EXAMENSARBETE INOM TEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2020

Portfolio Optimization: An Evaluation of the Downside Risk Framework on the Nordic Equity Markets

FABIAN PETTERSSON

OSKAR RINGSTRÖM

KTH SKOLAN FÖR TEKNIKVETENSKAP

Portfolio Optimization:

An Evaluation of the Downside Risk Framework on the Nordic Equity Markets

Fabian Pettersson

Oskar Ringström ROYAL

Degree Projects in Applied Mathematics and Industrial Economics (15 hp) Degree Programme in Industrial Engineering and Management (300 hp) KTH Royal Institute of Technology year 2020 Supervisor at KTH: Johan Karlsson Examiner at KTH: Sigrid Källblad Nordin

TRITA-SCI-GRU 2020:118 MAT-K 2020:019

Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Degree Project in Applied Mathematics and Industrial Economics

Portfolio Optimization: An Evaluation of the Downside Risk Framework on the Nordic Equity Markets

Authors Supervisor Fabian Pettersson Johan Karlsson Oskar Ringstrom¨

Abstract

Risk management in portfolio construction is a widely discussed topic and the tradeoff between risk and return is always considered before an investment is made. Modern portfolio theory is a mathematical framework which describes how a rational investor can use diversifica- tion to optimize a portfolio, which suggests using variance to measure financial risk. However, since variance is a symmetrical metric, the framework fails to correctly account for the loss aversion preferences most investors exhibit. Therefore, the use of downside risk measures were proposed, which only measures the variance of the portfolio below a certain threshold, usually set to zero or the risk-free rate. This thesis empirically investigates the differences in performance between the two risk measures when used to solve a real world portfolio optimization problem. Backtests using the different measures on all major Nordic equity markets are performed to highlight the dynamics between the frameworks, and when one should be preferred over the other. It is concluded that the optimization frameworks indeed provides a useful tool for investors to construct great performing portfolios. However, even though the downside risk framework is more mathematically rigorous, implementing this risk measure instead of variance seems to be of less importance for the actual results.

Keywords: Downside risk, Mean-variance optimization, Modern portfolio theory, Semi-variance

Examensarbete inom Till¨ampadMatematik och Industriell Ekonomi

Portf¨oljoptimering:En Utv¨arderingav Riskm˚attet Downside Risk p˚ade Nordiska Aktiemarknaderna

F¨orfattare Handledare Fabian Pettersson Johan Karlsson Oskar Ringstrom¨

Abstrakt

Riskhantering f¨or aktieportf¨oljer ¨ar mycket centralt och en avv¨agning mellan risk och avkastning g¨orsalltid innan en investering. Modern Portf¨olteori¨arett matematiskt ramverk som beskriver hur en rationell investerare kan anv¨andadiversifiering f¨oratt optimera en portf¨olj. Centralt f¨ordetta ¨aratt anv¨andaportf¨oljensvarians f¨oratt m¨atarisk. Dock, eftersom varians ¨arett symmetriskt m˚attlyckas inte detta ramverk korrekt ta h¨ansyntill den f¨orlustaversion som de flesta investerare upplever. D¨arf¨orhar det f¨oreslagitsatt ist¨alletanv¨andaolika m˚att p˚anedsiderisk (downside risk), som endast tar h¨ansyntill portf¨oljensvarians under en viss avkastningsgr¨ans, oftast satt till noll eller den riskfria r¨antan. Denna studie unders¨oker skillnaderna i prestation mellan dessa tv˚ariskm˚attn¨arde anv¨andsf¨oratt l¨osaett verkligt portf¨oljoptimerinsproblem. Backtests med riskm˚attenhar genomf¨ortsp˚ade olika nordiska aktiemarknaderna f¨oratt visa p˚alikheter och skillnader mellan de olika riskm˚atten,samt n¨ar det enda ¨aratt f¨oredraframf¨ordet andra. Slutsatsen ¨aratt ramverken ger investerare ett anv¨andbartverktyg f¨oratt smidigt optimera portf¨oljer.D¨aremotverkar den faktiska skillnaden mellan de tv˚ariskm˚attenvara av mindre betydelse f¨orportf¨oljernasprestation. Detta trots att downside risk ¨armer matematiskt rigor¨ost.

Nyckelord: Downside risk, Variansoptimering, Modern Portf¨oljteori,Semi-varians

Contents

1 Introduction 2 1.1 Background ...... 2 1.2 Aim ...... 2 1.3 Limitations ...... 3

2 Theoretical Framework 3 2.1 Mean-Variance Framework ...... 3 2.2 Downside Risk Framework ...... 4 2.3 Portfolio Selection ...... 5 2.4 Previous Research ...... 6

3 Methodology 7 3.1 Data Gathering ...... 7 3.2 Calculating the Efficient Frontier ...... 8 3.2.1 Parameter Estimations ...... 8 3.2.2 Application of Quadratic Programming ...... 8 3.2.3 Position Constraints ...... 9 3.2.4 Optimization Problems ...... 9 3.3 Backtesting ...... 10 3.3.1 Corporate Actions ...... 10

4 Results 11 4.1 OMX Copenhagen ...... 11 4.2 OMX Helsinki ...... 12 4.3 OMX Stockholm ...... 13 4.4 Summary of Results ...... 14

5 Discussion 15 5.1 Performance for Entire Period ...... 15 5.2 Performance during Bear Market ...... 16 5.3 Performance during Bull Market ...... 16 5.4 Portfolio Selections ...... 16 5.5 Performance against Benchmarks ...... 17 5.6 Implications for Investors ...... 17

6 Conclusion 17

References 18

Appendices 20

A Positive Semi-Definite Matrices 20

B OMX Index Constituents 21

1 Introduction 1.1 Background In 1952, Markowitz introduced Modern Portfolio Theory (MPT ) in his paper “Portfolio selec- tion”. According to this theory a risk-averse investor can construct an efficient frontier contain- ing optimal portfolios that minimizes the risk for a given level of expected return. MPT was the first model to describe portfolio selection and management from a mathematical point of view. However, since the MPT was restricted to the mean-variance framework, which not always represent the whole picture of financial risk, the theory has received considerable amount of criticism.1 One reason behind this is investor’s tendencies to prefer avoiding losses to obtain equivalent gains. This principle is referred to as loss aversion and was first identified by Kah- neman and Tversky [1979]. What differentiates this kind of aversion from risk aversion is that the utility of a monetary return depends on what was previously experienced or was expected to happen; a loss is more painful than an equal gain. An investor’s intuitive concept of risk is asymmetrical in nature. To reconcile this, Harlow [1991] proposed using lower partial moments as an asymmetrical measure of downside risk, which only measures the variance of the portfolio below a certain threshold. Secondly, the mean-variance framework has limited generality since it has been shown that the selection will only lead to optimal decisions if the utility function is assumed quadratic or the probability distribution is restricted.2 Pratt [1964] argued that the assumption of a quadratic utility function is unrealistic since it implies increasing absolute risk aversion over the whole domain and a negative marginal utility after a cut-off point. Further- more, the MPT has been criticized by Doganoglu et al. [2007] because of the assumption that the asset returns can be represented by the normal distribution. According to Doganoglu et al. asset returns often show skewness as well as kurtosis bigger than 3, which is indications of dis- tributions differentiate from Gaussian. Moreover, investments have unlimited upside potential but can only yield limited losses.

1.2 Aim This thesis aims to further contribute to the research body of MPT by comparing its performance when using different risk measures. Even thought this thesis investigates the performance of two different risk measures the aim is not to provide investors with a solution of how to allocate their portfolio in one optimal way, but to investigate the different characteristics of using both variance and downside risk as risk measures. The comparison will be based on how the risk measures perform from a risk adjusted return basis and what risk level an investor will be exposed to. Although higher risk adjusted returns is not always directly indicative of better performance, it offers a good starting point for analysing the portfolios. Most investors measure their performance on a risk adjusted basis, which is why that is relevant to evaluate. The mathematical theory behind the two risk measures are also discussed and evaluated. One risk measure might have desirable properties, that are not directly reflected in the portfolios performance. These are still relevant to examine. The study aims to highlight of the different characteristics of the risk measures, so that investors can adapt what fits their strategy and portfolio. This thesis also investigates the performances in different market conditions with the aim of identifying differences in the performance of the risk measures. Considering that downside risk is asymmetric to daily returns, it can be relevant to investigate how its performance differs in an upward and downward trending market. Furthermore, most of the research on MPT is published by

1See e.g. Doganoglu et al. [2007] and Taleb [2007]. 2See in particular Borch [1969] and Feldstein [1969].

2 Americans. This means that most empirical evaluations of the framework only consider American markets. Although some general conclusions certainly can be drawn from this, Nordic investors also would benefit from evaluating the framework on the Nordic markets. Therefore, this thesis will also contribute by empirically evaluate the performance of MPT on the Nordic equity markets, something which has not been excessively studied. The thesis seeks to answer the following questions:

• Does Modern Portfolio Theory offer Nordic investors a useful tool for improving portfolio performance? • Is downside risk a better measure of financial risk compared to variance on the Nordic markets? • Which market conditions favors the use of each risk measure?

1.3 Limitations The study will be restricted to three different universes. These are the Nasdaq indices OMXC20, OMXH25 and OMXS30. These are widely regarded to be an indicator of the performance of the entire stock market in Denmark, Finland and Sweden, respectively. The study will consider data from the last 13 years. It does not consider any real world trading constructions, such as commission or slippage. Although there are many ways of constructing portfolios using Modern Portfolio Theory, this study is limited to only consider the main two. These are the minimum risk and maximum risk adjusted return portfolios. These will be discussed in depth in the coming section.

2 Theoretical Framework

In this section the framework for portfolio optimization first developed by Markowitz is in- troduced. It is shown how the framework can be adapted to accommodate different investor preferences. Then, a new risk measure proposed by Harlow that better reflect investors’ views on risk is introduced. It is explained why this risk measure should be preferred. A method for computing this risk measure is described and finally, the optimization framework is adapted to be used for real world empirical studies.

2.1 Mean-Variance Framework The mean-variance optimization developed by Markowitz is an application of a quadratic pro- gramming problem. As with all quadratic optimization problems, the goal is to optimize a certain objective function given a set of constraints. For the context of portfolio optimization, Markowitz chose the variance of the portfolio as the function to be minimized. In mathematical terms, this function is defined by the covariance matrix (C) between the assets considered for the portfolio. The variance of the portfolio is obtained by multiplying the covariance matrix with a vector x representing the weights for the different assets. This gives a function for the variance, xT Cx, for all possible portfolios that can be constructed within the universe. However, not all portfolios match investor preferences and should not be considered. Therefore, the variance need to be optimized under a set of constraints. Markowitz identified two basic constraints for the portfolio construction. Firstly, all the investor’s funds should be invested at all times, i.e. that the investor is not allowed to have cash in its portfolio. This means that the sum of the weight vector x should be unit. This is represented by the constraint eT x = 1, where e is an all-ones

3 vector with the same dimension as x. Secondly, there needs to be a constraint for the expected return of the portfolio. This is represented by µT x = r, where µ are the expected returns for each asset and r is some return threshold. This gives a quadratic optimization problem formulated on the standard form

minimize xT Cx subject to µT x = r (1) eT x = 1 as described by Sasane and Svanberg [2019]. This optimization problem can then be modified to accommodate different investor preferences. An investor who only wants to take long positions can add the constraint x > 0. Another investor who wants to hold a fraction of their portfolio in cash can modify the second constraint to eT x ≤ 1. Solving equation 1 for different values of r generates a curve which Markowitz referred to as the efficient frontier. All optimal portfolios (i.e. with the highest expected return for a given level of risk) lie on the efficient frontier, and no portfolios can exist to its left. Without leverage, the upper and lower bounds for r are given by the asset with the highest and lowest expected return, respectively.

Figure 1: A visualization of the efficient frontier, and how constituents of the considered indices relate to it.

2.2 Downside Risk Framework To reflect the previously discussed asymmetrical nature of investor’s risk preferences, Harlow [1991] proposed using Lower Partial Moments (LMPn) to only measure downside risk. The n:th Lower Partial Moment is defined as Z t n LP Mn = (τ − R) dF (R) (2) −∞ where τ is the target rate and dF (R) is the probability density function for the continuous return function for the assets. He also noted that using n = 2 is analogous to variance, which is what this thesis will use to compute downside risk. This only measures variance below a certain target rate τ, often set to the risk-free rate or 0, as illustrated in figure 2.

4 Figure 2: A visualization of which returns contribute to downside risk.

Estrada [2008] describes how this risk measure is not widely used by practitioners, even though it is often considered a better approach to estimating risk than using variance. This is because this semi-variance optimization has no closed form solution, due to the fact that the semi-covariance matrix as defined in equation 2 is endogenous. Estrada therefore proposes a different, simpler, approach to estimating the semi-covariance matrix S as

T X Sij = min(Rti − rf , 0) · min(Rtj − rf , 0) (3) t=1 where Ri,t,Rj,t are the expected returns for assets i, j for the period t and rf is the chosen risk-free rate. T is the total period of data used for the calculation. This is the matrix that will be used for estimating the downside risk of the portfolios. Equation 1 is easily modified to accommodate for this, by substituting C with S.

2.3 Portfolio Selection Once the efficient frontier has been generated, different portfolios can be selected from it. Every point on the efficient frontier corresponds to an optimal portfolio, and selecting which portfolio to invest in comes down to the investor’s preferences. However, the portfolios on the lower end of the frontier (below the minimum risk portfolio) should not be invested in, since for the same expected risk a higher expected return can be achieved. This is due to diversification. This study focuses on two different portfolio selections. The portfolio with the minimum expected risk corresponds to the point on the efficient frontier with a vertical slope, i.e. furthest to the left. The risk adjusted returns S for each risk/return pair (R, σ2) on the efficient frontier can also be calculated, as R − r S = f . (4) σ

In the mean-variance framework this is called the sharpe ratio Sv as defined by Sharpe [1966], and in the downside risk framework it is called sortino ratio Sdr as defined by Sortino and Price [1994]. Then, the portfolio with the maximum risk adjusted returns can be chosen. Furthermore, Tobin [1958] describes how by introducing the risk free rate to the portfolio optimization problem, the efficient frontier becomes a straight line often referred to as the capital allocation line. This line tangents the normal efficient frontier at the maximum risk adjusted portfolio, and it is therefore also called the tangency portfolio.

5 Figure 3: Portfolio selection from the efficient frontier

2.4 Previous Research Studies on portfolio optimization with regards to risk minimization were first published in 1952 when Markowitz proposed a quantitative framework for how equity portfolios are allocated based on the assumption of investor’s willingness to minimize portfolio risk for any given expected rate of return. Scott and Horvath [1980] discuss how different directions of the moment impact the purpose of the investment. Boudt et al. [2008] stated that the literature on portfolio risk measurement seems to agree that in existence of distributions of return that is not normal or utility functions that is of higher order than the quadratic, a downside risk measure should be preferred over variance. Bawa [1975] introduced an analysis of downside risk measure for different utility functions with regards to the Lower partial moment, (LMPn), of return distribution. Harlow [1991] developed the theory of downside risk. He came up with the conclusion that by using LMPn, an investor can lower the risk while maintaining or increasing the expected return of the portfolio offered by the mean-variance framework. Furthermore, he stated that the LMPn approach gives the investor the opportunity to from a target rate define risk for a specific portfolio by the way each asset deviate below this rate. Grootveld and Hallerbach [1998] analyzed the differences between some US asset allocation portfolios based on variances and semi-variances. They found that by using LMPn as the risk measure the downside risk approach produced, on average, a slightly higher bond allocations than the mean-variance approach. Numerous studies have also investigated the performance of the variance and downside risk frameworks empirically. The general result seems to be that downside risk yields slightly better performing portfolios. A summary of the investigated studies can be found below, in table 1.

6 Authors Title Market Results Mean-downside risk and mean- Bourachnikova Symmetric vs. Downside Risk: Does Euronext Paris variance framework yield simi- and Yusupov It Matter for Portfolio Choice? lar results. Exchange- Portfolio optimization in a mean- Downside risk yields the same Boasson et al. traded index semivariance framework or slightly better results. funds Hong Kong, Portfolio optimization: The Downside London, New The results vary between mar- Ren and Sig- risk framework versus the Mean- Vari- York and ket/sample data mundsd´ottir ance framework Shanghai stock exchanges

Table 1: Previous research on the performance of the downside risk framework.

3 Methodology

To empirically evaluate the performance of both the variance and downside risk framework in the Nordics, backtest on the major Nordic equity markets were conducted. The backtests were constructed to operate in an ideal market environment, without considering commissions, slippage, liquidity etc. The theory described in previous sections were used to derive optimal portfolio weights. Some parameters needed for the calculations were selected in such a way to yield stable and reliable backtest results, without direct support from academia.3 Although there is no way of knowing that the selected values for the parameters are good, the goal was only to get a stable performance from the backtest. Then, as long as the parameters are the same for all calculations, any inefficiencies should cancel out when comparing the results between the frameworks. The parameters for the backtests are chosen to resemble those of the benchmark indices as closely as possible. For example, an index constituent can not have a negative weight in the index, so the backtests will only use long positions. Furthermore, indices can be restricted in how large weights one constituent can have, and this will also be considered.

3.1 Data Gathering In order to produce reliable backtest results, the universe considered by the backtesting algorithm needs to be consistent with the reality at the time. Historical price data for all assets and indices are gathered from the Nasdaq OMX Nordic website. Daily price data was collected for all currently listed companies. However, no price data were available for companies that was delisted during the considered time period. Data for these companies were obtained through various sources, such as the company website or the Refinitiv Eikon database. This database was also used to obtain the correct historical constituents for each index, as can be seen in appendix B. It should be noted that this data in not perfect. Specifically for OMX Copenhagen and Helsinki, the number of constituents in the indices drop below what it should be (e.g. 20 for OMX Copenhagen) in the last years of available data. This introduces some uncertainty towards the comparison with the actual index. However, that is not the main focus of this study, and the dynamics between the frameworks should not be affected by this. The risk free rate is estimated to τ = 0. 3This is mainly regarding the lookback period used for historical data, as well as how the expected returns for the assets are calculated.

7 3.2 Calculating the Efficient Frontier In order to generate optimal portfolios, the efficient frontier needs to be calculated. This is done by solving equation 1, for different values of the return threshold r. When using the variance framework, the objective function is given by equation 6. When using the downside risk framework, the objective function is given by the semi-variance matrix defined in equation 3. In this section it is described how the parameters needed for the optimization are calculated. It is shown how a quadratic optimizer can be adapted to solve the optimization problem. Lastly, it is discussed how to adapt the optimization to account for constraints in the position sizes.

3.2.1 Parameter Estimations As previously discussed, the optimization problem requires the estimation of several parameters. These are the expected returns µ, the covariance C and the semi-covariance S. Estimation of these parameters are not trivial, but there exists no academic consensus for how it should be done. Forecasting expected returns for stocks probably have as many methods as there are investors in the market. To limit the scope of this thesis, simple methods for estimating these parameters are therefore used. The basic method is to use historical data to estimate the parameter values for the coming period. To estimate the variance and semi-variance matrices as well as the expected returns, daily datapoints for a specified lookback period T is used. When using τ = 0 as the risk free rate, the expected returns µi can simply be estimated by

T 1 X µ = r (5) i T i t=1 which is the arithmetic mean of the historical data. The covariance matrix C is calculated with entries Cij as

T 1 X C = (R − µ )(R − µ ) (6) ij T − 1 ti i tj j t=1 which is the sample covariance between asset i and j. As previously discussed, the downside risk will be estimated by the semi-covariance matrix S. This matrix is calculated as described by Estrada [2008], with entries Sij as

T 1 X S = min(R − r , 0) · min(R − r , 0). (7) ij T − 1 ti f tj f t=1

3.2.2 Application of Quadratic Programming Sasane and Svanberg [2019] describes that the objective function for the quadratic optimization should be convex, i.e. that the covariance and semi-covariance matrices as defined by equations 6 and 7 need to be positive semi-definite. This is shown in appendix A. To solve equation 1, the numerical solver CVXOPT [2020] was used. It solves quadratic programming problems with linear constraints on the form 1 minimize xT P x + qT x 2 subject to Gx ≤ h (8) Ax = b

8 which can then be adapted to solving the optimization problem described by equation 1 by the substitutions P = 2C, q = 0,A = [µ, e]T , b = [r, 1]T ,G = h = 0, or G = −I, h = 0 when short positions are not allowed. The optimization methods used by the numerical solver are outlined by Sra et al. [2011]. The solver uses primal-dual path-following algorithms based on Nesterov-Todd scaling. These are described in depth by Vandenberghe [2010].

3.2.3 Position Constraints As discussed in section 2.1, the bounds for the possible expected returns on the efficient frontier are given by the maximum and minimum expected returns for the constituents of the universe, respectively. Without allowing a cash position or leverage, the portfolio with the highest expected return the investor can construct is to allocate 100% of the capital to the asset with the highest expected return. Similarly, the lowest performing portfolio is given by allocating 100% of the capital to the asset with the lowest expected return. When introducing constraints on the maximum allowed position size, the investor can no longer allocate all capital towards one asset, and the above therefore does not hold. Instead, for a given position size constraint xm < 1, the highest expected return possible Rmax is given by

b1/xmc X 1 1 R = ( µ x ) + ( − b c)µ (9) max i m x x b1/xmc+1 i=1 m m where µ1 > µ2 > ... > µn are the expected returns for the assets in the universe. To illustrate, consider xm = 0.3 and µ = [0.6 0.5 0.4 0.3 0.2]. Then, Rmax = 0.3 · 0.6 + 0.3 · 0.5 + 0.3 · 0.4 + (3.33 − 3) · 0.3 = 0.3(1.5 + 0.33) ≈ 0.55. Similarly, the lowest expected return possible Rmin is given by equation 9, but with µ1 < µ2 < ... < µn. This gives the bounds for which the efficient frontier should be calculated, i.e. that Rmin ≤ r ≤ Rmax in equation 1. In this study, a maximum allowed position size of xm = 0.1 was used. This is the position constraint impemented by Nasdaq OMX [2020] for the index OMXH25. To implement this, the matrix G and vector h in equation 8 need to be modified. To restrict the position sizes to maximum xm, n×n n×n T n×1 n×1 T set G = [−I ,I ] and h = [1 xm ] .

3.2.4 Optimization Problems To summarize the above theory, the complete optimization problems used to calculate the efficient frontiers are now described. The portfolios are not allowed to have a cash position. Furthermore, no individual position in the portfolios are allowed to surpass 10 % of the total portfolio. The efficient frontier is then calculated by solving

minimize f(x) subject to µT x = r (10) eT x = 1

0 ≤ xi ≤ xm T for discrete values Rmin ≤ r ≤ Rmax, and xm = 0.1. In the variance framework, f(x) = x Cx where C is calculated as described by equation 6. In the downside risk framework, f(x) = xT Sx where S is calculated as described by equation 7. The parameter µ is calculated as described by equation 5, Rmin and Rmax are calculated as described by equation 9, and xm = 0.1. In sections 3.2.2 and 3.2.3 it is outlined how to adapt this optimization problem to be solved by the numerical solver used in this study.

9 3.3 Backtesting To backtest the optimization strategies, a Python library developed by Quantopian Inc. [2018] was used. It provides the user with powerful and flexible tools for backtesting trading strategies. The basic structure of the library is a function that is called each day of the backtest, which the user can input trading logic to. The logic is then executed, and the portfolio value is calculated. In the conducted backtests, the portfolio was recalculated on a monthly basis. Firstly, the trading universe for that particular day was determined, i.e. all assets that was a part of the relevant index at the time. The backtest only should consider assets that was in the chosen index at that particular day. This was done by filtering the full list of index constituents by the ones who joined the index before, and left after, the current date. Historical data for a period leading up to the current date, as specified by the lookback period, was then used for calculating the efficient frontiers, and subsequently the portfolios. To make sure that sufficient data was available for all assets, only assets where ≤ 20% of data for the period is missing were considered. After the data was cleaned, it was used by the optimization algorithm, as previously described.

Backtest parameters Backtest start 31 Dec 2006 Bactest end 31 Dec 2019 Lookback period 3 · 252 Days Data frequency Daily Rebalancing frequency Monthly Starting capital 10M local ccy Cash position not allowed. Only long positions ≤ 10%. Limitations Only consider assets where ≤ 20% data is missing.

Table 2: Parameters used for evaluating the optimization frameworks.

3.3.1 Corporate Actions The data gathered from Nasdaq OMX Nordic [2020] is generally adjusted for splits and divi- dends. However, other corporate actions such as divestments, are not accounted for. These can have a substantial impact on the performance and reliability of the backtest. Furthermore, the backtesting library used is not able to account for such event either. Therefore, manual solutions needed to be implemented to produce a reliable backtest. In June 2017, SCA AB [2017] divested a part of their business into another listed company, Essity. All existing shareholders then received shares in Essity, resulting in the divestment not affecting shareholders’ portfolio values. However, this resulted in a radical price drop for SCA-B shares. Similarly in September NKT A/S [2017] demerged into two separate companies; NKT and Nilfisk. This also resulted in a sharp price drop for NKT shares which is not adjusted in the used data. To account for this during the backtest, a manual restriction was implemented for the considered universe during Jun and Sep 2017. All shares of SCA-B and NKT were sold at the end of May and August, and were not considered for the portfolio during June and September, respectively. After these divestments, and subsequent artificial price drops, SCA, Essity, NKT and Nilfisk shares were all considered by the backtest again.

10 Event Date Universe Action taken SCA divestment of Essity 12 Jun 2017 OMXS30 No trading in SCA-B during Jun 2017 NKT demerger with Nilfisk 18 Sep 2017 OMXC20 No trading in NKT during Sep 2017

Table 3: Corporate actions considered during the backtests.

4 Results

In the below sections the results from the backtests are presented. The cumulative performance of the portfolios and benchmarks are visible in the graphs. In the tables summary statistics for the time period are displayed. These are cumulative return at the end of the backtest (RT ), risk measures in the variance and downside risk frameworks (σv and σdr) and the risk adjusted performance measures sharpe ratio (Sv) and sortino ratio (Sdr). Each section contains the results for a market over the whole period, as well as broken down into the bear market during the 2008 financial crisis and the subsequent bull market. This will allow for an analysis of the performance of the risk frameworks in different market conditions, which could be of interest since the downside risk measure is asymmetric in nature. The bear market is defined as the dates for the highest peak of 2007 and the lowest trough of 2009. The bull market is defined as starting at the end date of the bear market, and ending at 31 Dec 2019.

4.1 OMX Copenhagen

RT σv σdr Sv Sdr Nasdaq OMXC20 index 1.5380 0.2072 0.1515 0.3587 0.4905 Variance fw, min risk 1.6304 0.1624 0.1182 0.4759 0.6538 Downside fw, min risk 1.5011 0.1627 0.1176 0.4495 0.6217 Variance fw, max sharpe 1.8418 0.1899 0.1444 0.4408 0.5797 Downside fw, max sortino 1.6785 0.1939 0.1473 0.4063 0.5348

Table 4: Summary statistics of the performance of the different port- folio selections and risk frameworks.

Figure 4: Cumulative performance of the dif- ferent portfolio selections and risk frameworks during the period 31 Dec 2006 - 12 Dec 2019. Performance during bear market

RT σv σdr Sv Sdr Nasdaq OMXC20 index -0.5728 0.3611 0.2457 -1.2631 -1.8563 Variance fw, min risk -0.5028 0.2589 0.1816 -1.5207 -2.1676 Downside fw, min risk -0.5103 0.2618 0.1781 -1.5289 -2.2481 Variance fw, max sharpe -0.4951 0.3376 0.2437 -1.1462 -1.5879 Downside fw, max sortino -0.5130 0.3519 0.2494 -1.1443 -1.6147 Correlation between max risk adjusted portfolios 0.9951 Correlation between min risk portfolios 0.9881

Table 5: Summary statistics of the performance of the different port- Figure 5: Cumulative performance of the dif- folio selections and risk frameworks the period 11 Oct 2007 - 6 Mar ferent portfolio selections and risk frameworks 2009. during the period 11 Oct 2007 - 6 Mar 2009.

11 Performance during bull market

RT σv σdr Sv Sdr Nasdaq OMXC20 index 4.2983 0.1803 0.1233 0.9251 1.3528 Variance fw, min risk 4.3634 0.1458 0.1000 1.1530 1.6809 Downside fw, min risk 4.2679 0.1450 0.0994 1.1463 1.6713 Variance fw, max sharpe 4.2435 0.1623 0.1125 1.0210 1.4724 Downside fw, max sortino 4.1822 0.1638 0.1142 1.0037 1.4399 Correlation between max risk adjusted portfolios 0.9971 Correlation between min risk portfolios 0.9927

Table 6: Summary statistics of the performance of the different port- Figure 6: Cumulative performance of the dif- folio selections and risk frameworks the period 6 Mar 2009 - 31 Dec ferent portfolio selections and risk frameworks 2019. during the period 6 Mar 2009 - 31 Dec 2019. 4.2 OMX Helsinki

RT σv σdr Sv Sdr Nasdaq OMXH25 index 0.4368 0.2266 0.1598 0.1249 0.1770 Variance fw, min risk 0.9207 0.2038 0.1440 0.2528 0.3579 Downside fw, min risk 0.8004 0.2054 0.1460 0.2254 0.3171 Variance fw, max sharpe 1.0276 0.2405 0.1698 0.2325 0.3293 Downside fw, max sortino 1.0320 0.2411 0.1698 0.2326 0.3303

Table 7: Summary statistics of the performance of the different port- folio selections and risk frameworks.

Figure 7: Cumulative performance of the dif- ferent portfolio selections and risk frameworks during the period 31 Dec 2006 - 12 Dec 2019. Performance during bear market

RT σv σdr Sv Sdr Nasdaq OMXH25 index -0.6464 0.3454 0.2208 -1.3590 -2.1258 Variance fw, min risk -0.5860 0.2863 0.1799 -1.4531 -2.3125 Downside fw, min risk -0.5919 0.2975 0.1855 -1.4152 -2.2698 Variance fw, max sharpe -0.6560 0.3641 0.2233 -1.3135 -2.1420 Downside fw, max sortino -0.6592 0.3631 0.2228 -1.3255 -2.1602 Correlation between max risk adjusted portfolios 0.9979 Correlation between min risk portfolios 0.9847

Table 8: Summary statistics of the performance of the different port- folio selections and risk frameworks during the period 13 Jul 2007 - Figure 8: Cumulative performance of the dif- 9 Mar 2009. ferent portfolio selections and risk frameworks during the period 13 Jul 2007 - 9 Mar 2009.

12 Performance during bull market

RT σv σdr Sv Sdr Nasdaq OMXH25 index 2.3453 0.2048 0.1415 0.5774 0.8356 Variance fw, min risk 3.1626 0.1902 0.1346 0.7416 1.0479 Downside fw, min risk 2.9172 0.1896 0.1341 0.7105 1.0041 Variance fw, max sharpe 3.4448 0.2174 0.1522 0.6809 0.9728 Downside fw, max sortino 3.4925 0.2184 0.1526 0.6831 0.9776 Correlation between max risk adjusted portfolios 0.9990 Correlation between min risk portfolios 0.9928

Table 9: Summary statistics of the performance of the different port- Figure 9: Cumulative performance of the dif- folio selections and risk frameworks during the period 9 Mar 2009 - ferent portfolio selections and risk frameworks 31 Dec 2019. during the period 9 Mar 2009 - 31 Dec 2019. 4.3 OMX Stockholm

RT σv σdr Sv Sdr Nasdaq OMXS30 index 0.5221 0.2175 0.1560 0.1511 0.2106 Variance fw, min risk 0.6221 0.1766 0.1273 0.2148 0.2979 Downside fw, min risk 0.5754 0.1747 0.1253 0.2039 0.2841 Variance fw, max sharpe 0.2092 0.2195 0.1608 0.0671 0.0916 Downside fw, max sortino 0.1347 0.2255 0.1657 0.0433 0.0590

Table 10: Summary statistics of the performance of the different portfolio selections and risk frameworks.

Figure 10: Cumulative performance of the different portfolio selections and risk frame- works during the period 31 Dec 2006 - 12 Dec 2019. Performance during bear market

RT σv σdr Sv Sdr Nasdaq OMXS30 index -0.5480 0.3348 0.2156 -1.3304 -1.8812 Variance fw, min risk -0.4766 0.2739 0.1877 -1.3931 -2.0093 Downside fw, min risk -0.4598 0.2727 0.1801 -1.3454 -1.8737 Variance fw, max sharpe -0.6131 0.3632 0.2467 -1.3929 -2.0175 Downside fw, max sortino -0.6451 0.3778 0.2570 -1.4204 -2.0680 Correlation between max risk adjusted portfolios 0.9936 Correlation between min risk portfolios 0.9875

Table 11: Summary statistics of the performance of the different Figure 11: Cumulative performance of the portfolio selections and risk frameworks during the period 16 Jul different portfolio selections and risk frame- 2007 - 21 Nov 2008. works during the period 16 Jul 2007 - 21 Nov 2008.

13 Performance during bull market

RT σv σdr Sv Sdr Nasdaq OMXS30 index 1.8284 0.1976 0.1403 0.4970 0.7164 Variance fw, min risk 1.7315 0.1604 0.1121 0.5908 0.8503 Downside fw, min risk 1.5683 0.1578 0.1112 0.5624 0.8178 Variance fw, max sharpe 1.5034 0.1936 0.1361 0.4452 0.6585 Downside fw, max sortino 1.5315 0.1976 0.1388 0.4419 0.6541 Correlation between max risk adjusted portfolios 0.9965 Correlation between min risk portfolios 0.9701

Table 12: Summary statistics of the performance of the different Figure 12: Cumulative performance of the portfolio selections and risk frameworks during the period 21 Nov different portfolio selections and risk frame- 2008 - 31 Dec 2019. works during the period 21 Nov 2008 - 31 Dec 2019. 4.4 Summary of Results In the below tables, the results from the backtests are summarized. A portfolio is considered better or less risky if both the variance and downside risk framework agree that there is more than one percentage point difference in the performance between the frameworks. Otherwise the results are considered similar. This is in order to only extract the significant results, and not over-analyze small differences in performance. The risk adjusted returns for the bear markets are intentionally left blank, since these ratios does not convey any useful information when the cumulative returns are negative.

Min portfolio Max portfolio Risk adj. Risk adj. Market Risk Return Risk Return return return Variance Variance Variance Variance OMXS30 Similar risk Similar risk better higher better higher Variance Variance Similar Downside OMXH25 Similar risk Similar risk better higher results higher Variance Variance Variance Variance OMXC20 Similar risk Similar risk better higher better higher

Table 13: Summary of the performance of the different portfolio selections and risk frameworks.

Min portfolio Max portfolio Risk adj. Risk adj. Market Risk Return Risk Return return return Downside Downside Variance OMXS30 Similar risk higher riskier higher Similar Similar OMXH25 Similar risk Similar risk returns returns Variance Variance OMXC20 Similar risk Similar risk higher higher

Table 14: Summary of the performance of the different portfolio selections and risk frameworks during the bear markets.

14 Min portfolio Max portfolio Risk adj. Risk adj. Market Risk Return Risk Return return return Variance Variance Similar Downside OMXS30 Similar risk Similar risk better higher returns higher Variance Variance Similar Downside OMXH25 Similar risk Similar risk better higher returns higher Similar Variance Variance Variance OMXC20 Similar risk Similar risk results higher better higher

Table 15: Summary of the performance of the different portfolio selections and risk frameworks during the bull markets.

5 Discussion

In the following section, the results will be explained and discussed. The discussions are mainly regarding the difference in characteristics between the variance and downside risk frameworks. The comparisons between the frameworks are always considering the same portfolio construction, e.g. that the minimum risk portfolio in the variance framework is only compared to the minimum risk portfolio in the downside risk framework, unless otherwise stated. Some conclusions are also drawn regarding the frameworks performance to the market. However, discussions regarding this needs to consider the fact that the data used for index constituents is not perfect. First, it can be concluded that both of the risk frameworks always agree on which portfolio performed best, on a risk adjusted basis. That is, if the sharpe ratio is higher for one framework, the sortino is as well. This is good because it removes any ambiguity in the results that could be caused by evaluating the downside risk framework using variance as the risk measure, and vice versa.

5.1 Performance for Entire Period Considering the backtest results for the whole period, the variance framework seems to produce better performing portfolios, on a risk adjusted basis. Both the minimum risk and maximum risk adjusted return portfolios in the variance framework outperform their peers in the downside risk framework. For OMX Copenhagen, the variance framework performs approx. three to four percentage points better than the downside risk framework. On the Helsinki market, only the minimum risk portfolio in the variance framework outperform its peer, by about three percentage points. The maximum risk adjusted return portfolios are performing very similar. On the Stockholm market, the variance framework outperforms, but only slightly. Now, the sharpe and sortino ratios used to evaluate risk adjusted returns are of course dependent on both risk and return. Comparing the risks between the markets and portfolios it is clear that the frameworks produce portfolios with similar risk profiles. The risk between the frameworks never differ more than half a percentage point, and often much less than that. Therefore, the difference in the risk adjusted performance seems to be mainly due to the difference in cumulative returns over the period. This in interesting, since it could suggest that the downside risk framework accurately measures risk, but fails to identify assets with the highest upside potential. Although it could be expected that the downside risk framework should perform better when markets trend upward, it is also reasonable to believe that some information about an asset’s performance is lost when not considering all volatility when calculating risk.

15 5.2 Performance during Bear Market Since the downside risk framework is asymmetric in nature, it is relevant to compare the perfor- mances during periods of falling returns on the market, and subsequent upturns. To do this, the drawdown periods of the 2008 financial crisis are analyzed separately. During this period, the three indices all lost more than half their values. Considering the results over the whole period, it is not reasonable to expect any positive returns from the optimal portfolios. This is indeed what the results show. With negative cumulative returns, risk adjusted metrics such as sharpe and sortino ratios does not convey any useful information. A low negative ratio would seem to be favorable, since it is closer to zero. However, one could achieve this by simply increasing the risk, which is not good. Conversely, a larger negative ratio in not favorable either, since that could be achieved by higher negative returns. As for the whole period, the risk of the portfolios are very similar between the frameworks. On the Copenhagen market, the cumulative returns for the downside risk framework are worse than for the variance framework, with a difference of about one or two percentage point. Therefore, we can conclude that the variance framework is performing better. On the Helsinki and Stockholm markets, the results are not as clear. On the Finnish market, the frameworks does not perform significantly different. On the Swedish market, the variance framework only outperforms for the minimum risk portfolios, while the downside risk framework performs better for the maximum risk adjusted return portfolios. Also, the correlations between the portfolios are higher during the bear market than the bull market.4 This is logical, since the difference between the frameworks would be less noticeable in a down- ward trending market. The downside risk measure is positively asymmetrical, meaning that only negative returns are considered risky, and so, the more negative returns the less difference be- tween the frameworks. This can also be seen in the cumulative returns. During the bear market, the difference in returns between the frameworks are much lower than the same during the bull market.

5.3 Performance during Bull Market Considering how downside risk is measured, and that it was specifically designed to not penalize high performing assets, one could expect that the downside risk framework would outperform in a upwards moving market. However, the results presented in section 4 seems to disagree. During the 10 year bull run, where the indices all grew substantially, the difference between the frameworks are not as great. It is interesting that although the correlations between the portfolios are lower during the bear market, the variance framework still outperforms the downside risk framework. It shows that the portfolio selections differs more between the frameworks in an upwards trending market, but that the variance framework still selects better portfolios.

5.4 Portfolio Selections Although not the main focus of this study, it is clear that there are some differences in perfor- mance between the portfolio selections. Considering the whole backtesting period, the minimum risk portfolios usually outperform the maximum risk adjusted returns, on a risk adjusted ba- sis.5 This means that even though the maximum sharpe and sortino portfolios should produce better risk adjusted portfolios, they do not. This could be because of the relatively simplistic parameter estimations used in this study. The maximum risk adjusted portfolios rely more on accurate estimation of the expected returns of the assets, while the minimum risk portfolio are

4The only exception is the minimum risk portfolio on the Stockholm market. 5Except for the downside risk portfolios on the Helsinki market, where the maximum sortino portfolio outper- forms the minimum risk portfolio.

16 more dependent on accurate calculations of the risk parameter. It is obvious that the historical mean returns of an asset are not a great prediction for future returns. Furthermore, the difference in performance between the frameworks for the maximum risk adjusted portfolios are smaller than for the minimum risk portfolios. This could be because of the position size constraint used. Since the minimum risk portfolios has lower return constraints, more combinations of assets could satisfy the constraint, and so, more difference between the performance could be expected. Conversely, when the return threshold is higher, only a small combination of assets can satisfy constraint, and the maximum adjusted return portfolios should therefore be more alike between the frameworks. This can also be seen by noting that the correlations between the maximum risk adjusted return portfolios are always higher that the minimum risk portfolios.

5.5 Performance against Benchmarks It can be interesting to compare the performance of the optimized portfolios against their bench- mark, in order to determine if the discussed optimization frameworks are useful for investors when construction portfolios. Considering the whole backtesting period, all optimized portfolios outperform their respective index, except for the maximum risk adjusted return portfolios on the Stockholm market. This would indicate that the optimization frameworks can indeed increase the risk adjusted returns of a portfolio, by optimizing the portfolio weights. However as noted before, the data used for constituents in each benchmark index is not entirely complete, and therefore one should be cautious to draw to strong conclusions when comparing the performances of the portfolios against their benchmarks.

5.6 Implications for Investors The results presented in this study seems to suggest that, overall, there is little difference in portfolio performance between the frameworks. The variance framework produces slightly higher returns. However, that does not necessarily mean that all investors should prefer it. It is clear that mean-variance optimization can be one great tool in an investor’s toolbox, but the choosing the best risk measure can depend upon other factors than pure performance. The downside risk framework is constructed to account for loss aversion, something that investors might value different. Some investors view a volatile market as risky, no matter the direction. They of- ten want to invest in underlying trends and want to filter out short term noise. These types of investors might prefer the variance framework. On the other hand, from a purely financial perspective, the downside risk measure represent a more true perspective of the risk. For the comparable portfolios, the downside risk is always much lower than the variance. Although risk is seldom measured in absolute terms, investors choosing to utilize the downside risk measure should at least be aware that the framework always will perceive assets as less risky by con- struction. Furthermore, the variance framework offers a clear advantage in that there exists a close form solution for the optimal portfolio. This means that much less computational power is needed for the variance framework, which can be relevant for investors looking to optimize portfolios with many hundred or thousands of assets. Although the portfolios were optimized using purely numerical solutions in this study, the authors noticed that calculating downside risk is significantly more computationally intense.

6 Conclusion

From its introduction, Modern Portfolio Theory has proven to be a useful tool for portfolio optimization. Although it has received much criticism, it provides investors with a relatively

17 simple framework for optimizing their portfolios. The results of this study shows that even though not perfect, Modern Portfolio Theory can still be successfully used to produce high performing portfolios, on the Nordic equity markets. The selection of risk framework, between variance and downside risk, seems to be of less importance for the overall performance, although it should be noted that the variance framework usually did produce better performing portfolios. In a bear market, where the benchmark indices drop greatly in value, the selection of risk framework is less important than in a bull market. However, there is a discrepancy between the results provided in this study and what academia seems to agree on. This study shows that the variance framework outperforms the downside risk framework in an upwards trending market, whereas the the mathematical theory and researchers agrees that this market condition should favour the downside risk framework. However, when examining other empirical studies of the frameworks, it is clear that the real difference between them are small or negligible. This is because estimating downside risk is not trivial, and investors looking to utilize the risk measure should implement more sophisticated methods for calculating it. As a final note, the optimization frameworks discussed in this thesis are only as good as the assumptions behind the underlying parameters. There is no doubt that the results can vary greatly depending on e.g. how well the expected returns of the assets are predicted. For future studies on this topic, more sophisticated methods for estimating expected returns and expected risk, variance or downside, should be implemented. However, for an investor looking for a simple way to optimize their portfolio, mean-variance optimization still remains an excellent alternative.

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19 Appendices

A Positive Semi-Definite Matrices

Lemma 1. A covariance matrix C is always positive semi-definite. Proof. Consider yT Cy for y ∈ Rn.

yT Cy = yT E[(X − E[X])(X − E[X])T ]y = E[yT (X − E[X])(X − E[X])T y] = E[(yT (X − E[X]))2] ≥ 0, since yT (X − E[X]) is a scalar.

Lemma 2. The semi-covariance matrix S as defined by equation 3 is always positive semi- definite.

Proof. It follows directly from noting that S is a sum of Gramian matrices, which always are 6 positive semi-definite. Specifically, let σt = [min(Rt1 − rf , 0) . . . min(Rtn − rf , 0)] and consider T n y Sy for σt, y ∈ R .

T ! 1 X yT Sy = yT σT · σ y T − 1 t t t=1 T 1 X = yT · σT · σ · y
T − 1 t t t=1 T 1 X = σ · y)2 ≥ 0 T − 1 t t=1

6See e.g. Kuptsov [1988].

20 B OMX Index Constituents

Company Ticker Joined date Leaved date ALK-Abello ALK-B 1997-04-23 1997-12-22 Radiometer A/S RADI-B 1997-04-23 1997-12-22 RPC Superfos SFSC 1997-04-23 1998-12-21 Sas Danmark SASD 1997-04-23 1998-12-21 Nordea Companies UNI 1997-04-23 2000-04-19 Ratin RATN-B 1998-01-28 2000-07-27 Novozymes NZYM-B 2000-11-17 2000-12-21 I-Data IDATA 2000-12-18 2001-06-18 Berendsen DK SOPH 1997-04-23 2002-06-21 Navision NAVI 1999-12-20 2002-06-24 Dampskibsselskab D1912-B 1997-04-23 2003-06-16 Vestas India NEG 2000-06-21 2004-03-09 Carlsberg CARL-N 2004-03-30 2004-04-27 Vestas Wind VWS-N 2004-05-19 2004-06-16 Falck FALCK 1997-12-22 2004-12-22 Lufthavne KBHL 1997-04-23 2006-01-02 EAC Invest EACI 2005-12-19 2006-12-18 G4S G4S 2004-07-21 2006-12-18 Maersk MAERSK-B 1997-04-23 2006-12-19 Bang & Olufsen BO 1997-12-22 2007-06-18 Torm A TRMD-A 2006-06-19 2007-12-27 Coloplast COLO-B 1997-04-23 2010-06-21 GN Store Nord GN 1997-04-23 2010-12-21 Dupont Nutrition DCO 1997-04-23 2011-05-25 D/S Norden DNORD 2007-12-27 2011-12-19 Pandora PNDORA 2012-06-18 2011-12-19 Sydbank SYDB 2006-12-18 2012-06-18 NKT NKT 2000-06-21 2012-12-27 Genmab GMAB 2007-06-18 2013-12-23 Topdanmark TOP 1998-12-21 2014-06-23 H.Lundbeck LUN 1999-12-20 2016-06-20 Nets NETS 2016-12-19 2017-06-19 Demant DEMANT 1999-12-20 2017-12-18 Tdc TDC 1997-04-23 2018-04-12 Maersk MAERSK-A 2006-12-19 2018-06-18 Nordea Bnk NDA-DK 2000-04-19 2018-06-18 FLSmidth FLS 1997-04-23 2019-06-24 Jyske Bank JYSK 1997-04-23 2019-06-24 Tryg TRYG 2005-12-19 2019-06-24 Ambu AMBU-B 2018-06-18 present Carlsberg CARL-B 1997-04-23 present Chr Hansen Hldg CHR 2010-12-21 present DSV Panalpina DSV 2002-06-24 present Danske Bk DANSKE 1997-04-23 present Iss DK ISS 1997-04-23 present Novo Nordisk NOVO-B 1997-04-23 present Orsted ORSTED 2016-12-19 present Royal Unibrew RBREW 2018-12-27 present Simcorp SIM 2019-06-24 present Vestas Wind VWS 1999-12-20 present

Table 16: Nasdaq OMXC20 index constituents

21 Company Ticker Joined date Leaved date Cultor CUL1S 1996-08-01 1997-02-01 Finnair FIA1S 1996-08-01 1997-08-01 Fortum Oil Gas NEST 1996-08-01 1998-02-01 Tamro TRO1V 1997-08-01 1998-08-01 Merita MERI-A 1996-08-01 1998-11-05 Cultor CUL2V 1996-08-01 1999-02-01 Partek PAR1S 1998-02-01 1999-02-01 Nokia NOKKV 1996-08-01 1999-04-12 Rauma Oyj RAM1V 1996-08-01 1999-07-01 Finnlines FLG1S 1996-08-01 2000-02-01 Merita MTA1V 1998-11-05 2000-02-01 Stora Enso STEAV 1999-08-01 2000-08-01 Outokumpu OUT1V 1996-08-01 2001-02-01 Raisio RAIVV 1996-08-01 2001-02-01 Talentum TTM1V 2000-08-01 2001-02-01 F-Secure FSC1V 2000-08-01 2001-08-01 Foxco EIMAV 2000-02-01 2001-08-01 Innofactor IFA1V 2001-02-01 2001-08-01 Wartsila WRT1V 1996-08-01 2001-08-01 Bittium BITTI 1999-08-01 2002-02-01 Stonesoft SFT1V 2001-02-01 2002-02-01 Teleste TLT1V 2000-08-01 2002-02-01 Oy Hartwall HARAS 1999-02-01 2002-07-02 Comptel CTL1V 2000-08-01 2002-08-01 Kesko KESKOB 1996-08-01 2002-08-01 Outokumpu Europe AVP1V 2002-02-01 2002-08-01 SymphonyEYC ALD1V 2000-08-01 2002-08-01 Telia Finland SRATLS 2002-11-21 2002-12-05 Telia Company TELIA1 1999-02-01 2002-12-09 Instrumentarium INS1V 2001-08-01 2003-01-31 Elcoteq ELQAV 2000-08-01 2005-02-01 Kone FI KONBS 1996-08-01 2005-06-01 Fortum Oyj FORTUM 1999-02-01 2005-08-01 Lite-On POS1V 2000-02-01 2005-08-01 Vakuutusosakeyh POH1V 1996-08-01 2005-10-20 Orion ORNBS 1996-08-01 2006-07-03 Konecranes KCRA 1997-02-01 2006-08-01 Orion ORNBV 2007-02-01 2009-08-13 Uponor UPONOR 1997-08-01 2009-08-13 Ahtium AHTIUM 2009-08-12 2012-02-01 Huhtamaki HUH1V 1996-08-01 2013-02-01 Sanoma SAA1V 2005-08-01 2013-02-01 OP Yrityspankki POH1S 2006-02-01 2014-04-08 Rautaruukki RTRKS 1996-08-01 2014-07-28 Valmet VALMT 2014-01-02 2014-08-01 Metsa Board METSB 1996-08-01 2016-08-01 Amer Sports AMEAS 1996-08-01 2019-03-12 Kemira KEMIRA 1996-08-01 2019-08-01 TietoEVRY TIETO 1997-02-01 2019-08-01 Yit YIT 1997-08-01 2019-08-01 DNA Oyj DNAO 2018-08-01 2019-10-01 Cargotec CGCBV 2005-08-01 present Elisa ELISA 1999-02-01 present Kojamo KOJAMO 2020-02-03 present Kone KNEBV 2005-08-01 present Metso METSO 1996-08-01 present Neste NESTE 2005-08-01 present Nokia NOKIA 1996-08-01 present Nokian Tyres TYRES 2002-08-01 present Nordea Bnk NDA-FI 2000-02-01 present Outotec OTE1V 2007-02-01 present Sampo SAMPO 1996-08-01 present Stora Enso STERV 1996-08-01 present UPM Kymmene Oyj UPM 1996-08-01 present

Table 17: Nasdaq OMXH25 index constituents

22 Company Ticker Joined date Leaved date Investor INVE-A 1996-06-24 1997-07-01 Stora Kopparberg STOR-B 1996-06-24 1997-12-22 Avesta Sheffield AVES 1996-06-24 1998-06-23 Stora Kopparberg STOR-A 1996-06-24 1998-12-30 AGA AGA-B 1996-06-24 1999-10-18 Trelleborg TREL-B 1996-06-24 2001-01-02 LB Icon ICON 2000-01-01 2001-07-02 CGI IT konsulter WM-B 2000-01-01 2003-01-02 Pharmacia PHA 1996-06-24 2003-01-02 Telenor Sverige EURO 2001-07-02 2003-03-05 Fabege Fastighet FABG 2003-07-01 2004-11-01 FAB Skandia SDIA 1996-06-24 2006-03-15 Holmen HOLM-B 1996-06-24 2007-01-02 Stora Enso STE-R 1999-01-04 2007-07-02 Vostok Gas VGAS-SDB 2006-07-03 2009-01-05 Eniro ENRO 2001-07-02 2009-07-01 Scania SCV-B 1997-07-01 2014-05-16 Modern Times MTG-B 2009-07-01 2016-01-04 Autoliv ALIV-SDB 1997-12-22 2017-01-02 Nokia NOKIA 1997-07-01 2017-01-02 LundinPetroleum LUPE 2008-01-03 2018-01-02 FPC FING-B 2016-01-04 2018-07-10 ABB ABB 1999-06-23 present Alfa Laval AB ALFA 2003-01-02 present Assa Abloy ASSA-B 2001-01-02 present AstraZeneca AZN 1996-06-24 present Atlas Copco ATCO-A 1996-06-24 present Atlas Copco ATCO-B 1996-06-24 present Boliden BOL 2006-07-03 present Cellulosa SCA SCA-B 1996-06-24 present Electrolux ELUX-B 1996-06-24 present Ericsson ERIC-B 1996-06-24 present Essity ESSITY-B 2017-06-12 present Getinge GETI-B 2009-07-01 present Hennes & Mauritz HM-B 1996-06-24 present Hexagon AB HEXA-B 2018-07-02 present Investor INVE-B 1996-06-24 present Kinnevik KINV-B 1996-06-24 present Nordea Bnk NDA-SE 1998-06-23 present Sandvik SAND 1996-06-24 present Securitas SECU-B 2000-01-01 present Skand Ensk Bank SEB-A 1996-06-24 present Skanska AB SKA-B 1996-06-24 present SKF SKF-B 1996-06-24 present SSAB SSAB-A 2007-07-02 present Svenska Hndlsbnk SHB-A 1996-06-24 present Swedbank AB SWED-A 1996-06-24 present Swedish Match SWMA 2003-01-02 present Tele2 TEL2-B 1999-07-02 present Telia Company TELIA 2000-06-15 present Volvo VOLV-B 1996-06-24 present

Table 18: Nasdaq OMXS30 index constituents

23

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