Measuring Performance of a Stochastic Dominance Based Portfolio Selection Model in Nordic Stock Markets

MEASURING PERFORMANCE OF A STOCHASTIC DOMINANCE BASED PORTFOLIO SELECTION MODEL IN NORDIC STOCK MARKETS

Master’s Thesis Joel Kinnunen Aalto University School of Business Information and Service Management Spring 2018

Aalto University, P.O. BOX 11000, 00076 AALTO www.aalto.fi

Abstract of master’s thesis

Author Joel Kinnunen Title of thesis Measuring Performance of a Stochastic Dominance Based Portfolio Selection Model in Nordic Stock Markets Degree Master of Science in Economics and Business Administration Degree programme Information and Service Management Thesis advisors Timo Kuosmanen, Juuso Liesiö Year of approval 2018 Number of pages 55 Language English Abstract Constructing stochastically dominating stock portfolios is attractive, since a stochastically dominating portfolio would mean higher returns without higher risk, lower risk without lower returns, or both higher returns and lower risk simultaneously. There is a lack of studies measuring the out-of-sample performance of stochastic dominance based portfolio selection models, when investing directly in stocks, or in Nordic markets. Therefore, there is a need for this study, which aims to increase knowledge about that topic. The research approach of this study is mainly quantitative. The data used includes 8 years of daily return data of OMX Nordic 40 index and its constituents. 84 different 1-year formation periods and 30-day out-of-sample holding periods are used. The portfolio selection model used is a stochastic dominance based model by Kopa & Post (2015). Empirical findings are that – in this empirical case – returns are higher and risk lower for the optimized portfolios compared to the index. Especially, cumulative returns over all holding periods combined are considerably higher for the optimized portfolios than for the index. Therefore, the optimized portfolios have excellent performance compared to the index. However, the portfolio turnover for this investment strategy is relatively high. Even though the empirical findings are promising, the statistical evidence of these findings being generalizable across samples is mostly weak, with the exception of strong evidence for lower variance of daily returns. Moreover, the high turnover can lead to high transaction costs and taxes in real-world investing, which can lower returns for such stochastic dominance based investments strategies that have high turnover. Stochastic dominance based portfolio selection models might be an effective way to systematically increase returns and lower risks in real-world investing. Yet, more research, testing and development of such models in real-world context is needed. Also, academically, more similar studies are needed to conclusively determine how such models perform in various stock investing contexts.

Keywords stochastic dominance, portfolio selection, portfolio choice, stock investing

Aalto-yliopisto, P.O. BOX 11000, 00076 AALTO www.aalto.fi

Maisterintutkinnon tutkielman tiivistelmä

Tekijä Joel Kinnunen Työn nimi Stokastiseen dominanssiin perustuvan portfolionvalintamallin suorituskyvyn mittaus pohjoismaisilla osakemarkkinoilla Tutkinto Kauppatieteiden maisteri Koulutusohjelma Information and Service Management Työn ohjaajat Timo Kuosmanen, Juuso Liesiö Hyväksymisvuosi 2018 Sivumäärä 55 Kieli Englanti Tiivistelmä Stokastisesti dominoivien osakeportfolioiden rakentaminen on houkuttelevaa, koska tällainen portfolio tarkoittaisi korkeampia tuottoja ilman korkeampaa riskiä tai alempaa riskiä ilman alentuneita tuottoja tai sekä korkeampia tuottoja että matalampaa riskiä samanaikaisesti. On pulaa tutkimuksista, jotka mittaavat stokastiseen dominanssiin perustuvien portfolionvalintamallien otoksen ulkopuolista suorituskykyä suoraan osakkeisiin sijoitettaessa tai pohjoismaisilla markkinoilla. Siksi on olemassa tarve tälle tutkimukselle, joka pyrkii lisäämään tietoa tästä aiheesta. Tutkimusote on pääosin kvantitatiivinen. Tutkimuksen data sisältää OMX Nordic 40 -indeksin ja sen sisältämien osakkeiden päivätuotot 8 vuoden ajalta. Tutkimuksessa käytetään 84 eri 1 vuoden portfolionmuodostusperiodia sekä 30 päivän omistusperiodia. Käytetty portfolionvalintamalli on stokastiseen dominanssiin perustuva Kopa & Post (2015). Empiiriset löydökset ovat, että tässä tapauksessa tuotot ovat korkeammat ja riski alempi optimoiduille portfolioille indeksiin verrattuna. Erityisesti kumulatiiviset tuotot kaikkien omistusperiodien aikana yhteensä ovat huomattavasti korkeammat optimoiduille portfolioille kuin indeksille. Näin ollen optimoitujen portfolioiden suorituskyky on erinomainen indeksiin verrattuna. Portfolion vaihtuvuus tässä sijoitusstrategiassa on kuitenkin suhteellisen korkea. Vaikka empiiriset löydökset ovat lupaavia, tilastollinen näyttö siitä, että nämä löydökset olisivat yleistettäviä eri otosten välillä, on pääosin heikkoa. Poikkeuksena on vahva näyttö päivätuottojen alemmasta varianssista. Lisäksi korkea vaihtuvuus voi johtaa korkeisiin transaktiokustannuksiin ja veroihin käytännön sijoittamisessa, mikä voi alentaa tuottoja sellaisille stokastiseen dominanssiin perustuville investointistrategioille, joilla on korkea vaihtuvuus. Stokastiseen dominanssiin perustuvat portfolionvalintamallit saattavat olla tehokas tapa järjestelmällisesti parantaa tuottoja ja alentaa riskiä käytännön sijoittamisessa. Lisää tällaisten mallien tutkimusta, testausta ja kehitystä käytännön kontekstissa kuitenkin tarvitaan. Myös akateemisesti lisää samankaltaisia tutkimuksia tarvitaan, jotta saataisiin lopullinen selvyys siitä, kuinka tällaiset mallit suoriutuvat erilaisissa osakesijoittamisen konteksteissa.

Avainsanat stokastinen dominanssi, portfolion valinta, osakesijoittaminen

Acknowledgements

I would like to thank my thesis advisors Timo Kuosmanen and Juuso Liesiö for helping me to decide the thesis topic, helping me to get started in the research area of stochastic dominance based portfolio selection, and guiding me through the whole research process. I am also grateful for Peng Xu, who especially helped me with the practicalities related to the implementation of the portfolio selection model used in this study.

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Contents

Acknowledgements ...... iii

1 Introduction ...... 1 1.1 Background and Motivation ...... 1 1.2 Research Question and Aim of the Study ...... 2 1.3 Research Approach and Scope ...... 2 1.4 Structure of the Study ...... 3

2 Theory ...... 4 2.1 Stochastic Dominance ...... 4 2.2 Portfolio Selection ...... 5 2.3 Stochastic Dominance Based Portfolio Selection Models ...... 6

3 Research Process and Data ...... 12 3.1 Data ...... 12 3.2 Investment Strategy and Empirical Test Setup ...... 12 3.3 Research Process ...... 13 3.4 Data Processing ...... 15 3.5 Data Analysis ...... 18

4 Empirical Findings ...... 19

5 Results ...... 30

6 Conclusions ...... 35 6.1 Answer to Research Question ...... 35 6.2 Evaluation of the Study ...... 35 6.3 Managerial Implications ...... 36 6.4 Theoretical Implications ...... 38

References...... 39

Appendix A: Time Periods Used in Data Analysis ...... 41

Appendix B: Portfolio Weights for Time Periods 7-14 ...... 45

Appendix C: Heatmap of All Portfolio Weights ...... 46

List of Tables

Table 1: Out-of-sample annual returns and other performance metrics of Kuosmanen (2004) and Kopa & Post (2011) models compared to the index and other investment strategies, as measured by Hodder et al.(2014) ...... 9

Table 2: Returns of Kopa Post portfolios and the index for all 84 out-of-sample, 30-calendar day holding periods ...... 20

Table 3: Key metrics for OMXN40-index and Kopa Post portfolio 30-day returns during 84 out-of-sample holding periods ...... 26

Table 4: Key metrics for OMXN40-index and Kopa Post portfolio daily returns during all holding periods ...... 28

Table 5: Portfolio turnover metrics for Kopa Post portfolios ...... 28

Table 6: Statistical significance of results for 30-day returns ...... 31

Table 7: Statistical significance of results for daily returns ...... 32

List of Figures

Figure 1: Research process for this and similar studies...... 14

Figure 2: Returns of the index and (out-of-sample returns of) Kopa Post portfolios...... 21

Figure 3: Frequency distribution of returns for OMXN40-index ...... 22

Figure 4: Frequency distribution of returns for Kopa Post portfolios ...... 23

Figure 5: Cumulative returns of OMXN40-index and Kopa Post Portfolios ...... 25

Introduction 1

1 Introduction

1.1 Background and Motivation

When portfolio G dominates portfolio F stochastically, it means that portfolio G has higher expected returns (without increased risk) or lower expected risk (without lower returns) or both simultaneously. Therefore, in theory, constructing stochastically dominating stock portfolios is attractive, since a stochastically dominating portfolio would mean higher returns, lower risk, or both simultaneously. Some portfolio selection models based on stochastic dominance have been developed (e.g. Kuosmanen, 2004; Kopa & Post, 2011; Kopa & Post, 2015). However, the third-party research on out-of-sample performance of such models is still limited, Hodder and his colleagues being a rare example of such study (Hodder et al., 2014). Furthermore, both in-sample and out-of-sample studies of this topic have often focused on weights of different industries in portfolio, instead of directly weights of specific companies. Several of these studies have used daily returns of different industries, often Fama-French’s return data of 49 industries (e.g. Kuosmanen, 2004; Hodder et al. 2014). Therefore, there is a lack of existing studies measuring the out-of-sample performance of stochastic dominance based portfolio selection models, when investing directly in stocks. This creates the need to study the out-of-sample performance of such models in this context. This study aims to partly fulfill that need. With better understanding of the performance of such optimization models in this context, stochastic dominance-based models – if proving to be an effective way to improve portfolio performance – can be utilized more effectively in investing and consequently provide higher returns and/or lower risks to investors.

Introduction 2

1.2 Research Question and Aim of the Study

The research question is: How well does the stochastic dominance based portfolio selection model by Kopa and Post (2015) perform, when applied to a portfolio consisting of ~40 stocks of OMX Nordic 40 stock market index?

