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Contents lists available atScienceDirect

Research in International Business and Finance

journal homepage:www.elsevier.com/locate/ribaf

Does bitcoin co-move and share risk with Sukuk and world and regional Islamic stock markets? Evidence using a time-frequency approach

Walid Mensi

a

, Mobeen Ur Rehman

b

, Debasish Maitra

c

, Khamis Hamed Al-Yahyaee

a,

*, Ahmet Sensoy

d

aDepartment of Economics and Finance, College of Economics and Political Science, Sultan Qaboos University, Muscat, Oman

bSouth Ural State University, 76, Lenin Prospekt, Chelyabinsk, Russian Federation

cIndian Institute of Management Indore, Indore, India

dFaculty of Business Administration, Bilkent University, Ankara, Turkey

A R T I C L E I N F O

Keywords:

Bitcoin

Islamic stock market Sukuk

Co-movement

VaR-based wavelet approach

A B S T R A C T

This paper examines the co-movements between Bitcoin (BTC) and the Dow Jones World Stock Market Index, regional Islamic stock markets, and Sukuk markets. We apply cross wavelet transform and wavelet coherence analysis with a wavelet-based measure of value at risk. The co- movement is stronger and in the same direction at lower frequencies, suggesting the benefits from diversification with BTC are relatively less for long-term investors compared to short-term investors. Co-movement in the opposite direction at high frequencies implies better benefits of hedging in the short run through diversification in BTC and Islamic equity markets. Robustness tests show that the correlations increase as we increase from an investment horizon of two days to one of 64 days. The frequency-domain causality test shows significant causality flow from BTC to the Islamic market of Asia-Pacific, Japan, and Sukuk markets in the short term. Additionally, BTC is found to lead Asia-Pacific Islamic stock markets in the long term. Finally, we note that the benefits of portfolio diversification with BTC and Islamic assets vary across time and frequencies.

1. Introduction

Co-movements among financial assets are of primary concern for investors, portfolio managers, and policymakers alike. Co- movements amongfinancial markets differ during downturn and upturn markets, indicating asymmetric relations. By accounting for co-movements, market participants benefit from additional information on asset allocation and portfolio design. Understanding financial asset price interconnectedness due to co-movements is critical not only to explain relations between different asset prices but also to determine whether the price of an asset or market reacts to changes in the prices of other assets or markets (Giudici and Abu-Hashish, 2019). Such an understanding helps investors tofind hedging opportunities.

Of late, cryptocurrency has gained attention from both investors and academic researchers; given the diversification benefits, it can offer against adverse movements in the price of other financial assets (Akhtaruzzaman et al., 2019). Among all the crypto- currencies, Bitcoin (BTC) is the largest in terms of market capitalization. More precisely, BTC’s market capitalization more than

https://doi.org/10.1016/j.ribaf.2020.101230

Received 6 July 2019; Received in revised form 9 March 2020; Accepted 24 March 2020

Corresponding author.

E-mail addresses:walid.mensi@fsegt.rnu.tn(W. Mensi),Mobeenrehman@live.com(M. Ur Rehman),debasishm@iimidr.ac.in(D. Maitra), yahyai@squ.edu.om(K. Hamed Al-Yahyaee),ahmet.sensoy@bilkent.edu.tr(A. Sensoy).

Available online 19 April 2020

0275-5319/ © 2020 Elsevier B.V. All rights reserved.

T

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doubled over 2016, increasing from 6.5 billion USD on December 31, 2015, to 15.5 billion USD on December 31, 2016. BTC accounts for more than 80 % of the total cryptocurrency capitalization during the same period (Al-Yahyaee et al., 2018). Bitcoin has given rise to a series of alternative coins such as Ethereum, Ripple, Litecoin, among others (Beneki et al., 2019). It has received a lot of publicity, given its innovative features, simplicity, transparency, and its increasing use (Eross et al., 2019). In addition, in December 2017, the Chicago Board Options Exchange (CBOE) introduced BTC futures, making the cryptocurrency an official investable asset in the eyes of both investors and policymakers. However, BTC futures are considered one of the culprits behind the recent crash in BTC returns in 2018.1Liu et al. (2019)found that, within thefirst 45 days after the futures launch, BTC suffered a 26.50 % loss, whereas other cryptocurrencies remained resilient and provided positive returns during the same period. BTC is highly volatile, and consequently, it is crucial to evaluate appropriate risk metrics and its dependence on thefinancial markets (Caporale and Zekokh, 2019).

Since the globalfinancial crisis in 2008–2009, investors around the world have been in search of alternative investment assets that can offer better diversification compared to equity markets. During this period, Sharia-compliant equities have emerged as an alternative investment class (Rahim and Masih, 2016;Narayan and Phan, 2017), given the different institutional characteristics of Islamic stock markets (Narayan et al., 2019a,b). More importantly, the systematic risk of Islamic equity markets is lower than that of their conventional counterparts (Sensoy, 2016). Empirical studies show that these Islamic equity assets serve as a safe haven during financial meltdowns, suggesting investors’ flight to quality during these turbulent periods (Mensi et al., 2015). On the other hand, Sukuk, as an Islamic bond market, has exhibited a significant increase in both Islamic and non-Islamic economies. Sukuk involves entitlement to rights of ownership of a given class of assets that the borrower provides to a lender as proof of ownership (Haque et al., 2017). In contrast to conventional bonds, Sukuk pays profits instead of interest.Abdul Halim et al. (2017)document that Sukuk deters managerial expropriation through a set of contractual arrangements that earmark the cashflows of the project to the un- derlying assets.

The evaluation of BTC in terms of Sharia rules is controversial, with Sharia scholars conflicted about the acceptance of BTC. The first group of scholars and fatwa are against the permissibility of BTC (i.e., it is haram).Abdul Halim et al. (2017)document that BTC is used in illegal activities and allows for money laundering and fraud. The second group supports BTC trading and considers it halal.

This group justifies BTC by the fact that people treat it as a valuable thing, it is accepted as a medium of exchange by a significant group of people, and it is a measure of value.2Cryptocurrencies werefirst considered under Sharia rule in July 2018, when the Sharia Review Board certified the first cryptocurrency in Bahrain, followed by an agreement between Ripple and Saudi Arabia’s National Commercial Bank to enable a system to send remittances via RippleNet. Indonesian Islamic scholar Muhammad Abu Bakarfirst commented on the legitimacy of cryptocurrency under Sharia, suggesting that BTC be accepted as a means of payment under the category of“customary money.” BTC is thus better aligned with the principles of Sharia than other formal banknotes.3

The theoretical argument that motivates the inclusion of BTC in the Islamic equity and Sukuk market is more indirect than direct.

