MASTER´S THESIS IN MATHEMATICS /APPLIED

MATHEMATICS

Using Beta as an Investment Strategy

(A study of the Swedish Equity Market)

by

Ojeabulu Godspower

Okoye Chukwuemeka

Kandidatarbete i matematik / tillämpad matematik

DIVISION OF APPLIED MATHEMATICS

MÄLARDALEN UNIVERSITY

SE-721 23 VÄSTERÅS, SWEDEN

DIVISION of APPLIED MATHEMATICS

___________________________________________________________________________

Master‟s thesis in mathematics / applied mathematics

Date:

2010-06-03

Project name:

Using Beta as an Investment Strategy (A study of the Swedish Equity Market)

Authors:

Ojeabulu Godspower, Okoye Chukwuemeka

Supervisor:

Lars Pettersson

Examiner:

Anatoliy Malyarenko

Comprising:

30 ECTS credits

_______________________________________________________________________

Abstract

This study investigates the effect of using the different benchmarks stated above to calculate the

beta of some Swedish stocks and to form a high risk stock vis-a-vis a low risk stock. The stocks

will be combined in different forms (scenarios) i.e. High beta stocks, low beta stocks and a

mixture of both high and low beta stocks to form a portfolio of stocks and tested to see the

performance level of the individual scenarios.

Acknowledgement

We would like to thank our supervisor Lars Pettersson for his comments and support throughout

our study years and also to Anatoliy Malyarenko for his inspiration and guidance.

Table of Contents

1. INTRODUCTION ................................................................................................................................. 1

1.1 Purpose .............................................................................................................................................. 2

1.2 Data & Methodology ....................................................................................................................... 2

1.3 Limitations of this Study ................................................................................................................... 3

2. THEORETICAL FRAMEWORK .......................................................................................................... 4

2.1 Capital Asset Pricing Theory (CAPM) ............................................................................................... 4

2.2 Capital Market Line ............................................................................................................................ 6

2.3 Security Market Line .......................................................................................................................... 7

2.4 Beta ..................................................................................................................................................... 9

2.4.1 Beta Volatility & Correlation ..................................................................................................... 10

2.4.2 Benchmark ............................................................................................................................. 10

2.4.3 Beta Estimation .......................................................................................................................... 10

2.4.4 What Index to use ...................................................................................................................... 11

3. MEASURING PORTFOLIO PERFORMANCE .................................................................................. 12

3.1. Treynor Index ................................................................................................................................... 12

3.2 The Jensen Measure .......................................................................................................................... 14

3.3 Sharpe Ratio ...................................................................................................................................... 15

4. PORTFOLIO SELECTION &ANALYSIS .......................................................................................... 16

4.1 Stock selection .................................................................................................................................. 16

4.2 Choice of benchmark ........................................................................................................................ 16

4.3 The Risk Free Rate ............................................................................................................................ 17

4.4 Portfolio Selection ............................................................................................................................ 17

4.5 Calculating the Return ...................................................................................................................... 17

4.6 Calculating the beta ........................................................................................................................... 18

4.7 Ranking of Stocks according to their Betas ...................................................................................... 18

4.8 Portfolio composition ........................................................................................................................ 25

6. CONCLUSION ...................................................................................................................................... 32

References ................................................................................................................................................... 33

Appendix A: Derivation of Capital Asset Pricing Model (CAPM) ............................................................ 34

Appendix B ................................................................................................................................................. 39

List of Figures

FIGURE 1: GRAPH SHOWING THE CAPITAL MARKET LINE .............................................................................................. 7

FIGURE 2: GRAPH SHOWING THE SECURITY MARKET LINE ............................................................................................ 8

FIGURE 3: RELATIONSHIP BETWEEN TR AND INVESTORS UTILITY ................................................................................ 13

FIGURE 4: CALCULATING BETA IN EXCEL .................................................................................................................... 18

List of Tables

TABLE 1: RANKING OF BETAS IN OMXS PI ................................................................................................................. 19

TABLE 2: RANKING OF BETAS IN OMXS 30 ................................................................................................................. 20

TABLE 3: RANKING OF BETAS IN MSCI GROWTH ........................................................................................................ 20

TABLE 4:RANKING OF BETAS IN MSCI VALUE ............................................................................................................ 21

TABLE 5: HIGH BETA STOCKS WITH OMXS PI AS THE INDEX ...................................................................................... 22

TABLE 6: LOW BETA STOCKS WITH OMXS PI AS INDEX ............................................................................................... 22

TABLE 7: HIGH BETA STOCKS WITH OMXS 30 AS INDEX ............................................................................................. 22

TABLE 8: LOW BETA STOCKS WITH OMXS 30 AS INDEX .............................................................................................. 23

TABLE 9: HIGH BETA STOCKS WITH MSCI GROWTH AS INDEX .................................................................................. 23

TABLE 10: LOW BETA STOCKS WITH MSCI GROWTH AS INDEX ................................................................................ 23

TABLE 11: HIGH BETA STOCKS WITH MSCI VALUE AS INDEX ................................................................................... 24

TABLE 12: LOW BETA STOCKS WITH MSCI VALUE AS INDEX ................................................................................... 24

TABLE 13: SHOWING PORTFOLIO MANAGER 1WITH HIGHER BETA STOCKS CONSTRUCTED WITH OMXS PI AS INDEX .. 26 TABLE 14: SHOWING PORTFOLIO MANAGER 3 WITH LOWER BETA STOCKS CONSTRUCTED WITH MSCI GROWTH AS

INDEX .................................................................................................................................................................. 27

TABLE 15: HIGH BETA PORTFOLIOS ............................................................................................................................. 28

TABLE 16: LOW BETA PORTFOLIOS .............................................................................................................................. 29

TABLE 17: COMPARISON OF BETA PORTFOLIOS ........................................................................................................... 30

1

1. INTRODUCTION

“An integral part of any decision-making should be the evaluation of the decision”. (Elton,

Gruber, Brown, & Goetzmann, 2007).

Evaluation of a stock or portfolio performance is an important issue for every portfolio manager

or investor. It is a process in which an investor is able to determine how well a fund has

performed in relation to another. The ability to estimate the expected return on any investment

that is made in the stock market is a huge challenge. Most investors tend to base their success on

just returns and not the risk involved while investing. Before 1950, investors or portfolio

managers did their evaluations on the basis of only returns. In 1952, this process changed when

Harry Markowitz introduced the paper “Portfolio Selection” (Markowitz, 1952) in which the

idea of Modern Portfolio Theory was born. He proposed that investors should allow for the rate

at which they capitalize returns from particular returns to vary with the risk i.e. investors should

be compensated for every additional risk.

