The impact of including Bitcoin to a global market portfolio

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The impact of including Bitcoin

to a global market portfolio

Abstract

This paper investigates whether the inclusion of Bitcoin to global market portfolio leads to diversification benefits. This is analyzed for the period mid-2010 till mid-2019. With using an advanced model of portfolio optimization different portfolios are constructed and afterwards compared. The results conclude that Bitcoin is indeed an interesting addition to an investors’ portfolio because the inclusion results in better risk-return trade-offs. Due to earlier empirical research and a change in the target return used in this study the results are robust. Besides this, Bitcoin is replaced with assets who have some similar characteristics (Gold and the US Dollar) and it can be concluded that Bitcoin is a better diversifier than those assets.

By: M. (Mark) Schutte

Student number: S2784718

Email address: mark.schutte.4@student.rug.nl Supervisor: dr. C.G.F. van der Kwaak

2nd_Supervisor: _{dr. I. Souropanis}

Course: Master’s Thesis Finance Course code: EBM866B20

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1. Introduction

In 2008 the global financial industry crashed. This was the worst financial crisis since the Great Depression of the 30s. Perhaps propitiously, the anonymous person Nakamoto Satoshi outlined during this financial crisis a new protocol for a peer-to-peer electronic cash system using a cryptocurrency. Cryptocurrencies or digital currencies differ from the traditional fiat currencies (like the US dollar, the Euro and the Japanese Yen) because they are not created or controlled by countries and central banks. Bitcoin is the most popular cryptocurrency (70% of the total market capitalization in September 2019 – Coinmarketcap, 2019). Since Nakamoto published the paper the growth and media attention has exploded. This resulted in increasing trading volume and capital gains and losses in a highly volatile environment. The paper of Nakamoto describes a set of rules in which people can exchange goods and services without the intervention of a trusted third party (banks or governments).

The reason to outline the paper is that according to Nakamoto (2008) fiat currencies are not an appropriate exchange medium because of the fact that a big part of the world has no access to the banking industry and the presence of the transaction costs (Nakamoto, 2008). Besides this, Nakamoto argues that fiat currencies are not a good store of value. Nakamoto claims that Bitcoin can act as a good store of value and proper exchange medium because of the following facts: the cryptocurrency supply is pre-determined, constant, finite and deflationary. A possible question that might pop up is: if Bitcoin is indeed a good store of value, should the cryptocurrency be included in an investment portfolio?

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3 In the last couple of years, academic research on Bitcoin and cryptocurrencies as a financial asset is growing (Corbet et al., 2019). According to Eisl et al. (2015) Bitcoin can be an interesting financial asset for an optimal portfolio because the cryptocurrency has low correlations with the more traditional financial assets like stocks, bonds, commodities and fiat currencies.

Portfolio optimization is an important part of finance. Harry Markowitz, winner of the Noble Prize in Economics in 1990, was one of the first scholars who developed a mean variance framework (1952) in which an optimal portfolio is constructed. In the ideal world, a portfolio should generate the highest possible returns given a certain level of risk. Diversification according to Markowitz means combining several assets with less-than-perfect positive correlations. In that way the risk of the portfolio can be reduced without giving up the return (Francis and Kim, 2013). To study and combining the different worlds of traditional finance and Bitcoin, the following research question is formulated:

‘’Can Bitcoin be used as a diversifier of an already well-diversified global market portfolio?”

To analyze whether the inclusion of Bitcoin generates higher returns and/or reduces the overall risk in a portfolio a global market portfolio needs to be constructed in this study. The global market portfolio will consist of all the traditional financial assets like stock indices (MSCI World Index, MSCI Emerging Markets Index), fixed income indices (world, emerging markets, and high yield), commodities, a real estate index, a private equity index and a hedge fund index following Eisl et al. (2015) and Brière et al. (2015). This portfolio will be considered as well-diversified and of this portfolio the return, risk and the correlations of the assets within the portfolio will be calculated. The portfolio return and risk are calculated with and without Bitcoin and in this way the the different portfolios are analyzed. To test the robustness of the results, Bitcoin will be replaced with assets who have some similar characteristics namely Gold and Cash (see section 2 - literature review) and a different target return will be used.

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2. Literature overview

In this section the known literature about the study will be grounded. Portfolio theory in general will be discussed in the first subsection. The second part of this section will describe the characteristics, the risks and the technology behind Bitcoin. As one of the robustness checks, Bitcoin will be replaced with Gold and the US dollar, the characteristics of these assets will also be described in the second part. Finally, the academic research on Bitcoin as an investment asset will be discussed.

2.1 Portfolio Theory

According to Nawrocki (2000) portfolio theory can be defined as the application of decision-making tools under risk to the problem of managing risky investment portfolios. Portfolio theory says: if we understand the statistical properties of the investment opportunities in the market and if we know what the investors’ attitudes towards the asset characteristics are it can be told what portfolio the investor should hold (Kaplan and Siegel, 1994). Mean-Variance Optimization (Markowitz) and Downside Risk Optimization (for example the model of Fishburn used in this study) are applications of portfolio theory. In this subsection the fundamentals of portfolio theory are described and in the following subsections it will be analyzed in what way Bitcoin can (possibly) contribute to the optimization problem for global market investors in modern times.

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6 In practice, this is often not the case and therefore the variance might be an unrealistic measure of risk. In the case of nonnormality of the stock returns, downside risk measures would help the investors better to make proper decisions when they are not risk neutral. An example of such a downside risk measure is the model of Fishburn (1977). Risk of an investment portfolio is in this model measured via a function of deviations below a specified target return (see section 3 - Methodology). In general, portfolio optimization problems can be seen and described as a sequence of mathematical programming problems.

First, the efficient frontier of the different portfolios needs to be constructed (see figure 1 below). On the horizontal axis the risk of a portfolio is given and on the vertical axis the return of the concerned portfolio is presented. Depending on the risk tolerance of the investors, the bold line represent the efficient frontier: a mix of portfolios that are superior to all other portfolios in the investment universe (regarding risk and return). The point where the line bends from the left to the right represents the Minimum Variance Portfolio (MVP): the most optimal portfolio with the lowest level of risk. The area below the line contains the inefficient portfolios. An investor will then choose the portfolio which maximizes his or her utility. The model of Fishburn (1977) is a Lower Partial Moment (LPM) risk measure (Nawrocki, 2000). One of the challenges of using this risk measure as constructor of efficient portfolios is the computational intensity. Therefore, in this research the optimal portfolio will be constructed using the Microsoft Excel Solver. In section 3 the formulas for calculating the returns, risks and the construction of the optimal portfolios are presented.

