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Simulation-Based Portfolio Optimization with Coherent Distortion Risk Measures

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Simulation-Based Portfolio Optimization with Coherent Distortion Risk Measures DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2020 Simulation-Based Portfolio Optimization with Coherent Distortion Risk Measures ANDREAS PRASTORFER KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Simulation Based Portfolio Optimization with Coherent Distortion Risk Measures ANDREAS PRASTORFER Degree Projects in Financial Mathematics (30 ECTS credits) Degree Programme in Applied and Computational Mathematics KTH Royal Institute of Technology year 2020 Supervisors at SAS Institute: Jimmy Skoglund Supervisor at KTH: Camilla Johansson Landén Examiner at KTH: Camilla Johansson Landén TRITA-SCI-GRU 2020:005 MAT-E 2020:05 Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Abstract This master's thesis studies portfolio optimization using linear programming algorithms. The contribu- tion of this thesis is an extension of the convex framework for portfolio optimization with Conditional Value-at-Risk, introduced by Rockafeller and Uryasev [28]. The extended framework considers risk mea- sures in this thesis belonging to the intersecting classes of coherent risk measures and distortion risk measures, which are known as coherent distortion risk measures. The considered risk measures belong- ing to this class are the Conditional Value-at-Risk, the Wang Transform, the Block Maxima and the Dual Block Maxima measures. The extended portfolio optimization framework is applied to a reference portfolio consisting of stocks, options and a bond index. All assets are from the Swedish market. The re- turns of the assets in the reference portfolio are modelled with elliptical distribution and normal copulas with asymmetric marginal return distributions. The portfolio optimization framework is a simulation-based framework that measures the risk using the simulated scenarios from the assumed portfolio distribution model. To model the return data with asymmetric distributions, the tails of the marginal distributions are fitted with generalized Pareto dis- tributions, and the dependence structure between the assets are captured using a normal copula. The result obtained from the optimizations is compared to different distributional return assumptions of the portfolio and the four risk measures. A Markowitz solution to the problem is computed using the mean average deviation as the risk measure. The solution is the benchmark solution which optimal solutions using the coherent distortion risk measures are compared to. The coherent distortion risk measures have the tractable property of being able to assign user-defined weights to different parts of the loss distribution and hence value increasing loss severities as greater risks. The user-defined loss weighting property and the asymmetric return distribution models are used to find optimal portfolios that account for extreme losses. An important finding of this project is that optimal solutions for asset returns simulated from asymmetric distributions are associated with greater risks, which is a consequence of more accurate modelling of distribution tails. Furthermore, weighting larger losses with increasingly larger weights show that the portfolio risk is greater, and a safer position is taken. Sammanfattning Denna masteruppsats behandlar portf¨oljoptimeringmed linj¨araprogrammeringsalgoritmer. Bidraget av uppsatsen ¨aren utvidgning av det konvexa ramverket f¨orportf¨oljoptimeringmed Conditional Value-at- Risk, som introducerades av Rockafeller och Uryasev [28]. Det utvidgade ramverket behandlar riskm˚att som tillh¨oren sammans¨attningav den koherenta riskm˚attklassen och distortions riksm˚attklassen.Denna klass ben¨amnssom koherenta distortionsriskm˚att.De riskm˚attsom tillh¨ordenna klass och behandlas i uppsatsen och ¨arConditional Value-at-Risk, Wang Transformen, Block Maxima och Dual Block Maxima m˚atten. Det utvidgade portf¨oljoptimeringsramverket appliceras p˚aen referensportf¨oljbest˚aende av aktier, optioner och ett obligationsindex fr˚anden Svenska aktiemarknaden. Tillg˚angarnasavkastningar, i referens portf¨oljen,modelleras med b˚adeelliptiska f¨ordelningaroch normal-copula med asymmetriska marginalf¨ordelningar. Portf¨oljoptimeringsramverket ¨arett simuleringsbaserat ramverk som m¨aterrisk baserat p˚ascenarion simulerade fr˚anf¨ordelningsmodellen som antagits f¨orportf¨oljen.F¨oratt modellera tillg˚angarnasavkast- ningar med asymmetriska f¨ordelningarmodelleras marginalf¨ordelningarnas svansar med generaliserade Paretof¨ordelningaroch en normal-copula modellerar det ¨omsesidigaberoendet mellan tillg˚angarna.Re- sultatet av portf¨oljoptimeringarnaj¨amf¨orssinsemellan f¨orde olika portf¨oljernasavkastningsantaganden och de fyra