Sustainable Investment Strategies: A Quantitative Evaluation of Sustainable Investment Strategies For Index Funds

Master Thesis, 30 ECTS

Master of Science in Industrial Engineering and Management Department of Mathematics and Mathematical Statistics

Spring 2019

SUSTAINABLE INVESTMENT STRATEGIES

A QUANTITATIVE EVALUATION OF SUSTAINABLE INVESTMENT STRATEGIES

FOR INDEX FUNDS

John Erikmats Johan Sjösten

Abstract

Modern society is faced with the complex and intractable challenge of global warming, along with other environmental issues that could poten- tially alter our way of life if not managed properly. Is it possible that financial markets and equity investors could have a huge part to play in the transformation towards a greener and more sustainable world?

Previous studies about investment strategies regarding sustainabil- ity have for the most part been centered around possibly less objective ESG-scores or around carbon and GHG-emissions only, with little or no consideration for water usage and waste management. This thesis aims to amend to the previous work on carbon reducing strategies and ESG-investing with the addition of water usage and waste management, specifically using raw data of these measures instead of ESG-ratings.

Index replicating portfolios have become more and more popular as it proves harder and harder to beat index, offering good returns along with cheap and uncomplicated portfolio construction and management.

In a trending market, the fear of missing out and the demand for market return can make an index replicating strategy a way for investors to have market exposure but still remain diversified and without confusion about which horses to bet on.

This thesis studies the relationship between tracking-error and the

increase of sustainability in a portfolio through reduction of the intensity

of carbon emissions, water usages and poor waste management. To be

able to make a fair comparison, these measures are normalized by divid-

ing each measure by the reported annual revenue. These three obtained

intensities are then implemented individually, as well as all together into

index replicating portfolios in order to study the effect from decreasing

them. First and foremost we study the effect on the tracking-error, but

also the effects on returns and volatility. We also study the effect on

liquidity and turnover in the portfolios to show that it is possible to im-

plement extensive sustainability increasing methods into an index replica-

tion equity portfolio. We follow the UCITS-directory to avoid overweight

in specific companies and only allow the portfolios to overweight a sector

with maximum 2%, in order to avoid an unwanted exposure to sectors

with naturally lower intensities. The portfolios are obtained by using a

multi-factor risk model to predict the expected statistical behaviour in

relation to the chosen factors. Followed by applying Markowitz Modern

Portfolio Theory through a convex optimization problem with the ob-

jective function to minimize tracking-error. All displayed portfolios had stable and convex optimization and were compliant with the UCITS- directory. We limited our study to only North American stocks and chose the index “MCSI NA” to replicate. Only stocks that were a part of the index were allowed to invest in and we did not allow negative weights for any stocks. The portfolios were constructed and backtested for the period 2014-12-01 until 2019-03-01 with rebalancing quarterly at the same points in time that the index is rebalanced by MCSI.

We found that it was possible to implement extensive sustainability considerations into the portfolios and still keep a high correlation with the index whilst keeping low tracking-errors. We believe that most index replicating investors should be able to implement reductions of above mentioned intensities of about 40-60% without compromising tracking- errors, returns and volatility too much. We found evidence that during this time and in this market our low-intensities portfolios would have over performed the index. We also found that returns increase and volatility decreases both as we increase the reduction of each individual measure and all three collectively. Reducing carbon intensity seemed to drive positive returns and lower volatility the most, but we also observed a positive effect from reduction of all intensities.

Our belief before conducting this study was that sustainability should

have a negative effect on returns due to the limitation of the feasible area

of investing. This motivated us to build portfolios with intent to make

up for these lesser returns and hopefully ”beat the index”. This failed in

almost all cases and the only way we were able to beat the index were

through implementing sustainability in our portfolios.

Sammanfattning

Det moderna samh¨ allet m˚ aste hitta praktiska l¨ osningar p˚ a det sv˚ arl¨ osliga problemet med global uppv¨ armning och alla andra milj¨ orelaterade prob- lem som potentiellt kan st¨ ora och hota v˚ ar m¨ anskiliga existens. Ar ¨ det m¨ ojligt att finansiella marknader och aktieinvesterare kan ha en avg¨ orande roll att spela i omst¨ allningen mot en gr¨ onare och mer h˚ allbar v¨ arld?

Tidigare studier p˚ a omr˚ adet investeringsstrategier f¨ or h˚ allbarhet har f¨ or det mesta varit centrerade runt m¨ ojligen mindre objektiva ESG-scores och runt koldioxid och v¨ axthusgaser, d¨ ar lite eller ingen h¨ ansyn har tag- its f¨ or vattenf¨ orbrukning och avfallshantering. Vi vill med denna upp- sats bidra och utvidga det tidigare arbetet med ESG-investeringar och koldioxidreducerande investeringsstrategier genom att ¨ aven ta h¨ ansyn till vattenf¨ orbrukning och avfallshantering. Vi g¨ or detta genom att anv¨ anda r˚ adata ist¨ allet f¨ or ESG-scores.

Indexreplikerande portf¨ oljer har blivit alltmer popul¨ ara eftersom det visar sig allt sv˚ arare att sl˚ a index. En indexportf¨ olj kan erbjuda god avkastning kombinerat med billig och okomplicerad portf¨ oljbyggnad och portf¨ oljf¨ orvaltning. I en marknad med tydlig trend kan ”fear of miss- ing out” och behovet av marknadsexponering g¨ ora en indexreplikerande strategi till ett bra alternativ, framf¨ orallt f¨ or investare som vill ¨ oka sin diversifiering och slippa fundera ¨ over vilka aktier som kommer bli vinnare och f¨ orlorare p˚ a b¨ orsen.

Denna uppsats studerar f¨ orh˚ allandet mellan aktiv risk (tracking-error) och att ¨ oka h˚ allbarheten i en portf¨ olj genom att minska intensiteten av koldioxidutsl¨ app, vattenanv¨ andande och bristf¨ allig avfallshantering.

F¨ or att kunna g¨ ora en r¨ attvis j¨ amf¨ orelse delas ovann¨ amnda m˚ att med

den rapporterade ˚ arliga oms¨ attningen. De tre intensitetsm˚ atten imple-

menterades sedan b˚ ade var f¨ or sig och alla samtidigt i olika indexrep-

likerande portf¨ oljer f¨ or att kunna studera effekten av att minska inten-

sitetsm˚ atten p˚ a f¨ orst och fr¨ amst p˚ a den aktiva risken, men ocks˚ a f¨ or

avkastning och volatilitet. Vi studerar ocks˚ a effekten p˚ a likviditet och

oms¨ attningen i portf¨ oljerna f¨ or att kunna visa att det ¨ ar praktiskt m¨ ojligt

att implementera en stor portion h˚ allbarhet i en indexreplikerande ak-

tieportf¨ olj. Vi f¨ oljer UCITS-reglerna f¨ or fonder f¨ or att undvika ¨ overvik-

tning av enskilda aktier och till˚ ater bara portf¨ oljerna att ¨ overvikta en

sektor med max 2 % j¨ amf¨ ort med index. Detta g¨ ors f¨ or att undvika en

o¨ onskad exponering mot sektorer som av naturliga sk¨ al har generellt l¨ agre

intensitetsm˚ att. Vi konstruerar portf¨ oljerna genom att f¨ orst anv¨ anda en multifaktorriskmodell f¨ or att f¨ ors¨ oka prediktera statistiska beteenden f¨ or alla tillg˚ angar och applicerar sedan Markowitz Moderna Portf¨ oljteori genom konvex optimering d¨ ar m˚ alfunktionen ¨ ar att minimera aktiv risk.

