Data-Driven Portfolio Management using Exchange Traded Funds

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Copenhagen, May 2017 Graduate Diploma Thesis Graduate Diploma in Business Administration (Finance)

Data-Driven Portfolio Management using Exchange Traded Funds

Emil Ahlmann Østergaard

Supervisor:

Niklas Kohl, Dept. of Finance, Copenhagen Business School Co-supervisor:

Kourosh Marjani Rasmussen, Dept. of Management Engineering, Technical University of Denmark

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Copenhagen Business School

The Department of Finance Campus Solbjerg Plads 3, Building A 4th and 5th floor DK-2000 Frederiksberg, Denmark Phone +45 3815 3601

fi@cbs.dk www.cbs.dk

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Abstract

Actively managed funds have long been accused of being both expensive and unable to continuously outperform simple cheap passive index funds after fees has been paid. This study investigates whether a happy medium exists, beneting from the best of the both worlds.

A data-driven generic framework for smart beta investing is proposed and tested on exchange traded funds.

An asset ltration approach is used to screen data, feature selection with agglomerative hierarchical clustering and a selection criteria ensures diversication and selection of instruments with desired properties. Conditional value-at-risk optimization with moment matching for scenario generation is used for asset allocation. A betting- against-beta factor, which is long leveraged low-beta ETFs and short high-betas ETFs yields positive risk- adjusted returns.

This study extends previous work by presenting an end to end framework considering funding and transaction cost applied to exchange traded funds. Hierarchical clustering is used to dene the market portfolio and the evidence of a betting-against-beta factor is tested on index level.

Keywords:·Asset Allocation·Active vs. Passive Investing· Betting Against Beta Factor·Exchange Traded Funds·Feature Selection ·Portfolio Construction·Portfolio Management ·Portfolio Optimization·

Smart-Beta· Trading Strategies

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Preface and Acknowledgments

This thesis was prepared at the Department of Finance at Copenhagen Business School (CBS) in partial fulllment of the requirements for acquiring the Graduate Diploma in Business Administration degree in nance (DK: HD nansiering).

The thesis deals with asset allocation and portfolio construction and how data can be used as a primary driver.

The idea for this project started a couple of years ago, when I attended a course, Financial Optimization, at the Technical University of Denmark (DTU), taught by Associate professor Kourosh Marjani Rasmussen, whom I have been working with afterwards within FinTech and life-cycle wealth management. Over the years, I have gained more insight in the nancial sector, especially personal nance and investment from a combination professional life, the study and a personal interest. In my opinion there is an obvious need for a reformation within the nancial sector or there are at least opportunities for businesses doing it dierently from the classical nancial institutions. In my opinion the classic approach to asset allocation and portfolio management is somewhat outdated and can to some extend be substituted by a simpler and cost ecient approaches, targeting, among other, novices with a very little interest in personal nance.

Lately the topic of using exchange traded funds and the discussion of active vs. passive investment has gained a lot more attention that earlier in the media Høie [6 March 2017], Knuthsen [2 March 2017], Buet [2017], Flood[24 April 2017] and in the academic worldPedersen[2016]. This has just motivated me even more to dig into this eld, as more people starts to realize that there might be room for changes.

I want to thank my supervisor Niklas Kohl (Ph.D. fellow at Dept. of Finance, Copenhagen Business School) and Co- supervisor Kourosh Marjani Rasmussen (Associate professor at Dept. of Management Engineering, Technical University of Denmark and Chief Optimization Architect at Schantz A/S) for interesting and inspir- ing discussions, which has given me a lot of input to this study. Furthermore, I am grateful for the opportunity I had to talk with Lasse Heje Pedersen (Professor at Dept. of Finance, Copenhagen Business School), who suggested testing the Betting-Against-Beta hypothesis on ETF data.

Emil Ahlmann Østergaard Copenhagen, May 2017

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Nomenclature

BAB Betting against beta CAPM Capital asset pricing model CVaR Conditional value-at-risk EMH Ecient market hypothesis ES Expected shortfall - see also CVaR ETF Exchange traded fund

ETL Expected tail loss - see also CVaR MPT Modern portfolio theory

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CHAPTER 1

Introduction

Financial markets are important factors in the global economy and its well-being, and has been expanding heavily in recent decades, which is illustrated in gure 1.1 where the global market capitalization of all listed companies from 1975 to 2016 is plotted along with the value of stocks traded globally - a tendency of a growing and expanding market. In 1975 the total value of the stock market was $1.22 Trillion and by 2016 it has grown to $64.8 Trillion. In table 1.1 the growth in total net assets of investment companies globally are listed by type.

Mutual funds are by far largest type of funds, but Exchange traded funds (ETFs) have shown a rapid growth in the period and counts for more than 10% of the total value by the end of 2015. The total value has more than tripled in the period, but ETFs has grown more than 100 times in value over the same period. Obviously, the nancial market is growing, both by value and value of the trades made. The investments opportunities are also growing, and low cost index-funds are getting more popular. The investment universe has never been bigger and more complex.

Year Mutual funds Closed-end funds ETFs Unit Investment Trusts (UITs) Total

1998 5525 156 16 94 5790

2003 7402 214 151 36 7803

2009 11113 223 777 38 12151

2013 15051 279 1675 87 17091

2014 15875 289 1974 414 18240

2015 15652 261 2100 94 18107

Table 1.1: Total net assets by fund type in billions USD year-end value.

Source: The Investment Company InstituteICI[2016]

Asset allocation, i.e. allocating funds to assets in a portfolio the right way to match the investors risk/return appetite, has been a subject of great interest to economist and researchers for decades. Both private investors and corporations, are interested in getting value for money, also when it comes to their investments. Asset allocation is probably the most important part of the investment process. The strategy itself and timing is by many professionals seen to be more important that the asset picking, i.e. the selection of individual assets.

Therefore, a lot of research has been done within the eld of portfolio and utility optimization. Harry Markowitz is the father of Modern Portfolio Theory (MPT), and his theories and models has for many years been used and discussed, both in the academic and nancial world, but also in the media because it aects all of us.

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2

1980 1990 2000 2010

0e+002e+134e+136e+138e+131e+14

The global stock market

Year

USD ($)

Total value of stocks traded globally Global market capitalization of listed companies

0e+002e+134e+136e+138e+131e+14

1980 1990 2000 2010

Figure 1.1: The value of stocks traded globally (CM.MKT.TRAD.CD) is the total number of shares traded globally, multiplied by their respective matching prices. The global market capitalization of all listed companies (CM.MKT.LCAP.CD) is the aggregated is the number of shares outstanding times the asset prices for listed companies globally. All data are converted to U.S. dollars using corresponding year-end foreign exchange rates.

