DEGREE PROJECT, IN APPLIED MATHEMATICS AND INDUSTRIAL, FIRST LEVELECONOMICS

STOCKHOLM, SWEDEN 2015

The influence of financial ratios ondifferent sectors

A MULTIVARIATE REGRESSION OF OMXSSTOCKS TO DETERMINE WHAT FINANCIALRATIOS INFLUENCE STOCK GROWTH INDIFFERENT SECTORS MOST

SAN-SAN MA, PATRICK TRUONG

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCI SCHOOL OF ENGINEERING SCIENCES

The influence of financial ratios on different sectors

A Multivariate Regression of OMXS stocks to determine what financial ratios influence stock growth in different sectors most

P A T R I C K T R U O N G A N D S A N - S A N M A

Degree Project in Applied Mathematics and Industrial Economics (15 credits)

Degree Progr. in Industrial Engineering and Management (300 credits) Royal Institute of Technology year 2015

Supervisors at KTH: Boualem Djehiche, Anna Jerbrant Examiner: Boualem Djehiche

TRITA-MAT-K 2015:02 ISRN-KTH/MAT/K--15/02--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Summary

1 Introduction 51.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Limitation and feasibility . . . . . . . . . . . . . . . . . . . . . . 6

2 Method 72.1 Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Model improvement . . . . . . . . . . . . . . . . . . . . . 82.2.2 Outlier analysis . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 Stepwise regression . . . . . . . . . . . . . . . . . . . . . . 92.2.4 Regression with stock growth adjusted for economic fun-

damentals . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Chosen Sectors and Covariates . . . . . . . . . . . . . . . . . . . 102.4 Systematic Literature review . . . . . . . . . . . . . . . . . . . . 11

3 Mathematical and statistical theory 113.1 Multivariate Linear Regression . . . . . . . . . . . . . . . . . . . 113.2 Gauss-Markov Assumptions . . . . . . . . . . . . . . . . . . . . . 123.3 Ordinary Least Squares . . . . . . . . . . . . . . . . . . . . . . . 123.4 R2 statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.5.1 Hypothesis testing of several βi with F-test . . . . . . . . 143.6 Possible Complications . . . . . . . . . . . . . . . . . . . . . . . . 14

3.6.1 Heteroscedasticity . . . . . . . . . . . . . . . . . . . . . . 143.6.2 Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . 15

3.7 Transformations and interpretations . . . . . . . . . . . . . . . . 153.7.1 Level-level regression . . . . . . . . . . . . . . . . . . . . . 153.7.2 Log-level regression . . . . . . . . . . . . . . . . . . . . . . 163.7.3 Log-log regression . . . . . . . . . . . . . . . . . . . . . . 16

3.8 Analysis of observational data . . . . . . . . . . . . . . . . . . . . 173.8.1 Endogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . 173.8.2 Remedy for endogeneity . . . . . . . . . . . . . . . . . . . 17

3.9 Model selection and improvement . . . . . . . . . . . . . . . . . . 183.9.1 Stepwise regression . . . . . . . . . . . . . . . . . . . . . . 183.9.2 Akaike information criterion (AIC) . . . . . . . . . . . . . 203.9.3 Bayesian Information Criterion (BIC) . . . . . . . . . . . 203.9.4 Mallows’s Cp . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.10 Analysis of variance (ANOVA) . . . . . . . . . . . . . . . . . . . 203.10.1 Tukey’s range test . . . . . . . . . . . . . . . . . . . . . . 21

3.11 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.11.1 Outliers, winsorising och trimming . . . . . . . . . . . . . 213.11.2 Quantile-Quantile plot (QQ plot) . . . . . . . . . . . . . . 223.11.3 Cooks Distance . . . . . . . . . . . . . . . . . . . . . . . . 223.11.4 Residual plot . . . . . . . . . . . . . . . . . . . . . . . . . 23

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4 Economical and Financial Theory 234.1 Fundamental analysis . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Efficient market hypothesis . . . . . . . . . . . . . . . . . . . . . 244.3 Financial ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3.1 Price ratios . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3.2 Share-based ratios . . . . . . . . . . . . . . . . . . . . . . 284.3.3 Liquidity ratio . . . . . . . . . . . . . . . . . . . . . . . . 304.3.4 Debt ratios . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3.5 Other financial ratios . . . . . . . . . . . . . . . . . . . . 314.3.6 Stock growth - what influences the stock prices . . . . . . 31

5 Result 335.1 Initial models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1.1 Bank sector . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1.2 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.1.3 Investment Company . . . . . . . . . . . . . . . . . . . . . 355.1.4 Real estate . . . . . . . . . . . . . . . . . . . . . . . . . . 365.1.5 Retail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.1.6 Summary of the initial models . . . . . . . . . . . . . . . 38

5.2 Initial model diagnostics . . . . . . . . . . . . . . . . . . . . . . . 385.2.1 Multicollinearity reduction . . . . . . . . . . . . . . . . . 435.2.2 Outlier analysis and removal . . . . . . . . . . . . . . . . 45

5.3 Automated stepwise procedure for model selection . . . . . . . . 465.4 Final models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.4.1 Bank model . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4.2 Energy model . . . . . . . . . . . . . . . . . . . . . . . . . 495.4.3 Investment Company model . . . . . . . . . . . . . . . . . 515.4.4 Real Estate Company model . . . . . . . . . . . . . . . . 545.4.5 Retail model . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.5 ANOVA - Difference in between sectors . . . . . . . . . . . . . . 595.6 Regression with stock growth adjusted for economic fundamentals 61

5.6.1 Bank model adjusted for economic growth . . . . . . . . . 615.6.2 Energy model adjusted for economic growth . . . . . . . . 625.6.3 Investment model adjusted for economic growth . . . . . 635.6.4 Real estate model adjusted for economic growth . . . . . 645.6.5 Retail model adjusted for economic growth . . . . . . . . 65

6 Analysis 676.1 Most Prominent Key Ratio . . . . . . . . . . . . . . . . . . . . . 676.2 Implications of difference between industry . . . . . . . . . . . . 686.3 Regression comparison of economical growth adjusted models . . 696.4 Analysis about general stock growth . . . . . . . . . . . . . . . . 69

7 Discussion 707.1 Model discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.2 Critisim of stepwise . . . . . . . . . . . . . . . . . . . . . . . . . . 707.3 Investor consideration and what financial ratios can indicate . . . 707.4 Explorative opportunities and future research areas . . . . . . . . 71

8 Conclusion 72

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9 Acknowledgement 749.1 Patrick Truong . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749.2 San San Ma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

10 Appendix 8110.1 Appendix for initial model . . . . . . . . . . . . . . . . . . . . . . 81

10.1.1 Bank Sector - Initial model - Summary and Box plots . . 8110.1.2 Energy Sector - Initial model - Summary and Box plots . 8410.1.3 Investment Company - Initial model - Summary and Box

plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8710.1.4 Real Estate Company - Initial model - Summary and Box

plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9010.1.5 Retail - Initial model - Summary and Box plots . . . . . . 93

10.2 Most successful stocks . . . . . . . . . . . . . . . . . . . . . . . . 9610.2.1 Summary and boxplot for these stocks . . . . . . . . . . . 97

10.3 Correlation matrix from the initial model . . . . . . . . . . . . . 10010.3.1 Bank correlation matrix . . . . . . . . . . . . . . . . . . . 10010.3.2 Energy correlation matrix . . . . . . . . . . . . . . . . . . 10010.3.3 Investment correlation matrix . . . . . . . . . . . . . . . . 10010.3.4 Real Estate correlation matrix . . . . . . . . . . . . . . . 10110.3.5 Retail correlation matrix . . . . . . . . . . . . . . . . . . . 101

3

A multivariate regression analysis of Swedish

OMX stocks to determine what variables

determine the stock values

Patrick TruongSan-San Ma

May 27, 2015

Abstract

Financial ratios are used to indicate a stocks performance. This thesisaims to clarify if there are any differences in how sectors respond to finan-cial ratios. By doing an deductive research this thesis establishes that themost prominent financial ratios are different in the various sectors whilealso establishing that financial ratios account only for a small part of thestock growth. The thesis also contains a qualitative study which attemptsto discuss the forces behind stock growth. The results indicate that thethe stock growth are mainly caused by fundamental factors, which is anotion also supported by previous research.

4

1 Introduction

An investment in the stock market has the intention of making as high profit aspossible and with limited resources, the investor tries to find the best investmentoptions available in the market, which is why investors analyzes financial infor-mation available. When buying stocks from an online bank you are presentedwith different kinds of information. Generally, you can divide the informationto three parts. First there is description, news and recommendations about thecompany, then there are graphs of the historical price trend of the stock andlastly there are the financial ratios which this analysis will focus on. Studieshave shown that investment strategies that are based on financial ratios producelarge abnormal returns on average.[56][80]

Financial ratios are used to evaluate a company which will help the investorto determine if it is a good decision to invest or not, which is a common prac-tice in fundamental analysis. Fundamental analysis is one of the most commonmethods to analyze stocks and one of the core concepts of fundamental analysisis to analyze a firms financial results and market data to be able to determinewhether it is a good investment or not.[90] By analyzing financial ratios, aninvestor who wishes to do long-term investments may get a better grasp of afirms future performance and make better choice of investments. Since there aremany different financial ratios, investors don’t have time to fully analyze everyfirm and all of their financial ratios and might also draw wrong or inaccurateconclusions because of focusing on less important financial ratios which thatis why it is important to know which financial ratios are the best indicatorsfor future growth. There have even been studies about how effective financialratios are to predict which firm will go bankrupt.[46] This analysis will try todetermine which financial ratios are the most important ones for different in-dustries in OMX Stockholm with the help of multivariate regression analysis asexplaining why they serve as good indicators. The results of this study couldprove to be helpful when performing any kind of analysis on financial ratios,such as fundamental analysis or quick evaluation of a stocks price.

1.1 Problem

Acquiring financial information has never been easier in today’s digital age. Thechallenge an investor faces is how to interpret the financial information availablein order to make good investment decision. Since time is a valuable resource,spending time to only analyze the most relevant information would be ideal.It is also common practice for investors to spread their risks and invest firmsin different industries.[10] Different industries work in different ways and theirfinancial ratios will likely vary which could mean some financial ratios are moreimportant than others in some industries.

This thesis to answer the following questions:

• Which financial ratios are the best indicators of a firms future growth?

• Are there any difference of which financial ratios are better indicators forgrowth in different industries?

• What are the underlying reasons for the results?

5

To answer these questions, a multivariate linear regression will be done of fivedifferent industries in OMX Stockholm. It will use data that is historical datafrom 2006 up to 2014 that was available publicly at that time.

1.2 Purpose

One of the most commonly practices methods to analyze a financial securityis fundamental analysis since it evaluates the securities financial results andmarket data.[90] This thesis attempts to identify which financial ratios are themost important variables to a stocks price in different industries with the helpof multivariate regression analysis. The results may serve as a complement forvarious methods such as fundamental analysis or any method which analyses afirms financial data.

Thus the purpose of this essay is to use regression analysis to determine theimportance of various financial ratios to a stocks price in different industries,and then if possible use the weight in combination with the given financial ratiosto determine if a specific stock will increase its value or not in the long term andthus predict whether to go long or short a stock. Furthermore this thesis aimsto research the importance of different financial ratios in at least five differentindustries, since it’s commonly known that one should at least have 3-4 differentstock in 4-5 different industries for a good diversification.[77]

Lastly, this thesis will perform a qualitative analysis of the results of themultivariate regression in order to try and explain why they serve as the bestindicators for stock growth. This is to understand the reasons behind the resultsin order to confirm the results were not a fluke.

1.3 Limitation and feasibility

This essay does not aim to determine if the conclusion is actually applicable inthe stock market if the findings shows that it might work. Of course knowingat the initial stage of operations the problematics of regression analysis andstochastic elements. This thesis does not try to directly predict stock pricing- as the time series method - but rather tries to identify which financial ratiosare of most importance to the stock value growth. The time frame, fundingand scope of this essay does not allow for extensive real life testing and thusempirical elements will be excluded and this essay will be solely and solemnlybe theoretical and rely on historical data.

The data will only consist from OMX Stockholm 2006-2014 of five differentindustries which means the results will only reflect the Swedish stock market.The choice of time period could mean the result are only relevant in the nearfuture since the market behaviour could change in the distant future.

The feasibility of this analysis will likely be able to determine which stockshas a high probability to profit but the question about how much profit in whattime frame could be a harder question to answer. This method will identifywhich financial ratios are most prevalent when evaluating stocks for differentindustries and could potentially determine which information is more relevantfor an investor to focus on.

6

2 Method

This thesis is a deductive study which analysis whether or not the financial ratioshave significant impact on the stock growth and if the impact differs amongdifferent sectors. Regression analysis is used to determine which financial ratioshave significant impact on the stock growth, while ANOVA and Tukey HSD isused to determine whether or not there exists a difference between sectors. Tobe able to perform the statistical analysis data need to be collected.

After the statistical analysis has been done a qualitative analysis of theresults will be performed. This analysis is performed to check what might causethe results of the regression. To do this a systematic literature review wasperformed. Sectors below will go though each step towards the results morethoroughly.

2.1 Data collection

This essay aimed to get the same data that any investor has. This meansthat the information used is the same that the equity market provide. Thefirst source that was considered for data in this thesis was Finbas which is thedatabase of Swedish House of Finance and Stockholm School of Economics.This database was donated to them by Nasdaq OMX in 2011 and has dailydata point values of all OMX companies since 1997, which is why this waschosen for first consideration. However due to untimely data delivery fromtheir representative other sources had to be considered. Due to time constraintssome financial ratios data was collected from a third party data provider, namelyborsdata.se.[17] However a lot of the data could not be collected from this sightdue to membership requirements. Most of the firm specific data was collectedfrom the firms financial statements, which are available in the firms website,and then processed in an excel spreadsheet to give the desired financial ratios.

Yahoo finance was used for historical values of stock prices due to theirsynoptic format and long data series. The data chosen did not have as longtime series as the data from Finbas, however, it did have sufficient data pointfor statistical analysis.

2.2 Data processing

Multivariate linear regression was used to analyze the data for a structural in-terpretation of stock growth and ANOVA was used to test whether or not thereexisted a difference between sectors. Separate regression was used for each sec-tor. The dependent variable was the growth of the firm and the independentvariables was the chosen financial ratios. Interaction effect between the covari-ates could exists and as such a model should test for interactions covariates,but realising the limitations of stepwise regression procedures imposed by thelimited observations. The effort to identify and include such effects were nottaken. The chosen sectors and covariates are described in detail below. To getthe correct regression model an initial model with all the chosen financial ratioswas set up. The purpose of this is to be able to select the right covariates forthe final model.

Non significant variables interfered with the regression. The first thing donewas to check how well the initial model holds the Gauss-Markov assumptions

7

with model diagnostics. The assumptions are never perfectly met in realityhowever it was possible to check if they were reasonable to work with usingdiagnostics tools. The methods for checking the validity of the Gauss-Markovassumptions can be classified into two types; graphical methods and formalsignificance testing methods. The graphical methods involve looking at theappropriate plots that are constructed by fitting residuals with respective fittedvalues, histograms, normal probability plots etc.[23, 6, 22] The graphical testsused in this thesis were residual plots for checking linearity; scale-location plotfor checking heterokedasticity; correlations matrix for identifying correlationsbetween covariates and QQ-plot for normality. Also Cook’s D was used tocheck the leverage. The formal significance tests used in this thesis are student-t testing (P-value and) and F-tests for model significance testing; Breusch-pagantest for heteroskedasticity and VIF for identifying multicollinearity.

Before any of this was done an initial test of how well the assumptions heldwas done by using the automated procedure in the ”gvlma”-package (Globalvalidation of Linear Model Assumptions) in R. This package tests for kurtosis,heteroskedasticity and skewness amongst other.

The statistical computing was done mostly in R, but as mentioned somecalculations and data manipulation utilized Microsoft Excel.

2.2.1 Model improvement

When the initial model had been diagnosed a model improvement procedurewas implemented. This procedure involved both model improvements using thediagnostics results and an automated stepwise procedure. A manual selectionfrom variables was first preferred before using automated selection methods.This was due to the fact that the automated selection methods can be affected bythe input model. The manual selection was performed with regards to the resultsachieved from the diagnostics procedure. The manual process consisted of aVIF-selection of the covariates. It is not desirable to let multicollinearity affectan automated stepwise procedure and as such variables which are collinear needsto be remedied before a stepwise regression.[72] The thesis utilizes a thresholdof V IF = 10. This is the most common rule of thumb threshold for VIF.[74, 73,48, 68] The covariates with VIF values above the threshold values was excludedif they logically could be excluded. The process was performed by removing thevalue with highest VIF and evaluating the VIF changes at every step until allthe VIF-values was under the threshold level.