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7 In general investors are concerned about two types of risk: systematic risk and unsystematic risk. Systematic risk is also known as market- or non-diversifiable risk, while unsystematic risk is the opposite and known as firm-specific or diversifiable risk. As suggested by the different names of the risks, investors can optimize their portfolios by eliminating the unsystematic risks. This are the risks of possible company- or industry-specific hazards. Fama and French (1993) presented in their paper some of those common firm specific risk factors: Small minus Big (related to firm size) and High minus Low (related to market-to-book ratio).

By increasing the number of assets in a portfolio the unsystematic risks will decrease. In the world there are always risks that go beyond an industry or firm, like for example macroeconomic risks or political risks that affect the overall market. This are the systematic risks which cannot be lowered by any amount and is also known as the volatility. Market risk cannot be diversified away but there are possibilities to hedge it with offsetting investment positions. For this study, market risk is not important and it will be analyzed whether the including of Bitcoin (and the US dollar and Gold as robustness checks) decreases the unsystematic risks of a portfolio.

2.2 Bitcoin

According to Nakamoto (2008) Bitcoin can act as an excellent store of value. A possible question that might pop up is: if this is indeed the case, should the cryptocurrency be included in an investment portfolio? Because Bitcoin is complex and unique further explanation is required: first in this subsection the risks, characteristics and technology behind Bitcoin will be explained. After that the case of Bitcoin as a potential asset in a portfolio will be discussed.

2.2.1 Bitcoin: characteristics, risks and the technology behind it

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8 The designer(s) of Bitcoin have limited the stock to 21 million coins and just like gold, the supply grows relatively slow on a yearly basis (the stock-to-flow ratio is low). Also inspired by gold, Bitcoins need to be ‘mined’. Miners of Bitcoin will take care of the network security though Proof of Work (PoW) and they need to verify the legitimacy of the transactions (Spithoven, 2019). Miners will be rewarded with additional coins. Simplified a transaction will take place in the following order: first a user will send bitcoins to another user in the network. The system will then generate a cryptographic puzzle to the network. Once a block of more transactions is full, nodes (members of the system), simultaneously perform Proof of Work (PoW) - mathematical operations that are hard to solve but the correct solution is easy to verify (Ammous, 2016). The fastest solution provided by miners or nodes will print a time stamp on the transaction which will indicate that the transaction is safe and unique. This technology, the blockchain, is thus a distributed ledger representing a network of transactions which is impossible to rewrite because of the length of the chain.

There are lot of other cryptocurrencies originated in the last couple of years. Those “alt coins” were especially popular in 2017 and 2018. But, Bitcoin dominates the cryptocurrency market and the capitalization of Bitcoin at the time of writing is around 70% of all the market capitalization in cryptocurrencies (CoinMarketCap, 2019). Bitcoin is the most mature cryptocurrency and might benefit from the first-mover advantage. Also is the supply growth rate of Bitcoin the lowest compared to other cryptocurrencies. According to Ammous (2018) currencies with such a reliably low level of supply increase attract safe haven demand, especially from citizens of countries whose currencies are experiencing hyperinflation or high inflation. Also based on the algorithm of the blockchain behind Bitcoin the annual growth rates of the supply will be significantly lower compared to the currencies of the developed countries. Because the reasons mentioned above, only in this study Bitcoin will be analyzed.

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Figure 2: Price Bitcoin (USD) on a log-scale

Since its origin in 2008 Bitcoin has grown spectacularly, in 2009 one Bitcoin was worth $0.07 cents and at its peak in 2017 one Bitcoin was worth $17.900. The total market capitalization in September 2019 is $188 billion (CoinMarketCap, 2019). In the figure above the peak at the end of 2017 and the ‘rebirth’ in the beginning of 2019 can be seen.

2.2.2 Bitcoin as potential asset in a portfolio

Brière et al. (2015) were one of the first scholars who analyzed the inclusion of Bitcoin in an investment portfolio. During that time, they found that Bitcoin was generating high returns and was very volatile compared to the traditional and alternative assets. In their sample period (2010-2013) spanning tests confirmed that already small proportions of Bitcoin in a portfolio led to higher returns/lower risks and therefore offers over that period significant diversification benefits. Because of the short sample period the results may reflect early-stage behavior with over-enthusiasm and should be taken with chariness.

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10 As stated in the introduction, Eisl et al. (2015) conclude that Bitcoin should be included in an optimal portfolio. They indicated that an investment in Bitcoin increases their risk measure (CVaR), but this additional risk is overcompensated by the higher returns which lead to better risk-return ratios. They advise to allocate weights ranging from 1.65% to 7.69% to be invested in Bitcoin based on the different portfolios constraints they use. Baumöhl (2018) focused on the correlation of Bitcoin with other forex currencies and concluded that the correlation is relatively low and the cryptocurrency can be used as a diversifier.

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3. Methodology, Data and Hypothesis

In this section the methodology, the data and the hypothesis will be discussed. In the first subsection it will be explained how the research question will be answered. In this research an extension of the original mean-variance model developed by Markowitz (1952) will be used (the model of Fishburn, 1977). The portfolio strategy will be elucidated in the second subsection. As mentioned in the introduction, the global market portfolio following Eisl et al. (2015) and Brière et al. (2015) is relatively broad and will contain a lot of asset classes. In the third subsection those asset classes will be explained in detail. Based on the literature review of section 2, the hypothesis of this research are formulated in the last part of this section. Also in the last part of this section the research objective will be grounded.

3.1 Methodology

As stated in the previous two sections, in this study several portfolios will be constructed: the global market portfolio alone and the global market portfolio with the inclusion of one of the following assets: Bitcoin, Gold and Cash (US Dollar). Besides this, portfolios with certain constraints will be constructed. In the main analysis twelve portfolios are formed. The different risk-return metrics will be calculated over the sample period of 2010 till mid-2019. Afterwards the differences can be compared and analyzed. This paper will use a general mean-risk dominance model developed by Fishburn (1977). In this model, the expected returns are calculated as the mean of the outcomes and the risk of the portfolios are measured by an empirical function of dispersions below a specified target return 𝑡. Fishburn (1977) showed how the rigour of stochastic dominance can be combined with portfolio theory by using a downside framework. The target return will be denoted by ℎ. Fishburn (1977) exploited a framework that is suitable for the individual investor and which can be used for different levels of investor risk tolerance (Sortino & Satchell, 2001).

