ent treasury bill, where as the phrase “ risk-free interest rate” suggests the variance is zero M arkowitz model gives the opportunity to the asset m anager to combine a risky asset or a set o f risky assets (a portfolio o f risky assets) with a risk-less asset Y et, this research is only looking the possible sets o f com binations am ong risky assets, i.e ííguring out the efficient frontier, thereby locating the portfolios lying

on that very curve

4 G ru b e reta l

s OíTer m aximum expected retum for varying levels o f risk and,

s Offer minim um risk for varying levels o f expected retum

are known as “ efficient sets” The effícient portfolio lies along the efficient írontier E íĩícient frontier poses unique risk and retum characteristics The investor will choose portíòlios from these efficient portfolios This concept falls under M odem Portíòlio Theory The theory assum es, am ong other things, that investors whole-heartedly attem pt to minim ize risk w hile being motivated for the highest retum attainable T he ứieory suggests rational investors always make decisions aim ed at m axim izing their retum for their acceptable level o f risk

Harry M M arkowitz described his portíolio in 1952 and it shows that it is possible for d iíĩeren t portíolios to have varying levels o f risk and retum Each investor m ust decide how m uch risk they can handle, accordingly allocating their investments The optimal risk portíòlio is usually determ ined to be somevvhere in the middle o f the curve since as one go higher up the curve; s/he takes on proportionately more risk for a lower incremental retum B ut low risk/low retum portfolios are pointless as s/he can achieve a sim ilar retum by investing in risk-free retum s like govem m ent securities

Investors can choose how m uch volatility s/he is w illing to bear in their portfolios by picking any other point that falls on the eữ ícien t frontier This will gi ve her/him maximum retum for a risk level they w ish to accept To select the optimal portíòlio, an investor should plot her/his ìndiữerence curves

C h a p t c r 2 l.iterature R e v ịe u

(1C) on the efíìcient set and then choose the portíblio that is on the indiữerence curve íầrthest northeast These portíblios vvill correspond to the point at which

an inditYerence curve 1S just tangent to the eíĩicient set.

In this figure, the point o f tangency between IC2 and efficient curve is A Though investor will prefer ICi which represent higher preference but such portíolio is unattainable Here, portfolio A is the dom inant portíòlio set - the optimal choice

M arkow itz model w as a brilliant innovation in the S c i e n c e o f portíòlio selection M arkowitz showed us that all the iníorm ation needed to choose the best portíolio for any given level o f risk is contained in three simple statistics: mean, Standard deviation and correlation In short, Harry M arkowitz Ãindamentally altered how investment decisions w ere made Virtually, every

m ajor portfolio m anager today consults an optim ization program They may not exactly follow its recommendations; hovvever, ửiey use it to evacuate basic risk and retum trade-offs (Goetzmann, 1999)

E ffic ie n t F ro n tie r

ơ p

F ig u r e 5 : E f f ic ie n t F r o n t i e r a n d I n d ií ĩ e r e n c e C u r v e

s the time and cost necessary to generate efficient portíòlios (solve a

quadratic programming problem ); and

s the diữículty o f educating portfolio m anagers to relate to risk retum

tradeoffs expressed in term s o f covariance as well as retum s and Standard deviations T his resulted from the traditional investing approach prior to M arkowitz th e o ry 5

It is these challenges o f the model that inspired a num ber o f researchers to develop new m athem atics so as to sim pliíying it W illiam Sharpe is a typical one

5 Edwin J Elton, M artin J Gruber, M anữ ed w Padberg 1976 ‘ S I M P L E

C R I T E R I A F O R O P T I M A L P 0 R T F O L l O S E L E C n O N , Joumaí o f F in a n c e

C h a p t c r 2 l.iteralure R c v ic u

2.3 Sharpe Model ofPortíolio Optimisation

W illiam Sharpe, who among others has tried to sim pliíy the process o f data inputs, data tabulation, and reaching a solution, has developed a sim pliíìed variant o f Markowitz model that substantially reduces its data and computational requirements M arkowitz Portfolio Selection Theory is sound by itself; however, its serious limitations were the sophistication and volume o f work far beyond the capacity o f all except a few analysts

W illiam F Sharpe’s pioneering achievement in this ĩield was contained in his essay entitled “ Capital Asset Prices: A Theory o f M arket Equilibrium under Conditions o f Risk (Sharpe, 1964) As per Sharpe model o f Portfolio