Alla portf¨ oljer som vi visar upp hade stabila och konvexa optimeringar som uppfyllde UCITS-reglerna f¨ or fonder. Vi begr¨ ansade studien till bara nordamerikanska aktier och valde att replikera indexet ”MSCI NA” som

¨

ar MSCIs Nordamerikanska index. Portf¨ oljerna till¨ ats bara att investera i aktier som ocks˚ a var en del av indexet och vi till¨ at inte negativa vikter f¨ or n˚ agra aktier i portf¨ oljen. Portf¨ oljerna konstruerades och backtestades f¨ or perioden 2014-12-01 till 2019-03-01 med kvartalsvis rebalansering vid samma tidpunkt som indexet rebalanseras av MCSI.

Vi fann att det ¨ ar fullt m¨ ojligt att implementera utbredd h˚ allbarhet i portf¨ oljerna men samtidigt bibeh˚ alla en stark korrelation med indexet och l˚ ag aktiv risk. Vi tror att de flesta indexreplikerande investerarna borde kunna implementera totala reduceringar runt 40-60% utan att av- sev¨ art f¨ ors¨ amra sin aktiva risk, avkastning eller volatilitet. Vi fann ocks˚ a att v˚ ara portf¨ oljer med l˚ aga intensiteter under denna tid givet den valda marknaden hade ¨ overpresterat gentemot index och dessutom ¨ okat avkast- ningen. Vidare minskade volatiliteten n¨ ar vi ¨ okade reduktionen av alla tre intensitetsm˚ att enskilt och tillsammans. Att reducera koldioxid framst˚ as driva positiv avkastning mest och minska volatiliteten mest, men vi s˚ ag ocks˚ a en positiv effekt av att reducera samtliga intensitetsm˚ att.

P˚ a f¨ orhand trodde vi att implementeringen av h˚ allbarhet i portf¨ oljerna

skulle medf¨ ora s¨ amre avkastning. Mest p˚ a grund av att vi krymper

investingsuniverset men kanske p˚ a grund av en smula pessimism. Vi

f¨ ors¨ okte d¨ arf¨ or bygga portf¨ oljer med m˚ alet att generera b¨ attre avkast-

ning och kompensera f¨ or vad vi trodde skulle bli s¨ amre avkastning. Detta

misslyckades rej¨ alt i n¨ astan alla fall och det enda viset vi lyckades sl˚ a in-

dexet p˚ a var genom implementeringen av bara h˚ allbarhet.

Acknowledgements

There are many people and organizations to thank for all the help and inspiration for this thesis, and on that note, we have a great deal to be grateful and thankful for.

First and foremost we would like thank Oscar Blomquist and Nils Everling at AP4, without them there simply would not be any thesis.

They have been extraordinarily helpful and supportive. At time and time again they have gone out of their way to assure that the thesis work was progressing as expected. Their help and supportive attitude have been absolutely crucial, hence, our immense gratefulness. We would also like to extend our deepest gratitude towards AP4 as an organization, who have provided us with both the practical resources necessary and the inspiration to further investigate the subject of sustainable invest- ing. They are a role model for other funds and should continue to be considered one of the very best in the business.

Secondly we would like to thank Trucost for providing the data neces- sary to perform our analysis and tests. Furthermore for the opportunity to explore different ways to use their superior data sets, in order to create more sustainable and impacting equity portfolios. Their contribution to this thesis, have also been crucial, since we would not be able to perform the analysis about water usage and waste management without their data, at least not in such an effortless and credible way.

We would of course also like to address a significant ”thank you”

to associate professor Lisa Hed, our supervisor at the Department of Mathematics and Mathematical Statistics. Her help and guidance in important decisions of the thesis has been immensely valuable, along with her help with structuring the thesis.

We want to thank Ume˚ a University and the Department of Math- ematics and Mathematical Statistics for giving us a competitive foun- dation, specifically one of a mathematical and statistical approach to finance.

Lastly we would like to thank our classmates for these wonderful five years that comes to an end with this thesis. You have been a wonderful support and a true inspiration for us, we will truly miss you and we wish you all the best in life.

Johan Sj¨ osten

John Erikmats

Stockholm, 29th May, 2019

Nomenclature

bp Basis Point

ESG Environmental, Social and Governance

Go long In this thesis going long means to over-weight a stock or sector compared to the index.

Go short In this thesis going short means to under-weight a stock or sector compared to the index.

UCITS Undertakings for Collective Investments in Transfer- able Securities

GICS The Global Industry Classification Standard AP4 The Fourth Swedish National Pension Fund MPT Markowitz Modern Portfolio Theory

FLAM The Fundamental Law of Active Management Tracking Error The standard deviation of the difference between a

portfolio’s return and the return of a benchmark Bullish market A market that is trending upwards, with higher and

higher prices of related assets

Bearish market A market that is trending downwards, with lower and lower prices of related assets

Risk Potential downfall in the portfolio, often defined as the standard deviation of portfolio return

Return The money made or lost on an investment defined as the current value divided by the previous value Revenue The total income for a company’s normal business

activites

α Alpha - The excess return compared to the market (or benchmark used).

β Beta - A coefficient that signals the volatility, or sys- tematic/market risk intrinsic to the market, of an in- dividual asset in contrast to the idiosyncratic risk

