Investing with Cryptocurrencies -- evaluating their potential for portfolio allocation strategies
Alla Petukhina, Simon Trimborn, Wolfgang Karl Härdle, Hermann Elendner
IInvesting with Cryptocurrencies – evaluatingtheir potential for portfolio allocationstrategies ∗ Alla Petukhina † Simon Trimborn ‡ Wolfgang Karl Härdle § Hermann Elendner ¶ September 18, 2020
Abstract
Cryptocurrencies (CCs) have risen rapidly in market capitalization over thelast years. Despite striking price volatility, their high average returns have drawnattention to CCs as alternative investment assets for portfolio and risk management.We investigate the utility gains for different types of investors when they considercryptocurrencies as an addition to their portfolio of traditional assets. We considerrisk-averse, return-seeking as well as diversification-preferring investors who tradealong different allocation frequencies, namely daily, weekly or monthly. Out-of-sample performance and diversification benefits are studied for the most popularportfolio-construction rules, including mean-variance optimization, risk-parity, andmaximum-diversification strategies, as well as combined strategies. To accountfor low liquidity in CC markets, we incorporate liquidity constraints via theLIBRO method. Our results show that CCs can improve the risk-return profileof portfolios. In particular, a maximum-diversification strategy (maximizing the ∗ Financial support from IRTG 1792 “High Dimensional Non Stationary Time Series,” Humboldt-Universitätzu Berlin, Czech Science Foundation under grant no.19/28231X and NUS FRC grant R-146-000-298-114“Augmented machine learning and network analysis with applications to cryptocurrencies and blockchains”,and the Yushan Scholar Program is gratefully acknowledged. The work of the authors is receiving supportfrom the European Union’s Horizon 2020 training and innovation programme ”FIN-TECH”, under thegrant No. 825215 (Topic ICT-35-2018, Type of actions: CSA) † Humboldt-Universität zu Berlin, School of Business and Economics, Dorotheen Str. 1, 10117 Berlin,Germany, tel: +49 (0)30 2093-99469, e-mail: [email protected] ‡ Department of Mathematics, National University of Singapore, Risk Management Institute, 21 Heng MuiKeng Terrace, I3 Building 04-03, 119613 Singapore tel: +65 6516-1245, e-mail: [email protected] § Humboldt-Universität zu Berlin, IRTG 1792, Dorotheen Str. 1, 10117 Berlin, Germany; Wang YananInstitute for Studies in Economics, N114, Economics Building, Xiamen University Xiamen, 361005 China;School of Business, Singapore Management University, 50 Stamford Road, Singapore 178899; Faculty ofMathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Prague, Czech Republic; Departmentof Information Management and Finance, National Chiao Tung University, Taiwan, ROC, tel: +49 (0)302093-99592, e-mail: [email protected] ¶ Weizenbaum-Institut, Hardenbergstraße 32, 10623 Berlin, Germany, e-mail: [email protected] a r X i v : . [ q -f i n . P M ] S e p ortfolio Diversification Index, PDI) draws appreciably on CCs, and spanning testsclearly indicate that CC returns are non-redundant additions to the investmentuniverse. Though our analysis also shows that illiquidity of CCs potentiallyreverses the results. Keywords : cryptocurrency, CRIX, investments, portfolio management, asset classes,blockchain, Bitcoin, altcoins, DLT
JEL Classification : C01, C58, G11 2
Introduction
Cryptocurrencies (CCs) have exhibited remarkable performance in the decade since Nakamoto(2008) invented the blockchain. Accompanied by huge inflows of capital into the market andstrong swings in prices, CCs have gained strongly in market value. Accordingly, indices likeCRIX (Trimborn and Härdle, 2018, thecrix.de ) were introduced to capture the marketevolution and provide a basis for ETFs. Driven by these developments, cryptocurrencymarkets became increasingly attractive to investors, who have started to consider CCs asa novel class of alternative investments. However, investors differ with regard to their riskprofiles, investment targets, individual trading behaviors, and generally their diverse motivesand preferences, and thus the perspective to include CCs into financial portfolios raises anumber of questions:1. For whom is investing in the CC market valuable? Is the benefit derived from addingCCs to a portfolio dependent on the investor’s objectives (e.g., return-oriented ordiversification seeking)?2. To which type of investor are CC investments most useful? Only professional traderswho rebalance their portfolio frequently, or also less actively trading retail investors?3. Should investors focus on one particular coin (e.g., Bitcoin), a selected few, or ratherbuild a portfolio of a broad selection of CCs?When an investor does decide to include CCs in the portfolio, further questions arise aboutthe choice of CCs for investment and their portfolio weights:4. What exposure to each CC should be held in the portfolio? How informative arepast prices, how stable are positions when re-balancing the portfolio? Do model-freestrategies like equal-weighting provide reasonable results?5. Can these strategies be implemented in practice? In particular, are all CCs liquidenough for inclusion in an investment portfolio? If not, how can investors still profit3rom promising CCs with little trading volume without exposing their portfolio toomuch to illiquidity? Moreover, how is performance affected by honoring such portfoliorestrictions?6. Overall, how do the properties of CC returns affect portfolios? Is a certain type ofportfolio-allocation method more suitable to manage and simultaneously exploit theirproperties?While we review the literature extensively in Section 2, clearly numerous studies haveinvestigated the properties referred to in Question 6, and agree that CCs exhibit remarkablyhigh average realized returns by the standards of traditional financial assets —and correspond-ingly high risk and uncertainty. Not only is price volatility high; also unfavorable propertiesobtrude, including frequent pricing bubbles (Fry and Cheah, 2016; Hafner, 2018; Chen andHafner, 2019; Núñez et al., 2019), accumulation of jumps (Scaillet et al., 2018), even evidenceof price manipulation (Gandal et al., 2018).At the same time, there is evidence of low correlations of CC returns with those of traditionalfinancial assets and other CCs. Therefore, the high risk of CC positions may be compensatedby appropriate returns as well as provide an opportunity to increase portfolio diversification.Results to that effect have been spearheaded by Brière et al. (2015) and Eisl et al. (2015), thefirst to include Bitcoin (BTC) in a portfolio of traditional assets, and subsequently bolsteredby Elendner et al. (2018), who include a broad cross-section of CCs, Chuen et al. (2017),who instead add CRIX, and lately Platanakis and Urquhart (2019) and Akhtaruzzamanet al. (2019), who include Bitcoin in advanced portfolio optimization and find it enhances therisk-return profile.So evidence exists that CCs can be beneficial for investors (Pele et al., 2020). However,taking the investor’s perspective, we see that while prior studies have covered cruciallyimportant aspects of investing with CCs, the outlined questions 1–5 remain fundamentally We do not compare CCs to derivatives, as they clearly constitute underlyings—in fact, a common complaint,albeit ignorant of their economic role, laments that CCs “do not derive their value from any real asset.”CC derivative markets still remain quite nascent. both a broad range of traditional assetstogether with a broad cross-section of CCs. Therein, we test the performance of an extensiveset of common investment strategies and thus consider different types of investors, whilewe employ the LIBRO method to handle liquidity concerns. We consider risk-oriented,return-oriented, risk-return-oriented, and combined strategies; see Table 1 for a full list ofstrategies under consideration. We estimate extending-window and rolling-window approachesoptimizations for a sizable breadth of different common objective functions. Finally, wecompare all strategies based on three different re-allocation frequencies, namely daily, weeklyand monthly, providing results for investors trading at different frequencies. To the best ofour knowledge, we thus present the broadest study on investing with CCs conducted so far.Closest related to our paper are Akhtaruzzaman et al. (2019) and Platanakis and Urquhart(2019), both also studying the influence of CC investment on optimal portfolio composition.5owever, both include only Bitcoin, whereas we consider a broad cross-section of 52 distinctCC price series. Moreover, both consider fewer traditional assets: industry portfolios (so equityonly) in the former paper, US equity and bond investments in the latter, plus commoditiesin a robustness test. In contrast, our set of traditional assets is critically broader: first, asCCs trade globally, our international approach includes equity returns for each of the 5 majoreconomic areas (Europe, USA, Japan, UK, China), as well as region-specific bond returns.Second, we always include alternative investments, namely gold, real estate, commodities,and the returns to FX trades between the five regions’ fiat currencies. Table 2 lists thetraditional assets all our portfolios include. As we have pointed out, this emphatically goesbeyond quantitatively extending prior studies: unless both a broad cross-section of CCs and of traditional assets are included, it remains impossible to determine the magnitude ofdiversification benefits, and more critically, also impossible to distinguish whether apparentbenefits of CCs are indeed present, or if CCs merely proxy for alternative assets.Moreover, we cover a longer time horizon, and can thus include more than 2 years afterpeak CC prices; also, we consider more allocation strategies. Most importantly, since we takethe investor’s perspective, we implement LIBRO and contrast portfolios with weights thatobserve the liquidity constraints with otherwise identical portfolios which do not: it turns outto cricitally affect performance for several popular trading strategies.Our study contributes to answering questions 1–6. Spanning tests show that more than50% of the CCs considered can improve the efficient frontier of a portfolio containing even ourbroad set of traditional assets. We show that purely risk-minimizing investors will optimallychoose to mostly forego CC investment; however, for investors with higher target returns theiraddition seriously expands the efficient frontier. Diversification-oriented investors benefit most,even in terms of maximizing cumulative wealth. We also document that a lower rebalancingfrequency (monthly) of the portfolios generally enhances cumulated returns. As mentioned, Platanakis and Urquhart (2019) do run a robustness test replacing Bitcoin with CRIX, acknowledging theimportance of altcoins. Naturally, diversification across
CCs necessitates an optimization including theirindividual, distinct return series.
6e confirm that several CCs exhibit low liquidity, which can be tackled with the LIBROapproach. Our results highlight the severity of low-liquidity risk, and how analyses that donot take this risk into account will compute investment returns that are infeasible for any butthe smallest personal portfolios.The paper is organized as follows. First, Section 2 reviews the related literature. Section 3provides an overview of the asset-allocation models under consideration, with a focus onconnections between them; therein Section 3.2 explains the approach of model averagingacross investment strategies. Section 4 reviews the LIBRO method. In Section 5 we explainthe methodology for comparing the performance of the models considered. Our dataset ofportfolio components is described in Section 6, and Section 7 presents the results of ouranalyses of out-of-sample performance of all portfolio strategies with CCs and traditionalassets. We conclude in Section 8.Code to produce the results of this paper is available via . Modern portfolio theory builds on the CAPM (Markowitz, 1952; Sharpe, 1964; Lintner, 1965),both a theoretical equilibrium model and a directly applicable statistical approach. Yet,financial markets do not meet its assumptions, so it lacks empirical accuracy. Asset pricingand portfolio optimization address this lack in one of two ways.The first we call the financial-economics approach: it follows Ross’s (1976) arbitrage-pricing theory which keeps the linear structure and adds more factors to capture systematicpatterns in returns. Popularized by Fama and French (1992, 1993), it was extended tofactors for momentum (Jegadeesh and Titman, 1993; Carhart, 1997) or profitability andinvestment (Fama and French, 2015). In principle, the approach renders portfolio optimizationstraightforward and unidimensional: a portfolio is better, the higher its alpha (the interceptafter accounting for all factors’ loadings). In practice, the choice of factors depends on the This approach puts the emphasis on the equilibrium model and is thus often preferred by theorists. quantitative-finance approach, due to its statistical nature. Its idea, inessence, says: if we can capture the (joint) return distribution (and its dynamics) of allinvestable assets (and parameters affecting them), then we can directly estimate portfolioweights to optimize the desired performance metric. Owing to the abundance of statisticaltechniques for the variety of modelling choices and investment objectives, this approachis most common in fund management. However, the easy customization has precluded astandard, unique approach. A portfolio’s optimal allocation thus depends crucially on threeelements: the investment universe, the investment strategy, and the investment objective asdefined by the metric of optimization.Most fundamental is the determination of the investment universe.