Source: The World BankWor[2017a,b]

In Markowitz [1952], he analyses the mean-variance structures and showed how diversication can be used to minimize the risk, which, the investor is exposed to. At that time researchers, including Markowitz, were focusing on one-period optimization, which is more useful in the academic world than in the real world, where it is interesting to optimize over a longer period with non-static cash ows. Merton[1969,1971,1973],Samuelson [1969] are all early examples of studies of multi-period models, but needs for closed-form models limited the models from reecting the true options and needs an investor faces in the real world. Often the complexity of the input to Merton intertemporal model (ICAPM) was reduced heavily, such that in practice was a one- period analysis. Scientist were aware of the solutions to one-period models were very dierent from multi-period models, since the up and down saving phases thru life can change several times. In the recent decades, where computational power, the selection of numerical methods and advanced modeling has grown, it has become easier to study real world situations. Brennan [1998], Barberis [2000], Lynch [2001] all set up multi-period model and by discretizing the states makes it possible to nd a solution to the optimization problem.

Wealth management for private is more important than ever, as our wealth grow, the average life expectancy gets higher, expenses for retirement, e.g. hospital and personal care grows and public benets are reduced for people with decent income. The actions and decisions taken today have large impact on the wealth at retirement, especially to the younger part of the population, due to the long-time horizon. Optimal wealth management is essential and valuable to all types of families. Every individual has plans and wishes of how they want to live their life in the future, and economy plays an important role. Wealth management is about nancial planning and optimization under the constraint of risk appetite and future goals.

A lot of attention within asset allocation has been given to the tactical level, where the weighting of individual securities is adjusted for a shorter period, such that they diverge from the long-term weighting, known as strategic asset allocation, in order benet in the short run from e.g. market anomalies or known business cycles eects. However, strategic allocation is at least as important when considering wealth management, but has not received as much attention. The reason for this might be related to the interests from the large nancial institutions and banks which earns money on transactions, therefore encourage to a tactical active investment prole. A good tactic asset allocation can be identied by a high risk adjusted return, whereas good wealth management cannot be identied as easily. A lower tax bill has the same value as an asset return of same amount, but might not seem equally exciting and attractive. The lack of well-founded methods for wealth management with a broad application and a complex issue with a lot of variables with dependencies results in several non-standardized approaches to wealth management. Often the nancial advisers own personal

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1.1 Portfolio theory 3

developed rule of thumbs is used for strategical allocating, resulting in very subjective advice. Bodie et al.[1992]

is one of the studies that tries to focus on the long-term asset allocation where they formulate a model on which they conclude that younger persons with a long investment horizon should accept higher risk that older persons with a shorter investment horizon. This nding is broadly accepted and used among nancial advisors, but the models does not say anything about how much should be put into risky assets or what a reasonable level for the beta of the portfolio is, hence subjective rules of thumb is used for this.

1.1 Portfolio theory

Portfolio theory is a major eld with in nance and investment theory and includes disciplines within nance, investments, mathematical modelling, statistics and accounting among other. When constructing and managing portfolios several aspects must be considered, e.g. risk-tolerance, expected return, investment horizon, asset classes, diversication, investment style and strategic and tactic asset allocation.

1.1.1 Modern portfolio theory

Modern Portfolio Theory (MPT) is the discipline of allocating and diversifying assets in a portfolio to maximize the expected return at a given risk-appetite. Harry Markowitz introduced MPT in 1952 (Markowitz[1952]) and showed how diversication can eliminate idiosyncratic risk and mean-variance optimal portfolios is represented on the ecient frontier. Risk and return of an investment shall not be viewed individually, but in the context of the portfolio, as correlations has a major inuence. Combining assets reduces the overall risk of the portfolio. Not all risk can be diversied, but in an optimal combination all the idiosyncratic (unsystematic) risk is eliminated, such that the portfolio risk only is aected by systematic risk. Investors are assumed to be risk averse and seeks therefore portfolios on the ecient frontier. MPT was a revolution of the nancial theories at the time, and its concepts is still fundamental of new theories, and is broadly used in the academic world and among nancial institutes using it in practice. However it has also gained a lot of criticism, especially in the recent decades where nancial crashes has proven that variance is not an appropriate risk measure because returns are not normally distributed which was rst discussed inMandelbrot[1963] andFama[1963], who proposed the use of stable distributions instead. However, variance, in the sense of value at risk (VaR) measure, and therefore also the normal assumption, has often been (and still is) used in risk management. Rockafellar and Uryasev [2000]

proposes conditional value at risk (CVaR) as a risk measure because it has better properties than VaR, e.g.

it is a coherent risk measure (Pug [2000]). CVaR is the average of the events exceeding VaR, and therefore considers the extreme evens in fat tailed distributions.

In the years after Markowitz introduced MPT, several studies contributed to the eld, and especially the capital asset pricing model (CAPM) has become popular. It states that the expected rate of return of an asset is a linear combination of the market risk premium and its beta, which is the volatility of the asset relative to the market portfolio. Fama and French[1993] criticizes this model by stating that the expected return cannot alone be explained by the value of beta. They propose the very famous FamaFrench three-factor model, which is one of many extensions of the CAPM model with two more factors, the SMB (company size) and HML (company price-to-book ratio), besides the market risks. This model has been extended subsequently in various studies e.g. inFama and French[2015]. Pedersen[2015] nd evidence of a BAB-factor (funding constrains) and shows that zero-beta portfolios made from long leverage low beta asset and short high beta assets yield signicant positive risk adjusted returns.

1.1.2 Portfolio style: active vs. passive

I the time between the CAMP model and the development of multi-factor models, Eugene Fama proposed the Ecient-market hypothesis (EMH) in Fama [1965] and Fama [1970]. The idea of EMH is that a price on an asset is fully reected by the information available, hence it is impossible to outperform the market

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1.1 Portfolio theory 4

consistently adjusting for risk, because the market adjust immediately as new information is available. The direct consequence of this is that active investing should not be protable, as an active strategy imply trading related costs, hence on average the strategy would underperform on a risk adjusted basis. Robert J. Shiller challenges EMH inShiller[1980], where he nds evidence of behavior in the US stock market that diverges for the expectations accordingly to EMH. He assumes that investors are valuing a stock based on the present value of its future dividends, which he nds to be more stable than the actual stock prices based of an ecient market model.

