THE TIME-VARYING PROFITABILITY OF CONVERGENT AND DIVERGENT INVESTMENT STRATEGIES
Aantal woorden/ Word count: 19.934
Jeroen Vroman Stamnummer/ Student number: 01202221
Promotor/ Supervisor: Prof. Dr. Michael Frömmel Co-promotor/ Co-supervisor: Prof. Dr. Gert Elaut
Masterproef voorgedragen tot het bekomen van de graad van: Master’s Dissertation submitted to obtain the degree of:
Master of Science in Business Engineering
Academiejaar/ Academic year: 2016 - 2017
THE TIME-VARYING PROFITABILITY OF CONVERGENT AND DIVERGENT INVESTMENT STRATEGIES
Aantal woorden/ Word count: 19.934
Jeroen Vroman Stamnummer/ Student number: 01202221
Promotor/ Supervisor: Prof. Dr. Michael Frömmel Co-promotor/ Co-supervisor: Prof. Dr. Gert Elaut
Masterproef voorgedragen tot het bekomen van de graad van: Master’s Dissertation submitted to obtain the degree of:
Master of Science in Business Engineering
Academiejaar/ Academic year: 2016 - 2017
PERMISSION
Ondergetekende verklaart dat de inhoud van deze masterproef mag geraadpleegd en/of gereproduceerd worden, mits bronvermelding.
Naam student: Jeroen Vroman
Nederlandstalige Samenvatting Achtergrond Diversificatie is een probleem van alle tijden. Vooral tijdens crisis periodes worden managers beoordeeld op basis van hun mogelijkheden om een return premium te realiseren, (crisis) alfa genaamd. Naast conventionele methoden van diversificatie, zoals spreiden in aandelen, obligaties en afgeleide producten, zoeken meer investment managers naar zinvolle alternatieven. Een kader aangereikt door Rzepczynski (1999) laat ons naar risicovolle investering strategieën kijken door een convergente en divergente bril. Convergente strategen geloven dat de markt stabiel is, ze zijn aanhangers van de Efficient Market Hypothesis en geloven in de fundamentele waarderingsmethoden. Divergente strategen onderkennen hun eigen onwetendheid omtrent de echte structuur van risico’s, ze zijn aanhangers van technische analyses en leunen dichter aan bij behavioural biases. De Adaptive Market Hypothesis (Lo, 2004) probeert beide lenzen te combineren. De literatuur (Chung, Rosenberg, & Tomeo, 2004) suggereert dat een gecombineerde aanpak tot betere resultaten kan leiden. We kijken aan de hand van een geconstrueerde index (MDI) naar verschillende portefeuilles en of we aan de hand hiervan gewenstere statistische kenmerken kunnen bekomen (zoals skewness en kurtosis). Verder vergelijken we performance measures voor de verschillende portefeuilles.
Resultaten Deze thesis onderzoekt wat en hoe een divergente aanpak kan bijdragen aan een convergente aanpak op dagelijkse basis. Vooreerst worden de convergente en divergente aanpak afzonderlijk bekeken tijdens verschillende perioden van divergentie. Hierna wordt gekeken of een portefeuille, gebouwd op basis van verschillende divergentie periodes in de tijd, een extra risicoaangepaste return kan opleveren. De data suggereert dat een gecombineerde aanpak sommige zwaktes van een volledig convergente of divergente aanpak kan maskeren voor de volledige sample. In crisis periodes zijn deze resultaten minder robuust en niet altijd significant. Verder zijn de resultaten van de convergente en divergente aanpak vrij gelijkaardig, enkel tijdens de Global Financial Crisis presteert de convergente aanpak ondermaats.
Conclusie Convergent en divergent beleggen kan een interessante manier zijn van diversifiëren. De Adaptive Market Hypothesis rijkt hier een mooi kader voor aan. Toch zal elke belegger voor zichzelf moeten uitmaken hoe hij zijn risico zal spreiden. De gecombineerde aanpak slaagt erin enkele sterktes van beide aanpakken te combineren, wat op hetzelfde moment ook een verzwakking impliceert, afhankelijk van uit welk standpunt gekeken wordt. De negatieve black swans van de convergente aanpak worden verminderd, terwijl de crisis alfa component van de divergente aanpak kan worden behouden. De beste performance measures voor de full sample worden gevonden voor de gecombineerde aanpak. In afzonderlijke crisis periodes zijn de resultaten minder robuust.
Sleutelwoorden Convergent; Divergent; Hefboomfonds; CTA; Risicogedrag
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Abstract Background Diversification is a problem of all times. Especially during crisis periods, managers are judged at how well they are able to earn a return premium, which is called (crisis) alpha. Next to conventional methods of diversification, for example investing in different assets, like shares, bonds and derivatives, more and more managers try to find useful alternatives. A framework first introduced by Rzepczynski (1999) gives us the opportunity to look at risky investment strategies through a convergent and divergent lens. Convergent strategists believe that the market is stable and well structured, they are followers of the Efficient Market Hypothesis and have faith in fundamental valuation methods. Divergent strategists profess their own incomprehension to the true structure of risks, they are followers of the behavioural finance discipline and use technical analysis methods. The Adaptive Market Hypothesis (Lo, 2004) tries to combine both lenses. Literature (Chung, Rosenberg, & Tomeo, 2004) suggests that a combined approach could yield in better results. We check different portfolios, built using a constructed index (MDI), and try to find more wanted statistical characteristics (like skewness and kurtosis). Furthermore we compare different performance measures for the different portfolios.
Results This thesis explores what and how a divergent approach could contribute to a convergent approach using a daily dataset. A first look is taken at the performance of the convergent and divergent approach on their own during different periods of divergence. Hereafter a portfolio, built using different periods of divergences over time, is constructed and we check if it is able to deliver a risk- adjusted return. The data suggests that a combined approach could mask some of the weaknesses from a full convergent or divergent approach for the full sample. In crisis periods these results are less robust and not always significant. Furthermore we find that the results for the convergent and divergent approach are quite similar, with the exception during the Global Financial Crisis, where the convergent strategy performs substandard.
Conclusion Investing using a convergent and divergent framework could be interesting. The Adaptive Market Hypothesis offers a nice theoretical background to do this. Nevertheless, each investor will have to choose how he will diversify his risk. The combined approach succeeds in combining some strengths of both approaches, implying a reduction at the same time of these strengths, depending on the viewpoint. Negative black swans from the convergent approach are reduced, while at the same time the crisis alpha component can be maintained. The combined approach results in the best performance measures for the full sample. These results are less robust in separate crises periods.
Keywords Convergent; Divergent; Hedge Fund; CTA; Risk-taking
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Preface Before you lies the dissertation “The Time-Varying Profitability of Convergent and Divergent Investment Strategies”, where I outlined what these two strategies encompass and where I had the ambitious target to find an indicator to switch between them. It has been written to fulfil the graduation of the Master Program at the University of Ghent.
The project and research question were formulated together with my supervisor, Prof. Dr. Michael Frömmel. The research was difficult, but conducting extensive investigation has allowed me to answer the questions we identified.
I would like to thank my supervisor, Prof. Dr. Frömmel, who offered me the opportunity to participate in a Summer School where I was given the chance to listen and talk with Kathryn Kaminski and Alexander Mende, two experts on the field. Furthermore he and my commissioner, Dr. Gert Elaut, for their excellent guidance and support during this process. Also, I thank the professors of the Business Engineering program in general, as without the absorption of the knowledge and insights they provided during the lectures, I would not have managed to write this master thesis.
I also benefitted from debating issues with my friends and family. If I ever lost interest, you kept me motivated. My parents deserve a particular note of thanks: your wise counsel and kind words have, as always, served me well. Your personal experience, confidence and supportive guidance will help me succeed in my aim to graduate as a Master of Science in Applied Economics – Business Engineering.
I hope you enjoy your reading.
Jeroen Vroman
18th of May 2016, Ghent
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Table of Contents
Nederlandstalige Samenvatting ...... i Abstract ...... ii Preface ...... iii Table of Contents ...... iv List of Used Abbreviations ...... vi List of Tables ...... vii List of Figures ...... viii Introduction ...... 1 1 Adaptive Market Hypothesis ...... 2 1.1 Alternative Investments ...... 2 1.2 Efficient Market Hypothesis ...... 5 1.3 Behavioural Finance ...... 6 1.4 Adaptive Market Hypothesis ...... 7 2 A new perspective: Convergent and Divergent ...... 8 2.1 Convergent and Divergent Worldviews ...... 9 2.2 Convergent and Divergent Trading Behaviour ...... 12 3 Time-varying Profitability ...... 13 3.1 Performance and Risk ...... 13 3.1.1 Sharpe Ratio and its Hidden Risks ...... 14 3.1.2 Crisis Alpha ...... 15 3.1.3 Sortino & MAR Ratio ...... 16 3.2 Convergent Performance ...... 17 3.3 Divergent Performance ...... 18 4 Hypotheses Development ...... 20 4.1 Market Divergence Index ...... 20 4.2 Why and When Convergent Strategies Work ...... 22 4.3 Why and When Divergent Strategies Work ...... 22 4.4 We need to be both! ...... 23 4.5 Crisis Alpha Opportunities ...... 25 5 Data ...... 26 6 Construction of MDI ...... 27 6.1 Methodology ...... 27 6.2 Empirical Results ...... 29 Correlation with the MDI ...... 29 iv
Performance During Periods with High or Low Divergence ...... 30 7 Building of Optimal Portfolio: Combined Strategy ...... 32 7.1 Methodology ...... 32 7.2 Empirical Results ...... 36 Portfolio 2 During Full Sample ...... 36 Portfolio 2 During Crisis Periods ...... 37 Crisis alpha ...... 40 8 Results ...... 41 9 Future Research ...... 42 10 Conclusion ...... 42 References ...... 44 Appendix ...... 48
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List of Used Abbreviations AMH – Adaptive Market Hypothesis
AuM – Assets under Management
CTA – Commodity Trading Advisors
EMH – Efficient Market Hypothesis
HF – Hedge Funds
MDI – Market Divergence Index
SNR – Signal to Noise Ratio
TF – Trend Following
VAMI – Value Added Monthly Index
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List of Tables Table 1: Divergent and Convergent Worldviews (Rzepczynski, 1999) ...... 12 Table 2: Divergent and Convergent Trading/Return Behaviour (Rzepczynski, 1999) ...... 13 Table 3: Descriptive Statistics Convergent Monthly Performance...... 18 Table 4: Descriptive Statistics Divergent Monthly Performance ...... 19 Table 5: Descriptive Statistics Daily Returns ...... 26 Table 6: Correlation Table MDI - C – D ...... 30 Table 7: Descriptive Statistics linked to Market Divergence ...... 31 Table 8: Descriptive Statistics Combined Portfolios ...... 32 Table 9: Performance Measures Combined Portfolios ...... 33 Table 10: Switching Portfolios built using the MDI ...... 34 Table 11: Descriptive Statistics Portfolios built on MDI ...... 34 Table 12: Performance Measures Portfolios built on MDI ...... 35 Table 13: Descriptive Statistics Final Portfolios ...... 36 Table 14: Performance Measures Final Portfolios ...... 37 Table 15: Performance during Crises Periods ...... 39 Table 16: Yearly and Cumulative Return Portfolio 2 ...... 40 Table 17: Return Decomposition Portfolio 2 ...... 41
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List of Figures Figure 1: Asset under Management in $ billions of Hedge Funds and CTA Funds ...... 4 Figure 2: A convergent strategy has an expected positive outcome and negative skewness...... 10 Figure 3: A divergent strategy has an expected negative return and positive skewness...... 11 Figure 4: VAMI S&P500 - Convergent...... 17 Figure 5: Histogram Convergent Monthly Returns ...... 18 Figure 6: VAMI S&P500 - Divergent ...... 19 Figure 7: Histogram Divergent Monthly Returns ...... 20 Figure 8: VAMI Convergent and Divergent Daily Indices ...... 27 Figure 9: The Market Divergence Index (2003-2017) ...... 28 Figure 10: Histogram of MDI ...... 28 Figure 11: MDI with rolling 100-day returns ...... 29 Figure 12: VIX & MDI ...... 30 Figure 13: Conditional Performance as a function of MDI ...... 31 Figure 14: VAMI Combined Portfolios ...... 32 Figure 15: Portfolios built using MDI ...... 34 Figure 16: VAMIs Optimal Portfolios ...... 35 Figure 17: Crisis Alpha (Portfolio 2) ...... 40 Figure 18: Return Decomposition (Convergent, Portfolio 2, Divergent) ...... 41 Figure 19: Histogram VIX and MDI ...... 48 Figure 20: Convergent Performance as a function of Low or High Divergence ...... 48 Figure 21: Divergent Performance as a function of Low or High Divergence ...... 49 Figure 22: VAMIs during Global Financial Crisis ...... 49 Figure 23: VAMIs during Eurodebt Crisis ...... 50 Figure 24: VAMIs during Chinese Bubble ...... 50 Figure 25: VAMIs during Brexit ...... 51 Figure 26: Crisis Alpha (Convergent) ...... 51 Figure 27: Crisis Alpha (Divergent) ...... 52
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Introduction The interest in alternative investments has been growing steadily, with a small relapse in the aftermath of the Global Financial Crisis occurring around 2008. Two big groups of active traders can be distinguished in the alternative investment universe, Hedge Funds and Commodity Trading Advisors. They both pool capital from sophisticated investors to invest in different kinds of assets. We can divide them into two core camps on how they try to make money. Hedge Funds sell overvalued assets and buy undervalued ones. CTAs make money when markets move, and are unable to do this when markets reverse or are in a whipsaw situation. (Greyserman & Kaminski, 2014) These two types in the alternative investment field can be looked at through a convergent and divergent lens. Convergent strategists, the Hedge Funds, believe that they can make money using fundamental analysis, because in the long run markets will be somewhat efficient. Divergent strategist, the CTAs, on the other hand, profess their own incomprehension of markets. They use technical analysis and put more faith in the behavioural biases discipline. (Rzepczynski, 1999; Chung, et al., 2004)
A combination of the two strategies might also be a good shot to look at, because the way we look at the world, convergent or divergent, will depend on our beliefs, but maybe even more on the situations we are faced with. This combined approach can be explained under the Adaptive Market Hypothesis, an extension to the Efficient Market Hypothesis, but with more notion to things (like biases or bubbles) that seemed conflicting with the efficiency of markets. (Lo, 2004)
Using a simple portfolio measure, called the Market Divergence Index (MDI), we have the ambitious idea to build a portfolio who is able to grasp the best characteristics of the convergent and divergent approach. We explore different portfolios using our switching indicator and evaluate them using performance measures. We evaluate if the portfolio is able to deliver crisis alpha, a concept introduced by Kaminski & Mende (2011) to evaluate a manager’s performance in times of market distress. The funds in our sample report returns on daily basis, which will allow us a more efficient measure of managerial risk-taking. We will evaluate the portfolios based on statistical characteristics and talk about their risk and return distributions.