The aim of this study is to increase the knowledge about the out-of-sample performance of stochastic dominance based portfolio selection models when investing in public stocks. This goal can be achieved by measuring the performance of such model in selecting stock portfolios consisting of OMX Nordic 40 -index constituents, and by comparing this performance to the corresponding index performance.

1.3 Research Approach and Scope

The data used in this study is quantitative. Also, the concept of stochastic dominance, and stochastic dominance-based models used in this study have quantitative nature. Therefore, the research approach and the research methods of this study are mainly quantitative. The scope of this study is limited to applying one such stochastic dominance based portfolio selection model (Kopa & Post, 2015) to one specific context and data sample: OMXN40-index, its constituents, and 84 data and evaluation periods with the combined time period of approximately eight (8) years. The length of each data period (based on which a portfolio is formed) is one (1) year, and the length of each evaluation period is 30 days. This scope was selected, because it is focused enough to give meaningful, valuable results, that the existing theory has not provided. Measuring performance across several contexts or measuring performance of several optimization models would have been a vast task for a master’s thesis, considering how much time various data acquiring, data processing, model implementation and data analysis with the current scope took. Those parts of research process took considerable amount of time, because of the new context for a study in this field (OMXN40-index and its constituents, instead of CRSP all-share index and Fama-French industry returns (French, 2018)), for which data was not readily available in the correct format.

Introduction 3

1.4 Structure of the Study

This study consists of six main chapters. Introduction chapter gives background and motivation for the study, presents the research question, and aim of the study and describes the research approach and scope. Theory chapter reviews some literature relevant to the research topic: more specifically, theory about the concept of stochastic dominance, portfolio selection, and stochastic dominance based portfolio selection models. Research process and data chapter describes the research process and explains in detail some phases such as data gathering, data processing and data analysis. Empirical Findings chapter describes the empirical results such as portfolio weights recommended by stochastic dominance based models, performance of such portfolios, and compares this performance to Nordic OMX 40-index. Results-chapter summarizes the main empirical results and also examines them from theoretical perspective. Finally, Conclusions chapter answers the research question, presents managerial as well as theoretical implications including potential future research areas, and evaluates the study.

Theory 4

2 Theory

2.1 Stochastic Dominance

The concept of stochastic dominance is relevant when dealing with uncertain choices, such as different investment options. Stochastic dominance is based on ordering probabilities related to such prospects. There are several degrees of stochastic dominance: first-order stochastic dominance (FSD), second-order stochastic dominance (SSD), third-order stochastic dominance (TSD), and higher than third order stochastic dominance. However, TSD and higher order stochastic dominance are less relevant for this study, so they are excluded from our scope. (Hadar & Russell, 1969) FSD can be defined the following way. When G(x) and F(x) are cumulative probability distributions for random variable x, then G FSD dominates F if and only if

퐺(푥) ≤ 퐹(푥) 푓표푟 푎푙푙 푥. (1)

In other words, the cumulative probability distribution G is partly or completely under F, which means that G is at least as large as F in the sense of FSD. This means that if investment portfolio G FSD dominates F, then the cumulative distribution of G’s probability function is always lower or the same as F’s probability function. Consequently, the return probabilities for G are concentrated more towards higher returns than corresponding probabilities of F. Moreover, when portfolio G FSD dominates portfolio F, then G has higher expected return than F. Yet, not all portfolios that have higher expected return are FSD dominating portfolios towards ones with lower expected return. (Hadar & Russell, 1969) SSD is a weaker degree of stochastic domination than FSD. When G FSD dominates F, then G also SSD dominates F. The practical meaning of SSD in investing can be summarized the following way: if portfolio G SSD dominates portfolio F, then G has lower risk, while maintaining at least as high expected return as G. More formally, for G’s probability function to SSD dominate F, the area (integral) under cumulative probability distribution G(y) has to be smaller or equal to the corresponding area of F(y) in the following way (Hadar & Russell, 1969):

Theory 5

푥 푥 ∫ 퐺(푦)푑푦 ≤ ∫ 퐹(푦)푑푦 푓표푟 푎푙푙 푥. (2) 푥1 푥1

The portfolio selection models presented in subchapter 2.3 – such as the Kopa & Post (2015) model used in our empirical study – rely heavily on algorithms that identify and choose portfolios that SSD dominate other portfolios. SSD based portfolio selection criteria assume nondecreasing and concave utility function, which in the context of investing translates to risk-averse investors who prefer high returns over low returns. Therefore, SSD based portfolio selection models have potential to be valuable in the field of finance and investing. (Post, 2003) More generally, the concept of stochastic dominance has been utilized in fields such as economics, finance, statistics, marketing, and operations research since about 1970. However, the major drawback of stochastic dominance in finance has been the lack of methods to choose efficient diversification strategies or portfolios using it. (Levy, 1992) The next subchapter briefly introduces the more traditional approach – that does not utilize stochastic dominance – to portfolio selection in investing.

2.2 Portfolio Selection

Markowitz (1952) discusses various rules for portfolio selection in his widely cited article – the article that laid the foundation for the modern portfolio theory, and for the still popular mean-variance approach to investing. He presents the still widely used investment principle or rule: that investors should consider (high) expected return a desirable thing and (high) variance of return an undesirable thing. He also suggests that the expected rate of return could be allowed to vary with risk: lower expected return may be acceptable if the risk is also lower. He rejects the idea that investors should try to simply maximize discounted returns, because according to this idea a well-diversified portfolio seems to never be preferable to non-diversified portfolios with high expected return. Markowitz suggests that selecting a well-diversified portfolio is the

Theory 6 best way to select portfolios. He also suggests that straightforward maximization of expected value, regardless of exact definition of expected value (e.g. discounted or not discounted future returns), tends to lead to investors placing all funds into just one security with the highest expected returns. (Markowitz, 1952) Williams (1938) proposes that investors should diversify all funds among all those securities which have the maximum expected return. According to him, the rule of large numbers ensures that the actual yield of the portfolio will be close to the expected return. However, Markowitz (1952) rejects this proposition too, citing that the returns from different securities are too intercorrelated. The expected return of a portfolio can be calculated using the formula

푁

퐸 = ∑ 푋푖 휇푖, (3) 푖=1 where E= Expected return of the portfolio,

푋푖= weight of the security i of the whole portfolio, and 휇푖 = the expected return of security i. (Markowitz, 1952)

The so-called E-V (Expected value – Variance) rule – can be used for the selection of a stock portfolio. However, to use it, one must be able to determine expected value and variance of each stock. (Markowitz, 1952) Since its publication, approaches related to E-V rule have been common and popular methods for portfolio selection in investing. However, the next subchapter presents a different approach: stochastic dominance based portfolio selection. Interestingly, despite their differences, both E-V rule (and mean-variance approach), and stochastic dominance based portfolio selection aim at the same goal: to find portfolios with simultaneously with high expected return and low risk.

2.3 Stochastic Dominance Based Portfolio Selection Models

The lack of ways to select efficient portfolios based on stochastic dominance used to significantly hinder the utilization of stochastic dominance in finance (Levy, 1992).

Theory 7

Fortunately, some stochastic dominance based portfolio selection models have since been developed (e.g. Kuosmanen, 2004; Kopa & Post, 2011; Kopa & Post, 2015). Because such models are a relatively new field, the number of them is still limited. Furthermore, the research related to the performance – especially out-of-sample performance – of such models in stock investing is even more limited. The empirical part of this study examines the performance of Kopa & Post (2015) portfolio selection model. Therefore, it is important to describe that model in this literature review. Kopa and Post (2015) develop and present and SSD-based portfolio selection model, in the form of a linear programming test for finding the optimal portfolio weights. It is a dual SSD portfolio efficiency test, which means that the second part of the test identifies a second portfolio that dominates the first evaluated portfolio in the case the first portfolio is found SSD-inefficient. Consequently, some additional improvement possibilities for portfolio weights of an inefficient portfolio can be identified and executed. (Kopa & Post, 2015) Moreover, Kopa & Post (2005) model allows the user to set various linear constraints based on user’s more specific needs and optimization goal. For example, a mutual fund manager could add a constraint that limits the maximum weight of one stock to 10% of the portfolio. Also, constraints related to transaction costs could be set to be part of the utility function when using the model. This way, such costs could be considered in the model, and consequently the net investment returns could be increased. (Kopa & Post, 2015) Kopa and Post (2015) also apply their model to CRSP all-share index and conclude that the index is significantly SSD-inefficient, and that therefore no rational or risk-averse investor should hold the index instead of better performing more concentrated portfolios. The linear programming procedure of Kopa & Post (2015) dual SSD portfolio efficiency model that was used in our study is

Theory 8

(4)

In Model (4) N is the number of assets, T is the number of return periods, w refers to portfolio weights, τ is the evaluated portfolio, and λ is the second portfolio that SSD dominates τ, if it is SSD inefficient. The Kopa & Post (2015) dual SSD model maximizes the weighted sum of differences in cumulative returns between τ and λ. (Kopa & Post, 2015) Before the development of Kopa & Post (2015) model, Hodder, Jackwerth and Kolokolova (2014) examined the out-of-sample performance of Kuosmanen (2004) and Kopa & Post (2011) SSD-based portfolio selection models. Instead of direct stock return data like in our study, they used daily return data of different industries from year 1927 to 2012. Their benchmark index was the value-weighted CRSP all-share index. This index represents all stocks listed in stock exchanges in the USA. Their data was gathered from the French data library (French, 2018). They used 85 different time periods: each 1-year data periods, and the subsequent 1-year out-of-sample evaluation periods. (Hodder et al., 2014)

Theory 9

The results by Hodder et al. (2014) are shown in Table 1. The table describes the annual out-of-sample returns of Kuosmanen (2004) and Kopa & Post (2011) models, compared to the index and other investment strategies. Also, annual standard deviation, skewness and kurtosis are shown. Both Kuosmanen (2004) (12.16%) and Kopa & Post (2011) (14.77%, 14.93%, 14.96%) models had higher mean returns than the index (11.38%). The returns by the Kopa & Post models were higher than the return of Kuosmanen model. However, the Kopa & Post models also had high standard deviations (22.05%, 22.34%, 22.32%) – higher than the index (20.40%) – whereas Kuosmanen had lower standard deviation (19.59%) than the index. In other words, based on these results the Kopa Post models beat the index in terms of returns, but at the cost of higher risk, whereas Kuosmanen managed to beat the index by providing higher returns and lower risk simultaneously. (Hodder et al., 2014) Kuosmanen (2004) portfolio selection model can be implemented by using the following linear programming procedure:

Theory 10

(5)

Similarly to the previous model, in Model (5) N is the number of assets and T is the number of return periods – in our study return days. The Kopa & Post (2011) model can be implemented using the following linear programming procedure (Hodder et al., 2014):

(6)

Different versions of this model (6) can be done by assigning different weights into vector w. Hodder et al. (2014) created three different versions of Kopa & Post (2011) model such way: (1) a model focusing on risk reduction, (2) a model focusing on maximizing average return, and (3) a model focusing on improvement compared to the power utility function. (Hodder et al., 2014)

Theory 11

Finally, in a recent study by Liesiö, Xu and Kuosmanen (2017), robust stochastic dominance based linear programming models were developed. Unlike most stochastic dominance based portfolio selection models, their models do not assume that historical asset returns are equally probable as future returns. Therefore, their models are developed to be used under incomplete probability information. When using these models, the probabilities can be specified, for example by using estimates from multiple experts. Like many studies in this field, they applied their models by using CRSP all-share index as the benchmark portfolio, and the Fama-French daily return data of 49 industry portfolios as asset data (French, 2018). Their results indicate that such approach can help to identify portfolios that perform better out-of-sample, compared to their index. (Liesiö et al., 2017)

Research Process and Data 12

3 Research Process and Data

3.1 Data

The empirical data used in this study were the following: (1) historical daily returns of OMX Nordic 40 constituents and the index, (2) historical daily SEK (Swedish Krona)/EUR and DKK (Danish Krona)/EUR currency rates and (3) Leavers & Joiners data of OMX Nordic 40-index. Daily return data is from June 22th 2009 to June 20th 2017. Daily returns of OMX Nordic 40 constituents were obtained from Thomson Reuters Datastream financial database. Daily prices of OMX Nordic 40-index were obtained from the website of Nasdaq OMX Nordic (Nasdaqomxnordic.com, 2018). The SEK/EUR and DKK/EUR daily currency rates were obtained from Thomson Reuters Datastream using WM Company’s Closing Spot Rate at 4:00PM UK time. In addition, Leavers & Joiners data of OMX Nordic 40 were downloaded from Thomson Reuters Eikon database, to determine which stocks were constituents of the index during different time periods.

3.2 Investment Strategy and Empirical Test Setup

The investment strategy is a buy-hold trading strategy with no short positions, with 1-year historical formation periods, and 30-day out of sample holding periods, during which the portfolio performance is measured. This length of formation and holding periods is common when testing stochastic dominance models. Also, these lengths were chosen, because previous researchers have shown than these lengths are effective. Moreover, 1-month holding periods have shown good performance – because of momentum – when using stochastic dominance based portfolio selection (e.g. Moskowitz & Grinblatt, 1999; Liesiö et al., 2017; Hodder et al., 2014). All calendar days are counted as ‘days’ for that 30-day period, not only trading days. This way returns for evaluation periods are fixed duration of time, instead of changing in duration based on which month it is, which increases the validity and reliability of the study.

Research Process and Data 13

The first data period is from June 22th 2009 to June 21th 2010. The performance of the corresponding portfolio – made from stock weights given by the model based on using data of the first data period – is measured during the first 30-day holding period following that (from June 22th 2010 to July 21th 2010). The second forming and holding periods ended and started one month after the first one, and so on. In total there are 84 different data and holding periods up until 2017. This way the performance of models is measured using many different time periods, which improves the reliability of the results of this study. The exact list of time periods used in the study is presented in Appendix A at the end of this thesis. Between and during these time periods, the composition of Nordic OMX 40-index constituents changed many times. The portfolios selected should consist only of stocks that were OMX Nordic 40-index constituents at the start moment of corresponding holding period. Therefore, Appendix A shows also which version of the index composition was used for which data and holding period. In total, there were 14 different compositions of index constituents used in this study. Even though at each moment the index had approximately 40 constituents, during all different holding periods analyzed in this study, there were in in total 62 different stocks, each of which was included in the index at least during one of the 84 holding periods.

3.3 Research Process

The research process used in this study is shown in Figure 1. This process is suitable for other similar studies too. Moreover, this process may be used by investment fund managers and other professionals, who consider utilizing stochastic dominance based portfolio selection models in investing. The first phase is data gathering. Daily return data of index constituents must be retrieved in a way, that considers dividends and share splits too. Financial databases can be good data sources. Also, daily values of the benchmark index are needed. If the stocks included in the index are traded in different stock markets with different currencies, then daily currency rate data is necessary to calculate returns in index currency. If the composition of the benchmark index had changed during the time periods used, then data about leavers and joiners of the index is needed too to determine which companies were included in the index during different time periods.

Research Process and Data 14

After the data is gathered, the next phase is data processing. The data must be transformed into such form, that data analysis can be performed. This happens by transforming data into datasets that consist of daily total returns of each index constituent during all used time periods. Also, similar index returns are needed. These returns should be in percentages and in the index currency. Moreover, these returns should include potential dividends and share splits. Thirdly, after the data is processed into a suitable form, it can be analyzed. The data is included in the implementation of the used stochastic dominance portfolio selection model(s). Then, the model(s) are run, which returns the portfolio weights for each holding period. After all weights are known, the returns of optimized portfolios can be calculated for each out-of-sample holding period. The corresponding returns are calculated for the benchmark index too. Finally, based on returns of holding periods, various metrics can be calculated. In addition, metrics can be calculated also directly based on daily returns during holding periods.

• Stock and benchmark index daily return data • Currency rate data (if necessary) DATA • Leavers & Joiners data (if necessary) GATHERING

• Transform data into suitable form DATA • Daily total %-returns in index currency PROCESSING

• Implement model(s) with the data • Run model(s) for each time period • Calculate returns of portfolios formed with stock weights given by the DATA model(s) ANALYSIS • Calculate other metrics

Figure 1: Research process for this and similar studies

Research Process and Data 15

The data analysis gives information about how the model(s) and the data used are likely to perform out-of-sample. Based on these results, it can be decided how to either implement the model(s) in real investment practices, or how to further study or develop such models. In the other subchapters, these phases of this study are described in more detail. The data and data sources used in this study are described in subchapter 3.1 Data. The exact data processing phase is described in subchapter 3.4 Data Processing. The data analysis methods are described in subchapter 3.5Data Analysis.

3.4 Data Processing

To utilize the data gathered in this study, the data needed some processing first. It had to be changed into suitable forms so that the stochastic dominance-based models could analyze it. Therefore, it was processed in the ways presented in this subchapter. Daily EOD (End-of-day) values and returns of OMX Nordic 40-index are calculated in Euro currency (Nasdaqomxnordic.com, 2018). However, many of the index constituents are not: Finnish stocks are in Euros, but for Swedish stocks the local currency is SEK and for Danish stocks DKK. To calculate the daily EUR returns and values of OMX Nordic 40- index, the index itself uses WM Company’s Closing Spot Rate at 4:00PM UK time daily currency rates (SEK/EUR, DKK/EUR) (Nasdaqomxnordic.com, 2018). The daily historical return data of the index constituents was in the form of ‘Total Return Index (RI)’ by Thomson Reuters DataStream. RI shows the theoretical growth in value of a share holding over a time period. RI assumes that dividends are re-invested to purchase additional units of the equity at the closing price on the ex-dividend date (Datastream, 2018). The formula for calculating RI is

푃푡 푅퐼푡 = 푅퐼푡−1 ∗ , (7) 푃푡−1

except when t = ex-date of the dividend payment 퐷푡, then the formula is

푃푡 + 퐷푡 푅퐼푡 = 푅퐼푡−1 ∗ , (8) 푃푡−1

Research Process and Data 16 where:

푃푡 = price on ex-date

푃푡−1 = price on previous day

퐷푡 = dividend payment related to ex-date t. RI calculation ignores tax and re-investment charges (Datastream, 2018). In other words, RI is a form of data that considers dividends and potential share splits so that by comparing daily values of RI the (mostly) real daily stock returns can be easily calculated. However, RI is calculated in local currency of the stock, whereas in this study the portfolios are selected based on expected EUR (%) performance. Therefore, to calculate the daily EUR (%) returns of each Danish and Swedish stock, beforementioned daily currency rate data was used. These data processing calculations were made in Microsoft Excel. The currency rates used are the exact same the OMX Nordic 40-index uses to calculate its own EUR values and returns. As the result, the calculated EUR (%) returns of portfolios selected using stochastic dominance-based model can be fairly compared to OMX Nordic 40-index returns. Picture 1 demonstrates how daily EUR (%) returns of stocks were calculated in excel based on daily RI-values and currency rates.

Picture 1: Demonstration how daily EUR (%) returns of stocks were calculated in Excel

The original data source for OMXN40-index daily values did not include days when OMX Helsinki stock exchange was closed, but Danish or Swedish exchanges were not, because the index itself is trades in Euro currency. Also, some of such days were not real trading days for Danish or Swedish exchanges either, but because of currency rate changes

Research Process and Data 17 on those days the constituents listed in other currencies than Euro, were able to have (EUR %) returns in those days. In other words, the constituent return data included these days too, and the (EUR) returns of stocks listed in Helsinki for such days were rightfully marked as zero already in the original data. Therefore, to make index return data comparable to the constituent return data, such missing days were manually added as zero-return days into index return dataset. In total, there were 24 such days. This process is demonstrated in Picture 2.

Picture 1: Adding Missing Days Into Index Returns Dataset

Furthermore, such days in the original constituent return data, which had zero (EUR) returns for all companies (in all stock exchanges) – meaning that there were neither trading nor currency fluctuations those days – were removed from the constituent return data. In addition, there were approximately five (5) zero-return days (in reality, these were not trading days for the index) marked in the original index-data, but these days were completely lacking from the constituent data because they were not real trading days for the constituents either. Therefore, these days were removed from the index return dataset. These two data processing steps made the number of return data days identical – 2067 days of return data – for both Index data and constituent data, which made accurate data analysis significantly easier. These steps did not in any way lower the validity or accuracy of this study, but rather improved them.