In a recent study,Narayan et al. (2019a,b) found that BTC prices affect the monetary system of an economy. The three channels through which BTC influences the monetary system and overall stock markets consist of monetary aggregates, inflation, and ex- change rates. If the adoption of BTC can substitute for conventional money, it can then not only change the role of money but also reduce the circulation of money. This phenomenon thus leads to the demise of the quantitative theory of money.

Similarly, BTC trading and its inclusion in the mainstream economy can influence inflation and the marginal cost of production through wealth effects, since BTC is considered not only an investment asset but also an instrument that stores value. Hence, through value and wealth effects, BTC triggers the demand for goods and services and puts upward pressure on prices. Unlike conventional money, BTC is not controlled by monetary policy. Therefore, policy rates determined by the central bank to contain inflation might not work. BTC has wealth effects on trade that are reflected in the exchange rate, as in the other two channels. The exchange rate is thus considered to appreciate or depreciate because of trading and investment activity in BTC.

In another recent study bySchilling and Uhlig (2019)analyzed the evolution of BTC prices and the consequences of monetary policy. The authors mentioned that BTC, which is intrinsically worthless, is not only a medium of exchange, and storable but also stays out of the stabilization or manipulation mechanisms of the central bank. These characteristics of Bitcoin make it different from conventional currencies.

The literature concludes that Islamic stock markets experienced lower volatility and less spillover during the lastfinancial crisis (Arouri et al., 2013). The stricter norms of Islamic law lead tofirms’ lower interest burden because of lower leverage. The smaller amount of accounts receivable provides less of a chance of incurring bad debt in the balance sheet (Hoque et al., 2016). These specific features of Islamic equities not only distinguish them from their conventional counterparts but also make them attractive for portfolio diversification with other assets such as BTC, which has become more integrated with the conventional financial system and markets.

Financial integration, including that of equity markets with other assets, such as cryptocurrencies, offers fewer hedging opportunities.

The fundamental factors that determine price formation in BTC and other assets are different. Hence, a bubble cycle in the BTC markets can be expected to differ from that of other assets, as in Islamic equity markets, which makes both the BTC and Islamic equity markets suitable for portfolio diversification (Kang et al., 2019).

Price changes in one asset due to price movements in another are explained by the gradual information diffusion hypothesis and investor conservatism (Narayan et al., 2019a,b). These two theories suggest that Islamic stocks can behave differently in the face of

1The CBOE has suspended the trading of BTC futures and stopped listing BTC futures contracts in March 2019 (https://www.cnbc.com/2019/03/

18/cboe-to-stop-listing-Bitcoin-futures-as-interest-in-crypto-trading-cools.html).

2SeeAbu-Bakar (2018)for more information.

3See“Islamic finance and cryptocurrency: Convergence at last?” athttps://www.reuters.com/brandfeatures/venture-capital/article?id=62709.

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cryptocurrencies, compared to conventional stocks. The information diffusion hypothesis proposed byHong and Stein (1999)can be applied to our study. Since Islamic stocks differ from conventional stocks, they will react differently to price changes in crypto- currencies.

We contribute to the literature in three main ways. First, we assess whether there are any co-movements between the BTC and Shariafinancial markets. Given BTC’s nature, it offers not only hedging opportunities for traditional financial assets but also Islamic equity and bond investments. These could be of interest to investors who trade in alternative investment investments such as Islamic equity markets. The results of time-scale co-movements between BTC and Islamicfinancial markets provide greater insight for investors into the potential diversification benefits of frequency-based portfolio strategies.

Second, the results of our examination of risk-sharing between the BTC and Islamic equity markets in a portfolio with or without co-movement have direct implications for risk management and portfolio diversification. The identification of time scales for which the value at risk (VaR) of portfolios between BTC and the Islamic equity and Sukuk markets is lower could benefit investors seeking alternative asset classes and investment opportunities (Benhmad, 2013). Moreover, wavelet-based VaR can indicate when investors can expect a greater potential loss, that is, in the short or long run (Mensi et al., 2017).

Third, we examine the relation between BTC and the Islamic stock and Sukuk markets by employing the frequency domain causality test ofBreitung and Candelon (2006; henceforth BC). Frequency domain causality tests between BTC and the Islamic equity and Sukuk markets could provide evidence that thefinancial crisis has changed the asset market linkage. Frequency domain causality captures whether the cross-market linkage between BTC and Islamic stock and Sukuk markets is determined by temporary or per- manent shifts indicative of contagion or interdependence, respectively (Bodart and Candelon, 2009). From a hedging perspective, the examination of causality at different frequencies could help investors determine whether transitory and permanent causal relations in the short and long run, respectively, can help short-term traders and long-term investors (Bouri et al., 2019b).

In this paper, wefirst examine the co-movements between BTC prices and the Dow Jones (DJ) Sukuk index and eight major DJ Islamic stock market indexes by using a wavelet approach. This time-frequency domain based wavelet approach allows us to detect low-frequency (high-frequency) movements when the wavelet transform stretches (compresses) into a long (short) wavelet function (Aguiar-Conraria and Soares, 2011a,b). Time–frequency-based co-movements provide different perspectives to investors and pol- icymakers regarding short- and long-term investment and policy interventions for overall macroeconomic stability (Mensi et al., 2019). Frequencies corresponding to (short, medium, and long term) periods offer a better approach to the examination of short- and long-run co-movements and their implications for risk assessment and hedging strategies. The trading and investment activities of speculators and investors depend on investment horizons, which vary due to different expectations based on fundamental or utili- tarian characteristics (i.e., risk appetite, expected return, risk premium, available information) and psychological or value-expressive characteristics such as sentiment (Statman, 1999). The differences in trading activities and market expectations change from one group of traders to the other, making the market very heterogeneous. Therefore, short investment horizons (high-frequency short- term traders and speculators) and long investment horizons (low-frequency long-term investors) can have different implications for the relation between cryptocurrency and Islamic equity markets.

Gençay et al. (2003)document that wavelet offers a natural platform to analyze the systematic risk at different time horizons without losing any data points. It differentiates seasonalities, reveals structural breaks and volatility clusters, and identifies local and global dynamic properties of a process at these timescales. The wavelet approach can handle nonstationary data, localization in time, and the resolution of the signal in terms of the time scale of analysis (Ramsey and Lampart, 1998;Ramsey, 2002).

Existing literature (Aguiar-Conraria and Soares, 2011a, 2014,Aguiar-Conraria et al., 2018) emphasize that wavelet transfor- mation is an appropriate method for handling the noisy, nonstationary, and nonlinear data offinancial time series. The wavelet method can also better determine portfolio diversification strategies between assets (Benhmad, 2013).Kang et al. (2019)emphasized that wavelet decomposition helps capture the behavior of two time series not only in the time domain, but also across different investment scales, to investigate the risk-level preferences of different categories of investors (speculators, noise traders, and long- term investors) and their portfolio diversification strategies at a more granular level.