The goal of an investor may be long term or short term. Haim et al (1994) asks this basic

question “how should the results of current investment research that question the appropriate

measure of risk be translated into an appropriate investment strategy for an individual

investor?”.

To measure the performance of a portfolio, three sets of measurements tools are commonly used;

Treynor, Sharpe and Jensen ratios. This three combine the risk-return performance into a single

value, but with each being slightly different from the other (Pareto).

Treynor (1965) was the first to create a means of measuring portfolio performance. He measured

the risk of a portfolio with beta. Brenner & Smidt (1977 cited in Alexander & Chervany, 1980,

p.123) states that the beta coefficients must be accurately estimated for the following reasons; to

helps for an easier understanding of the risk-return relationships in the capital market and also

for making investment decisions. Antony & Jeevanand (2007) claim that the measure beta is an

inevitable tool in portfolio management and that any asset pricing without beta is inconceivable.

2

In investigating the performance of a portfolio, the primary purpose is to compare the returns

from one portfolio with the return on another portfolio. According to Elton et al (2007) there are

different processes. Most investors would want to invest in a security that has a high beta,

because a high beta means high return. So determining the beta of stocks is very important as

they can be ranked from high beta to low beta, these enable the investor to see at a glance which

stocks are to be selected into making a portfolio.

In doing this, it is very important to acknowledge some of the difficulties in determining the beta

of stocks. Baesel (1974 cited in Alexander & Chervany, 1980, p.123) found that “the stability of

beta is dependent upon both the estimation interval used and the extremity of the beta chosen”.

One of the problems we are going to be looking at was emphasized by Roll who pointed out that

Beta is an ambiguous risk measure. The Capital Asset Pricing Model (CAPM) model claims that

a high beta generates a high return than the Low beta. Is this applicable to the Swedish Equity

market? The Beta when using the OMX Stockholm 30 is not the same as the Beta when using the

OMX Stockholm mid cap in the market. What does this mean? Is it that different benchmark

gives different beta for the same stock? Is there a relation between historical Beta and the

expected return on the Swedish equity market?

1.1 Purpose

The purpose of this study is to investigate the effect of using the different benchmarks stated

above to calculate the beta of some Swedish stocks and to form a high risk stock vis-a-vis a low

risk stock. The stocks will be combined in different forms (scenarios) i.e. High beta stocks, low

beta stocks and a mixture of both high and low beta stocks to form a portfolio of stocks and

tested to see the performance level of the individual scenarios.

1.2 Data & Methodology

The data to investigate consists of 16 large cap stocks listed on the Swedish stock exchange.This

study uses the historical returns of the monthly adjusted closing prices of this 16 stocks within a

time frame of 16years from 1994 to 2010 (see appendix) and 4 indexes (OMXS PI, OMXS 30,

3

MSCI Growth and MSCI Value). Sveriges Riksbank Treasury bills with a 1 month maturity with

a time period from May 1994 to May 2010, has been used.

There are different ways to calculate the beta of a stock but this study is going to be using the

slope function in excel. The Treynor index will be used to measure the performance of the

portfolio.

1.3 Limitations of this Study

There is a short time to come with a comprehensive report as expected. With this in mind, this

study will skip the basic concepts of Financial Mathematics and will refrain from defining or

explaining certain terms in this report.

4

2. THEORETICAL FRAMEWORK

In this section of the study, the theories guiding the research and analysis of this study will be

presented. It will include the description of the Capital Asset Pricing Model (CAPM), Beta, the

concepts of measuring the portfolio performance and the risk adjusted performance measure

based on CAPM.

2.1 Capital Asset Pricing Theory (CAPM)

The theory behind the Capital Asset Pricing model (CAPM) was first introduced by Jack Treynor

(1961) with subsequent work by William Sharpe (1964) and John Lintner (1965). The CAPM by

Sharpe & Lintner marked the birth of the asset pricing model (resulting in the Noble prize for

Sharpe in 1990). CAPM is widely used in estimating the costs of capital for firms and evaluating

the performance of managed portfolios. “The theory of the model creates a way of measuring

risk and the relation between risk and rate of return” (Fama & French, 2004).In the CAPM

model, beta serves as an index of a security‟s systematic risk.

The model also claims that a positive relation between risk (beta) and expected return would

show that the model is a good model for estimating future expected returns.

CAPM builds on the theory expounded by Harry Markowitz (1959) which assumes that investors

are risk averse and when they decide to choose a portfolio, what is of concern to them is the

mean and the variance of the one-period investment return.

CAPM can calculate the investment risk and the return on that investment. Returns can be

defined as the rate which the value of assets changes within a defined time period. Risk also as

volatility, is the uncertainty in what a security price or return will be at a certain point in the

future. An investment contains two types of risk; “Systematic Risk and Unsystematic Risk”.

Haim et al defines the Systematic risk (also known as market risk) as the risk associated with the

market as a whole. This risk cannot be diversified away. “Beta” is the measure, which captures

the influence of the systematic risk. Haim et al. states that the unsystematic risk is the unique or

firm-specific risk associated with an individual security or stock. This risk can be diversified

away or eliminated by an investor. Investing in a wide range of securities can lead to the

5

elimination of the negative performance of one security by the positive performance of other

securities or stocks.

The capital asset pricing model (CAPM) is based on the following set of assumptions1:

All investors have identical expectations about the assets with inputs into the portfolio

decision.

There is limitless borrowing and lending of risk free assets.

The rates for the risk free borrowing and lending is equal.

The market has a fixed number of assets.

Perfectly efficient capital markets.

There are no arbitrage opportunities.

There are no restrictions on short selling.

Returns (assets) are distributed by normal distribution.

CAPM comprises the risk and return where risk is characterized through variance. Every point

on a security market line can be attained for different combinations of risk and return and the

combination of securities or stocks with different risk –return traits. In CAPM, there is a positive

connection between risk and return. The more risk one takes the more returns he should expect.