The structure of this paper will be as follows: in Section 1, we will take a closer look at the Adaptive Market Hypothesis and initiate where the convergent and divergent framework finds it origin. In Section 2 the differences between a convergent and divergent approach are more deeply tackled. In Section 3, we will have a quick look at the performance over time. We will use existing literature on convergent and divergent risk taking to develop hypotheses in Section 4. Section 5 describes the used dataset. The empirical results are described in Section 6, 7 & 8. Section 9 provides avenues for future research and Section 10 concludes.
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1 Adaptive Market Hypothesis 1.1 Alternative Investments In the last decades a lot has been written about the world of institutional investments and their managers. With two very turbulent decades just behind us, academics wondered if institutional managers are able to deliver performance. Kosowski, Naik, & Teo (2007) examined for example if Hedge Funds deliver alpha – the excess return over the predicted theoretical return. Others (Basak, Pavlova, & Shapiro, 2007) wanted to confirm or neglect that some of these managers have time management skills. This boom in financial papers on the topic of institutional investments and their managers is understandable because the asset under management (AuM) is growing. The mutual funds industry amounts to $33.4 trillion (Investment Company Fact Book, 2016) and the Hedge Fund industry manages close to $3 trillion of this (Herbst-Bayliss, 2017). Investors want to know if the fees they pay are worth it (Vukovic, 2004) and where they get the best returns.
Most investors are looking for the best performing manager and or strategy. They search for new vehicles of diversification and more and more research is being done to deliver the biggest returns or to limit the risks. Taking the growth in AuM in consideration, alternative investments appear to become a bigger part in people’s portfolios. Alternative investments are assets that are not conventional, such as cash, bonds and equity stocks. They include private equity, Hedge Funds, managed futures, commodities, real estate and derivatives contracts (Fung & Hsieh, 1999). Because investors are looking for the next big winning strategy, we encounter an eventual degradation of alpha. What once was alpha, will become beta due to competition and innovation. (Greyserman & Kaminski, 2014). In a basic risk/reward relationship the expected excess return of a portfolio is composed of alpha plus beta times the expected return premium plus an error term. Alpha expresses the return a manager performs above the normal speculative risk premium. Beta measures the exposure to market fluctuations (Frömmel, 2013).
퐸(푟푝,푡 − 푟푓) = 훼푝 + 훽푝퐸(휋푝) + 휀푝,푡
Eq. 1: Risk/reward Relationship
The second important takeaway for investors is of course the risk they are willing to take. The way people handle risk is about their beliefs, perspectives and in combination with their past experiences. (Greyserman & Kaminski, 2014) People are constantly confronted with risk and risk taking. It might happen that you end up being late for something and you decide to drive faster than the speed limit. This is an example where you take risk yourself. But it could also happen that you are the good responsible driver but someone else is driving too fast and brings you into danger. This is a different example where you are confronted with risk. We are exposed to risks almost daily in our personal lives.
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But we also encounter risk and risk taking in our financial lives. “Risk taking is at the core of human experience” (Greyserman & Kaminski, 2014, p. 93). If we mirror the first example to our financial lives, where someone makes the decision himself, someone might choose where to allocate his funds to. The second example might occur if we encounter market risk. This market risk is mostly divided in two parts: systematic risk and unsystematic risk. The general belief is that the latter risk can be diversified away if your portfolio consists of enough different asset types or if your portfolio consists of enough assets. (Montgomery & Singh, 1984)
Alternative investment managers also try to tackle risk. We will first make a distinction in the asset class which we will use further through this paper. Two big subcategories can be formed if we look at the way alternative investment managers handle risk. The first being Hedge Funds and the second one being managed futures managers or CTAs.
Hedge Funds pool money from investors to invest in a wide range of assets, often using sophisticated techniques to manage their portfolios. They can operate with greater flexibility compared to other investment funds due to less regulation. (Fung & Hsieh, 1999) They are only available to accredited individuals and organisations. A Hedge Fund has managers who usually invest a portion of their personal wealth to ensure the alignment of interest among the partners. The other partners are of course the investors, which have to pay a management fee which is mostly performance based (Kouwenberg & Ziemb, 2007). The name “hedge fund” is believed to come from Alfred Winslow Jones. This man started the strategy around 1950 by the key insight that a manager could buy stocks who were undervalued and sell short stocks that he believed were overvalued. He was the first to combine long and short positions in order to hedge market risk (Jones A. W., 2017). Nowadays there is an extreme variety amongst different Hedge Funds. A classification commonly used distinguishes four categories: equity hedge (= long/short), event driven, macro or global trading and relative value. In the first category the Hedge Fund has the freedom to short securities (as compared to traditional investment vehicles who are only allowed to go long). The second category, event-driven funds try to profit when prices change due to certain corporate actions, like mergers, acquisitions or bankruptcy. Macro or global funds try to make tactical decisions about an optimal global asset allocation mix. Finally, the fourth category, relative value, aims at mispriced financial assets. They rely on market- efficiency and believe that prices will revert to equilibrium levels (Morningstar, 2007). A second classification is possible taking the location into account. The location here represents the asset class where the fund will invest in. We can discern equity, commodities, currencies, bonds or diversified, a combination of all of these (Fung & Hiesh, 1997).
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The second subcategory in the alternative investments class are CTAs. CTA stands for Commodity Trading Advisor and stems from the underlying assets in which they trade, which were in the beginning commodities. Nowadays they trade within different kinds of managed futures classes. They trade in futures on a futures exchange. A future is regulated, settlement will be done daily and the underlying can be sold anytime during the duration of the contract. The big advantage is that there is no counterparty risk, there is no fee to short it. A margin has to be put in and losses or gains will be taken from your futures account (Schneeweis, 2001). They also trade in forward markets, options, foreign exchange and derivatives. They apply leverage using their margin or via the use of derivatives. Two main strategies can be disclosed: systematic (fully automated) or discretionary strategies, in which the latter present a strategy were the manager has a level of manager discretion. The former consists of 4 subcategories: medium/long-term trend following, short-term trading, fundamental trading and relative value/non-trend. The first category searches for trends and uses signals from historical prices to determine when these trends happen. The second category focusses on a shorter term but does the same. The third category, fundamental managers, looks at mispricing. And the final category tries to find mispricing between different markets or across time (Greyserman & Kaminski, 2014). Trend Following is the core strategy, we will talk about this more in detail in Section 2.
Although the line between the two subcategories might not always be clear, Fung & Hsieh (1997) talk about a blurred distinction, we will present an approach in Section 2 where the differences will become clearer. Figure 1 graphs the Asset under Management (AuM) for Hedge Funds as well as for CTAs since 1988 (BarclayHedge, 2017).
AuM
$3 000,00
$2 500,00
$2 000,00
$1 500,00
$1 000,00
$ 500,00
$-
2008 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2009 2010 2011 2012 2013 2014 2015 2016
HF($ Billions) CTA($ Billions)
Figure 1: Asset under Management in $ billions of Hedge Funds and CTA Funds
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A closer look at the two risk taking strategies is necessary because risk is at the core of human experience. How we handle and form our expectations will have an impact on our financial performance. Before we do this, we will use existing literature to push these two alternative investment strategies in a box. The Hedge Fund approach will be explained using the Efficient Market Hypothesis and the CTA approach will be explained using technical analysis and behavioural biases.
1.2 Efficient Market Hypothesis ‘A market in which prices always "fully reflect" available information is called "efficient."’
– Fama (1970, p. 383)
The Nobel Prize of 2013 was awarded to two groups of researchers who investigated empirical asset pricing. These two groups both had a very different financial worldview. One worldview was the EMH and the other group believed in behavioural biases. Because they both received the Nobel Prize we can hardly ignore them. Might it be that there is some truth in both financial worldviews?
Eugene Fama (1970) developed the EMH because he argued that market prices incorporate at any time their fair value, using all information rationally and instantaneously. This implicates that it would be impossible to “beat the market”. Market prices should only react to new information and therefor it is not possible to outperform the market either using stock selection or market timing. An investor can only obtain a higher return by pure chance or by picking an investment with a higher risk (Malkiel, 2005). The EMH is highly discussed upon because it would mean that technical analysis and portfolio diversification would not add value. Malkiel (1989) and Fama (1970) argued that there were three variants of his hypothesis. Ranging from a “weak”, to “semi-strong” to “strong” form of efficiency. The weakest form claims that all past information is already incorporated. The semi-strong form adds all publicly available information and the necessity that all prices change instantly when new information becomes public. In the “strong” form of the EMH inside information is added as well. The emerging disciplines of behavioural economics have challenged this hypothesis, but Fama (1970) and his supporters defended the hypothesis stating that the EMH is a simplification of the world and that the market is close to efficient for most individual investors. In his review, Fama (1991) expressed and emphasized that the strong form is surely false and only useful as a benchmark. Two problems occurred when Fama (1991) wondered what fully reflected mean. The first being an empirical problem and the second a theoretical one. The empirical problem or joint hypothesis problem cannot be solved. The hypothesis tries to answer if the prices we have valued are correct and in the case we find a deviation if either there is a misalignment of the asset or if we just used a wrong model to price the asset (Frömmel, 2013). Grossman & Stiglitz (1980) describe the theoretical problem with the EMH, namely the information paradox. This paradox wonders if extra analysis generates extra profits,
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because if markets are truly efficient, all the information is already included in the prices. There is no incentive for financial institutions to collect more information in a strict efficient market.
This short introduction to the EMH relates to Hedge Fund managers as well, they believe the market is most of the time efficient (weakly or semi-strong), but also that doing fundamental value analysis is still worth it. Hedge Funds exploit some of the mispricing in the market, because not everybody has always the same level of information they do. They believe that they can outperform the market in the short run and handle faster to short and small deviations from equilibriums and therefore are able to deliver a positive alpha.
1.3 Behavioural Finance On the other side of the winning table at the Nobel Prize Ceremony sat Robert Shiller. He is said to be the father of behavioural finance. Shiller (1981) challenged the EMH, which was the dominant view in economics back then. As stated above we can say that proponents of the EMF believe that the market incorporates all available information. Saying this would mean that the market is efficient. If we quickly look back at for example the past 20 years, markets have behaved rather inefficient (e.g. Dotcom Bubble (1997-2000) and Global Financial Crisis(2007-2009)). In the EMH framework gaining an abnormal return is said to be impossible. Behavioural finance stands in contradiction with the efficient markets theory. It incorporates a psychological and sociological approach to look at finance. It uses these social sciences in order to counter the wishful thinking which was proposed by the EMH (Shiller, 2003). Behavioural finance helps us explaining why markets are not (always) efficient. They state that humans trade with behavioural biases, which causes them to act on emotion. The markets behave inefficient because of human interference. In the EMH all agents are believed to be rational, biases give examples where agents forget their rational framework. We will not try to fully grasp all biases, but four examples are explained.
Examples of biases are overconfidence, reducing regret, limited attention span and chasing trends. Overconfidence happens when people overestimate their knowledge and think they are better than the average. This happens for example when investing in something you are familiar with. Reducing regret (or conservatism) occurs when an agent is confident of his strategy and refuses to admit that he or she was wrong, often building up more losses compared to a rational agent. Limited attention span occurs when people only pay attention to stocks that come to their attention. They do not take enough time to go over all the alternatives. Finally chasing trends (or herding) occurs when a lot of people start buying or selling the same assets (Ritter, 2003).
Behavioural finance further helps us explaining certain anomalies. “An empirical result is anomalous if it is difficult to "rationalize," or if implausible assumptions are necessary to explain it within the
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paradigm.” (Thaler, 1987, p. 169) Examples are calendar anomalies, fundamental anomalies or technical anomalies. The January effect is a calendar anomaly, where prices increased during the first half of January. When a positive relation between a value-related variable (like book-to-market) exists and the security’s returns, we talk about a value effect. The value effect is a fundamental anomaly. Finally, the momentum effect is a technical anomaly. It is the empirical observation that winners in the past also seem to be future winners (Frömmel, 2013).
