disposition subsection section

#  Investing with Cryptocurrencies – evaluating their potential for portfolio
allocation strategies111Financial support from IRTG 1792 “High Dimensional Non
Stationary Time Series,” Humboldt-Universität zu 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 support from the
European Union’s Horizon 2020 training and innovation programme ”FIN-TECH”,
under the grant No. 825215 (Topic ICT-35-2018, Type of actions: CSA)

Alla Petukhina 222Humboldt-Universität zu Berlin, School of Business and
Economics, Dorotheen Str. 1, 10117 Berlin, Germany, tel: +49 (0)30 2093-99469,
e-mail<EMAIL_ADDRESS>Simon Trimborn 333Department of
Mathematics, National University of Singapore, Risk Management Institute, 21
Heng Mui Keng Terrace, I3 Building 04-03, 119613 Singapore tel: +65 6516-1245,
e-mail<EMAIL_ADDRESS>Wolfgang Karl Härdle 444Humboldt-Universität
zu Berlin, IRTG 1792, Dorotheen Str. 1, 10117 Berlin, Germany; Wang Yanan
Institute for Studies in Economics, N114, Economics Building, Xiamen
University Xiamen, 361005 China; School of Business, Singapore Management
University, 50 Stamford Road, Singapore 178899; Faculty of Mathematics and
Physics, Charles University, Ke Karlovu 3, 121 16 Prague, Czech Republic;
Department of Information Management and Finance, National Chiao Tung
University, Taiwan, ROC, tel: +49 (0)30 2093-99592, e-mail: haerdle@hu-
berlin.de Hermann Elendner 555Weizenbaum-Institut, Hardenbergstraße 32, 10623
Berlin, Germany, e-mail<EMAIL_ADDRESS>
###### Abstract

Abstract

Cryptocurrencies (CCs) have risen rapidly in market capitalization over the
last years. Despite striking price volatility, their high average returns have
drawn attention to CCs as alternative investment assets for portfolio and risk
management. We investigate the utility gains for different types of investors
when they consider cryptocurrencies as an addition to their portfolio of
traditional assets. We consider risk-averse, return-seeking as well as
diversification-preferring investors who trade along different allocation
frequencies, namely daily, weekly or monthly. Out-of-sample performance and
diversification benefits are studied for the most popular portfolio-
construction rules, including mean-variance optimization, risk-parity, and
maximum-diversification strategies, as well as combined strategies. To account
for low liquidity in CC markets, we incorporate liquidity constraints via the
LIBRO method. Our results show that CCs can improve the risk-return profile of
portfolios. In particular, a maximum-diversification strategy (maximizing the
Portfolio Diversification Index, PDI) draws appreciably on CCs, and spanning
tests clearly indicate that CC returns are non-redundant additions to the
investment universe. Though our analysis also shows that illiquidity of CCs
potentially reverses the results.

Keywords: cryptocurrency, CRIX, investments, portfolio management, asset
classes, blockchain, Bitcoin, altcoins, DLT

JEL Classification: C01, C58, G11

##  1 Introduction

Cryptocurrencies (CCs) have exhibited remarkable performance in the decade
since [(81)] invented the blockchain. Accompanied by huge inflows of capital
into the market and strong swings in prices, CCs have gained strongly in
market value. Accordingly, indices like CRIX [(108), thecrix.de] were
introduced to capture the market evolution and provide a basis for ETFs.
Driven by these developments, cryptocurrency markets became increasingly
attractive to investors, who have started to consider CCs as a novel class of
alternative investments. However, investors differ with regard to their risk
profiles, investment targets, individual trading behaviors, and generally
their diverse motives and preferences, and thus the perspective to include CCs
into financial portfolios raises a number of questions:

  1. 1.

For whom is investing in the CC market valuable? Is the benefit derived from
adding CCs to a portfolio dependent on the investor’s objectives (e.g.,
return-oriented or diversification seeking)?

  2. 2.

To which type of investor are CC investments most useful? Only professional
traders who rebalance their portfolio frequently, or also less actively
trading retail investors?

  3. 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?

When an investor does decide to include CCs in the portfolio, further
questions arise about the choice of CCs for investment and their portfolio
weights:

  1. 4.

What exposure to each CC should be held in the portfolio? How informative are
past prices, how stable are positions when re-balancing the portfolio? Do
model-free strategies like equal-weighting provide reasonable results?

  2. 5.

Can these strategies be implemented in practice? In particular, are all CCs
liquid enough for inclusion in an investment portfolio? If not, how can
investors still profit from promising CCs with little trading volume without
exposing their portfolio too much to illiquidity? Moreover, how is performance
affected by honoring such portfolio restrictions?

  3. 6.

Overall, how do the properties of CC returns affect portfolios? Is a certain
type of portfolio-allocation method more suitable to manage and simultaneously
exploit their properties?

While we review the literature extensively in Section 2, clearly numerous
studies have investigated the properties referred to in Question 6, and agree
that CCs exhibit remarkably high average realized returns by the standards of
traditional financial assets666We 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.—and correspondingly
high risk and uncertainty. Not only is price volatility high; also unfavorable
properties obtrude, including frequent pricing bubbles [(45), (49), (23),
(83)], accumulation of jumps [(96)], even evidence of price manipulation
[(46)].

At the same time, there is evidence of low correlations of CC returns with
those of traditional financial assets and other CCs. Therefore, the high risk
of CC positions may be compensated by appropriate returns as well as provide
an opportunity to increase portfolio diversification. Results to that effect
have been spearheaded by [(18), (37)], the first to include Bitcoin (BTC) in a
portfolio of traditional assets, and subsequently bolstered by [(39)], who
include a broad cross-section of CCs, [(30)], who instead add CRIX, and lately
[(87), (5)], who include Bitcoin in advanced portfolio optimization and find
it enhances the risk-return profile.

So evidence exists that CCs can be beneficial for investors [(85)]. However,
taking the investor’s perspective, we see that while prior studies have
covered crucially important aspects of investing with CCs, the outlined
questions 1–5 remain fundamentally unanswered, for at least two reasons.
First, the key result of a diversification benefit of CCs (or BTC) cannot be
established unless a broad set of non-CC alternative investments are included.
For simplicity, in this paper we refer to all non-CC investments as
“traditional assets,” including alternative investments like gold or real
estate, in order to focus on the potential of adding CCs to well-diversified
portfolios. Second, it remains unclear which strategies can actually be
implemented in practice unless specific care is taken to address the
frequently extremely dry liquidity in CC markets.

The importance of addressing liquidity concerns is pinpointed by [(109)], who
introduce LIquidity Bounded Risk-return Optimization (LIBRO) when considering
a large sample of CCs added to a portfolio consisting of the S&P100, US bonds
and commodities. Given the low liquidity of CCs as compared to traditional
markets, LIBRO is designed to protect investors from an inability to trade a
CC in necessary amounts due to low trading volume.

Against this background, we address the questions above by performing a large-
scale comparative investment-strategy study including _both_ a broad range of
traditional assets together with a broad cross-section of CCs. Therein, we
test the performance of an extensive set of common investment strategies and
thus consider different types of investors, while we 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 of
strategies under consideration. We estimate extending-window and rolling-
window approaches optimizations for a sizable breadth of different common
objective functions. Finally, we compare all strategies based on three
different re-allocation frequencies, namely daily, weekly and monthly,
providing results for investors trading at different frequencies. To the best
of our knowledge, we thus present the broadest study on investing with CCs
conducted so far.

Closest related to our paper are [(5)] and [(87)], both also studying the
influence of CC investment on optimal portfolio composition. However, both
include only Bitcoin,777[(87)] do run a robustness test replacing Bitcoin with
CRIX, acknowledging the importance of altcoins. Naturally, diversification
_across_ CCs necessitates an optimization including their individual, distinct
return series. whereas we consider a broad cross-section of 52 distinct CC
price series. Moreover, both consider fewer traditional assets: industry
portfolios (so equity only) in the former paper, US equity and bond
investments in the latter, plus commodities in a robustness test. In contrast,
our set of traditional assets is critically broader: first, as CCs trade
globally, our international approach includes equity returns for each of the 5
major economic 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 the traditional assets all
our portfolios include. As we have pointed out, this emphatically goes beyond
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 of diversification benefits, and more critically, also
impossible to distinguish whether apparent benefits 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 after peak CC prices; also, we consider more allocation strategies. Most
importantly, since we take the investor’s perspective, we implement LIBRO and
contrast portfolios with weights that observe the liquidity constraints with
otherwise identical portfolios which do not: it turns out to cricitally affect
performance for several popular trading strategies.

Our study contributes to answering questions 1–6. Spanning tests show that
more than $50\%$ of the CCs considered can improve the efficient frontier of a
portfolio containing even our broad set of traditional assets. We show that
purely risk-minimizing investors will optimally choose to mostly forego CC
investment; however, for investors with higher target returns their addition
seriously expands the efficient frontier. Diversification-oriented investors
benefit most, even in terms of maximizing cumulative wealth. We also document
that a lower rebalancing frequency (monthly) of the portfolios generally
enhances cumulated returns. As mentioned, we confirm that several CCs exhibit
low liquidity, which can be tackled with the LIBRO approach. Our results
highlight the severity of low-liquidity risk, and how analyses that do not
take this risk into account will compute investment returns that are
infeasible for any but the smallest personal portfolios.

The paper is organized as follows. First, Section 2 reviews the related
literature. Section 3 provides an overview of the asset-allocation models
under consideration, with a focus on connections between them; therein Section
3.2 explains the approach of model averaging across investment strategies.
Section 4 reviews the LIBRO method. In Section 5 we explain the methodology
for comparing the performance of the models considered. Our dataset of
portfolio components is described in Section 6, and Section 7 presents the
results of our analyses of out-of-sample performance of all portfolio
strategies with CCs and traditional assets. We conclude in Section 8.

Code to produce the results of this paper is available via www.quantlet.de
missing.

##  2 Literature review

Modern portfolio theory builds on the CAPM [(74), (103), (69)], 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 pricing and portfolio optimization address this lack in one of
two ways.

The first we call the _financial-economics approach:_ it follows [(94)]’s (94)
arbitrage-pricing theory888This approach puts the emphasis on the equilibrium
model and is thus often preferred by theorists. which keeps the linear
structure and adds more factors to capture systematic patterns in returns.
Popularized by Fama_French_1992 (42, 40), it was extended to factors for
momentum (Jegadeesh_Titman_1993, 58, 21) or profitability and investment
(Fama_French_2015, 41). In principle, the approach renders portfolio
optimization straightforward and unidimensional: a portfolio is better, the
higher its alpha (the intercept after accounting for all factors’ loadings).
In practice, the choice of factors depends on the investment universe, and
also for given asset classes controversy remains about factors (the “zoo” of
Cochrane_2011, 33), how to choose them (Feng_Giglio_Xiu_2020, 43), even basic
methodology (NovyMarx_2014, 82).

A strand of the literature on cryptocurrencies (CCs) is devoted to finding and
using factors in CC markets (see Liu_Tsyvinski_2019, 72, 38, 56, 106, 104);
however, in this paper we pursue the second approach.

We term it the _quantitative-finance approach,_ due to its statistical nature.
Its idea, in essence, says: if we can capture the (joint) return distribution
(and its dynamics) of all investable assets (and parameters affecting them),
then we can directly estimate portfolio weights to optimize the desired
performance metric. Owing to the abundance of statistical techniques for the
variety of modelling choices and investment objectives, this approach is most
common in fund management.999An additional benefit is how it links potential
empirical shortcomings to insufficiently captured statistical properties,
offering remedy via more refined methods.  However, the easy customization has
precluded a standard, unique approach. A portfolio’s optimal allocation thus
depends crucially on three elements: the investment universe, the investment
strategy, and the investment objective as defined by the metric of
optimization.

Most fundamental is the determination of the _investment universe._ Our paper
focuses on its role by analyzing it for extensive sets of common strategies
and objective functions; concretely, on the potential of adding CCs.
Historically, starting from stocks101010markowitz_portfolio_1952 (74). and a
risk-free interest rate,111111Sharpe_1964 (103). the diversification benefits
to adding bonds (Liu_2016, 70), foreign exchange
(Kroencke_Schindler_Schrimpf_2013, 66, 9, 2), real estate
(Benjamin_Sirmans_Zietz_2001, 12, 3), and commodities
(Belousova_Dorfleitner_2012, 11) including gold (Hoang_et_al_2015, 52) have
been established in the literature.121212In fact, already Roll_1977 (93) had
stressed the “market portfolio” ought to include _all_ wealth. Naturally, his
critique has led to innumerous suggestions for further asset classes that
cannot all be part of our analysis, including private equity
(Gompers_et_al_2010, 48), fine art (Mei_Moses_2002-AER, 76, 20), or even fine
wine (Fogarty_2010, 44, 29). We term _all_ these assets “traditional
investments”, and we include proxies for all of them in our benchmark
portfolio. This breadth is key, as our goal is to investigate the effect of
including _additionally_ CCs. Only the broad traditional portfolio ensures we
assess the diversification potential _of CCs_ as investments: otherwise, CCs
might merely substitute for other alternative investments.

Considering CCs as investments contains subtle irony, as Nakamoto_2008 (81)
pseudonymously introduced the blockchain as a technology to serve as
money,131313More precisely, the intent was a protocol with the emphasis on the
tokens’ role as medium of exchange, not as stores of value. not a profitable
investment opportunity. Thus, initially doubt and debate shrouded the economic
role141414Some debate centered on the question whether investments in CCs play
an economic role similar to gold: See Dyhrberg_2016 (36, 102) for affirmative
views, and walther2018bitcoin (64) for a dissenting one. of CCs
(Glaser_et_al_2014, 47, 115, 14, 10).151515Generally, mainstream economics has
joined the research effort on CCs deplorably late; it is now catching up, see
for instance Schilling_Uhlig_2019 (100, 1). Game-theoretic modelling has been
more active, including Houy_2016 (54, 35, 19, 15).

However, after more than a decade of increasing demand, market
capitalizations, and trading volumes for a multiplying number of CCs,161616At
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 capitalization close to 250
billion USD (almost two thirds of which are due to Bitcoin), with a 24-hour
trading volume surpassing 150 billion USD. the recently flourishing academic
literature converges to the consensus that CCs constitute investments,
generally, and a distinct asset class in particular. This literature can be
categorized along two dimensions: first, which CC investment is considered?
Only Bitcoin, also a fistful of other highly visible CCs like Ethereum or
Ripple, or a broad cross-section of tradable CCs?171717Note that the commonly
reported thousands of CCs include mostly such with extremely low liquidity: As
of 2020-04-28, only 10 CCs exhibit daily trading volumes exceeding 1 billion
USD; volume below 100,000 USD exists already among the top 200 CCs. Second,
which portfolio allocations are considered? Only the CC(s), or also
traditional markets? If the latter, only equity markets, or a broad range of
traditional investments?