William F. Sharpe shares his point of view in the discussion with two statements in inSharpe[1991] (1) before costs, the return on the average actively managed dollar will equal the return on the average passively managed dollar and (2) after costs, the return on the average actively managed dollar will be less than the return on the average passively managed dollar. The implications of EMH and Sharpe's statement have been the foundation of a countless numbers of discussions, studies and articles and still are. One of the main reason is that the nancial industry make a lot of their earnings from trading related activities, i.e. active investing generates higher earnings that passive investing due to higher turnover, and for obvious reasons they do not share the same view as Sharpe et al. Some investors, portfolio managers, and investment funds, e.g. Warren Buet seems to be able to beat the market consistently and critics often uses this argument as falsication of EMH.

Warren Buet himself also takes part in the discussion. In the 2016 edition of the annual letter to the Share- holders of Berkshire Hathaway Inc.¹ (Buet [2017]), he discusses a bet he made in 2007 with a hedge fund manager that his investment in a simple S&P 500 index fund will outperform the hedge fund manager selection of funds-of-funds after fees. The bet was made because Buet in the Berkshire's 2005 annual report argued that managers with an active strategy over a period would underperform compared to a passive strategy because of various fees. The results are clear Buet won that bet. In the period from 2008 to 2016, none of the funds had an aggregated return that could match the return of the S&P 500 index fund.

Lately, Lasse H. Pedersen states inPedersen[2015] the market to be Eciently Ineciently i.e. that markets are in a near-ecient equilibrium due to professional investors, but keeps being so inecient that they are compensated for their risk and costs. InPedersen[2016] he challenges Sharpe's prepositions mentioned above, and states that active investment managers can beat the market after costs, however importantly - not everyone does.

The subject has also been discussed in the media lately. In a chronicle in Børsen March 2017 (Knuthsen [2 March 2017]), Teis Knuthsen, CIO at Saxo Privatbank, points out why he thinks that passive index funds are favorable for private investors and discusses the critique Professor at Copenhagen Business School, Steen Thomsen has toward passive investment. Steen Thomsen says that passive investments are a consequence of bad/lazy leadership. Head of Asset Management for Wealth Management Business Division at Danske Bank A/S, Henrik Gade Jepsen explains in in Børsen , March 2017 (Høie[6 March 2017]), that Danske Bank is going shut down approximately 50 % of their investment funds, especially the traditional active funds, because the demand has moved toward passive funds. In an article in Financial Times from April 2017 (Flood [24 April 2017]), CEO of Nordea Asset Management, Nils Bolmstrand, states that Nordea, also going forward, believes in active strategies, but also indicates that they might be looking in the direction of smart-beta strategies.

1.1.3 Smart beta

Smart beta strategies have in the recent couple of years gained a lot of attention, but smart beta is just a new term used for a broad group of strategies used for decades, e.g. factor investing. They can be categorized somewhere between index funds and active funds. A loose denition of smart beta is a type of strategies that tries to capture return drivers using systematic and transparent rules for screening and weighting. Smart beta portfolios have an alternative asset weighting that traditional market capitalization based indices with the aim to outperform the market, by reduce risk and take advantage of diversication. Examples of smart beta are fundamental weighting, momentum weighting and dividend weighting. In 2005 Arnott, et al. (Arnott et al.

[2005]) presented fundamental indexation, which a type smart beta strategy based on cap-indierent measures

1American multinational company where Warren buet is Chairman of the Board, President, and Chief Executive Ocer]

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1.2 Thesis Statement 5

of company size, and conclude that these indexes consistently outperform standard cap-weighted indexes. The strategies are somewhere in between active and passive investment styles.

Smart beta exchange traded funds are relatively new instruments in the nancial market. They are like regular ETFs, which tracks a given index but with smart beta rules applied to adjust the weightings of the assets. They are often more expensive in terms of cost than regular ETFs, but also cheaper than most actively managed funds.

Lemmon and Kahn [2014] recommend smart beta for investors that believe that the markets are in general ecient, but inecient in certain points or large investors who believe in active investing but are limited due to capacity constraints.

1.2 Thesis Statement

The purpose of this project is to focus on how private investors can achieve a well-diversied portfolio of nancial asset with acceptable returns without paying expensive management fees. The classic active vs. passive investing style subject is unavoidable and a discussion well be given based on the ndings in this study and other recent studies.

Exchange traded funds will represent the asset universe, because they have some interesting features that are desirable in cost-ecient portfolio management and which would be interesting to study the eects of.

Furthermore, the ETFs are accessible to both private and professional investors which is necessary to make the use of this study applicable in a real-world setting. The starting point of many studies within portfolio optimization and asset allocation, has been the use of classic asset i.e. stocks, bonds and commodities. Using ETFs also makes this study dier from the masses.

A generic framework to construct and manage portfolios will be used to test and illustrate the idea of a simple and cost ecient approach to portfolio management. All steps in the cycle will be covered, starting from the initial analysis of the huge amount of potential investment opportunities that exists, a step that is very time consuming and incomplete if done manually, until the nal portfolio maintenance, asset allocation and performance evaluation.

The ideas and concepts are targeting all investors, but especially private investors might benet from this, because they cannot, or are not, willing to spend the time it takes to get fully informed of the nancial markets and manage existing investments while looking for new potential securities. Furthermore, they do not have the same access and resources as nancial institutions do, when they are making their decisions of how their portfolio should be constructed and managed. The methods used in the frame work must either be data-driven, i.e. that the decisions are made upon a thoroughly analysis of historical data, or build on common knowledge, hence there is need for forecasts or analysis of what might happen in the future.

The framework consists of three major steps:

1. Top: Identify assets that might be relevant 2. Middle: Select assets include in the portfolio 3. Bottom: Asset allocation

The rst step, is as mentioned above the process of nding needles in a haystack. The investment universe is enormous and all its constituents cannot be analyzed in-depth. An initial screening process must be implemented based on data, common sense and due diligence such that a recent subset of the total investment universe is covered and reduced. Only interesting assets that passes the screening are considered in the next step.

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1.2 Thesis Statement 6

Only the most interesting assets must be identied in the second step using advanced data analysis. In this step, the properties that the manager wants the asset to possess is as a selection criterion based on historical data.

In the last step, the nal allocations are made based on decision models that either might be sophisticated mathematical models or more simple strategies

The second and third step in the frame work can be viewed as a smart beta type of approach, but whereas smart-betas strategies often are considering the allocation within an index the approach in this study is to focus on weighting of various indices in the portfolio.

The concept must be evaluated to alternative approaches. The evidence of a betting against beta factor is tested on index level, which previously only has been done on single-asset-level.