Second, we analyze systemic risk by calculating the wavelet-based VaR, a popular risk evaluation tool. This risk management measure provides a full picture of risks that are reflected in extreme co-movements during bearish and bullish market states. The VaR within the time scale accounts for the exposure to risk for different frequencies and time horizons. This information is crucial for the decision making of both short- and long-term investors. It predicts potential losses in shorter time horizons (or for higher frequencies) and longer time horizons (or for lower frequencies).

Third, we augment our analysis with robustness tests. More specifically, we carry out the BC (Breitung and Candelon, 2006) causality test to investigate the relation between BTC and the Islamic stock and Sukuk markets. The use of the frequency domain causality test is motivated by the fact that thefinancial crisis is expected to have altered the asset market linkage and direction. Thus, the use of frequency-domain causality can differentiate between temporary and permanent shifts in cross-market linkages, indicating contagion and interdependence, respectively (Bodart and Candelon, 2009). Furthermore, the decomposition of causality at different frequencies will help short-term speculators or noise traders and long-term investors understand the transitory and permanent causal relations between these two markets for better portfolio diversification (Bouri et al., 2019b).

Ourfindings reveal that the co-movements between BTC and both Sukuk and Islamic equity price returns vary in time-frequency space. The co-movements between the markets considered are stronger and in the same direction at low frequencies. This result suggests that diversification benefits are relatively less available to long-term investors compared to short-term investors. We note relatively weak co-movements in the opposite directions at high frequencies, which implies better benefits of hedging through diversification in BTC and Islamic equity markets in the short term. Also, the strength of the co-movements between BTC and Islamic equity returns varies across countries. In terms of VaR-based risk analysis, the portfolio diversification benefits of BTC and Islamic

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equity/Sukuk/Titans 100 vary across time and frequencies. Portfolio risk is more likely to increase at low frequencies than at high frequencies (except the Islamic equity index returns for Canada and Titans 100). Co-movement with Sukuk decreases portfolio risk.

Moreover, wefind a sudden increase in short-term portfolio risk from 2015 to 2016. In the long term, Islamic equity market returns of the United States, the United Kingdom, Europe, Asia-Pacific, and Japan might not offer portfolio diversification benefits. In contrast, the equity returns of Canada and Sukuk can continue to be part of the portfolio. Robustness tests show that the correlations are significantly different from zero across years and within time investment horizons. Besides, the correlations among the markets considered increase as we move away from a shorter investment horizon of two days to a longer one of 64 days. Finally, the frequency domain causality test shows causalityflow from BTC to the Islamic market of Asia-Pacific, Japan, and Sukuk in the short term. BTC is also found to cause Asia-Pacific Islamic stock markets in the long term. Unlike wavelet-based co-movement analysis, frequency domain causality addresses only frequencies. Besides, ourfindings with the optimal hedge ratio between Islamic equity, Sukuk, and BTC show that BTC offers diversification benefits. However, Sukuk has the lowest hedge ratio with BTC as compared to Islamic equity markets.

The remainder of this article is structured as follows. Section2reviews the literature. Section3outlines the methodology. Section 4describes the data and descriptive statistics. Section5presents and discusses the empirical results. Section5summarizes the main findings and draws our primary conclusions.

2. Literature review

The literature on cryptocurrencies has covered a range of different aspects, starting with the technical and investment properties of BTC with respect to risk and returns (Dwyer, 2015;Bariviera, 2017;Caporale and Zekokh, 2019;Eross et al., 2019;Katsiampa, 2019), market efficiency (Bariviera, 2017;Jiang et al., 2018;Tiwari et al., 2018;Al-Yahyaee et al., 2018;Sensoy, 2019;Shrestha, 2019), and risk and return spillover to other assets (Corbet et al., 2018;Guesmi et al., 2018;Selmi et al., 2018;Matkovskyy and Jalan, 2019;Pal and Mitra, 2019;Salisu et al., 2019;Shahzad et al., 2019). A significant number of studies are centered on BTC’s suitability as an asset to diversify the risks of other conventional assets. Consistent with the goal of this article, we limit our focus on the literature concerned with the relation between BTC and other conventional assets.

Bouri et al. (2017a)suggested that BTC can provide hedging benefits. They employed a dynamic conditional correlation gen- eralized autoregressive conditional heteroskedasticity (DCC-GARCH) model to determine whether BTC can act as a hedge and safe haven for major world stock indexes. Their results revealed that BTC could only serve as a strong safe haven against extreme weekly down movements in Asian stocks. The hedging property of BTC varies between horizons.Bouri et al. (2017b)stated that the BTC crash in 2013 changed the dynamics of the BTC markets. BTC appeared to be a hedge against commodities, including energy, as well as a safe haven before 2013, whereas, after 2013, BTC became merely a diversifier.Corbet et al. (2018)later examined the co- movements between cryptocurrencies andfinancial assets in time-frequency space.Katsiampa (2019), andBeneki et al. (2019) investigated the asymmetric relationships among leading cryptocurrencies (BTC, ETH, XRP, and LTC), and volatility transmission between BTC and Ethereum, respectively. The evidence suggests that there exists a positive and time-varying conditional correlation, and major events influence the volatility transmission mechanism among the cryptocurrencies under investigation.

In a different study,Bouri et al. (2018)proposed that volatility and spillover between BTC and other assets are not only time- varying but also dependent on underlying market conditions. The return spillover between BTC and other asset classes is more apparent than the volatility spillover. There is positive return spillover from world markets and emerging markets to BTC during bull markets, but the sign reverses during bear markets. On the contrary, BTC provides positive return spillover to the world, developing, and Chinese stock markets under bear market conditions. Commodity, energy, and bonds exhibit positive return spillover to BTC under both bull and bear market conditions. The fact that BTC receives more spillover than it transmits does not make it a risky asset in the globalfinancial system.Bouri et al. (2019a)mentioned the beneficial role of cryptocurrencies as a risk diversifier against equity markets. Using four MSCI equity markets of the United States, Europe, Asia-Pacific (excluding Japan), and Japan, the authors showed that cryptocurrencies play an important role in diversifying the risk in Asia-Pacific and Japan.Bouri et al. (2019b)investigated volatility surprise among cryptocurrencies by employing frequency domain causality to differentiate between transitory and per- manent causalities. They suggested that the behavior of causality changes from short-run, or temporary, causality to long-run, or permanent, causality.

Similarly,Selmi et al. (2018)investigated the effects of BTC and gold on the oil markets by employing a conditional quantile-on- quantile regression model and suggested that BTC provides opportunities to hedge against oil price risk.Guesmi et al. (2018), using a vector autoregressive moving average model DCC-GARCH model, analyzed the role of BTC in portfolio management with other assets such as the stock, foreign exchange, CBOE Volatility Index, and commodity markets. They stated that the DCC-GARCH model captures volatility spillover better, but also suggested that BTC can significantly reduce portfolio risks.