The formula for the CAPM:

NiRRERRE iMfMfi ,,1,

Where,

E(Ri) = Expected return on any asset or security (i)

Rf = risk-free interest rate (f)

E(RM) = Expected return of the market (M)

E(RM) Rf = the premium per unit of beta risk

βiM = the sensitivity of the anticipated return of asset (f) to changes in the anticipated

return of the market (M).

1 (Fama & French, 2004)

6

The market beta of asset i is the slope in the regression of its return on the market return. Beta

measures the sensitivity of the asset´s return to that of the market return. Another interpretation

of Beta that suits the portfolio paradigm which is the genesis of CAPM, is that the “risk of the

market portfolio as measured by the variance of its return (beta of the market) is a weighted

average of covariance risks of the assets” (Fama & French, 2004).

2.2 Capital Market Line

The capital market line is derived from CAPM and it shows the rates of return for efficient

portfolios in relation to the market portfolios‟ beta (risk).The capital market line illustrates the

rates of return for efficient portfolios based on the risk-free rate of return and the level of risk

(standard deviation) for a particular portfolio. Derived by drawing a tangent line from the

intercept point on the efficient frontier to the point where the expected return equals the risk-free

rate of return.

The CML is viewed to be better than the efficient frontier since it considers the addition of a

risk-free asset in the portfolio. The equation for the CML is:

e

M

fM

fe

RRERRE

Where,

E(Re) = the expected return on portfolio e

Rf = the risk free rate

E(RM) = the expected return on the market portfolio M

σe = the standard deviation of portfolio e

σ M = the standard deviation of the market portfolio M

7

Figure 1: Graph showing the Capital Market Line2

2.3 Security Market Line

The relationship between beta (risk) and the expected return is known as the security market line

(SML). The SML is a graph of the capital asset pricing model. The slope of the line is given by

(RM – Rf) i.e. the units of return over the risk-free rate per unit of systematic risk. This linear

relationship shows that an investor can increase his expected return by increasing the risk as

according to CAPM securities with higher risk must have a higher return in order to compensate

the investor for the risk which confirms the CAPM risk aversion assumption. The capital asset

pricing model expresses the risk-reward structure of assets.

The graph below shows the risk-return relationship in the form of Beta and also it can be seen

that the market portfolio to the point beta is equal to one.

The equation for the security market line is:

M

iM

M

fM

fi

rrrR

2 http://www.investopedia.com/study-guide/cfa-exam/level-1/portfolio-management/cfa9.asp

8

Where the slope of the Security Market line for the is given as

i

fi rR

Then

fMifi rrrR

Where;

= the expected rate of return of the asset or portfolio i

= the risk free interest rate

= the asset´s beta

= the expected excess rate of return on the market M

= the standard deviation of the portfolio or asset

= the standard deviation of the market portfolio M

This linear correlation between the systematic risk and return is the basis for of the CAPM. The

plot below shows the relationship between risk, the expected return, and the security market line.

Figure 2: Graph showing the Security Market Line3

3 http://www.investopedia.com/study-guide/cfa-exam/level-1/portfolio-management/cfa3.asp

iR

fr

i

fM rr

i

M

9

2.4 Beta

Beta (β) also known as the “beta coefficient” denotes different meanings but the underlying

significance is that it has a relationship with risk and the return of the market. The beta (β) of a

stock or portfolio could be defined as the response of the stock to the changes in stock market.

To put this differently, Beta value is a measure of a stock‟s volatility with respect to market

volatility. Sharpe (1963 cited in Alexander & Chervany, 1980, p.123) defined beta (β) as the

slope term in the simple linear regression function where the rate of return on a market index was

the independent variable and a security or stock´s return is the dependent variable

Haim et al. states that the beta (β) of a stock measures the volatility of a stock or security in

relation to the market. Beta is a measure of the volatility, or systematic risk, of a security or a

portfolio in comparison to the market as a whole. The risk measured by beta is undiversified. It is

a popular indicator used by many traders and investors to facilitate their trades. Beta is expressed

by taking the market volatility 1, and beta values of a stock are calculated as a measure of how

much the stock price moved from this market volatility. An asset or stock that has a beta of 0 is

uncorrelated with the market i.e. it does not follow the market trend. A positive beta indicates the

asset is correlated to the market. An asset with a negative beta shows that the asset increases or

decreases in worth depending on the upward and downward movement of the market. In like

manner, a beta of 1.0 indicates that the investment's price will move in correlation with the

market. A beta of less than 1.0 indicates a less volatile investment and, correspondingly, a beta of

more than 1.0 indicates that the investment is more volatile than the market. For example, if a

fund portfolio's beta is 1.4, in theory it is 40% more volatile than the market.

The beta coefficient measures the part of the asset's statistical variance that cannot be mitigated

by the diversification provided by the portfolio of many risky assets, because it is correlated with

the return of the other assets that are in the portfolio. Beta can be estimated for individual

companies using regression analysis against a stock market index.

The formula for the Beta Coefficient is;

Beta (β) = Covariance (stock versus market returns) / Variance of the Stock market

=

m

ms rrCov2

,

,

10

Where;

rs = Stock return

rm = Market return

σ2 m = Market variance

2.4.1 Beta Volatility & Correlation

The relationship between beta and volatility is denoted by the following formula:

rm

Where r denotes the risk, σ denotes volatility of the asset and σm denotes the market volatility.

Beta shows the relationship between volatility and correlation. Beta can determine between two

stocks which is more risky based on their level of volatility and correlation vis-vis low-high and

high-low.

2.4.2 Benchmark

In the securities or stock market, market indices such as the MSCI World, the FTSE 100 are used

by beta as a benchmark. The asset and the benchmark selected should be alike. The decision to

select an index should not be based on the ability of the index to replicate the portfolio under

consideration.

2.4.3 Beta Estimation4

To estimate the beta of a stock, two sets of data are necessary;

Returns or closing prices for the stock or asset being examined.

Returns or closing prices for the index you're choosing as a proxy for the stock market.

These returns can be daily, weekly or any period. Many a time, the beta value is calculated using

the month-end stock price for the security being examined and the closing price at the end of the

4 http://www.money-zine.com/Investing/Stocks/Stock-Beta-and-Volatility/

11

month of the “given” index being used. Then the standard formulas from linear regression is

used to obtain the slope. The slope of the line is the estimated Beta.