Research Process and Data 18

3.5 Data Analysis

After data processing, it was time to run the stochastic dominance-based model with the data. The stochastic dominance-based model by Kopa and Post (2015) was implemented through Python programming language using Gurobi solver. Running the models would give the optimal weights – according to Kopa and Post (2015) model – of each stock in the portfolio. First, Kopa and Post (2015) model was implemented. The researcher received a Python Gurobi implementation of this model that was made by other researchers. To specify, the implementation was written in Python language and it utilized Gurobi’s optimization libraries for implementing the optimization related code for the optimization model. However, the researcher had to edit the implementation so that it used the data of this study. This required additions to the Python code, to enable reading the correct daily returns of OMXN40-index and its constituents from excel files – for each of the 84 data periods. This process was repeated 84 times using different data time period each time. Each of the 84 runs had approximately 110-135 second CPU processing duration, using a laptop with two- core CPU and 8GB of RAM. After each run, the results – the weights of each stock in the portfolio for that holding period – were saved into an Excel-file. After finishing all 84 runs for the Kopa Post optimization model, all portfolio weights for all time periods were known. At this point, the returns for all 62 stocks and the OMXN40- index itself for all 84 holding periods were calculated. Then, the 84 Kopa Post portfolio returns – one for each holding period – were calculated using portfolio weights and returns of stocks as inputs. Finally, various statistical metrics were calculated based on the returns. Also, descriptive graphs were drawn. The returns and these metrics and graphs are presented in the next chapter.

Empirical Findings 19

4 Empirical Findings

A sample of some of some portfolio weights (for time periods 7-14) is shown at Appendix B. Also, a heatmap that shows all portfolio weights during all time periods in a visual way is shown in Appendix C. As seen from these appendices, the weights given by the model tended to change only moderately between consequent time periods. The weights of time periods far away from each other were often very different from each other. However, there were certain stocks that the model selected much more often than the others. Some of such companies were Swedish Match (tobacco and snus industry, included in portfolio during 69/84 time periods) (swedishmatch.com, 2018), Astrazeneca (healthcare sector, drug manufacturing industry, 68/84), Novo Nordisk B (healthcare sector, biotechnology industry, 64/84), Carlsberg B (consumer defensive sector, beverages and beers industry, 60/84), Sampo A (financial services sector, insurance – property & casualty industry, 55/84) and Fortum (utilities sector, utilities – diversified industry, 49/84). (Nastaqomxnordic.com, 2018) Based on the model often selecting these companies, the model seems to favor companies operating in less volatile industries. Typically, healthcare, consumer products such as beverages and tobacco products, insurance, and utilities (such as electricity) tend to be less volatile, less cyclical (defensive) industries, because the demand for such products and services does not heavily fluctuate based on macroeconomic cycles. Consequently, it seems that these stocks were determined to have low risk but good returns – in a way that fulfills stochastic dominance based criteria of the Kopa & Post (2015) model – and were thus often selected into the portfolio by the model. The returns of optimal portfolios – based on Kopa & Post (2015) model – for each of the 84 holding periods, and the corresponding returns of OMXN40-index are presented in Table 2. The returns given by the optimized portfolios were out-of-sample returns. This is an important distinction, because it is relatively easy to get great in-sample returns for optimized models by overtraining the model for that data sample. However, such overtrained model would not necessarily give good real-world, out-of-sample returns. Therefore, it is important to measure the performance of this kind of models specifically using out-of- sample evaluation periods like in this study.

Empirical Findings 20

Table 2: Returns of Kopa Post portfolios and the index for all 84 out-of-sample, 30-calendar day holding

periods

Time Period Time OMXN40-Index KopaPostPortfolios Period Time OMXN40-Index KopaPostPortfolios Period Time OMXN40-Index KopaPostPortfolios Period Time OMXN40-Index KopaPostPortfolios 1 -2.5% -1.5% 22 -2.7% 4.5% 43 5.0% 3.1% 64 -1.7% -3.0% 2 -1.0% 1.9% 23 -8.5% -4.5% 44 0.0% 4.2% 65 7.5% 10.8% 3 8.8% 5.0% 24 3.7% 2.5% 45 -0.3% -2.8% 66 -4.2% -1.8% 4 2.0% -0.7% 25 6.5% 10.0% 46 -1.9% -1.0% 67 -9.0% -9.3% 5 -1.6% 2.4% 26 7.0% 1.2% 47 4.4% 10.3% 68 -0.2% 4.5% 6 8.4% 3.3% 27 -0.4% -2.3% 48 0.7% 4.8% 69 3.7% 1.8% 7 -1.8% 0.3% 28 -1.7% 0.2% 49 -1.4% -1.4% 70 2.8% 2.2% 8 -0.4% 6.0% 29 -2.1% -6.1% 50 0.7% -2.0% 71 -4.1% 3.8% 9 -2.1% -2.5% 30 4.8% 1.5% 51 3.4% 1.7% 72 -0.7% -1.3% 10 2.8% 5.1% 31 3.0% 3.6% 52 -6.1% -5.7% 73 2.3% 5.6% 11 0.7% 1.9% 32 5.8% 1.4% 53 4.7% 6.6% 74 -2.4% 0.2% 12 -11.5% -7.0% 33 0.0% 5.8% 54 -1.2% 0.2% 75 1.0% 1.1% 13 -0.2% 2.0% 34 -4.8% 0.8% 55 3.8% 4.3% 76 0.1% -0.5% 14 -19.5% -11.2% 35 8.5% 7.3% 56 9.0% 5.4% 77 -4.7% -7.2% 15 4.3% 2.5% 36 -9.3% -8.3% 57 5.5% 5.5% 78 6.3% 3.4% 16 4.4% 1.8% 37 7.5% 5.9% 58 0.8% 3.2% 79 1.3% 2.5% 17 -0.9% -1.0% 38 2.0% -2.0% 59 -3.1% -2.1% 80 3.1% 2.0% 18 5.7% 8.3% 39 7.3% 1.9% 60 -4.9% -5.5% 81 -0.9% 1.1% 19 10.4% 3.1% 40 -0.7% 2.6% 61 4.7% 5.3% 82 0.5% 0.7% 20 8.4% 4.4% 41 0.0% 1.9% 62 -6.8% -8.3% 83 3.9% 3.8% 21 -3.3% 1.8% 42 1.1% 3.7% 63 -0.5% -0.1% 84 2.7% 4.3%

Even though Table 2 includes all data about the returns of the model compared to the index, it is not intuitive to understand the results or make any conclusions just by looking at this table. Therefore, the same returns are visualized in Figure 2. This graph shows the same returns of the index and the optimized portfolios. The graph is (only) slightly more intuitive

Empirical Findings 21 than the table, perhaps giving the impression that the optimized portfolios may have performed slightly better than the index on average.

Figure 2: Returns of the index and (out-of-sample returns of) Kopa Post portfolios. (During each 84 30-day holding period. In this graph, the 100% means 0% return and e.g. 105% refers to 5% return during a holding period.)

The frequency distributions of the same returns as in the previous table and the previous graph are presented in Figure 3 and Figure 4. First, Figure 3 shows the frequency distribution of OMXN40-index returns. Looking at the graph, it resembles the normal distribution that has its peak close to zero. The “0%” bracket of the graph includes the returns

Empirical Findings 22 that were from -2% to 0%. This distribution is negatively skewed, meaning that the left tail of the distribution is longer than the right tail. Also, there is one negative ‘outlier’ 30-day return that is in -20% to -18% bracket (more specifically -19.5%) – significantly worse than return for any other holding period.

Figure 3: Frequency distribution of returns for OMXN40-index

Then, the Figure 4 shows the corresponding frequency distribution of returns for the optimized portfolios. Compared to the index returns, this distribution looks to have its peak (and mean) at the 2% bracket (returns from 0% to 2%) instead of 0% that it was in the index return distribution. Also, the returns in 4% and 6% brackets of this distribution, especially for the 6% one, look more frequent compared to the index returns and the normal distribution. In addition, the portfolio returns do not have any ‘outlier’ returns – unlike the index returns – which makes it less negatively skewed than the index returns. Based on these figures, it again looks like the optimized portfolios had slightly better performance than the index.

Empirical Findings 23

Figure 4: Frequency distribution of returns for Kopa Post portfolios

However, to really see the relevant real-world difference between the index returns and the portfolio returns, it is helpful to look at the cumulative returns during all 84 holding periods. The cumulative returns of both the index and the optimized portfolios are presented in Figure 5. These returns are calculated using the following formula for each of the time periods:

퐶푢푚푢푙푎푡푖푣푒 푟푒푡푢푟푛

= 1 ∗ (푟푒푡푢푟푛 푡1 + 1) ∗ (푟푒푡푢푟푛 푡2 + 1) … ∗ (푟푒푡푢푟푛 푡푛 + 1) (9) ∗ 100% − 100%.

In the formula (9), n is the (chronological) number of a holding period for each holding period. After calculating the product of returns added with 1 until the corresponding time period, the formula converts the value into percentages and subtracts 100% to show the cumulative return (instead of cumulative value) in percentages at the end of the corresponding time period. In other words, these cumulative returns are the total return that would result from the following investing process: First, investing in the optimized portfolio of the first time period, and holding it for the duration of the first holding period. Then selling it at the end of the first holding period. After that waiting (typically 1 or 2 days) until the second holding

Empirical Findings 24 period is about to start and then buying the stock at the end of the last trading day before the start of that holding period: thus, the purchasing price being the previous closing price. Then, before the beginning of the second holding period, investing all the money – received from selling the first portfolio – in the optimized portfolio corresponding to the second time period, and holding that portfolio for the duration of the second holding period and then selling it at the end of that period. This buying and selling process would be repeated identically for all 84 holding periods. The calculation of cumulative returns is similar for the index too, with the obvious difference of using the index returns instead of the returns of optimized portfolios. For simplicity and clarity, this calculation assumes there are no taxes or transaction costs. These cumulative index returns are almost identical to the continuous, cumulative return of the index itself: or the scenario of investing at the beginning of the holding period 1 into a theoretical OMXN40-index tracker-ETF (which has identical returns compared to the index, and no fees), and continuously holding until the end of holding period 84. Such total continuous holding period would be approximately seven (7) years from June 22, 2010 to June 20, 2017. When looking at the graph of cumulative returns in Figure 5, it is obvious that the optimized portfolios had significantly higher returns than the index. At the end of the last holding period (84), the cumulative return of the index was only 66.20%, whereas the corresponding return for the optimized portfolios was lucrative 168.98%. This means that the cumulative return of optimized portfolios was 2.55 times the cumulative index return, or 155% higher than the cumulative index return.