2.2.2 Outlier analysis

After the VIF-selection, an outlier analysis was performed. It is natural thatsome outliers exists among the data, either because of the specific stock hasextreme under- or overperformance or due to human errors during data collec-tion or processing. The outliers can have an impact on the linearity of a modelan thus also affect the outcome of the automated stepwise procedure. To getthe most accurate stepwise regression the outliers thus had to be removed.[92]To identify the outliers, a comparison of the covariates medians and means wasmade. If they deviate excessively they were considered outliers. However beforeremoving the outliers each of them had to be evaluated, because it is impor-tant to understand why the outliers have the outlying qualities before deciding

8

whether or not the data point should be trimmed from the set.[37] When choos-ing which outliers to be removed in this thesis special care was taken to thespecific business and sector that the outlier was in. The fact that the moststocks follow stock index and that the interesting stocks often are the winningoutliers which outperform others in the same sector was also considered. Thusthe covariates for the stocks that consistently outperformed the stock marketwas not altered. The stocks that have very high volatility however, i.e. thathave short periods of extravagant stock growth followed by large downfalls orhave only been noted in the stock market for a short period of time are con-sidered not to be consistent, and as such are individually assessed to determinewhether they should be trimmed, winsorised or kept. A minimum requirementfor stocks to be considered successful and not alterable is that is outperformsthe 30% growth that the index grew between 2006 and 2013. [1] Also the finan-cial crisis between 2007-2009 allows for negative stock return during these yearswithout being considered ”too volatile”.[7]

Another rule used in this thesis was that valid observations was not adjusted.Such observations include observations of financial statement per share basis,current ratio and Capex. Ratios converging to zero significantly skewed themean upwards. For example earnings converging to zero gave very high PEratios. Likewise applied to all other ratios. These factors can have an impacton linearity which is why extremely high values had to be adjusted for.

Because of the limited amount of observations in this thesis winsorisingmethod was preferred if possible. However if specific stocks were consideredto be non representative stocks those data points were trimmed.

As mentioned before care had to be taken in selecting which data had to beadjusted for. Thus extreme values which were plausible was kept.

2.2.3 Stepwise regression

Automated selection of the final covariates could be done after the initial modelhad been adjusted to hold the Gauss-Markov assumptions as well as possible.The method used was the stepwise regression procedure with AIC and BIC.In a stepwise forward procedure, a covariate that have been added in earlierstages might become redundant because of the relationship between it and thecovariates that has been added afterwards, which is why the forward procedurewas not used. The best results for stepwise regression is generally conceivedwhen the backward elimination and stepwise methods converge.[49, p. 75-83] Ifthey do not converge there might be some problems with the initial model.

The reason both AIC and BIC are used is because the AIC model has betterperformance than BIC when the sample size is small. The BIC model howeveris constant in variance meaning that it gives a true model if the sample size islarge enough.[87] Due to the larger penalty size of log(n) if the observations islarge the BIC also tends to give a simpler model with lower p-values.[42] Thepenalization in AIC (i.e a factor of 2) allows for a critical p-value of 15.7%.[89]Lastly the Mallows’s Cp was not used as a criterion for stepwise selection dueto the results being similar to AIC.[13]

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2.2.4 Regression with stock growth adjusted for economic funda-mentals

Another regression was performed after the initial regression. This regressionhas stock growth minus the OMX30 index as dependent variable. This was doneto see if the financial ratios showed any surplus in explanation when the stockgrowth was adjusted for the economic fundamentals. The reason sector indexwas not used was due to lack of data points. The process for performing thisregression was the same as above mentioned.

The following sections will contain information about the industries andcovariates that was chosen for this thesis.

2.3 Chosen Sectors and Covariates

The thesis analyzed stocks in the five chosen sectors to try to determiningwhether or not there exists a difference in how much impact a financial ra-tio has on a specific sector. The financial ratios are the same ratios mentionedbelow in the theory section along with detailed explanation of each financialratio.”.

The following five sectors were chosen for this analysis: real estate, oil, retail,bank and investment firms. The five sectors that were chosen were selected witha few thoughts in mind. The main idea was to have sectors with different traitsand qualities. One thought was the size of the industry. Oil and retail arerelatively small industries in OMX Stockholm compared to investment and realestate firm. Another thought was to include industries that are heavily affectedor macro-economical factors or not. Oil firms share common risks due to oilprices while retail firms likely tend to vary from firm to firm. Lastly, it was ofinterest to study two industries that should be similar to each other to see ifthey are similar in terms of their financial ratios such as investment and realestate firms.

A short description of the five industries:

Real estate Real estate firms core business is about land and building prop-erties. Common business practices consists of investing in buildings, rent prop-erties such as office places while others focus on property development.

Retail Retail firms sell goods and/or services to customers to earn money.Retail is a very vast industry since the goods or services which a firm sellsare very different and have different supply and demand which means that thedifferent components of retail industry might not correlate with itself as muchas other industries.

Investment firm Investment firms business idea is to invest in other stockfirms. There are several types of investment firms. Some investment firmsonly tries to achieve the best rate of return for the stockholders while otherinvestment firms may influence or control other firms.

Oil Oils firms business consist of exploration, extraction, refining, transportingand marketing petroleum products as well as gas. The oil industry has as of

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late faced great turmoil and the stock price development has been a similardownward trend for all oil industries.

Bank Banks business concerns are of various financial services such as lendingmoney, transactions, asset management and more. A banks customers may beboth private customers or other companies. Bank may also operate pensionsavings and insurance.

2.4 Systematic Literature review

To be able to understand the underlying causes and usefulness of the regressionresults a qualitative analysis was done. To do this, a systematic literaturereview was performed. The purpose of the systematic literature review was tofind out what affected the stock growth. It consisted of scientific articles andliterature within finance, macroeconomics and statistics which had theoreticaland empirical explanations of what affects stock growth. By finding generalcausation to stock growth and combining it with sector specific explanation tostock growth the regression analysis results can be explained.

Most of the scientific articles where collected from university library such asKTHB or google scholar. Search word such as ”stock growth”, ”stock return”,”stock growth economic return”, ”economic growth”, ”stock growth causality”,”financial ratio”, ”performance ratio” etc. was used in the searching process.The scientific articles were then chosen based on title and abstract relevance.

3 Mathematical and statistical theory

3.1 Multivariate Linear Regression

Multivariate linear regression tries model the relationship between the outcomeof a random dependent variable y to the covariates xi. The covariates areconsidered the explanatory variables and each covariate attempts to explain theoutcome of the dependent variable y. The covariates can be either observational(outcomes which are uncontrollable) or experimental (results which may becontrolled). The study of the interpreted relation can be either predictive orstructural. The multivariate linear regression model is the following:

yi = β1xi1 + ...+ βnxin = XTi β+i (1)

orY = Xβ + ε

Where

Y = (y0, y1, y2, ...yi), X =

XT

1

XT2...XTn

=

x11 . . . x1px21 . . . x2p

.... . .

...xn1 . . . xnp

, β =

β1β2...βp

, ε =

= ε1ε2...εn

Here X is a matrix with all the covariates xij , Y is a matrix with the

dependent variable yi, β is the vector with all the slopes of the line for theirrespective covariate and ε is the error term.[62, p. 3-5]

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3.2 Gauss-Markov Assumptions

In order to use linear regression there are a few assumptions which needs to befulfilled to make sure the model is accurate and those assumptions are calledGauss-Markov Theorem. There are four principal assumptions in order to justifythe use of linear regression models

1. Linearity and additivity of the relationship between dependent and inde-pendent variables: y = β0 + β1x1 + β2x2 + ...+ βixi

• The expected value of dependent variable y is a linear function ofeach covariate, when all else is fixed.

• The slope of the line does not depend of the values of the othervariables.

• The effects of all the covariates on the expected value of the depen-dent variable are additive.

2. Random Sampling.

3. Sample variation in explanatory variables.

• This mean not all samples of the same xi will have the same value,namely xi : i = 1, 2, 3..., i does not all have the same value.

4. Zero conditional mean.

• The error term ei has expected value of zero despite the value of Xi,namely E[ei|Xi] = 0.

The less these assumptions are satisfied, the less efficient the model is and atworst the results may be extremely biased or misleading.[8]

3.3 Ordinary Least Squares

The Ordinary Least Squares (OLS) is a method for estimating variables inlinear regression. It is the Best Linear Unbiased Estimator (BLUES). The OLSestimate of β is

β = (XtX)−1XtY

Here β is the unbiased estimate of β, hence E[β] = β. β is the estimate of βwhich minimizes the sum of the squares of the residuals ete=|e2|. The estimationof the residual is given by

e = Y −Xβ

and satisfies the normal equation

Xte = 0

for β. This means the covariates X are orthogonal to the residuals e. Thecovariance matrix for β is calculated in the following way

Cov(β) = (XtX)−1σ2

12

σ2 is the variance. It is estimated in the following way

s2 =1

n− k − 1· |e|2

where n is the number of observations and k is the number of covariates. Thestandard error for any β is

SE(βj) =

√diag(Cov(βj |X)

i.e. the square root of the diagonal elements of (XtX)−1. The standard erroris often calculated by programs that can perform linear regression.[62, p. 5-8 ]

3.4 R2 statistic

The dependent variables sample variance, V ar(y), can be expressed in two terms

V ar(y) = V ar(xβ) + V ar(e)

A statistic which is of interest is R2 and is defined as

R2 =V ar(xβ)

V ar(y)= 1− V ar(e)

V ar(y)R2ε[0, 1]

R2 is a measure of goodness of fit where a higher value means it fits the observeddata well.[62, p. 8]

3.5 Hypothesis testing

Hypothesis testing is a method for testing a hypothesis or claim by using datafrom a statistical sample size to determine the probability of the hypothesisbeing true. Hypothesis testing can be summarized into four steps:

1. First identify and formulate a null-hypothesis H0 that should be tested aswell as the alternative hypothesis H1 if H0 would be false.

2. Identify a test statistic and select a criterion to decide if the hypothesisprove to be true or not.

3. Collect a sample, measure the sample mean and compute the p-value forthe null-hypothesis H0. The p-value is the probability that a variates valuewill be equal or greater than the observed value.

4. Compare the p-value from the measured sample size to an acceptablesignificance value α in order to see if the null-hypothesis is acceptable ornot.

[62, p. 8-9]

13

3.5.1 Hypothesis testing of several βi with F-test

A F-test will tell if a group of variables are collectively significant in a regressionmodel. Consider a case when the null-hypothesis is that a number r of β:s areall equal to zero. Let e be the residuals from the unrestricted regression and e∗be the residuals from the restricted regression. Consider that the residuals e arenormally distributed, then the F-test will be an exact test and the test variableis:

F =1

r

|e∗|2 − |e|2

s2=n− k − 1

r(|e∗|2

|e|2− 1)

The null-hypothesis will be rejected if F is large. In special cases when thenull-hypothesis is that all βi coefficients are zero, it is possible to only estimatethe full regression model and the F -statistic may be expressed in terms of R2

in the following way:

F =n− k − 1

k

R2

1−R2

[5] [100]

3.6 Possible Complications

3.6.1 Heteroscedasticity

The ideal situation is that all residuals ei have the same standard deviation,which means the residuals are homoscedastic. Homoscedasticity is the idealsituation but most of the time everything is not perfect and we have to dealwith heteroscedasticity which is that the residuals ei have different standarddeviation in different data points.

Figure 1: Homoscedasticity and heteroscedasticity graphs[55]

In the picture of homoscedasticity, the average distance between the pointsand line are around the same but in the picture of heteroscedasticity the pointsdistance between the line and the points decreases the further right it goes. Ifthe residuals are significantly heteroscedastic, OLS will no longer be BLUESand the standard errors can be biased which in turn leads to bias in confidenceintervals and test statistics. There are many ways to fix heteroscedasticity andone of them is simply to reformulate the model which consists of using dummyvariables, choice of covariates and transformation of variables. Another optionis to use the “robust errors” option built into various computer programs Thisoption uses White’s Consistent Variance Estimator which gives correct variance-covariance-matrix in case of heteroscedasticity. [99][62, p. 15-20]

14

3.6.2 Multicollinearity

There are cases where there exists correlations between covariates and thus canbe describes as a linear combination of one or several other covariates. This cancause mismatching to the data due to the regression cannot distinguish how thecorrelated covariates affects the dependent variable y. This is commonly spottedby that the estimated standard deviations for some coefficients beta are verylarge. A common method to identify multicollinearity and see how inflated thecoefficients are is to calculate the Variance Inflation Factor (VIF):

VIFi =1

1−R2i

The VIF value is a measure of how inflated the coefficient is.[62, p. 15]

3.7 Transformations and interpretations

Transformation is used when the the assumption of linearity and homoscedastic-ity is violated i.e. the linear regression model is non-linear and heteroscedastic.To deal with non-linearity one can either choose to do polynomial transforma-tion of the covariates or do a log transformation, which will be explained below.

Polynomial or log transformation of the covariates can be used to deal withthe problem of non-linearity. If there is heteroscedasticity, a transformation ofthe Y variable may help to mitigate the problem. If a transformation is neededone can do the following:

1. Plot the dependent variable against each independent in the model tocheck for non-linearity or heteroscedasticity.

2. Use residual diagnostics plots to spot models with bad fit.

3. Trial and error to see how good the models fits.

There are certain rules of thumb for when to use transformations:

1. Natural logarithm of Y - If the dependent variable is positive valuedand could be skewed (e.g. sales and salaries etc)

2. Square root of Y - If the dependent variable is a count. I.e. Y takes ononly small numbers of integers, such as the number of plane crashes in amonth.

3. Logistic transformation of Y - If the dependent variable is betweenzero and one. This could be percentages such as market share.

Besides reducing non-linearity and heteroscedasticity, as mentioned above, atransformation can also reduce skewness.[50]

3.7.1 Level-level regression

The plain modely = β0 + β1x1 + ε

15

is a level-level regression because of the raw values (levels) of y are being re-gressed on raw values of x. The interpretation of β in this case is given bydifferentiation of y with respect to x

dy

dx1= β1.

Thus this means that the β is the marginal effect.[44]

3.7.2 Log-level regression

A log-level regression is given when the natural log transformed values of Y arebeing regressed on raw values of x

log(y) = β0 + β1x1 + ε

This model is used when increase in X leads to a constant percentage increasein y. Examples are when regressing wage on education and forest volume onyears. To get the interpretation of β, the regression equation has the be solved

log(y) = β0 + β1x1 + ε⇒

Y = eβ0+β1x1+ε

and differentiateddy

dx= βeβ0+β1x1+ε

which then gives

β1 =dy

dx1

1

y

This means that the marginal effect depends on y while β is the growth rate. Sofor example if β1 = 0.1 this would mean another year increases the somethingby 10%. Note this form is also called semi-log (dependent) regression.[44]

3.7.3 Log-log regression

To get a log-log regression (also knwon as log-linear regression) both the thevalues of y and x are natural log transformed

log(y) = β0 + β1 log(x1) + ε

This model is best suited when the percentage increases in x and leads to per-centage increases in y. Examples include constant demand elasticity models.The interpretation of this case is given by

log(y) = β0 + β1 log(x1) + ε =

y = eβ0+β1 log(x1)+ε

dy

dx1=β1x1eβ0+β1 log(x1)+ε = β1

y

x1So this means that the marginal effect, β1, is an elasticity. E.g. If x1 is priceand y is demand and an estimation of β1 = −0.2 is given. It means that a10% increase in the price of the good will lead to 2% decrease in demand.[45]In the case of a multivariate linear regression it is acceptable to use the log ofsome of the covariates while leaving other in their original form. For exampleit would not make sense to log a dummy variable. This would give a semi-log(independent) regression.[44]

16

3.8 Analysis of observational data

3.8.1 Endogeneity

OLS estimation requires that the residuals are uncorrelated with the chosencovariates. Endogeneity occurs when E[ei] = 0 is violated. This is likely becauseat least one of the covariates chosen is correlated with the residual ei which isa problem when using OLS since it will not produce consistent estimates. Thisproblem is often occurs in structural interpretations of a regression equation,but not when it is for prediction.

Some common reasons for endogeneity are:

1. Sample selection Bias: This occurs when he data is biased due to othercriterion than the choice of covariates. A common selection bias is selfselection bias which occurs when non-probability related sampling is made,such as when asked if tarot customers if they are satisfied.

2. Simultaneity : This occurs when the dependent variable y also influencesone or several of the covariates. An example of this is the relation betweendepression and unemployment. On one hand depressed people are moreeasily laid off but on the other hand people who get fired are more likelyto become depressed. The cause and effect correlation between those twovariable may go more than one way.

3. Missing relevant covariates: An example of this is that low fuel consump-tion is a sought after attribute for cars and thus should increase a carsprice. However, a regression shows that higher fuel consumption correlateswith higher prices on cars. This is because higher horse power is also asought after attribute for a car and was a missing covariate.

4. Measurement error.

[62, p. 24-29]

3.8.2 Remedy for endogeneity

One of the most common remedy for endogeneity is by including instrumentalvariables (2SLS). When there is endogeneity, at least one of the covariates areendogenous, which means they correlate with the residual. Using instrumentalvariables means finding new covariates that correlates well with the endogenouscovariates but not with the residual. The original exogenous variables plus thenew covariates are the instrumental variable and they must be at least as manyas the original covariates.