CO

Carbon Dioxide

Contents

Abstract . . . . i

Sammanfattning . . . . iii

Acknowledgements . . . . v

Nomenclature . . . . vi

1 Introduction . . . . 1

1.1 Background . . . . 1

1.2 Purpose . . . . 3

1.3 The Fourth Swedish National Pension Fund(AP4) . . . . 3

1.4 Investment Universe & Data . . . . 4

1.5 Equity Funds . . . . 4

1.6 Index Funds . . . . 5

1.7 Previous studies related to sustainable finance . . . . 6

1.8 Sustainability . . . . 8

1.8.1 ESG . . . . 8

1.8.2 Carbon Reducing Strategies . . . . 9

1.8.3 Our Measures of Sustainability . . . . 9

1.9 Outline . . . . 11

2 Theory . . . . 12

2.1 Portfolio Considerations . . . . 12

2.1.1 UCITS Directory . . . . 13

2.1.2 GICS Sectors . . . . 14

2.2 Portfolio Theory . . . . 14

2.2.1 Markowitz Modern Portfolio Theory . . . . 15

2.3 Portfolio Optimization . . . . 16

2.3.1 Sharpe Ratio . . . . 16

2.3.2 Tracking Error . . . . 17

2.3.3 Minimize Tracking Error . . . . 17

2.3.4 Convex Optimization Theory . . . . 18

2.4 Portfolio Measures . . . . 19

2.4.1 Correlation . . . . 20

2.4.2 Beta . . . . 20

2.4.3 Alpha . . . . 21

2.4.4 Multi-Factor Risk Model . . . . 21

2.4.5 Price to Earnings Ratio (P/E Ratio) . . . . 22

2.5 Portfolio Management . . . . 23

2.5.1 Volume . . . . 23

2.5.2 Trading Cost . . . . 24

2.5.3 Rebalancing . . . . 24

2.5.4 Turnover . . . . 25

2.6 Portfolio Evaluation . . . . 25

2.6.1 Returns . . . . 25

2.6.2 Risk . . . . 26

2.6.3 Fundamental Law of Active Management . . . . . 26

2.6.4 Sustainability . . . . 27

3 Method . . . . 28

3.1 General Algorithm . . . . 28

3.2 Data Collection . . . . 29

3.2.1 Index Data . . . . 29

3.2.2 Sustainability Data . . . . 29

3.2.3 Carbon Intensity . . . . 29

3.2.4 Water Intensity . . . . 30

3.2.5 Waste Intensity . . . . 30

3.2.6 Market Data . . . . 31

3.2.7 Fundamental Data . . . . 31

3.3 Data Management . . . . 31

3.3.1 Filling NaN’s . . . . 31

3.3.2 Winsorizing . . . . 32

3.3.3 Managing Negative P/E-Ratios . . . . 33

3.3.4 Non-disclosed information . . . . 33

3.4 Portfolio Construction . . . . 33

3.4.1 MSCI NA Replication . . . . 34

3.4.2 Long Only . . . . 34

3.4.3 Sector Neutrality . . . . 34

3.4.4 Prediction Using a Multi-Factor Risk Model . . . 35

3.4.5 Adjusting Returns for trading costs . . . . 35

3.5 Portfolio Definitions . . . . 35

3.5.1 Portfolio 1 (P1) . . . . 36

3.5.2 Portfolio 2 (P2) . . . . 37

3.5.3 Portfolio 3 (P3) . . . . 38

3.5.4 Portfolio 4 (P4) . . . . 39

3.5.5 Portfolio 5 (P5) . . . . 41

3.5.6 Portfolio 6 (P6) . . . . 43

3.5.7 Portfolio 7 (P7) . . . . 46

3.5.8 Portfolio 8 (P8) . . . . 49

3.6 Validation . . . . 51

3.6.1 Backtesting . . . . 51

3.6.2 Benchmark . . . . 51

3.6.3 Stability . . . . 51

3.6.4 UCITS Compliance . . . . 52

3.6.5 Convexity . . . . 52

4 Results . . . . 53

4.1 Clusters in Sustainability Data . . . . 53

4.1.1 Scatter plots of Intensities . . . . 54

4.1.2 Scatter plot of Market Capitalization & Carbon Intensity . . . . 56

4.2 Index . . . . 57

4.3 Portfolio 1 (P1) . . . . 59

4.4 Portfolio 2 (P2) . . . . 61

4.5 Portfolio 3 (P3) . . . . 63

4.6 Portfolio 4 (P4) . . . . 65

4.7 Portfolio 5 (P5) . . . . 67

4.8 Portfolio 6 (P6) . . . . 69

4.9 Portfolio 7a (P7a) . . . . 72

4.10 Portfolio 7b (P7b) . . . . 74

4.11 Portfolio 8 (P8) . . . . 75

5 Discussion and Conclusions . . . . 78

5.1 Summary of Observed Results . . . . 78

5.2 Potential Reason For Observed Results . . . . 79

5.3 Conclusion of Results . . . . 80

5.4 Real World Implementation . . . . 81

5.5 Shortcomings . . . . 82

6 Further Studies . . . . 84

6.1 Extension . . . . 84

6.2 Further Research of Sustainable Finance . . . . 84

References . . . . 86

Appendices . . . . 90

1 Introduction

1.1 Background

Global warming and climate change are more present in both media and in everyday life now than ever before. On a global level we are experiencing a more unstable climate, with heat waves, wildfires, floods and devastating storms occurring worldwide[1]. In the last couple of years the western hemisphere have received its fair share of catastrophes including Hurricane Katrina, Hurricane Sandy, the wildfires in California and heat waves in Europe. It ought to be clear to all investors in Europe and North America that these are not issues of far-distant countries, and that global warming is not a myth nor an exaggeration.

Global warming is largely fueled by human activities and mainly by past or ongoing emissions. The greenhouse gas that contributes the most to global warming and which emissions are easily connected to human activities is CO

, also named carbon dioxide. Reducing or limiting such a gas is crucial if the trend of global warming is to be turned around.

This have been the main focus in the debate about environmental issues the last decade [1].

The warmer climate naturally has other implications that are less obvious than catastrophes, and perhaps the most important one is the weather’s impact on water supply. The warmer climate reduces the water supply for farming, drinking, plants and wildlife, hence the importance of sustainable water usage [1].

Furthermore, the climate threat is not only about carbon emissions, both global warming and human consumption in general poses great threats to other important functions in society. Apart from oxygen, wa- ter is by far the most important utility for humans, and access to clean drinkable water is absolutely crucial for survival. According to the World Wildlife Fund (WWF), 1.1 billion people worldwide are lacking the access to clean water, and that an additional 2.7 billion have a scarce access to water for a minimum of one month per year [2]. As scary as those num- bers may sound, Mekonnen and Hoekstra finds evidence that 0.5 billion people live under constant water scarcity, but that 4.0 billion people are experiencing severe water scarcity for at least one month of the year [3].

According to WWF, Agriculture is currently using approximately 70%

of the accessible fresh water worldwide, but the wastage is substantial

[2]. Other industries are also using fresh water which could have been

used for drinking or for agriculture, making sustainable and responsible

water usage an actual matter of life-and-death.

Sadly the negative impact caused by human activities on the envi- ronment, human life and wildlife are not limited to water scarcity and natural disasters. The oceans are affected in disastrous way by poor waste management around the world. Giant island-like hoardings of rub- bish and plastics are gathering in oceans around the world and their negative impact is increasing [4]. Directly they affect their surroundings by interfering with the wildlife of the oceans, causing death and crippling for living creatures [4]. The rubbish ruins beautiful beaches and hot spots for tourism, surfing and alters the conditions for people who might have inhabited an island for centuries. Indirectly fishing and the access to food along with the livelihood it provides are also affected. Since fishing nets account for a large part of the biggest garbage islands (yes, it has a name) “The Great Pacific Garbage Patch”, it is very important that the indirect and direct effects from big companies on waste management is monitored and quantified [4].