Empirical Findings 25

Figure 5: Cumulative returns of OMXN40-index and Kopa Post Portfolios

By any standard, the optimized portfolios clearly beat the index, when examining the combined performance of all 84 holding periods. The performance of optimized portfolios is excellent, and vastly higher than the index, especially when looking at the cumulative returns. Individual returns of each holding period (Table 2, Figure 2, Figure 3, Figure 4) and cumulative returns of holding periods (Figure 5) have been examined earlier in this chapter. However, there are also other statistical metrics that can be calculated from the returns of the index and the optimized portfolios. These metrics are: the mean return during a 30-day holding period, the median return during a 30-day holding period, standard deviation, variance, minimum return, maximum return, skewness of the return distribution, kurtosis of the return distribution, annualized cumulative return, and annualized mean return. The values of these metrics are listed for both the index and the optimized portfolios in Table 3.

Empirical Findings 26

Table 3: Key metrics for OMXN40-index and Kopa Post portfolio 30-day returns during 84 out-of-sample holding periods

Metric OMXN40-Index KopaPostPortfolios Mean 30-day return 0.74% 1.28% Median 30-day return 0.61% 1.88% Standard Deviation 5.02% 4.33% Variance 0.25% 0.19% Min Return -19.48% -11.18% Max Return 10.37% 10.81% Skewness -0.84 -0.56 Kurtosis 2.23 0.59 Cumulative Return over 84 periods 66.20% 168.92% Cumulative return annualized 7.53% 15.18% Mean annual return 8.83% 15.35%

The mean 30-day holding period return was significantly higher for the optimized portfolios compared to the index (1.28% vs 0.74%): it was 74%, or 0.54 percentage points higher than the index. To highlight the significance of this difference in real-world investing, when converting these 30-day mean returns into approximate 1-year returns (by simply multiplying by 12), the yearly mean return would be 8.83% for the index, but 15.35% for the optimized portfolios. To put this into perspective, 8.83% yearly mean return is close to typical yearly mean return for stock market indexes and low cost index investing. For example, the annualized total return for the S&P 500 index – consisting of 500 biggest stocks listed in American stock exchanges – over the past 90 years was 9.8% when calculated recently (CNBC, 2017). However, mean yearly return of 15.35% would be an exceptionally high return, especially if maintained over extended period of time (e.g. 10 years or more). Similarly, the median 30-day holding period return was vastly higher for the optimized portfolios compared to the index (1.88% vs 0.61%): it was 207%, or 1.27 percentage points higher than the index. When converting these 30-day median returns into approximate 1- year returns (by simply multiplying by 12), the yearly median return would be 7.33% for the index, but 22.53% for the optimized portfolios. Here the difference is even larger between the index and the optimized portfolios, than when looking at mean returns. Another way to estimate the yearly returns of the optimized portfolios compared to the index is converting the cumulative return over all 84 periods (66.20% for the index and 168.92% for the optimized portfolios) into annualized returns. This can be done by using a

Empirical Findings 27 simple 7-year time period (only one day longer than the duration from the beginning of our holding period 1 to the last day of the holding period 84), and the calculation

퐴푛푛푢푎푙푖푧푒푑 퐶푢푚푢푙푎푡푖푣푒 푅푒푡푢푟푛 = (퐶푢푚푢푙푎푡푖푣푒 푅푒푡푢푟푛 + 1)1/7 − 1. (10)

The annualized cumulative return (formula 10) is the annual return, with which the cumulative return over our seven (7) year time period would be what it is now. The annualized cumulative return for the index was 7.53% and for the optimized portfolios 15.18%. By this measure too, the return was vastly higher for the optimized portfolios. Moreover, the simple annual mean return (mean 30-day return multiplied by 12) was 8.83% for the index and 15.35% for the optimized portfolios. Not only were the returns of optimized portfolios higher, also the standard deviation and variance of returns for optimized portfolios were lower than for the index returns. The standard deviation was for the optimized portfolios 4.33%, and 5.02% for the index. The variance was for the optimized portfolios was 0.19%, and 0.25% for the index. The standard deviation for the optimized portfolios was 86%, and the variation for optimized portfolios only 74% those values for the index returns. These findings imply that the optimized portfolios had lower risk than the index. Moreover, the worst 30-day holding period was significantly worse for the index (- 19.48% return) than for the optimized portfolios (-11.18% return). The loss for the optimized portfolio during the worst holding period was only 57% of the worst loss for the index. Also, the maximum 30-day return for the optimized portfolios (10.81%) was slightly higher than for the index (10.37%). In addition, the skewness of the returns of the optimized portfolios was less negative (-0.56) than for the index (-0.84). This means that the left tail of the return distribution was shorter for the optimized portfolios. Also, the returns of the optimized portfolios had vastly lower kurtosis (0.59) compared to the index (2.23). Distributions with high kurtosis values have heavy tails, or outliers, whereas distributions with low kurtosis values have only light tails, or lack of outliers. As described earlier, the index returns had one clear, negative outlier, whereas the optimized portfolios did not have any outlier returns. This partly explains these significant differences in skewness and kurtosis between the index and the optimized portfolios.

Empirical Findings 28

Table 4: Key metrics for OMXN40-index and Kopa Post portfolio daily returns during all holding periods Metric OMXN40-Index KopaPostPortfolios Mean daily return 0.0357% 0.0592% Median daily return 0.0470% 0.0764% Standard Deviation 1.21% 0.96% Variance 0.0147% 0.0092% Min Return -6.92% -5.54% Max Return 6.50% 4.75% Skewness -0.298 -0.227 Kurtosis 3.44 3.01

Similar key metrics for daily returns, instead of 30-day returns, are shown in Table 4. In short, these metrics looks mostly similar to 30-returns: mean, median and minimum daily returns are higher, and standard deviation and variance lower for the optimized portfolios compared to the index. However, the maximum daily return is lower for the optimized portfolios than for the index. Finally, metrics related to portfolio turnover for optimized portfolios are presented in Table 5. These metrics are important, because high turnover increases transaction costs and often tax costs too. The mean and median 1-year turnover were 190.6% and 193%, respectively. The smallest 1-year turnover was 84.6%, while the biggest one was 320.9%. Furthermore, the mean and median 30-day turnover were 15.5% and 14.2%, respectively. The smallest 30-day turnover was only 3.5%, while the biggest one was approximately 15 times higher: 51.4%.

Table 5: Portfolio turnover metrics for Kopa Post portfolios Mean 1-year turnover 190.6% Median 1-year turnover 193.0% Min 1-year turnover 84.6% Max 1-year turnover 320.9% Mean 30-day turnover 15.5% Median 30-day turnover 14.2% Min 30-day turnover 3.5% Max 30-day turnover 51.4%

Empirical Findings 29

Usually portfolio turnover is measured as yearly turnover, but 30-day turnover metrics were included too to give better understanding about the degree of portfolio changes between two sequential holding periods. The 1-year turnovers were calculated as sums of 12 previous 30-day turnovers. Obviously, it was assumed that a portfolio is directly transformed into the portfolio of the next holding period without unnecessary selling of all assets between holding periods. In the next chapter, the significance of the empirical findings in this chapter is examined. Also, the meaning of the findings is interpreted and their relation to relevant theory is examined.

Results 30

5 Results

To summarize the key empirical findings examined in the previous chapter, the returns for the optimized portfolios were substantially higher compared to the index returns. Also, the risk – based on standard deviation and variance – was lower for the Kopa & Post (2015) optimized portfolios than for the index. In order words, in this empirical study the optimized portfolios performed considerably better than the index. The optimized portfolios easily beat the index in this empirical case. However, this raises questions related to the generalizability of these findings: Was the higher performance of the optimized portfolios compared to the index mainly luck? Would the results be similar with other data, such as with another stock market index and its constituents? And how would the optimization model perform with a very long time period – longer than seven years in our study? To estimate what the answers to such questions are likely to be, it is helpful to evaluate the generalizability of our empirical results by measuring their statistical significance. The strength of evidence that such generalizability exists, can be measured with some statistical tests. T-test is suitable for estimating the statistical significance of difference between two sets of values. In this case, these two sets are 1) the returns for the index and 2) the returns for the optimized portfolios during the 84 different holding periods. In t-tests, the result is a p-value. When using a two-tailed t-test, p-value measures the probability that the two data sets measured are not from populations with significantly different mean values. When using a one-tailed t-test, p-value measures the probability that a specific one set of the two data sets comes from a population with significantly higher mean value compared to the population of the other data sample. Furthermore, a t-test can be either paired or unpaired. In an unpaired test, the order of a data sample does not matter, because the order of the observations does not have any relation to the order of the other data sample. On the other hand, paired test is used when each observation of one sample is in a relation to a specific observation from the other sample. In our case, t-test is used to answer the question: ‘Are returns statistically significantly higher for Kopa & Post (2015) portfolios compared to the index?’ Answering this question requires a one-tailed t-test. Moreover, our data is paired, because each observation (return)

Results 31

of the index return data corresponds to one specific observation (return) of the return sample from optimized portfolios. In other words, each index return is paired with the return of the optimized portfolio of the same holding period. The resulting p-value from such t-test is presented in Table 5. The p-value is 0.0647. This is slightly over 0.05, which is often considered the standard limit for statistical significance. Since our p-value is slightly higher than 0.05, this leads to the following conclusion: There are weak statistical evidence that returns of Kopa & Post (2015) portfolios are systematically higher (out-of-sample) compared to their corresponding stock market index, also when using different data than the data in our study. However, based on this test the evidence of higher returns for such optimized portfolios compared to index, when generalized across samples, is not necessarily strong. F-test is suitable for estimating whether there is a difference in variances between populations of two samples. By default, F-test is two-tailed and consequently tends to result in higher p-values than one-tailed tests. In this study, F-test is used to answer the question: ‘Are variances statistically significantly different between the optimized portfolios and the index?’ The answer to this question – in the form of a two-tailed p-value – is 0.181. This question and test-result related to it are also presented in Table 6. This value too is over 0.05. Even if the p-value was halved (trying to justify that with argument that then it would have similar meaning to one-tailed p-value and answer the question: ‘Are variances statistically significantly lower for the optimized portfolios compared to the index?’), it would still be over 0.05 (0.09). Consequently, there are no strong statistical evidence of systematically lower variance in 30-day return distributions of Kopa & Post (2015) portfolios, compared to the index, when generalized across samples. At most, there is weak evidence of that based on this empirical study.