If we denote the matrix of instrumental variables as Z then Z will have atleast as many columns as in the matrix X. If Z has as many columns as X,then we have

Zte = 0

which is the natural normal equation. But if Z has more columns than Xthen there are more equations than coefficients which means we have an overdetermined system. To reduce the number of equations, we project X onto Zin the following way:

X = Z(ZtZ)−1ZtX

17

which gives the normal equation:

Xte = 0

The point estimate of β then becomes:

β = (XtX)−1XtY

and the heteroscedasticity robust covariance matrix estimator is:

Cov(β =n

n− k − 1(XtX)−1X(XtX)−1

β is the 2SLS estimate of β. It is not unbiased, but it is consistent.[62, p. 24-29]

3.9 Model selection and improvement

3.9.1 Stepwise regression

Stepwise regression is a common method of building a model by successivelyadding or removing variables based on hypothesis testing methods or qualitativetests of the model such as Akaike information criterion (AIC), Bayesian infor-mation criterion (BIC), Mallows’s Cp, PRESS, or false discovery rate. Thesewill be explained further down. If the stepwise regression is used correctly itcould be a efficient tool for getting a model with good power. However if usedincorrectly the stepwise method could give a poor model that at first glancelooks valid. The stepwise regression can be performed with by beginning withonly dependent variable and successively adding covariates (foward stepwise),by beginning with a general model and successively removing covariates (back-wards stepwise) or by successively adding and removing covariates. In eithercase it is important to check each stage for if the automated steps taken arelogical or if some variables that are removed or entered should not have beenand tweak the model as necessary. Note that tweaking the model can triggera new chain of events. When tweaking the model it is also important to checkthe adjusted R2 to ensure it gets larger. The largest R2 is desired.

There are certain precautions that should be taken when using the stepwisemethod:

1. One should be careful of including variables with fewer observations thanother variables because the stepwise method uses a correlation matrix.

2. If the number of variables tested is large compared to number of observa-tional data (more than 1 variable for every 10 observations), or there existsexcessive multicollinearity among variables, then the there is a chance thatthe stepwise includes all the variables.

3. Stepwise method might exclude dummy variables, even though they logi-cally should be in the model.

4. The computer does not gave intuition and the user thus have to use ownintuition to determine if final model is correct

5. It is important to have good quantity and quality of data for a goodstepwise regression.[72]

18

However using automated models such as the stepwise model has been criticizedin by many academics. Harrell (2001) comments that some of the problem withstepwise regression are:[harrel]

1. It yields R2 values that are badly biased to be high.

2. The F and χ2 tests quoted next to each variable on the printout do nothave the claimed distribution.

3. The method yields confidence intervals for effects and predicted valuesthat are falsely narrow.[4, p. 771-783]

4. It yields p-values that do not have the proper meaning, and the propercorrection for them is a difficult problem.

5. It gives biased regression coefficients that need shrinkage (the coefficientsfor remaining variables are too large.[97]

6. It has severe problems in the presence of multicollinearity.

7. It is based on methods (e.g., F tests for nested models) that were intendedto be used to test pre-specified hypotheses.

8. Increasing the sample size does not help very much.[26]

9. It allows us to not think about the problem.

The conclusion that Harrell draws is that

1. The multicollinearity affected whether or not the correct covariates endedup in the final model.

2. The number of initial covariates affected how many noise covariates endedup in the final model.

3. Observations size did not particularly affect the number of correct covari-ates in the final model.

4. The R2 is determined by the initial model instead of the final model.[51]

Other Academics agree with Harrell.[34]

Stepwise method with t-statistics The stepwise regression is performed bycalculating the t-statistic for the variables estimated coefficient in the model andsquares it. This is the ”F-to-remove” statistic. Likewise the method calculatesthe t-statistic for estimated coefficients for the variables not in the model as ifthey were in the model and squares is. This is the ”F-to-enter” statistic. Theprogram then enters the variable with highest ”F-to-enter” and removes thevariables with lowest ”F-to-remove” statistic in accordance to specified criterias.This is the standard method for stepwise regression. [72]

19

3.9.2 Akaike information criterion (AIC)

One of the more common selection methods is the Akaike information criteriontest. It is used to evaluate whether a covariate should enter or leave a model.It can be used in a stepwise regression as mentioned above. To do the test onechooses the model that minimises

AIC = n ln(|e|2) + 2k

where k is the number of covariates (including the intercept) and n the numberof observations.[2]

3.9.3 Bayesian Information Criterion (BIC)

Another common selection method is the Bayesian Information Criterion test.It is used in the same way as AIC however the model for minimising is different

BIC = n ln(|e|2) + k ln(n)

where k is the number of covariates (including the intercept) and n the numberof observations. [75]

3.9.4 Mallows’s Cp

Mallows Cp is another selection test which is used in the manner as AIC. Like theAIC a smaller Mallows Cp is preferred. The formula for Mallows’s Cp statisticis

Cp =SSEpS2

−N + 2P

whereSSEp = ΣNi=1(Yi − Ypi)2

which is the error sum of squares for the model with P covariates. Ypi is the ofthe ith observation of Y , S2 is the residual sum of squares and N the samplesize.

Boisbunon et al. (2013) showed that Mallows’s Cp is equivalent to AIC inthe special case of Gaussian linear regression.[14]

3.10 Analysis of variance (ANOVA)

In many cases, it is of interest to compare whole groups with each other tofind any significant differences between their means. When there are only twosamples with needs to be compared a t-test will suffice. But often, the analysisrequires a comparison between more than two groups, other similar methodsare preferred such as Tukey’s range test.

ANOVA assumptions Anova has four main assumptions:

1. The expected value of the errors are zero: E[eij] = 0.

2. The variance of all errors are the same.

3. The errors are independent.

4. The errors are normally distributed.

[93]

20

3.10.1 Tukey’s range test

Tukey’s range test may be used to compare two groups of data in order to findany significant differences between the groups. The test compares all possiblepairs of means of each group to the other group, namely µi−µj and determinesif there are any significant differences between two means. The ideal case ofTukey’s range test should have all sizes of each sample being equal.

Tukey’s range test is similar to the t-test with the difference being that itcorrects for experiment-wise error rate. The Tukey’s test formula is:

qs =YA − YBSE

YA and YB are two means with YA being the bigger of the two. SE is thestandard error of the data. Here qs is the value which will be compared to aq-value from the studentized range distribution in order to identify if there is asignificant difference between the two means or not. Studentized range is thedifference between the largest and smallest data in a sample.[81]

3.11 Diagnostics

3.11.1 Outliers, winsorising och trimming

Outliers There are cases when a few data points are distant from all otherdata points. This may have occurs by chance because of the distribution butit often suggest measurement error or possibly not being normal distributed.If it is measurement error, it is advisable to remove those data points. But ifit happens to be the population has a heavy-tailed distribution, one should becareful when when using methods, tools or intuition that assumes somethingis normal distributed. Outliers are points that are far distant from other datapoints, but how much is far enough to be a outlier depends on the experiment,data, and such.[83]

Trimming A method for handling outliers is called trimming. This methodinvolved removing the outliers from the sample. There are various ways trim-ming is done with different procedure and different conditions of which datapoints should be removed.[37]

Winsorising Winsorising is a method to limit extreme values in the data bytransforming the data in the highest and lowest percentile of the data when itis ordered from the lowest to the highest values. For example, a 95% percentilewould mean the 2, 5% highest and lowest percentile of the data will be trans-formed to the highest and lowest values in the 95% percentile in the middle.This will decrease the standard error and might increase or decrease the meanvalue depending on how symmetrical the data is. Winsorising is often appro-priate for ratios and measures when the denominator can take on very smallvalues. However, winsorising is not appropriate for valid observations such asinvestment returns or RD expenditures.[3]

21

3.11.2 Quantile-Quantile plot (QQ plot)

QQ plot may be used to identify if two data sets are from populations withsimilar distribution. The QQ plot technique plots the quantiles of the first dataset against the quantiles of the second data set along with a reference line. Thecloser the data points are to the reference line, the more likely it is that the twodata sets populations have the same distribution and less likely if they the datapoints are far away from the reference line. Here is an example.

Figure 2: Quantile-Quantile plot of two data sets[32]

In the picture above, batch 1 has significantly higher value than batch 2since almost all the points are above the reference line by a significant margin,especially between the values 525 and 625. Some advantages of QQ plot thatthe sample sizes does not need to be equal and that many distributional aspectscan be tested at the same time.[96, p. 21-23][32]

3.11.3 Cooks Distance

Cooks distance may help determine outliers in the covariates. It may be usedto estimate data points and their influence in a regression analysis which helpsto find outliers in the data sample. Cooks distance measures how much changeoccurs when an observation is removed. Let the Cooks distance of observationi be Di, then we have:

Di =

∑nj =1(yj − yj(i))2

p ·MSE

Here yj is the jth fitted response value (the dependent variable) and yj(i) is thejth fitted response value without observation i. p is the number of coefficientsin the regression model and MSE is the mean squared error. If an observationsCooks distance is larger than three times the mean then that observation likelyis an outlier.[33]

22

3.11.4 Residual plot

A residual plot shows residuals compared to the independent variables, withresiduals being in the y-axis and independent variables in x-axis. By doing so,it is possible to determine if a linear or nonlinear model is appropriate for thedata. If the data points are dispersed around a horizontal line, a linear modelwould be appropriate and a nonlinear model if otherwise.[11] Below are threeexamples of residual plots:

Figure 3: Residual plots[11]

The first plot indicates that the data is appropriate for a linear model. Theother two plots suggests a more appropriate model would be a non-linear model.

4 Economical and Financial Theory

4.1 Fundamental analysis

The core concept of fundamental analysis is to use real data to try and deter-mine a stocks true value. It is one of the cornerstones of investing. Fundamentalanalysis analyses the financial data that is considered ”fundamental” to a firmsfuture performance such as earnings, sales, revenue, market share, financial re-serves and more. Fundamental analysis can be performed of more than singlestocks such as whole industries or economies. The biggest part of fundamentalanalysis is the study of quantitative factors, which are the factors with numer-ical values and can be measured. The other part is the qualitative factors areless tangible factors which is related or based on the quality or character ofsomething such as board members, patents, brand-name recognition. Financialratios are considered quantitative factors.

There are some core assumptions to fundamental analysis. One of the as-sumptions in fundamental analysis is that the prices on the stock does not fullyreflect the ”true” value of the stock. The second assumption is that the stockmarket will reflect the fundamental in the long term. These assumptions areoften criticized by believers of the efficient market hypothesis since they believethat it is impossible to beat the market in the long run and that the market isefficiently pricing all stocks at all times.[90][20]

23

4.2 Efficient market hypothesis

The efficient market hypothesis states that the prices of each security fullyreflects the available information completely unbiased and therefore are unbiasedestimate of the underlying value of the security. This also implies that it isimpossible to beat the market since all information and reflections are alreadyincorporated into the securities price. the efficient market hypothesis has threeassumptions:

1. Investors act rationally and value each security to their fundamental value.

2. Some investors may act irrational but their actions are uncorrelated andrandom.

3. In case of highly correlated activities, which means their actions will notcancel each other out, the correlated activities will be eliminated by pro-fessional arbitragers which makes a profit of it.

There are several ”states” of the market efficiency which reflects how much ofthe information is presented in the securities price. When only past informationis reflected into the price it is considered weak-formed. A state which reflects allpublicly available information is considered semi-strong form and when all in-formation, including private information is reflected in the price, it is consideredto be in a strong form.[102]

The efficient market hypothesis used to be widely accepted but by thestart of the twenty-first century, many investors, economists, statisticians andsuch started to believe that the price movement of a security is partially pre-dictable.[67]

4.3 Financial ratios

Financial ratios are ratios of a firms financial statements and serve as a way toevaluate a firms financial condition. There are financial ratios that only serve aspecific niche usage but some financial ratios are frequently used and have beenempirically successful in measuring a firms financial condition. Financial ratiosare seldom evaluated by itself, but is often used combined with other financialratios and other firms financial ratios.

4.3.1 Price ratios

The price ratios gives an investor the notion of whether a stock’s price is justi-fiable. These finacnial ratios are the most intuitive and easiest to use. Howeverthese are relative performance indicators. This means that they are only usefulwhen comparing to other stocks ratios, itself over time or benchmarks.[40][39]

Price/Earnings Ratio (P/E) The formula for P/E is

P/E =price per share

earning per share

Note that the earnings are not the realized returned but rather the earnings thatthe firm makes in the financial statement. This ratio is of interest because it

24

tells how many times the earnings an investor is paying for the share. A P/E of12 means that the investor is paying 12 times the earnings and that it takes 12years for the investor to get the paid price in earni-ngs. Some general guidelinesfor the swedish stock market is that the P/E should be somewhere between 5 and30 to even be a considerable investment. If the P/E is to high an investor mightsuspect that the investment is overvalued in relations to the earnings. Thismeans that the investor is paying too much money for the potential earnings,and that it would take too many years for the investor to recoup the investment.Likewise if the P/E is low the investment might be undervalued and might beworthy to consider. It is important to know that even though the P/E gives aquick look into the earnings potential and valuation it does not cover the wholeaspect of the company. Because P/E is a relative performance indicator it isdifferent across sectors and businesses.[59][101][69, p. 82] The following thingsare worth considering when using P/E-ratio:

1. Mature vs high growth business - A low P/E might be typical for aspecific sector buy considered buy worthy in another. Generally maturesectors and businesses have higher P/E while companies with high growthpotential will have lower P/E. The reason for this is that the mature busi-nesses are cash cows and that the businesses with high growth potentialwill want to reinvest in their business. As a result a high P/E should beaccepted for businesses with high growth potential.

2. Companies in cyclical industries - The revenues for these companiesare very dependant on the business cycle. Generally reveneues are higherin the periods of economic prosperity and lower in periods of economicdownturns. Airline, raw material and heavy equipment manufacturers arecyclical industries.[19] These companies have lower P/E-ratio at the end ofeconomic prosperity periods due to market expectation of declining futurerevenue.

3. Real estate and investment companies - P/E does not tell muchabout these companies. The price/book-ratio (P/B-ratio) is a better in-dicator for these companies because the assets are of substantial value inthese companies. The large asset value motivates higher P/E in these in-dustries, because much of the stock value is in the assets, which also cangrow or decline in value.

Because the P/E is determined by the stock value, which is determined byvaluation of current earnings and the markets future expectation. It could beinaccurate if the market expectation is wrong. Another complication with P/Eis that the earnings can be calculated using different methods such as actualearning, predicted future earnings, earnings before interest and taxes and earn-ings after taxes. But most often earnings is calculated by using predicted futureearnings on upcoming quarter or historical earnings. It is worth considering thiscomplication when buying stocks from different stock markets.[101]

Price/Earning-Growth Ratio PEG-ratio is a modified form of P/E ra-tio. It is the P/E-ratio divided by the growth rate of earnings per share, whichis estimated. The PEG-ratio takes future earning growth into account, which

25

makes it easier to compare mature and high-growth businesses. The PEG hasthe following general guidelines:

• PEG < 1 - Undervalued

• PEG = 1 - Correctly valued

• PEG > 1 - Overvalued

• PEG > 1.5− 2 - High overvaluation

The growth estimate can be based on the previous year growth in earnings orbe weighted average of the previous year growth and predicted future earnings.

For example if company X has a P/E = 5 with the growth estimate g = 2%and company Y has P/E = 30 with the growth estimate g = 45% then A hada PEG = 5/2 = 2.5 and company B has a PEG = 30/45 = 0.75. Even thoughthe P/E for company A is low it is overvalued because it has very low growthpotential. Likewise company B has a very high P/E, but this is justified by thehigh growth potential, which even makes it undervalued.[79][35]

The PEG is not as convenient as the P/E. The P/E is prevelant in manystock markets in the basic information sheet about company performance. ThePEG however will require a potential investor to estimate the growth by goingthrough the financial statements.

Price/Sales-ratio (P/S) The P/S formula is the following

P/S =Price per share

Annual sales (revenue) per share

an alternative formula is

P/S = P/E · profit margin

This is thus the price an investor pays for every cash amount of sales that thecompany makes. The P/S is useful when the P/E is negative or the earningsare low och non existent. P/S specifies how the revenue is valued in the market.A company whose earnings are not performing good does not necessarily implythat it is a bad investment. The earnings might be generated later years becauseof large investments or because the company has that kind of business model.

A low P/S means that an investor is paying low amount for the generatedrevenue, so generally a low P/S is better. Although a low P/S also indicatesthat the market does not have high expectations of the company. A high P/Showever indicated that the market expects potential for revenues to increase,because the stock has a high value in relation to its’ revenues.

P/S if often used in combination with the profit margins. A company withhigh P/S and high profit margins indicates that it is correctly valued. Likewisea company with low P/S and low profit margins indicates that it is correctlyvalued. The reason for is that high margins motivates a higher stockprice andvice versa. Note that the profit margins vary across industries. Retail often haslower profit margins but inventory turnover, which generated revenue, whileheavy equipment industries have very low turnover and high profit margins.Thus the profit are different across sectors given a fixed revenue. Herein lies

26

also the P/S weakness i.e. that it is not very useful as an individual financialratio, but has to be related to other performance measures.[88][69, p. 85]

Other things to consider when using P/S is

• Companies with high debt will generally have lower P/S due to interestsand installments.

• P/S is good to use in high growth companies without profits

• P/S does not account for the capital structure

Because the profit can be affected by short-term company strategies suchas marketing campaigns and sales or investments the P/E-ratio is more volatilethan the P/S-ratio. The P/S-ratio thus gives the investor a more accurateperformance measure over time. A high growth company might have varyingP/E due to reinvestments but the P/S will remain stable.

The historical mean for P/S in Sweden is approximately 1 and it usuallyfluctuates between 0.5 and 5.