Regarding the first point, the literature started by investigating the
properties of the Bitcoin price process (Kristoufek_2015, 65, 28, 22, 110, 13,
8, 84, 71), establishing, in essence, the presence of all critical properties
of equity returns, yet often up to an order of magnitude stronger: CCs exhibit
exceptionally high mean returns, and likewise volatility and drawdowns;
returns also feature extremely heavy tails and high heteroskedasticity.
Correlations with common return series turn out extremely low to non-existent.

Given such low correlations, it is intuitive why including Bitcoin can enhance
a portfolio of traditional assets. Whereas the high riskiness leads to low
portfolio weights for risk-averse investors, an inclusion is beneficial as it
improves diversification.

At the same time, exploding interest in blockchain led to an explosion in the
number of investable CCs.181818Quick 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. This
kickstarted investigations into the joint-return properties of a broad cross-
section of CCs (elendner_cross-section_2017, 39, 112, 16, 116, 113), which
confirmed the return characteristics of Bitcoin to be representative for the
entire asset class;191919The 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
they constitute the only means to provide real resources to decentralised
apps. yet generally so-called _altcoins_ exhibit still higher risk and mean
returns. (Even more extreme were returns of Initial Coin Offerings, ICOs, in
particular during their peak in 2017—see, for instance,
Adhami_Guidici_Martinazzi_2018 (4, 78, 80, 79). However, despite the important
economic role of ICOs 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 of our paper.)

A key finding is that correlations are low even among CCs, as long as they are
no close substitutes or forks. This implies a potential diversification
benefit from a broad basket of CCs (chuen2017cryptocurrency, 30).
Alessandretti2018 (6), optimizing CCs-only portfolios with LSTMs and decision
trees, also find enhanced return performance.

As one consequence, CC indices were developed: The CRIX (trimborn_crix_2016,
108) captures the broad CC market movement with a statistically optimized
varying number of constituents; CCI30 (rivin2018cci30, 91) is a simple, close
analogue to stock-market indices; F5 (Elendner_2018, 38) is a momentum-factor-
based, transaction-cost-optimised basis for an exchange-traded portfolio; C20
(crypto20, 101) is an on-chain crypto-asset itself. VCRIX (Kim2019, 63) is a
volatility index for option pricing. A first paper on option pricing of
cryptos is Hou_et_al_2020 (53).

The second key finding of cross-sectional analyses is that CCs beyond the most
prominent exhibit considerably low liquidity. Portfolio calculations ignoring
liquidity might suggest trades which are impossible without extreme price
impact. trimborn_investing_2017 (109) introduce LIquidity Bounded Risk-return
Optimization (LIBRO) to account for illiquidity in CC portfolio formation.
Since our focus is to evaluate the potential of adding CCs to traditional
portfolios, i.e., we take the investor’s perspective, we provide results both
without and with the inclusion of LIBRO constraints.

In summary, the literature on CCs so far has solidly established potential
benefits of holding CCs in investment portfolios (foremost high returns and
low correlations), as well as certain difficulties (critically low liquidity).
Yet open questions remain; prime among those whether CCs “only” proxy for
alternative (non-CC) assets, or provide investment opportunities that cannot
be realized without CC positions. We close this gap by evaluating a wide range
of common asset-allocation models with and without CC positions.

##  3 Asset-allocation models

Consider a matrix $X\in\mathbb{R}^{P\times N}$ of log returns of $N$ assets
for $P$ days. In our comparative analysis we rely on a moving-window approach.
Specifically, we choose an estimation window of 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.202020We also test strategies on extending windows as in
trimborn_investing_2017 (109); since the insights are similar, these results
are not reported. For each rebalancing period $t$ ($t=1,\dots,T$, with $T$ the
number of moving windows, defined as $T=\frac{P-K}{k}$), starting on date
$K+1$, we use the data in the previous $K$ days to estimate the parameters
required to implement a particular strategy. These parameter estimates are
then used to determine the relative portfolio weights $w$ in the portfolio of
risky assets. Based on these weights, we compute the strategy’s return in
rebalancing period $t+1$. This process is iterated by adding the $k$ daily
returns for the next period in the dataset and dropping the corresponding
earliest returns, until the end of the dataset is reached. The outcome of this
rolling-window approach is a series of $P-K$ daily out-of-sample returns
generated by each of the portfolio strategies listed in Table 1. To simplify
notation, we omit the index $t$ for moving window or rebalancing period.

The traditional evaluation literature (e.g., demiguel2009optimal, 34, 97)
considers an investor whose preferences are specified in terms of utility
functions and fully described by the portfolio mean $\mu_{P}$ and variance
$\sigma_{P}$. However, merton1980estimating (77) showed that a very 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 haugen1991efficient (51), chopra1993effect (24) and chow2011survey
(27), suggest to rely only on estimates of the covariance matrix as input of
the optimization procedure. Thus, investors assume that all stocks have the
same expected returns and under this strong assumption the optimal portfolio
is the global minimum-variance portfolio. The minimum-variance portfolio
strategy represents one of the so-called risk-based portfolios, i.e., the only
input used is the estimate 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 that we consider. In addition
totraditional approaches, we consider a decision maker with risk preferences
specified in percentile terms, and portfolio construction based on higher
moments of the portfolio return-distribution, such as skewness and kurtosis.
Therefore, in our comparative study we distinguish three groups of strategies:
return-oriented, risk-oriented (or risk-based, as in clarke2013risk (31)), as
well as a tangency portfolio with Maximum Sharpe Ratio (MV-S), which we
categorize as a risk-return-oriented strategy.

Taking into account that the ranking of models changes over time, and
motivated by the fact that in many fields a combination of models performs
well (see, e.g., clemen1989combining, 32, 7), we also extend our analysis to
include the combination of portfolio models based on a bootstrap approach
inspired by schanbacher2014combining (97, 98). The detailed methodology of
combined portfolio models is discussed in Section 3.2.

###  3.1 Common asset-allocation models

In this section we review those models that we consider in the empirical
analysis. We also discuss links between the strategies and give conditions
under which they are equivalent. In general, when bringing the theoretical
models to the data, we employ in-sample moments of return distributions as
estimators of their theoretical counterparts; naturally, all evaluation then
concerns out-of-sample performance. As subsequent prices provide new
information about 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, given that—with the exception of bitcoin, for which futures are
traded since December 2017—taking short positions on CCs is at the very least
impractical, if not outright impossible.

####  3.1.1 Equally-weighted portfolio

The most naïve portfolio strategy sets equal weights (EW) for all
constituents: every asset gets a weight $w_{i}=1/N$ for $i=1,\dots,N$. If all
constituents have the same expected returns and covariances, the EW portfolio
is mean-variance optimal. However, there is no need for assumptions or
estimates regarding the distribution of the assets’ returns to implement EW.
Moreover, as demiguel2009optimal (34) show, EW allocations can actually
perform well, in particular in settings of high uncertainty, i.e., parameter
instability—the model-free approach avoids overfitting. This is also the
reason why the F5 crpto strategy builds on an EW baseline benchmark.

####  3.1.2 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 (hardle_applications_2015, 50). 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 and convenience accepts the drawbacks widely
discussed in the literature: high portfolio concentration, i.e., high
portfolio weights for a limited subset of the investment universe, and high
sensitivity to small changes in parameter estimates of $\mu$ and $\sigma$, see
jorion1985international (60), simaan1997estimation (105), kan2007optimal (62).
In a Gaussian world, portfolio weights $w$ are obtained by solving the
following optimization problem:

$\displaystyle\underset{w\in\mathbb{R}^{p}}{\text{min}}$
$\displaystyle\sigma_{P}^{2}(w)\stackrel{{\scriptstyle\operatorname{def}}}{{=}}w^{\top}\Sigma
w$ (1) s.t. $\displaystyle\mu_{P}(w)=r_{T},$ $\displaystyle
w^{\top}1_{N}=1,\hskip 8.53581ptw_{i}\geq 0$

where
$\Sigma\stackrel{{\scriptstyle\operatorname{def}}}{{=}}{{\operatorname{\text{E}}}}_{t-1}\\{(X-\mu)(X-\mu)^{\top}\\}$
and
$\mu\stackrel{{\scriptstyle\operatorname{def}}}{{=}}{{\operatorname{\text{E}}}}_{t-1}(X)$
are the sample covariance matrix and vector of mean returns respctively,
$\mu_{P}(w)\stackrel{{\scriptstyle\operatorname{def}}}{{=}}w^{\top}\mu$, is
the portfolio mean and $r_{T}$ the _target return,_ ranging from minimum
return to maximum return to trace out an efficient frontier.
${{\operatorname{\text{E}}}}_{t-1}$ is the expectation operator conditional on
the information set available at time $t-1$.

We include three benchmark Mean-Variance portfolios in our analyses: first,
the global minimum variance portfolio (“MinVar” in Table 1); second, the
tangency portfolio (“MV-S”), and third the portfolio with the highest in-
sample return (“RR-MaxRet”). In our classification approach, MinVar is a risk-
based decision rule, since it is the most averse to risk and accepts the
lowest target portfolio return. At the opposite end of Markowitz’ efficient
frontier lies the return-orientated RR-MaxRet portfolio, accepting any risk to
choose the (currently) highest possible reward. In between these two
endpoints, the MV-S portfolio occupies middle-ground: it maximizes the Sharpe
ratio (18), in this way involving both risk and return estimation for the
portfolio construction. We characterise MV-S as a risk-return-based strategy.

####  3.1.3 Optimal Conditional-Value-at-Risk portfolio

A strong limitation of Markowitz-based portfolio strategies lies in the
assumption of Gaussian distributions of assets’ log-returns. Absent those, for
investors whose preferences are not fully described 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
of CCs have even heavier tails as compared to those of equities, as detailed
in chuen2017cryptocurrency (30) and elendner_cross-section_2017 (39). The
descriptive statistics of our investment universe in Figure 7 and Table 9 in
Appendix 9.2 again provide strong evidence of this heavy-tailed distributions
for CCs. Therefore, we include a strategy that accounts for higher moments via
Conditional Value at Risk (CVaR): we include a Mean-CVaR-optimized portfolio
as in rockafellar2000optimization (92, 67).

For a given $\alpha<0.05$ risk level, the CVaR-optimized portfolio weights $w$
are derived as:

$\displaystyle\underset{w\in\mathbb{R}^{N}}{\text{min}}\text{CVaR}_{\alpha}(w),\hskip
5.69054pt\text{s.t. }\mu_{P}(w)=r_{T},w^{\top}1_{p}=1,w_{i}\geq 0,$ (2)
$\displaystyle\text{CVaR}_{\alpha}(w)=-\frac{1}{1-\alpha}\int\displaylimits_{w^{\top}X\leq-\text{VaR}_{\alpha}(w)}w^{\top}Xf(w^{\top}X|w)dw^{\top}X,$
(3)

with $\frac{\partial}{\partial w^{\top}X}F(w^{\top}X|w)=f(w^{\top}X|w)$ the
probability density function of the portfolio returns with weights $w$.
$\text{VaR}_{\alpha}(w)$ is the corresponding $\alpha$-quantile of the
cumulative distribution function, defining the loss to be expected in
$(\alpha\cdot 100)\%$ of the times.

As for the MV portfolio, we construct the efficient frontier, from which to
derive the portfolios to add to our analyses. As a risk-oriented strategy, we
add the MinCVaR strategy, minimizing the risk in terms of CVaR. As far as a
return-oriented strategy is concerned, given our methodology, the maximal
expected return arises in the same way as in the maximum-return portfolio
(“RR-MaxRet” in Table 1), by investing in the riskiest asset only. Thus, we
report this portfolio only as RR-MaxRet.

####  3.1.4 Risk-parity portfolio (with equal risk contribution, ERC)

One traditional risk-based portfolio strategy is based on the concept of risk
parity. The underlying idea is to set weights such that each asset has the
same contribution to portfolio risk, see qian2005financial (90).
maillard2010properties (73) derive properties of such portfolios and rename
them “equal-risk-contribution” (ERC) instruments. The Euler decomposition of
the portfolio volatility $\sigma_{P}(w)=\sqrt{w^{\top}\Sigma w}$
(hardle_applications_2015, 50) allows to present it in the following form:

$\displaystyle\sigma_{P}(w)\stackrel{{\scriptstyle\operatorname{def}}}{{=}}\sum_{i=1}^{N}\sigma_{i}(w)=\sum_{i=1}^{N}w_{i}\frac{\partial\sigma_{P}(w)}{\partial
w_{i}},$ (4)

where $\frac{\partial\sigma_{P}(w)}{\partial w_{i}}$ is the marginal risk
contribution and $\sigma_{i}(w)=w_{i}\frac{\partial\sigma_{P}(w)}{\partial
w_{i}}$ is the risk contribution of the $i$-th asset. So, to construct the ERC
portfolio, we calibrate:

$\displaystyle\sigma_{i}(w)=\frac{1}{N}\quad\forall i$ (5)

The ERC portfolio can be compared to the EW portfolio: instead of allocating
capital equally 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 become identical to EW portfolio. The ERC
portfolio is also comparable to the MinVar portfolio, which focuses on parity
of marginal contributions of all assets.

####  3.1.5 Maximum-diversification portfolio (based on the Portfolio
Diversification Index, PDI)

Originally, the Maximum Diversification portfolio (MD) uses an objective
function introduced in choueifaty2008toward (25) that maximizes the ratio of
weighted average asset volatilities to portfolio volatility or diversification
ratio as in Equation (22). In our study, instead of the diversification ratio
we maximize the Portfolio Diversification Index (PDI) proposed by
rudin2006portfolio (95). It consists in assessing a Principal Component
Analysis (PCA) on the weighted asset returns’ covariance matrix, i.e.,
identifying orthogonal sources of variation. In its original form, PDI does
not account for the actual portfolio weights, here we incorporate weighted
returns. We optimize:

$\displaystyle\underset{w\in\mathbb{R}^{N}}{\text{max}}\text{
PDI}_{P}(w),\quad\text{s.t. }w^{\top}1_{p}=1,\quad w_{i}\geq 0$ (6)
$\displaystyle PDI_{P}(w){=}2\sum_{i=1}^{N}iW_{i}-1,$ (7)

where $W_{i}=\frac{\lambda_{i}}{\sum_{i=1}^{N}\lambda_{i}}$ are the normalised
covariance eigenvalues $\lambda_{i}$ in decreasing order, i.e., the relative
strengths. Thus, an “ideally diversified” portfolio, i.e., when all assets are
perfectly uncorrelated and $W_{i}=1/N$ for all $i$, then $PDI={N}$. On the
contrary a $PDI\approx 1$ indicates diversification is effectively impossible.
Thus, in case of perfectly uncorrelated assets the MD portfolio will be
exactly the EW portfolio. The PDI summarises the diversification of a large
number of assets with a single statistic, and can compare the diversification
across different portfolios or time periods.