2.4.4 What Index to use

Traditionally, the evaluation of a Portfolio performance involves using an index or benchmark to

which the portfolio return will be compared. The following weighting methods are used in

computing the market index;

The price-weighted method: An example is the Dow Jones Industrial Average

The market value-weighted index: Examples include the Nasdaq Composite Index,

S&P 500, Wilshire 5000 Equity Index, Hang Seng Index, and EAFE Index. To calculate

this index, the market value of the securities included in the index at time t, is divided by

the market value of the securities at time 0.This is then added to the index base at time 0.

The equal-weighted method: This index consists of companies whose stocks in the

index have the same weight. The size of the company does not matter. Examples of this

kind of index include the S&P 500 EWI Every stock in the index has the same weight,

regardless how large or small the company is.

To evaluate the performance of a portfolio, the benchmark (index) to be used has to have the

same return measurement method for the index and the portfolio under evaluation.

12

3. MEASURING PORTFOLIO PERFORMANCE

To measure the performance of a portfolio, the adjustments are based on the security market line

and the capital market line. This measures look at the risk–return relationship. The Capital

Market line performance measures is the Sharpe ratio while the Jensen´s Alpha and the Treynor

index are the security market line (SML) based performance measures.

3.1. Treynor Index

Jack L. Treynor (1965) was the first researcher to provide investors with a composite measure of

portfolio performance that also included risk. Before then, investors mostly measured portfolio

performance based on the returns of the portfolio. We know that some investors are risk takers

while others are risk adverse but Jack L. Treynor found a performance measure that could apply

to all investors regardless of their risk preferences. This risk adjusted measure was used to rank

the performance of mutual funds. Treynor also suggested that risk has two components: the risk

arising from fluctuations in the market (Systematic risk) and the risk arising from the fluctuations

of individual securities (unsystematic risk).

The concept of the security market line (SML) was also introduced by Treynor. The SML

defines the relationship between portfolio returns and the rates of returns of the market.The slope

of the line measures the relative volatility between the portfolio and the market (as represented

by beta). The Treynor Index uses the SML as a benchmark.

The treynor measure can be defined as the difference between portfolio return and the risk free

rate divided by the Beta (Portfolio Return- Risk-Free Rate)/ Beta.

Represented mathematically as:

p

fp

P

rrT

where

13

rp = portfolio rate of return

rf = risk free rate of return

βp= portfolio beta or sensitivity of portfolio to market changes.

When rp > rf and βp > 0, a larger Treynor value is gotten implying a better portfolio for the

investors regarding their individual risk performance.

When rp < rf and βp > 0, Treynor becomes negative since the portfolio performance is poor.

When βp < 0 implying a negative beta and the portfolio performs superbly.

Figure 3: Relationship between TR and investors utility

The Treynor ratio (TR) is affected by different borrowing and lending rates and by the variation

of beta across periods. The Treynor index gives the slope of the security market line. The graph

above shows that the higher the Treynor, the better the ranking of the portfolio and can be seen if

one introduces indifference curves of a risk-averse investor. With a greater TR, higher

indifference curves of a risk-averse investor can be attained and his utility will be greater.

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3.2 The Jensen Measure

Proposed by Michael C. Jenson, this measure uses CAPM as its underlying theory and also

calculates the excess return that a portfolio generates over its expected return. It is a measure of

the positive or negative abnormal return relative to the return predicted by the CAPM. This

measure is also known as Alpha. “The Jensen ratio measures how much of the portfolio's rate of

return is attributable to the manager's ability to deliver above-average returns, adjusted for

market risk” (Pareto, n.d). The higher ratio means better risk-adjusted returns. A portfolio with a

frequently positive excess return will have a positive alpha, while a portfolio with a consistently

negative excess return will have a negative alpha.

The formula is broken down as follows:

Jensen's Alpha = Portfolio Return – Benchmark Portfolio Return

Where: Benchmark Return (CAPM) = Risk Free Rate of Return + Beta (Return of Market –

Risk-Free Rate of Return).

Mathematically, this can be written as:

fMpfpp rrrr

Where

p = Jensen Alpha

pr = expected total return of an asset or portfolio

fr = risk free rate of return

Mr = expected return of the market

βp = beta or systematic risk of the asset or portfolio

Alpha represents the return gap between the return of the portfolio and the return for the

systematic risk of portfolio (p).

The Treynor index and the Jensen measure are both related because they both assume that

investments are fully diversified and therefore taking into account only the systematic risk.

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3.3 Sharpe Ratio

The Sharpe Ratio shares an identity with the Treynor measure, except that in the case of the

Sharpe Ratio, the risk measure is the standard deviation of the portfolio instead of the systematic

risk, as represented by beta. In contrast with the Treynor measure, the Sharp Ratio evaluates the

portfolio on the basis of both rate of return and diversification. The Sharpe Ratio is more

appropriate for well diversified portfolios, because it more accurately takes into account the risks

of the portfolio. The Sharpe Ratio measures a portfolios average performance over the risk-free

rate per unit of total risk of the portfolio. It can be defined as:

(Portfolio Return-Risk free rate) / Standard deviation

Mathematically, it can be represented as

p

fRpRpSR

Where,

R(p) = expected return of the portfolio (p)

R(f) = risk free rate (f)

σ (p) = Standard deviation of portfolio (p)

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4. PORTFOLIO SELECTION &ANALYSIS

Most investments are carried out by observing the return and risk. There is an ambiguity when

risk. The beta of a stock is what most investors use to identify the risk involves in an investment

but beta in its self has become an ambiguous measure.

Richard Roll pointed out this problem in 1977 when he said that beta is not unambiguous

measure; he said that the beta when using the S&P 500 is not the same as the beta when using the

Dow Jones index as the market. This means that changing the definition of the market portfolio

can change the beta and the ranking of the portfolio. So in evaluating portfolio performance, it‟s

not just enough to look at the beta of the portfolio but to consider some other factors in ranking

or selecting a portfolio

4.1 Stock selection

All stocks are large caps and are all adjusted for dividends. It was provided by Nasdaq OMX and

MSCI BARRA websites. We will also be considering equally weighted portfolios and these

portfolios will be divided into eight parts each assigned a portfolio manager to make our work

easier.