A benefit of using the P/S is that the revenue is calculated using the samemethod for different companies. If the revenue goes down the price of the stockwill follow.[82][88]

Price/Book-ratio (P/B) The P/B-ratio is sometimes also called Market-to-book ratio. It is the amount of money an investor has to pay for every cashunit of equity. The formula for P/B-ratio is

P/B =Price per share

Book Value per share

another way to view this ratio is as

P/B =Market Value of equity

Book value of equity

The book value is the assets minus the debts.The P/B-ratio indicates how the market is currently valuing the company.

It answers the question ”Is the company high och low valued in relation to thefinancial statements?”. Generally a low P/B-ratio indicates that the companyis undervalued and vice versa. However when an investor is using the P/B-ratiothe following has to be considered:

• Companies usually do own valuation of assets and the valuation of someassets are complicated, such as intangible assets (patents, brandvalue etc.)and raw material (oil and wood). If the company has wrongfully valuedits’ assets then the P/B-ratio will be biased.

• Companies that are service intensive have value in their know-how whichis not represented in the P/B-ratio, such companies include consultantand lawfirms.

• In times of economic prosperity the P/B is often higher.[78][31][69, p. 83]

General guidelines for P/B-ratio is the following

27

• P/B < 1 - Either the company is undervalued or the market expects thatthe book value will decrease in the coming future. This could also implythat the market thinks the assets in the company is unprofitable and thusworth less than their book value.

• P/B = 1 - The market considers the stock correctly valued

• P/B > 1 - Either the company is overvalued or the market expects thatthe book value will increase in the coming future.

However the P/B-ratio for most successful firms substantially exceeds 1,which indicates that the value of the companies assets when put to use exceedstheir historical value. Low P/B-ratio stocks are often called value stocks andhigh P/B-ratio stocks are often called growth stocks. The P/B ratio will varyacross businesses due to differences in fundamental characteristics and how muchvalue the management adds. Thus this ratio also serves the management byproviding them information about what the market thinks about them.[10, p.23-63][31]

4.3.2 Share-based ratios

Dividend per share The amount of money a firm pay out of their profitto every share decided during the annual general meeting. A higher dividenddoes not necessarily mean better since it could show that the firm does not havemuch to reinvest into and therefore might not have much potential to grow. Afirm may also choose to pay out smaller dividend and instead use the money toreinvest into the stock.[53][69, p. 44]

Dividend yield A part of the companies profits will be paid out as dividendsto the investors. The dividend yield is the relation between the companiesdividend per share and price of the share

Dividend yield =Dividend per Share

Price per Share

When analyzing the dividend yield it is important view the historical values.With large mature companies the investor would expect a steady dividend yieldwithout compromising a steady share value increase. This simply implies thatthe mature company is stable and can thus maintain growth and payouts.

Note that the company board decides the dividend payout. This means thatthere is a possibility of the company paying out more than what is healthyfor it. It is even possible for companies to take loans to payout to the stockowners which of course is not a viable long-term strategy. An investor shouldbe suspicious to companies with low equity ratio that keeps paying out highdividends.

Furthermore a company can have high dividend yield due to many causes.The best possible reason is that the stock is undervalued, another reason howeveris that the market expects bad future performances. Likewise a low dividendyield can be motivated by high market expectations for the growth of a company.Naturally a company that does not have positive earnings should not pay outdividends.[27][69, p. 33]

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Earnings per share Earnings per share is an important financial ratio toevaluate a stocks price since it measures a firms profit. It is calculated as:

Earnings per share =Net income-Dividend

Number of shares

It evaluates how much profit a firm makes allocated to each share but it is notenough information by itself since two firms can generate the same earningsper share but one firm needs to use less equity to do so, which shows that onefirm is more efficient with their equity. Earnings per share should therefore becomplemented along with other financial ratios when evaluating a stock.[28][69,p. 55]

Revenue per share Revenue per share is the total revenue allocated to eachshare in one year:

Revenue per share =Total revenue

number of shares

It is helpful to measure a firms business activity. A higher revenue per sharemeans a higher activity.[86]

Equity per share Equity per share is the total value of the common equityof a firm allocated to each share:

Equity per share =Total value of common equity

number of shares

It is a helpful indicator of the minimum value a share has for the firm.[15]Profitability ratio Profitability ratios indicates how effective the firm is in

turning operations into profit. These performance measures are relevant becauseprofit is the main reason investors buy stocks.

Profit margin Profit margin tells the investor how much of the sales thatbecomes profits. Naturally investors seek higher profits. The formula for profitmargin is

Profit Margin =Net Income

Sales

The profit margin shows the investor how much of the revenue that is going tobe theirs after all the expenses. The differences in the profit margin can be dueto different efficiency. A company with high inventory turnover, such as retailcompanies, have lower profit margins but can be profitable because they turnover large volumes of their inventory. The profit margines can also be affectedby the amount of leverage in the company.[10, p. 23-63][69, p. 76]

EBITDA margin EBITDA margin stands for earnings before interest, taxes,depreciation and amortisation and is a measurement of a firms operating prof-itability. The higher EBITDA margin, the less operating expenses absorbs theprofit a company makes. It helps an investor measure a firms efficiency andperformance.[29][69, p. 48]

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Return on Equity (ROE) ROE indicated how good the company is atcompensating the investor for the investment. The formula for ROE is

Profit Margin =Net Income

Book Value of Equity

The ROE tells us how efficient the company is at returning profits from its’equity. If the ROE is constant and while the equity increases it means thatthe company is good at reinvesting. It is wise to compare this financial ratiobetween years. A high ROE may indicate that the firm is able to find investmentopportunities that are very profitable. Because this financial ratio is dependenton equity, a weakness is that it is hard to determine book value of equity.[10, p.23-63][69, p. 90]

Return on Asset (ROA) ROA tells us how good the company is at usingits’ assets to generate income. It is the ratio of net income and the total bookvalue of the firms assets

ROA =Net Income

Book Value of Assets

A firm must earn both positive ROE and ROA to grow.[10, p. 23-63][69, p. 91]

4.3.3 Liquidity ratio

The liquidity ratios indicate how well a company can meet its short-term needs.A good liquidity means that a company can withstand tougher times. Withoutliquidity a company can be forced to make unfavorable decisions, such as takingunfavorable loans, during economic downfall.[69, p. 3]

Current Ratio This is the ratio of current assets to current liabilities

Current ratio =Current Assets

Current Liabilities

It measures the company’s ability to pay short-term liabilities with its short-term assets. General guidelines for the current ratio are:

• Current ratio > 1 - The firm has more short-term assets than liabilitiesand thus are able to withstand economic hardship.

• Current ratio < 1 - The firm has less short-term assets than liabilities andis thus vulnerable.

A high current ratio can also imply that the firm has difficulty selling of inven-tory.[10, p. 23-63][69, p. 32]

4.3.4 Debt ratios

The debt ratios indicates the long-term financial health of a company and tellsus the effect of the companies capital decisions.

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Debt-Equity ratio The debt to equity ratio is the ratio of a company’s totalamount of debt to the value of its equity

Debt-Equity Ratio =Total debt

Total market value of equity

It tells the investor how leveraged the company is. The leverage is a measureof the extent to which a firm relies on debt as a source of financing. Generallya more debt-financed a company is the more risky it is, due to the risk of it notbeing able to meet its debt obligations. [25][69, p. 41]

Equity ratio Equity ratio indicates how big proportion of the firms assets isfinanced with the firms own equity:

Equity ratio =Total shareholder’s equity

Total asset

. Equity ratio is related to debt-equity ratio and is also a good indicator of howleveraged a company is.[30][69, p. 13]

4.3.5 Other financial ratios

CAPEX CAPEX stands for capital expenditure and is how much a firm hasspent on fixed assets, which could be for maintenance, upgrade or expansion.A negative CAPEX means the firm sold more than what they spent on fixedassets. CAPEX can not be deducted from income for tax purposes since it isadded to the firms assets. CAPEX is a very industry-specific financial ratio andshould only be compared with other similar firms.[18][10, p. 678]

4.3.6 Stock growth - what influences the stock prices

There are three main forces that move stock prices; fundamental factors, tech-nical factors and market sentiment. The fundamental factors are the financialstatements of the company, which could be expressed as earnings per share ochany financial ratio which are explained above. These express the future claimsthat the owner of the stock have and as such should are the primary determinantof stock prices in an efficient market. Because the financial ratios are affectedby expected growth and discount rate (which is used to calculate present valueof future earnings) these wo factors also affect the stock growth. Lastly the dis-count rate is determined by the perceived risk, so thus the fundamental factorsthat affect stock growth are

• Financial ratios

• Expected growth

• Discount rate

• Perceived risk

[60, p. 291-357][52, p. 3-14]However because stock is are traded in a market where supply and demand

influences the stock price, there are external factors that affect the stock growth.

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These factors are called technical factors. The technical factors can effect affectthe fundamentals by altering the perceptions of risk and growth. The technicalfactors that can influence a stock are the following

• Inflation - Inflation is the decreasing time value of money. Low inflationhas shown to have strong inverse correlation with valuation, i.e. low infla-tion increases financial ratios and vice versa. Deflation however is generallybad for stock growth because lowers the pricing accuracy of companies.

• Economic strength of market - Company stocks tend to follow sectorindices. Companies within the same sector are often highly correlated.

• Substitutes - The yields on other investment products can affect thestock growth. A higher interest rate gives investors less incentive to takerisky assets and vice versa.

• Incidental transactions - Transactions that are caused by high profilecharacters can influence the stock price. Such as an insider trading stocksor investment companies buying high amount of stocks. These transac-tions can have affect on stock price even though fundamentals have notchanged.

• Demography - A demography of many middle-aged high earners willwant to invest their money for the future which will increase the demandson stock and thus increase the stock prices. Likewise a population con-sisting mostly of older investors would likely want to realize their savingswhich will decrease the demand of stock, and thus decreasing the stockprices.

• Trends - Stocks are often following short-term trends. This is a caseof stock psychology, where collective behaviour determines stock prices.These trends however are only ever noticed in hindsight which make trendsrather inaccurate for predicting the future.

• Liquidity - If a investor puts money on a risky asset, they would wantit to be liquid. Large cap stocks are almost exclusively liquid, howeversmall cap and penny stocks are not always so. These stock thus sufferfrom ”liquidity discount”.

[24, p. 7-15][64, chap. 6]Lastly the psychology that affects the market participants is called the mar-

ket sentiment. This matter is subjective and as such hard to predict. It is arelatively new field in behavioural finance. Assumptions that are made for themarket sentiment is that the market is not efficient most of the time. These in-efficiencies are caused by human psychology and subjective behaviour.[21, chap.1]

Previous research about stock growth

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The relations of economic growth and stock returns It is suggestedin a the research paper ”Economic Growth, Expected Stock Returns and Volatil-ity: A case of Indian Stock Market” that investors are not sensitive to economicgrowth in the short-term. However they are very sensitive to economic growth inthe long-term. This is due to the stock market volatility. High economic growthrates stabilizes decision and create certainty among investors, under such cir-cumstances investors are more reluctant to spontaneously change their invest-ment decisions. A low growth rate however makes investment decisions morevolatile. The study has shown that volatility is invariable to economic growthrate in short-term, while in the investors with long-term horizons are largelyaffected by economic growth rate. The study concludes that high volatility isassociated with low economic growth rate and vice versa.[61] Another study inthe Tunisian stock market (Observation of the Disconnection between Real andFinancial Spheres: The Case of the Tunisian Stock Market during the Period1969-2008) confirms the what is suggested. In this study it is explained thatvolatility is caused by disparity between stock prices and their fundamental val-ues. It is concluded that the stock market is not efficient and thus the stockvalues are not determined by economic fundamentals.[71]

Another study indicates that macroeconomic growth at the fourth quarterstrongly influences the stock returns, however the effects are not as noticeablethe rest of the year.[70]

Other studies has interestingly suggests that economic growth is positivelyaffected by well functioning stock markets.[63]

Relationships between stock returns and real economic growth induring crisis The study ”Casual Relationship between stock returns and realeconomic growth in the pro- and post-crisis period: evidence from China” hasshown that during periods of crisis there is a unidirectional causality in meanand variance from real economic growth to stock returns.[43]

5 Result

The following section will present the results achieved in this thesis. First itwill go through the regression analysis and with the results from the regressionfurther financial, economical and managerial implication will be discussed.

5.1 Initial models

In the following the initial models for the chosen sectors will be presented.

5.1.1 Bank sector

Obs. df R2 Adjusted R2 Std. Error F Sig.

69 51 0.2222 -0.03704 44.74 0.8571 0.6235

Table 1: Bank sector model summary

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Estimate Std. Error t value Pr(>|t|) Sign.

(Intercept) -6.0211 41.9963 -0.14 0.8866

PE -0.0221 0.0234 -0.94 0.3494PS 3.3442 7.1583 0.47 0.6424PB -5.8187 6.8611 -0.85 0.4004Dividend per share 1.5976 3.4605 0.46 0.6463Earnings per share -4.2107 2.8207 -1.49 0.1417Revenue per share 0.2380 0.7796 0.31 0.7613Equity per share 0.1759 0.3938 0.45 0.6570Dividend yield 2.3011 2.4819 0.93 0.3582Profit margin 0.2705 1.4662 0.18 0.8543ROE -0.2346 1.3590 -0.17 0.8636ROA -0.5034 1.5358 -0.33 0.7444Current ratio -2.5707 4.0385 -0.64 0.5273debt equity ratio 0.5857 0.7486 0.78 0.4376equity ratio 0.7317 0.6054 1.21 0.2324ebitda margin -0.0123 1.1935 -0.01 0.9918PEG 1.9522 1.9310 1.01 0.3168Capex 0.0459 0.0491 0.94 0.3540

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 2: Bank initial model

The R2 and the significans level of the model indicates that the initial model isweak for the bank sector. Also notice that the initial model yield no significantcovariates.

5.1.2 Energy

Obs. df R2 Adjusted R2 Std. Error F Sig.

63 45 0.1479 -0.174 1974 0.4595 0.9584

Table 3: Energy sector model summary

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Estimate Std. Error t value Pr(>|t|) Sign.

(Intercept) -426.6275 1641.2563 -0.26 0.7961

PE 0.1518 2.0480 0.07 0.9413PS 3.1336 11.7777 0.27 0.7914PB 158.0429 209.1256 0.76 0.4537Dividend per share -448.2888 1193.8162 -0.38 0.7090Earnings per share 114.2852 103.9310 1.10 0.2773Revenue per share 13.4039 65.4063 0.20 0.8385Equity per share 0.7642 34.7023 0.02 0.9825Dividend yield 109.4508 226.6512 0.48 0.6315Profit margin 0.6189 2.9911 0.21 0.8370ROE 6.4899 34.2133 0.19 0.8504ROA -52.7870 44.2932 -1.19 0.2396Current ratio -3.7070 18.0467 -0.21 0.8382debt equity ratio -357.3298 530.0365 -0.67 0.5037equity ratio 2.7084 21.8216 0.12 0.9018ebitda margin 0.0071 4.4125 0.00 0.9987PEG 24.4127 190.0786 0.13 0.8984Capex 0.0459 0.1725 0.27 0.7914

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 4: Energy initial model

Again we notice that no significant results were returned by the regression.

5.1.3 Investment Company

Obs. df R2 Adjusted R2 Std. Error F Sig.

163 145 0.3709 0.2971 26.78 5.028 1.768e-08

Table 5: Investment Company Model summary

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Estimate Std. Error t value Pr(>|t|) Sign.

(Intercept) -8.1607 20.7685 -0.39 0.6949PE 0.0189 0.0124 1.52 0.1299PS 0.1191 0.0961 1.24 0.2173PB 0.3242 4.2107 0.08 0.9387Dividend per share 1.2094 1.2311 0.98 0.3275Earnings per share 0.1616 0.1838 0.88 0.3809Revenue per share 0.0021 0.0759 0.03 0.9775Equity per share -0.0116 0.0486 -0.24 0.8124Dividend yield -0.8554 0.4779 -1.79 0.0756 ·Profit margin 0.0104 0.0086 1.21 0.2282ROE 0.1862 0.2606 0.71 0.4761ROA 0.2158 0.3698 0.58 0.5604Current ratio -0.0320 0.0370 -0.87 0.3875debt equity ratio -3.8158 7.4574 -0.51 0.6097equity ratio 0.1777 0.2629 0.68 0.5003ebitda margin 0.0046 0.0120 0.38 0.7047PEG 1.3070 0.8670 1.51 0.1339Capex -0.0020 0.0013 -1.55 0.1235

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 6: Investment Company inital model

The model summary indicates that the initial model is significant. Althoughthe R2 indicates the model is weak. Also notice in the regression model thatdividend yield is near the accepted significance level.

5.1.4 Real estate

Obs. df R2 Adjusted R2 Std. Error F Sig.

169 151 0.1319 0.03418 50.95 1.35 0.1698

Table 7: Real Estate Model summary

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Estimate Std. Error t value Pr(>|t|) Sign.

(Intercept) 10.8160 45.0161 0.24 0.8104PE 0.0049 0.0269 0.18 0.8554PS -0.1521 0.2806 -0.54 0.5886PB -9.7929 4.7106 -2.08 0.0393 *Dividend per share -0.8182 2.0524 -0.40 0.6907Earnings per share 1.7874 0.8625 2.07 0.0399 *Revenue per share -0.1768 0.2563 -0.69 0.4913Equity per share -0.0156 0.0129 -1.22 0.2261Dividend yield 0.1701 2.0454 0.08 0.9338Profit margin -0.0416 0.0267 -1.56 0.1216ROE 0.1206 0.1720 0.70 0.4842ROA -0.1147 0.3043 -0.38 0.7068Current ratio 0.3906 1.5125 0.26 0.7966debt equity ratio 2.7313 8.9173 0.31 0.7598equity ratio 0.2504 0.6747 0.37 0.7111ebitda margin 0.0075 0.0232 0.32 0.7479PEG 7.0220 4.2471 1.65 0.1003Capex 0.0110 0.0073 1.51 0.1332

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 8: Real Estate initial model

The R2 is low and the model is not significant.

