MASTER THESIS IN MATHEMATICS/ APPLIED MATHEMATICS A ...

Department of Mathematics and Physics

MASTER THESIS IN MATHEMATICS/ APPLIED MATHEMATICS A Quantitative Risk Optimization of Markowitz Model

An Empirical Investigation on Swedish Large Cap List by Amir Kheirollah Oliver Bj?rnbo Magisterarbete i matematik/till?mpad matematik

Code: MDH.IMA.MAT.0088 (2007) 20p ? AF

ABSTRACT

This paper is an empirical study on Harry Markowitz work on Modern Portfolio Theory. The model introduced by him assumes the normality of assets' return. We examined the OMX Large Cap List1 by mathematical and statistical methods for normality of assets' returns. We studied the effect of the parameters, Skewness and Kurtosis for different time series data. We tried to figure it out which data series is better to construct a portfolio and how these extra parameters can make us better informed in our investments.

1 We have chosen 42 stocks from this list from different sectors of length 10 years. The complete Large Cap list is available at appendix X.

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Table of Contents

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

DATA AND METHODOLOGY ................................................................................................................................ 2

THEORY ON MODERN PORTFOLIO THEORY .......................................................................................... 3

MODERN PORTFOLIO THEORY (MPT)................................................................................................................. 3 RISK AND REWARD (MEAN AND VARIANCE ANALYSIS) ..................................................................................... 4

A Short Note on Mean Calculation................................................................................................................ 5 Arithmetic Mean ............................................................................................................................................ 5 Geometric Mean ............................................................................................................................................ 6 Geometric Versus Arithmetic Mean............................................................................................................... 6 When to use Geometric Mean........................................................................................................................ 7 Variance and Standard Deviation ................................................................................................................. 7 Description of Standard Deviation in Portfolio theory ................................................................................. 8 Annualizing Returns and Standard deviation ................................................................................................ 8 MATHEMATICS OF THE MARKOWITZ MODEL...................................................................................................... 9 DIVERSIFICATION.............................................................................................................................................. 14 Diversification in Markowitz model ............................................................................................................ 16 The Risk Free Asset ..................................................................................................................................... 18 THE SECURITY MARKET LINE ? (SML) ............................................................................................................ 18 THE CAPITAL MARKET LINE ? (CML) .............................................................................................................. 19 THE SECURITY CHARACTERISTIC LINE ? (SCL)................................................................................................ 19 THE CAPITAL ASSET PRICING MODEL ? (CAPM) ............................................................................................. 20 THE EFFICIENT FRONTIER AND MARKET PORTFOLIO........................................................................................ 21 THE SHARPE RATIO........................................................................................................................................... 22 The Sharpe ratio in Portfolio theory ........................................................................................................... 23 SKEWNESS ........................................................................................................................................................ 23 KURTOSIS.......................................................................................................................................................... 24

CONSTRUCTION OF THE MODEL ON EXCEL......................................................................................... 25

EXCEL MODULES FOR PORTFOLIO MODELLING .................................................................................................. 25 PORTFOLIO RISK AND RETURN ......................................................................................................................... 25 USING SOLVER TO OPTIMIZE EFFICIENT POINTS................................................................................................. 26 FURTHER EXCEL IMPLEMENTATIONS................................................................................................................. 27

Implementing the Portfolio VaR in Excel .................................................................................................... 28

EMPIRICAL INVESTIGATION...................................................................................................................... 30

THE JARQUE BERA TEST OF NORMALITY .......................................................................................................... 30 The Result of Jarque-Bera Test on Our Portfolio Assets............................................................................. 31

USING PLOTS TO MOTIVATE THE NON-NORMALITY OF ASSET'S RETURN DATA .............................................. 32 Normal Probability Plot for Determining Non-Normality .......................................................................... 33 The Result of Normal Probability Plot on OMX Large Cap List................................................................. 35

THE PROBLEM WITH SHARPE RATIO AND THE REASON..................................................................................... 36 Adjustments to the Risk Regarding Higher Moments' Effects ..................................................................... 38

CONSTRUCTION OF THE PORTFOLIO WITH NEW ADJUSTMENTS TO SHARPE RATIO ........................................... 39 ANALYSIS OF THE EMPIRICAL INVESTIGATION.................................................................................................. 39

Analysis for the different type of time series for constructing a portfolio ................................................... 44 SUMMARY OF THE RESULTS TOUCHED BY EMPIRICAL INVESTIGATION ............................................................ 50

CONCLUSION ................................................................................................................................................... 54

LIST OF REFERENCES ................................................................................................................................... 55

APPENDICES ..................................................................................................................................................... 57

APPENDIX 1 ? PROOF OF EXPECTED VALUE (MEAN) ........................................................................................ 57 APPENDIX 2 ? PROOF OF VARIANCE.................................................................................................................. 58 APPENDIX 3 ? TABLE OF SKEWNESS ................................................................................................................. 59 APPENDIX 4 ? TABLE OF KURTOSIS................................................................................................................... 60

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APPENDIX 5 ? TABLE OF BETA FOR INDIVIDUAL STOCKS.................................................................................. 61 APPENDIX 6 ? TABLE OF JARQUE BERA TEST ................................................................................................... 62 APPENDIX 7 ? NORDIC LARGE CAP LIST ........................................................................................................... 63 LIST OF FIGURES ............................................................................................................................................ 66 LIST OF TABLES .............................................................................................................................................. 67 LIST OF ABBREVIATIONS............................................................................................................................. 68

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Acknowledgement

We thank our supervisor Lars Pettersson, Asset Manager at IF Metall, for his support and comments on our study.

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