This short introduction to behavioural finance relates to CTAs as well, they care less about market efficiency. CTAs exploit mispricing in the market, and even thrive when certain anomalies occur or when other investors suffer from biases (like herding). Now that we have given short overviews of the EMH and behavioural finance, it is time to go to the Adaptive Markets Hypothesis, which was presented and developed by Andrew Lo (2004).
1.4 Adaptive Market Hypothesis ‘The Adaptive Market Hypothesis (AMH) is a framework for explaining how markets behave using principles from evolutionary biology.’ - Greyserman & Kaminski (2014, p. 65)
Andrew Lo presented in 2004 a new perspective that reconciles the two biggest opposing schools in finance, the EMH and behavioural finance. Financial practitioners and academics had by then started to agree that the EMH often neglected certain aspects of financial markets. They also agreed that markets are not always rational, “but can be driven by fear and greed instead” (Lo, 2004). Still, no consensus has been reached on how efficient markets actually are. Andrew Lo added a biological and evolutionary perspective to the EMH on top of the sociological and psychological perspectives already added by the behavioural finance discipline. He realised – amongst others – that there was some truth in both worldviews. With this lens he explained the adaptive and evolving environment of financial markets (Lo, 2005).
Lo (2004) started explaining his new synthesis by adopting terms from evolutionary biology. He explains the different drivers of evolution in the market, being competition, mutation, reproduction and natural selection. Individuals will learn from their past experiences and when competition increases, the agents who are stronger and have a competitive advantage over others, will adapt and avoid extinction (Greyserman & Kaminski, 2014). If the market environment changes, the heuristics might change as well. It might occur that we observe behavioural biases in this context. But behavioural biases and other counterexamples to economic reality are just examples of individuals who are changing their behaviour to the new environment. They are not irrational, but should be called “maladaptive”. Lo (2004, p.17) explains it as follows: “The flopping of a fish on dry land may seem strange and unproductive, but under water, the same motions are capable of propelling the fish away
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from its predators”. Through the lens of evolutionary biology, contradictions between the EMH and behavioural biases can be better understood. Agents in the market are limited in their abilities and won’t always make optimal decisions, but rather satisfactory ones (Lo, 2010).
The AMH has five implications. The first implication is that from an investment perspective the relation between risk and reward exists, but varies due to the population of market agents and the market environment, it is not constant over time. The second implication is very important and states that market efficiency might change from time to time, it is a continuum. This would mean that arbitrage opportunities exist from time to time, and gives a motivation for portfolio management (Lo, 2011). The third one implies that “investment strategies will wax and wane, performing well in certain environments and performing poorly in other environments” (Lo, 2004, p. 22). This means that due to changing business conditions and the number of agents entering and exiting the market, an investment strategy could thrive or wither. The fourth implication tells us that because the risk/reward relationship varies over time, adaptation is key to survival. Your alpha may be positive from time to time, but your unique investment opportunity will be discovered and adopted by others resulting in the decay of alpha (Lo, 2011). The fifth and final implication is that “survival is the only thing that matters. While profit maximization, utility maximization, and general equilibrium are certainly relevant aspects of market ecology, the organizing principle in determining the evolution of markets and financial technology is simply survival”. (Lo, 2004, p.24)
Dislocations as well as anomalies get a place in this theory and can be explained by the evolutionary perspective. There is place for alternative investments in the academic world. These alternative investments have long been ignored and neglected by proponents of the EMH. The key takeaway is that the Adaptive Market Hypothesis is consistent with both the Efficient Market Hypothesis and the behavioural finance discipline. Using this evolutionary biological theory it is now time to link the AMH from Andrew Lo (2004) with the terminology by Mark Rzepczynski (1999).
2 A new perspective: Convergent and Divergent Using the AMH we can understand how the human species has developed a variety of strategies to handle the risk it faces. We might have to make choices on where to allocate our resources in, which investment style we are going to follow or when to enter and exit markets. In all these decisions we encounter some level of risk or uncertainty. The difference is not always clear, but Greyserman & Kaminski (2014, p. 94) describe risk as “the chance things will not turn out as expected. Uncertainty, perhaps ominous, is the situation where the consequences, extent or magnitude of circumstances, conditions or events is unknown.” They represent the concept of risk by an Urn A filled with 100 balls, of which 50 red and 50 black. Betting on which colour you would pick involves taking risk. Uncertainty
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is represented by Urn B, where you do not know the distribution of the colours. You engage in an uncertain decision. The uncertainty in Urn B makes most of us uncomfortable (Greyserman & Kaminski, 2014). Fortunately, we can partially reduce this uncertainty in finance. Unfortunately, this also means that models fail sometimes due to unquantifiable parts. On a spectrum where physics and mathematics are on the left side, i.e. where uncertainty and risk are reducible, and religion on the right side, i.e. where uncertainty and risk are irreducible and unquantifiable, finance, psychology and economics would be right in the middle (Lo & Mueller, 2010). The choices we make daily in our lives, the amount of risk we are willing to take and the behaviour driving these decisions depends on our (financial) worldview. Rzepczynski (1999) introduced a two folded worldview: a convergent one and a divergent one. The first worldview is a convergent one. The world of a convergent risk taker or manager is stable, somewhat knowable and well structured. The markets they trade in may be temporarily over- or undervalued. The second worldview is a divergent one. Their world is rather nonstationary and quite uncertain (Rzepczynski, 1999).
2.1 Convergent and Divergent Worldviews Rzepczynski (1999) made the first distinction using these worldviews. According to him the convergent worldview contains of a stationary and stable world. The convergent strategist believes his world is static, knowable and that errors are random. In case there are divergences, they are short and the markets adapt quickly, this is why the link can be made with the EMH, where markets were said to be efficient. In the short run a convergent strategist believes that the fundamentals do not change. He forms rational expectations (Rzepczynski, 1999). Let us play a game to explain how a convergent strategist would handle. Imagine that you invest some amount X in a game. In the case you win, you take your money and play another game. In the case you lose, you double up, because you believe that you were right from the beginning. You keep doing this until you finally win. In this convergent strategy you will experience many small wins and occasionally a devastating loss. If we translate this to the distribution, we can state that it is heavily left, negatively skewed with a fat tail. People who use this strategy have trust in their fundamentals, they have strong conviction that in the long run they will be correct. In case their strategy does not pay off in the short run, they wait until their beliefs are reconfirmed. A player playing this strategy is certain about the outcome and expects to win small amounts every single time. In the case of a loss, he might lose a lot (Greyserman & Kaminski, 2014). The payoff of this strategy is given in Figure 2.
Before we go to a divergent risk taker we will give two examples of convergent risk taking from our daily lives. If we cross the street, we believe (and hope) that we will make it safely to the other side. We take a convergent risk. In the unlikely event that we get run over by a car, the outcome could be catastrophic. A second example can be found in social networking. If we only talk to people we already
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know and have a relationship with, we will not create a new potential interesting network, we might miss a big opportunity to meet someone famous for example. (Greyserman & Kaminski, 2014)
Convergent risk taking 70,0%
60,0%
50,0%
40,0%
30,0% Probability 20,0%
10,0%
0,0% -120 -60 -30 -15 -5 -1 1 Outcome
Figure 2: A convergent strategy has an expected positive outcome and negative skewness.
In the financial world most investors believe in investing in equities. They have confidence in the long run that models can truly predict the fundamental values and will build a long/short portfolio. If we look at the distribution of an investment in equity markets, we see that their returns are positive in expectation, and we see fat left tails, meaning they are negatively skewed. Convergent investors believe in the existence of an equity risk premium and the efficiency of markets, following the EMH (Chung, Rosenberg, & Tomeo, 2004). Table 1 gives an overview of the two worldviews (page 12).
A divergent risk taker on the other hand believes in a dynamic world, where divergence can exist in a longer run. Their world is nonstationary and biases from behavioural finance find a place. Divergence is linked to volatility. Greyserman & Kaminski (2014, p.94) describe volatility as “the risk or uncertainty regarding a position in an underlying security or asset”. Rzepczynski (1999) stated that in a divergent world information takes time to be adjusted. The fundamental believes are less valuable and changes in these fundamentals are often unanticipated. A lot of uncertainty exists and market participants form rational beliefs but they can make mistakes and have biases. Let us now play the same game from above. If we are a divergent strategist, we believe that there are no strict rules, our world is uncertain and dynamic. In case we win, we double up. We profess our ignorance to the impeding risk. In case we lose, we cut our losses. This strategy will experience small losses and in the rare occasions we will win big. The return distribution of this divergent strategy is heavily right skewed (so positive) with fat tails. A person who uses this strategy has little faith when losing. If they lose they want to be out as soon as possible. In the other case, when their strategy does pay off, they follow their streak of luck and even
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double up. A player playing this strategy is quite comfortable with losing, he almost expects the losses. He never invests too much in one particular game and hopes for the euphoric wins (Greyserman & Kaminski, 2014). The payoff of this strategy can be found in Figure 3.
Divergent risk taking 70,0%
60,0%
50,0%
40,0%
30,0% Probability 20,0%
10,0%
0,0% -1 1 5 15 30 60 120 Outcome
Figure 3: A divergent strategy has an expected negative return and positive skewness.
If we mirror the social networking event to a divergent risk taker, he will talk to as many new people as possible and hope he meets the guy who wants to invest millions in his start-up (Greyserman & Kaminski, 2014). A second example can be found in the drug development business. Big pharmaceutical companies push thousands of potential drugs through the funnel and hope that one of them will pay off big in the long run. A financial example of divergent risk taking is found in growth companies and with venture capitalist. They make investments in start-ups and know from the beginning that not every start-up in their portfolio will become the next Google. But they hope they do have a new Spotify in their wallet. A second example is trend following. Trend followers guess that markets are driven by the animal spirits from the behavioural finance perspective and that periods of opportunities are apparent. They ignore the fundamentals and follow the trends. The distribution of trend following strategies has positive skewness with positive expectations. Divergent investors are closer to the technical side of making money (Chung, Rosenberg, & Tomeo, 2004).
Now that we have tackled the divergent and convergent worldview, we will give some more attention to their statistical properties and talk about their return distributions.
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Convergent financial worldview Divergent financial worldview Stationary and stable world Nonstationary World is knowable and static; structural World is uncertain and dynamic; knowledge structural ignorance Market participants generally form Market participants form rational rational expectations; errors are beliefs but may make mistakes and random have biases Markets adjust relatively to new Learning takes time; slower information adjustments to information Divergences from equilibrium are short Divergences exist and may be dramatic lived from time to time Fundamentals do not change Fundamental changes are often dramatically in the short run unanticipated
Table 1: Divergent and Convergent Worldviews (Rzepczynski, 1999) 2.2 Convergent and Divergent Trading Behaviour Now that we have a better grasp of the two worldviews we can adopt them to the trading and return behaviour of investors. Convergent risk takers will have a strong belief in fair value, they will use fundamental analysis to earn profits. Long-term risk premiums and reversion to the mean are examples of how profits are made. They will be achieved through arbitrage trading, contrarian or value trading. A divergent risk taker has no prediction about the fair value, he will make profits when extreme events take place. Trend following and momentum strategies are the best examples. If we focus on the trading and return characteristics of the convergent and divergent strategies, a convergent strategy focuses on small wins with an occasional big loss, resulting in a concave payoff with negative skewness. A divergent strategy on the other hand will have small losses with an occasional big win, resulting in a convex payoff with positive skewness (Rzepczynski, 1999). The differences in convexity between the two strategies are important. Convexity is the concept where the initial input or risk is magnified by a factor much larger than the input. A convergent strategy has negative convexity, meaning that most of the time it earns lots of small gains but with the occasional black swan where a gigantic loss appears. A divergent strategy has positive convexity, where a small amount can yield a significantly larger outcome. An example of a negative convexity event is for example a market crisis (Greyserman & Kaminski, 2014). The skewness is a measure of the symmetry in a distribution. A symmetrical dataset will have a skewness equal to 0. So, a normal distribution will have a skewness of 0. Skewness essentially measures the symmetry and relative sizes of the two tails. Important from an investor perspective is to obtain positive skewness to avoid the heavy left (negative) tails. Table 2 summarizes.
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Both risk taking strategies make sense, the different payouts and utilities depends on someone’s targets, our decision making and behaviour surrounding risks includes both convergent and divergent approaches. It is clear that the risk structure of an asset is important to define if the strategy will be successful or not. The level of risk or uncertainty may indicate the potential success of the strategies. Divergent strategies flourish during times with high uncertainty. This relationship is of course time varying. We will discuss the performance over time in the next subsection.
The AMH believes that both the EMH and behavioural biases tell a part of the bigger truth, which is why a combination of divergent and convergent strategies can yield a better payoff. Greyserman & Kaminski (2014, p 100) summarize it nicely “convergent risk taking allows us to compete and maintain value over time while exposing us to hidden risks (the so called black swans) whereas divergent risk taking allows us to adapt, innovate, and hopefully survive during periods of markets distress”.
Convergent behaviour Divergent behaviour Strong sense of fair value No prediction of fair value Arbitrage trading, value trading, Trend following, momentum contrarian Profits made from reversion to the Profits made from the extremes, the mean or long-term risk premiums mean fleeting events Negative convexity Positive convexity Concave payoffs Convex payoffs Negative skewness Positive skewness
Table 2: Divergent and Convergent Trading/Return Behaviour (Rzepczynski, 1999) 3 Time-varying Profitability Now that we have a clearer view on what convergent and divergent strategies are, it is time to have a look at their performance over the years. Before we make a distinction between the two worldviews, we talk about four metrics that allow to quantify the performance of an alternative investment strategy. We introduce the concept crisis alpha and end this paragraph with a monthly performance overview for both a convergent and divergent strategy.