It is not easy to make an impact and help the situation as an aver- age person, except reducing ones personal footprint and in the process contribute with consumer power. What one can do however is to make sure that ones pension and savings are used in away that contributes to a more sustainable world. Exactly how big of an impact a mutual fund (or in this case an equity fund) actually has is debated, but according to Nordea ones choice of pension fund could have as high of an impact as 27 times the impact of other consumer related adjustments [5]. It should at least be safe to say that the manner one chooses to invest ones sav- ings and pensions has an impact, even though the size of the impact is hard to prove. Large institutional investors worldwide have already been looking at sustainable investments for some time now. The Institutional Investors Group on Climate Change (IIGCC) is a forum created with the aim to have a positive impact on the climate and to put pressure on cor- porations, policymakers and other investors. IIGCC’s goal is to change market signals to encourage corporations to become more sustainable and make sustainable investments more accessible. Their members represent over €23 trillion in assets, which only consist of big investors committed to sustainable investments today. Therefore, sustainable equity invest- ments cannot simply lack impact [6].

So if you are a large investor, such as a large pension fund and are com-

mitted to sustainable investments, how do you make sure that you have

the impact you desire but still reach your goals regarding risk-adjusted

returns and tracking-error? This thesis will quantitatively evaluate in- vestment strategies, in order to identify how your fund can have the best possible climate impact whilst maintaining acceptable returns and tracking-errors.

1.2 Purpose

The purpose of this thesis is to study investment strategies for an index fund manager with the objective to track an index as closely as possible, but with the implementation of sustainability constraints in the portfolio.

This approach signifies that the main objective of this thesis is to study the relationship between tracking-errors and constraints. More specifi- cally when reducing certain intensities related to sustainability that will be explained later.

The goal of this thesis work is to observe if there exists a superior investment strategy that has a greater positive impact on sustainability with consideration for tracking-errors in first hand, but also for returns and volatility. Our hope is that our findings can show how big of an impact index fund managers can have without interfering excessively with their other objectives. How big that impact is of course depending on individual investors willingness to deviate from the index.

Since actual returns, volatility and other measures of the portfolio is studied, the results will hopefully also be applicable to investors who care less about following an index and more about the general performance of their portfolio.

1.3 The Fourth Swedish National Pension Fund(AP4)

The Fourth Swedish National Pension Fund (AP4) is one out of six pen- sion funds run as a government agency. As of June 30, 2018, AP4 had a total fund of SEK 367 billion. With a responsibility of managing a part of the Swedish pensioners future capital (national pension system’s buffer capital) AP4 responsibly commits using a long-term- and active investing perspective with a strong commitment to sustainability and ESG focus.

AP4 self-evidentially strives to yield high returns at a low cost in order

to secure expected pensions. The fund strongly believes a sustainability

perspective is crucial and a prerequisite in order to maintain long-term

success. On account of it AP4 has acted as somewhat of a role model

at the front line of responsible investing [7]. According to the first half

of 2018, AP4’s portfolio consisted of approximately: 40% global equities,

21% global interest rates, 16% Swedish equities, 12% Swedish interest rates and 10% real assets [7].

1.4 Investment Universe & Data

This thesis is limited to study only North American equities, exclusively equities because of the nature of the project received from AP4, but also to have large and complete data sets available. Furthermore, this limitation was also done in order to be able to perform a credible analysis and stable optimizations, as more constraints are added. Because of the intention of performing a fair comparison to the benchmark index, the model is constrained to only invest in equities that are a part of the index

“MCSI NA”.

The environmental data was delivered from Trucost and the thesis will focus on more “raw” environmental data instead of ESG-scores (more on ESG in Section 1.8.1), since the “raw” data rarely lie and some ESG- scores are based on gradings set by actual persons in the different insti- tutions and therefore might lack objectivity. It is also convenient to work with raw data to be able to construct our own definitions for sustainabil- ity, and to have the ability of analyzing Carbon Intensity, Water Usage and Waste Management as individual factors.

Historical data for the estimation of parameters was extracted from the AP4 database. For the purpose of being able to benchmark against the index ”MSCI NA” without any discrepancies, all market data in terms of returns during the backtesting periods were retrieved from MSCI.

A more in-depth description and analysis of the data sources and the col- lection of data is supplied in the section called ”Data Collection” (Section 3.2).

1.5 Equity Funds

Equity funds are mutual funds or exchange-traded funds that collect capital from several, or few investors and invests solely in equities, such as publicly traded common stock to generate returns. The benefits of Equity Funds are in general the lower trading costs by economies of scale.

Thus because of its size it is easier to diversify the investments (this can sometimes be a problem for very large funds too). Case in point, if a retail investor were to invest their savings in stocks, they would probably not be able to follow their investment strategy, or have a big investment universe, hence, they have to buy at least 1 unit of stock, not 0.01.

On the contrary, the equity fund manages a greater deal of equity and

can borderline invest seamlessly according to a specific strategy with a considerable investment universe [8].

In general there are two major types of equity funds, actively managed funds that try to beat a benchmark such as an index and index funds that seek to track a benchmark or index by replicating its statistical properties. The portfolios in this thesis is something in between those two, since we in the study in general is trying to track an index, but occasionally with constraints that intend to create excess returns. Thus managing it rather actively in comparison to than an index portfolio [8], [9].

This thesis is limited to investing in equities only, but since most mutual funds and pension funds invest in stocks this could be seen a report specifically for an equity division at a large fund like AP4.

1.6 Index Funds

This thesis will as mentioned above focus mainly on tracking an index, hence being an index fund, but why be or invest in an index fund instead of manage actively and hopefully reach better returns?

A pure index fund simply buys all of the stocks in the chosen index according to their specific weight in the index. Stocks receive weights according to their corresponding company’s market capitalization (their valuation), giving the stocks of highly valued companies bigger weights and smaller to smaller companies [8].

By doing this a fund manager does no stock-picking or weighting, since the index have already decided how the portfolio should be built.

This makes the stock-picking process very cheap, since no or very little research is needed in order to make investment decisions. On the opposite an actively managed fund either has to do a lot of research and know a lot about the company or develop clever quantitative trading algorithms to create better returns, both of which are quite expensive.

An active portfolio manager will in most cases have to rebalance their portfolio by buying more of some stocks and selling some in order to follow a quantitative strategy, or if you are investing fundamentally you might want to buy a stock on some great news or sell on some bad.

This rebalancing process leads to transaction costs which adds to the

total costs of keeping an active portfolio. The index replicating manager

only has to make changes in the portfolio when a stock enters or leaves the index, since any change in value of the stocks in the index will be replicated by owning the same stocks with the same weighting in the index replicating portfolio [9], [10].

Both these factors forces the active portfolio manager to deliver at least as much return as the research and transactions costs in order to break-even, and in a lot of cases they do not.