###  3.2 Averaging of portfolio models

Additional to individual allocation models, we also consider combinations of
models. After all, 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 in forecasting (avramov2002stock,
7). Traditional model-averaging methods use information criteria—like AIC or
BIC—to identify relative shares of models. Across portfolio-allocation models
the likelihood is unknown, however, therefore we calculate model shares with
the loss function $l$, defined as

$l(w)=w^{\top}\hat{\mu}-\frac{\gamma}{2}w^{\top}\hat{\Sigma}w.$ (8)

The parameter $\gamma$ reflects the investor’s risk aversion, with $\gamma$
being large (small) for a risk-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 a bootstrap procedure
described in schanbacher2014combining (97). However, in order to account for
possible time series dependencies at a daily frequency, we apply the
stationary bootstrap algorithm of politis1994stationary (88) with automatic
block-length selection proposed by politis2004automatic (89).

Consider a set of $m$ asset allocation models. The corresponding portfolio
weights per model are given by $W=(w^{1},\dots,w^{m})$. Relative shares of (or
beliefs in) individual models are $\pi=(\pi^{1},\dots,\pi^{m})$, such that
$\pi^{\top}1_{m}=1$. Then the asset weights for the combined portfolio are
given by:

$w^{comb}=\sum_{i=1}^{m}\pi^{i}w^{i}$ (9)

The Naïve combination over all asset allocation models just assigns equal
shares, i.e., $\pi^{i}_{t}=\frac{1}{m}$ for all $i=1,\dots,m$.

The alternative, more sophisticated approach is to set the share $\pi^{i}_{t}$
equal to the probability that model $i$ outperforms all other models. We apply
a bootstrap method to estimate these probabilities. For every period $t$ we
generate a random sample (with replacement) of $k$ returns using returns
$X_{k(t-1)+1}\dots X_{k(t-1)+1+K}$, i.e., $K$-long returns vectors of the
$t-1$ rolling window. We apply all $m$ asset allocation models to these
bootstrapped returns. The procedure is repeated $B$ times. Let $s_{i,b}=1$ if
model $i$ outperforms in terms of the loss function other models in the $b$-th
bootstrapped sample, otherwise $s_{i,b}=0$. The probability of model $i$ being
best is then estimated as

$\hat{\pi}^{i}_{t}=\frac{1}{B}\sum_{b=1}^{B}s_{i,b}$ (10)

where $B=100$ is our number of independent bootstrap samples, and $s_{i,b}=1$
if model $i$ is the best model in the $b$-th sample.

Model | Reference | Abbreviation
---|---|---
Model-free strategies
Equally weighted | demiguel2009optimal (34) | EW
Risk-oriented strategies
Mean-Variance – min Var | merton1980estimating (77) | MinVar
Mean-CVaR – min risk | rockafellar2000optimization (92) | MinCVaR
Equal Risk Contribution | maillard2010properties (73) | ERC
(Risk-parity) |  |
Maximum Diversification | rudin2006portfolio (95) | MD
Return-oriented strategies
Risk-Return – max return | markowitz_portfolio_1952 (74) | RR-MaxRet
Risk-Return-oriented strategies
Mean-Variance – max Sharpe | jagannathan_risk_2003 (57) | MV-S
Combination models
Naïve Combination | schanbacher2015averaging (98) | CombNaïve
Weight Combination | schanbacher2014combining (97) | Comb

Table 1: List and categorization of all asset allocation models we implement,
including their abbreviations and references.

##  4 Liquidity constraints with the LIBRO framework

In this section, we review the LIBRO framework for portfolio formation, which
prevents too high portfolio weights for low-liquidity assets, by introducing
weight constraints in the portfolio optimization which depend on liquidity.

Liquidity, however, does not have a unique definition; different concepts and
measures abound. wyss_measuring_2004 (114, 111) survey the extensive
literature on liquidity measures in equity markets; the literature on CC
liquidity is still scarce, with notable exceptions of Brauneis_et_al_2020 (17,
99). Due to the highly fragmented market structure of CC exchanges (no
dominant or central exchange is trading all assets), we employ Trading Volume
(TV) as our proxy for liquidity. TV is also the basis for the widely used
illiquidity measure, and proved suitable for the LIBRO methodology. In
principle, alternative measures like the bid-ask spread would also be
applicable, as many exchanges 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 CC markets, and aggregated
without problems. For these reasons, we follow trimborn_investing_2017 (109)
and employ TV as our liquidity measure. TV is defined as

$TV_{ij}=p_{ij}\cdot q_{ij},$ (11)

where $p_{ij}$ is the closing price212121Technically, CC markets never close;
the terminology “closing price” is still used in reference to the last price
of a day, where days are customary defined on UTC time. of asset $i$ at date
$j$, and $q_{ij}$ is the volume traded at date $j$ of asset $i$. The liquidity
of asset $i$ in period $t$ can then be measured with the sample median of
trading volume,

$TV_{i}=\frac{1}{2}(TV_{i,up}+TV_{i,lo}),$ (12)

where $TV_{i,up}=TV_{i,\lceil\frac{l+1}{2}\rceil}$ and
$TV_{i,lo}=TV_{i,\lfloor\frac{l+1}{2}\rfloor}$.

Define $M$ as the total amount invested in all $N$ assets, so that $Mw_{i}$
denotes the market value held in asset $i$. trimborn_investing_2017 (109)
formulate the constraint on the weight of asset $i$ as

$\displaystyle Mw_{i}\leq TV_{i}\cdot f_{i},$ (13)

where $f_{i}$ captures the speed with which an investor intends to be able to
clear the current position in asset $i$ via multiples of TV. For example,
$f_{i}=0.5$ implies the position in asset $i$ must not exceed 50% of median
trading volume. It results in a boundary for the weight on asset $i$ as

$\displaystyle w_{i}\leq\frac{TV_{i}\cdot f_{i}}{M}=\widehat{a}_{i}.$ (14)

The beauty of this approach lies in its ease to include it into any portfolio
optimization.

##  5 Evaluating the performance of portfolios

While Section 3 presents the set of common asset-allocation models we
implement, no unique metric exists to evaluate and compare them. In order to
draw conclusions about the effect of adding CCs to broadly diversified
portfolios, we pursue three dimensions: First, we calculate a range of widely
used performance measures in Section 5.1. Second, in Section 5.2 we run direct
tests for differences between strategies on a pair-wise basis. Third and
finally, in Section 5.3 we address the diversification effect of CCs directly
by calculating three well-known measures of portfolio concentration.

###  5.1 Performance measures

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 well as by practitioners. Performance measures
are computed based on the time series of daily out-of-sample returns generated
by each strategy.

First, we measure the _cumulative wealth (CW)_ generated by each strategy $i$

$W_{i,t+1}=W_{i,t}+{\hat{w}_{i,t}^{\top}X_{t+1}},$ (15)

starting with an initial portfolio wealth of $W_{0}=\$1$. Cumulative wealth,
while naturally of high interest as a measure of performance achieved over the
period considered, is not sufficient to rank our allocation approaches.
Therefore why we also compute two traditional measures 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 and
the non-Gaussian nature of the return distributions.

The _Sharpe Ratio_ (SR) of strategy $i$ is defined as the sample mean of out-
of-sample excess returns (over the risk-free rate), scaled by their respective
standard deviation. This definition presumes 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 zero lower
bound on interest rates; so we can sidestep the question by implicitly setting
the riskless rate to 0 and defining

$\widehat{SR_{i}}=\frac{\hat{\mu_{i}}}{\hat{\sigma_{i}}^{2}}.$ (16)

The _Certainty Equivalent_ (CEQ) captures, for an investor with a given risk
aversion $\gamma$, the riskless return that said investor would consider of
equal utility as the risky return under evaluation. For the case $\gamma=1$,
it is equivalent to the close-form solution of markowitz_portfolio_1952 (74)
portfolio optimization problem in Equation (1).

$\widehat{CEQ_{i,\gamma}}=\hat{\mu_{i}}-\frac{\gamma}{2}\hat{\sigma_{i}}^{2}$
(17)

While there is debate about the risk-averion coefficient best describing
investors going back to Mehra_Prescott_1985 (75), we argue that current CC
investors are unlikely to be characterized by extremely high risk aversion,
and calculate the CEQ in the empirical part of our paper with a $\gamma$ 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 assets exhibit normally distributed returns. To address this drawback,
pezier2006relative (86) propose the _Adjusted Sharpe Ratio_ (ASR). ASR
explicitly incorporates skewness and kurtosis:

$\widehat{ASR_{i}}=\widehat{SR_{i}}\left[1+\left(\frac{S_{i}}{6}\right)\widehat{SR_{i}}-\left(\frac{K_{i}}{24}\right)\widehat{SR_{i}}^{2}\right]$
(18)

where $SR_{i}$ 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 and negative excess kurtosis, as
it contains a penalty factor for negative skewness and positive excess
kurtosis.

To assess the impact of potential transaction costs associated with asset
rebalancing, we also calculate two measures for turnover. _Portfolio turnover_
is computed to capture the amount of trade necessary on rebalancing dates as

${TO_{i}}=\frac{1}{T-K}{\sum_{t=1}^{T-K}}{\sum_{j=1}^{N}}|\hat{w}_{i,j,t+1}-\hat{w}_{i,j,t+}|$
(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 change over the period,
as one needs to execute trades in order to rebalance the portfolio towards the
$w_{t}$ target. High turnover will imply significant transaction costs;
consequently, the lower Turnover of a strategy, the better it performs.

_Target turnover,_ the second turnover-related measure, captures the amount of
change in target weights between two consecutive rebalancing dates as

${TTO_{i}}=\frac{1}{T-K}{\sum_{t=1}^{T-K}}{\sum_{j=1}^{N}}|\hat{w}_{i,j,t+1}-\hat{w}_{i,j,t}|$
(20)

In contrast to Equation (19), here the difference between weights spans the
time interval of one rebalancing period, instead of the (conceptually
infinitesimal) duration of rebalancing trades. Therefore, the realized price
paths of the assets affect the measure only insofar as they lead to different
parameter estimates and thus a revision in target weights. The difference
between the two turnover measures is best illustrated by considering the EW
strategy: it may require high turnover to return to exactly equal weights per
asset every rebalancing date; yet by definition it will never exhibit and
target turnover.

###  5.2 Testing for performance differences between strategies

To test if strategies are significantly different from each other, we provide
the $p$-values of pairwise tests. The common approach by jobson1981performance
(59) is widely used in the performance evaluation literature (e.g., also in
demiguel2009optimal, 34). However, this test is not appropriate when returns
have tails heavier than the normal distribution. Therefore, as a testing
procedure we rely on the ledoit2008robust (68) test with the use of robust
inference methods. We test for difference of both CEQ and SR, and report
results for the HAC (heteroskedasticity and autocorrelation) inference
version. The procedure is described in Appendix 9.1.

###  5.3 Measuring diversification effects

To evaluate portfolio concentration and portfolio diversification effects, we
calculate three measures:  a) the _Portfolio Diversification Index_ (PDI) as
in introduced in Equation (7),  b) Effective N as introduced by
strongin2000beating (107), and  c) the Diversification Ratio.

_Effective N_ is defined as

$N_{\textit{eff}}(w_{t})=\frac{1}{\sum_{j=1}^{N}{w^{2}_{j,t}}}$ (21)

with $j=1,\dots,N$ indexing assets. Effective N varies from 1 in the case of
maximal concentration, i.e., the portfolio entirely invested in a single
asset, to $N$—its maximum achieved by an equally-weighted portfolio. The
design of Effective N is related to other traditional concentration measures,
e.g., the Herfindahl Index, the sum of squared market shares to measure 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 under consideration.

The _diversification ratio,_ suggested by choueifaty2011properties (26),
measures the proportion of a portfolio’s weighted average volatility to its
overall volatility:

$DR(w_{t})=\frac{w_{t}^{\top}\sigma_{t}}{\sqrt{w_{t}^{\top}\Sigma_{t}w_{t}}}=\frac{w_{t}^{\top}\sigma_{t}}{\sigma_{P,t}(w_{t})}$
(22)

Thus, the diversification ratio has the form of the Sharpe Ratio in Equation
(18), with the sum of weighted asset volatilities replacing the expected
excess return. In case of perfectly correlated assets, the DR equals 1; in
contrast, in a situation of “ideal diversification,” i.e., perfectly
uncorrelated assets, $DR=\sqrt{N}$. Hence, in our empirical study we report
the results for $DR^{2}$, for two reasons: First, to make it comparable to the
other two used metrics, and second, because choueifaty2008toward (25)
demonstrate that for a universe of $N$ independent risk factors, the portfolio
that weighted each factor by its inverse volatility would have a $DR^{2}$
equal to $N$. Hence $DR^{2}$ can be viewed as a measure of the effective
degrees of freedom within a given investment universe.

##  6 Data

For the empirical analysis, we collect daily price data on a sample of CCs and
traditional financial 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 is acquired from Bloomberg.
Many CCs were established only after January 2015, or ceased to trade prior to
the period we study. Since investors who apply rules-based optimization
techniques usually only consider assets with sufficient price histories, we
require CCs to have a continuous return time-series over the period of our
study in order to be included. By excluding coins that did not already
circulate in January 2015, went extinct before December 2019, or have only
patchy price series, we effectively focus on solid CCs, of interest to
investors considering positions in this novel asset class.222222We also run
our entire analysis for a sample period extending until end of December, 2017.
For this shorter period, 55 CCs fulfil our criteria, and with minor exceptions
only for combined strategies, all our results remain qualitatively unchanged.
We also sidestep ICOs. Hence, our final data sample for portfolio construction
includes 52 CCs next to 16 traditional assets. In order to cover 3 different
reallocation frequencies (daily, weekly, monthly), we calculate with daily,
weekly and monthly return series for all assets treated equally.

We employ a rolling-window approach for the portfolio construction. The
initial portfolio weights are determined from estimations based on the first
year (2015), after which we ‘roll’ through the dataset by estimating new
portfolio weights at the reallocation frequency. Depending on the employed
frequency approach, this adds one day, week, or month of data to the
estimation set and leaves out the oldest day, week or month of data, in order
to capture potentially time-varying parameters.232323As a robustness test, we
also calculate with extending windows, where no historical data is dropped and
only new observations added as they become observable. The results are
qualitatively the same.