3.1 Performance and Risk The performance of a portfolio can be evaluated based on lots of statistical parameters. In general there is a threefold way to break the return of a portfolio down. It encompasses the risk free rate, a speculative risk premium and the alpha generated by the manager (recall Equation 1) (Frömmel, 2013). The last parameter is becoming more and more important. A manager who has the ability to deliver a good return, higher than the expected return (e.g. calculated by the CAPM) will be highly valued. One
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approach to tackle the performance of a portfolio is by breaking it down using the portfolio beta and alpha. If the alpha is positive, the manager has obtained his target in reaching a bigger return than expected. But it is of course not that simple, a manager with a higher alpha compared to another manager is not always better, we have to take the risk into account as well (Markovitz, 1991).
3.1.1 Sharpe Ratio and its Hidden Risks A good measure, widely accepted to compare returns over different portfolios and asset classes is the Sharpe ratio. It enables us to examine the performance of an investment by adjusting it for risk. It measures the risk premium per unit of deviation and is named after William F. Sharpe (1994). To calculate the Sharpe ratio we simply take the average return 푟̅ of an asset or portfolio minus the risk free rate rf and divide it by the standard deviation 휎. The higher the Sharpe ratio, the better. The Sharpe ratio will give correct results if returns are normally distributed and only if the investments being considered are independent of the rest of our portfolio. It does not take into account correlations, it ignores the interactions between the additional investment and the ones already in the portfolio (Dowd, 2000).
푟̅ − 푟푓 푆푅 = 휎
Eq. 2: Sharpe Ratio
Generally Hedge Fund’s Sharpe ratios (convergent strategist) range from 0.5 to 1.5, whereas the typical Sharpe ratio for equities (long-only) is 0.25 to 1.5. Sharpe ratios for trend following strategies (divergent strategist) range from 0.5 to 0.7 (Greyserman & Kaminski, 2014).
As suggested by Dowd (2000), the Sharpe ratio is not always a good measure to compare investments. Kaminski & Mende (2011) discuss four main sources of risk in alternative investments, and apply the term “hidden risks” to risks that the Sharpe ratio does not fully grasps and even underestimates. Price risk, credit risk, liquidity risk and leverage risk pertain to those four. We will encounter price- or market- risk when the price of an asset or portfolio moves in an unfavourable direction in the future. It is mostly found in traditional investments and can be measured by the mean reversion. Credit risk is associated with a counterparty who is unable to meet the requirements on their side of the contract, the correlation with the Ted spread1 is useful as a proxy. Liquidity risk, which can be measured by autocorrelation, is defined as the risk when an asset cannot quickly enough be sold (or bought) to avoid a loss or minimize it. It occurs when there is a lack of marketability (Pástor & Stambaugh, 2003). The fourth hidden risk is leverage risk. It is defined “as taking exposure based on borrowed funds. One
1 “The difference between the interest rate at which the US Government is able to borrow on a three month period (T-bill) and the rate at which banks lend to each other on a three month period.” (Financial Times, 2017) 14
tricky thing with leverage is that it can be achieved by either borrowing funds directly or using derivatives that have implicit leverage” (Greyserman & Kaminski, 2014, p. 194). A final note is that it is important to only compare Sharpe ratios for data with the same frequency. Because the Sharpe ratios for relative value investments mainly grasp the price risk, which is mean reverting over the long run, they will decrease over the long run going from daily to yearly data (Lo, 2002).
3.1.2 Crisis Alpha To suffer devastating losses during an equity market crisis is the nightmare of most investors. The dream in true diversification is to find an investment that can overcome these periods of market distress. A measure to track these opportunities is crisis alpha. It “represent the difference between the original alternative investment strategy and the strategy without crisis periods where the performance of the strategy is substituted with an investment in the short-term debt rate.” (Kaminski & Mende, 2011, p. 2) Most of the times a 3-month T-bill is used as a proxy for the short-term rate.
A crisis period can be defined by many alternative definitions, but before we define crisis, we explain in brief how an equity crisis could occur (of course many other reasons exist and we see this as a major generalization). Hedge Funds, as a part of the vast majority of investors, are long biased and when equity markets go down (regardless of the way how or why), they will realize losses. When people realize losses they will be more susceptible by emotional based decision-making and behavioural biases. Factors of adaptation as well as institutional restrictions may come into play. Leverage and risk limits – which will be triggered by the losses and increased volatility – will drive the herd into action. Liquidity disappears and credit issues arise, people will go into crisis mode. They will meet their limits, get less money and will come to a point where they are forced to sell. Panic will spread, fundamental valuation becomes less relevant and persistent trends occur across markets. At the same time, investors will try to change their positions and meet the borders of liquidity. CTAs might even extend these trends because they will try to exploit it (Campello, Graham & Harvey, 2010; Greyserman & Kaminski, 2014; Kaminski, 2011; Taylor, 2009). The best performance for CTAs emerge during the best and worst moments for equities. This so called “CTA-smile” is driven by CTAs ability to remain competitive during periods of market stress as well as grasp a part of the bullish markets (Bradshaw, Hutton, Marcus, & Tehranian, 2010). The performance of trend followers (which is the biggest subclass in CTAs) and CTAs in general is best “during the best and worst moments for equities” (Greyserman & Kaminski, 2014, p. 15), they are able to exploit these trends.
What a crisis period exactly is, depends on the definition. There is no generally accepted formal definition of bull and bear markets in finance literature (Lunde & Timmerman, 2004). Greyserman & Kaminski (2014, p.81) say that a crisis “can be defined as a sequence of negative returns, which represents a drawdown below some threshold” or “months when the index return is lower than a
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specific threshold (Greyserman & Kaminski, 2014, p. 148). Pagan & Sossounov (2002) use an algorithm (built on the work of Lunde & Timmerman (2004)) to define a turning point when to go from a bullish to a bearish market. Maheu & McCurdy (2000) use increased volatility to identify bear markets. The Financial Times defines a crisis as “a situation when, for reasons that may not [be] necessarily grounded in accurate information or apparent logic, parties to financial contracts in many nations simultaneously conclude that the contracts they hold are unlikely be honoured by counterparties or that the financial assets that they hold are likely to be worth substantially less than previously thought.” What hopefully became clear is that many definitions exist and that many of them are qualitative of nature, defining a crisis quantitatively is not easy and will depend on the definition of the reader.
Important to remember is that crisis alpha captures the positive performance of an investment (above the risk free rate) during such a crisis. Crisis alpha yields profits which are gained by exploiting the trends that occur during these periods of crisis, we will call them crisis alpha opportunities (Greyserman & Kaminski, 2014).
3.1.3 Sortino & MAR Ratio Some risks cannot be detected by the traditional Sharpe Ratio, which is why we also look at other alternative investment ratios. The first being the Sortino Ratio and the second one being the Managed Account Reports Ratio (MAR Ratio). Both put more stress on “bad volatility”, the underlying idea is that deviation to the upside is harmless and deviation to the downside hurts the most. (Bacmann & Scholz, 2003)
The Sortino ratio measures the risk-adjusted return of an investment, just like the Sharpe Ratio. But it replaces the standard deviation in the denominator by the downside standard deviation, which results in an adjustment just for downside volatility. It is therefore more able to grasp some of the skewness and kurtosis of returns as opposed to the Sharpe Ratio which only takes the first and second moment of the return distribution into consideration. Despite the fact that they both measure risk-adjusted returns, their significant difference will often lead to different conclusions. Only when the return distributions are close to a normal one (skewness close to 0, kurtosis around 3 and the mean not too far away from the median), they will produce similar results. (Chaudhry & Johnson, 2008) To calculate the Sortino ratio we simply take the average return 푟̅ of an asset or portfolio minus the risk free rate rf
(as we did for the Sharpe ration) and divide it by the downside standard deviation 휎푑. The higher the Sortino ratio, the better.
A final measure is the MAR Ratio, it also looks at downside risk but in a different way than the Sortino ratio. To calculate the MAR Ratio, we need to find the maximum drawdown of the portfolio. A drawdown is the peak-to-trough drop during a certain period. It is expressed as a percentage and
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measures the worst case scenario you could have when buying at the highest point (peak) and waiting without selling it, until it reaches the trough. To calculate the MAR Ratio we simply take the average annual return and divide it by the absolute value of the maximum drawdown. Literature is mixed on which performance measures result in the most honest conclusions (Eling & Schuhmacher, 2007).
3.2 Convergent Performance For the performance of a convergent risk taking strategy we will take a closer look at a Hedge Fund index as done by Chung, Rosenberg, & Tomeo (2004). The objective is not to derive strict statistical conclusions but rather to describe the theory in a more illustrative way. We also justify briefly the choice of a Hedge Fund index as a convergent approach. Monthly data from Barclay/GHS Hedge Index (Barclay) is used, it was provided by RPM Risk & Portfolio Management (Sweden). The sample covers the period December 1996 to February 2017.
Convergent - S&P500 600
500
400
300
200
100
0
jun/98 jun/01 jun/04 jun/07 jun/10 jun/13 jun/16
sep/12 sep/97 sep/00 sep/03 sep/06 sep/09 sep/15
dec/96 dec/99 dec/02 dec/05 dec/08 dec/11 dec/14
mrt/99 mrt/02 mrt/05 mrt/08 mrt/11 mrt/14
S&P500 Convergent Index
Figure 4: VAMI S&P500 - Convergent
In Figure 4 we compare the Value Added Monthly Index (VAMI) of the convergent investment with the S&P500. At first sight we discover a quite high correlation with the S&P index. This may support in some way the thinking of a convergent strategist, who believes that market prices are correct in the long run. Although they seem to have managed to outperform the market, which may presume that the AMH is correct and that some players can outperform the market. Rzepczynski (1999, p.80) states that in a convergent strategy, “there is the underlying belief that the market has mispriced the value of a firm and that through strong research an information advantage can be created. […] profits for convergent traders will be made against the averages of a price distribution. Increased volatility or significant market moves suggest uncertainty and instability, which should be avoided. Under changing market conditions, structural knowledge will decline and return generation may dissipate.” We also see a drop occurring around October 2008 which might suggest the notes we made about risks and
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that “Convergent risk taking strategies, as a class, tend to hold hidden risks, which come out during periods of stress” (Greyserman & Kaminski, 2014, p.188). Table 3 summarizes the descriptive statistics, compared to the S&P500. Finally we will take a look at the Histogram from the returns. Figure 5 shows the Histogram. We see a small negative skewness. This is in line with the trading and return behaviour from Table 2 by Rzepczynski (1999).
Return S&P500 Return C Annualized Mean 5,61% 8,61% Monthly Mean 0,5760% 0,7083% Median 0,0100 0,0079 Standard Deviation 0,0439 0,0203 Kurtosis 1,0586 3,4750 Skewness -0,6288 -0,6179 Minimum -0,1694 -0,0841 Maximum 0,1077 0,0773 Observations 243 243 Table 3: Descriptive Statistics Convergent Monthly Performance2
Return C
32
28
24
20
y
t
i
s 16
n
e
D 12
8
4
0 -.10 -.08 -.06 -.04 -.02 .00 .02 .04 .06 .08 .10 .12
Histogram Normal Kernel Figure 5: Histogram Convergent Monthly Returns 3.3 Divergent Performance For the performance of a divergent risk taking strategy we will take a closer look at an index composed of CTAs. The choice for a CTA index will be explained hereafter. Monthly data from Barclay CTA Index (Barclay) is used, it was provided by RPM Risk & Portfolio Management (Sweden). The sample covers the period December 1996 to February 2017. Figure 5 compares the VAMI of the divergent strategy with the S&P500. It becomes quite clear that there is no real correlation between the two, and sometimes the correlation might even be negative. This indicates that divergent strategist put less
2 Kaminski & Mende (2011) find an annual mean of 10.7% for the period January 1997 to January 2011 using monthly data. 18
value in the underlying value and valuation methods, they care less about the fundamentals. To further discuss this approach we will talk about the biggest group in this index, being trend followers.
A trend follower will systematically cut his losses when they occur. They take lots of small positions in different markets and are agnostic to the reasons why things happen. They just follow the trends when they occur. Ideologically they have no biases, they are completely systematic and not discretionary. Greyserman & Kaminski (2014) also look at empirical results of trend followers and they see the ideological reasons confirmed as well. Trend followers have returns with positive skew, the wins are concentred by extremely large events and drawdowns are correlated with moments of small losses. All these results suggest that trend following is consistent with a divergent strategy.
Divergent - S&P500 350 300 250 200 150 100 50
0
apr/98 apr/00 apr/02 apr/04 apr/06 apr/08 apr/10 apr/12 apr/14 apr/16
dec/96 dec/98 dec/00 dec/02 dec/04 dec/06 dec/08 dec/10 dec/12 dec/14 dec/16
aug/97 aug/99 aug/01 aug/03 aug/05 aug/07 aug/09 aug/11 aug/13 aug/15
S&P500 Divergent Index
Figure 6: VAMI S&P500 - Divergent
Return S&P500 Return D Annualized Mean 5,61% 3,94% Mean 0,5760% 0,3397% Median 0,0100 0,0018 Standard Deviation 0,0439 0,0192 Kurtosis 1,0586 0,6583 Skewness -0,6288 0,3976 Minimum -0,1694 -0,0461 Maximum 0,1077 0,0645 Observations 243 243 Table 4: Descriptive Statistics Divergent Monthly Performance3
3 Kaminski & Mende (2011) find an annual mean of 5.9% for the period January 1997 to January 2011 using monthly data. 19
What finally becomes clear from Figure 6 is that a divergent approach can be our seatbelt in times of market distress. If we look at the months surrounding October 2008 we see a big drop in the S&P500 value, but the divergent index remains quite competitive. Table 4 gives the descriptive statistics forthe divergent index. In Figure 7 we see the Histogram, with a positive skewness. Return D
25
20
15
y
t
i
s
n
e D 10
5
0 -.08 -.06 -.04 -.02 .00 .02 .04 .06 .08
Histogram Normal Kernel Figure 7: Histogram Divergent Monthly Returns 4 Hypotheses Development Now that we hopefully have a clearer understanding of what convergent and divergent strategies are and how we can bring them to life using a Hedge Fund index and a CTA index, we will turn to the development of hypothesis based on common sense and existing literature. We will kick off by introducing a portfolio measure to estimate divergence. Thereafter we will outline the core strengths and weaknesses of both worldviews and finally see if the MDI could be a good indicator to combine both worldviews.