Since you never deviate from the index you will never carry a big risk of a single asset tanking or underperforming, nor do you rely too heavily on a specific sector, hence lowering the sector risk and if you track a global index you do even carry less country specific risk. It should be said that there is a debate whether or not you should invest to scale or if you should invest equally across countries, sectors and even single assets in order to reduce risk the most, but the general consensus is that an index replication portfolio is a great way to diversify your portfolio [10].

So if you are a big pension fund like AP4 or another big investor with the objective of investing long-term and do not want to take on too much risk, keeping an index replicating portfolio is cost-efficient and also provides good diversification.

1.7 Previous studies related to sustainable finance

Previous studies have to a large degree been focused around investment strategies build around ESG-scores (read about ESG and ESG-scores in Section 1.8.1 below) or solely around minimizing or reducing carbon emissions. The most recent master theses written by students from Ume˚ a University we read before our own study where “Greenhouse Gas Foot- print Minimization of Credit Default Swap Baskets” by Britse & Jarnmo (2018) [11] and “ESG Investing In Nordic Countries: An analysis of the Shareholder view of creating value” by Dahlberg & Wiklund (2018) [12].

The first is a study aimed to investigate different credit default swap (CDS) baskets with a goal function of minimizing greenhouse gas foot- print given an acceptable level of deviation from a CDS-index. This study is similar to ours but implemented on CDS’s and without any constraints for water usage and waste management [11].

The second study investigates if the market performance of Nordic

companies is enhanced by receiving a better ESG-score, and finds that

better rated ESG-companies sometimes receive higher valuations. This

study is not aimed to investigate stock performance or return on invest- ment for an investor, but investigates the effect on valuation by ESG- scores [12].

In the study “Integrating ESG in Portfolio Construction” by Henriks- son et al. (2019) [13] ESG-scores is again in focus. The authors try to expand the concept of ESG-classification by ranking them according to their material usage instead of reported scores, which is in a way what we did, but again centered around ESG-scores and ESG-rating instead of using raw data. The authors find some positive results in terms of returns from investing according to material usage [13].

In “ESG Preference and Market Efficiency: Evidence from Mispricing and Institutional Trading” by Cao et al. (2019) [14] the authors evaluate the effect on ESG on returns for public U.S firms between 2003 and 2013.

The authors find that good ESG-companies seemed to be overvalued and generated less excess returns during the period, and suggests that the preference of large institutional investors for good ESG-companies causes this overvaluation. This is one of the reasons for why we choose to deviate from investing according to ESG-scores [14].

In the study “The Effect of Socially Responsible Investing on Portfo- lio Performance” by Kempf & Osthoff (2007) [15], the effect of investing in socially responsible companies are examined. The authors find that investing in companies with the absolute best social ranking yields the biggest alphas (abnormal returns). Although deviating as far as these authors do from an index is not an option for us, their results hints that there is value to be found in sustainable and responsible companies [15].

The study that we found most alike our project was “Tracking error minimization under varying sustainability criterion stringency: environ- mental ratings and U.S. stock portfolios” by Olsson (2010) [16], where the author uses a undisclosed risk-model to generate ratings regarding environmental values for companies and then constructs index replicating portfolios like ours. The author finds that good environmental portfolios with low tracking-errors can be found and that those companies have a small downside and bigger upside in terms of returns. Although we suspect again that the used environmental scores are similar to the En- vironmental score in a composite ESG-score, and that the study is not focused on raw sustainability data [16].

These previous studies show differing results regarding the relation-

ship between sustainable investments and different performance mea- sures. This is one of the reasons for why we find this topic interesting and important to study in our thesis project.

1.8 Sustainability

Sustainability is a hot topic, and an important topic for us the authors. A lot of investors agree that sustainability within finance and investments is important. However, sustainability is difficult to define and quantify in a fair way and in a fashion that everyone can agree upon. This is besides blatant denial of climate change one of the greatest critiques towards sustainable finance and sustainable investments, since it could tilt investments in an unfair direction and in a possibly less sustainable direction.

1.8.1 ESG

One of the main definitions that tries to define sustainability from dif- ferent perspectives is ESG. ESG is short for Environmental, Social and Governance, and is a way to describe how “good” a company is in terms of sustainability.

The Environmental score (ENV) evaluate the company’s impact on the environment in terms of emissions, pollution and waste, use of re- sources and innovation. The Social score (SOC) evaluate the social re- sponsibility of the company and this includes their level of human capi- tal awareness, the company’s compliance with human rights, their work within their community, stakeholders and their product responsibility.

The Governance score (GOV) evaluate how the company is governed and accounts for corporate governance, corporate behavior and Corpo- rate Social Responsibility [17], [18].

The different areas (Environmental, Social and Governance) are of- ten evaluated and graded with a score in terms of how the company is performing, and altogether these three scores are evaluated as what is called an ESG-score. Refinitiv (previously Thomson Reuters) does also provides an “ESG controversies score” that evaluates how many and how serious scandals a company is involved in and together with the ESG- scores they provide an “ESG Combined Score”.

Previous work with sustainable finance and sustainable investments

has as stated earlier often been carried out with maximizing ESG-scores

as the objective, and there is a lot of talk about “ESG-investing” circu- lating in the financial sector.

1.8.2 Carbon Reducing Strategies

Another approach is to invest according to how much carbon emissions a company causes. There has been done work with and without normal- izing for company size, revenue, market capitalization etc. In general we feel like not normalizing would lead to unfair biases against big compa- nies and give the portfolio a tilt towards smaller companies that is not justifiable. As we try to stress in the background there are other areas that relate to sustainable finance that should be considered in investment decisions, and we believe that only looking at carbon and GHG-emissions is a narrow perspective now that additional data is available.

1.8.3 Our Measures of Sustainability

In order to broaden the existing work on Carbon reducing strategies and ESG investments, this thesis will add water usage and waste management to the constraints of some the portfolios. Since ESG data is voluntary disclosed and in some cases graded by the institute and therefore exposed to personal biases and model biases this thesis will instead look at the raw data. ESG investing is the primary goal of this thesis, but without ESG ratings from institutes, instead we will use raw data and a com- bination between definitions of sustainability from Trucost and our own definitions. The primary goal of this thesis is to evaluate the effect on the portfolio by reducing carbon intensity, water intensity, and waste inten- sity. On order to capture the exposure to non-renewable energy sources we use the statistical factor exposure to oil prices as an experiment and to hopefully add a new perspective on sustainability in equity portfolios.

Carbon Intensity To compute the emissions part of the sustainability

we use carbon emission intensity, calculated as total carbon emissions in

tons over annual dollar revenue in millions. The scope for emission is

called “Direct and first tier indirect emissions” and are very similar to

the GHG protocol definition of scope 1 and scope 2 emissions (learn about

Scope 1-3 in Section 2.6.4) but with the amendment of direct emissions

that can easily be tied to the companies’ operations. The companies first

tier upstream supply chain are also amended, which corresponds to a

part of the definition of scope 3 emissions, although not the whole scope

3 to avoid an unfair double counting of emissions.