Our paper focuses on itsrole by analyzing it for extensive sets of common strategies and objective functions; concretely,on the potential of adding CCs. Historically, starting from stocks and a risk-free interestrate, the diversification benefits to adding bonds (Liu, 2016), foreign exchange (Kroenckeet al., 2013; Barroso and Santa-Clara, 2015; Ackermann et al., 2017), real estate (Benjaminet al., 2001; Addae-Dapaah and Loh, 2005), and commodities (Belousova and Dorfleitner,2012) including gold (Hoang et al., 2015) have been established in the literature. We term all these assets “traditional investments”, and we include proxies for all of them in our benchmark An additional benefit is how it links potential empirical shortcomings to insufficiently captured statisticalproperties, offering remedy via more refined methods. Markowitz (1952). Sharpe (1964). In fact, already Roll (1977) had stressed the “market portfolio” ought to include all wealth. Naturally, hiscritique has led to innumerous suggestions for further asset classes that cannot all be part of our analysis,including private equity (Gompers et al., 2010), fine art (Mei and Moses, 2002; Campbell, 2008), or evenfine wine (Fogarty, 2010; Chu, 2014). additionally
CCs. Only the broad traditional portfolio ensures we assess the diversification potential ofCCs as investments: otherwise, CCs might merely substitute for other alternative investments.Considering CCs as investments contains subtle irony, as Nakamoto (2008) pseudonymouslyintroduced the blockchain as a technology to serve as money, not a profitable investmentopportunity. Thus, initially doubt and debate shrouded the economic role of CCs (Glaseret al., 2014; Yermack, 2015; Böhme et al., 2015; Baur et al., 2018). However, after more than a decade of increasing demand, market capitalizations, andtrading volumes for a multiplying number of CCs, the recently flourishing academic literatureconverges to the consensus that CCs constitute investments, generally, and a distinct assetclass in particular. This literature can be categorized along two dimensions: first, whichCC investment is considered? Only Bitcoin, also a fistful of other highly visible CCs likeEthereum or Ripple, or a broad cross-section of tradable CCs? Second, which portfolioallocations are considered? Only the CC(s), or also traditional markets? If the latter, onlyequity markets, or a broad range of traditional investments?Regarding the first point, the literature started by investigating the properties of the Bitcoinprice process (Kristoufek, 2015; Chu et al., 2015; Cheah and Fry, 2015; Urquhart, 2017; Blau,2018; Bariviera et al., 2017; Osterrieder, 2017; Liu and Tsyvinski, 2018), establishing, inessence, the presence of all critical properties of equity returns, yet often up to an order of More precisely, the intent was a protocol with the emphasis on the tokens’ role as medium of exchange, notas stores of value. Some debate centered on the question whether investments in CCs play an economic role similar to gold:See Dyhrberg (2016) and Shahzad et al. (2019) for affirmative views, and Klein et al. (2018) for a dissentingone. Generally, mainstream economics has joined the research effort on CCs deplorably late; it is now catchingup, see for instance Schilling and Uhlig (2019) and Abadi and Brunnermeier (2019). Game-theoreticmodelling has been more active, including Houy (2016), Dimitri (2017), Caginalp and Caginalp (2019),and Bolt and van Oordt (2020). At the latest update of this writing, in 2020-Q2, the leading dedicated information platform coinmarketcap.com records more than 5000 CCs traded at more than 21,000 markets, totalling a market capitalizationclose to 250 billion USD (almost two thirds of which are due to Bitcoin), with a 24-hour trading volumesurpassing 150 billion USD. Note that the commonly reported thousands of CCs include mostly such with extremely low liquidity: Asof 2020-04-28, only 10 CCs exhibit daily trading volumes exceeding 1 billion USD; volume below 100,000USD exists already among the top 200 CCs. This kickstarted investigations into the joint-return properties of a broadcross-section of CCs (Elendner et al., 2018; Wang and Vergne, 2017; Brauneis and Mestel,2018; Zhang et al., 2018; Wei, 2018), which confirmed the return characteristics of Bitcoin tobe representative for the entire asset class; yet generally so-called altcoins exhibit still higherrisk and mean returns. (Even more extreme were returns of Initial Coin Offerings, ICOs, inparticular during their peak in 2017—see, for instance, Adhami et al. (2018), Momtaz (2018),Momtaz (2019b), and Momtaz (2019a). However, despite the important economic role ofICOs and STOs (Security Token Offerings) as novel channels of venture-capital investment,they are unsuitable for rules-based portfolio allocation, and hence fall outside the scope ofour paper.)A key finding is that correlations are low even among CCs, as long as they are no closesubstitutes or forks. This implies a potential diversification benefit from a broad basket ofCCs (Chuen et al., 2017). Alessandretti et al. (2018), optimizing CCs-only portfolios withLSTMs and decision trees, also find enhanced return performance.As one consequence, CC indices were developed: The CRIX (Trimborn and Härdle, 2018)captures the broad CC market movement with a statistically optimized varying number ofconstituents; CCI30 (Rivin and Scevola, 2018) is a simple, close analogue to stock-market Quick growth in the number of traded CCs was mostly driven by the free-software nature of Bitcoin,allowing forks, and to a lesser degree by development of new (sometimes blockchainless) CCs. The reasons to consider CCs an asset class naturally go beyond the similarity of their return processes;the major reason is that their economic rationale differs decisively from all other asset classes, as theyconstitute the only means to provide real resources to decentralised apps.
Consider a matrix X ∈ R P × N of log returns of N assets for P days. In our comparativeanalysis we rely on a moving-window approach. Specifically, we choose an estimation windowof length K = 252 days (corresponding to the number of trading days in a calendar year).We investigate the performance of strategies for three rebalancing frequencies k : monthly,with k = 21 days, weekly, with k = 5 days, and daily with k = 1 day. For each rebalancingperiod t ( t = 1 , . . . , T , with T the number of moving windows, defined as T = P − Kk ), starting We also test strategies on extending windows as in Trimborn et al. (2019); since the insights are similar,these results are not reported.
11n date K + 1, we use the data in the previous K days to estimate the parameters requiredto implement a particular strategy. These parameter estimates are then used to determinethe relative portfolio weights w in the portfolio of risky assets. Based on these weights, wecompute the strategy’s return in rebalancing period t + 1. This process is iterated by addingthe k daily returns for the next period in the dataset and dropping the corresponding earliestreturns, until the end of the dataset is reached. The outcome of this rolling-window approachis a series of P − K daily out-of-sample returns generated by each of the portfolio strategieslisted in Table 1. To simplify notation, we omit the index t for moving window or rebalancingperiod.The traditional evaluation literature (e.g., DeMiguel et al., 2009; Schanbacher, 2014)considers an investor whose preferences are specified in terms of utility functions and fullydescribed by the portfolio mean µ P and variance σ P . However, Merton (1980) showed that avery long time series is required in order to receive accurate estimates of expected returns.Due to this high margin of error of expected-return estimates some authors, including Haugenand Baker (1991), Chopra and Ziemba (1993) and Chow et al. (2011), suggest to rely only onestimates of the covariance matrix as input of the optimization procedure. Thus, investorsassume that all stocks have the same expected returns and under this strong assumption theoptimal portfolio is the global minimum-variance portfolio. The minimum-variance portfoliostrategy represents one of the so-called risk-based portfolios, i.e., the only input used is theestimate of the variance-covariance matrix. In this paper we consider the most popular ones:Maximum Diversification, Risk-Parity, Minimum Variance and Minimum CVaR portfolio.In Section 3.1 we describe the individual strategies from the portfolio-choice literature thatwe consider. In addition totraditional approaches, we consider a decision maker with riskpreferences specified in percentile terms, and portfolio construction based on higher moments ofthe portfolio return-distribution, such as skewness and kurtosis. Therefore, in our comparativestudy we distinguish three groups of strategies: return-oriented, risk-oriented (or risk-based,as in Clarke et al. (2013)), as well as a tangency portfolio with Maximum Sharpe Ratio12MV-S), which we categorize as a risk-return-oriented strategy.Taking into account that the ranking of models changes over time, and motivated by thefact that in many fields a combination of models performs well (see, e.g., Clemen, 1989;Avramov, 2002), we also extend our analysis to include the combination of portfolio modelsbased on a bootstrap approach inspired by Schanbacher (2014) and Schanbacher (2015). Thedetailed methodology of combined portfolio models is discussed in Section 3.2. In this section we review those models that we consider in the empirical analysis. We alsodiscuss links between the strategies and give conditions under which they are equivalent. Ingeneral, when bringing the theoretical models to the data, we employ in-sample moments ofreturn distributions as estimators of their theoretical counterparts; naturally, all evaluationthen concerns out-of-sample performance. As subsequent prices provide new informationabout assets’ returns, all estimates are updated before any rebalancing trades.In all models we rule out short selling, a standard assumption in the CC literature, giventhat—with the exception of bitcoin, for which futures are traded since December 2017—takingshort positions on CCs is at the very least impractical, if not outright impossible.
The most naïve portfolio strategy sets equal weights (EW) for all constituents: every assetgets a weight w i = 1 /N for i = 1 , . . . , N . If all constituents have the same expected returnsand covariances, the EW portfolio is mean-variance optimal. However, there is no need forassumptions or estimates regarding the distribution of the assets’ returns to implement EW.Moreover, as DeMiguel et al. (2009) show, EW allocations can actually perform well, inparticular in settings of high uncertainty, i.e., parameter instability—the model-free approachavoids overfitting. This is also the reason why the F5 crpto strategy builds on an EW baselinebenchmark. 13 .1.2 Optimal mean-variance portfolio Many portfolio managers still rely on Markowitz’ risk-return or mean-variance (MV) rule,combining assets into an efficient portfolio offering a risk-adjusted target return (Härdle andSimar, 2015). MV portfolios are optimal if the financial returns follow a normal distribution(which, generally, they do not), or if risk can be fully captured via volatility (which, generally,it cannot). Otherwise, MV serves as an approximation, which in favor of tractability andconvenience accepts the drawbacks widely discussed in the literature: high portfolio concen-tration, i.e., high portfolio weights for a limited subset of the investment universe, and highsensitivity to small changes in parameter estimates of µ and σ , see Jorion (1985), Simaan(1997), Kan and Zhou (2007). In a Gaussian world, portfolio weights w are obtained bysolving the following optimization problem:min w ∈ R p σ P ( w ) def = w > Σ w s.t. µ P ( w ) = r T ,w > N = 1 , w i ≥ def = E t − { ( X − µ )( X − µ ) > } and µ def = E t − ( X ) are the sample covariance matrix andvector of mean returns respctively, µ P ( w ) def = w > µ , is the portfolio mean and r T the targetreturn, ranging from minimum return to maximum return to trace out an efficient frontier. E t − is the expectation operator conditional on the information set available at time t − A strong limitation of Markowitz-based portfolio strategies lies in the assumption of Gaussiandistributions of assets’ log-returns. Absent those, for investors whose preferences are not fullydescribed by a quadratic utility funcion, variance or volatility is an insufficient risk measure,leading the MV strategy to give a non-optimal portfolio composition. Importantly, returns ofCCs have even heavier tails as compared to those of equities, as detailed in Chuen et al. (2017)and Elendner et al. (2018). The descriptive statistics of our investment universe in Figure 7and Table 9 in Appendix 9.2 again provide strong evidence of this heavy-tailed distributionsfor CCs. Therefore, we include a strategy that accounts for higher moments via ConditionalValue at Risk (CVaR): we include a Mean-CVaR-optimized portfolio as in Rockafellar andUryasev (2000) and Krokhmal et al. (2002).For a given α < .