Table 6: Statistical significance of results for 30-day returns

Are 30-day returns statistically significantly higher for Kopa Post Portfolios compared to the index? T-Test, p-value (1-tailed, paired) 0.0647 Are variances of 30-day returns statistically significantly different between Kopa Post portfolios and the index? F-test, p-value (2-tailed) 0.181

Results 32

However, the statistical significance is different when t-test and F-test are applied directly to daily returns during all holding periods, instead of 30-day returns. The results of these tests for daily returns are shown in Table 7. The p-value of t-test for daily returns is 0.0809, which is similar the corresponding value for 30-day returns: both are slightly over 0.05. Interestingly, the p-values are extremely low (7.12*10−23) for the F-test applied to daily variances. This means that the model effectively lowered daily variances, in other words the daily volatility, of the optimized portfolios compared to the index. Furthermore, this finding indicates that lower daily variances can be generalized across samples when using the same model.

Table 7: Statistical significance of results for daily returns Are daily returns statistically significantly higher for Kopa Post portfolios compared to the index? T-test, p-value (1-tailed, paired) 0.0809 Are daily variances statistically significantly different between Kopa Post portfolios and the index? F-test, p-value (2-tailed) 7.12E-23

The lower volatility – especially daily volatility – is remarkable, because the optimized portfolios in this study had much less diversification compared to the index itself. The optimized portfolios had 5 to 12 different stocks – depending on the holding period – whereas the index always had ~40 stocks. Traditionally, less diversified portfolios – that consist only of a few different stocks – are considered riskier than portfolios – or indexes – consisting of plenty of different stocks (e.g. Markowitz, 1952). In the existing relevant theory, there are some – but few – studies that have measured the out-of-sample performance (return and/or risk) of stochastic dominance based portfolio selection models. In 2014, Hodder et al. tested the performance of some such models that had been created by then. Even though the total data period in this study was shorter than Hodder at al. (2014) period (8 vs 86 years), the number of evaluation periods was similar (84 vs 85). None of the stochastic dominance based portfolio selection models (Kuosmanen, 2004; Kopa & Post, 2011) tested by them performed as well compared to the index as the Kopa & Post (2015)

Results 33 model in our study. The Kuosmanen (2004) model in their study had 0.78%-points higher annual mean return than the index (12.16% vs 11.38%) and Kopa Post (2011) portfolios (3.39, 3.55, and 3.58%-points higher returns than the index (14.77%, 14.93% and 14.96% vs 11.38 %). Yet, the Kopa & Post (2015) optimized portfolios in our study had 6.52%- points higher mean annual return compared to the index (15.35% vs 8.83%). (Hodder et al., 2014) Simultaneously, the Kuosmanen model had lower standard deviation than the index (19.59% vs 20.40%), approximately 96% of the index standard deviation, and the Kopa Post portfolios had higher (1-year return) standard deviation than the index: 22.05%, 22.34% and 22.32% vs 20.40%, which are 108%, 110% and 109% of the index standard deviation. In our study, Kopa & Post (2015) portfolios had lower (30-day return) standard deviation than the index (4.33% vs 5.02%), which is only 86% of the index value. (Hodder et al., 2014) Furthermore, Kuosmanen model had slightly better minimum return (-41.50% vs - 44.14%) and a vastly better maximum return (88.67% vs 56.89%) compared to the index. Kopa & Post (2011) models also had better minimum returns (-37.60%, -36.91% and - 37.59% vs -44.14%) and substantially higher maximum returns (89.70%, 92.21% and 90.42% vs 56.89%) than the index. The optimized portfolios in our study too had both better minimum return and better maximum return compared to the index. However, in our study the min return was vastly better and maximum return was only slightly better than the minimum and maximum index returns, whereas models in Hodder et al. study had vastly better maximum returns, but only slightly better minimum returns compared to their index. Based on these results, the Kopa & Post (2015) portfolios in our study performed better in relation to their index than any stochastic dominance based model in Hodder et al.’s (2014) study. The optimized portfolios in our study had simultaneously higher return and lower risk (standard deviation) compared to the index than any model in that study. (Hodder et al., 2014) Consequently, our empirical findings seem even more promising than the earlier results by Hodder et al. (2014). Our results indicate, that stochastic dominance based portfolio selection models can significantly improve portfolio performance compared to the index. There were some differences in our study compared to the setting in Hodder et al. study. Firstly, in this study the duration holding periods was only 30 days, whereas it was 1 year in Hodder et al. study. Secondly, the benchmark index and geographical area of the stock market was different (Nordic countries vs USA). Thirdly, in this study the optimized

Results 34 portfolios invested directly into stocks, whereas in Hodder et al. study the portfolios invested in industries, instead of directly investing in stocks. Also, the total time period examined in this study was only 8 years compared to 86 years in Hodder et al., study. Consequently, the 84 data periods in this study were overlapping in contrast to nonoverlapping data periods in Hodder et al. study. Finally, this study measured a different portfolio selection model (Kopa & Post, 2015) than the models tested in Hodder et al. study. These differences may explain the different and better performance in this study compared to Hodder et al. study. (Hodder et al., 2014) Kopa and Post (2015) suggested that the CRSP market index – to which they applied their model – is highly inefficient, and that therefore more concentrated, SD-efficient portfolios perform better and are better investment choices for a risk-averse investor. The findings – after applying their model to OMXN40-index – indicate similar results: that OMXN40-index is significantly inefficient, and that better performance can be achieved by investing in more concentrated and more SD-efficient portfolios. Thus, the findings of this study indicate that the suggestion of Kopa and Post holds true in our empirical case too. Finally, the mean 1-year turnover of 191% in this study is high compared to most funds and portfolios. Studies have found that increased portfolio turnover tends to increase trading costs and consequently decrease performance of a mutual fund. Additional 100% of annual turnover is associated with lowering the annual return of a fund by 0.95-1.24% (Bogle, 1994; Carhart, 1997; Sharkansky, 2001). In some cases, for example when trading small-cap equities, these costs can be significantly higher, such as 2.55% (Sharkansky, 2001). Moreover, there are studies about the tax impact of portfolio turnover: higher turnover is associated with lower after-tax investment returns (e.g. Longmeier & Wotherspoon, 2006). (Blanchett, 2007)

Conclusions 35

6 Conclusions

6.1 Answer to Research Question

How well does the stochastic dominance based portfolio selection model by Kopa and Post (2015) perform, when applied to a portfolio consisting of ~40 stocks of OMX Nordic 40 stock market index?

The portfolios selected using Kopa & Post (2015) model had excellent performance compared to the OMX Nordic 40 index. These optimized portfolios had substantially higher returns than the index, and yet also lower variance of returns. Mean return, median return, and cumulative returns were all considerably higher for the optimized portfolios than for the index. In addition, both minimum and maximum returns of optimized portfolios were higher than those for the index. Even though the Kopa & Post (2015) portfolios clearly beat the index in this empirical case, based on the weak statistical significance of these results, the evidence that similar performance could be reliably generalized across samples, is only weak. The exception to this is the lower variance of daily returns for the optimized portfolios compared to the index, for which the statistical evidence is very strong. Overall, the results of our empirical study are promising, and encourage to further develop, measure, and utilize the Kopa & Post (2015) model, as well as other stochastic dominance based portfolio selection models. In addition, the relatively high portfolio turnovers when applying stochastic dominance based investment strategies can lower real investment returns compared to the index.

6.2 Evaluation of the Study

This study has several limitations. Firstly, there is only one stock market index and its constituents examined. Secondly, the time period observed was relatively short – seven (7)

Conclusions 36 years for holding periods combined and eight (8) years when the first data period is included too. The results might be very different with other stock market index or with other time period. Another indication that the results of this study likely cannot be reliably generalized across samples, are the only weak statistical significance of our empirical results, with the exception of strong statistical evidence of lower variance in daily returns for the optimized portfolios. Furthermore, this study tested only one stochastic dominance based portfolio selection model – Kopa & Post (2015) model. Therefore, based on our results it is impossible to know, what the results would be for other stochastic dominance based models. Another limitation of this study is, that it does not consider taxes or transaction costs, even though in real-world investing they have an important effect on real returns the investor gets. Therefore, this may limit the direct applicability of the findings in real-world investing. Finally, stock market returns tend not to accurately follow normal distribution. However, the t-test and F-test used in this study assume that returns are normally distributed. Therefore, the p-values received from those tests in this study may not be entirely valid but should rather be interpreted as approximate estimations of statistical significance.

6.3 Managerial Implications

The performance of Kopa & Post (2015) model in this study is promising for real-world investing. Consequently, stochastic dominance based portfolio selection models are an interesting research and development area for real-world investors and various financial organizations. For big investors such as banks and funds, even slightly increasing their returns or lowering their risks would be very valuable financially. This might be possible by developing and implementing stochastic dominance based portfolios selection models into their investment practices. However, these empirical results should be interpreted cautiously, because the statistical evidence of better returns and lower risk during a holding period is only weak. The Models like these might improve returns or lower risks in real-world investing, but it cannot be guaranteed, at least not based on current evidence. However, the empirical evidence is

Conclusions 37 strong that the Kopa & Post (2015) can be very effective in decreasing volatility of daily returns. Another factor to consider, when applying stochastic dominance based portfolio selection to real-world investing, is that the results of this study assume no taxes and no transaction costs. In practice, trading a significant part of a portfolio each month (even when transforming the existing portfolio directly into the portfolio of the next holding period) leads to high portfolio turnovers, which can lead to high related transaction costs and taxes. These costs could be lowered to some degree by lengthening the holding period – unless it significantly lowers the performance of the portfolio selection model. Another potential way to lower such costs could be to include constraints related to transaction costs in the utility function when using the optimization model. This way, the model might give higher net – after transaction and tax costs – investment returns. (Kopa & Post, 2015) However, especially for actors who are able to apply stochastic dominance based portfolio selection models to real-world investing without significantly increasing their taxes or transaction costs, developing and implementing such models might be an attractive option. Moreover, in practice the implementation of stochastic dominance based portfolio selection models into one’s investment strategy may be challenging. Especially for a typical small individual investor, the direct usage of such models is likely infeasible. However, if such models prove effective in real-world investing, there may be demand for various financial services and funds that utilize such models. By investing in stochastic dominance based fund, even a typical small investor could potentially benefit from stochastic dominance based portfolio selection. Also, for financial institutions and other actors with more resources and large investment capital, the utilization of stochastic dominance based portfolio selection may be feasible and profitable. Additionally, in this study the optimized portfolios consisted of vastly lower number of different stocks than the index itself, while retaining lower risk than the index. In real investing this would be a very attractive proposition – especially when simultaneously receiving higher returns compared to the index. To sum up, stochastic dominance based portfolio selection models are interesting to real-world investors, because such models might be an effective way to systematically increase returns and lower risks in investing. Yet, more research, testing and development of such models in real-world context is needed. Until then, investors should remain cautious.

