Table Of ContentAsset Allocation Based on Shortfall Risk
Von der Fakultät für Wirtschaftswissenschaften der
Technischen Universität Chemnitz
genehmigte
Dissertation
zur Erlangung des akademischen Grades
Doctor rerum politicarum
(Dr. rer. pol.)
vorgelegt von
Dipl.-Kffr. Denisa Čumova
geboren am 07.09.1976 in Košice
eingereicht am 18. November 2004
Gutachter: Prof. Dr. F. Thießen
Prof. Dr. Th. Burkhardt
Prof. Dr. B. Hoffmann
Tag der mündlichen Prüfung: 09. Februar 2005
Gewidmet meiner lieben Mutter
Vorwort
Die vorliegende Arbeit entstand während meines Promotionsstudiums an der Professur der
Finanzwirtschaft und Bankbetriebslehre der Technischen Universität Chemnitz.
Mein besonderer Dank gilt Herrn Prof. Dr. F. Thießen, dem Leiter der Professur
Finanzwirtschaft und Bankbetriebslehre der Fakultät Wirtschaftswissenschaften an der TU
Chemnitz. Durch sein entgegengebrachtes Vertrauen in das Gelingen dieser Arbeit, durch
seine sowohl professionelle als auch persönliche Förderung und nicht zuletzt durch
zahlreiche Diskussionen wurde das Erarbeiten der Dissertation erst ermöglicht. Herrn Prof.
Dr. Th. Burkhardt, Leiter der Arbeitsgruppe Finanzen am Institut für Management der
Universität Koblenz-Landau, und Herrn Prof. Dr. B. Hoffman, Leiter der Professur Inverse
Probleme der Fakultät für Mathematik an der TU Chemnitz, danke ich für ihr Interesse an
dieser Arbeit und für die Übernahme des Mitberichts.
Weiterhin möchte ich mich herzlichst bei Herrn Prof. D. Nawrocki, Ph.D., College of
Commerce & Finance, Villanova University in Pennsylvania, für die Durchsicht des
Manuskriptes und für viele konstruktive Anregungen und Diskussionen, die wesentlich
zum Gelingen der Arbeit beigetragen haben, bedanken.
Herrn Dr. Th. Unger bin ich verbunden für die Hilfestellung bei der Lösung von
Optimierungsproblemen.
Bei Herrn P. Belsky und J. Intorsureanu möchte ich mich bedanken für die Hilfe bei der
Programmierung von Applikationen.
Den Mitarbeitern der Professur der Finanzwirtschaft und Bankbetriebslehre der TU
Chemnitz, Herrn V. Weber, Frau G. Viehweg, bedanke ich mich für fachliche
Diskussionen in einer freundlichen Arbeitsatmosphäre.
Mein besonderer Dank gilt dem HOST-Programm und der Gesellschaft der Freunde der
TU Chemnitz, die die Arbeit durch ein Stipendium freundlich unterstützt haben.
Meinem Mann Reiko Thiele danke ich von ganzem Herzen für die liebevolle
Unterstützung, für seine Geduld und Verständnis während der Anfertigung dieser Arbeit.
An meine Mutter möchte ich ein ganz besonderes Dankeschön richten. Für die stets von ihr
erfahrene große persönliche Unterstützung bin ich sehr dankbar.
