Bitcoin, gold, and commodities as safe havens for stocks: New insight through wavelet analysis

Bitcoin, gold, and commodities as safe havens for stocks: New insight through wavelet analysis

G Model QUAECO-1350; No. of Pages 9 ARTICLE IN PRESS The Quarterly Review of Economics and Finance xxx (2020) xxx–xxx Contents lists available at Sc...

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G Model QUAECO-1350; No. of Pages 9

ARTICLE IN PRESS The Quarterly Review of Economics and Finance xxx (2020) xxx–xxx

Contents lists available at ScienceDirect

The Quarterly Review of Economics and Finance journal homepage: www.elsevier.com/locate/qref

Bitcoin, gold, and commodities as safe havens for stocks: New insight through wavelet analysis Elie Bouri a,∗ , Syed Jawad Hussain Shahzad b,c , David Roubaud d , Ladislav Kristoufek e , Brian Lucey f a

USEK Business School, Holy Spirit University of Kaslik, Jounieh, Lebanon Montpellier Business School, Montpellier, France c South Ural State University, 76, Lenin prospekt, Chelyabinsk, Russia d Center for Energy and Sustainable Development, Montpellier Business School, Montpellier, France e Faculty of Social Sciences, Institute of Economic Studies, Charles University in Prague, Czech Republic f Trinity Business School, Trinity College Dublin, Dublin 2, Ireland b

a r t i c l e

i n f o

Article history: Received 1 August 2019 Received in revised form 13 November 2019 Accepted 21 March 2020 Available online xxx JEL classifications: C52 G11 G17 Keywords: Gold Commodities Bitcoin Stock indices Safe haven Wavelets VaR

a b s t r a c t In this study, we compare the safe-haven properties of Bitcoin, gold, and the commodity index against world, developed, emerging, USA, and Chinese stock market indices for the period 20 July 2010–22 February 2018. We apply the wavelet coherency approach and show that the overall dependence between Bitcoin/gold/commodities and the stock markets is not very strong at various time scales, with Bitcoin being the least dependent. We study the diversification potential at the tail of the return distribution through wavelet value-at-risk (VaR) and reveal that the degree of co-movement between gold and stock returns affects the portfolio’s VaR level. Specifically, the benefits of diversification vary in the timefrequency space, with Bitcoin exhibiting a superiority over both gold and commodities. Our findings are useful for investors and financial advisors searching for the best asset among Bitcoin, gold, and commodities to hedge extreme negative movements in stock market indices, while accounting for the heterogeneity in the horizons of investors. © 2020 Board of Trustees of the University of Illinois. Published by Elsevier Inc. All rights reserved.

1. Introduction Both gold, and its larger family, commodities, are considered effective diversifiers against stock market returns in several developed and emerging economies (Conover, Jensen, Johnson, & Mercer, 2010; Wen & Cheng, 2018; Henriksen, 2018). Gold especially is considered a safe-haven property, given its ability to hedge stock movements in times of crisis (Baur & Lucey, 2010). However, recent studies have questioned whether that property holds in the post global financial crisis (GFC) period (see, among others, Baur & Glover, 2012; Bekiros, Boubaker, Nguyen, & Uddin, 2017; Klein,

2017), when most leading central banks have adopted an ultraloose monetary policy and the popularity and financialization of commodity investing has increased substantially. Since the emergence of the first cryptocurrency in early 2009, under the name Bitcoin, the financial press has become attracted by this digital asset. Previous studies point to the effect of monetary policy on Bitcoin returns and/or volatility (e.g., Corbet, McHugh, & Meegan, 2017; Nguyen, Nguyen, Nguyen, & Pham, 2019), while ˜ other studies find no significant effect (e.g., Vidal-Tomás & Ibanez, 2018). Furthermore, financial headlines often compare the virtues of gold and Bitcoin,1 claiming that the latter is also a safe-haven

∗ Corresponding author. E-mail addresses: [email protected] (E. Bouri), [email protected] (S.J.H. Shahzad), [email protected] (D. Roubaud), [email protected] (L. Kristoufek), [email protected] (B. Lucey).

1 Bitcoin is classified as a commodity by the Commodity Futures Trading Commission (CFTC), and has non-political and (virtual) commodity attributes that make it quite similar to gold. In fact, Selgin (2015) argues that Bitcoin has features of commodities (i.e. gold) and can be regarded as synthetic commodity money.

https://doi.org/10.1016/j.qref.2020.03.004 1062-9769/© 2020 Board of Trustees of the University of Illinois. Published by Elsevier Inc. All rights reserved.