To evaluate the performance of each of the strategies we consider, our
research question studies 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 in
nature, 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 diversification without CCs, ensuring any
relevance of CCs we find is genuine, and at the same time is still narrow
enough to allow us to add each CC individually as an asset without leading to
high-dimensionality issues in covariance estimation. The full list of
traditional constituents of the investment universe is provided in Table 2.
Tables 9 and 8 in Appendix 9.2 report summary 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., chuen2017cryptocurrency (30): CCs outperform traditional
asset classes in terms of average daily realised returns, their returns
exhibit higher volatility, with means mostly positive while the medians are
mostly negative, positive movements occur less frequently than negative ones,
but with higher magnitudes (absolute values of minima and lower deciles are
less than of maxima and higher deciles for the majority of CCs). Correlation
analysis of the top 5 CCs by market capitalization with traditional asset
classes shows the potential of CCs to increase diversification: As can be seen
from Table 7, correlation coefficients with none of the traditional assets
exceed $0.1$.

Name | Asset class
---|---
EURO STOXX 50 | Equity
S&P100 | Equity
NIKKEI225 | Equity
FTSE100 | Equity
SSE (Shanghai Stock Exchange) index | Equity
MSCI ACWI COMMODITY PRODUCERS | Commodities
GOLD | Commodities
FTSE EPRA/NAREIT DEV REITS | Real Estate
EUR/USD | Fiat currency
GBP/USD | Fiat currency
CNY/USD | Fiat currency
YEN/USD | Fiat currency
Eurozone 10Y Gov Bonds | Fixed income
UK 10Y Gov Bonds | Fixed income
USA 10Y Treasuries | Fixed income
Japan 10Y Gov Bonds | Fixed income
Table 2: List of traditional constituents of the investment universe. Note
that we term all these asset classes, including alternative assets,
“traditional” in order to contrast them with investments in cryptocurrencies
(CCs). We obtain price series for all traditional investments from Bloomberg.

##  7 Empirical results

In this section we evaluate the out-of-sample performance of the portfolio
allocation strategies in order to address Questions 1–6. We analyze two
dimensions: First, how does risk-adjusted performance compare across different
strategies and performance measures? Second, which diversification benefits
are generated by each method?

###  7.1 Including CCs in portfolios: performance effects

The first step of our performance analysis examines how adding CCs to a
portfolio affects efficient frontiers. In principle, the efficient frontier is
unique, thus identical for all allocation strategies. However, it depends on
the risk measure (variance or CVaR in this paper), as well as on whether
liquidity constraints are enforced (via LIBRO in this paper) or not.

Our second step then addresses the performance comparison across portfolio
strategies, in terms of cumulative wealth as well as popular risk-adjusted
measures.

Figure 1: Efficient frontiers surfaces: the first column displays the
frontiers for portfolios with only traditional assets (including alternative
investments, but no cryptocurrencies, CCs) as constituents, the second column
adds CCs without liquidity constraints, and the third column instead adds only
CCs up to a liquidity constraint (via the LIBRO approach with an investment
sum of USD 10 mln). The top row depicts frontiers from mean-variance
optimization, the lower one from mean-CVaR optimization. All frontiers are
built on a daily basis and plotted over the period from 2016-01-01 to
2019-12-31.

CCPEfficient_surface

####  7.1.1 Efficient frontiers

Figure 1 plots efficient frontiers for three groups of assets: only
traditional assets, traditional assets & CCs without liquidity constraints,
and traditional assets & liquid CCs, up to the constraint defined via the
LIBRO approach with an investment sum of USD 10 mln. The top row depicts
frontiers from mean-variance optimization, the lower three panels are based on
mean-CVaR-optimal allocations. All panels show frontiers built on a daily
basis, evolving over 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, but much higher expected returns can be
sought when CCs are included. The second important observation is that mean-
variance frontiers, in most cases, are shorter than mean-CVaR frontiers (the
same level of returns has lower variance than CVaR), evidence of risk not
being inadequately captured by variance, in line with expectations. The LIBRO
approach shortens the frontiers especially in the beginning of the investment
period, because it limits the influence of turbulently growing CCs with low
trading volumes. At the same time, it is visible 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 trading
volumes 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 the trough, series of strongly negative returns amidst high
volatility and evaporating liquidity lead to CCs playing close to no role in
optimal portfolios. As the market consolidates, in 2019 CCs again pick up
their role in extending the efficient frontier: however, until today the
discrepancy between portfolios with and without concern for liquidity
considerations remains pronounced. Consequently, the importance of limiting
exposure to illiquid CCs remains high. Portfolio optimization without
liquidity constraints may promise an attractive performance in theory which it
cannot realize in the market.

####  7.1.2 Comparing strategies via performance metrics

First we examine cumulative wealth, produced by the allocation strategies we
study. Figures 2 and 3 display the dynamics of cumulative wealth for eight of
the strategies considered, with and without enforcing liquidity constraints,
respectively. As benchmarks we also plot S&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
discussed subsequently. Table 3 summarizes all performance indicators.

The following conclusions can be drawn regarding final Cumulative Wealth (CW)
over the entire period of our study when ignoring liquidity: despite CCs
trading far below their historical peaks at the end of our time span, most
portfolios with CCs generally outperform benchmark portfolios with only
conventional constituents. However, the discrepancies across strategies are
huge, and the worst-performing strategy RR-MaxRet, which invests always in the
asset with the highest expected return (and thus most often in a CC), ends up
with what can be called a catastrophic result: over the four years of our
study, it loses 97% of its initial wealth by the end of 2019. Critically, the
strategy did provide stellar results during the boom 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-
performing CCs, 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 accumulated final wealth of 275%. This amounts to an annualized rate of
return of just below 30% over a 4-year period in which the S&P100 lost 10%.
Critically, this result is also achieved by investing in small CCs (and
therefore also follows the boom-and-bust cycle to a comparable degree): the
difference is driven by the very strong diversification the MD strategy
pursues by design. 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 extremes during 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 the benchmark strategies. However, Figure 4 shows how EW
performance exhibits high variation over the time span, similar in nature to
MaxRet and MD. The figure displays MD and EW separately, to elucidate two
important points: first, how disproportionately the performance of small coins
exceeded the gains of established CCs in the 2017 price explosion; second, how
seriously calculated results of portfolio allocation rules can diverge from
returns achievable by investors if lack of liquidity is not taken into
account.

Generally, LIBRO portfolios have mixed results in terms of cumulative wealth.
Most importantly, MD underperforms when enforcing LIBRO constraints. Of
course, this implies that the high performance of unconstrained optimization
can only be reached for very small investment sums. For larger portfolios,
when the liquidity constraints turn binding, performance need not necessarily
suffer. By limiting the exposure to individual (and thus also 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 it retains 59.1%. Also the combined strategy COMB, which provides a
positive performance without LIBRO, further improves by 8.6% when protecting
the portfolio from liquidity risk.

Figure 2: _Performance_ in terms of cumulative wealth of portfolio strategies
_without liquidity constraints_ with 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 strategy from Table 1.
“TrA” denotes only traditional, i.e., non-CC assets are included. Note that
the date axes are aligned, but the wealth axes are not, due to large disperion
in scales.

CCPPerformance

Figure 3: _Performance_ in terms of cumulative wealth of portfolio strategies
_with liquidity constraints_ (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 strategy from Table 1. “TrA” denotes only
traditional, i.e., non-CC assets are included. Note that the date axes are
aligned, but the wealth axes are not, due to large disperion in scales.

CCPPerformance

Allocation | Portfolio performance measures: monthly rebalancing |  |  |  |  |
---|---|---|---|---|---|---
Strategy | CW | SR | ASR | CEQ | TO | TTO
No 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 | $0.900$ | $0.900$ | $-0.016$ | $-0.016$ | $-0.016$ | $-0.016$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$
EW TrA | $1.102$ | $1.102$ | 0.033 | 0.033 | 0.033 | 0.033 | $-0.001$ | $-0.001$ | $3.615$ | $3.615$ | $0.000$ | $0.000$
MV-S TrA | $1.076$ | $1.076$ | $0.028$ | $0.028$ | $0.028$ | $0.028$ | $0.000$ | $0.000$ | $2.199$ | $2.199$ | $0.274$ | $0.274$
EW | $0.877$ | $0.877$ | $-0.003$ | $-0.003$ | $-0.003$ | $-0.003$ | $-0.001$ | $-0.001$ | $4.345$ | $4.345$ | $0.000$ | $0.000$
Risk-oriented strategies |  |  |  |  |
MinVaR | $0.990$ | $0.990$ | $-0.041$ | $-0.011$ | $-0.041$ | $-0.011$ | $-0.007$ | $-0.002$ | $8.541$ | $7.672$ | $0.056$ | $0.056$
MinCVaR | $1.021$ | $1.018$ | $0.026$ | $0.023$ | $0.026$ | $0.023$ | $0.000$ | $0.000$ | $14.884$ | $8.093$ | 0.112 | 0.114
ERC | $1.224$ | $1.035$ | 0.033 | $0.009$ | 0.033 | $0.009$ | $0.000$ | $0.000$ | $4.193$ | $9.840$ | $0.058$ | $0.064$
MD | 2.751 | $0.858$ | $0.020$ | $-0.003$ | $0.020$ | $-0.003$ | $0.000$ | $-0.001$ | $20.315$ | $48.707$ | $0.391$ | $0.209$
Return-oriented strategies |  |  |  |  |
RR-MaxRet | $0.030$ | $0.591$ | $-0.015$ | $-0.016$ | $-0.015$ | $-0.016$ | $0.000$ | $0.000$ | 0.687 | 0.731 | $0.687$ | $0.479$
Risk-Return-oriented strategies |  |  |  |  |
MV-S | $1.090$ | $1.096$ | $0.024$ | $0.028$ | $0.024$ | $0.027$ | $0.000$ | $0.000$ | $4.021$ | $8.591$ | $0.291$ | $0.290$
Combination of models |  |  |  |  |
COMB NAÏVE | $0.716$ | $0.908$ | $-0.015$ | $-0.005$ | $-0.015$ | $-0.005$ | $-0.001$ | $0.000$ | $3.553$ | $36.731$ | $0.211$ | $0.156$
COMB | $1.048$ | 1.134 | $0.010$ | $0.029$ | $0.010$ | $0.029$ | $0.000$ | $0.000$ | $6.758$ | $5.759$ | $0.148$ | $0.145$

Table 3: Performance measures for all investment strategies as well as
benchmarks over the entire 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 in Table 1. Highest
results are highlighted in red.

Figure 4: _Performance_ in terms of cumulative wealth of portfolio strategies
of the maximum-diversification strategy (MD) without (left panel) and with
(right panel) liquidity constraints (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 the wealth axes are not, due to large
disperion in scales.

Allocation strategy |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |  10 |  11
---|---|---|---|---|---|---|---|---|---|---|---
1 | S&P100 |  |  |  |  |  |  |  |  |  |  |
2 | EW-TrA |  |  |  |  |  |  |  |  |  |  |
3 | EW |  |  |  |  |  |  |  |  |  |  |
4 | RR Max Ret |  |  |  |  |  |  |  |  |  |  |
5 | MV-S |  |  |  |  |  |  |  |  |  |  |
6 | MinVar |  |  |  |  |  |  |  |  |  |  |
7 | ERC |  |  |  |  |  |  |  |  |  |  |
8 | MinCVaR |  |  |  |  |  |  |  |  |  |  |
9 | MD |  |  |  |  |  |  |  |  |  |  |
10 | COMB NAÏVE |  |  |  |  |  |  |  |  |  |  |
11 | COMB |  |  |  |  |  |  |  |  |  |  |

Table 4: Tests for difference between the Sharpe ratio SR (lower triangle) and
the certainty equivalent (CEQ, upper triangle) of all strategies with respect
to each other: color-coded $p$-values with significance at the 0.01, 0.05 and
0.1 level (without liquidity constraints).

CCPTests

Figure 5: Evolution of the _portfolio composition_ (i.e., relative weights) of
all allocation strategies (without liquidity constraints) with monthly
rebalancing over the period from 2016-01-01 to 2019-12-31: the black line
separates conventional assets (“TrA,” upper yellow part of the spectrum) from
cryptocurrencies (CCs, lower green-blue part of the spectrum).

CCPWeights

Figure 6: Evolution of the _portfolio composition_ (i.e., relative weights) of
all allocation strategies with a position limit of USD 10 mln (via the LIBRO
approach) with monthly rebalancing over the period from 2016-01-01 to
2019-12-31: the black line separates conventional assets (“TrA,” upper yellow
part of the spectrum) from cryptocurrencies (CCs, lower green-blue part of the
spectrum).

CCPWeights

Next, we analyse risk-adjusted performance for all portfolios. While MD
demonstrates superior absolute performance, ERC dominates in terms of risk-
adjusted performance, in particular in terms of its (adjusted) Sharpe Ratio of
$0.033$. Importantly, turnover is much lower at $4.2$ (unconstrained,
constrained: $9.8$), slightly below that of EW and above MV-S. Turnover and
target turnover per strategy are reported in the last four columns in Table 3,
which show that the only strategy with appreciably lower trading is RR-MaxRet
at $0.69$ (constrained: $0.73$), with the above-mentioned harsh result. This
is expected, given that the strategy is by construction the most concentrated
one, consisting of the one asset with the highest return (see also Figures 5
and 6).

It is interesting to note that for the strategies with strong diversification,
in particular MD and ERC, but also MV-S, enforcing the LIBRO constraints leads
to _higher_ turnover. This is of concern to investors, as it prompts higher
transaction costs. At first sight this observation appears 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 the last two columns, reporting target turnover: clearly, changes
_in target weights_ are mitigated via the liquidity constraints, corresponding
to intuition. At the same time, it is exactly small and illiquid CCs which
exhibit the largest volatility, and thus prompt _larger_ trades when at the
next rebalancing date positions are brought back to target weights. Enforcing
LIBRO constraints leads to positions in more (and prone to be smaller) CCs,
triggering larger rebalancing needs in terms of portfolio turnover.

Finally, Table 4 reports when the differences between strategies in terms of
CEQ or SR are significant, based on tests described in Appendix 9.1. Although
MD, COMB, and in particular MV-S have SR and CEQ higher than the EW strategy,
tests do not support significance of this difference. In constrast, the ERC
portfolio exhibits a higher SR and this difference is significant. The
comparison of risk-adjusted metrics for MinVar and MinCVaR reveals that they
differ significantly from each other—testament to the strong deviation of CC
returns from the normal distribution. MinCVaR also differs significantly from
the diversifying strategies MD and ERC.

As a robustness check, we also conduct all analyses for weekly and daily
rebalancing of portfolios. 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.

