Cryptocurrencies as speculative assets: extension on portfolio diversification in combination with testing for bubbles

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Cryptocurrencies as speculative assets: extension on portfolio

diversification in combination with testing for bubbles

December 2018

Merel Engberink

11903988

Msc. Finance - Asset Management Prof. Jeroen Ligterink

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Abstract

This paper examines the potential of portfolio diversification when adding cryptocurrencies to a well-diversified portfolio. Furthermore, the bubble behavior on the cryptocurrency market will be investigated. These bubbles are suggested by the dramatically increases in

the cryptocurrency prices, which are repeatedly in the news. The four most common cryptocurrencies in circulation today will be used in this study. These are Bitcoin, Litecoin, Ripple and Ethereum. Next to that, a cryptocurrency portfolio, which will be created out of

these four cryptocurrencies, will be included in the research. First, we will use a mean-variance spanning test to show that the Bitcoin, Ripple, Ethereum and the cryptocurrency portfolio do not coincide with the initial asset’s portfolio. This well-diversified initial asset’s

portfolio will be created out of indices of the S&P 500, FTSE and FAZ. The outcome of the mean-variance spanning suggests that diversification benefits could follow from including these cryptocurrencies into a well-diversified portfolio. Secondly, with the Phillips-Perron methodology, the expectancy of bubbles for the Ripple exchange prices can be rejected with

significance. Lastly, the GSADF test helps to identify multiple bubble periods reaching from 2013 up and until 2018 for the Bitcoin, Litecoin, Ethereum and the cryptocurrency portfolio.

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Statement of originality

This document is written by Merel Engberink who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for

the supervision of completion of the work, not for the contents.

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1. Introduction

In October 2014, the New York Times headed: “Bitcoin’s Price in Dollars Sustains Another Wild Swing, Falling Nearly 20 Percent”. Fear arose about the drops in prices of the cryptocurrency. While cryptocurrencies are initially developed as payment instruments, the use of them as speculative assets became the main feature (Baur et al., 2018). Therefore, investors may be interested in the consequences of involving cryptocurrencies in their portfolios.

This study tests whether there is mean-variance spanning of various cryptocurrencies on a well-diversified asset’s portfolio created out of the indices of the S&P 500, FTSE and FAZ. Next to that, this research tests for the existence of explosiveness of the prices of these cryptocurrencies. Lastly, there will be researched whether there are bubble periods for these cryptocurrencies. The cryptocurrencies of interest for this research are Bitcoin, Litecoin, Ripple, Ethereum and a cryptocurrency portfolio combining these four.

Since cryptocurrencies are rather new to the investment field, the research done after this type of assets is limited. This paper therefore can provide insights about the gains and losses of investing in cryptocurrencies.

The diversification benefits that cryptocurrencies possibly can provide are tested with the use of a mean-variance spanning. In this test, the minimum-variance frontier of the returns of the separate cryptocurrencies and the cryptocurrency portfolio will be spanned against the original portfolio. The original well-diversified asset’s portfolio is created with the indices of the S&P 500, FTSE and FAZ. In case both minimum-variance frontiers coincide, no diversification benefits can be gained from adding the cryptocurrency of interest to the initial portfolio. The concept of mean-variance spanning is developed by Huberman & Kandel (1987). De Roon et al. (1968) and De Roon et al. (2001) show the use of mean-variance spanning with their researches. In their studies, they apply spanning on futures contracts and nontraded assets (De Roon et al., 1968), as well as on short sales constraints and transaction costs (De Roon et al., 2001). Later, Briere et al. (2015) apply mean-variance spanning on the Bitcoin, while Dorfleitner & Lung (2018) do this for eight different cryptocurrencies. There is strong cohesion between results of the studies that research the diversification benefits of cryptocurrencies. Both Briere et al. (2015) and Dorfleitner & Lung (2018) find diversification benefits when adding the cryptocurrencies to a well-diversified asset’s portfolio. Briere et al. (2015) find portfolio diversification benefits for the Bitcoin, which is in line with the findings

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2 of this study. Furthermore, Corbet et al. (2017) use a GARCH approach (Engle, 2001) to show that Bitcoin and Litecoin offer diversification benefits for investors with short investment horizons. While our study finds diversification benefits for Bitcoin and Litecoin as well, it also adds the suggestion of diversification benefits of Ethereum and the cryptocurrency portfolio to cryptocurrency portfolio to this existing literature.

To investigate whether there are bubbles in the cryptocurrency markets, the Phillips-Perron (1988) tests will be conducted. Based on the current prices and the delayed prices of the cryptocurrencies and the cryptocurrency portfolio, estimations can be made regarding the explosiveness of the assets. Stein (1987) and Shiller (1981) show that when bubbles are present, prices show explosive behavior. The existence of bubbles on the prices of Bitcoin, Litecoin, Ethereum and the cryptocurrency portfolio cannot be rejected. For the prices of Ripple, no bubbles appear in this investigation. These results are in line with the study of Cheah & Fry (2016). They find explosiveness for the prices of Bitcoin, Litecoin and Ethereum. According to them, the explosivity of the prices of Bitcoin reduces the occurrence of the existence of explosivity in Ripple, which addresses this research as well. This study is able to amplify this statement with the contribution of an extended sample period. This research finds the existence of bubbles for Bitcoin, Litecoin, Ethereum and the cryptocurrency portfolio. In line with the expectations of Cheah & Fry (2016), this research can reject the existence of bubbles with significance for Ripple. As the research of Cheah & Fry (2015) uses a time sample until 2015, this paper extents that period until November 2018.

To test the hypothesis that aims to identify multiple bubbles in the cryptocurrency market, a GSADF methodology will be adopted (Phillips et al., 2011a; Phillips et al., 2011b; Phillips et al. 2012). This Generalized Augmented Dickey-Fuller methodology uses a rolling window in which the Augmented Dickey-Fuller test will run backwards in order to detect different periods in which a bubble might have happened. Cheung et al. (2015) use this test in order to detect bubble periods on the Mt. Gox Bitcoin exchange market. The first bubble they detected was that of June 2011, which was at the same moment a big theft has took place on the Mt. Gox Bitcoin exchange market. This created a loss of trust, and therefore prices dropped dramatically. Furthermore, the largest bubble they have found was the one starting in November 2013 and ending in 2014, when China put a ban on cryptocurrencies. This bubble period is detected in this study as well. The study of Li et al. (2018b) extended the paper of Cheung et al. (2015) by researching the entire market of Bitcoin. Li et al. (2018b) find the

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3 second bubble for Bitcoin as well as this study does, taking place in the second halve of 2017. Even though they include the research from the Mt. Gox market to the entire Bitcoin market, other cryptocurrencies are left out. This research extends the studies of Cheung et al. (2015) and Li et al. (2018b) by applying the GSADF approach on Litecoin, Ethereum and the cryptocurrency portfolio as well as on Bitcoin and Ripple. In line with the papers of Cheung et al. (2015) and Li et al. (2018b), our research finds bubble periods for Bitcoin as well. Furthermore, this research is able to detect multiple bubble periods for Litecoin and Ethereum.