Al-Yahyaee et al. (2019)recently examined the downside risk and diversification benefits of BTC by applying bivariate DCC- GARCH models and confirmed that portfolios with BTC could deliver greater effectiveness.Pal and Mitra (2019)later confirmed that BTC has hedging ability to reduce risk in asset portfolios. However,Shahzad et al. (2019)stated that, with the help of the cross- quantilogram, BTC could be considered a weak safe-haven asset in some cases, its safe-haven property changing over time and across markets.Salisu et al. (2019)studied the role of BTC in predicting stock returns in G7 countries. They usedflexible generalized least squares and found that BTC is a better predictor of stock market returns than macro variables are. Using a regime-switching model, Matkovskyy and Jalan (2019)examined the contagion effects between BTC and financial markets. They observed that BTC return co- movements had changed significantly since the launch of BTC futures.

The evidence from the literature shows that interconnectedness and co-movements between cryptocurrencies and other

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conventional assets have been thoroughly investigated. However, we cannotfind any study that has examined the relation between BTC and Islamic stocks and Sukuk markets. Our study thus differs from the literature in two major aspects. First, we consider the co- movements and portfolio implications between BTC and Islamic stocks and Sukuk markets. Understanding the nature of their co- movements is essential for investors interested in including BTC and Islamic stocks and Sukuk in their portfolios. Second, we apply both cross wavelet transform (XWT) and wavelet coherence (WTC) analytical approaches. The wavelet is a popular instrument for analyzing nonlinear correlations betweenfinancial series. Since the behavior of cross-market or cross-asset linkages with crypto- currencies is dynamic and varies under both bullish and bearish markets, the literature has employed different linear (Corbet et al., 2018;Salisu et al., 2019) and nonlinear approaches (Bouri et al., 2017a,b;Selmi et al., 2018;Matkovskyy and Jalan, 2019;Pal and Mitra, 2019;Shahzad et al., 2019). The dynamic nonlinear and asymmetric behavior offinancial assets makes the use of linear correlation unsuitable for analyzing their co-movements and leads to spurious estimations in asset allocation and regulatory decision making. The wavelet-based time-frequency domain approach gives us the unique advantage of taking care of the heterogeneity created by the diverse set of market players in terms of investment horizons or time scales (Dash and Maitra, 2018). Unlike the GARCH-based model used in the literature, time-frequency domain analysis is suitable for detecting co-movements under different frequencies and takes into account the concerns of short-term investors (speculators) and long-term investors (institutional investors).

GARCH-based models and the linear Granger causality test ignore the heterogeneous market hypothesis. Our study analyzes the co- movements between BTC and Islamic market indexes over time and across different frequencies ranging from lowest to highest frequency.

3. Methodology

3.1. Wavelet coherence (WTC)

To investigate the relationship between BTC and both Sukuk and Islamic equity market returns, we employ the wavelet technique developed byHudgins et al. (1993)andTorrence and Compo (1998). Later we use a cross XWT approach with a phase difference to analyze the time-frequency relation between BTC and the sampled Islamic indexes.

The specification of wavelet coherency between the two series as proposed byTorrence and Webster (1999)is

=

R S S s W s

S s W s S s W s

( ) | ( ( ))|

( | ( )| ). ( | ( )| )

n nxy

nx

ny

2 1 2

1 2 1 2 (1)

where S is a smoothing operator. This definition can also be considered a traditional correlation coefficient that explains wavelet coherency as a local correlation coefficient in time-frequency space. We can, therefore, rewrite Eq.(4)as follows, for the case in which the smoothing function value is equal to one with a time-scale complication:

=

W S S W s

( ) Scale( Time( n( ))) (2)

where SScaleand the complication highlight smoothing along the wavelet axis and time, respectively. A regular window and Gaussian function are used for the scale and time convolution, respectively (Torrence and Compo, 1998). We can articulate the smoothing power corresponding to the Morlet wavelet as follows (Grinsted et al., 2004):

=⎛

Stime(W) s W s cn( )*

t

s s

12 2 2

(3)

=

Sscale(W)|n (W s cn( )* Π(0.6 ))|2 s n (4)

wherec1andc2represent normalized constants andΠindicates a regular function.

We can also determine co-evolutions and normalized coefficients directly and indirectly, respectively. We use Monte Carlo si- mulations to analyze the wavelet coherency distribution and estimate the phase difference through the mean and the confidence interval of the associated time series. Therefore, the expression for the mean phase with different angles (ai, i = 1,…, n) is

∑ ∑

= = =

= =

am arg( ,X Y)withX cos( )anda X sin( )a

i n

i

i n

i

1 1 (5)

To estimate the reliable confidence interval for the mean angle, the independence of the phase angles can be rather helpful. We can set the scale resolution to a variety of angles, where greater resolution means higher angles. We can specify the circular standard deviation as

= −

s 2ln( / )R n (6)

where R equals (X2+Y2) and the circular standard deviation has a similar meaning as the traditional standard deviation. To identify the statistical level of significance, we use Monte Carlo simulations in which the lag length in a phase can be defined as

= ∈ −

ϕ tan I W

R W{ } ϕ π π

{ }, [ , ]

x y

nxy nxy x y

, 1

, (7)

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whereI and R indicate the real and imaginary parts, respectively.

We can characterize the phase difference between two variables using the path difference. A phase difference with zero value indicates that both variables co-move with the stated frequency. Thefirst series, x, lags behind the time series y ifϕx y, ∈[0, π/2]. On the other hand, the series x leads whenϕx y, ∈ −[ π/2, 0]. A negative association between the two series, that is, an anti-phase relation or a phase difference ofπ or( −π), would suggestϕx y, ∈ −[ π/2, π] U [−π π, /2]. However, ifϕx y, ∈[ /2,π π], then the variable x leads, and variable y leads ifϕx y, ∈ −[ π/2, −π].

3.2. Cross-wavelet transformation (XWT)

The cross wavelet approach involves a correlation coefficient in time–frequency space, whereas phase difference gives in- formation about the synchronization or delay in the co-movement of two time series (Aguiar-Conraria and Soares, 2011a,b). Ac- cording toAguiar-Conraria and Soares (2011a,b), the cross wavelet coherency approach is also defined by using the ratio of the cross spectrum to the product of the spectrum of the two series, and can be considered the correlation between the two series in both the time and frequency domains. As for traditional correlation coefficients, the coefficient of wavelet coherency equals one in the case of higher correlation between the two time series approaches, and zero in the case of no correlation or association between them. The wavelet power spectrum depicts variance in the series, and greater variance in the wavelet power spectrum is described by higher power. The cross wavelet power spectrum thus highlights the high covariance value between the two series across different time and frequency arrangements. This continuous family of wavelets allows us to examine the relations between macroeconomic andfi- nancial variables (e.g.,Bhanja et al., 2012;Dar et al., 2014;Tiwari et al., 2014). A wavelet function with both frequency () and time (dt) dimensions is a function with a zero mean in both frequency and time.

