Re-evaluating cryptocurrencies' contribution to portfolio diversification -- A portfolio analysis with special focus on German investors
RRe-evaluating cryptocurrencies’ contribution toportfolio diversification
A portfolio analysis with special focus on German investors (First Draft: June 12, 2020)
Tim Schmitz a, ∗ , Ingo Hoffmann a a Financial Services, Faculty of Business Administration and Economics,Heinrich Heine University Düsseldorf, 40225 Düsseldorf, Germany
Abstract
In this paper, we investigate whether mixing cryptocurrencies to a Germaninvestor portfolio improves portfolio diversification. We analyse this researchquestion by applying a (mean variance) portfolio analysis using a toolbox con-sisting of (i) the comparison of descriptive statistics, (ii) graphical methods and(iii) econometric spanning tests. In contrast to most of the former studies weuse a (broad) customized, Equally-Weighted Cryptocurrency Index (EWCI) tocapture the average development of a whole ex ante defined cryptocurrencyuniverse and to mitigate possible survivorship biases in the data. Accordingto Glas/Poddig (2018), this bias could have led to misleading results in somealready existing studies. We find that cryptocurrencies can improve portfoliodiversification in a few of the analyzed windows from our dataset (consistingof weekly observations from 2014-01-01 to 2019-05-31). However, we cannotconfirm this pattern as the normal case. By including cryptocurrencies in theirportfolios, investors predominantly cannot reach a significantly higher efficientfrontier. These results also hold, if the non-normality of cryptocurrency returnsis considered. Moreover, we control for changes of the results, if transactioncosts/illiquidities on the cryptocurrency market are additionally considered.
Keywords:
Capital Allocation, Portfolio Diversification, Spanning-TestsStepdown-Tests, Cryptocurrencies, Transaction Costs
JEL Classification:
C12, C13, C32, E22, G11.
ORCID IDs: ∗ Corresponding author. Tel.: +49 211 81-11515; Fax.: +49 211 81-15316
Email addresses:
[email protected] (Tim Schmitz),
[email protected] (IngoHoffmann) a r X i v : . [ q -f i n . S T ] A ug cknowledgement: We thank Coinmarketcap.com for generously providing thecryptocurrency time series data for our research. Moreover, very specialthanks go to Raymond Kan (University of Toronto) and Christoph J. Börner(Heinrich Heine University Düsseldorf), as well as to participants of the 2019Unicredit/ HypoVereinsbank (HVB) Doctoral Seminar in Hannover(Germany) for providing helpful advices.
Management Summary / Key Messages: • To capture the cryptocurrency market development as unbiased as possi-ble, we construct an Equally-Weighted Cryptocurrency Index (EWCI) ofthe whole cryptocurrency market at one reference date. • An equal weighting scheme of cryptocurrency returns reduces the expectedreturn of cryptocurrency investments (compared to the usual market capweighting schemes) • Consequence: in most of the analyzed periods cryptocurrencies do notplay a significant role in the optimal portfolio any more • The (remaining) diversification effects of cryptocurrencies do not neces-sarily fade away, if transaction cost and illiquidity issues are included inthe considerations
1. Introduction
In the aftermath of the subprime crisis, Nakamoto (2008) developed thetechnical foundations of Bitcoin as the first cryptocurrency in history. Accord-ing to the definition of He et al. (2016), cryptocurrencies are virtual currencies,which are additionally characterized by their convertibility to real goods andservices, their decentralized transaction networks and the use of cryptographictechniques.After the introduction of the Bitcoin, the cryptocurrency market grew fast,reaching a number of 2,216 cryptocurrencies (effective date: 2019-06-01) exist-ing on the market and a market capitalization (market cap) of 243.98 billioneuro (Coinmarketcap.com (2019)).Although cryptocurrencies are intended as substitutes for national (fiat) cur-rencies as alternative media of exchange, recent studies in the literature, such asYermack (2015) or Baur et al. (2018), find that cryptocurrencies do not act as aform of money because they do not fulfill the classical three functions of money(store of value, medium of exchange, unit of account). Instead, cryptocurrencies’high attention in the media and social networks in combination with their inno-vative technology and the expectation of a high return potential make investorsrather use them as (speculative) investment vehicles. This development led2o a rising speculative value in cryptocurrency prices ( → bubbles: Cheah/Fry(2015)) – especially in December 2017 – and the rise of several indirect formsof cryptocurrency investments (e.g. futures, certificates, funds, etc.) besides thetraditional direct investments in cryptocurrencies (via specialized, mostly un-regulated exchanges). Since december of 2017, where the Chicago MerchantileExchange (CME) and the Chicago Board Options Exchange (CBOE) as classical(regulated) exchanges started trading the first Bitcoin futures, the cryptocur-rency market was also opened for institutional investors (Corbet et al. (2018)).To put it more precisely, cryptocurrencies can also be classified as a stand-aloneasset class, because Krückeberg/Scholz (2018) find, that they meet the assetclass criteria defined by Sharpe (1992). Because the (emerging) asset class ofcryptocurrencies is currently not widely spread in – institutional and private– investors’ portfolios, it is useful to investigate their (possible) diversificationbenefits, if cryptocurrencies are added to an already (well) diversified investmentportfolio. For the means of our paper, we define diversification as the consciouscombination of different assets with varying risk-return-characteristics in theinvestors portfolio – aiming at a reduction of portfolio risk (Markowitz (1952)),as it is shown in the Portfolio Selection Theory by Markowitz (1952, 1959).Therefore, we analyse, whether there is a measurable improvement of portfoliodiversification, if N cryptocurrencies (called test assets hereafter) are mixed intoa reference portfolio of K exemplarily selected assets (called benchmark assetshereafter).In the contentually-related economic literature , we find different approachesto analyze those possible diversification benefits of cryptocurrencies.On the top level, we have to differentiate between two different literature strands:the analysis of diversification benefits on average (means: in normal times ) andin times of financial distress (means: in times of crises ) . Following the clas-sical Portfolio Selection Theory, where the average correlation is used as aninput factor in the portfolio optimization framework (Markowitz (1952, 1959)),our paper contributes to the analysis of the diversification benefits on average(first literature strand). Within this literature strand, the diversification bene-fits of the integration of cryptocurrencies in investor portfolios can be analyzedin three different ways • using an approach based on descriptive statistics , • using a graphical approach, • using a regression-based approach. Baur/Lucey (2010) use the term safe haven to describe assets which are uncorrelated ornegatively correlated to other assets or portfolios in times of crises. Building on their quantileregression approach (in combination with a GARCH (1,1) specification), which was originallyinvented for a save haven analysis in the context of gold investments, or using extensions likedynamic models (e.g. DCC-GARCH (1,1) models by Engle (2002), there is also a growingliterature with focus on a possible save haven status of cryptocurrencies (e.g. Bouri et al.(2017); Smales (2019); Shahzad et al. (2019); Urquhart/Zhang (2019)). first approach , the so-called descriptive-statistics-based approach , com-pares descriptive statistics of portfolios (such as optimal portfolio weights anddifferent return-, risk and performance measures) with and without cryptocur-rencies being mixed in the portfolio (e.g. used by Eisl et al. (2015)).The second approach , the so-called graphical approach , covers methods whichcompare one type of graphic in two situations – with and without the use ofcryptocurrencies in the investment opportunity set of the investors. One possi-ble comparison could be, whether the use of cryptocurrencies in the investors’portfolios leads to an upward shift of the efficient frontier or not (e.g. used byLee Kuo Chuen et al. (2017)). Such an upward shift of the efficient frontiermeans reaching a higher return for a given risk level.The third approach , the so-called regression-based approach , uses econometrictests to evaluate the contribution to portfolio diversification of different assetclasses. Spanning tests are some of the most important econometric tests fordiversification issues. Those spanning tests are linked to the graphical approachmentioned above and check, whether possible shifts of the efficient frontier arestatistically significant (Kan/Zhou (2012)).To create a preferably holistic approach, our analyis combines elements ofall the three approaches to analyse the research question from different pointsof view and to compare the different results. This combination allows us toevaluate, (i) to which extent cryptocurrencies should be included in the opti-mal portfolios (optimal portfolio weights), (ii) whether these optimal portfolioweights lead to a shift in the efficient frontier and (iii) whether this shift is sta-tistically significant.Independent of the approach used in their analysis, former studies – exceptfor Glas/Poddig (2018) – consistently find, that considering cryptocurrencies ininvestors’ portfolios leads to a significant upward shift of the efficient frontier,which is (sometimes) confirmed by spanning tests and (sometimes) by descrip-tive statistics (higher portfolio returns, higher portfolio performance). In themethodologically related literature diversification issues are mostly analysedusing a traditional Markowitz Mean-Variance-Framework (Markowitz (1952,1959)) – independent of the analyzed asset class. In the context of cryptocur-rencies, we identify different approaches for the analysis of diversification ef-fects. First, there are some existing studies, just like Wu/Pandey (2014) orEisl et al. (2015), which do not implement such a traditional Mean-Variance-Framework, but a Mean-Conditional-Value-at-Risk-Framework, as it is proposedby Rockafellar and Uryasev (2000) because of the non-normality of cryptocur-rency returns (Osterrieder et al. (2017)). On the other hand, there is also a hugenumber of studies, such as Borri (2019), Glas/Poddig (2018), Brauneis/Mestel(2019) and Liu (2019), which nevertheless stick to the (widely spread) Markowitzframework for simplicity reasons, e.g. because the scope of their analysis is onadditional restrictions (just as the impact of transaction costs or illiquidity is-sues).Focusing on the above-mentioned spanning tests, there are a lot of appli-cations in the literature to show the diversification impact of a specific asset4lasses. A famous application of spanning tests in the context of commodity in-vestments is given by Belousova/Dorfleitner (2012). Moreover, there is alreadysome literature on spanning tests in the cryptocurrency context (e.g. Anyfantakiet al. (2018), Brière et al. (2015), Chowdhury (2014), Glas/Poddig (2018), LeeKuo Chuen et al. (2017)). Except for Glas/Poddig (2018), all of these studiesalso find an additional diversification effect, if cryptocurrencies are consideredin an investor’s portfolio, which is consistent with the findings of the other ap-proaches mentioned above.With regard to our regional focus, we find that Glas/Poddig (2018) is theonly study which analyses diversification effects of cryptocurrencies for Germaninvestors. Most of the earlier studies dealing with cryptocurrencies’ contribu-tion to diversification (explicitly or implicitly) use a global or an US-investor’spoint of view (e.g. Anyfantaki et al. (2018); Lee Kuo Chuen et al. (2017); Brièreet al. (2015); Eisl et al. (2015); Wu/Pandey (2014)). But neither the investorsnor their financial market behavior are completely comparable in Germany andthe US because of cultural differences (e.g. different levels of risk aversion) andstructural differences, such as different regulatory frameworks, different status offinancial markets in the economy (bank-oriented financial systems in Germanyvs. financial-market-oriented financial system in the USA) and – as a conse-quence – different relationships to financial market investments in the contextprivate retirement provisions (Vitols (2005)). Thus, changing the geographicalfocus of the analysis is interesting and well justified in this context.From a more technical point of view, Glas/Poddig (2018) find, that the timeseries of nearly all existing studies on diversification in the cryptocurrency con-text are (possibly) biased (e.g. survivorship bias) because of some distortingrestrictions in their datasets causing misleading results. Consistently, they findthat investing in cryptocurrencies – in reality – does not lead to measurable di-versification advantages for investors. To prevent biased results, we follow theirtrend-setting approach and use a customized Equally-Weighted CryptocurrencyIndex (EWCI) to capture the average development of the cryptocurrency marketmore accurately.Moreover, Glas/Poddig (2018) state, that those results would become worse,if – in addition – the possibility for investors to lose their whole invested amount(because of dying cryptocurrencies) or considering liquidity restrictions (as theyare observable on the market) in the optimization framework would be consid-ered. While the latter effect was already analyzed by Borri (2019) from the A brief look at scores of the cultural dimensions by Hofstede (2001) for Germany andthe USA, which were more recently published in Hofstede et al. (2010), shows, that there ise.g. a higher degree of uncertainty avoidance in Germany than in the USA. Although thisterm is not completely congruent to the term of risk aversion, it is often interpreted as theapproximate inverse of "risk tolerance" (Frijns et al. (2013)) or the approximate inverse of"cultural appetite for risk" (Aggarwal et al. (2012)) in the literature, which both come closeto the definition of risk aversion. death ’ can only be a temporary phenomenon for somecryptocurrencies, because they can recover after a substantial revamp (systemupdates, new programming codes, etc.). As a consequence, even though a cer-tain coin is (temporarily) dead, they can still have a neglegible trading volumeon the market. This phenomenon is driven by investors who try to monetarizeparts of their initial investments on the one hand, and by investors who buythose coins because they bet on a future revocery (revamp) on the other hand.Feder et al. (2018) try to define a concrete rule for cryptocurrencies, to be clas-sified as ’ dead ’ and ’ recovered ’. Due to their definition, a cryptocurrency can beclassified as dead, if its average daily trading volume for a given month is loweror equal to 1% of its historical peak. On the other hand, a dead cryptocurrencyis classified as ’resurrected’, if this average daily trading volume reaches a valueof more or equal to 10% of its historical peak again (Feder et al. (2018)).To capture those effects the best as possible, we adjust the calculationmethodology of the cryptocurrency index (in comparison to other studies). Be-cause of the limited data availability, our study does not follow the methodologyof Feder et al. (2018), but uses a more simplified approach to identify temporaryinactiveness. Hence, in the following, a cryptocurrency is marked as temporarilyinactive , if we observe data gaps (with respective to price- and market cap data)in the time series of the respective coins at the rebalancing dates of the index.In a nutshell, this paper analyses, whether considering cryptocurrencies (rep-resented by an Equally-Weighted Cryptocurrency Index (EWCI)) in the invest-ment opportunity set of German investors leads to diversification benefits (anda higher portfolio efficiency) under the assumption of a classical mean-variance6ramework. The study combines three different analytical (non-econometric andeconometric) approaches, two different data aggregation levels (whole sample,subsamples) and two different scenarios (firstly without transaction costs andilliquidity issues and secondly with the consideration of those additional restric-tions).We find, that considering cryptocurrencies in the investment opportunityset does not lead to (significant) diversification benefits in most of the analysedwindows of our dataset (2014-01-01/2019-05-31). However, there were also someobservation windows, in which a significant diversification effects were measur-able. Moreover, the additional assumption of illiquidity issues and transactioncosts (especially on the cryptocurrency market) in the portfolio optimization didnot always worsen the diversification benefits of cryptocurrencies, as it mightbe expected at a first glance (e.g. by Glas/Poddig (2018)).The remainder of the paper is structured as follows: In section 2, we describethe methodological framework of our study – with special regard to the underly-ing portfolio optimization framework and the fundamentals of the implementedspanning tests. In section 3, we give information about the derivation of therespective dataset and the calculation of our customized cryptocurrency index.Section 4 shows the results of the different econometric and non-econometricapproaches and derives implications for German investors. In section 5, we ad-just our portfolio optimization framework to additionally control for the effectsof transaction costs and illiquidity considerations on the efficient frontier. Thelast section discusses the results and summarizes the key points.