Water Intensity The measure for water usage is similar to carbon intensity the companies’ direct and indirect usage of water in tons divided by annual dollar revenue in millions. Both the companies’ own direct usage of water is included, as well as water that is being purchased.

Waste Intensity To quantify waste management and not punish com- panies that are good at recycling we are using the costs in million dollars that arise for external parties due to the operations of the companies. In this fashion we believe that we can capture true sustainable behavior by the companies. These costs are divided by the annual dollar revenue in millions as for the previous measures.

Exposure to oil IPCC, short for “The Intergovernmental Panel on Climate Change”, is as the name suggest an intergovernmental organi- zation for cooperation in regards to climate research and assembles the brightest and most eminent climate researchers on this planet [19]. IPCC releases a big climate report every 8th year, but recently released an ex- tra report as a way to demonstrate the impact of a global increase of temperature of 1.5 degrees Celsius as agreed upon as a goal in the Paris Agreement [1], [20].

The report aims to predict the impact of an increase of 1.5 degrees Celsius on everything from wildlife to corporations but also includes a section where increases and decreases in certain commodities are pre- dicted, in order to meet the “1.5 goal”. This describes how much the usage of mainly coal, oil and gas would have to decrease in order for the world to meet the goal. That information is very interesting in the eyes of an equity investor since one can hope that politicians and policy makers will at least try to achieve this goal by actions as passing legisla- tion, increase taxes and other policies that effect corporations and their earnings.

We built an equity portfolio that would, or at least could, benefit from these actions by trying to find corporations that would benefit from cheaper oil. We do this by increasing the negative statistical exposure to oil for one of the portfolios. The statistical negative exposure we are increasing is the arithmetic sum of our investment weights times the individual assets risk-coefficient against oil prices from the multi-risk factor model described in Section 2.4.4.

This measure has a distinct difference to the previous ones since it

has no direct relation to the companies operations and actual work with

sustainability and is not normalized by the size of the revenue. It is purely a historical statistical measure of how the companies share price has been exposed to the price of oil. This is hopefully a way for us to quantify how sustainable each specific business model is, and how they will hold up in the future if a big change in politics is coming.

1.9 Outline

This thesis will try to have as chronological structure as possible, in order to make the thesis friendly for readers that are neither students nor professionals of finance and financial markets.

Section 2 goes further into more statistical and mathematical areas, and describe the mathematics of the portfolio optimization problems that we performed.

In Section 3 the applied methodology is disclosed, to give the reader a better view of how the actual work and modelling have been carried out.

This section also provides a motivation and specification for all portfolios we built. We are also giving each portfolio a number to refer to and we provide a background of why a strategy would work or be interesting to evaluate and what our expectations were about their performance.

Section 4 contains our findings and results, a concise and objective description on how our portfolios acted and performed during the back- testing period.

In Section 5 we go through the results and what they implicate for implementing a more sustainable equity strategy and which strategy we consider the most practically doable.

Section 6 contains our recommendations for further studies in the yet

quite uninvestigated field of sustainable finance and ESG-investments,

both for studies comparable to ours, but also for sustainable finance in

general.

2 Theory

This section of the report presents the theoretical mathematical founda- tion of which this thesis is based upon. It intends to improve understand- ing of our methods and results by describing the theoretical background regarding the optimization of portfolios, various key indicators and mod- els. Beyond the glossary chapter, certain key terms might be described directly in this section, in order to enhance the legibility.

2.1 Portfolio Considerations

In order to make our findings applicable for investment decisions in the real world there are certain rules and practical assumptions that need to be considered to make our findings more credible.

First we need to comply with the most used rules and regulations for mutual funds in the EU, the UCITS directory. Not all funds have to or choose to follow the UCITS-directory but since most do, we want to make our findings as widely applicable as possible. The UCITS-directory also restricts the investment weight in a single asset of the fund, which decreases the concentration risk of having too much weight in some assets.

This is very desirable for a portfolio and it also provides more a stable optimization.

Secondly we will talk a lot about sectors, since many business are very different and have different challenges and conditions regarding sus- tainability it is important to be able to define and tell companies apart from each other. We order the companies according to sectors and use the GICS-definitions of sectors, since it is widely used and considered to be good enough to define a company’s operations. We will then use these sectors to keep the weighting between sectors in our portfolios close to the weighting of the index we are trying to replicate. This is to be able to say that our portfolios are actually more sustainable, and not just tilted towards sectors with naturally low intensities, like the financial sector for example.

It is also important to keep sectors within a reasonable weighting since

overweighting some sectors leads to higher sector specific risk. Sometimes

you want higher sector specific risk when you have an investment idea

or a focus, but for the general case and for an index replication portfolio

this is unwanted.

2.1.1 UCITS Directory

UCITS is an abbreviation for “Undertakings for the Collective Invest- ment in Transferable Securities” which is a regulatory framework for the management and sale of mutual funds. The UCITS directory is there- fore the main European directory housing collective investment schemes.

Funds regulated by UCITS account for 75% of all investments by small investors in Europe. Therefore, the perception by the general public is that UCITS funds are perceived to be secure investments. The history of UCITS span back to the adoption of the first UCITS Directive, on December 20th, 1985. Since then modifications and additions have been made. The most recent change known as UCITS V took place in March 2016 [21]. The UCITS rules are often used as constraints in an optimiza- tion. Definition 2.1 below states 9 of the rules set up for UCITS funds [22].

Definition 2.1. The UCITS rules contain, but are not restricted to:

1. UCITS can invest in an absolute minimum of 16 assets: 4 holdings of up to 10% each plus 12 holdings of up to 5% each.

2. A UCITS fund may invest no more than 5% of its value in approved securities or money market instruments issued by any one body.

This limit can be increased to 10% provided that the total value of any holdings between 5% and 10% does not exceed 40% of the fund.

3. No more than 20% of the fund as deposits with any one bank 4. No more than 20% of the fund invested in any one other fund 5. Up to 35% of the fund in any one bond issue provided the rest of

the fund is invested in other types of assets; or a minimum of six issues if the fund is over 35% invested in Government bonds.

6. No more than 10% exposed in derivatives with another bank as counterpart

7. Hold no more than 20% of the voting shares of a company

8. Hold no more than 10% of the bonds issued by a company

9. Hold no more than 20% of the value of another fund

2.1.2 GICS Sectors

In short the GICS (The Global Industry Classification Standard) is a sector classification system, developed in 1999. MSCI and SP Dow Jones Indices announced GICS to provide an efficient tool for investors, in or- der to caption the breadth, depth and evolution of industry sectors. In 1999, GICS had 10 sectors which later became 11, subsequent to the sep- aration of the ”Real Estate” sector and the ”Financials” sector. Today GICS consists of 11 sectors, which are:

• Energy

• Materials

• Industrials

• Consumer Discretionary

• Communication Services

• Utilities

• Consumer Staples

• Health Care

• Financials

• Information Technology

• Real Estate

GICS consists of a four-tiered, hierarchical classification system. How- ever, we used the GICS sectors when constructing our portfolios and formulating constraints, in order to ensure that no sector receives a dis- proportional allocation weight. In the decision of a recipients principal business activity, the MSCI and SP 500 uses revenues as a key indicator.