Chemnitz, im Februar 2005
CONTENT
Abbreviations…………………………………………………………….. VII
Symbols……………………………………………………………………. IX
Figures……………………………………………………………………. XII
Tables………………………………………………………………….. XVIII
1. INTRODUCTION
1.1. Objectives………………………………………………...…………... 1
1.2. Proceeding……………………………………………………………. 2
2. PORTFOLIO SELECTION BASED ON MEAN AND VARIANCE... 5
2.1. Fundamentals of the (µ, σ)- portfolio model…………………..…… 6
2.1.1. Measures of the portfolio value, risk and return dependency…………….... 7
2.1.2. The (µ, σ)- efficient frontier……………………………………………...… 8
2.1.2.1. The (µ, σ)- efficient portfolios without the risk-free asset………...….. 9
2.1.2.2. The (µ, σ)- efficient portfolios with the risk-free asset……………… 10
2.1.3. Selection of the (µ, σ)- optimal portfolio…………………………………. 11
2.2. Analysis of the (µ, σ)- portfolio model…………………………….. 13
2.2.1. Investor’s risk and value preferences……………………………..………. 14
2.2.2. Asset universe……………………………………………...……………… 15
2.2.3. Congruence of the (µ, σ)- decision principle and decision principles
based on comprehensive evaluation of random variables………………… 18
2.2.3.1. Congruence of the (µ, σ)- decision principle and expected utility
maximization………………………………………………………... 18
2.2.3.2. Congruence of the (µ, σ)- decision principle and stochastic
dominance………………………………………………………...….23
2.2.4. Risk diversification with the covariance…………………………….…….23
2.2.4.1. Examples of risk diversification with the covariance……….………. 25
2.2.4.2. Asymmetry in the risk diversification and the covariance…….…….. 28
I
2.2.4.2.1. Empirical evidence of asymmetry in return dependence
on global capital market………………………...…………... 30
2.3. Optimization algorithm for the (µ, σ)- portfolio model………….. 35
2.3.1. Formulation of the optimization problem…………………...…………….35
2.3.2. Optimum conditions……………………………….……………………… 37
2.3.3. Efficient segment………………………………………………………….. 41
2.3.4. Adjacent efficient segments……………………………………...………..43
3. GENERAL RISK AND VALUE ANALYSIS IN PORTFOLIO
CONTEXT………………………………………………...……………. 48
3.1. Portfolio models as decision models under risk………………….. 48
3.2. Risk analysis………………………………………………………... 50
3.2.1. Risk understanding………………………………………………………... 50
3.2.2. Risk measurement………………………………………………………… 51
3.2.2.1. Both–side risk measures…………………………………………..… 53
3.2.2.2. Lower partial measures……………………………………………… 56
3.2.2.3. Market risk measures………………………………………………... 58
3.2.2.4. Other risk measures………………………………………………….. 59
3.2.3. Coherent risk measures…………………………………..……………….. 62
3.2.4. Empirical studies of risk understanding and measuring………………….. 63
3.3. Value analysis………………………………………………………. 65
3.3.1. Measures of central tendency……………………………………………... 65
3.3.2. Upper partial measures……………………………………………………. 67
3.4. Main approaches in the portfolio optimization according to the
risk and value………………………………………………………. 69
4. PORTFOLIO SELECTION BASED ON THE MEAN AND
SHORTFALL RISK………………………………………………….. 72
II
4.1. Overview of the literature…………………………………………. 73
4.2. Safety-first criteria…………………………………………………. 77
4.2.1. Roy’s criterion…………………………………………………………….. 77
4.2.2. Kataoka’s criterion………………………………………………………... 78
4.2.3. Telser’s criterion…………………………………………………………... 79
4.3. Fundamentals of the portfolio selection based on mean and
lower partial moment………………………………………………. 81
4.3.1. Measures of portfolio risk, value and return dependency………………… 81
4.3.2. The (µ, LPM)- efficient frontier…………………………………………... 86
4.3.2.1. The (µ, LPM)- efficient portfolios without the risk-free asset………. 86
4.3.2.2. The (µ, LPM)- efficient portfolios with the risk-free asset………….. 88
4.3.3. Selection of the (µ, LPM)- optimal portfolio……………………………... 89
4.4. Analysis of the (µ, LPM)- portfolio model………………………… 91