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12 Of the different assets used in the optimal portfolios first the individual returns will be calculated using the following formula (Siew & Hoe, 2016):

𝑅_𝑖𝑡 = 𝑃𝑖,𝑡− 𝑃𝑖,𝑡−1

𝑃_{𝑖,𝑡−1} (1)

Where:

𝑅_𝑖𝑡 = total return asset 𝑖 at time t 𝑃_𝑖,𝑡 = closing price asset 𝑖 at time t 𝑃𝑖,𝑡−1 = closing price asset 𝑖 at time 𝑡 − 1

The mean return of the individual assets over the entire sample period is calculated using equation 2 (Siew & Hoe, 2016).

𝑟_𝑖 = 1 𝑇 ∑ 𝑅𝑖𝑡 𝑇 𝑡 =1 (2) Where:

𝑟_𝑖 = mean return of asset 𝑖

𝑅𝑖𝑡 = total return asset 𝑖 at time t

𝑇 = number of periods

If the individual mean returns are calculated and the optimal portfolio composition (see following pages) is known the mean return of the optimal portfolio can be calculated as follow (Siew & Hoe, 2016):

𝑟_𝑝 = ∑ 𝜔_𝑖

𝑁

𝑖=1

𝑟_𝑖 (3)

Where:

𝑟_𝑝 = mean return of the optimal portfolio 𝑟_𝑖 = mean return of asset 𝑖 at time t

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13 In the general model of mean-risk dominance distribution 𝐹 dominates distribution 𝐺 if and only if 𝜇(𝐹) ≥ 𝜇(𝐺) and 𝜌(𝐹) ≤ 𝜌(𝐺), besides this there must be at least one strict inequality. 𝜇 is the mean return and 𝜌 (𝐹) can be defined as (Fishburn, 1977):

𝜌 (𝐹) = ∫ 𝜑(ℎ − 𝑥)𝑑𝐹(𝑥)

ℎ

−∞

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𝜑 (𝑦) is a function that represents the riskiness of a return ℎ that is below target. It is important to describe the basics of the general model, but in this paper the focus will be on the model of Fishburn (1977). The model uses a Lower Partial Moment (LPM) measure to calculate the risk involved in an investment. For 𝑇 observations the following equation can be used (Wojt, 2010): 𝐿𝑃𝑀_{𝑎,ℎ,𝑖}∗ = 1 𝑇 ∑(max(0, ℎ − 𝑅𝑖𝑡)) 𝑎 𝑇 𝑡=1 (5)

In which 𝑎 > 0, and where: ℎ = target return

𝑅_𝑖𝑡 = total return asset 𝑖 at time t

𝑎 = level of investor risk tolerance - the relative impact of large and small deviations 𝑇 = number of periods

The empirical distribution of the investment returns will be used (Grootveld and Hallerbach, 1999). The level of investor risk tolerance (𝑎) can be seen as a measure of risk aversion where the risk aversion increases with 𝑎. Risk as measured by the 𝑎-LPM reflects the skewness and asymmetry of the cumulative probability distribution of the investment returns (Sing & Ong, 2000). 𝑎 = 1 is the point where the investor is risk neutral.

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14 In this research Siew and Hoe are followed because when 𝑎 is bigger than 3 the LPMs will be very small. If the target return ℎ will be set at the risk-free rate of return, the dichotomy between risk and return is vanished as it is known in the mean-variance approach (Grootveld and Hallerbach, 1999). The target return ℎ in this research will be set equal to 0% and stays fixed at 0% during the sample period (Wojt, 2010). This reference level is chosen due to the low current interest rates in the world. As robustness check the analysis will be done in the exact same way but now with a target return ℎ of 2% based on the Federal Funds Rate in September 2019 (Federal Reserve, 2019).

Equation 5 on the previous page can be characterized as Fishburn’s risk equation. As stated, the equation is intended to be a function of dispersions below target return ℎ. It is not Fishburn’s purpose to ignore all the returns who are above ℎ but with using this framework the focus will be on the probability of falling below the target return and how far the return falls below this target return. To analyze this all the observations need to be measured and those returns above ℎ are taken into account but their value not. Equation 5 results in a percentage which represents the average riskiness over the sample period of an individual asset. The higher the Lower Partial Moment measure, the higher the risk in the investment and the greater the degree of negative skewness in the return distribution (Nawrocki, 1991).

As stated in the literature section, the portfolio theory optimization of Markowitz assumes that the distributions of the returns are normal. In the Mean-Lower Partial Moment portfolio optimization this assumption can be relaxed and the investors now cares only about the means and the lower partial moments. In Mean-Lower Partial Moment portfolio optimization co-lower partial moments needs to be introduced. As seen, the lower partial moments of the individual assets returns are calculated using equation 5 and can be analyzed on risk tolerance with the different 𝑎’s.

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15 Based on this assumption, the following portfolio optimization algorithms appear. The co-lower partial moments of the return on asset 𝑖 with asset 𝑗 can be expressed in terms of LPM as (Wojt, 2010): 𝐶𝐿𝑃𝑀𝑎,ℎ,𝑖.𝑗 = (𝐿𝑃𝑀𝑎,ℎ,𝑖∗ 𝐿𝑃𝑀𝑎,ℎ,𝑗∗ ) 1 𝑎_𝜌 𝑖,𝑗 (6) Where:

𝐶𝐿𝑃𝑀_{𝑎,ℎ,𝑖.𝑗} = Co-Lower Partial Moment between asset 𝑖 and asset 𝑗 𝐿𝑃𝑀_{𝑎,ℎ,𝑖}∗ = Lower Partial Moment asset 𝑖

𝐿𝑃𝑀_{𝑎,ℎ,𝑗}∗ = Lower Partial Moment asset 𝑗 𝜌_𝑖,𝑗= correlation between asset 𝑖 and asset 𝑗

𝑎 = level of investor risk tolerance - the relative impact of large and small deviations

Knowing the co-moments between the individual assets the LPM of the total portfolio can calculated as follow (Wojt, 2010):

𝐿𝑃𝑀_{𝑎,ℎ,𝑝} = ∑ ∑ 𝜔_𝑖 𝜔_𝑗 𝐶𝐿𝑃𝑀_{𝑎−1,ℎ,𝑖,𝑗} 𝑁 𝑗=1 𝑁 𝑖=1 (7) Where:

𝐿𝑃𝑀_{𝑎,ℎ,𝑝} = Risk/Lower Partial Moment portfolio

𝐶𝐿𝑃𝑀_{𝑎−1,ℎ,𝑖,𝑗} = Co-Lower Partial Moment between assets 𝜔_𝑖 = weight allocated to asset 𝑖 (amount of funds invested) 𝜔_𝑗 = weight allocated to asset 𝑗 (amount of funds invested)

When all the calculations of the individual assets are done, the matrix 𝐿 can be introduced. The optimal weights of the portfolios can be calculated with this matrix (using Microsoft Excel Solver). As stated, the matrix 𝐿 is symmetric and therefore the optimization problem is comparable to the original Markowitz mean-variance optimization, the difference in Mean-Lower Partial Moment optimization is that the covariance matrix is changed with the matrix 𝐿 using LPMs instead of variances. This matrix can be expressed as (Wojt, 2010):

𝐿 = (

𝐶𝐿𝑃𝑀_{𝑎−1,ℎ,1,1} ⋯ 𝐶𝐿𝑃𝑀_{𝑎−1,ℎ,1,𝑇}

⋮ ⋱ ⋮

𝐶𝐿𝑃𝑀_{𝑎−1,ℎ,𝑇,1} ⋯ 𝐶𝐿𝑃𝑀_{𝑎−1,ℎ,𝑇,𝑇}

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16 The Mean-Lower Partial Moment portfolio selection problem contains three stages. First the relevant data (see subsection 3.3) needs to be gathered and the relevant parameters (𝑎 and ℎ ) needs to be determined. Then the portfolio analysis stage appear, where the efficient sets of portfolios are computed. This stage forms an efficient frontier that results in the highest level of expected returns given a certain level of risk. To find the efficient portfolio with expected mean return 𝜇 from the opportunity set P of N assets, the non-linear optimization problem below must be solved (Grootveld and Hallerbach, 1999):

min {𝜔} 𝐿𝑃𝑀𝑎,ℎ,𝑝 𝑠. 𝑡. {𝑟𝑝} = 1 𝑁∑{ 𝑟𝑖} 𝑁 𝑖=1 = 𝜇 (9) ∑ 𝜔_𝑖 = 1, 𝑎𝑛𝑑 ∀𝑖 ∈ 𝑃. 𝑁 𝑖=1

The final stage is the selection of the optimal portfolios but this depends on the specific preferences of the investor (target return and the level of investor risk tolerance - 𝑎). To compare the different optimal portfolios and to see whether the inclusion of Bitcoin has positive effects on the return and/or on the risks the previous steps are done in the exact same way including Bitcoin as a possible investment asset. If the Kappa ratio (see following page - Kaplan and Knowles, 2004) of the portfolio with Bitcoin is higher than the portfolio without Bitcoin, it can be stated that Bitcoin is an effective diversifier in an optimal portfolio. The Kappa ratio is similar to the Sharpe ratio, but specific used as measure of portfolio performance when risk is expressed in terms of LPMs.

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17 The Kappa ratio can be calculated as follows (Kaplan and Knowles, 2004):

𝐾𝑎 (ℎ) =

𝑟_𝑝− ℎ √𝐿𝑃𝑀𝑎,ℎ,𝑝

𝑎 (10)

Where:

𝑟_𝑝 = mean return of the optimal portfolio ℎ = target return

𝑎 = level of investor risk tolerance - the relative impact of large and small deviations 𝐿𝑃𝑀_{𝑎,ℎ,𝑝} = Risk/Lower Partial Moment optimal portfolio

3.2 Portfolio Strategy

This research requires to build efficient portfolios where Bitcoin is included in the portfolios and ceteris paribus, Bitcoin is not included in the portfolios. The efficient frontier gives all the possible portfolios that can be constructed from the different assets where the return is the highest given the risks the investor wants to take (for the necessary formulas see previous subsection 3.1). In this study the different optimization contexts of Eisl et al. (2015) are followed. This means that four different scenarios will be analyzed and in total twelve portfolios will be constructed in the main analysis (three different levels of risk tolerance times four optimization strategies). A backtesting technique will be used to calculate the returns and form the optimal portfolios (with the most efficient assets weights – 𝜔𝑖 ). The following four

strategies will be used in this research:

Scenario 1: Naïve or equally-weighted portfolio (𝜔𝑖 = 1 𝑁 ∀ 𝑖)

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Scenario 2: Long-only portfolio (𝜔𝑖 ∈ ℝ0+: 0 ≤ 𝜔𝑖 ≤ 1)

In the literature section it became clear that Bitcoin is very a volatile asset relative to the other traditional and alternative assets. Short selling of Bitcoin is even more risky because with short selling the investor is speculating that the asset will decline in price with the use of borrowed funds. In this strategy the individual weights are limited to 100% and cannot go below 0%.

Scenario 3: Semi-constrained (-100%/+100%) portfolio (𝜔_𝑖 ∈ ℝ ∶ −1 ≤ 𝜔_𝑖 ≤ 1)

In the optimization process of this scenario it is not possible to place weight-related constraints on assets who not allow leverage. This scenario should generate higher returns than scenario 1 and 2 because it is now also possible to short potential assets who performed relatively bad during the past 9 years. This scenario is more advanced and might not be used by individual investors but more by institutional investors.

Scenario 4: Unconstrained portfolio (𝜔𝑖 ∈ ℝ)

The scenario with the unconstrained portfolio should generate the highest risk-returns ratios of all the different scenarios because, as the name stated, this portfolio is unconstrained and therefore there are no weight constraints to certain assets. This means that is possible to short and/or leverage assets. A consequence of this scenario might be that given its unconstrained character there will be large differences in the weights allocated to the assets (in both ways: short or long). This scenario will be less used in the real world because it is also the most risky scenario (due to the possible large long and short positions).