###  7.2 Including CCs in portfolios: diversification effects

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 the other hand, the most diversifying strategies
MD and ERC performed best. First, we examine the composition of the optimal
portfolios over time. Second, we run mean-variance spanning tests in order to
establish if CCs are a valuable addition to broadly diversified portfolios of
traditional assets. Third, we analyse diversification across the portfolio
strategies by means of dedicated diversification measures.

####  7.2.1 Portfolio composition

Figures 5 and 6 plot the evolution of portfolio constituents across time,
without and with liquidity constraints, respectively. At each date on the
abscissa, the simplex of weights is color-coded vertically, with traditional
assets on the light end of the spectrum and CCs towards the dark end; a black
lines indicates the boundary between the two groups. We can see wide variation
in the extent to which the strategies rely on CCs: MaxRet and MD are prone to
invest heavily in CCs, while risk-oriented strategies like MinVar and MinCVaR
hardly include any. The risk-return-oriented strategy MV-S employs CCs
conservatively, yet it does reach at times noteworthy allocations even against
the background of such a well-diversified portfolio 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,
significantly affects 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 in the first half of the
investment period.

The weights distribution of the COMB portfolio undergoes quite pronounced
changes over the investment period: from high concentration of traditional
assets to high concentration of CCs, 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 volatility structure of CCs leads to
disproportionate risk contributions relative to their capital weights:
traditional assets affect changes in portfolio values to a visibly lower
degree.

####  7.2.2 Mean-variance spanning

In order to investigate the impressions from the efficient-frontier plots in
Section 7.1.1, we conduct two mean-variance spanning tests on each of the 52
CCs: first, the corrected test of (HK, huberman1987mean, 55), second the step-
down test by kan2012tests (61).

Cryptocurrency | F-Test | F-Test1 | F-Test2
---|---|---|---
BCN | 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 frontier constructed from all traditional investment assets,
including alternative assets (see Table 2 for a complete list; $p$-value in
parentheses). F-Test refers to the corrected test of huberman1987mean (55),
$F_{1}$ and $F_{2}$ to step-down tests by kan2012tests (61), testing for
spanning of tangency portfolios and for global minimum portfolios,
respectively. Only CCs for which at least one test rejects spanning at the 10%
level are reported.

Table 5 lists only CCs with at least one test rejecting the hypothesis that
traditional assets span the frontier at the 10% level. Recall that our
definition of traditional assets includes a broad set of alternative
investments, all but CCs. The corrected HK test rejects spanning for 3 CCs. In
contrast, the step-down test provides information on the source for spanning
rejection: $F_{1}$ tests for spanning of tangency portfolios, whereas $F_{2}$
tests spanning for global minimum portfolios. From Table 5, we see that the
$F_{1}$ test rejects spanning for only 2 CCs, pointing out that tangency
portfolios which include CCs are significantly different from the benchmark
tangency portfolio, but also that the inclusion of the two years 2018–19 has
dramatically reduced that number from previously 27 CCs, which included
Bitcoin (BTC), Ripple (XRP), Dash (DASH) and Litecoin (LTC). $F_{2}$ rejects
spanning for 5 CCs for the entire time period, still including one of the
coins with the highest market capitalisation, XRP. Thus, we conclude there
still exists evidence that a MV-S portfolio can be improved by 7 out of 52
CCs, but that the integration of CC with financial markets has progressed
markedly. Anecdotal evidence in line with this finding comes from the recent
outbreak of the corona-virus pandemic, when initially CC markets moved for the
first time with strong positive correlation together with financial markets,
driven by institutional investors rebalancing in favor of cash holdings,
before CCs resumed their diversifying role in subsequent weeks.

Also, but there is little evidence that a MinVar portfolio can be improved.
This result is supported by the dynamics of the portfolios’ composition
presented in Figures 5 and 6 for unconstrained and LIBRO portfolios,
respectively: MinVar portfolios in both cases are constructed entirely from
traditional assets, whereas MV-S portfolios have a (varying) CC component
throughout the whole investment period.

In sum, the results imply that investors should consider a broader selection
of CCs (see Question 3), not only BTC. However, only a small fraction of CCs
continue to improve the efficient frontier.

####  7.2.3 Diversification metrics across portfolio strategies

Table 6 reports results on our three chosen diversification metrics (detailed
in Section 5.3). As expected, for the RR-MaxRet strategy there are no
diversification benefits—by definition it consists of only one asset at a time
(unless LIBRO forces it into more than one asset). The range of values across
diversification metrics emphasizes that diversification has different aspects
and its quantification depends on the definition used. Consequently, different
measures do not always provide identical conclusions about the diversification
effects of CCs in portfolios. For instance, in terms of a $DR^{2}$ of $13.73$
($13.44$),242424Here and henceforth we provide the values of the performance
metric for LIBRO portfolios in parentheses. MinCVaR is characterized as most
diversified strategy. Slightly lower measures pertain for MinVar and ERC
portfolios with $12.02$ ($11.72$) and $8.71$ ($9.41$), respectively. The MD
portfolio is a special case regarding this type of diversification, with
$DR^{2}$ of $2.64$ ($2.07$) and at the same time a PDI of $21.06$ ($21.01$).
Clearly PDI highest for the MD portfolio because of its objective function,
maximizing diversification via the number of independent sources of variation
in the portfolio.

Allocation | Portfolio diversification effects: monthly rebalancing |
---|---|---
Strategy | $DR^{2}$ | Effective $N$ | PDI
No const | 10 mln | No const | 10 mln | No const | 10 mln
Benchmark strategies |
MV-S TrA | $5.39$ | $5.39$ | $3.11$ | $3.11$ | $4.92$ | $4.92$
Return oriented strategies |
RR-MaxRet | $1.00$ | $1.00$ | $1.00$ | $1.48$ | $1.00$ | $1.00$
Risk-oriented strategies |
MinVar | $12.02$ | $11.72$ | $2.68$ | $2.69$ | $20.60$ | $20.60$
ERC | $8.71$ | $9.41$ | 17.16 | 13.61 | $20.62$ | $20.62$
MinCVaR | 13.73 | 13.44 | $3.15$ | $3.15$ | $20.60$ | $20.61$
MD | $2.64$ | $2.07$ | $3.99$ | $3.17$ | 21.06 | 21.01
Risk-Return-oriented strategies |
MV-S | $8.03$ | $7.48$ | $3.26$ | $3.35$ | $20.62$ | $20.62$
Combination of models |
COMB NAÏVE | $3.43$ | $3.62$ | $11.60$ | $10.92$ | $20.65$ | $20.67$
COMB | $10.00$ | $9.44$ | $3.39$ | $3.40$ | $20.61$ | $20.61$

Table 6: Measures of diversification for all investment strategies and a
benchmark, over the entire time period from 2016-01-01 to 2019-12-31, with
monthly rebalancing ($l=21$). $DR^{2}$ 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 without liqudity
constraints (columns “No const”) are contrasted with those when applying LIBRO
with a threshold of USD 10 mln (column “10 mln”). Highest results are
highlighted in red.

The ERC portfolio is characterized by the highest Effective $N$ of $17.16$
($13.61$) by a large margin, also a typical result (see, e.g., clarke2013risk,
31) due to its nature: it includes all assets by definition. Apart from
MaxRet, the lowest Effective $N$ of $2.68$ ($2.69$) arises for the MinVar
portfolio, containing only traditional assets, showing that fewer than $3$
equally-weighted stocks would provide the same diversification by this
measure. All other individual strategies also exhibit Effective $N$ ranging
between 3 and 4. One more remarkable result concerns the combined portfolios’
concentration: While COMB’s Effective $N$ lies in the range of individual
strategies, COMB Naïve exceeds $10$ both in constrained and unconstrained
portfolios. In terms of $DR^{2}$, the combined strategies rank inversely,
reaching $3.43$ ($3.62$) for COMB Naïve and $10.0$ ($9.44$) for COMB; their
PDIs are similar to those of the other risk-oriented portfolios MinVar,
MinCVaR and ERC.

Note that with the exception of ERC liquidity constraints do not strongly
affect the diversification features of portfolios: all metrics display only
minor changes. This result is due to the fact that LIBRO generally lowers the
weight of constituents, but does not completely exclude them.

We particularly highlight the difference of diversification of the MV-S
portfolios with and without CCs: diversification measured by $DR^{2}$
increases through the inclusion of CCs from $5.39$ ($5.39$) to $8.03$
($7.48$), and PDI scales up even more strongly, from $4.92$ ($4.92$) to
$20.62$ ($20.62$). Thus, the inclusion of CCs remarkably improves portfolio
diversification, especially in terms of the distribution of principal
portfolio variances.

###  7.3 Interpretation of the results

In this section we relate our empirical results to the Questions 1–6, along
which the contribution of this paper is structured.

_Question 1: For whom is investing in the CC market valuable? Is the benefit
derived from adding CCs to a portfolio dependent on the investor’s objectives
(e.g., return-oriented or diversification seeking)?_

As the efficient frontiers in Figure 1 clearly show, the main benefit of CCs
accrue to investors who make use of the high-risk/high-return character of
their returns; investors with low risk tolerance benefit least. While it is
not surprising that CCs constitute risky investments, 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 a risk-
return orientation is necessary for CCs to play a noteworthy and permanent
role in portfolios. 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 end of 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 time period 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 precisely the ones that gained
disproportionately in value during the price rally, and subsequently suffered
the severest. Therefore, over our complete time span the EW portfolio in fact
lost $12.3\%$ 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 broader cross-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 depends on the investor’s objective. In particular investors
targeting a well-diversified portfolio while willing to bear some risk are
advised to consider CCs for their investments.

_Question 2: To which type of investor are CC investments most useful? Only
professional traders who rebalance their portfolio frequently, or also less
actively trading retail investors?_

The rebalancing frequency (whether portfolio positions are traded daily,
weekly, or monthly to react to market developments by updating estimates and
to revert positions to target weights) does influence the performance of
investors’ portfolios. For instance, over our study period cumulative wealth
for the MV-S strategy grows by 5% when readjusting the portfolio on a daily
basis, by 12% with weekly, and 9% with monthly position changes. These
difference become more pronounced when transaction costs are deducted, as
turnover is naturally higher at a higher rebalancing frequencies. For the RR-
MaxRet strategey, the loss attenuates with weekly and exacerbates with daily
reallocations.

However, the overall picture does not change across rebalancing frequencies:
Even with a more frequent reallocations, still diversification-seeking
investors (ERC and MD) significantly outperform the other investment
strategies. Therefore, our general conclusions about the effect of adding CCs
into investment portfolios do not change qualitatively between daily traders,
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 is beneficial. At the same time, investors could diversify too
much. As Table 6 shows, the MD strategy, which had the highest return,
showcases an Effective $N$ of only $3.26$. ERC has much higher Effective $N$
of $17.16$, still it features considerably lower final cumulative wealth, at
least in unconstrained optimization. Judged by PDI, MD is the most successful
strategy, which of course is driven by the fact that the target-weight
allocation of MD is derived precisely by maximizing PDI. However, this also
indicates that including as many assets in the portfolio as possible is not
necessary to adequately represent the covariance matrix, and not beneficial in
terms of cumulative wealth.

Figures 5 and 6 caution the interpretation of MD dominating in terms of
accumulated returns. Both Figures show that MD includes a broad range of CCs,
whereas MinVar and MinCVaR—both with comparable Effective N and PDI—almost
entirely exclude them, giving weight only to traditional assets. In this sense
we do find evidence that CCs can substitute for traditional assets in
portfolio optimization.

Regarding the ERC strategy, while it reaches optimal diversification for the
alternative metric of Effective $N$, it provides sizable gains in cumulative
wealth and at the same adequately diversifies the portfolio. Figures 5 and 6
indicate that CCs and traditional assets are mixed in the portfolio, while the
PDI is close to the one of MD and $DR^{2}$ only second to the pure risk-
oriented strategies MinVar and MinCVaR.

Therefore, including CCs to diversify the portfolio is beneficial to achieve
high target returns, and balancing traditional assets and CCs is advisable.

_Question 4: What exposure to each CC should be held in the portfolio? How
informative are past prices, how stable are positions when re-balancing the
portfolio? Do model-free strategies like equal-weighting provide reasonable
results?_

Even though CCs are highly volatile, the past pricing series are informative
for portfolio allocation. As such, quantitative methodologies for portfolio
allocation are applicable and one 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, strategies
exclusively targeted at lowering risk at all cost do not benefit from CCs.
This is of course unsurprising, since lower risk must go at the expense of
lesser expected return, most clearly visible already in the efficient
frontiers in Figure 1.

_Question 5: Can these strategies be implemented in practice? In particular,
are all CCs liquid enough for inclusion in an investment portfolio? If not,
how can investors still profit from promising CCs with little trading volume
without exposing their portfolio too much to illiquidity? 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
LIBRO (trimborn_investing_2017, 109) turn out to bind indicates that several
CCs are not sufficiently liquid for investors with deeper pockets. Still, the
approach allows the inclusion of illiquid CCs up to restricted amounts. This
has the positive effect that investors can still perform diversification
strategies to quite some degree—strategies that rank among the most
profitable. However, the impressive results by strategies with broad CC
exposure turn out not to be very scalable. For instance, the MD strategy shows
excellent performance without liqudity constraints ($+175\%$), yet the
application of LIBRO pushes final CW below initial wealth ($-14\%$).

Clearly, the performance of unconstrained MD profits from unreasonably high
weights on small and illiquid CCs. Table 6 illustrates this in terms of
Effective $N$ and PDI: MD reaches a very low Effective $N$ of $3.99$, although
it includes only CCs, compare the weight composition in Figures 5 and 6. PDI
is clearly higher than for other strategies, in line with the objective of 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) singular CC(s), driving the high returns, which
cannot be traded sufficiently for a portfolio of USD 10 mln. At the other end
of the spectrum, the mininum-risk strategies focus on traditional assets 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 certain type of portfolio-allocation method more suitable to manage and
simultaneously exploit their properties?_

Regarding CC return properties, a lot of prior research exists (see also our
literature review in Section 2). We confirm that the patterns of generally
high means, high volatilities, excess kurtosis, and low correlations with
traditional assets are also present in our sample (see the descriptive
statistics in Appendix 9.2). Our contribution addresses the effect of
including CCs in already broadly diversified portfolios: Beyond what we have
established in the answers to the preceding five questions, our central
finding is that the key conclusion of prior studies—that CCs are valuable
additions to the investment universe—holds true, but it is critical to mind
the limits of quantitative results derived from simplified frameworks. While
diversification strategies prove most promising, including only the top CCs
foregoes diversification potential. Most importantly, returns of broad CC
portfolios that are calculated without accounting for liquidity remain
virtual: they cannot be realized by professional investors.

Finally, for certain types of investors, namely those highly risk averse, the
benefits can prove too risky to pursue.