The definition of Morlet wavelet with least uncertainty dt dω( . )is defined as follows.

=

ψ α0( ) π 14eω tce 12α2 (8)

whereωcand t are the dimensionless frequency and time, respectively. Financial and macroeconomic time series decompositions are mostly carried out with a Morlet wavelet of angular frequencyωc= 6, which strikes a good balance between time and frequency localization. We use the same angular frequency in the present study.

Cross wavelet transformation is applied to the time series as a bandpassfilter. The time series is stretched by scale variation such thatα=s t. and is normalized to unit energy. We can see that, for a Morlet wavelet withωc= 6, the Fourier period (λWt) is ap- proximately equal to the scale ofλWt=1.03s. For a given signal spread over time sα t1, =1,…,N−1, Nwith uniform time steps δt, the CWT is defined as the convolution of xt with the scaled and normalized wavelet:

= ⎡

⎣ − ⎤

=

W s δt

s x ψ t n δt

( ) ( )s

tA

t N

1 t 0

(9) The wavelet power is defined as W s| tA( )|2. The wavelet cannot be completely localized in time and hence induces artifacts.

We now separate the edge effect cone of influence, which shows the area within which the relation is distorted. The background power spectrum (Pk) also generates the statistical significance of wavelet power and coherencies. As shown byTorrence and Compo (1998), we test the wavelet power significance to assess whether the data-generating process is an AR(0) or AR(1) stationary process (the null hypothesis) with a background power spectrum (Pk). However, more general processes require Monte Carlo simulations.

The following expression helps generate the distribution of the power spectra:

⎣⎢ < ⎤

⎦⎥= D W s

σ p P χ p

| ( )| 1

2 ( )

tA

A k v

2 2

2

(10)

3.3. Wavelet value at risk (VaR)

To reduce portfolio risk through diversification strategy, the VaR is a useful measure in management practices and risk assess- ments. The VaR defines the worst possible loss predicted within the specified confidence interval over a period. To estimate the impact of decreasing portfolio risk, that is, a portfolio’s VaR,Gençay et al. (2005)extended their previous study by including a discrete wavelet transform to test the validity of the capital asset pricing model. This work was then further extended byFernandez (2006), who calculated the wavelet’s VaR. We estimate the VaR of an equally weighted portfolio with a confidence interval 1 − α as

= ∅ − ≤ ≤

α V α σ

VaR( ) 0 1(1 ) , 0p α 1 (11)

where V0denotes the level of an initial investment with a cumulative normal distribution∅ andσp denotes the total risk or the portfolio’s standard deviation, with the portfolio’s total risk consisting of unsystematic and systematic risk and measured through the covariance (cov r r( ,i j)) of the assets, as follows:

∑ ∑ ∑

= +

=

σp ω σ ω ω cov r r( , )

i k

i i i k

i j k

i j i j

2 1

2 2

(12) with weightωi, standard deviation σi, and returnri for equity market i. The risk of an equal-weighted portfolio represents a

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combination of two risks: each stock’s individual risk and the degree of co-movement among the set of stocks.

Following the portfolio VaR approach ofGençay et al. (2005),Rua and Nunes (2009)used continuous wavelet counterparts of variance and covariance in analyzing wavelet VaR. They analyzed the equity co-movement between international markets in a portfolio VaR model and concluded that the ratio of the portfolio VaR with equity co-movement to the portfolio VaR without equity co-movement, namely, cov r r( ,i j)= 0, quantifies the decrease/increase in the portfolio VaR due to stock market return co-movement.

Therefore, a ratio equal to one implies no effect on the portfolio variable; however, a ratio greater (less) than one indicates an increase (decrease) in the VaR attributable to stock return co-movement. The authors are also of the view that positive co-movement between equity markets implies greater VaR. Therefore, the interpretation and application of wavelet VaR are counterintuitive. Thus, the addition of new assets with less than perfect correlations with existing assets, that is,cov r r( ,i j)>0, should reduce total portfolio risk and, therefore, VaR.

Although the ratio of the VaR of the portfolios with and without co-movement highlights the change in VaR attributable to co- movement, this does not mean that the increase in VaR is attributable only to co-movement. In this case, a higher value of the VaR is estimated (as the ratio of the VaR of a portfolio with co-movement to the VaR of a portfolio without co-movement) because of the addition of a new asset to the existing portfolio, where the covariance between portfolio assets is zero and not the actual VaR of the portfolios. Therefore, it is important for an investor to know the increase or decrease in the VaR of a portfolio if a new asset is added to it. The wavelet VaR would help investors realize that an increase in the VaR is associated with a decrease in the diversification benefits between BTC and Islamic equity and Sukuk markets. For example, if BTC and Islamic equity and Sukuk markets are found to co-move in opposite directions in the short term but tend to co-move in the same direction in the long run, then the diversification benefits of long-term investors will be lower compared to those of short-term traders. Hence, VaR is expected to increase in the low- frequency scale (i.e., in the long term).

4. Data and descriptive statistics

We consider daily closing spot prices for BTC and Islamic stock markets from July 9, 2010, through March 23, 2018. We use BTC (XBTUSD currency) and the DJ Islamic Market World Index (DJIM), US Islamic Index (IMUS), Europe Islamic Index (DJIEU), Asia- Pacific Islamic Index (DJIAP), UK Islamic Index (DJIUK), Japan Islamic Index (DJIJP), Canada Islamic Index (DJICA), DJ Islamic Market Titans 100 Index (IMXL), and DJ Sukuk Index (DJSUKUK), with data compiled by Bloomberg. The data availability of BTC influences the sample periods. Specifically, we consider the DJIM, the IMXL, and six DJ Islamic country-based markets for the United States, Europe, Asia-Pacific, United Kingdom, Japan, and Canada. The DJIM is weighted by free-float market capitalization and excludes any stocks that do not follow Sharia rules, prohibiting business activities involving, for example, alcohol, tobacco, pork- related products, conventionalfinancial services (banking, insurance, etc.), weapons and defense, and entertainment. Additionally, shares are removed from the DJIM if thefinancial ratios (which should not be more than 33 %) are not suitable for Islamic investment purposes. The IMXL assesses the performance of the largest 100 stocks traded globally that pass rules-based screens for adherence to Sharia investment guidelines.

Fig. 1plots the time variations of BTC with the respective Islamic Sukuk and stock indexes, showing an uptrend in 2017 for BTC.

The Islamic stock market indexes and Sukuk index exhibit an increasing trend during the sample period.Fig. 2shows evidence of volatility clustering in all markets. We observe important fat tails distributions in 2011 and 2016 for the Islamic stock markets; in 2011, 2013, and 2016 for Sukuk; and in 2010, 2013, and 2014 for BTC.

Table 1presents the descriptive statistics of the return series and shows positive mean returns for all markets except for the DJICA.