Please cite this article in press as: Bouri, E., et al. Bitcoin, gold, and commodities as safe havens for stocks: New insight through wavelet analysis. The Quarterly Review of Economics and Finance (2020), https://doi.org/10.1016/j.qref.2020.03.004

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asset given its resilience to crisis periods such as the European debt crisis of 2010–20132 and the Cypriot banking crisis of 2012–2013 (Luther & Salter, 2017). Academic literature applies standard models (correlation coefficient, linear regression models, and GARCH-based techniques3 ) and highlights the very weak correlation between Bitcoin and stock market returns (Baur, Hong, & Lee, 2018; Brière, Oosterlinck, & Szafarz, 2015; Bouri, Molnár, Azzi, Roubaud, & Hagfors, 2017; Bouri, Jalkh, Molnár, & Roubaud, 2017; Dyhrberg, 2016; Fang, Bouri, Gupta, & Roubaud, 2019; Ji, Bouri, Gupta, & Roubaud, 2018). However, there is still a lack of empirical comparison between the safe-haven properties of Bitcoin, gold, and other commodities against world, developed, emerging, and country stock markets.4 This is where we aim to contribute. This study moves the debate on the similarity/dissimilarity between the hedging and safe-haven properties of gold, commodities, and Bitcoin forward by applying a combination of methods that consider investments horizon variety, and related portfolio implications. Specifically, we follow the approach of Grinsted, Moore, and Jevrejeva (2004) that ascertains the best time-frequency localization for an asset to act as a hedge or strong safe haven for stock market returns, i.e. to display respectively no relation or a negative relationship across a range of frequencies that, in financial terms, correspond to the inverse of investment horizons. This is important as market participants can be characterized as having heterogeneous investment horizons.5 Then, we build on the wavelet results and provide portfolio implications by quantifying the diversification ability of each of the three assets (Bitcoin, gold, and commodities) with the stock index via the value-at-risk (VaR) measure in the time-frequency space (Rua & Nunes, 2009). Our sample period covers the post-GFC period (20 July 2010−22 February 2018), making our empirical analysis interesting. Several studies of the post-GFC period: (1) challenge the safe-haven role of gold (e.g., Bekiros et al., 2017; Klein, 2017), or (2) point to Bitcoin’s role as an investment shelter during recent crises in Europe (Greek and Cypriot). The current study is useful for individual investors, portfolio managers, and financial advisors searching for the best asset, of Bitcoin, gold, and commodities, to hedge extreme negative movements in stock market indices while accounting for the heterogeneity in the horizons of investors. The rest of the paper is structured as follows. Section 2 provides the background to our research. Section 3 presents the wavelet coherence and the wavelet VaR methods. Section 4 describes the dataset. Section 5 presents and discusses the empirical results. Section 6 provides concluding remarks. 2. Research background and related studies After being proposed by Nakamoto (2008), Bitcoin emerged on January 3, 2009 as a medium of exchange and an alternative

2 https://www.cnbc.com/2015/07/01/greece-is-in-crisis-why-no-love-for-goldcommentary.html. 3 Potential heterogeneity in the relationship between Bitcoin and stock markets suggests the inappropriateness of using standard linear models. 4 Corbet et al. (2018) study the return and volatility transmission in the short and long run between leading cryptocurrencies and other assets (GSCI commodity index, US$ Broad Exchange Rate, S&P500 Index, gold price, US VIX, and the Markit ITTR110 index). However, their focus is on the connectedness spillover, not on a comparison between the safe-haven abilities of Bitcoin, gold, and other commodities versus stock indices and the portfolio implications. 5 Short-term investment horizons are related to speculators (chartists, day traders, or event-driven traders), who are involved with trading activities spanning from several minutes to several days. In contrast, long-term investment horizons are related to fundamentalist investors (portfolio managers and institutional investors), who are involved with investment activities spanning from several months to several years.