05 risk level, the CVaR-optimized portfolio weights w are derived as:min w ∈ R N CVaR α ( w ) , s.t. µ P ( w ) = r T , w > p = 1 , w i ≥ , (2)CVaR α ( w ) = − − α Z w > X ≤− VaR α ( w ) w > Xf ( w > X | w ) dw > X, (3)with ∂∂w > X F ( w > X | w ) = f ( w > X | w ) the probability density function of the portfolio returnswith weights w . VaR α ( w ) is the corresponding α -quantile of the cumulative distributionfunction, defining the loss to be expected in ( α · One traditional risk-based portfolio strategy is based on the concept of risk parity. Theunderlying idea is to set weights such that each asset has the same contribution to portfoliorisk, see Qian (2006). Maillard et al. (2010) derive properties of such portfolios and renamethem “equal-risk-contribution” (ERC) instruments. The Euler decomposition of the portfoliovolatility σ P ( w ) = √ w > Σ w (Härdle and Simar, 2015) allows to present it in the followingform: σ P ( w ) def = N X i =1 σ i ( w ) = N X i =1 w i ∂σ P ( w ) ∂w i , (4)where ∂σ P ( w ) ∂w i is the marginal risk contribution and σ i ( w ) = w i ∂σ P ( w ) ∂w i is the risk contributionof the i -th asset. So, to construct the ERC portfolio, we calibrate: σ i ( w ) = 1 N ∀ i (5)The ERC portfolio can be compared to the EW portfolio: instead of allocating capitalequally across all assets, the ERC portfolio allocates the total risk equally across all assets.Consequently, if variances of log-returns were all equal, the ERC portfolio would becomeidentical to EW portfolio. The ERC portfolio is also comparable to the MinVar portfolio,which focuses on parity of marginal contributions of all assets. Originally, the Maximum Diversification portfolio (MD) uses an objective function introducedin Choueifaty and Coignard (2008) that maximizes the ratio of weighted average assetvolatilities to portfolio volatility or diversification ratio as in Equation (22). In our study,16nstead of the diversification ratio we maximize the Portfolio Diversification Index (PDI)proposed by Rudin and Morgan (2006). It consists in assessing a Principal ComponentAnalysis (PCA) on the weighted asset returns’ covariance matrix, i.e., identifying orthogonalsources of variation. In its original form, PDI does not account for the actual portfolio weights,here we incorporate weighted returns. We optimize:max w ∈ R N PDI P ( w ) , s.t. w > p = 1 , w i ≥ P DI P ( w )=2 N X i =1 iW i − , (7)where W i = λ i P Ni =1 λ i are the normalised covariance eigenvalues λ i in decreasing order, i.e., therelative strengths. Thus, an “ideally diversified” portfolio, i.e., when all assets are perfectlyuncorrelated and W i = 1 /N for all i , then P DI = N . On the contrary a P DI ≈ Additional to individual allocation models, we also consider combinations of models. Afterall, every individual model is subject to estimation risk; the idea of combining (or averaging)models in order to reduce such risk received attention in various areas, and particularly inforecasting (Avramov, 2002). Traditional model-averaging methods use information criteria—like AIC or BIC—to identify relative shares of models. Across portfolio-allocation models thelikelihood is unknown, however, therefore we calculate model shares with the loss function l ,defined as 17 ( w ) = w > ˆ µ − γ w > ˆΣ w. (8)The parameter γ reflects the investor’s risk aversion, with γ being large (small) for arisk-averse (risk-seeking) investor. We use two approaches to construct combined strategies:Naïve averaging of the portfolio weights, as well as the combination method based on abootstrap procedure described in Schanbacher (2014). However, in order to account forpossible time series dependencies at a daily frequency, we apply the stationary bootstrapalgorithm of Politis and Romano (1994) with automatic block-length selection proposed byPolitis and White (2004).Consider a set of m asset allocation models. The corresponding portfolio weights per modelare given by W = ( w , . . . , w m ). Relative shares of (or beliefs in) individual models are π = ( π , . . . , π m ), such that π > m = 1. Then the asset weights for the combined portfolio aregiven by: w comb = m X i =1 π i w i (9)The Naïve combination over all asset allocation models just assigns equal shares, i.e., π it = m for all i = 1 , . . . , m .The alternative, more sophisticated approach is to set the share π it equal to the probabilitythat model i outperforms all other models. We apply a bootstrap method to estimate theseprobabilities. For every period t we generate a random sample (with replacement) of k returnsusing returns X k ( t − . . . X k ( t − K , i.e., K -long returns vectors of the t − m asset allocation models to these bootstrapped returns. The procedure isrepeated B times. Let s i,b = 1 if model i outperforms in terms of the loss function othermodels in the b -th bootstrapped sample, otherwise s i,b = 0. The probability of model i beingbest is then estimated as ˆ π it = 1 B B X b =1 s i,b (10)18 odel Reference Abbreviation Model-free strategies
Equally weighted DeMiguel et al. (2009) EW
Risk-oriented strategies
Mean-Variance – min Var Merton (1980) MinVarMean-CVaR – min risk Rockafellar and Uryasev (2000) MinCVaREqual Risk Contribution Maillard et al. (2010) ERC(Risk-parity)Maximum Diversification Rudin and Morgan (2006) MD
Return-oriented strategies
Risk-Return – max return Markowitz (1952) RR-MaxRet
Risk-Return-oriented strategies
Mean-Variance – max Sharpe Jagannathan and Ma (2003) MV-S
Combination models
Naïve Combination Schanbacher (2015) CombNaïveWeight Combination Schanbacher (2014) CombTable 1: List and categorization of all asset allocation models we implement, including theirabbreviations and references.where B = 100 is our number of independent bootstrap samples, and s i,b = 1 if model i is thebest model in the b -th sample. In this section, we review the LIBRO framework for portfolio formation, which preventstoo high portfolio weights for low-liquidity assets, by introducing weight constraints in theportfolio optimization which depend on liquidity.Liquidity, however, does not have a unique definition; different concepts and measuresabound. Wyss (2004) and Vayanos and Wang (2013) survey the extensive literature onliquidity measures in equity markets; the literature on CC liquidity is still scarce, with notableexceptions of Brauneis et al. (2020) and Scharnowski (2020). Due to the highly fragmentedmarket structure of CC exchanges (no dominant or central exchange is trading all assets), we19mploy Trading Volume (TV) as our proxy for liquidity. TV is also the basis for the widelyused (Amihud2002) illiquidity measure, and proved suitable for the LIBRO methodology. Inprinciple, alternative measures like the bid-ask spread would also be applicable, as manyexchanges report bid and ask prices; however, reliable order-book data aggregated across exchanges and for all CCs is lacking. TV, in contrast, is available for practically all CCmarkets, and aggregated without problems. For these reasons, we follow Trimborn et al.(2019) and employ TV as our liquidity measure. TV is defined as
T V ij = p ij · q ij , (11)where p ij is the closing price of asset i at date j , and q ij is the volume traded at date j ofasset i . The liquidity of asset i in period t can then be measured with the sample median oftrading volume, T V i = 12 ( T V i,up + T V i,lo ) , (12)where T V i,up = T V i, d l +12 e and T V i,lo = T V i, b l +12 c .Define M as the total amount invested in all N assets, so that M w i denotes the marketvalue held in asset i . Trimborn et al. (2019) formulate the constraint on the weight of asset i as M w i ≤ T V i · f i , (13)where f i captures the speed with which an investor intends to be able to clear the currentposition in asset i via multiples of TV. For example, f i = 0 . i must not exceed 50% of median trading volume. It results in a boundary for the weight on Technically, CC markets never close; the terminology “closing price” is still used in reference to the lastprice of a day, where days are customary defined on UTC time. i as w i ≤ T V i · f i M = b a i . (14)The beauty of this approach lies in its ease to include it into any portfolio optimization. While Section 3 presents the set of common asset-allocation models we implement, no uniquemetric exists to evaluate and compare them. In order to draw conclusions about the effect ofadding CCs to broadly diversified portfolios, we pursue three dimensions: First, we calculate arange of widely used performance measures in Section 5.1. Second, in Section 5.2 we run directtests for differences between strategies on a pair-wise basis. Third and finally, in Section 5.3we address the diversification effect of CCs directly by calculating three well-known measuresof portfolio concentration.