4.2 Choice of benchmark

For the purpose of this thesis, the authors will be looking at 4 indexes in Sweden, the OMXS PI,

OMXS 30, MSCI GROWTH and MSCI VALUE. MSCI Barra is a leading provider of

investment decision tools.They have product ranging from index to portfolio risk and

performance analysis. The authors decided on this as a benchmark because MSCI Barra

calculates more than 120,000 Equities and indices from around the world and has formed a

global equities benchmark, this has enabled the authors to expand their scope and to get a more

accurate result. Other indexes considered by the authors are the OMXS price index (PI) and the

OMXS 30. The OMXS 30 indexes are made up of mainly large cap stock.

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4.3 The Risk Free Rate

The Swedish Treasury bill with one month maturity was used as a proxy to the risk free asset. It

was taken from Riksbank website and under the same period under consideration i.e. June 1994

to May 2010.

4.4 Portfolio Selection

Before we look into how the portfolio were selected, we want to consider the stock selections

first and the various calculations that will result into the portfolio selections. The authors took

series of large cap stocks from the Swedish stock market, these stocks are listed below:

ABB Ltd Handelsbanken A Gunnebo

Atlas Copco A Fabege Skanska B

ASSAB SEB A

Atlas Copco B Volvo B

Scania B Securitas

Electrolux B Hoganas B

Ericsson B Hennes & Mauritz B

4.5 Calculating the Return

The return on the investment of each individual stock were calculated using excel and since all

data were daily data except the data from MSCI, we took day 2 divided by day 1 and then

subtracting it from 1 to get the return of day2. Then the monthly average was taken, this is to

make sure that all data are on the same time period and format. Since we had our MSCI indexes

in months, the authors had to convert the daily returns to monthly returns using the pivot table

function in excel. After which all returns were converted to percentage.

See Appendix B.

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4.6 Calculating the beta

In order for the authors to be able to rank these portfolios, the betas of each stock had to be

calculated. There are different ways to calculate the beta of stock. Using regression analysis

where by the equation of the line will be given is one way of going about it. To make the work

easy and faster, the authors decided to use the slope function in excel to calculate the beta. The

slope of each individual stock was calculated in relation to the four indexes mention earlier on

this paper.

Figure 4: Calculating Beta in excel

4.7 Ranking of Stocks according to their Betas

In order for the authors to separate high beta stock s from low beta stocks, they had to rank them

in descending order. A stock is said to have a high beta if its value is more than 1. The market

index has a beta of 1 so any stock that has a beta greater than the market index is said to have a

high beta. Same argument can be said about stocks with betas less than one but this time we say

that the stock has a low beta. Having said that, it should be noted that only few of our stock had a

19

beta that is more than one and so the authors took the betas that are close to 1 to be a high beta

and the once that are far less than one to be low betas. For example, a stock of beta 0.6 would be

said to have a high beta compared to stock with beta 0.07. Based on this argument, the authors

ranked the stocks in a descending order.

To avoid confusion, the authors divided these ranking into the number of stock market

indexes, i.e. 4 groups with each having a high and low beta values.

See the tables below

Table 1: Ranking of Betas in OMXS PI

RANKING OF BETAS IN OMXS PI

HOGANANS B 1

ELECTOTLOUX 0.668416306

ATLAS COPCO B 0.397987341

ATLAS COPCO A 0.396797816

GUNNEBO 0.396639782

SKF B 0.377803174

SEB 0.369621569

SKANSKA B 0.329104308

VOLOV 0.318990622

SSAB A 0.308664134

ERICSSON B 0.303613375

SCA B 0.243799057

¨HANDLES BANK 0.187025554

FABERG 0.168183153

H&M 0.13461153

SECURITAS 0.068001539

20

Table 2: Ranking of Betas in OMXS 30

RANKING OF BETAS IN OMXS 30

ERICSSON B 1.948661565

ATLAS COPCO B 1.0326395

ATLAS COPCO A 1.022235579

SEB 0.955262168

VOLOV 0.943447255

SKANSKA B 0.880995557

GUNNEBO 0.854025625

SKF B 0.844757806

ELECTROLUX 0.82943699

SSAB A 0.707782248

H&M 0.645677635

SECURITAS 0.624388131

HOGANAS B 0.614634706

HANDLES BANK 0.544271145

SCA B 0.51958394

FABEGE -0.143240329

Table 3: Ranking of Betas in MSCI Growth

RANKING OF BETAS IN MSCI GROWTH

ERICSSON B 0,070237261

ATLAS COPCO B 0.024215556

ATLAS COPCO A 0.023953977

GUNNEBO 0.022536376

SEB 0.020283407

SKANSKA B 0.020256054

VOLOV 0.020184116

H&M 0.019894235

SKF B 0.019129467

SECURITAS 0.018542184

ELECTOLUX B 0.018455545

21

SSAB A 0.016499671

HOGANAS 0.014009842

SCA B 0.01038242

HANDLES BANK 0.009009131

FABEGE -0.014186589

Table 4:Ranking of Betas in MSCI Value

The aim of the author once again is to construct a portfolio with high beta stocks and also a

portfolio with low beta stocks. Since investors prefer stocks that have a higher beta, the authors

constructed a portfolio with high beta and also portfolio with low beta and compare their

performances. Since we now have the beta of the various stocks and have subsequently ranked

them in a decreasing order, it would be wise to just draw out a benchmark separating the higher

RANKING OF BETAS IN MSCI VALUE

ERICSSON B 0.06819832

ATLAS COPCO B 0.050645907

ATLAS COPCO A 0.050249385

VOLOVO 0.047622779

SEB 0.046463257

SKANSKA B 0.039652898

SKF B 0.038811435

ELECTOLUX B 0.038779991

GUNNEBO 0.033963939

SSAB A 0.033310032

HANDLES BANK 0.033098575

HOGANAS B 0.031129065

SCA B 0.029376906

H&M 0.024639499

SECURITAS 0.020989545

FABEGE -0.023073421

22

betas from the lower betas under each indexes and hence construct a portfolio based on the

result.

The result from dividing the 16 stocks into to the various beta levels are shown below.

