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Evaluation of a Portfolio in Dow Jones Industrial Average Optimized by Mean-Variance Analysis

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Evaluation of a Portfolio in Dow Jones Industrial Average Optimized by Mean-Variance Analysis EXAMENSARBETE INOM TEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2020 Evaluation of a Portfolio in Dow Jones Industrial Average Optimized by Mean-Variance Analysis DANIEL LIU ALEXANDER STRID KTH SKOLAN FÖR TEKNIKVETENSKAP Evaluation of a Portfolio in Dow Jones Industrial Average Optimized by Mean-Variance Analysis Daniel Liu Alexander Strid 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 2020:103 MAT-K 2020:004 Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Abstract This thesis evaluates the mean-variance analysis framework by comparing the perfor- mance of an optimized portfolio consisting of stocks from the Dow Jones Industrial Average to the performance of the Dow Jones Industrial Average index itself. The results show that the optimized portfolio performs better than the corresponding in- dex when evaluated on the period between 2015 and 2019. However, the variance of the returns are high and therefore it is difficult to determine if mean-variance analysis performs better than its corresponding index in the general case. Furthermore, it is shown that individual stocks can still influence the movement of an optimized portfolio significantly, even though the model is supposed to diversify firm-specific risk. Thus, the authors recommend modifying the model by restricting the amount that is allowed to be invested in a single stock, if one wishes to apply mean-variance analysis in real- ity. To be able to draw further conclusions, more practical research within the subject needs to be done. 2 Utv¨ardering av en portf¨olj i Dow Jones Industrial Average optimerad genom mean-variance analysis Alexander Strid, Daniel Liu May 22, 2020 Sammanfattning Denna uppsats utv¨arderar ramverket "mean-variance analysis" genom att j¨amf¨ora pre- standan av en optimerad portf¨olj best˚aendeav aktier fr˚anDow Jones Industrial Ave- rage med prestandan av indexet Dow Jones Industrial Average sj¨alvt. Resultaten vi- sar att att den optimerade portf¨oljen presterar b¨attre ¨an motsvarande index n¨ar de utv¨arderas p˚aperioden 2015 till 2019. Dock ¨ar variansen av avkastningen h¨og och det ¨ar d¨arf¨or sv˚artatt bed¨oma om mean-variance analysis generellt sett presterar b¨attre ¨an sitt motsvarande index. Vidare visas det att individuella aktier fortfarande kan p˚averka den optimerade portf¨oljens r¨orelser, fast¨an modellen antas diversifiera f¨oretagsspecifik risk. P˚agrund av detta rekommenderar f¨orfattarna att modifiera modellen genom att begr¨ansa m¨angden som kan investeras i en individuell aktie, om man ¨onskar att till¨ampa mean-variance analysis i verkligheten. F¨or att kunna dra vidare slutsatser s˚akr¨avs mer praktisk forskning inom omr˚adet. 3 Contents 1 Introduction 6 1.1 Background . .6 1.2 Problem Statement . .7 1.3 Earlier Research . .8 1.4 Scope and Delimitations . .8 1.5 Purpose . .9 2 Theoretical Framework 10 2.1 Modern Portfolio Theory . 10 2.1.1 E − V Rule . 10 2.1.2 The Efficient Frontier . 11 2.2 Risk Aversion Reconciled with Mean-Variance Analysis . 12 2.3 Quadratic Optimization . 13 2.4 Solution to the Portfolio Optimization Problem . 14 2.5 Introduction of the Risk-Free Asset . 16 2.5.1 Equation of Capital Market Line . 17 2.5.2 Sharpe Ratio . 18 2.6 Firm-Specific Versus Systematic Risk . 18 3 Methodology 19 3.1 Collection and Processing of Data . 19 3.2 Beliefs about Future Market Performance . 20 3.2.1 Calculation of Expected Returns . 21 3.2.2 Calculation of Covariances . 22 3.3 Determination of Optimal Portfolio and Adjustment to Risk . 22 3.4 Expansion of Problem with Allocation Limits . 23 3.5 Simulation and Evaluation of Portfolio Performance . 24 4 Results 24 4.1 Results when Investing in the Tangency Portfolio . 25 4.2 Results when Weighting to Expected Risk . 28 4.3 Limitations of Asset Allocation . 30 4.4 Validation of Results . 32 5 Discussion and Analysis 33 5.1 Analysis of Results . 33 5.1.1 Explanation of the Difference in Results . 34 5.1.2 Comparison of the No-Limit Portfolio and the 20%-Limit Portfolio . 34 5.1.3 Simplifications . 35 5.2 Discussion on Reliability of Results . 36 4 5.3 Reconciliation with Utility Theory . 36 5.4 Discussion on Validity of Investigation . 37 5.5 Criticism of Mean-Variance Analysis . 