##  8 Conclusion

This study investigates cryptocurrencies as new investment assets available to
portfolio management. We investigate the utility gains for different types of
investors when they consider cryptocurrencies as an addition to a well
diversified portfolio of traditional assets. We consider risk-averse, return-
seeking as well as diversification-preferring investors who trade along
different allocation frequencies, namely daily, weekly or monthly. To conduct
this study, we analyze the performance of commonly used asset-allocation
models based on historical prices and trading volumes of 52 cryptocurrencies,
combined with 16 traditional assets. The rules-based investment methods cover
a broad spectrum of investor objectives, from the classical Markowitz
optimization to recent strategies aiming to maximize portfolio
diversification. Along with individual portfolio allocation strategies, we
also include combined strategies from model averaging. The performance of
portfolios is evaluated with a range of different measures, including
cumulative wealth, risk-adjusted performance and diversification effects
produced by portfolios.

We find that due to the volatility structure of cryptocurrencies, the
application of traditional risk-based portfolios strategies, such as equal-
risk contribution, minimum-variance and minimum-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 or
broader cryptocurrency exposure for investors. As for diversification
benefits, we demonstrate an effect beyond well-diversified, global portfolios
of conventional assets without CCs. We also document how various rules have
different effects on portfolio diversification, depending on the concept of
diversification and the chosen measure of its quantification.

Furthermore, following the idea of model averaging and diversification across
models, we show that both naive and bootstrap-based combined portfolios
exhibit robust high risk-adjusted returns. Portfolios with model-averaged
weights achieve significantly higher performance than purely risk-oriented
strategies and not significantly lower than the best performing strategies.

We also show how different rebalancing frequencies affect performance, as well
as how constraints mitigating liquidity risks of cryptocurrencies (LIBRO) can
significantly affect the outcome of strategies that rely on a larger cross-
section of CCs. The results remain coherent across all frameworks. Further
extensions can be made along three main lines: first, more involved estimators
of expected returns and the covariance matrix could be employed; second, more
performance measures could be used to evaluate the investment strategies’
results; and third, additional portfolio-allocation strategies could be
included in the comparison. In particular, factor-based APT (arbitrage price
theory) models would constitute the complementary approach to statistical-
optimisation techniques studied in this paper.