Conclusions 38

6.4 Theoretical Implications

The main theoretical contribution of this study was testing the out-of-sample performance of a stochastic dominance based portfolio selection model (Kopa & Post, 2015) empirically, in a way relevant for real-world investing. Also, in this study the model was applied directly to selecting individual stocks, instead of choosing weights of different industries in a portfolio. The results of this study are promising, but mostly not conclusive (since the statistical significance was not strong except for daily variances). Consequently, this opens up many potential future research areas: Firstly, the out-of-sample performance of stochastic dominance based models – such as Kopa & Post (2015) model – should be measured in other stock investing contexts too. These studies could apply such models to different indexes and their constituents. For example, indexes and their constituents may behave differently in different geographical areas. In addition, it would be interesting to see how well such models perform when applied also to indexes that have low or high number of constituents: such as index with only ten constituents or index with up to thousands of constituents – although the latter may be computationally demanding. Also, many different time periods should be studied. Especially studies that include long total time period – longer than seven to eight years in this study – might be able to provide more statistically significant results than this study. Such research could help to conclusively determine, how effective approach stochastic dominance based portfolio selection would be in real-world investing. Moreover, this study measured the performance of only one stochastic dominance based portfolio selection model (Kopa & Post, 2015). Other models – such as Kuosmanen (2004) – could be studied similarly too. In brief, conducting the future research suggested in this subchapter would help to conclusively determine how well the promising performance of a stochastic dominance based portfolio selection model in this empirical study can be generalized across different data samples, portfolio selection models and investing contexts. This would result in valuable understanding about how effective approach stochastic dominance based portfolio selection truly can be in investing.

References 39

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Appendix A: Time Periods Used in Data Analysis 41

Appendix A: Time Periods Used in Data Analysis

Data Period Holding period Time period Start End Start End VersionOfConstituents 22 06 21 06 22 06 21 07 Time Period 1 2009 2010 2010 2010 1 22 07 21 07 22 07 20 08 Time Period 2 2009 2010 2010 2010 1 22 08 21 08 22 08 20 09 Time Period 3 2009 2010 2010 2010 1 22 09 21 09 22 09 21 10 Time Period 4 2009 2010 2010 2010 1 22 10 21 10 22 10 20 11 Time Period 5 2009 2010 2010 2010 1 22 11 21 11 22 11 21 12 Time Period 6 2009 2010 2010 2010 1 22 12 21 12 22 12 20 01 Time Period 7 2009 2010 2010 2011 2 22 01 21 01 22 01 20 02 Time Period 8 2010 2011 2011 2011 2 22 02 21 02 22 02 23 03 Time Period 9 2010 2011 2011 2011 2 22 03 21 03 22 03 20 04 Time Period 10 2010 2011 2011 2011 2 22 04 21 04 22 04 21 05 Time Period 11 2010 2011 2011 2011 2 22 05 21 05 22 05 20 06 Time Period 12 2010 2011 2011 2011 2 22 06 21 06 22 06 21 07 Time Period 13 2010 2011 2011 2011 2 22 07 21 07 22 07 20 08 Time Period 14 2010 2011 2011 2011 2 22 08 21 08 22 08 20 09 Time Period 15 2010 2011 2011 2011 3 22 09 21 09 22 09 21 10 Time Period 16 2010 2011 2011 2011 3 22 10 21 10 22 10 20 11 Time Period 17 2010 2011 2011 2011 3 22 11 21 11 22 11 21 12 Time Period 18 2010 2011 2011 2011 3 22 12 21 12 22 12 20 01 Time Period 19 2010 2011 2011 2012 4 22 01 21 01 22 01 20 02 Time Period 20 2011 2012 2012 2012 4 22 02 21 02 22 02 22 03 Time Period 21 2011 2012 2012 2012 4 22 03 21 03 22 03 20 04 Time Period 22 2011 2012 2012 2012 4

Appendix A: Time Periods Used in Data Analysis 42

22 04 21 04 22 04 21 05 Time Period 23 2011 2012 2012 2012 4 22 05 21 05 22 05 20 06 Time Period 24 2011 2012 2012 2012 4 22 06 21 06 22 06 21 07 Time Period 25 2011 2012 2012 2012 5 22 07 21 07 22 07 20 08 Time Period 26 2011 2012 2012 2012 5 22 08 21 08 22 08 20 09 Time Period 27 2011 2012 2012 2012 5 22 09 21 09 22 09 21 10 Time Period 28 2011 2012 2012 2012 5 22 10 21 10 22 10 20 11 Time Period 29 2011 2012 2012 2012 5 22 11 21 11 22 11 21 12 Time Period 30 2011 2012 2012 2012 5 22 12 21 12 22 12 20 01 Time Period 31 2011 2012 2012 2013 5 22 01 21 01 22 01 20 02 Time Period 32 2012 2013 2013 2013 6 22 02 21 02 22 02 23 03 Time Period 33 2012 2013 2013 2013 6 22 03 21 03 22 03 20 04 Time Period 34 2012 2013 2013 2013 6 22 04 21 04 22 04 21 05 Time Period 35 2012 2013 2013 2013 6 22 05 21 05 22 05 20 06 Time Period 36 2012 2013 2013 2013 6 22 06 21 06 22 06 21 07 Time Period 37 2012 2013 2013 2013 6 22 07 21 07 22 07 20 08 Time Period 38 2012 2013 2013 2013 7 22 08 21 08 22 08 20 09 Time Period 39 2012 2013 2013 2013 7 22 09 21 09 22 09 21 10 Time Period 40 2012 2013 2013 2013 7 22 10 21 10 22 10 20 11 Time Period 41 2012 2013 2013 2013 7 22 11 21 11 22 11 21 12 Time Period 42 2012 2013 2013 2013 7 22 12 21 12 22 12 20 01 Time Period 43 2012 2013 2013 2014 7 22 01 21 01 22 01 20 02 Time Period 44 2013 2014 2014 2014 8 22 02 21 02 22 02 23 03 Time Period 45 2013 2014 2014 2014 8 22 03 21 03 22 03 20 04 Time Period 46 2013 2014 2014 2014 8 22 04 21 04 22 04 21 05 Time Period 47 2013 2014 2014 2014 8

Appendix A: Time Periods Used in Data Analysis 43

22 05 21 05 22 05 20 06 Time Period 48 2013 2014 2014 2014 9 22 06 21 06 22 06 21 07 Time Period 49 2013 2014 2014 2014 9 22 07 21 07 22 07 20 08 Time Period 50 2013 2014 2014 2014 10 22 08 21 08 22 08 20 09 Time Period 51 2013 2014 2014 2014 10 22 09 21 09 22 09 21 10 Time Period 52 2013 2014 2014 2014 10 22 10 21 10 22 10 20 11 Time Period 53 2013 2014 2014 2014 10 22 11 21 11 22 11 21 12 Time Period 54 2013 2014 2014 2014 10 22 12 21 12 22 12 20 01 Time Period 55 2013 2014 2014 2015 10 22 01 21 01 22 01 20 02 Time Period 56 2014 2015 2015 2015 11 22 02 21 02 22 02 23 03 Time Period 57 2014 2015 2015 2015 11 22 03 21 03 22 03 20 04 Time Period 58 2014 2015 2015 2015 11 22 04 21 04 22 04 21 05 Time Period 59 2014 2015 2015 2015 11 22 05 21 05 22 05 20 06 Time Period 60 2014 2015 2015 2015 11 22 06 21 06 22 06 21 07 Time Period 61 2014 2015 2015 2015 12 22 07 21 07 22 07 20 08 Time Period 62 2014 2015 2015 2015 12 22 08 21 08 22 08 20 09 Time Period 63 2014 2015 2015 2015 12 22 09 21 09 22 09 21 10 Time Period 64 2014 2015 2015 2015 12 22 10 21 10 22 10 20 11 Time Period 65 2014 2015 2015 2015 12 22 11 21 11 22 11 21 12 Time Period 66 2014 2015 2015 2015 12 22 12 21 12 22 12 20 01 Time Period 67 2014 2015 2015 2016 13 22 01 21 01 22 01 20 02 Time Period 68 2015 2016 2016 2016 13 22 02 21 02 22 02 22 03 Time Period 69 2015 2016 2016 2016 13 22 03 21 03 22 03 20 04 Time Period 70 2015 2016 2016 2016 13 22 04 21 04 22 04 21 05 Time Period 71 2015 2016 2016 2016 13 22 05 21 05 22 05 20 06 Time Period 72 2015 2016 2016 2016 13

44 Appendix B: Portfolio Weights for Time Periods 7-14

22 06 21 06 22 06 21 07 Time Period 73 2015 2016 2016 2016 13 22 07 21 07 22 07 20 08 Time Period 74 2015 2016 2016 2016 13 22 08 21 08 22 08 20 09 Time Period 75 2015 2016 2016 2016 13 22 09 21 09 22 09 21 10 Time Period 76 2015 2016 2016 2016 13 22 10 21 10 22 10 20 11 Time Period 77 2015 2016 2016 2016 13 22 11 21 11 22 11 21 12 Time Period 78 2015 2016 2016 2016 13 22 12 21 12 22 12 20 01 Time Period 79 2015 2016 2016 2017 14 22 01 21 01 22 01 20 02 Time Period 80 2016 2017 2017 2017 14 22 02 21 02 22 02 23 03 Time Period 81 2016 2017 2017 2017 14 22 03 21 03 22 03 20 04 Time Period 82 2016 2017 2017 2017 14 22 04 21 04 22 04 21 05 Time Period 83 2016 2017 2017 2017 14 22 05 21 05 22 05 20 06 Time Period 84 2016 2017 2017 2017 14