to cash payments. Thanks to its blockchain technology,6 Bitcoin does not involve intermediaries such as clearinghouses and is thus independent of sovereign risk. Given the tradability of Bitcoin on specialized exchanges (Polasik, Piotrowska, Wisniewski, Kotkowski, & Lightfoot, 2015), it has become an investment asset,7 although its returns are most often associated with high levels of volatility. Recent developments provide legitimacy to the largest cryptocurrency, making it more difficult for individual or institutional investors to ignore as an alternative investment option. Bitcoin investment has become more accessible with the availability of several Bitcoin-linked funds offered by global investment banks (e.g., Falcon Private Bank and ARK Investment Management). Interestingly, in December 2017, the CME Group and the CBOE launched futures contracts with Bitcoin as an underlying asset, legitimizing it further. This has allowed Bitcoin to join commodities, including gold, in futures trading, and eventually move from the margins to the centre of the financial world. While gold differs from Bitcoin in several respects, such as history, tangibility, intrinsic value, low volatility, consumption and use in production processes, the two assets share several important characteristics. Firstly, both gold and Bitcoin are regulated as commodities, especially in the US where Bitcoin has recently been classified as a commodity by the CFTC. Secondly, Bitcoin has non-political and (virtual) commodity attributes that make it quite similar to gold. As such, the absence of a central authority to control and adjust Bitcoin mining and transaction makes Bitcoin independent of inflation (Baur et al., 2018), an important feature commonly associated with gold. Thirdly, like gold, Bitcoin does not generate cash-flow and is produced in a process called ‘mining’, meaning it has a limited supply and a precisely specified creation process, as dictated by Bitcoin protocol,8 which ensures that a maximum of 21 million coins can ever be mined. Fourthly, previous studies provide evidence of the inverted asymmetric reactions to positive and negative news in both gold (Bouri, Azzi, & Dyhrberg, 2017) and Bitcoin (Baur, 2012). In emerging countries such as China, Bitcoin represents a solution to taking cash out of the country while circumventing the strict regulations regarding capital flows. Furthermore, the recent scrutiny by the Chinese government over the physical gold market has made Bitcoin an ideal alternative to gold. Bitcoin is often seen as a panacea, replacing financial institutions and providing shelter from sovereign risk and weakness in the global financial system (Bouri, Molnár et al., 2017; Luther & Salter, 2017). Its price spiked during the European debt crisis of 2010–20139 and the Cypriot banking crisis of 2012–2013 (Luther & Salter, 2017), as some investors found shelter in Bitcoin against political and sovereign risks. Importantly, the literature argues that Bitcoin is very weakly related to traditional assets, which makes it a valuable diversifier (Brière et al., 2015; Bouri, Molnár et al., 2017; Bouri, Jalkh et al., 2017; Dyhrberg, 2016), and in some cases a hedge or a safe haven against equities (Bouri, Molnár et al., 2017).10 Lean and Wong (2015) argue that the difference between the determinants of gold prices and other financial assets (e.g. stocks) is responsible for the negative relation or absence of relation between gold and those assets, and thus for gold’s hedging and safe-haven

6 A detailed explanation of Bitcoin and its underlying technology, the blockchain, is given by Selgin (2015). 7 Other studies point to the speculative nature of Bitcoin (Glaser, Zimmermann, Haferkorn, Weber, & Siering, 2014; Yermack, 2015). 8 The physical mining of gold is entirely different from the digital-based mining of Bitcoin (Bouri, Jalkh et al., 2017). 9 https://www.cnbc.com/2015/07/01/greece-is-in-crisis-why-no-love-for-goldcommentary.html. 10 These roles being played by Bitcoin are documented, despite its extremely high price return and volatility (Brière et al., 2015).

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properties. Based on the same logic, previous studies argue that the very weak correlation of Bitcoin with most other financial assets might be because Bitcoin does not share a lot of common price determinants with those financial assets (Bouoiyour, Selmi, Tiwari, & Olayeni, 2016; Kristoufek, 2015). Bitcoin prices depend less on economic and financial variables (Ciaian, Rajcaniova, & Kancs, 2016; Kristoufek, 2015) and more on a unique set of characteristics such as attractiveness (Kristoufek, 2015), energy prices (Li & Wang, 2017), user anonymity (Ober, Katzenbeisser, & Hamacher, 2013), computer programming enthusiasm, and illegal activity (Yelowitz & Wilson, 2015).11 The limited literature on Bitcoin sometimes includes gold and other commodities (e.g. crude oil) within its empirical analysis, with the aim of studying the relationship between gold and Bitcoin (Al-Khazali, Bouri, & Roubaud, 2018; Bouoiyour & Selmi, 2015; Bouri, Molnár et al., 2017; Ciaian et al., 2016; Dyhrberg, 2016; Ji et al., 2018; Kristoufek, 2015). Corbet, Meegan, Larkin, Lucey, and Yarovaya (2018) apply the generalized variance decomposition approach and a frequency domain method to study connectedness measures between leading cryptocurrencies and other assets (GSCI commodity index, US$ Broad Exchange Rate, S&P500 Index, gold price, US VIX, and the Markit ITTR110 index). They reveal that leading cryptocurrencies such as Bitcoin are isolated from financial and economic assets. While the authors point to the possibility that cryptocurrencies offer diversification benefits for investors, they focus only on connectedness spillovers. In that sense, they neither conduct a comparative analysis between the safe-haven abilities of Bitcoin, gold, and commodities for stock indices, nor quantify extreme connectedness spillovers. In parallel, it is well documented that commodity investments represent a valuable addition to a portfolio of stocks (Henriksen, 2018),12 and that gold provides protection to a portfolio during stress periods (Areal, Oliveira, & Sampaio, 2015; Baur & Lucey, 2010; Klein, 2017). However, it is unclear whether Bitcoin and gold (commodities) share safe-haven roles against extreme movements in a sample of global and country stock market indices. We aim to clarify the role of Bitcoin in this matter and compare its properties with gold and the general commodity index as a standard safe-haven investment. The sample of stock indices represents the countries that have the most influential institutional investors, the longest history in stock market activities, and the largest market capitalization and economic output. The sample allows for potential heterogeneity between developed and emerging stock indices and between the largest developed economy (USA) and the largest emerging economy (China). The latter is a large player in the Bitcoin market although it has recently adopted much stricter regulatory measures against Bitcoin exchanges. 3. Wavelets and wavelet VaR The empirical methodology applies wavelet coherence (Grinsted et al., 2004) and wavelet VaR (Rua & Nunes, 2009) to assess the safe-haven role of three assets (Bitcoin, gold, and commodities) against global, regional, and country stock market indices. The application of the wavelet coherence approach uncovers the co-movement between potential safe-haven assets and stock market indices at various frequencies. Bitcoin, gold, and commodities markets are complex systems consisting of a vari-