3.3 Data

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Table 1 Investment assets included in the sample:

Name Asset Class Symbol/Ticker

Bitcoin USD Price Index Cryptocurrency BTCUSD

DXY US Dollar Currency Index* Cash DXY

MSCI World Index Equity MSCIWORLD

MSCI Emerging Markets Index Equity MSCIEM

Solactive Global Developed Government Bond TR Index Fixed Income SOLDSOV Invesco Emerging Markets Sovereign Debt ETF Fixed Income PCY SPDR Bloomberg Barclays High Yield Bond ETF Fixed Income JNK

Reuters Commodity Index Commodities RTCI

COMEX Gold Composite Commodity Commodities GCc1

Reuters Private Equity Buyout Index Private Equity TRPEI FTSE EPRA/NAREIT Developed Real Estate Index Real Estate FTEPRA Wells Fargo Hedge Fund Manager Holdings Index Alternative HEDGEPX

3.4 Research objective and Hypothesis

The main objective of this research is to study whether the inclusion of Bitcoin in a global market portfolio has diversification benefits. Can Bitcoin really not be ignored these days or is it (as in the view of some academics) a scam and failure? Is Bitcoin despite its volatile character a valuable addition to a portfolio? To answer these questions and achieve the main objective a hypothesis is formulated. In the literature section it became clear that in most empirical studies Bitcoin shows diversification benefits but those papers are analyzing relatively short time frames. In this research the time frame is longer and it will be tested if those earlier findings and conclusions still apply. According to those findings the following hypothesis is formulated:

H0: The including of Bitcoin in a global market portfolio does give diversification benefits H1: The including of Bitcoin in a global market portfolio does not give diversification benefits

* The DXY US Dollar Currency Index is a measure of the value of US dollar relative to the value of a basket of currencies of

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4. Results

In this section the results of the study will be presented. In the first subsection the results of the analysis regarding the inclusion of Bitcoin to the different portfolios will be described. In order to say something about the robustness of the results, as stated in the previous sections, Bitcoin is replaced by assets who have some similar characteristics: Gold and Cash. As second robustness check the target return is increased to 2%. The results of the robustness checks are reviewed in the second subsection.

4.1 Results research

Firstly, the individual assets in the portfolios are separately analyzed. Table 2 below shows the summary statistics of monthly returns in US dollars of Bitcoin, the traditional and the alternative assets over the time period 31-08-2010 till 31-08-2019. Due to the use of monthly returns there are in total 108 observations in the sample period.

Table 2 Summary statistics individual assets

*In table 2 abbreviations are used for the assets names. Total names are: Bitcoin USD Price Index, DXY US Dollar Currency Index, MSCI World Index, MSCI Emerging Markets Index, Solactive Broad Global Developed Government Bond TR Index, Invesco Emerging Markets Sovereign Debt ETF, SPDR Bloomberg Barclays High Yield Bond ETF, Reuters Commodity Index, COMEX Gold Composite Commodity, Thomson Reuters Private Equity Buyout Index, FTSE EPRA/NAREIT Developed Real Estate Index and Wells Fargo Hedge Fund Manager Holdings Index.

Name Mean Median Max Min Volatility Skewness Kurtosis

Bitcoin 21,03% 6,93% 450,02% -39,75% 65,63% 4,21 22,06

US Dollar Index 0,23% 0,17% 5,99% -5,39% 2,07% 0,28 0,31

MSCI World Index 0,70% 1,03% 10,26% -8,99% 3,63% -0,27 0,68

MSCI EM Index 0,13% 0,13% 13,08% -14,78% 4,92% 0,00 0,66

Gl. Dev. Gov. Bond Index 0,15% 0,04% 3,67% -5,41% 1,56% -0,60 1,37 EM Sovereign Bond Index 0,09% 0,20% 4,89% -6,74% 2,32% -0,53 0,54 High Yield Bond Index -0,05% 0,18% 7,68% -6,70% 1,99% 0,07 2,23

Commodity Index -0,14% 0,00% 8,55% -10,21% 2,74% 0,12 2,33

Gold 0,29% 0,19% 12,30% -12,12% 4,66% 0,02 0,20

Private Equity Index 1,53% 1,62% 15,79% -12,65% 5,00% -0,34 1,16 Dev. Real Estate Index 0,59% 0,16% 13,20% -8,25% 3,89% 0,23 0,34

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21 As can be seen in table 2 on the previous page, Bitcoin’ monthly mean return outperforms the other mean returns clearly. As measure of the volatility first the standard deviation of the monthly returns is given. In the tables on the following pages volatility is measured as a Lower Partial Moment. A possible investment in Bitcoin include of course risks, which is represented by the high volatility (65,63% monthly). However, compared to the second riskiest investment opportunity, private equity, this is ‘only’ 13 times more volatile while, as stated, the returns are outperforming the other assets. Therefore, Bitcoin is very interesting. During the past 9 years also private equity, hedge funds, stocks (MSCI World Index) and real estate performed relatively well in terms of mean return.

Table 2 shows that the Bitcoin return distribution exhibits large excess kurtosis (22,06) and is positively skewed (4,21). As mentioned in the literature section a downside risk measure (Lower Partial Moment) is than a more appropriate measure of risk compared to the standard deviation (volatility). Based on the high kurtosis, skewness and the correlation matrix (table 3) the results of this research are in line with the findings of Eisl et al. (2015) and Brière et al. (2015). In table 3 below the correlation matrix is presented. This matrix shows the correlation coefficient between all the assets. The correlation coefficient can take a value between -1 and 1 at each extremes. An asset is for example highly negatively correlated with another asset if the value is -0,90.

Table 3 Correlation matrix

Bitcoin USD MSCI

World MSCI EM Dev Gov Bond EM Sov Bond HY Bond

Commodity Gold Private

Equity Real Estate Hedge Fund Bitcoin 1,00 USD -0,14 1,00 MSCI World 0,16 -0,54 1,00 MSCI EM 0,15 -0,63 0,82 1,00

Dev Gov Bond 0,03 -0,65 0,11 0,31 1,00

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22 The correlations between assets in the matrix on the previous page needs to be taken with caution because they are known to be unstable and can change dramatically during crises (Brière et al, 2015). However, based on the historical data, it still might be useful for investment analysis as for example demonstrated by a popular quote of Mark Twain: “history may not repeat itself but it often rhymes”. In the matrix in table 3 it becomes clear that Bitcoin rarely correlates with the other assets. All correlation coefficients of Bitcoin and the other assets lay in the of -0,14 (US Dollar) and +0,16 (MSCI World Index/Hedge Fund Index) range. As explained in the introduction, if an asset does not correlated that much with another asset, it might contribute to the diversification benefits of constructing a portfolio. The assets used for the first robustness check, the US Dollar and Gold, are more correlated with the other assets (as can been seen through the higher correlation coefficients).

In this study three different levels for 𝑎 are used (level of investor risk tolerance), namely 1, 2 and 3. This is represented in the table 4 below by LPM1, LPM2 and LPM3. Also the average LPMs are calculated based monthly returns of the different assets (108 observations in total). As can be seen in table 4, if the level of investor risk tolerance (𝑎) increases, the LPM risk measure decreases. This is because the risk aversion increases with 𝑎. Based on these three levels of 𝑎 the optimal portfolios are calculated.