## References

  * (1) Joseph Abadi and Markus K. Brunnermeier  “Blockchain Economics” Available at https://scholar.princeton.edu/markus/publications/blockchain-economics, 2019
  * (2) Fabian Ackermann, Walt Pohl and Karl Schmedders  “Optimal and Naive Diversification in Currency Markets”  In _Management Science_ 63.10, 2017, pp. 3347–3360  DOI: 10.1287/mnsc.2016.2497
  * (3) Kwame Addae-Dapaah and Hwee Loh  “Exchange Rate Volatility and International Real Estate Diversification: A Comparison of Emerging and Developed Economies”  In _Journal of Real Estate Portfolio Management_ 11.3, 2005, pp. 225–240  DOI: 10.5555/repm.11.3.l8g742127472h533
  * (4) Saman Adhami, Giancarlo Giudici and Stefano Martinazzi  “Why do businesses go crypto? An empirical analysis of initial coin offerings” FinTech – Impact on Consumers, Banking and Regulatory Policy  In _Journal of Economics and Business_ 100, 2018, pp. 64–75  DOI: https://doi.org/10.1016/j.jeconbus.2018.04.001
  * (5) Md Akhtaruzzaman, Ahmet Sensoy and Shaen Corbet  “The influence of Bitcoin on portfolio diversification and design”  In _Finance Research Letters_ , 2019, pp. 101344  DOI: https://doi.org/10.1016/j.frl.2019.101344
  * (6) Laura Alessandretti, Abeer ElBahrawy, Luca Maria Aiello and Andrea Baronchelli  “Anticipating Cryptocurrency Prices Using Machine Learning”  In _Complexity_ 2018 Hindawi Limited, 2018, pp. 1–16  DOI: 10.1155/2018/8983590
  * (7) Doron Avramov  “Stock return predictability and model uncertainty”  In _Journal of Financial Economics_ 64.3 Elsevier, 2002, pp. 423–458
  * (8) Aurelio F. Bariviera, María José Basgall, Waldo Hasperué and Marcelo Naiouf  “Some stylized facts of the Bitcoin market”  In _Physica A: Statistical Mechanics and its Applications_ 484, 2017, pp. 82–90  DOI: https://doi.org/10.1016/j.physa.2017.04.159
  * (9) Pedro Barroso and Pedro Santa-Clara  “Beyond the Carry Trade: Optimal Currency Portfolios”  In _Journal of Financial and Quantitative Analysis_ 50.5 Cambridge University Press, 2015, pp. 1037–1056  DOI: 10.1017/S0022109015000460
  * (10) Dirk G. Baur, KiHoon Hong and Adrian D. Lee  “Bitcoin: Medium of exchange or speculative assets?”  In _Journal of International Financial Markets, Institutions and Money_ 54, 2018, pp. 177–189  DOI: https://doi.org/10.1016/j.intfin.2017.12.004
  * (11) Julia Belousova and Gregor Dorfleitner  “On the diversification benefits of commodities from the perspective of euro investors”  In _Journal of Banking and Finance_ 36 Elsevier, 2012, pp. 2455–2472  URL: https://epub.uni-regensburg.de/24267/
  * (12) John Benjamin, Stacy Sirmans and Emily Zietz  “Returns and Risk on Real Estate and Other Investments: More Evidence”  In _Journal of Real Estate Portfolio Management_ 7.3, 2001, pp. 183–214  DOI: 10.5555/repm.7.3.b27x8337007445g5
  * (13) Benjamin M. Blau  “Price dynamics and speculative trading in Bitcoin”  In _Research in International Business and Finance_ 43, 2018, pp. 15–21  DOI: https://doi.org/10.1016/j.ribaf.2017.07.183
  * (14) Rainer Böhme, Nicolas Christin, Benjamin Edelman and Tyler Moore  “Bitcoin: Economics, Technology, and Governance”  In _Journal of Economic Perspectives_ 29.2, 2015, pp. 213–38  DOI: 10.1257/jep.29.2.213
  * (15) Wilko Bolt and Maarten R.C. van Oordt  “On the Value of Virtual Currencies”  In _Journal of Money, Credit and Banking_ , 2020  DOI: 10.1111/jmcb.12619
  * (16) Alexander Brauneis and Roland Mestel  “Price discovery of cryptocurrencies: Bitcoin and beyond”  In _Economics Letters_ 165, 2018, pp. 58–61  DOI: https://doi.org/10.1016/j.econlet.2018.02.001
  * (17) Alexander Brauneis, Roland Mestel and Erik Theissen  “What Drives the Liquidity of Cryptocurrencies? A Long-Term Analysis”  In _Finance Research Letters_ , 2020, pp. 101537  DOI: https://doi.org/10.1016/j.frl.2020.101537
  * (18) Marie Brière, Kim Oosterlinck and Ariane Szafarz  “Virtual currency, tangible return: Portfolio diversification with bitcoin”  In _Journal of Asset Management_ 16.6, 2015, pp. 365–373
  * (19) Carey Caginalp and Gunduz Caginalp  “Establishing cryptocurrency equilibria through game theory”  In _AIMS Mathematics_ 4.3, 2019, pp. 420–436  DOI: http://dx.doi.org/10.3934/math.2019.3.420
  * (20) Rachel. Campbell  “Art as a Financial Investment”  In _The Journal of Alternative Investments_ 10.4 Institutional Investor Journals Umbrella, 2008, pp. 64–81  DOI: 10.3905/jai.2008.705533
  * (21) Mark M. Carhart  “On Persistence in Mutual Fund Performance”  In _The Journal of Finance_ 52.1 [American Finance Association, Wiley], 1997, pp. 57–82  URL: http://www.jstor.org/stable/2329556
  * (22) Eng-Tuck Cheah and John Fry  “Speculative bubbles in Bitcoin markets? An empirical investigation into the fundamental value of Bitcoin”  In _Economics Letters_ 130, 2015, pp. 32–36  DOI: https://doi.org/10.1016/j.econlet.2015.02.029
  * (23) Cathy Yi-Hsuan Chen and Christian M Hafner  “Sentiment-induced bubbles in the cryptocurrency market”  In _Journal of Risk and Financial Management_ 12.2 Multidisciplinary Digital Publishing Institute, 2019, pp. 53
  * (24) Vijay Kumar Chopra and William T Ziemba  “The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice”  In _The Journal of Portfolio Management_ 19.2 Institutional Investor Journals Umbrella, 1993, pp. 6–11
  * (25) Yves Choueifaty and Yves Coignard  “Toward maximum diversification”  In _Journal of Portfolio Management_ 35.1 Euromoney Institutional Investor PLC, 2008, pp. 40–51
  * (26) Yves Choueifaty, Tristan Froidure and Julien Reynier  “Properties of the most diversified portfolio”  In _Journal of Investment Strategies_ 2.2, 2011, pp. 49–70
  * (27) Tzee-man Chow, Jason Hsu, Vitali Kalesnik and Bryce Little  “A survey of alternative equity index strategies”  In _Financial Analysts Journal_ 67.5 CFA Institute, 2011, pp. 37–57
  * (28) Jeffrey Chu, Saralees Nadarajah and Stephen Chan  “Statistical Analysis of the Exchange Rate of Bitcoin”  In _PLOS ONE_ 10.7 Public Library of Science, 2015, pp. 1–27  DOI: 10.1371/journal.pone.0133678
  * (29) Patrick Chu  “Study on the Diversification Ability of Fine Wine Investment”  In _The Journal of Investing_ 23, 2014, pp. 123–139  DOI: 10.3905/joi.2014.23.1.123
  * (30) David LEE Kuo Chuen, Li Guo and Yu Wang  “Cryptocurrency: A New Investment Opportunity?”  In _The Journal of Alternative Investments_ 20.3 Institutional Investor Journals Umbrella, 2017, pp. 16–40
  * (31) Roger Clarke, Harindra De Silva and Steven Thorley  “Risk parity, maximum diversification, and minimum variance: An analytic perspective”  In _Journal of Portfolio Management_ 39.3 Euromoney Institutional Investor PLC, 2013, pp. 39–53
  * (32) Robert T Clemen  “Combining forecasts: A review and annotated bibliography”  In _International Journal of Forecasting_ 5.4 Elsevier, 1989, pp. 559–583
  * (33) John H. Cochrane  “Presidential Address: Discount Rates”  In _The Journal of Finance_ 66.4, 2011, pp. 1047–1108  DOI: 10.1111/j.1540-6261.2011.01671.x
  * (34) Victor DeMiguel, Lorenzo Garlappi and Raman Uppal  “Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy?”  In _The Review of Financial Studies_ 22.5 Oxford University Press, 2009, pp. 1915–1953
  * (35) Nicola Dimitri  “Bitcoin Mining as a Contest”  In _Ledger_ 2, 2017, pp. 31–37  DOI: 10.5195/ledger.2017.96
  * (36) Anne Haubo Dyhrberg  “Hedging capabilities of bitcoin. Is it the virtual gold?”  In _Finance Research Letters_ 16, 2016, pp. 139–144  DOI: https://doi.org/10.1016/j.frl.2015.10.025
  * (37) Alexander Eisl, Stephan M. Gasser and Karl Weinmayer  “Caveat Emptor: Does Bitcoin Improve Portfolio Diversification?” Available at SSRN 2408997, 2015
  * (38) Hermann Elendner  “F5: optimised crypto-currency investment strategies” Available at f5crypto.com, 2018  URL: %5Curl%7Bf5crypto.com%7D
  * (39) Hermann Elendner, Simon Trimborn, Bobby Ong and Teik Ming Lee  “The Cross-Section of Crypto-Currencies as Financial Assets”  In _Handbook of Digital Finance and Financial Inclusion: Cryptocurrency, FinTech, InsurTech, and Regulation_ 1 Academic Press, 2018
  * (40) Eugene F. Fama and Kenneth R. French  “Common risk factors in the returns on stocks and bonds”  In _Journal of Financial Economics_ 33.1, 1993, pp. 3–56  DOI: https://doi.org/10.1016/0304-405X(93)90023-5
  * (41) Eugene F. Fama and Kenneth R. French  “A five-factor asset pricing model”  In _Journal of Financial Economics_ 116.1, 2015, pp. 1–22  DOI: https://doi.org/10.1016/j.jfineco.2014.10.010
  * (42) Eugene Fama and Kenneth French  “The Cross-Section of Expected Stock Returns”  In _The Journal of Finance_ 47.2 [American Finance Association, Wiley], 1992, pp. 427–465  DOI: 10.1111/j.1540-6261.1992.tb04398.x
  * (43) Guanhao Feng, Stefano Giglio and Dacheng Xiu  “Taming the Factor Zoo: A Test of New Factors”  In _The Journal of Finance_ , 2020  DOI: 10.1111/jofi.12883
  * (44) James J. Fogarty  “Wine Investment and Portfolio Diversification Gains”  In _Journal of Wine Economics_ 5 Cambridge University Press, 2010, pp. 119–131  DOI: 10.1017/S1931436100001401
  * (45) John Fry and Eng-Tuck Cheah  “Negative bubbles and shocks in cryptocurrency markets”  In _International Review of Financial Analysis_ 47, 2016, pp. 343–352  DOI: https://doi.org/10.1016/j.irfa.2016.02.008
  * (46) Neil Gandal, JT Hamrick, Tyler Moore and Tali Oberman  “Price manipulation in the Bitcoin ecosystem”  In _Journal of Monetary Economics_ 95, 2018, pp. 86–96  DOI: https://doi.org/10.1016/j.jmoneco.2017.12.004
  * (47) Florian Glaser, Kai Zimmermann, Martin Haferkorn, Moritz Christian Weber and Michael Siering  “Bitcoin – Asset or Currency? Revealing Users’ Hidden Intentions”  In _ECIS 2014_ , 2014
  * (48) Paul Gompers, Anna Kovner, Josh Lerner and David Scharfstein  “Performance persistence in entrepreneurship”  In _Journal of Financial Economics_ 96.1, 2010, pp. 18–32  DOI: https://doi.org/10.1016/j.jfineco.2009.11.001
  * (49) Christian M Hafner  “Testing for Bubbles in Cryptocurrencies with Time-Varying Volatility”  In _Journal of Financial Econometrics_ 18.2, 2018, pp. 233–249  DOI: 10.1093/jjfinec/nby023
  * (50) Wolfgang Karl Härdle and Léopold Simar  “Applications in Finance”  In _Applied Multivariate Statistical Analysis_ Springer, 2015, pp. 487–499
  * (51) Robert A Haugen and Nardin L Baker  “The efficient market inefficiency of capitalization-weighted stock portfolios”  In _The Journal of Portfolio Management_ 17.3 Institutional Investor Journals Umbrella, 1991, pp. 35–40
  * (52) Thi-Hong-Van Hoang, Hooi Hooi Lean and Wing-Keung Wong  “Is gold good for portfolio diversification? A stochastic dominance analysis of the Paris stock exchange”  In _International Review of Financial Analysis_ 42, 2015, pp. 98–108  DOI: https://doi.org/10.1016/j.irfa.2014.11.020
  * (53) Ai Jun Hou, Weining Wang, Cathy Y H Chen and Wolfgang Karl Härdle  “Pricing Cryptocurrency Options”  In _Journal of Financial Econometrics_ 18.2, 2020, pp. 250–279  DOI: 10.1093/jjfinec/nbaa006
  * (54) Nicolas Houy  “The Bitcoin Mining Game”  In _Ledger_ 1, 2016, pp. 53–68  DOI: 10.5195/ledger.2016.13
  * (55) Gur Huberman and Shmuel Kandel  “Mean-variance spanning”  In _The Journal of Finance_ 42.4 Wiley Online Library, 1987, pp. 873–888
  * (56) Stefan Hubrich  “Know when to hodl them, know when to fodl them: An Investigation of Factor Based Investing in the Cryptocurrency Space” Available at SSRN 3055498, 2017  DOI: 10.13140/RG.2.2.35090.96969
  * (57) Ravi Jagannathan and Tongshu Ma  “Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps”  In _The Journal of Finance_ 58.4, 2003, pp. 1651–1683
  * (58) Narasimhan Jegadeesh and Sheridan Titman  “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency”  In _The Journal of Finance_ 48.1 [American Finance Association, Wiley], 1993, pp. 65–91  URL: http://www.jstor.org/stable/2328882
  * (59) J Dave Jobson and Bob M Korkie  “Performance hypothesis testing with the Sharpe and Treynor measures”  In _The Journal of Finance_ 36.4 Wiley Online Library, 1981, pp. 889–908
  * (60) Philippe Jorion  “International portfolio diversification with estimation risk”  In _Journal of Business_ 58 JSTOR, 1985, pp. 259–278
  * (61) Raymond Kan and GuoFu Zhou  “Tests of Mean-Variance Spanning”  In _Annals of Economics & Finance_ 13.1, 2012, pp. 145–193
  * (62) Raymond Kan and Guofu Zhou  “Optimal portfolio choice with parameter uncertainty”  In _Journal of Financial and Quantitative Analysis_ 42.3 Cambridge University Press, 2007, pp. 621–656
  * (63) Alisa Kim, Simon Trimborn and Wolfgang K Härdle  “VCRIX—A Volatility Index for Crypto-Currencies” Available at SSRN 3480348, 2019
  * (64) Tony Klein, Hien Pham Thu and Thomas Walther  “Bitcoin is not the New Gold – A comparison of volatility, correlation, and portfolio performance”  In _International Review of Financial Analysis_ 59 Elsevier, 2018, pp. 105–116
  * (65) Ladislav Kristoufek  “What Are the Main Drivers of the Bitcoin Price? Evidence from Wavelet Coherence Analysis”  In _PLOS ONE_ 10.4 Public Library of Science, 2015, pp. 1–15  DOI: 10.1371/journal.pone.0123923
  * (66) Tim A. Kroencke, Felix Schindler and Andreas Schrimpf  “International Diversification Benefits with Foreign Exchange Investment Styles”  In _Review of Finance_ 18.5, 2013, pp. 1847–1883  DOI: 10.1093/rof/rft047
  * (67) Pavlo Krokhmal, Jonas Palmquist and Stanislav Uryasev  “Portfolio optimization with conditional value-at-risk objective and constraints”  In _Journal of Risk_ 4, 2002, pp. 43–68
  * (68) Oliver Ledoit and Michael Wolf  “Robust performance hypothesis testing with the Sharpe ratio”  In _Journal of Empirical Finance_ 15.5 Elsevier, 2008, pp. 850–859
  * (69) John Lintner  “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets”  In _The Review of Economics and Statistics_ 47.1 The MIT Press, 1965, pp. 13–37  URL: http://www.jstor.org/stable/1924119
  * (70) Edith X. Liu  “Portfolio Diversification and International Corporate Bonds”  In _Journal of Financial and Quantitative Analysis_ 51.3 Cambridge University Press, 2016, pp. 959–983  DOI: 10.1017/S002210901600034X
  * (71) Yukun Liu and Aleh Tsyvinski  “Risks and Returns of Cryptocurrency” Available at SSRN 3226952, 2018
  * (72) Yukun Liu, Aleh Tsyvinski and Xi Wu  “Common Risk Factors in Cryptocurrency” Available at SSRN 3379131, 2019
  * (73) Sébastien Maillard, Thierry Roncalli and Jérôme Teïletche  “The properties of equally weighted risk contribution portfolios”  In _The Journal of Portfolio Management_ 36.4, 2010, pp. 60–70
  * (74) Harry Markowitz  “Portfolio Selection”  In _The Journal of Finance_ 7.1, 1952, pp. 77–91  DOI: 10.2307/2975974
  * (75) Rajnish Mehra and Edward C. Prescott  “The equity premium: A puzzle”  In _Journal of Monetary Economics_ 15.2, 1985, pp. 145–161  DOI: https://doi.org/10.1016/0304-3932(85)90061-3
  * (76) Jianping Mei and Michael Moses  “Art as an Investment and the Underperformance of Masterpieces”  In _American Economic Review_ 92.5, 2002, pp. 1656–1668  DOI: 10.1257/000282802762024719
  * (77) Robert C Merton  “On estimating the expected return on the market: An exploratory investigation”  In _Journal of Financial Economics_ 8.4 Elsevier, 1980, pp. 323–361
  * (78) Paul P Momtaz  “Initial coin offerings” Available at SSRN 3166709, 2018
  * (79) Paul P. Momtaz  “The Pricing and Performance of Cryptocurrency”  In _The European Journal of Finance_ Routledge, 2019  DOI: 10.1080/1351847X.2019.1647259
  * (80) Paul P Momtaz  “Token sales and initial coin offerings: Introduction”  In _The Journal of Alternative Investments_ 21.4 Institutional Investor Journals Umbrella, 2019, pp. 7–12
  * (81) Satoshi Nakamoto  “Bitcoin: A peer-to-peer electronic cash system” Available at: https://bitcoin.org/bitcoin.pdf, 2008  URL: https://bitcoin.org/bitcoin.pdf
  * (82) Robert Novy-Marx  “Predicting anomaly performance with politics, the weather, global warming, sunspots, and the stars”  In _Journal of Financial Economics_ 112.2, 2014, pp. 137–146  DOI: https://doi.org/10.1016/j.jfineco.2014.02.002
  * (83) José Antonio Núñez, Mario I. Contreras-Valdez and Carlos A. Franco-Ruiz  “Statistical analysis of bitcoin during explosive behavior periods”  In _PLOS ONE_ 14.3 Public Library of Science, 2019, pp. 1–22  DOI: 10.1371/journal.pone.0213919
  * (84) Joerg Osterrieder  “The Statistics of Bitcoin and Cryptocurrencies”  In _2017 International Conference on Economics, Finance and Statistics (ICEFS 2017)_ Atlantis Press, 2017  DOI: https://doi.org/10.2991/icefs-17.2017.33
  * (85) Daniel Traian Pele, Niels Wesselhöfft, Wolfgang K. Härdle, Michalis Kolossiatis and Yannis G. Yatracos  “A Statistical Classification of Cryptocurrencies” Available at SSRN 3548462, 2020
  * (86) Jacques Pezier and Anthony White  “The relative merits of investable hedge fund indices and of funds of hedge funds in optimal passive portfolios”  In _The Journal of Alternative Investments_ 10.4, 2008, pp. 37–49  DOI: https://doi.org/10.3905/jai.2008.705531
  * (87) Emmanouil Platanakis and Andrew Urquhart  “Should investors include Bitcoin in their portfolios? A portfolio theory approach”  In _The British Accounting Review_ , 2019, pp. 100837  DOI: https://doi.org/10.1016/j.bar.2019.100837
  * (88) Dimitris N. Politis and Joseph P. Romano  “The stationary bootstrap”  In _Journal of the American Statistical Association_ 89.428 Taylor & Francis Group, 1994, pp. 1303–1313
  * (89) Dimitris N. Politis and Halbert White  “Automatic block-length selection for the dependent bootstrap”  In _Econometric Reviews_ 23.1 Taylor & Francis, 2004, pp. 53–70
  * (90) Edward E Qian  “On the financial interpretation of risk contribution: Risk budgets do add up”  In _Journal of Investment Management_ 4.4, 2006, pp. 41–51
  * (91) Igor Rivin and Carlo Scevola  “The CCI30 Index” Available at https://arxiv.org/abs/1804.06711, 2018  arXiv:1804.06711 [q-fin.GN]
  * (92) R. Rockafellar and Stanislav Uryasev  “Optimization of conditional value-at-risk”  In _Journal of Risk_ 2, 2000, pp. 21–42
  * (93) Richard Roll  “A critique of the asset pricing theory’s tests Part I: On past and potential testability of the theory”  In _Journal of Financial Economics_ 4.2, 1977, pp. 129–176  DOI: https://doi.org/10.1016/0304-405X(77)90009-5
  * (94) Stephen A Ross  “The arbitrage theory of capital asset pricing”  In _Journal of Economic Theory_ 13.3, 1976, pp. 341–360  DOI: https://doi.org/10.1016/0022-0531(76)90046-6
  * (95) Alexander M. Rudin and Jonathan S Morgan  “A portfolio diversification index”  In _The Journal of Portfolio Management_ 32.2 Institutional Investor Journals, 2006, pp. 81–89
  * (96) Olivier Scaillet, Adrien Treccani and Christopher Trevisan  “High-Frequency Jump Analysis of the Bitcoin Market”  In _Journal of Financial Econometrics_ 18.2, 2018, pp. 209–232  DOI: 10.1093/jjfinec/nby013
  * (97) Peter Schanbacher  “Combining Portfolio Models”  In _Annals of Economics and Finance_ 15.2 WUHAN UNIV JOURNALS PRESS DONGHU RD, 155, WUHAN, HUBEI 430072, PEOPLES R CHINA, 2014, pp. 433–455
  * (98) Peter Schanbacher  “Averaging across asset allocation models”  In _Jahrbücher für Nationalökonomie und Statistik_ 235.1 Lucius & Lucius, 2015, pp. 61–81
  * (99) Stefan Scharnowski  “Understanding Bitcoin liquidity”  In _Finance Research Letters_ , 2020, pp. 101477  DOI: https://doi.org/10.1016/j.frl.2020.101477
  * (100) Linda Schilling and Harald Uhlig  “Some simple bitcoin economics” SPECIAL CONFERENCE ISSUE: “Money Creation and Currency Competition” October 19-20, 2018 Sponsored by the Study Center Gerzensee and Swiss National Bank  In _Journal of Monetary Economics_ 106, 2019, pp. 16–26  DOI: https://doi.org/10.1016/j.jmoneco.2019.07.002
  * (101) Daniel Schwartzkopff, Luke Schwartzkopff, Raymond Botha, Matthew Finlayson and Frans Cronje  “CRYPTO20: The First Tokenized Cryptocurrency Index Fund” Available at: https://cdn.crypto20.com/pdf/c20-whitepaper.pdf, 2017  URL: https://cdn.crypto20.com/pdf/c20-whitepaper.pdf
  * (102) Syed Jawad Hussain Shahzad, Elie Bouri, David Roubaud, Ladislav Kristoufek and Brian Lucey  “Is Bitcoin a better safe-haven investment than gold and commodities?”  In _International Review of Financial Analysis_ 63, 2019, pp. 322–330  DOI: https://doi.org/10.1016/j.irfa.2019.01.002
  * (103) William F. Sharpe  “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk”  In _The Journal of Finance_ 19.3 [American Finance Association, Wiley], 1964, pp. 425–442  URL: http://www.jstor.org/stable/2977928
  * (104) Dehua Shen, Andrew Urquhart and Pengfei Wang  “A three-factor pricing model for cryptocurrencies”  In _Finance Research Letters_ , 2019  DOI: 10.1016/j.frl.2019.07.021
  * (105) Yusif Simaan  “Estimation risk in portfolio selection: The mean variance model versus the mean absolute deviation model”  In _Management Science_ 43.10 INFORMS, 1997, pp. 1437–1446
  * (106) Yhlas Sovbetov  “Factors Influencing Cryptocurrency Prices: Evidence from Bitcoin, Ethereum, Dash, Litcoin, and Monero”  In _Journal of Economics and Financial Analysis_ 2, 2018, pp. 1–27
  * (107) Steve Strongin, Melanie Petsch and Greg Sharenow  “Beating benchmarks”  In _The Journal of Portfolio Management_ 26.4 Institutional Investor Journals Umbrella, 2000, pp. 11–27
  * (108) Simon Trimborn and Wolfgang Karl Härdle  “CRIX an Index for cryptocurrencies”  In _Journal of Empirical Finance_ 49, 2018, pp. 107–122
  * (109) Simon Trimborn, Mingyang Li and Wolfgang Karl Härdle  “Investing with Cryptocurrencies–a Liquidity Constrained Investment Approach”  In _Journal of Financial Econometrics_ 18.2, 2019, pp. 280–306  DOI: 10.1093/jjfinec/nbz016
  * (110) Andrew Urquhart  “Price clustering in Bitcoin”  In _Economics Letters_ 159, 2017, pp. 145–148  DOI: https://doi.org/10.1016/j.econlet.2017.07.035
  * (111) Dimitri Vayanos and Jiang Wang  “Market Liquidity—Theory and Empirical Evidence”  In _Handbook of the Economics of Finance_ 2 Elsevier, 2013, pp. 1289–1361  DOI: https://doi.org/10.1016/B978-0-44-459406-8.00019-6
  * (112) Sha Wang and Jean-Philippe Vergne  “Buzz Factor or Innovation Potential: What Explains Cryptocurrencies’ Returns?”  In _PLOS ONE_ 12.1 Public Library of Science, 2017, pp. 1–17  DOI: 10.1371/journal.pone.0169556
  * (113) Wang Chun Wei  “Liquidity and market efficiency in cryptocurrencies”  In _Economics Letters_ 168, 2018, pp. 21–24  DOI: https://doi.org/10.1016/j.econlet.2018.04.003
  * (114) Rico Wyss  “Measuring and Predicting Liquidity in the Stock Market”, 2004
  * (115) David Yermack  “Is Bitcoin a Real Currency? An Economic Appraisal”  In _Handbook of Digital Currency_ San Diego: Academic Press, 2015, pp. 31–43  DOI: http://dx.doi.org/10.1016/B978-0-12-802117-0.00002-3
  * (116) Wei Zhang, Pengfei Wang, Xiao Li and Dehua Shen  “Some stylized facts of the cryptocurrency market”  In _Applied Economics_ 50.55 Routledge, 2018, pp. 5950–5965  DOI: 10.1080/00036846.2018.1488076

##  9 Appendix

###  9.1 Test for difference of SR or CEQ between two strategies

We employ the test by ledoit2008robust (68). Let
$\nu=(\mu_{i},\mu_{j},\sigma_{i},\sigma_{j})$ denote the vector of the moments
of two strategies $i$ and $j$.

