EXAMENSARBETE INOM TEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2020 Portfolio Optimization: An Evaluation of the Downside Risk Framework on the Nordic Equity Markets FABIAN PETTERSSON OSKAR RINGSTRÖM KTH SKOLAN FÖR TEKNIKVETENSKAP Portfolio Optimization: An Evaluation of the Downside Risk Framework on the Nordic Equity Markets Fabian Pettersson Oskar Ringström ROYAL Degree Projects in Applied Mathematics and Industrial Economics (15 hp) Degree Programme in Industrial Engineering and Management (300 hp) KTH Royal Institute of Technology year 2020 Supervisor at KTH: Johan Karlsson Examiner at KTH: Sigrid Källblad Nordin TRITA-SCI-GRU 2020:118 MAT-K 2020:019 Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Degree Project in Applied Mathematics and Industrial Economics Portfolio Optimization: An Evaluation of the Downside Risk Framework on the Nordic Equity Markets Authors Supervisor Fabian Pettersson Johan Karlsson Oskar Ringstrom¨ Abstract Risk management in portfolio construction is a widely discussed topic and the tradeoff between risk and return is always considered before an investment is made. Modern portfolio theory is a mathematical framework which describes how a rational investor can use diversifica- tion to optimize a portfolio, which suggests using variance to measure financial risk. However, since variance is a symmetrical metric, the framework fails to correctly account for the loss aversion preferences most investors exhibit. Therefore, the use of downside risk measures were proposed, which only measures the variance of the portfolio below a certain threshold, usually set to zero or the risk-free rate. This thesis empirically investigates the differences in performance between the two risk measures when used to solve a real world portfolio optimization problem. Backtests using the different measures on all major Nordic equity markets are performed to highlight the dynamics between the frameworks, and when one should be preferred over the other. It is concluded that the optimization frameworks indeed provides a useful tool for investors to construct great performing portfolios. However, even though the downside risk framework is more mathematically rigorous, implementing this risk measure instead of variance seems to be of less importance for the actual results. Keywords: Downside risk, Mean-variance optimization, Modern portfolio theory, Semi-variance Examensarbete inom Till¨ampadMatematik och Industriell Ekonomi Portf¨oljoptimering:En Utv¨arderingav Riskm˚attet Downside Risk p˚ade Nordiska Aktiemarknaderna F¨orfattare Handledare Fabian Pettersson Johan Karlsson Oskar Ringstrom¨ Abstrakt Riskhantering f¨or aktieportf¨oljer ¨ar mycket centralt och en avv¨agning mellan risk och avkastning g¨orsalltid innan en investering. Modern Portf¨olteori¨arett matematiskt ramverk som beskriver hur en rationell investerare kan anv¨andadiversifiering f¨oratt optimera en portf¨olj. Centralt f¨ordetta ¨aratt anv¨andaportf¨oljensvarians f¨oratt m¨atarisk. Dock, eftersom varians ¨arett symmetriskt m˚attlyckas inte detta ramverk korrekt ta h¨ansyntill den f¨orlustaversion som de flesta investerare upplever. D¨arf¨orhar det f¨oreslagitsatt ist¨alletanv¨andaolika m˚att p˚anedsiderisk (downside risk), som endast tar h¨ansyntill portf¨oljensvarians under en viss avkastningsgr¨ans, oftast satt till noll eller den riskfria r¨antan. Denna studie unders¨oker skillnaderna i prestation mellan dessa tv˚ariskm˚attn¨arde anv¨andsf¨oratt l¨osaett verkligt portf¨oljoptimerinsproblem. Backtests med riskm˚attenhar genomf¨ortsp˚ade olika nordiska aktiemarknaderna f¨oratt visa p˚alikheter och skillnader mellan de olika riskm˚atten,samt n¨ar det enda ¨aratt f¨oredraframf¨ordet andra. Slutsatsen ¨aratt ramverken ger investerare ett anv¨andbartverktyg f¨oratt smidigt optimera portf¨oljer.D¨aremotverkar den faktiska skillnaden mellan de tv˚ariskm˚attenvara av mindre betydelse f¨orportf¨oljernasprestation. Detta trots att downside risk ¨armer matematiskt rigor¨ost. Nyckelord: Downside risk, Variansoptimering, Modern Portf¨oljteori,Semi-varians Contents 1 Introduction 2 1.1 Background . 2 1.2 Aim . 2 1.3 Limitations . 3 2 Theoretical Framework 3 2.1 Mean-Variance Framework . 3 2.2 Downside Risk Framework . 4 2.3 Portfolio Selection . 5 2.4 Previous Research . 6 3 Methodology 7 3.1 Data Gathering . 7 3.2 Calculating the Efficient Frontier . 8 3.2.1 Parameter Estimations . 8 3.2.2 Application of Quadratic Programming . 8 3.2.3 Position Constraints . 9 3.2.4 Optimization Problems . 9 3.3 Backtesting . 10 3.3.1 Corporate Actions . 10 4 Results 11 4.1 OMX Copenhagen . 11 4.2 OMX Helsinki . 12 4.3 OMX Stockholm . 13 4.4 Summary of Results . 14 5 Discussion 15 5.1 Performance for Entire Period . 15 5.2 Performance during Bear Market . 