The hypothesis to be tested in this study is:

Alternative investment opportunities exist, which are simple to apply, cheap and competitive with both expensive actively managed funds but also the passive managed funds suggested by many, including William F. Sharpe

This study is structured in the following steps, and the question associated with them will be addressed:

• A simple approach to portfolio construction

Is it possible to set up at framework to portfolio construction that both can be generic, simple, data- driven but also competitive? How can an investor consider as many potential assets for the portfolio as possible without spending an extensive amount of time on analyzing individual securities, and how can data mining and data analysis be used in this process? Are data screening and data analysis sucient to make well-performing smart 1/N strategies which are applicable in the real world. Can a decision model improve the performance of the strategies?

• Portfolio optimization

What is conditional value at risk optimization, and how can uncertainty be modeled?

• Beta estimation and the betting against beta factor

What is a market portfolio, and how can a beta of an index fund be estimated? Is there any evidence of a betting against beta in the index fund market?

• Performance Evaluation

Which performance measures can be used to evaluate and compare portfolios performances? How are the strategies suggested performing? What issues might we face when using a data-driven approach and what benets are there?

• Active vs. passive investing

Is an active strategy worth it, and what might the future of portfolio management look like?

In this study a framework for portfolio construction and management is proposed. The steps include hierarchical ltering, clustering and optimization, which all are based on the properties found in historical data. A alternative approach for constructing market portfolios are suggested and the betting against beta hypothesis is tested on exchange traded funds.

The mathematical modelling language R (v. 3.3.3)² is used for data mining, analysis, calculations, portfolio construction etc. GAMS (v. 24.8)³, a software language for algebraic modelling is used to formulate the

2https://www.r-project.org/

3https://www.gams.com/

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1.3 Limitations and delimitations 7

scenario generation problem and the portfolio optimization problem and solved with CPLEX and CONOPT solvers. Data are provided by Thomson Reuters Datastream.

1.3 Limitations and delimitations

In this study taxation is not considered for two reasons. Dierent tax rules apply around the world, and it would be an extensive work just make an analysis covering a reasonable subset of countries. Furthermore, taxation rules are subjected to change frequently. Regarding the Danish pensions system, investment types are taxed equally, but ETFs must be UCITS regulated. This is neither considered, but it is likely that an alternative substitute exists for those ETF that are not under the regulations.

An extensive data analysis of the result is beyond the scope for this project, as it is the approach to portfolio and asset management which should be in focus, but the results will be examined satisfactorily enough to have a good discussion that can be used to answer the questions in this thesis statement.

Actively managed funds are not included in the performance comparison, as it is hard get sucient information to perform a fair evaluation.

1.4 Overview

The thesis is structured as follows: First a presentation of data used is given in chapter 2. Next, in chapter 3 to 5 the methodology of this study is presented along with the theory behind and the outcome. The betting against beta hypothesis is tested in chapter 6. An analysis of the results is given in chapter 7 and the results are discussed and summarized in chapter 8 and 9. At last, ideas for further work are suggested in chapter 10 and at the very end the appendix is listed, where additional graphs and tables are found.

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CHAPTER 2

Data

2.1 Exchange traded funds

Exchange traded funds, ETFs, are index based funds that have grown rapidly in the recent years since the rst ETF was created in Canada in 1990 and the rst in the US in 1993. ETFs tries to replicate a benchmark, e.g.

an index by passive fund management to lover the expense ratio. They distinguish from other funds by being traded on a stock exchange, and the traded price is based on the bid and ask prices.

The global ETF market

Year Number of ETFs 0500100015002000

29 30 80 102 113119 152 204

359 629

728797 923

11341194 1294

1411 1594

0500100015002000

1998 2000 2002 2004 2006 2008 2010 2012 2014

0.0e+005.0e+111.0e+121.5e+122.0e+12

EFT Tota Net Assets Numbers of ETFs worldwide

USD ($)

Figure 2.1: The development in the global ETF market. Source: The Investment Company Institute ICI [2016]

In the beginning, ETFs were mostly traded by institutional investors to execute exotic trading strategies, but soon afterwards they were adopted by individual investors and nancial advisors. The rst bond ETF was introduced in 2002 and the rst smart-beta (alternatively weighted) ETF landed on the market in 2003. The development since 1998 until now is illustrated in gure 2.1. In 1998 29 ETFs was listed with total net assets of

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2.1 Exchange traded funds 9

$16billion and by the end of 2015 1594 ETFs were listed with total net assets of $2.1trillion ($2.100.000.000.000).

By the end of 2015 the United States counted for72% of the ETF market, whereas Europe counted for17%, Africa and Asia-Pacic for9%and other Americas for 2%(ICI[2016]).

ETFs have been selected for this study because of their favorable properties, e.g. they do in general have low expense ratios, available to both private and professionals, source of diversication, and can easily be implemented in investment strategies.

2.1.1 ETF data

ETF data is provided by Thomson Reuters Datastream. Daily total return index prices (TR) converted to euro, volume (VO), ISIN (ISIN) and type description (TRAD) are used from 01-01-1997 to 31-12-2017. Datastream is usually known to deliver high quality data. The tickers used are provided by Datastream as predened list of almost all ETFs available. Any symbol of ETFs mentioned in this thesis are Datastream codes. Data is accessed thru the API either by an Excel add-in or directly thru R, dependent on the type needed. Total return index prices are preferred because they include fees, dividends etc.

02468Cumulative return on 1 EUR investment

Cumulative ETF Returns

2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

Cumulative Returns Median 1. & 3. quantile Min & Max

Year

Figure 2.2: Cumulative return on 1 EUR investment of 559 ETFs series used for this study.

The data-set used for in-depth analysis and modelling consists of 559 ETFs with daily total return data from 01-01-2007 until 31-12-2016. The series are plotted in gure 2.2. Is clear that this data-set represents a great variety the nancial investment opportunities with very dierent performance and statistical properties. The period from 2008 to 2017 are in general used as out of sample period, and data for 2007 are used for model building, however, rolling windows are also used in some modelling techniques.

The portfolio out of sample tests are all based on realized return from 2008 to 2017, hence both the nancial crisis and the subsequent recovery is covered is a full economic cycle. This is, very powerful since the portfolio performance is measured under dierent circumstances is hard not to make a prot when the general market trend is pointing upwards, but how will the portfolio perform when the general market trend is pointing in the opposite direction? On gure 2.2 it is seen that the portfolios are constructed just before nancial crisis and the broad markets starts to plunge. It becomes very important how the portfolios handle risk thru this period.

If they perform very badly in this period, it will be very hard to recover from.

The ETFs are listed by underlying asset classes in table 2.1. The equity ETFs are heavily represented, but it reects the products available on the market. Importantly, also the commodity and bond markets along with alternatives are well represented reecting the potential markets for private investors. Ideally money market would have been represented with more ETFs.