4.4.1. Investor’s risk and value preferences……………………………………... 91
4.4.1.1. Decomposition of the lower partial moment………………………… 92
4.4.1.2. Parameter of risk aversion…………………………………………… 92
4.4.1.3. Minimal aspiration return level……………………………………… 93
4.4.2. Asset universe………………………………………………….…………. 97
4.4.3. Congruence of the (µ, LPM)- decision principle and decision principles
based on comprehensive evaluation of random variables………………..102
4.4.3.1. Congruence of the (µ, LPM)- decision principle and expected
utility maximization……………………………………………..… 102
4.4.3.2. Congruence of the (µ, LPM)- decision principle and stochastic
dominance…………………………………………………………. 104
4.4.4. Risk diversification with the co-shortfall risk…………………………… 107
4.4.4.1. Comparison of the covariance and co-shortfall risk………………... 109
4.4.4.2. Asymmetry in the risk diversification and the co-shortfall risk……. 116
4.5. Optimization algorithm for the (µ, LPM)- portfolio model…….. 117
4.5.1. Optimization algorithm for the portfolio model based on mean and
semivariance……………………………………………………………... 120
4.6. Empirical tests…………………………………………………….. 123
4.6.1. Objectives and conception………………………………………………. 123
III
4.6.2. Procedure……………………………………………………………….... 124
4.6.3. Asset universe…………………………………………………………… 126
4.6.3.1. Statistical characteristics of the examined return time series……… 127
4.6.4. Asset allocation with the (µ, σ)- and (µ, LPM )- portfolio model……… 131
2,τ
4.6.4.1. Hypothesis …………………………………………. ……………. 132
…
4.6.4.2. Definition of the portfolio types……………………………………. 132
4.6.4.3. The (µ, σ)- and (µ, LPM )- portfolio optimization………………... 133
2,τ
4.6.4.4. Analysis of the ex-ante portfolio differences………………………. 136
4.6.5. Realized performance with the (µ, σ)- and (µ, LPM )- portfolio model.. 147
2,τ
4.6.5.1. Hypotheses and objectives…………………………………………. 148
4.6.5.2. Determination of realized performance of the (µ, σ)- and (µ, LPM )
2,τ
-portfolio model……………………………………………………. 150
4.6.5.2.1 Computation of the realized performance………………….. 150
4.6.5.2.2 The test of statistical significance and standard errors of the
realized performance………………………………….. ……..152
4.6.5.2.3 Results of the realized performance………………………….154
4.6.5.3. Analysis of realized performance…………………………………... 156
4.6.5.3.1. Analysis of realized performance ratios……………………... 156
4.6.5.3.2. Analysis of realized returns and risks………………………... 159
4.6.5.4 Confirmation (or refutation) of the hypothesis…………………... 161
4.6.5.4 Influences on the differences in the realized performance ………...162
4.6.6 Simulation……………………………………………………………….....163
4.6.6.1 Definition of the asset universe and portfolio types ……………… 164
4.6.6.2 Estimation of the parameters of return distributions for all
assets as an input for the generation of random returns…...……….165
4.6.6.3 Generation of random returns………………………………………165
4.6.6.4 Optimization with the (µ, σ)- and (µ, LPM )- portfolio for given
2,τ
expected return……………………………………………………..165
4.6.6.5 Realized performance of the (µ, σ)- and (µ, LPM )- portfolio
2,τ
model………………………………………………………………168
4.6.6.5.1 Computation of the realized performance…………..……….168
4.6.6.5.2 Standard error of the results…………………………………169
4.6.6.6 Analysis of realized simulated performance………………………169
IV
4.6.6.7 Influences on the differences in the realized simulated
performance……………………………………………………….176
4.6.6.8 Conclusion of the simulation………………………….…………..185
4.6.7 Conclusion of the empirical study……………………………………….. 185
5. PORTFOLIO SELECTION BASED ON THE UPSIDE POTENTIAL
AND SHORTFALL RISK……………………………………………. 187
5.1. Fundamentals of the portfolio selection based on upper and lower
partial moment……………………………………………………. 188
5.1.1. Measures of portfolio value, risk and return dependency……………….. 188