Table 5: High Beta stocks with OMXS PI as the index

HOGANANS B 1

ELECTOTLOUX 0.668416306

ATLAS COPCO B 0.397987341

ATLAS COPCO A 0.396797816

GUNNEBO 0.396639782

SKF B 0.377803174

SEB 0.369621569

SKANSKA B 0.329104308

Table 6: Low beta stocks with OMXS PI as index

VOLVO 0.318990622

SSAB A 0.308664134

ERICSSON B 0.303613375

SCA B 0.243799057

¨HANDLES BANK 0.187025554

FABERG 0.168183153

H&M 0.13461153

SECURITAS 0.068001539

Table 7: High beta stocks with OMXS 30 as index

ERICSSON B 1.948661565

ATLAS COPCO

B

1.0326395

ATLAS COPCO

A

1.022235579

SEB 0.955262168

23

VOLOV 0.943447255

SKANSKA B 0.880995557

GUNNEBO 0.854025625

SKF B 0.844757806

Table 8: Low beta Stocks with OMXS 30 as index

ELECTROLUX 0.82943699

SSAB A 0.707782248

H&M 0.645677635

SECURITAS 0.624388131

HOGANAS B 0.614634706

HANDLES

BANK

0.544271145

SCA B 0.51958394

FABEGE -0.143240329

Table 9: High beta stocks with MSCI GROWTH as index

ERICSSON B 0.070237261

ATLAS COPCO

B

0.024215556

ATLAS COPCO

A

0.023953977

GUNNEBO 0.022536376

SEB 0.020283407

SKANSKA B 0.020256054

VOLOV 0.020184116

H&M 0.019894235

Table 10: Low Beta stocks with MSCI GROWTH as index

SKF B 0.019129467

SECURITAS 0.018542184

24

ELECTOLUX B 0.018455545

SSAB A 0.016499671

HOGANAS 0.014009842

SCA B 0.01038242

HANDLES

BANK

0.009009131

FABEGE -0.014186589

Table 11: High Beta Stocks with MSCI VALUE as index

ERICSSON B 0.06819832

ATLAS COPCO

B

0.050645907

ATLAS COPCO

A

0.050249385

VOLOVO 0.047622779

SEB 0.046463257

SKANSKA B 0.039652898

SKF B 0.038811435

ELECTOLUX B 0.038779991

Table 12: Low Beta Stocks with MSCI VALUE as index

GUNNEB 0.033963939

SSAB A 0.033310032

HANDLES

BANK

0.033098575

HOGANAS B 0.031129065

SCA B 0.029376906

H&M 0.024639499

SECURITAS 0.020989545

FABEGE -0.023073421

25

Noticed that we have different ranking with different indexes. Like Richard Roll pointed out,

beta is an ambiguous measure. The various betas with different indexes have given rise to

different ranking, i.e. Eriksson B was ranked number one on the MSCI VALUE chat but was

ranked number eleven on the OMXS PI chat, this is because, the beta of Eriksson on the MSCI

VALUE is different from the beta was have on the OMXS PI. This problem poses a question to

investors, what value of Beta should they choose in order to make their investment decision?

To answer this question we would have to form a portfolio on the various betas to see how they

perform, if it really matters the beta we use in any investment.

4.8 Portfolio composition

The composition of the portfolio will be determined by the various betas, this is so because the

aim of the study is to see if a higher beta portfolio performs better than lower beta portfolio. As

we well know from portfolio theory and from the theory of CAPM, higher beta stock gives rise

to higher returns but does this really give a higher performance?

The authors constructed eight equally weighed portfolios consisting of 8 stocks for each beta

type and benchmark (indexes). The risk free interest rate represented by the Swedish Treasury

bill was taken from Riksbanken from the period under review.The figures below (figures 1 and

2) describe how our portfolio looks like. It was tagged PORFOLIO MANAGER 1 and 3 to make

it a lot easy to understand. The authors assumed different managers with different approach to

investment. We are going to be comparing the performances of the various managers to see if

high beta stocks actually performed better than low beta stocks. These selected portfolio

managers constructed their portfolios based on the OMXS PI and MSCI GROWTH with higher

and lower beta stocks in it respectively. It should be noted that these selected managers are just

examples taken from 4 managers each constructing a portfolio with high and low betas ad with

the 4 different market indexes.

Notice that we also have calculated the Treynor ratio. This ratio will be discussed in details

shortly.

26

Table 13: Showing Portfolio manager 1with higher beta stocks constructed with OMXS PI as index.

HIGHER BETAS

OMXS PI

PORTFOLIO MANAGER 1

Stocks RETURN WEIGHTED

RETURN

Weighted Beta Risk free

HOGANANS B 0.22 0.028 0.0088 0.24

ELECTOTLOUX 0.03 0.004 0.0063 0.24

ATLAS COPCO B 0.05 0.006 0.0063 0.24

ATLAS COPCO A 0.05 0.006 0.0060 0.24

GUNNEBO 0.04 0.005 0.0058 0.24

SKF B 0.07 0.009 0.0050 0.24

SEB 0.05 0.006 0.0049 0.24

SKANSKA B 0.03 0.004 0.0048 0.24

Portfolio Return 0.0675 0.0478 0.24

Treynor ratio -3.608

Excess Return -0.1725

LOWER BETAS

MSCI GROWTH

Portfolio Manager 3

STOCKS AVERAGE

RETURN

WEIGHTED

AVE.RET

Weighted Beta Risk Free

SKF B 0.07 0.070 0.0191 0.24

SECURITAS 0.02 0.020 0.0185 0.24

ELECTOLUX B 0.03 0.030 0.0185 0.24

SSAB A 0.08 0.080 0.0165 0.24

HOGANAS 0.04 0.040 0.0140 0.24

SCA B 0.04 0.040 0.0104 0.24

27

HANDLES

BANK

0.05 0.050 0.0090 0.24

FABEGE 0.15 0.150 -0.0142 0.24

Portfolio Return 0.48 0.0918 0.24

Treynor ratio 2.61

Excess Return 0.24

Table 14: showing portfolio manager 3 with lower beta stocks constructed with MSCI GROWTH as

index

28

5. RESULTS AND DISCUSSION

In this section we present the general result of our thesis, it consists of a condensed form of our

calculations put in one piece. First we look at the result from all the portfolio managers and there

various indexes. The table below shows at a glance the result gotten from the combination of all

high beta portfolio and their various indexes. It should be noted that these managers were not

ranked but was taken at random.