This research does not solely contributes to existing literature with the application of cryptocurrencies other than Bitcoin to the mean-variance spanning (Huberman & Kandel, 1987), Phillips-Perron (Phillips & Perron, 1988) and GSADF (Phillips et al., 2011a; Phillips et al., 2011b; Phillips et al. 2012) tests, it also creates a cryptocurrency portfolio, in which Bitcoin, Litecoin, Ripple and Ethereum are combined. For this cryptocurrency portfolio, our research finds diversification effects, as well as the existence of a bubble and the detection of the bubble periods. Furthermore, this research adds a complete overview of the consequences of investing in four different cryptocurrencies as well as investing in a cryptocurrency portfolio to the existing literature. While Chowhury (2014) and Dorfleitner & Lung (2018) focus on testing for mean-variance spanning of cryptocurrencies and the papers of Cheung et al. (2015) and Li et al. (2018b) solely pay attention to the existence of cryptocurrency bubbles and their bubble periods, our paper contributes in a way that it combines both fields. Both diversification benefits and bubble detection are addressed in our research. This paper, therefore, provides a complete overview of the consequences investors face when adding cryptocurrencies to their portfolios.

The paper will be structured as follows. Section two will start with a description of cryptocurrencies, next to a review on existing literature. Both the subjects of portfolio diversification using cryptocurrencies as the bubble behavior of cryptocurrencies will be addressed. Section three will describe the methodology, in which the mean-variance spanning test will be set out. This model tests for spanning with of cryptocurrencies and a cryptocurrency portfolio on a well-diversified asset’s portfolio. Furthermore, the Phillips-Perron and Generalized Supremum Augmented Dickey-Fuller test will be applied to test for bubbles on the four cryptocurrencies and the portfolio of them. In section four the data used to identify spanning, the existence for bubbles and identifying bubble periods will be

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4 described. Section five will discuss the results, after which section six will contain the robustness controls. This research will end with the conclusion and a discussion on this.

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2. Literature review

The new assets cryptocurrencies have recently gained a lot of interest from investors (Clements, 2018). Especially the use of cryptocurrencies as speculative assets appears to be attractive. Bauer et al. (2017) show this for Bitcoin. A third of Bitcoin is held by investors, while only a minority of the users appear to use Bitcoin as medium of exchange. Investing in these cryptocurrencies is contagious, as investors see others earning high margins on their investments. Because of that, investors are interested in the value that cryptocurrencies can add to their existing portfolio. Based on existing literature, this literature review will explain what cryptocurrencies are, where their value is coming from and it investigates whether investors can gain from adding Bitcoin, Litecoin, Ripple, Ethereum and a cryptocurrency portfolio to their existing well-diversified portfolio. Furthermore, the market price behavior of these cryptocurrencies will be researched. The market price appears to be highly volatile compared to other assets (Cheah and Fry, 2015) and this suggest the existence of pricing bubbles (Shiller,1981). Even though portfolio diversification benefits are wanted, investing in assets that show bubble behavior is discouraged. Combining these fields of interest, will lead to the development of the three hypotheses this paper researches.

2.1 Cryptocurrencies

Cryptocurrencies are defined as digital currencies that use cryptography at the base (Gandal & Haraburda, 2014). Initially, cryptocurrencies were created to improve the established electronic payment system. Cryptocurrencies differentiate theirselves from other payment means with the use of digital signatures, assuring the proof of ownership (Böhme et al., 2015). The proof of ownership is installed to create trust between buyer and seller (Nakamoto, 2008). Next to that, the proof of ownership contributes to protect from fraud with this digital currency. Cryptocurrencies have the ability to substitute online payment methods, such as PayPal and credit and debit cards (Kerner, 2014). Even though they can be compared to cash (Evans-Pughe, 2012) due to their anonymity, they differ because of their purely digital characteristics. The supply of for example Bitcoin, Litecoin and Ripple are limited to a fixed amount and the path of money creation is prescribed over time. The supply of this transaction medium is not managed by any bank or government (Blau, 2018). No financial third party is able to freeze or lock an account or influence the economy and deflation. The peer-to-peer

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6 framework is namely used for payments directly between two users. No third party is involved in these transactions. Therefore, financial institutions are not able to interfere in the transactions. However, the lack of a financial institution as third party raises the threat of double spending. Double spending is defined as the situation in which it is problematic to verify whether the owner of the coin is trustworthy and has sold his coin only once, and not more than that. The irreversible aspect of transactions that is included into the design of the cryptocurrency can prevent this event. The transaction information of cryptocurrencies is included in a block, which is linked to the previous block. This creates a blockchain, linking all blocks to each other, back to the initial block from when the cryptocurrency network started (Nakamoto, 2008). This blockchain carries the Proof of Work, which contains the whole transaction history that cannot be changed unless all the Proof of Work of all blocks in the chain are changed. In order to create a change to a cryptocurrency’s sequence of transactions, the majority of the network needs to adapt the change. Assuming that the majority of the systems in the network are righteous, attackers cannot cause any harm to the entire cryptocurrency’s Blockchain (Barber et al., 2012). According to Cheah & Fry (2015) cryptocurrencies have a fundamental value of zero. The valuation of most assets is based on the cashflow, cost of production, assets or final consumptive demand of its underlying company. Cryptocurrencies do not have any of these. Like all other currencies, the only value given to cryptocurrencies is due to confidence in its use as unit of exchange (Stokes, 2012). The confidence in the cryptocurrency creates a relationship between supply and demand (Bouoiyour & Selmi, 2015).

The first cryptocurrency transaction was that of a Bitcoin, in January 2009 (Blau, 2017). Thereafter, other cryptocurrencies such as Litecoin, Ripple and Ethereum followed rapidly (Alam, 2017). Litecoin is released in 2011, Ripple launched its cryptocurrency protocol in 2012. Thereafter, Ethereum released its cryptocurrency in 2015. In Graph D in the Appendix the developments of the prices of the Bitcoin, Litecoin, Ripple and Ethereum are showed. As can be seen in Graph D in the Appendix, the prices of the cryptocurrencies are highly volatile. What does this mean for speculators? This research investigates what effects cryptocurrencies have on the portfolio of investors.

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7 2.2 Diversification benefits of adding cryptocurrencies to a well-diversified portfolio One of the purposes of this research is to test whether it is beneficial for an investor to add cryptocurrencies to its already existing well-diversified asset portfolio. Prior research has been done after the effect of adding new asset classes to an existing asset’s portfolio.

Anyfantaki & Topaloglou (2018) use a stochastic dominance approach for this. The stochastic dominance approach allows for the ranking of investments, based on their general conditions for decision making under risk (Hadar & Russell, 1969; Hanoch & Levy, 1696; Rothschild & Stiglitz, 1970). This in order to test whether a given combination of assets is stochastically efficient relative to all other combinations of a discrete set of alternatives (Kuosmanen, 2004; Roman et al., 2006). Therefore, it can be used in order to test for diversification benefits of adding assets to an existing portfolio. The performances are compared both in- and out-of-sample. For the initial portfolio they create one with indices of assets, bonds and the Libor rate, which is used as proxy for the traditional asset market, the bond market and the risk-free market. This portfolio is used to test the diversification benefits of the four most common cryptocurrencies. According to Anyfantaki & Topaloglou (2018), adding cryptocurrencies to a portfolio provides better investment opportunities, due to the diversification benefits. This especially applies to out-of-sample tests. For their in-sample test, the outcome is that only risk averse investors would benefit from this specific assets’ extension.

Bouri et al. (2017) apply the dynamic conditional correlation approach (Engle, 2002). This model captures the time-varying and dynamic relations across return series of the Bitcoin, in this case, and earlier established assets. This is done by measuring the dynamic conditional correlation (DCC) between both returns. With this model, Bouri et al. (2017) show that Bitcoin is an effective diversifier against movements in the assets such as the S&P 500, FTSE 100, oil, gold, and currencies.