Nonetheless, earnings and market perception are also used as indicators for classification intents [23].

2.2 Portfolio Theory

Even though a portfolio’s behavior in terms of volatility and returns often is talked about as if the portfolio were a single entity, it is a product of the behavior of the multiple underlying assets that builds up the portfolio.

Since all assets co-exist on the market and are dependent of each other

we can’t simply add each assets expected return and volatility scaled by

the investment weight and expect our portfolio to behave according to

this. The correlation between all assets and the effect of weighting them

gives the portfolios actual expected behavior. This is described and de-

fined by Harry Markowitz and this subsection aims to give the necessary

conditions to understand how the different measures of a portfolio are

computed.

2.2.1 Markowitz Modern Portfolio Theory

In 1952 the economist Harry Markowitz introduced Modern Portfolio Theory, also known as “MPT” or “mean-variance analysis”. MPT is a theory describing how one can construct portfolios to optimize expected return based on a certain level of market risk. MPT allude to the view that the characteristics of an investment’s risk and return should not be judged alone, but rather how an investment influence the entire portfolio, in terms of both risk and return. The MPT theory is based on the theory that a rational person would not invest in a portfolio, if another portfolio exists that generate a better expected return given the same level of risk.

Hence, this theory supports that diversification can generate the better expected portfolio return, with the same level of risk. The MPT model uses variance as a measure of risk [24].

Expected Return E(R

) is in simple terms the return a portfolio is expected to generate. Given that R

is the return on the portfolio p, R

is the return on asset i and w

is the weighting of asset i (that is, the proportion of asset i in the portfolio), then the expected return of the portfolio is defined as:

E(R

) = X

i

w

E(R

) (2.1)

Portfolio Return Variance σ

measures how far the portfolio is from the mean value. Given that σ is the (sample) standard deviation of the periodic returns on an asset, and p

is the correlation coefficient between the returns on assets i and j. Then with respect to this, the portfolio return variance is defined as:

σ

= X

i

w

σ

+ X

i

X

j6=i

w

w

σ

σ

p

(2.2)

Portfolio Return Volatility σ

is measured by using the stan-

dard deviation or variance between returns from index. High volatility is

usually considered high risk. Given that we know the variance σ

from

Equation 2.2, σ

is simply the square root of the variance defined as:

σ

= q

σ

(2.3)

2.3 Portfolio Optimization

In order to know what to do with the information about the possible choices of portfolios described in the previous sub-section, we need to clarify what an investor could prefer and what is desirable properties for a portfolio. In general you can say that there is two ways to optimize your portfolio given your preferences.

The first way is that you want to maximize your return given the risk you are willing to take, ranging from the portfolio with the least amount of risk possible to the one with the highest expected return but with a larger amount of risk. If this is your preference you are picking the portfolio with the best risk-adjusted return. This can be quantified with the Sharpe-ratio, which is described in the upcoming sub-section.

The other way is that you want to replicate and track an index, for reasons that are laid out in Section 1.6 about index funds. If this is your preference you want to choose your portfolio so that you expect the least amount of tracking-error, the standard deviation of the difference in returns between the portfolio and the index. How one can compute both these kinds of portfolios will be described in this sub-section.

2.3.1 Sharpe Ratio

When building a portfolio, and especially with equities, most investors must account for the associated risk of the portfolio, usually measured as the portfolio volatility, denoted σ

and defined previously in Equation 2.3.

More volatile portfolios carry greater risks but can sometimes provide better expected returns, from now denoted µ

and defined in Equation 2.1. As an investor, the general idea is that you want to combine different positions in equities (or other instruments) to maximize your portfolios expected returns µ

and minimize your volatility σ

, hence finding the largest ratio between µ

and σ

, called the Sharpe-ratio. This is the foundation for portfolio optimization theory and states that an investor should be concerned with both the potential return and potential loss of a portfolio, creating a risk-return perspective of investing [25], [26].

Subtracting the risk-free rate from the average return simplifies an

investors understanding of which profits are associated with risk-taking

activities. Hence, understanding the Sharpe Ratio is important in order

to grasp whether a portfolio’s alpha (roughly equalling excess returns, learn mote about alpha in Section 2.4.3) are due to superior investing or by virtue of greater risk taking. It is only superior if the excess returns generated are not on behalf of extra risk. Given that R

is the portfolio return, R

is the risk-free rate and σ

is the standard deviation of the portfolio’s excess return, then the Sharpe Ratio is defined as [27]:

Sharpe Ratio = (R

− R

) σ

It should be mentioned that we use a risk-free rate of zero in our calcu- lations to describe the performance of the portfolios.

2.3.2 Tracking Error

Tracking error (abbreviated TE) is the divergence between the price ac- tion of an individual position or portfolio and the price action of a bench- mark, i.e. an index. The tracking error is therefore effectively the dif- ference between the realized return and the benchmark it was intended to track or beat. TE is also known as the active risk. To calculate the tracking error, two methods are common. One is simply subtracting the return of the benchmark from the portfolio’s return, however we decided to use the method below. Then if TE is the tracking error, R

is the portfolio return, R

is the return of the benchmark/index and N is the number of return periods, then TE is defined as [28]:

TE = s

P

i=1

(R

− R

)

N − 1

(2.4)

2.3.3 Minimize Tracking Error

In certain investment strategies, the objective is to optimize a portfolio

whilst trying to minimize the tracking error against a particular bench-

mark. A tracking error is usually measured as volatility, in other words,

standard deviation from the index/benchmark. Thus, by minimizing

the tracking error, one manages to enhance stability, therefore reducing

the risk of extreme swings in value. Assume that P ∈ R

, c ∈ R

, A ∈ R

, b ∈ R

, A

∈ R

, b

∈ R

and {l

, u

} ∈ R

, and that n denotes the number of stocks in the selected benchmark and the number of constraints used in the minimization problem.The optimization prob- lem is convex since the covariance matrix is positive semi-definite and since the weight vector is as well. Then the convex optimization problem can be expressed as:

min

f (w) := 1

2 w

X

w + c

w s.t. w ∈ R

A · w ≤ b A

· w = b

l

≤ w ≤ u

2.3.4 Convex Optimization Theory

Convex optimization stems from mathematical optimization of minimiz- ing convex functions over convex sets. Applications of convex optimiza- tion is frequently used in a range of fields, including data analysis and modeling, finance and statistics. General optimization means finding the maxima or minima of functions, feasibly subject to constraints [29]. In terms of investing, fund manager’s occasionally use optimization in order to find the optimum portfolio given a specific investment strategy.

Portfolio optimization variables may be amounts invested in different assets. Constraints may include elements such as: budget, weight, mini- mum return etc.