5.1.5 Retail

Obs. df R2 Adjusted R2 Std. Error F Sig.

62 44 0.5144 0.3268 28.97 2.742 0.003684

Table 9: Retail Sector Model Statistic

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Estimate Std. Error t value Pr(>|t|) Sign.

(Intercept) -9.2777 18.0327 -0.51 0.6095PE -0.1217 0.1495 -0.81 0.4202PS -2.4706 1.2108 -2.04 0.0473 *PB -4.0042 1.8735 -2.14 0.0382 *Dividend per share -5.7473 2.6491 -2.17 0.0355 *Earnings per share 0.8482 0.8724 0.97 0.3362Revenue per share -0.2244 0.1064 -2.11 0.0406 *Equity per share 0.3784 0.3593 1.05 0.2979Dividend yield 6.1089 2.2722 2.69 0.0101 *Profit margin 0.3741 0.4789 0.78 0.4389ROE 0.1626 0.1403 1.16 0.2526ROA 1.8802 1.1110 1.69 0.0977 ·Current ratio 1.5941 2.1432 0.74 0.4610debt equity ratio 6.9012 3.2493 2.12 0.0393 *equity ratio 0.2101 0.3701 0.57 0.5730ebitda margin -0.3382 0.5624 -0.60 0.5507PEG 1.5470 0.7694 2.01 0.0505Capex 0.0004 0.0012 0.32 0.7475 ·

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 10: Retail Sector Inital model

The model summary indicates that the initial model is better than for the othermodels. It has R2 of 0.5144 and p-value of 0.003684 which means that the modelis significant. However the F-value is 2.742 which is low. The initial model showsthat the covariates PS, PB, Dividend per share, Revenue per share, Dividendyield and Debt-equity ratio are significant.

5.1.6 Summary of the initial models

The R2 of all the inital models are low, which means that they are all weak.Two of the models (investment companies and retail) are significant. Howeveronly the retail model has significant covariates. It is clear that all the initalmodels have redundant variables and are thus suitable for model improvementmeasures.

5.2 Initial model diagnostics

The initial screening of how well the Gauss-Markov Assumptions held with”gvlma”-package (Global Validation of Linear Model Assumptions) in R showedthat the assumptions hold reasonably well for the bank, investment and retailsector, but that all models can be improved.

The results for the more detailed analysis with graphical and formal signifi-cance test is shown below.

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Figure 4: Diagnostic plots for the inital bank sector model

Figure 5: Diagnostic plots for the inital energy sector model

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Figure 6: Diagnostic plots for the inital investment company model

Figure 7: Diagnostic plots for the inital real estate sector model

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Figure 8: Diagnostic plots for the inital retail sector model

From the residual plot and the Scale-location plots of the inital model it ispossible to tell that the residuals of bank, investment companies, real estate acompanies, and retail sector are fairly homoskedastic if the tails are ignored.This will need to be adjusted for. However the plots for the energy sector doesnot show constant variance and indicates some linear dependencies.

To check for heteroskedasticity with formal significance Breusch-pagan testwas employed.

Breusch-pagan - Bankχ2 df p-value12.543 17 0.7662

Breusch-pagan - Energyχ2 df p-value11.43 17 0.8334

Breusch-pagan - Investmentχ2 df p-value12.014 17 0.7993

Breusch-pagan - Real Estateχ2 df p-value7.5912 17 0.9745

Breusch-pagan - Retailχ2 df p-value4.6178 17 0.9987

The null hyopothesis for the p-value is that the variance does not change.[16]For 17 degrees of freedom the critical value for χ2 is 27.587 for α = 0.05. A lowp-value in combination with a χ2 over the critical value indicates heteroskedas-ticity. It is clear that there is no indications for heteroskedasticity.

The QQ-plot shows that retail sector is reasonably normal distributed whilethe other sectors show signs of log-normal distribution. This means that thedependant variable for these sectors should be transformed using log-function.However due the dependant variable containing negative values such transfor-mations can not be performed to improve linearity without involving complex

41

values. This is not desired and as such the log distribution has to be handledin an other way. Likely these end tails is caused by outliers, which would meanthat removing these outliers from the sample would reduce the non-linearity.Note also that in all sectors the QQ-plot is skewed toward the tails. The bank,energy, investment company and real estate QQ-plots have tails twisting coun-terclockwise which indicates leptokurtosis and the retail sector has tails twistingclockwise which indicates platykurtosis. QQ-plot for the retail sector also indi-cates slight right skewness.

Cooks distance indicates that the leverage and residuals are fine for the mostof the models. The bank sector however show high leverage and high residual.

One never want to much multicollinearty in the models. To check this acorrelation matrix for the covariates was analysed.

Figure 9: Correlation matrix plot for the financial ratios:s

From the figure it is clear that general correlation problems are PS and profitmargin and PS and ebitda margin is highly negatively correlated. It also showsthat profit margin and ebitda margin is highly positively correlated. A goodthing is that stock growth generally is not correlated to any of the covariateswhich imply that there should not be any simultaneity.

As a price ratio is preferred in the model due to it being the most reviewablein stock markets, both profit margin and ebitda margin should be left out of themodel to reduce multicollinearty. Also note that many variables have some cor-relation i.e. the small blue dots. Correlation matrix for each sector is prevalentin the appendix.

One can also perform a VIF-test to measure multicollinearity with formalsignificance test.

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VIF table

Covariate Bank Energy Investment Real Estate Retail

PE 1.639 8.672 2.065 1.106 1.515PS 9.392 141.992 23.779 1.768 3.520PB 9.125 107.072 7.716 2.183 68.223Dividend per share 6.047 87.781 2.472 6.684 9.133Earning per share 13.319 4.219 5.025 1.837 5.146Revenue per share 6.912 3.705 2.752 1.112 4.409Equity per share 11.450 2.991 2.879 1.016 15.757Dividend yield 1.869 11.403 1.891 5.660 2.443Profit margin 24.495 430.216 14.187 2.911 16.525ROE 34.962 27.390 15.962 7.413 19.788ROA 27.264 10.713 20.613 37.256 8.487Current ratio 2.196 1.334 1.220 1.921 4.188debt equity ratio 3.755 115.180 14.294 8.004 37.395equity ratio 6.462 3.896 7.861 43.133 9.728ebitda margin 16.779 714.419 9.519 1.436 14.969PEG 1.087 8.454 1.930 1.049 1.166Capex 1.149 53.230 1.048 1.070 1.220

The correlation matrix presented above was made for the all the sectorsas sample whereas the VIF-table was constructed by individual sectors (thereare correlation matrix for all sectors in appendix). The VIF-values show thatmany of the variables have multicollinearity. Using these two in combinationit is easy to detect which covariates correlates. VIF-values over 10 indicatesmulticollinearity.[73, 48, 68] These covariates affect the significance of othervariables. For instance one can assume that earning per share and revenue pershare tells roughly the same thing for an investor and thus affects the stockgrowth in the same manner. One of these should thus be removed. Likewisethe covariates ROE ROA and profit margin ebitda margin has very similarmeaning.

So the initial model diagnostics has shown that there significant model im-provements that could be made.

5.2.1 Multicollinearity reduction

Earning per share, Equity per share, profit margin, ROE, ROA and ebitdamargin exhibits high VIF values. From the correlation matrix one can see thatEarnings per share, Equity per share, dividend per share, revenue per share,ebitda margin and profit margin correlates. One can conclude that earningsand equity tell roughly the same thing as dividend and revenue per share. ROEand ROA shows high VIF and the correlation matrix tells that they are stronglypositively correlated. They can also be assumed to indicate the same thing.

In the bank sector ROE has the highest VIF of 34.962. Removing thiscovariate yields new VIF-scores and this process is iterated until all the VIF arebelow treshold value as mentioned above. This leads to profit margin havingthe highest VIF (22.41). Profit margin has high correlation with ebitda margin,which also has high VIF (14.85). They also indicate roughly the same thing.

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So ebitda margin was removed instead of profit margin at step 2 because profitmargin has higher beta and is more intuitive to read for most. At step 3 earningsper share is being removed(VIF = 12.89). A similar process was applied for allthe other sectors.

In the energy sector ebitda margin was first removed (VIF = 714.42). Atstep 2 debt equity ratio was removed (VIF = 107.113). At step 3 dividend pershare was removed (VIF = 86.29). At step 4 PS was removed (VIF = 75.158).Lastly at step 5 ROE was removed (VIF = 23.47).

For the investment companies PS was first removed (VIF = 23.79). Notethat this covariate actually does not tell much about investment companies sincethey defacto do not generate sales, but act as incubators or sum such. At step2 ROA was removed (VIF = 20.61). Lastly debt equity ratio was removed (VIF= 13.43).

For the real estate companies equity ratio which has a high VIF of 43.133have a high correlation with ROA and slight correlation with many other co-variates. This is thus removed.

Lastly for the retail sector PB was first removed (VIF =68.22). At the stepebitda margin (VIF = 14.89) was removed even though profit margin exhibitshigher VIF-value for the same reason as stated above and at step 3 equity pershare (VIF = 12.73) was removed.

VIF table - values under treshold

Covariate Bank Energy Investment Real Estate Retail

PE 1.535 8.286 1.931 1.064 1.298PS 5.372 - - 1.759 3.392PB 6.436 1.553 1.33 2.182 -Dividend per share 5.791 4.331 2.3272 6.683 3.564Earning per share - 2.907 3.893 1.772 3.977Revenue per share 3.638 2.968 2.492 1.096 2.792Equity per share 7.495 2.361 2.689 1.014 -Dividend yield 1.811 - 1.869 5.589 2.017Profit margin 3.326 1.143 5.071 2.775 2.357ROE - - 3.329 3.222 7.998ROA 2.484 3.358 - 1.304 5.562Current ratio 1.950 1.260 1.184 1.284 3.846debt equity ratio 3.364 - - 1.483 7.938equity ratio 4.988 2.786 2.784 - 3.113ebitda margin - - 4.693 1.435 -PEG 1.058 8.023 1.893 1.045 1.142Capex 1.140 4.324 1.036 1.059 1.190

The table above shows the new VIF-valued after the most collinear covariateshave been removed. However doing performing an regression and looking at thenew diagnostics plots still indicates that further model improvement can bemade. There are still redundant covariates and indications of outliers.

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5.2.2 Outlier analysis and removal

A comparison of the covariates mean and median is made to check for outliers.If they deviate to much one might suspect that there are outliers. Also boxplots and histograms where made for each covariate in each sector to visualizethese outliers. A summary of the quantiles and box plots is prevalent in theappendix section. The following outliers is noticed:

• Bank Sector - ROE, ROA and Capex

• Energy Sector -PE, PS, Dividend per share, Revenue per share, Equityper share, Profit margin, ROE, ROA, Current ratio, Ebitda margin, Capex

• Investment Company - PE, PS, Earning per share, Revenue per share,Equity per share, Profit margin, ROE, Current ratio, Ebitda margin,Capex

• Real Estate Company - PE, Equity per share, ROE, ROA, Ebitdamargin, Capex

• Retail Sector - Revenue per share, Equity per share, Capex

The appendix has a short-list and model summary for the most successfulstocks that were not altered. Looking at these however indicates that the mostsuccessful are not burdened by outliers, which means that the restraints thatwill be put forth will not affect these stocks.

The main adjustments were done to the price ratios. For the bank sectorthe PE, ROE and ROA have been adjusted. Even though the mean and medianof the PE does not indicate any outlier. The data point that cause this outlieris the stock ”FX International” which is a non-consisent stock, which has onlybeen in the time-frame for three years and as such should not be allowed toinfluence too much. This data point was thus trimmed from the sample.

The energy sector in OMX Stockholm is not very extensive. There are onlythree large cap companies (Africa oil corp., EnQuest, Lundin Petroleum). Therest of the companies are from smaller caps which is why this sector is lessconsistant and has more outliers than the other sectors. In this sector the PEwas assigned the value 0 in accordance with previous criteria. Also ”Blackpearlresources” has an earning converging to zero year 2012. This PE was winsorizedto 3rd quantile. ”Taurus Energy” has a sales converging to zero year 2010 Alsoprofit margins and ebitda margin for ”Taurus Energy” is very negative (-13582.5-2418.6 and -11837.5 -2178.66 respectively) year 2010 and 2011. Since ”TaurusEnergy” is not representative either in stock growth or in its covariates it theobservation is trimmed from the sample. ”Crown Energy” has a very negativeROE of -244.06 during 2011. But this value was not altered. Lkewise was ROAof ”PetroGrand” 2009 was an outlier that was not altered..

Among the investment companies all the outliers are caused by ”yield life sci-ence” a small company listed on ”Aktietorg”. The data points for this companyis thus trimmed.

For the real estate companies the PE was adjusted like above. Many of theoutlying points were caused by ”ChronTech Pharma” this datapoint was thustrimmed from the sample. ”Victoria Park” had an ebitda margin of -1462.67year 2007 and ”Corem Property Group” had an ebitda margin of -2085.71 year2010. These values were not altered.

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Lastly the retail sector did not suffer from as many outliers as the othersectors.

5.3 Automated stepwise procedure for model selection

Because outlier adjustment is somewhat controversial the stepwise procedureis performed for both the outlier adjusted and unadjusted data.1 The finalmodel that hold the Gauss-Markov assumptions best is favoured. If both theoutlier adjusted and unadjusted final model holds the Gauss-Markov assumptionequally well the most logically constructed is favoured.

5.4 Final models

In the following section the the final models will be presented. First the AICmodel will be presented and afterward the BIC model will be presented. Thiswill give a quick overview of which model is the best. For both the models thesame data set will be used i.e. either the outlier adjusted or the unadjusted.Both models will also be checked for model convergence when using backwardselimination versus stepwise procedure.

5.4.1 Bank model

For this model the unadjusted data was used because the adjusted data did notproduce a final model with as high power as the unadjusted.

AIC model The stepwise procedure converged with the backwards elimina-tion as expected for the bank sector. The following results was acquired.

Obs. df R2 Adjusted R2 Std. Error F Sig.

69 60 0.2903 0.1957 39.4 3.068 0.005794

Table 11: AIC Bank Model summary

Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) 72.4805 17.8648 4.06 0.0001 ***PE -1.3818 0.3264 -4.23 0.0001 *** 1.458PS -6.5543 3.7804 -1.73 0.0881 · 3.401PB -6.8275 4.1941 -1.63 0.1088 4.302Dividend per share 7.8654 2.4186 3.25 0.0019 ** 3.451Equity per share -0.6900 0.2086 -3.31 0.0016 ** 4.073Profit margin -0.7712 0.4084 -1.89 0.0638 · 2.473Current ratio -4.4827 3.1588 -1.42 0.1610 1.735debt equity ratio 1.0533 0.5452 1.93 0.0581 · 2.526

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 12: Bank AIC model

1outlier

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Breusch-pagan - AIC Bank modelχ2 df p-value7.0253 8 0.5339

Figure 10: QQ-plot for AIC Bank

The bank model shows significant improvement over the initial model. Themodel is now significant and the effect size has increased. The R2 is 0.29 whichmeans that the linear model predicts 29 % of the variance in this model. Themodel still has low VIF-values which means that there are no indications ofmulticollinearity. The χ2 is lower than the critical value (14.067 for df = 8) andthe p-value is high, thus the model does not show any sign for heteroskedasticityeither. Looking at the QQ-plot one can see that the model hold normalityreasonably well.

BIC model Backward procedure converge with AIC model, while the step-wise model yield the following results.

Obs. df R2 Adjusted R2 Std. Error F Sig.

69 64 0.2134 0.1643 40.16 4.342 0.003608

Table 13: BIC Bank Model summary

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Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) 59.6695 16.2618 3.67 0.0005 ***PE -1.2436 0.3190 -3.90 0.0002 *** 1.340PB -9.4548 3.3330 -2.84 0.0061 ** 2.615Dividend per share 5.2286 2.1027 2.49 0.0155 * 2.510Equity per share -0.5972 0.2066 -2.89 0.0052 ** 3.845

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 14: Bank BIC model

Breusch-pagan - BIC Bank modelχ2 df p-value3.9931 4 0.4069

Figure 11: QQ-plot for BIC Bank

The BIC is significant and holds the the Gauss-Markov assumptions. Itis simpler than the AIC model as expected. However it explains less of thevariance with R2 of 0.21. The model shows more significant variables. Bothindicate negative influence on stock growth by PE and positive stock growthby dividend per share. They both also indicate slight negative influence fromequity per share.

The BIC model shows that the covariate PB has a negative influence on stockgrowth. This covariate is near significant in the AIC model with corresponinginfluence.

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5.4.2 Energy model

In the energy sector the adjusted data was used due to the unadjusted modelnot fullfilling the normality assumption.

AIC model The energy AIC model converged using the two methods.

Obs. df R2 Adjusted R2 Std. Error F Sig.