4.1 Market Divergence Index As outlined in the above paragraphs, divergent risk-taking strategies perform well when extreme events occur. This often goes hand in hand with high volatility. Practitioners praise a divergent strategy’s ability to perform good during turbulent periods and gave it the name “CTA Smile” (Sansin & Kissko, 2012; Git & Robertson, 2015) It is related to the “smirk curve” which can be observed in option pricing (Duffie, Pan, & Singleton, 2000; Bradshaw, Hutton, Marcus, & Tehranian, 2010).
A concept linked to this volatility is market divergence. Greyserman & Kaminski (2014) state that Trend Following strategies deliver a convex performance and that this demonstrates the role of divergence in the market. Market divergence can be measured empirically using the Market Divergence Index (MDI). The MDI is the average of the absolute price move over a certain lookback period. It is first
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important to realize that volatility represents both risk and uncertainty (recall Urn A and Urn B from section 2). Risk is most of the time represented by standard deviation. We outlined above that it is more complex than that. The same is true for market divergence, the MDI proposed by Kaminski is also simpler than the concept of market divergence in general. To measure market divergence in prices or in a portfolio, a first measure has to be composed on the individual price series level. This signal to noise ratio “is a ratio between the trend an individual price changes over a specific period. For any specific day, at time (t), the Signal to Noise Ratio, SNRt for a particular price series with lookback period (n) can be calculated mathematically using the following formula:
|푃푡 − 푃푡−푛| 푆푁푅푡 = 푛−1 ∑푘=0 |푃푡−푘 − 푃푡−푘−1|
Eq. 3: Signal to Noise Ratio where (Pt) is the price at time (t) and (n) is the lookback window for the signal, or the signal observation period.” (Greyserman & Kaminski, 2014, p. 107) A common lookback window for medium-to long-term trend followers is 100 days. If the SNR is high, the quality of the trend is better or the individual market has a higher price divergence. If noise or volatility comes into play the SNR becomes lower, and the trend is harder to notify. For each individual market a SNR can be calculated, if we aggregate these at the portfolio level, the Market Divergence Index (MDI) can be composed. The MDI is a simple portfolio level measure. “It is a simple aggregate measure of “trendiness” in prices taking into account the level of volatility (or noise) in the price series. When the MDI is higher, this corresponds to a market environment with higher trendiness across markets in a portfolio” (Greyserman & Kaminski, 2014, p. 109). This is for simplicity just the average of the used signal to noise ratios. It can be quantified for a given period (n)
푀 1 푀퐷퐼 (푛) = ∑ 푆푁푅푖(푛) 푡 푀 푡 푖=1
Eq. 4: Market Divergence Index
where SNRt is the signal to noise ratio for the individual markets of which the portfolio consists, (n) is the observation window and (M) is the number of markets included. Greyserman & Kaminski (2014) find that both market divergence and the speed of market divergence are stationary over time. This suggests that market divergence is a normal phenomenon in financial markets. They find a correlation of 0.74 between the 100-day MDI and the rolling 100-day return for a trend following program. We will check if this is in general true for a divergent portfolio represented by a CTA index. The following hypotheses can be derived and tested:
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Hypothesis 1a: A divergent approach has a high correlation with the MDI
Hypothesis 1b: A convergent approach has a low correlation with the MDI
4.2 Why and When Convergent Strategies Work A convergent strategist believes that fundamental value approaches (like the dividend discount model) can be used to get an idea of the intrinsic value of an asset. The price will eventually converge to this intrinsic value. They agree with the AMH, which relaxes the EMH, that prices fluctuate randomly around their intrinsic values and that market mispricing is possible in the short run. They try to exploit this mispricing using tools and detailed information.
Two conditions are needed for a convergent strategy to work. The first conditions which allows a convergent strategy to earn abnormal returns is that “the information generated by fundamental analysis predicts future economic variables that will eventually be priced by the market” and the second is that “the market temporarily underuses the information in the fundamental signals about future economic variables” (Chung, Rosenberg, & Tomeo, 2004, p. 45).
Literature is mixed on the fact that active convergent management pays off, but performance is often evaluated “on average”. In some periods Hedge Funds are able to realize a positive excess return (Fung, Hsieh, Naik, & Ramadorai, 2008; Ackermann, McEnally, Ravenscraft, 1999; Berk & Green, 2004). Greyserman & Kaminski (2014) find a conditional performance between the 100-day rolling return for a trend following program (divergent) as a function of the MDI. Higher market divergence values result in a higher conditional mean for the rolling return. Intuition suggest that values below the average of the MDI - when market divergence is low - might result in good results for a convergent approach. Andrew Lo (2004, 2005, 2007, 2010) used the terminology proposed by Mark Rzepczynski (1999), and connected the AMH with the convergent and divergent thinking method. He states that convergent risk takers believe in a stable world in the long run. We hereby test the following hypothesis:
Hypothesis 2: A convergent approach performs best during periods with a low MDI
4.3 Why and When Divergent Strategies Work A divergent strategist believes that past patterns can predict future patterns. He will use technical analysis methods to price an asset. He agrees with the AMH because he believes that past patterns are due to changing and adaptive actions by investors. These reactions are driven by investors’ emotions and irrational factors. A divergent investor is able to benefit from past prices when gathering information is costly or when investors over- or underreact to the available information. This is possible when not everybody has access to the same fundamental information (especially in volatile and uncertain markets) or when some investors get access quicker than others. Secondly, divergent strategies might work if the information is vague or sketchy. Thirdly, a lot of factors non-quantitative
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effects due to government intervention, war or politics, are not quantifiable in fundamental analysis methods. Furthermore factors of adaptation, as well as behavioural biases, may influence prices which differ substantially from their intrinsic value (Chung, Rosenberg, & Tomeo, 2004).
Literature is mixed on the fact that active divergent management pays off. Some find that CTAs exhibit no skills and even argue that people can be fooled by CTAs performance (Bhardwaj, Gorton, & Rouwenhorst, 2014). Elaut, Frömmel, & Mende (2017) on the other hand, find that CTAs “on average” can time the market (either bullish or bearish). Greyserman and Kaminski (2014) also suggest that if the MDI is high, the performance for a trend following strategy is higher. They split the Market Divergence Index in three buckets. The first bucket (where the MDI is lower than 0.10) is called “Nontrend”, the middle one (MDI between 0.10 and 0.15) is called “Moderate Divergence” and the third bucket “Financial Crisis” (MDI higher than 0.15). Rzepczynski (1999) argues that profits for a divergent strategy can be made when trends occur or when things are extreme. We hereby test the following hypothesis:
Hypothesis 3: A divergent approach performs best during periods with a high MDI
4.4 We need to be both! “The traditional investment framework is not wrong, but merely incomplete. The sooner we adapt, the more likely we are to survive.”
- Lo (2011, p. 4)
Harry Markowitz (1991, p. 470) laid out the foundations of portfolio theory. He stated that “diversification is a common and reasonable investment practice. Why? To reduce uncertainty! Clearly, the existence of uncertainty is essential to the analysis of rational investment behaviour”.
In standard portfolio theory diversification benefits are based on correlation. In general, the lower the correlation the better for the overall portfolio. If assets have a similar level of volatility, correlation remains a trustworthy measurement. The style an active manager eventually follows will be based on the level of model uncertainty he perceives. In the aftermath of the financial crisis of 2008, finding such diversification options has not been that simple. Investments which are able to deliver performance during such crisis periods were and are highly valued. Although evolution from agents is believed to be stable in the long run, short periods with mutation and rapid change exist. (Lo, 2004) Market divergences and dislocations occurred quite frequently in the past years (with for example during the Euro debt crisis, the Brexit and more recently the devaluation of the Yuan or even the election of Donald Trump). Learning about such dislocations and market changes takes time, and using the convergent and divergent framework it might be time now to combine these two worldviews.
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Building on the AMH, Greyserman & Kaminski (2014, p. 100) suggested that “if markets are efficient, convergent strategies may be prudent in assets that maintain a core fundamental structure. […] On the other hand, we must also profess our ignorance to the true structure of financial markets yielding to the “animal spirits” of the market. In this case, divergent risk-taking strategies that respond well to the unknown and the uncertain. The optimal combination will depend of course on the state of the financial markets and the players who participate in them. The best overall strategy will be to combine some of both risk-taking approaches.”
Switching between those trading styles is not easy. An eclectic manager might try to switch between these strategies based on his perceived information and market conditions. The strengths of the chosen strategy will depend on the probability of market disturbances as well as the volatility in the market. Rzepczynski (1999, p. 81), the man who thought out convergent and divergent thinking, sums it all up clearly: “If there is a desire for broad diversification, holding only one type of worldview will expose a portfolio to unique and powerful risks that can have a significant effect on performance. For example, if only convergent traditional managers and Hedge Funds are held, there is greater exposure to divergent risks. Because a divergent risk is an unknown event that may come from structural change, it may not be diversified away by holding more than one manager of the same trading style. The convergent managers would not be able to exploit the change and produce underperformance. Similarly, holding only divergent traders will allow a portfolio to capture the maximum return from major events but will put a portfolio at risk if there is an extended period of stable or calm market behaviour.” Chung et al. (2004) build different portfolios with different weights of convergent and divergent approaches. They find empirical evidence that combining both strategies results in increased returns and reduced risks relative to both strategies on a standalone base. Their resulting portfolio has lower volatility, a higher kurtosis, smaller negative outliers and a shift in skewness to the positive side. Furthermore they find increased returns in turbulent economic periods where the convergent strategy alone suffers. Finally they obtain the maximum risk-return trade-off and find significant higher risk- adjusted performance measures. Greyserman & Kaminski (2014) investigate that trend following strategies might improve the Sharpe ratio when added to a buy-and-hold strategy. Trend following is a systematic strategy, which goes against the green and is not focused on conviction in a specific style or approach. It is a natural complement to a convergent strategy. In their multi strategy approach they find empirical evidence that adding some convergent might indeed increase the Sharpe ratio, but they also warn that other desirable statistical characteristics (skewness, crisis alpha) might get lost. Based on the empirical evidences from Chung et al. (2004), we will investigate if the Market Divergence Index from section 4.1 might be a good indicator to switch between the two worldviews. We will test the following hypotheses:
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Using the MDI as an indicator to switch between a convergent and a divergent index, a combined approach …
Hypothesis 4: … reduces the volatility of the individual strategy, and increases the kurtosis of the return distribution
Hypothesis 5: … reduces the negative outliers, and shifts the skewness to the positive side
Hypothesis 6: … enhances returns in economic environments where the individual strategy alone offers limited return opportunities
Hypothesis 7: … increases the risk-adjusted performance measures significantly, obtaining the maximum risk-return trade-off
4.5 Crisis Alpha Opportunities We explained in section 3 what crisis alpha is. In short it is a measure to represent the performance of a certain strategy during crisis periods. It was a concept introduced by Kaminski & Mende (2011) to represent and highlight the ability of Trend Following strategies to remain competitive during periods of market distress. A lot of definitions can be used to define such a period of market distress: we could look at a world index and find crisis periods where the monthly returns are less than minus 5 percent, or when the VIX4 has a move greater than 20 percent over the past month. (Greyserman & Kaminski, 2014). A lot of (alternative) investment strategies are regulated and subjected to rules which need to be followed also during these crisis periods. Kaminski & Mende (2011) argue that convergent strategies (with a long equity bias) will be more exposed to these regulatory rules and as a result often take more losses. Furthermore such strategies also carry more hidden risks (like credit and liquidity risk) which will limit them to battle these periods. The leveraged positions Hedge Funds take, could magnify these effects. On the other hand they find empirical evidence that Managed Futures are not susceptible to these extreme losses. They list different risks in the various sub-strategies of alternative investments and find a positive relationship between CTAs and crisis alpha. The relationship between a Hedge Fund and crisis alpha is negative. Keeping in mind that adding a divergent approach to a convergent one could be a seatbelt for our investments (Sundt, 2012), we could ask ourselves if a combined portfolio is able to keep this desired divergent property. The following hypothesis can be derived:
Hypothesis 8: A combined approach is able to deliver crisis alpha
4 “The VIX Index is a key measure of market expectations of near-term volatility conveyed by S&P500 stock index option prices.” (CBOE Volatility Index VIX, 2017) 25
5 Data We will investigate the hypotheses developed in Section 4 by using a database of Hedge Funds and CTAs. The dataset is provided by RPM Risk & Portfolio Management from Sweden. The sample covers the period March 2003 to April 2017 and reports on a daily basis. The Hedge Funds index will be used to represent the convergent strategy. The CTA sample will represent the divergent strategy. For the Hedge Fund index the HFR Hedge Fund Index was chosen, it reports “daily indices utilizing a rigorous quantitative selection process to represent the larger hedge fund universe” (HFR Indices, 2017). The CTA sample is represented by the SocGen CTA Index, which “is designed to track the largest 20 (by AuM) CTAs and be representative of the managed futures space” (Societe Generale, 2017).