2. Methodological Framework
The starting point to understand diversification is the Portfolio SelectionTheory by Markowitz (1952, 1959). He showed that the mix of assets with acorrelation coefficient ρ < R t = [ R t , R t ] as a transposed returnvector of the total returns of the K + N risky assets, where R t is a trans-posed K -vector containing the returns of the K benchmark assets, and R t isa transposed N -vector containing the returns of the N test assets (Kan/Zhou(2012)).In the next step, we can use this information to calculate vectors of theexpected returns µ and variance-covariance-matrices of this returns V . As aresult, we have µ = E[ R t ] ≡ (cid:20) µ µ (cid:21) (1)as an expected return vector of the dimension ( K + N ) ×
1, which consists of allexpected returns of the K considered benchmark assets µ and another vectorcontaining the expected returns of the N test assets µ . Moreover, we have V = Var[ R t ] ≡ (cid:20) V V V V (cid:21) (2)as the corresponding (non-singular) variance-covariance-matrix, which is seg-mented in the four sub-matrices V , V , V , V (block matrix). This blockmatrix contains the variances and covariances of the benchmark- and test assets– within ( V , V ) and between this groups ( V , V ).In this framework, we calculate two kinds of portfolios – the Global Mini-mum Variance Portfolio (GMVP), as it is proposed by Markowitz (1952, 1959),and the Tangency Portfolio (TP), which is the result of a maximization of theSharpe-Ratio by Sharpe (1966, 1994) (Chapados (2011)). In this first stage ofthe study, we additionally assume, that there are no transaction costs and thereare no liquidity issues for the cryptocurrencies to be considered in the sampleof the optimization problem.In the next step, we follow the notations by e.g. Roncalli (2011) and Chapa-dos (2011) and define the investors’ optimization problems using the Lagrangeformalismsmax ω L TP = max ω (cid:20) ω µ − r F √ ω V ω + λ (1 − ω ) (cid:21) (3)min ω L GMVP = min ω (cid:20) ω V ω + λ (1 − ω ) (cid:21) (4)8ith L as Lagrange function, ω as portfolio weights vector, µ as expected re-turn vector, r F as risk-free interest rate (assumption: r F = 0%), as vector,wherby every element is equal to 1, and λ as the typical Lagrange multiplier.Under the consideration of a budget restriction (1 − ω = 0) in both cases, wemaximize the portfolio’s Sharpe Ratio (see Eq. (5)) for the calculation of thetangency portfolio and minimize the variance of the portfolio returns for thecalculation of the GMVP (see Eq. (6)). Please note, that we apply the standardversion of the optimization model without the introduction of additional shortsale constraints (e.g. long-only constraints) for consistency reasons, because thisunrestricted framework will also be assumed for the spanning tests following in alater part. This optimization framework is frequently used in the portfolio op-timization and spanning test literature (e.g. Glas/Poddig (2018)). Nevertheless,as far as it is possible, we additionally calculate the results for the introductionof a long-only assumption ( ω i ≥ ∀ i ) as a robustness check.Under the assumption of an unconstrained portfolio, it is possible to cal-culate the optimal portfolio analytically by solving the respective optimizationproblems (Eq. (3) and (4)). This calculation generates the (optimal) weightsvectors ω TP for the tangency portfolio (Eq. (5) with r F = 0%) and ω GMVP forthe Global Minimum Variance Portfolio (GMVP) (Eq. (6)): ω TP = V − µ K + N V − µ (5) ω GMVP = V − K + N K + N V − K + N (6)where K + N is a K + N vector of ones. For the means of our analysis, we use the abovementioned portfolio opti-mization framework (cf. Sec. 2.1.1) and calculate the optimal portfolio weightsfor the respective asset classes (with and without the consideration cryptocur-rencies) for two cases: • Case A:
The first case should be more general, which means that it con-tains only one optimization for the whole dataset (Observation period This assumption might not be problematic, because e.g. the introduction of Bitcoin futurecontracts makes it possible to short sell Bitcoins on the CME and CBOE: e.g. investors cansell the promise to deliver units of Bitcoins at a future date for a previously defined priceand speculate, that they can buy those units for an even lower price to make profits (Hale etal. (2018)). We assume, that those future contracts will be traded on traditional (regulated)exchanges for a growing number of cryptocurrencies in the future. Nevertheless, a parallelapplication of a (more feasible) long-only portfolio might mitigate possible objections withregard to the final results. Obs = 65 months). Subsequently, we compute only one respective (opti-mal) weight vector for the situations with and without the considerationof cryptocurrencies. • Case B:
The second case should control for the changing data base (e.g.caused by market trends) more in detail. In this course, we stick to thesetting of Eisl et al. (2015) and limit the data observation period to thefirst year of observations in the dataset. In this first year, the investors areassumed to allocate their capital between all benchmark assets in equalportions. After one year of observation time, the portfolio is constructedon the basis of this observed data (Observation period T Obs = 12 months).Hence, the investors have to decide, whether they want to include the testassets in their existing portfolio of benchmark assets. Afterwards, we rollthis one-year-window through-out the dataset in monthly steps to assumea monthly rebalancing of the portfolio considering the last year of ob-servations in the optimization. The resulting optimal portfolio allocation(based on the data of the last year) is implemented for the whole monthfollowing on that observation window. Because we assume the same ini-tial portfolio with and without cryptocurrencies and no transaction costsin this stage, the initial portfolio allocation is not relevant for the advan-tageousness of one alternative (with/without cryptocurrencies) until weintroduce further assumptions in Sec. 5.1 concerning transaction costs.After the calculation of the respective optimal portfolios for both cases, wefinally focus on Case B more in detail and give the following portfolio metricsfor the resulting portfolio returns: mean return, standard deviation, CVaR,maximum drawdown, Sharpe Ratio and – at last – the resulting portfolio valueat the end of the planning horizon (end value), if 100 .
00 EUR are invested inthis (frequently rebalanced) portfolio at the portfolio construction date. Thisbacktesting procedure is not applicable to Case A, because the whole dataset isalready used as an observation period.At this stage of our analysis, we do not apply an additional portfolio smooth-ing technique (such as the usual 3-month EWMA smoothing used by Eisl et al.(2015)) for consistency reasons, because we refer to the resulting optimal port-folios later on in the context of the spanning tests. Those techniques are usedto prevent extreme changes of the portfolio weights (Eisl et al. (2015)), whichoccur in the course of portfolio rebalancing, because Markowitz portfolios tendto have extreme (positive or negative) portfolio weights (Härdle et al. (2018)).Applying such a technique would lead to biased results, so that the results of theportfolio-optmization-approach and the regression-based-approach would not becomparable any more. However, by introducing additional constraints (for thejoint capturing of transaction cost and liquidity issues) in Sec. 5.1, the shift inthe (optimal) portfolio weights will be smoothed indirectly by an adjustment ofthe respective optimization frameworks in a later stage of the analysis.10 .2. Econometric Approach: Spanning Tests2.2.1. Mean-Variance-Spanning
Besides the comparison of portfolio statistics with and without mixing N cryptocurrencies (test assets) in a portfolio of K benchmark assets, cf. Sec.2.1.1, it is also possible to test, whether the consideration of those N test assetsleads to a (significant) shift of the investors’ efficient frontier. Therefore, weuse the econometric approach of spanning tests as a suitable measure for those(possible) shifts. Note, that the usual spanning tests, which are also selectedfor this paper, do not allow for long-only restrictions, so that we refer to theunconstrained optimization framework in Sec. 2.1.1 in this stage of the analysis.Mathematically, the underlying concept of spanning , which is part of the lin-ear algebra, is fulfilled, if a vector can be replicated by a linear combination ofother vectors of the same vector space (Mac Lane/Birkhoff (1999)). This linearcombination is called a span . With regard to this mathematical definition, wecan use the term spanning , if the same efficient frontier (as a linear combina-tion of the single assets’ risks and returns) results – independent of the fact,whether investors include the test assets in their portfolio (Huberman/Kandel(1987); Kan/Zhou (2012)). Consequently, every efficient portfolio (and also theoptimal portfolio) will remain the same, even if there are new investment op-portunities available in the investment opportunity set. In other words: the testassets added to the investment opportunity set are not relevant for the construc-tion of an efficient portfolio (means: zero portfolio weights of the test assets), sothat they do not have a significant contribution to portfolio diversification. Butif we observe a shift in the efficient frontier after the test assets are included inthe investment opportunity set, there is no spanning any more.The first group of spanning tests, the so-called mean variance spanning testsbuilding on the work of Huberman/Kandel (1987), assume a mean varianceportfolio optimization framework, as it is defined in Sec. 2.1.1. Moreover, theyare the foundation of different extensions and practical applications in the lit-erature (cf. Kan/Zhou (2012) and others for an exemplary overview). Theirtheoretical foundation will be presented in the notation of Kan/Zhou (2012)afterwards.The foundation of the mean variance spanning tests is the (linear) regressionmodel R t = α + βR t + (cid:15) t , (7)where the test assets’ returns R t are regressed on the returns of the bench-mark assets R t . In this context, we assume that the requirements E[ (cid:15) t ] = N (unbiasedness) and E[ (cid:15) t R t ] = N × K (no endogeneity) hold. The intuition ofthe linear regression is, that the test asset returns should be generated by alinear combination of the benchmark asset returns. Moreover, we can rewritethe regression model in Eq. (7) as α = µ − βµ and define the N × K matrix β – in accordance with its Ordinary Least Squares solution – as β = V V − .11n addition we introduce the vector δ with δ = N − β K .In sum, by using a linear regression model, we assume a linear relation-ship between benchmark and test assets in a first step. In other words, weassume that the test assets’ returns are replicable by a linear combination ofthe benchmark assets’ returns (spanning). In a second step, we try to falsifythis relationship using different hypothesis tests, which check the fit of thisassumption. Therefore, we formulate the null hypothesis H : α = N , δ = N , (8)which simultaneously checks whether the test assets have a zero portfolio-weightin the tangency portfolio (first condition) and in the GMVP (second condition).In other words, the null hypothesis checks whether the test assets are irrele-vant in the optimal portfolio. In this context it is important to remind that,if we know two portfolios on the efficient frontier (here: TP, GMVP), we canconstruct the whole efficient frontier using linear combinations (Glas/Poddig(2018)). Hence, if the null hypothesis holds for both portfolios, we can concludethat it will also hold for the other portfolios on the efficient frontier.For a better understanding of the null hypothesis, it is necessary to derivethe null hypothesis from the classical Lagrange optimization formalism for thetangency portfolio in Eq. (3) and for the GMVP in Eq. (4). Solving these opti-mization problems generates the (optimal) weights vectors ω TP for the tangencyportfolio in Eq. (5) and ω GMVP for the GMVP in Eq. (6).Inverting the covariance matrix Eq. (2) by using the block matrix inversionformula and substituting β = V V − (= Ordinary Least Squares solution)and Σ = V − V V − V (= N × N Schur complement) leads to V − = " V − + β Σ − β − β Σ − − Σ − β Σ − , (9)which is used in the work of Kan/Zhou (2012) (Lemma 2 therein).We multiply the weight vectors of Eq. (5) and (6) by Q = [ N × K , I N × N ](with I N × N is the identity matrix) to get a vector only consisting of the testassets’ weightings. Simultaneously, we insert V − as defined in Eq. (9), so thatwe have Qω TP = QV − µ K + N V − µ = [ − Σ − β , Σ − ] [ µ , µ ] K + N V − µ = Σ − ( µ − βµ ) K + N V − µ = Σ − α K + N V − µ (10)12 ω GMVP = QV − K + N K + N V − K + N = [ − Σ − β , Σ − ] [ K , N ] K + N V − K + N = Σ − ( N − β K ) K + N V − K + N = Σ − δ K + N V − K + N (11)Now we can show that α = N and δ = N is a signal for zero weights inthe test assets (spanning), as we assumed in Eq. (8): α = N → Qω TP = N (12) δ = N → Qω GMVP = N (13)To estimate the regression model, we additionally assume that α and β areconstant over time, so that we can re-write Eq. (7) R t = α + R t β + (cid:15) t , with t = 1 , , . . . , T (14)or in a simple matrix notation as Y = XB + E (15)where Y is a T × N matrix containing the test asset returns R t . The T × ( K +1)matrix X contains rows, which are formatted as follows: [1 , R t ]. The matrix B = [ α , β ] pools the regression parameters α and β in one joint matrix withdimension ( K + 1) × N . The T × N matrix E contains rows, which containall the error terms (cid:15) t of the different time series regressions from Eq. (14) –one regression for each test asset. At least, it is assumed that the condition T ≥ K + N + 1 is fulfilled and that the matrix ( X X ) is non-singular. Bothconditions are fulfilled in what follows.For the estimation of the regression model, we calculateˆ B = [ ˆ α , ˆ β ] = ( X X ) − ( X Y ) (16)ˆ Σ = 1 T ( Y − X ˆ B ) ( Y − X ˆ B ) (17)as the estimators of the matrix B and the covariance-matrix Σ , respectively. Ifthere is an additional matrix Θ = [ α , δ ] , both requirements ( α = N , δ = N )of the null hypothesis Eq. (8) can be tested simultaneously, if the null hypothesisis rewritten as H : Θ = × N . Since the matrix Θ can be calculated as Θ = AB + C using the (projection) matrices A = (cid:20) K − K (cid:21) and C = (cid:20) N N (cid:21) (18)13he maximum likelihood estimator (MLE) of the matrix Θ would be ˆ Θ =[ ˆ α , ˆ δ ] = A ˆ B + C . For the calculation of the test statistics, we need the addi-tional 2 × G and H withˆ G = T A ( X X ) − A = " µ ˆ V − ˆ µ ˆ µ ˆ V − K ˆ µ ˆ V − K K ˆ V − K (19)ˆ H = ˆ Θ ˆ Σ − ˆ Θ = " ˆ α ˆ Σ − ˆ α ˆ α ˆ Σ − ˆ δ ˆ α ˆ Σ − ˆ δ ˆ δ ˆ Σ − ˆ δ . (20)If λ and λ denote the eigenvalues of the matrix ˆ H ˆ G − (with λ ≥ λ ≥ LR ), the Wald Test ( W ) andthe Lagrange Multiplier Test ( LM ) are LR = T P i =1 ln(1 + λ i ) A ∼ X N (21) W = T ( λ + λ ) A ∼ X N (22) LM = T P i =1 λ i λ i A ∼ X N . (23)Despite the fact of equal asymptotic distributions and the existence of similartest statistics, Berndt/Savin (1977) and Breusch (1979) show that there is anorder in the result of the different test statistics W ≥ LR ≥ LM , meaningthat (under asymptotic distributions) there is the possibility, that those testsprovide conflicting results: while the Wald test faster favors a rejection, theLM test longer favors the acceptance of the null hypothesis. Because noneof the beforementioned tests is dominant with regard to its predictive power,we decide to follow Belousova/Dorfleitner (2012) and perform all three testssimultaneously to achieve more reliable results. The above-mentioned Mean-Variance-Spanning-Tests (as MLE-based tests)assume that the returns of the single assets in the portfolio follow a normaldistribution and that the error terms in equation (14) are homoscedastic (Be-lousova/Dorfleitner (2012)). But, as we will show in the descriptive statisticsof our data sample (cf. Sec. 3.2), cryptocurrency returns do not (necessarily)follow such a normal distribution, so that Mean-Variance-Spanning-Tests maybe inefficient in that context. Moreover, the multivariate Ljung-Box test for This test, which is discussed more in detail by Tsay (2014), is a multivariate version ofthe univariate version by Ljung/Box (1978). and leads to highly significant results ( p ≤ . (cid:15) t ) exhibit indications of conditionalheteroscedasticity, which leads to the problem, that the test statistics of the for-mer tests (Eq. (21)-(23)) do not follow a X N distribution any more.But spanning tests are also applicable outside the mean-variance-framework– especially in the context of non-normal returns and conditional heteroscedas-ticity. As a first example, we refer to Ferson et al. (1993) (FFE), who proposea specialized spanning test, the so called GMM-based Wald-test (also: GMM-Wald test) , which is not dependent of the underlying distribution (Glas/Poddig(2018)). Assuming the stationarity of R t and R t (which is fulfilled for ourdataset ), Ferson et al. (1993) test H by calculating a similar test statisticcompared to the Wald test used for the Mean-Variance-framework before (cf.Eq. (22) and Kan/Zhou (2012)). The main difference is, that the GMM-Waldtest builds on a different regression model, a so-called Generalized Method ofMoments (GMM) estimation, as it is proposed by Hansen (1982). Those GMMestimations are dependent of the formulation of the different moment conditionsof the regression model (Kan/Zhou (2012)).Besides this regression approach , which is represented by the FFE spanningtest, there is also another kind of GMM-Wald test used in the literature, whichwas (among others) proposed by Bekaert/Urias (1996) (BU). In contrast to theFFE spanning test introduced before, the BU spanning test does not use re-gression coefficients (like FFE), but statistical discount factors (SDF) based onthe work of Hansen/Jagannathan (1991) instead. Thus, the BU approach isrefered to the literature strand of the so-called SDF approaches (e.g. like Ferson(1995); DeSantis (1993)). The BU spanning test is available in two differentversions (i) with adjustment of
Errors in Variables (EIV) and (ii) without thisadjustment. For a more precise estimation of the model parameters, we rely onthe alternative with EIV adjustment in addition to the FFE spanning test inour study.The resulting test statistics of both GMM-Wald test approaches (FFE, BUwith EIV adjustment), which we use but are not introduced here in detail, are(asymptotically) χ N distributed. We do not focus on a deeper methodologicalfoundation here, but refer to Ferson et al. (1993), Hansen (1982), Bekaert/Urias(1996), Hansen/Jagannathan (1991) and especially Kan/Zhou (2012) for a moredetailed description of the necessary fundamentals of both approaches and thenecessary theoretical fundamentals. There is no need to additionally use a GMM-based version of the LR- and LM- test,because Newey/West (1987) show, that these tests have the same form as the GMM-Waldtest (cf. Kan/Zhou (2012) for a more detailed discussion). We used an Augmented Dickey Fuller (ADF) test by Dickey/Fuller (1979) to check thestationarity of our return time series. As a result, the p -value for every time series was p ≤ .
01, so that the assumption of stationarity has not to be rejected in this use case.
All the formerly introduced spanning tests are joint test, which test bothconditions of the null hypothesis Eq. (8) simultaneously. An important short-coming of these tests is that the estimation ˆ δ can be estimated more preciselythan ˆ α which influences the weighting of both conditions of the null hypothesisin the test results (Kan/Zhou (2012)). As a consequence, it might be possi-ble, that the spanning tests’ results indicate significant changes in the tangencyportfolio, even though there are no significant changes in reality. On the otherhand, it is possible that the spanning tests do not lead to significant results withregard to the GMVP, even though, there are significant changes in reality.Subsequently, we follow Kan/Zhou (2012), implementing an additional step-down procedure as an additional robustness check, which allow us to control forthe acceptance or rejection of both conditions seperately. In other words, thisprocedure allows us to account for different diversification impacts for investorsdepending on their optimization strategy (the global minimum variance portfo-lio, tangency portfolio).From the methodological point of view, those stepdown tests apply twostages of F -tests: In the first stage we use an F -test (so called F -test) to check,whether the condition α = N can be accepted. Thus, we compute the teststatistics F = T − K − NN ! | ¯ Σ || ˆ Σ | − ! ∼ F N,T − K − N (24)with ˆ Σ denoting the unconstrained estimate of Σ and ¯ Σ denoting the respectiveconstrained estimate of Σ , where the constraint is defined as α = N .In the second stage , we run an additional F -test (so called F -test) to check,whether the condition δ = N can be accepted, if the first condition α = N holds. In this context significant results for the F -test indicate, that we have asignificant change in the tangency portfolio, whereas significant result in the F test indicate a significant change in the GMVP. The respective test statistics is F = T − K − N + 1 N ! | ˜ Σ || ¯ Σ | − ! ∼ F N,T − K − N +1 (25)with ˜ Σ as the constrained estimate of Σ by imposing both constraints of (i) α = N and (ii) δ = N . In what follows, we denote with ξ the significance level16f the F -test and ξ denotes the respective significance level of the F -test. Un-der the step-down procedure, the spanning hypothesis has to be accepted, if bothtests lead to significant results. Therefore, the significance level of this step-downtest (in general) can be calculated as ξ joint = 1 − (1 − ξ )(1 − ξ ) = ξ + ξ − ξ ξ .In the further analyzes, the results of this stepdown-tests give us a deeperinsight in the reasons, why a joint spanning test leads to significant or insignifi-cant results. Moreover, it is possible to use different significance levels ( ξ = ξ )for the different stages, if the investors put more focus on a certain optimizationstrategy (compared to the other). Kan/Zhou (2012) propose to use this instru-ment to choose a ξ higher than ξ to capture, that it is more difficult to detectthe significance of F . Therefore, we set the a significance level of ξ = 0 .
10 for F and ξ = 0 .
05 for F (and all the other spanning tests mentioned before) assuitable significance levels. The conception of the spanning test analysis follows the conception of theportfolio optimization analysis. This means: before we can run those before-mentioned spanning tests, we need to differentiate two different cases (consis-tent to the portfolio optimization approach), which are both relevant for ourmethodology: • Case A:
The first case should be – again – more general, which meansthat it contains only one optimization for the whole dataset (Observationperiod T Obs = 65 months). Subsequently, we compute one efficient frontierfor each case – one with cryptocurrencies, one without – and also oneregression result for each applied spanning test. In this case, the resultscan be easily reported in detail afterwards. • Case B:
The analysis should control for the changing data base (e.g.caused by market trends), as it was already done in the non-econometricanalysis above. Thus, the spanning tests need to be re-run every monthbased on the updated observation window (Observation period T Obs = 12months), which is consistent to the setting in Sec. 2.1.1. In the latter case,we have 54 efficient frontiers and 54 corresponding regression results, sothat – for simplicity – we only give the share of the significant resultscompared all results afterwards. This kind of subsample analysis wasalready implemented beyond the specialized cryptocurrency literature, e.g.by Kan/Zhou (2012), but in a less granular way.For the means of our study, we decide to compute the results for both of thebeforementioned cases (A and B). We use case A to firstly demonstrate thefunctioning of the applied spanning tests. By additionally focusing on a moregranular analysis (as in case B), we will get a more detailed insight in the di-versification effects of crpyoturrencies than former studies (such as Glas/Poddig(2018)), which only report aggregated results (as in case A). Furthermore, caseB allows for rebalancing, as we did in the non-econometric analysis before.17 . Dataset
The dataset used in this study consists of historical weekly total return in-dex data of different asset classes, which are relevant from a German investors’perspective, from 2014-01-01 to 2019-05-31. A weekly data frequency, like e.g.Wu/Pandey (2014) already used for their analysis, has the advantage, not tobe prone to weekday effects (less noisy) and to exclude weekends, because onlycryptocurrency data is available for weekends (Gibbons/Hess (1981); Yermack(2015)).Therefore, as a first step, we need to define an exemplary benchmark port-folio for German investors without cryptocurrencies ( benchmark assets only).On the superordinate level, we start with the asset class categories Horn/Oehler(2019) and Oehler/Horn (2019) identified in their analysis of German house-holds’ balances: stocks, cash equivalents, bonds, real estate and luxury goods.Because of the heterogeneity of these different categories, we use a more gran-ular classification. We subsume money market investments and internationalcurrencies under the cash equivalents. The category bonds is divided in the as-set classes sovereign bonds, covered bonds and corporate bonds. Finally, for thedefinition of the category of luxury goods, we do not only refer to luxury goodsin the narrow sense (such as jewellery, paintings, oldtimer-cars, etc.), but alsoto commodities (especially precious metals) because of their use in jewellery,fashion (haute couture), for collectors’ coins, etc.We identify benchmark indices for all the benchmark asset classes, which areshown in Table 1 below. All the relevant indices are provided by Bloomberg,Thomson Reuters Eikon and the Frankfurt Stock Exchange (see Appendix for adetailed line-up of the indices and the related data sources). Besides the alreadyexisting benchmark indices, we also use an own (customized) currency basketindex based on exchange rate data provided by Bloomberg. The customized equally-weighted currency basket is an index, which tracks the develop-ment of the Euro exchange rates of important international currencies: US-Dollar (USD),Swiss Franc (CHF), Japanese Yen (JPY), Australian Dollar (AUD), New Zealand Dollar(NZD), Canadian Dollar (CAD), Norwegian Krone (NOK), Danish Krone (DKK), SwedishKrona (SEK), British Pound (GBP), Turkish Lira (TRY), South African Rand (ZAR), RussianRuble (RUB), Polish Zloty (PLN), Mexican Peso (MXN), Indian Rupee (INR) and ChineseYuan (CNY). sset Class IndexStocks Stoxx Europe 600 TRMoney Market iBoxx EUR Jumbo TR 1-3Currencies (FX) Equally-weighted Currency Index (customized)Sovereign Bonds iBoxx Euro Eurozone Sovereign Overall TRCovered Bonds iBoxx Euro Covered TRCorporate Bonds iBoxx Euro Liquid Corporates Diversified TRReal Estate RX REIT Performance IndexLuxury Goods Solactive Luxury and Lifestyle Index TRCommodities / Gold GSCI Commodity Index TR Table
1: Dataset (Benchmark-Assets).