60 57 0.1235 0.09277 42.62 4.016 0.02334

Table 15: AIC Energy Model summary

Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) -1.2818 5.5915 -0.23 0.8195Earnings per share 1.8348 1.1518 1.59 0.1167 1.111Capex 0.0009 0.0005 1.72 0.0909 · 1.111

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 16: Energy AIC model

Breusch-pagan - AIC Energy modelχ2 df p-value1.0201 2 0.6005

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Figure 12: QQ-plot for AIC Energy

Looking at the model summary one can see that the model is significant.However the model does not show any significant covariates.

BIC model The backward elimination converges with the AIC model, whilethe stepwise BIC procedure gives following results

Obs. df R2 Adjusted R2 Std. Error F Sig.

60 58 0.0845 0.06872 43.19 5.353 0.02425

Table 17: BIC Energy Model summary

1

Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) -2.4373 5.6172 -0.43 0.6660Capex 0.0012 0.0005 2.31 0.0242 * -

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 18: Energy BIC model

Breusch-pagan - Final Energy modelχ2 df p-value0.4997 1 0.4796

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Figure 13: QQ-plot for BIC Energy

The BIC model is even weaker than the AIC model. Nonetheless it has asignificant covariate - Capex. Although it is significant the estimator is nearzero, which tells that it does not indicate anything. Thus one can conclude thatthe covariate does not tell anything of the enegy sector.

5.4.3 Investment Company model

The unadjusted data was used due to this model having higher power.

AIC model Again the stepwise procedure converges.

Obs. df R2 Adjusted R2 Std. Error F Sig.

163 155 0.3443 0.3146 26.44 11.62 7.793e-12

Table 19: AIC Investment Model summary

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Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) -15.1188 7.5743 -2.00 0.0477 *PE 0.0252 0.0113 2.24 0.0268 * 1.749Earnings per share 0.2147 0.1371 1.57 0.1193 2.864Dividend yield -0.4886 0.3527 -1.39 0.1680 1.056ROE 0.3348 0.1078 3.11 0.0022 ** 2.799equity ratio 0.2488 0.0949 2.62 0.0096 ** 1.051PEG 1.5308 0.8228 1.86 0.0647 · 1.782Capex -0.0020 0.0013 -1.58 0.1158 1.032

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 20: Investment AIC model

Breusch-pagan - AIC Investment modelχ2 df p-value4.766 7 0.6885

Figure 14: QQ-plot for AIC Investment

The model is significant and explains 0.344 of the variations of stock growth.The VIF values for the model is within the treshold and Breush-pagan test(χcrit

2 = 14.067 for df = 7) does not indicate any heteroskedasticity. Howeverthe significant covariates in this model only have low β. The QQ-plot indicatesthat the model is slightly positively skewed.

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BIC model Here the BIC converges for backward elimination and stepwiseprocedure.

Obs. df R2 Adjusted R2 Std. Error F Sig.

163 158 0.3142 0.2968 26.78 18.09 2.97e-12

Table 21: BIC Investment Model summary

Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) -15.9182 7.5694 -2.10 0.0371 *PE 0.0276 0.0113 2.44 0.0158 * 1.717ROE 0.4677 0.0655 7.14 0.0000 *** 1.009equity ratio 0.2541 0.0953 2.67 0.0085 ** 1.033PEG 1.8534 0.8105 2.29 0.0235 * 1.686

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 22: BIC final model

Breusch-pagan - Final Investment modelχ2 df p-value0.78924 4 0.9399

Figure 15: QQ-plot for BIC Investment

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The BIC model shows higher F and almost the same R ∗ 2 (0.3142). Thereare no indications of multicollinearity or heteroskedasticity and again the modelis slightly right skewed. Both models show that PE, ROE and equity ratio issignificant with slight positive impact on stock growth. The BIC model indicatesthat PEG has positive impact while this covariate is near significant in the AICmodel.

5.4.4 Real Estate Company model

For the Real estate companies the trimmed data set was used.

AIC model As expected the AIC selection convereged using both backwardsand stepwise methods.

Obs. df R2 Adjusted R2 Std. Error F Sig.

165 156 0.2092 0.1687 46.78 5.159 1.011e-05

Table 23: AIC Real Estate Model summary

Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) 38.1489 8.2204 4.64 0.0000 ***PS -0.4935 0.2324 -2.12 0.0353 * 1.402PB -20.6847 5.8035 -3.56 0.0005 *** 1.636Dividend per share 3.5841 2.0195 1.77 0.0779 · 7.648Dividend yield -3.4337 1.9516 -1.76 0.0805 · 6.039Profit margin -0.1072 0.0325 -3.30 0.0012 ** 2.889ROE -0.5538 0.2332 -2.38 0.0188 * 4.286ROA 5.7564 1.2228 4.71 0.0000 *** 5.463PEG 6.9496 3.8226 1.82 0.0710 · 1.001

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 24: Real estate AIC model

Breusch-pagan - AIC Real estate modelχ2 df p-value13.221 8 0.1045

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Figure 16: QQ-plot for AIC Real estate

The model is significant and it holds the Gauss-Markov assumptions withright skew. However the explanatory power is low (0.21). The AIC model showstha PB has strong negative influence on stock growth, while PS, profit marginand ROE has slight negative influende on it. ROA has positive influence onstock growth.

BIC model The BIC stepwise procedure and backward elimination converged.

Obs. df R2 Adjusted R2 Std. Error F Sig.

165 162 0.1188 0.1079 48.46 10.92 3.556e-05

Table 25: BIC Real Estate Model summary

Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) 24.0570 6.8994 3.49 0.0006 ***PB -13.0075 4.7002 -2.77 0.0063 ** 1.000021ROA 2.0478 0.5419 3.78 0.0002 *** 1.000021

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 26: Real estate BIC model

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Breusch-pagan - BIC Real estate modelχ2 df p-value2.1533 2 0.3407

Figure 17: QQ-plot for BIC Real estate

The model is significant and holds the Gauss-Markov assumptions. Just likethe AIC model it has a right skew in its distribution. The explanatory poweris much less than for the AIC model (0.11) and the model is much simpler.There are only two significant variables, namely, PB and ROA. PB indicatesstrong negative influence and ROA indicates positive influence just as in theAIC model. As noted the PS, ROE and profit margin had only slight influenceand as such has been included in the BIC model.

5.4.5 Retail model

The retail sector did not require outlier adjustment as such only unadjusteddata set was regressed.

AIC model The stepwise procedures converge for this dataset.

Obs. df R2 Adjusted R2 Std. Error F Sig.

70 54 0.3696 0.2879 29.8 4.523 0.0005013

Table 27: AIC Retail Model summary

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Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) 9.7057 7.2558 1.34 0.1866PS -2.0065 0.8151 -2.46 0.0171 * 1.508Dividend per share -2.9783 1.4338 -2.08 0.0426 * 2.530Earnings per share 1.6223 0.6541 2.48 0.0163 * 2.735Revenue per share -0.1709 0.0823 -2.08 0.0426 * 2.496Dividend yield 5.4438 1.9096 2.85 0.0062 ** 1.632Profit margin 0.2751 0.1469 1.87 0.0665 · 1.469PEG 1.5461 0.7498 2.06 0.0440 * 1.047

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 28: Retail AIC model

Breusch-pagan - AIC Retail modelχ2 df p-value4.7949 7 0.685

Figure 18: QQ-plot for AIC Retail

Lastly the AIC retail sector model is also significant and and holds theGauss-Markov assumptions. The QQ-plot however shows that the model isslightly light tailed. The covariates PS and dividend per share shows negativeinfluence on stock growth, while earning per share, dividend yield and PEGshows positive influence on stock growth.

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BIC model The BIC stepwise procedure and backward elimination converged.

Obs. df R2 Adjusted R2 Std. Error F Sig.

70 56 0.285 0.2212 31.16 4.465 0.001709

Table 29: BIC Retail Model summary

Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) 6.0249 7.4284 0.81 0.4208PS -2.0224 0.8076 -2.50 0.0152 * 1.354Earnings per share 1.1716 0.5656 2.07 0.0429 * 1.870Revenue per share -0.2290 0.0831 -2.76 0.0079 ** 2.234Dividend yield 5.9474 1.8801 3.16 0.0025 ** 1.446PEG 1.7842 0.7779 2.29 0.0256 * 1.030

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 30: Retail BIC model

Breusch-pagan - BIC Retail modelχ2 df p-value4.9797 5 0.4184

Figure 19: QQ-plot for BIC Retail

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Comparing the BIC and AIC model again it is possible to see that R2 ishigher at the AIC model. The AIC model has a R2 of 0.36 while the BIChas a R2 of 0.29. Thus the retail sector is the one that is most influenced byfinancial ratios. The BIC model holds the Gauss-markov assumptions howeverthis model is right skewed instead of light tailed. In both models PS has negativeinfluence on stock growth; earnings per share, dividend yield and PEG haspositive influence on stock growth; PS has negative influence on stock growthand revenue per share has slight negative influence on stock growth. Profitmargin and dividend per share was not included in the BIC model selection.

5.5 ANOVA - Difference in between sectors

To statistically see if there was any underlying differences between the industriesgrowth an one-way ANOVA was performed with stock growth as dependent andsector as independent. The corresponding data for the final model was used inthe ANOVA.

Df Sum Sq Mean Sq F value Pr(>F)Sector 5 23932.58 4786.52 2.70 0.0204Residuals 518 919765.10 1775.61

Table 31: One-way ANOVA table

The ANOVA table tells that the difference between industries is significant,so Tukey’s Honest Significant Difference(HSD) is performed to identify whichindustries has significant differences and which do not.

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Tukey’s HSD

Sector Diff. Lower Upper P. adj. Sign.

Bank- -1.04e+01 -131.829 110.98 1.000Energy- -1.95e+01 -141.146 102.06 0.997Invest- -1.49e+01 -135.850 105.95 0.999Real- -1.08e+00 -121.979 119.81 1.000Retail- -1.49e+0.1 -136.338 106.44 0.999Energy-Bank -9.12e+00 -30.795 12.56 0.836Invest-Bank -4.52e+00 -21.833 12.79 0.976Real-Bank 9.34e+00 -7.936 26.62 0.634Retail-Bank -4.52e+00 -24.969 15.93 0.989Invest-Energy 4.59e+00 -14.076 23.26 0.981Real-Energy 1.85e+01 -0.181 37.10 0.054 ·Retail-Energy 4.59e+00 -17.016 26.20 0.990Real-Invest 1.39e+01 0.555 27.18 0.036 *Retail-Invest 9.73e-05 -17.224 17.22 1.000Retail-Real -1.39e+01 -31.059 3.33 0.193

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 32: Tukey HSD intervals

Figure 20: Tukey HSD Confidence intervals60

Tukey-HSD show that most sectors do not have any underlying difference instock growth. The only significant difference is between the sectors real estateand investment (p = 0.036). This difference is substantial indicating 13.9%higher stock growth for real estate sector than investment companies. There isalso a near significant difference between real estate and energy stock growth,which if considered is also substantial.

5.6 Regression with stock growth adjusted for economicfundamentals

The results for the regression adjusted for economic growth is presented below.These models where conceived in the same manner as the final model, with VIFselection, outlier analysis and stepwise regression. Only the stepwise regressionperformed with the most relevant criteria is presented in this section.

5.6.1 Bank model adjusted for economic growth

Obs. df R2 Adjusted R2 Std. Error F Sig.

69 60 0.4122 0.3226 30.47 3.226 0.0001281

Table 33: AIC Bank Model adjusted for Economic fundamentals summary

Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) 23.2672 16.5452 1.41 0.1649PE -1.2329 0.2478 -4.98 0.0000 *** 1.404PB -10.1687 2.6929 -3.78 0.0004 *** 2.967Dividend per share 3.1518 1.8082 1.74 0.0865 · 3.226Equity per share -0.4378 0.1768 -2.48 0.0162 * 4.893ROA -0.4188 0.2533 -1.65 0.1036 1.599debt equity ratio 1.2225 0.4346 2.81 0.0067 ** 2.685equity ratio 0.8312 0.3051 2.72 0.0085 ** 3.568PEG 1.7031 1.2878 1.32 0.1911 1.037Capex 0.0437 0.0321 1.36 0.1784 1.055

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 34: Bank AIC model adjusted for Economic fundamentals

Breusch-pagan - AIC Bank adjusted for Economic fundamentals modelχ2 df p-value18.336 9 0.03147

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Figure 21: QQ-plot for AIC Bank model adjusted for Economic fundamentals

5.6.2 Energy model adjusted for economic growth

Obs. df R2 Adjusted R2 Std. Error F Sig.

60 57 0.1549 0.1253 41.77 5.225 0.008252

Table 35: AIC Energy Model adjusted for Economic fundamentals summary

Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) -3.1438 5.4992 -0.57 0.5698Earnings per share 3.0944 1.0716 2.89 0.0055 ** 1.001PEG 2.1597 1.3854 1.56 0.1245 1.001

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 36: Energy AIC model adjusted for Economic fundamentals

Breusch-pagan - AIC Energy adjusted for Economic fundamentals modelχ2 df p-value0.71509 2 0.6994

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Figure 22: QQ-plot for AIC Energy model adjusted for Economic fundamentals

5.6.3 Investment model adjusted for economic growth

Obs. df R2 Adjusted R2 Std. Error F Sig.

163 158 0.2033 0.1831 24.16 10.08 2.729e-07

Table 37: BIC Investment Model adjusted for Economic fundamentals summary

Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) -18.9479 6.8268 -2.78 0.0062 **PE 0.0249 0.0102 2.44 0.0156 * 1.716ROE 0.2674 0.0591 4.52 0.0000 *** 1.009equity ratio 0.2293 0.0860 2.67 0.0084 ** 1.033PEG 1.8093 0.7310 2.48 0.0144 * 1.686

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 38: Investment BIC model adjusted for Economic fundamentals

Breusch-pagan - BIC Investment adjusted for Economic fundamentals modelχ2 df p-value0.29062 4 0.9904

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Figure 23: QQ-plot for BIC Investment model adjusted for Economic funda-mentals

5.6.4 Real estate model adjusted for economic growth

Obs. df R2 Adjusted R2 Std. Error F Sig.

165 161 0.1298 0.1136 43.74 8.003 5.264e-05

Table 39: BIC Real estate Model adjusted for Economic fundamentals summary

Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) 17.2456 6.2319 2.77 0.0063 **PB -9.6823 4.2543 -2.28 0.0242 * 1.006Profit margin -0.0635 0.0216 -2.95 0.0037 ** 1.452ROA 2.4401 0.5881 4.15 0.0001 *** 1.445

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 40: Real estate BIC model adjusted for Economic fundamentals

Breusch-pagan - BIC Real estate adjusted for Economic fundamentals modelχ2 df p-value2.1798 3 0.5359

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Figure 24: QQ-plot for BIC Real Estate model adjusted for Economic funda-mentals

5.6.5 Retail model adjusted for economic growth

Obs. df R2 Adjusted R2 Std. Error F Sig.

70 55 0.3091 0.2337 28.02 4.1 0.001803

Table 41: AIC Retail Model adjusted for Economic fundamentals summary

Estimate Std. Error t value Pr(>|t|) Sign. VIF

(Intercept) -14.5890 6.5335 -2.23 0.0296 *Dividend per share -2.4984 1.1243 -2.22 0.0304 * 1.759Dividend yield 4.4927 1.7781 2.53 0.0144 * 1.600ROE -0.0877 0.0616 -1.42 0.1604 4.080ROA 1.4001 0.5216 2.68 0.0096 ** 2.000debt equity ratio 1.8447 1.0731 1.72 0.0912 · 4.36PEG 0.9986 0.7039 1.42 0.1617 1.04

Sign. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’·’ 0.1 ’ ’ 1

Table 42: Retail AIC model adjusted for Economic fundamentals

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Breusch-pagan - AIC Retail adjusted for Economic fundamentals modelχ2 df p-value6.7681 6 0.3428

Figure 25: QQ-plot for AIC Retail model adjusted for Economic fundamentals

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6 Analysis

6.1 Most Prominent Key Ratio

Sector(obs) Bank(69) Energy(60) Invest(163) Real(165) Retail(70)

β AIC BIC AIC BIC AIC BIC AIC BIC AIC BIC

PE - - (+) (+)PS - · (-) - -PB - - -DPS + + + · -EPS + +RPS (-) (-)EqPS (-) (-)Div. Y - · + +Profit (-) · (-) (+)·ROE (+) (+) (-)ROA + +Cur. rat.D/E rat. + ·Eq. rat. (+) (+)Ebitda m.PEG + · + + · + +Capex (-) · (-)

’+’ pos influence, ’-’ negative influence, ’( )’ = β < 1 , ’·’ near significance.

Table 43: The most prominent key ratios for each sector

From the table above one notices that financial ratios do not tell so much abouta specific sectors performance. This is argument is further enhanced by the lowR2 that the final models exhibits. However there are some financial ratios thatmatter and some results that can be distinguished from the table. Generallyhigher price ratios seem to influence stock growth negatively. Good financialstatements per share basis and good liquidity ratios seem to influence stock pricepositively. Which is all very general and very intuitive. It is worth to mentionthat some of the covariates have high correlations which why for example ifPE is selected for the final model, an model selection procedure should excludePEG.