O ptimisation, the linearity o f security should be found The beta o f security

represents the m arket linearity o f the stock The m arket iníluences each stock Negative beta deĩines that security is not linear to the market The security having negative beta coeíĩicient is rẹịected as investment altemative Similarly, security that provides lower rate o f retum than risk-free rate o f retum is rẹịected as investment altemative because such stocks entail some investment risk but they are not compensating the investm ent risk

T he construction o f an optimal portlblio is simplifíed i f a single number

m easures the desirability o f including a stock in ứie optimal portíòlio I f accept the single-index model, such a number exists In this case, the desirability o f any stock is directly related to its excess retum -to-beta ratio

I f the stocks are ranked by excess retum to beta (from highest to lowest), the ranking represents the desirability o f any stock’s inclusion in a portĩolio The num ber o f stocks selected depends on a unique c u to íĩ rate such ửiat all stocks

w ith higher ratios o f (Rị - Rj)/0i will be included and all stocks w ith lower

ratios be excluded To determine vvhich stocks are included in the optimal portfolio, ửie following síeps are necessary:

e' = variance o f a stock's m ovem ent that is not associated w ith the

R i = expected retum o f stock i

R ị = risk-free rate o f retum

After getting the Ci o f each security, investors select highest Ci value that is c*

among all the securities and develop a ranking on all securities Then investors com pare c* with excess retum to beta o f each security Then, the securities

Where,

^ m = variance in the m arket index

m ovem ent o f the m arket index; this is the stock's unsystematic risk

C h a p t e r 2 Literaturc Reviexv

having value greater than c* are selected O nce investors know which securities are to be included in the optimum portíblio, investors must calculate the percent invested in each security The percentage invested in each security

G a = unsystematic risk o f stock /

The above expression determines the relative investment in each security The íírst expression simply scales the weights on each security so that they sum to 1 (ensurefull investment) The residual variance on each security plays an important role in determining how much to invest in each security Then, ứie portfolio retum and

S ta n d a rd d e v ia t io n c a n b e o b t a in e d b y u s i n g t h e e q u a t io n s w h ic h a re u s e d in

Markowitz theory

6 This formula was proved by Edwin J Elton, M artin J G ruber and Manfred w

Padberg in ửieir article ‘Simple Criteria for Optiraal Portíòlio Selection’, Joumal of Fũiance.l976

C h a p t e r 3: R e s e a rc h M c th o d o lo g y

3 RESEARCH METHODOLOGY

3.1 Review otstudies in Vietnamese Stock Market

M arkowitz’s theory o f E ữicient Portfolio Selection is no longer a new topic from the end o f 2006 - the exceptionally boom ing year o f Vietnamese stock market Since then, there have been a few articles and studies about this very theme Typically, there are one article posted on Saga w ebsite about using

M arkowitz model to form optimal portíòlio with the running o f Crystal Ball -

an add-in for Excel; another article o f quite sim ilar content syndicated on giaiphapExcel.com; other than articles, som e universities also have chosen this theme to be one o f the thesis topic for their students

Indeed, this research paper is also about M arkow itz model - M odem Portfolio Selection but it explores a very different aspect o f the model In particular, the study is to íígure out optimal portíòlios by putting into practice M arkowitz model under the circumstance o f each portíolio including only tw o securities;

in addition, the study tries to find out th e very optim um portíolio o f various securities by applying Sharpe model o f portíolio optim isation These two attempts are purported to illustrate the fact that Sharpe model is really a sim pliííed variant o f M arkow itz’s - that th e requirem ents o f inputs & steps to

be taken by Sharpe’s are less tíjiat\hose o f M arkow itz’s theory But these two mcxiels do have som e limitations w hich will be discussed in the later part; thereíbre, the m ain focus - and also, the very purpose o f this study - is on applying Excel into forming diíĩerent optim al portfolios subject to different level o f risk and retum , which is likely to suit a variety o f investors’ preferences All data analysis is done in Excel Excel itselí is sừong enough to run any regression or econometric equations necessary to do the research Recently, an add-in “Crystal Ball” has been popularly used to solve the optim isation problem o f M arkowitz model Some other research has even utilised some program m ing language like FORTRAN to solve the same

C h a p t e r 3 R e s e a rc h M c ih o d o lo i’)

problem It, in fact, is the language vvhich was written based on the

m athem atics that Markowitz applied in his problem o f optimisation in 1959 Again, this study only uses Excel 2003 in solving that very optimisation problem