To assess the performance of the investment strategies we consider as it develops over time,we employ the following five common performance criteria widely used in literature, as wellas by practitioners. Performance measures are computed based on the time series of dailyout-of-sample returns generated by each strategy.First, we measure the cumulative wealth (CW) generated by each strategy iW i,t +1 = W i,t + ˆ w > i,t X t +1 , (15)starting with an initial portfolio wealth of W = $1. Cumulative wealth, while naturallyof high interest as a measure of performance achieved over the period considered, is notsufficient to rank our allocation approaches. Therefore why we also compute two traditional21easures of risk-adjusted returns: the Sharpe ratio, and the certainty equivalent. Moreover,we provide the Adjusted Sharpe Ratio (ASR) in order to address the MinCVaR strategy andthe non-Gaussian nature of the return distributions.The Sharpe Ratio (SR) of strategy i is defined as the sample mean of out-of-sample excessreturns (over the risk-free rate), scaled by their respective standard deviation. This definitionpresumes an unambiguous risk-free rate, inexistent in the global context of CCs. Fortunately,our sample period is characterized by most of the global economy at or very close to the zerolower bound on interest rates; so we can sidestep the question by implicitly setting the risklessrate to 0 and defining d SR i = ˆ µ i ˆ σ i . (16)The Certainty Equivalent (CEQ) captures, for an investor with a given risk aversion γ , theriskless return that said investor would consider of equal utility as the risky return underevaluation. For the case γ = 1, it is equivalent to the close-form solution of Markowitz (1952)portfolio optimization problem in Equation (1). (cid:92) CEQ i,γ = ˆ µ i − γ σ i (17)While there is debate about the risk-averion coefficient best describing investors going back toMehra and Prescott (1985), we argue that current CC investors are unlikely to be characterizedby extremely high risk aversion, and calculate the CEQ in the empirical part of our paper witha γ of 1. As can be noted, the CEQ corresponds to the loss function l defined in Equation (8).The CEQ and in particular the SR are more suitable to assess of strategies when assetsexhibit normally distributed returns. To address this drawback, Pezier and White (2008)propose the Adjusted Sharpe Ratio (ASR). ASR explicitly incorporates skewness and kurtosis: (cid:92)
ASR i = d SR i (cid:20) (cid:18) S i (cid:19) d SR i − (cid:18) K i (cid:19) d SR i (cid:21) (18)22here SR i denotes the Sharpe Ratio, S i the skewness, and K i the excess kurtosis of asset i .Thus, the ASR accounts for the fact that investors generally prefer positive skewness andnegative excess kurtosis, as it contains a penalty factor for negative skewness and positiveexcess kurtosis.To assess the impact of potential transaction costs associated with asset rebalancing, wealso calculate two measures for turnover. Portfolio turnover is computed to capture theamount of trade necessary on rebalancing dates as
T O i = 1 T − K T − K X t =1 N X j =1 | ˆ w i,j,t +1 − ˆ w i,j,t + | (19)where w i,j,t and w i,j,t +1 are the weights assigned to asset j for periods t and t + 1 and w i,j,t + denotes its weight just before rebalancing at t + 1. Thus, we account for the price changeover the period, as one needs to execute trades in order to rebalance the portfolio towardsthe w t target. High turnover will imply significant transaction costs; consequently, the lowerTurnover of a strategy, the better it performs. Target turnover, the second turnover-related measure, captures the amount of change intarget weights between two consecutive rebalancing dates as
T T O i = 1 T − K T − K X t =1 N X j =1 | ˆ w i,j,t +1 − ˆ w i,j,t | (20)In contrast to Equation (19), here the difference between weights spans the time interval ofone rebalancing period, instead of the (conceptually infinitesimal) duration of rebalancingtrades. Therefore, the realized price paths of the assets affect the measure only insofar as theylead to different parameter estimates and thus a revision in target weights. The differencebetween the two turnover measures is best illustrated by considering the EW strategy: it mayrequire high turnover to return to exactly equal weights per asset every rebalancing date; yetby definition it will never exhibit and target turnover.23 .2 Testing for performance differences between strategies To test if strategies are significantly different from each other, we provide the p -values ofpairwise tests. The common approach by Jobson and Korkie (1981) is widely used in theperformance evaluation literature (e.g., also in DeMiguel et al., 2009). However, this testis not appropriate when returns have tails heavier than the normal distribution. Therefore,as a testing procedure we rely on the Ledoit and Wolf (2008) test with the use of robustinference methods. We test for difference of both CEQ and SR, and report results for theHAC (heteroskedasticity and autocorrelation) inference version. The procedure is describedin Appendix 9.1. To evaluate portfolio concentration and portfolio diversification effects, we calculate threemeasures: a) the Portfolio Diversification Index (PDI) as in introduced in Equation (7), b) Effective N as introduced by Strongin et al. (2000), and c) the Diversification Ratio. Effective N is defined as N eff ( w t ) = 1 P Nj =1 w j,t (21)with j = 1 , . . . , N indexing assets. Effective N varies from 1 in the case of maximal concen-tration, i.e., the portfolio entirely invested in a single asset, to N —its maximum achievedby an equally-weighted portfolio. The design of Effective N is related to other traditionalconcentration measures, e.g., the Herfindahl Index, the sum of squared market shares tomeasure the amount of competition. Effective N can be interpreted as the number of equally-weighted assets that would provide the same diversification benefits as the portfolio underconsideration.The diversification ratio, suggested by Choueifaty et al. (2011), measures the proportion of24 portfolio’s weighted average volatility to its overall volatility: DR ( w t ) = w > t σ t q w > t Σ t w t = w > t σ t σ P,t ( w t ) (22)Thus, the diversification ratio has the form of the Sharpe Ratio in Equation (18), with thesum of weighted asset volatilities replacing the expected excess return. In case of perfectlycorrelated assets, the DR equals 1; in contrast, in a situation of “ideal diversification,” i.e.,perfectly uncorrelated assets, DR = √ N . Hence, in our empirical study we report theresults for DR , for two reasons: First, to make it comparable to the other two used metrics,and second, because Choueifaty and Coignard (2008) demonstrate that for a universe of N independent risk factors, the portfolio that weighted each factor by its inverse volatility wouldhave a DR equal to N . Hence DR can be viewed as a measure of the effective degrees offreedom within a given investment universe. For the empirical analysis, we collect daily price data on a sample of CCs and traditionalfinancial assets (including alternative investments) over the period 2015-01-01 to 2019-12-31(1304 daily log-returns). CC prices are provided by CoinGecko, data for traditional assets isacquired from Bloomberg. Many CCs were established only after January 2015, or ceasedto trade prior to the period we study. Since investors who apply rules-based optimizationtechniques usually only consider assets with sufficient price histories, we require CCs to havea continuous return time-series over the period of our study in order to be included. Byexcluding coins that did not already circulate in January 2015, went extinct before December2019, or have only patchy price series, we effectively focus on solid CCs, of interest to investorsconsidering positions in this novel asset class. We also sidestep ICOs. Hence, our final data We also run our entire analysis for a sample period extending until end of December, 2017. For this shorterperiod, 55 CCs fulfil our criteria, and with minor exceptions only for combined strategies, all our resultsremain qualitatively unchanged. To evaluate the performance of each of the strategies we consider, our research questionstudies the effects of including CCs as an addition to classical, well-diversified portfolios.Therefore, our investment universe always includes 16 traditional assets from 5 asset classes:equity, fixed-income, fiat currencies, commodities, and real estate. Since CCs are global innature, our traditional assets cover the 5 main economic areas around the globe (Europe, USA,UK, Japan, China). In this way, the asset space is sufficiently broad to allow diversificationwithout CCs, ensuring any relevance of CCs we find is genuine, and at the same time isstill narrow enough to allow us to add each CC individually as an asset without leading tohigh-dimensionality issues in covariance estimation. The full list of traditional constituentsof the investment universe is provided in Table 2. Tables 9 and 8 in Appendix 9.2 reportsummary statistics of all constituents considered in our empirical study.The main properties of our data correspond to the findings of the prior literature, e.g.,Chuen et al. (2017): CCs outperform traditional asset classes in terms of average dailyrealised returns, their returns exhibit higher volatility, with means mostly positive whilethe medians are mostly negative, positive movements occur less frequently than negativeones, but with higher magnitudes (absolute values of minima and lower deciles are less thanof maxima and higher deciles for the majority of CCs). Correlation analysis of the top 5 As a robustness test, we also calculate with extending windows, where no historical data is dropped andonly new observations added as they become observable. The results are qualitatively the same. . In this section we evaluate the out-of-sample performance of the portfolio allocation strategiesin order to address Questions 1–6. We analyze two dimensions: First, how does risk-adjustedperformance compare across different strategies and performance measures? Second, whichdiversification benefits are generated by each method?27 .1 Including CCs in portfolios: performance effects
The first step of our performance analysis examines how adding CCs to a portfolio affectsefficient frontiers. In principle, the efficient frontier is unique, thus identical for all allocationstrategies. However, it depends on the risk measure (variance or CVaR in this paper), as wellas on whether liquidity constraints are enforced (via LIBRO in this paper) or not.Our second step then addresses the performance comparison across portfolio strategies, interms of cumulative wealth as well as popular risk-adjusted measures.Figure 1: Efficient frontiers surfaces: the first column displays the frontiers for portfolios withonly traditional assets (including alternative investments, but no cryptocurrencies,CCs) as constituents, the second column adds CCs without liquidity constraints, andthe third column instead adds only CCs up to a liquidity constraint (via the LIBROapproach with an investment sum of USD 10 mln). The top row depicts frontiersfrom mean-variance optimization, the lower one from mean-CVaR optimization. Allfrontiers are built on a daily basis and plotted over the period from 2016-01-01 to2019-12-31. CCPEfficient_surface
Figure 1 plots efficient frontiers for three groups of assets: only traditional assets, traditionalassets & CCs without liquidity constraints, and traditional assets & liquid CCs, up to the28onstraint defined via the LIBRO approach with an investment sum of USD 10 mln. The toprow depicts frontiers from mean-variance optimization, the lower three panels are based onmean-CVaR-optimal allocations. All panels show frontiers built on a daily basis, evolvingover time.For both optimization rules, including CCs leads to a distinct extension of the frontiers:for low levels of risk, portfolios with CCs give a similar level of return as without them, butmuch higher expected returns can be sought when CCs are included. The second importantobservation is that mean-variance frontiers, in most cases, are shorter than mean-CVaRfrontiers (the same level of returns has lower variance than CVaR), evidence of risk notbeing inadequately captured by variance, in line with expectations. The LIBRO approachshortens the frontiers especially in the beginning of the investment period, because it limitsthe influence of turbulently growing CCs with low trading volumes. At the same time, it isvisible that starting roughly in January 2017, the difference between frontiers with (LIBRO)and without constraints all but vanishes—a change driven by the extreme growth of tradingvolumes together with capitalisation of the entire CC market during that boom period.The CC market crash in early 2018 is also clearly visible as the frontiers collapse. At thetrough, series of strongly negative returns amidst high volatility and evaporating liquiditylead to CCs playing close to no role in optimal portfolios. As the market consolidates, in2019 CCs again pick up their role in extending the efficient frontier: however, until today thediscrepancy between portfolios with and without concern for liquidity considerations remainspronounced. Consequently, the importance of limiting exposure to illiquid CCs remains high.Portfolio optimization without liquidity constraints may promise an attractive performancein theory which it cannot realize in the market.