General Optimization Problem When picturing a general optimiza-

tion problem one can imagine the action of trying to minimize or maxi-

mize an objective function with respect to specified constraints. In our

thesis we aim to minimize the tracking error with respect to our given

constraints on sustainability and our reductions on carbon dioxide, wa-

ter, waste and UCITS. Assume that x is the optimization variable, in our

case the variables the tracking error is dependant upon. f

is the objec-

tive function, which is minimizing TE from index and f

is the constraint

function, then the general optimization problem can be defined as:

minimize f

(x)

subject to f

(x) ≤ b

, i = 1, . . . , m

Convex Optimization Problem We use convex optimization in our thesis in order to retrieve optimal allocation weights in order to optimize our portfolios. A typical example of a convex optimization problem where you minimize the objective function with respect to given certain convex constraints can be expressed as [29]:

minimize f

(x)

subject to f

(x) ≤ b

, i = 1, . . . , m

Objective and constraint functions are convex if they are on the form below f

(αx + βy) ≤ αf

(x) + βf

(y)

if α + β = 1, α ≥ 0, β ≥ 0

2.4 Portfolio Measures

We now know how to compute the portfolios we are looking for, but in order to actually find these portfolios we need to estimate and predict the variables of the portfolios, the correlation, volatility, expected returns etc.

of each available assets. One could do this with just simple rolling moving averages for each variable but since markets have no memory one cannot expect the future market to behave statically exactly as the past did. In order to capture how different market conditions affect individual assets and the portfolio as a whole we are using a multi-factor risk model that uses different dependencies of different factors to predict the behavior of each asset. From this we can then find the expected behavior of the available portfolios.

We will also be looking at the Price-to-Earnings-ratio, shortened P/E,

as we use it to make investment decisions for one of our portfolios. Ratios

that are derived from a company’s balance sheet is called fundamentals,

and investing according to those is called fundamental investing. This

differs from the otherwise more quantitative investing that is based on

estimating variables and trying to predict the outcome of the future, or

at least estimate the odds. Fundamental investing is more about knowing

what you are actually buying, since it is derived from the balance sheet,

and less about the statistical behavior of the stocks.

2.4.1 Correlation

In the financial sector, correlation is a statistical measure that indicates the degree to which two separate assets move in relation to one another.

Correlation is commonly used in advanced portfolio management, and it is implied by its definition a coefficient value in between -1 and +1.

Correlation is thus often a measure of beta, in other words, how an as- set/security move in relation to a specified benchmark. A perfect positive correlation imply a coefficient value equal to exactly 1. For example if Amazon’s stock price had a correlation of 1 to the SP 500 Index, the price of Amazon’s stock should rise all days that price of the index increases.

Similarly if the correlation was -1 Amazon’s stock price would move in the opposite direction of the SP 500 Index. If restricted in the ability to short stocks, an investment manager can instead go long in a specific instrument or stock with a negative correlation to the stock/instrument intended to short. If we have two variables X and Y , X is the mean of observations of variable X and Y is the mean of observations of variable Y the correlation coefficient r is defined as [30]:

r = P(X − X)(Y − Y ) q

P (X − X)

q

(Y − Y )

2.4.2 Beta

Beta, denoted as β is a coefficient that signals the volatility, or system-

atic/market risk intrinsic to the market, of an individual asset in contrast

to the unsystematic risk of the market overall. Hence, beta is widely used

by investors in order to grasp whether an investment correlates positively

or negatively to the market. It is therefore imperative that the bench-

mark used as the market is related to the invested asset. An asset with

a beta of around 1.0 barely diverges from the overall benchmark, indi-

cating a small additional risk to the portfolio. The give-up of this, is

a greater chance of generating alpha (roughly viewed as the excess re-

turn, learn about alpha in Section 2.4.3). In this study, since we try to

minimize the active risk (tracking error), we will in most cases end up

with a beta close to 1.0. If we compare our portfolio return to the index

and assume that R

is the return on our portfolio, R

is the return of

the index, covariance is the joint variability between a stock’s/portfolio’s

return to the market/index and variance is the spread from the average

value. Then the beta is defined as: [31]

β = Covariance(R

, R

) V ariance(R

)

2.4.3 Alpha

Alpha, denoted as α, is most commonly used within investing to describe the excess return compared to the market (or benchmark used).

Alpha may be positive or negative, it is solely the return of active investing. In general terms, alpha is the return on an allocation that is not caused by the overall movement in the particular market or index related [32]. The mathematical formula for alpha is defined and derived from CAPM (the capital asset pricing model) [33]. Assume that R

is the security’s or portfolio’s return, R

is the risk-free rate of return, β is the systematic risk of a portfolio and R

is the return of overall market.

Then α is derived from the CAPM model and defined as [34]:

α = R

− R

− β · (R

− R

)

In this thesis we are using a risk-free rate of 0.

2.4.4 Multi-Factor Risk Model

The usage of a multi-factor model is convenient when investors are look-

ing to understand the true drivers of their assets. In simple terms, a

multi-factor model is a model commonly used in finance that utilizes

multiple predefined factors in order to clarify market phenomena. A

multi-factor model is often implemented by performing a large-scaled

statistical regression, where a number of factors are regressed during a

given time span to see how these factors correlate and affect the volatil-

ity of returns in individual asset. When done for a multiple number of

assets a matrix of each assets coefficients to the individual factors is at-

tained. This method is often used by funds attempting to track an index,

however, it can be used on both individual securities and portfolios. A multi-factor risk model can also be used to see the risk attribution from each factor for an asset and thus which factors that drive the risk. For one asset i and k factors we can define the multi-factor risk model. As- sume that r

is the return of security i, f

, f

, f

...f

is the rate of return for each of the predefined factors implemented where f m is the rate of return for the market, x

is the exposure coefficient between asset i and factor k and U

is the specific return of asset i not explained by the factors. Then the formula for a multi-factor model can be defined as follows in Equation 2.5:

r

= x

· f m + x

· f

+ x

· f

+ . . . + x

· f

+ S

(2.5) This is used in our minimization of tracking-errors denoted Ω (and de- fined in Equation 2.4), as can be seen in Equation 2.6. We let X be the coefficient matrix between all assets and all factors, P the covariance matrix between the factors, w the vector of investment weights in each of the assets and U the vector of the specific return of each asset. Then our minimization of the tracking-error Ω is defined as:

min Ω = (X

w)

X

(X

w) + w

U w w ∈ R

(2.6)

2.4.5 Price to Earnings Ratio (P/E Ratio)

The P/E ratio is a commonly used measure to value a company. It mea- sures a company’s current share price relative to its earnings per share, often abbreviated “EPS”. Therefore P/E ratios are commonly used by investors and analysts only when valuing public companies on the stock market. However the P/E ratio is occasionally used when valuing pri- vate companies, primarily as a benchmarking tool to the competition [35].

If we simple let the share price be defined as the currently observed price for one unit of stock and earnings per share be the annual reported earnings divided by the amount of outstanding shares, the P/E-ratio can be defined as:

P/E = Share Price

Earnings per Share