3.2 The Research’s Hypotheses

T he research hypothesises that:

s Investors are rational and behave in a m anner as to maximise their utility

w ith a given level o f income or money

s Investors have free access to fair and correct iníòrm ation on the retum s

s Investors base decisions on expected retum s and variance or Standard

deviation o f these retum s from the mean

s Investors prefer higher retum s to lower retum s for a given level o f risk

T hey are actually the assumptions on which M arkowitz model is based on A portíolio o f assets is under the above assumptions for a given level o f risk

O ther assets o r portfolio o f assets offers a higher expected retum with the same

o r low er risk o r lower risk w ith ứie same or higher expected retum

D iversification o f securities is one method by vvhich tíie above objectives can

be secured The unsystematic and company related risk can be secured The unsystem atic and company related risk can be reduced by diversiíícation ữito various securities and assets w hose variability is different and oỡsetting o r put

in diíĩerent words which are negatively correlated o r not correlated at all

O ne additional hypothesis is that the data on past retum s are normally distributed If it is violated, the Sharpe model may not tum out a true outcome

C h a p t e r 3 R e s e a rc h M e th o d o lo g y

as it should be This is limitation o f the traditional Sharpe ratio vvhich will be elaborated in the chapter o f the research weaknesses Yet, as long as a model can explain an econom ic phenomenon under certain conditions o f assumptions,

it is still a model o f value

3.3 Sampling and Method oíAnalysis

The study is based on the risk and retum data o f a sam ple o f 26 stocks listed in

HO SE (refer to Annex 1) The selection o f sample is based on the follow ing criteria:

s Since the study covers a period o f ĩiscal year 2005 - 2007, only those

com panies whose common stocks vvere listed before 31/12/04 are seiected as sample companies

s O nly those stocks offering m ean retum above the risk free rate are

selected as samples The securities that provide lover retum than the risk-free rate are excluded because such stocks entail som e investment risk but their revvards do not com pensate for the risk exposure (Sharpe, 1956)

G iven these criteria, 26 stocks are selected to be the sam ples for the study This

is th e total num ber o f listed stocks on H O SE at the tim e T h eừ m ean retum s are all calculated and all are greater than the risk ữ e e rate In this research, ửie prim e rate syndicated by State B ank o f V ietnam is chosen to be the proxy for risk-free rate SBV prime rate for 2005, 2006 and 2007 is 7.8%, 8.25% and 8.25% respectively T he risk-free rate o f interest used in this paper is the average o f those three interest rates; thereíore, the risk-less rate used is 8.1%

pa The num ber o f total listed com panies and num ber o f sam ple stocks in each industrial category are gi ven in Table belovv

vndirect.com But the source o f data used in the rese

3.4 Procedures offorming optimal portíotios under

Markowitz model

It is w orth to note that M arkowitz’s Portíòlio Selection Theory is designed to select a large num ber o f securities to portíolios B ut his original & empirical research was done by forming two-stock portíòlios am ong 9 securities and cash T hus, with the aim to test the m odel’s effectiveness in Vietnamese stock market, the quite sim ilar procedures are períbrm ed, i.e fm ding optimal two- stock portíòlios from 26 listed securities Following is the steps taken:

s M ean retum , Standard deviation, beta o f each stock and the m arket has

been calculated The retum here is the sim ple retum , i.e r = (r2- riVr i- Som e one m ight w onder why log retum is not used There might not exist a signiíícant difference between them then; however, ứie simple retum is used ju st because HOSE has not totally applied the contữiuous order-m atching system yet

^ CZ26 (= 325) two-stock portíolios have been formed from 26 sample stocks and correlations o f these 435 sets have also been determined.split and dividends These data can be found in : ite as

C h a p te r 3: R e s e a rc h M e th o d o lo g y

s Out o f these 325 com er portíblios, 50 sets o f two-stock portíolios with

least correlation have been selected for consideration

■/ Risk m inim ising weights, portíòlio retum and Standard deviation have

been calculated for these 50 sets o f two-stock portíolio W eights are determ ined by th e weight calculation formula for minimum variance portíolio:

The steps here do make sense M arkowitz does em phasise that the quality o f a portíolio is absolutely different ữom th at o f individual securities His model is concem ed w ith the analysis o f portíolio containing large num ber o f securities And in such a portíolio o r even the one involving large num ber s o f coưelated securities, the importance o f variance o f individuals shrinks in size, Ieaving room for the role o f covariance or correlation am ong the securities to be selected into a portíolio A security adds m uch or little to th e variability o f a iarge portíolio, not according to the size o f its ow n variance, but according to the sum o f all its covariance w ith the other securities o f the portfolio