This index selection deliberately ignores, that a global diversification couldbe more rational in the context of asset allocation. To capture the investmentbehavior of German investors more realisticly, the chosen indices mostly havea European focus, which is a slightly different perspective compared to otherstudies in the literature. This index selection does not only reflect the typical,significant home bias of German investors’ portfolios, as it is reported in the re-lated literature (e.g. by Baltzer et al. (2015); Horn/Oehler (2019); Oehler et al.(2007, 2008)). Instead, we refer to Balli et al. (2010), who identify, that this orig-inal home bias already became a euro bias in the course of (European) financialintegration. Thus, we follow this approach and implement mostly European in-dices – except for commodities, luxury goods, currencies and real estate. Whilethe first three asset classes are traded on global markets, we propose that forreal estate investments, we should restrict the focus on Germany. Most of thehouseholds own real estate properties in their own environment. As an example,consider owner-occupied real estate, (rented-out) heritages or memberships inGerman housing cooperatives.For the further analysis, we follow Horn/Oehler (2019) and assume, thatinvestors use exchange-traded funds (ETFs) or certificates to easily and cheaplytrace the development of those selected indices without the necessity to buy allthe index constituents as individual direct investments.
Besides those benchmark assets, we also need to define the cryptocurrenciesunder study as test assets. In the related literature, there are three differentapproaches to capture the cryptocurrency market: • First Approach: Individually-Selected Cryptocurrency Portfo-lios
The first approach to capture the asset class of cryptocurrencies is theindividual definition of a customized cryptocurrency portfolio. Especiallyearly studies on diversification effects of cryptocurrencies used a specialcase of this approach – a Bitcoin-only Approach (e.g. Wu/Pandey (2014)or Eisl et al. (2015)). The idea of those studies is the use of Bitcoin as a19epresentative cryptocurrency for the whole market because of its domi-nant market position, its superior level of awareness and the biggest datahistory available for all cryptocurrencies.Besides the Bitcoin-only approach, there are also studies, which slightlyexpand the size of the cryptocurrency universe under study, but still selectthose analyzed cryptocurrencies individually, e.g. a certain number of themost popular ones such as Borri (2019). But we interpret the results ofElBahrawy et al. (2018), Glas/Poddig (2018), Glas (2019) and Sovbetov(2018) in a way, that there are remarkable differences of the cryptocur-rency performance – probably driven by their position in the marketcapranking (e.g. because of their higher visibility, a higher trading volume,the access of institutional investors, etc.).Therefore, the individual selection of a customized cryptocurrency port-folio would lead to a selection bias in the data in both cases, so that weneed to reject this approach. • Second Approach: Market-Cap-Weighted Cryptocurrency Index
More recent studies, such as Trimborn et al. (2018b), use Market-Cap-weighted cryptocurrency indices to capture a broader, more representativesample of the cryptocurrency market. One index, which is mostly usedin this context, is the so called CRIX index developed by Trimborn etal. (2018a), covering a group of the most important cryptocurrencies withregard to their market cap. Market-cap-weighting is a usual weightingscheme which is also implemented in other well-established indices, suchas the stock indices DAX, EUROSTOXX50, S&P500 and NASDAQ. • Third Approach: Equally-Weighted Cryptocurrency Index
Glas/Poddig (2018) criticize former approaches and add a third approachusing a customized equally-weighted cryptocurrency index. They argue,that former indices – related to the first approaches – do not use data onthe whole cryptocurrency market, but too small samples of the crypto-universe only including the biggest (= large-cap) cryptocurrencies withregard to their marketcap. But this crypto-universe is changing fast,so that there is still no consistent data base existing. In this context,the development of small (possibly dying) cryptocurrencies is ignored bythe development of the index, which leads to a survivorship bias in thedata. Moreover, the index development is driven by the cryptocurren-cies with the biggest market capitalization – especially by the Bitcoin,which has shares of around 50 −
90% of the total market cap during theobservation window. But if we assume, that the development of the cryp-tocurrencies returns depends on their market cap, there would be anotherbias in the index data, if this problematic weighting scheme would stillbe used. More general, the European regulations of UCITs (containing:exchange-traded funds (ETFs)) (cf. Sec. XIII. No. 49 in ESMA (2014), e.g.20§ 207-209 KAGB ) and Alternative Investment Funds (AIF) (e.g. § 221KAGB) consistently forbid such high weightings of individual assets tosecure diversification within the portfolios of those investment products.This fact would make those indices not be suitable for e.g. cryptocurrencyETFs. In contrast, the equal weighting scheme of Glas/Poddig (2018)captures the average development of all included cryptocurrencies. Onthe other hand, this weighting scheme ignores the smaller market dephtsof those cryptocurrencies with smaller market caps. As a consequence,(especially institutional) investors could have problems to realize thoseequally-weighted cryptocurrency portfolios – especially with rising invest-ment volumes.In this study, we follow Glas/Poddig (2018) (Third Approach) and implementa (customized) Equally-Weighted Cryptocurency Index (EWCI), which tries tocapture the average development the whole cryptocurrency market at the ref-erence date 2014-01-01. To mitigate those liquidity issues mentioned above, wewill add an adjusted portfolio optimization approach after the regular analysis,which will also capture transaction costs and market liquidity in a combinedapproach (see Sec. 5.1). For index creation, we now need to define a suitablecryptocurrency universe.Our cryptocurrency universe consists of N = 66 cryptocurrencies listed inthe CoinMarketCap Cryptocurrency Market Cap Ranking as at 2014-01-01 (cf.Tab. 2). For consistency reasons, we consider weekly closing prices for eachcryptocurrency ( T = 283 weeks), as we also did for the benchmark assets.Contrary to most of the other existing approaches, such as the CRIX byTrimborn et al. (2018a), the EWCI index tries to capture the market as broadas possible to prevent survivorship biases (Glas/Poddig (2018)), not just a small(allegedly) representative segment. On the other hand the use of a closed cryp-tocurrency universe has the disadvantage, that some of the included cryptocur-rencies become temporarily inactive or even so-called dead coins, which leads The Kapitalanlagegesetzbuch (KAGB) is a German capital investment law implement-ing European directives for both UCITS (as codified in Directive 2009/65/EC) and AIF (ascodified in the Directive 2011/61/EU). In this context ElBahrawy et al. (2018) use a Wright-Fisher model of neutral evolution toanalyse the patterns of the cryptocurrency market. They state, that the (technical) similarityof the cryptocurrencies under study and the randomness of the next generation of dead-coins leads to the absence of a logical order of all the existing cryptocurrencies (no "selectiveadvantage" ) to some extent, because there is no consensus between the investors, which cryp-tocurrency could survive the cryptocurrency competition. If investors are not able to bringthe investment alternatives into a logical order, an 1/N weighting would be obvious becauseof Laplace criterion Börner et al. (2020)). We do not follow this argumentation in total, butrather partly, because we are of the opinion, that there are different criteria, such as the qualityof the whitepaper, or the degree of innovation (compared to existing cryptocurrencies), whichmake investors at least be able to differentiate between serious (emerging) cryptocurrenciesand e.g. junk coins or parody coins. Our motivation to use the equal weighting scheme is tocapture the average market development, because the research question is not only limited tolarge-cap cryptocurrencies, but to cryptocurrencies in general.
21o the necessity of additional index creation rules in this context to challengethose additional hurdles. As a reaction, the created EWCI index uses a flexibleindex size (such as Trimborn et al. (2018a), which is adjusted at the beginningof each month in steps of five (or even multiples).
Table
2: Creation of the Dataset (Test-Assets), Data Source: CoinMarketCap.
At a rebalancing date, we count the members of the active cryptocurrencyuniverse in this point of time to compute the new index size ( N Ind ), which is theclosest multiple of 5 and not bigger than the size of the active cryptocurrencyuniverse ( N Act ). To mitigate liquidity issues, we rank all the active cryptocur-rencies by their market capitalization and choose those N Ind cryptocurrencieswith the largest market capitalization. Now, if a current index constituent be-comes inactive during the month, we follow Trimborn et al. (2018a) and usethe Last-Observation-Carried-Forward (LOCF) method until the active cryp-tocurrency universe is updated again. The LOCF method is a frequently usedtechnique for index calculation in practise, especially with regard to illiquid as-22ets or the consideration of holidays (Brown (1994); Morgan Stanley CapitalInternational (2018); STOXX (2018)).At those rebalancing dates, we use additional normalization factors to en-sure, that the index value at this date is the same for the old and the new indexcomposition. This rebalancing technique is frequently used in index calculation(cf. exemplary Trimborn et al. (2018a) and CCI30.com (2020)).Note, we do not use further data cleaning techniques in the sense of Ince/Porter(2006), because there are (still) no such techniques available for cryptocurrencies(as already concluded by Glas/Poddig (2018)).
After we defined both benchmark assets and test assets as subsets of the finaldataset, we now calculate log returns r i,t = log h P i,t P i,t − i for this whole dataset.As a first impression of the result, we report descriptive statistics (cf. Tab. 3).According to these descriptive statistics (cf. Tab. 3), the sampling and weight-ing of cryptocurrencies affects the measured performance of the cryptocurrencymarket, as it was expected before (as in Glas/Poddig (2018)). This means: Inthis sample, cryptocurrency returns are not extraordinary high compared toboth cryptocurrency returns in other studies (with marketcap weighted cryp-tocurrency indices) and to the other asset classes’ returns in this study. Al-though cryptocurrencies have the second highest buy-and-hold (B&H) return per annum for the whole planning horizon, the expected returns (on a weeklybasis) do not significantly differ from 0, as an additional t -test indicates. In ad-dition to this (quasi non-existent) expected returns, cryptocurrency returns aremuch more volatile (riskier) than returns of the other assets under study, whichis also reflected by the extreme values of the respective returns. In contrast tothe results of most of the existing studies before, cryptocurrency returns are nownot high enough to offset their large volatility, leading to smaller performancemeasures (Sharpe Ratio, Sortino Ratio) compared to most of the other assetclasses.Skewness and Kurtosis indicate that cryptocurrencies are not normally-distributed as already shown by Krückeberg/Scholz (2018). This indication isconfirmed by additional empirical normal distribution tests (Jarque-Bera-Test,Shapiro-Wilk-Test, Anderson-Darling-Test, Cramer-von-Mises-Test), which allled to significant results (cf. Tab. 3), so that the normality assumption (nullhypothesis) must be rejected.
4. Results of the (Standard) Portfolio Analysis
As a first indication of cryptocurrencies’ contribution to portfolio diversifi-cation it is necessary to focus on the (static) correlations of cryptocurrencies23 ata ID Min Mean Median MaxTest-Assets: CryptocurrenciesEWCI EWCI -0.3884 0.0024 -0.0141 0.7708Benchmark-Assets:Money Market MON -0.0022 -0.00006 0.0000 0.0014Currencies CUR -0.0243 -0.0002 0.000002 0.0296Sovereign Bonds SOV -0.0222 0.0008 0.0013 0.0157Covered Bonds COV -0.0083 0.0005 0.0007 0.0058Corporate Bonds COR -0.0137 0.0005 0.0010 0.0105Stocks STO -0.0691 0.0011 0.0044 0.0499Real Estate RES -0.0508 0.0027 0.0030 0.0584Luxury Goods LUX -0.0879 0.0022 0.0038 0.0625Commodities / Gold COM -0.0899 -0.0017 0.0005 0.0574ID t-Test B & H- St.-Dev. Maximum Sharpe Sortino( p -value) Return Drawdown Ratio RatioTest-Assets: CryptocurrenciesEWCI 0.2669 0.1319 0.1498 0.9630 0.0159 0.0253(0.7898)Benchmark-AssetsMON 2.2405 0.0029 0.0004 0.0046 0.1334 0.1955(0.0258)CUR -0.5319 -0.0125 0.0076 0.1791 -0.0317 -0.0428(0.5952)SOV 2.5392 0.0402 0.0050 0.0585 0.1512 0.2156(0.0117)COV 3.6380 0.0243 0.0021 0.0237 0.2166 0.3278(0.0003)COR 2.8247 0.0285 0.0032 0.0364 0.1682 0.2458(0.0051)STO 0.9221 0.0589 0.0200 0.2385 0.0549 0.0752(0.3572)RES 2.5786 0.1509 0.0176 0.1766 0.1536 0.2457(0.0104)LUX 1.6614 0.1201 0.0220 0.2561 0.0989 0.1406(0.0978)COM -1.0872 -0.0838 0.0260 0.5631 -0.0647 -0.0815(0.2779)ID Skewness Kurtosis Jarque- Shapiro- Anderson- Cramer-Bera Wilk Darling von Mises( p -value) ( p -value) ( p -value) ( p -value)Test-Assets: CryptocurrenciesEWCI 0.9172 3.5608 188.520 0.9448 250.00 0.644- - (< 0,001) (< 0,001) (< 0,001) (< 0,001) Table
3: Descriptive Statistics, Source: Own Calculations.