Table 5 summarizes the descriptive statistics for the daily returns in our sample. We see a lower mean for the convergent strategy compared to the divergent strategy (although not significant). This is unlike the reported returns found in Chung et al. (2004)5.
RETURN_C RETURN_D Annualized Mean 1,4408% 3,6887% Cumulative Return 22,1730% 66,0503% Mean 0,0059% 0,0153% Median 0,0258% 0,0266% Maximum 1,8860% 2,2659% Minimum -1,9465% -2,7782% STD 0,2375% 0,4777% Skewness -1,1149 -0,3748 Kurtosis 10,6659 4,9579 Observations 3580 3580 Table 5: Descriptive Statistics Daily Returns
At first sight, if we look at Figure 8, we see that from inception to 2008, the VAMIs run quite similar, but the convergent strategy takes a major hit around 2008, while the divergent strategy remains competitive. We do see a significant difference (on a 1 percent significance level) in standard deviation. The convergent strategy has a lower standard deviation. This is consistent with findings of Chung et al. (2014), who find that divergent strategies are prone to take more risk. Furthermore we see more negative skew in the convergent returns compared to the divergent returns. The skewness for the divergent portfolio is not positive, but this could be due to our dataset and the high frequency of the data.6
5 They find that “the returns of convergent and divergent strategies are relatively similar in the long run, while there exist differences in sub-periods” (Chung, Rosenberg, & Tomeo, 2004, p. 46) Their total sample covers the period 1994-2002. 6 Li (2017) finds positive skewness in monthly returns and small negative skewness in daily returns. 26
Daily VAMI 200 180 160 140 120 100 80 60
HFR Hedge Fund index Soc Gen CTA Index
Figure 8: VAMI Convergent and Divergent Daily Indices 6 Construction of MDI This section will start by explaining how the MDI for our portfolio is built and thereafter certain hypotheses will be tested.
6.1 Methodology A lot of variants of the Market Divergence Index exist.7 Greyserman & Kaminski (2014), for example, build a MDI for a trend following system using an absolute value indicator which includes all asset classes (commodities, equity indexes, fixed income and currencies). We will take a totally different approach and will compose our portfolio only using the convergent and divergent approach. We will see the convergent strategy as a market to invest in, as well as the divergent strategy. We therefore first calculate the SNR for the convergent index. Secondly, we calculate the SNR for the divergent index.8 After this, we take the average of both SNRs and thereby get the MDI for our portfolio. Using only two markets has the advantage that the MDI will not flatten out easily (the more markets you add, the more averaged the MDI will become), but as a disadvantage it might be less robust. The underlying idea is that the MDI should only be composed of the markets one trades in. In a convergent and divergent framework, we could see these as the only two possible markets an investor could choose from. Figure 9 depicts the MDI for our portfolio (the first 100 days in our dataset are lost due to calculation of the SNR).
7 The absolute value indicator is used here. We could also opt to use squares (higher impact of extreme moves). 8 For both calculations a 100-day lookback window was used, in accordance to Greyserman & Kaminski (2014) who suggest that most trend followers are medium- to long-term. 27
MDI Portfolio
.5
.4
.3
.2
.1
.0 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 Figure 9: The Market Divergence Index (2003-2017)
If we take a look at Figure 10, where the histogram of the MDI is given, we see that the MDI looks to have fat right tails. It is therefore positively skewed. The mean of 0.15 is quite high compared to a mean of 0.11 found by Greyserman & Kaminski (2014). They report a standard deviation of 0.03 as opposed to our standard deviation of 0.08. This was to be expected since we only took two markets in our portfolio. The higher mean could be explained by the very turbulent period the last two decades have been. If we split the MDI into three buckets – as Greyserman & Kaminski (2014) did, using 0.10 and 0.15 as cut-off values – we find that in 30% of the time (1045 observations) there is no clear trend. In 24% (826 observations) of the time there is moderate divergence and in 46% (1610 observations) of the time we are in a financial crisis.
500 Series: MDI_PORTFOLIO Sample 3/31/2003 4/05/2017 400 Observations 3481
Mean 0.151944 300 Median 0.140253 Maximum 0.423166
200 Minimum 0.001776 Std. Dev. 0.082447 Skewness 0.640513 100 Kurtosis 3.100181
Jarque-Bera 239.4731 0 Probability 0.000000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Figure 10: Histogram of MDI
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6.2 Empirical Results Correlation with the MDI Trend following strategies have a high correlation with the MDI, if the MDI is high, their profitability is also high (Greyserman & Kaminski, 2014). Interesting is to check if a general divergent approach also has a high correlation with our MDI. We see in Figure 11 an indication that there is no clear relationship between the MDI and the 100-day rolling returns of the divergent approach nor with the convergent approach (as was expected).9
Correlations Rolling Returns / MDI 1
0,5
0
-0,5
-1
-1,5
MDI Portfolio Rolling Return D Rolling Return C
Figure 11: MDI with rolling 100-day returns
To empirically test Hypothesis 1, we calculate the correlations. Table 6 gives the results. We see that a low correlation exists between the MDI of our portfolio and the rolling 100-day returns of the divergent strategy. A high correlation might be possible between trend followers, but we do not see this clear link with our CTA sample. We do see a very low (and even slightly negative) correlation between the rolling-100 day returns of the convergent strategy. This suggests that balancing a portfolio based on a MDI could be an interesting diversification method from the viewpoint of a convergent risk taker. We do not find evidence for Hypothesis 1a, this could be due to our broader index. On the other hand, we do confirm Hypothesis 1b, there is no correlation between the MDI and a convergent risk taking approach.
9 The rolling 100-day returns where multiplied by 5 to work with similar axes. 29
MDI Portfolio Rolling Return D Rolling Return C MDI Portfolio 1 Rolling Return D 0,3468 1 Rolling Return C -0,0226 0,03270 1 Table 6: Correlation Table MDI - C – D
Performance During Periods with High or Low Divergence Greyserman & Kaminski (2014) report that divergence is a normal phenomenon in markets. We graph the MDI of our portfolio and compare it to the VIX index, a proxy for market volatility. It is considered “the world’s premier barometer of investor sentiment and market volatility” (CBOE Volatility Index VIX, 2017). It is calculated based on options on the S&P500 Index and represents the market’s expectations of stock market volatility over the next month. We see in Figure 12 that our MDI (which is purely built on our portfolio which only consist of a convergent and divergent asset) has a small correlation (0.2867) with the VIX. This is in accordance with what was suggested in section 2.1, divergence is linked to volatility, but it is not the same.
VIX - MDI 100 80 60 40 20 0
VIX MDI*100
Figure 12: VIX & MDI
Table 7 summarizes the performance of our two strategies in different periods. The period with the lowest divergence is called “Non-trend”, where we expected the convergent strategy to perform best (30% of the time). The middle bucket is called “Moderate Divergence”, where similar returns could be expected (26%). And finally the third bucket, which is called “Financial crisis”, where the divergent strategy is expected to outperform the convergent strategy (46% of the time). If we look at the returns in the different periods, we see that the divergent strategy performs best during a “Financial Crisis”, and worst, with even a negative return, during “Non-trend” periods. The convergent strategy has surprisingly as well its highest returns during the “Financial Crisis” period and we note a small negative return during the “Moderate Divergence” period. In Figure 12, the conditional performance of both strategies as a function of the MDI is graphed.
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NONTREND MODERATE DIVERGENCE FINANCIAL CRISIS RETURN_C RETURN_D RETURN_C RETURN_D RETURN_C RETURN_D Mean 0,0035% -0,0017% -0,0033% 0,0086% 0,0088% 0,0270% Median 0,0194% 0,0011% 0,0149% 0,0135% 0,0360% 0,0467% Maximum 0,8170% 2,2659% 0,7607% 1,4849% 1,8860% 1,8706% Minimum -1,3843% -2,4054% -1,1007% -2,7782% -1,9465% -2,2110% STD 0,2262% 0,4494% 0,2197% 0,4432% 0,2568% 0,5025% Skewness -0,8367 -0,1945 -0,6730 -0,5708 -1,3636 -0,3914 Kurtosis 6,4945 5,3673 4,8897 5,4321 13,4695 4,5423 Observations 1045 1045 826 826 1610 1610 Table 7: Descriptive Statistics linked to Market Divergence
For a convergent strategy, the best performance was expected during low divergence periods, the return statistics and Figure 13 are an indication that this hypothesis is not correct and indeed no statistical significant difference (p- value of 0.5833) was found between the returns of the convergent strategy between the low and high divergence periods. (See Figure 20 in the Appendix for the returns of the convergent strategy during low and high divergence periods.) We thereby fail to confirm Hypothesis 2.
For a divergent strategy on the other hand, we hypothesised that the best performance for a divergent strategy would be when market divergence is high. Table 7 and Figure 13 seem to suggest that Hypothesis 3 is correct. Nevertheless, no statistical evidence is found to confirm this (p-value of 0.1343), we hereby cannot confirm Hypothesis 3 as well. (Figure 21 gives the returns of the divergent strategy for the different divergence periods.)
Conditional Performance on MDI 0,0300%
0,0250%
0,0200%
0,0150%
0,0100%
Meanreturns 0,0050%
0,0000% Low Moderate High -0,0050% Divergence Convergent Divergent Convergent Trendline Divergent Trendline
Figure 13: Conditional Performance as a function of MDI
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7 Building of Optimal Portfolio: Combined Strategy 7.1 Methodology To check if a combination of a convergent and divergent strategy pays off and is still able to maintain the most wanted characteristics of both strategies, we start by building five portfolios. They are constructed by assigning different weights to the convergent or divergent approach. Weightings for the convergent strategy range from 100% / 80% / 60% / 50% / 40% / 20% / 0%. The VAMIs for the five different portfolios and the full convergent and divergent proxy indices are given in Figure 14. Table 8 summarizes the descriptive statistics for the five portfolios.
200
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160 100% C 80% C 60% C 140 50%-50% 40% C 20% C 120 100% D
100
80 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17
Figure 14: VAMI Combined Portfolios
Mean Median Maximum Minimum STD Skew Kurtosis RETURN_100_C 0,0059% 0,0258% 1,8860% -1,9465% 0,2374% -1,12 10,67 RETURN_80_C 0,0078% 0,0230% 1,3276% -1,5377% 0,2354% -0,86 6,62 RETURN_60_C 0,0096% 0,0202% 0,9764% -1,7060% 0,2685% -0,67 5,54 RETURN_50__50_ 0,0106% 0,0202% 1,0994% -1,7902% 0,2950% -0,60 5,36 RETURN_40_C 0,0115% 0,0243% 1,2223% -1,9119% 0,3262% -0,55 5,20 RETURN_20_C 0,0134% 0,0279% 1,6010% -2,3450% 0,3981% -0,45 5,00 RETURN_100_D 0,0153% 0,0264% 2,2659% -2,7782% 0,4776% -0,37 4,96 Table 8: Descriptive Statistics Combined Portfolios10
To build our final optimal portfolio, we will go even further and use our MDI from section 6.1 (which is still calculated only using a convergent and divergent market) to try to find an optimal switching situation. Although the findings that were expected from Hypothesis 2 and 3 and suggested by Figure 13 were not confirmed statistically, we rely on the theory that convergent strategies perform best in stable periods and divergent strategies best in turbulent periods (with high divergence).
10 All portfolios have 3581 observations. 32
To decide which weights we will put in our final portfolio, we first take a closer look at Table 9. For each portfolio from Table 8 we calculated some performance measures. The “Efficiency” is calculated by taking the mean and dividing it by the STD11, as done by Chung et al. (2004). After annualizing both mean and STD12, we calculated the Sharpe Ratios.13 Furthermore we also calculated the downside STD and annualized this, in order to calculate the Sortino Ratios. Looking at “Efficiency” we would be encouraged to take a portfolio which contains 60% convergent. Taking the annual Sharpe ratio calculated on the daily data, we would go for only 20% convergent. After annualizing the mean and STD, we would even opt for a full divergent portfolio. Looking at downside risk, 80% convergent would be best. The Sortino ratio, which takes into account the downside risk, tells us that the 100% divergent approach is best. But taking a closer look we see that from the 50%-50% portfolio onwards the differences become rather small. Looking at Table 8 we see that the skewness for the 50%-50% portfolio is shifted considerably to the positive side as well.
Efficiency Annual Annual Downside Annual Annual Sharpe on Sharpe on STD Downside Sortino Daily Annual STD Returns Mean RETURN_100_C 2,4745 0,1954 0,1343 0,0021 0,0330 0,1542 RETURN_80_C 3,2975 0,3250 0,2657 0,0019 0,0303 0,3301 RETURN_60_C 3,5931 0,3971 0,3476 0,0020 0,0325 0,4584 RETURN_50__50 3,5892 0,4124 0,3687 0,0022 0,0352 0,4934 RETURN_40_C 3,5350 0,4191 0,3809 0,0024 0,0385 0,5152 RETURN_20_C 3,3710 0,4191 0,3903 0,0029 0,0462 0,5368 RETURN_100_D 3,2042 0,4124 0,3906 0,0034 0,0548 0,5434 Table 9: Performance Measures Combined Portfolios
Now that we have an idea on which portfolios to start from, we will use the MDI to switch at certain points in time. Four more portfolios are build. The weights in the portfolios are based on high or low divergence. Portfolio 1 for example will be 100% convergent in low and moderate divergence times (MDI < 0.15), and switches to full divergent in high divergence times (MDI > 0.15). Portfolio 2 is 50%- 50% balanced in moderate divergent times (0.10 < MDI < 0.15), switches to 80% convergent/20% divergent when divergence is low (MDI < 0.10) and switches to 20% convergent/80% divergent in turbulent times (MDI > 0.15). Portfolio 3 and 4 are similar to Portfolio 2 but put different weightings in. Table 10 summarizes. In portfolios 2 to 4 we are never fully exposed to one of the two strategies.