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2.2 Money market rates. 10

Asset class Number of Assets

Alternative Exchange-Traded Fund 14

Bond Exchange-Traded Fund 36

Closed-Ended Fund 1

Commodity Exchange-Traded Fund 16

Equity Exchange-Traded Fund 485

Mixed Asset Exchange-Traded Fund 2 Money Market Exchange-Traded Fund 1

Other Exchange-Traded Fund 4

Total 559

Table 2.1: ETFs represented by asset class (security type description) dened by Thomson Reuters classication system.

Statistical properties of nancial data have been analyzed in several studies, see Oestergaard [2012], Bulla and Bulla[2006], Cont[2001],Taylor[2008]. Returns shows properties like autocorrelation, heteroscedasticity, fat-tailed (leptokurtic) skewed distributions, all properties which have been observed in data for this study.

Log-returns are favored over raw returns for its statistical properties. If prices are log-normal distributed, then log(1 +r_t)is normal distributed, which is desirable since a lot of statistics is assume normality. Obviously, the index prices are autocorrelated, but this is not the case for the log-returns due to its nature:

logR_t= log P_t

Pt−1

= log (P_t)−log (P_t−1) =∇log(P_t).

When calculating cumulated returns, log-returns are additive, whereas raw returns are multiplicative, hence only cumulative log-returns are normal distributed since sum of normally-distributed variables is normal assuming i.i.d. but the products of normally-distributed variables is not.

1-day log-returns are used in this study, except when correlations are estimated, 3-day log-returns are used instead. The reason for this is that the assets used are traded globally on dierent exchanges with dierent trading hours. Some assets are known to load on each other, but if one is traded on an exchange which is closed, when the other is traded, the reaction will occur the next trading day and the correlation is only observed if it's made time-dependent or that data covers the lagged eect, e.g. 3-day log-returns, see also Frazzini and Pedersen[2014].

2.2 Money market rates.

Money market rates are used in dierent contexts. The zero-coupon Euro-bond yield with 1-yearto maturity ECB[2017] is used as a proxy for the return on the risk-free investment, RF. The Euro bond is made from AAA- rated euro area central government bonds, and can therefore more or less be seen as a risk-free rate related to the euro, which the asset prices are denominated in. A maturity of one year is chosen the reect the investment horizon where portfolios are rebalanced annually. In gure 2.3 the daily rates are plotted from 2007 to 2017.

The risk-free rate,r_f is quoted annually (January 1), and as a proxy for the borrow rate,r_b the risk-free rate plus 50 BPS (annually) is used. In the later years, after the nancial crisis, the rate has become negative, which in classic nancial theory was not imaginable. Therefore, a lot of procedures and models has been revised and changed to t the new regime. The borrow rate used in this study is ensured to be non-negative.

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2.3 Portfolio related costs 11

−0.000050.000000.000050.000100.000150.00020Daily rate

Money market Rates

1Y Euro bond yield Borrow rate Risk−free rate

2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

Year

Figure 2.3: Money market rates, based on 1Y Euro-bonds.

2.3 Portfolio related costs

The portfolios must from time to time be rebalanced for several reasons. The desired allocation has changed over time due to dierent growth among the asset, and therefore the risk has changed as well. Another reason is that other assets has become more attractive than the ones in the portfolio. Rebalancing requires trading, and trading is associated with costs. Most studies neglect the cost of trading, as it makes the problems even more complicated. In this study, transaction cost is considered for all portfolios, as cost related to trading is an important in the active vs. passive investment style discussion. The transactions costs are assumed to be linear with a factorγ, e.g. no minimum fee or discount if your trade for at least a certain amount money. A γ= 0.001or 10 BPS is used throughout the study, and it reects fairly the trading costs of private investors.

Each rebalancing period and initially the related transaction costs is withdrawn from the subsequent periodical portfolio return. Bid/ask spread is not considered in this study as the hierarchical asset lter is calibrated to leave out the most illiquid ETFs, which minimizes the eect of bid/ask spread. However, it will have an impact on portfolios with high turnover, but this is considered to negligible, asγ has been set a little high to account for this.

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CHAPTER 3

Portfolio construction I

In this chapter the rst step in the portfolio construction framework is presented, which is general for all the portfolios presented. The fundamental idea underlying the framework is to let the data speak, meaning that the properties of data will have a large impact on how the portfolio is constructed. The framework consists of three steps, to which several methods can be formulated. The rst step is a raw reduction in the number of assets considered using common sense and due diligence analysis. The middle step is an in-depth analysis of data, yet again reducing the number of asset considered. The nal step is to use a decision model to determine the nal allocation based on historical data. This way of doing it, should reect an alternatively and hopefully favorable approach compared to classic portfolio management strategies, which are widely used in the nancial sector today, even though they build on assumptions which has been questioned lately, especially since the nancial crisis. In this chapter an approach for initial reduction of the investment universe is presented. In chapter 4 the feature selection approach is presented, which reduces the smaller investment universe to potential investment subjects. The feature selection algorithm might be used as it is, for portfolio construction, but can also serves as input to a portfolio optimization model, i.e. a decision model which is explained in chapter 5. In chapter 6 the rst two steps in the framework are combined with a more hedge-fund-type-of-strategy presented in Frazzini and Pedersen [2014]. This strategy is more academic, and might not be directly reproducible for a private investor in the real world. However, it is still highly relevant as the nding inFrazzini and Pedersen[2014] is tested on ETF universe, and the outcome is good input for the active/passive investment style discussion.

3.1 Hierarchical asset ltration

Hierarchical asset ltration is somewhat like the classic top-down approach for asset picking, but includes less analysis of exogenous variables, e.g. macro-economic factors and starts at higher level. Furthermore, it allows the manager to consider an unlimited amount of asset for his/her portfolio, if sucient data is available. The framework consists of some prespecied lters, which the portfolio manager applies in a certain (hierarchical) order. This method is intended to be used as the rst step in asset-picking, where a huge number of investment opportunities must be reduced to a subset. Therefore, hierarchical asset ltration serves well as a pre-process to asset allocation on strategical level. The portfolio manager species the lters and the order of use, but the intention is that the lters work on data, only allowing assets with the desired features to pass. The lters are often based on common sense and due diligence analysis. In this study the lters has been set up to ensure data

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3.1 Hierarchical asset filtration 13

quality, that sucient amount data is available for the mid-level analysis, reduce assets that are exposed to risk factors which is not market risk, e.g. liquidity risk and capture a variety of asset classes represented from all parts of the world.