ety of market participants (speculators and investors) operating over different time horizons. For example, speculators, such as chartists, day traders, or event-driven traders, have short-term investment horizons spanning from several minutes to several days. In contrast, portfolio managers and institutional investors have long-term investment horizons and are involved with investment activities that span several months to years. It follows that the degree of co-movement between potential safe-haven assets and stock indices may be frequency-dependent and thus may differ across the investment horizons of various market participants. Accordingly, wavelet analysis captures slow and persistent comovements, allowing for a more nuanced understanding of the interdependence between markets than standard methods that only consider the time domain perspective (Reboredo & RiveraCastro, 2013). Importantly, wavelet analysis ascertains the best time frequency for the three assets under study to act as a hedge or safe haven for stock markets, i.e. to display, respectively, no relationship or a negative relationship across a range of frequencies. To provide portfolio implications, we quantify the diversification ability of each of the three potential safe-haven assets (Bitcoin, gold, and commodities) with the stock markets under study via the value-at-risk (VaR) measure in the time-frequency space (Rua & Nunes, 2009). This so-called wavelet VaR captures portfolio risk while allowing it to vary both in time and across frequencies. 3.1. Wavelet coherence analysis We are interested in studying at which optimal time scale each of the three assets (gold, commodities, and Bitcoin) can act as a safe haven for stock markets. This is important as financial market participants often operate with different investment horizons (i.e. short- and long-term). The wavelet coherency approach of Grinsted et al. (2004) offers this possibility by decomposing the economic relationship into time-frequency components. Based on wavelets, we consider the risk reduction due to weak or no co-movement between the safe-haven asset and the stock index via the VaR measure. The wavelet technique (i) decomposes the return series into time-scale components, and (ii) represents the variability and structure of the stochastic processes on a scale-by-scale basis. The wavelet function is a small wave and can be manipulated (stretched or squeezed over time) to extract the frequency components from a complex signal. The mother wavelet13 is used to produce small waves. It is expressed as a function of time t and scale s as: ,s

t −  

1 (t) = √ s

where , s and

1 √ s

represent the time position (translation

parameter), scale (dilation parameter related to frequency) and normalization factor, respectively. The normalization factor ensures that the transformation remains comparable across scales and over time.14 The literature provides various types of wavelets for the decomposition of time series depending on the research topic. To examine

13 14

A real or complex value function ␺ (.) is defined over the real axis. The mother (t) should also have certain properties so that it can be used

to unity, A recent study by Gerritsen, Bouri, Ramezanifar, and Roubaud (2019) indicates that technical trading rules contain significant forecasting power for Bitcoin prices. 12 The authors indicate that commodities’ valuable diversification attributes are most often time-varying.

(1)

s

for decomposition. It must have zero mean,

11

3

 +∞

2

−∞

 +∞ −∞

(t) dt = 1, which means that

(t) dt = 0; its square integrates (t) is limited to an interval of

time; and it should also satisfy the so-called admissibility condition, 0 < C =  +∞ | (ω)|2 dω < +∞ where ˆ (ω) is the Fourier transform of (t), that is, ˆ (ω) = ω

0+∞ −∞

(t) e−iω dt.

Please cite this article in press as: Bouri, E., et al. Bitcoin, gold, and commodities as safe havens for stocks: New insight through wavelet analysis. The Quarterly Review of Economics and Finance (2020), https://doi.org/10.1016/j.qref.2020.03.004

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the wavelet coherence among stock markets and the market of Bitcoin/gold/commodities, we use the Morlet wavelet, which provides the best balance between time and frequency localization (for further details on the Morlet wavelet, see e.g., Addison, 2002). Grinsted et al. (2004) show that the Fourier period for the Morlet wavelet is almost equal to the scale used: M