First we examine cumulative wealth, produced by the allocation strategies we study. Figures 2and 3 display the dynamics of cumulative wealth for eight of the strategies considered,29ith and without enforcing liquidity constraints, respectively. As benchmarks we also plotS&P100, EW, MV-S and MinVar portfolios built only from traditional investment constituents(Traditional Assets, “TrA”). The EW strategy is displayed separately in Figure 4, and discussedsubsequently. Table 3 summarizes all performance indicators.The following conclusions can be drawn regarding final Cumulative Wealth (CW) overthe entire period of our study when ignoring liquidity: despite CCs trading far below theirhistorical peaks at the end of our time span, most portfolios with CCs generally outperformbenchmark portfolios with only conventional constituents. However, the discrepancies acrossstrategies are huge, and the worst-performing strategy RR-MaxRet, which invests alwaysin the asset with the highest expected return (and thus most often in a CC), ends up withwhat can be called a catastrophic result: over the four years of our study, it loses 97% of itsinitial wealth by the end of 2019. Critically, the strategy did provide stellar results during theboom phase of 2017, exceeding a multiple of 20 times initial wealth at its peak. Yet clearly,historical returns were no long-term predictor of expected returns for the best-performingCCs, and the lack of diversification hurt this strategy badly.On the other end of the spectrum, the highest result is achieved by MD, with an accumulatedfinal wealth of 275%. This amounts to an annualized rate of return of just below 30% overa 4-year period in which the S&P100 lost 10%. Critically, this result is also achieved byinvesting in small CCs (and therefore also follows the boom-and-bust cycle to a comparabledegree): the difference is driven by the very strong diversification the MD strategy pursues bydesign. It is therefore not surprising that ERC turns out the second-best strategy, with a+22% return over the period. Its construction successfully limits its exposure to the extremesduring 2017/18 to about an order of magnitude lower than MD.Regarding the combined strategies, the naïve version is strongly susceptible to RR-MaxRet,while the bootstrapped version performs quite well.Finally, the model-free EW strategy with CCs underperforms with a final loss of 13%,while equal weighting across only traditional assets achieves the best performance among the30enchmark strategies. However, Figure 4 shows how EW performance exhibits high variationover the time span, similar in nature to MaxRet and MD. The figure displays MD and EWseparately, to elucidate two important points: first, how disproportionately the performanceof small coins exceeded the gains of established CCs in the 2017 price explosion; second, howseriously calculated results of portfolio allocation rules can diverge from returns achievable byinvestors if lack of liquidity is not taken into account.Generally, LIBRO portfolios have mixed results in terms of cumulative wealth. Mostimportantly, MD underperforms when enforcing LIBRO constraints. Of course, this impliesthat the high performance of unconstrained optimization can only be reached for verysmall investment sums. For larger portfolios, when the liquidity constraints turn binding,performance need not necessarily suffer. By limiting the exposure to individual (and thusalso small) coins, some strategies, including RR-MaxRet, are positively affected by LIBRO.When ignoring the liquidity risk, this strategy retained 3% of its initial value; with LIBRO itretains 59.1%. Also the combined strategy COMB, which provides a positive performancewithout LIBRO, further improves by 8.6% when protecting the portfolio from liquidity risk.Next, we analyse risk-adjusted performance for all portfolios. While MD demonstratessuperior absolute performance, ERC dominates in terms of risk-adjusted performance, inparticular in terms of its (adjusted) Sharpe Ratio of 0 . . . . . higher turnover. This isof concern to investors, as it prompts higher transaction costs. At first sight this observation31 C u m W ea l t h RR-MaxRet C u m W ea l t h MV-S C u m W ea l t h MinVar C u m W ea l t h ERC C u m W ea l t h MinCVaR C u m W ea l t h MD Jan16 Jan17 Jan18 Jan19
Date C u m W ea l t h COMB NAIVE
Jan16 Jan17 Jan18 Jan19
Date C u m W ea l t h COMB
Figure 2:
Performance in terms of cumulative wealth of portfolio strategies without liquidityconstraints with monthly rebalancing ( l = 21) over the period from 2016-01-01 to2019-12-31 with the following colour code: S&P100, EW–TrA, RR-MaxRet–TrA andthe corresponding allocation strategy from Table 1. “TrA” denotes only traditional,i.e., non-CC assets are included. Note that the date axes are aligned, but the wealthaxes are not, due to large disperion in scales. CCPPerformanceappears counterintuitive, as restricted weights could be expected to reduce trading needs(due to positions partially remaining at their binding limits). The puzzle is explained by thelast two columns, reporting target turnover: clearly, changes in target weights are mitigatedvia the liquidity constraints, corresponding to intuition. At the same time, it is exactlysmall and illiquid CCs which exhibit the largest volatility, and thus prompt larger tradeswhen at the next rebalancing date positions are brought back to target weights. EnforcingLIBRO constraints leads to positions in more (and prone to be smaller) CCs, triggering largerrebalancing needs in terms of portfolio turnover.32 C u m W ea l t h RR-MaxRet C u m W ea l t h MV-S C u m W ea l t h MinVar C u m W ea l t h ERC C u m W ea l t h MinCVaR C u m W ea l t h MD Jan16 Jan17 Jan18 Jan19
Date C u m W ea l t h COMB NAIVE
Jan16 Jan17 Jan18 Jan19
Date C u m W ea l t h COMB
Figure 3:
Performance in terms of cumulative wealth of portfolio strategies with liquidityconstraints (based on LIBRO at the level of USD 10 mln) and monthly rebalancing( l = 21) over the period from 2016-01-01 to 2019-12-31 with the following colour code:S&P100, EW–TrA, RR-MaxRet–TrA and the corresponding allocation strategyfrom Table 1. “TrA” denotes only traditional, i.e., non-CC assets are included. Notethat the date axes are aligned, but the wealth axes are not, due to large disperionin scales. CCPPerformanceFinally, Table 4 reports when the differences between strategies in terms of CEQ or SR aresignificant, based on tests described in Appendix 9.1. Although MD, COMB, and in particularMV-S have SR and CEQ higher than the EW strategy, tests do not support significance ofthis difference. In constrast, the ERC portfolio exhibits a higher SR and this difference issignificant. The comparison of risk-adjusted metrics for MinVar and MinCVaR reveals thatthey differ significantly from each other—testament to the strong deviation of CC returns fromthe normal distribution. MinCVaR also differs significantly from the diversifying strategiesMD and ERC. 33 llocation Portfolio performance measures: monthly rebalancingStrategy CW SR ASR CEQ TO TTONo const 10 mln No const 10 mln No const 10 mln No const 10 mln No const 10 mln No const 10 mln Benchmark strategies
S&P100 0 .
900 0 . − . − . − . − .
016 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . .
102 1 .
102 0.033 0.033 0.033 0.033 − . − .
001 3 .
615 3 .
615 0 .
000 0 . .
076 1 .
076 0 .
028 0 .
028 0 .
028 0 .
028 0 .
000 0 .
000 2 .
199 2 .
199 0 .
274 0 . .
877 0 . − . − . − . − . − . − .
001 4 .
345 4 .
345 0 .
000 0 . Risk-oriented strategies
MinVaR 0 .
990 0 . − . − . − . − . − . − .
002 8 .
541 7 .
672 0 .
056 0 . .
021 1 .
018 0 .
026 0 .
023 0 .
026 0 .
023 0 .
000 0 .
000 14 .
884 8 .
093 0.112 0.114ERC 1 .
224 1 .
035 0.033 0 .
009 0.033 0 .
009 0 .
000 0 .
000 4 .
193 9 .
840 0 .
058 0 . .
858 0 . − .
003 0 . − .
003 0 . − .
001 20 .
315 48 .
707 0 .
391 0 . Return-oriented strategies
RR-MaxRet 0 .
030 0 . − . − . − . − .
016 0 .
000 0 .
000 0.687 0.731 0 .
687 0 . Risk-Return-oriented strategies
MV-S 1 .
090 1 .
096 0 .
024 0 .
028 0 .
024 0 .
027 0 .
000 0 .
000 4 .
021 8 .
591 0 .
291 0 . Combination of models
COMB NAÏVE 0 .
716 0 . − . − . − . − . − .
001 0 .
000 3 .
553 36 .
731 0 .
211 0 . .
048 1.134 0 .
010 0 .
029 0 .
010 0 .
029 0 .
000 0 .
000 6 .
758 5 .
759 0 .
148 0 . Table 3: Performance measures for all investment strategies as well as benchmarks over theentire time period from 2016-01-01 to 2019-12-31, with monthly rebalancing ( l = 21).The performance measures are final cumulative wealth (CW), the Sharpe ratio (SR),the adjusted Sharpe ratio (ASR), the certainty equivalent (CEQ), and turnover. “TrA”denotes only traditional, i.e., non-CC assets are included. Strategies are detailed inTable 1. Highest results are highlighted in red.As a robustness check, we also conduct all analyses for weekly and daily rebalancing ofportfolios. Results are provided in Appendix 9.4, generally confirming the conclusions so far,and show that the qualitative results are robust with regard to the rebalancing frequency. We separately analyse diversification characteristics of the allocation rules for two reasons:On the one hand, CCs are known from the literature for their diversifying properties; on theother hand, the most diversifying strategies MD and ERC performed best. First, we examinethe composition of the optimal portfolios over time. Second, we run mean-variance spanningtests in order to establish if CCs are a valuable addition to broadly diversified portfolios oftraditional assets. Third, we analyse diversification across the portfolio strategies by meansof dedicated diversification measures. 34 an16 Jan17 Jan18 Jan1905101520253035404550 C u m W ea l t h MD Jan16 Jan17 Jan18 Jan19024681012141618 C u m W ea l t h MD Figure 4:
Performance in terms of cumulative wealth of portfolio strategies of the maximum-diversification strategy (MD) without (left panel) and with (right panel) liquidityconstraints (based on LIBRO at the level of USD 10 mln), with monthly rebalancing( l = 21) over the period from 2016-01-01 to 2019-12-31. For reference, the equally-weighted EW strategy is displayed. Note that the date axes are aligned, but thewealth axes are not, due to large disperion in scales.Allocation strategy 1 2 3 4 5 6 7 8 9 10 111 S&P1002 EW-TrA3 EW4 RR Max Ret5 MV-S6 MinVar7 ERC8 MinCVaR9 MD10 COMB NAÏVE11 COMBTable 4: Tests for difference between the Sharpe ratio SR (lower triangle) and the certaintyequivalent (CEQ, upper triangle) of all strategies with respect to each other: color-coded p -values with significance at the , and level (without liquidityconstraints). CCPTests35igure 5: Evolution of the portfolio composition (i.e., relative weights) of all allocation strate-gies (without liquidity constraints) with monthly rebalancing over the period from2016-01-01 to 2019-12-31: the black line separates conventional assets (“TrA,” upperyellow part of the spectrum) from cryptocurrencies (CCs, lower green-blue part ofthe spectrum). CCPWeights Figures 5 and 6 plot the evolution of portfolio constituents across time, without and withliquidity constraints, respectively. At each date on the abscissa, the simplex of weights iscolor-coded vertically, with traditional assets on the light end of the spectrum and CCstowards the dark end; a black lines indicates the boundary between the two groups. We cansee wide variation in the extent to which the strategies rely on CCs: MaxRet and MD areprone to invest heavily in CCs, while risk-oriented strategies like MinVar and MinCVaR hardly36igure 6: Evolution of the portfolio composition (i.e., relative weights) of all allocation strate-gies with a position limit of USD 10 mln (via the LIBRO approach) with monthlyrebalancing over the period from 2016-01-01 to 2019-12-31: the black line separatesconventional assets (“TrA,” upper yellow part of the spectrum) from cryptocurrencies(CCs, lower green-blue part of the spectrum). CCPWeightsinclude any. The risk-return-oriented strategy MV-S employs CCs conservatively, yet it doesreach at times noteworthy allocations even against the background of such a well-diversifiedportfolio of traditional assets. The share of CCs is lower in the last 2 years of the time period,but does not drop to zero.Most importantly, the figures point out how the LIBRO approach, as expected, significantlyaffects portfolio weights; the most visible difference arises for models with a high share of CCs,namely MD and RR-MaxRet, but also ERC, where it mitigates the exposure particularly in37he first half of the investment period.The weights distribution of the COMB portfolio undergoes quite pronounced changes overthe investment period: from high concentration of traditional assets to high concentration ofCCs, and back—confirming that no individual model outperforms its competitors permanently.To shed more light on how these weights affect the performance of each strategy’s portfolio,we also compare the risk structures for all strategies in Figures 8 and 9. After all, the volatilitystructure of CCs leads to disproportionate risk contributions relative to their capital weights:traditional assets affect changes in portfolio values to a visibly lower degree.
In order to investigate the impressions from the efficient-frontier plots in Section 7.1.1, weconduct two mean-variance spanning tests on each of the 52 CCs: first, the corrected test ofHuberman and Kandel (HK, 1987), second the step-down test by Kan and Zhou (2012).