De Roon et al. (1998) and De Roon et al. (2001) use mean-variance spanning to test for portfolio diversification. The concept of mean-variance spanning has been outlined by Huberman and Kandel (1987). They state that spanning is the coincidence of the Minimum-variance Frontier (MVF) of the initial portfolio assets with the MVF of the portfolio including the assets of interest. This minimum-variance frontier is better known as the Markowitz efficient frontier, based on Markowitz portfolio theory (Markowitz, 1991), representing the maximum level of return for its level of risk (Das et al., 2011). The minimum-variance thus can

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8 be adopted to examine the distance (in mean-variance space) between a given market index and the minimum variance frontier of a given set of assets (Huberman and Kandel, 1987). For an investor with mean-variance preferences, the set of portfolios on the efficient frontier is optimal. This set of portfolios namely offers the highest level of expected return for a given level of risk or, in contrast, the lowest level of risk for a set level of expected return (Merton, 1972). Since the aim is to find the most optimal portfolio of initial assets and assets from new asset classes, applying this with mean-variance spanning is appropriate. When the minimum-variance frontiers of two assets coincide, mean-minimum-variance spanning takes place. In this case, adding the specific assets to the existing portfolio, does not gain any diversification benefits for an investor. This test is therefore able to identify whether diversification benefits can follow from adding new asset classes to an existing portfolio. De Roon et al. (2001) use this framework in order to test for mean-variance spanning regarding market frictions investors face, such as short sales constraints and transactions costs. De Roon et al. (1998) do this for futures contracts and nontraded assets. The mean-variance spanning methodology provides a model that is specifically designed to test for diversification benefits of adding different asset classes to the investor’s portfolio. Since investors treat cryptocurrencies as speculative assets, replacing the assets of interest of the research of De Roon et al. (1987) and the research of De Roon et al. (2001) with cryptocurrencies as the variables of interest seems appropriate. This means that in case that the mean-variance frontier of the set of assets in the original investor’s portfolio coincides with the MVF of the new investor’s asset portfolio including the cryptocurrency or the cryptocurrency portfolio that is added, the investor will be indifferent with respect to holding the additional assets in the form of cryptocurrencies. In case the two mean-variance frontiers have one point in common only, so-called intersection takes place (De Roon et al., 1998). Only investors from whom the intersection portfolio is optimal do not need to revise their portfolio and gain new investment opportunities regarding the additional asset. Briere et al. (2015) have practiced this in their research regarding spanning the Bitcoin on their diversified portfolio including both traditional assets and alternative investments. They find that BTC investments gain significant diversification benefits when added to the diversified portfolio. Moreover, the risk-return trade-off of the initial portfolio dramatically improves by adding even a small percentage of Bitcoins. Dorfleitner & Lung (2018) also apply mean-variance spanning tests. They find significant diversification benefits for eight different cryptocurrencies separately, as well as in a combination. They state that the improvement of

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9 the yield purely descend from the increase in the portfolio returns, not from any reduction in risk. The research adds though, that the gain only holds in periods of bullish market phases. The data used for this tests stem from augustus 11th_{2015 up and until august 7}th_2018.

As previously stated, diversification benefits are highly appreciated by portfolio managers. However, investors should be aware of the danger of pricing bubbles of the assets kept in their portfolio. When the asset’s bubble burst, the whole investment in that asset can evaporate. Therefore, this study shines a light on cryptocurrency’s pricing bubbles as well.

2.3 Bubbles in the prices of cryptocurrencies

This research investigates the existence of cryptocurrency bubbles. In this section, a definition of asset’s bubbles in the financial market is provided. Furthermore, the models used to test for asset’s bubbles in the financial market are set out, next to the existing literature on asset’s bubbles. In this research, the Phillips-Perron model (Phillips & Perron, 1988) and GSADF statistic (Phillips et al., 2012; 2013) are used for the detection of bubbles regarding cryptocurrencies.

According to Jensen (1978) given the assumption of rational behavior and expectations, the price of an asset must reflect its market fundamentals. The current market price reflects information concerning current and future returns from this specific asset (Bariviera, 2017; Nadarajah & Chu, 2017; Urquhart, 2016; Tiwari et al., 2018). In a stochastic bubble, according to Blanchard and Watson (1982), investors are irrational in the expectations of the profit that they will gain from buying or selling an asset that is overpriced. Speculators buy financial assets at a price that is above its fundamental value, with the expectancy they will gain from it. When the current asset market prices are driven by arbitrary, self-fulfilling expectations, the risk of a bubble might raise. A bubble, therefore, is a period in which the price of an asset increases sharply. As a consequence, the bubble can burst. In that case, prices will decrease dramatically (Li et al., 2018b).

The research of Wheatley et al. (2018) is confident about the existence of bubbles within the cryptocurrency market. The research states that the bubbles can be described by a deterministic nonlinear trend called the Log-Periodic Power Law Singularity (LPPLS) model. This LPPLS model combines two features of bubbles that have been empirically proven before

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10 (Sornette et al., 2015). The papers of Sornette (2017) and Ide & Sornette (2002) show these features. They state that both the price exhibits a transient faster-than-exponential growth and that accelerating periodic volatility fluctuations are present. These accelerating log-periodic volatility fluctuations express spirals of competing expectations of higher returns and an impending crash (Johansen, 1999; Sornette, 1998). Briefly, there is thus a process characterized in the model in which the bubble matures towards its endogenous critical point. Therefore, it becomes increasingly unstable. This causes a situation in which any small disturbance can trigger a crash (Wheatley, 2018).

Another model that investigates asset’s bubbles is based on the volatility of prices. According to Stein (1987), price instability is critical for the rise of the probability of bubbles. Stein (1987) points out that price instability can be caused due to the presence of speculation. When more speculators enter the market, the informativeness to existing traders of the price of the asset goes down. This results in price destabilization, which in turn restrains the arbitrage on the market. This causes the supply-demand principle to be disturbed, resulting in the price to turn out to be unstable. This price instability of assets has been researched by Shiller (1981) as well. His study observes disproportionate levels of volatility in speculative prices. The research of Shiller (1981) connects the speculation and destabilization of the prices in the asset market in such a way that it shows that the high rates of volatility counteract the efficiency of the market, which is assumed to stabilize the asset prices. Williams (2014) finds a significantly high volatility of Bitcoin. In his study, he finds a volatility of Bitcoin that is even eight times greater than that of S&P 500 and 18 times greater than that of the US dollar. The high volatility and speculative characteristics (Dowd, 2014; Selgin, 2015; Baek & Elbeck, 2015) found for the cryptocurrencies suggests a bubble on the cryptocurrency market (Dale et al., 2005).