In this project the initial asset universe considered9676tickers, all ETF data. One might notice that, it is way more that there exists cf. gure 2.1. The tickers are withdrawn from prespecied lists in Datastream, which turned out to include a lot of junk, e.g. duplicates which was removed using the lters. Datastream are in general known to provide high quality data, hence it is assessed that the nal data used (after ltration and manually clean-up) has a sucient quality for this project. The hierarchical asset ltration in this study has the following structure:

1. Remove suspicious data, i.t. only zeros, NA or dead series (9613) 2. ETFs must be available from 01-01-2007 (938)

3. Removing ETFs without turnover information (856)

4. Removing ETFs with has been traded for less than 100 days in 2007 which had251 US ocial trading days. (681)

5. Removing ETFs with an average turnover larger than100.000EUR/day in 2007, only days with turnover information considered (606)

6. ETFs with a coecient of variation, σ¯a/µ¯a ≥4 based on 2007 turnover are removed. Only stock-based ETF turned out to be removed in this lter. (580)

The numbers in parenthesis indicates the numbers of ETF that passes the lter.

The order of the lters have been chosen such that the properties of data extracted from the high-level lters are absolutely required. The properties that the lower order lters can be evaluated ex-post ind order control the mask size of the lter.

The rst step simply removes, ETFs that obviously are junk. The second step ensures that there is enough data to make sensible data analysis. The period is rather long, but an even longer period would have been even better for the back testing of strategies. A compromise between having enough ETFs and long enough data history ended up being 01-01-2007.

The third and fourth steps ensure that only commonly traded ETFs pass, and the fth steps ensures that the volume traded is suciently large. The sixth step are added to handle situations where an ETF has a low turnover on daily basis, but at some rare occasions has a very large turnover. This could happen if most of the ETF are owned by a few large investors, liquidating their investments in large portions. Together, these four steps ensure that liquidity risk is kept at a minimum.

It is important to note, that the ltering is based on historical data. The situation might change in the future, therefor, the portfolio manager must put new data through the lter frequently and adjust to the reduced asset universe if necessary. The lters must also be maintained and updated such that they keep their characteristics event if the market condition changes.

After the ltering, data has been analyzed in the period from 2008 to 2016 to ensure the quality of data used in the rest of the study. Series including bad data was removed was removed, but could also have been corrected for e.g. splits. This is not a step in the ltration process, as it would have been corrected by our data provider or by our self in a real-life situation. Finally, 559 ETFs are ready to be analyzed in the next step of the framework the inn-depth data analysis.

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3.2 1/N portfolios 14

3.2 1/N portfolios

A classic approach to test trading strategies is to see if you can beat a benchmark like 1/N in a universe of N assets. This portfolio is over-weighted whit initially more than 85% equity based ETFs due to the nature of the data-set used.

3.2.1 1/N without rebalancing (1/N)

A portfolio containing all 559 ETFs, each initially weighted withw= ₅₅₉¹ on 01-01-2008 is made to reect the average return an investor could expect in this universe. This portfolio was never rebalanced through the period ending 31-12-2016, however is was tested with both monthly and yearly rebalancing, but it did not make a big dierence, hence the results are left out. The cumulated returns are plotted in gure 3.1.

3.2.2 Asset class portfolio (AC)

A common approach to portfolio construction is to determine the strategic asset allocation. A 50-40-10 strategy would be equal to an allocation of 50% to stocks, 40% to bonds and 10% among other asset classes. This strategy is made with annual balancing, and the 10% dedicated to the others equally divided among alternative, commodity, mixed asset and other Exchange-Traded Funds. Comparing the result in gure 3.1 portfolios, this portfolio performs better during the nancial crisis due to higher exposure to bonds, which also is reected in the bond-only portfolio with annual rebalancing. It takes several years for the portfolios exposed to stocks to reach the same level due to the severe losses during 2008 and 2009, however at the end of 2016 the portfolios including stocks has a higher cumulative return. This illustrates they idea of portfolios with a long investment horizon should be exposed to portfolios with shorter investment horizon.

0.51.01.52.0Cumulative return on 1 EUR investment

1/N portfolio returns

0.51.01.52.00.51.01.52.0

2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

Year 1/N

AC (50/40/10) AC (0/100/0)

Figure 3.1: Cumulative (geometric aggregated) returns on 1 EUR investment made on 01-01-2008 of 1/N and AC portfolios.

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CHAPTER 4

Portfolio construction II - Feature selection

The next step to reduce the investment universe to a subset of potential candidates, requires a more in-depth analysis of data. The approach suggested here is feature selection, which is very exible and does not require much from data.

4.1 A feature selection algorithm

Feature selection is a mathematical concept of selecting a subset of (statistical) features for model building, often in the context of machine learning. A great variety of approaches exists, but this study will be limited to the general principles of the approach presented in Bjerring et al. [2016], where hierarchical clustering is used to divide the assets into groups, and then within each group use a criterion to select the assets that enters the portfolio. The result is a diversied portfolio consisting of assets which has been picked purely based on features in historical data and some general prespecied strategies. Therefore, the portfolio manager has no directly inuence on which assets is chosen, and the portfolio is therefore not directly subjected to the issues known from behavioral nance. On the other hand, the portfolio manager might be informed with knowledge which is not present in data, and would therefore like to overrule the outcome of the algorithm. This issue is discussed further in chapter 8.

4.1.1 Agglomerative hierarchical clustering

Within data mining, clustering is to congregate or classify data into groups reecting specic properties of data.

Hierarchical clustering seeks to classy hierarchically either from a bottom-up approach known as agglomerative hierarchical clustering, where each observation starts in its own cluster and gradually cluster together as we move up in the hierarchy, or from a top-down approach, known as divisive hierarchical clustering where all observations starts in the same cluster.

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4.1 A feature selection algorithm 16

The clustering of observations is based on a measure of dissimilarities between the sets. The dissimilarities must be specied by a linkage criterion which determines the distance between sets of observations as a function of the pairwise distances between observations. To measure the distance, a proper metric is required, e.g. the Euclidean distance¹or the maximum distance². An example of a linkage criterion is the Unweighted Pair Group Method with Arithmetic Mean (UPGMA) which is looking at the average distance between observations of each cluster. Another criterion is the complete-linkage criteria which denes the distance between two clusters as at the largest distance between two observations from each cluster.

In this study, the agglomerative approach is used with Euclidean distances and Ward's minimum variance criterion which seeks to minimizes the total within-cluster variance (Murtagh and Legendre[2011]). The reason for choosing this approach is related to the features that is desired to be expressed in the clusters. A portfolio managers job is to construct and maintain a portfolio with a high risk-adjusted return, hence he or she must pick assets with high expected returns but also benets from the diversication eect. Therefore, it is obvious to use the clustering to classify assets, such that the manager directly will be well diversied if an asset from each cluster is picked. As a measure of similarities, the correlation between the assets are used. The correlation matrix is estimated from 3-days log-returns, see chapter 2.