Cryptocurrency F-Test F-Test1 F-Test2BCN 3.28 1.23 5.32(0.04) (0.27) (0.02)DOGE 1.73 0.01 3.46(0.18) (0.92) (0.06)EAC 1.70 0.09 3.32(0.18) (0.76) (0.07)NLG 2.79 4.31 1.26(0.06) (0.04) (0.26)PPC 3.19 0.61 5.78(0.04) (0.44) (0.02)XMG 1.86 3.44 0.28(0.16) (0.06) (0.60)XRP 1.88 0.83 2.93(0.16) (0.36) (0.09)
Table 5: Spanning Tests for individual cryptocurrencies with respect to the efficient frontierconstructed from all traditional investment assets, including alternative assets (seeTable 2 for a complete list; p -value in parentheses). F-Test refers to the correctedtest of Huberman and Kandel (1987), F and F to step-down tests by Kan andZhou (2012), testing for spanning of tangency portfolios and for global minimumportfolios, respectively. Only CCs for which at least one test rejects spanning at the10% level are reported.Table 5 lists only CCs with at least one test rejecting the hypothesis that traditional assets38pan the frontier at the 10% level. Recall that our definition of traditional assets includes abroad set of alternative investments, all but CCs. The corrected HK test rejects spanning for 3CCs. In contrast, the step-down test provides information on the source for spanning rejection: F tests for spanning of tangency portfolios, whereas F tests spanning for global minimumportfolios. From Table 5, we see that the F test rejects spanning for only 2 CCs, pointingout that tangency portfolios which include CCs are significantly different from the benchmarktangency portfolio, but also that the inclusion of the two years 2018–19 has dramaticallyreduced that number from previously 27 CCs, which included Bitcoin (BTC), Ripple (XRP),Dash (DASH) and Litecoin (LTC). F rejects spanning for 5 CCs for the entire time period,still including one of the coins with the highest market capitalisation, XRP. Thus, we concludethere still exists evidence that a MV-S portfolio can be improved by 7 out of 52 CCs, but thatthe integration of CC with financial markets has progressed markedly. Anecdotal evidence inline with this finding comes from the recent outbreak of the corona-virus pandemic, wheninitially CC markets moved for the first time with strong positive correlation together withfinancial markets, driven by institutional investors rebalancing in favor of cash holdings, beforeCCs resumed their diversifying role in subsequent weeks.Also, but there is little evidence that a MinVar portfolio can be improved. This resultis supported by the dynamics of the portfolios’ composition presented in Figures 5 and 6for unconstrained and LIBRO portfolios, respectively: MinVar portfolios in both cases areconstructed entirely from traditional assets, whereas MV-S portfolios have a (varying) CCcomponent throughout the whole investment period.In sum, the results imply that investors should consider a broader selection of CCs (seeQuestion 3), not only BTC. However, only a small fraction of CCs continue to improve theefficient frontier. 39 .2.3 Diversification metrics across portfolio strategies Table 6 reports results on our three chosen diversification metrics (detailed in Section 5.3). Asexpected, for the RR-MaxRet strategy there are no diversification benefits—by definition itconsists of only one asset at a time (unless LIBRO forces it into more than one asset). The rangeof values across diversification metrics emphasizes that diversification has different aspectsand its quantification depends on the definition used. Consequently, different measures do notalways provide identical conclusions about the diversification effects of CCs in portfolios. Forinstance, in terms of a DR of 13 .
73 (13 . MinCVaR is characterized as most diversifiedstrategy. Slightly lower measures pertain for MinVar and ERC portfolios with 12 .
02 (11 . .
71 (9 . DR of 2 .
64 (2 .
07) and at the same time a PDI of 21 .
06 (21 . N of 17 .
16 (13 .
61) by a largemargin, also a typical result (see, e.g., Clarke et al., 2013) due to its nature: it includes allassets by definition. Apart from MaxRet, the lowest Effective N of 2 .
68 (2 .
69) arises forthe MinVar portfolio, containing only traditional assets, showing that fewer than 3 equally-weighted stocks would provide the same diversification by this measure. All other individualstrategies also exhibit Effective N ranging between 3 and 4. One more remarkable resultconcerns the combined portfolios’ concentration: While COMB’s Effective N lies in the rangeof individual strategies, COMB Naïve exceeds 10 both in constrained and unconstrainedportfolios. In terms of DR , the combined strategies rank inversely, reaching 3 .
43 (3 . . .
44) for COMB; their PDIs are similar to those of the otherrisk-oriented portfolios MinVar, MinCVaR and ERC.Note that with the exception of ERC liquidity constraints do not strongly affect thediversification features of portfolios: all metrics display only minor changes. This result is due Here and henceforth we provide the values of the performance metric for LIBRO portfolios in parentheses. llocation Portfolio diversification effects: monthly rebalancingStrategy DR Effective N PDINo const 10 mln No const 10 mln No const 10 mln
Benchmark strategies
MV-S TrA 5 .
39 5 .
39 3 .
11 3 .
11 4 .
92 4 . Return oriented strategies
RR-MaxRet 1 .
00 1 .
00 1 .
00 1 .
48 1 .
00 1 . Risk-oriented strategies
MinVar 12 .
02 11 .
72 2 .
68 2 .
69 20 .
60 20 . .
71 9 .
41 17.16 13.61 20 .
62 20 . .
15 3 .
15 20 .
60 20 . .
64 2 .
07 3 .
99 3 .
17 21.06 21.01
Risk-Return-oriented strategies
MV-S 8 .
03 7 .
48 3 .
26 3 .
35 20 .
62 20 . Combination of models
COMB NAÏVE 3 .
43 3 .
62 11 .
60 10 .
92 20 .
65 20 . .
00 9 .
44 3 .
39 3 .
40 20 .
61 20 . Table 6: Measures of diversification for all investment strategies and a benchmark, over theentire time period from 2016-01-01 to 2019-12-31, with monthly rebalancing ( l = 21). DR denotes the squared diversification ratio, PDI the portfolio diversification index;all three measures are detailed in Section 5.3. “TrA” denotes only traditional, i.e.,non-CC assets are included. Strategies are detailed in Table 1. Results withoutliqudity constraints (columns “No const”) are contrasted with those when applyingLIBRO with a threshold of USD 10 mln (column “10 mln”). Highest results arehighlighted in red.to the fact that LIBRO generally lowers the weight of constituents, but does not completelyexclude them.We particularly highlight the difference of diversification of the MV-S portfolios with andwithout CCs: diversification measured by DR increases through the inclusion of CCs from5 .
39 (5 .
39) to 8 .
03 (7 . .
92 (4 .
92) to 20 . . .3 Interpretation of the results In this section we relate our empirical results to the Questions 1–6, along which the contributionof this paper is structured.
Question 1: For whom is investing in the CC market valuable? Is the benefit derivedfrom adding CCs to a portfolio dependent on the investor’s objectives (e.g., return-oriented ordiversification seeking)?
As the efficient frontiers in Figure 1 clearly show, the main benefit of CCs accrue toinvestors who make use of the high-risk/high-return character of their returns; investorswith low risk tolerance benefit least. While it is not surprising that CCs constitute riskyinvestments, Figure 5 shows how the risk-oriented strategies (minimizing variance or CVaR)consist almost entirely of traditional assets, so CCs have no influence on them; at least arisk-return orientation is necessary for CCs to play a noteworthy and permanent role inportfolios. At the other end of the specturm, the extremely CC-affine MaxRet strategy,despite stellar performance during the boom phase of 2017, was all but wiped out by the endof our period (ultimately retaining only 3% of its initial value).The model-free EW strategy is a special case: its performance in the middle of our timeperiod was extraordinary, and so was its collapse when the 2017 price rally in CCs disintegrated.As with MaxRet, both parts are driven by the high weight of small CCs—these were preciselythe ones that gained disproportionately in value during the price rally, and subsequentlysuffered the severest. Therefore, over our complete time span the EW portfolio in fact lost12 .
3% in value, whereas other types of investors ended up with gains.By far the best performance was achieved by investors who target strong (or even maximal)diversification. These strategies, ERC and MD, lead to sizeable exposures to a broadercross-section of CCs, while they limit the risks the EW strategy incurs.The general conclusion is that the utility from adding CCs to a portfolio strongly dependson the investor’s objective. In particular investors targeting a well-diversified portfolio whilewilling to bear some risk are advised to consider CCs for their investments.42 uestion 2: To which type of investor are CC investments most useful? Only professionaltraders who rebalance their portfolio frequently, or also less actively trading retail investors?
The rebalancing frequency (whether portfolio positions are traded daily, weekly, or monthlyto react to market developments by updating estimates and to revert positions to targetweights) does influence the performance of investors’ portfolios. For instance, over our studyperiod cumulative wealth for the MV-S strategy grows by 5% when readjusting the portfolioon a daily basis, by 12% with weekly, and 9% with monthly position changes. These differencebecome more pronounced when transaction costs are deducted, as turnover is naturally higherat a higher rebalancing frequencies. For the RR-MaxRet strategey, the loss attenuates withweekly and exacerbates with daily reallocations.However, the overall picture does not change across rebalancing frequencies: Even with amore frequent reallocations, still diversification-seeking investors (ERC and MD) significantlyoutperform the other investment strategies. Therefore, our general conclusions about theeffect of adding CCs into investment portfolios do not change qualitatively between dailytraders, weekly rebalancing and monthly reallocation (retail investors).
Question 3: Should investors focus on one particular coin (e.g., Bitcoin), a selected few,or rather build a portfolio of a broad selection of CCs?
Most importantly, our findings clearly indicate that diversification also across CCs isbeneficial. At the same time, investors could diversify too much. As Table 6 shows, the MDstrategy, which had the highest return, showcases an Effective N of only 3 .
26. ERC has muchhigher Effective N of 17 .
16, still it features considerably lower final cumulative wealth, at leastin unconstrained optimization. Judged by PDI, MD is the most successful strategy, which ofcourse is driven by the fact that the target-weight allocation of MD is derived precisely bymaximizing PDI. However, this also indicates that including as many assets in the portfolioas possible is not necessary to adequately represent the covariance matrix, and not beneficialin terms of cumulative wealth.Figures 5 and 6 caution the interpretation of MD dominating in terms of accumulated43eturns. Both Figures show that MD includes a broad range of CCs, whereas MinVar andMinCVaR—both with comparable Effective N and PDI—almost entirely exclude them, givingweight only to traditional assets. In this sense we do find evidence that CCs can substitutefor traditional assets in portfolio optimization.Regarding the ERC strategy, while it reaches optimal diversification for the alternativemetric of Effective N , it provides sizable gains in cumulative wealth and at the same adequatelydiversifies the portfolio. Figures 5 and 6 indicate that CCs and traditional assets are mixedin the portfolio, while the PDI is close to the one of MD and DR only second to the purerisk-oriented strategies MinVar and MinCVaR.Therefore, including CCs to diversify the portfolio is beneficial to achieve high targetreturns, and balancing traditional assets and CCs is advisable. Question 4: What exposure to each CC should be held in the portfolio? How informative arepast prices, how stable are positions when re-balancing the portfolio? Do model-free strategieslike equal-weighting provide reasonable results?
Even though CCs are highly volatile, the past pricing series are informative for portfolioallocation. As such, quantitative methodologies for portfolio allocation are applicable andone is not restricted to non-quantitative or model-free investment schemes. The EW strategy,which we discussed in the answer to Question 1, can exhibit phases of extraordinary returns,but does not manage risk well. At the other end of the spectrum, however, strategiesexclusively targeted at lowering risk at all cost do not benefit from CCs. This is of courseunsurprising, since lower risk must go at the expense of lesser expected return, most clearlyvisible already in the efficient frontiers in Figure 1.
Question 5: Can these strategies be implemented in practice? In particular, are all CCsliquid enough for inclusion in an investment portfolio? If not, how can investors still profitfrom promising CCs with little trading volume without exposing their portfolio too much toilliquidity? Moreover, how is performance affected by honoring such portfolio restrictions?
This question is addressed by LIBRO: The fact that the bounds on CC weights by LIBRO44Trimborn et al., 2019) turn out to bind indicates that several CCs are not sufficiently liquidfor investors with deeper pockets. Still, the approach allows the inclusion of illiquid CCs up torestricted amounts. This has the positive effect that investors can still perform diversificationstrategies to quite some degree—strategies that rank among the most profitable. However, theimpressive results by strategies with broad CC exposure turn out not to be very scalable. Forinstance, the MD strategy shows excellent performance without liqudity constraints (+175%),yet the application of LIBRO pushes final CW below initial wealth ( − N and PDI: MD reaches avery low Effective N of 3 .
99, although it includes only CCs, compare the weight compositionin Figures 5 and 6. PDI is clearly higher than for other strategies, in line with the objectiveof MD, and the PDI only shrinks marginally when incorporating LIBRO, whereas Effective N drops by about 1. This implies that the strategy focuses disproportionately on (a) singularCC(s), driving the high returns, which cannot be traded sufficiently for a portfolio of USD10 mln. At the other end of the spectrum, the mininum-risk strategies focus on traditionalassets with high trading volume, therefore they are little affected by LIBRO. Question 6: Overall, how do the properties of CC returns affect portfolios? Is a certaintype of portfolio-allocation method more suitable to manage and simultaneously exploit theirproperties?