BTC has the highest average returns, whereas Sukuk has a null average return. The average returns for all Islamic stock market indexes are very similar. In addition, the standard deviations are the same for all the DJ Islamic indexes, while the BTC exhibits the highest risk level. It is worth noting that Sukuk (Islamic bonds) has no risk. All returns series deviate from normal distributions, as detected by the results of skewness, kurtosis, and Jarque–Bera tests. The linear correlation between BTC and Islamic stock indexes is positive for all the Islamic stock market indexes, except Asia-Pacific and Japan. The highest correlation with BTC is that with Europe, followed by the United States and the United Kingdom, while the lowest correlation is that with DJIM. BTC is positively correlated with Sukuk. These results motivate us to study co-movements not only in time but also across frequencies.

5. Results and discussions

This section presents the results of the wavelet approach. Second, we use ourfindings to analyze portfolio risk by applying the VaR-based wavelet. Finally, we augment our work with robustness tests for more in-depth analysis.

5.1. Time-frequency space analysis

Fig. 3plots the Cross wavelet transform (XWT and Wavelet coherence (WTC values for all BTC–index pairs. We report the cross wavelet and WTC in panels A to H, with BTC and DJ stock indexes of the world, the United States, Europe, Asia-Pacific, the United Kingdom, Japan, and Canada in panels A to G, and for BTC and the DJ Islamic Sukuk index in panel H. Panel I shows the results for BTC and the Islamic Market Titans 100 index.

It is interesting to note that inFig. 3, for the XWT, the directions of the arrows at different scales and over the period are not the same for the pairs of BTC and stock indexes. We see that BTC and the stock indexes are in phase, however, if we consider the significant region. A strong, continuous island of co-movement is observed until 2012 for Islamic equity returns, and the co-

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movement continues until 2014 for Asia-Pacific, Japan, and Canada. An in-phase relation is observed in the lower frequency band of 128–256 days; the arrows are pointed toward the upper right, suggesting that BTC is leading, while BTC and the Islamic indexes are out of phase for 64- to 128-day cycles; the arrows are directed toward the upper left, indicating that the Islamic equity markets are leading. We also note that the BTC–Islamic stock index pairs occupy a significant area in the higher-frequency cycles of four to 32 days; however, no phase difference is visible. It is worth mentioning that, if we consider high-power regions, the BTC–stock index pairs are out of phase for the 256-day to 512-day cycles.

Similar to the XWT value for the BTC–Islamic Sukuk pair, we find that the BTC–Islamic Sukuk pair is in phase for cycles of 128–256 days until 2014; the arrows are pointing toward the upper right, suggesting BTC is leading. The pair is in and out of phase for cycles of 64 and 128 days, respectively, from 2010 to 2011, when BTC is leading, and from 2013 to 2014, when Sukuk is leading.

The pair is also out of phase for cycles of more than 512 days from 2013 to 2015. To examine is there are any relations between the top 100 leading companies complying with Sharia adjustments and BTC, we perform the same analysis for the BTC–IMXL pair. BTC and the IMXL are in phase for cycles of 128–256 days until 2014, and out of phase for the frequency band of 64–128 days during 2010–2011.

The wavelet cross-spectrum can explain the common power of two variables without normalization to the single wavelet power spectrum. Sometimes, this approach can lead to ambiguous results, because, if one of the spectra is local and the other shows a very high jump, the jump generated in the cross-spectrum, which is the product of the continuous wavelet transformations of two series, cannot be attributed to the relation between the two series. Consistent with the literature (Andrieș et al., 2014), we conclude that the wavelet cross-spectrum is not suitable for testing the significance of the relation between BTC and Islamic equity and Sukuk markets.

We, therefore, perform the analysis using wavelet coherency. Wavelet coherency helps identify significant co-movements between two series in both frequency bands and time intervals.

We present the results of squared wavelet coherency on the right-hand side of the panels inFig. 3. For the BTC–Islamic equity index pairs, we witness significant regions of co-movement between BTC and Islamic equity markets over 256- to 512-day cycles in the long run, from 2011 to 2013. The pairs are out of phase, and the arrows pointed toward the upper left suggest Islamic equity markets are leading. However, we also notice that a few arrows are directed toward the upper right in the co-movement between BTC Fig. 1. Time trends of BTC and Islamic stock and Sukuk price indexes.

Notes: The x- and y-axes indicate, respectively, time (in days) and Bitcoin, the Sukuk, and the Islamic stock market indexes (world, US, Europe, Asia- Pacific, UK, Japan, and Canada), the Sukuk index, the Islamic equity index of Titans 100, and BTC (in USD).

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and DJIM, suggesting BTC leads the Islamic equity markets. A visible pattern of lead-lag relations is not observed in the wavelet coherency between BTC and Islamic equity markets. Unlike for the cross wavelet spectrum, we see no significant out-of-phase movement over 128- to 256-days cycles, in the medium term, from 2011 to 2013. We also note small islands of significant co- movement for BTC–Islamic equity market pairs during 2015–2017, over cycles of 32–64 days. The overall co-movements between BTC and the Islamic equity markets are not strong. In comparison to the United Kingdom, Asia-Pacific, Europe, and Canada, we find that the world, US, and Japanese Islamic indexes show strong coherency between BTC and the Islamic equity markets. Thefindings of strong coherency in the long run from 2011 to 2013 are supported by the results ofBouri et al. (2017b), whofind that 2013 is a watershed year in the BTC markets since the BTC crash in 2013 changed the dynamics of BTC markets.

For the BTC–Sukuk pair, we see a significant region of co-movement during 2011–2014 over 128- to 256-day cycles. The arrows are pointed toward the right and sometimes up, suggest the two series are in phase, and BTC is leading. We also observe a small but Fig. 2. Volatility clustering of the BTC and Islamic stock and Sukuk indexes.

Notes: The x- and y-axes indicate, respectively, time (in days) and the realized volatility of Islamic stock market indexes, Sukuk, and BTC.

Table 1

Descriptive statistics.

Statistic World US Europe Asia-Pacific UK Japan Canada Sukuk Titans100 BTC

Mean 0.0003 0.0004 0.0002 0.0003 0.0001 0.0003 −0.0001 0.0000 0.0004 0.0058

Minimum −0.0529 −0.0641 −0.0648 −0.0468 −0.1003 −0.0684 −0.0605 −0.0121 −0.0484 −0.6009

Maximum 0.0375 0.0459 0.0526 0.0402 0.0554 0.0676 0.0629 0.0081 0.0368 0.5170

Variance 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0000 0.0001 0.0047

Std. dev. 0.0079 0.0091 0.0110 0.0085 0.0119 0.0113 0.0126 0.0012 0.0079 0.0686

Skewness −0.5924 −0.4875 −0.3624 −0.4530 −0.5314 −0.2425 −0.2518 −1.4881 −0.5074 −0.3354

Kurtosis 4.9494 4.6851 3.6403 2.8796 5.3163 3.7911 2.3812 21.0884 4.2912 13.6394

Jarque–Bera 2167.1* 1916.3* 1153.0* 762.75* 2459.2* 1222.5* 495.8* 37935.3* 1627.0* 15558.0

ρBTC IM, 0.0388* 0.0419* 0.048* −0.015* 0.041* −0.011* 0.0385* 0.0247* 0.0441* 1

Notes: This table presents the descriptive statistics of BTC, Sukuk, and Islamic stock market indexes during the sample period, from 2010 to 2018.

ρBTC IM, stands for the linear correlations between BTC price returns and Islamic market returns. * denotes significance at the 1% level.

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significant island of coherency between BTC and Sukuk during 2017–2018 for the frequency band from eight to 32 days.

For the BTC–IMXL pair, we observe strong coherency during 2011–2013 for the frequency of 256–512 days, with arrows pointing to the upper left, which suggests they are out of phase and IMXL is leading. However, significant regions between BTC and IMXL are also seen during 2016–2017, where they are in phase, and IMXL is leading (the arrows are pointed to the upper right). We see that Fig. 3. Cross-wavelet transform (XWT) and wavelet coherence (WTC) analysis of BTC and Islamic markets.

Notes: The dense black outlines surrounding red patches indicate the 5% level of significance. The area outside the cone of influence (at the bottom of every image) highlights the edge effect. Note the color code on the right of each image, where blue indicates low power and red indicates high power. Greater color density indicates greater wavelet power. The x-axis presents the time line for each cryptocurrency pair, whereas the y-axis measures the scale or frequency. The phase difference between the two series is indicated by the arrow directions, as follows: →, both variables are in phase (cyclical effect on each other); ↗, BTC is leading; ↘, BTC is lagging; ⟵, the variables are out of phase (anticyclical effect), ↖, BTC is lagging; ↙, BTC is leading. A value of zero suggests that both variables co-move with the stated frequency. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article).

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Fig. 3. (continued)

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Islamic equity markets and Titans 100 lead, whereas, in co-movement with Sukuk, BTC leads. This phenomenon is due to the nature of Islamic equity markets, which differ from Sukuk. Sukuk is an Islamic financial instrument similar to a bond in conventional finance. We note that co-movements between BTC and Islamic stock markets and Sukuk are not very strong, in line with the literature (Dyhrberg, 2016;Bouri et al., 2017a,b;Corbet et al., 2018), where BTC remains isolated. This holds not only for traditionalfinancial and economic assets but also for other, alternative investment assets, such as Islamic stocks and Sukuk. Consistent with thefindings of Corbet et al. (2018), our results exhibit little evidence of volatility spillover between BTC and otherfinancial markets, such as Islamic stocks and Sukuk, at low frequencies.

It is evident that the BTC–Islamic equity/Sukuk/Titans 100 pair co-varies in the opposite direction in the lower-frequency scale, that is, over 256- to 512-day cycles during 2011–2013. However, the co-movement changes to the same direction in the high- frequency scales, that is, 128- to 256-day cycles, during the same period. Interestingly, we alsofind, from 2015 to 2017, a very small but significant island of coherency between BTC and Islamic equity/Sukuk/Titans 100 for frequencies of 32–64 days. The reason behind the relatively strong coherency between BTC and Islamic equity markets in the United States and Japan could be that the United States and Japan have shown a positive attitude toward virtual currencies and blockchain technology. In 2016, Japan le- gitimized virtual currencies by officially recognizing BTC and other cryptocurrencies as money. Our findings for Japan are consistent with those ofBouri et al. (2019a)that cryptocurrencies play an important role in diversifying risk in Asia-Pacific and Japan.

Most of the bans and new regulations to block the legitimacy of BTC started appearing after 2016, which is why we do not see strong coherency between BTC and Islamic stocks/Sukuk/Titans 100 after 2016. It is clear that negative co-movement exists mostly for lower frequencies (256–512 days), where BTC offers an opportunity for diversification. The results are in agreement with those of Dyhrberg (2016)andBouri et al. (2017a), in that isolation emerges in the short run, due to lesser or no co-movement between BTC and Islamic stock markets and Sukuk markets in the short term. Therefore, evidence of the long-term diversification benefits is not conclusive. Besides, our results support the argument that Islamic stocks can be considered a risk diversifier only for short periods, because, in the long run, the systematic risks of Islamic stocks converge with those of conventional markets (Sensoy, 2016). The results are also consistent with thefindings ofCorbet et al. (2018), in that cryptocurrencies are more connected in the long run than in the short-run.

5.2. Portfolio risk analysis

In this section, we study the systemic risk by analyzing the VaR for a portfolio assuming co-movement and no co-movement between BTC and Islamic equity/Sukuk/Islamic Titans 100, respectively. To do so, we calculate portfolio variances as the ratio of the VaR based on co-movement to the VaR based on no co-movement. This method allows us to detect the percentage of increase or decrease due to co-movement. A ratio equal to one suggests no effect of co-movement on the VaR, whereas a ratio of more (less) than one indicates co-movements have a higher (lower) impact on the VaR.

The results are reported inFig. 4. Panels A to D shows that the ratio is higher than one, implying that the co-movements between BTC and Islamic equity returns have a higher VaR across frequencies and time. The corresponding VaRs exist at all frequencies and are predominantly between 2010 and 2017. However, co-movements have differential impacts on portfolio risk over frequencies and time. The co-movements have more significant effects at low frequencies over the entire period. This result suggests that potential losses are more significant at low frequencies than at high frequencies (Mensi, 2018). Higher ratios are observed in the portfolio with the Islamic world, US, Europe, and Asia-Pacific indexes, and lower ratios are seen with the UK and Japan indexes. On the contrary, we find that co-movements have a lower VaR for the portfolio with the Islamic Canada index, except in 2016, when the co-movement impact is noted to be higher for the frequency band of 32–128 days. For the portfolio of BTC and Sukuk, we show that co-movement results in lower VaR across frequencies and over time, except during 2014–2016 for the frequency bands of 32–64, 64–128, and 128–256 days. For the portfolio of BTC and Islamic Titans 100, we also see that co-movements decrease the VaR during 2010–2013 and 2017–2018 for cycles of 32–64 days, while higher impacts of co-movements are noted during 2014–2016 for cycles of 32–64 days and 64–128 days.