The third model that detects asset’s bubbles is that of Phillips & Perron (1988). This model is able to test the stationarity of an asset. In case an asset cannot be proven to be stationary, prices are not stable, and a stochastic irrational bubble is likely to arise (Li et al., 2018b; Corbet et al., 2018). The Phillips-Perron test (Phillips & Perron, 1988) is designed to detect explosiveness in the prices of assets, which suggest the existences of an asset’s pricing bubble. The GSADF statistic (Phillips et al., 2012; 2013) is an addition to that, which allows for the date stamping of multiple bubbles in the time series of assets. This combination of the Phillips-Perron test (Phillips & Perron, 1988) and the GSADF statistic (Phillips et al., 2012; 2013)

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11 is relatively the most statistical model to test for bubbles. Since cryptocurrencies have a fundamental value of zero (Cheah & Fry, 2015), testing for cryptocurrency bubbles requires a fully statistical model. Because of that, it is appropriate to apply the Phillips-Perron test (1998) in combination with the GSADF statistic (Phillips et al. 2012; 2013) to cryptocurrency bubble tests. Gomez-Gonzalez et al. (2017) test for assets bubbles with the help of the Phillips-Perron test (Phillips & Perron, 1988) and the application of the GSADF statistic (Phillips et al., 2012; 2013). They apply these two to be able to identify bubbles in three markets. Their investigation tests for bubbles in the housing, equity and currency market in South Africa, Colombia, the Netherlands, the United Kingdom, Portugal and Canada. Price explosiveness is mainly found in housing market, which suggests that this market is particularly prone to bubble behavior. Explosiveness in the prices of currency and stock markets are less frequent and their duration is shorter. Gomez-Gonzalez et al. (2017) expect that this is mainly due to the fact that housing is subject to more regulations than the stock and currency market. Also, the imperfections of the housing market are an important characteristic. Housing namely depends strongly on individual preferences. Next to that, the transaction costs in the more standardized financial markets are lower than the transaction costs in housing markets. Stocks and currencies are often transacted at online platforms. Therefore, arbitrage opportunities are gained relatively faster and with higher frequencies. This decreases the time in which a pricing bubble can develop. Therefore, the possibility of a bubble decreases as well.

This methodology is applied to the cryptocurrency market as well. Cheung et al. (2015) have used the GDSAPF test (Phillips et al., 2012; 2013) to provide significant evidence for several bubble periods on the Mt. Gox Bitcoin exchange market. These bubbles can be largely explained due to the economic environment at those moments. The first downfall in exchange prices for the Bitcoin where found in June 2011 (Bradbury, 2013). This was also the moment in which a big theft has taken place on the Mt. Gox exchange. This caused the trust in the Bitcoin market to collapse, causing a deflating bubble. The burst of the second bubble shows a connection with the Chinese ban on cryptocurrency use by financial institutions with their ‘’Notice on the Prevention of the Risk of Bitcoin’’, stating that BTC is not a real currency (Li et al., 2018b) and a suspension of trading on the Mt. Gox exchange. The loss of confidence, as is stated by Cheah & Fry (2015), clarifies the drop in the exchange rate of Bitcoin.

Li et al. (2018b) apply the same research method, using the Generalized Sup Augmented Dickey-Fuller test (Phillips et al., 2012; 2013), but the angle of approach of the

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12 study is different. Their aim is to find the long-term value of the cryptocurrency. The study tests for Bitcoin bubbles. Li et al. (2018b) find extreme fluctuations in prices. According to them, these fluctuations cause asset’s bubbles, as they are created by the speculation behavior on the asset market.

This paper is initiated to test whether it is beneficial for an investor to add Bitcoin, Litecoin, Ripple, Ethereum or a cryptocurrency portfolio to its already existing well-diversified asset portfolio. First, this paper shines a light on the possibilities of portfolio diversification benefits that arise when adding various cryptocurrencies and the cryptocurrency portfolio to a well-diversified initial asset’s portfolio. In order to test for the diversification benefits of adding cryptocurrencies to a well-diversified asset’s portfolio mean-variance spanning (Huberman & Kandel 1987) is applied. This is in accordance with the papers of Briere et al. (2015) and Dorfleitner & Lung (2018), who apply this test as well to detect diversification benefits for different cryptocurrencies. Furthermore, our paper does not only shine a light on the diversification benefits of adding cryptocurrencies to a portfolio, it also pays attention to the risks.

Diversification benefits are wanted, as long as the investment is not enormously risky. Therefore, this paper adds the investigation into the existence of explosiveness in the prices of Bitcoin, Litecoin, Ripple, Ethereum and the cryptocurrency portfolio. This price explosiveness addresses bubbles of the cryptocurrency prices. In accordance to Chueng et al. (2015), this study tests for the price explosiveness and aims to detect the bubble periods. In order to do a test for bubbles, this research applies the Phillips-Perron test (1988), as well as the GSADF statistic (Phillips et al. 2012; 2013). Since cryptocurrencies have no fundamental value (Cheah & Fry, 2015), applying these tests, that are fully statistical, is appropriate for our reserach.

In addition to the study of Cheung et al. (2015), this paper investigates the existence of cryptocurrency bubbles for Bitcoin, Litecoin, Ripple, Ethereum and the cryptocurrency portfolio, which is created out of these four cryptocurrencies. Next to that, the time periods for the bubbles of these cryptocurrencies will be researched. As Bouri et al. (2018) find multiple bubble periods for Bitcoin, Litecoin and Ethereum with the GSADF statistic (Phillips et al. 2012; 2013), our research applies the same method. This leads to the following hypotheses.

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13 Hypothesis I : There is mean-variance spanning of Bitcoin, Litecoin, Ripple, Ethereum and a cryptocurrency portfolio on a well-diversified initial asset’s portfolio.

Hypothesis II : Bitcoin, Litecoin, Ripple, Ethereum and a cryptocurrency portfolio show explosiveness in their prices.

Hypothesis III : For the various cryptocurrencies and a cryptocurrency portfolio there will be no bubble periods.

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3. Methodology

First, we test whether Bitcoin, Litecoin, Ripple, Ethereum and the cryptocurrency portfolio span a well-diversified asset’s portfolio. This will be tested in this research using mean-variance spanning for portfolio testing and the Phillips-Perron test and the GSADF statistic will be applied in order to test for bubbles in cryptocurrency prices. This section sets out the methodology as the fundament for these three tests.

3.1 Portfolio diversification testing for cryptocurrencies

This research investigates whether there are diversification benefits when entering several different cryptocurrencies and the portfolio of these cryptocurrencies to a well-diversified asset’s portfolio. This is formulated into a hypothesis that tests for mean-variance spanning. This model is developed by Huberman & Kandel (1987) and further applied by De Roon et al. (1998), De Roon et al. (2001) and many others.

As Huberman & Kandel (1987) show in their paper, a regression analysis based on the spanning theory is in line with this research. The Minimum-Variance Frontier (MVF) of the initial well-diversified portfolio will be spanned against the MVF of four different cryptocurrencies as well as a portfolio created with them. According to Huberman & Kandel (1987) monthly data is the most appropriate for this test. The monthly indices from the S&P 500, the FAZ and the FTSE, transferred into USD, will be used to create the original well-diversified portfolio. From these indices the market capitalization per year will be calculated. Their weight in the portfolio is dependent on their market capitalization compared to each other. As expressed in the following equation (1):

𝐼𝑛𝑑𝑒𝑥 𝑤𝑒𝑖𝑔ℎ𝑡 𝑖𝑛 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 = 𝑚𝑎𝑟𝑘𝑒𝑡 𝑐𝑎𝑝𝑖𝑡𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡

𝑡𝑜𝑡𝑎𝑙 𝑚𝑎𝑟𝑘𝑒𝑡 𝑐𝑎𝑝𝑖𝑡𝑎𝑙𝑖𝑧𝑎𝑖𝑡𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑡ℎ𝑟𝑒𝑒 𝑖𝑛𝑑𝑖𝑐𝑒𝑠 (1)

With that, the initial asset’s portfolio will be created as a portfolio of the three indices with weight based on their market capitalization. This is expressed in equation (2). With Pm as the price of the index for that month and Wy as the weight of the index for that year.