Regarding CC return properties, a lot of prior research exists (see also our literature reviewin Section 2). We confirm that the patterns of generally high means, high volatilities, excesskurtosis, and low correlations with traditional assets are also present in our sample (see thedescriptive statistics in Appendix 9.2). Our contribution addresses the effect of including CCsin already broadly diversified portfolios: Beyond what we have established in the answers tothe preceding five questions, our central finding is that the key conclusion of prior studies—thatCCs are valuable additions to the investment universe—holds true, but it is critical to mindthe limits of quantitative results derived from simplified frameworks. While diversification45trategies prove most promising, including only the top CCs foregoes diversification potential.Most importantly, returns of broad CC portfolios that are calculated without accounting forliquidity remain virtual: they cannot be realized by professional investors.Finally, for certain types of investors, namely those highly risk averse, the benefits canprove too risky to pursue.
This study investigates cryptocurrencies as new investment assets available to portfoliomanagement. We investigate the utility gains for different types of investors when theyconsider cryptocurrencies as an addition to a well diversified portfolio of traditional assets.We consider risk-averse, return-seeking as well as diversification-preferring investors whotrade along different allocation frequencies, namely daily, weekly or monthly. To conductthis study, we analyze the performance of commonly used asset-allocation models based onhistorical prices and trading volumes of 52 cryptocurrencies, combined with 16 traditionalassets. The rules-based investment methods cover a broad spectrum of investor objectives,from the classical Markowitz optimization to recent strategies aiming to maximize portfoliodiversification. Along with individual portfolio allocation strategies, we also include combinedstrategies from model averaging. The performance of portfolios is evaluated with a range ofdifferent measures, including cumulative wealth, risk-adjusted performance and diversificationeffects produced by portfolios.We find that due to the volatility structure of cryptocurrencies, the application of tradi-tional risk-based portfolios strategies, such as equal-risk contribution, minimum-variance andminimum-CVaR, does not boost the performance of investments significantly. In contrast,approaches such as the maximum-return strategy (or strategies with high target returns),and also the maximum-diversification portfolio reach higher expected returns via higher orbroader cryptocurrency exposure for investors. As for diversification benefits, we demonstratean effect beyond well-diversified, global portfolios of conventional assets without CCs. We46lso document how various rules have different effects on portfolio diversification, dependingon the concept of diversification and the chosen measure of its quantification.Furthermore, following the idea of model averaging and diversification across models, weshow that both naive and bootstrap-based combined portfolios exhibit robust high risk-adjustedreturns. Portfolios with model-averaged weights achieve significantly higher performance thanpurely risk-oriented strategies and not significantly lower than the best performing strategies.We also show how different rebalancing frequencies affect performance, as well as howconstraints mitigating liquidity risks of cryptocurrencies (LIBRO) can significantly affectthe outcome of strategies that rely on a larger cross-section of CCs. The results remaincoherent across all frameworks. Further extensions can be made along three main lines:first, more involved estimators of expected returns and the covariance matrix could beemployed; second, more performance measures could be used to evaluate the investmentstrategies’ results; and third, additional portfolio-allocation strategies could be included in thecomparison. In particular, factor-based APT (arbitrage price theory) models would constitutethe complementary approach to statistical-optimisation techniques studied in this paper.47 eferences
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Appendix9.1 Test for difference of SR or CEQ between two strategies
We employ the test by Ledoit and Wolf (2008). Let ν = ( µ i , µ j , σ i , σ j ) denote the vector ofthe moments of two strategies i and j .Then we can test for a difference of the strategies’ CEQs or SRs via the test statisticsdefined as the differences of those measures, f CEQ ( ν ) = µ i − γ σ i − µ j + γ σ j , (23)or f SR ( ν ) = µ i σ i − µ j σ j , (24)respectively.Applying the delta method yields that if √ T − M (ˆ ν − ν ) d −→ N (0 , Ψ), then √ T − M ( ˆ f − f ) d −→ N (0 , ∇ f ( ν )Ψ ∇ f ( ν )) , (25)where ∇ f stands for the derivative of f .The standard error for such a test statistic ˆ f then amounts to: SE ( ˆ f ) = s ∇ f ( ν )Ψ ∇ f ( ν ) T − M , (26)so we require a consistent estimator ˆΨ for Ψ.The standard method to provide such an esimator is to apply heteroskedasticity- andautocorrelation-robust kernel estimation to obtain the estimateΨ T − M = T − MT − M − T − M − X j = − T + M +1 Ker jS T − M ! ˆΓ T − M ( j ) , (27)where a kernel function Ker ( · ) and a bandwidth S T − M need to be chosen.Then a two-sided p -value for the hypothesis H : f = 0 is given as:ˆ p = 2Φ | ˆ f | SE ( ˆ f ) . (28)56 .2 Descriptive statistics of portfolio components For completeness, we present descriptive statistics both for our traditional assets as well asall 52 CCs in our sample. Table 7 shows that, as expected, correlations between CCs andtraditional assets are low to non-existent, also in our sample. Tables 8 and 9 show univariatedistributional properties of daily log returns on traditional assets and CCs, respectively. Thegenerally elevated magnitude for CCs is clear; Figure 7 visually confirms the strong leptokurticnature of CC returns.
DOGE ZET XMG SYS POT DGC DMD RBY START EMC2CNY –0.03 –0.01 –0.02 –0.04 0.01 –0.03 –0.01 0.04 0.00 –0.04REIT 0.01 0.02 –0.01 0.05 0.01 0.01 –0.01 0.03 0.03 –0.02EUR 0.02 0.00 –0.01 0.00 0.01 0.01 0.01 0.00 –0.05 –0.02GBP 0.02 0.02 –0.02 –0.02 –0.03 0.01 0.00 0.01 –0.02 –0.04JPY 0.00 0.01 0.02 0.00 –0.01 0.03 –0.01 0.01 0.00 0.00MSCI CP 0.04 0.05 0.04 0.06 0.04 0.00 0.00 0.02 0.00 –0.04GOLD 0.05 0.00 0.02 0.01 –0.01 –0.01 0.00 –0.02 0.01 0.02NIKKEI225 0.00 0.02 0.00 0.01 –0.03 0.04 0.03 –0.04 0.02 –0.02SSE –0.06 0.00 0.00 0.04 –0.06 0.02 0.03 0.02 0.03 0.02S&P100 0.04 0.03 0.05 0.05 0.03 –0.02 –0.01 0.03 0.00 –0.04EURO STOXX 50 0.04 0.05 0.02 0.02 0.01 0.01 –0.01 0.02 –0.01 0.00FTSE 100 0.04 0.04 0.04 0.07 0.05 0.01 –0.02 0.02 0.01 –0.01UK 10Y 0.00 –0.05 0.00 0.01 0.02 –0.02 0.00 0.00 0.02 0.02Japan 10Y 0.02 0.02 –0.02 –0.02 0.02 –0.04 –0.01 0.02 0.00 0.08USA 10Y 0.00 –0.02 –0.02 –0.02 –0.01 –0.03 –0.01 –0.03 0.00 0.01EURO 10Y 0.01 –0.04 0.02 –0.02 –0.01 –0.01 0.02 0.01 0.01 0.04
Table 7: Correlation coefficients of daily log returns of the top ten CCs with all conventionalfinancial assets in our analysis (detailed in Table 2) over the entire sample periodfrom 2016-01-01 to 2019-12-31. 57 sset name Max P Med Mean P Min SDCNY 1.84 0.24 0.00 0.01 -0.23 -1.20 0.23JPY 2.22 0.58 0.00 -0.01 -0.61 -3.78 0.53EUR 3.02 0.59 0.00 -0.01 -0.63 -2.38 0.52GBP 3.00 0.67 -0.02 -0.01 -0.65 -8.40 0.61FTSE REIT 4.14 1.04 0.03 0.02 -1.08 -9.38 0.96GOLDS 4.58 0.90 0.02 0.02 -0.84 -3.38 0.77MSCI CP 2.67 0.78 0.04 0.02 -0.76 -4.88 0.69NIKKEI225 7.43 1.24 0.01 0.02 -1.22 -8.25 1.19SSE 5.60 1.47 0.01 -0.00 -1.33 -8.87 1.46S&P100 4.84 0.97 0.03 0.03 -0.84 -4.18 0.83EURO STOXX50 4.60 1.18 0.04 0.01 -1.20 -9.01 1.07FTSE100 3.51 0.96 0.02 0.01 -0.95 -4.78 0.86UK 10Y 2.00 0.30 0.00 0.00 -0.29 -1.43 0.28Japan 10Y 0.74 0.09 0.00 -0.00 -0.08 -0.63 0.10USA 10Y 1.29 0.30 0.00 0.00 -0.28 -1.55 0.27EURO 10Y 0.85 0.22 0.00 -0.00 -0.21 -1.90 0.22
Table 8: Descriptive statistics for daily log returns (in %) of all conventional assets in ourbaseline portfolio (detailed in Table 2) over the entire sample period from 2016-01-01to 2019-12-31. P and P denote the first and ninth decile, respectively, “Med” themedian, and “SD” standard deviation.58 C Max P Med Mean P Min SDABY 35.10 14.18 -0.19 0.01 -13.65 -29.69 12.13AUR 29.85 12.50 -0.12 -0.09 -12.68 -27.22 11.00BCN 21.80 10.40 -0.20 -0.12 -10.93 -20.65 8.77BLK 22.88 9.61 -0.25 -0.07 -9.59 -22.44 8.55BTC 9.58 5.01 0.22 0.24 -4.03 -9.88 3.80BTS 17.26 8.04 -0.32 -0.07 -8.03 -16.64 6.71BURST 21.70 10.51 0.24 0.09 -10.44 -20.09 8.57BYC 30.16 12.10 0.00 -0.14 -11.80 -26.84 10.57CANN 37.87 12.56 -0.06 0.18 -12.44 -28.67 12.00CURE 25.31 11.55 -0.26 -0.05 -11.22 -20.68 9.35DASH 15.67 7.01 -0.16 0.21 -6.04 -12.56 5.59DGB 22.92 10.36 -0.56 0.09 -9.22 -18.47 8.23DGC 54.02 16.36 -0.33 -0.53 -17.31 -62.84 18.93DMD 17.83 9.57 -0.13 -0.01 -9.16 -19.58 7.62DOGE 14.99 6.75 -0.25 0.05 -5.62 -12.58 5.34EAC 41.30 12.35 -0.07 -0.04 -13.06 -37.18 13.11EMC2 25.40 11.60 -0.35 -0.01 -10.82 -23.49 9.68FTC 27.50 11.43 -0.77 -0.20 -10.64 -21.49 9.55GRC 37.14 13.80 -0.50 0.21 -13.23 -23.57 11.61HUC 28.23 12.92 0.00 0.05 -12.93 -22.96 10.43IOC 28.86 14.69 -0.07 0.25 -13.03 -28.26 11.55LTC 15.71 6.36 -0.07 0.12 -5.99 -12.70 5.33MAX 80.52 21.21 -0.44 -0.24 -22.02 -81.86 26.40NAV 26.75 12.16 -0.34 0.10 -11.07 -20.07 9.53NEOS 28.82 12.37 0.00 -0.22 -12.43 -25.50 10.36NLG 18.09 8.89 -0.20 0.04 -8.53 -14.55 6.91NMC 16.64 7.34 -0.20 -0.07 -7.27 -16.26 6.27NOTE 27.56 11.89 -0.35 -0.39 -12.44 -25.80 10.47NVC 20.18 7.19 -0.17 -0.08 -7.92 -15.06 6.61NXT 17.15 8.26 -0.54 -0.22 -7.92 -15.68 6.56POT 20.16 9.77 -0.07 -0.03 -10.39 -19.77 8.13PPC 16.94 6.98 -0.19 -0.07 -7.47 -15.07 6.26QRK 37.18 11.97 -0.38 -0.05 -12.22 -30.88 12.15RBY 25.58 12.34 0.00 0.10 -12.33 -28.14 10.41RDD 32.46 14.26 -0.05 0.08 -13.67 -28.14 11.90SLR 25.15 11.21 -0.23 0.01 -11.01 -21.65 9.41START 29.14 14.48 -0.66 -0.16 -12.99 -26.97 11.36SYS 24.83 10.61 -0.14 0.20 -10.24 -19.16 8.72UNO 22.07 11.13 -0.01 0.16 -9.47 -23.61 8.79VIA 25.06 10.68 -0.00 -0.01 -11.35 -20.53 9.22VRC 33.85 14.29 -0.51 0.04 -13.28 -28.46 11.86VTC 27.29 10.98 -0.41 0.03 -10.74 -19.54 9.09WDC 29.58 11.66 0.00 -0.42 -12.39 -33.05 11.24XCN 59.83 20.82 -0.42 -0.16 -22.36 -51.64 20.06XCP 22.41 10.77 -0.48 -0.19 -10.48 -21.09 8.84XDN 24.72 11.73 -0.29 -0.13 -12.12 -22.17 9.58XMG 31.69 12.70 -0.18 0.24 -11.33 -23.38 10.54XMR 16.29 8.54 -0.05 0.26 -7.22 -14.12 6.36XPM 21.99 9.61 -0.25 -0.20 -10.58 -19.71 8.45XRP 16.18 6.70 -0.39 -0.06 -6.05 -13.14 5.52XST 29.94 13.98 -0.43 -0.04 -13.15 -26.17 11.24ZET 32.50 16.43 -0.30 -0.13 -15.80 -33.04 13.16