HIGH BETA PORTFOLIOS

PORTFOLIOS -MARKET

INDEX RETURN TREYNOR BETA

EXCESS

RETURN

MANAGER 1-OMXS PI 0.0675 -3.608 0.0478 -0.1725

MANAGER 2-OMXS30 0.049 -0.18 1,0603 -0.191

MANAGER 3-MSCI

GROWTH 0.0475 -6.951 0.0277 -0.1925

MANAGER 4-MSCI VALUE 0.0475 -4.048 0.0476 -0.1925

Table 15: High Beta Portfolios

From the result above, it could be seen that portfolio manager 2 has a higher Treynor value and

the highest beta value. It could also be noted that the excess return or the risk premium is not the

highest when you compare it with the excess return from manager 1. Considering the definition

of the treynor, it would be wise to say that manager 2 has the best portfolio performance even

though it has a lower excess return compared to manager 1. For investors that are only looking at

return, they will choose to go for portfolio manager 1 because he has the highest rate of return

and also excess return. Note that since we have negative values on the excess return column, a

lower negative value means a higher return as compared to others.

Portfolio manager 2 has performed better with the OMXS 30 than any other market index. It

could also be seen that the beta of portfolio manager 2 is also the highest, and its above 1, which

makes it more volatile than the market.

The theory of CAPM did not hold here i.e., a higher beta value did not give rise to a higher

return. This is one of the inadequacies of CAPM and makes it difficult for it to be tested. Richard

29

Roll suggested that using beta as a risk measure can pose a problem in testing CAPM; he said

how you can test CAPM when different benchmark gives different Betas. In an economy like

Sweden, where you have different market index, it shows from this paper that you are bound to

achieve different Betas because Beta is calculated based on a given benchmark or market index

and would pose problems in ranking of portfolio. Portfolio manager 2 had the highest beta but

not the highest return and this was determined by the choice of the market index. Put generally,

using beta as a choice for investment is not really the ideal because the choice of beta is

determined by the index used which was what Roll pointed out.

Treynor ratio in the authors view is the best performance measure; it helps investors decided on where to

put their money. Since Portfolio manager 2 performed better than all other managers, investors should put

their money on that portfolio.

Let‟s consider lower beta portfolios.

Managers 1 to 4 decided to construct a portfolio that is made up of lower beta stocks and to see what the

outcome would be. The table below represents all the results at a glance from the various constructions.

LOW BETA PORTFOLIOS

PORTFOLIOS-MARKET INDEX RETURN TREYNOR BETA EXCESS RETURN

MANAGER 1- OMXS PI 0.0625 -0.819 0.2166 -0.1775

MANAGER 2- OMXS30 0.0590 0.334 0.5428 0.181

MANAGER 3-MSCI GROWTH 0.0600 -15.68 0.0115 0.18

MANAGER 4- MSCI VALUE 0.0600 -7.850 0.0229 -0.18

Table 16: Low Beta Portfolios

The result above does not quite differ from the result we got from all high beta stock(Table 13).

We could see that Manager 1 has the highest return, Manager 2 has the highest beta value and

the highest excess return. And managers 3 and 4 have a slightly higher return.

Once again, manager 2 has a better performance measure (Treynor ratio) and the beta is the

highest in all the portfolios. Even though manager 1 has a good return compared to other,

manager 2 has a better performance ratio and also the highest beta. The risk premium for

manager 2 is slightly higher than other portfolio managers; very close to it is managers 3 and 4.

30

Higher beta stocks for portfolio did not actually bring about higher return in this case and the

OMXS 30 index performed better than all other stock indices that were considered.

Investors that look at return alone will invest in manager 1 because of its higher return but the

portfolio that performs better is from portfolio manager 2.

Let‟s now compare the result of high beta stocks to low beta stock to see if there are differences.

HIGH BETA PORTFOLIOS

PORTFOLIOS -MARKET

INDEX RETURN TREYNOR BETA

EXCESS

RETURN

MANAGER 1-OMXS PI 0.0675 -3.608 0.0478 -0.1725

MANAGER 2-OMXS30 0.049 -0.18 1,0603 -0.191

MANAGER 3-MSCI

GROWTH 0.0475 -6.951 0.0277 -0.1925

MANAGER 4-MSCI VALUE 0.0475 -4.048 0.0476 -0.1925

LOW BETA PORTFOLIOS

PORTFOLIOS-MARKET

INDEX RETURN TREYNOR BETA

EXCESS

RETURN

MANAGER 1- OMXS PI 0.0625 -0.819 0.2166 -0.1775

MANAGER 2- OMXS30 0.0590 0.334 0.5428 0.181

MANAGER 3-MSCI

GROWTH 0.0600 -15.68 0.0115 0.18

MANAGER 4- MSCI VALUE 0.0600 -7.850 0.0229 -0.18

Table 17: Comparison of Beta Portfolios

The tables above show the combination of both scenarios and could be seen that portfolio

manager 2 has performed better than any other managers have. There is an interesting twist, the

lower beta stock preformed best in the two scenarios (portfolio manager 2 in lower beta

portfolio), It has a Treynor value of 0.334 and this is the highest value if you put both scenarios

31

together even though high beta portfolios has the highest beta value (1.0603) .The highest return

was gotten from high beta portfolio with portfolio manager 1 having a beta of 0.0675. While the

highest beta was 1.0603 from manager 2, manager 1 with a beta of 0.0478 had the highest return

and this again has disproved the theory that high beta portfolio yield higher return.

32

6. CONCLUSION

It is not all about the beta of a stock that determines the return on a portfolio, the beta is a good

risk measure but it is better to look at other factors before making a decision on where to invest

our money. The authors had considered two scenarios where low and high beta portfolios were

formed. It could be gathered that the best portfolio does not rely on the highest beta only but on

the performance of that portfolios. It should also be gathered that before investing in a portfolio,

investors should draw a similar scenarios and see what portfolio performs better given the index

used. Performance is what every investor should look upon since beta cannot be relied upon.

This is so because beta depends on the market index used and its varies from one market index to

another. Having a lower beta portfolio does not mean portfolios will perform less; one may have

a higher return but may not perform so well. Our results show that the best portfolios based on its

performance comprised of low Beta stocks. Ordinarily, investors would over look a low beta

portfolio and take their money somewhere else, perhaps to a portfolio consisting of high betas,

but those high Betas did not perform as much as low beta. So why invest in a portfolio with less

performance? To investors performance as well as return should be the driving force to investing

in a given portfolio because when we considered both risk and return as given by Jack L.

Treynor, they stand a good chance of succeeding in their investments all things being equal.