11 Multiplied by 100 for readability. 12 Mean is annualized by adding 1 and taking it to the power of 255, which is the average number of days in a year for our sample, before subtracting 1 again. Annualized Mean = (1+Daily Mean)^255-1 STD is annualized by multiplying it by the square root of 255. Annualized STD = Daily STD * √255 13 For the calculation of both the Sharpe and Sortino ratio, a risk-free rate of 1% is used (Federal Reserve Bank of St Louis, 2017).
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We always start from the 50%-50% situation in moderate times because the theory suggests that combining both approaches might be a good seatbelt, even though Table 9 suggested that the divergent approach is optimal (if we look at the Sharpe and Sortino ratios).
Low Divergence Moderate Divergence High Divergence Portfolio 1 100% Convergent 100% Convergent 100% Divergent Portfolio 2 80% Convergent 50%-50% 20% Convergent 20% Divergent 80% Divergent Portfolio 3 70% Convergent 50%-50% 30% Convergent 30% Divergent 70% Divergent Portfolio 4 60% Convergent 50%-50% 40% Convergent 40% Divergent 60% Divergent Table 10: Switching Portfolios built using the MDI
Figure 15 plots the performance of the four portfolios which were built using the MDI.
180
170
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150 100% C 140 Portfolio1 Portfolio2 130 Portfolio3 Portfolio4 120 100% D
110
100
90 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 Figure 15: Portfolios built using MDI
Now that we have built these four portfolios we ask ourselves for the final time if it is possible to obtain a better portfolio using a convergent and divergent approach, taking into account the MDI to lead us on the weights convergent and divergent should carry in our portfolio. We first try to find out which of these four is optimal. Table 11 summarizes the descriptive statistics.
Mean Median Maximum Minimum STD Skewness Kurtosis RETURN_100_C 0,0043% 0,0254% 1,8860% -1,9465% 0,2394% -1,1052 10,5800 PORTFOLIO1 0,0127% 0,0249% 1,8706% -2,2110% 0,3791% -0,4321 6,8134 PORTFOLIO2 0,0121% 0,0245% 1,4681% -2,0427% 0,3383% -0,5300 5,8883 PORTFOLIO3 0,0111% 0,0261% 1,3452% -1,9585% 0,3187% -0,5665 5,6471 PORTFOLIO4 0,0102% 0,0219% 1,2223% -1,8744% 0,3036% -0,5940 5,4680 RETURN_100_D 0,0140% 0,0252% 2,2659% -2,7782% 0,4734% -0,3686 4,9547 Table 11: Descriptive Statistics Portfolios built on MDI
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We see that all four builded portfolios have higher means compared to the 100% convergent one. All four portfolios have a lower standard deviation compared to the 100% divergent one. The kurtosis is shifted to the lower side for all four and skewness is shifted to the positive side. For portfolio 4 the minimum is lower compared to both 100% convergent and 100% divergent. For the other three, the minimum is reduced compared to the divergent portfolio. We now calculate the same performance measures as done in Table 9, to find our optimal final portfolio. Table 12 gives the results.
Efficiency Annual Annual Downside Annual Annual Sharpe Sharpe STD Downside Sortino on Daily on STD Returns Annual Mean RETURN_100_C 1,8070 0,0904 0,0285 0,0021 0,0333 0,0328 PORTFOLIO1 3,3565 0,4109 0,3796 0,0030 0,0479 0,4802 PORTFOLIO2 3,5905 0,4331 0,3972 0,0026 0,0414 0,5186 PORTFOLIO3 3,4986 0,4099 0,3702 0,0024 0,0386 0,4886 PORTFOLIO4 3,3440 0,3778 0,3347 0,0023 0,0364 0,4460 RETURN_100_D 2,9555 0,3718 0,3482 0,0034 0,0542 0,4859 Table 12: Performance Measures Portfolios built on MDI
The results from Table 12 are clear. Both the Sharpe and Sortino ratio guide us in the direction of Portfolio 2, where we start from our 50%-50% portfolio in moderate divergence periods and switch to 80% convergent / 20% divergent in low divergent times and to 20% convergent/80% divergent in times of high divergence. Figure 14 gives the VAMI for the four portfolios.
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170
160
150
140 VAMI 100%C VAMI P2 130 VAMI 50%-50% VAMI 100% D 120
110
100
90 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17
Figure 16: VAMIs Optimal Portfolios
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7.2 Empirical Results Portfolio 2 During Full Sample Now that we have our optimal Portfolio 2 – which seems to be the best one – we will compare it to the full convergent and full divergent portfolios, as well as the more “passive” 50%-50% portfolio where the MDI is not used to switch, but just a combined approach is followed. To empirically test Hypothesis 4 to 8, we will take a look at Table 12 where the descriptive statistics are given for the four remaining portfolios.
Mean Median Maximum Minimum STD Skew Kurtosis RETURN_100_C 0,0043% 0,0254% 1,8860% -1,9465% 0,2394% -1,1052 10,5800 RETURN_P2 0,0121% 0,0245% 1,4681% -2,0427% 0,3383% -0,5300 5,8883 RETURN_50__50_ 0,0092% 0,0198% 1,0994% -1,7902% 0,2938% -0,6094 5,4066 RETURN_100_D 0,0140% 0,0252% 2,2659% -2,7782% 0,4734% -0,3686 4,9547 Table 13: Descriptive Statistics Final Portfolios
We will start by comparing the standard deviations between the different portfolios. We already found that the standard deviations for the full convergent and full divergent portfolios are significantly different (p-value 0.000), with the divergent portfolio having the highest standard deviation. Portfolio 2 has a significant higher standard deviation (p-value 0.000) compared to the convergent portfolio, but also a significant lower standard deviation compared to the divergent portfolio (p-value 0.000). Furthermore we take a look at the kurtosis14 and see clearly that it has decreased compared to a full convergent portfolio, reducing the heavy tails of the convergent portfolio alone. We hereby confirm Hypothesis 4, the volatility for our Portfolio 2 is significantly lower compared to the divergent strategy alone and the heavy tails from the convergent strategy are reduced as well.
To further check the hypotheses, we will calculate the performance measures as above. Table 14 summarizes. To check Hypothesis 5, if negative outliers are reduced, we calculated another performance measure: the MAR Ratio, which takes into account the maximum drawdown. It is measured by dividing the average return over the maximum drawdown. The higher the ratio, the better. This ratio tells us the same as the Sortino ratio, being that Portfolio 2 is able to reduce the negative outliers and from Table 13, we see that the skewness is shifted considerably to the positive side (as compared to the convergent strategy). This is consistent with findings from Greyserman &
14 Kurtosis measures the combined size of the tails of a distribution. It measure the probability in those tails. The value of a normal distribution is equal to 3. If we observe a greater kurtosis, than the dataset has heavier tails than a normal distribution. Kurtosis is often said to measure the peaks of a distribution. This is a misconception, it is a measure for the tails and their probability (Wheeler, 2011).
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Kaminski (2014), who find that the maximum drawdown is higher for return distributions with more negative skewness. We hereby confirm Hypothesis 5.
RETURN_100_C RETURN_P2 RETURN_50__50_ RETURN_100_D Efficiency 1,8070 3,5905 3,1168 2,9555 Annual Sharpe on Daily 0,0904 0,4331 0,3363 0,3718 Returns Annual Sharpe on Annual 0,0285 0,3972 0,2904 0,3482 Mean Annual Sortino on 0,0328 0,5186 0,3891 0,4859 Downside STD Max Drawdown -26,30% -10,11% -11,22% -12,43% MAR Ratio 0,04 0,31 0,21 0,29 Table 14: Performance Measures Final Portfolios
Although we do not find significant differences in the mean returns across the different portfolios, we do find higher Sharpe and Sortino ratios for Portfolio 2. This suggests that our combined approach is able to increase the risk adjusted performance measures and suggests that we have an optimal risk/return trade-off. We validate Hypothesis 7.
Portfolio 2 During Crisis Periods Finally, we want to check how our combined approach performs in crisis periods. We start by defining crisis periods, we do not look at divergence periods now to avoid dealing with autocorrelation in our portfolios. We choose to take a more qualitative approach and find crises by looking at highly regarded papers and reports. Four periods discern. The first period starts around October 2007 and lasts until March 2009. The main drivers are the US housing bubble, the subprime mortgage crisis and the US recession in general. These are even enforced with a recession period in Russia (July 2008 – 2009) and a global crisis in the automotive sector. We will call this period the “Global Financial Crisis”. A second period, which follows shortly hereafter, is linked to the banking crisis in Ireland, enforced by the Greek Government crisis and we will call it the “Eurodebt Crisis”. It started around December 2009 and last until 2014. To focus not only on Europe and the US, we will look at a bubble in the Chinese stock market who occurred from the middle of June 2015 to February 2016. We call this the “Chinese Bubble”. A last crisis period we will take a look at, is the “Brexit”, driven by investors’ uncertainty and unbelief about the UK leaving the European Union by April 2019 (The Economist, 2014; Swagel, 2013; Anderson, 2013; Financial Times, 2017).
Table 15 gives the descriptive statistics and the calculated performance measures for our full sample and the four crisis periods. We already found that over the full sample Portfolio 2 performs best if we look at the risk-adjusted performance measures (Sharpe, Sortino, worst drawdown and MAR). Looking at the first sample period (Global Financial Crisis), we see that the convergent period took heavy hits
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and realised heavy losses. The divergent strategy on the other hand, performed superb, resulting in the best performance measures for that period. Portfolio 2 stood strong and managed to realise a positive return during this heavy equity crisis, it realised acceptable Sharpe and Sortino ratios, as opposed to the convergent strategy. The worst drawdown is close to the one from the divergent strategy. Compared to the 50%-50% portfolio, Portfolio 2 is also more optimal.
RETURN_100_C RETURN_P2 RETURN_50__50_ RETURN_100_D
3481 Observations3481 (Oct 2003 Full Mean 0,0043% 0,0121% 0,0092% 0,0140%
Sample STD 0,2394% 0,3383% 0,2938% 0,4734% Skew -1,1052 -0,5300 -0,6094 -0,3686
-
Apr 2017) Kurtosis 10,5800 5,8883 5,4066 4,9547 Sharpe 0,0285 0,3972 0,2904 0,3482
Sortino 0,0328 0,5186 0,3891 0,4859 Max -26,30% -10,11% -11,22% -12,43% Drawdown MAR Ratio 0,0422 0,3111 0,2105 0,2922
379 Observations (Oct 2007 Global Financial Crisis Mean -0,0668% 0,0094% -0,0152% 0,0364% STD 0,3820% 0,3268% 0,2857% 0,4839% Skew -0,7572 -0,2100 -0,3240 -0,2152
-
March 2009) Kurtosis 8,4756 3,9770 3,7635 4,0875 Sharpe -2,7307 0,2717 -1,0505 1,1304 Sortino -3,1634 0,3899 -1,4551 1,6410
Max -25,21% -6,76% -11,11% -6,52% Drawdown MAR Ratio -0,6212 0,3574 -0,3414 1,4920
1050 Observations1050 (Dec 2009 Eurodeb Mean 0,0062% 0,0014% 0,0030% -0,0001% STD 0,1871% 0,2958% 0,2731% 0,4410%
t Crisist Skew -1,1971 -0,6017 -0,5962 -0,4578
-
Dec2014) Kurtosis 9,0496 6,9627 5,9192 5,6476
Sharpe 0,1952 -0,1360 -0,0510 -0,1451
Sortino 0,2335 -0,1719 -0,0676 -0,1969
Max -10,59% -7,59% -8,70% -12,43% Drawdown MAR Ratio 0,1495 0,0471 0,0894 -0,0017
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154 Observations (June2015 Chinese Bubble Mean -0,0619% -0,0027% -0,0011% 0,0597% STD 0,2856% 0,3896% 0,2957% 0,5909% Skew -0,5689 -0,4026 -1,2000 -0,5965
-
Feb 2016) Kurtosis 4,0415 6,5700 8,8871 6,4489
Sharpe -3,4214 -0,2718 -0,2713 1,6354 Sortino -4,4386 -0,3560 -0,3444 2,1873
Max -10,10% -4,96% -3,69% -4,03% Drawdown MAR Ratio -1,4455 -0,1392 -0,0760 4,0778
206 Observations (June2016 Brexit Mean 0,0250% 0,0061% 0,0079% -0,0092%
STD 0,1947% 0,2836% 0,2825% 0,4782% Skew -0,6815 0,1838 -0,1367 0,2113
-
Mar 2017)Mar Kurtosis 7,6244 3,3083 3,8190 5,4852 Sharpe 1,7949 0,1247 0,2281 -0,4358 Sortino 2,4203 0,2201 0,3751 -0,6902
Max -2,04% -4,15% -4,67% -9,92% Drawdown MAR Ratio 3,2192 0,3769 0,4341 -0,2346 Table 15: Performance during Crises Periods
In the second sample (Eurodebt crisis period), the convergent strategy realizes the best return and performance measures. The divergent period takes a loss, this might be due to the aftermath of the Global Financial crisis, with the world economy slightly picking up again, this period might have been to turbulent to discover clear trends without noise. Portfolio 2 has the lowest worst drawdown, but due to the small return it marks small negative Sharpe and Sortino ratios. They are still better than the divergent strategy alone. During the third sample period (“Chinese Bubble”), the divergent approach realised again the best results and performance measures. But we clearly see that Portfolio 2 counters the heavy losses suffered by the convergent strategy. The Sharpe and Sortino Ratios are way higher than for the full convergent strategy. In the fourth and final sample period (“Brexit”), the convergent strategy performs best again, but Portfolio 2 is again much more stable compared to the divergent strategy. For Portfolio 2 we note an overall shift in skewness to the positive side. Looking at these four different periods, we see that the performance of the convergent and divergent strategy alone can be very turbulent. Portfolio 2 on the other hand is much more consistent and over the full sample the better one. The VAMIs for the four different periods are plotted and can be found in the Appendix (Figure 22 – Figure 24).