Table 9: Descriptive statistics for daily log returns (in %) of all 52 CCs eligible for our portfoliostrategies (detailed in Table 1) over the entire sample period from 2016-01-01 to2019-12-31. P and P denote the first and ninth decile, respectively, “Med” themedian, and “SD” standard deviation.59 D en s i t y Density of Top 10 Cryptos
Figure 7: Density of daily log returns of the top 10 CCs (DOGE, ZET, XMG, SYS, POT,DGC, DMD, RBY, START, EMC2) against a normal distribution with same meanand variance. The time span is from 2016-01-01 to 2019-12-31.CCPHistReturnsDensity60 .3 Dynamics of risk contributions for portfolio strategies
The outcome of portfolio optimization can be viewed in two different ways: first, in termsof the weights the chosen strategy assigns to each asset; second, in terms of the risk eachconstituent contributes to the portfolio. While flip sides of the same coin, with stronglydivergent statistical properties across assets, as in our case, relative risk contributions candiffer noticably from relative portfolio shares. For instance, if a portfolio were to hold thesame percentage of its value in UK bonds and in bitcoin, the changes in portfolio valueover time driven by BTC will amount to a multiple of those stemming from the same-sizedfixed-income position.While we reported weigts in Figures 5 and 6 in the main text, for completeness we presentthe risk contributions as a function of time in Figures 8 and 9 for portfolio optimizationswithout and with enforced liquidity constraints, respectively.61igure 8: Evolution of risk contributions (i.e., fraction of portfolio value changes driven byeach constituent) of all allocation strategies (without liquidity constraints) withmonthly rebalancing over the period from 2016-01-01 to 2019-12-31: the black lineseparates conventional assets (“TrA,” upper yellow part of the spectrum) fromcryptocurrencies (CCs, lower green-blue part of the spectrum).CCPRisk_contribution62igure 9: Evolution of risk contributions (i.e., fraction of portfolio value changes driven byeach constituent) of all allocation strategies with a position limit of USD 10 mln (viathe LIBRO approach) with monthly rebalancing over the period from 2016-01-01to 2019-12-31: the black line separates conventional assets (“TrA,” upper yellowpart of the spectrum) from cryptocurrencies (CCs, lower green-blue part of thespectrum). CCPRisk_contribution63 .4 Results for daily and weekly rebalanced portfolios
While our main analysis maintained the industry standard of rebalancing on a monthly basis,we deem it important to also consider higher trading frequencies in the CC market. Wetherefore report the performance results based on weekly rebalancing in Table 10, as well asfor daily reallocations in Table 12.Since diversification effects can also be affected by the rebalancing frequencies, Tables 11and 13 display the diversification measures for a weekly and daily frequency, respectively.
Allocation Portfolio performance measures: weekly rebalancingStrategy CW SR ASR CEQ TO TTONo const 10 mln No const 10 mln No const 10 mln No const 10 mln No const 10 mln No const 10 mln
Benchmark strategies
S &P100 0 .
901 0 . − . − . − . − .
016 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . .
103 1 .
103 0 .
033 0.033 0 .
033 0.033 − . − .
001 9 .
557 9 .
557 0 .
000 0 . .
069 1 .
069 0 .
028 0 .
028 0 .
027 0 .
027 0 .
000 0 .
000 7 .
631 7 .
631 0 .
195 0 . .
889 0 . − . − . − . − . − . − .
001 6 .
136 6 .
136 0 .
000 0 . Risk-oriented strategies
MinVaR 0 .
99 0 . − .
019 0 . − .
019 0 . − . − .
001 15 .
370 11 .
352 0.035 0.035MinCVaR 1 .
014 1 .
014 0 .
017 0 .
017 0 .
017 0 .
017 0 .
000 0 .
000 32 .
224 8 .
301 0 .
087 0 . .
213 1 .
036 0 .
032 0 .
009 0 .
032 0 .
009 0 .
000 0 .
000 9 .
951 7 .
050 0.035 0 . . − . − .
002 0 . − .
002 0 . − .
001 36 .
707 34 .
133 0 .
252 0 . Return-oriented strategies
RR-MaxRet 0 . . − . − . − . − .
016 0 .
000 0 .
000 0.454 2.952 0 .
454 0 . Risk-Return-oriented strategies
MV-S 1 .
124 1.107 0.034 0 .
032 0.034 0 .
032 0 .
000 0 .
000 6 .
662 6 .
475 0 .
202 0 . Table 10: Performance measures for all investment strategies as well as benchmarks over theentire time period from 2016-01-01 to 2019-12-31, with weekly rebalancing ( k = 5).The performance measures are final cumulative wealth (CW), the Sharpe ratio (SR),the adjusted Sharpe ratio (ASR), the certainty equivalent (CEQ), and turnover.“TrA” denotes only traditional, i.e., non-CC assets are included. Strategies aredetailed in Table 1. Highest results are highlighted in red.Highest results are highlighted in red. 64 llocation Portfolio diversification effects: weekly rebalancingStrategy DR Effective N PDINo const 10 mln No const 10 mln No const 10 mln
Benchmark strategies
MV-S TrA 5 .
310 5 .
310 3 .
120 3 .
120 4 .
870 4 . Return oriented strategies
RR-MaxRet 1 .
000 1 .
000 1 .
000 1 .
500 1 .
000 1 . Risk-oriented strategies
MinVar 12 .
030 11 .
730 2 .
670 2 .
680 20 .
520 20 . .
710 9 .
400 17.200 13.600 20 .
540 20 . .
160 3 .
150 20 .
520 20 . .
650 2 .
060 4 .
000 3 .
170 20.970 20.920
Risk-Return-oriented strategies
MV-S 8 .
010 7 .
430 3 .
320 3 .
390 20 .
540 20 . Table 11: Measures of diversification for all investment strategies and a benchmark, over theentire time period from 2016-01-01 to 2019-12-31 for weekly rebalancing ( k = 5). DR denotes the squared diversification ratio, PDI the portfolio diversificationindex; all three measures are detailed in Section 5.3. “TrA” denotes only traditional,i.e., non-CC assets are included. Strategies are detailed in Table 1. Results withoutliqudity constraints (columns “No const”) are contrasted with those when applyingLIBRO with a threshold of USD 10 mln (column “10 mln”).65 llocation Portfolio performance measures: daily rebalancingStrategy CW SR ASR CEQ TO TTONo const 10 mln No const 10 mln No const 10 mln No const 10 mln No const 10 mln No const 10 mln Benchmark strategies
S&P100 0 .
900 0 . − . − . − . − .
016 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . .
102 1.102 0.033 0.033 0.033 0.033 − . − .
001 10 .
557 10 .
557 0 .
000 0 . .
029 1 .
029 0 .
012 0 .
012 0 .
012 0 .
012 0 .
000 0 .
000 6 .
832 6 .
832 0 .
096 0 . .
877 0 . − . − . − . − . − . − .
001 19 .
400 19 .
400 0 .
000 0 . Risk-oriented strategies
MinVaR 0 .
985 0 . − . − . − . − . − . − .
002 19 .
930 18 .
083 0.013 0.013MinCVaR 1 .
003 1 .
004 0 .
004 0 .
005 0 .
004 0 .
005 0 .
000 0 .
000 15 .
207 14 .
233 0 .
026 0 . .
216 1 .
031 0 .
032 0 .
008 0 .
032 0 .
008 0 .
000 0 .
000 13 .
144 12 .
347 0 .
014 0 . .
967 0 . − .
001 0 . − .
001 0 . − .
001 55 .
774 58 .
244 0 .
121 0 . Return-oriented strategies
RR-MaxRet 0 .
006 0 . − . − . − . − .
023 0 .
000 0 .
000 0.245 1.639 0 .
240 0 . Risk-Return-oriented strategies
MV-S 1 .
053 1 .
040 0 .
015 0 .
012 0 .
015 0 .
012 0 .
000 0 .
000 8 .
432 6 .
713 0 .
099 0 . Table 12: Performance measures for all investment strategies as well as benchmarks over theentire time period from 2016-01-01 to 2019-12-31, with daily rebalancing ( k = 1).The performance measures are final cumulative wealth (CW), the Sharpe ratio (SR),the adjusted Sharpe ratio (ASR), the certainty equivalent (CEQ), and turnover.“TrA” denotes only traditional, i.e., non-CC assets are included. Strategies aredetailed in Table 1. Highest results are highlighted in red.66 llocation Portfolio diversification effects: daily rebalancingStrategy DR Effective N PDINo const 10 mln No const 10 mln No const 10 mln
Benchmark strategies
MV-S TrA 5 .
340 5 .
340 3 .
130 3 .
130 4 .
870 4 . Return oriented strategies
RR-MaxRet 1 .
000 1 .
000 1 .
000 1 .
500 1 .
000 1 . Risk-oriented strategies
MinVaR 12 .
060 11 .
750 2 .
670 2 .
680 20 .
490 20 . .
720 9 .
400 17.220 13.620 20 .
510 20 . .
160 3 .
160 20 .
500 20 . .
650 2 .
060 4 .
010 3 .
180 20.950 20.900
Risk-Return-oriented strategies
MV-S 8 .
030 7 .
460 3 .
310 3 .
400 20 .
510 20 . Table 13: Measures of diversification for all investment strategies and a benchmark, over theentire time period from 2016-01-01 to 2019-12-31 for daily rebalancing ( k = 1). DR2