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

i

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.

ii

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

3

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

4

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

5

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.

6

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).

7

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

8

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 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.

Graph 2, An example of the efficient frontier with CAL

(Bodie et al, 2018, 209)

9

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.

𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 =𝑅𝑝 − 𝑅𝑓

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.)

10

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).

11

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)

12

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).

𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 =𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝐵𝑒𝑓𝑜𝑟𝑒 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑎𝑛𝑑 𝑇𝑎𝑥𝑒𝑠

(𝑁𝑒𝑡 𝑊𝑜𝑟𝑘𝑖𝑛𝑔 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 + 𝑁𝑒𝑡 𝑓𝑖𝑥𝑒𝑑 𝑎𝑠𝑠𝑒𝑡𝑠)

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.

𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑌𝑖𝑒𝑙𝑑 = 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝐵𝑒𝑓𝑜𝑟𝑒 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑎𝑛𝑑 𝑇𝑎𝑥𝑒𝑠

𝐸𝑛𝑡𝑒𝑟𝑝𝑟𝑖𝑠𝑒 𝑣𝑎𝑙𝑢𝑒

13

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.

14

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:

𝐶𝐴𝑃𝑀 = 𝐸(𝑅) = 𝑅𝑓 + 𝛽 ∗ (𝑅𝑚 − 𝑅𝑓)

(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:

𝐴𝑙𝑝ℎ𝑎 = 𝑅(𝑝) − 𝐶𝐴𝑃𝑀

(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).

15

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

16

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.

17

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


H03: The mean return of the GMV portfolios is equal to the mean return of the


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.

18

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.

19

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.

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.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4

Ret

urn

Standard deviation

2010-2011

20

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,

21

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

on average.


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.

22


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.

23

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.

24

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.

25

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.

26

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.

27

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.

28

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.

29

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

0

200

400

600

800

1000

1200

1400

1600

2008 2010 2012 2014 2016 2018 2020 2022

Cumulative Returns

Optimal

EQW

OMXSPI

GMV

30

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

31

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.

32

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.

33

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.

34

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.


35

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

https://doi.org/10.1111/j.1467-9965.2006.00260.x

https://doi.org/10.1111/j.1467-9965.2006.00260.x


http://links.jstor.org/sici?sici=0022-1082%28195203%297%3A1%3C77%3APS%3E2.0.CO%3B2-1

http://links.jstor.org/sici?sici=0022-1082%28195203%297%3A1%3C77%3APS%3E2.0.CO%3B2-1

I

Appendices

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.

0

0,10,2

0,30,4

0,5

0,60,7

0 0,1 0,2 0,3 0,4

Ret

urn

Standard deviation

2010-2011

II

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.

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.

0

0,1

0,2

0,3

0,4

0,5

0,6

0 0,1 0,2 0,3 0,4R

etu

rnStandard deviation

2011-2012

0

0,1

0,2

0,3

0,4

0,5

0 0,1 0,2 0,3 0,4

Ret

urn

Standard deviation

2012-2013

III

The standard deviation of the equally weighted portfolio was expected to be 23,4% and


portfolio had an expected return of 41%, standard deviation of 25,6% and a Sharpe ratio

of 1,367.

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.




ratio of 1,183.

0

0,1

0,2

0,3

0,4

0,5

0 0,1 0,2 0,3 0,4

Ret

urn

Standard deviation

2013-2014

0

0,1

0,2

0,3

0,4

0,5

0 0,05 0,1 0,15 0,2 0,25

Ret

urn

Standard deviation

2014-2015

IV

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.




ratio of 2,051.

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.




ratio of 1,703.

0

0,2

0,4

0,6

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

Ret

urn

Standard deviation

2015-2016

V

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 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.

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.

0

0,2

0,4

0,6

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

Ret

urn

Standard Deviation

2016-2017

0

0,5

1

1,5

2

0 0,2 0,4 0,6 0,8 1 1,2

Ret

urn

Standard Deviation

2017-2018

VI




a sharpe ratio of 1,77.

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 optimal portfolio had an Expected return of 25,6%, standard deviation of 12,7%

and a sharpe ratio of 1,53%.

0

0,2

0,4

0,6

0,8

1

1,2

0 0,2 0,4 0,6 0,8

Ret

urn

Standard Deviation

2018-2019

VII

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%.

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

0

0,2

0,4

0,6

0,8

1

0 0,2 0,4 0,6 0,8R

etu

rn

Standard Deviation

2019-2020

0

0,2

0,4

0,6

0,8

0 0,2 0,4 0,6 0,8

Ret

urn

Standard Deviation

2020-2021

VIII

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



a Sharpe ratio of 0,94%.

Documents

Modern Portfolio Theory combined with Magic Formula