If we look at the two periods where the convergent period took heavy losses, we see that Portfolio 2 is able to remain competitive. During the Global Financial Crisis a mean of -0.0668% was found for the convergent strategy, and Portfolio 2 realized a small positive mean of 0.0094%, this is significantly
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higher (p-value 0.0033). During the Chinese Bubble a mean of -0.0619% was found for the convergent strategy, this is not significantly different (p-value 0.1295) from the returns realised by our Portfolio 2 (-0.0027%). This mixed evidence and the performance measures from Table 15 necessitate us to reject Hypothesis 6. Although Table 15 and the first crisis periods (Global Financial Crisis) suggest that our Portfolio 2 enhances the returns in economic environments in which the convergent strategy alone offers limited opportunities, this cannot be confirmed statistically for equity crises in general.
Crisis alpha The final Hypothesis we would like to check is if our builded portfolio is able to deliver a crisis alpha component over the full sample. We will calculate crisis alpha by taking the performance of the portfolio during the whole sample set and break it down into a risk premium component and a risk free component. We then compare this performance with the performance of the same portfolio but without crisis periods, as if we would only look at normal periods. Comparing these two will bring us crisis alpha. For the crisis periods, we took the same approach as above and will use the Global Financial Crisis, shortly followed by the long Eurodebt crisis and the two more recent crises (Chinese Bubble and the period surrounding the uncertainty about the Brexit).
Table 16 reports the cumulative return for Portfolio 2, the Risk Free rate and Portfolio 2 without crises periods. Figure 17 graphs Portfolio 2; the T-bill rate as a proxy for the risk free rate and Portfolio 2 without the crises periods.
Portfolio 2 Risk Free P2 (w/o crisis) Cumulative Return 48,99% 13,96% 43,66% Average Yearly Return 2,97% 0,96% 2,70% Table 16: Yearly and Cumulative Return Portfolio 2
Crisis Alpha Decomposition 170 160 150 140 130 120 110 100 90
VAMI P2 T bill VAMI P2 (w/o crisis)
Figure 17: Crisis Alpha (Portfolio 2)
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Table 17 reports the decomposition of crisis alpha in our sample. And Figure 18 graphs it.
Return Decomposition Risk Free 0,96% = Risk Free Risk Premium 1,73% = P2 - Risk Free Crisis Alpha 0,27% = P2 - P2 (w/o crisis) TOTAL 2,97% Table 17: Return Decomposition Portfolio 2
Crisis Alpha Decompostion 4,00%
3,00% Crisis Alpha Risk Premium Crisis Alpha 2,00% Risk Premium Risk Premium 1,00% Risk Free Risk Free Risk Free 0,00%
-1,00% Crisis Alpha -2,00%
-3,00% Convergent Portfolio 2 Divergent Risk Free Risk Premium Crisis Alpha
Figure 18: Return Decomposition (Convergent, Portfolio 2, Divergent)
We see that Portfolio 2 is able to hold a small portion of the crisis alpha which was generated mostly by the divergent strategy. Our convergent approach has a negative crisis alpha, which is consistent with findings from Kaminski & Mende (2011), who find a negative crisis alpha of – 5.0% using monthly data from January 1997 to January 2011. They also find a positive crisis alpha for the divergent strategy of 2.4% for the same period. Using this method, we find that our combined portfolio is able to deliver a small amount of crisis alpha during the full sample set, we hereby confirm Hypothesis 8.
8 Results Taking these results together, we recapitulate our findings. The MDI we calculated and used for our portfolios does not have a high correlation with the divergent approach. A correlation between the convergent approach and this MDI was expected to be very low, and indeed we found a small (negative) correlation coefficient, which suggested good diversification between our approaches. Looking at the performance of our full convergent and full divergent strategies we did not find significant differences for the returns between the low and high divergence periods. We rejected
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Hypothesis 2 and 3. This suggested that our MDI (built on only two assets in our portfolio) was not the best indicator.
Nevertheless, we used it to build different portfolios. Portfolio 2 seemed to be the best one. We found that during our full sample the combined approach had significant lower volatility compared to the full divergent approach and a lower kurtosis compared to the full convergent approach. Secondly negative outliers compared to the full convergent approach were reduced and the skewness was shifted to positive side. Furthermore we found better risk-adjusted performance measures for the combined portfolio. We therefor confirmed Hypothesis 4, 5 and 6.
Looking at equity crises periods (defined qualitatively) we only found one period (the Global Financial Crisis) where the combined approach realized significantly higher returns. In the other periods no significant differences were found. Taking these crises periods into account we did find a small crisis alpha. We rejected Hypothesis 7, but confirmed Hypothesis 8.
9 Future Research Our analysis of the combined approach of convergent and divergent profitability suggests a number of avenues for further research. First, we conducted our empirical research on two indices who may not be fully representative for the convergent and divergent approach. Looking at different indices (or aggregating them) might an interesting approach. Secondly, we used daily data, needed to calculate the MDI, this high frequency data has the advantage of high transparency, but might result in lower performance, which we found in our sample dataset (Shi, 2011). Future researches might want to use monthly or bi-weekly data. Third, as we only disposed of daily returns, we did not take the cost of switching between the low and high divergence periods into account. Using net-of-fee data or other approaches might be looked at, to take transaction costs into account. Finally, we already suggested that the MDI can be calculated in many different ways. Taking more markets into account or looking at other alternatives to calculate it, should be investigated.
10 Conclusion There is a broad variety to look at risks and uncertainties in life. Not only in our daily lives, but also and perhaps even more in the financial choices we make. Lots of papers on portfolio management try to find strategies to diversify these risks and uncertainties away. Two specific ways to look at risks where presented in this paper, being a convergent and a divergent lens. This two-way worldview depends on the underlying beliefs, ideas and past experiences we have encountered. We shortly reviewed some main thinking concepts behind the Efficient Market Hypothesis and the behavioural biases discipline to resume the Adaptive Market Hypothesis. This AMH relaxed the strict assumptions of the EMH and explained the biases in markets by adopting terminology from evolutionary biology.
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Using this AMH we divide two similar, but still totally different based on their worldviews, ways of investing in the alternative investment universe. We explained how Hedge Funds can be put in the convergent box, looking at the way they trade and how CTAs can be put in the divergent box. Literature suggested that both worldviews have their advantages and carry their own specific risks. We discussed how to evaluate and compare these worldviews using different performance measures (the widely accepted Sharpe Ratio; the Sortino Ratio who puts more weight on the tails and symmetry of the distribution; the concept of crisis alpha and we looked at the worst drawdown moment by calculating the MAR Ratio).
We looked at market divergence and calculated a Market Divergence Index for our portfolios. This MDI consisted of only two markets where we invested in, the first being the convergent market and secondly the divergent market. We found that using this MDI, divergence in our portfolio is not reliable to invest in Hedge Funds or CTAs. Using this same MDI we constructed an optimal portfolio in terms of performance measures for the full sample period. This optimal portfolio always consisted of a combination of both convergent and divergent approaches. We checked if it could retain some benefits from the individual strategies. Overall for the full sample, starting from the convergent approach (and thus adding divergent), we found that the combined approach is able to protect our investment in turbulent periods. The overall skewness is shifted to the positive side and kurtosis is reduced, resulting in less fatter tails. Furthermore the worst drawdown moment is reduced strongly, avoiding the occasional negative black swans for our convergent approach. From a divergent approach (and thus adding convergent), we found that the standard deviation is reduced significantly without a loss in returns. Looking at the portfolios in qualitatively described crises periods, we found that the combined approach was able to maintain similar results, with a special note to the period surrounding 2008. This equity market crisis was extremely bad for the convergent strategy, while the divergent strategy boomed. The combined approach could have been a good seatbelt, resulting in a significant higher return compared to the convergent strategy. The combined approach is able to preserve a small crisis alpha component.
Although not all findings are significant, combining convergent and divergent thinking could have protected us from the extreme negative black swans during the Global Financial crisis occurring around 2008. At the same time, similar returns could be realised with the highest Sharpe and Sortino ratios for the full sample period. The convergent and divergent framework offers an interesting and trustworthy approach if you want to limit the risks carried by the individual approaches and if you are pleased with a midway return.
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e
u
u
q
q
e
r 400 e
r
F 200
F
200 100
0 0 0 10 20 30 40 50 60 70 80 90 0 5 10 15 20 25 30 35 40 45
MDI*100 Figure 19: Histogram VIX and MDI 500 .02
400
y 300
c .01
n
e
u
q
e
r 200
F
100 .00
0 0 5 10 15 20 25 30 35 40 45 -.01
-.02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17
RETURN_C_HIGHDIVERGENCE RETURN_C_LOWDIVERGENCE
Figure 20: Convergent Performance as a function of Low or High Divergence
48
.03
.02
.01
.00
-.01
-.02
-.03 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17
RETURN_D_HIGHDIVERGENCE RETURN_D_LOWDIVERGENCE Figure 21: Divergent Performance as a function of Low or High Divergence
Global Financial Crisis 125
115
105
95
85
75
65
4/24/2008 1/21/2009 1/03/2008 1/18/2008 2/05/2008 2/21/2008 3/07/2008 3/25/2008 4/09/2008 5/09/2008 5/27/2008 6/11/2008 6/26/2008 7/14/2008 7/29/2008 8/13/2008 8/28/2008 9/15/2008 9/30/2008 1/05/2009 2/05/2009 2/23/2009 3/10/2009 3/25/2009
10/01/2007 10/16/2007 10/31/2007 11/15/2007 12/03/2007 12/18/2007 10/15/2008 10/30/2008 11/14/2008 12/02/2008 12/17/2008
VAMI_100_C VAMI_P2 VAMI_50__50_ VAMI_100__D
Figure 22: VAMIs during Global Financial Crisis
49
Eurodebt Crisis 110 108 106 104 102 100 98 96 94 92
90
4/06/2010 1/14/2011 1/12/2010 2/24/2010 5/17/2010 6/25/2010 8/05/2010 9/15/2010 2/24/2011 4/06/2011 5/17/2011 6/28/2011 8/09/2011 9/19/2011 1/18/2012 2/28/2012 4/09/2012 5/18/2012 6/28/2012 8/09/2012 9/19/2012 1/24/2013 3/07/2013 4/18/2013 5/30/2013 7/11/2013 8/21/2013
10/28/2011 12/01/2009 10/26/2010 12/06/2010 12/08/2011 10/30/2012 12/11/2012 10/02/2013 11/12/2013 12/24/2013
VAMI_100_C VAMI_P2 VAMI_50__50_ VAMI_100__D
Figure 23: VAMIs during Eurodebt Crisis
China Bubble 115
110
105
100
95
90
85
9/07/2015 7/13/2015 7/20/2015 7/27/2015 8/03/2015 8/10/2015 8/17/2015 8/24/2015 8/31/2015 9/14/2015 9/21/2015 9/28/2015 1/05/2016 1/12/2016 1/19/2016 1/26/2016 2/02/2016 2/09/2016
11/16/2015 10/05/2015 10/12/2015 10/19/2015 10/26/2015 11/02/2015 11/09/2015 11/23/2015 11/30/2015 12/07/2015 12/14/2015 12/21/2015 12/28/2015
VAMI_100_C VAMI_P2 VAMI_50__50_ VAMI_100__D
Figure 24: VAMIs during Chinese Bubble
50
Brexit
109 107 105 103 101 99 97
95
6/01/2016 6/09/2016 6/17/2016 6/27/2016 7/05/2016 7/13/2016 7/21/2016 7/29/2016 8/08/2016 8/16/2016 8/24/2016 9/01/2016 9/09/2016 9/19/2016 9/27/2016 1/05/2017 1/13/2017 1/23/2017 1/31/2017 2/08/2017 2/16/2017 2/24/2017 3/06/2017 3/14/2017
11/08/2016 10/05/2016 10/13/2016 10/21/2016 10/31/2016 11/16/2016 11/24/2016 12/02/2016 12/12/2016 12/20/2016 12/28/2016
VAMI_100_C VAMI_P2 VAMI_50__50_ VAMI_100__D
Figure 25: VAMIs during Brexit
Crisis Alpha Decomposition Convergent 170 160 150 140 130 120 110 100 90
VAMI C T bill VAMI C (w/o crisis)
Figure 26: Crisis Alpha (Convergent)
51
Crisis Alpha Decomposition Divergent 180 170 160 150 140 130 120 110 100 90
VAMI D T bill VAMI D (w/o crisis)
Figure 27: Crisis Alpha (Divergent)
52