MON CUR SOV COV COR STO RES LUX COM EWCIMON 1.00(***)CUR 0.29 1.00(***) (***)SOV 0.55 0.17 1.00(***) (***) (***)COV 0.78 0.16 0.73 1.00(***) (***) (***) (***)COR 0.56 0.25 0.65 0.74 1.00(***) (***) (***) (***) (***)STO 0.01 0.51 0.02 -0.11 0.17 1.00(***) (**) (***)RES 0.09 0.29 0.11 0.10 0.26 0.48 1.00(***) (***) (**) (***) (***) (***)LUX 0.05 0.49 0.02 -0.08 0.18 0.80 0.45 1.00(***) (**) (***) (***) (***)COM -0.04 0.33 -0.10 -0.11 0.04 0.39 0.13 0.29 1.00(***) (***) (**) (***) (***)EWCI 0.06 -0.02 0.03 0.02 0.06 0.03 0.07 0.04 0.02 1.00(***)
Significance-Levels: *** ≤ ≤ ≤ Table
4: Pearson Correlation Matrix, Source: Own Calculations. tocurrencies have an insignificant correlation with the other asset classes, whichmeans their returns develop independently of the other asset classes. Accord-ing to Markowitz (1952, 1959) and Malkiel (2019), independent assets have amedium diversification potential. In the following, we want to further examine the results shown in the last lineof Tab. 4 (cryptocurrency correlations) by considering the changes of the corre-lation coefficients ρ over time. Fig. 1 shows the results of this analysis: we cal-culate 10-week rolling-window-correlations between cryptocurrencies and otherexemplary assets (here: stocks, corporate bonds and commodities) to check thestability of the correlation coefficient close to ρ = 0. With the probability den-sity p ( r ; n, ρ ) of the correlation coefficient (Eq. 2.1 in Olkin/Pratt (1958) for thecase of unknown parameters) and the assumed correlation ( ρ ≈ n < ∞ ) due to statistical errors. Because of the non-normality of cryptocurrency returns, we follow Krückeberg/Scholz(2018) and re-run our calculations using the two non-parametric correlation measures Spear-man’s rho and Kendall’s tau. Because the results of the different correlation measures remainstable (only incremental changes in the coefficients), we do not report those results seperately. r ] = n − . Since the cryptocurrencies do not have normally distributedreturns, this calculation is an indicative estimate in the present case. For the10-week rolling window ( n = 10), the estimate of the standard deviation, as thesquare root of the variance σ = p Var[ r ] = q − = , is visualized by an errorband (black) in Fig. 1. It becomes obvious, that the majority of the correlationcoefficients lies within the error band. In addition, no clear trends can be iden-tified that permanently lead the correlation coefficient out of the band. Thisresults lead to the strong assumption that the correlation coefficient betweencryptocurrencies and other asset classes is stable over time in the observationperiod, without trends and close to zero. Fig.
1: 10-Week Rolling Correlations (EWCI to selected Asset Classes), Source: Own Cal-culations.
After the correlation analysis gave a first indication of cryptocurrencies’ di-versification potential, we now show the results for the graphical and descriptivestatistics based approaches of the portfolio analysis. Therefore, we need to dif-ferentiate the results for the Cases A and B (Sec. 2.1.1):
Results of Case A:
As a first step, we focus on the results of the portfolio optimization, if the whole26ataset is used as an observation period (here: in the unrestricted case). Fromnow on, we differentiate between the results for • the (Global) Minimum Variance Portfolio (GMVP) and • the Tangency Portfolio (TP)for both cases – with and without the consideration of cryptocurrencies. It be-comes obvious, that cryptocurrencies only have a small (to neglegible) portfolioweight (close to 0 . of unconstrained Mean-Varianceoptimizations (Härdle et al. (2018)) fades away after the introduction of theportfolio weight restrictions (Fan et al. (2012); Jagannathan/Ma (2003)), as theresults of the long-only portfolios confirm. Asset Portfolio Weights
Minimum-Variance-Portfolio Tangency-Portfolio
No EWCI With EWCI No EWCI With EWCIUnconstrained Portfolio: No Transaction Costs (Full Sample: 2014-01/2019-05)MON 1.156794 1.156753 -17.085677 -15.793628CUR -0.008207 -0.008307 -3.130671 -2.934682SOV 0.003906 0.003891 -2.524664 -2.348019COV -0.172364 -0.017240 25.849404 23.980406COR 0.019226 0.019418 -3.832146 -3.504868STO -0.002055 -0.002065 0.228977 0.209941RES 0.002053 0.002067 0.645085 0.603059LUX -0.000234 -0.000217 1.078560 1.006219COM 0.000883 0.000895 -0.228868 -0.209167EWCI 0.000000 -0.000034 0.000000 -0.009261
Table
5: (Optimal) unconstrained Portfolio for Case A (with and without cryptocurren-cies), Source: Own Calculations.
Results of Case B:
As a second step, we re-run the analysis under the assumption of monthly re- Although previous studies did not necessarily disclose all of their optimal portfolio weights,we expect similar patterns for the unconstrained optimization results in at least some ofthose studies due to the comparability of the applied optimization approaches. According toGreen/Hollifield (1992) such effects can be driven by a single dominant factor in the covariancematrix. Because of the high dimensionality and the limited data availability, we checked allof our covariance matrices, whether they are well-conditioned. As a result, we do not seesubstantial problems with regard to a suitable condition of the covariance matrices used inthis paper. However, if a substantial problem with a suitable condition of the covariancematrices arises in practise, it is proposed to select an alternative covariance matrix estimatore.g. by Ledoit/Wolf (2004). .
00% – except for some still small, but slightly more extreme outliers in theTP (Min: 0.029514; Max: 0.098583) (cf. Fig. 2). An introduction of an addi-tional long-only constraint ( ω i ≥ ∀ i ) leads to comparably small cryptocurrencyweightings and to less extreme outliers (Min: 0.000000; Max: 0.064390) in theTP (cf. Fig. A.2 in the Appendix.).This first indication is confirmed by an additional look at the average port-folio weights of the respective asset classes (cf. Tab. 6, Tab. A.4), where theaverage portfolio weight of cryptocurrencies is – again – close to 0 .
00% with themost extreme cryptocurrency weight in the tangency portfolio of the long-onlymodel (1 . (a) GMVP: Benchmark assets only(b) TP: Benchmark assets only (c) GMVP: Considering Cryptocurrencies(d) TP: Considering Cryptocurrencies Fig.
2: Portfolio Weight Heatmap for Case B ( T = 54, 1-Year Rolling Windows): Uncon-strained Portfolios (No Transaction Costs), Source: Own Calculations. sset Average Portfolio Weights Minimum-Variance-Portfolio Tangency-Portfolio
No EWCI With EWCI No EWCI With EWCIUnconstrained Portfolio: No Transaction Costs (1-Year Rolling Windows)MON 1.160374 1.157118 0.891922 1.823775CUR -0.007018 -0.007123 0.220982 -0.124588SOV 0.009944 0.010162 0.013924 0.092893COV -0.168913 -0.162026 0.164702 -0.810179COR 0.004965 0.000874 -0.179664 -0.019372STO -0.001195 -0.001144 0.091388 -0.081516RES 0.000871 0.001178 -0.023147 0.033939LUX -0.000067 0.000002 -0.138160 0.051076COM 0.001039 0.000937 -0.041947 0.029609EWCI 0.000000 0.000022 0.000000 0.004364
Table
6: Unconstrained Portfolio: Average Portfolio Weights for Case B (with and withoutcryptocurrencies), Source: Own Calculations.
As a result of the Cases A and B, these (mostly) low portfolio weights ofcryptocurrencies lead to the following first indications:(i) The benchmark assets might span the test assets in most of the analyzedwindows, so that there would be no significant diversification potentialsin most of the analysed windows.(ii) The differentiation of the Cases A and B discloses, that a too aggregatedanalysis of the diversification potentials of cryptocurrencies (case A) canlead to a (to some extent misleading) conclusion, that there are no di-versification potentials of cryptocurrencies at all (like Glas/Poddig (2018)concluded), whereas a more granular analysis (Case B) indicates, thatthere are also a few important exceptions from this conclusion.(iii) The diversification effects seem to depend on the respective optimizationframework (unconstrained, long-only) in detail – even though most ofthe results of both frameworks seem to be comparable. Moreover, theadvantageousness of a consideration of cryptocurrencies is also dependentof the investors’ risk attitude (means: preference of the GMVP over theTP or vice versa).These first conclusions might be surprising at a first glance, because the in-dependency of cryptocurrency returns (as shown in the correlation analysis)and former results in the literature may drive the expectation of existing di-versification potentials. A comprehensible reason for that largely contradictoryresult could be the high volatility and comparably low expected returns of cryp-tocurrencies in the observation window, which is notoriously also driven by theequal weighting scheme of the EWCI (cf. Glas/Poddig (2018) for similar re-sults). Thus, it is interesting to run the additional spanning tests in a later stepto check the significance of the test assets’ potential diversification effects moreprecisely. 29 .2.2. Portfolio Metrics and Wealth Development
In the former analysis we found that the optimal portfolio weights of cryp-tocurrencies (and therefore their diversification potentials) for a certain obser-vation window also depend on the assumed optimization model (e.g. uncon-strained or long-only). Besides those optimal portfolio weights, which gaveinteresting results as preliminary works for the following spanning tests, it isalso interesting to stress this differences by comparing divergent metrics of theresulting portfolios and the development of how the wealth invested in theseoptimal portfolios develops in time.Rebalancing the portfolios in accordance to the abovementioned optimalportfolio weights of Case B leads to the following portfolio metrics for the un-constrained portfolio (cf. Tab. 6):
Measure Portfolio Statistics
Minimum-Variance-Portfolio Tangency-Portfolio
No EWCI With EWCI No EWCI With EWCIMinimum Return -0.002498 -0.002504 -0.057127 -0.249672Mean Return -0.000003 -0.000004 0.000048 -0.003051Maximum Return 0.001036 0.001061 0.111843 0.017646Standard Deviation 0.000353 0.000358 0.011902 0.021174Max Drawdown 0.005660 0.006424 0.214632 0.536954Sharpe Ratio -0.007686 -0.011475 0.003994 -0.144067Sortino Ratio -0.009840 -0.014770 0.007129 -0.144041End Value ( C T in EUR) 99.94 99.90 99.51 46.50 Table
7: Key Statistics of the Unconstrained Portfolio for Case B (with and without Cryp-tocurrencies, 1-Year Rolling Window), Source: Own Calculations.
In this context, we find that mixing cryptocurrencies in the benchmark port-folios leads to lower mean returns, higher risks and lower performance measures(cf. Tab. 7). Moreover, an initial investment of C = 100 .
00 EUR in the (con-tinuously rebalanced) portfolios leads to lower end values C T at the end of theplanning horizon, if cryptocurrencies are included in the investment opportu-nity set of the investors. This results hold for both – the GMVP and the TP. Incontrast to the unconstrained case, the consideration of cryptocurrencies in theinvestment opportunity set of the long-only model now leads to a remarkablyhigher return, (again) to a higher risk and – as a consequence – to a higherperformance. Moreover, the consideration of cryptocurrencies now leads to (re-markably) higher end values C T for both portfolios (GMVP, TP; cf. Tab. A.4in the Appendix).These two examples demonstrate, that the consideration of cryptocurrenciescannot be called ’ efficient ’ or ’ inefficient ’ in general, but the results dependon the optimization framework and the current market trends, instead. In ourexample, we can observe, that the disadvantegeousness of considering cryptocur-30encies for the TP in the unconstrained framework is mainly driven by window46 with an extremely negative portfolio return (caused by extreme short salesand extreme returns). As a consequence, we can justify our granular subsampleanalysis by the obvious impact of a few windows on the overall portfolio de-velopment and therefore the evaluation of the efficiency of cryptocurrencies ingeneral. After this typical portfolio optimization analysis, we now change the pointof view and repeat the analysis for the unconstrained portfolio framework usingthe spanning tests introduced above (as a regression-based approach). Again,we differentiate two different cases (in analogy to the portfolio optimizationanalysis).