5.1.2. The (UPM, LPM)- efficient frontier……………………………………... 190
5.1.2.1. The (UPM, LPM)- efficient frontier without the risk-free asset…… 190
5.1.2.2. The (UPM, LPM)- efficient frontier with the risk-free asset………..192
5.1.3. Selection of the optimal portfolio……………………………………….. 194
5.2. Analysis of the (UPM, LPM)- portfolio model…………………... 194
5.2.1. Investor’s risk and value preferences……………………………………. 194
5.2.2. Asset universe……………………………………………………………. 198
5.2.3. Congruence of the (UPM, LPM)- decision principle and decision
principles based on comprehensive evaluation of random variables…….. 200
5.2.3.1. Congruence of the (UPM, LPM)- decision principle and expected
utility maximization……………………………………………….. 200
5.2.3.1.1. Basic types of the (UPM, LPM)- utility function……………. 202
5.2.3.1.2. Estimation of the grade of shortfall risk aversion and
chance potential exposure…………………………………….207
5.2.4. Risk diversification with the co-shortfall risk and co-chance potential…..208
5.3. Geometric analysis of the (UPM, LPM)- efficient sets………….. 209
5.3.1. Iso-UPM curves………………………………………………………….. 210
5.3.2. Iso-LPM curves………………………………………………………….. 214
5.3.3. Efficient portfolios………………………………………………………. 216
5.4. Optimization algorithm for the (UPM, LPM)-portfolio model… 221
5.4.1. Generalized reduced gradient method…………………………………… 224
5.4.2. Augmented Lagrangian algorithms……………………………………… 228
V
5.5. Application of the (UPM, LPM)- portfolio model……………….. 229
5.5.1. Introduction……………………………………………………………… 229
5.5.2. Application of the (UPM, LPM)- portfolio model on optioned portfolios. 231
5.5.2.1. Characteristics of optioned assets…………………………………... 231
5.5.2.2. Procedure…………………………………………………………… 233
5.5.2.3. Asset universe………………………………………………………. 233
5.5.2.4. Estimation of the input parameters…………………………………. 235
5.5.2.5. Optimization………………………………………………………... 240
5.5.2.6. Investment strategies……………………………………………….. 241
5.5.2.6.1. Investment strategy of the shortfall risk aversion and chance
potential seeking……………………………………………... 242
5.5.2.6.2. Investment strategy of the shortfall risk aversion and chance
potential aversion……………………………………………. 254
5.5.2.6.3. Investment strategy of the shortfall risk seeking and chance
potential aversion……………………………………………. 261
5.5.2.6.4. Investment strategy of the shortfall risk seeking and chance
potential seeking…………………………………………….. 272
5.5.2.7. Conclusion…………………………………………………………. 278
6. SUMMARY…………………………………………………………… 280
APPENDIX A Bernoulli’s decision principle and decision rules of
stochastic dominance…………………………………………...… 284
APPENDIX B Example of differences between the formulae LPM 4.12 and
ij
4.13.………………………………………………………… ……..291
APPENDIX C Input data used in the empirical test (4.6)……………….……… 293
APPENDIX D Standard error of the Sharpe-ratio……………….……… ……..302
APPENDIX E Example of differences between the (µ, σ)- and (µ, LPM)-
efficient portfolios for symmetrical asset return distributions... 304
LITERATURE…………………………………………………………………… 305
VI
Abbreviations
abs. absolute
Answ. answer
Arg Argentina
Can Canada
CAPM Capital Asset Pricing Model
CC covered call written
CLA Critical line Algorithm
const. constant
DAX Deutscher Aktienindex
diff. difference
E expectation
e.g. exempli gratia
eds. editor
et al et alii
etc. et cetera
GB Great Britain
Ger Germany
Gr Greece
I indifference curve
i.e. id est
ind. indifferent
Indon Indonesia
Ital Italy
Jap Japan
Kor Korea
log logarithm
Mal Malaysia
Max, max maximal, maximize
MaxUPM maximum chance potential portfolio
Mex Mexico
Min, min minimal, minimize
MinLPM minimum shortfall risk portfolio
VII
Description:Asset Allocation Based on Shortfall Risk. Von der Fakultät für Wirtschaftswissenschaften der. Technischen Universität Chemnitz genehmigte.