The elements of the correlation matrix are converted to Euclidean distances and used as input for the clustering algorithm.

The correlation matrix of the log-returns is computed based on Spearman correlation which is a non-parametric measure of rank correlation, i.e. does not assume normality. This is very convenient since log-returns often are assumed to be normal, but rarely are when tested. As stated by Ledoit and Wolf [2004] a sample covariance matrix should not be used directly for portfolio optimization due to estimation errors. The inuence of the error can be reduced by shrinking the covariance/correlation matrix before usage, seeLedoit and Wolf [2004].

The shrinkage estimate of the correlation matrix is calculated by shrinking the empirical estimates towards the identity matrix. The shrinkage estimate of the correlation matrix ensures some desirable properties e.g. that it is positive denite and well-conditioned.

The correlation are converted using equation 4.1, such that such that assets with a correlation of one get a parameter of zero, whereas uncorrelated or fully negative correlated assets gets parameters of respectively zero and 2, hence after conversion of C to Euclidean distances the cluster algorithm tries to classify such that negative correlated assets ends up in dierent clusters. Alternatively, one might have used the absolute value of the correlation coecient in equation 4.1 which is a less Markovian way of thinking, indicating that one might not get the full diversication eect from negatively correlated assets. Both approaches were tested empirically in this study, and the dierences turned out to be negligible, hence the algorithm taking advantage of negatively correlated assets is used in the data results presented here.

C=I−corr(X) (4.1)

4.1.2 Selection criterion

When the clustering has nished, as subset of each cluster must be selected. Again, the portfolio managers can add their own avors, by specifying the selection criterion. In this study the results of using two criteria are presented. Other criteria might yield higher risk adjusted returns, however it is not the scope of this study to nd the best criterion, but to show the conceptual idea, and that it might be an alternative approach within portfolio management. The criteria presented are the Sharpe ratio, and STARR ratio, which both are risk adjusted performance measures. These are both good criteria in relation to the previous step which was intended to ensure diversication. Each cluster consists of highly correlated assets, but might still have very dierent beta values. If the selection criteria were purely highest return, one might end up with a highly volatile

1The Euclidean norm: ka−bk₂

2The Chebyshev norm: ka−bk_∞

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4.1 A feature selection algorithm 17

portfolio as the diversication might not o-set the risk factors completely. Finally, beta is also used as criterion, inspired by the nding inFrazzini and Pedersen[2014].

Sharpe ratio

The Sharpe is named after William F. Sharpe who developed it in 1966, see Sharpe [1966, 1994]. It can be found using formula 4.2, wherer¯ais the expected return of asseta,rf is the risk-free rate andσa is the standard deviation of the returns of asset a. In this study, rf is set to zero regarding calculations of Sharpe ratio and STARR ratio as the actual value of the ratio is less important, only its relative size.

S_a =r¯a−rf

σ_a (4.2)

Using Sharpe ratio to compare two assets or portfolios, requires the return distributions to be normal, and if it is not the case, the result might lead to a misleading interpretation. It has already been stated that returns often fails test of normality, even log-returns. The Sharpe ratios might still be useful as selection criterion because the assets within a cluster possess similar statistical properties, hence also similar distributions and therefore the error might very well be insignicant. Another argument is that, it might not be the be best asset which is picked, even though it has the highest Sharpe ration, but it is probably among the best assets.

STARR

Sharps ratio is a broad accepted tool to measure risk adjusted returns, but also has some drawbacks one mentioned above. Another drawback is the use of standard deviation as risk measure, as it punishes equally for up- and downside volatility.

The Stable Tail Adjusted Return Ratio (STARR) has been suggested as an alternative, seeMartin et al.[2003], which takes this into account. STARR uses CVaR(1−α) as a proxy for risk, see formula 4.3.

ST ARR_a= r¯a−rf

CVaR0.05(¯r_a) (4.3)

The Conditional Value-at-Risk³at condence levelα(CVaR(1−α)) is dened as the average loss in the100 (1−α) % worst scenarios. Theα-quantile of the loss distribution is called the Value-at-Risk (VaR(1−α)). In this study a 95%condence level is used, i.e. α= 95%.

Beta

Beta is measure of an assets volatility relative to the market portfolio. A description of how beta is estimated for ETFs in this study is given in section 6.1.

3Also known as expected shortfall (ES) or expected tail loss (ETL )

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4.2 1/n portfolios using feature selection 18

4.2 1/n portfolios using feature selection

4.2.1 Clustering of ETFs

The rst step in the feature selection algorithm is to analyze the results of the hierarchical clustering, which is illustrated in gure 4.1. The cluster dendrogram shows at which level (Spearman's distance between clusters) the clusters are merges. The sucient number of clusters needed can be found by manually inspecting the dendrogram, or just dene a level for the clustering. The purpose of the clustering here a pragmatic way of getting diversied and considering that correlation structure changes over time the numbers of cluster is also chosen rather pragmatic, but analysis of this could be a subject for further work.

Spearman Distance

Dendrogram (n=7)

02468Spearman Distance

Instruments

Figure 4.1: Cluster dendrogram based on 3-day log return data from 01-01-2007 to 31-12-2007 The data-set consists of ve asset classes (plus other, mixed and the closed-ended fund), so at least 6 clusters would be appropriate to reect this. ETFs are already diversied to some extent, so a rule of thumb like no assets must have a weight of more than 5% , i.e. the must me at least 20 assets in the portfolio to get diversied does not t ETF portfolios, hence a lower number of ETFs is acceptable. The clustering should reect an overall classication and small clusters are not desirable for the next step, the selection. If the clusters are too small, the cluster might not process the features that you want from your asset, but the algorithm must pick one anyway. Taking that into consideration and based on a manual inspection n= 7 clusters is chosen for all strategies and all subsequent years. The numbers of ETFs within the clusters are suciently large, and merging clusters from n= 7increases Ward's criterion signicant, hence the variance within the clusters increases i.e.

asset that are dierent starts being clustered together.

The assets selected within each cluster for a given selection criterion is listed in table 4.1, and the cluster dendrogram are found in appendix B.

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Criterion ETF selected

Sharpe F:MTA(BO) H:IBZL(EQ) U:FVL(EQ) C:CBQ(EQ) CN:EFD(EQ) C:XSB(BO) IN:GSL(MM) STARR IGLT(BO) H:EGB(BO) U:FXS(ALT) C:CDZ(EQ) S:DJEU(EQ) U:KXI(EQ) U:DXJ(EQ)

Beta J:ISNI(EQ) S:LYCR(CO) U:FXB(ALT) U:PGF(EQ) C:CDZ(EQ) S:UBS(CEF) F:MTB(BO) Table 4.1: ETFs selected using feature selection based on data from 01-01-2007 to 31-12-2007.

4.2.2 1/n sample portfolios (1/n)

The 1/N portfolio can be used as benchmark to the feature section portfolios, but it is desirable to reect the diversication by having portfolios with the same numbers of instruments. 10.000 portfolios with 7 ETFs equally weighted are randomly sampled with annual re-balancing. A plot of the result of a back-test of the rst 100 portfolios is found in gure 4.2. The min, max, and 1. 2. and 3. quantile portfolios will be used as reference going forward.

01234Cumulative return on 1 EUR investment

1/n sample portfolios

2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

Year Random 1/n

Median 1. & 3. quantile Min & Max

Figure 4.2: Cumulative return on 1 EUR investment of the rst 100 1/n portfolios, along with the min, max, median and 1. and 2. quantile portfolios.

4.2.3

_z¹_i_n

portfolio (

_z¹_i_n

)

An alternative benchmark is suggested, because the underlying risk of1/Nand1/nmight be very dierent due the high exposure towards stocks. zi is the numbers of instruments in each cluster. In the average portfolio within cluster i, each instrument must have a weighting of1/zi, hence the instruments in the average feature selection portfolio must have a weighting of _z¹_i_n

A back-test shows as expected the average feature selection portfolio behave like the median1/n portfolio, see gure4.3, but performs slightly worse maybe because of lower (beta) risk. This portfolio serves as benchmark for the feature selection portfolios in the rest of this study.

4.2.4 Feature selection portfolios with min STARR criterion (F-STARR)

The cluster dendrogram of the instrument using the STARR selection criterion is in listed in appendix B and the correlation matrix found below. All the ETFs are very low correlated or not even correlated at all, hence the portfolios might benet from a diversication eect if the correlations are constant over time.

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Average feature selection portfolio

2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

Year 1/zn

Median 1/N 1. & 3. quantile 1/N

Figure 4.3: Back-test: cumulative return on 1 EUR investment of the average feature selection portfolio along with the median and 1. and 2. quantile1/n portfolios.

Cor=







U:KXI C:CDZ U:DXJ U:FXS IGLT S:DJEU H:EGB

U:KXI 1.00 0.43 0.56 0.00 −0.10 0.43 −0.04

C:CDZ 0.43 1.00 0.42 0.08 −0.05 0.59 −0.23

U:DXJ 0.56 0.42 1.00 −0.11 0.00 0.49 −0.01

U:FXS 0.00 0.08 −0.11 1.00 −0.19 0.09 −0.04

IGLT −0.10 −0.05 0.00 −0.19 1.00 −0.17 0.18

S:DJEU 0.43 0.59 0.49 0.09 −0.17 1.00 −0.20

H:EGB −0.04 −0.23 −0.01 −0.04 0.18 −0.20 1.00







Three feature selection portfolios with minimum STARR criterion is constructed. The rst one (F-STARR-AR) considers the ETFs found by clustering on the 2007 data. These are equally weighted and the is re-balanced annually (AR). The second and third portfolio (F-STARR-AC and F-STARR-MC) are re-clustered annually (AC) or monthly (MC), using data from the previous year or month. New instruments are entering, and the ones that are not leaving, are re-balanced such that all asset are weighted equally. The re-clustering is supposed to keep the portfolio diversied, and adapt to recent information. The assets selected previously might not keep on being the best asset in the cluster according to the selection criterion.

STARR feature selection portfolios

0.51.01.52.00.51.01.52.00.51.01.52.0

2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

Year 1/zn

Median 1/N 1. & 3. quantile 1/N F−STARR−AR F−STARR−MC F−STARR−AC

Figure 4.4: Back-test: cumulative return on 1 EUR investment in the F-STARR portfolios.

The result of a back-test are plotted in gure 4.4. If risk is not taken into consideration, the strategies performs rather poorly throughout the period, especially F-STARR-AC seems to be performing badly. Interestingly

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4.3 Partial conclusion 21

all strategies do not seem to experience as big losses as the benchmark portfolios during the nancial crisis 2008/2009. The reason for this can be a combination of better diversication and risk control directly beneting from feature-selection algorithm. The F-STAR-AR yield the same return over the period as the benchmark _z¹_i_n.

4.2.5 Feature selection portfolios with max Sharpe criterion (F-Sharpe)

Three feature selection portfolios with max Sharpe criterion portfolios are constructed (F-Sharpe-AR, F-Sharpe- AC, F-Sharpe-MC). The cumulative returns of a back-test are plotted in gure 4.5 and again, on a non-risk- adjusted basis, the strategies underperform compared to the benchmark, and F-Sharpe-MC has a negative return throughout the period. Only the strategy with monthly clustering seems to handle the nancial crisis better than the benchmark. Comparing this, with the result of the F-STARR strategies, one see that using STARR captures the downside volatility better that Sharpe and that the STARR algorithm ensures a portfolio that is less risky during a down trend, but is also takes a while before it realizes that the market has changed to an uptrend.

Sharpe feature selection portfolios

0.51.01.52.00.51.01.52.00.51.01.52.0

2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

Year 1/zn

Median 1/N 1. & 3. quantile 1/N F−Sharpe−AR F−Sharpe−MC F−Sharpe−AC

Figure 4.5: Back-test: cumulative return on 1 EUR investment in the F-Sharpe portfolios.

4.2.6 Feature selection portfolios with min beta criterion (F-beta)

The minimum beta criterion selects the asset in clusterci with the lowest beta values, and tries to benet from the BAB hypothesis. Instruments with negative CVaR is not considered.

Three feature selection portfolios with minimum beta criterion portfolios are constructed (F-beta-AR, F-beta- AC, F-beta-MC) and the results of back-tests are plotted in gure 4.6.The strategies seem to benet from being less exposed to the broad market risk during the nancial crisis, and they are in general less volatile. The portfolio with annual rebalancing and the portfolio with annual re-clustering performs reasonable compared to the benchmark, and the AR-strategy has a slightly lager total return over the period. The strategy with monthly clustering does certainly not impress.

4.3 Partial conclusion

The general observation of the Feature selection portfolios is that, they on a not-risk-adjust basis, earns a lover return, however they are less volatile and are doing better through the nancial crisis. The reason for this is probably that the clustering ensures diversication and that the selection criteria takes risk into consideration.

Data-Driven Portfolio Management using Exchange Traded Funds