Notice that for all the models except for retail does AIC model include theBIC selected covariates. Also notice that the real estate companies BIC modelis much simpler than the AIC model. This is likely due some multicollinearityproblem. In this thesis a VIF cut-off value of 10 was used, which is the recom-mended maximum level of VIF. [48, 58, 73] However some scientist recommendVIF threshold at a value of 5 and even 4.[see p. ;421-431; 84, 76] In the bankingand real estate sector there are VIF values over 4; 4.3 for PB (insignificant) and4.1 for equity per share (significant) in bank sector and 7.6 for dividend pershare (near significant), 6.0 for dividend yield (near significant), 5.5 for ROA(significant) and 4.3 for ROE (significant). This means that their standard er-rors are inflated with a factor corresponding thier VIF value, which means that

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they could interfere with each other during a multiple regression.Anyhow looking at the bank sector one can see that the most prominent key

ratios are PE and dividends per share. There are some discrepancies betweenthe models of whether PB is a significant key ratios or not. The BIC modelusually gives true model for large sample as stated before, however the samplein this sector is quite small (69 observation). Which is why the AIC model willbe preferred. Thus the most prominent key ratios for bank is PE and dividendper share.

For the energy sector the covariate capex was near significant for AIC andsignificant for BIC. However in both models it has very marginal impact on thestock growth. The sample is also limited which means that AIC is favoured overBIC model, which further strengthens the argument that there is no influenceby financial ratios on the energy sector. This could be either because of lack ofdata points (60 observations) or because of other causes such as if the energysector stock growth is more influenced by oil prices. [see p. 3327-3333; 38]

In the investment sector both models indicate that PE, ROE and equity ratiois significant. However these covariates have only marginal effect on the stockgrowth. In the AIC model PEG is near significant while in the BIC model it issignificant. This section has a relatively large samle (163 observations) whichmeans that the BIC model is favoured. Thus PEG is the most prominent keyratio. Note that PE and PEG should be somehow infuencing each other(notethe VIF values of these in relation to the VIF values of the other covariates)removing PEG might makes PE not significance and vice versa.

In the real estate sector the the AIC models indicate that PS, profit mar-gin and ROE are significant. However their influence on the stock growth ismarginal. In addition dividend per share, dividend yeild and PEG is considerednear significant in AIC. Both the AIC model and BIC model agree on negativeinfluence from PB and positive influence from ROA on stock growth. The sam-ple size is large (165 observations) which indicates that BIC should be preferred.Thus the only clear prominent key ratios are PB and ROA.

Lastly The retail sector models agree on PS being a negative influence andrevenue per share being a marginal negativ influence, while earnings per share,dividend yield and PEG being positive influence for stock growth. In the AICmodel dividends per share also is a negative influence but reading interpretingboth dividend per share and dividend yield could be erronoeus due to factorsmentioned above. Thus the most prominent key ratios are PS, earnings pershare, dividend yield and PEG.

6.2 Implications of difference between industry

The analysis showed that there did not exist differences between the most sec-tors. This is in accordance with the general perception that stock returns followeconomic growth.[see p. 1429-45; 289-315; 65, 9] However Yao et al. argu-ments that economic growth is determined entirely by technological progressalong with the balanced growth path of standard growth models. They arguethat economic growth is separate from stock growth except from during periodsof high output volatility. During those periods there are a positive correlationbetween economic growth and stock growth. Nonetheless generally the stockgrowth follow the risk-free interest rate plus the cost of bearing equity i.e. theprice of time and risk.[see p. 1257-1271 66] Even though Yao disputed the gen-

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eral conception, both his view and the general conception supports the idea thatstock growth follow a pattern discern from company fundamentals.

However the real estate sector seem to outperform the other sectors. Thiscould be due to increased investment in real estate from foreign actors.[103]But it is also likely because of changes in the Swedish fundamentals whichhas increased the disposable income and as such increased the valuation ofhousing.[91, 94] Another factor that might have influenced the real estate sectorgrowth performance is the low interest rates of Sweden. [36] All of these factorssupports the notion that general stock growth follows patterns that are discernfrom financial ratios.

6.3 Regression comparison of economical growth adjustedmodels

Looking at the retail model one can see that although the R2 is higher and theQQ-plot is more normal distributed the fundamentally adjusted model indicatedheteroscedasticity.

The fundamentally adjusted energy model shows slightly higher R2 and hasa significant variable. Everything else are similar.

The fundamentally adjusted investment model shows lower R2 and howevereverything else is similar.

In the real estate fundamentally adjusted model the R ∗ 2 is slightly higherand it has an additional covariate - profit margin.

Lastly the fundamentally adjusted retail model shows slightly lower R2 andvariables are slightly changed. The non-adjusted and the adjusted models aresimilar, thus the financial ratios did not explain much of the surplus from thegeneral growth.

6.4 Analysis about general stock growth

The systematic literature study implies that the stock growth is determined byeconomic fundamentals. However the regression adjusted for economical funda-mentals showed no clear difference from the non-adjusted models as mentionedin previous sector. This means that even though the long-term stock should bevalued primarily by fundamentals they are not, which goes in line with stud-ies indicating that the stock market is not an efficient market and are highlyaffected by volatility and trends in market.[71]

A study suggested that stock returns are not sensitive to economic growthin short-term and very sensitive to economic growth in long-term.[61] Anotherstudy suggested that there existed relationship between economic growth andstock returns during economic crisis.[43] However this thesis had data set be-tween 2006 and 2013. During this period the stock market experienced botha crisis and a period of low and high economic growth. This means that thedata set distorts the results conceived by regression due to the many differentstock environment, which is why it is not clear that if this thesis validates theseexpressions or not.

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7 Discussion

7.1 Model discussion

The BIC model is consistently simpler than the AIC model. The covariates inthe BIC always indicate the same thing as in the AIC model. This of courseis logical since the AIC and BIC model are similar in thier structure. Thedifference lies which factor they pentalize. The AIC penalizes with a factor thatis almost always smaller than the BIC penalization i.e. a factor of 2 versus afactor of log(n) where n is the number of observations. Because a simple modelaccount for less factors it is also natural for the BIC model to explain less ofthe variance. In this case one could use the BIC model to get a overview ofthe most important covariates and look further in the AIC model to see if thereis anything of interest. In most cases the BIC has removed covaraites withvery small β values which would not indicate any significant impact anyhow.Nonetheless in the retail sector the BIC selection did not include ”dividend pershare” covariates which has negative impact on stock growth but a not so smallβ. Since the R2 for most models is relatively small the linear models accountonly for small amount of the variance in the model. This does however not meanthat the models are useless. As Birnbaum (2006) preposterously expressed it:

The ballpark is ten miles away, but a friend gives you a ride for thefirst five miles. You’re halfway there, right? Nope, you’re actuallyonly one quarter of the way there.

[12]The non-linear correlation of R2 to real world impact however means that

the R2 underestimates the practical significance of the model.[see p. 289-291;54] This means that even though the R2 is statistically correct it may be flawedfor explaining substance in a intuitive manner in real life applications. Otherresearchers agree.[see p. 216; ;306-308; 85, 95, 57] It is important to rememberthat the R2 is the variance of residuals explained and not the residuals explained,and as such even though the R2 is small the models could indicate significanteffect on stock growth.

7.2 Critisim of stepwise

As mentioned in the theory section the stepwise procedures have had a lot ofcriticism and the drawbacks are well documented in statistical literature, yetit is common practice to use this method. Many of these criticism comes fromthe usage of mathematical stepwise procedure in behavioural and humanitariansciences. In these sciences soft variables influence a lot more than in financeand as such require.[see p.1182-1189 98] As such the of stepwise regression ismore validated this thesis, however one cannot forecome all the shortcomings.So they are important to consider nonetheless.

7.3 Investor consideration and what financial ratios canindicate

The analysis above hints that stock growth follow sector specific macroeconomictrends better than financial ratios. However it is clear from the analys that in

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the short-term stock returns are volatile due to investors disparity. Investorsmust thus always consider specific macroeconomic trends when investing, butalso know the risks of volatility. Note that there might be positive relationshipbetween financial ratios and macroeconomic trends, since positive sector trendsbenefit company fundamentals, and vice versa.

Investors should always first know the implications of macroeconomic eventsbefore analysing the financial ratios. However suppose that an investor knowsthis, then the financial ratios can serve as diagnostics tools for evaluating whichstock that will perform best. This study indicates that certain key ratios tellmore about a specific sector than others. Knowing this will help the investorput more weight on the significant financial ratios found in this thesis than theothers.

Besides this financial ratios can also tell how a specific firm is doing. Astudy conducted by Hagberg suggests that a company default can be predictedusing financial ratios. He suggests that the capital structure significantly differsup to a period of 5 year prior to default, profitability ratios differs up to aperiod of 4 years before default and liquidity ratios differs up to a period of 3years prior to default. The difference become more pronounced the nearer thedefault the companies come. The capital structure is shown by the financialratios; debt-quity ratio, equity ratio. For companies that are going to defaultthe equity is lower and the debts are higher, and are increasing with time. Theprofitability ratios will be significantly worse and so will the liquidity ratios. Forexample the current ratio for companies that do not default is over 1.3 and hasa mean value of 1.96. The companies that are going to default however havesignificantly lower current ratios. [47] Thus the financial ratios are a vital toolfor assessing specific firms and their well-being even if they do little to predictthe stock growth.

Simply by knowing that marcoeconomic fundamentals are more importantand considering the value of time and that unsystematic risk is not compensatedfor makes index investing the most lucrative for the vast majority of investors.However many investors still choose to invest in stocks and by many investorslose relative to index. This is due to many investors tend to chase positivetrends in return, at the same time as they over-react to negative returns, whichresults in liquidation of funds in those positions.[see p. 28-29 41] This phe-nomenon however can be explained by the utility functions. These investors areexperiencing positive utility from being a part of the market and selecting andchoosing specific stocks, which is why investors still engage in stock selectioneven though in most cases it is best to choose index funds.

Lastly investors that are risk-loving might use short-term volatility effectsto his or her advantage.

7.4 Explorative opportunities and future research areas

The insight gained from this thesis has opened a few suggestions for futureexplorative opportunities.

Looking at interaction effects of financial ratios - this thesis did notconsider the possibility of interaction effects between the financial ratios andhow these would influence the stock growth in specific sectors, such analysis canfor example be performed with two-way ANOVA. This could be a future study

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because interaction between certain financial ratios might indicate somethingthat an individual financial ratios does not.

Another area of potential future research lies in examining the financialratios influence on stock growth that is adjusted for general growthin specific sectors. By adjusting the stock growth to remove the influence ofmacroeconomic effects or at least mitigating stock growth caused by these asmuch as possible one can have a clearer look at how the financial ratios influencestock growth in specific sector. This thesis evaluated the subject by adjustingwith OMXS30 as indicator for general economic growth, however it would beinteresting to see results for adjustment by sector specific growth (which thelimited data set of this thesis did not allow).

Also one could analys during specific market environments, such as onlyhigh economic growth, low economic growth och crisis. By doing this it couldbe easier to identify what which event fundamentals affect stock return moreand which event other factors influence more.

A thought for future studies is also to try to use other regression methodsto see if they yield better results.

8 Conclusion

This thesis has shown that financial ratios do affect stock growth. It has showedthat the most prominent key ratios varies in the different sectors.

The bank sector has an R2 of 0.29. and it has negative influence from PE,and positive from dividends per share.

The energy sector was the only sector that no financial ratios indicatedanything, which suggests that the growth of this sector is mostly affected byfundamental growth. This claim is backed by other scientists.[38]

The investment company sector has an R2 of 0.31. It is positively influencedby PEG.

The real estate sector has an R2 of 0.12 is negatively influenced by PB andpositively influenced by ROA.

The retail sector has an R2 of 0.37 is negatively influenced by PS and div-idend per share and positively influenced by earnings per share, dividend yieldand PEG. Is is thus the sector that is most influenced by financial ratios.

In this thesis the notion of R2 was also discussed. Even though the R2 waslow. It could be of practical significance. This claim has scientific support-ing.[see p. 289-291; 54] The influence on stock growth by financial ratios aremarginal compared to the influence on stock growth by general macroeconomictrends in the long term. This is illustrated by the real estate sector over per-forming, due to macroeconomic effects, compared to the other sectors. Theothers however showed no difference in stock growth depending on sector. Thisis one of the reason for why the R2 is so low in the models that was conceived.

The systematic literature review indicated that fundamentals should be theprimary drivers of stock return in an efficient market. However studies hasshown that the stock market is not efficient. Instead stock growth is highlyaffected by volatility and disparity between investors. A regression analysisadjusted for economic growth with OMXS30 showed that indeed compensatingfor general economic growth did show little difference between the models, and

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as such fundamentals did little to display any surplus stock growth beyond thegeneral economic growth.

By showing that stock growth is mostly influenced by macroeconomic effectindirectly and suggesting that many investors are biased by the utility function,it also shows that one of the most profitable investment strategy for many privateinvestors in the stock market is by investing in index funds in the long-term. Italso suggest that short-term positions are largely affected by volatility.

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9 Acknowledgement

9.1 Patrick Truong

I would like to take this acknowledgement to give my gratitude to all my fam-ily and friends for their support. Special gratitudes for my girlfriend MariePetrusson for cooking food and being genuinely supportive for me while I wasfranatically writting this thesis during the semester. I want to thank MehdiLahlou and Filip Falkenborn for the support and insight in how to go about inthis thesis, recommendations on how to use the statistical tools and for winningthe Swedish championship. I also want to thank my thesis partner San San Mafor staying up late during the hours and being a real asian workhorse. Last butnot least i would also like to show my appreciations for Boualem Djehiche for be-ing very helpful and last minute appointments on a very timely and professionalmanner.

9.2 San San Ma

A special thanks to Patrick Truong for holding out every night and workingalong with me with this thesis. A big thanks and happy birthday to Yusi whosebirthday party I missed this year due to schoolwork and for understandingthat. Thanks to Bjorn, Alexander and Axel for entertaining me and giving mesuggestions through Skype. Thanks to my parents Rong and Yun for givng mesuggestions and supporting me throughout this thesis.

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10 Appendix

10.1 Appendix for initial model

10.1.1 Bank Sector - Initial model - Summary and Box plots

Stock growth Stock value PE PS PB1 Min. :-83.33 Min. : 0.47 Min. :-95.90 Min. : 0.000 Min. :0.5002 1st Qu.:-15.24 1st Qu.: 24.00 1st Qu.: 8.99 1st Qu.: 2.380 1st Qu.:1.3303 Median : 5.57 Median : 77.50 Median : 11.55 Median : 3.170 Median :2.3704 Mean : 11.00 Mean : 87.95 Mean : 11.15 Mean : 3.702 Mean :3.1595 3rd Qu.: 35.61 3rd Qu.:131.00 3rd Qu.: 15.70 3rd Qu.: 4.620 3rd Qu.:4.4706 Max. :159.78 Max. :294.40 Max. : 60.51 Max. :14.180 Max. :9.890

Dividend per share Earnings per share Revenue per share Equity per share1 Min. : 0.000 Min. :-10.660 Min. : 0.00 Min. : 0.192 1st Qu.: 1.500 1st Qu.: 1.250 1st Qu.: 6.80 1st Qu.: 7.323 Median : 4.000 Median : 6.160 Median :21.03 Median : 32.304 Mean : 4.524 Mean : 7.107 Mean :25.15 Mean : 47.875 3rd Qu.: 7.170 3rd Qu.: 10.190 3rd Qu.:41.83 3rd Qu.: 79.356 Max. :16.500 Max. : 24.840 Max. :58.12 Max. :175.17

Dividend yield Profit margin ROE ROA Current ratio1 Min. : 0.000 Min. :-53.67 Min. :-133.28 Min. :-130.0000 Min. : 0.00002 1st Qu.: 3.070 1st Qu.: 15.24 1st Qu.: 12.03 1st Qu.: 0.5000 1st Qu.: 0.00003 Median : 4.350 Median : 27.65 Median : 16.49 Median : 0.6700 Median : 0.00004 Mean : 4.243 Mean : 23.66 Mean : 14.49 Mean : 0.9598 Mean : 0.72725 3rd Qu.: 5.120 3rd Qu.: 35.55 3rd Qu.: 24.17 3rd Qu.: 2.0500 3rd Qu.: 0.81006 Max. :16.810 Max. : 57.17 Max. : 48.16 Max. : 28.7800 Max. :14.9000

debt equity ratio equity ratio ebitda margin PEG Capex1 Min. : 0.07 Min. : 0.07 Min. :-36.36 Min. :-14.3000 Min. :-811.6202 1st Qu.: 1.99 1st Qu.: 3.97 1st Qu.: 25.77 1st Qu.: -0.5500 1st Qu.: -0.0103 Median :20.58 Median : 4.47 Median : 38.78 Median : 0.1500 Median : 0.9604 Mean :17.77 Mean :16.44 Mean : 35.33 Mean : -0.2575 Mean : 6.1735 3rd Qu.:24.05 3rd Qu.:32.44 3rd Qu.: 49.51 3rd Qu.: 0.6100 3rd Qu.: 26.9306 Max. :70.33 Max. :83.32 Max. : 57.93 Max. : 6.2000 Max. : 211.750

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10.1.2 Energy Sector - Initial model - Summary and Box plots