𝑃𝑟𝑖𝑐𝑒 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 A𝑠𝑠𝑒𝑡:_{𝑠 P𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜}

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15 Next to that, the monthly exchange rates of the cryptocurrencies in USD will be used separately, but also in the form of a cryptocurrency portfolio to test for spanning. This because in case that the returns of the separate cryptocurrencies do not perfectly correlate with each other, they might diversify away a part of the risk of investing in cryptocurrencies when they are added together in a cryptocurrency portfolio. Since the Bitcoin has a significant higher price than the prices of the other three cryptocurrencies, weights will not be dependent of price or market capitalization. A market weighted portfolio namely would mean a situation in which cryptocurrency is overrepresented. More than 90% of the total cryptocurrency portfolio would than exist out of Bitcoin. The effect that other cryptocurrencies would have on the diversification in the cryptocurrency portfolio would mostly fade away. Therefore, all cryptocurrencies bear the same weight in the cryptocurrency portfolio. This is expressed in equation (3). In this equation P is the price of the cryptocurrency per month.

𝑃𝑟𝑖𝑐𝑒 𝐶𝑟𝑦𝑝𝑡𝑜𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑦 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 = 0.25 ∗ 𝑃_UHV + 0.25 * 𝑃_XHV + 0.25 ∗ 𝑃_YZ? + 0.25 * 𝑃_IH[ (3)

In the following model (4), K will represent the new assets added to the portfolio and N will represent the original assets. K thus will be the cryptocurrencies separately as well as together in a portfolio. N will be the original asset’s portfolio created out of the indices of S&P 500, FTSE and FAZ. The K X 1 vector stands for the returns on assets K and N X 1 denotes the returns on the original N assets (Huberman & Kandel, 1987). Next to that, iK is the K X 1 vector with all elements being equal to one and iN is the N X 1 vector with all element equal to one.

The regression showed in equation (4) is used for the mean-variance spanning. This model is based on the research of De Roon et al. (2001):

rt+1 = a + bRt+1 + et+1, t = 1, 2, …., T (4)

Where t is the length of time series and rt+1 represents the gross returns on the additional assets, K, and Rt+1 as gross return on the initial portfolio created out of the three indices, N. In this model, et+1, is the error term and a is the N-dimensional vector of Jensen’s alphas of the assets rt+1 relative to the mean-variance efficient portfolio of Rt+1 with zero-beta return.

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16 b here is an N X K matrix. Intersection implies that a + (biK – iN) = 0 and spanning holds when a = 0 and biK - iN = 0 with i as the vector with all elements equal to 1. In the case of intersection, the MVF’s only unify at one point. In this case, both K and N have the same gross returns for the same level of risk. Therefore, no diversification benefits can be gained from adding the new assets to the existing portfolio. An efficient frontier has been found when there is intersection of the cryptocurrencies and the original portfolio. In this case, it provides a maximum return (Chen et al., 2010). This means that the ratio of cryptocurrencies with respect to the initial assets at which the MVF’s intersect, should be added to the well-diversified portfolio to create a situation in which diversification benefits can occur.

3.2 Identifying bubbles

Next, this research is searching for the existence of bubbles within the price of four different cryptocurrencies and the cryptocurrency portfolio. In order to do so, two bubble tests will be conducted. The Phillips-Perron tests (Phillips & Perron, 1988) is able to detect the existence of bubbles within the price of an asset. The downfall of this test is that when it is the case that several bubbles exists in the price, the model is not able to detect them. Therefore, the Generalized Sup Augmented Dickey-Fuller statistic (Phillips et al., 2011b) will be applied.

3.2.1 The Phillips-Perron test

As the researches of Li et al. (2018a) and Li et al. (2018b) use the sup ADF test (Phillips et al., 2012; 2013) based on the Phillips-Perron test (1988) in order to test for bubbles, this test will also be applied in this research testing for bubbles. The model is designed to test for unit roots. In case of a unit root, there is a trend in the price of a cryptocurrency. When this happens, no bubble will be expected. When this hypothesis can be rejected, bubbles can occur in the cryptocurrency price. Bubbles are stated as periods of 20 or more days of explosive prices. Since bubbles are measured in days, daily pricing data of the four cryptocurrencies will be used in this test. Equation (5) represents the model of Phillips & Perron (1988):

(5)

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17 With DRt as the change in the cryptocurrency’s price in USD and k as the number of lags, which is determined by both the AIC and BIC test (Liew, 2004). For t we take the number of time periods. NID denotes the independent and normal distribution. The variable a represents the intercept. Under the null hypothesis, H0: b = 1, which means that the cryptocurrency or portfolio of interest contains a unit root. This is equal to the assumption that that cryptocurrency or portfolio is not stationary, providing evidence for a bubble. The alternative states H1: b <1, that the cryptocurrency of interest does not contain a unit root. Therefore, the cryptocurrency or the cryptocurrency portfolio has a stochastic trend and can be seen as stationary. Prior literature on the existence of bubbles states that if there is a bubble present, prices show explosive behavior in the form of instable prices (Stein, 1987; Shiller, 1981), which this test is able to show.

The main drawback that comes up with using this test is that when multiple bubbles exist, the test fails for the detection of the bubbles (Evans, 1991). This caused Phillips et al. (2011a) to develop this test further into the generalized sup ADF (GSADP) test, which is able to indicate multiple irrational stochastic bubbles, under a certain time period.

3.2.2 The GSADF test

As in Phillips et al. (2011a) and Phillips et al. (2011b), we also apply the GSADF test in order to test for explosive behavior. Since this research is looking for explosiveness within the prices of different cryptocurrencies as well as those studies, applying this statistic is very appropriate. The GSADF statistic is the supremum value of the Augmented Dickey-Fuller statistic. In addition, it allows for more flexible estimation windows (Caspi, 2016). The GSADF test is designed to repeatedly run the ADF regression. The GSADF statistic is defined as the largest ADF statistic over the feasible ranges it contains (Phillips et al., 2012).

The equation for the null hypothesis is as in equation (6):

𝑦_e = µ_e+q𝑦_efg+ e_e with 𝜇e = 𝛼𝑇fn (6)

With gt as the daily closing price of the cryptocurrencies and p as the drift parameter. This drift parameter controls for the magnitude of the drift. The drift is referred to the description of a unit root, which is being defined as a random walk with a drift. In a situation in which p > 0,

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18 the drift will be seen as small compared to the linear trend. When p > 0,5 the drift is small relative to the stochastic trend. Under the null hypothesis, there is the expectancy of the existence of a bubble. With the H0: q = 1, containing a unit root. The alternative, H1: q > 1, states that the sample contains an explosive root. The ADF test will run repeatedly in a rolling window with size n, with n is e(r0, T), where r0 is the smallest window size possible. For each window, the following OLS regression (7) will be estimated, according to Cheung et al. (2015):

𝛾_e = 𝜇_e+ 𝜃_q_rst+ u ∆𝑦_efw

x

wyg

+ 𝜀_e

(7)

In this equation, J is the lag order, from which the regression will be computed for a sliding window from the last observation to the first observation. In this OLS regression with a rolling window, r1 will be the start point of the rolling window and r2 is the end point for this. This is showed in the ADF t-test statistic (8), which will be showed below:

𝐴𝐷𝐹_~_t_~_•= 𝜃~t~• 𝑆𝐸 (𝜃~t~•) (8)

In this case, n = T for the standard ADF. As the Generalized Supremum Augmented Dickey-Fuller statistic is defined as the largest ADF out of all repetitions in the window size and place, varying r2 from n to T, a series of GSADF statistics can be found. The first starting point of a bubble will be indicated as the first observation where the backward GSADF statistic is greater than the GSADF statistic defined by a Monte Carlo simulation (Cheung et al., 2015). The end of the same bubble will be the point where backward GSADF is below this Monte Carlo simulation value. Testing this statistic also allows for data price data of the four cryptocurrencies and the portfolio of the cryptocurrencies. In order to filter out the large outliers of price differences between relatively small periods of a view days, a minimum period for defining a bubble will be installed. As is common in this type of study, this is when the GSADF value is above the Monte Carlo simulation value for 20 days or more (Stukart, 2018).