Then we can test for a difference of the strategies’ CEQs or SRs via the test
statistics defined as the differences of those measures,

$f_{CEQ}(\nu)=\mu_{i}-\frac{\gamma}{2}\sigma^{2}_{i}-\mu_{j}+\frac{\gamma}{2}\sigma^{2}_{j},$
(23)

or

$f_{SR}(\nu)=\frac{\mu_{i}}{\sigma_{i}}-\frac{\mu_{j}}{\sigma_{j}},$ (24)

respectively.

Applying the delta method yields that if
$\sqrt{T-M}(\hat{\nu}-\nu)\xrightarrow{\text{d}}N(0,\Psi)$, then

$\sqrt{T-M}(\hat{f}-f)\xrightarrow{\text{d}}N(0,\nabla^{\prime}f(\nu)\Psi\nabla
f(\nu)),$ (25)

where $\nabla f$ stands for the derivative of $f$.

The standard error for such a test statistic $\hat{f}$ then amounts to:

$SE(\hat{f})=\sqrt{\frac{\nabla^{\prime}f(\nu)\Psi\nabla f(\nu)}{T-M}},$ (26)

so we require a consistent estimator $\hat{\Psi}$ for $\Psi$.

The standard method to provide such an esimator is to apply
heteroskedasticity- and autocorrelation-robust kernel estimation to obtain the
estimate

$\Psi_{T-M}=\frac{T-M}{T-M-4}\sum_{j=-T+M+1}^{T-M-1}Ker\left(\frac{j}{S_{T-M}}\right)\hat{\Gamma}_{T-M}(j),$
(27)

where a kernel function $Ker(\cdot)$ and a bandwidth $S_{T-M}$ need to be
chosen.

Then a two-sided $p$-value for the hypothesis $H_{0}$: $f=0$ is given as:

$\hat{p}=2\Phi\frac{\lvert\hat{f}\rvert}{SE(\hat{f})}.$ (28)

###  9.2 Descriptive statistics of portfolio components

For completeness, we present descriptive statistics both for our traditional
assets as well as all 52 CCs in our sample. Table 7 shows that, as expected,
correlations between CCs and traditional assets are low to non-existent, also
in our sample. Tables 8 and 9 show univariate distributional properties of
daily log returns on traditional assets and CCs, respectively. The generally
elevated magnitude for CCs is clear; Figure 7 visually confirms the strong
leptokurtic nature of CC returns.

| DOGE | ZET | XMG | SYS | POT | DGC | DMD | RBY | START | EMC2
---|---|---|---|---|---|---|---|---|---|---
CNY | –0.03 | –0.01 | –0.02 | –0.04 | 0.01 | –0.03 | –0.01 | 0.04 | 0.00 | –0.04
REIT | 0.01 | 0.02 | –0.01 | 0.05 | 0.01 | 0.01 | –0.01 | 0.03 | 0.03 | –0.02
EUR | 0.02 | 0.00 | –0.01 | 0.00 | 0.01 | 0.01 | 0.01 | 0.00 | –0.05 | –0.02
GBP | 0.02 | 0.02 | –0.02 | –0.02 | –0.03 | 0.01 | 0.00 | 0.01 | –0.02 | –0.04
JPY | 0.00 | 0.01 | 0.02 | 0.00 | –0.01 | 0.03 | –0.01 | 0.01 | 0.00 | 0.00
MSCI CP | 0.04 | 0.05 | 0.04 | 0.06 | 0.04 | 0.00 | 0.00 | 0.02 | 0.00 | –0.04
GOLD | 0.05 | 0.00 | 0.02 | 0.01 | –0.01 | –0.01 | 0.00 | –0.02 | 0.01 | 0.02
NIKKEI225 | 0.00 | 0.02 | 0.00 | 0.01 | –0.03 | 0.04 | 0.03 | –0.04 | 0.02 | –0.02
SSE | –0.06 | 0.00 | 0.00 | 0.04 | –0.06 | 0.02 | 0.03 | 0.02 | 0.03 | 0.02
S&P100 | 0.04 | 0.03 | 0.05 | 0.05 | 0.03 | –0.02 | –0.01 | 0.03 | 0.00 | –0.04
EURO STOXX 50 | 0.04 | 0.05 | 0.02 | 0.02 | 0.01 | 0.01 | –0.01 | 0.02 | –0.01 | 0.00
FTSE 100 | 0.04 | 0.04 | 0.04 | 0.07 | 0.05 | 0.01 | –0.02 | 0.02 | 0.01 | –0.01
UK 10Y | 0.00 | –0.05 | 0.00 | 0.01 | 0.02 | –0.02 | 0.00 | 0.00 | 0.02 | 0.02
Japan 10Y | 0.02 | 0.02 | –0.02 | –0.02 | 0.02 | –0.04 | –0.01 | 0.02 | 0.00 | 0.08
USA 10Y | 0.00 | –0.02 | –0.02 | –0.02 | –0.01 | –0.03 | –0.01 | –0.03 | 0.00 | 0.01
EURO 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 conventional financial assets in our analysis (detailed in Table 2) over
the entire sample period from 2016-01-01 to 2019-12-31.

Asset name | Max | $P_{90}$ | Med | Mean | $P_{10}$ | Min | SD
---|---|---|---|---|---|---|---
CNY | 1.84 | 0.24 | 0.00 | 0.01 | -0.23 | -1.20 | 0.23
JPY | 2.22 | 0.58 | 0.00 | -0.01 | -0.61 | -3.78 | 0.53
EUR | 3.02 | 0.59 | 0.00 | -0.01 | -0.63 | -2.38 | 0.52
GBP | 3.00 | 0.67 | -0.02 | -0.01 | -0.65 | -8.40 | 0.61
FTSE REIT | 4.14 | 1.04 | 0.03 | 0.02 | -1.08 | -9.38 | 0.96
GOLDS | 4.58 | 0.90 | 0.02 | 0.02 | -0.84 | -3.38 | 0.77
MSCI CP | 2.67 | 0.78 | 0.04 | 0.02 | -0.76 | -4.88 | 0.69
NIKKEI225 | 7.43 | 1.24 | 0.01 | 0.02 | -1.22 | -8.25 | 1.19
SSE | 5.60 | 1.47 | 0.01 | -0.00 | -1.33 | -8.87 | 1.46
S&P100 | 4.84 | 0.97 | 0.03 | 0.03 | -0.84 | -4.18 | 0.83
EURO STOXX50 | 4.60 | 1.18 | 0.04 | 0.01 | -1.20 | -9.01 | 1.07
FTSE100 | 3.51 | 0.96 | 0.02 | 0.01 | -0.95 | -4.78 | 0.86
UK 10Y | 2.00 | 0.30 | 0.00 | 0.00 | -0.29 | -1.43 | 0.28
Japan 10Y | 0.74 | 0.09 | 0.00 | -0.00 | -0.08 | -0.63 | 0.10
USA 10Y | 1.29 | 0.30 | 0.00 | 0.00 | -0.28 | -1.55 | 0.27
EURO 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 our baseline portfolio (detailed in Table 2) over the
entire sample period from 2016-01-01 to 2019-12-31. $P_{10}$ and $P_{90}$
denote the first and ninth decile, respectively, “Med” the median, and “SD”
standard deviation.

CC | Max | $P_{90}$ | Med | Mean | $P_{10}$ | Min | SD
---|---|---|---|---|---|---|---
ABY | 35.10 | 14.18 | -0.19 | 0.01 | -13.65 | -29.69 | 12.13
AUR | 29.85 | 12.50 | -0.12 | -0.09 | -12.68 | -27.22 | 11.00
BCN | 21.80 | 10.40 | -0.20 | -0.12 | -10.93 | -20.65 | 8.77
BLK | 22.88 | 9.61 | -0.25 | -0.07 | -9.59 | -22.44 | 8.55
BTC | 9.58 | 5.01 | 0.22 | 0.24 | -4.03 | -9.88 | 3.80
BTS | 17.26 | 8.04 | -0.32 | -0.07 | -8.03 | -16.64 | 6.71
BURST | 21.70 | 10.51 | 0.24 | 0.09 | -10.44 | -20.09 | 8.57
BYC | 30.16 | 12.10 | 0.00 | -0.14 | -11.80 | -26.84 | 10.57
CANN | 37.87 | 12.56 | -0.06 | 0.18 | -12.44 | -28.67 | 12.00
CURE | 25.31 | 11.55 | -0.26 | -0.05 | -11.22 | -20.68 | 9.35
DASH | 15.67 | 7.01 | -0.16 | 0.21 | -6.04 | -12.56 | 5.59
DGB | 22.92 | 10.36 | -0.56 | 0.09 | -9.22 | -18.47 | 8.23
DGC | 54.02 | 16.36 | -0.33 | -0.53 | -17.31 | -62.84 | 18.93
DMD | 17.83 | 9.57 | -0.13 | -0.01 | -9.16 | -19.58 | 7.62
DOGE | 14.99 | 6.75 | -0.25 | 0.05 | -5.62 | -12.58 | 5.34
EAC | 41.30 | 12.35 | -0.07 | -0.04 | -13.06 | -37.18 | 13.11
EMC2 | 25.40 | 11.60 | -0.35 | -0.01 | -10.82 | -23.49 | 9.68
FTC | 27.50 | 11.43 | -0.77 | -0.20 | -10.64 | -21.49 | 9.55
GRC | 37.14 | 13.80 | -0.50 | 0.21 | -13.23 | -23.57 | 11.61
HUC | 28.23 | 12.92 | 0.00 | 0.05 | -12.93 | -22.96 | 10.43
IOC | 28.86 | 14.69 | -0.07 | 0.25 | -13.03 | -28.26 | 11.55
LTC | 15.71 | 6.36 | -0.07 | 0.12 | -5.99 | -12.70 | 5.33
MAX | 80.52 | 21.21 | -0.44 | -0.24 | -22.02 | -81.86 | 26.40
NAV | 26.75 | 12.16 | -0.34 | 0.10 | -11.07 | -20.07 | 9.53
NEOS | 28.82 | 12.37 | 0.00 | -0.22 | -12.43 | -25.50 | 10.36
NLG | 18.09 | 8.89 | -0.20 | 0.04 | -8.53 | -14.55 | 6.91
NMC | 16.64 | 7.34 | -0.20 | -0.07 | -7.27 | -16.26 | 6.27
NOTE | 27.56 | 11.89 | -0.35 | -0.39 | -12.44 | -25.80 | 10.47
NVC | 20.18 | 7.19 | -0.17 | -0.08 | -7.92 | -15.06 | 6.61
NXT | 17.15 | 8.26 | -0.54 | -0.22 | -7.92 | -15.68 | 6.56
POT | 20.16 | 9.77 | -0.07 | -0.03 | -10.39 | -19.77 | 8.13
PPC | 16.94 | 6.98 | -0.19 | -0.07 | -7.47 | -15.07 | 6.26
QRK | 37.18 | 11.97 | -0.38 | -0.05 | -12.22 | -30.88 | 12.15
RBY | 25.58 | 12.34 | 0.00 | 0.10 | -12.33 | -28.14 | 10.41
RDD | 32.46 | 14.26 | -0.05 | 0.08 | -13.67 | -28.14 | 11.90
SLR | 25.15 | 11.21 | -0.23 | 0.01 | -11.01 | -21.65 | 9.41
START | 29.14 | 14.48 | -0.66 | -0.16 | -12.99 | -26.97 | 11.36
SYS | 24.83 | 10.61 | -0.14 | 0.20 | -10.24 | -19.16 | 8.72
UNO | 22.07 | 11.13 | -0.01 | 0.16 | -9.47 | -23.61 | 8.79
VIA | 25.06 | 10.68 | -0.00 | -0.01 | -11.35 | -20.53 | 9.22
VRC | 33.85 | 14.29 | -0.51 | 0.04 | -13.28 | -28.46 | 11.86
VTC | 27.29 | 10.98 | -0.41 | 0.03 | -10.74 | -19.54 | 9.09
WDC | 29.58 | 11.66 | 0.00 | -0.42 | -12.39 | -33.05 | 11.24
XCN | 59.83 | 20.82 | -0.42 | -0.16 | -22.36 | -51.64 | 20.06
XCP | 22.41 | 10.77 | -0.48 | -0.19 | -10.48 | -21.09 | 8.84
XDN | 24.72 | 11.73 | -0.29 | -0.13 | -12.12 | -22.17 | 9.58
XMG | 31.69 | 12.70 | -0.18 | 0.24 | -11.33 | -23.38 | 10.54
XMR | 16.29 | 8.54 | -0.05 | 0.26 | -7.22 | -14.12 | 6.36
XPM | 21.99 | 9.61 | -0.25 | -0.20 | -10.58 | -19.71 | 8.45
XRP | 16.18 | 6.70 | -0.39 | -0.06 | -6.05 | -13.14 | 5.52
XST | 29.94 | 13.98 | -0.43 | -0.04 | -13.15 | -26.17 | 11.24
ZET | 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 portfolio strategies (detailed in Table 1) over the entire
sample period from 2016-01-01 to 2019-12-31. $P_{10}$ and $P_{90}$ denote the
first and ninth decile, respectively, “Med” the median, and “SD” standard
deviation.

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 mean
and variance. The time span is from 2016-01-01 to 2019-12-31.

CCPHistReturnsDensity

###  9.3 Dynamics of risk contributions for portfolio strategies

The outcome of portfolio optimization can be viewed in two different ways:
first, in terms of the weights the chosen strategy assigns to each asset;
second, in terms of the risk each constituent contributes to the portfolio.
While flip sides of the same coin, with strongly divergent statistical
properties across assets, as in our case, relative risk contributions can
differ noticably from relative portfolio shares. For instance, if a portfolio
were to hold the same percentage of its value in UK bonds and in bitcoin, the
changes in portfolio value over time driven by BTC will amount to a multiple
of those stemming from the same-sized fixed-income position.

While we reported weigts in Figures 5 and 6 in the main text, for completeness