Stock growth Stock value PE PS PB1 Min. : -87.211 Min. : 0.90 Min. :-219.18 Min. : 0.00 Min. : 0.4202 1st Qu.: -31.928 1st Qu.: 8.75 1st Qu.: -11.81 1st Qu.: 1.16 1st Qu.: 1.0603 Median : -4.803 Median : 17.70 Median : -1.75 Median : 2.67 Median : 1.7304 Mean : 222.485 Mean : 31.39 Mean : 62.85 Mean : 48.17 Mean : 3.8755 3rd Qu.: 21.088 3rd Qu.: 43.47 3rd Qu.: 27.82 3rd Qu.: 6.16 3rd Qu.: 3.4156 Max. :14455.556 Max. :155.40 Max. :2806.12 Max. :1980.28 Max. :99.7807 NA’s :2 NA’s :2 NA’s :2

Dividend per share Earnings per share Revenue per share Equity per share1 Min. : 0.0000 Min. :-27.4600 Min. : 0.000 Min. : 0.072 1st Qu.: 0.0000 1st Qu.: -1.1350 1st Qu.: 0.185 1st Qu.: 7.143 Median : 0.0000 Median : -0.1100 Median : 4.830 Median :12.444 Mean : 0.3698 Mean : -0.4202 Mean : 6.946 Mean :16.735 3rd Qu.: 0.0000 3rd Qu.: 1.6550 3rd Qu.:10.280 3rd Qu.:26.896 Max. :15.5500 Max. : 11.0900 Max. :28.910 Max. :41.217 NA’s :2 NA’s :2 NA’s :2 NA’s :2

Dividend yield Profit margin ROE ROA Current ratio1 Min. : 0.00 Min. :-13582.50 Min. :-244.06 Min. :-84.480 Min. : 0.1202 1st Qu.: 0.00 1st Qu.: -16.35 1st Qu.: -23.52 1st Qu.:-14.265 1st Qu.: 0.7753 Median : 0.00 Median : 0.00 Median : -2.36 Median : -0.850 Median : 1.9104 Mean : 1.19 Mean : -282.67 Mean : -12.18 Mean : -7.042 Mean : 6.0135 3rd Qu.: 0.00 3rd Qu.: 13.09 3rd Qu.: 4.73 3rd Qu.: 2.875 3rd Qu.: 4.0606 Max. :26.33 Max. : 723.56 Max. : 49.96 Max. : 22.850 Max. :114.2107 NA’s :2 NA’s :2 NA’s :2 NA’s :2 NA’s :2

debt equity ratio equity ratio ebitda margin PEG Capex1 Min. : 0.010 Min. : 2.40 Min. :-11837.50 Min. :-28.6600 Min. :-8824.312 1st Qu.: 0.160 1st Qu.:51.06 1st Qu.: 0.00 1st Qu.: -0.1200 1st Qu.: 0.003 Median : 0.670 Median :59.75 Median : 24.83 Median : 0.0000 Median : 39.414 Mean : 1.269 Mean :65.66 Mean : -217.71 Mean : -0.6556 Mean : 1263.165 3rd Qu.: 0.960 3rd Qu.:86.30 3rd Qu.: 48.39 3rd Qu.: 0.0000 3rd Qu.: 144.856 Max. :40.700 Max. :99.26 Max. : 69.30 Max. : 4.7900 Max. :83061.007 NA’s :2 NA’s :2 NA’s :2 NA’s :2 NA’s :2

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10.1.3 Investment Company - Initial model - Summary and Boxplots

Stock growth Stock value PE PS PB1 Min. :-78.379 Min. : 0.8197 Min. :-945.12 Min. : 0.000 Min. : 0.30452 1st Qu.:-16.024 1st Qu.: 18.9000 1st Qu.: -2.22 1st Qu.: 0.000 1st Qu.: 0.68883 Median : 5.747 Median : 70.2500 Median : 3.25 Median : 0.000 Median : 0.82264 Mean : 4.416 Mean : 73.1453 Mean : 12.81 Mean : 15.346 Mean : 1.08985 3rd Qu.: 23.274 3rd Qu.:111.5290 3rd Qu.: 9.71 3rd Qu.: 1.165 3rd Qu.: 0.99726 Max. :151.796 Max. :257.8395 Max. :2913.64 Max. :1167.884 Max. :16.9950

Dividend per share Earnings per share Revenue per share Equity per share1 Min. : 0.000 Min. :-97.940 Min. : 0.00 Min. : 0.88982 1st Qu.: 0.000 1st Qu.: -0.465 1st Qu.: 0.00 1st Qu.: 26.63793 Median : 1.750 Median : 2.660 Median : 0.00 Median : 65.66004 Mean : 2.483 Mean : 7.066 Mean : 24.93 Mean : 83.06565 3rd Qu.: 4.500 3rd Qu.: 15.810 3rd Qu.: 31.32 3rd Qu.:121.68006 Max. :16.350 Max. : 78.850 Max. :209.34 Max. :320.5900

Dividend yield Profit margin ROE ROA Current ratio1 Min. : 0.000 Min. :-8503.450 Min. :-224.447 Min. :-170.068 Min. : 0.0002 1st Qu.: 0.000 1st Qu.: 0.000 1st Qu.: -3.037 1st Qu.: -2.404 1st Qu.: 1.0303 Median : 2.672 Median : 0.000 Median : 6.250 Median : 3.790 Median : 1.7204 Mean : 3.764 Mean : -119.703 Mean : 1.765 Mean : 1.522 Mean : 19.5475 3rd Qu.: 4.800 3rd Qu.: 4.595 3rd Qu.: 18.310 3rd Qu.: 13.685 3rd Qu.: 6.4616 Max. :50.590 Max. : 549.510 Max. : 57.515 Max. : 57.229 Max. :443.530

debt equity ratio equity ratio ebitda margin PEG Capex1 Min. : 0.001229 Min. : 7.749 Min. :-6548.28 Min. :-29.0000 Min. :-14711.182 1st Qu.: 0.048950 1st Qu.:65.362 1st Qu.: 0.00 1st Qu.: -0.0900 1st Qu.: -80.623 Median : 0.190000 Median :83.492 Median : 0.00 Median : 0.0000 Median : 0.004 Mean : 0.504070 Mean :76.542 Mean : -57.07 Mean : -0.1594 Mean : -69.695 3rd Qu.: 0.532179 3rd Qu.:95.390 3rd Qu.: 10.95 3rd Qu.: 0.0000 3rd Qu.: 97.736 Max. :11.905020 Max. :99.877 Max. : 57.84 Max. : 25.7200 Max. : 7870.00

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10.1.4 Real Estate Company - Initial model - Summary and Boxplots

Stock growth Stock value PE PS PB1 Min. :-89.65 Min. : 0.1134 Min. :-1973.440 Min. : 0.000 Min. : 0.27022 1st Qu.:-13.09 1st Qu.: 14.4243 1st Qu.: 2.780 1st Qu.: 2.380 1st Qu.: 0.78923 Median : 12.19 Median : 39.9880 Median : 6.610 Median : 4.089 Median : 1.02004 Mean : 16.10 Mean : 42.8694 Mean : -6.647 Mean : 7.088 Mean : 1.24595 3rd Qu.: 31.83 3rd Qu.: 65.0000 3rd Qu.: 10.460 3rd Qu.: 6.130 3rd Qu.: 1.21006 Max. :324.82 Max. :118.7500 Max. : 84.670 Max. :225.750 Max. :12.7767

Dividend per share Earnings per share Revenue per share Equity per share1 Min. : 0.000 Min. :-14.336 Min. : 0.000 Min. : -0.0452 1st Qu.: 0.000 1st Qu.: 0.290 1st Qu.: 5.339 1st Qu.: 15.0373 Median : 1.000 Median : 4.260 Median : 8.156 Median : 43.2904 Mean : 1.915 Mean : 4.532 Mean : 11.912 Mean : 64.1765 3rd Qu.: 2.450 3rd Qu.: 8.070 3rd Qu.: 14.880 3rd Qu.: 62.1006 Max. :59.000 Max. : 30.150 Max. :165.076 Max. :4031.000

Dividend yield Profit margin ROE ROA Current ratio1 Min. : 0.000 Min. :-2085.710 Min. :-608.333 Min. :-985.000 Min. : 0.0093662 1st Qu.: 0.000 1st Qu.: 6.841 1st Qu.: 2.332 1st Qu.: 0.870 1st Qu.: 0.1800003 Median : 2.392 Median : 44.820 Median : 11.080 Median : 4.020 Median : 0.4800004 Mean : 3.201 Mean : 53.137 Mean : 4.645 Mean : -3.353 Mean : 1.5101085 3rd Qu.: 4.527 3rd Qu.: 90.713 3rd Qu.: 17.670 3rd Qu.: 7.260 3rd Qu.: 1.3399116 Max. :44.240 Max. : 1465.450 Max. : 243.210 Max. : 81.522 Max. :30.310000

debt equity ratio equity ratio ebitda margin PEG Capex1 Min. :-1.247 Min. :-405.00 Min. :-2085.71 Min. :-9.8000 Min. :-2081.982 1st Qu.: 0.830 1st Qu.: 29.52 1st Qu.: 25.10 1st Qu.:-0.2000 1st Qu.: 0.003 Median : 1.820 Median : 35.42 Median : 54.20 Median : 0.0000 Median : 94.224 Mean : 1.841 Mean : 38.29 Mean : 22.71 Mean :-0.1917 Mean : 169.305 3rd Qu.: 2.373 3rd Qu.: 52.66 3rd Qu.: 64.17 3rd Qu.: 0.0300 3rd Qu.: 249.706 Max. : 9.498 Max. : 98.24 Max. : 120.91 Max. : 1.4000 Max. : 5760.93

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10.1.5 Retail - Initial model - Summary and Box plots

Stock growth Stock value PE PS PB1 Min. :-62.953 Min. : 5.75 Min. :-74.240 Min. : 0.200 Min. :-81.7602 1st Qu.:-21.210 1st Qu.: 23.73 1st Qu.: 8.107 1st Qu.: 0.360 1st Qu.: 1.3023 Median : -1.817 Median : 66.95 Median : 15.015 Median : 1.025 Median : 2.7804 Mean : 4.365 Mean :130.91 Mean : 14.484 Mean : 2.939 Mean : 2.7745 3rd Qu.: 27.153 3rd Qu.:227.43 3rd Qu.: 18.795 3rd Qu.: 2.638 3rd Qu.: 5.0006 Max. :100.000 Max. :471.00 Max. :153.500 Max. :30.140 Max. : 49.300

Dividend per share Earnings per share Revenue per share Equity per share1 Min. : 0.000 Min. :-25.090 Min. : 1.77 Min. :-10.1102 1st Qu.: 0.000 1st Qu.: 0.160 1st Qu.: 21.17 1st Qu.: 7.1053 Median : 1.950 Median : 2.500 Median : 49.94 Median : 15.9004 Mean : 3.456 Mean : 5.069 Mean : 77.89 Mean : 34.4065 3rd Qu.: 5.875 3rd Qu.: 8.258 3rd Qu.:137.40 3rd Qu.: 53.2826 Max. :14.000 Max. : 49.680 Max. :386.58 Max. :157.3107 NA’s :8 NA’s :8 NA’s :8 NA’s :8

Dividend yield Profit margin ROE ROA Current ratio1 Min. :0.000 Min. :-86.1500 Min. :-613.4900 Min. :-20.940 Min. : 0.58002 1st Qu.:0.000 1st Qu.: 0.6975 1st Qu.: 0.9175 1st Qu.: 1.032 1st Qu.: 0.95753 Median :3.220 Median : 3.9750 Median : 9.4700 Median : 6.390 Median : 1.76004 Mean :2.880 Mean : 5.1436 Mean : 7.2450 Mean : 6.137 Mean : 3.04345 3rd Qu.:4.855 3rd Qu.: 11.6125 3rd Qu.: 31.9950 3rd Qu.: 13.960 3rd Qu.: 3.13256 Max. :8.520 Max. :159.7000 Max. : 348.3900 Max. : 21.440 Max. :14.5900

debt equity ratio equity ratio ebitda margin PEG Capex1 Min. :-31.5800 Min. :-14.27 Min. :-71.440 Min. :-11.4800 Min. : -848.132 1st Qu.: 0.2225 1st Qu.: 22.00 1st Qu.: 5.075 1st Qu.: -0.2200 1st Qu.: 3.713 Median : 1.3750 Median : 38.63 Median : 9.805 Median : 0.0000 Median : 32.724 Mean : 1.6926 Mean : 44.17 Mean : 11.796 Mean : 0.9397 Mean : 404.405 3rd Qu.: 2.9125 3rd Qu.: 68.97 3rd Qu.: 18.117 3rd Qu.: 0.9375 3rd Qu.: 60.726 Max. : 21.7500 Max. : 95.19 Max. :136.060 Max. : 31.0600 Max. :26015.63

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10.2 Most successful stocks

The following are the most successful stocks in growth order:

Taurus Energy 3611.048159Fast. Balder 863.2247914Melker Schorling 346.6910998Tethys Oil 257.4777309Wallenstam 216.1592506Intrum Justitia 176.3768116Fast Partner 162.262233Klovern 146.5277778Latour 127.6672878Kinnevik B 122.3300971ICA Gruppen 117.3399015Atrium Ljungberg 111.6241393Swedish Match 98.75555556Dios Fastigheter 94.33345441Investor B 73.95189003Bure Equity 70.76499244Hufvudstaden A 65.60344828Axfood 65.46898638Avanza Bank 64.28571429Heba 63.99768474Castellum 56.36363636Lundbergforetagen 53.86934673Handelsbanken A 53.22352941Traction 51.74096117Oresund 43.02315187

These are the same stocks in alphebetical order. *indicates consistent growthand are those not to be altered in the outlier management.Atrium Ljungberg*, Avanza Bank, Axfood*Bure EquityCastellumDios FastigheterFast Partner*, Fast. Balder*Handelsbanken A*, Heba, Hufvudstaden A*ICA Gruppen, Intrum Justitia*, Investor BKinnevik B, KlovernLatour*, LundbergforetagenMelker Schorling*Swedish MatchTaurus Energy, Tethys Oil*, Traction*Wallenstam*Oresund

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10.2.1 Summary and boxplot for these stocks

Stock growth Stock value PE PS PB1 Min. :-41.428 Min. : 9.454 Min. :-29.260 Min. : 0.000 Min. :0.32692 1st Qu.: -1.962 1st Qu.: 44.645 1st Qu.: 3.562 1st Qu.: 1.419 1st Qu.:0.86963 Median : 11.455 Median : 66.805 Median : 8.210 Median : 3.127 Median :1.12584 Mean : 17.877 Mean : 78.590 Mean : 6.450 Mean : 5.008 Mean :1.65815 3rd Qu.: 34.541 3rd Qu.: 90.900 3rd Qu.: 12.953 3rd Qu.: 5.008 3rd Qu.:2.07756 Max. :166.983 Max. :294.400 Max. : 46.880 Max. :121.970 Max. :6.1300

Dividend per share Earnings per share Revenue per share Equity per share1 Min. : 0.000 Min. :-69.400 Min. : 0.000 Min. : 5.872 1st Qu.: 1.060 1st Qu.: 3.765 1st Qu.: 6.725 1st Qu.: 29.983 Median : 2.000 Median : 6.930 Median : 11.924 Median : 52.524 Mean : 2.737 Mean : 9.085 Mean : 31.536 Mean : 64.305 3rd Qu.: 3.000 3rd Qu.: 11.075 3rd Qu.: 42.838 3rd Qu.: 82.436 Max. :16.500 Max. : 78.850 Max. :183.370 Max. :296.10

Dividend yield Profit margin ROE ROA Current ratio1 Min. : 0.000 Min. :-61.29 Min. :-139.979 Min. :-103.3570 Min. : 0.00002 1st Qu.: 1.042 1st Qu.: 2.55 1st Qu.: 9.423 1st Qu.: 0.9125 1st Qu.: 0.13253 Median : 3.349 Median : 33.45 Median : 14.529 Median : 6.3350 Median : 0.46004 Mean : 3.216 Mean : 50.91 Mean : 12.647 Mean : 4.9899 Mean : 2.25695 3rd Qu.: 4.577 3rd Qu.: 67.05 3rd Qu.: 22.694 3rd Qu.: 10.4529 3rd Qu.: 1.03256 Max. :18.370 Max. :723.56 Max. : 57.084 Max. : 46.7272 Max. :60.6667

debt equity ratio equity ratio ebitda margin PEG Capex1 Min. : 0.007547 Min. : 3.47 Min. :-46.04 Min. :-14.3000 Min. :-2081.982 1st Qu.: 0.762500 1st Qu.:33.27 1st Qu.: 23.81 1st Qu.: -0.2100 1st Qu.: 31.753 Median : 1.515000 Median :39.73 Median : 52.88 Median : 0.0000 Median : 93.164 Mean : 3.372292 Mean :46.41 Mean : 41.99 Mean : 0.3861 Mean : 96.485 3rd Qu.: 2.007488 3rd Qu.:56.63 3rd Qu.: 62.18 3rd Qu.: 0.3325 3rd Qu.: 179.566 Max. :27.800000 Max. :95.43 Max. : 68.65 Max. : 31.0600 Max. : 1215.25

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10.3 Correlation matrix from the initial model

10.3.1 Bank correlation matrix

10.3.2 Energy correlation matrix

10.3.3 Investment correlation matrix

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10.3.4 Real Estate correlation matrix

10.3.5 Retail correlation matrix

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TRITA -MAT-K 2015:02

ISRN -KTH/MAT/K--15/02--SE

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