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19 The previously named critique of Evans (1991) with respect to the standard ADF test is tackled with the extension done in the GSADF test. That is, due to running rolling windows, the model is able to detect multiple bubbles.

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20

4. Data

This section will describe the data that is used for this study. Furthermore, the data sources will be outlined. Next to that, the descriptive statistics of the cryptocurrencies will be presented.

4.1 Variables used for portfolio testing

As historical literature primarily utilizes the gross return of assets as the main variable for mean-variance spanning (Markowitz, 1991; Das et al., 2011, Briere et al., 2015; Dorfleitner & Lung, 2018) this research will use this variable likewise. In this research, returns of the cryptocurrencies and its portfolio will be accounted for. A graph of the returns for cryptocurrencies and the portfolios can be found in Graph C in the Appendix.

4.2 Variables used for detecting bubbles

Since asset’s prices and the lags of these prices have been the variable of choice for measurements regarding bubbles in prior literature (Fatazizini et al., 2017; Chueng et al., 2015; Li et al., 2018b), this research will apply the prices of the cryptocurrencies as the main dependent variable as well in order to test the two models for bubble testing. The justification of using variables in the form of asset’s prices for this research lies in the definition of bubbles. As an asset’s bubble is indicated as a period of an explosive price of the asset, it is suggested that the models that test for pricing bubbles aim to detect explosive prices. Therefore, the use of cryptocurrency prices will be applied to this research as well. The graph of the cryptocurrency prices is showed in Graph D in the Appendix.

4.3 Descriptive statistics

In this section, the variables and descriptive statistics for this study will be set out.

This research uses mean-variance spanning in order to detect diversification benefits. Different types of cryptocurrencies and a portfolio of all of them will be spanned against an initial portfolio, created out of stock indices. Table A in the Appendix presents the sources of the prices of the cryptocurrencies and the indices.

The variables added to the original portfolio are the four most common cryptocurrencies Bitcoin, Litecoin, Ripple and Ethereum. Furthermore, a cryptocurrency

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21 portfolio will be created with these four cryptocurrencies. Since Bitcoin’s exchange rate is significantly higher than that of the other cryptocurrencies, the cryptocurrency portfolio will not be created out of market capitalization weighted averages. When this would apply, Bitcoin would still be dramatically overrepresented. Therefore, within this cryptocurrency portfolio, all the four cryptocurrencies will have an equal weight. The formula for this portfolio has been set out in equation (3) in the methodology. The exchange rates from the cryptocurrencies in USD are obtained from Coinmetrics. These will be interpreted as the price of the cryptocurrencies in USD. From these prices, the monthly returns of the assets will be calculated. According to Huberman & Kandel (1987) monthly returns are most appropriate for this test. The equation (9) for the asset’s return is calculated as follows:

𝑅𝑒𝑡𝑢𝑟𝑛 𝐶𝑟𝑦𝑝𝑡𝑜𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑦 (𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜)< =?~„…† V~‡ê‰…Š~~†‹…‡_{?~„…† V~‡ê‰…Š~~†‹…‡}BŒ ?~„…† V~‡ê‰…Š~~†‹…‡Bst

Bst (9)

With Return Cryptocurrencym as the return of the cryptocurrency of interest at the first of that specific month and Price Cryptocurrencym as the price of that cryptocurrency at the first of the same month. Price Cryptocurrencym-1 represents the price of the cryptocurrency of interest at the first of the month before. This calculation can also be adapted for the cryptocurrency portfolio. As Table B in the Appendix shows, for the Bitcoin (BTC), return rates are included monthly from the first of June 2013 up and until the first of November 2018. The time span for the return of Litecoin (LTC) is the same. For Ripple (XRP), the return rates included start a bit later, at the first of October in 2013. The last return rates included for Ripple is the same as for Bitcoin and Litecoin, at the first of November 2018. Ethereum (ETH) returns have a time span from the first of October in 2015 up and until the first of November 2018. Since the cryptocurrency portfolio is created out of these four cryptocurrencies, it can be included from the first of October 2015 also. The last return included is the same as for the others, the rate at the first of October 2018. A graph of the returns of the cryptocurrencies and its portfolio can be found in Graph C in the Appendix. As the graph shows, the greatest peak of return is between the last quarter of 2013 and the first quarter of 2014 for Litecoin. During this period, the return of all cryptocurrencies and the cryptocurrency portfolio peak. The second largest peak is between the last quarter of 2017 and the first quarter of 2018. This is the peak in the return of Ripple. After the first quarter of 2018, the returns for all

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22 cryptocurrencies and the cryptocurrency portfolio show negative returns, apart from the small positive returns at the second quarter of 2018. The descriptive statistics of the returns of the cryptocurrencies and the cryptocurrency portfolio is given in Table 4.1 below. In the table can be seen that there are large differences between the minimum and the maximum for the cryptocurrencies. Especially the returns of Litecoin stands out. For this cryptocurrency, a minimum of -0.424% is detected and a maximum of 16.338% has been found. Furthermore, the standard deviation of the cryptocurrency portfolio appears to be lower than that of the cryptocurrencies separately. This suggests that the returns of the cryptocurrencies have relatively less outliers than that of the cryptocurrencies separately. Especially the standard deviations of the returns of the Litecoin, 2.060, and the Ripple, 1.423, creates concerns regarding the large differences in returns. Cocco et al. (2017) show a relatively lower standard deviation of Bitcoin, namely 0.057. Furthermore, the study of Cocco et al. (2017) finds a mean value of the return of Bitcoin of 0.007, while this research finds a mean value of 0.129. Their study is published in July 2017. As can be seen in both Graph C and Graph D, a lot has happened to the prices and returns of Bitcoin and the other cryptocurrencies. Therefore, it can be expected that the standard deviations differ between this paper and the paper of Cocco et al. (2017) This paper adds an extension of the sample period to the existing literature, causing different descriptive statistics. The sample period of Liu & Tsyvinski (2018) is broader compared to the research of Cocco et al. (2017), just as the sample period of this paper. The descriptive statistics that they find are therefore more in line with the descriptive statistics this study finds. The research of Liu & Tsyvinski (2018) presents a standard deviation of the monthly returns of Ripple of 1.4331, which is close to 1.423, which this research shows. Furthermore, this study finds a value for the standard deviation of the monthly returns of Ethereum of 0.659. In the paper of Liu & Tsyvinski (2018) this was 0.6756. The mean return of Ripple in this paper is 0.359, as they show a mean return of 0.3620 for Ripple. Next to that, there are some differences in the descriptive statistics between their paper and this study as well. The mean of the monthly returns of Ethereum is namely 0.121, while it is 0.3026 in the investigation of Liu & Tsyvinksi (2018). Also, the standard deviation of Bitcoin in this paper is 2.060, while it is 0.6946 in their study. These differences might come from the fact that the sample period from this paper is seven months longer than the sample period in their study. While their paper account for the period until the first of May 2018, this paper extends the sample period to the first of November 2018. As can be seen in Graph D, large differences in

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23 the Bitcoin price appeared in the last period of 2018. This had its effect on the returns of the cryptocurrency. This can clarify the change in the differences between the descriptive statistics.