(t) =

1 1/4

eiω0 t e−t

2 /2

(2)

where ω0 indicates the central frequency of the wavelet. We use ω0 = 6, as the Morlet wavelet with this central frequency provides good localization between time and frequency.15 Wavelet analysis can be performed using either the continuous wavelet transform (CWT) or the discrete wavelet transform (DWT). The former has many advantages over the latter. It provides freedom to select wavelets according to the length of data, and the redundancy in CWT makes the interpretation and discovery of patterns or hidden information easier (Aguiar-Conraria & Soares, 2011). A continuous wavelet transform Wx of a discrete time series (x (t), t = 0,1,. . .,n) with respect to (t), can be represented as:



+∞

Wx (, s) =

x (t)

∗ ,s

−∞

1 (t) dt = √ s



+∞

x (t) −∞



t −   s

dt

(3)

where * denotes the complex conjugate. Notably, the wavelet transform preserves the energy of a time series that can be used to analyze the power spectra. Accordingly, the variance is given by: 1 x = C







+∞

2

   Wx (, s)2  d ds s2

−∞

0

(4)

Torrence and Compo (1998) define the cross-wavelet power | Wxy (, s) | of two time series x(t) and y(t) with the continuous transforms Wx (, s) and Wy (, s) as: Wxy (, s) = Wx (, s) . Wy∗ (, s)

(5)

The cross-wavelet power shows the areas of high common power between two time series in the time-frequency space. The wavelet squared coherence16 between the two time series is given by:



 S s−1 Wxy (, s) 2 2 R (, s) =   2    2  S s−1 Wx (, s) .S s−1 Wy (, s)

(6)

where R2

and S (.) represent the wavelet squared coherency and the smoothing operator, respectively. The value of the wavelet squared coherence ranges between zero (no co-movement) and one (high co-movement) and can be seen as a scale-specific squared correlation between series. In addition, the wavelet coherence framework allows us to study the lead-lag relationship between series while avoiding the issue of the squared coherence not being able to distinguish between the positive and negative relationship between series. Specifically, the phase difference is defined as: x,y = tan

−1



, x,y ∈ [−, ]

I Wxy (, s)

R Wxy (, s)

(7)

15 The Morlet wavelet, with ω0 = 6, is a good choice when using wavelets for feature extraction purposes, because it is reasonably localized in both time and frequency (For comparison among different wavelet filters and frequencies, see Aguiar-Conraria, Azevedo, & Soares, 2008; Grinsted et al., 2004; Torrence & Compo, 1998). Furthermore, ω0 = 6 is often used in economic applications (see, among others, Shahzad, Nor, Kumar, & Mensi, 2017). 16 Torrence and Compo (1998) define wavelet transform coherence as the squared absolute value of the smoothed cross wavelet spectra that is normalized by the product of the smoothed individual wavelet power spectra.

where the parameters I and R give the imaginary and real parts of the smooth power spectrum, respectively. The phase difference is standardly represented by a directional arrow in a chart of timescale wavelet coherence. The arrow points to the right if the series are positively correlated, and to the left if negatively correlated. If the arrow points upwards, the first series leads the other by ␲/2 (the actual period is based on the specific frequency/scale of the wavelet coherence chart), and the opposite for a downward-pointing arrow. The wavelet coherence results are standardly shown on a chart with time and scale (or frequency) on the respective axes and the coherences are represented by a colour scale. The hotter the colour the higher the coherence, meaning cold colours represent weak or no relationship between series. The statistical significance is based on Monte Carlo simulations assuming a red noise (an AR(1) process) and statistically significant regions are marked by a thick black curve. As the wavelets are more stretched at large scales, the validity of the results suffers. For easier orientation, the cone of influence is shown in these charts to distinguish between the reliable (top) and less reliable (bottom) regions of the time-frequency space. From the perspective of safe-haven characteristics, Bitcoin/gold/commodities are considered to be safe havens either if there is no strong relationship (represented by blue) or the relationship is negative and the Bitcoin/gold/commodities series are led by the stock market (which would be represented by a hot colour and an arrow pointing South-West). The former is a sign of a weak safe haven and the latter of a strong safe haven. 3.2. Wavelet VaR To provide financial insight for risk management from the wavelet analysis, we conduct a VaR analysis of a portfolio containing the safe-haven asset and the stock index. The VaR quantifies the largest likely loss of a portfolio in a specific timeframe, which helps identify the risk reduction arising from weak or no co-movement between the safe-haven asset and the stock index. For a portfolio of k assets, the VaR is defined as: VaR(˛) = V0 ˚−1 (1 – ˛)p

(8)

where ˛, V0 , ˚(.), and  p denote confidence level, value of the initial investment, cumulative distribution function of the standard normal distribution, and square root of the portfolio variance, respectively. The portfolio variance is given by: p2 =

k i=1

ωi2 i2 +

k

k i

/ j i=

ωi ωj cov(ri rj )

(9)

where ωi and ri denote the weight of asset i in the portfolio and the return of asset i, respectively. Clearly, the right side of Eq. (9) involves two terms. The first reflects the variance of the return of each stock index (or safe-haven asset), while the second reflects the degree of co-movement.17 Following Rua and Nunes (2009), we compute the VaR of a portfolio assuming no co-movement between the stock market index and the safe-haven asset, and the VaR of the same portfolio assuming co-movement. For further insight we compare the two VaR measures by computing the ratio between them – called the ratio of portfolio VaR. Accordingly, we capture the percentage increase/decrease in the VaR due to comovement. A ratio equal to one means the VaR level is insensitive to co-movement while a ratio higher (lower) than one indicates that the co-movement implies a higher (lower) VaR. Therefore, a ratio below one suggests the asset can be considered a safe haven whereas a ratio above one suggests that the portfolio risk does not

17 Following Rua and Nunes (2009), we use the wavelet (continuous) counterparts of the variance and the covariance to study the wavelet VaR.

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Table 1 Summary statistics and unit root tests.

Bitcoin Gold Commodities World Developed Emerging USA China

Mean (%)

Std. Dev. (%)

Sharp ratio

Skewness

Kurtosis

ADF

KPSS

0.591 0.006 −0.023 0.034 0.019 0.012 0.077 0.023

6.549 1.011 1.174 0.813 0.945 0.951 1.650 1.261

0.090 0.006 −0.020 0.041 0.020 0.013 0.047 0.019

−0.017 −0.820 −0.197 −0.606 −0.534 −0.415 −3.377 −0.183

12.044 11.174 5.639 8.114 7.887 6.110 7.192 6.182

−42.80*** −45.10*** −46.02*** −39.19*** −39.19*** −35.15*** −48.17*** −41.89***

0.137 0.157 0.259 0.159 0.065 0.053 0.110 0.054

Notes: The sample period is from 20 July 2010 to 22 February 2018. ADF and KPSS tests present empirical statistics of the augmented Dickey-Fuller unit root test (Dickey and Fuller, 1979) and KPSS (Kwiatkowski et al., 1992) stationarity test, respectively. *** Indicates significance at 1 % level or better.

decrease by adding this specific asset. As this specific VaR variant is based on the wavelet analysis, its representation is parallel to the wavelet coherence chart. 4. Data We consider daily prices of three potential safe-haven assets (Bitcoin, gold, and commodities) and the stock market indices of world, developed, and emerging markets, China, and the USA. Bitcoin price data are extracted from CoinDesk (https://www. coindesk.com) which computes the average price of Bitcoin from leading trading exchanges (Bouri, Molnár et al., 2017). Price data on the rest of the variables are collected from DataStream. Gold prices are represented by the spot price of one ounce of gold, the commodity index is the S&P Goldman Sachs Commodity Index (S&P GSCI) that represents the most widely used benchmark index in the commodity market, and the five stock indices (world markets, developed markets, emerging markets, China, and the USA) are represented by the corresponding Morgan Sanely Capital International (MSCI) indices. The empirical analyses are conducted with daily log return series covering the period 20 July 2010−22 February 2018, where the starting point is dictated by data availability of Bitcoin prices. Looking at the summary statistics and unit root tests of the return series (Table 1), Bitcoin appears to have both the highest average return and standard deviation (Brière et al., 2015).18 Bitcoin’s Sharpe ratio is also the highest (i.e. the highest excess return adjusted for the (high) volatility of Bitcoin). The opposite is true for gold and commodities, which give the lowest performance with the lowest Sharpe ratio. Results from the ADF unit root test and KPSS stationarity test indicate that all return series are stationary, and thus the empirical analysis is conducted using the return series. 5. Results 5.1. Wavelet coherence Fig. 1 presents the results from the wavelet coherence. It captures the co-movement between the stock markets and the three potential safe-haven assets in the time-frequency space. The figure shows the wavelet coherence through a contour plot where the horizontal axis represents the time domain and the vertical

18 For all cases, the excess positive values for the kurtosis dominate, especially for Bitcoin and gold. Negative skewness values are present in all cases, meaning that all return series have longer left tails that represent more dominant negative extreme values. However, Bitcoin possesses a negative skewness that is very close to a symmetric distribution, i.e. the highest losses are of almost the same level as the highest gains. However, in the case of gold, the extreme movements are of a higher magnitude than for Bitcoin, commodities, and the stock market indices, except for the USA.

represents the frequency domain. The frequency domain is shown as the scale (number of days), with a higher frequency indicating a longer investment horizon. The elements of the chart are described in Section 3.1.1. The colour pallet represents the strength of wavelet squared coherence – the hotter the colour the higher the coherence and thus the dependence between series. The directional arrows show the lead-lag relationship as well as the sign of the dependence. If the arrow points to the right, the two series are positively correlated, and if to the left, negatively correlated. If the arrow points upwards, the first series leads the other, and the opposite for a downward-pointing arrow. In the results presented in Fig. 1, the ‘first’ series is always a respective stock index (i.e. the rows) and the ‘other’ series is the potential safe-haven asset (i.e. the columns). The results are quite straightforward and stable across the potential safe havens, and do not vary much for the stock indices either. Let us examine each potential safe-haven asset separately. For Bitcoin, we can see that most of the time-frequency plane is dominated by cold colours (blue) for each stock index. This indicates that Bitcoin is not correlated to the stock indices and such a statement seems valid for the whole period analyzed over almost all frequencies. There are only several small islands of significant dependence, which could easily be due to the finite sample and noise (Grinsted et al., 2004). For the world and developed stock market indices, we observe that for very high scales (above 2 years), there is a significant connection between the indices and Bitcoin. However, most of these significant parts are below the cone of influence and are thus not considered valid here (more details about the cone of influence is given in Section 3.1.1). Therefore, Bitcoin can be considered a safe-haven asset from the perspective of wavelet coherence and this is true for the whole period analyzed and all analyzed stock indices. The results are rather similar for gold which seems to have close yet lower potential as a safe-haven asset compared to Bitcoin based on the wavelet coherence profile. Apart from significant coherence at high scales (again around 2 years) between 2014 and 2016, there are only several small (albeit bigger than those of Bitcoin) islands of significance. The overall perspective is that gold keeps its high potential as a safe-haven asset and portfolio diversifier. This cannot be said for the commodities, which show significant coherence with the stock indices in a much more pronounced way. This is true mainly for the world and global stock indices but for the others, the significant coherence is still apparent. We see that the coherence is majorly positive and the stock indices mostly lead the commodities. The diversification potential of commodities is thus quite limited, at least with respect to the stock indices analyzed.

5.2. Wavelet value-at risk We extend the results of the wavelet coherence analysis and provide a more practical analysis of the safe-haven role of Bitcoin, gold, and commodities through their respective association with

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Fig. 1. Wavelet coherence between potential safe havens (Bitcoin, gold, and commodities) and global stock markets. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Note: The value of squared wavelet coherence is depicted in colour and the value of relative phase by arrows. The colour code for the coherence ranges from blue (low coherence - close to zero) to red (high coherence - close to one). The area affected by edge effects is the semi-transparent region at the left and right boundary separated by the black U-shaped curve which is the cone of influence (CoI). The thick black contours within CoI are the regions of significant coherence (at 5% level calculated using 1000 Monte Carlo simulations).

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Fig. 2. Diversification potential of potential safe havens through wavelet value-at-risk. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Notes: This figure shows the time and frequency domain ratio of value-at-risk (VaR – 5%) with and without co-movements of a portfolio formed of the respective stock market and safe-haven asset. The vertical axis represents the frequency range while the horizontal axis shows the time dimension. The increase/decrease in VaR ratio is shown through the colour code shown at the bottom of the figure. Lighter colours (blue) indicate decreases in wavelet VaR while darker colours (red) show otherwise.

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the various stock markets using the VaR framework. Such portfolio risk analysis is crucial for investors and portfolio managers. However, instead of capturing the risk from a standard VaR perspective, we apply the wavelet VaR technique that combines both time and frequency dimensions simultaneously. The results of the wavelet coherence analysis are confirmed but are given more solid ground as VaR can be directly connected to the safe-haven properties of the assets whereas the wavelet coherence on its own can do so only indirectly. The graphs of the estimated wavelet VaR for the Bitcoin-stock pairs, gold-stock pairs, and commodities-stock pairs are given in Fig. 2. The representation here is parallel to the wavelet coherence graphs. Here the cooler colours represent lower wavelet VaR, and the hotter colours represent higher wavelet VaR. The former suggests that the asset under study has a safe-haven property whereas the latter suggests otherwise. For Bitcoin, we observe that green and blue colours dominate the VaR ratio chart. This suggests that the risk of the portfolio does not increase by adding Bitcoin, it may even decrease. This is true across frequencies and across stock markets. The only exception seems to be around the year 2016 where the stock markets and Bitcoin were apparently hit by the same negative news. For gold, the results are much more informative than those based solely on the wavelet coherence. Specifically, we see that in the short term, i.e. at low scales, the VaR ratio is quite stably above one, which suggests that gold is not a good diversifier. The results vary somewhat for higher scales, i.e. longer investment horizons, but it is clear that the risk contribution to portfolios is much higher than for Bitcoin, which strongly dominates gold in this respect. The wavelet coherence results are confirmed by the VaR ratios for the commodities. We can see that the charts are dominated by hot colours implying that commodities are tightly connected to the stock markets during the negative extreme events on the markets and they do not contribute to reducing risk. The overall ranking of the possible safe havens based on the wavelet VaR is thus Bitcoin (the most promising safe-haven asset), gold, and commodities (the least appropriate).