Table 4 Lower Partial Moments individual assets (target return = 0%)

Name LPM1 LPM2 LPM3

Bitcoin 6,006% 0,361% 0,022%

US Dollar Index (Cash) 0,726% 0,005% 0,000%

MSCI World Index 1,068% 0,011% 0,000%

MSCI EM Index 1,819% 0,033% 0,001%

Gl. Dev. Gov. Bond Index 0,535% 0,003% 0,000% EM Sovereign Bond Index 0,854% 0,007% 0,000% High Yield Bond Index 0,751% 0,006% 0,000%

Commodity Index 1,040% 0,011% 0,000%

Gold 1,661% 0,028% 0,000%

Private Equity Index 1,218% 0,015% 0,000% Dev. Real Estate Index 1,242% 0,015% 0,000%

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23 As mentioned in the methodology section, in the main analysis all the calculations are done based on a target return ℎ of 0%. The inclusion of Bitcoin in four different portfolio strategies is analyzed. In total there are 10 different assets in the optimal portfolio with Bitcoin and 9 different assets in the optimal portfolios without Bitcoin. In table 5 the results of the equally-weighted portfolio strategy are presented.

Table 5 Optimal portfolio with/without BTC – Equally-weighted portfolio strategy

The naïve diversification strategy means that the portfolio weights of the different assets are held constant during the sample period. For a detailed overview of the different assets in the optimal portfolios see appendix 1. In the first row the inclusion of Bitcoin in the optimal portfolio is analyzed. In this portfolio there are in total 10 assets and therefore the weight allocated to Bitcoin is set fixed at 10,00%. This results in a monthly optimal portfolio return of 2,515% and because of the fixed character of the weights of the assets in the optimal portfolio this the same for all levels of risk tolerance.

As predicted by theory, due to the increase in the risk tolerance level (𝑎), the mean portfolio Lower Partial Moment decreases. In the second row the results of the optimal portfolios without Bitcoin are presented. The optimal portfolio mean return for those portfolios are 0,457% and this is far below the returns of portfolios with Bitcoin. To analyze the portfolio performance of the different assets regarding the risk-return characteristics the Kappa ratio is introduced (see methodology section).

Equally-Weighted Portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3

BTC Weight

Mean Portfolio Return BTC Mean Portfolio LPM BTC Kappa ratio 10,00% 2,515% 0,160% 0,629 10,00% 2,515% 0,005% 3,690 10,00% 2,515% 0,0002% 10,988 BTC Weight

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24 As can be seen in table 5 the optimal portfolios with Bitcoin all have a higher Kappa ratio compared to the portfolios without Bitcoin but in relative terms also the risks in terms of the LPMs of those portfolios are higher. In table 6 the results of the long-only portfolio strategy are presented. In this strategy it is not possible to take short positions in the assets. For the total portfolio compositions with the different 𝑎’s see appendix 2. The investor is risk neutral when 𝑎 = 1. This risk neutrality can be seen in the optimal portfolio with Bitcoin when 𝑎 = 1 because in this scenario the weight allocated to Bitcoin is 48,96%. This is very high but due to the risk neutrality understandable.

The monthly mean return of this scenario is spectacular resulting in an annual mean return of 129,80%. When the risk tolerance of the investor decreases and 𝑎 increases, it is straightforward that the weights allocated to the riskiest asset (Bitcoin) decreases, as illustrated in table 6. In the second row the optimal portfolios without Bitcoin are given. When the risk tolerance decreases, the mean returns of those portfolios decreases slightly but the mean LPMs decreases proportionally more. Also in this portfolio strategy the Kappa ratios are higher in the portfolios with Bitcoin than the portfolios without Bitcoin. But again, also the risks measured by the LPMs are higher.

Table 6 Optimal portfolio with/without BTC – Long-only portfolio strategy

Long-Only Portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3

BTC Weight

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25 In table 7 below the optimal portfolios of the semi-constrained portfolio strategies are given. In this optimization process the weights of the assets must lay between -1 and +1 (-100%/+100% portfolios). Therefore, very large short positions (below -1) or using leverage on certain assets is not possible. However, in this scenario short selling is possible and some assets in the optimal portfolio composition will have negative weights. Those negative weights (i.e. short selling the asset) are placed on assets who performed badly during the past 9 years (High Yield Bond Index and Commodity Index).

The exact weights of the different portfolios regarding this strategy and the different levels of risk tolerance are given in appendix 3. Due to more possibilities for the investor (short selling) there are more options and this resulted in better optimal portfolios (in terms of risk-return metrics). In the first scenario with 𝑎 = 1 more weight to Bitcoin is allocated compared to the long-only portfolio strategy. This is made possible by short selling assets who performed badly. Therefore, the mean portfolio returns and mean portfolio LPMs are higher than the long-only portfolio strategy. In general, which also applies in this portfolio strategy, when 𝑎 increases, the mean returns of the optimal portfolios decreases slightly but the mean LPM decreases proportionally more. The Kappa ratios are higher in the optimal portfolio with Bitcoin compared to the optimal portfolios without Bitcoin.

Table 7 Optimal portfolio with/without BTC – Semi constrained portfolio strategy

-100%/+100% Portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3

BTC Weight

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26 In table 8 below the results of the unconstrained portfolio strategy are given. In this strategy it is possible to short and/or leverage assets. However, the results are exactly the same as with the semi-constrained portfolio strategy which indicate that it is not optimal to take large short positions or to leverage certain assets. In appendix 4 the weights of all the assets in the optimal portfolios are presented.

Table 8 Optimal portfolio with/without BTC – Unconstrained portfolio strategy

4.2 Robustness checks 4.2.1 Bitcoin vs. Cash and Gold

As robustness check Bitcoin is replaced with assets who have some similar characteristics: Gold and the US dollar (Cash). In appendix 5 first the robustness check for Gold is performed, this is done in exact the same way as with Bitcoin, so four portfolio strategies and three different levels of risk tolerance (𝑎). In the equally-weighted portfolio strategy the portfolios with Gold resulted surprisingly in lower portfolio returns (0,440% monthly mean return), but also lower risks. The portfolios without Gold performs better in terms of returns and Kappa ratios. This is striking because in general a portfolio with more assets (who also had a positively monthly mean return) should lead to better optimal portfolios. However, the equally weighted character of the strategy is the reason for this inefficiency. In the long-only portfolio strategy (appendix 5, table 2) the weight to Gold allocated in the optimal portfolios is between 4,33% and 1,62%. The returns of those optimal portfolios are lower, but the Kappa ratio is higher. This is also the case for the semi-constrained and unconstrained portfolio strategies (which weight allocated to Gold is between 4,60% and 1,72%).