To be able to test the effect of adding these variables in a well-diversified portfolio, an original portfolio is necessary. As in the research of De Roon et al. (1998), this study will create a base portfolio from three international stock indices. According to them, these are able to construct a well-diversified portfolio for an US investor. This portfolio is created in such a way that all risk is diversified away, resulting in a portfolio with relatively constant returns. The indices used are the S&P 500 (United States), the FAZ (Germany) and the FTSE (United Kingdom), which are obtained from Datastream (see Table A from the Appendix). The FAZ and FTSE will be converted into monthly US Dollar prices with the help of monthly exchange rates. With these price data, weighted averages of the indices, based on market capitalization, can be calculated. These equation (1) for this calculation is showed in the methodology. The so-called initial portfolio then will be constructed as a combination of these three indices, based on their weighted average. This calculation is set out in equation (2) in the methodology. The returns for these tests are set from the first of June 2013 up and until the first of November 2018 (see Table B in the Appendix). The graph representing the returns of the original portfolio can be found in Graph C in the Appendix. As can be expected from a well-diversified portfolio, the returns of the initial asset’s portfolio remain stable during the entire period of 2013 to 2018. No remarkable peaks appear in the returns of the asset’s portfolio. This is in line with the paper of De Roon et al. (1998). De Roon et al. (1998) have created the initial portfolio in order to test for the effect on portfolio returns of adding any new asset. Therefore, they choose to develop a portfolio with no remarkable peaks in returns. This creates a situation in which the effect of adding the new asset to the portfolio can be measured. Therefore, this original asset’s portfolio is appropriate for this study as well. The descriptive statistics for the original asset’s portfolio returns are presented in Table 4.1. The relatively low standard deviation of 0.035 consolidates the statement that the returns for this portfolio are rather stable. Both Graph C and the descriptive statistics show that the original asset’s portfolio does not have any peaks in returns.

Next, two bubble tests will be conducted. These require daily data. The period of interest is namely expressed in days. A bubble is defined as a period in which the GSADF for the cryptocurrencies is higher than the GSADF statistic according to the Monte Carlo Simulation

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24 during a period of 20 days. Therefore, it is convenient to do the bubble test based on daily data. The bubble tests use the price of the cryptocurrencies of interest. The exchange price to US Dollar of the four most common cryptocurrencies and of the portfolio that combines them will be used as the cryptocurrency’s price. In these bubble tests, price data from one day will be compared to another moment in order to detect the existence of exponential growth. The data for the currency exchange rates is obtained from Coinmetrics. As Table B in the Appendix shows, the price from Bitcoin is included in this research from April 28th_{, 2013, for Litecoin this} is the same. The price from Ripple is accounted for since the fourth of August 2013. For Ethereum this is since August seventh, 2015. The cryptocurrencies portfolio price will be included from the seventh of August 2015 as well. The end date is for all the four cryptocurrencies and the portfolio the same, to with the 31st of October 2018. Graph D in the Appendix represents the prices for the cryptocurrencies and the cryptocurrency portfolio. It therefore extends the graph of Ciaian et al. (2015), who exhibit the prices of the Bitcoin. Furthermore, Table 4.1 below represents some of the descriptive statistic for the cryptocurrencies and the portfolios used. As can be seen from Table 4.1, the price from Bitcoin started at $86.50 and shows a peak of $19475.80. For the Ethereum this difference in price is remarkable as well. With a start price of $0.43 it has gone up, showing its highest level at $1397.48. The relatively high standard deviation of 3439.09 for the prices of Bitcoin suggests that there are relatively more outliers. The standard deviation of the price of Litecoin, 54.606, Ethereum, 214.36, and the cryptocurrency portfolio, 1078.6, are relatively high as well. The price of Ripple is the only one that does not show a standard deviation of above 1. With a standard deviation of 0.343, Ripple shows relatively less outliers. This indicates that the prices for Ripple do not fluctuate as much as they do for the other cryptocurrencies and the cryptocurrency portfolio.

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25 Table 4.1: The descriptive statistics

The following table presents a selection of important descriptive statistics. For the four most common cryptocurrencies as well as for its portfolio and the original well-diversified asset’s portfolio the statistics are shown. The variables variate from the price of the cryptocurrencies and its portfolio to returns of the cryptocurrencies and both portfolios. Since the prices are accounted for on daily basis and the returns on monthly basis, a larger amount of observations can be seen for the prices of cryptocurrencies and the cryptocurrency portfolio than for the returns of the cryptocurrencies and the portfolios.

Variables N Mean SD Min Max Kurtosis Skewness

Price BTC 2013 2238.818 3439.09 68.5 19475.8 7.090 2.071 Price LTC 2013 30.940 54.606 1.15 359.13 10.722 2.689 Price XRP 1915 0.159 0.343 0.003 3.36 26.347 4.0548 Price ETH 1182 214.36 281.5 0.432 1397.48 4.727 1.475 Price Cryptocurrency Portfolio 1182 953.101 1078.6 53.572 5118.644 4.105 1.290 Monthly Return Original Portfolio 69 0.0003 0.035 -0.109 0.070 3.307 -0.582 Monthly Return BTC 65 0.129 0.604 -0.336 4.537 45.170 6.111 Monthly Return LTC 65 0.304 2.060 -0.424 16.338 58.214 7.422 Monthly Return XRP 61 0.359 1.423 -0.496 8.187 19.008 3.853 Monthly Return ETH 37 0.2820 0.659 -0.536 2.156 4.192 1.344 Monthly Return Cryptocurrency Portfolio 37 0.121 0.246 -0.343 0.763 3.198 0.627

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26 A matrix between the cryptocurrencies and both portfolios is interesting as well. The correlation for the returns of the original asset’s portfolio, the Bitcoin, the Litecoin, the Ripple, the Ethereum and the cryptocurrency portfolio is given in Table 4.2 below.

Table 4.2: Correlation Matrix of the Returns

This table shows the correlation between the returns of Original Asset’s Portfolio, the Bitcoin (BTC), Litecoin (LTC), Ripple (XRP), Ethereum (ETH) and the Cryptocurrency Portfolio. The full sample of this correlation matrix runs from October 2015 until November 2018.