Results of Case A:
To demonstrate the (general) functioning of the introduced spanning tests, westart with a graphical analysis of the efficient frontier in the (unconstrained)mean-variance-model (cf. Fig. 6a in Sec. 5.2), where the observation periodcontains the whole dataset (Case A).As we can see, the efficient frontier is only shifted minimally, if cryptocurren-cies (test assets) are considered in the investment opportunity set. This canbe interpreted as the next indication, that the diversification effect of cryptocur-rencies is small to neglegible in this context. To analyze, whether this graphicalimpression is a significant effect, we firstly focus on the regression results ofthe mean-variance-spanning tests summarized in Tab. 8. In addition to thismean-variance-model, we also run the further spanning-tests introduced above(GMM-Wald-Tests, Stepdown Tests) to gain deeper insights, if these resultsremain robust, if the methodology is changed (robustness checks).
Test- MV-Spanning-Tests GMM-Wald-Tests Stepdown-TestAssets W LR LM FFK BU (EIV Adj.) F1 F2( p -val.) ( p -val.) ( p -val.) ( p -val.) ( p -val.) ( p -val.) ( p -val.)EWCI 0.1138 0.1138 0.1138 0.0937 0.0899 0.0385 0.0715(0.945) (0.945) (0.945) (0.954) (0.956) (0.845) (0.789) Table
8: Results of the Spanning-Tests (Case A), Source: Own Calculations.
The econometric results confirm the former conjecture, that the implemen-tation of cryptocurrencies (test assets) in German investors’ portfolios does notlead to a (significant) shift of the efficient frontier – and therefore significant di-versification effects. This result does not only hold for the mean variance span- The key results also hold, if we do not assume an unconstrained portfolio, but a long-only portfolio instead (cf. Fig. 6b). But the spanning tests only refer to the unconstrainedportfolios.
Results of Case B:
If this spanning test analysis is now repeated for smaller subsamples of a 1-year rolling-window (Case B), we can observe results as in Tab. 9. For a betteroverview, the results are aggregated, so that we count all regression results, inwhich we find significant diversification effects for a given significance level ξ (with ξ = (0 . , . , . , . Afterwards, we compute their shares to thetotal number of regression results (54) for each test (cf. Tab. 9).
Test- MV-Spanning-Tests GMM-Wald-Tests Stepdown-TestAssets W LR LM FFK BU (EIV Adj.) F1 F2Significant 0.204 0.185 0.148 0.222 0.167 0.204 0.074Thereof:10% ≤ ξ <
5% — — — — — 0.074 —5% ≤ ξ <
1% 0.111 0.093 0.074 0.093 0.093 0.111 0.0741% ≤ ξ < .
1% 0.037 0.093 0.074 0.074 0.056 0.019 0.0000 . ≤ ξ Table
9: Results of the 1-Year Rolling-Window-Spanning-Tests (Case B), Source: OwnCalculations.
We find that the question, whether considering cryptocurrencies in Germaninvestors’ portfolios leads to a (significant) diversification effect or not, must beanswered more differentiated than it was done in case A (and the former studiesin the literature). Our results reflect, that the (joint) spanning tests’ do notreject the null hypothesis most of the time (in around 80 −
85% of the cases) The significance level of α = 0 .
10 is only used to detect significant results in the coursestepdown procedure ( F ), as it is described above. All other tests only use significance levelsup to α = 0 .
05 for this purpose. −
20% significant results, anyway. This shows, that neglectingthe diversification effects of cryptocurrencies in general would be misleading.
Fig.
3: Rolling-Window-Spanning-Tests (Case B): p -values, Source: Own Calculations. As a next step, we present the results of the Rolling-Window-Spanning-Testsmore in detail (Fig. 3). These disclose three interesting perspectives:(i) The observation windows with significant diversification effects do not ap-pear randomly, but rather in clusters instead, as the green (or yellow)blocks in the heatmaps signalize. Moreover we find widely consistent clus-ters and only small differences in the results of all the tests applied in thiscontext. If we now refer to Fig. 4, it is obvious, that the first bigger cluster33windows 1-5) contains a remarkable market downturn of the cryptocur-rency market, while the second bigger cluster (windows 31-38) containsthe cryptocurrency boom in 2017 and the bust in January 2018, whichwas a consequence of the introduction of Bitcoin futures. But this resultshould not lead to the hasty conclusion, that extreme movements of thecryptocurrency market are the only premise for significant diversificationbenefits. The – in some parts unexpected – negative portfolio weights forthe windows 31-38 (cf. Fig. 2) point out, that other factors beyond the(expected) cryptocurrency returns, such as the (co-)variances of the dif-ferent benchmark- and test assets also seem to be relevant drivers in thiscontext. Moreover, the respective portfolio weights for the windows withsignificant spanning tests confirm, that even in those times with extrememarket movements on the cryptocurrency market the optimal cryptocur-rency weights still remain small.(ii) The relevance of cryptocurrencies for the optimal portfolio mostly dependson the optimization strategy. This means: if we measure a significantdiversification effect of cryptocurrencies for the GMVP at a certain rebal-ancing date, we do not necessarily find significant results for the TP (andvice versa). This pattern indicates, that the efficient frontier is not reallyshifted, but rather rotated in such cases. Therefore, it is theoreticallypossible, that the joint test still accept the null hypothesis, whereas thestepdown procedure indicates significant diversification effects (Kan/Zhou(2012)).(iii) Comparing the share of significant results of both F and F , we findthat F has a remarkably higher share of significant results – which alsoholds if we (temporarily) assume the same significance levels for bothtests ( ξ = ξ = 0 .
05) against the recommendation of Kan/Zhou (2012).Thus, cryptocurrencies are more relevant in the tangency portfolio. Fol-lowing the interpretation of the p -values in the stepdown procedure byBelousova/Dorfleitner (2012), cryptocurrencies are then more able to im-prove the return of a portfolio than reducing its level of risk, which is anexpectable result with regard to their high level of price volatility. Then itis likely, that e.g. extreme expected cryptocurrency returns during remark-able market movements outweigh their high volatility, which was identi-fied as at least one possible driver of significant cryptocurrency shares(besides the covariance matrix) in the investors’ tangency portfolios in (i).However, according to Fig. 4, cryptocurrencies predominantly do not leadto significant diversification benefits in normal times, when their returnsare presumably not able to outweigh the respective volatilites and/or the(co-)variances do not lead to remarkable risk reduction potentials. Thisobvious difference to most of the existing cryptocurrency literature, whichobserves a more general diversification benefit of cryptocurrencies, can beexplained by the different cryptocurrency weighting scheme and (possibly)by different observation windows.34 ig.
4: Comparison of the EWCI index development and obvious clusters of observationwindows with significant cryptocurrency weights (grey), Source: Own Calculations.
5. Effects of Additionally Accounting for Transaction Costs and aReduced Market Liquidity on the Efficient Frontier
The analysis above assumed that there are no transaction costs for investorswhen (re-)allocating their capital between the given investment alternatives – inanalogy to the optimization framework by Markowitz (1952, 1959). Moreover,the optimization framework assumed, that the market depth is high enough sothat no illiquidities must be considered.But according to Borri (2019) and Borri/Shaknov (2018), investors defini-tively suffer transaction costs and illiquidity issues on the cryptocurrency mar-ket. A short look on the development of Bitcoin’s historical transaction costsconfirms that point, because it shows, that there seems to be a positive re-lation between investment demand and transaction costs (BitcoinVisuals.com(2020a,b)). The main reason for that positive relationship is, that investors caninfluence the probability of their transactions to be added to the blockchainfaster by bidding higher transaction fees to the miners, which becomes moreimportant, if the trading volume rises and more transactions compete with eachother (Choi (2018); Dwyer (2015)). Looking back to the cryptocurrency boomin 2017, the Bitcoins’ transaction fees exploded up to over 40.00 EUR per trade,as the development of the daily mean of all median transaction fees per tradeand block (cf. Fig. 5; blue) indicates. In contrast, the average of these dailyaverage transaction fee data for this observation window (1.29 EUR; black)indicates, that the transaction fees are much lower or even neglegible (as therespective minimum of 0.02 EUR shows) in normal times . We used the daily median transaction fee data (per transaction and block, converted inUSD and excluding coinbase transactions) by BitcoinVisuals.com (2020a) and converted it toEUR by using the daily USD-EUR exchange rate from Thomson Reuters Eikon. ig.
5: Bitcoin: Historical Daily Median Transaction Fees per Trade versus Daily Transac-tions. Data Sources: BitcoinVisuals.com (2020a,b) and Thomson Reuters Eikon.
The beforementioned liquidity issues can be confirmed by a closer look onthe market depths of those cryptocurrencies: if a cryptocurrency has a highermarket cap (and therefore a higher ranking in the CoinMarketCap Cryptocur-rency Ranking) than other cryptocurrencies, it tends to have more attention(visibility) and is traded more frequently, which leads to a higher market depth.The fact, that we use an equally-weighted cryptocurrency index to capturethe average development of an ex-ante defined cryptocurrency market, mightlead to a problem: For high portfolio weights of the cryptocurrency index orvery huge investment amounts, it is possible that we are forced to buy morecoins of a single cryptocurrency than it is offered on the cryptocurrency market(Trimborn et al. (2018b)). In this context, beeing forced to buy (nearly) all ofthe available coins on the market would lead to extreme market movements atfirst. On the other hand, if more investors would use this strategy, the smallercryptocurrencies’ prices, trading volumes and market depth would rise, so thatthis problem would probably fade in time. On the other hand, those smaller cryptocurrencies are prone to so-called pump and dumpschemes, where investors appoint to buy a certain cryptocurrency to boost its market price(because of its small market capitalization and trading volume), which is followed by a waveof selling transactions to realize profits from speculation or to stop potential losses (Li etal. (2019)). But this phenomenon is distinctive for the cryptocurrency market and can bescheduled for a (random) cryptocurrency. It is useless (and nearly impossible) to identify allhistorical attempts of pump and dump schemes and to exclude all cryptocurrencies from the
36o account for those liquidity constraints mentioned above, Trimborn et al.(2018b) propose a Liquidity Bounded Risk Return Optimization (LIBRO) ap-proach based on the methodological foundation of Darolles et al. (2015). In thiscourse, they add a liquidity constraint for the individual portfolio weights onthe cryptocurrency level (based on their respective turnover values) as a furtherequation in the portfolio optimization framework. But this approach is harderto apply, if not only single cryptucurrencies, but (aggregated) indices represent-ing dozens of single titles of different asset classes are used in the optimization.Borri (2019) and Borri/Shaknov (2018) use an alternative approach to cap-ture both transaction costs and illiquidity issues. They define a portfolio op-timization framework assuming unilateral (or: asymmatric) transaction costs.This means: transaction costs are assumably not equal for all asset classes and– at least in their study – limited to cryptocurrency trades, which can be rea-soned by the above-mentioned explosion of cryptocurrency transaction costs inbooms. As a (positive) side-effect, those transaction costs on cryptocurrencytrades cause, that investors are penalized for extreme shifts in their cryptocur-rency positions (leading to consistently small portfolio shares of cryptocurren-cies in their framework). This effect also reduces the probability, that there areless cryptocurrency units available on the exchanges than the number of unitsdemanded (illiquidity issues).Other works, just as Anyfantaki et al. (2018) and Topaloglou/Tsomidis(2018), also assume asymmetric transaction costs (means: penalizing transac-tions of a certain asset class more than the trades of other asset classes) to thedisadvantage of cryptocurrencies in another methodological context, but in con-trast to Borri (2019) and Borri/Shaknov (2018) they do not set the benchmarkassets’ transaction costs to zero. Moreover, they do not distinguish differenttransaction cost levels for taker- and maker- transactions. Instead, they assumetransaction costs of 35 BP for all benchmark asset transactions and 50 BP for allcryptocurrency transactions. However, for their final asset allocation, investorsprimarily focus on the difference between both transaction cost levels, so thatthere is no big difference between these approaches mentioned before.In order to control for (possible) changes of the results, if transaction costsare implemented and illiquidity issues are reduced, we refine the general ideasof the beforementioned studies and build a customized alternative optimiza-tion framework, which also takes a more realistic (asymmetric) transaction costscheme into consideration. dataset, which were related to this attempt. But it is noteworthy, that an exclusion is notnecessary, because those pump and dump scenarios only last just a few seconds, which could(more likely) bias intra-day data, but not our weekly observation windows. τ Benchmark i = c Benchmark i | ∆ ω i | V , (26)where c Benchmark i is a fixed transaction cost factor for asset i , ∆ ω i is the changein the portfolio weights of asset i and V is the wealth invested in the initialportfolio. We set c Benchmark i = 35 BP, as it is proposed by Anyfantaki et al.