Our results with VaR-based risk analysis suggest overall that the benefits of portfolio diversification with BTC and Islamic equity/

Sukuk/Titans 100 vary across time and frequencies. Co-movements increase portfolio risk for Islamic equity indexes at low fre- quencies, with exceptions such as the Islamic equity index returns of Canada and Titans 100, where co-movements decrease the portfolio risk. Co-movements with Sukuk also reduce portfolio risk. However, a common period from 2015 to 2016 for cycles of 32–64 and 64–128 days shows an increase in portfolio risk, even for co-movements with Sukuk and Titans 100. The period from 2015 to 2016 coincides with thefinancial turmoil in Russia, a crash in the commodity markets, the economic crisis in Brazil, and the stock market crash in China. Overall, our portfolio results are consistent with those ofMensi et al. (2018)and traditional portfolio management theory, in that portfolio risk rises with increases in co-movements between BTC and Islamic equity markets.

BTC offers a better hedging mechanism to diversify the risks of the Islamic equity markets of Canada and Titans 100 and Sukuk.

Our results also corroborate thefindings ofMensi (2019), where frequency and time-wise portfolio balancing and rebalancing offer better insurance against risks since diversification benefits change across time-frequency space. Our portfolio results are in line with those ofBouri et al. (2017b), in that BTC can serve as a useful diversifier; however, its hedging properties differ between horizons, due to change in co-movements with, for example, Islamic stock markets. Furthermore, the results are also in agreement with those of Sensoy (2016), who show that the systematic risks of Islamic equity markets vary over time. The increases in VaR at the low frequencies that correspond to long-term investors indicate that systematic risks arise more in the long run than in the short run.

We add to the argument ofSensoy (2016)that the systematic risks of Islamic equity and Sukuk markets and the sharing of this risk with cryptocurrency are not only time-variant but also frequency-dependent. The higher VaR, in the long run, is supported by the

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(caption on next page)

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habitat view of the theoretical argument that investors who select Islamic equity markets and cryptocurrencies as their preferred habitats for investment and inclusion in their portfolio for diversification are motivated by transaction costs, trading restrictions, or religious beliefs or lack information. Thus, over time, the habitat-based view of co-movement is supported, since a set of common information exists in the returns of Islamic equities and cryptocurrencies, which are held and traded by this specific subset of investors (Barberis et al., 2005). Investors’ constant search for a common factor between Islamic equity markets and cryptocurrencies determines the equilibrium prices in both markets and leads to integration in the long run (Hoque et al., 2016).

Furthermore, we also estimate a dynamic optimal hedge ratio between Islamic equity, equity, and BTC markets.Fig. 5displays the optimal hedge ratio considering a long position in Sharia equity and Sukuk markets and a short position in BTC. The optimal hedge ratio is estimated using a dynamic conditional correlation GARCH method. We notice that the optimal hedge ratio is not constant, becomes very high during 2010–2011, and fell sharply during 2015–2016. The period 2010–2011 and 2015–2016 overlap with the introduction of Bitcoin, 19,000 BTCs were lost ($ 5 million), respectively. However, during 2017 the hedge ratio again increased; this is on account of the price hike of BTC beyond 1000 mark. The optimal hedge ratio shows that BTC offers hedging and diversification benefits to Islamic equity and Sukuk markets. Nevertheless, as compared to Islamic equity, Sukuk has the lowest hedge ratio with BTC.

5.3. Robustness tests

We augment our analysis with robustness tests. More precisely, we split the robustness tests into two steps. First, we assess whether the correlations between BTC and the Islamic stock and Sukuk markets vary statistically within years and across different frequencies (investment horizons). Second, we examine the relation between BTC and the Islamic stock and Sukuk markets by employing the BC causality test. This BC causality approach allows us to quantify theflow of causality at various frequencies and to cross-check thefindings with the time-frequency domain approach.

Table 2reports the correlations between BTC and Islamic stocks and Sukuk returns across investment horizons of two days to more than 64 days. The last column shows the significance level of the analysis of variance (ANOVA) F-test to compare the hypothesis of homogeneity—that the correlations are constant within a particular year and across time horizons—against the alternative hy- pothesis, in which the correlations vary within a specific year and over time horizons. We apply maximal overlap discrete wavelet transform decomposition to the frequency correlations. This approach helps investors determine whether the risk correlations be- tween BTC and the Islamic stock market and Sukuk market are significantly different from zero. The results inTable 2confirm that the correlations are significantly different from zero across years and within time investment horizons. Also, the correlations increase as we move away from the shorter investment horizon of two days to a longer horizon of 64 days.

Moreover, upon comparing the degree of correlations across years and across horizons, wefind that the correlations increase significantly from 2010 to 2018, except for the Islamic stock market of Canada. This result indicates that the correlations evolve over the years and investment horizons (Al-Yahyaee et al., 2019). We document the highest number of negative correlations between BTC and the Islamic stock markets and Sukuk markets for an investment horizon of 32 days.

Furthermore,Fig. 6displays the frequency domain causality between BTC and the Islamic stock and Sukuk markets. The wavelet co-movements, which indicate one asset leading the other, do not, however, necessarily mean there is specific causality between the two (Dewandaru et al., 2017). To observe any causation, we need to employ the proper causality tests. In this context,Fig. 6shows no significant causality between BTC and the Islamic stock markets, except from BTC to the Islamic markets of Asia-Pacific and Japan and Sukuk in the short term. BTC is also found to cause Asia-Pacific Islamic stock markets in the long term. Unlike wavelet-based co- movement analysis, frequency domain causality only addresses frequencies. The results are consistent with the argument that the short-run behavior of causality is different from long-run causality (Bouri et al., 2019b). There are transitory causalityflows from BTC to Asia-Pacific and Japan and Sukuk markets; however, interdependency is found between BTC and the Asia-Pacific Islamic equity markets. BC causality confirms evidence from the time–frequency domain analysis in Section5.1that BTC is isolated from most of the Islamic stock markets and, hence, offers hedging opportunities.

6. Conclusion

Islamic stock markets, Sukuk, and cryptocurrency have been extremely popular since their creation. Co-movement between international markets is an open topic that has attracted the attention of portfolio managers, investors, academicians, and the press alike. This paper sheds light on co-movements and portfolio risk assessment, which not only vary over time but also change according to the investor’s horizon (i.e., short- and long-term investors), which is the frequency domain (Rua and Nunes, 2009). We use a cross wavelet spectrum, wavelet coherency, and wavelet-based VaR between BTC and Islamic asset returns from different regions, that is, the DJ Islamic world, US, UK, Europe, Asia-Pacific, Japan, and Canada indexes, as well as the Sukuk and Titans 100 indexes. We augment our analysis by robustness tests, including homogeneity and frequency domain causality tests.

Fig. 4. Wavelet VaR analysis.

Notes: Thisfigure shows the WTC of BTC with the DJ Islamic stock markets and Sukuk indexes. We label frequency and time on the vertical axis (from January 1, 2010, to June 30, 2018) and horizontal axis, respectively. Frequency is denoted in days. Lighter shades of color indicate greater coherence between BTC and the other index of the pair. The solid black lines are the boundaries isolating regions with a significance level of 5% or better. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article).

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Fig. 5. Optimal hedge ratio between Islamic equity/Sukuk and BTC.

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