riskm˚atten.Problemet l¨oses ¨aven med Markowitz optimering d¨ar"mean average deviation" anv¨andssom riskm˚att.Denna l¨osningkommer vara den "benchmarkl¨osning"som kommer j¨amf¨orasmot de optimala l¨osningarna vilka ber¨aknasi optimeringen med de koherenta distortionsriskm˚atten. Den speciella egenskapen hos de koherenta distortionsriskm˚attensom g¨ordet m¨ojligtatt ange anv¨andar- specificerade vikter vid olika delar av f¨orlustf¨ordelningenoch kan d¨arf¨orv¨arderamer extrema f¨orluster som st¨orrerisker. Den anv¨andardefineradeviktningsegenskapen hos riskm˚attenstuderas i kombina- tion med den asymmetriska f¨ordelningsmodellen f¨oratt utforska portf¨oljersom tar extrema f¨orlusteri beaktande. En viktig uppt¨ackt ¨aratt optimala l¨osningartill avkastningar som ¨armodellerade med asym- metriska f¨ordelningar¨arassocierade med ¨okad risk, vilket ¨aren konsekvens av mer exakt modellering av tillg˚angarnasf¨ordelningssvansar. En annan uppt¨ackt ¨ar,om st¨orrevikter l¨aggsp˚ah¨ogref¨orlusters˚a ¨okar portf¨oljrisken och en s¨akrareportf¨oljstrategiantas. i Acknowledgements I want to start by expressing my deepest gratitude to my supervisor Jimmy Skoglund at SAS Institute, for his guidance and advice. I would also like to thank him for the inspiring conversations and the words of encouragement when I experienced difficulties. I want to thank my colleagues and friends at SAS Institute for making me feel welcome at the office and for making this project a joyful experience. I want to thank my supervisor Camilla Land´en,my academic supervisor at the Royal Institute of Technology, for valuable feedback and her helpful hand when most needed. Finally, I would like to thank my family and friends for their love and support and for always believing in me. Stockholm, January 6, 2020 ii Contents 1 Introduction 1 1.1 Background . .1 1.2 Project Goals . .1 1.3 Disposition . .2 2 Mathematical Background 2 2.1 Risk Measure Theory . .2 2.1.1 Coherent Risk Measures . .2 2.1.2 Distortion Risk Measures . .4 2.2 Elliptical Distributions . .7 2.3 Financial Time Series . .7 2.4 GARCH models . 10 2.5 Extreme Value Theory . 12 2.6 Copulas . 14 3 Method 16 3.1 Introduction to Portfolio Analysis . 16 3.2 Introduction to Portfolio Optimization . 17 3.3 Portfolio Models . 18 3.3.1 Remark: Elliptically Distributed Returns in Portfolio Selection . 18 3.3.2 Asymmetric Distributions . 19 3.4 Linear Programming Methods . 22 3.4.1 Markowitz Linear Program . 22 3.4.2 Rockafellar and Uryasev's CVaR Optimization . 23 3.4.3 Extension of Rockafellar and Uryasev CVaR Optimization . 24 3.5 Risk contributions . 27 4 Analysis and Conclusion 28 4.1 Reference Portfolios and Benchmark Solution . 28 4.2 Mean Average Deviation Benchmark Solution . 32 4.3 Optimization With Coherent Distortion Risk Measures . 33 4.3.1 Conditional Value-At-Risk . 34 4.3.2 Wang Transform . 36 4.3.3 Block Maxima . 39 4.3.4 Dual Block Maxima . 41 4.4 Robustness of solution for Mean Variation . 44 4.5 Comparison of CDRM Optimization . 46 4.6 Analysis of Euler Risk Contributions . 47 5 Final Conclusion and Further Investigation 50 A Appendix 51 A.1 Fitting GARCH and GDP parameters . 51 A.2 Estimated Parameters . 55 A.3 Payoff Functions and Profit-and-Loss distributions for Derivatives . 56 iii 1 Introduction 1.1 Background The trade-off between risk and reward is the main focus for investors. It is well known that with higher potential reward comes increased risk. Choosing an investment strategy often narrows down to the risk appetite of the investor. An investment portfolio may consist of assets with various risk levels. The possible combinations are endless but not all are optimal. Harry Markowitz [24] demonstrated this with the efficient frontier, which illustrates the set of optimal investment in the risk-reward spectrum. Any portfolio on the efficient frontier represents an optimal investment portfolio, while any portfolio below the frontier is sub-optimal. The seminal work by Markowitz introduced a new portfolio selection theory that has had an enormous effect on how investment opportunities are analyzed. He called it the modern portfolio theory. Upon introduction, the modern portfolio theory focused on linear portfolio assets, and the theory relied on the central assumption that the assets could be modeled using a normal distribution. Markowitz used variance as a quantifier of risk. Since the introduction of the modern portfolio theory, more advanced methods of risk measurement have been developed. Mainly to acknowledge the stylized fact that returns are not normally distributed in practice. Artzner et al. [2] proposed the coherent risk measures via an axiomatic approach, in which the mathematical properties of the risk measures were derived from a set of intuitive principles. This axiomatic approach was extended further by Wang et al. [37], introducing
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