A íter all param eters o f those 50 sets o f tw o-stock portfolios including w eights, portfolio retum and risk are calculated, tìiey will b e plotted on a M ean - SD plane A nd hence, the eữ ícient frontier will be íígured out The portíolios among them w hich lie on the curve would be the efficient ones The others are attainable but inefficient choices

3.5 Procedures offorming an optimal portíolio under Sharpe model

ư n lik e th e case above, on putting Sharpe mcxiel into operation aimed at solving this very same optim isation problem , only one optimum portíolio vvill be

7 The formula is taken from Investment Portfolio Management textbook, page 231

C h a p t e r 3 R e s e a rc h M ethodoloux

worked out It might include a great number o f securities, not just two The steps are as follows:

s Excess retum -to-beta ratio has been calculated for each stock under

revievv and they have been ranked from highest to lowest

s A cu t-o ff point has been determined.

s Optim um portfolio has been íorm ed from those stocks that have higher

excess retum -to-beta ratio than the cut-off point

A fter such stocks have been found out, the last step is to find the weights for them T he íòrm ula applied is the one mentioned in the literature revievv

C alculation o f portfolio retum and SD is similar to the M arkowitz model case

3.6 Procedures ofForming Optimal Portíoiios by Using Excel

The Excellence o f Excel has brought about quite m uch com fort and convenience The M arkowitz and Sharpe m odels do have some inherent lim itations which will be discussed in the limitations chapter below Here states the steps that should be followed so as to come up with the satisíĩed outcome:

First, go to ToolsVAdd-ins:

wa«ttxẩ-i D - V N h É i t'w k đ « ự i ►ơ < ■TI BTC CAN M U BTC ca u o UAf UÀS Q U

7 » u i 0 » u LA/ 0 3 l « B * 0 O O H 0M 7 t t 4 3 # n u Q.4MK01I’ •J**i51TO o m ề o t s 0441MTMÌ O M M Ỉ3 5

Mần \l T77 CliOOOMI n

O ì i s m m 0C 7M P92

A MT7TM71 0073*4*0 02XMI«14 ữiìinmn

m ' » a * 0 « 2 KH a i n r o * * 0 j m * 0 4 í O ỉ u m m u m u n ? txmiiM c a a o M T i eM O K T T

l õ n a t M . 0TJ n rm 0 « * G » e « U K D » 04Ì(*X77| • 4M 0Q27 « 1 tman 0 U X 7 N 7

IM TW • A 0 « Mm 0J2J<P4M •Itma* 0 X 3 KOI 71 n I m i u • S T O M 9 oasirum 0 ( 7 2 1 ( 7 « 0 1 M 0 D I7

B j n* B*ơ» ► 0 « 2* T * 0 1®JWỒJ tí oimsm 0 ja » fM ỉ4 C2XứáKtỉ 01M M M Ỉ o x m M > j O JlO C i»4

0 »

M i m 01»

M T m

0 »

n « « w

i a m

0 1 * 6 * 7

N*

F igure 6: Excel A dd-ins

Next, put a tick to “Analysis ToolPak” and “ Solver A dd-in” They are two built-in add-ins in Excel 2003 Then click OK

&dd-Ins avalablo:

Analysts TodPak - VBA

Condtional Sum Wfeard Euro Currency Toob

Internet Asâstant VBA

C h a p te r 3 R o s c a rc h M c th o d o lo g y

The next step is to create a Correlation matrix Make sure that the daily retums

o f 26 stocks have been calculated Then, go to Tools\D ata Analysis

31MB34M1

axmrai aousviM

F igure 8: Excel D ata Analysis

Click to choose Correlation M atrix in the box “ Data A nalysis” and enter

Díil <i Analysis

A n d p is T o o b

Covariance

Descrlptìve Stabsttcs Exponenttàl Smoothing

F-Test Two-Sampte fo r V âriânces

J

C ancd

tWP

F igure 9: Data Analysis Box

After that, the follow ing box w ill appear:

C h a p t e r 3: R e s e a rc h M e th o d o lo g y

L o r r e l a t i o n

In p u t InputR anọc:

In this box, ju st deííne the Input and O utput Range and click ok A Correlation

M atrix will com e out, like in the case o f this research w ith 26 stocks: