Modern Portfolio Theory Combined with Magic Formula

Bachelor Thesis

Modern Portfolio Theory combined with Magic Formula

A study on how Modern Portfolio Theory can improve an established investment strategy.

Authors: Axel Ljungberg & Anton Högstedt Supervisor: Maziar Sahamkhadam Examiner: Håkan Locking Term: Spring 21 Subject Finance Level: Bachelor Course code: 2FE32E

Abstract

This study examines whether modern portfolio theory can be used to improve the Magic Formula investment strategy. With the assets picked by the investment strategy we modify the portfolios by weighting the portfolios in accordance with modern portfolio theory. Through the process of creating efficient frontiers and weighting the portfolios differently we will create two alternative portfolios each year. One portfolio that aims for maximum Sharpe ratio and one that aims for minimum variance. These weighted portfolios produce higher risk-adjusted returns consistently during the examined period of 2010-2020. We conclude that the Magic Formula can be improved by using modern portfolio theory.

Keywords

Magic Formula, Modern portfolio theory, efficient frontier, Efficient market hypothesis, Sharpe ratio, risk-adjusted returns, Jensen’s alpha, beta, CAPM, OMXSPI

Acknowledgements

We want to thank our supervisor Maziar Sahamkhadam for his highly appreciated help throughout the development of this thesis as well as our examiner Håkan Locking for constructive discussions.

Table of Contents

Introduction ______3 Background ______3 Problem definition ______5 Research question ______6 Limitations ______6

Theory ______7 Modern Portfolio Theory ______7 Efficient Market Hypothesis ______10 The Magic Formula ______11 Measures of risk ______13

Methodology ______15 Study approach ______15 Data collection ______15 Significance testing ______17 Risk-free rate ______18

Empirical Results ______19 Efficient frontier and illustration of results ______19 EQW, Optimal and GMV portfolio results ______20 Cumulative returns and Measures of risk ______22 Significance tests ______24

Analysis ______27 Efficient market hypothesis ______27 Comparing empirical results ______28 Risk measures ______30

Conclusion ______32

Future research ______33

References______34 Literature ______34 Articles ______34

Appendices ______I

Introduction

Background

The stock market provides an opportunity for individuals to get an increased value of assets over time. By allocating resources on the stock market individuals can acquire both positive and negative returns. To assist investors with allocating their resources several investment strategies has been developed to support individuals when investing on the stock market. The investment strategies are there to support individuals with making strategic and wise investment decisions.

A popular value-based strategy is the “Magic Formula”. The Magic Formula was first described by Joel Greenblatt in the book “The little book that beats the market” which was published in 1980. The purpose of the strategy is to help investors find undervalued stocks and beat the market. The Magic Formula does this with the two key figures Return on Capital (ROC) and Earnings Yield (EY). The strategy helps both unexperienced and experienced investors with building their individual portfolios and only needs a small amount of effort from the investor. According to Joel Greenblatt the Magic Formula presents an easy way to make money for any individual investor, with or without any knowledge of the stock market.

Recent research of the Magic Formula proved that by using certain adjustments and improvements to the formula an investor can increase their returns (Sjöbäck & Verngren, 2019). However, this research does not always consider the amount of risk the investors are exposed to when using the method. Joel Greenblatt’s research uses the 30 highest ranked companies to decide for the investor which companies the investor should invest in. The investor is asked to weigh the stocks equally, meaning the investor should invest the same amount of money into each company the investor buys (Greenblatt, 2006, p. 131-132).

When reading previous studies about investment strategies, several discuss that it would be interesting to weigh the companies based on their performance (Eriksson & Svensson, 2020; Sjöbäck & Verngren, 2019). The value investing strategy Magic Formula could easily be weighted based on each stocks rank. Since the method involves ranking each stock based on RoC and EY, an investor could decide to weight the

highest ranked stock the highest, the second highest ranked stock the second highest etc. A big disadvantage of this weighting based on ranks would be the amount of risk the investor is exposed to. Since the Magic Formula strategy does not take any measure of risk into account, there would be nothing that would keep the investor from losing most or all his/her money if the highest ranked stocks would decrease in value.

Since the Magic Formula is directed towards individuals who are both experienced and inexperienced it is simplified and there should be room for improvement. If we weight the stocks that are picked by the Magic Formula strategy with a method that includes risk in its calculations, the performance of the strategy should be improved. By using Harry M. Markowitz “Modern Portfolio theory” (MPT) we can narrow down the number of companies used, include a measure of risk, and calculate the expected returns. MPT is used by analysing historical data and looking at its historical returns and volatility. It can be used to visualize and describe different portfolio options available to an investor and provide estimations of how much risk as well as returns that are expected of the different portfolios.

The study will be conducted by constructing three portfolios each year. The three portfolios will have the same stocks available for investment but the method for their construction will differ. One of the portfolios will be constructed by dividing the available capital equally between the assets selected. This portfolio will be referred to as the equally weighted portfolio (EQW). The second and third portfolio will share the stocks available for investment with the EQW portfolio each year but will not split the capital equally and will instead be weighted differently each year. These portfolios will be referred to as the optimal portfolio and the Global Minimum Variance portfolio (GMV).

Our contribution is to combine MPT, the ideas first proposed by Markowitz with a well- established investment strategy such as Joel Greenblatt’s Magic Formula. The investment strategy suggests investing in an arbitrary number of stocks, but why not use well established theories to reduce the risk for similar return, or increase return for similar risk. To our knowledge, there is little to no research that combines MPT with any investing strategy. By combining MPT with Magic Formula, we discuss a possible

way MPT can be used in practice and whether the theory can contribute to an improved strategy on the stock market.

Problem definition

Joel Greenblatt claims that the Magic Formula strategy can make anyone a master on the stock market (2011, p.39). He claims that anyone can beat the market, beat professional investors and most trustees with the help of two simple rules. This claim is also supported by historical data . However, the efficient market hypothesis (EMH) states that investors cannot consistently beat the market since share prices already reflects all information. For this paper to be viable and contribute to research, the market cannot be completely effective.

The Magic Formula strategy ranks companies based on RoC and EY and suggests investors should buy the highest 20-30 ranked stocks, and weight the stocks equally. Harry M. Markowitz, the father of MPT claims that there is (at least in theory) a possibility to create an optimal portfolio which maximizes the amount of return in relation to the amount of exposed risk. The theory suggests that in every portfolio, there is an optimal way of weighting the stocks.

In our essay the foundation of the Magic Formula is used together with MPT where the stocks found by Greenblatt’s strategy is weighted with Markowitz’s theory to create optimally weighted portfolios, with the highest possible return-risk ratios as well as minimum variance. The optimal portfolio and the GMV is compared to the EQW. The Magic Formula is according to Joel Greenblatts an approach that can be used by investors to achieve returns that in the long-term consistently beats the market. If any investor with access to Magic Formula can consistently beat the market despite the EMH, MPT and the supposedly consistent investment strategy should create more efficient portfolios with better risk-return ratios.

If Markowitz’s theory can improve the performance of the Magic Formula strategy by weighting the portfolios and taking the amount of risk into account, there should be little to no reason to use the EQW Magic Formula strategy over the combined approach.

Research question

- Can the most efficient combination of assets in a portfolio, based on Modern portfolio theory increase the risk-adjusted return of an investment strategy such as Joel Greenblatts Magic Formula?

Limitations

The study will be made on the Swedish stock exchange and is based on the assumptions that there are no transaction costs or taxes for assets. Greenblatt describes in his book that not all stocks should be used when practicing his strategy. Both the industry and size of the company matter and small companies specifically are not well suited. According to Greenblatt the size matters as the companies must be large enough so that an investor is able to purchase “a reasonable number of shares without pushing prices higher” (Greenblatt, 2011, p. 59). He recommends to either use a limit of 50-, 200-, or 1000-, million USD (Greenblatt, 2011, p. 134). As such we will only be using mid to large-cap sized companies’ stocks since it guarantees that all companies are of sufficient size. Greenblatt also suggests that companies in specific industries should be disregarded, specifically: banks, investment companies and real estate companies will not be considered for investment regardless of if they are of a sufficient size (Greenblatt, 2011, p.136).

Theory

Modern Portfolio Theory

MPT was first established by Markowitz in his 1952 article “Portfolio Selection” and his continued work on the subject eventually led to him winning the 1990 Nobel Prize in economics for his development of portfolio choice theory which since has been built on by several economists. At its core MPT exists to help investors build and combine portfolios for maximization of returns while minimizing risk.

In MPT risk is the same as volatility and to measure risk MPT uses the expected return of the portfolio, the variance and standard deviation of the expected returns as well as covariance and correlation between assets (Mangram, 2013). The more varied or volatile portfolios or assets returns are, the riskier it is deemed as variance of returns is deemed something investors want to avoid by Markowitz (Markowitz, 1952). In his 1952 paper Markowitz discusses the importance of diversification of assets as the main way for investors to counteract this risk, specifically diversification in stocks and assets that do not have a high covariance (Elton & Gruber, 1977). As described earlier the main way investors can reduce this type of risk is by diversifying and holding multiple assets in their portfolio as some assets decreasing in value will be met by other assets in the portfolio increasing in value, this effect means that a portfolio consisting of several different assets will have a lower variance and risk than the assets that comprise it even if those assets have the same variance (Markowitz, 2013), and as the number of assets increase towards infinity the variance of the portfolio will move towards zero (Frantz et al, 2011).

The efficient frontier is a key concept of MPT. The efficient frontier is a curved line that displays the most efficient combinations of assets in a portfolio risk-return wise. The different combinations of assets on the frontier are the ones with the best expected return for a given risk level, and as Markowitz states; for higher risk levels investors should require a higher return to counteract the risk (Markowitz 1952). The frontier is comprised of the assets expected return and the standard deviation of those returns. The expected return used to construct the frontier is the same as the average historical return of the asset. In this study we will use the average monthly return from 5 years prior to

the purchase of the stock. The risk measured by the frontier is the standard deviation of the same returns.

The frontier is traced out by identifying the combinations of asset weights in the portfolio that provide the highest return for a given risk level. This frontier can then be combined with the Capital Allocation Line to identify the optimal portfolio out of all the efficient ones, the optimal portfolio will provide the highest risk adjusted return on the frontier.

The Global Minimum Variance portfolio is the portfolio with the lowest standard deviation possible. On the efficient frontier of risky assets (blue line) we can find both the GMV portfolio and Optimal Portfolio, where the Optimal can be found where the efficient frontier is tangent with CAL and the GMV can be found the furthest to the left on the efficient frontier (Bodie et al, 2018, 210-211).

Graph 1. An example of the efficient frontier(Bodie et al, 2018, 209) Graph 2, An example of the efficient frontier with CAL (Bodie et al, 2018, 209)

Graph 1 shows the efficient frontier in blue. The part of the frontier that curves “downwards” are inefficient portfolios. Which means that there exist different combinations of assets that provide better return for the same or lower standard deviation. The graph also shows us the global minimum variance portfolio. As the name implies it is located on the point of the frontier with the least variance, closest to the Y axis, all the portfolios located below the GMV are inefficient as they are riskier for the same or less return. Graph 2 shows the optimal portfolio on the tangent point of the CAL and frontier which will give the highest risk-reward ratio.

The Capital allocation line is the black straight line (graph 2) and displays the relationship of the return and risk of assets in the portfolio by plotting all the return – volatility combinations that provide the best risk-reward ratio or the highest Sharpe ratio. The tangent point of the efficient frontier and the CAL is the optimal portfolio as this is the one combination out of all the possible ones that the frontier plots with the highest Sharpe ratio. The CAL tangents the y axis at 0,06 as this is the return of the risk- free investment which as the name implies has a standard deviation of 0.

푅푝 − 푅푓 푆ℎ푎푟푝푒 푅푎푡𝑖표 = σp (Bodie et al. 2018, p. 203)

Rp = Return of the portfolio Rf = Risk free rate σp = Standard deviation of the portfolio’s excess return

The Sharpe ratio displays the excess return compared to the standard deviation of the portfolio and can be used to effectively compare different portfolios. Where the one with the highest Sharpe ratio offers better risk adjusted return.

A lot of the criticism and problematics with MPT lie with the assumptions that the theory is based on. MPT assumes that all investors receive the complete information regarding their investments at the same time, but this is also not the case. If the assumption of perfect information were true investment strategies like the Magic Formula that seeks out undervalued stocks would not work, and we would not need regulation regarding insider trading (Mangram, 2013). Criticism can also be directed at MPT’s way of measuring the risk with variance since variance punishes overperforming stocks as equally as those that underperform; thus, two portfolios can be equally risky despite one being comprised of stocks that only have positive returns while the other has stocks that are both positive and negative. (Jin et al. 2006.)

Efficient Market Hypothesis

“The notion that stocks already reflect all available information is referred to as the efficient market hypothesis.” -Bodie et al (2014, 350-351)

EMH suggests that investors cannot beat the market since all market participants has access to the same available information. As soon as new information about a company is available the prices will be adjusted and again reflect the available information (Eugene Fama, 1991).

If markets were efficient and the market price would reflect the true value of an investment, no investment strategy and no group of investors would be able to find undervalued stocks. Eugene Fama (1970) provided different classifications of levels of market efficiency by separating the efficiency into weak-form, semi-strong form and strong form. The weak-form efficiency includes all past prices to find the current value of the asset which means that technical analysis strategies that are based on previous asset prices would not be useful in finding undervalued assets. The semi-strong form efficiency claims that all available (public) information is considered in the current price of the asset, and that no fundamental or technical analysis can help investors find undervalued stocks. The strong form efficiency includes all information, including the non-public information which means that no investor could find undervalued stocks (Damodaran 2012, 112).

In an efficient market there would be no possible way to beat the market based on research, technical analysis, and fundamental analysis. All efforts of trying to find undervalued stocks would be a very costly task and, in the end, random. It is important to acknowledge that this theory claims that an investor cannot beat the market consistently. However, there are many individual investors that manage to make long- term profits (such as Warren Buffet), there are strategies such as the Magic Formula that claim to beat the market long-term. Historical data continue to provide evidence of surprisingly high gains of using the Magic Formula and Joel Greenblatt argues that it will continue to do so in his book “The little book that still beats the market” (Greenblatt, 2011, 155).

Adaptive market hypothesis (AMH) is a newer concept where human behaviour is included when discussing the EMH. Andrew W. Lo introduced this concept in 2004 to recognize and explain the behaviour of financial markets and investors. This addition to the EMH focuses on how behaviour responds to changing market conditions where investors are neither perfectly rational nor completely irrational. Investors change their minds whenever the conditions of the market change since they are intelligent and forward looking. Since AHM is a newer concept, it is not fully developed, however there is a lot of new information that can be collected regarding investor behaviour and the theory has become more relevant as well as accepted in recent years. The addition of behaviour means that investors are irrational and inefficient which contradicts EHM (Lo, 2012). Behavioural finance claims that investors are not always rational and that stocks therefore are not always traded at their fair value under stochastic environments. Adaptive market hypothesis considers both perspectives (Lo, 2012). Andrew W. Lo explains that EHM is a reasonably good approximation of reality under stable market conditions and that AMH emerge whenever there is a more dynamic and stochastic environment. Adaptive market hypothesis can be used to explain how the financial markets fluctuate away from the EMH, how behaviour affects the dynamics of the market and how it is affected by different market conditions (Lo, 2012, 27-28). In the article “Adaptive Markets and the New World Order” Andrew W. Lo states “By modelling the change in behaviour as a function of the environment in which investors find themselves, it becomes clear that efficient and irrational markets are two extremes, neither of which fully captures the state of the market at any point in time.” (Lo, 2012)

The Magic Formula

“Beating the market is not the same as making money!” – Joel Greenblatt (2011, p.159).

The Magic Formula is a strategy that is based on companies’ financial reports, and supposedly can find undervalued companies (Johansson & Werner, 2020). According to Joel Greenblatt the Magic Formula has generated returns that beat the market (which in his research is the S&P 500) consistently from 1988 to 2009. When choosing the 30 highest ranked companies with out of the 1000 largest companies (with enterprise value over $1 Billion) his strategy has generated 19,7% in average yearly returns (Greenblatt, 2011, 191). With the 3500 largest companies (with enterprise value over $50 Million)

his strategy has generated 23,8% in average yearly returns. This is compared to S&P 500’s average yearly returns of 9,5% (Greenblatt, 2011, 160).

By using the investment strategy an investor gets several stocks to invest in based on a ranking system. The ranking system is based on two key figures, Return on Capital and Earnings Yield. The ranking system ranks the two figures separately and will consider mid to large cap sized companies. Since Joel Greenblatt claims that the size of the companies an investor chooses to invest in should be for example either over $50 million or $200 million in market capitalization, we can guarantee reaching at least these values by sorting the companies as mid to large cap. When measuring these the company with the highest Return on Capital will be rank 1, the company with the second highest ROC will be rank 2 and so on (Greenblatt, 2011, p) The same approach will be done with Earnings Yield, where the company with the highest Earnings Yield will be rank 1 and so on. After the ranks of the two key figures has been done the final ranking of the companies can be concluded. By adding the ranks together each company will gain a new number where the companies with the lowest numbers will be considered the best (Sjöbäck & Verngren, 2019).

If for example, a company is rank 5 when it comes to ROC and rank 255 when it comes to EY, the “value” for this company will be 260 (255+5). If no other company has a lower value than 260, this specific one will be considered rank 1 and the most attractive company according to Magic Formula (Greenblatt, 2011, p 54).

퐸푎푟푛𝑖푛푔푠 퐵푒푓표푟푒 퐼푛푡푒푟푒푠푡 푎푛푑 푇푎푥푒푠 푅푒푡푢푟푛 표푛 퐶푎푝𝑖푡푎푙 = (푁푒푡 푊표푟푘𝑖푛푔 푐푎푝𝑖푡푎푙 + 푁푒푡 푓𝑖푥푒푑 푎푠푠푒푡푠)

Joel Greenblatt claims in the book “The little book that still beats the market” that ROC gives the best indication of capital efficiency and a company’s profitability. This key figure measures how much of the capital invested by a company that converts into profits.

퐸푎푟푛𝑖푛푔푠 퐵푒푓표푟푒 퐼푛푡푒푟푒푠푡 푎푛푑 푇푎푥푒푠 퐸푎푟푛𝑖푛푔푠 푌𝑖푒푙푑 = 퐸푛푡푒푟푝푟𝑖푠푒 푣푎푙푢푒

The key figure Earnings Yield compares the earnings before interest and taxes of a company (EBIT) to the enterprise value (EV). The main purpose of this figure is to obtain how much the company earns in comparison to how much is costs to buy the company (Greenblatt, 2011, p 141).

Measures of risk

The Beta of a portfolio measures the risk that can be explained by systematic risk in a portfolio or asset (Bodie et al, 2014, p. 284). The systematic risk represents the risk that cannot be removed by diversifying and is common throughout the stock market. If an asset or portfolio has a beta of 1.0 it means that the asset or portfolio is equally as volatile as the whole market, should the market experience a price increase it will reflect a similar movement in the price of the asset. A beta of more or less than 1.0 means that the volatility is higher or lower in the asset compared to the market. A beta of 1.5 is 50% more volatile than the market, while a beta of 0.5 means that the volatility is 50% lower. betas can also be negative (Bodie et al, 2018, p. 257). A beta of -1.0 means that the asset or portfolio will move in opposite directions of the market with similar volatility. Gold and gold mining securities for example can have negative beta values, the price of gold increasing when the stock markets see downwards movement (Bodie et al, 2018, p. 261).

The Capital Asset Pricing Model (CAPM) is used to calculate the required rate of return from a security for investors to be willing to commit to investing in it. The CAPM is built on specific assumptions that consider both investors behaviour and market structure, the assumptions are (Bodie et al, 2014 p. 277-278):

• All investors are rational and will optimize their portfolios mean-variance wise. • They have the same single investing period. • Homogenous expectations – All investors have access to the same publicly relevant information. • All assets are publicly held and traded at stock exchanges. • All investors can borrow and lend at a risk-free rate and are free to take both long and short positions on assets. • There are no transaction costs or taxes.

Even though not all assumptions will hold on the actual markets it is still a valuable tool for measuring risk and return for individual assets or between portfolios. The CAPM is calculated as follows:

퐶퐴푃푀 = 퐸(푅) = 푅푓 + 훽 ∗ (푅푚 − 푅푓) (Bodie et al, 2018, p. 284) Where: E(R) = Expected Return of the investment Rf = Risk-Free rate β = Beta of the investment Rm-Rf = Market risk premium.

Jensen’s alpha describes the return of an investment compared to the market overall using the CAPM and beta. It measures the return from the investment that is excess the required return of the CAPM and is simply calculated as follows:

퐴푙푝ℎ푎 = 푅(푝) − 퐶퐴푃푀 (Bodie et al, 2018, p. 814) Where: R(p) = Return of the portfolio

A positive Jensen’s alpha means that the return of the asset or portfolio is larger than the return described by the beta value. For example, an asset has a beta of 1,5, the market sees a price increase of 10 percent and if the asset sees a larger price increase than 15% it has a positive Jensen’s alpha (Bodie et al, 2014 p.819).

Methodology

Study approach

Previous research has tested the magic formula and it has resulted in the magic formula beating indexes on several markets (including the Swedish stock market). To find the most efficient combination of assets in portfolios, based on MPT and to examine whether it in fact can improve the magic formula strategy, we base the research on historical data and performance. By comparing the yearly results of the magic formula to the yearly results of a “weighted Magic Formula” we can evaluate our methods performance. By using an investment strategy such as Magic Formula an investor does not need to have any extensive knowledge about the stock market other than how to buy stocks. Without needing any knowledge about the stock market, the Magic formula can pick the stocks for the investor. We show that if desired, an investor could use MPT to weight the stocks, which would help the investor with how much to put into each stock. We will use MPT to find the combination of assets that provide the highest Sharpe ratio and the combination of assets that provide the minimum variance.

Data collection

The first step will be to identify the “best” stocks using the Magic Formula and pick the 30 stocks that score the highest. To sort the companies, we will rank each mid to large cap sized company based on the key figures Return on Capital and Earnings Yield. By using Börsdata.se we can select the best 30 stocks for each year. The collection of stock prices was done using Thomson Reuters DataStream and will be compared to Börsdata.se to make sure the numbers were correct. Neither Börsdata.se nor DataStream includes dividends in the stock prices but includes splits which makes them comparable. We will start off by collecting the yearly stock prices for each of the 30 stocks that was selected for each year. The yearly stock prices will be used to calculate the percentage change in price and therefore the return of an EQW for every year. The process of picking stocks and building the portfolio will be done once every year where we will pick the 30 highest ranking stocks according to the Magic Formula that year. Based on these stocks we will construct three different portfolios each year. The EQW, optimal and GMV portfolio based on different weighting of the individual stocks. As described earlier the EQW will have the capital split equally among all

assets, the optimal will weigh the assets into the combination that yields the highest Sharpe ratio and the GMV portfolio will be weighted to yield minimum variance.

To use the MPT approach, we need data before the year the stocks (in theory) are to be purchased. MPT is based on historical data and uses the historical volatility of an asset to measure whether an investment is risky or not. To measure the expected returns and volatility of each stock we will calculate the monthly difference in stock prices during a five-year period prior to the investment date. By using MPT and measuring the assets risk by the volatility of its historical returns as well as its development in price level we can calculate which out of the 30 stocks that has the best expected performance. We will use the historical data to estimate the future expected developments and will collect the monthly stock prices five years prior to the investment date. For the portfolios that will be created in 2010, we need the monthly historical returns between 2005-2010 of the 30 stocks picked, in the portfolios that will be created in 2011, we need the monthly historical returns between 2006-2011 etc.

The issue with this approach is that it is necessary to have historical data prior to the investment date to measure the volatility. If an asset does not have historical data available five years prior to the construction of the portfolio the earliest available date will be used. This can result in some stocks having incorrectly projected values as there was not enough datapoints available or that stocks tend to be more volatile closer to their IPO date (Lowry et al, 2010). The only year where the portfolios does not consist of 30 stocks is 2019 where only 29 stocks were available. This is due to one of the ranked stocks not having any historical data available. This discrepancy is a limitation of how the Efficient frontiers are constructed using historical data to predict future values, when no historical data exists one cannot calculate the expected return and standard deviation. Therefore, the decision to exclude the stock from the 2019 portfolios was made.

After the frontiers have been plotted and the portfolios that result in the maximum Sharpe ratios have been constructed, we will apply the weights to stocks to calculate the return that would have occurred had the stocks been purchased on March 1st year X and sold on March 1st year X+1. For the years that March 1st was a weekend or other day where the stock markets were closed the next possible weekday would be used.

When the tables and graphs show the year 2010 it is the period of March 1st, 2010 to March 1st, 2011 that is being referred to. The weighted portfolios return will be compared to the EQW portfolios. This means that while the portfolios have the same base of stocks available to choose from the weighted optimal and GMV portfolios can end up consisting of less than 30 stocks if that would result in a higher expected Sharpe ratio or lower variance respectively.

From the data we will compare the yearly, average and cumulative returns of the EQW, optimal, and GMV portfolios. We will also compare the portfolios with the overall Stockholm stock exchange and will use the index OMXSPI which covers the entire stock exchange. The data for this index has also been gathered using Thomson Reuters DataStream. The returns will be used to calculate the risk measures of standard deviation, beta, CAPM, alpha and the Sharpe ratios. The portfolios standard deviation will be computed using the Excel command “STDEV.S” and the Beta will be calculated with the “SLOPE” command.

Significance testing

To increase legitimacy of our results we will make several t-tests at a 95% significance level. We will use a paired two sample test for means to test whether the mean returns of the different portfolios are statistically significantly different compared to the Stockholm stock exchange, with the following hypothesises:

H01: The mean return of the Optimal portfolios is equal to the mean return of the Stockholm stock exchange.

H02: The mean return of the EQW portfolios is equal to the mean return of the Stockholm stock exchange.

H03: The mean return of the GMV portfolios is equal to the mean return of the Stockholm stock exchange.

H04: The mean return of the Optimal portfolios is equal to the mean return of the EQW portfolios.

H05: The mean return of the GMV portfolios is equal to the mean return of the EQW portfolios.

Each null hypothesis H01-H03 has the corresponding alternate hypothesis:

H11: The mean return of the portfolios is not equal to the mean return of the Stockholm stock exchange.

Each null hypothesis H04-H05 has the corresponding alternate hypothesis:

H12: The mean return of the portfolios is not equal to the mean return of the EQW portfolios. The t-tests will be made in excel using the data analysis function to perform two sample paired t-tests for means.

Risk-free rate

The reason a risk-free rate is needed is that it is one of the foundations for several calculations. The risk-free rate is the rate of return of an investment without any risk. A risk-free rate is a theoretical investment where investors can achieve returns without exposing him/herself to any risk. This investment should never be expected to generate higher returns than any investment with risk, since it is risk-free and would be preferred by any investor. Since it is a hypothetical rate, there are several ways to determine the value. Damodaran (2012) explains that the most common approach is the three-month treasury bill. This approach should provide a risk-free rate for every year which we can include in our calculations. However, since we focus on the Swedish stock market, it would be accurate to use the Swedish three-month treasury bill. This will not be used in our case. The reason we cannot use the Swedish three-month treasury bill is that it is negative for several years. A risk-free investment that is negative, is not an accurate representation of a risk-free rate since it would not be attractive to an investor since it guarantees negative returns. In order to keep our results consistent, it would also be preferred with a consistent risk-free rate since we build new portfolios every year. We assume a yearly risk-free rate of 6%. The reason we assume a yearly risk-free rate of 6% is that we want to have the risk-free investment as a legitimate option for investors. Since we suggest a (to us considered) high risk-free rate we force our methods to generate significantly higher returns as well as not-to-high risk in order to be a good investment option. This should add legitimacy to our study.

Empirical Results

Efficient frontier and illustration of results

For the portfolio supposedly held the first year from 2010-2011 the efficient frontier (blue line) was as following.

2010-2011

0,7

0,6

0,5

0,4

Return 0,3

0,2

0,1

0 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 Standard deviation Graph 3 shows the efficient frontier (blue line) and CAL (red line) of portfolio built in 2010.

Out of each stock picked by Magic Formula in 2010, the monthly data between 2005- 2010 was collected. With the data we calculated an EQW portfolio where each company was assigned to 1/30th of the investment. With this data we could conclude a Standard deviation (Std. Dev, Table 1) of 18%, return of 18,8% and a Sharpe ratio of 0,714. The portfolio of 2010 was thereafter weighted with Markowitz theory to find the forecasted optimal weighted portfolio (Table 3) with the highest Sharpe ratio. The expected standard deviation was 17,1% and the expected return was 31,0%. This resulted in a Sharpe ratio of 1,463. The GMV portfolio in 2010 gave us forecasted results of 7,81% standard deviation, 7,65% returns and a Sharpe ratio of 0,211 (Table 5).

These calculations were made with the five year monthly historical data of AstraZeneca, Addtech B, Skanska B, Betsson B, AAK, KnowIt, Duni, Sweco, ABB ltd n, BTS Group G, Fenix outdoor, NCC B , Lagercrantz Group B, af Poyry B, Axfood, Proact it Group, AQ Group, Alfa Laval, Tele2 B, Millicom intl.celu.sdr, Securitas B, Beijer alma B, Vitec Software Group B, Hennes & Mauritz B, Systemair, Swedish match, Eolus vind B, Kindred Group sdr, Telia company, Midsona b.

The above companies where the highest ranked according to the Magic Formula strategy, on the Swedish stock market in 2010. The same process were made for 2011- 2020.

The standard deviation, return and Sharpe ratios are all forecasted calculations which are used to try to “predict” the future. The forecasted calculations of each year are summarized in the tables below, whereas the graphs with the efficient frontiers of each year (like graph 3) and companies are sorted under Appendix. The forecasted results are compared to the actual results in order to compare each year’s performance of an EQW portfolio, an Optimal portfolio and GMV portfolio.

EQW, Optimal and GMV portfolio results

The Actual EQW (Table 2) calculations are based on monthly returns between 2010-03- 01 and 2011-03-01 of the EQW portfolios when the stocks would have been owned and represents the standard deviation, return and Sharpe ratio had the portfolios been held each year.

Table 1 (Orange) shows the forecasted results, Table 2 (Blue) shows the actual results.

Based on the forecasted EQW portfolios (Table 1) we conclude that the expected standard deviation was 18,2%, return 18% and Sharpe 0,70. As can be seen in Table 2,

the EQW portfolios actual standard deviation was 16,2%, return 18,6% and Sharpe 0,87 on average.

Table 3 (Orange) shows the forecasted results, Table 4 (Blue) shows the actual results.

When maximizing the Sharpe ratios each year we found that the Optimal Portfolio would generate an expected average standard deviation of 17,5%, return of 31,4% and Sharpe ratio of 1,48. The actual results of the Optimal portfolio from 2010 to 2020 was on average a standard deviation of 16,3%, return of 30,2% and Sharpe ratio of 1,46%.

By using the stocks picked by Magic Formula, the same way as when building Optimal Portfolios as foundation of our portfolios we created another option for an investor based on MPT. By minimizing the standard deviation, we built Global Minimum Variance portfolios as an option for risk-averse investors.

Table 5 (Orange) shows the forecasted results, Table 6 (Blue) shows the actual results.

The GMV portfolios were expected (as can be seen in Table 5) to generate, on average a standard deviation of 11,74%, returns of 17,8% and a Sharpe ratio of 0,81. The actual results of the GMV portfolios where on average a standard deviation of 14,30%, return of 21,92% and Sharpe ratio of 1,16.

Cumulative returns and Measures of risk

Table 7 displays the yearly-, average and total returns for OMXSPI and the portfolios.

Table 7 displays the total cumulative returns for the Stockholm stock exchange, the optimal portfolio, the EQW portfolio and the GMV portfolio all starting with an initial investment of 100 SEK. The years represent the same periods as the portfolios were held, 2010 thus refers to the period of 1/3-2010 – 1/3-2011. The returns for each year have been reinvested in the following years. Stock dividends has not been included. The optimal portfolio when sold was worth 1412 SEK resulting in an overall return of 1312%. The EQW was worth 585 SEK, an overall return of 485% and the GMV was sold for 772SEK. The capital invested in OMXSPI grew to 272 SEK, an overall return of 272%.

The years of 2016 and 2020 saw extraordinary growth for both the EQW and optimal portfolios, with the latter having returns of 99,8% and 89% respectively.

Table 8 displays the beta, CAPM and Jensen’s alpha for the portfolios.

With the returns and standard deviations calculated we could now calculate the risk measurements to compare the portfolios as displayed in table 8. The Beta was calculated with the returns of the EQW, GMV and Optimal portfolios and compared with the Stockholm stock exchange’s returns for the same periods. The optimal portfolio has “higher” risk measures overall with the Alpha being especially large compared to the EQW portfolio.

Significance tests

Table 8. t-test Paired two sample for means. Optimal - OMSXPI The T test for the optimal portfolios shows that the mean returns are significantly different from the Stockholm stock exchange at a 95% significance level since the two tale p-value is less than 0,05.

Table 9. t-test Paired two sample for means. EQW - OMSXPI The T test for the EQW portfolio shows that the mean returns are not significantly different from the Stockholm stock exchange at a 95% significance level.

Table 10. t-test Paired two sample for means. GMV - OMSXPI The t-test for the GMV portfolio also shows that the mean returns are not significantly different at a 95% significance level.

Table 11. t-test Paired two sample for means. Optimal - EQW The t-test for the Optimal portfolios shows that the mean returns are significantly different from the EQW portfolios at a 95% significance level.

Table 13. t-test Paired two sample for means. GMV – EQW The t-test for the GMV portfolios shows that the mean returns are significantly different from the EQW portfolios at a 95% significance level.

Analysis

Efficient market hypothesis The EMH argues that investors cannot consistently beat the market since stock prices reflects all available information. The Magic Formula proves the opposite. By using this method investors can find undervalued stocks which according to EMH should not be possible. Behavioural finance claims that stocks are not always traded at their fair value which several of the stocks in our analysis probably are not. Andrew W. Lo stated that completely efficient and completely irrational markets are two extremes. The market is not fully efficient in our studies since the Magic Formula consistently could find what we see as undervalued stocks, beat the market, and end up with higher returns during the period.

The Magic Formula strategy beats the market in the long-term as Joel Greenblatt claims, when equally weighted. The combination of MPT and Magic Formula increases the returns. It beats the market, but more importantly it beats the Magic Formula strategy, consistently. Out of the portfolios created in this study the optimally weighted beats the EQW eight out of eleven times. The optimal beats the market every year, while the EQW portfolios does not. The GMV portfolios beats market nine out of eleven years and beats the EQW seven out of eleven years. To an investor, the weighted Magic Formula options should be a lot more attractive.

With a 95% confidence interval we tested to see if the mean return of the three different portfolios where equal to the mean return of the Stockholm stock exchange. We found that at a 95% confidence interval, we reject the null hypothesis for the optimal portfolio

(H01). This means that the mean return of the optimal portfolio is significantly different from the mean return of the Stockholm stock exchange. Since we reject the null hypothesis for the optimal portfolio, we conclude that our testing with the optimal portfolio beats the market at a 95% confidence interval.

We cannot reject the null hypothesis of the EQW (H02) and GMV (H03) portfolios. This means that we cannot prove that the mean return of the EQW and GMV portfolios are significantly different from the mean return of the Stockholm stock exchange. At a 95% confidence interval we cannot conclude that these two portfolios beat the market.

Comparing empirical results By estimating the efficient frontiers as well as the CAL’s we constructed the optimal portfolios by maximizing the Sharpe ratio. For example, in the year 2010 the highest possible forecasted Sharpe ratio 1.463 with a forecasted standard deviation of 17,1% and a forecasted return of 31%. This is where the efficient frontier is tangent to the CAL. The optimal portfolios weighted with Markowitz theory had a forecasted sum of 192,0% standard deviation and 344,6% return. The actual results were 179,1% standard deviation and 338,1% return. This shows that the optimal portfolio had higher returns in comparison standard deviation both forecasted and actual.

When minimizing the standard deviation and building the GMV portfolios we found a forecasted sum of 129,10% standard deviation and 189,03% return. The actual results were 157,33% standard deviation and 241,14% return. These portfolios, also weighted with Markowitz theory generated returns that where higher than the EQW (Magic Formula) returns. The purpose with this approach is to minimize the standard deviation, and it generated 20,58 percentage points less standard deviation compared to the EQW.

According to the calculations made with Markowitz theory the EQW was expected to have a much worse average Sharpe ratio than the Optimal portfolio and GMV portfolio. This means that an investor would expect to be exposed to a higher amount of risk for the same return with the EQW portfolio. This was also the case when looking at the actual results. Although an investor might be more diversified with EQW the person’s risk-return ratio is worse.

Cumulative Returns 1600

1400

1200

1000 Optimal

800 EQW OMXSPI 600 GMV 400

200

0 2008 2010 2012 2014 2016 2018 2020 2022

Graph 4 visualises the cumulative returns of the portfolios.

To put the percentages into perspective we created graph 4 that shows the cumulative results with an initial investment of 100SEK in 2010. If the investor followed the EQW, Magic Formula method the investor would end up with 585SEK after the whole period. If the investor would buy the stocks of the GMV portfolios the investor would end up with 772SEK. If the investor bought the optimally weighted portfolios which we argue would be the best option, the investor would end up with 1412SEK. Although this might not sound as much, it is still a development of 1312% (Optimal) instead of 672% (GMV) or 485% (EQW). Imagine this with a starting investment of 1000SEK, 10 000SEK or even 100 000SEK instead.

Based on the t-test in table 12 we reject H04. This means that the optimal and EQW mean returns are significantly different from each other proving that by using MPT to solve for maximum Sharpe ratio we managed to statistically improve the average return of the portfolio. We cannot reject H05, however. Effectively this suggests that we cannot on a 95% confidence interval conclude that the GMV portfolios mean returns where significantly different from the EQW portfolios for the period studied.

If the investor in question is risk-averse, the results suggests that the GMV portfolios should be recommended. It generates the lowest both forecasted and actual standard

deviation, therefore the least amount of risk. While it at the same time generates the second highest returns (after the optimal).

For any rational and wise investor, the highest amount of return and the lowest amount of standard deviation should be preferred. However, it is not possible to get the best of both worlds. We can however, by using MPT, create an option that generates the highest return-risk ratio. The optimal Portfolios, with the highest return-risk ratios should attract rational investors, that are not risk-averse, and be preferred over the other portfolios.

Risk measures As described in the theory part the risk of MPT refers to the volatility, this however does not mean that variance and standard deviation are the only measures of risk that should be used. The beta describes the systematic risk, and it differs quite a lot between the three methods. The EQW portfolio has a beta value 0,92 which suggests that it is slightly less volatile than the market., and the GMV portfolio with its beta of 0,7 is even less volatile. The optimal portfolio has a beta value of 1,53 which means it should be roughly 50% more volatile than the Stockholm stock exchange and be considerably more volatile compared to the other two portfolios. This can be expected partly because the return of the optimal portfolio is higher than the returns of the GMV and EQW portfolios and partly because the weighting process reduced the number of stocks in the portfolio. Markowitz describes the risk-reward trade off as investors who are willing to take on more volatile and riskier portfolios require higher returns. The optimal portfolio having higher returns than the less volatile EQW- and GMV portfolios cannot come for free and must be “paid” for by being a more risky, more volatile investment compared to the market.

The CAPM values for the three portfolios do not differ as much as the beta with the optimal portfolio having 2 percentage points higher CAPM compared to the EQW, and 3 points compared to the GMV. This means that the required expected return is 2 percentage points higher than for the EQW. The CAPM for the EQW means that investors should require an expected return of 10% to be willing to invest in the EQW portfolio and 9% for the GMV portfolio. For the optimal weighted portfolio this

required expected return is 12%. The CAPM value for the EQW portfolio is equal to the average return of the Stockholm stock exchange despite being less volatile.

To measure if MPT can improve the strategy that beats the market, we used Jensen’s Alpha. The Jensen’s Alpha value for the optimal portfolio is twice as large as the value for the EQW portfolio, the values being 0,09 for the EQW and 0,18 for the optimal portfolio. The Jensen’s Alpha measures the excess returns above the required one set by the CAPM and here we can clearly tell that the optimal portfolio offers greatly improved excess return to investors and this is supported by Sharpe ratios for the optimal portfolios being larger than the Sharpe ratios for the EQW portfolios. The same can be said for the GMV portfolio which also has a higher Jensen’s Alpha than the EQW but lower than the optimal portfolio, this relation is also repeated in the Sharpe values meaning that the GMV portfolio not only offers better risk-adjusted return, but better overall return while still being less volatile less risk than the EQW portfolio.

By weighting the stocks by maximizing the Sharpe ratio the portfolios became more volatile and riskier compared to the EQW portfolio, but this does not need to be considered too bad. In his 1952 paper Markowitz states that investors want to minimize risk and that diversification is the main way for them to do so. But this does not mean that the optimal portfolios should be less attractive for investors despite being less diversified. The Sharpe ratio tells us that it lies in investors’ interest to take on this extra risk as it results in greater excess returns for them. The average Sharpe for the EQW portfolio is 0,87 while it for the optimal portfolio averages at 1,46. This represents an improvement of 67% for the optimal portfolio proving that the extra risk incurred by the weighting process is “outweighed” by the excess returns. Weighting can also be used to decrease the volatility of the portfolio. The EQW and optimal portfolios had 16,2% and 16,3% standard deviation each while the GMV portfolios standard deviation was slightly lower with 14,3% while still yielding higher returns on average and overall compared to the EQW portfolio. This results in a Sharpe ratio for the GMV at 1,16 compared to the 0,87 of the EQW.

Although we do not take transaction costs into consideration in our results, we argue that since less stocks are bought with the optimal and GMV portfolios (each year) the different in actual returns would be even higher.

Conclusion

By using Markowitz’s theory, we could increase the risk adjusted return of the investment strategy Magic Formula. The optimal portfolio yielded higher returns compared to the EQW eight out of the eleven years studied and yielded a higher Sharpe ratio for seven of the years. The GMV portfolio yielded higher returns than the EQW seven out of the eleven years, while at the same time reducing risk. The EQW portfolios average Sharpe ratio was 0,87 compared to 1,16 (GMV) and 1,46 (optimal) meaning that the risk adjusted returns were improved by weighting the stocks.

We conducted t-tests that proves the mean returns from the optimal portfolio are significantly different from the market at the 95% confidence interval, this indicates that the market can be beaten despite the claims of EMH. The EQW and GMV portfolios t- tests were not statistically significant, but still beat the market during our testing period. T-tests also shows that the optimal portfolio saw significantly increased average return compared to the EQW portfolio.

When estimating the systematic risk through the Beta we found that the optimal portfolio is more volatile than the market, meaning it is riskier but also has the possibility for higher returns compared to the other portfolios. The Jensen’s Alpha is higher for the GMV and optimal portfolios in comparison to the EQW portfolios, with the optimal portfolio especially having an alpha twice as high as the EQW portfolio. Further proving that the risk adjusted return is higher.

Finally, we conclude that the risk of the optimal portfolio was increased to allow for higher returns, while the GMV portfolio sought to minimize the risk and yielded higher returns compared to the EQW. Weighting the portfolios is thus recommended for both risky and risk averse investors. According to Markowitz and MPT any rational investor should prefer the optimal portfolio. We could, by combining MPT and the Magic Formula strategy create more efficient combinations of assets in most of the portfolios and therefore improve the investment strategy.

Future research

We found that the Optimally weighted Magic Formula method managed to achieve better performance. For future research it would be interesting to see how the same strategy would perform under unstable market conditions, such as under a financial crisis. By looking at the different performances between the two portfolios during unstable market conditions, the research might find that the Optimal is exposed to even higher risk because of the systematic risk or that it less risky than the EQW since it includes less stocks.

References

Literature

Bodie, Zvi., Alex Kane, and Alan J. Marcus. Investments Global edition, International ed. New York: McGraw-Hill Education, 2014.

Bodie, Zvi., Alex Kane, and Alan J. Marcus. Investments. 11. ed. New York: McGraw- Hill Education, 2018.

Damodaran, Aswath. Investment Valuation Tools and Techniques for Determining the Value of Any Asset. 3rd ed. Hoboken, N.J.: Wiley, 2012. Wiley Finance Ser. Web.

Greenblatt, Joel. En Liten Bok Som Slår Aktiemarknaden (fortfarande). Vaxholm: Sterner I Samarbete Med Ekerlid, 2011.

Articles

Bakircioglu Eriksson, B., & Svensson, K. (2020). Magic Formula och Graham Screener på Small, Mid och Large Cap : Hur investeringsstrategier presterar på Stockholmbörsens undergrupperingar (Dissertation). Retrieved from http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-97041

Elton, Edwin J & Gruber, Martin J. 1997. Modern portfolio theory, 1950 to date. Journal of Banking & Finance. Vol 21: 1743-1759.

Fama, Eugene F. "EFFICIENT CAPITAL MARKETS: A REVIEW OF THEORY AND EMPIRICAL WORK." The Journal of Finance (New York) 25.2 (1970): 383-417.

Fama, Eugene F. "Efficient Capital Markets: II." The Journal of Finance (New York) 46.5 (1991): 1575-1617.

Frantz, P; Payne, R; Favilukis, J. 2011. Corporate Finance. University of London. London: University of London.

Jin, Hanqing; Markowitz, Harry; Zhou, Xun Yu. A note on Semivariance. Mathematical Finance. Vol 16, No. 1: 53-61. https://doi.org/10.1111/j.1467- 9965.2006.00260.x

Johansson, V., & Werner, D. (2020). Den Magiska Formeln : En studie om magiska formeln och effekterna av olika portföljstorlekar på avkastningen (Dissertation). Retrieved from http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-95588

Lo, Andrew W. "Adaptive Markets and the New World Order." Financial Analysts Journal 68.2 (2012): 18-29.

Markowitz, Harry. 1952. Portfolio Selection. The Journal of Finance. Vol 7, no. 1: 77- 91. http://links.jstor.org/sici?sici=0022- 1082%28195203%297%3A1%3C77%3APS%3E2.0.CO%3B2-1

Mangram, Myles E. 2013. A simplified perspective of the Markowitz Portfolio Theory. Global Journal of Business Research. Vol 7, no. 1: 59-70

Appendices

2010-2011 0,7 0,6 0,5 0,4 0,3 Return 0,2 0,1 0 0 0,1 0,2 0,3 0,4 Standard deviation

The highest ranked companies of 2010 when using the Magic Formula strategy were: astrazeneca, addtech b, skanska b, betsson b, aak, knowit, duni, sweco, abb ltd n, bts group b, fenix outdoor, ncc b, lagercrantz group b, af poyry b, axfood, proact it group, aq group, alfa laval, tele2 b, millicom intl.celu.sdr, securitas b, beijer alma b, vitec software group b, hennes & mauritz b, systemair, swedish match, eolus vind b, kindred group sdr, telia company, midsona b.

We created an EQW out of the 30 stocks above in 2010 where each stock was assigned to 1/30th of the investment. With the monthly historical data of 5 years the standard deviation of the EQW was expected to be 18,0% and the expected return was 18,8%. This resulted in a Sharpe ratio of 0,714 with a yearly risk-free rate of 0,06.

The portfolio of 2010 was thereafter weighted with Markowitz theory to find the optimal weighted portfolio (with the highest Sharpe ratio). The expected standard deviation was 17,1% and the expected return was 31%. This resulted in a Sharpe ratio of 1,463.

2011-2012 0,6 0,5 0,4 0,3

Return 0,2 0,1 0 0 0,1 0,2 0,3 0,4 Standard deviation

In 2011 the stocks were: Astrazeneca, AF poyro B, Kindred group SDR, Fenix outdoor B, Autoliv SDB, Skanska B, Betsson B, Boliden ORD SHS, OEM international B, Enea, Millicom INTL.CELU. SDR, BTS, group B, Billerudkorsnas, Beijer Alma B, Clas Ohlson B, Axfood, Electrolux B, Nolato B, Bilia A, Hennes & Mauritz B, Tele2 B, Rottneros, Swedish Match, NCC B, Eolus Vind B, Knowit, Loomis, Duni, Sweco B, Mekonomen.

The standard deviation of the equally weighted portfolio was expected to be 21,6 % and the expected return was 12,0%. This resulted in a Sharpe ratio of 0,277. The optimal portfolio had an expected return of 27,5%, standard deviation of 16,2% and a Sharpe ratio of 1,326.

2012-2013 0,5 0,4 0,3

0,2 Return 0,1 0 0 0,1 0,2 0,3 0,4 Standard deviation

In 2012 the stocks were: Haldex, Astrazeneca, Skanska B, Enquest, JM, Autoliv SDB, SAAB B, Holmen B, Eolus Vind B, OEM International B, Bulten, Beijer Alma B, BTS group B, Fenix outdoor, Vitec Software group B, AQ group, Billerudskorsnas, Knowit, Betsson B, Kindred group SDR, Rejlers B, Millicom INTL.CELU SDR, Lundin energy, Xano Industri B, ABB LTD N, Axfood, Byggmax group, Concentric, Bilia A.

The standard deviation of the equally weighted portfolio was expected to be 23,4% and the expected return was 12,1%. This resulted in a Sharpe ratio of 0,261. The optimal portfolio had an expected return of 41%, standard deviation of 25,6% and a Sharpe ratio of 1,367.

2013-2014 0,5 0,4 0,3

0,2 Return 0,1 0 0 0,1 0,2 0,3 0,4 Standard deviation

In 2013 the stocks were: Biogaia B, Tethys oil, Enquest, JM, Modern Times Group MTG B, Astrazeneca, Enea, OEM International B, Autoliv SDB, Sectra B, Nolato B, SAAB B, Clas Ohlson B, Addtech B, BTS group B, Swedish match, Atlas Copco A, Fenix Outdoor B, Byggmax Group, Millicom INTL.CELU. SDR, Sweco B, Concentric, Axfood, Vitec Software group B, Hexpol B, Probi, Addnode group B, Telia company, Sandvik, Beijer Alma B.

The standard deviation of the equally weighted portfolio was expected to be 21,2% and the expected return was 15,5%. This resulted in a Sharpe ratio of 0,448. The optimal portfolio had an expected return of 31,8%, standard deviation of 21,8% and a Sharpe ratio of 1,183.

0,5 2014-2015 0,4 0,3

0,2 Return 0,1 0 0 0,05 0,1 0,15 0,2 0,25 Standard deviation

III

In 2014 the stocks were: Lucara Diamond, ICA gruppen, SAS, Enea, Nolato B, JM, Eolus vind B, Tethys Oil, Swedish Match, Skanska B, Atlas Copco A, Autoliv SDB, Modern times group MTG B, Betsson B, Enquest, Clas Ohlson B, AQ group, BTS group B, Probi, NCC B, Axfood, Lagercrantz group B, Beijer Alma B, Byggmax group, Cellavision, Hexpol B, Fenix outdoor B, Kindred group SDR, OEM International B, Xano Industri B.

The standard deviation of the equally weighted portfolio was expected to be 17,2% and the expected return was 27,6%. This resulted in a Sharpe ratio of 1,251. The optimal portfolio had an expected return of 41,3%, standard deviation of 17,2% and a Sharpe ratio of 2,051.

2015-2016

0,6

0,4

Return 0,2

0 0 0,05 0,1 0,15 0,2 0,25 0,3 Standard deviation

In 2015 the stocks were: Lucara Diamond, Hexatronic group, Tethys oil, Kindred group SDR, Proact it group, JM, Nolato B, Modern times group MTG B, Clas Ohlson B, Enea, AQ group, Rottneros, Byggmax group, Biogaia B, Invisio, OEM International B, BTS group B, Swedish Match, Elanders B, Lagercrantz group B, Bergman & Beving, Addnode group B, ABB LTD N, Besqab, Mycronic, Axfood, Sweco, Beijer Alma B, Addtech B, Concentric.

The standard deviation of the equally weighted portfolio was expected to be 15,5% and the expected return was 19,7%. This resulted in a Sharpe ratio of 0,883. The optimal portfolio had an expected return of 28,8%, standard deviation of 13,4% and a Sharpe ratio of 1,703.

2016-2017 0,6

0,4

Return 0,2

0 0 0,05 0,1 0,15 0,2 0,25 0,3 Standard Deviation

In 2016 the stocks chosen were: Lucara diamond (ome), Sas, Rottneros, Besqab, Mycronic, Nolato b, Proact it group, Elanders b, Bilia a, Nokia (ome), Jm, Skanska b, Enea, Bts group b, Concentric, Eolus vind b, Clas ohlson b, Hexatronic group, Atlas copco a, Granges, Haldex, Probi, Oem international b, Bergman & beving, Tethys oil, Swedish match, Cellavision, Knowit, Nobia.

The standard deviation of the equally weighted portfolio was expected to be 17,7% and the expected return was 18,7%. This resulted in a Sharpe ratio of 0,71. The optimal portfolio had an Expected return of 31%, standard deviation of 16,5% and a sharpe ratio of 1,49.

2017-2018 2

1,5

1 Return 0,5

0 0 0,2 0,4 0,6 0,8 1 1,2 Standard Deviation

In 2017 the stocks were: Lucara diamond, Fingerprint cards b, Swedish match, Sas, Proact it group, Jm, Besqab, Nobia, Electrolux b, Mycronic, Rottneros, Axfood, Bts group b, Nordic waterproofing holding, Enea, Knowit, Betsson b, Oem international b, Kindred group sdr, Hennes & mauritz b, Autoliv sdb, Skanska b, Enquest (ome), Byggmax group, Tietoevry, Hexpol b, Bulten, Fenix outdoor international b, Concentric, Peab b.

The standard deviation of the equally weighted portfolio was expected to be 15,4% and the expected return was 24,1%. This resulted in a Sharpe ratio of 1,17. The optimal portfolio had an Expected return of 34%, standard deviation of 15,6% and a sharpe ratio of 1,77.

2018-2019

1,2 1 0,8 0,6

Return 0,4 0,2 0 0 0,2 0,4 0,6 0,8 Standard Deviation

The stocks chosen 2018 were: Besqab, Lucara diamond, Jm, Sas, Ferronordic, Proact it group, Mycronic, Nobia, Tethys oil, Clas ohlson b, Lundin mining, Electrolux b, Boliden ord shs, Knowit, Hennes & mauritz b, Sandvik, Swedish match, Peab b, Granges, Hexpol b, Oem international b, Betsson b Beijer alma b, Axfood, Skistar b, Fenix outdoor international b, Instalco, Concentric, Bilia a, Rottneros.

The standard deviation of the equally weighted portfolio was expected to be 13,7% and the expected return was 18,7%. This resulted in a Sharpe ratio of 0,92. The optimal portfolio had an Expected return of 25,6%, standard deviation of 12,7% and a sharpe ratio of 1,53%.

2019-2020 1 0,8 0,6

0,4 Return 0,2 0 0 0,2 0,4 0,6 0,8 Standard Deviation

The year 2019 differs from the rest as the portfolios consists of 29 stocks instead of 30. This is due to limitations of data available.

The chosen stocks 2019 were: eolus vind b, G5 entertainment, Tethys oil, Ferronordic, Sas, Concentric, Betsson b, Jm, Mycronic, Skf b, Lundin mining, Rottneros, Kindred group sdr, Nolato b, Boliden ord shs, Knowit, International petroleum, Fenix outdoor international b, Lundin energy, Sandvik, Ncab group, Stora enso r, Momentum group, Epiroc b, Proact it group, Atlas copco a, Vbg group b, Oem international b, Nobia.

Standard deviation of the equally weighted portfolio was expected to be 17,9% and the expected return was 18,8%. This resulted in a Sharpe ratio of 0,71. The optimal portfolio had an Expected return of 31,5%, standard deviation of 18,5% and a sharpe ratio of 1,36%.

2020-2021 0,8 0,6 0,4

Return 0,2 0 0 0,2 0,4 0,6 0,8 Standard Deviation

The stocks chosen 2020 were: Orexo, Ferronordic, Lundin energy, Electrolux b, Holmen b, Tethys oil, Rottneros, Concentric, Volvo b, Lucara diamond, Mycronic, Jm, Knowit, Betsson b, Ncab group, Boliden ord shs, G5 entertainment, International

VII

petroleum, Autoliv sdb, Oem international b, Nolato b, Skf b, Epiroc b, Fenix outdoor international b, Beijer alma b, Skanska b, Vbg group b, Swedish match, Alfa laval Hexpol b

The Standard deviation of the equally weighted portfolio was expected to be 18,1% and the expected return was 11,6%. This resulted in a Sharpe ratio of 0,31. The optimal portfolio had an Expected return of 24%, standard deviation of 18,5% and a Sharpe ratio of 0,94%.

VIII