16 5.3 Performance during Bull Market . 16 5.4 Portfolio Selections . 16 5.5 Performance against Benchmarks . 17 5.6 Implications for Investors . 17 6 Conclusion 17 References 18 Appendices 20 A Positive Semi-Definite Matrices 20 B OMX Index Constituents 21 1 Introduction 1.1 Background In 1952, Markowitz introduced Modern Portfolio Theory (MPT ) in his paper \Portfolio selec- tion". According to this theory a risk-averse investor can construct an efficient frontier contain- ing optimal portfolios that minimizes the risk for a given level of expected return. MPT was the first model to describe portfolio selection and management from a mathematical point of view. However, since the MPT was restricted to the mean-variance framework, which not always represent the whole picture of financial risk, the theory has received considerable amount of criticism.1 One reason behind this is investor's tendencies to prefer avoiding losses to obtain equivalent gains. This principle is referred to as loss aversion and was first identified by Kah- neman and Tversky [1979]. What differentiates this kind of aversion from risk aversion is that the utility of a monetary return depends on what was previously experienced or was expected to happen; a loss is more painful than an equal gain. An investor's intuitive concept of risk is asymmetrical in nature. To reconcile this, Harlow [1991] proposed using lower partial moments as an asymmetrical measure of downside risk, which only measures the variance of the portfolio below a certain threshold. Secondly, the mean-variance framework has limited generality since it has been shown that the selection will only lead to optimal decisions if the utility function is assumed quadratic or the probability distribution is restricted.2 Pratt [1964] argued that the assumption of a quadratic utility function is unrealistic since it implies increasing absolute risk aversion over the whole domain and a negative marginal utility after a cut-off point. Further- more, the MPT has been criticized by Doganoglu et al. [2007] because of the assumption that the asset returns can be represented by the normal distribution. According to Doganoglu et al. asset returns often show skewness as well as kurtosis bigger than 3, which is indications of dis- tributions differentiate from Gaussian. Moreover, investments have unlimited upside potential but can only yield limited losses. 1.2 Aim This thesis aims to further contribute to the research body of MPT by comparing its performance when using different risk measures. Even thought this thesis investigates the performance of two different risk measures the aim is not to provide investors with a solution of how to allocate their portfolio in one optimal way, but to investigate the different characteristics of using both variance and downside risk as risk measures. The comparison will be based on how the risk measures perform from a risk adjusted return basis and what risk level an investor will be exposed to. Although higher risk adjusted returns is not always directly indicative of better performance, it offers a good starting point for analysing the portfolios. Most investors measure their performance on a risk adjusted basis, which is why that is relevant to evaluate. The mathematical theory behind the two risk measures are also discussed and evaluated. One risk measure might have desirable properties, that are not directly reflected in the portfolios performance. These are still relevant to examine. The study aims to highlight of the different characteristics of the risk measures, so that investors can adapt what fits their strategy and portfolio. This thesis also investigates the performances in different market conditions with the aim of identifying differences in the performance of the risk measures. Considering that downside risk is asymmetric to daily returns, it can be relevant to investigate how its performance differs in an upward and downward trending market. Furthermore, most of the research on MPT is published by 1See e.g. Doganoglu et al. [2007] and Taleb [2007]. 2See in particular Borch [1969] and Feldstein [1969]. 2 Americans. This means that most empirical evaluations of the framework only consider American markets. Although some general conclusions certainly can be drawn from this, Nordic investors also would benefit from evaluating the framework on the Nordic markets. Therefore, this thesis will also contribute by empirically evaluate the performance of MPT on the Nordic equity markets, something which has not been excessively studied. The thesis seeks to answer the following questions: • Does Modern Portfolio Theory offer Nordic investors a useful tool for improving portfolio performance? • Is downside risk a better measure of financial