6. Discussion and conclusions In this study, we compare the safe-haven roles of gold, commodities, and Bitcoin against global and country stock market indices. We apply an approach that covers heterogeneity across scales and their time evolution. We consider the wavelet value-atrisk, which captures risk in the time-frequency space for portfolios containing individual stock indices and the potential safe-haven assets. This is important for investors who seek to minimize the likelihood of extreme losses. Our main results are summarized as follows. Firstly, the wavelet coherence approach shows that the overall dependence of Bitcoin/gold/commodities and the stock markets is not very strong, even though the ranking of Bitcoin as the least dependent and commodities as the most dependent emerges clearly. Secondly, the wavelet VaR analysis focusing on tail dependence indicates the superiority of Bitcoin over both gold and commodities in terms of diversification benefits and strongly supports the ranking found by the wavelet coherence analysis of Bitcoin as the most promising safe-haven asset, followed by gold, and the dynamics of commodities being quite close to the dynamics of the examined stock markets. The results are in line with some previous research arguing that Bitcoin is isolated from financial assets (e.g., Corbet et al., 2018) and can be seen as a new virtual gold by the logic of its hedging properties, i.e. specifically hedging against stocks and the US dollar (Dyhrberg, 2016), even though Bitcoin’s statistical properties

might be very different from gold itself. Its diversification benefits have been shown repeatedly (Bouri, Jalkh et al., 2017; Bouri, Molnár et al., 2017; Brière et al., 2015; Dyhrberg, 2016) but they have also been disputed (Chowdhury, 2016). The exponential price growth of Bitcoin raises doubts about Bitcoin’s true underlying value and the possibilities of an irrational bubble or Ponzi scheme (Bouoiyour et al., 2016; Kristoufek, 2013; Li & Wang, 2017). Even though both stock market and Bitcoin prices increased markedly during the period analyzed, they remain virtually uncorrelated which is tightly connected to the fact that Bitcoin price has been increasing superexponentially. While stock markets price inflation can be, at least partially, attributed to the various waves of quantitative easing, this is not the case for Bitcoin as most new money was issued to banks and financial institutions that do not invest in Bitcoin due to various legal, taxation and accounting issues (Tan & Low, 2017). The pools of Bitcoin and stock market investors are thus quite different (Filtz, Polleres, Karl, & Haslhofer, 2017; Kondor, Csabai, Szüle, Pósfai, & Vattay, 2014), and their price driving factors are different as well (Bouoiyour et al., 2016; Kristoufek, 2015). Their uncorrelatedness is then not surprising. What distinguishes our analysis from previous studies is the focus on time variability and non-linearity of the relationship between Bitcoin (and gold/commodities) and the stock markets. The wavelet-based method offers a comprehensive view of the dynamics and connections between the assets. Even though the results are statistically strong and have far-reaching practical consequences for investors, it should not be taken as a given that investors consider or make use of Bitcoin as a fully-fledged alternative to gold/commodities (or other safe-haven assets) in their portfolios or investment strategies. This is connected to the notyet-solved status of Bitcoin in the international financial market. Even though some financial institutions have started considering it, they aim at the blockchain technology rather than Bitcoin itself, which might complicate further expansion of the cryptocurrency. With the launch of Bitcoin futures contracts by the CME and CBOE in December 2017, Bitcoin moved closer to the centre of the financial world making it harder for policymakers, institutional investors, and bankers to ignore its role as an investment. This has pushed Bitcoin towards legitimacy, and, importantly, this might ultimately help manage its price volatility. Our findings are important for investors and traders who now have empirical evidence that Bitcoin has some of the virtues of gold against extreme down movements in the global stock market indices. Our findings also indicate that gold has lost some of its glow, but not to the detriment of Bitcoin, as many claim. However, Bitcoin still has a long way to go in order to catch up to gold in terms of history, price stability, and accessibility. There are several issues that need to be taken into consideration when interpreting the results for general policy and investment considerations. Firstly, the Bitcoin market was very shallow until around 2013 when trading volumes started increasing and Bitcoin started to become better known to the public. Therefore, the results before 2013 need to be taken with a grain of salt. This does not change our main findings a great deal, but rather explains why some of the results are very different for the period before 2013. Secondly, the biggest Bitcoin losses occur not during rallies of the stock markets, they are rather independent. Negative news, such as closures of cryptocurrency exchanges and political decisions, has been shown to drag down the prices in the early stages of Bitcoin development (Kristoufek, 2013, 2015) and the present market situation (Philippas, Rjiba, Guesmi, & Goutte, 2019). Such negative news is standardly not connected to the traditional financial markets. Market manipulation as a price driver in either direction is exogenous with respect to the traditional financial markets as well (Gandal, Hamrick, Moore, & Oberman, 2018). The highest losses of Bitcoin and stock indices thus occur at different times, which improves Bit-

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