39 6 Conclusion 40 References 41 Appendices 44 A Stocks used for calculation 44 B Most Invested Stocks in Tangency Portfolio per Month 45 C Summary of Monthly Results, Adjusted to Actual Risk 46 D Summary of Monthly Results, Weighted to Expected Risk 47 5 1 Introduction In 2016, 49.3 percent of the households in the United States stated that they kept some part of their savings in the stock market [1]. Saving money for the future is beneficial, whether it is for retirement, to buy a house or to send ones children to college. Investing in the stock market is one way to make the money grow and potentially be worth more in the future. While the market might grow in the long run, there is no guarantee that a given investment will yield a positive return during a fixed time period. Very risk averse investors may choose to save all their money in a practical risk-free rate, but such an investment usually has a low yield. It is reasonable to assume that most investors want a high return on their investment, but at the same time are reluctant to expose themselves to unnecessary risk. One method that could be considered is to use a framework called mean-variance analysis (MV analysis). In theory, mean-variance analysis minimizes the risk the investors expose themselves to, while still maximizing the return they expect to receive. This thesis is an empirical investigation of the performance of the mean-variance analysis framework on the Dow Jones Industrial Average. 1.1 Background The process of investing among various assets to gain a positive return is a game of uncer- tainty. In 1952, Harry Markowitz marked the beginning of modern portfolio theory with his article "Portfolio Selection" [2]. For the first time, the problem of portfolio selection is clearly stated and solved [3]. The essence of the theory does not only imply diversification, but the "right kind" of diversification for the "right reason" [2]. For instance, the adequacy of diversification should not solely be assessed by the number of securities in the portfolio but rather by low covariance of returns across different types of securities. Furthermore, as with all models and frameworks, assumptions need to be made regarding the nature of the actors in question. In mean-variance analysis it is assumed that the investors are rational and risk averse with concave utility functions [3]. It is also shown that some conventional "truths" within finance must be rejected. One way of diversifying a portfolio without extensive research is to invest in mutual funds. Ever since the bull market began in 2009, index funds have grown substantially in pop- ularity and in 2019, assets invested in U.S. passive equity funds topped those in actively managed funds [4]. This is mainly because of the increasingly substantiated belief that actively managed funds on average do not appear to provide higher returns than pas- sively managed funds [5]. Considering the mean variance analysis and the superiority of index mutual funds, an intriguing question that could be asked is whether it is possible to achieve positive alpha, excess return of a benchmark index at the same risk, by optimizing the diversification of the stocks for a given index. 6 One index that is appropriate for this kind of investigation is the Dow Jones Industrial Average (DJIA). DJIA is an index consisting of 30 blue chip stocks in the U.S. stock market. At least historically, the index has been used to reflect the U.S. market as a whole [6], but the S&P 500 might be a better indicator since it encompasses more stocks [7]. Despite not representing the whole market, the DJIA serves as a good "test" index for portfolio optimization because of primarily three reasons: 1. It is a stable index with few company changes and the changes that have occurred are well documented. 2. All the stocks that are, or have been a part of the index in recent years, have been listed for at least 10 years (at the time of writing). 3. It only contains 30 stocks which limits the required computational power when cal- culated in practice. The first two reasons facilitate the process of eliminating the so-called survivorship bias which is the logical error of concentrating, in this case, on the wrong types of companies. An example of the survivorship bias would be to exclude companies that have ceased to exist but were active during the backtesting period. This is elaborated later on in the thesis. A list of the stocks that are used in the calculations is presented in Appendix A. 1.2 Problem Statement The aim of this thesis is to investigate whether a portfolio optimized according to the theory of MV analysis can, in practice, perform better than its corresponding index. In other words, this thesis investigates if it is possible for an investor to construct a portfolio consisting of stocks in a defined market index so that the actual return of the portfolio is greater than that of the index, while exposing themselves to the same risk.
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