45 Appendix B: Portfolio Weights for Time Periods 7-14

Appendix B: Portfolio Weights for Time Periods 7-14

Stock/Time period 7 8 9 10 11 12 13 14 A P MOLLER - MAERSK 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% ABB LTD N (OME) - DAILY RETURN % IN EUR 0.00% 1.68% 0.00% 0.00% 0.00% 6.28% 6.04% 9.63% ALFA LAVAL - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% ASSA ABLOY 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% ASTRAZENECA (OME) - DAILY RETURN % IN EUR 19.24% 16.94% 28.45% 25.79% 24.27% 20.51% 20.46% 22.06% ATLAS COPCO 'A' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% CARLSBERG 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.06% 0.00% 0.00% 0.29% 5.11% 3.66% DANSKE BANK - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% ELECTROLUX 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% ERICSSON 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% FLSMIDTH & CO.'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% FORTUM - DAILY RETURN % IN EUR 9.68% 14.59% 8.27% 19.18% 18.32% 18.31% 15.34% 9.78% GETINGE - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% HENNES & MAURITZ 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% HEXAGON 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% INVESTOR 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% KONE 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% METSO - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% NOKIA - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% NORDEA BANK - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% NOVO NORDISK 'B' - DAILY RETURN % IN EUR 29.56% 27.06% 25.60% 23.98% 22.59% 17.95% 16.94% 15.43% NOVOZYMES - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 3.44% 3.56% 7.89% SAMPO 'A' - DAILY RETURN % IN EUR 17.89% 17.21% 17.89% 12.55% 14.29% 14.83% 13.98% 13.09% SANDVIK - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% SCA 'B' (Cellulosa SCA)- DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% SCANIA 'B' DEAD - 06/06/14 - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% SEB 'A' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% SKANSKA 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% SKF 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% SSAB 'A' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% STORA ENSO 'R' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% SVENSKA HANDBKN.'A' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% SWEDBANK 'A' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% SWEDISH MATCH - DAILY RETURN % IN EUR 23.63% 22.51% 19.73% 18.49% 20.53% 18.39% 18.57% 18.45% TELE2 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% TELIA COMPANY - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% UPM-KYMMENE - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% VESTAS WINDSYSTEMS - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% VOLVO 'B' - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% WARTSILA - DAILY RETURN % IN EUR 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

Appendix C: Heatmap of All Portfolio Weights 46

Appendix C: Heatmap of All Portfolio Weights

The data visualization on the next page shows all portfolio weights during all 84 time periods. All zero weights are shown as empty cells. The cells with weight have numbers in them. That number refers to the portfolio %-weight of that stock during that holding period. Also, the heavier the weight is, the darker the cell is.

Stock name /Time Period Stock /Time name MILLICOM INTL.CELU.SDR A A P MOLLER MAERSK- 'B' SVENSKA HANDBKN.'A' SVENSKA VESTAS VESTAS WINDSYSTEMS HENNES & MAURITZ 'B' SCA 'B' (Cellulosa SCA) 'B'SCA (Cellulosa ASTRAZENECA ASTRAZENECA (OME) LUNDIN PETROLEUM NOVO NORDISK 'B'NORDISK NOVO FLSMIDTH & CO.'B' NOKIAN RENKAAT NOKIAN ABB LTD N (OME) SWEDISH MATCH VOSTOK GAS VOSTOK SDB ATLAS COPCO 'A' RAUTARUUKKI 'K' ATLAS COPCO 'B' GN STORE NORD TELIA COMPANY OUTOKUMPU 'A' UPM-KYMMENE STORA STORA ENSO 'R' ELECTROLUX 'B' ASSA ABLOY ASSA 'B' NORDEA BANK NORDEA BANK SWEDBANK 'A' DANSKE BANK DANSKE BANK COLOPLAST 'B' CARLSBERG 'B' SECURITAS 'B' AUTOLIV SDB NOVOZYMES 'B'INVESTOR HEXAGON 'B' ERICSSON ERICSSON 'B' KINNEVIK 'B' KINNEVIK SKANSKA 'B' SKANSKA ALFA LAVAL LAVAL ALFA SCANIA 'B' SCANIA SAMPO 'A' WARTSILA PANDORA VOLVO 'B'VOLVO ELEKTA 'B' SANDVIK SANDVIK GENMAB BOLIDEN GETINGE FORTUM KONE KONE 'B' TELE2 'B' ESSITY B SSAB SSAB 'A' ORSTED METSO DSV DSV 'B' NOKIA NOKIA SEB 'A' SKF 'B'SKF NESTE

ELISA TDC 23% 22% 35% 15% 5% 1 25% 17% 36% 17% 5% 2 18% 13% 35% 24% 8% 1% 3 21% 14% 34% 23% 5% 0% 3% 1% 4

16% 14% 30% 11% 23% 3% 4% 5 15% 16% 31% 12% 23% 2% 6 24% 18% 30% 10% 19% 7 23% 17% 27% 15% 17% 2% 8 20% 18% 26% 28% 8% 0% 9 18% 13% 24% 19% 26% 10 21% 14% 23% 18% 24% 11 18% 15% 18% 18% 21% 12 3% 0% 6% 19% 14% 17% 15% 20% 13 4% 5% 6% 18% 13% 15% 10% 22% 10% 14 8% 4% 33% 16% 23% 15 1% 7% 5% 8% 8% 33% 16% 23% 16 0% 8% 2% 9% 8% 33% 10% 17% 32% 17 2% 0% 6% 29% 19% 34% 18 2% 8% 9% 26% 17% 36% 19 6% 8% 7% 26% 19% 38% 20 1% 8% 1% 6% 27% 23% 13% 29% 21 2% 1% 5% 27% 23% 13% 32% 22 5% 28% 23% 10% 34% 23 6% 28% 28% 30% 24 8% 6% 31% 23% 34% 25 0% 6% 6% 31% 22% 36% 26 4% 7% 20% 27% 41% 27 2% 3% 0% 7% 15% 29% 41% 28 1% 2% 1% 6% 5% 14% 36% 30% 29 3% 2% 2% 2% 2% 5% 6% 10% 18% 18% 24% 28% 30 5% 4% 0% 3% 1% 7% 17% 17% 23% 27% 31 4% 4% 8% 1% 3% 1% 3% 8% 19% 25% 10% 26% 32 6% 2% 7% 0% 0% 2% 2% 10% 25% 17% 26% 33 2% 1% 3% 5% 1% 1% 9% 26% 16% 27% 34 2% 2% 6% 5% 2% 4% 2% 1% 7% 15% 27% 15% 29% 35 5% 1% 3% 0% 1% 4% 11% 22% 10% 15% 32% 36 1% 0% 1% 2% 3% 3% 31% 15% 10% 22% 37 0% 4% 2% 1% 3% 6% 0% 4% 0% 18% 28% 21% 12% 38 2% 6% 2% 1% 4% 1% 4% 0% 16% 23% 10% 17% 12% 39 0% 5% 4% 3% 1% 3% 7% 12% 15% 19% 14% 40 8% 2% 6% 9% 3% 0% 5% 9% 15% 16% 10% 16% 11% 11% 41 4% 1% 9% 3% 4% 12% 12% 12% 13% 15% 10% 42 1% 8% 2% 8% 7% 0% 19% 11% 12% 11% 17% 10% 43 8% 2% 9% 3% 17% 10% 11% 21% 11% 44 2% 9% 3% 4% 9% 4% 14% 13% 11% 25% 11% 45 1% 3% 1% 4% 2% 9% 6% 12% 13% 30% 46 0% 3% 2% 5% 3% 4% 6% 9% 7% 6% 15% 14% 10% 11% 19% 47 0% 3% 7% 9% 2% 0% 8% 19% 20% 12% 10% 48 0% 2% 7% 9% 2% 0% 8% 2% 9% 17% 12% 20% 11% 10% 11% 49 2% 1% 7% 1% 1% 2% 0% 5% 15% 10% 21% 12% 50 1% 3% 9% 2% 3% 3% 9% 7% 4% 12% 11% 18% 10% 12% 10% 51 0% 8% 3% 2% 1% 5% 3% 3% 1% 14% 16% 15% 11% 11% 52 7% 1% 1% 5% 0% 4% 4% 1% 0% 7% 3% 1% 12% 16% 14% 11% 53 9% 3% 1% 4% 8% 3% 3% 6% 8% 1% 12% 16% 13% 11% 11% 54 7% 2% 6% 1% 4% 6% 6% 6% 19% 12% 10% 17% 55 7% 9% 1% 3% 6% 7% 6% 4% 12% 23% 10% 16% 56 8% 0% 5% 7% 2% 7% 1% 4% 7% 2% 24% 20% 57 9% 9% 3% 7% 8% 4% 2% 8% 3% 0% 11% 18% 10% 11% 18% 58 8% 4% 6% 5% 8% 5% 0% 1% 12% 12% 10% 21% 59 1% 3% 6% 4% 9% 1% 6% 3% 7% 1% 0% 11% 21% 13% 21% 60 6% 0% 3% 7% 3% 7% 2% 1% 3% 14% 16% 10% 18% 61 7% 8% 2% 2% 4% 9% 2% 2% 3% 2% 1% 15% 19% 11% 10% 13% 62 5% 8% 4% 6% 6% 1% 2% 13% 29% 12% 11% 63 3% 9% 4% 5% 9% 0% 4% 0% 10% 26% 14% 13% 64 2% 6% 5% 1% 6% 8% 8% 15% 11% 11% 11% 12% 65 6% 5% 5% 3% 0% 7% 5% 6% 3% 18% 15% 12% 10% 66 9% 2% 0% 3% 7% 8% 7% 4% 6% 14% 10% 33% 15% 67 3% 7% 9% 5% 0% 0% 3% 17% 15% 28% 12% 68 3% 4% 6% 5% 4% 5% 36% 16% 11% 17% 69 5% 5% 4% 1% 4% 38% 16% 11% 12% 70 2% 2% 3% 5% 1% 5% 5% 39% 10% 17% 12% 10% 71 0% 6% 6% 39% 10% 18% 72 5% 6% 8% 6% 8% 40% 16% 15% 73 3% 1% 8% 1% 2% 8% 6% 44% 16% 12% 74 6% 4% 1% 9% 8% 36% 13% 15% 15% 75 5% 3% 4% 9% 37% 12% 25% 16% 76 6% 4% 34% 20% 17% 77 4% 2% 1% 1% 7% 9% 6% 34% 16% 24% 15% 78 1% 2% 3% 5% 29% 15% 22% 23% 79 1% 4% 6% 21% 13% 27% 30% 80 1% 0% 2% 6% 22% 17% 10% 11% 15% 81 0% 5% 9% 6% 5% 19% 10% 16% 12% 15% 82 3% 4% 8% 2% 6% 6% 18% 17% 17% 12% 83 1% 1% 7% 3% 9% 7% 7% 4% 17% 16% 21% 16% 84 2% 2% 1% 5% 5% 6% 3% 6%