The result from our study shows that basing investment decisions on the beta of a portfolio is not

adequate. One should look more extensively into how the portfolios perform given the various

benchmarks. Portfolios should be ranked based on their performance and not on the betas

however if we had used the standard deviation (Sharp Ratio), this conclusion might have been

different since the Sharp ratio deals with both systematic risk and the unsystematic risk in the

portfolio.

Therefore, we recommend that investors or would be investors should invest in portfolios that

has the highest overall performance regardless of the Beta.

33

References

Alexander, G. J., & Chervany, N. L. (1980, March). On the Estimation and Stability of Beta. The Journal

of Financial and Quantitative Analysis , 123-137.

Antony, C., & Jeevanand, E. S. (2007). The Elasticity of the Price of a Stock and Its Beta. Journal of

Applied Quantitative Methods , Vol. 2, 334-341.

Baesel, J. B. (1971). On the Assessment of Risk:Some further considerations. Journal of Finance , 29,

1491-1494.

Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2007). Modern Portfolio Theory and

Investment (7th ed.). New York: Wiley.

Fama, E. F., & French, K. R. (2004). The Capital Asset Pricing Model:Theory and Evidence. Journal of

Economic Perspectives , 18, 25-46.

Levy, H., Gunthorpe, D., & Wachowicz, J. J. (1994). Beta and an Investor´s Holding Period. Review of

Business , Vol.15 (No. 3), 32-35.

Markowitz, H. (1952). Portfolio Selection. The Journal of Finance , 7, 77-91.

Pareto, C. (n.d.). Measure your Portfolio´s Performance . Retrieved May 04, 2010, from

http://www.investopedia.com/articles/08/performance-measure.asp.

Roll, R. (1977). A Critique of the Asset Pricing Theory´s Part 1:On Past and Potential Testability of the

Theory. Journal of Financial Economics , 4, 129-176.

Sharpe, W. (1994). The Sharpe Ratio. The Journal of Portfolio Management , 1-19.

Treynor, J. L. (1965, Jan/feb). How to Rate Management of Investment Funds. Harvard Business Review

, 63-75.

34

Appendix A: Derivation of Capital Asset Pricing Model (CAPM)

imi rxErxE )()(

where

E(xi) = expected return on the asset i

r = risk free rate

E(xm)= expected return on the market portfolio (S&P index)

i = i-th asset‟s systematic risk (a proportion of market risk)

Optimal investment proportions - each individual investor on the market attempts to reach the

highest feasible market line. The Market line can be found by minimising the standard deviation

o for any given portfolio„s expected return E(xo).

n

i

n

i

iiio rpxEpxE1 1

1)()(

(1)

n

i

n

iji

jijiiio xxCovppp1 1

22 )(2

(2)

Where

iP is the proportion of the portfolio invested in i-th asset.

The function C is defined as follows;

n

i

n

i

iiioo rpxEpxEC1 1

1)()(

(3)

where is a Lagrange multiplier, and the expression in brackets equals zero.

We are trying to find the optimal proportion of each asset, which minimises the risk of optimal

portfolio. The Market line can be found analytically by differentiating the equation above with

35

respect to each pi and with respect to Lagrange multiplier and setting the first derivatives equal

to zero. This yields n+1 equations.

0)(),(222

11

2

1

2

11

2/12

1

rxExxCovpp

p

C n

j

jjo

(4)

0)(),(222

12

21

2

2

22

2/12

2

rxExxCovpp

p

C n

jj

jjo

(5)

0)(),(222

1 1

1

22/12

rxExxCovpp

p

Cn

n

j

jnjnno

n

(6)

01)()(1 1

n

i

n

i

iiio rpxEpxEC

(7)

Let‟s now take the set of equations (4), (5), and (6) and multiply them by p1, p2 , pn ,

We will obtain:

rxEpxxCovpppn

j

jj

o

)(),(1

11

2

11

2

1

2

1

(8)

(9)

rxEpxxCovppp nn

n

j

jnjnnn

o

)(),(1 1

1

22

(10)

n

i

n

i

iiio rpxEpxE1 1

1)()(

(11)

rxEpxxCovpppn

jj

jj

o

)(),(1

22

21

22

2

2

2

2

36

If we sum up equations 8,9 and 10 above , we will obtain

n

i

ii

n

i

n

ijj

jiji

n

i

ii

o

rxEpxxCovppp11 11

22 )(),(1

(12)

The term in the bracket on the left-hand side is a variance of the optimal portfolio o2.

Substituting it for the bracketed expression on the LHS and multiplying the RHS expression

through we obtain:

n

i

i

n

i

iio

o

rpxEp11

2 )(1

(13)

Thus the standard deviation of the optimal portfolio is

rrpxEpn

i

n

i

iiio

1 1

1)(

(14)

At a specific point where 11

n

i

ip we obtain:

rxE mm )( (15)

And hence,

m

m rxE

1

(16)

37

Where m denotes market portfolio, which is optimal for all investors, and m is a standard

deviation of the market portfolio.

Expression

m

m rxE

)( defines the slope of the market line.

Reciprocal of Lagrange multiplier 1/ measures the price of a unit of risk or required increase in

expected return when one unit of risk is added to the portfolio.

Now we can derive the equilibrium relationship between an individual asset‟s expected return

and its risk. Risk reflects the standard deviation of returns on the asset itself and also the

covariance with the returns of all other risky assets in the market.

We can use the set of equations (8), (9), and (10) to derive the general relationship among the

expected returns of all shares and their risk.

In general, the i-th equation of (7) at the point 11

n

i

ip can be rewritten as:

0)(),(1

1

2

rxExxCovpp i

n

ijj

jijii

m

(17)

Solving this equation for the expected return of i-th asset E(xi) we obtain:

n

ijj

jijii

m

i xxCovpprxE1

2 ),(1

)(

(18)

Substituting for from the equation (16), we obtain

n

ijj

jijii

m

m

i xxCovpprxE

rxE1

2

2),(

)()(

(19)

38

Now, recall, that by definition, the covariance of the i-th asset with the market portfolio can be

written as;

n

ijj

jijiimi xxCovppxxCov1

2 ),(),(

Thus the expected return on any risky asset can be written as:

),()(

)(2 im

m

m

i xxCovrxE

rxE

Or alternatively

imi rxErxE )()(

Where

2

),(

m

im

i

xxCov

.

39

Appendix B