(2018) and Topaloglou/Tsomidis (2018).But with regard to the cryptocurrencies (still our test assets), we use adivergent approach. In the light of demand-dependent transaction cost factors,a (linear) transaction cost function τ Test (Lin.) i including a fixed transaction costfactor c Test i , such as τ Test (Lin.) i = c Test i | ∆ ω i | V , (27)which is (implicitly) proposed in the former studies in the literature (e.g. Borri(2019); Borri/Shaknov (2018); Anyfantaki et al. (2018); Topaloglou/Tsomidis(2018)), would not be suitable, because of the overproportional rise of the trans-action costs, which was observed in Fig. 5. We now implement, that the trans-action cost factor c Test i is not fixed any more, but now depends on the shifts | ∆ ω i | in the portfolio allocation and a marginal cost contribution factor ˜ c Test i ,which consists of the fixed transaction cost factor c Test i (from the linear case)and a special scalability factor Ψ. The latter has to be chosen in a further step.Consequently, these changes lead to c Test (new) i = 12 Ψ c Test i | {z } ˜ c Test i | ∆ ω i | . (28)If Eq. (28) is inserted in Eq. (27), we have a quadratic transaction cost function,so that we can simply write τ Test (Quad.) i = 12 ˜ c Test i | ∆ ω i | V . (29)This beforementioned form of the transaction cost function can also be derivedfrom the Taylor series (see Appendix A.3). The function now captures risingtransaction cost factors c Test (new) i , if investors buy/sell high shares of cryptocur-rencies. Furthermore, it penalizes those great shifts of the portfolio allocationin favor of cryptocurrencies, which is desirable with regard to the illiquidityissues on the cryptocurrency market. A more practical rationale for this for-mula could be: if investors face extreme shifts of their optimal portfolio to the(dis)advantage of cryptocurrencies, this is probably caused by a significant mar-ket upturn (downturn). In both situations investors are assumed to bid highertransaction fees to accelerate the current transactions.On the other hand, for small changes of the respective portfolio weight, wehave smaller transaction costs – sometimes even smaller than the transaction38osts of other asset classes (calculated with the linear case formula), which isconsistent with the low (average) transaction cost level on the cryptocurrencymarket in normal times.As a generalization of the cryptocurrency case, the comparison of both trans-action cost functions (linear/quadratic) discloses, that we need to seperate threedifferent cases:(i) τ Test (Quad.) i < τ Test (Lin.) i ( ∀ | ∆ ω i | < | ∆ ω ∗ i | )(ii) τ Test (Quad.) i = τ Test (Lin.) i ( ∀ |∆ ω i | = | ∆ ω ∗ i | )(iii) τ Test (Quad.) i > τ Test (Lin.) i ( ∀ | ∆ ω i | > | ∆ ω ∗ i | ).The calculation of the intersection point | ∆ ω ∗ i | is demonstrated in the Appendixin detail. But this calculation depends on the individual parameterization ofthe transaction cost function (with regard to Ψ). In this study, we aim ata parametrization, which ensures, that we have the same expected transac-tion cost factor as Anyfantaki et al. (2018) and Topaloglou/Tsomidis (2018):E[ τ Test (Quad.) i ] = E[ τ Test (Lin.) i ] = 50 BP. This result is reached, if the integralbetween both cost functions is zero : Z maxmin d ( | ∆ ω i | ) (cid:20)
12 ˜ c Test i | ∆ ω i | V − c Test i | ∆ ω i | V (cid:21) ! = 0 . (30)Note, that the interval [min , max] can be set exogeneously depending on theoptimization strategy, e.g. [0 ,
1] for a long-only strategy. In this case, the pa-rameter ˜ c Test i must be set to ˜ c Test i = Ψ c Test i = 3 c Test i to fulfill this condition in thelong-only case. For the unconstrained portfolio, it is important to mention,that max is not allowed to reach infinity for mathematical reasons. However,this restriction does not mean, that it is not possible to calculate the uncon-strained portfolio: we do not restrict the portfolio weights to a maximum value,but only changes in the portfolio weights, so that building up a high positionof a certain asset with gradual changes in the portfolio would still be allowed.Moreover, in the further analysis we introduce a transaction cost budget ˜ K ,which limits the maximum amount of transactions in any case, so that the re-striction of max will not be binding any more. Thus, we can consistently assume[min , max] = [0 ,
1] also for the unconstrained portfolios.We are now able to adjust our previous Lagrange functions (Eq. (3) and (4)) This relationship only holds, if the distribution of the ∆ ω i is an equal distribution inthe interval [min , max]. This assumption is based on the Laplace critereon, because we donot know the empirical distribution of this variable. If further research has a bigger database, which makes estimations of this distributions more reliable, our approach can easily beadjusted. See Appendix for a detailed derivation of this conditions.
39y an additional transaction costs restriction, which leads tomax ω L T P = max ω (cid:20) ω r − r F √ ω V ω + λ (1 − ω ) + λ (cid:0) ˜ K ≥ C (cid:1)(cid:21) (31)min ω L GMV P = min ω (cid:20) ω V ω + λ (1 − ω ) + λ (cid:0) ˜ K ≥ C (cid:1)(cid:21) (32)in the unconstrained case with ˜ K as a transaction cost budget (for the cur-rent portfolio shift) and C = T as the cumulated transaction costs, where T is a ( K + N )-vector containing all the individual transaction costs T = (cid:20)(cid:0) τ Benchmark K × (cid:1) , (cid:16) τ Test (Quad.) N × (cid:17) (cid:21) caused by trades of the test assets τ Test (Quad.) N × and benchmark assets τ Benchmark K × . According to the additional transaction costrestriction, the (cumulated) transaction costs caused by the shift of the assets’weightings is not allowed to exceed the exogeneously given transaction cost bud-get. This budget is here set to ˜ K = 10 BP at first (and varied afterwards asdifferent budget levels). In this context, it is important to mention, that ouroptimization model does not aim at a transaction cost optimal portfolio (suchas a transaction cost minimal portfolio; cf. e.g. Lobo et al. (2007) for examples),but uses the same optimization strategies as introduced in the sections before– under the additional restriction, that portfolio shifts are only allowed to acertain extent. Like in Sec. 2.1.1, it is also possible to introduce an additionallong-only constraint ( ω i ≥ ∀ i ) as a robustness check, if the possibility of shortsales should be excluded from the analysis.We run this optimization model using a dataset covering the whole rangeof the sample (2014-01-01 to 2019-05-31) as an observation period, like we didin Sec. 4.3 in the course of the spanning tests. As a starting point for the op-timization, we stick to the assumption an (equally-weighted) initial portfolioonly containing the benchmark assets (like in Sec. 2.1.1), which is shifted tothe calculated (optimal) portfolio after a certain observation period. The re-sults give some indication, whether cryptocurrencies should considered in theoptimal portfolio, if liquidity- and transaction cost considerations restrict theoptimal portfolio choice. This composition of the initial portfolio is feasible inthe light of the fact, that investors first have to get in touch (and gain experi-ences) with the emerging asset class of cryptocurrencies, while they are moreexperienced with the other traditional asset classes. Moreover, this compositiondirectly translates the usual research question "How does the possibility of mix-ing cryptocurrencies into an investor’s portfolio of benchmark assets affect theefficiency and diversification potentials of this portfolio?" into an optimizationmodel. If cryptocurrencies would be already included in the initial portfolio,there is no need to add them to the portfolio in a further step.In this context, Glas/Poddig (2018) expected that the results generatedby the spanning tests become even worse, if transaction costs would be intro-40uced, but do not analyse the effects on the efficient frontier in detail. Borri(2019)) consistently showed a negative shift of the efficient frontier after theintroduction of transaction costs, but did not seperate between the efficientfrontiers with and without the consideration of cryptocurrencies. Other works,just as Anyfantaki et al. (2018) and Topaloglou/Tsomidis (2018) directly in-cluded transaction costs in their spanning tests, but their results (significantportfolio diversification potentials) are presumably driven by their limited, in-dividually selected cryptocurrency portfolio, which might lead to biased results(Glas/Poddig (2018)). As a consequence, it is interesting to observe, whetherthe expectations of Glas/Poddig (2018) can be verified in this analysis. Applying the portfolio optimization under transaction costs for the given ob-servation period, we find that the portfolio weights of cryptocurrencies remainat a small level at the first glance, which is not a surprising result: the cryp-tocurrency weightings were already small before the trading of huge amounts ofcryptocurrencies was penalized with high transaction costs. On the other hand,low shifts in the cryptocurrency weightings cause lower transaction costs thanshifts of the other asset classes’ weightings, which might have favorable effectson the cryptocurrency weightings. A closer look on the optimization results con-firms this prediction: the introduction of transaction costs leads changing signsin the TP (before: -0.009261; after: 0.006391) and to (slightly) more extreme(but still very small) cryptocurrency weights in the GMVP (before: -0.000034;after: 0.000640) in the unconstrained framework. These shifts in the cryptocur-rency weights are also observable for the TP (before: 0.000000; after: 0.004317)and the GMVP (before: 0.000103; after: 0.000544) in the long-only framework( ω i ≥ ). Consistent tothe small weights of cryptocurrencies in all cases (with and without transactioncosts), we can conclude, that this shift is not only due to the consideration ofcryptocurrencies. Instead, the main part of the shift is driven by the transactioncosts caused by trades of the benchmark assets.Nevertheless, we can observe, that the consideration of cryptocurrenciesleads to small shares of cryptocurrencies in the GMVP and TP (and there- Note, that the underlying transaction cost function remains the same for both optimiza-tion frameworks (unconstrained, long-only, so that the resulting curve is only printed once inFig. 6a as representative example for all the other graphical analyses hereafter. (a) Base Case: Unconstrained Portfolio. (b) Robustness Check: Long-Only Portfo-lio.
Fig.
6: Efficient frontiers with and without the consideration of cryptocurrencies and trans-action costs (Full Sample: 2014-01/2019-05), Source: Own Calculations.
Although we did not detect significant effects in the spanning tests’ resultsof the global analysis (Case A) before transaction costs, it does not mean, thatcryptocurrencies did not play a significant role in smaller subsamples. In theresults of the rolling-window spanning tests before transaction costs (cf. Fig. A.3in the Appendix), we identified smaller subsamples, in which cryptocurrenciesplayed a significant role in the unconstrained portfolio. A view on the portfolioweight heatmaps (cf. Fig. 2 and A.2) will strenghten this point, because thissignificant role also becomes manifest in high cryptocurrency weights. Finallya more visible positive shift of the efficient frontier has to be expected for thesesubsamples (without transaction costs). In this situation, it would be interestingto observe, how the efficient frontier and the portfolio allocation is affected bythe introduction of transaction cost. In other words, the question arises: doesthe introduction of transaction costs lead to smaller cryptocurrency weights (andtherefore smaller diversification potentials) of cryptocurrencies in those periods,where we actually identified significant results in the spanning test framework?To answer this research question, we choose a subsample from 2014-01 to2014-12 and repeat the calculation of the same efficient frontier figure under the42onsideration of transaction costs. This subsample has the advantage, that italready led to significant results in the spanning tests (especially for the tangencyportfolio) and is also the first possible window for the optimization. Thus, wecan – again – observe the first shift from the initial portfolio (see above) to thefirst optimal portfolio.In this course, considering cryptocurrencies in the investment opportunityset has a (again small, but now more visible) positive impact on the efficient fron-tier at first, if no transaction costs are assumed (gross efficient frontier), whichalso manifests in slightly different TPs (cf. Fig. 7). After the introduction oftransaction costs, the efficient frontier for the situation with cryptocurrencies is– again – shifted downwards (net efficient frontier), but still remarkably abovethe net efficient frontier without cryptocurrencies, which can be interpreted asa remaining diversification effect. However, because of the lack of a suitablespanning test for our customized optimization model, it could be an interestingfuture research project to check, whether this remaining effect stays significant(as in the traditional spanning tests before). On the other hand, a closer lookon the portfolio weights shows some considerable cryptocurrency weights in theTP, as it was expected. Moreover, it can be observed, that the cryptocurrencyweights become less extreme (with a changing sign) after the additional con-sideration of transaction costs for the TP (before: 0.098583; after: -0.047093)and more extreme (with a changing sign) the GMVP (before: -0.000797; after:0.012052). As a consequence, if cryptocurrencies have a (too) high weight inthe situation without cryptocurrencies, the higher transaction costs lead to alower portfolio weight and therefore a less importance with regard to portfoliodiversification, just as Glas/Poddig (2018) expected.The contrary effect of more extreme cryptocurrency weights (and sign changes)for small portions of cryptocurrencies and more restrictive transaction cost bud-gets can also confirmed – at least by tendency – by Tab. 10, which comparesthe (optimal) cryptocurrency weights for the global analysis (whole sample)and different transaction cost budgets ˜ K = 2 , , , , , , , , ,
100 BPs.However, despite this obvious effect, the cryptocurrency weightings remain com-paratively small (as it was expected for this restrictive weighting scheme). ˜ K EWCI Weight (GMVP) EWCI Weight (TP)100 BP 0.000001 -0.00079430 BP -0.000033 -0.00033820 BP -0.000039 0.00195015 BP -0.000957 0.00345312 BP 0.000541 0.00338210 BP 0.000640 0.0063918 BP 0.000708 0.0075926 BP 0.000810 0.0089894 BP 0.000888 0.0103962 BP 0.001192 0.011813
Table
10: Cryptocurrency Portfolio Weights for varying transaction cost budgets (Uncon-strained Scenario, Full Sample: 2014-01/2019-05), Source: Own Calculations. (a) Variation of the Observation Period:Subset (2014-01/2014-12), Unconstrained. (b) Variation of the Transaction CostBudget: Full Sample (2014-01/2019-05),Unconstrained.