Unconstrained portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3

BTC Weight

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27 In appendix 6 cash (US Dollar) is analyzed as robustness check. In the equally-weighted portfolio strategy the mean portfolio returns are lower but in general the optimal portfolios including cash are better, as represented by a higher Kappa ratio. Also in the other three portfolios strategies the optimal portfolios with Cash resulted in a higher Kappa ratio. An interesting point in the robustness check analysis of the US dollar is that when the risk tolerance decreases the proportion of weight allocated to the US dollar increases. In the long-only portfolio strategy: from 8,71% at 𝑎 = 1 to 16,44% when 𝑎 = 3. The same appears for the semi-constrained and unconstrained portfolio: from 9,21% at alpha = 1 to 17,33% when 𝑎 = 3.

4.2.2 Different target return: 0% vs. 2%

As second robustness check the target return is changed from 0% to 2% and it is analyzed if Bitcoin in this scenario shows diversification benefits regarding increased returns/decreased risks. As stated in the methodology section the 2% target return is chosen because in September 2019 this was the Federal Reserve Fund Rate (Federal Reserve, 2019). The robustness analysis is done in the exact same ways as the main analysis. Therefore, also here the four different portfolio strategies are used and the same levels of investor risk tolerance: (𝑎) = 1, 2 and 3.

The exact results of this robustness check are presented in appendix 7. As can be seen in the first table of appendix 7 the returns are logically the same with the equally-weighted scenario compared to the main analysis with the 0% target return (due to equally-weighted character of the portfolios). However, the risks in terms of LPMs are higher with the 2% target return robustness check and therefore the Kappa ratios are lower compared to the main analysis. Either way, also in this robustness check the Kappa ratios of portfolios including Bitcoin are higher than the portfolios without Bitcoin and this can be applied to all the other portfolio strategies as well. Therefore, this robustness check with a different target return backs the main results of this research.

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5. Discussion

In this section the results presented in the previous section will be discussed and interpreted. In the results section it is analyzed if the inclusion of Bitcoin to the global market portfolio lead to better risk return metrics in terms of the Kappa ratio compared to the portfolios without Bitcoin. In table 5 till table 8 of the previous section the results of the main analysis are presented. For the four portfolio strategies the Kappa ratio is here the most important result to focus on. As stated in the methodology section if the Kappa ratio is higher in the optimal portfolios including Bitcoin it can be concluded that Bitcoin is indeed an effective diversifier in an optimal portfolio and shows diversification benefits.

In the previous section it can be seen that in every portfolio strategy the optimal portfolio including Bitcoin leads to higher Kappa ratios than the optimal portfolio without Bitcoin (table 5 till table 8). This means that it can be stated that in the past 9 years Bitcoin is an effective diversifier of an optimal portfolio. The results are in line with previous studies of Brière et al (2015), Kajtazi and Moro (2019), Eisl et al. (2015) and Carpenter (2016) who also concluded all that Bitcoin is an effective diversifier of an optimal portfolio. Some authors of previous studies analyzing Bitcoin as a possible diversifier has chosen to exclude some periods of massive price increases of Bitcoin, for example excluding the year 2013. For this research it is chosen not to do this because then there is need to exclude more periods in the sample, for example the bull runs in 2011 and 2017. This results in an inconsistent sample period.

To check the robustness of the main analysis Bitcoin is replaced with Gold and Cash and a different target return is used, namely 2%. Again to see if there are diversification benefits the Kappa ratios of the different scenarios are compared to each other. In every portfolio strategy the inclusion of Gold and Cash to a global market portfolio also leads to higher Kappa ratios, expect for the equally-weighted portfolio strategy with Gold. The reason for this is the equally weighted character because of the relatively higher risk of Gold. Therefore, it can be stated that in almost all portfolio strategies also Gold and Cash are effective diversifiers in a global market portfolio based on the past 9 years.

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6. Conclusion

Bitcoin and the underlying blockchain technology are clearly one of the most important financial innovations in the past decade. In this paper the performance of Bitcoin compared to the more traditional financial assets is analyzed and it is studied whether an inclusion of Bitcoin to an optimal portfolio leads to better risk-return trade-offs. In the literature section it became clear that other researchers concluded that Bitcoin showed over their respective time frames signification diversification benefits. The question is if this is also the case for the sample period of this research (mid-2010 till mid-2019). To analyze this an advanced model of portfolio optimization is used namely a downside risk framework developed by Fishburn (1977). This model is chosen because the original portfolio optimization model of Markowitz had some flaws as explained in the methodology section.

In the results section first the individual assets are separately analyzed. In this section it became clear that Bitcoin’ monthly mean returns outperforms the other assets’ mean returns clearly. As already mentioned in the introduction, in finance there is no such at thing as a free lunch. This outperforming monthly mean monthly comes at a cost: also the risk involving an investment in Bitcoin is much higher compared to the other assets. The higher risk is represented by the highest standard deviation and the highest Lower Partial Moments. In the results section it is also analyzed if Bitcoin correlates much with the other assets. It can be concluded that this is not case: the correlation coefficients of Bitcoin with all the other assets are in the -0,14 and +0,16 range. This implicates that Bitcoin is a good asset to include in a portfolio if the investor only looks at the correlations.

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30 As studied by other researchers due to the volatile character of Bitcoin, the benefits regarding the better risk-return trade-offs might come irregularly. Therefore, for portfolios concentrating on a relatively short term period, like for example 1 till 3 years, an addition of Bitcoin to the portfolio might lead to inefficient portfolios. However, on the longer run (9 years in this paper), as demonstrated in this paper, an addition of Bitcoin to an optimal portfolio leads to better risk-return trade-offs. Therefore it can be recommended to include Bitcoin to an investors’ portfolio, but in relative small proportions (the exact proportions depends on the risk tolerance for the investor - 𝑎 ).

As robustness checks Bitcoin is replaced with assets who have some similar characteristics: Gold and the US dollar (Cash). An addition of Gold to the optimal portfolios resulted in higher Kappa ratios relative to the portfolios without Gold for the following three portfolio strategies: long-only, semi constrained and unconstrained. However, the increases in the Kappa ratios are relatively small compared to increases with the addition of Bitcoin to the optimal portfolios. The addition of Cash to the optimal portfolios resulted in all the portfolio strategies to higher Kappa ratios. Either way, also with Cash the increases in the Kappa ratios are relatively small. Therefore it can be concluded that Bitcoin is a better diversifier than Gold and Cash.

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Appendix 1: Equally-weighted portfolio strategy

Table 1: Weights assets included in the optimal portfolios

Bitcoin

MSCI World Index MSCI EM Index

Gl. Dev. Gov. Bond Index EM Sovereign Bond Index High Yield Bond Index Commodity Index Private Equity Index Dev. Real Estate Index Hedge Fund Index Total 0,10 0,10 0,10 0,10 0,10 0,10 0,10 0,10 0,10 0,10 1 0,10 0,10 0,10 0,10 0,10 0,10 0,10 0,10 0,10 0,10 1 0,10 0,10 0,10 0,10 0,10 0,10 0,10 0,10 0,10 0,10 1 MSCI World Index

MSCI EM Index

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35

Appendix 2: Long-only portfolio strategy

Long-only Portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3

Bitcoin

Gl. Dev. Gov. Bond Index EM Sovereign Bond Index High Yield Bond Index Commodity Index Private Equity Index Dev. Real Estate Index Hedge Fund Index Total 0,490 0,083 0,002 0,037 0,014 0,000 0,000 0,169 0,062 0,142 1 0,135 0,142 0,009 0,118 0,029 0,000 0,000 0,239 0,088 0,239 1 0,025 0,146 0,006 0,242 0,037 0,000 0,000 0,215 0,078 0,252 1 MSCI World Index

MSCI EM Index

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36

Appendix 3: Semi constrained portfolio strategy

Semi-constrained Portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3

Bitcoin

Gl. Dev. Gov. Bond Index EM Sovereign Bond Index High Yield Bond Index Commodity Index Private Equity Index Dev. Real Estate Index Hedge Fund Index Total 0,509 0,087 0,003 0,039 0,014 -0,016 -0,025 0,176 0,065 0,148 1 0,142 0,150 0,010 0,124 0,031 -0,020 -0,032 0,251 0,093 0,251 1 0,026 0,155 0,006 0,257 0,040 -0,029 -0,034 0,228 0,083 0,268 1 MSCI World Index

MSCI EM Index

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37

Appendix 4: Unconstrained portfolio strategy

Unconstrained Portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3

Bitcoin

Gl. Dev. Gov. Bond Index EM Sovereign Bond Index High Yield Bond Index Commodity Index Private Equity Index Dev. Real Estate Index Hedge Fund Index Total 0,509 0,087 0,003 0,039 0,014 -0,016 -0,025 0,176 0,065 0,148 1 0,142 0,150 0,010 0,124 0,031 -0,020 -0,032 0,251 0,093 0,251 1 0,026 0,155 0,006 0,257 0,040 -0,029 -0,034 0,228 0,083 0,268 1 MSCI World Index

MSCI EM Index

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38

Appendix 5: Robustness check - Gold

Table 1 Optimal portfolio with/without Gold – Equally-weighted portfolio strategy

Table 2 Optimal portfolio with/without Gold – Long-only portfolio strategy

Table 3 Optimal portfolio with/without Gold – Semi constrained portfolio strategy

Gold Weight

Mean Portfolio Return Gold Mean Portfolio LPM Gold Kappa ratio 10,00% 0,440% 0,116% 0,129 10,00% 0,440% 0,001% 1,189 10,00% 0,440% 0,00002% 10,246 Gold Weight

Mean Portfolio Return No Gold Mean Portfolio LPM No Gold Kappa ratio 0% 0,457% 0,122% 0,131 0% 0,457% 0,001% 1,239 0% 0,457% 0,0002% 10,988 Long-Only Portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3 Gold Weight

Mean Portfolio Return No Gold Mean Portfolio LPM No Gold Kappa ratio 0% 1,010% 0,265% 0,196 0% 0,929% 0,002% 1,879 0% 0,816% 0,00002% 17,956 -100%/+100% Portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3 Gold Weight

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39 Table 4 Optimal portfolio with/without Gold – Unconstrained portfolio strategy

Gold Weight

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40

Appendix 6: Robustness check – Cash (US Dollar)

Table 1 Optimal portfolio with/without Cash – Equally-weighted portfolio strategy

Table 2 Optimal portfolio with/without Cash – Long-only portfolio strategy

Table 3 Optimal portfolio with/without Cash – Semi constrained portfolio strategy

Cash Weight

Mean Portfolio Return Cash Mean Portfolio LPM Cash Kappa ratio 10,00% 0,435% 0,105% 0,134 10,00% 0,435% 0,001% 1,284 10,00% 0,435% 0,00001% 11,489 Cash Weight

Mean Portfolio Return No Cash Mean Portfolio LPM No Cash Kappa ratio 0% 0,457% 0,122% 0,131 0% 0,457% 0,001% 1,239 0% 0,457% 0,0002% 10,988 Long-Only Portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3 Cash Weight

Mean Portfolio Return No Cash Mean Portfolio LPM No Cash Kappa ratio 0% 1,010% 0,265% 0,196 0% 0,929% 0,002% 1,879 0% 0,816% 0,00002% 17,956 -100%/+100% Portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3 Cash Weight

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41 Table 4 Optimal portfolio with/without Cash – Unconstrained portfolio strategy

Cash Weight

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42

Appendix 7: Robustness check Bitcoin – Target return = 2%

Table 1 Optimal portfolio with/without BTC – Equally-weighted portfolio strategy

Table 2 Optimal portfolio with/without BTC – Long-only portfolio strategy

BTC Weight

Mean Portfolio Return No BTC Mean Portfolio LPM No BTC Kappa ratio 0% 0,457% 0,268% 0,088 0% 0,457% 0,006% 0,610 0% 0,457% 0,00013% 4,010 Long-Only Portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3 BTC Weight

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Table 3 Optimal portfolio with/without BTC – Semi constrained portfolio strategy

Table 4 Optimal portfolio with/without BTC – Unconstrained portfolio strategy

-100%/+100% Portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3

BTC Weight

Mean Portfolio Return No BTC Mean Portfolio LPM No BTC Kappa ratio 0% 1,150% 0,629% 0,145 0% 1,105% 0,011% 1,032 0% 1,108% 0,00024% 7,225 Unconstrained portfolios 𝒂 = 1 𝒂 = 2 𝒂 = 3 BTC Weight