Return Original

Portfolio Return BTC Return LTC Return XRP Return ETH

Return

Cryptocurrency Portfolio

Return Original Portfolio

1.000 Return BTC 0.1929 1.000 Return LTC 0.2271 0.5624 1.000 Return XRP 0.2654 0.3617 0.7663 1.000 Return ETH 0.0893 0.3032 0.4651 0.4675 1.000 Return Cryptocurrency Portfolio 0.2085 0.9967 0.5913 0.3974 0.3567 1.00

The main interest of the correlation table above is the correlation of the return of the original portfolio with the returns of the cryptocurrencies and the cryptocurrency portfolio. Since the correlation coefficient shows the degree of linear cohesion, perfect correlation states that the minimum-variance frontier of one of the cryptocurrencies would move in exactly the same direction as the minimum-variance frontier of the original portfolio. When the correlation coefficients of the cryptocurrencies with the original asset’s portfolio show no perfect correlation, mean-variance spanning of the minimum-variance frontiers of a cryptocurrency and the original asset’s portfolio is not likely. This is because the minimum-variance frontiers do not more in the same direction, and therefore, perfect coinciding of the MVF’s is not possible. Since the table shows that the correlation between the return of the original portfolio and the cryptocurrencies separately as together in a portfolio is relatively low, mean-variance spanning would not be likely. Therefore, diversification benefits can be expected. Goetzmann & Kumar (2008) explain this occurrence. In their paper, they show that when

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27 correlation estimates between two different assets are not 1, both assets capture different characteristics. This suggests that adding assets that are not perfectly correlated to the original asset’s portfolio, provides different characteristics to the original portfolio. And, therefore, portfolio diversification increases. Guesmi et al. (2018) show this for Bitcoin in their research. They state that due to the low correlation between Bitcoin and financial assets, Bitcoins can be used as hedging and diversification tools. This is in line with the findings in this research. This research adds these findings for Litecoin, Ripple, Ethereum and the cryptocurrency portfolio to the existing literature

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5. Results

In this section, the results for the hypotheses will be elaborated on.

5.1 Mean-variance spanning results

The results of the mean-variance spanning in order to test for the first hypothesis are showed in this section. This hypothesis is developed in order to test whether four different cryptocurrencies and a portfolio of them are able to span a well-diversified original asset’s portfolio. The results of hypothesis I are presented in Table 5.1 below.

Table 5.1: Results for mean-variance spanning

This table shows the results of the mean-variance spanning of four cryptocurrencies and a portfolio of these cryptocurrencies on a well-diversified original portfolio created out of the S&P 500, FTSE and FAZ indices. The dependent variables in this table are the monthly returns of the cryptocurrencies and the cryptocurrency portfolio. The independent variable is the monthly return of the original asset’s portfolio. Model (1) shows the spanning of Bitcoin on the original portfolio, model (2) shows the spanning of Litecoin on the original portfolio, model (3) shows the spanning of Ripple on the original portfolio, model (4) shows the spanning of Ethereum on the original portfolio and model (5) shows the spanning of the cryptocurrency portfolio on the original portfolio. The t-values of these estimates are reported in parenthesis, with * indicating a t-value with a confidence interval of 95% and ** indicating a t-value with confidence interval of 99%.

(1) (2) (3) (4) (5)

Return BTC Return LTC Return XRP Return ETH Cryptocurrency Return Portfolio Return Original Portfolio 1.941 3.382 7.272 1.846 1.611 (2.06)* (1.34) (1.40) (0.55) (1.54) a 0.130 0.306 0.364 0.280 0.119 (1.72) (1.19) (2.01)* (2.55)* (2.99)** R2 0.01 0.00 0.03 0.01 0.04 N 65 65 61 37 37

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29 As can be seen in Table 5.1 above, neither one of the cryptocurrencies nor the cryptocurrency portfolio is spanned by the original asset’s portfolio. Since neither Bitcoin, Litecoin, Ripple, Ethereum nor the cryptocurrency portfolio has an a of 0, the first condition for mean-variance spanning does not hold. For Bitcoin and Litecoin, this conclusion should be interpreted with care, since their t-values do not bear any significance. The a of Ripple, which is 0.364, is significant at 5% level, as well as Ethereum, that has an of a 0.280. For the estimate of the a of spanning of the cryptocurrency portfolio on the original portfolio, which is 0.119 in this case, the significance level is 1%. The a of Ripple, Ethereum and the cryptocurrency portfolio thus is significantly different from 0. Therefore, we can reject the null hypothesis for Ripple and Ethereum with 5% significance and the cryptocurrency portfolio with 1% significance based on the assumption of a = 0.

As is shown in Table 5.1 above in the first row, none of the cryptocurrencies has a b(ik – iN) of 0 when they are being spanned against the original asset’s portfolio. This applies to the cryptocurrency portfolio as well. However, only the estimate of b(ik – iN) Bitcoin appears to be significant. The b(ik – iN) estimate for Bitcoin is 1,941 at a 95% confidence interval. Therefore, when solely looking at this assumption, this research rejects the null hypotheses for Bitcoin at 5% significance level. This means that we can reject the null hypothesis for Bitcoin that there is mean-variance spanning with the initial asset’s portfolio. Furthermore, we fail to reject the null hypothesis with respect to spanning of Litecoin, Ripple, Ethereum and the cryptocurrency portfolio on the original asset’s portfolio.

We can state with significance, that for Bitcoin, Ripple, Ethereum and the cryptocurrency portfolio one of the assumptions does not hold. Therefore, we can reject the null hypothesis with significance that there is mean-variance spanning for Bitcoin, Ripple, Ethereum and the cryptocurrency portfolio on the original indices portfolio. For Litecoin this study fails to reject the null hypothesis that there is mean-variance spanning between the cryptocurrency and the original portfolio. In terms of economical meaning, this means that the variance frontier of the original portfolio does not coincide the minimum-variance frontiers of Bitcoin, Ripple, Ethereum and the cryptocurrency portfolio with significance. This suggests that adding these cryptocurrencies or the cryptocurrency portfolio to the original well-diversified portfolio creates diversification benefits, from which an investor could gain. This is suggested by Briere et al. (2015) as well. They find a spanning

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30 estimate for Bitcoin on a combination of traditional assets and alternative assets of 6.93, at 1% significance level. Even though the study of Briere et al. (2015) uses another asset’s portfolio as base portfolio, both their paper and our paper suggest that investors could gain form diversification benefits arising from adding Bitcoin to their well-diversified portfolio.

5.2 Phillips-Perron test results

The results of the Phillips-Perron test for the four cryptocurrencies as well as for the cryptocurrency portfolio are presented in this section. With this methodology, the second hypothesis has been tested. This in order to detect any price bubbles within the four cryptocurrencies or the cryptocurrency portfolio. The results of the tests are presented below in Table 5.2.

Table 5.2 Results of the Phillips-Perron test

This table presents the test statistic for Z(t) of the Phillips-Perron test for the cryptocurrencies and cryptocurrency portfolio. Furthermore, the 5% significance value for Z(t) and the p-value for Z(t) are provided. The significance levels are given as * indicating p<0.05 and ** indicating P<0.01

Variable Test statistic for Z(t) 5% Critical Value for Z(t) p-value

BTC 2.311 3.410 0.4280 LTC 2.710 3.410 0.2318 XRP 3.833* 3.410 0.0150 ETH 1.656 3.140 0.7697 Cryptocurrency Portfolio 2.046 3.140 0.5762

Table 5.2 above presents the results for the Phillips-Perron test for the four cryptocurrencies and the cryptocurrency portfolio. The Z(t) value here represents the asymptotic distribution of the variables. An asymptotic distribution indicates that there is a sequence of distributions, which is the case, since this test is designed in order to compare up following prices with each other in order to detect any explosiveness. As the theory states, the null hypothesis can be rejected when the test statistic of Z(t) is greater than the 5% critical value of Z(t). This is solely the case for the Ripple. Here, the value of Z(t) is 3.833, whereas the 5% critical value for Z(t) is 3.410. This suggests that there are no explosions within the prices of the Ripple. This estimate

Cryptocurrencies as speculative assets: extension on portfolio diversification in combination with testing for bubbles