Fig.
7: Robustness Checks: Effects of a changing optimization framework (Subsample Anal-ysis, Variation of Transaction Costs) on the (unconstrained) efficient frontier, Source: OwnCalculations.
In toto, the beforementioned results are an indication, that parts of therecognized diversification effects seem to remain even after an introduction oftransaction costs (and illiquidity reflections). The results of the Markowitzframework and the respective spanning tests are therefore biased to some extent– just as Glas/Poddig (2018) expected before. But in contrast to their expecta-tions, it could be demonstrated, that it is also possible, that the cryptocurrencyweightings become more extreme after the introduction of transaction costs bytendency.
6. Conclusion
In this paper, we analyzed, whether cryptocurrencies are able to improvethe diversification of German investors’ portfolios.44n contrast to most of the studies in the literature, we used a customizedEqually-Weighted Cryptocurrency Index (EWCI) as a measure for the devel-opment of the cryptocurrency market to reduce the usual survivorship bias incryptocurrency datasets and to measure the (average) development of the cryp-tocurrency market more accurately than already existing cryptocurrency indices(CRIX, CCI30).The analysis combined three different methodological approaches, whichwere all applied in previous studies in the related literature: (i) an approachbased on descriptive statistics , (ii) a graphical approach, (iii) an econometric approach based on spanning tests.We have shown, that in the absence of transaction costs cryptocurrenciesdo not improve portfolio diversification for German investors significantly inmost of the analysed cases for our dataset including weekly observations fromthe window 2014-01-01 to 2019-05-31. By including cryptocurrencies in theirportfolios, investors predominantly cannot reach a (significantly) higher efficientfrontier, which means that they cannot realize (significantly) higher returns ata given risk level. However, the results of the rolling-window spanning testsshowed, that there were also a few periods, in which the consideration of cryp-tocurrencies improved portfolio diversification, anyway.By additionally demonstrating the effects of introducing liquidity- and trans-action cost considerations, we show that the the beforementioned results of atraditional Mean-Variance-Optimization framework seem to be biased to someextent. In contrast to the expectations of e.g. Glas/Poddig (2018), we demon-strated, that cryptocurrencies do not necessarily become less attractive for in-vestors after the introduction of transaction cost budgets, which would manifestin shrinking portfolio weights. Instead, the comparatively low transaction costsof smaller cryptocurrency trades in our model led to a higher attractiveness insome of the analyzed contexts. Only in the case of high portfolio weights inthe scenario without cryptocurrencies the introduction of (demand-dependent)transaction costs led to a reduction of the portfolio weight.Because of the lack of a suitable spanning test for our customized optimiza-tion model, our research could be a starting point for future research to check,whether the introduction of transaction costs leads to more periods with signif-icant shifts of the (net) efficient frontier.In toto, cryptocurrencies must be characterized as extremely speculativeand risky assets with many hazards (hacker attacks, fraud, etc.), which maynot be suitable for inexperienced private investors and do not really fit to theirrisk attitude, too. Moreover, even if they would fit to the risk attitude an(experienced) investor, they are often not relevant for portfolio diversification,as the optimization results indicated. Nevertheless, we also gave some examplesof periods, where cryptocurrencies had at least a small (but significant) portfolioweight, so that a complete exclusion from the investors’ investment opportunity45et could be misleading to some extent.46 . Appendix
A.1 Tabular Appendix D a t a I nd e x F r e q u e n c y F r o m ... t o ... S o u r ce S t o ck s : S t o xx E u r o p e T R I nd e x w ee k l y - - - - B l oo m b e r g M o n e y M a r ke t : i B o xx E U R J u m b o T R - I nd e x w ee k l y - - - - B l oo m b e r g S o ve r e i g n B o n d s : i B o xx E u r o E u r o z o n e S o v e r e i g n O v e r a ll T R I nd e x w ee k l y - - - - B l oo m b e r g C o ve r e d B o n d s : i B o xx E u r o C o v e r e d T R I nd e x w ee k l y - - - - B l oo m b e r g C o r po r a t e B o n d s : i B o xx E u r o L i q u i d C o r p o r a t e s D i v e r s i fi e d T R I nd e x w ee k l y - - - - B l oo m b e r g R e a l E s t a t e : R X R E I T P e r f o r m a n ce I nd e x I nd e x w ee k l y - - - - F r a n k f u r t S t o c k E x c h a n g e L u x u r y G ood s : S o l a c t i v e L u x u r y a nd L i f e s t y l e I nd e x T R I nd e x w ee k l y - - - - T h o m s o n R e u t e r s E i k o n C o mm od i t i e s / G o l d : G S C I C o mm o d i t y I nd e x T R I nd e x w ee k l y - - - - T h o m s o n R e u t e r s E i k o n Table
A.1: Dataset (Benchmark-Assets except for Currencies). urrency ID Frequency From ... to ... SourceUS-Dollar USD weekly 2014-01-01 / 2019-06-01 BloombergSwiss Franc CHF weekly 2014-01-01 / 2019-06-01 BloombergJapanese Yen JPY weekly 2014-01-01 / 2019-06-01 BloombergAustralian Dollar AUD weekly 2014-01-01 / 2019-06-01 BloombergNew Zealand Dollar NZD weekly 2014-01-01 / 2019-06-01 BloombergCanadian Dollar CAD weekly 2014-01-01 / 2019-06-01 BloombergNorwegian Krone NOK weekly 2014-01-01 / 2019-06-01 BloombergDanish Krone DKK weekly 2014-01-01 / 2019-06-01 BloombergSwedish Krona SEK weekly 2014-01-01 / 2019-06-01 BloombergBritish Pound GBP weekly 2014-01-01 / 2019-06-01 BloombergTurkish Lira TRY weekly 2014-01-01 / 2019-06-01 BloombergSouth African Rand ZAR weekly 2014-01-01 / 2019-06-01 BloombergRussian Ruble RUB weekly 2014-01-01 / 2019-06-01 BloombergPolish Zloty PLN weekly 2014-01-01 / 2019-06-01 BloombergMexican Peso MXN weekly 2014-01-01 / 2019-06-01 BloombergIndian Rupee INR weekly 2014-01-01 / 2019-06-01 BloombergChinese Yuan CNY weekly 2014-01-01 / 2019-06-01 Bloomberg Table
A.2: Dataset (Benchmark-Assets: Currencies). sset Portfolio Weights Minimum-Variance-Portfolio Tangency-Portfolio
No EWCI With EWCI No EWCI With EWCIUnconstrained Portfolio: No Transaction Costs (Full Sample: 2014-01 / 2019-05)MON 0.988127 0.987901 0.000000 0.000000CUR 0.007469 0.007721 0.000000 0.000000SOV 0.000000 0.000000 0.081421 0.081421COV 0.000000 0.000000 0.710920 0.710920COR 0.000000 0.000000 0.151744 0.151744STO 0.003397 0.003259 0.000000 0.000000RES 0.000000 0.000000 0.055916 0.055916LUX 0.000000 0.000774 0.000000 0.000000COM 0.001007 0.001016 0.000000 0.000000EWCI 0.000000 0.000103 0.000000 0.000000
Table
A.3: (Optimal) Long-Only Portfolio for Case A (with and without cryptocurrencies),Source: Own Calculations.Asset Average Portfolio Weights
Minimum-Variance-Portfolio Tangency-Portfolio
No EWCI With EWCI No EWCI With EWCILong-Only Portfolio: No Transaction CostsMON 0.997254 0.997008 0.192668 0.227877CUR 0.000437 0.000555 0.000000 0.000000SOV 0.000000 0.000000 0.046024 0.046591COV 0.000000 0.000000 0.291057 0.276957COR 0.000225 0.000274 0.154598 0.129275STO 0.000585 0.000531 0.000849 0.000849RES 0.000209 0.000201 0.072647 0.062880LUX 0.000006 0.000007 0.212714 0.215571COM 0.001284 0.001225 0.029441 0.029255EWCI 0.000000 0.000198 0.000000 0.010745Measure Portfolio Statistics
Minimum-Variance-Portfolio Tangency-Portfolio
No EWCI With EWCI No EWCI With EWCIMinimum Return -0.002284 -0.002324 -0.044861 -0.044861Mean Return 0.000024 0.000025 0.000386 0.000433Maximum Return 0.001134 0.001163 0.035554 0.051399Standard Deviation 0.000412 0.000419 0.008823 0.010340Maximum Drawdown 0.004555 0.004659 0.177805 0.199183Sharpe Ratio 0.057894 0.059393 0.043771 0.041882Sortino Ratio 0.078063 0.080314 0.059972 0.060333End Value ( C T in EUR) 100.55 100.57 108.35 109.17 Table
A.4: Key Statistics of the Long-Only Portfolio for Case B (with and without Cryp-tocurrencies, 1-Year Rolling Window), Source: Own Calculations. .2 Graphical Appendix (a) Index Development: Test Assets (b) Index Development: Benchmark As-sets Fig.
A.1: Historical Development of the Indices considered in the Dataset, Source:Bloomberg, Thomson Reuters Eikon, CoinMarketCap. (a) GMVP: Benchmark Assets only(b) TP: Benchmark Assets only (c) GMVP: Considering Cryptocurrencies(d) TP: Considering Cryptocurrencies
Fig.
A.2: Portfolio Weight Heatmap for Case B ( T = 54, 1-Year Rolling Windows): Long-Only-Portfolios (No Transaction Costs), Source: Own Calculations. .3 Mathematical Appendix | ∆ ω i | , we can express a general transaction cost function f as f ( | ∆ ω i | ) = f (0) + f | | ∆ ω i | + 12 f | | ∆ ω i | + .... (A.1)We set f (0) = 0. Hence, if no assets are traded, no transaction fee has tobe paid. Furthermore, we also set f | = 0, because transaction costs in thecryptcurrency context do not follow a linear form. Instead, the behavior of themarket participants in times of historic cryptocurrency booms makes us assume,that transaction cost rise disproportional, if the trading volume exceeds a cer-tain limit. Because the market trading volume is not equal to the individualtrading volume, we have to give a short intuition, why changes in the individ-ual portfolio allocations lead to changes in the transaction cost structure: weassume that (rational) investors follow the optimal portfolio, so that a risingnumber of extreme changes in the individual portfolio weights would lead tosignificant market changes. As a consequence, the transaction cost dynamicscan better be captured by the quadratic term f | | ∆ ω i | in the Taylor series.If we interpret f | as ˜ c Test i , this also leads to Eq. (29).2.) Derivation of the parametrization of Ψ in the long-only [0 ≤ ω ≤
1] scenario:At first, we aim at equal expectation values of both cost functions leadingto the equationE[ τ Test (Quad.) i ( | ∆ ω i | )] ! = E[ τ Test (Lin.) i ( | ∆ ω i | )] . (A.2)In graphical terms, this condition can be converted into: Z d ( | ∆ ω i | ) "
12 ˜ c Test i | ∆ ω i | V ρ ( | ∆ ω i | ) ! = Z d ( | ∆ ω i | ) " c Test i | ∆ ω i | V ρ ( | ∆ ω i | ) . (A.3)For ρ ( | ∆ ω i | ) = 1 as the probability density function of the continuous uniformdistribution of | ∆ ω i | and ˜ c Test i = Ψ c Test i , the expression simplifies to Z d ( | ∆ ω i | ) "
12 Ψ | ∆ ω i | V ! = Z d ( | ∆ ω i | ) " | ∆ ω i | V . (A.4)Solving this equation with respect to Ψ leads to ⇔ Z d ( | ∆ ω i | ) (cid:20)
12 Ψ | ∆ ω i | V − | ∆ ω i | V (cid:21) ! = 0 (A.5)51
12 Ψ | ∆ ω i | − | ∆ ω i | (cid:12)(cid:12)(cid:12)(cid:12) = 0 (A.6) ⇔
16 Ψ −
12 = 0 (A.7) ⇔ Ψ = 3 . (A.8)3.) Calculation of the intersection of the linear and quadratic transaction costfunctions in the long-only scenario:To calculate the intersection of both transaction cost functions, we need toset τ Test (Quad.) i ! = τ Test (Lin.) i . (A.9)Inserting the respective formulas for the transaction cost functions leads to12 ˜ c Test i | ∆ ω i | V = c Test i | ∆ ω i | V . (A.10)Solving this equation with respect to | ∆ ω i | leads to12 ˜ c Test i | ∆ ω i | = c Test i . (A.11)Under the assumption of Ψ = 3, the variable ˜ c Test i can now be substituted bythe expression Ψ c Test i = 3 c Test i , which leads to the final solution | ∆ ω i | = | ∆ ω ∗ i | = 23 , (A.12)where | ∆ ω ∗ i | is the calculated amount of changes in the portfolio weights, forwhich both transaction cost regimes lead to the same result. References
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